Grupo de Dinámica Espacial (SDG–UPM)

Departamento de Física Aplicada a las Ingenierías Aeronáutica y Naval

Escuela Técnica Superior de Ingenieros Aeronáuticos

Universidad Politécnica de Madrid

Regularization in Astrodynamics: applications to relativemotion, low-thrust missions, and orbit propagation

Ph.D. Dissertation

Javier Roa VicensIngeniero Aeronáutico

Supervisedby

Prof. Jesús Peláez ÁlvarezDoctor Ingeniero Aeronáutico

MadridSeptember 2016

Tribunal nombrado por el Sr. Rector Magfco. de la Universidad Politécnica de Madrid, eldía ...... de .................... de 2016.

Presidente: Dr. José Manuel Perales

Vocal: Dr. Michael Efroimsky

Vocal: Dr. Anastassios Petropoulos

Vocal: Dr. Hodei Urrutxua

Secretario: Dra. Ana Laverón Simavilla

Suplente: Dr. Martín Lara Coira

Suplente: Dr. Manuel Sanjurjo Rivo

Calificación: ...................................................

Realizado el acto de defensa y lectura de la Tesis el día 30 de septiembre de 2016 en la E.T.S.I.Aeronáuticos.

EL PRESIDENTE LOS VOCALES

EL SECRETARIO

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“The true voyage of discovery consists not inseeking new landscapes, but in having new eyes.”

— Marcel Proust

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Regularization in Astrodynamics: applications to relativemotion, low-thrust missions, and orbit propagation

Abstract

Regularized formulations of orbital motion provide powerful tools for solving various problems in orbitalmechanics, both analytically and numerically. They rely on a collection of dynamical and mathematicaltransformations that yields a more convenient description of the dynamics. The goal of the present thesisis to recover the foundations of regularization, to advance the theory toward practical applications, and touse this mathematical contrivance for solving three key challenges in modern astrodynamics: the dynamicsof spacecraft formations, the design of low-thrust trajectories, and the high-performance numerical propa-gation of orbits.

The introduction of a fictitious time is a typical practice when regularizing the equations of motion.This technique leads to a new theory of relative motion, called the theory of asynchronous relative mo-tion. It improves the accuracy of the linear propagation by introducing nonlinear terms through simpledynamical mechanisms, and simplifies significantly the derivation of analytic solutions. In addition, it ad-mits any type of orbital perturbation. Themethod is compact and seems well suited for its implementationin navigation filters and control laws. Universal and fully regular solutions to the relative dynamics follownaturally from this theory. They are valid for any type of reference orbit (circular, elliptic, parabolic, or hy-perbolic) and are not affected by the typical singularities related to the eccentricity or inclination of the orbit.The nonlinear corrections proposed by the method are generic and can be applied to existing solutions toimprove their accuracy without the need for a dedicated re-implementation.

We present a novel shape-basedmethod for preliminary design of low-thrust trajectories: the family ofgeneralized logarithmic spirals. This new solution arises from the search for sets of orbital elements in theaccelerated case. It is fully analytic and involves two conservation laws (related to the equations of the energyand angular momentum) that make the solution surprisingly similar to the Keplerian case and simplify thedesign process. The properties of the solution to the Keplerian Lambert problem find direct analoguesin the continuous-thrust case. An analysis of the dynamical symmetries in the problem shows that theperturbing acceleration can be generalized and provides two additional families of analytic solutions: thegeneralized cardioids and the generalized sinusoidal spirals.

As the complexity of space missions increases, more sophisticated orbit propagators are required. Inorder to integrate flyby trajectories more efficiently, an improved propagation scheme is presented, exploit-ing the geometry ofMinkowski space-time. Themotion of the orbital plane is decoupled from the in-planedynamics, and the introduction of hyperbolic geometry simplifies the description of the planar motion.General considerations on the accuracy of the propagation of flyby trajectories are presented. In the con-text of N-body systems, we prove that regularization yields a simplified Lyapunov-like indicator that helpsin assessing the validity of the numerical integration. Classical concepts arising from stability theories areextended to higher dimensions to comply with the regularized state-space. In this thesis, we present, for thefirst time, the gauge-generalized form of some element-based regularized formulations.

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Regularización en Astrodinámica: aplicaciones al movimientorelativo, misiones de bajo empuje, y propagación de órbitas

Resumen

Las formulaciones regularizadas del movimiento orbital son potentes herramientas para resolver diversosproblemas en mecánica orbital, tanto analítica como numéricamente. Se basan en un conjunto de trans-formaciones físico-matemáticas que proveen una descripción más conveniente de la dinámica. El objetivode esta tesis es recuperar las bases de la regularización, avanzar en sus fundamentos teóricos para abordarcuestiones prácticas, y emplear este artificio matemático para resolver tres retos fundamentales en la astrod-inámica moderna: el movimiento de formaciones de satélites, el diseño de trayectorias de bajo empuje, y lapropagación numérica de órbitas de alta precisión.

Introducir un tiempo ficticio es una práctica habitual cuando se regularizan las ecuaciones delmovimiento. Esta técnica da lugar a toda una nueva teoría del movimiento relativo, denominada teoríadel movimiento relativo asíncrono. Mejora la precisión de la propagación al introducir términos no linealesmediante mecanismos dinámicos sencillos, y simplifica notablemente la obtención de soluciones analíti-cas. Además, la teoría admite cualquier tipo de perturbación. El método es compacto y adecuado para suimplementación en algoritmos de navegación y leyes de control. Soluciones universales y completamenteregularizadas surgen de forma natural al emplear esta teoría. Dichas soluciones son válidas para cualquiertipo de órbita de referencia (circular, elíptica, parabólica, hiperbólica) y no se ven afectadas por las típicassingularidades relacionadas con la excentricidad o la inclinación de la órbita. Las correcciones no linealesintroducidas por este método son generales, y pueden aplicarse a soluciones ya existentes para mejorar suprecisión sin necesidad de reimplementarlas.

Se ha desarrollado un nuevo método basado en la forma para el diseño preliminar de trayectorias conbajo empuje: las espirales logarítmicas generalizadas. Esta solución surge de buscar conjuntos de elementosorbitales que permanezcan constantes en el caso acelerado. Es completamente analítica y admite dos leyes deconservación (relacionadas con las ecuaciones de la energía y elmomento angular) que hacen que la soluciónsea sorprendentemente parecida al caso kepleriano, lo que simplifica el proceso de diseño. Las propiedadesde la solución al problema de Lambert kepleriano pueden traducirse al problema con empuje continuo,donde aparecen propiedades similares. Un análisis detallado de las simetrías dinámicas del problema revelaque la aceleración de perturbación puede generalizarse, dando lugar a dos familias de soluciones adicionales:los cardioides generalizados, y las espirales sinusoidales generalizadas.

Conforme aumenta la complejidad de las misiones espaciales, se necesitan propagadores orbitales másavanzados. Para integrar trayectorias con flybys de forma más eficiente, se presenta un esquema de propa-gación mejorado que se sirve de la geometría sobre la que se fundamenta el espacio-tiempo de Minkowski.El movimiento del plano orbital se desacopla de la dinámica dentro del plano, que se ve simplificada alemplear la geometría hiperbólica. La discusión incluye consideraciones generales sobre la precisión de lapropagación. En el contexto de sistemas de N-cuerpos, se demuestra que la regularización da lugar un indi-cador de Lyapunov simplificado que ayuda a evaluar la validez de la integración numérica. Los conceptosclásicos que se derivan de las teorías de estabilidad serán extendidos a dimensiones más altas de acuerdo conel espacio de fases regularizado. En esta tesis se presenta, por primera vez, la generalización gauge de ciertasformulaciones regularizadas basadas en elementos.

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Acknowledgments

Gracias, Jesús. I feel like I would never say this enough. I first contacted Prof. Jesús Peláez by chance back in2012 (referred by Prof. Rafael Ramis, towhom Iwill always be grateful), and I completedmymaster’s thesisunder his supervision. Not only did he teach me all I know about astrodynamics, but I also owe him mypassion for orbital mechanics. That is why I did not hesitate about pursuing a Ph.D. under his supervision.This was to become one of the best decisions of my life, and I want to thank him for accepting me as hisstudent. He has truly been a maestro, in all its Spanish meaning, more than a mere teacher. If this werenot enough, he was also who arranged my trip to the Jet Propulsion Laboratory during the second year ofmy doctorate (I still remember that first email he sent me on August 7, 2014). Then, he gave me his fullsupport when I suggested the possibility of finishing what remained ofmy Ph.D. from theU.S., and agreedto supervise my work remotely. I will miss those Skype calls. Thanks to him I am living my dream, and Ijust hope that I will be able to repay him, somehow, in the future.

Gracias, Juan. If it were not for Dr. Juan Senent, I would have never come to JPL in the first place.And for that I am indebted to him. He convinced me of the importance of solving actual problems inengineering, as opposed to just developing abstract methods, and helped me complete this thesis with real-world applications. Moreover, he motivated me to look into continuous-thrust trajectories, which resultedin the new analytic solution and design strategy presented in Chapters 9–12. I could not have felt morewelcomed: in fact, have I had a different advisor, I may not have considered extending my stay.

Gracias, Fernando. I feel truly lucky to have met Fernando Abilleira while at JPL. Specially because Imet him almost by chance. He had no obligation toward me: still, he became a constant support both inthe professional and personal sides. For me, he has been an example of motivation and perseverance, andhe always found the right encouraging words for the not-so-good times. He welcomed Marta and me intohis family, and I will never forget that. Thanks to himCalifornia felt a little more like home. I feel fortunatethat this adventure has just started. But I am sure that I would not have made it without his help.

I want to thank Dr. Michael Efroimsky for his selfless help and guidance. To me, he has been theperfect example of how a researcher should be: willing to collaborate, to discuss exciting concepts, and al-ways open to new ideas. He referred me to the gauge freedom in celestial mechanics, a topic that I foundfascinating from the first time I looked into it. In addition, he read this dissertation thoroughly and madevery valuable suggestions. A good part of this thesis focuses on the numerical integration of orbits. And Iowe almost all I know about this subject to Dr. Hodei Urrutxua. He was always there to answer my ques-tions, providing a detailed explanation. I really enjoyed our conversations, and I look forward to workingtogether in so many projects that we discussed. I would also like to thank Dr. Sebastien Le Maistre for hiscompany, for sharing his office with me, and for all he taught me about planetary science. The trailer is notthe same without him… Dr. Anatoliy Ivantsov was kind enough to share his expertise on rotation modelswith me. Let me also thank Jordi Paredes for transmitting me his resolve, and for showing me that it paysto persevere.

Dr. Claudio Bombardelli got me involved in very exciting projects that helped me learn many things,and he taught me many others. His door was always open to me, and I cannot wait to recover some of theideas that we discussed and to collaborate in the future. Gracias, Claudio. Dr. Anastassios Petropoulos andDr. Nitin Arora’s help was invaluable when developing the continuous-thrust theory. Many ideas came upduring our meetings. I would like to thank Dr. Vladimir Martinuşi for our conversations about relativemotion, which contributed to improving the present work. Similarly, I would like to thankDr. Giulio Baùfor many interesting debates about orbit propagation, and I hope that we will find the time in the future tofinish our projects. I had a great time discussing historical references with Dr. Aaron Rosengren. I thankCarolina Silva, Kim Jackson, Arpine Margallan, and Claudia Tobar for their assistance while at JPL. I waslucky enough to participate in the Caltech Space Challenge with Team Voyager, and to meet so interestingpeople. I am also grateful to Prof. Carlos Blanco-Pérez for our exciting conversations.

I can only be grateful to the people Iworkedwith inMadrid: to JuanLuisGonzalo, for all his helpwithcoding, and everything related to computers; to Dr. Manuel Sanjurjo and David Morante, for their workon trajectory optimization using generalized logarithmic spirals; to Davide Amato and Javier Hernando,

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for their help when facing the doctoral courses; to Mari Carmen Velasco, María Jesús de Andrés, and SilviaMuñoz, for solving each and every administrative problem I had. And to Ana, for welcoming me everymorning with a smile. I can say that this Ph.D. would not have been possible without their constant assis-tance.

Many people helped me in one way or the other during these years. Dr. Rodney Anderson referredme to very interesting problems involving periodic orbits, and helpedme prepare several proposals. I wouldalso like to thankDr. Jon Sims, Dr. RobyWilson, Dr. RalphRoncoli, andDr. JoeGuinn for their patienceand council, and I hope did not get too persistent. This Acknowledgments section would not be completewithout thanking Dr. Ryan Park, Dr. Paul Chodas, and Dr. Steve Chesley for believing in me and foroffering me the opportunity to work with them. Dr. Mar Vaquero was a constant support to me, andI thank her for referring me to this group in the first place. I would like to thank Prof. Jeremy Kasdin forinvitingme to Princeton, and Prof. SimoneD’Amico for his time and consideration; I hopewewill have thechance to work on some of the ideas we discussed. I am deeply grateful to all themembers of the committeeevaluating this thesis (Prof. José Manuel Perales, Dr. Michael Efroimsky, Dr. Anastassios Petropoulos, Dr.Hodei Urrutxua, Prof. Ana Laverón, Dr. Martín Lara, and Dr. Manuel Sanjurjo) and also to Prof. TerryAlfriend and Prof. Massimiliano Vasile for reviewing this manuscript in great detail.

This past year I have asked for a lot of recommendation letters, for which I would like to acknowledgeDr. Martín Lara, Dr. MartinOzimek,Dr. JohanKnapen (also for hostingme at the Instituto deAstrofísicade Canarias in 2013), Prof. Ryan Russell, and Prof. Roberto Furfaro. Dr. Ozimek also made valuablecontributions to the low-thrust work, and Dr. Lara helped me with various proposals. I could not forgetthat Prof. Peláez, Dr. Senent, F. Abilleira, and Dr. Efroimsky acted as references many times too.

My Ph.D. was possible thanks to the financial support of Obra Social “La Caixa”. I am grateful notonly for the doctoral fellowship, but for all the help that I received, the activities of the society, the network-ing events, etc. I thank Ainhoa Martín, Anna Mauri, María Inmaculada Jiménez, Elisabet Linero, and therest of the team at Fundación “La Caixa”. I am also grateful to Prof. Carmina Crusafón for including mein the board.

So many emotions and memories come to my mind when thanking my parents. They have alwaysbeen there for me, specially in the most stressful moments. Always supportive, always understanding mydedication, and always puttingmyneeds before theirs. I hope that they realize that all I achievedwas becauseof them, and that they are proud of who I have become. With their example and dedicated lives they taughtme the most important lessons one can learn. They are the best models I can think of. In my brother Ihave always found the right piece of advice, the perfect words of encouragement, and I thank for him forhis patience and for teachingme with his brilliant example. My sister has been able to transmit her illusionsto me, her will for living, and her constant joy. She has the gift of cheering up every situation, a gift that wasmost valuable in the difficult moments.

Einstein once said that love is power, because it multiplies the best we have. And this is his only theorythat I have been able to prove, or at least to experience myself. And for that I thank Marta. She has alwaysbeen there, she has been my constant support, encouraging me to keep going. She made so many sacrificesfor me… It has not been an easy path. But, precisely because it was hard, we can now look back happilyand enjoy the fact that we finally made it. With time one realizes that the best of achieving your goals is tohave someone to share them with. I could not have been luckier. And this thesis is my gift to you. Gracias,Marta.

Javier Roa VicensPasadena, CA, August 2016

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Dissertation Overview

This dissertation consists in two parts. The first part focuses on the theoretical aspects of regularization,and the second part exploits the properties and advantages of regularized formulations to tackle differentproblems in astrodynamics. When reading a dissertation for the first time, it is often hard to know whatsections are reviews of existing work, and what sections introduce new and truly original material. Forthis reason, the following lines describe briefly the contents of each chapter, stating clearly where the newconcepts and results are presented.

Chapter 1: this is the introduction to the thesis. It discusses the relevance of the three problems that will beaddressed in the second part of the dissertation: spacecraft relative motion, low-thrust mission design,and orbit propagation.

Chapter 2: the second chapter is devoted to the theory of regularization. It starts by justifying why regular-ization is useful, and revisits various methods and techniques that improve the description of orbitalmotion. This chapter introduces several concepts that will be recovered throughout the dissertation.

Chapter 3: a completely new theory of stability is presented in this chapter, based on the topology ofKustaanheimo-Stiefel (KS) space. It leads to a novel Lyapunov indicator. The second part of the chap-ter provides the elements attached to KS space with an innovative geometric interpretation. Finally,the gauge-generalized form of the equations of motion in this formalism is derived for the first time.

Chapter 4: the first part of this chapter is a review of the Dromo formulation, a special-perturbationmethod. The second part presents new results related to the singularity of Dromo, and the gauge-generalized version of the equations of motion.

Chapter 5: the chapter presents a novel propagation method for hyperbolic orbits, based on the geometryofMinkowski space-time and recovering thedynamical structure of theDromopropagator. The resultis a more stable integration scheme, which is more accurate because of not being affected by periapsispassage.

Chapter 6: this chapter explains the structure of the high-fidelity orbit propagator developed during thisdoctoral work. The force models are discussed in detail. The main goal of the software is to comparethe performance of various numerical propagation methods.

Chapter 7: this is the first chapter in the second part of the dissertation, and focuses on spacecraft relativemotion. After an introduction to the problem, the theory of asynchronous relative motion is pre-sented. It is a novel concept that approaches the problem from the perspective of regularization. Itimproves the accuracy of the purely linear solution without complicating the algorithm significantly.This new theory can be applied in many other scenarios apart from relative motion (as long as theyinvolve the variational equations of motion) and admits any source of perturbation.

Chapter 8: making use of the theory of asynchronous relative motion, this chapter models the relative dy-namics using various regularized formulations. The result is a fully universal and regular solution,more accurate than the linear one and valid for any type of orbit.

Chapter 9: a new analytic solution with continuous thrust is derived: the generalized logarithmic spirals.This new family of curves has interesting properties for preliminary design of low-thrust missions.There are two integrals ofmotion thatmake the solution very similar to theKeplerian case. The thrustdecreases with the square of the radial distance and it is directed along the velocity vector.

Chapter 10: using the new analytic solution presented in the previous chapter, Chapter 10 solves Lambert’sproblem with continuous thrust. The versatility of the solution is improved by introducing a controlparameter. It affects both the magnitude and direction of the thrust vector: it is no longer tangential.The properties of the ballistic Lambert problem are then translated to the continuous-thrust case.Examples of application to mission design can also be found.

Chapter 11: advancing on the solution to Lambert’s problem, in Chapter 11 the reader will find a new de-sign strategy for generating preliminary continuous-thrust transfer solutions, including examples. Al-

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though the original solution is planar, a fully three-dimensional approach is derived in this chapter. InAppendix G the potential of Seiffert’s spherical spirals for mission design is explored.

Chapter 12: in this chapter, it is demonstrated that the family of generalized logarithmic spirals can be gen-eralized by adding an extra degree of freedom to the perturbing acceleration in order to yield new fam-ilies of orbits. The symmetries of Kepler’s problem are exploited to render integrals of motion withcontinuous thrust. Keplerian orbits and generalized logarithmic spirals appear naturally as particularinstances of this new and more general integrable system.

Chapter 13: the final chapter includes the conclusions to the thesis and suggests future lines of research.

The paper based on Chapter 7 won the Best Paper Award at the 25th AAS/AIAA Space Flight Me-chanics Meeting in Williamsburg, VA, January 11-15, 2015. The next section includes the complete list ofpublications based on the present dissertation.

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List of Publications

During my doctoral studies I have been involved in the preparation of the following research papers, and Ipresented several others in international conferences:

Publications in Journals of the JCR

1. Roa, J. (2016): “Nonconservative extension of Keplerian integrals and a new class of integrable sys-tem,”Monthly Notices of the Royal Astronomical Society. Available online,doi: 10.1093/mnras/stw2209.

2. Roa, J. and Peláez, J. (2016): “The theory of asynchronous relative motion I: Time transformationsand nonlinear terms,” Celestial Mechanics and Dynamical Astronomy. Accepted for publication.

3. Roa, J. and Peláez, J. (2016): “The theory of asynchronous relative motion II: Universal and regularsolutions,” Celestial Mechanics and Dynamical Astronomy. Available online,doi: 10.1007/s10569-016-9730-z.

4. Roa, J., Peláez, J. and Senent, J. (2016): “New analytic solution with continuous thrust: generalizedlogarithmic spirals,” Journal of Guidance, Control and Dynamics. Available online,doi: 10.2514/1.G000341.

5. Roa, J., Peláez, J. and Senent, J. (2016): “Spiral Lambert’s problem,” Journal of Guidance, Control andDynamics. Available online, doi: 10.2514/1.G000342.

6. Bombardelli, C., Gonzalo, J.L. and Roa, J. (2016): “Approximate solution of nonlinear circular orbitrelativemotion in curvilinear coordinates,”Celestial Mechanics and Dynamical Astronomy. Availableonline, doi: 10.2514/1.G000341.

7. Roa, J., Urrutxua, H., and Peláez, J. (2016): “Stability and chaos in Kustaanheimo-Stiefel space in-duced by the Hopf fibration,”Monthly Notices of the Royal Astronomical Society, (459) 2444–2454.doi: 10.1093/mnras/stw780.

8. Roa, J. and Peláez, J. (2015): “Orbit propagation inMinkowskian geometry,”Celestial Mechanics andDynamical Astronomy, (123) 13–43. doi: 10.1007/s10569-015-9627-2.

9. Roa, J., Sanjurjo-Rivo, M. and Peláez, J. (2015): “Singularities in Dromo formulation. Analysis ofdeep flybys,”Advances in Space Research, (56) 569–581. doi: 10.1016/j.asr.2015.03.019.

10. Roa, J. and Peláez, J. (2015): “Frozen-anomaly transformation for the elliptic rendez-vous problem,”Celestial Mechanics and Dynamical Astronomy, (121) 61–81. doi: 10.1007/s10569-014-9585-0.

11. Knapen, J.H., Erroz-Ferrer, S., Roa, J., Bakos, J., Cisternas, M., Leaman, R. and Szymanek, N. (2014):“Optical imaging of galaxies from the Spitzer Survey of Stellar Structure in Galaxies.” Astronomy andAstrophysics, (569) A91. doi: 10.1051/0004-6361/201322954

Conference Proceedings

1. Roa, J. and Peláez, J.: “Spiral Lambert’s problem with generalized logarithmic spirals,” 26thAAS/AIAA Space Flight Mechanics Meeting in Napa, CA, USA. February 14-18, 2016. AAS 16-316.

2. Roa, J. and Peláez, J.: “Introducing a degree of freedom in the family of generalized logarithmic spi-rals,” 26th AAS/AIAA Space Flight Mechanics Meeting in Napa, CA, USA. February 14-18, 2016.AAS 16-317.

3. Roa, J. and Peláez, J.: “Three-dimensional generalized logarithmic spirals,” 26th AAS/AIAA SpaceFlight Mechanics Meeting in Napa, CA, USA. February 14-18, 2016. AAS 16-323.

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4. Urrutxua, H., Roa, J., Gonzalo, J. L., Peláez, J. and Bombardelli, C.: “Quantification of the Perfor-mance of Numerical Orbit Propagators,” 26th AAS/AIAA Space FlightMechanicsMeeting in Napa,CA, USA. February 14-18, 2016. AAS 16-351.

5. Bombardelli, C., Roa, J. and Gonzalo, J.L.: “Approximate Analytical Solution of the Multiple Rev-olution Lambert’s Problem,” 26th AAS/AIAA Space Flight Mechanics Meeting in Napa, CA, USA.February 14-18, 2016. AAS 16-212.

6. Roa, J. and Peláez, J.: “Generalized logarithmic spirals for low-thrust trajectory design,” 2015AAS/AIAA Astrodynamics Specialist Conference in Vail, CO, USA. August 9-13, 2015. AAS 15-729.

7. Roa, J. and Peláez, J.: “Efficient trajectory propagation for orbit determination problems,” 2015AAS/AIAA Astrodynamics Specialist Conference in Vail, CO, USA. August 9-13, 2015. AAS 15-730.

8. Bombardelli, C., Gonzalo, J.L. and Roa, J.: “Compact solution of circular orbit relative motion incurvilinear coordinates,” 2015 AAS/AIAA Astrodynamics Specialist Conference in Vail, CO, USA.August 9-13, 2015. AAS 15-661.

9. Roa, J. and Peláez, J.: “Regularized formulations in relative motion,” 25th AAS/AIAA Space FlightMechanics Meeting in Williamsburg, VA, USA. January 11-15, 2015. AAS 15-210.

10. Roa, J., Gómez-Mora, J.I. and Peláez, J.: “Error propagation in relative motion,” 25th AAS/AIAASpace Flight Mechanics Meeting in Williamsburg, VA, USA. January 11-15, 2015. AAS 15-272.

11. Roa, J. and Peláez, J.: “Orbit propagation in Minkowskian geometry,” 25th AAS/AIAA Space FlightMechanics Meeting in Williamsburg, VA, USA. January 11-15, 2015. AAS 15-209.

12. Roa, J., Sanjurjo-Rivo,M. and Peláez, J.: “A nonsingular Dromo formulation,” 2nd KEPASSAWork-shop, La Rioja, Spain. April 14-17, 2014.

13. Roa, J. and Peláez, J.: “The elliptic rendezvous problem in Dromo formulation,” 24th AAS/AIAASpace Flight Mechanics Meeting in Santa Fe, NM, USA. January 26-30, 2014. AAS 14-382.

Non-Refereed Publications

1. Roa, J. andHandmer, C. (2015): “Quantifyinghazards: asteroid disruption in lunar distant retrogradeorbits,” arXiv preprint arXiv:1505.03800.

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Contents

Abstract v

Resumen vii

Acknowledgments ix

Dissertation Overview xi

List of Publications xiii

Contents xvi

1 Introduction. Current challenges in space exploration 11.1 Accessible space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Distributed space systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Efficient orbit transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Orbit propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 The aim of the present thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I Regularization 7

2 Theoretical Aspects of Regularization 92.1 Why bother? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 The Sundman transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Stabilization of the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Sets of orbital elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Gauge-freedom in celestial mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 The Kustaanheimo-Stiefel space and the Hopf fibration 25

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3.1 The need for an extra dimension: fibrations of hyperspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 The KS transformation as a Hopf map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Stability in KS space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Order and chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Topological stability inN -body problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Gauge-generalized elements in KS space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.7 Orthogonal bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 The Dromo formulation 434.1 Hansen ideal frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Dromo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Improved performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Variational equations and the noncanonicity of Dromo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.5 Gauge-generalized Dromo formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.6 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.7 Modified Dromo equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.8 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Orbit propagation in Minkowskian geometry 555.1 Orbital motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Hyperbolic rotations and the Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Orbital plane dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.5 Time element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.6 Numerical evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 PERFORM: Performance Evaluation of Regularized Formulations of Orbital Motion 696.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.2 PERFORM as a high-fidelity propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.3 Evaluating the performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

II Applications 83

7 The theory of asynchronous relative motion 857.1 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.2 Synchronism in relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.3 Generalizing the transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.4 The circular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.5 Numerical evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8 Regularization in relative motion 1018.1 Relative motion in Dromo variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.2 Relative motion in Sperling-Burdet variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038.3 Relative motion in Kustaanheimo-Stiefel variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.4 On the fictitious time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.6 Generic propagation of the variational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9 Generalized logarithmic spirals: a new analytic solution with continuous thrust 1139.1 The Equations of Motion. First Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1149.2 Elliptic Spirals (K1 < 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199.3 Parabolic Spirals (K1 = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.4 Hyperbolic Spirals (K1 > 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.6 Osculating Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.7 In-Orbit Departure Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269.8 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289.9 Continuity of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

10 Lambert’s problem with generalized logarithmic spirals 13110.1 Introduction to Lambert’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13110.2 Controlled generalized logarithmic spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13210.3 The two-point boundary-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13410.4 Fixing the time of flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.5 Repetitive transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13910.6 Evaluating the performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14010.7 Additional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14010.8 Additional dynamical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14110.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

11 Low-thrust trajectory design with extended generalized logarithmic spirals 14511.1 Orbit transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14511.2 Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15011.3 Multinode transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15111.4 Three-dimensional motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15211.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15611.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

12 Nonconservative extension of Keplerian integrals and new families of orbits 16112.1 The role of first integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16112.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16312.3 Case γ = 1: conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16612.4 Case γ = 2: generalized logarithmic spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16612.5 Case γ = 3: generalized cardioids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16612.6 Case γ = 4: generalized sinusoidal spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16912.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17212.8 Unified solution in Weierstrassian formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17212.9 Physical discussion of the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17312.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

13 Conclusions to the thesis 17713.1 Outlook and advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17713.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Appendices 180

A Hypercomplex numbers 183A.1 Complex and hyperbolic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184A.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

B Formulations in PERFORM 187

C Stumpff functions 189

D Inverse transformations 193D.1 Inverse transformations in equinoctial variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193D.2 Cartesian to Dromo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194D.3 Linear form of the Hopf fibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

E Elliptic integrals and elliptic functions 197E.1 Properties and practical relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

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E.2 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198E.3 Jacobi elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199E.4 Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

F Controlled generalized logarithmic spirals 201F.1 Elliptic spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202F.2 Parabolic spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203F.3 Hyperbolic spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203F.4 Osculating elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

G Dynamics in Seiffert’s spherical spirals 207G.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207G.2 The geometry of Seiffert’s spherical spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210G.3 Groundtracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210G.4 Relative motion between Seiffert’s spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211G.5 The significance of Seiffert’s spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

List of Figures 213

References 217

Index 229

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“I would like to die on Mars. Just not on impact.”

—Elon Musk

1Introduction.

Current challenges in space exploration

S pace programs have undergone profound changes, fromeconomic restructuring to the implementation of novelmission architectures. During the ColdWar, space projects

enjoyed an apparently unlimited funding that led to unprece-dented technical and scientific breakthroughs in just a few years.In fact, between 1958 and 1966, when it reached its historicalmax-imum, the budget of NASA raised from 0.1% up to 4.4% of thetotal US federal budget. However, after the Apollo program thisfigure has been going down progressively until reaching a steady0.5%, which has remained almost constant in the last decade. Inorder to maintain the highest scientific standards with a shrink-ing budget, the efficiency of the mission concepts must be maxi-mized.

From a scientific perspective, the exciting goals set for the fu-ture require pushing the boundaries of the current technologylevels, mission architectures, and even changing the paradigm ofspace exploration in general. In the coming decades we may seeprobes investigating the subsurface oceans of the icymoons, aster-oid rocks brought toEarth, lunar bases, and evenhumanswalkingon the surface of Mars. The way to go is more or less clear, butmuch work needs to be done to reach such objectives.

Leaving the Earth is one of the most expensive phases of amission, with the launch costs becoming as high as 30-40% of thetotal cost. It is also the most violent part, and malfunctions atthis stage are catastrophic. Once in orbit, the probe still needs totravel to its final destination. The amount of propellant that re-mains in the spacecraft after reaching its target orbit determinesthe mission lifetime. Planning an efficient space travel requireshigh-fidelity physical models to ensure that the probe will followan orbit that is at least close to the optimized nominal trajectory.

A poor model will lead to unexpected maneuvers to correct thecourse, reducing the amount of fuel available for nominal opera-tions and, consequently, shortening the duration of the mission.

1.1 Accessible space

An exciting advance in the last decade has been the rise of privateventures to satisfy the increasing demand for efficient launch ve-hicles. With the end of the Space Shuttle Program approaching,NASA needed to guarantee its launch capabilities to supply theInternational Space Station (ISS). In December 2008, two pri-vately owned companies, SpaceX andOrbital Sciences (currentlyOrbital ATK),were awardedCommercialResupply Services con-tracts to conduct unmanned cargo launches to the ISS. Russiahad been in charge of this task for more than 20 years, with over50 launches. On May 22, 2012, Elon Musk’s SpaceX became thefirst company to send commercial cargo to the ISS, followed byOrbital Sciences’ Antares rocket launch in September 2013. In2014, NASA awarded $4.2B and $2.6B contracts to Boeing andSpaceX, respectively, in order to regain the capability of launch-ing astronauts to the ISS from US soil. Boeing is developing theOrion capsule, whereas SpaceX is focusing on the crewed versionof the Dragon capsule. In the future, the Orion capsule will sendastronauts to an asteroid as part of the Asteroid Redirect Mis-sion (ARM),whereas theDragon capsulewill evolve into theRedDragon, designed to carry humans to Mars.

These initiatives have improved the competitiveness betweenlaunch providers significantly, resulting in reductions of the costof the launchers. Table 1.1 compares the cost-per-kilogram of

1

2 1 Introduction. Current challenges in space exploration

launches to low Earth orbit (LEO) with different vehicles fromvarious providers. It is worth noticing that the ultimate goal ofthe Space Shuttle Program was to reduce the launch costs thanksto partially reusing the launch vehicle. Due to several factors likechanges in the design or high maintenance requirements, the fi-nal cost was around 18,000 USD/kg to LEO, much higher thanthe Russian Proton. This factor, together with safety concerns,resulted in the cancellation of the program in 2011. SpaceX fol-lowed a different approach toward reusability: the first stage ofthe rocket lands autonomously and, after refueling, it will beready for the next launch. Given the success of Falcon 9, the fu-ture FalconHeavywill eventually reduce the launch cost down to2,500 USD/kg, the most affordable option to date.

Table 1.1: Launch cost to LEO (in USD/kg).

Ariane 5 Delta IV-H Falcon 9 Falcon Heavy Vega ProtonM

10,500 13,800 4,100 2,500 15,600 4,300

A direct consequence of the launch cost reductions is spacebecoming more accessible. This factor, combined with the ad-vances in miniaturization, has led to the flourishing of cube-sats. Cubesats are smaller, cheaper, and easier to build thanregular spacecraft and have many applications, ranging frommerely university experiments to astronomical observations, re-mote sensing, and communications. Their reduced size andmassmake them perfect candidates for secondary payloads, or evenfor exploring alternative deployment strategies. The NanoRacksCubeSat Deployer aboard the ISS is capable of deploying 6Ucubesats, which canbe sent as cargowith the supply capsules. An-other popular concept is the use of rocoons, the combination ofa rocket and a balloon. The Spanish company Zero 2 Infinity hasalready conducted successful experiments based on this concept.The rapid development in these areas have turned small satellitesinto key players in space industry.

1.2 Distributed space systems

The full potential of miniaturized spacecraft relies on operatingseveral satellites that perform cooperative tasks, forming a dis-tributed space system. The relative motion between small space-craft (down to the pico-scale, weighing around 100 g) has receivedmuch attention for industrial applications in recent years. Someadvanced concepts promoted by DARPA, among others, evenconsider the use of spacecraft swarms, with tens of thousands ofspacecraft.

Formation flying introduces a new paradigm of space mis-sion design, and it is rapidly replacing monolithic solutions inmany scenarios. Even the startup world is taking advantage ofthe wide range of possibilities provided by this concept. The SanFrancisco-based Planet Labs, for example, currently operates aconstellation of almost a hundred 4 kg-spacecraft, which imagethe Earth continuously.

But the potential of distributed space systems is not merelyeconomic. It also opens a whole new world of possibilities froma scientific and operational perspective. The German TanDEM-X formation flying mission (launched in 2010) served as a proof

concept, and it also generated a high-accuracy digital elevationmodel of the Earth using synthetic aperture radar. The GravityRecovery and Climate Experiment mission (GRACE) consists intwo spacecraft flying in formation taking precise measurementsof their relative states. This data allows scientists to generate themost precise gravity model of the Earth to date. This same con-cept was exploited by the Gravity Recovery and Interior Lab-oratory (GRAIL), which mapped the Moon’s gravity field. InFebruary 2016, the LIGO team announced the detection of grav-itational waves for the first time, confirming Einstein’s theory ofgeneral relativity. To extend these pioneering results the evolvedLaser Interferometer Space Antenna (eLISA), consisting in threespacecraft in wide formation, will be launched in the 2030s to de-tect more accurately the ripples in space-time. LISA Pathfinderwas launched in 2015 to test the key technologies required byeLISA, in particular the formation-keeping capabilities.

When designing a telescope, the aperture of the instrumentis determined by the diameter of the mainmirror. This is limitedby obvious practical constraints, like the diameter of the launchvehicle. For instance, the diameter of the mainmirror aboard theHubble Space Telescope is 2.4 m, just enough to fit in the spaceshuttle cargo bay. The James Webb Space Telescope (JWST) willreplace Hubble in 2018, and the new observatory mounts a mir-ror of 6.5 m in diameter. The mirror cannot fit inside the Ari-ane 5 fairing, so the engineering team opted for a foldable mir-ror composed by hexagons. Although more flexible, this solu-tion is still limited by the launcher and future concepts explorethe use of multiple satellites. The Terrestrial Planet Finder con-cept (Fig. 1.1) involvedmultiple small infrared telescopes flying inprecise formation, simulating an unprecedentedly large apertureobservatory. In addition, having multiple collectors allows theastronomers to apply sophisticated reduction techniques in thedata pipeline to subtract bright stars. Unfortunately, thismissionwas canceled in 2011 due to budget issues.

Figure 1.1: Artistic view of the Terrestrial Planet Finder. Source:

NASA.

The Kepler space telescope has discovered up to 2,327 exo-planets, improving our knowledge about extrasolar worlds andunveiling astonishing examples of bizarre planetary systems. Thetelescope only instrument is a photometer, which transmitslightcurves of almost 150,000 stars back to Earth. The search forexoplanets is based on the transit method: if the apparent bright-ness of a star decreases periodically, this could mean that a planetis orbiting it. This method has some disadvantages, like the high

§1.3 Efficient orbit transfers 3

number of false positives (around 40% for Kepler) and the factthat the only planets that can be detected are those in edge-on or-bits.

An interesting alternative to transit detection is the directimaging of the planets. The telescope observes the planets them-selves, insteadof focusingon the central star. The challenge is thatthe stars are much brighter than the planets, and light reflectedon them is typically lost. To solve this problem, coronographscan be placed between the telescope and the source to block thelight of the star, and then the telescope is pointed to the plan-ets. The most flexible solution is to design an occulter that fliesin formation with the telescope, and blocks the light accordingly.The New Worlds mission features an occulter that will possiblyfly with the JWST enabling direct observations of exoplanets. Aproof of concept mission is also under development at Stanford,theMiniaturizedDistributedOcculter/Telescope (mDOT). Therequired precision in both the relative positioning and pointingof the spacecraft make formation flying for astronomical applica-tions one of the major challenges in future missions.

Relative motion between spacecraft will soon have inter-planetary applications. The InSight mission incorporates twocubesats, which will become the first to fly in deep space. Pro-videdwith bothUHF andX-band antennas, theMars CubeOne(MarCO) will serve as a communication relay for InSight, spe-cially during entry, descent and landing. The cubesats will followtheir own orbits to Mars, and this poses an important challengein regards to the navigation and operation of multiple spacecraft.Even the design team of the Europamission, the latest of NASA’sflagship program, is considering the use of two cooperative space-craft. After reaching Jupiter’s moon Europa along a low-energytransfer, the main spacecraft will deploy a lander that will main-tain a communication link with the mothership. When the lan-der completes its experiments on the surface ofEuropa, itwill takeoff, rendezvous, and dock to the carrier. These are just some ap-plications showing the relevance of distributed space systems infuture mission concepts.

1.3 Efficient orbit transfers

Advances in launch systems have cut down the cost of leaving theEarth, and the use of spacecraft formations reduces the unitarycost of the craft and improves the scientific capabilities. But, oncein space, the problem of reaching the final orbit still remains. Amission comes to an end when the spacecraft runs out of fuel.Thus, the propellant expenditures during the cruise phase needto be minimized, so the craft has as much fuel as possible whenstarting its nominal operations.

Interplanetary travels are the best examples of the need forsmart transfer strategies. Finding the adequate launch window isone of the key elements. For example, a 154-day transfer to MarslaunchedonApril 18, 2018will require amass of propellant that is55% of the total mass of the spacecraft. However, if the launch isdelayed by only 20 days, the propellant fractionwill rise to 93% tokeep the same time of flight, because of the configuration of theplanets. Trajectories are usually very sensitive to changes, whatcomplicates the design process. This is particularly critical dur-ing the preliminary phases, in which there aremany iterations be-

tween the different engineering and scientific teams.

1.3.1 Low-thrust missions

The specific impulse (Isp) of a propulsion system can be regardedas a measure of its efficiency: the higher the specific impulse, theless propellant is needed to exert the same change in the velocity ofthe spacecraft. Table 1.2 compares the specific impulse and maxi-mum thrust of different propulsion systems. Solid rocket boost-ers deliver high-thrust levels with low specific impulses, and aregood choices for launch vehicles. Once ignited, the combustioncannot be stopped. Rockets with liquid fuel (bipropellant) canbe switched on and off, and they have been the preferred choicefor spacemaneuvers. New concepts based on electric propulsion,like ion engines and the variable specific impulse magnetoplasmarocket (VASIMR), increase the specific impulse significantly, atthe cost of producing smaller thrust forces.

Table 1.2: Specific impulse and thrust of different propulsion systems.

Solid rocket Bipropellant rocket Ion engine VASIMR

(STS booster) (RS-25) (NEXT) (VX-50)

Isp [s] 250 350-450 4,000 5,000

Thrust [N] 13,800,000 1,900,000 0.250 0.500

A typical space maneuver consists in switching on a chemicalengine for a few minutes, which yields a change of velocity thatis almost instantaneous given the total duration of the mission.But its low specific impulse makes it less efficient than electricpropulsion systems. For these reason, missions like Deep Space 1,Hayabusa, or Dawn used the latter instead of the usual chemicalthrusters. Given the low thrust levels of ion engines, the thrusterneeds to be on for months instead of minutes, and the spacecraftfollows a spiral trajectory that slowly takes it to its final orbit. Themass of propellant can be reduced significantly, at the cost of in-creasing the time of flight.

In the industrial side, low-thrust electric propulsion has greatpotential for placing geostationary satellites into orbit. Thehigh altitude of the geostationary orbit (GEO), which is almost36,000 km, makes it impossible for a rocket to insert a satellite di-rectly into this orbit. The rocket puts the spacecraft into a highlyelliptical orbit with its apogee atGEOaltitude, and then the satel-lite’s engines are used to circularize the orbit. TheABS-3Ageosta-tionary satellite (Boeing 702SP bus) was launched inMarch 2015and used a revolutionary transfer strategy, as it was the first GEOsatellite using electric propulsion. The spacecraft spiraled for oversix months until it reached its operational orbit. Less propellantwas spent in this phase and, as a result, there is more fuel availablefor the required station-keeping maneuvers, increasing its opera-tional life. These corrective maneuvers compensate the drift ofthe satellites at GEO due to the Earth’s nonspherical gravity field.

The design of a transfer using chemical propulsion consistsin finding the optimal distribution of the maneuvers along theorbit, and in characterizing such maneuvers. When using electricthrusters, the engine is on for almost the entire duration of thetransfer. The steering and throttling of the engine needs to be

4 1 Introduction. Current challenges in space exploration

determined, and this process turns out to be a complex optimiza-tion problem.

1.3.2 Solar sailing

The Russian space pioneer Konstantin Tsiolkovsky envisionedpotential alternatives to rockets like the use of solar sails that, bythe effect of the solar radiation pressure, accelerate the spacecraftwithout spending fuel. The concept has been around for decades,but it involves some technical difficulties that have hindered itsimplementation in actual missions. In 2010, the JapanAerospaceExplorationAgency (JAXA) launched IKAROS, the first success-ful interplanetary mission provided with a solar sail. The space-craft deployed a 200m2 sail that propelled it to Venus. Later thatyear, NASA launched the NanoSail-D2 cubesat equipped witha 10 m2 solar sail. Figure 1.2 shows an artistic conception of theLightSail cubesat, launched in 2015 and promoted by the Plane-tary Society.

Figure 1.2: The LightSail spacecraft. Source: The Planetary Society.

The dynamics of spacecraft using solar sails are similar to themotion of probes with low-thrust electric propulsion systems.The force due to the solar radiation is small in magnitude, and itis accelerating the spacecraft continuously. This acceleration canbe controlled by changing the attitude of the sail. The techniquesused for low-thrust trajectory optimization canbe extrapolated tothe design of solar sail missions. The advantage of the latter is thefact that no propellant is required.

1.3.3 Gravity-assist trajectories

In 1966,Gary Flandropublished an interesting discovery hemadeduring a summer he spent at the Jet PropulsionLaboratory (JPL).Between 1977 and 1978, Jupiter, Saturn, Uranus, and Neptunewould be aligned in such a way that a flyby at Jupiter would senda spacecraft to Saturn, a second flyby at Saturnwould aim the tra-jectory toUranus, and a third flyby would send the probe towardNeptune. He coined the term Planetary Grand Tour, and thissequence was exploited during the design the trajectory of Voy-ager 2 (see Fig. 1.3). During the close approach there is an en-ergy exchange between the probe and the planet that is flown by.Because of the difference in their masses the change in the orbitof the planet is negligible, but the spacecraft experiences an im-portant change in its velocity. Although Pioneer 10 was the firstmission to use this technique, it showed all its potential with theVoyager program.

Figure 1.3: The orbit of Voyager 2 from 08/21/1977 (launch) to

12/01/1992, represented in the ICRF/J2000 frame.

Gravity-assist maneuvers have been widely used since the1970s. In fact, most interplanetary trajectories involve flybys ofintermediate planets in order to reduce theuse of its engines. Thismakes the design process even more complicated for the missionanalyst: determining the optimal trajectory is not about findingthemost direct way, but rather to come upwith the best sequenceof flybys, dates, impulsive maneuvers, etc. The preliminary de-sign becomes a counterintuitive task, with millions of possiblecombinations and solutions. Adequate analytical and numericaltools need to be implemented as the mission goals become moreambitious. Specially if the trajectory involves not only planetaryflybys but also arcs with low-thrust electric propulsion, whichneed to be optimized as well.

1.4 Orbit propagation

The Pioneer 10 and 11 spacecraft were the first to visit Jupiter andSaturn, respectively, in 1973 and 1979. After reaching the two gasgiants, they initiated their journey to interstellar space. But thespacecraft were not moving as they were expected to. An anoma-lous acceleration was reducing their speed faster than the modelspredicted. This phenomenon, known as the Pioneer anomaly,puzzled the scientific community for many years. In fact, it wasnot until 2012 when scientists agreed that the most likely causewas the effect of the emission of thermal radiation from the space-craft. The magnitude of the anomalous acceleration was just ofthe order of 10−10 m/s2, which shows how extremely accurateforce models and simulations need to be.

A careful analysis of Doppler data taken during the geo-centric flybys of Galileo (December 8, 1990), NEAR (January23, 1998), and Rosetta (March 4, 2005) revealed an unexpectedchange in their velocities, of the order of the millimeter per sec-ond. The origin of this energy gain remains a mystery, and hasmotivated many theories: from the existence of dark matter ha-los around the Earth to relativistic effects, issues with the sensorsat the DSN stations, propagation errors…These anomalies are, inthe end, unexpected differences between themotion predicted bythe models and the actual measurements.

The flyby anomaly and the Pioneer anomaly are famous ex-

§1.5 The aim of the present thesis 5

amples, but discrepancies between the computed trajectory andthe actual measurements always appear. In fact, measuring thesedifferences can provide valuable scientific data. For example, theJuno science team, which recently completed the Jupiter orbit in-sertion, will study the structure of Jupiter’s interior and windsbased on the deviations between the true and predicted orbits,which are obtained with numerical simulations. Numerical er-rors are always present, and minimizing this error is a critical taskbecause errors in the propagation may lead to wrong scientificconclusions.

From an operational point view, accurate and reliable orbitpropagators are used to compute the nominal trajectory. Thenavigation and control teams will be in charge of making surethat the spacecraft stays on its design course. If the nominal orbitis constructed with a poor force model, adjusting maneuvers willbe constantly needed, because the probewill never follow the pre-dicted path. This is even more critical when multiple flybys arepresent. A close encounter has an amplifying effect on the nu-merical error. Therefore, the propagation must be as accurate aspossible for the trajectory solution to be trusted. With exampleslike the Cassinimission, which has completed 120 flybys of Titan,22of Enceladus, andothers of Iapetus, Rhea andDione, ensuringthe accuracy of the propagation becomes a complicated problem.Figure 1.4 depicts the orbit of Cassini during its first 18months ofoperation, and shows the complexity of the trajectory.

Figure 1.4: The orbit of Cassini in the ICRF/J2000 frame centered at

Saturn (06/20/2004-12/01/2005).

Every effort toward more efficient and accurate propagationmethods benefits almost every application in celestial mechanics.Trajectory optimization problems are interesting examples. Inpractice, optimization algorithms will be given a cost function tominimize. This cost function typically includes orbit propaga-tions. Since the function will be evaluated thousands of times,the runtime needs to be as short as possible. In addition, numer-ical errors will make the optimizer converge to erroneous solu-tions, or it simplywill not converge to any solution. On the otherhand, the relative motion between spacecraft is sensitive to thedifferential effect of orbital perturbations. Numerical models arerequired for high-precision formation flying applications, whichagain involve numerical propagations.

1.5 The aim of the present thesis

The physics that govern the motion in space have not changedsince the time Kepler formulated his celebrated laws. It is onlyour perception, our models, what have evolved. And depend-ing on how a physical problem is described using a mathematicalmodel, different conclusions may be derived. This thesis focuseson the theory of regularization, which was born in an attemptto eliminate the singularities in the equations of orbital motion.This theory can be regarded as a collection of mathematical anddynamical contrivances that provide amore adequate descriptionof the dynamics. In the second chapter of his bookThe Prisoner,Marcel Proust writes:

“The only true voyage of discovery, the only fountain ofEternal Youth, would be not to visit strange lands but topossess other eyes [...]”

This quote captures the motivation of the dissertation. Takingregularized formulations as “new eyes”, this thesis approachesthree of the main challenges in modern astrodynamics: forma-tion flying, low-thrust mission design, and high-performance or-bit propagation. The use of an unconventional formulation ofthe dynamics has yielded new results in these three areas, pre-sented in the following chapters.

The main application of regularization for the last 60 years,basically the age of space engineering, has been the developmentof improved schemes for numerical integration. By recoveringthe basics of the theory, this dissertation shows how regulariza-tion canbe applied systematically to solve other problems of prac-tical interest, as well as how to exploit all the advantages of thesemethods for conducting numerical simulations.

The first part of the thesis is of theoretical nature, and focuseson regularization itself. Chapter 2 introduces the theory, justify-ing why it is worth seeking alternative representations of the dy-namics. The different methods and techniques that lead to regu-larization are presented, together with their advantages. Chap-ter 3 is focused on a specific formulation, the Kustaanheimo-Stiefel regularization (KS for short), and presents new results. Inparticular, a new theory of stability is developed resulting in anovel Lyapunov indicator for characterizing chaotic regimes. Thenext chapter, Chap. 4, analyzes the Dromo formulation and pro-poses some advances to the theory. In Chap. 5 an entirely newformulation is derived, based on the geometrical construction ofMinkowski space-time. It improves the numerical integration offlyby trajectories. The last chapter of the first part, Chap. 6, de-scribes themain software tool that has been developed alongwiththis doctoral research. It is a propagator with high-fidelity forcemodels anddifferent formulations, conceived for testing their nu-merical performance.

The second part of the dissertation includes the applicationsof regularization to spacecraft relativemotion and low-thrustmis-sion design. Chapter 7 presents the theory of asynchronous rel-ative motion, a new approach to the relative dynamics in space.In Chap. 8 this same problem is solved using different formula-tions, each presenting particular advantages. The search formoreconvenient descriptions of the problem of low-thrust transfersled to the discovery of a whole new family of spiral trajectories,

6 1 Introduction. Current challenges in space exploration

called generalized logarithmic spirals. The definition and mainproperties of the orbits can be found in Chap. 9. In Chap. 10,the spiral Lambert problem is solved using this new family of spi-rals, andChap. 11 presents a complete strategy for designing three-dimensional low-thrust transfers. These results can be general-ized to define other families of new orbits, with different physi-cal interpretations ranging from solar sailing to representing theSchwarzschild geodesics (Chap. 12).

Part I

Regularization

7

“Read Euler, read Euler, he is the master of us all”

—Pierre-Simon Laplace

2Theoretical Aspects of Regularization

P hillip Herbert Cowell (1870–1949) lived during an ex-citing period of dynamical astronomy, and was one of thevarious luminaries who contributed to the theory of the

motion of the Moon. As second chief assistant at the Royal Ob-servatory at Greenwich, he became intrigued by the discrepanciesbetween the observed trajectory of theMoon and the predictionsin Hansen’s tables. He corrected some coefficients of the peri-odic terms predicted by Hansen, and introduced new terms toaccount for long-period dynamics. With powerful perturbationtheories at hand, he then focused his attention on the imminentreturn of Halley’s comet (it approached the Earth in 1910). An-drew C. Crommelin noticed in 1906 a difference of almost threeyears in the predictions of the date of the approaching return ofthe comet and, being Cowell’s colleague at the Greenwich Obser-vatory, Crommelin suggested he recalculate the orbit.

In 1892, 1904 and 1905 three additional moons were discov-ered orbiting Jupiter. Astronomers were struck by these findings,because for almost 400 years the Galilean moons were thoughtto be the only Jovian satellites. But on the night of February 28,1908, an even more surprising discovery was made. A new moonwas observed with a period that seemed tens of times larger thanthat of the Galilean moons. A direct orbit with this character-istics could not be stable according to Laplace’s classical theory,so Crommelin made a revolutionary suggestion: the orbit mightbe retrograde.* He then teamed up with Cowell to explore theraremotion of this object. The perturbation from the Sun attrac-

*In 1975 thismoonwas calledPasiphae, and its orbit is indeed retrograde. Itsorbital period of 764.1 days is significantly longer than the period of Io (1.8 days),Europa (3.5 days), Ganymede (7.1 days) and Callisto (16.7 days). The moonwas discovered at the Royal Greenwich Observatory by astronomer Philibert J.Melotte, a colleague of Cowell and Crommelin.

tion was so large (between six and ten percent of the attractionfrom Jupiter), that they did not even consider relying on the an-alytical theories that worked so well for the Moon. Instead, theydecided to integrate the equations ofmotion in Cartesian coordi-nates bynumerical quadrature. Themethodpresented byCowelland Crommelin (1908) was later referred to as Cowell’s method,and it is possibly the most common method for propagating or-bits. It reduces to integrating the system of differential equations

d2rdt2 +

µ

r3 r = ap (2.1)

inwhich r = [x, y, z]⊤ is theposition vector in an inertial frame,µis the gravitational parameter, and ap are external perturbations.

Cowell himself was well aware of the power of this methodand he soon applied it to the propagation of the orbit of Halley’scomet (Cowell and Crommelin, 1910). It is interesting to notethat, in themean time, Tullio Levi-Civita was looking for alterna-tives to the use of Cartesian coordinates in the problem of threebodies (Levi-Civita, 1903, 1904, 1920). In the N-body problemcollisions may occur, meaning that the denominator in Eq. (2.1)will vanish. Thus, the equations of motion become singular andfail to reproduce the dynamics. An entire new branch of celestialmechanics was born from the pioneering studies by Levi-Civitaand Peter Hansen about the three-body problem: the theory ofregularization. In Sect. 2.4 we will discuss early contributions byLaplace.

Regularization is a collection ofmathematical and dynamicaltransformations that seek a more convenient formulation of theequations of motion. In the second half of the 19th century andthe first half of the 20th century regularizationwasmostly applied

9

10 2 Theoretical Aspects of Regularization

in the context of the three-body problem. With the rise of com-puters and the Space Age in the 1960s and 1970s, the numericalintegration of the equations of orbital motion received renewedinterest. Regularization was no longer a mere theoretical artifact,for scientists like Joachim Baumgarte, Eduard Stiefel, or AndréDeprit proved that regularized schemes were better suited for nu-merical integration. Different formulations were published inthose years with operational applications.

Reviewing all the availablemethods and techniques for regu-larizing the equations of motion (or for improving its numericalbehavior in general) will fill an entire book, and there will still besome missing formulations. This chapter is just devoted to pre-senting the foundations of regularization, together withmore de-tailed historical notions and a review of the state of the art in thefield. Section 2.1 deals with the question of whether it is worthor not to use sets of variables different from the Cartesian ones.The Sundman time transformation, one of the best known reg-ularizing transformations, is discussed in Sect. 2.2. Sections 2.3–2.5 present different techniques for improving the overall perfor-mance of the integration, including the stabilization of the equa-tions of motion by embedding first integrals, or the use of or-bital elements. Canonical transformations are briefly discussed inSect. 2.6, and the concept of gauge-freedom in celestial mechan-ics, developed by Efroimsky and Goldreich (2003), is reviewed inSect. 2.7.

2.1 Why bother?

Cowell’s method presents a series of issues related to its numericalbehavior and the form of the equations themselves. A singularityarises from the formof thepotential of a particle orbiting a centralmass with gravitational parameter µ = Gm,

V (r) = −µr

Indeed, at the origin of the potential well (r → 0) the functionV (r) takes infinite values. This is not only a theoretical singular-ity, because during close encounters and deep flybys the relativeseparation decreases rapidly.

This singularity is even more critical in the context of the N-body problem, where the potential takes the form

V (r) = −G∑i< j

mim j

||ri − r j||

andbecomes singularwhen anypair of particles (i, j) collide. Col-lisions in N-body simulations of stellar systems are usual, andthe singularity needs to be overcome. We refer to Aarseth (2003,chap. 5) for a more detailed discussion about the need for regu-larization in N-body systems.

The singularity also affects the velocity. The local circular ve-locity of a particle orbiting a central mass is simply

v =

õ

r

It increases as the orbit reduces its size. Thus, in the limit casewhen the orbit collapses to the origin (r → 0) the corresponding

orbital velocity becomes infinite. This phenomenon can also beunderstood in view of the conservation of the angular momen-tum, h = rv in a circular orbit. As the radial distance goes tozero, the velocity grows rapidly to compensate the angular mo-mentum.

Another issue related to the form of Eq. (2.1) is its behav-ior when integrated numerically. If one solves these differentialequations in the Keplerian case (ap = 0), which is the simplestcase possible, the evolution of the error is that shown in Fig. 2.1.It is computed as the difference between the solution propagatednumerically and the analytic solution. Evenwhen integratedwiththe Störmer-Cowell integrator and a fine tolerance (10-14), theerror grows exponentially in time, which yields a separation ofabout 500 m after 20,000 revolutions. The example correspondsto a geocentric orbit with e = 0.3 and a = 8000 km.

0 5 10 15 20

100

102

104

Thousands of revolutions

Err

or

[m]

Figure 2.1: Evolution of the propagation error of a Keplerian orbit.

The orbit is integratedwith a variable step-size Störmer-Cowell inte-

grator of order nine with a tolerance of 10-14.

The exponential growth of the error in Cowell’s method canbe understood by studying the relative dynamics between twoneighboring trajectories. Let r∗ define a Keplerian orbit, initiallyseparated a distance δr0 from a reference solution r. The relativemotion is governed by the equation

r∗ − r = − µ

r∗3r∗ +

µ

r3 r

Under the assumption that the separation is small, i.e. ||δr0|| ≪||r0||, the dynamics can be linearized about the reference trajec-tory. Retaining only the first order terms the problem reducesto

r∗ − r ≈ − µr5

[r2 I − 3(r ⊗ r)

] δr = A(r) δr

Here I is the second-order unit tensor (represented by the identitymatrix),⊗ is the dyadic product, anddenotes contraction.* Thissecond order dynamical system can be reduced to a first order sys-tem when introducing the state vector δx that contains both the

*Under this notation it is:

(a ⊗ b) c = a (b · c)

The dyadic product of two rank-one tensors (vectors) results in a rank-two tensor(matrix),

a ⊗ b =

a1b1 · · · a1bn...

. . ....

anb1 · · · anbn

It is often written in matrix form as a b⊤, with a and b column vectors.

§2.2 The Sundman transformation 11

relative position and velocity vectors, δr and δv:

d(δx)dt= L(x) δx

The linear operator L(x) is

L(x) =

0 I

A(x) 0

and models the linear dynamics of the system. The spectrum ofthe linear operator is

λ1 = −√

2µr3 , λ2 = +

√2µ

r3 , λ3,4 = −λ5,6 = i√µ

r3 (2.2)

Four eigenvalues are purely imaginary and yield oscillatorymodes. There are two eigenvalues with nonzero real parts, λ1and λ2; the real part of λ1 is negative, whereas the real part of λ2is positive. This yields an exponential growth of the separation,according to the law eλ2t. The system is unstable in the sense ofLyapunov. See Bond (1982) for a connection with the stabilityof the numerical integration methods. The stable and unstablemanifolds associated to the resulting saddle configuration relateto the right eigenvectors v1 and v2, associated to λ1 and λ2, re-spectively. If the components of the position vector in an inertialframe I are [x, y, z], the eigenvectors v1 and v2 read

v1 = [−kx,−ky,−kz, x, y, z]⊤

v2 = [+kx,+ky,+kz, x, y, z]⊤

with k = r3/2/√

2µ.The referred instability is closely related to the choice of the

independent variable. Consider two circular orbits of radii r andr + δr. Even if the relative separation is initially small, δr ≪ r,the evolution of the angular separation between the particles δϑcan be visualized thanks to a simple series expansion:

δϑ =

[õ

(r + δr)3 −√µ

r3

]t = −3

2

õ

r5 t δr + O(δr2) (2.3)

It grows secularly in time (the motion is Lyapunov unstable).However, the separation between the orbits, δr, is constant, andremains small. That is, the orbital motion is Poincaré (or or-bitally/structurally) stable. Let Σ be a Poincaré section that istransversal to the flow and let p1,p2, . . . be the consecutive in-tersections of the orbit with Σ. The Poincaré mapP : Σ 7→ Σ issuch that

pn+1 =P(pn)

and for a periodic orbit pi = p j ≡ p orPn(p) = p. When theseparation between the orbits ismeasured at a given time t it is de-fined as ||r∗(t) − r(t)||, and grows exponentially. However, whenthe separation is measured at a value of the true anomaly (whichis equivalent to analyzing the separation on a Poincaré section)it is ||p∗ − p||, and remains constant. This suggests that the in-dependent variable plays a critical role when formulating orbitalmotion.

Szebehely (1967, pp. 233–234) presents an elegant and uni-fied treatment of the Lyapunov and structural stabilities by intro-ducing the concepts of isochronous and normal correspondence.

The former yields a synchronization in physical time and there-fore Lyapunov stability, whereas the latter relates to the synchro-nism in an angular variable and involves structural stability.

2.2 The Sundman transformation

The three-body problem is the simplest among the N-body sys-tems, and collisions appear naturally. The Finnish astronomerKarl F. Sundman worked extensively on this problem and waswell aware of the importance of regularizing the equations ofmo-tion. He introduced the time transformation*

dt = r ds (2.4)

that was later named after him. The fictitious time s replacesthe physical time t as the independent variable, and behaves asan angle. For historical reasons we shall mention that Levi-Civita (1906) and Bohlin (1911) also considered this transforma-tion when regularizing the restricted three-body problem. Thelatter worked directly with the eccentric anomaly instead of con-sidering it a fictitious time.

The derivatives with respect to physical time transform into

ddt=

1r

dds

andd2

dt2 =1r3

(rd2

ds2 −drds

dds

)The typical notation for the derivatives with respect to fictitioustime is the prime ′, whereas dots refer to derivatives with re-spect to physical time. If Eq. (2.1) is rewritten in terms of the fic-titious time it transforms into

d2rds2 −

1rdrds

drds+µ

rr = r2ap (2.5)

This equation still presents problems for r → 0 although theorder of the singularity has been reduced, changing the denom-inator from r3 to r. Additional transformations are required toachieve the complete regularization of the system.

The Sundman transformation has a beneficial effect in thenumerical integration process. It yields what JörgWaldvogel calls“the slow motion movie” (Waldvogel, 2007); if the orbital mo-tion is discretized using a fixed step in physical time, most of thepoints will be gathered around the apoapsis of the ellipse. Unfor-tunately, at periapsis, where the dynamics are faster and requirea finer step size, there will be fewer points. This is shown by Ke-pler’s second law: given a differential dt the particle sweeps a dif-ferential of area which is constant,

dA =12

r2 dϑ =⇒ dϑ ∝ 1r2

*We refer to thememoir that themathematicianMagnusG.Mittag-Lefflerinvited Sundman towrite on the subject in 1913 (Sundman, 1913). The results pre-sented in that paper were a collection of previous results obtained by the authorand published in different works, mainly in Recherches sur le problème des troiscorps andNouvelles recherches sur le problème des trois corps. These two papers ap-peared inActa Societatis Scientiarum Fennicae, volumes 34 (1907) and 35 (1909)respectively. Although in this chapter we only recover the time transformation,wemust emphasize that themostmemorable result presented in Sundman’sMé-moire is the conclusion that all solutions to the regularized three-body problemcan be represented by convergent series. The time transformation from Sund-man (1913, eq. 71) yields a new form of the canonical system; Levi-Civita (1920,§2) referred to that transformation as the Darboux-Sundman transformation.

12 2 Theoretical Aspects of Regularization

anddϑ is angular separation in true anomalybetween consecutivepoints. When the particle is closer to the origin (periapsis) the an-gular separation grows. Conversely, when r is large (apoapsis) thepoints are close to each other, and yield the distribution shown inFig. 2.2(a). On the other hand, differentiating Kepler’s equationyields

dt = αr dE (2.6)

with E the eccentric anomaly and α a constant depending onthe semimajor axis and the mean motion. Identifying Eqs. (2.4)and (2.6) shows that the fictitious time s is in fact equivalent tothe eccentric anomaly E, except for a constant factor affecting itstime evolution. The discretization of the orbit with fixed steps infictitious time (equivalent to E) is shown in Fig. 2.2(b).

(a) Fixed step in t (b) Fixed step in s

Figure 2.2: Discretization of a highly elliptical orbit (e = 0.8) usingthe physical and the fictitious times and the same number of points.

The examples in Fig. 2.2 have been constructedwith the samenumber of points, meaning that the integrator will take the samenumber of steps. However, using the discretization in physicaltime (Fig. 2.2(a)) yields very large steps during periapsis passage.This introduces important numerical errors because the integra-tion fails to capture the rapid changes in the states. On the con-trary, a discretization in fictitious time (Fig. 2.2(b)) leads to a finerrepresentation of the dynamics around periapsis, and the resultwill bemore accurate. At apoapsis, where the dynamics aremuchslower, the wider step size of Fig. 2.2(b) reduces the number ofunnecessary steps, speeding up the integration.

2.2.1 Generalized Sundman transformation

The time transformation in Eq. (2.4) is often generalized to

dt = αrn ds (2.7)

where α is an arbitrary constant and n defines the order of thetransformation. Different values of n have been tested in the pastcentury. Before diving into the possible transformations we shallfirst borrow a theorem from Stiefel and Scheifele (1971, p. 78):

Theorem 1: (Stiefel and Scheifele, 1971, Thm. 3) Given theanalytic step-size adaption in Eq. (2.7), a full regularization ofthe equations of motion (2.1) can only be achieved for n < 3/2.

Theoriginal transformationbySundmancorresponds ton =1 and, according to this theorem, the complete regularizationof the equations of motion is possible. If the constant parame-ter α takes the value α = (a/µ)1/2, the fictitious time becomes

the eccentric anomaly. Many regularization methods are basedon the transformation n = 1. Some examples are the formula-tions byLevi-Civita (1904, 1906, 1920) and later extended byKus-taanheimo and Stiefel (1965), or alternative methods by Sperling(1961), Burdet (1967), Janin (1974), Bond and Gottlieb (1989),Bond (1990),Roa andPeláez (2015d) andBaù et al. (2015), amongothers.

The case n = 3/2 was explored by Nacozy (1977), whocoined the term intermediate anomaly for referring to the ficti-tious time in this case. The constant factorα readsα = µ−1/2(1+e2)1/2. The intermediate anomaly relates to the true anomaly byvirtue of

s = ek2 F(ϑ/2, k)

with k2 = 2e/(1 + e) and F(ϑ/2, k) the incomplete elliptic in-tegral of the first kind. Nacozy’s discovery was motivated by theconcept of the partial anomalies proposed by Hansen (1853), amethod that he had already applied to determine the orbit ofComet Encke (Nacozy, 1969).

Hansen (1853) derived a new general perturbation methodfor propagating cometary orbits. As the director of the SeebergObservatory he was concerned about the accuracy of the compu-tations. The orbits of comets are typically elliptical and cross thecircular orbits of the planets. Consequently, the planetary pertur-bations upon the comet vary in different time scales and involvemany frequencies. This phenomenon complicates the develop-ment of series solutions because they depend heavily on the nat-ural frequencies. In order to account for the different time scales,he introduced the inferior and superior partial anomalies relatedto the quadrants of the ellipse. In Hansen (1853, §3, p. 57) a de-tailed discussion of the possible situations regarding the in-orbitposition of the comet can be found. Later, Glydén (1870) im-proved the convergence of the series solution by introducing el-liptic functions. A review of the methods can be found in a dedi-cated publication by Baillaud (1876).

In the case n = 2 and α = 1/h, the fictitious time evolveslike the true anomaly. There are a number of regularized theoriesbased on this time transformation, for example those from Bur-det (1969); Vitins (1978); Palacios and Calvo (1996); Peláez et al.(2007) and Urrutxua et al. (2015b). It is interesting to note thatBurdet (1969) introduced the terms central and focal method, re-ferring to the regularization with n = 1 and n = 2 respectively.The names come from the fact that the eccentric anomaly rotatesaround the center of the conic section, whereas the true anomalyis referred to the focus. Vitins (1978) and Peláez et al. (2007)exploited the gyroscopic representation of the orbital plane, thelater taking advantage of the theory of ideal frames developed byHansen (1857). We refer to Jochim (2012) for a translation of themost important parts of the original manuscript, as well as fora detailed analysis of the fundamental results. It is worth notic-ing that this transformationwas already applied byLaplace (1799,book II, chap. 2, p. 150) when deriving his theory of orbital mo-tion.

The time transformation in Eq. (2.7) can be written in itsmost general form as

dt = g(r) ds (2.8)

§2.2 The Sundman transformation 13

where g(r) , 0 is an arbitrary function. Many different formsof the function g(r) have been explored, specially for regularizingN-body systems (see Sect. 2.2.2). For example, Cid et al. (1983)addressed the dynamics of particlesmoving in a central force-fieldgoverned by the potential

V (r) =∑

i

ai

ri

and introduced a time transformation with

gn(r) =rn/2√

rn−2 + b1rn−3 + . . . + bn−2

in order to achieve the full regularization of the Hamiltonian.Ferrándiz et al. (1987) unified the transformation g3(r) and theone proposed by Belen’kii (1981), g(r) = r3/2(1 + br)−1/2, byintroducing the time transformation:

dt =r3/2

√b0 + b1r

ds

A similar equation played a key role in the regularization and lin-earization (in Stiefel and Scheifele’s sense) of the radial intermedi-ary introduced by Cid et al. (1986).

The last section of the celebratedwork byKustaanheimo andStiefel (1965) has not receivedmuch attention in the past; this sec-tion is devoted to the formal treatment of nonconservative per-turbations in KS formalism. They suggested the use of a generictime transformation, in an attempt to find an adequate form ofg(r) so that the angular frequency of the harmonic oscillator ispreserved.

In their book, Stiefel and Scheifele (1971, §19) introduced anew time variable that relates to the total energy of the particle.Thus, disturbing potentials are introduced in the time transfor-mation improving the stability of the integration. Baumgarteand Stiefel (1974) published a dedicated analysis of other possi-ble transformations and discussed their advantages.

Satō (1998) published an ingenuous regularization of colli-sional orbits by representing Keplerian motion in the projectivespace. Thanks to this construction he arrived to the concept ofthe projective anomaly, a new parameterization of orbital mo-tion that successfully regularizes the problem when r → 0. Byconnecting the projective anomaly with the eccentric anomaly,Satō introduced the generalized anomaly. This variable includesnot only the eccentric and projective anomaly as special cases, butalso the true anomaly. Kopeikin et al. (2011, §1.3.5 and §1.3.6) re-viewed this concept in detail, and took advantage of its propertiesfor exploring the gauge freedom of orbital motion (see Sect. 2.7).

2.2.2 Time transformations in N-body systems

It was Sundman (1913, §11) himself one of the first to realize thatthe transformation in Eq. (2.4) does not work well when thereare more than two bodies. For an N-body system r is not unique,because there are N(N − 1)/2 relative distances ri j between theparticles. He focused on the three-body problem and defined analternative transformation in terms of the regularizing function

g(ri j) =(1 − e−r12/ℓ

) (1 − e−r13/ℓ

) (1 − e−r23/ℓ

)

Here ℓ is a constant depending on the physical properties of thesystem.

Finding an adequate transformationof the independent vari-able is a critical part of the regularizing strategy. As shown bySundman, when there are more than two bodies it is more con-venient to define a transformation involving the relative states ofall the bodies. Aarseth and Zare (1974) arrived to a remarkableregularization of the three-body problem by introducing the KStransformation in order tomodel the dynamics of a single binary.Binary collisions are completely regularized, and the integrationof triple encounters is well behaved numerically. They found aregular form of the Hamiltonian in KS variables relying on thetime transformation

dt = αrikr jk ds (2.9)

Under this notation the regularized double encounters are rik →0 and r jk → 0. Later that year, Heggie (1974) advanced onAarseth and Zare’s method by generalizing the regularizationto any pair of particles, and then extending the formulation togeneric N-body systems. For that he considered the new timetransformation

dt =∏i< j

ri j ds

that involves the relative separation between all the particles. Inthe particular case N = 3 he explored an alternative transforma-tion

dt =r12r13r23

(r12 + r13 + r23)3/2 ds

and found slight improvements in the accuracy and numericalperformance. The special role of the exponent 3/2, which laterled to the intermediate anomaly (Nacozy, 1977), will be discussedin more detail in Sect. 2.3. It is again remarkable that a transfor-mation similar to the one in Eq. (2.9) can be found in a note byLevi-Civita.*

An important breakthrough in the definition of time trans-formations was the introduction of conserved quantities such asthe energy or the Hamiltonian in the regularizing function g.Zare and Szebehely (1975) approached the problem from a rig-orous perspective looking for values of g that makes∑

i

piqi =

∫g(2T − V ) ds

integrable, where qi and pi are the coordinates and themomenta,and T is the kinetic energy. Noting that 2T − V = H + T =

2H + V = (3H +L)/2, with L the Lagrangian, the previousexpression transforms into∑

i

piqi =

∫g(H + T) ds = H(t − t0)+

∫gT ds

∑i

piqi =

∫g(2H + V ) ds = 2 H(t − t0)+

∫gV ds

∑i

piqi =

∫g(3H +L) ds =

32H(t − t0) +

12

∫gL ds

*Levi-Civita, T. “Sul problema piano dei tre corpi. Caso limite in cui unadelle masse eè infinitesima”, Rendiconti del Lincei, 1915, 24, p. 559.

14 2 Theoretical Aspects of Regularization

The form of the last term suggests the use of the regularizingfunctions g = 1/T, g = 1/V or g = 1/L in order to arriveto the trivial integrals

∫ds. Recall that the use of the potential

function in the regularizing process had already been explored byLevi-Civita (1903, p. 66) and Stiefel and Scheifele (1971, §19). Thetransformation g = 1/L was introduced in Heggie’s method byMikkola (1985). Thanks to this modified transformation, the re-ferred method now allows to handle simultaneous binary colli-sions. In addition, theHamiltonian adopts amore compact formthat makes its differentiation with respect to canonical coordi-nates and momenta simpler. The resulting method is easier toimplement than the original version byHeggie. The transforma-tion g = 1/L has a remarkable implication: the fictitious time isdefined by

s =∫ t

t0L dt

and it is equivalent to the action integral S . Thus, the action isused as the new independent variable. The principle of least ac-tion states that δS = 0, and therefore δs = 0. The path followedby the particle is the one that minimizes the fictitious time be-tween the departure and arrival points.

2.3 Stabilization of the equations of mo-tion

Baumgarte (1972b) proved in a seminal work that embeddingconserved quantities in the system of differential equations to beintegrated improves significantly the stability of the integrationprocess. Stability in this case refers to how well the integrals ofmotion or dynamical constraints are conserved. This techniquecomes from noting that the numerical integration scheme has ingeneral no information about the conservation laws. Thus, ifthey are introduced in the system they behave as control termsand reduce the error-growth rate. This technique proves usefulnot only in the Keplerian case, but also in presence of conser-vative perturbations where the total energy is conserved (Baum-garte, 1972a).

Stabilization can be understood from two different perspec-tives, as Velez (1974) observed: the equations can be stabilized nu-merically and analytically. The techniques aiming for one or theother type of stabilization are similar, and reduce to:

1. Transforming the independent variable by introducing a fic-titious time.

2. Linearizing the equation of motion in order to transformEq. (2.1) into a linear oscillator.

3. Embedding the energy equation into the system of equa-tions.

An alternative to steps 2 and 3 is to integrate sets of elements in-stead of coordinates (Sect. 2.5 is devoted to the use of elements inorbit propagation).

In the 1970s, Dr. C. E. Velez’s group did a lot of work onnumerical integration of orbits at Goddard Space Flight Center,while developing the Goddard Trajectory Determination System

(GTDS, see Wagner and Velez, 1972). For example, Velez (1967)presented an innovative predictor-corrector algorithm to inte-grate multi-revolution orbits more efficiently. Velez and Hilinski(1978) analyzed the impact of time transformations on the trun-cation errors when integrating Cowell’s method through a num-ber of numerical experiments. Thequestion about the stability ofthe numerical integration was further developed by Velez (1975).

The system of equations (2.1) is unstable in the sense ofLyapunov. Under an adequate transformation these equationsmay be transformed to a harmonic oscillator, which is stable (seeSect. 2.4 for a detailed analysis of the linearization of the equa-tions). The Lyapunov instability of the system mainly affects thepropagation of the truncation error and is reduced when intro-ducing a fictitious time. Recall the example in Eq. (2.3), whichproved that the angular separation between twoneighboring par-ticles grows in time proportionally to −(3/2)r−5/2µ1/2. Consid-ering now a Sundman transformation of the form dt = r ds, theseparation between the particles is governed by the equation

δϑs =

[(r + δr)

õ

(r + δr)3 − r√µ

r3

]s

= −12

õ

r3 s δr + O(δr2)

In a normalized example (µ = 1 and r = 1) it is simply δϑs =

−(s/2) δr, whereas the separation given in Eq. (2.3) evolves asδϑt = −(3t/2) δr. The fact that δϑs/δϑt = 1/3 is what moti-vatedBaumgarte’s statement: the introductionof a fictitious timedt = r ds reduces the Lyapunov instability of orbital motion by afactor three (Baumgarte, 1972a). A partial analytic stabilization isachieved. Clearly, when the true anomaly is the independent vari-able δϑ does not change, meaning that for dt = r2 ds the systemis completely stabilized in Lyapunov’s sense.

Although orbital motion referred to the fictitious time is sta-bilized, the time transformation needs to be integrated to providethe physical time:

t − t0 =∫

g(s) ds

and reducing theLyapunov instabilitymight not improve the nu-merical conditioning of this integral. The error in the propaga-tion of the time variable is coupled with the total error of the tra-jectory, and it is difficult to assess its impact in the most generalcase. An interesting result fromVelez (1974) in the case of anorbitperturbed by the J2 gravity terms of the central body is that theerror growth rate is minimum for n ∼ 1.5, although the systemis not strictly stabilized.

The stabilizing effect of embedding first integrals in the equa-tions of motion can be seen in the following example. Let us re-cover the equations of motion in terms of the fictitious time —Eq. (2.5)—,

r′′ − r′

rr′ +

µ

rr = r2ap (2.10)

The equation of theKeplerian energy can be understood as a con-trol term; multiplying this integral of motion by r renders[(r′ · r′)

2r2 − µr− Ek

]r = 0

§2.4 Linearization 15

This control term appears recursively in the literature and itis sometimes referred to as the Poincaré term (Baumgarte andStiefel, 1974; Janin, 1974). Adding the previous two equationsyields the controlled equations of motion

r′′ − r′

rr′ +

[(r′ · r′)

2r2 − Ek

]r = r2ap (2.11)

Figure 2.3 shows the difference in accuracy between integratingEq. (2.10) and Eq. (2.11), which includes the control term. In or-der to isolate the effect of the control term, the nominal orbit isconsidered circular. The particle orbits around the Earth and theorbital radius is 8000 km. The equations ofmotion are integratedusing a RKF 5(4) scheme, with absolute and relative tolerancesε = 10−11. The error growth rate of the controlled equations issmaller than the uncontrolled case. As a result, after for example5,000 revolutions, the error in position is reduced by about threeorders of magnitude, from about 500 m down to 0.5 m.

Figure 2.3: Error in position for the integration of the uncontrolled

equations (Eq. (2.10), in black) compared to the controlled equations(Eq. (2.11), in gray).

Using the energy equation as a stabilizing mechanism is par-ticularly powerful when dealing with conservative perturbations.The perturbing acceleration derives from a disturbing potential

ap = −∇Vp

The total energy of the system is constant and reads

E = Ek + Vp

Thus, the control term can be referred to the total energy,[(r′ · r′)

2r2 − µr− E + Vp

]r = 0

so the controlled system adopts the form

r′′ − r′

rr′ +

[(r′ · r′)

2r2 − E + Vp

]r = r2ap

Thanks to using the total energy instead of the Keplerian energy,the constant of motion E is embedded in the system. This tech-nique is widely applied in regularization theories. It is advanta-geous not only from the numerical perspective, as it might alsosimplify analytic formulations. In Sect. 2.2 a number of timetransformations involving the potential or the energy were dis-cussed. For instance, using the eccentric anomaly as the fictitioustime, for a Keplerian orbit it is

dtdE=

r√−2Ek

In presence of conservative perturbations it is possible to replacethe Keplerian energy by the total energy, defining a generalizedeccentric anomaly, φ. It is the variable that evolves in time like

dφdt=

√−2E

r

and φ(0) = E0. The element formulations proposed by Stiefeland Scheifele (1971, §19) and Baù et al. (2015), for example, arebased on this construction.

2.4 Linearization

The linearization of the equations of motion is crucial in the reg-ularization procedure. In this context, linearization does not re-fer to expansions for small values of a given parameter. It meanstransforming a system of nonlinear equations into a system oflinear equations, without neglecting any terms in the equations.The book by Stiefel and Scheifele (1971) and the essay by Depritet al. (1994) are two of themost completeworks on this particularsubject.

The first approach to the linear form of the equations ofmo-tion is possibly due to Laplace. We credit Deprit et al. (1994) forwriting the original form of the equations by Laplace in a morefriendly way.* Let C = uρ,uθ,uz be a rotating frame with uz

normal to the reference plane, uρ defined by the projection of theposition vector onto the referenceplane, anduθ completing adex-tral frame, uθ = uz × ur. Figure 2.4 depicts the geometry of theproblem.

Figure 2.4: Geometric decomposition of Laplace's linearization.

The projected radius is ρ = r · uρ, the projected angular momen-tum reads λ = h · uz, and the transversal component of the mo-tion is z = r ·uz. Introducing the variables ϱ = 1/ρ, ζ = z/ρ andthe transformation ds = λϱ2 dt, the equations of orbital motion

*The system (K) presented by Laplace (1799, p. 151) is written in scalar formand the components of the perturbing acceleration are represented by the deriva-tives of a force function Q. The time transformation introducing a new indepen-dent variable is rather complicated since it involves the integral of a perturbingterm.

16 2 Theoretical Aspects of Regularization

take the form:

d2ϱ

ds2 + ϱ = −ϱ (ap · uρ) −dϱds

(ap · uθ)

d2ζ

ds2 + ζ = −ζ (ap · uρ) −dζds

(ap · uθ) + (ap · uz)

dλds= λ (ap · uθ)

The resulting differential equations are linear, and explicitly de-coupled. In addition, if there are no external perturbations act-ing on the system (ap = 0) then the previous equations reduce toharmonic oscillators of unit frequency.

2.4.1 Levi-Civita variables

Tullio Levi-Civita was particularly interested by the problem ofregularizing collisions between pairs of bodies. In 1906 he sug-gested the introduction of a new set of coordinates for definingthe planar motion, by means of his celebrated transformation inthe complex plane (Levi-Civita, 1906):

x + iy = (ξ + iη)2 (2.12)

which provides

x = ξ2 − η2 and y = 2ξη

and therefore

x2 + y2 = (ξ2 + η2)2 (2.13)

He achieved a global regularization of themotionwhen introduc-ing a fictitious time, dt = r ds. Special attention was paid to thecanonicity of the transformed system. His regularization schemecan be applied to the general perturbed two-body problem. Writ-ing r = [x, y]⊤ and u = [ξ, η]⊤ the Levi-Civita transformationL : u 7→ r reduces to

r = L(u) u with L(u) =[ξ −ηη ξ

]Matrix L(u) is the Levi-Civita matrix. Applying this transforma-tion to Eq. (2.5) yields:

d2uds2 + ω

2 u =r2L⊤(u) ap (2.14)

It is the equation of an oscillator of frequencyω2 = −E/2, withE the energy of the system (total or Keplerian), forced by the ex-ternal perturbations:

dω2

ds= −[L(u) u′] · ap (2.15)

The derivation of Eq. (2.14) and some interesting properties ofthe Levi-Civita matrix can be found in Chap. 3, and also in thededicated monograph by Stiefel and Scheifele (1971, §8). In theKeplerian case ap = 0 and Eq. (2.14) reduces to a harmonic os-cillator. The differential equations are linear and decoupled, animportant improvement with respect to Eq. (2.1). In the latter

the errors in one variable propagate rapidly to the others. In ad-dition, Eq. (2.14) has effectively embedded the energy. Its evolu-tion can be integrated separately in order to rake advantage of thestabilization mechanism described by Baumgarte (1972b). Fig-ure 2.5 compares the error growth rate of Cowell’s method (al-ready shown in Fig. 2.1) with that of Eq. (2.14). The dynamicalinstability has been weakened, as expected, and the final error inposition is reduced by two orders of magnitude. However, theerror still grows in time even for this simple case because of theinstability of the time transformation.

0 5 10 15 20

100

102

104

Thousands of revolutions

Err

or

[m]

Figure 2.5: Evolution of the propagation error of the Keplerian orbit

discussed in Fig. 2.1. The black line corresponds to the integration of

Eq. (2.1), and the gray line shows the accuracy of Eq. (2.14). The lat-ter is integratedwith a variable order and variable step-size Adams

scheme, setting the tolerance to 10-13.

The volume XXIV of Rendiconti del Lincei, published in1915, includes a total of five notes by Levi-Civita. In the first one,entitled “Sulla regolarizzazione del problema piano dei tre corpi”and presented to the Academy in July, he recovered the transfor-mation in Eq. (2.12) and refined the regularization. The secondnote dives into more profound concepts related to the dynamicsof a special type of holonomic system. The third (November 21)and fourth (December 5) notes focus on the canonical formalismof the three-body problem regularized bymeans of Eq. (2.12). Fi-nally, his fifth note “Sul problema piano dei tre corpi. Caso lim-ite in cui una delle masse è infinitesima” focuses on the restrictedproblem and explores more general forms of his transformation,namely

x + iy = f (ξ + iη)

He even introduced a set of elliptic coordinates thanks to thetransformation x + iy = r/2 cosh(ξ + iη). Most of these resultswere collected in his renowned 1920memoir. This memoir startsby reviewing the previous efforts by the author toward regular-ization of the three-body problem, and discusses the work doneby P. Painlevé, R. Ball, and K. Sundman. It is interesting to notethat Levi-Civita pointed out that KarlWeierstrass, famous for hiswork on elliptic functions, also contributed to the regularizationof the N-bodyproblem. Indeed, in a letter to SophieKowalevski*,he briefly discussed the topic and hoped to advance on it in thefuture.

There is an intriguing phenomenon related to the transfor-mation in Eq. (2.12): transformations of this type exist only fordimensions 1, 2, 4, and 8, as proved by Hurwitz (1898) and later

*The letter datedDecember 16, 1874, canbe found inWeierstrass’ biographyby Mittag-Leffler (1912, p. 30)

§2.4 Linearization 17

by Adams (1960). Thus, is it possible to extend Levi-Civita’s reg-ularization to three-dimensions? The answer to this problem re-quired the joint efforts of Paul Kustaanheimo and Eduard Stiefel(Kustaanheimo and Stiefel, 1965). We save the detailed explana-tion of the KS transformation for Chapter 3, where the need foran extra dimension is explained.

Transformations in the complex plane were also exploited byGeorge D. Birkhoff. In 1915 he published a novel regularizationof the restricted three-body problem. First, he sets the origin ofthe system not at the barycenter (the common practice) but atthemidpoint between the primaries. By introducing the complexvariable q = x+ iy+1/2−ν (with ν themass ratio of the primarybodies) he succeeded in reducing the problem to

q + 2iq = ∇qV (q)

A concise summary of the exposition by Birkhoff (1915) was pro-vided by Szebehely (1967, pp. 97–103). Birkhoff acknowledgesthat a fewmonths before the publication of his paper Levi-Civitareferred him to a previous global regularization admitting doublecollisions: it was the Thiele-Burrau regularization.* Other trans-formations like the Lemaître regularization are based on similarconcepts, and were beautifully unified by Szebehely (1967, §3.8).

2.4.2 Cartesian coordinates

Linearization can also be achieved relying on the usual positionand velocity vectors, as Laplace’s results suggest. A major break-throughwas that ofHans Sperling and Claude Burdet, whowereable to achieve the linear and regular form of the equations ofmotion (Sperling, 1961; Burdet, 1967). Sperling embedded theintegral of motion of the energy and transformed the time vari-able, seeking a generalized solution to Kepler’s problem. Burdet,who was Eduard Stiefel’s doctoral student and was deeply influ-enced by his work in regularization, followed a similar approachbut he also introduced the eccentricity vector in the formulation.We shall now derive this formulation as an illustrative exampleof the steps toward regularization described in Sect. 2.3: intro-ducing a fictitious time, linearizing the nonlinear equations, andembedding integrals of motion in the formulation.

First, the integrals of motion of Kepler’s problem are writtenin terms of the fictitious time, defined by dt = r ds. The energytakes the form

E =v2

2− µ

r=

(r′ · r′)2r2 − µ

r(2.16)

Similarly, the Laplace-Runge-Lenz vector is defined as

e =v × hµ− r

r=⇒ µe =

1r2

[(r′ · r′) r − (r′ · r) r′

] − µr

r

(2.17)*Thiele (1895) presented a transformation involving a complex transforma-

tion of the form ξ + iη = cos(E + iF), with ξ and η relating to the coordinates(x, y), and E and F defining the relative orientation of the bodies. This solu-tion assumed that the primaries had equal masses, as it emanated from the Eulerproblem of three-bodies. In 1906 C. Burrau extended the method in order toaccount for cases different from ν = 1/2. It is remarkable that Thiele alreadymade use of the time transformation dt = r12r13 ds that would be adopted laterby Levi-Civita.

Let us now recover Eq. (2.5) and write it in compact form:

r′′ − (r · r′)r2 r′ +

µ

rr = r2ap

The second term can be solved from Eq. (2.17), and when intro-duced in the previous expression yields

r′′ − 2[(r′ · r′)

2r2 − µr

]r = r2ap − µe

The terms in brackets correspond to the definition of the energygiven in Eq. (2.16), meaning that

r′′ − 2Er = r2ap − µe (2.18)

This equation needs to be integrated together with the Sundmantransformation in order to compute the physical time. To achievethe stabilization described by Baumgarte (1972b) one should alsointegrate the evolution of the Laplace-Runge-Lenz vector,

µe′ = 2(r′ · ap) r − (r · ap) r′ − (r′ · r) ap

together with

E ′ = (r′ · ap)

Practical comments about the method can be found in the bookby Bond and Allman (1996, §9.3). Baumgarte (1972a) and Janin(1974) followed almost the same procedure, although they didnot introduce the eccentricity vector.

2.4.3 Universal solutions

Transforming the equations of orbital motion to an oscillatorpresents additional advantages apart from regularization and nu-merical stability. In absence of perturbations the angular fre-quency of the oscillator is defined in terms of the Keplerian en-ergy Ek, which is constant. The sign of the frequency squared isgiven by the sign of the energy, or equivalently by the type of or-bit (elliptic, parabolic and hyperbolic). Because Eq. (2.14) reducesto

u′′ + ω2u = 0

with ω2 = Ek/2 constant, this equation is integrated easily andthe solution depends on the type of orbit:

u(s) =

u0 cos(ωs) + (u′0/ω) sin(ωs), Elliptic, ω2 > 0

u0cosh(ωs) + u′0 s, Parabolic, ω2 = 0

u0 cosh(ωs) + (u′0/ω) sinh(ωs), Hyperbolic, ω2 < 0

The solution can be unified by introducing the Stumpff func-tions Ck(z), with k their degree and z = (ωs/2)2 their argument.Namely

u(s) = u0C0(z) + s u′0C1(z)

A dedicated analysis of the Stumpff functions including some as-pects on their implementation can be found in Appendix C .

18 2 Theoretical Aspects of Regularization

Equation (2.18) can be solved in a similar way for the caseap = 0, resulting in

r(s) = r0C0(4z) + s r0C1(4z) v0 − µs2C2(4z) e (2.19)

It provides a universal vectorial solution to Kepler’s problem (seethe work by Condurache and Martinuşi, 2007a, for a thoroughstudy). In addition, the Sundman transformation dt = r ds canbe integrated in closed form and yields the universal form of Ke-pler’s equation

t − t0 = sr0C1(4z) + s2r′0C2(4z) + µs3C3(4z) (2.20)

This equation avoids the usual complications related to replacingthe eccentric anomaly by the hyperbolic anomaly, or switching toBarker’s equation. As an exercisewe shall prove that this equationindeed reduces to Barker’s equation in the parabolic case: the en-ergy vanishes, meaning that ω = 0; provided that Ck(0) = 1/k!—see Eq. (C.1)—, and assuming that the particle departs fromperiapsis (r0 = rp) it follows

t − tp =s6

(6rp + s2) (2.21)

The identity r′0 = r′0 · r0/r0 = r0 · v0 = 0 holds because of beingcomputed at periapsis. The time of periapsis passage is tp. Thefictitious time relates to the true anomaly thanks to

dϑdt=

hr2 =⇒

dϑds=

hr

Inverting this expression and integrating from periapsis to ϑ it is

s = (h/µ) tanϑ

2

Introducing the semilatus rectum p one can write rp = p/2 andh = (pµ)1/2, and Eq. (2.21) finally becomes

t − tp =16

√p3

µ

(3 tan

ϑ

2+ tan3 ϑ

2

)

which is, indeed, Barker’s equation (Battin, 1999, p. 150).

2.5 Sets of orbital elements

Elements are the integration constants in orbital motion. Stiefeland Scheifele (1971, p. 83) extended the concept of element bystating that an element is any variable that remains constant orgrows linearly with the independent variable in the Kepleriancase. Under this definition the mean anomaly, for example, isconsidered an element.

Consider the set of classical elements oe = a, e, i, ω, Ω, M0,namely the semimajor axis, eccentricity, inclination, argumentof periapsis, right ascension of the ascending node, and (initial)mean anomaly. Their time evolution is governed by the Gauss

form of Lagrange’s planetary equations:

dadt=

2a2

pr[re sinϑ (ap · i) + p (ap · j)

](2.22)

dedt= sinϑ (ap · i) +

1p[(p + r) cosϑ + re

](ap · j) (2.23)

didt=

rp

cos(ϑ + ω) (ap · k) (2.24)

dωdt= −1

ecosϑ (ap · i) +

p + rep

sinϑ (ap · j)

− rp

cot i sin(θ + ω) (ap · k) (2.25)

dΩdt=

rpsin(ϑ + ω)

sin i(ap · k) (2.26)

dM0

dt=

√1 − e2

ep[(p cosϑ − 2re)(ap · i) − (p + r) sinϑ(ap · j)

](2.27)

in which p is the semilatus rectum of the orbit. The basis L =i, j,k is the orbital frame, defined by i = r/r, k = h/h andj = k × i. The modified perturbing term reads

ap = h/(n2a3) ap

Equations (2.22–2.27) can be integrated numerically in lieuof Eq. (2.1), obtaining the osculating classical elements at each in-tegration step. The state vector is then solved analytically fromthe dynamics in a Keplerian orbit. When there are no perturba-tions acting on the system Eq. (2.1) reduces to

d2rdt2 = −

µ

r3 r (2.28)

whereas the system of equations (2.22–2.27) abides by

doed t= 0 (2.29)

Equation (2.29) is integrated trivially and the values of the ele-ments will not be affected by numerical errors. On the contrary,the right-hand side of Eq. (2.28) is non-zero and the solutionmaysuffer from truncation errors. The reduction of the computa-tional time is also a direct consequence of using Eq. (2.29) insteadof Eq. (2.28). Figure 2.6 shows this phenomenon with a trivialexample. A Keplerian orbit is integrated using Cowell’s method(with the Störmer-Cowell integrator of order nine) andusing a setof elements. The error in the integration of the Cartesian equa-tions of motion grows in time. Conversely, the only source of er-ror in the propagation using elements is the numerical resolutionof Kepler’s equation, and the round-off error in the transforma-tion of the elements to the state vector.

When perturbations are present the right-hand side ofEqs. (2.22–2.27) will no longer be zero. However, it scales withthe magnitude of the perturbation. In practical scenarios themagnitude of the perturbations will typically be smaller than theKeplerian term. Thus, the derivatives of the elements will besmall in magnitude compared to the derivatives of the coordi-nates. The evolution of the elements will be smoother, becausethey evolve with the time scale of the perturbation. All these fac-tors are highly beneficial from a numerical perspective.

§2.5 Sets of orbital elements 19

Figure 2.6: Propagation error along a Keplerian orbit in Cartesian

coordinates (black) and using a set of elements (gray).

The classical elements have been the preferred choice for cen-turies due to their simplicity and intuitive interpretation.* Theintrinsic singularities of the classical elements (for e = 0 and fori = 0) can be readily observed in Eqs. (2.22–2.27) and motivatedresearchers to propose alternative sets of elements. Broucke andCefola (1972) suggested the use of a new set of elements, calledthe equinoctial elements, that avoid the referred singularities ofthe classical elements. In the original notation proposed by theauthors they form the set a, h, k, λ0, p, q, and read

h = e sin(ω + Ω), k = e cos(ω + Ω), λ0 = M0 + ω + Ω,

p = sinΩ tani2, q = cosΩ tan

i2

A slightly improved version from the implementation point ofview was proposed by Walker et al. (1985).

Like Euler, Lagrange, Laplace, D’Alembert, Clairaut andmany other great names, Charles-EugèneDelaunay devoted him-self to improving the theory of lunar motion. In the very firstchapter of his book (Delaunay, 1860, pp. 1–13) he suggested theuse of a new set of elements for parameterizing orbital motion,ℓ, g, h, L,G,H (in the original order and notation). They relateto the classical orbital elements by means of

ℓ = M, g = ω, h = Ω

L =õa, G = L

√1 − e2, H = G cos i

Their evolution in terms of the disturbing potential is also givenexplicitly byDelaunay (1860, p. 13, eq. 9). The canonicity of theseelements in Kepler’s problem, proved by the simplified form ofthe resulting Lagrange and Poisson brackets, makes them appeal-ing for analytic studies. The interested reader is referred to workssuch as those byBrouwer (1959),Deprit (1981), Alfriend andCof-fey (1984) orGurfil andLara (2014) for examples of their use. TheDelaunay equations become singular for small eccentricities andinclinations, and the Poincaré elements Λ, ξ, p, λ, η, q are often

*As a historical comment we recall that it was Johannes Kepler the one whoparameterized the orbits using the eccentricity and the axes. “Ergo ellipſis eſtPlanetæ iter”. With this sentence he stated that the planets follow elliptic or-bits around the Sun. It is found in chapter 58 of hisAstronomia Nova, publishedin 1609. His reasoning was based on observations of the orbit of Mars. It wasnot until 1775 that Leonhard Euler published his theory about the rotation ofsolids, entitled “Dumouvement d’un corps solide quelconque lorsq’il tourne au-tour d’un axe mobile” and presented in Memoires de l’academie des sciences deBerlin, vol. 16, pp. 176–227. He introduced what were later called the Euler an-gles. Gauss (born just two years after) already made extensive use of the anglesi, ω, Ω in most of his work, and they also appear in the work of Laplace (1799,chap. 4).

used instead, being (Arnold et al., 2007, p. 260):

Λ = L, λ = ℓ + g + h

ξ = +√

2(L −G) cos(g + h), η = −√

2(L −G) sin(g + h)

p = +√

2(G − Θ) cos h, q = −√

2(G − Θ) sin h

With only 21 years, Bengt Strömgren published a special per-turbation method based on a vectorial set of elements. He pa-rameterized the orbit in terms of the instantaneous angular ve-locity associated to the rotation of the orbit (dΨ/dt in his origi-nal notation), the eccentricity, themeanmotion, andmean longi-tude at epoch (Strömgren, 1929). The angularmomentumvectorand the Laplace-Runge-Lenz vector, constant in magnitude anddirection in the Keplerian case, were exploited by Milankovitch(1939). In the work by Roy and Moran (1973) not only a morecompact formulation of Milankovitch’s method was presented,but they also derived a modified method for dealing with recti-linear and quasi-rectilinear orbits. In the introduction they dis-cussed the connectionwithHansen’s lunar theory and the param-eterization byHerrick (1953). The latter introduced the semima-jor axis and semiminor axis vectors as the elements defining theorbit, and the mean anomaly behaved as independent variable.Recently, Rosengren and Scheeres (2014) exploited the proper-ties of Milankovitch’s vectorial elements in Hamiltonian formal-ism. As an example of application they solved in closed-form thesecularmotion of a particle accelerated by the solar radiation pres-sure.

Laplace noted the potential of accounting for the radial mo-tion and the rotation of the radial direction separately (see for ex-ample the derivations leading to the system of equations labeled(H) in his “Traité de Mécanique Céleste”, p. 149). This tech-nique was developed in all its splendor by Deprit (1975), who re-covered the concept of Hansen ideal frames (see Sect. 4.1), andbyFerrándiz (1988), who introduced the concept of the projectivedecomposition of the dynamics when developing his dimension-rising canonical transformation. Deprit’s elements and later thoseproposed by Vitins (1978) used quaternions for modeling the dy-namics of the orbital plane. These concepts were later unifiedby Peláez et al. (2007), who published a new special perturbationmethod that they calledDromo. Chapter 4 is devoted to this par-ticular formulation.

Time elements

From a numerical point of view, the main advantage of element-based parameterizations of orbitalmotion is that the propagationin the Keplerian case is trivial. Moreover, in the presence of weakperturbations the problem is still well behaved since the deriva-tives are small in magnitude and evolve in the time scales of theperturbations.

When the physical time t is replaced by a fictitious time s theSundman transformation needs to be integrated numerically inorder to compute the physical time. In its simplest form the equa-tion to be integrated is

dtds= r

20 2 Theoretical Aspects of Regularization

The right hand-side does not vanish (except for collision trajecto-ries): the derivative is not necessarily small, it evolves in the timescale of orbital motion, and suffer from truncation errors even inthe Keplerian case.

In order the time transformation to benefit from the proper-ties of element-like variables, a time element is introduced. Stiefeland Scheifele (1971, §18) presented possibly the first dedicatedanalysis of the theory of time elements. They opened their pre-sentation explaining how the mean anomaly can be consideredan element, and somehow behaves as a simplified time element.They then defined a time element τ attached to the set of naturalelements in KS space.

Given a generic time element τ, if the perturbations vanishthe evolution of the time element will reduce to

dτds= γ

where γ is constant. If γ , 0 then τ is a linear time element,whereas γ = 0 yields a constant time element. The physical timeis recovered thanks to

t = τ + ft(s, r, r′)

Here ft(s, r, r′) denotes a generic nonlinear function. For exam-ple, using the eccentric anomaly as the independent variable andthe time of periapsis passage as the time element (τ ≡ tp), it is

t = τ +1n

(E − e sin E)

The introduction of a time element was a technique that Baum-garte (1976a) explored as part of his efforts toward stabilizing theequations of motion. Nacozy (1981) considered different defini-tions of the time element based on the eccentric, the true, and theintermediate anomalies. He supported his analysis with numer-ical examples showing the improvements in the numerical per-formance when propagating an orbit perturbed by the Earth’s J2.Numerical explorations byFukushima (2005) yielded two impor-tant conclusions about the use of time elements (in the contextof the KS regularization): first, the error growth rate is reducedfrom quadratic to linear; second, it removes the periodic compo-nents of the error in unperturbed problems. A detailed review ontime elements can be found in the work by Baù and Bombardelli(2014), where they also attached a constant and a linear time ele-ment to the set of Dromo elements.

2.6 Canonical transformations

Let q = [q1, q2, q3]⊤ and p = [p1, p2, p3]⊤ be the set of gener-alized coordinates andmomenta. Given the HamiltonianH andthe canonical forces (Q1,Q2,Q3) and (P1, P2, P3), the canonicalequations of motion read

dq j

dt= +

∂H

∂p j− Q j (2.30)

dp j

dt= −∂H

∂q j+ P j (2.31)

with j = 1, 2, 3. The canonical force P j relates to the non-conservative perturbations and Q j is introduced to preserve the

symmetry of the equations. In its most general form the Hamil-tonian can be written

H = H(t; q1, q2, q3, p1, p2, p3)

The time evolution of theHamiltonian along a solution in phasespace is governed by

dHdt=∂H

∂t+

∑j

(∂H

∂q j

dq j

dt+∂H

∂p j

dp j

dt

)Introducing Eqs. (2.30–2.31) in the convective term yields

dHdt=∂H

∂t+

∑j

(∂H

∂q jQ j −

∂H

∂p jP j

)(2.32)

This is the law of energy. If there are no canonical forces actingon the system (Q j = P j = 0) then this equation reduces to theconservation law:

dHdt=∂H

∂t

This expression proves that the Hamiltonian of a conservativesystem is constant if it does not depend explicitly on time.

Consider now a generic Sundman transformation

dtds= g(q,p)

that introduces a fictitious time s. The physical time t becomesa dependent variable. We shall attach the time to the coordi-nates q by writing q0 = t, so the extended vector results inq = [q0, q1, q2, q3]⊤. In order to preserve the symmetry of thesystem the conjugate momentum p0 is attached to p. FollowingStiefel and Scheifele (1971, chap. VIII), this construction yieldsthe homogeneous Hamiltonian

Hh = H + p0

Setting j = 0 in Eq. (2.30) leads to

dq0

dt=∂Hh

∂p0− Q0 = 1 − Q0

The fact that q0 ≡ t shows that it must be Q0 = 0. Similarly,Eq. (2.31) renders

dp0

dt= −∂Hh

∂q0+ P0 = −

∂H

∂t∂t∂q0+ P0 = −

∂H

∂t+ P0 (2.33)

If the canonical force P0 is defined as

P0 = −3∑

j=1

(∂H

∂q jQ j −

∂H

∂p jP j

)then Eq. (2.33) can be identified with Eq. (2.32) by making

p0 ≡ −H

and the canonical equation for p0 is the law of energy.Inwhat remains of the section the canonical forces Q j and P j

are neglected (Keplerian case). Using the coordinates (t, x, y, z)

§2.7 Gauge-freedom in celestial mechanics 21

and momenta (−H, x, y, z), the homogeneous Hamiltonian inthe Keplerian case reduces to

Hh =12

(p21 + p2

2 + p23) − µ√

q21 + q2

2 + q23

+ p0 (2.34)

The canonical equations readily lead to the equations of motionand the conservation of the energy,

dq0

dt= +

∂Hh

∂p0= 1,

dq j

dt= +

∂Hh

∂p j= p j

dp0

dt= −∂Hh

∂q0= 0,

dp j

dt= −∂Hh

∂q j= − µ

r3 q j

with j = 1, 2, 3.Let G be the time transformation

G : (t; q0, . . . , q3, p0, . . . , p3) 7→ (s; q0, . . . , q3, p0, . . . , p3)

The fictitious time s is the new independent variable, with dt =g(r) ds. The coordinates andmomenta are left unchanged. Con-sidering the transformed Hamiltonian

Hh = g(r)Hh(q0, . . . , q3, p0, . . . , p3)

the canonical equations take the form

dq0

dσ= +

∂Hh

∂p0= g(r),

dq j

dσ= +

∂Hh

∂p j= g(r)p j

dp0

dσ= −∂Hh

∂q0= 0,

dp j

dσ= −∂Hh

∂q j= − µ

r3 g(r)q j

with j = 1, 2, 3. The transformation is a trivial canonical trans-formationprovided that the coordinates andmomenta are invari-ant.

Since we have introduced a fictitious time, it is natural toexplore the canonicity of the transformation from the physicalvelocity v to the fictitious velocity r′, which leads to Eq. (2.5).There are different methods for checking the canonical nature ofa generic transformation of the coordinates and momenta,

T : (q, p) 7→ (q, p)

Denoting by A the matrix containing the partial derivatives ofthe transformed variables (q, p) with respect to the old variables(q, p), the transformationT is canonical if A is symplectic, i.e.

A⊤J A = J (2.35)

with

J =

[0 I

−I 0

]Taking q ≡ q and p = g(r)p the matrix A results in

A =1g(r)

g(r)I 0

−v ⊗ ∇rg(r) I

The form of the product

A⊤J A =1g(r)

[∇rg(r) ⊗ v − v ⊗ ∇rg(r) I

−I 0

]

shows that matrix A will only be symplectic for g(r) = 1. Con-sequently, transformations of this type are not canonical. If A issymplectic then its inverse

A−1 = −J A⊤J

is symplectic too. Thus, the canonicity of a transformation can beverified both with the Jacobian of the new variables with respectto the old ones and with the Jacobian of the old variables withrespect to the new ones.

The condition in Eq. (2.35) can be stated in terms of the La-grange brackets. This operator applies to any pair of variables(a, b) and is referred to a set of coordinates andmomenta. It reads

[a, b

]=

∑j

(∂p j

∂a∂q j

∂b−∂p j

∂b∂q j

∂a

)=∂p∂a· ∂q∂b− ∂p∂b· ∂q∂a

The Lagrange brackets satisfy[a, b

]= −[b, a

]Following the reasoning that yielded Eq. (2.35), the trans-

formation T is canonical if it leaves the Lagrange brackets un-changed, meaning that[a, b

]p,q =

[a, b

]p,q

A necessary and sufficient condition for the T-invariance of theLagrange brackets is[qk, qℓ

]= 0,

[pk, qℓ

]= δi j,

[pk, pℓ

]= 0 (2.36)

Introducing the Poisson brackets

a, b

=

∑j

(∂a∂q j

∂b∂p j− ∂a∂p j

∂b∂q j

)=∂a∂q· ∂b∂p− ∂a∂p· ∂b∂q

and exploiting the properties of symplecticmaps the condition inEq. (2.36) turns out to be equivalent toqk, qℓ

= 0,

pk, qℓ

= δi j,

pk, pℓ

= 0

2.7 Gauge-freedom in celestial mechanics

The elements are the constants of integration in Kepler’s prob-lem. Considering a particular set of elements oe, the position andvelocity vectorswill be definedbynonlinear functions of the form

r(t) = f(t; oe1, . . . , oen) and v(t) = g(t; oe1, . . . , oen)

The state vector is the solution to the unperturbed two-bodyproblem

d2rdt2 = −

µ

r3 r ≡ F

Here F denotes the acceleration on the particle.The motion under an external perturbation ∆F is governed

by the equation

d2rdt2 = F + ∆F (2.37)

22 2 Theoretical Aspects of Regularization

The solution to the perturbed motion can be obtained throughthe variation of parameters technique. The method was origi-nally developed by Euler in his works about the precession of theequinoxes (Euler, 1749) and further developed by Laplace in a se-ries of papers entitled “Théorie des variations séculaires des élé-mens des Planetes”,* in which he studied the evolution of the or-bital elements of the planets. In presence of perturbations theorbital elements are no longer constant, but functions of time.Thus, the solution follows from

r(t) = f(t; oe1(t), . . . , oen(t))v(t) = g(t; oe1(t), . . . , oen(t))

The time derivative of the position vector yields

drdt=∂f∂t+

∑ ∂f∂oei

doei

d t= g +Φ (2.38)

Note that it is the full derivative of the radius vector. It is definedin terms of g = ∂f/∂t and the convective terms, grouped under

Φ =∑

j

∂f∂oe j

doe j

d t(2.39)

In order for the velocity of the perturbed problem, dr/dt, to havethe same form than the velocity of the unperturbed problem, g,Lagrange imposed the constraint

Φ ≡∑

j

∂f∂oe j

doe j

d t= 0

This ensures that the computed elements define the osculating or-bit at every point. But this constraint can be relaxed in practice, asMichael Efroimsky proved in a series of works (Efroimsky, 2002,2005).

Let us transform the system (2.37) to a first order systemand assume that the right-hand side is uniformly Lipschitz con-tinuous, which is a fair assumption provided that typically theperturbing terms are continuously differentiable. The Picard-Lindelöf theorem readily shows that the solution (r, v) is unique.This proves that r and v are invariant to the selection of the func-tion Φ. Different forms of Φ render different sets of nonoscu-lating elements, but they will define the same state vector underthe corresponding transformation. Motivated by this freedom ofchoice, Efroimsky (2002) suggested that the Lagrange constraintis rather arbitrary and it can be relaxed making Φ , 0. He re-ferred to Φ as a gauge function, representing a reparameteriza-tion of the solution (r, v). The set of all possible definitions ofthe gauge function forms the gauge group, a symmetry group de-noted G .

The gauge-generalized equations of motion

The invariance of r(t) under the action of the symmetry groupG yields relations between different sets of elements emanating

*Published in Nouveaux Mémoires de l’Académie Royale des Sciences etBelles-lettres (Berlin) in three parts: the first one appeared in 1781 (pp. 199–276),the second in 1782 (pp. 169–292), and the third in 1783 (pp. 161–190).

from different definitions of the gauge. In particular, denotingthe osculating elements by oe j, it follows

f(t; oe) = f(t; oe) (2.40)

Similarly, the velocity matching condition renders

g(t; oe) = g(t; oe) +Φ(t; oe) (2.41)

These two systems of nonlinear algebraic equations can be solvedfor the nonosculating elements given the osculating ones, andvice-versa.

DifferentiatingEq. (2.38) oncemorewith respect to time ren-ders

d2rdt2 =

∂g∂t+

∑ ∂g∂oe j

doe j

d t+

dΦd t

where ∂g/∂t ≡ F. Equating this result to Eq. (2.37) leads to∑ ∂g∂oe j

doe j

d t+

dΦd t= ∆F (2.42)

If Eq. (2.39) is multiplied by ∂g/∂oen and Eq. (2.42) is multipliedby ∂f/∂oen, subtracting the resulting equations provides:

∂f∂oen

· dΦd t+

∂f∂oen

·∑

j

∂g∂oe j

doe j

d t− ∂g∂oen

·∑

j

∂f∂oe j

doe j

d t

=∂f∂oen

· ∆F − ∂g∂oen

·Φ

The left-hand side simplifies to∑j

[oen, oe j

]doe j

d t+

∂f∂oen

· dΦd t=

∂f∂oen

· ∆F − ∂g∂oen

·Φ

where[oen, oe j

]are the Lagrange brackets using the position and

velocity vectors as canonical variables in phase space (q ≡ f andp ≡ g),[oen, oe j

]=

∂f∂oen

· ∂g∂oe j− ∂f∂oe j· ∂g∂oen

(2.43)

This result yields the gauge-generalized Lagrange planetary equa-tions∑

j

[oen, oe j

]doe j

d t=

(∆F − dΦ

d t

)· ∂f∂oen

− ∂g∂oen

·Φ (2.44)

Making Φ = 0 yields the classical form of the planetary equa-tions. Note that this expression involves the full derivative of thegauge function, which reads

dΦd t=∂Φ

∂ t+

∑j

∂Φ

∂oe j

doe j

d t

Therefore, if the gaugedepends on the elements Eq. (2.44) shouldbe resorted to∑

j

([oen, oe j

]+∂Φ

∂oe j· ∂f∂oen

)doe j

d t

=

(∆F − ∂Φ

∂ t

)· ∂f∂oen

− ∂g∂oen

·Φ (2.45)

§2.8 Conclusions 23

The choice of a nontrivial gauge might have theoretical andpractical advantages. Efroimsky and Goldreich (2003, 2004)showed that the gauge can be defined in order to preserve thecanonicity of the formulation in presence of velocity-dependentperturbations. Gurfil and Klein (2006) found the optimal formof the gauge that minimizes the integration error of a Runge-Kutta scheme. Later, Gurfil (2007) exploited the gauge freedomin the problem of spacecraft relative motion in order to find newfamilies of frozen orbits. More details can be found in the bookby Kopeikin et al. (2011, §1.5.3).

2.8 Conclusions

Regularizationwas born from the necessity of avoiding the singu-larities in the three-body problem. At first, regularized theorieswere mathematical contrivances that helped researchers to ana-lyze the dynamics of the system. The formulations were based ondifferent principles, from Hamiltonian mechanics to topologicaltransformations.

During the design and operation of a space mission, how-ever, the dynamical regime is far from the singularity and it seemsthat regularization is not useful. But it was soon realized that thetransformations that worked so well in the three-body problemexhibited important advantages in the two-body problem too.Different techniques aim for an improved numerical scheme. Forexample, the introduction of special variables can potentially alle-viate the negative effects that the Lyapunov instability of orbitalmotion has on the numerical integration of the Cartesian equa-tions of motion. By replacing the time by an alternative indepen-dent variable one achieves an analytic step-size adaption that en-hances the performance of the numerical integrator. Similarly,embedding integrals ofmotion in the equations behaves as a con-trol term that slows down the error growth.

Theuse of orbital elements tomodel thedynamics is advanta-geous in many cases. Since weakly perturbed problems are closeto Keplerian orbits, in such cases the evolution of the elementswill be smoother than the evolution of the coordinates. From anumerical perspective, the right-hand side of the equations to beintegrated will be smaller, meaning that the problem is less sen-sitive to truncation errors. Recent discoveries prove that the ele-ments need not be referred to the osculating orbit (see, for exam-ple, the work by Dosopoulou and Kalogera, 2016, which derivesa physical model for mass-transferring binary systems using non-osculating elements). The freedom of choice in the dynamicalmodeling has great potential for analytic and numerical studies.

In the following chapters the foundations of regularizationare recovered and applied to different problems in astrodynam-ics. The techniques described in the present chapter will be ex-ploited for finding new analytic solutions to some problems, andfor deriving new formulations to describe particular phenomena.

R : S0 → S1 → S1

C : S1 → S3 → S2

H : S3 → S7 → S4

O : S7 → S15 → S8

Fibrations of hyperspheres and their correspondence to the fournormed algebras: real numbers, complex numbers, quaternions,and octonions.

3The Kustaanheimo-Stiefel space and

the Hopf fibration

I n the 1960s, Eduard Stiefel started to organize the Oberwol-fachMeetings onCelestialMechanics, in an attempt to drawthe interest of mathematicians into this subject. The first of

those meetings took place in 1964 and Paul Kustaanheimo pre-sented his work on describing Keplerian motion using spinors.His work on spinors combined with Stiefel’s experience in topol-ogy and numerical calculus gave birth to the celebrated exten-sion of the Levi-Civita transformation to the three-dimensionalcase, known as the Kustaanheimo-Stiefel transformation or KSfor short (Kustaanheimo and Stiefel, 1965). This extension hadeluded researchers since Levi-Civita (1920) presented his origi-nal regularization of the planar problem andHurwitz (1898) andAdams (1960) proved that transformations of this type only existin spaces of dimension n = 1, 2, 4, 8.

HeinzHopf discovered a particular transformation from theunit three-sphere S3 (a three-dimensional manifold embedded inR4) onto the unit two-sphere S2 (a two-dimensional manifoldembedded in R3) so that the preimage of each point in three-dimensional space turns out to be a circle on S3, called a fiber(Hopf, 1931). All points in this fiber transform into the samepoint in three-dimensional space. Such transformation is referredto as the Hopf fibration. In fact, the Kustaanheimo-Stiefel trans-formation can be understood as a particular Hopf map (Stiefeland Scheifele, 1971, §44). Hopf was Stiefel’s doctoral advisor andinfluenced other areas of his research, including the Hopf-Stiefelfunctions and the Stiefel manifolds. Davtyan et al. (1987) devel-oped the generalization of KS transformation to the case R8 →R5 in order to transform the problem of the five-dimensionalhydrogen atom into an eight-dimensional oscillator. They suc-cessfully rewrote the Hamiltonian of the hydrogen atom as the

Hamiltonian of an eight-dimensional isotrope oscillator. Depritet al. (1994) published an exhaustive treaty on the transforma-tions underlying KS regularization. They focused on the topicof linearization, connecting with prior work from Lagrange.

The KS transformation provides a robust regularizationscheme for dealing with close approaches or even impact trajecto-ries. Close encounters are one of the major challenges in N-bodysimulations. The first extensions of the KS transformation to N-body problems are due to Peters (1968), Aarseth (1971) and Bettisand Szebehely (1971). We refer to the work of Szebehely and Bet-tis (1971) for a review of the methods developed in those years.Aarseth and Zare (1974) focused on the three-body problem andtheir method was later generalized by Heggie (1974), who refor-mulated theHamiltonian for dealingwith an arbitrarynumber ofparticles. The shortcoming of Heggie’s method is that it fails toreproduce collisions of more than two particles. Mikkola (1985)discovered a technique for avoiding this singularity by rewritingthe Sundman time transformation in terms of the Lagrangianof the system. This method integrates 4N(N − 1) + 1 equa-tions of motion, so its use is recommended for few-body prob-lems. The formulations based on the KS transformation havebeen improved throughout the years (Mikkola andAarseth, 1998;Mikkola and Merritt, 2008), especially since the development ofthe chain regularization techniques (Mikkola and Aarseth, 1989,1993). The introduction of relativistic corrections in the mod-els has occupied different authors: Kupi et al. (2006) modifiedthe KS regularization for two-body close encounters in N-bodysimulations by introducing post-Newtonian effects; Funato et al.(1996) published a reformulation of the KS transformation fo-cused on time-symmetric algorithms. Aarseth (1999, 2003) pre-

25

26 3 The Kustaanheimo-Stiefel space and the Hopf fibration

sented several reviews of the evolution and keystones in the devel-opment of N-body simulations. The Levi-Civita variables haverecently been recovered by Lega et al. (2011) to detect resonantclose encounters in the three-body problem, and used by As-takhov and Farrelly (2004) in combination with an extension ofthe phase space to analyze the elliptic restricted three-body prob-lem.

The topological implications and numerical performance ofthe KS transformation has motivated many studies on the sub-ject. Stiefel was a committed teacher and supervised over sixtydoctoral students. Some of them made significant contributionsto regularization and celestial mechanics in general, names likeClaude Burdet, Jiri Kriz, Gerhard Scheifele, Michael Vitins, andJörg Waldvogel. Velte (1978) explored the representation of theKS transformation in the language of quaternions, a task that alsooccupied Vivarelli (1983) and Waldvogel (2006a,b). Differentrepresentations of KS regularization have been published by Vi-varelli (1986, 1994), including a representation in hypercomplexalgebra (Vivarelli, 1985). Hypercomplex numbers will be recov-ered in Chap. 5 to formulate a propagator based on the geometryof Minskowski space-time. Deprit et al. (1994) re-derived the KStransformation by doubling, using quaternions and octonions.They concluded by deriving the Burdet-Ferrándiz transforma-tion from the foundations of KS regularization (Ferrándiz, 1988).Saha (2009) has recently reformulated the problem by combin-ing quaternionswith theHamiltonian formalism. ElBialy (2007)approached the KS transformation from an alternative perspec-tive, focusing on the connection with the Hopf map and thetopological structure of the transformation. Moser (1970) de-termined the topological structure of Kepler’s problem by reg-ularizing the energy manifold. In a series of papers, Fukushima(2003, 2004, 2005) analyzed different numerical aspects of regu-larization, seeking scaling factors that guarantee the conservationof the integrals of motion and time elements for improving thestability of the time transformation. Recently, Roa et al. (2016c)discovered a topological phenomenon in KS space related to thetransition to chaos. By analyzing the structure supportingHopf’sfibration they proved that numerical errors related to the expo-nential divergence of the orbits may break down the KS trans-formation, as fibers will no longer be transformed to points inthree-dimensions.

The first part of this chapter explains the need for an addi-tional dimension when regularizing the three-dimensional prob-lem. It is combined with historical notes about the relevance ofthis problem in mathematics. The equations of the KS transfor-mation can be found in Sect. 3.2, with comments emphasizingthe connection with the Hopf fibration. The inverse transfor-mation is defined explicitly in terms of the Hopf map. The maincontributionof this chapter is thedevelopmentof anew theoryofstability, by translating classical concepts toKS space (see Sect. 3.3and Roa et al., 2016c). These new results serve as foundationfor introducing the new concept of manifold of solutions andtopological stability, which will lead to a novel Lyapunov indi-cator (Sect. 3.4). Numerical examples of N-body systems are pre-sented in Sect. 3.5 to show the consequences of topological insta-bilities. Finally, in Sect. 3.6 we derive, for the first time, the gauge-generalized equations for the elements attached to KS space, afterpresenting a new geometric interpretation of the elements. Sec-

tion3.7 serves as an appendixdiscussing geometrical and algebraicaspects of the KS transformation.

3.1 The need for an extra dimension: fibra-tions of hyperspheres

The Levi-Civita variables, introduced in Sect. 2.4.1, yield a globalregularization of the planar problem. Levi-Civita himself ad-mitted that the extension of his transformation to the three-dimensional problem was not straightforward (in fact, he neverachieved it):

“The problem in space has long resisted my efforts, as Itried to approach it by similar coordinate changes. Theusual canonical transformations relating to elliptical mo-tion do not regularize either.” Levi-Civita (1920, p. 2, inFrench)

Where is the difficulty coming from? Why did Kustaanheimoand Stiefel define a transformation in four dimensions?

The answer comes from a classical problem in number the-ory, later extended to more profound topological results, relatedto quadratic forms of the type

(x21 + . . . + x2

r )(y21 + . . . + y

2s) = z2

1 + . . . + z2n (3.1)

where the terms zk are bilinear functions of xi and y j. This iden-tity is said to be of dimension (r, s, n). It is uncertain how hearrived to this result, but in the third century Diophantus ofAlexandria stated (in words) that the identity (2, 2, 2) holds for

z1 = x1y1 + x2y2, z2 = x1y2 − x2y1 (3.2)

The reader might observe that taking two complex numbers p =x1 − i x2 and q = y1 + i y2 this identity reduces to

z = pq

with z = z1 + i z2. Under this formalism Eq. (3.1) is none otherthan the relation between the moduli of two complex numbers:

|p|2|q|2 = |z|2

Here |p|2 = pp† with p† the conjugate of p. Writing p = q =ξ+i η and z = x+i y one readily obtains the fundamental relationof the Levi-Civita transformation presented in Eq. (2.13), namely

x2 + y2 = (ξ2 + η2)2

Equation (3.2) becomes theLevi-Civita transformation itself, x =ξ2 − η2 and y = 2ξη.

On May 4, 1748, Leonhard Euler wrote a letter to ChristianGoldbach in which he noted that if p = x2

1 + x22 + x2

3 + x24 and

q = y21 + y

22 + y

23 + y

24 then pq = z2

1 + z22 + z2

3 + z24, with

z1 = x1y1 + x2y2 + x3y3 + x4y4

z2 = x1y2 − x2y1 − x3y4 + x4y3

z3 = x1y3 + x2y4 − x3y1 − x4y2

z4 = x1y4 − x2y3 + x3y2 − x4y1

§3.2 The KS transformation as a Hopf map 27

That is, he proved that Eq. (3.1) holds also for dimension (4, 4, 4).Just like the case (2, 2, 2) relates to multiplication in C, Eu-ler’s polynomial identity is equivalent to the quaternion product(although defined almost a century before Hamilton describedquaternions in 1843). Indeed, taking two quaternions p, q ∈ Hdefined as p = x1 − ix2 − jx3 − kx4 and q = y1 + iy2 + jy3 + ky4,Euler’s result reduces to

z = pq

where again |p|2|q|2 = |z|2 and z ∈ H.It took 150 years after Euler’s discovery until Adolf Hurwitz

proved that this kind of constructionwith dimension (n, n, n) ex-ists only for n = 1, 2, 4, 8 (Hurwitz, 1898). This is an extremelybeautiful result, since these are precisely the dimensions of thefour only normed algebras: real numbers, R, complex numbers,C, quaternions,H, and octonions,O.*

Provided that there are no normed algebras of dimensionn = 3, Kustaanheimo and Stiefel (1965) decided to move to thefour-dimensional case, and then sought a transformation to threedimensions. They had the right tools sinceHeinzHopf, whowasStiefel’s doctoral advisor, had already presented important resultsregarding the transformation of spheres from four dimensions tothree dimensions (Hopf, 1931). Consider the four dimensionalsphere S3,

x21 + x2

2 + x23 + x2

4 = 1

and identify R4 with C2, i.e. (x1, x2, x3, x4) ↔ (z1, z2) = (x1 +

ix2, x3 + ix4). Thus, S3 admits an alternative definition:

|z1|2 + |z2|2 = 1

Hopf introduced the mapping F : C2 7→ C × R

F(z1, z2) = (2z1z2 †, |z1|2 − |z2|2) = (y1 + iy2, y3) (3.3)

that is called theHopf fibrationor theHopf map. The first compo-nent of its image is a complex number, whereas the second com-ponent is a real number. Thus, the Hopf fibration maps four-dimensional spheres to three-dimensional spheres, S3 → S2. Inaddition, if we define a different point on a great circle of S3 interms of (w1,w2) = (b z1, b z2) and with |b| = 1 defining thegreat circle, it follows

F(w1,w2) =F(b z1, b z2) = |b|F(z1, z2) =F(z1, z2) (3.4)

Thismeans that every point on the great circle |b| = 1 of S3 trans-forms into the same point on S2. TheHopf fibration transforms

*Dickson (1919) tackled the problem described in Eq. (3.1) and referred toit as the eight square theorem. He was possibly the first to note that the resultingformulas are indeed the norm of complex numbers, quaternions, and octonions(Cayley numbers). This leads to the Cayley–Dickson construction, which gener-ates the higher-dimensional algebras by doubling the lower-dimensional ones.They allow to define additional algebras, like sedenions with n = 16, but theseare no longer normed algebras: the algebraC is commutative and associative un-der multiplication,H is associative but not commutative, andO is only alterna-tive. Beyond octonions these properties are lost. Mathematicians like Ferrándiz(1988) and Deprit et al. (1994) made extensive use of these observations when de-riving regularized theories. The latter connected the so called binary trees withthe standard basis in hypercomplex systems related to quaternions, octonions,and sedenions.

circles, S1, on the four-dimensional sphere to points on the three-dimensional sphere, defining the fiber bundle S1 → S3 → S2.The KS transformation is, in fact, a particular case of the Hopfmap.

For completeness we note that a direct consequence of thefundamental theorem by Adams (1960) is that there are four ad-missible fiber bundles,

Sp → Sn → Sn−p

which connect the spheres with (p, n) pairs (0, 1), (1, 3), (3, 7),and (7, 15). Their representation complies with R, C,H, andO,respectively.

3.2 The KS transformation as a Hopf map

Kustaanheimo and Stiefel (1965) introduced the four-dimensional vector u = [u1, u2, u3, u4]⊤, defined in theparametric space U4, in order to regularize the motion inthree-dimensional Euclidean space E3. The parametric space isidentified with C2 thanks to considering the pair of complexnumbers (v1, v2) = (u1 + iu4, u2 − iu3). Introducing this pair inthe Hopf fibration, Eq. (3.3) provides (y + iz, x), with:

x = u21 − u2

2 − u23 + u2

4

y = 2(u1u2 − u3u4) (3.5)z = 2(u1u3 + u2u4)

By identifying (x, y, z) with the components of the position vec-tor r in an inertial frame, Kustaanheimo and Stiefel defined theKS transformationK : U4 7→ E3. It can bewritten in a compactmatrix form just like the Levi-Civita transformation,

x =K(u) = L(u) u (3.6)

simply noting that x = [x, y, z, 0]⊤ is the trivial extension to R4

of the position vector. The KS matrix L(u) reads:

L(u) =

u1 −u2 −u3 u4u2 u1 −u4 −u3u3 u4 u1 u2u4 −u3 u2 −u1

Alternative representations of the KS transformation have beenprovided by Vivarelli (1983) and Waldvogel (2006b), in the lan-guage of quaternions, spinors, and hypercomplex numbers. Theproperties of the KSmatrix are similar to those of the Levi-Civitamatrix. To emphasize the connection with Hopf’s results we re-fer the reader toHopf (1931, eq. 1). He stated that the coordinates(x1, x2, x3, x4) inR4 relate to the coordinates (ξ1, ξ2, ξ3) inR3 bymeans of Eq. (3.5).

The KS matrix is r-orthogonal, i.e.

L−1(u) =1rL⊤(u) (3.7)

Every pointu is KS-mapped to one single point inCartesian spaceE3. The radial distance r relates to the KS variables by means of

r = u21 + u2

2 + u23 + u2

4 = ||u||2 (3.8)

28 3 The Kustaanheimo-Stiefel space and the Hopf fibration

The KS transformationmaps fibers on the three-sphere of radius√r inU4 to points on the two-sphere of radius r in E3.

Regularizing the equations of orbitalmotionbymeans of theKS transformation requires the time transformationdue toSund-man:

dt = r ds (3.9)

which was widely discussed in the previous chapter.

3.2.1 Defining the fibers

Hopf (1931) proved that the transformation from the three-sphere to the two-sphere maps circles to single points, definingthe structure S1 → S3 → S2. A direct consequence of the resultshown in Eq. (3.4) is the fact that Eq. (3.6) is invariant under thegauge transformation R : u 7→ w,

x = L(u) u = L(w) w (3.10)

Vector w = [w1, w2, w3, w4]⊤ takes the form:

w = R(ϕ; u) = R(ϕ) u (3.11)

where R(ϕ) is the matrix

R(ϕ) =

cos ϕ 0 0 − sin ϕ

0 cos ϕ sin ϕ 00 − sin ϕ cos ϕ 0

sin ϕ 0 0 cos ϕ

(3.12)

This matrix is orthogonal, and also

R⊤(ϕ) = R(−ϕ)

Being R(ϕ) orthogonal Eq. (3.11) can be inverted to provide

u = R−1(ϕ; w) = R(−ϕ) w (3.13)

The transformation R preserves the radius r, i.e.

r = u · u = w · w

Since the radius is invariant to the selection of the point inthe fiber it follows that the physical time, defined by Eq. (3.9),is R-invariant as well.* The identity in Eq. (3.10) and the r-orthogonality of matrix L furnish a useful relation:

w = L−1(w) L(u) u = R(ϕ) u =⇒ L⊤(w) L(u) = rR(ϕ)

The angular variable ϕ parameterizes the Hopf fibration infour-dimensional space. In fact, Eq. (3.11) defines explicitly thefiber F: changing the value of ϕ defines different points in U4

that are KS transformed to the same point in E3. This yields thedefinition of fiber as the subset of all points in four-dimensional

*Alternative forms of the time transformation can be found in the litera-ture, generalized as dt/ds = g(x, x). We refer to Sect. 2.2 for a survey of trans-formations involving different powers of the radial distance, the potential, theLagrangian, or combinations of the relative separations for the case of N-bodyproblems. The vectors x and x are R-invariant, so the uniqueness of the physicaltime is also guaranteed for more general transformations.

space that are mapped into the same point in E3 by means of theKS transformation,

F =w(ϕ) ∈ U4

∣∣∣ x =K(w), ∀ϕ ∈ [0, 2π)

Adifferent fiber transforms into a different point. Consequently,two fibers cannot intersect because the intersection point willthen be transformed into the same point in E3 despite belongingto two different fibers (Stiefel and Scheifele, 1971, p. 271). Thestereographic projection of the fibers onto E3 (see for exampleGriguolo et al., 2012) reveals that two fibers in KS space are con-nected by a Hopf link, as sketched in Fig. 3.1.

Figure 3.1: Hopf link connecting two different fibers in KS space. The

Hopf fibration is visualized bymeans of the stereographic projection

toE3.

3.2.2 The velocity and the bilinear relation

Let u,w ∈ U4. The KS matrix satisfies the property

L(u) w = L(w) u ⇐⇒ ℓ(u,w) = 0

where ℓ(u,w) denotes the bilinear relation

ℓ(u,w) = u1w4 − u2w3 + u3w2 − u4w1

In addition, if ℓ(u,w) = 0 then

(u · u) L(w) w − 2(u · w) L(u) w + (w · w) L(u) u = 0 (3.14)

This propertywas referred to as the “reciprocity ofKS-operators”by Deprit et al. (1994). It is easy to verify that ℓ(u, u) = 0 is re-sponsible for the fourth component of L(u) u being null, and theresult is the extension to U4 of a three-dimensional vector. Thefourth component of L(u) w is null if, and only if, ℓ(u,w) = 0.The operator L(u) is linear, meaning that

L(ku) = kL(u), L(u + w) = L(u) + L(w)

Differentiating Eq. (3.6) with respect to fictitious time andtaking into account the time transformation fromEq. (3.9) yields

x =2rL(u) u′ (3.15)

where x = [vx, vy, vz, 0]⊤ is the velocity vector extended to R4.Note that the fourth component is zero, which means

ℓ(u, u′) = 0

§3.2 The KS transformation as a Hopf map 29

Moreover, Stiefel and Scheifele (1971, p. 29) proved thatℓ(u, u′) = 0 is a first integral of orbital motion. Provided thatthe KS transformation is R-invariant, it follows that the bilinearrelation holds for all points in a given fiber,

ℓ(w,w′) = ℓ(u, u′) = 0 (3.16)

Let t(ϕ) ∈ U4 denote the vector that is tangent to a fiberF at w(ϕ). The direction of t can be obtained by differentiatingEq. (3.11) with respect to ϕ. It reads

t = R∗(ϕ) u

whereR∗(ϕ) = dR(ϕ)/dϕ is obtainedbydifferentiatingEq. (3.12).Taking as an exampleϕ = 0 yields the components of the tangentvector t,

t = [−u4, u3,−u2, u1]⊤ (3.17)

This unveils a geometric interpretation of the bilinear relationℓ(u, v) = 0: it can be understood as an orthogonality condition(Stiefel and Scheifele, 1971, §43), since

ℓ(u, v) = 0 ⇐⇒ v · t = 0

Two vectors u and v satisfy the bilinear relation ℓ(u, v) = 0 if vis orthogonal to the fiber through u. Provided that ℓ(u,u′) = 0holds naturally and it is an integral of motion it follows that thevelocity inKS space,u′, is always orthogonal to the fiber atu. Thefiber bundle S1 → S3 → S2 shows that the fibers constitutingthe three-sphere are circles, corresponding to points on the two-sphere. Indeed, the tangent vector t(ϕ) is always perpendicular tothe position vector w(ϕ),

w · t = [R(ϕ) u

] · [R∗(ϕ) u]= u ·

R⊤(ϕ)

[R∗(ϕ)u

]= 0

no matter the value of ϕ. Section 3.7 is devoted to the definitionof orthogonal bases and a cross product inU4.

3.2.3 The two-body problem

Equation (2.5) governs the evolution of the two-body problem interms of the fictitious time s. Using the notation introduced inthe present chapter it takes the form

x′′ − r′

rx′ +

µ

rx = r2ap

Vector ap denotes the perturbing acceleration, with a null fourthcomponent in order to extend it to four dimensions. Differenti-ating Eq. (3.10) with respect to fictitious time renders

rL(u′) u′ + L(u)(ru′′ − r′u′ +

µ

2u)=

r3

2ap (3.18)

The fictitious radial velocity r′ transforms into

r′ =r · r′

r= 2(u · u′)

Identifying w ≡ u′, Eq. (3.14) provides the relation

rL(u′) u′ = L(u)[2(u · u′) u′ − (u′ · u′) u

]

Introducing this expression in Eq. (3.18) and pre-multiplying theresult by L−1(u) yields

u′′ +1r

[µ

2− (u′ · u′)

]u =

r2L⊤(u) ap

The term in brackets turns out to be equivalent to the Keplerianenergy of the system, Ek:

ω2 =1r

[µ

2− (u′ · u′)

]=

12

(µ

r− v

2

2

)= −Ek

2

In sum, the perturbed two-body problem reduces to

u′′ + ω2u =(u · u)

2L⊤(u) ap (3.19)

under the KS regularization

3.2.4 The inverse mapping

The inverse KS transformation K−1 : x 7→ u maps points tofibers. Let us introduce the auxiliary vector v = [v1, v2, v3, v4]⊤,defined by the mapping

v1 = R sin θ

v2 =1

2R(y sin θ − z cos θ)

v3 =1

2R(y cos θ + z sin θ)

v4 = −R cos θ

(3.20)

Here R2 = (r+ |x|)/2. The angle θ is different from ϕ: the pointson the fiber are parameterized by θ, which is measured with re-spect to a certain axis; given two points u and w obtained by set-ting θ = θ1 and θ2 in Eq. (3.20), respectively, they relate by virtueof Eq. (3.11). This equation then provides the relation:

θ2 − θ1 = ϕ

meaning that the variable ϕ denotes the angular separation be-tween points along the same fiber. The value of θ depends onthe position of the reference axis, whereas ϕ is independent fromthe selection of the origin of angles.

The point u is finally defined as

u = [v1, v2, v3, v4]⊤ if x ≥ 0u = [v2, v1, v4, v3]⊤ if x < 0

(3.21)

Two alternative expressions are considered for avoiding potentialsingularities. They differ in the selection of the axes in KS space.From this result any point w0 in the initial fiber F0 can be ob-tained attending to

w0(ϕ) = R(ϕ) u0 if x0 ≥ 0w0(ϕ) = R(−ϕ) u0 if x0 < 0

(3.22)

so that x0 = L(w0)w0. The sign criterion complies with the dif-ferent definitions of axes in KS space. See Sect. 3.7 for a discus-sion about the orthogonal frames attached to the fiber. The linear

30 3 The Kustaanheimo-Stiefel space and the Hopf fibration

form of the inverse transformation has been defined by Roa andPeláez (2015f) and can be found in Sect. D.3.

The velocity inU4 is obtained by inverting Eq. (3.15), takinginto account the orthogonality relation in Eq. (3.7):

u′ =12L⊤(u) x

The geometry of the inverse KS transformation can be stud-ied from Fig. 3.2. The gray sphere is three-dimensional and of ra-dius r. The black arc corresponds to a set of initial conditions r j

on the sphere. The white dot represents one particular positionin E3, ri. The inverse KS transformation applied to ri yields thefiber Fi. The fiber is represented by means of its stereographicprojection to R3. The black surface consists of all the fibers F j

that are KS mapped to the points r j. In this figure it is possibleto observe the Hopf link connecting different fibers.

Figure 3.2: Stereographic projection toE3 of the Hopf fibration

corresponding to a set of initial positions on the three-dimensional

sphere of radius r. The black semi-torus consists of all the fibers thattransform into the semi-circumference on the two-sphere on the

bottom-left corner. One single fiberFi is plotted in white, corre-

sponding to ri.

3.3 Stability in KS space

The classical concepts of stability from Lyapunov and Poincarécan be translated to KS language by considering the topology ofthe transformation. First, we introduce an important new theo-rem regarding the geometry of the fibers (Roa et al., 2016c). Fromthis theorem the concept of the fundamental manifold arises nat-urally.

The stability concepts presented here are not based on nu-merical analyses; previous studies about the stability of KS trans-formation (Baumgarte, 1972a, 1976b; Arakida and Fukushima,2000) focused on the behavior of the numerical procedure. Weaim for a series of definitions that capture the physical behavior,which should be independent from the formulation of the dy-namics.

3.3.1 A central theorem

Two fibers can never intersect, as discussed when formally defin-ing a fiber. It is now possible to advance on this statement and to

formulate a fundamental property of the KS transformation:

Theorem 2: (Roa et al., 2016c) The angular separation betweentwo trajectories emanating from F0, measured along every fiber, isconstant. That is

w0 = R(ϕ0) u0 =⇒ w(s) = R(ϕ0) u(s)

for any value of ϕ0 and the fictitious time s. This is an intrinsicproperty of KS space and does not depend on the dynamics of thesystem.

Proof: Consider two trajectories in KS space, u = u(s) and w =w(s), departing from the same fiber F0. They relate by means ofEq. (3.11). In themost general case the angleϕ can be described bya function ϕ = ϕ(s) and initially it is ϕ(0) = ϕ0. The trajectoriesevolve according to

w(s) = R(ϕ; u(s)) = R(ϕ)u(s) (3.23)

Differentiating this equation with respect to fictitious time yields

w′(s) = R′(ϕ) u(s) + R(ϕ) u′(s) (3.24)

Equation (3.16) proved that the bilinear relation holds for any tra-jectory in KS space, meaning that ℓ(w,w′) = ℓ(u,u′) = 0. Thisrenders:

ℓ(w,w′) = ℓ(R(ϕ)u, R′(ϕ) u + R(ϕ) u′

)= 0

after substituting Eqs. (3.23) and (3.24). Expanding the bilinearrelation in the previous expression shows that

ℓ(R(ϕ)u, R′(ϕ) u + R(ϕ) u′

)= r

dϕds+ ℓ(u, u′) = 0 (3.25)

Assuming that r > 0 and considering that ℓ(u, u′) = 0 one gets

dϕds= 0 =⇒ ϕ(s) = ϕ0

so the angular separation along every fiber remains constant. Weemphasize that no assumptions about the dynamics have beenmade.

Adirect consequence of this result is the relation between thevelocities along the trajectories u(s) and w(s):

w′(s) = R(ϕ; u′(s))

3.3.2 The fundamental manifold Γ

A trajectory in Cartesian space, understood as a continuum ofpoints in E3, is represented by a continuum of fibers inU4. Eachfiber is KS transformed to a point of the trajectory. The fibersform the fundamental manifold, Γ, as Roa et al. (2016c) sug-gested.

Equation (3.22) defines the initial fiber F0, which yields awhole family of solutions parameterized by the angular variableϕ. Every trajectoryw(s) is confined to the fundamentalmanifold.Thanks to Thm. 2 themanifold Γ can be constructed following a

§3.3 Stability in KS space 31

Figure 3.3: Construction of the fundamental manifold. Themapping

gt : x0 7→ x(t) denotes the integration of the trajectory from t0 to t.Similarly, gs refers to the propagation using the fictitious time.

simple procedure: first, a reference trajectory u(s) is propagatedfrom any point inF0; then, mapping the transformation R overit renders a fiber Fi for each point u(si) of the trajectory. The set∪iFi defines Γ. Recall that∩

i

Fi = ∅

The fact that all trajectories emanating fromF0 are confined to Γis what makes an arbitrary choice of θ in Eq. (3.20) possible. Thediagram in Fig. 3.3 depicts the construction of the fundamentalmanifold Γ.

3.3.3 Fixed points, limit cycles and attractors

Points in E3 transform into fibers in U4. Thus, a fixed pointin Cartesian space, x0, translates into a fixed fiber in KS space,F0. Asymptotically stable fixed fibers (to be defined formally inthe next section) attract the fundamental manifold of solutions,Γ→ F0. Asymptotic instability is equivalent to the previous caseunder a time reversal.

Limit cycles are transformed to fundamental manifolds, re-ferred to as limit fundamental manifolds Γ0. A fundamentalmanifold Γ originating in the basin of attraction of a limit fun-damental manifold will converge to it after sufficient time. ForΓ → Γ0 convergence means that each fiber in Γ approaches thecorresponding fiber inΓ0. Correspondence between fibers is gov-erned by the t-synchronism.*

In a more general sense, attractors inU4 are invariant sets ofthe flow. The point-to-fiber correspondence connects attractorsin E3 with attractors in KS space. The basin of attraction of anattractive set Yu ⊂ U4 is built from its definition in three dimen-sions. Let X ⊂ E3 be the basin of attraction of Y . It can be trans-formed to KS space, X → Xu, thanks to

Xu = (R K−1)(X) = R(K−1(X)

)This construction transforms arbitrary sets in E3 to U4. Theinverse KS transformation constitutes a dimension raising map-ping, so in general dim(Xu) = dim(X) + 1.

*A dedicated discussion about the synchronism of the solutions can befound in Chapter 7. An interesting and closely related concept is the isochronouscorrespondence defined by Szebehely (1967, p. 233).

3.3.4 Relative dynamics and synchronism

The theories about the local stability of dynamical systems arebased on the relative dynamics between nearby trajectories. Theconcepts of stability formalize how the separation between two(initially close) trajectories evolves in time. But the concept oftime evolution requires a further discussion because of having in-troduced an alternative time variable via the Sundman transfor-mation.

Keplerian motion is known to be Lyapunov unstable (seeChap. 2 for details). Small differences in the semimajor axes oftwo orbits result in a separation that grows in time because ofhaving different periods. However, Kepler’s problem transformsinto a harmonic oscillator by means of the KS transformation,with the fictitious time being equivalent to the eccentric anomaly.The resulting system is stable: for fixed values of the eccentricanomaly the separationbetweenpoints in eachorbitwill be small,because of the structural (or Poincaré) stability of the motion.These considerations are critical for the numerical integration ofthe equations of motion. But in this chapter we seek a theory ofstability inU4 expressed in the language of the physical time t, be-cause of its physical and practical interest. The conclusions aboutthe stability of the system will be equivalent to those obtained inCartesian space.

The spectrum of the linearized and normalized form of Ke-pler’s problem written in Cartesian coordinates,

d2rd t2 = −

rr3

exhibits one eigenvalue with positive real part, λ = (2/r3)1/2 —see Eq. (2.2)—. Lyapunov’s theory of linear stability states thatthe system is unstable.

Under the action of the KS transformation Kepler’s problemtransforms into

d2uds2 = −

h2

u (3.26)

where h is minus the Keplerian energy. Although the linear anal-ysis is not useful in this case, selecting a candidate Lyapunov func-tionV(u,u′) = h(u·u)/4+(u′ ·u′)/2 the stability of the system isproved. In order to represent the Lyapunov instability of themo-tion with respect to time t the Sundman transformation needs tobe considered. Given two circular orbits of radii r1 and r2, thetime delay between both solutions reads

∆t = t2 − t1 = (r2 − r1)s

The time delay grows with fictitious time and small values of r2−r1 do not guarantee that ∆t remains small.

This phenomenon relates to the synchronism of the solu-tions (Roa et al., 2015a; Roa and Peláez, 2016c). Solutions to thesystem defined in Eq. (3.26) are stable if they are synchronized infictitious time, but unstable if they are synchronized in physicaltime. We adopt this latter form of synchronism for physical co-herence.

32 3 The Kustaanheimo-Stiefel space and the Hopf fibration

3.3.5 Stability of the fundamental manifold

Lyapunov stability

A trajectory r(t) in E3 is said to be Lyapunov stable if, for everysmall ε > 0, there is a value δ > 0 such that for any other solu-tion r∗(t) satisfying ||r(t0) − r∗(t0)|| < δ it is ||r(t) − r∗(t)|| < ε,with t > t0. In KS language trajectory translates into fundamen-tal manifold. In order to extend the definition of Lyapunov sta-bility accordingly, an adequate metric d to measure the distancebetween manifolds is required.

Let Γ1 and Γ2 be two (distinct) fundamental manifolds. Thefibers in Γ1 can never intersect the fibers in Γ2. But both man-ifolds may share certain fibers, corresponding to the points ofintersection between the two resulting trajectories in Cartesianspace. The distance between the manifolds at t ≡ t(s1) = t(s2) isthe distance between the corresponding fibers. Setting θ to a ref-erence value θref in Eq. (3.20) so that θ1 = θ2 ≡ θref, we introducethe metric:

d(t;Γ1,Γ2) =1

2π

∫ 2π

0||w1(s1; ϕ) − w2(s2; ϕ)|| dϕ (3.27)

with d(t;Γ1,Γ2) ≡ d(F1,F2). It is measured by computing thedistance between points in Γ1 and Γ2 with the same value of ϕ,and then integrating over the entire fiber. It is defined for givenvalues of physical time, and not fictitious time. The reason is thatthe goal of this section is to define a theory of stability such thatthe fundamental manifold inherits the stability properties of thetrajectory in Cartesian space. This theory is based on the physicsof the system, not affected by a reformulation of the equations ofmotion.

Consider a fundamental manifold Γ, referred to a nominaltrajectory r(t), and a second manifold Γ∗ corresponding to a per-turbed trajectory r∗(t). If the nominal trajectory is Lyapunov sta-ble, then for every εu > 0 there is a number δu > 0 such that

d(t0;Γ,Γ∗) < δu =⇒ d(t;Γ,Γ∗) < εu

If the initial separation between the manifolds is small it will re-main small according to the metric defined in Eq. (3.27).

The nominal solution r(t) is said to be asymptotically stableif ||r(t)−r∗(t)|| → 0 for t → ∞. Similarly, the fundamentalman-ifold Γ will be asymptotically stable if d(t;Γ,Γ∗) → 0 for suffi-ciently large times. The opposite behavior d(t;Γ,Γ∗) → ∞ cor-responds to an asymptotically unstable fundamentalmanifold. Itbehaves as if it were asymptotically stable if the time is reversed.

Poincaré maps and orbital stability

The notion of Poincaré (or orbital) stability is particularly rele-vant when analyzing the fundamental manifold due to its geo-metric implications. Kepler’s problem is unstable in the sense ofLyapunov but it is orbitally stable: disregarding the time evolu-tion of the particles within their respective orbits, the separationbetween the orbits remains constant.

The definition of the Poincaré map in E3 involves a two-dimensional section Σ that is transversal to the flow. Denotingby p1, p2,… the successive intersections of a periodic orbit with

Σ, the Poincaré mapP renders

P(pn) = pn+1

The generalization of the Poincaré section to KS space K : Σ→Σu results in a subspace embedded inU4. In Sect. 3.2.2we showedthat the trajectories intersect the fibers at right angles, providedthat the velocityu′ is orthogonal to the vector tangent to the fiber.Thus, every fiber defines a section that is transversal to the flow.The transversality condition for Σ translates into the section con-taining the fiber at u.

The Poincaré section Σu can be constructed by combiningthe set of fibers that are KS transformed to points in Σ. Letn = [nx, ny, nz]⊤ be the unit vector normal to Σ in E3, projectedonto an inertial frame. The Poincaré section takes the form

Σ ≡ nx(x − x0) + ny(y − y0) + nz(z − z0) = 0 (3.28)

where (x0, y0, z0) are the coordinates of the first intersectionpoint. Equation (3.28) can be written in parametric form asΣ(x(η, ξ), y(η, ξ), z(η, ξ)), with η and ξ two free parameters. Theextended Poincaré section Σu is obtained by transforming pointson Σ to KS space and then mapping the fibration R:

Σu = (R K−1)(Σ)

The choice of the Poincaré section Σ is not unique, and there-fore the construction of Σu is not unique either. The resultingPoincaré section Σu is a subspace of dimension three embeddedinU4. Indeed, the transformation (R K−1)(Σ) provides:

Σ 7→ Σu(u1(η, ξ, ϕ), u2(η, ξ, ϕ), u3(η, ξ, ϕ), u4(η, ξ, ϕ))

meaning that points in Σu are fixed by three parameters, (η, ξ, ϕ).The dimension is raised by (R K−1).

The intersection between a given fundamental manifold andthe Poincaré section Σu results in a fiber,

Γ ∩ Σu = F

Successive intersections can be denoted F1, F2,…The Poincarémap inU4,P : Σu → Σu, is

P(Fn) = Fn+1

Every point in a fiber intersects Σu simultaneously. Due to theR-invariance of the Sundman transformation the timeperiodbe-tween crossings is the same for every trajectory connectingFn andFn+1.

LetΓdenote a fundamentalmanifold representing anominalperiodic orbit, and let Γ∗ be a perturbed solution. They differ inthe conditions at the firstΣ-crossing,F1 andF∗1 respectively. Themanifold Γ is said to be Poincaré (or orbitally) stable if

d(F∗1,F1) < δu =⇒ d(Pn(F∗1),F1) < εu

If the separation between the fibers at the first crossing is small,the separation will remain small after n crossings.

§3.4 Order and chaos 33

3.4 Order and chaos

In the previous sectionwe generalized the key concepts of dynam-ical stability to KS space. The approach we followed aims for atheory that captures the physical properties of the system, insteadof focusing on its purely numerical conditioning. The next stepis the analysis of chaos inU4.

Chaotic systems are extremely sensitive to numerical errorsdue to the strong divergence of the integral flow. This is speciallyimportant in the vicinity of singularities, and it is precisely herewhere KS regularization exhibits all its potential. This section fo-cuses on characterizing the exponential divergence of trajectoriesinU4 due to highly unstable dynamics.

By definition the fundamental manifold is mapped to a tra-jectory in E3. The equations of motion inU4 are no more than areformulation of a dynamical system originally written inE3. Forsufficiently smooth perturbations the Picard-Lindelöf theoremensures the uniqueness of the solution. Thus, the correspond-ing fundamental manifold is also unique and its KS transformdefines only one trajectory. This means that any trajectory in thefundamental manifold is mapped to the same exact trajectory inE3, no matter the position within the initial fiber. An observerin three-dimensional space, unaware of the extra degree of free-dom introduced by the gauge R, will always perceive the sametrajectory no matter the values of ϕ.

3.4.1 The K-separation

In order to integrate the equations of motion numerically in U4

the initial values of u0 and u′0 need to be fixed. Thismeans choos-ing a point in the fiberF0. Since all the points inF0 are KS trans-formed to the same exact state vector in E3, the selection of thepoint is typically arbitrary. But for an observer in U4 differentvalues of ϕ yield different initial conditions, and therefore the ini-tial value problem to be integrated may behave differently. Ide-ally* all trajectories emanating fromF0 remain in the same funda-mentalmanifold, that is unique. However, numerical errors lead-ing to the exponential divergence of the trajectories can cause thetrajectories to depart from the fundamental manifold. In otherwords, after sufficient time two trajectories originating from thesame fiber F0, w0 = R(ϕ; u0), will no longer define the samefiber F(s), w(s) , R(ϕ; u(s)). In this case Thm. 2 will be vi-olated. Multiple fundamental manifolds will appear, obtainedby mapping the transformation R over each of the trajectories.The observer in E3 will see a collection of trajectories that departfrom the same exact state vector and they separate in time, as ifthe problem had a random component. This behavior can onlybe understood in four dimensions.

These topological phenomena yield a natural way ofmeasur-ing the error growth in KS space without the need of a precise

*Due to the limited precision of floating-point arithmetic, even the factthat all points generated with Eq. (3.22) and varying ϕ will be KS-transformedto the same exact point in E3 should be questioned. The loss of accuracy in thecomputation of the initial conditions in U4 will eventually introduce errors ofrandom nature. As a result, Eq. (3.22) provides points that are not exactly in thetrue fiber. Although the separation is small (of the order of the round-off error)and negligible in most applications, it may have an impact on the numericalintegration of chaotic systems.

solution. Let u(s) be a reference trajectory inU4, and let w(s) bea second trajectory defined by w0 = R(ϕ; u0). It is possible tobuild the fundamental manifold Γ from the solution u(s). Thesecond solution is expected to be w∗(s) = R(ϕ; u(s)) by virtueof Thm. 2. When numerical errors are present w(s) and its ex-pected value w∗(s) (the projection of the fundamental manifold)may not coincide. Note that w(s) = w∗(s) ensures the unique-ness of the solution, but says nothing about its accuracy. The sep-aration betweenw(s) and its projection onΓ is an indicator of thebreakdown of the topological structure supporting the KS trans-formation, meaning that the solutions can no longer be trusted.

Motivated by this discussionwe introduce the concept of theK-separation, dK

dK(s) = ||w(s) − w∗(s)|| = ||w(s) −R(ϕ; u∗(s))|| (3.29)

defined as the Euclidean distance between an integrated trajec-tory and its projection on the manifold of solutions. Monitoringthe growth of the K-separation is a way of quantifying the errorgrowth of the integration. In the context of N-body simulations,Quinlan and Tremaine (1992) discussed how the separation be-tween nearby trajectories evolves: the divergence is exponentialin the linear regime when the separation is small, but the growthrate is reduced when the separation is large. At this point the sep-aration might be comparable to the interparticle distance. TheK-separation will grow exponentially at first (for dK ≪ 1) un-til it is no longer small (dK ∼ O(1)), and then its growth slowsdown. Locating the transition point is equivalent to finding thetime scale tcr in which the solution in KS space can no longer betrusted: for t < tcr the topological structure of U4 is preserved,but for t > tcr the uniqueness of the manifold of solutions Γ isnot guaranteed.

For t < tcr the R-invariance of the Sundman transformationholds. The time for all the points in a fiber coincides. Thus, tcrand scr are interchangeable: at t < tcr it is also s < scr. Thebehavior of the solutions can be equally analyzed in terms of thephysical or the fictitious time.

In practice the K-separation is evaluated as follows:

1. Choosing a reference θ in Eqs. (3.20) and (3.21), for exampleθ = 0, integrate u∗(s).

2. Propagate a second trajectory w(s) generated fromEq. (3.22) with ϕ , 0.

3. Build the expected trajectory w∗(s) by mapping R(ϕ) overu∗(s), i.e. w∗ = R(ϕ)u∗.The K-separation is the Euclideandistance between w(s) and w∗(s).

3.4.2 Topological stability

The uniqueness of Γ can be understood as topological stability.KS space is said tobe topologically stable if all the trajectories ema-nating from the same fiber define a uniquemanifold of solutions,and therefore they are all KS-transformed to the same trajectoryinE3. For an observer inE3 a topologically unstable system seemsnon-deterministic, with solutions departing from the same initialconditions but separating in time with no apparent reason.

34 3 The Kustaanheimo-Stiefel space and the Hopf fibration

A system is topologically stable in the interval t < tcr. Thetrajectories diverge exponentially,

dK(t)/dK(0) ∼ eγt t or dK(s)/dK(0) ∼ eγs s

Here γ is equivalent to a Lyapunov exponent. For t > tcr thisequation no longer models the growth of the K-separation andthe system is topologically unstable. Simulations over the transi-tion time tcr integrated inU4 can no longer be trusted. Depend-ing on the integrator, the integration tolerance, the floating-pointarithmetic, the compiler, etc. the values of tcr for a given problemmight change. Thus, topological stability is a property of a certainpropagation, which requires all the previous factors to be defined.

The validity of the solution for an integration over the criticaltime tcr is not guaranteed. When tcr < tesc (with tesc denoting theescape time) not even the value of tesc can be estimated accurately.In such a case solutions initialized at different points in the fibermay yield different escape times.

Themethod presented in this section provides an estimate ofthe interval in which the propagation is topologically stable. Theexponentγ depends on the integration scheme and the dynamics,but it is not strongly affected by the integration tolerance. Anestimate of the value of γ provides an estimate of the critical timefor a given integration tolerance ε. Assuming dK(tcr) ∼ 1:

tcr ∼ −1γt

log ε (3.30)

Conversely, if the simulation needs to be carried out up to a givent f , the required integration tolerance is approximately

ε ∼ e−γt t f

This simple criterion proves useful for tuning and evaluating thenumerical integration. In the following examples of applicationthe values of γt are estimated by finding the slope of the exponen-tial growth of the K-separation in logarithmic scale. Althoughmore rigorous algorithms could be developed, this approxima-tion provides a good estimate of transition time between regimes.

3.5 Topological stability in N -body prob-lems

Two examples of N-body problems are analyzed in this section.The first example is the Pythagorean three-body problem. Thesecond example is a non-planar configuration of the four-bodyproblem. This problem simulates the dynamics of two field starsinteracting with a stellar binary. The experiments are designedfor showing the practical aspects of the new concept of stabilityintroduced in this chapter: the topological stability of KS space.

The problems are integrated using the regularization of theN-body problem based on the KS transformation proposed byMikkola (1985) as a reformulation of the method by Heggie(1974). The initialization of the method is modified so that dif-ferent points on the initial fiber F0 can be chosen. This meansgeneralizing the relative coordinates ui j to wi j by means of thetransformation R : u 7→ w. The trajectories depart from thesame initial conditions in E3. We inherit the normalization pro-posed in the referred paper, so that the gravitational constant isequal to one.

Heggie-Mikkola’s method is implemented in Fortran. TheHamiltonian nature of the N-body problem has motivated theuse of symplectic integrators. These algorithms do not sufferfrom the long-term accumulation of the error in the conservationof the energy (Wisdom and Holman, 1991; Sanz-Serna, 1992).Symplectic integrators rely on a fixed-step integration, which hin-ders the integration of close encounters. Although Duncan et al.(1998) developed SYMBA, a multi-step and truly symplectic inte-grator that adjusts the step size according to the perturbations,and Chambers (1999) invented a hybrid symplectic integratorthat deals with close encounters using a conventional integrator,conventional integrators with variable step size are preferred fornumerical explorations of binary encounters because handlingthe integration step size is critical (Mikkola, 1983; Hut and Bah-call, 1983; Bacon et al., 1996; Mikkola and Tanikawa, 1999). AsMikkola recommends, wewill integrate the problemwith theBu-lirsch and Stoer (1966) extrapolation scheme (Hairer et al., 1991,§II.9). See also the work by Murison (1989) for an analysis of theperformance of this integrator in the three-body problem. ThetotalK-separation is computedby combining theK-separationsfor the relative dynamics of each pair of bodies. Writing ui j ≡ uℓit is

dK =

√∑ℓ

d2K,ℓ

where dK,ℓ is the K-separation computed for uℓ. Murison(1989) for

3.5.1 The Pythagorean three-body problem

Originally developed by Burrau (1913), the Pythagorean problemconsists in three particles of masses m1 = 3, m2 = 4, and m3 =

5. The particles will be denoted (1), (2), and (3). At t = 0 thebodies are at rest and lying on the vertices of a Pythagorean righttriangle of sides 3, 4, and 5. The initial conditions are summarizedinTable 3.1. This problemhas been solved and discussed in detailby Szebehely and Peters (1967), so the solution to the problem isknown.

Table 3.1: Initial configuration of the Pythagorean problem. ``Id''

refers to the identification index of each body.

Id x y z vx vy vz

(1) 1.0000 3.0000 0.0000 0.0000 0.0000 0.0000

(2) -2.0000 -1.0000 0.0000 0.0000 0.0000 0.0000

(3) 1.0000 -1.0000 0.0000 0.0000 0.0000 0.0000

The solution is displayed in Fig. 3.4. Initially the bodies ap-proach the origin and after a number of close-approaches body(1) is ejected along a trajectory in the first quadrant, whereas (2)and (3) form a binary that escapes in the opposite direction. Theescape occurs at approximately tesc ∼ 60. The solution shownin the figure is obtained by setting the integration tolerance toε = 10−13.

The problem is first integrated from a set of initial conditionsobtained with θ = 0 in Eq. (3.20). Then, a second trajectory

§3.5 Topological stability inN -body problems 35

Figure 3.4: Solution to the Pythagorean three-body problem. The

thick dots represent the initial configuration of the system.

Figure 3.5: K-separation for the Pythagorean problem computed

from a reference trajectory with θ = 0 andϕ = 120 deg.

initialized with θ ≡ ϕ = 120 deg is integrated and their K-separation is shown in Fig. 3.5. After a transient the separationgrows exponentially with γt ∼ 5/12, and no transitions are ob-served until the escape time (tcr > tesc). As discussed in the pre-vious section this is equivalent to saying that the K-separationremains small, and consequently the integration in U4 is topo-logically stable. The transformed solution in E3 will be uniqueno matter the initial position in the fiber F0.

3.5.2 Field stars interacting with a stellar binary

This second example analyzes the gravitational interaction of abinary system (1,2), of masses m1 = m2 = 5, with two incomingfield stars (3) and (4) of masses m3 = m4 = 3. The initial condi-tions, presented in Table 3.2, have been selected so that both fieldstars reach the binary simultaneously.

Table 3.2: Dimensionless initial conditions for the binary system,

(1,2), and the field stars, (3) and (4). ``Id'' refers to the identification

index of each star.

Id x y z vx vy vz

(1) 0.6245 0.6207 0.0000 -0.7873 0.0200 -0.0100

(2) 0.6245 -0.6207 0.0000 0.7873 0.0200 0.0100

(3) 3.0000 3.0000 3.0000 -0.3000 -0.3000 -0.3000

(4) -5.0817 -3.0000 -3.0000 0.3000 0.2333 0.3000

Themanifold of solutions is constructed from a reference so-lution with θ = 90 deg. A second solution with θ = 120 deg (orϕ = 30 deg) is propagated and the corresponding K-separationis plotted in Fig. 3.6. The are two different regimes in the growthof the K-separation: the first part corresponds to the linearregime where the K-separation is small, whereas in the secondpart the separation is no longer small. Both regimes are separatedby tcr ∼ 42, when solutions in E3 no longer coincide. This re-sult is in good agreement with the value predicted by Eq. (3.30),which is tcr ∼ 40. Since for t = tcr the bodies have not yet es-caped and the integration continues, the solution is topologicallyunstable. The escape time associated to the reference solution,tesc ∼ 75, might not be representative because it corresponds tothe interval t > tcr.

Figure 3.6: K-separation for the four-body problem computed from

a reference trajectory with θ = 90deg and a second trajectory withϕ = 30deg.

A direct consequence of the topological instability of the in-tegration is the fact that solutions departing from the initial fiberF0 no longer represent the same solution in E3. Figure 3.7 showstwo solutions that emanate from different points of the initialfiber. Ideally they should coincide exactly; but because the inte-gration is topologically unstable for t > tcr the difference betweenboth solutions becomes appreciable and the accuracy of the inte-gration over tcr cannot be guaranteed.

The topological instability is not directly related to the con-servation of the energy. Although for t > tcr the integrationbecomes topologically unstable, Fig. 3.8 shows that the energyis conserved down to the integration tolerance until tesc, well be-yond tcr. This is a good example of the fact that the preservationof the integrals of motion is a necessary but not sufficient condi-tion for concluding that a certain integration is correct.

The evolution of the K-separation depends on the integra-

36 3 The Kustaanheimo-Stiefel space and the Hopf fibration

Figure 3.7: Two solutions to the four-body problem departing from

the same fiberF0: the top figure corresponds to θ = 90deg, and thebottom figure has been generatedwith θ = 120deg.

Figure 3.8: Relative change in the energy referred to its initial value,

(E(t) − E0)/E0.

tion scheme and the tolerance. In order to analyze this depen-dency Fig. 3.9 shows the results of integrating the problem withfour different tolerances and of changing fromdouble to quadru-ple precision floating-point arithmetic. It is observed that refin-ing the integration tolerance might extend the interval of topo-logical stability. However, the dynamics of the system remainchaotic and the solutions will eventually diverge for sufficientlylong times.

3.6 Gauge-generalizedelements inKSspace

Stiefel and Scheifele (1971, §19) attached a set of vectorial elementsto KS space, with important advantages from the numerical inte-gration perspective. In this sectionweprovide a newoutlook intothe geometry of the elements and present, for the first time, thegauge-generalized elements in KS space.

The perturbation can be decomposed in

ap = −∇V + p

Figure 3.9: K-separation for the four-body problem for different

integration tolerances. The solutions for ε = 10-15 and 10-17 are

computed in quadruple precision floating-point arithmetic.

whereV is a disturbingpotential andp refers to theperturbationsnot deriving from any potential. With this, Eq. (3.19) transformsinto

d2uds2 + ω

2u = −V

2u − r

4

[∂V

∂u− 2L⊤(u) p

](3.31)

under theKS transformation. The angular frequencyω relates tothe total energy E by means of

ω2 = −E

2=

12

(µ

r− v

2

2− V

)= −1

2(Ek + V )

It differs from theKeplerian energyEk by thedisturbingpotentialV . The relation in Eq. (3.14) combined with Eq. (3.15) renders auseful expression:

L(u′) u′ =(x · x)

2x − v

2

4x

between the velocity and the derivative of u.

3.6.1 Natural sets of elements: geometric interpretation

Differentiating Kepler’s equation in the unperturbed case yieldsthe time evolution of the eccentric anomaly. The result can beinverted to provide an alternative form of Sundman’s transfor-mation:

d tdE=

r2ωk

(3.32)

where ωk is the natural frequency referred to the Keplerian en-ergy, ωk = (−Ek/2)−1/2. The eccentric anomaly replaces the fic-titious time as the independent variable, E = 2ωk s. Derivativeswith respect to E, denoted by ∗, relate to the derivatives withrespect to s by means of

ddE=

12ωk

dds

The velocity, given in Eq. (3.15), transforms into

x =4ωk

rL(u) u∗ (3.33)

§3.6 Gauge-generalized elements in KS space 37

Introducing the eccentric anomaly as the new independent vari-able and neglecting the perturbations the right-hand side ofEq. (3.31) vanishes, and the solution is simply

u(E) = u0 cosE2+ 2u∗0 sin

E2

(3.34)

The initial conditions u0 = u(0) and u∗0 = u∗(0) are defined interms of the position and velocity at E = 0. The state at periapsis,written in the perifocal frame P = iP, jP,kP, reads

r0 = rp iP and v0 =

õ(1 + e)/rp jP

with rp the radius at periapsis. The radius at apoapsis will be de-noted ra.

Let a, b ∈ U4 be the representation of vectors iP and jP inKS space. The initial conditions r0 and v0 yield u0 =

√rp a andu∗0 =

√ra/4 b. These expressions transform Eq. (3.34) into

u(E) =√

rp a cosE2+√

ra b sinE2

(3.35)

This equation and its derivative can be inverted to provide

a =1√rp

(u cos

E2− 2u∗ sin

E2

)b =

1√

ra

(u sin

E2+ 2u∗ cos

E2

)

The vectors a and b satisfy the bilinear relation

ℓ(a, b) =a2

√1 − e2 ℓ(u0,u∗0) = 0

Equation (3.35) shows that the motion in the parametric spaceU4 is confined to the plane spanned by vectors a, b. This planeis in fact a plane of Levi-Civita’s type,* or an L-plane, providedthat the relation ℓ(a, b) = 0 holds. The L-plane attached to aand b is the representation of the orbital plane in U4. Vector adetermines the direction of the major axis, and b corresponds tothe minor axis.

Figure 3.10 depicts the geometry of the problem in the para-metric space U4. Equation (3.35) is in fact the equation of an el-lipse centered at the origin. It is worthy emphasizing that vectoru is not referred to the focus of the ellipse, but to its center. Theminor and major axes correspond to the square roots of the radiiat periapsis and apoapsis, respectively. The position in the ellipseis given by the half angle E/2. The ellipse is contained in the L-plane defined by the vector elements (a, b).

*Stiefel and Scheifele (1971, §43) defined an L-plane as the two-dimensional plane in U4 passing through the origin and spanned by two vec-tors a, b ∈ U4 satisfying the bilinear relation ℓ(a, b) = 0. This definition canbe extended to state that any pair of vectors in the L-plane satisfy the bilinearrelation. The transformation to planes in the Cartesian space is a Levi-Civitamapping, which doubles the angles and squares the distances.

Figure 3.10: Elliptic orbit in theL-plane spanned by (a, b) inU4.

By KS-transforming the expression defining u0 it follows

x0 = L(u0) u0 = rp L(a) a =⇒ iP = L(a) a

We have relaxed the notation by not distinguishing the unit vec-tor iP from its extension to U4, because the fourth componentvanishes. Similarly, vector u∗0 transforms into

x0 =4ωk

rpL(u0) u∗0 =

õ(1 + e)

rp

[L(a) b

]=⇒ jP = L(a) b

Provided that the bilinear relation ℓ(a, b) = 0holds, it is L(a) b =L(b) a. Note that vector b can be written b = 2r−1/2

a u∗0, andtherefore

L(b) b =4ra

L(u∗0) u∗0 = −x0

rp

This result yields a useful relation

L(b) b = −L(a) a

Finally, it is worth proving the orthogonality of a and b. It isshown by the inner product†

a · b = [L⊤(a) iP] · [L⊤(a) jP] = iP · [L(a) L⊤(a) jP]= iP · jP = 0

Since a and b are unit vectors the linear operators L(a) and L(b)are orthogonal —see Eq. (3.7)—. Equipping the KS space with across product as defined in Sect. 3.7.2 one can prove that kP =

iP × jP is given by the first three components of a × b. Select-ing, for example the definition from Eq. (3.58) given by Vivarelli(1987), the vectors in the triple a, b, a × b are mutually orthog-onal.

Consider now themore general problem in which p , 0 andV , 0. The eccentric anomaly E is replaced by a generalizedeccentric anomaly, φ. The new independent variable is given adynamical definition: it evolves with an angular velocity of the

†Recall the general property of the inner product involving vectors x, y ∈Rn and a matrix A ∈ Rn×n :

(A x) · y = x · (A⊤y)

38 3 The Kustaanheimo-Stiefel space and the Hopf fibration

same form than the time rate of change of the Keplerian eccentricanomaly —Eq. (3.32)—:

dtdφ=

r2ω

(3.36)

The Keplerian natural frequencyωk is replaced by the frequencyrelated to the total energy E

ω =

√−E

2=

√µ − rV

2(r + 4 ||u∗||2)(3.37)

Atdeparture the generalized eccentric anomaly coincideswith theeccentric anomaly. Both angles in the parametric space U4 relateby means of

φ = E + σ (3.38)

The angle σ/2 is the angle traversed by the osculating perifocalframe, defined by a and b, due to the perturbations. If no per-turbations are considered (V = 0 and p = 0) the generalizedeccentric anomaly coincides with the eccentric anomaly. Theirevolution relates by the expression

dσdt=

dφdt− dE

dt

where dE/dt is the full derivative of the osculating eccentricanomaly. It can be obtained by differentiating Kepler’s equationwith respect to time, taking into account the convective termsfrom the rate of change of the elements.

The generalized eccentric anomaly ismeasured from a partic-ular reference frame. This frame, referred to as the intermediateframeA, is the one that rotates with respect to the perifocal framePwith angular velocity

ωA/P =12

dσdt

(a × b)

The intermediate frame is defined dynamically. At departure itcoincides with the perifocal frame, so that iA has the directionof e0. Vectors (a, b) define the perifocal frame inU4, that differsfrom the intermediate frame in a rotation of magnitude σ/2. Inwhat remains of the chapter the generalized eccentric anomaly isused as the independent variable. For the sake of simplicity weshall use the symbol ∗ for derivatives with respect to φ.

Taking the generalized eccentric anomaly as thenew indepen-dent variable the equations of motion become

d2udφ2 +

u4= Q (3.39)

Here Q defines the forcing terms:

Q = − 116ω2

2V u + r

[∂V

∂u− 2L⊤(u) p

]− 1ω

dωdφ

dudφ

The evolution of the natural frequencyω is governed by the dif-ferential equation:

dωdφ= − r

8ω2

∂V

∂ t− 1

2ω[u∗ · L⊤(u)p] (3.40)

The first term vanishes when the disturbing potential does notdepend explicitly on time.

We seek a solution of the form

u(φ) = a(φ) cosφ

2+ b(φ) sin

φ

2(3.41)

where vectorsa,b ∈ U4 are elements of the unperturbedmotion.Introducing Eq. (3.38) and equating to Eq. (3.35) yields the lineartransformation between the elements (a, b) and (a,b):

a = √rp a cosσ

2− √ra b sin

σ

2

b = √rp a sinσ

2+√

ra b cosσ

2

Recall that in the unperturbed problem it is σ = 0. Thus, a andb will be equal to a and b multiplied by the square root of theradius at periapsis and apoapsis, respectively.

The determinant of the resulting (linear) transformation isa√

1 − e2. This result shows that the transformation is not a ro-tation, in general. Only when the orbit is circular vectors (a,b)will be obtained from (a, b) by a rotation of amplitude σ/2 anda dilation of magnitude

√a. This can also be understood geo-

metrically; let γa denote the angle between a and a, and γb theangle between b and a. They can be solved from the previousequations:

tan γa = −√

1 + e1 − e

tanσ

2, and tan γb = +

√1 + e1 − e

cotσ

2

to prove that γa , −σ/2 and γb , −(π+σ)/2 except for e = 0.In general, vectors a and b will not be orthogonal. The inner

product

a · b =rp − ra

2sinσ

shows that only ifσ = 0 or rp = ra the vectors (a,b) will form aset of orthogonal elements. The former occurs when no pertur-bations are considered. The latter requires the eccentricity to benull during the entire motion. This will only occur under veryspecific conditions, for example an initially circular orbit subjectonly to perturbations that are normal to the orbital plane.

The norm of a and b reduces to

||a|| = a +rp − ra

2sinσ and ||b|| = a −

rp − ra

2cosσ

showing that they are not unit vectors, in general. The geometryof the orbital problem in U4 is sketched in Fig. 3.11. The originof the generalized eccentric anomaly is the intermediate frameA.When mapped to the parametric space U4 frames P and A re-late by a rotation of magnitude σ/2. The osculating eccentricanomaly is measured with respect to the perifocal frame.

§3.6 Gauge-generalized elements in KS space 39

Figure 3.11: Geometrical interpretation of the elements a andb in

theL-plane.

The state of the particle in U4 has been parameterized interms of the vector elements a and b. The total energy or, moreprecisely, the natural frequencyω, is the ninth element. When noperturbations act on the system these variables are constant andtheir numerical integration is trivial. Note, however, that the timetransformation (3.36) needs to be integrated even in the pure Ke-plerian problem. For the sake of improving the numerical perfor-mance of the method the physical time is decomposed in (Stiefeland Scheifele, 1971, p. 92)

t = τ − 1ω

(u · u∗) (3.42)

where τ is a time element, governed by the perturbations. Differ-entiating this expression yields the equation of evolution of thetime element:

dτdφ=

1ω3 (µ − 2rV ) − r

16ω3

u ·

[∂V

∂u− 2L⊤(u) p

]− 2ω3

dωdφ

(u · u∗) (3.43)

When the disturbing terms vanish the time element grows lin-early in time.

Incorporating the time element the problem is completelydefined by a set of ten orbital elements, oe j (with j = 1, . . . , 10).They are the natural frequency ω, the time element τ, and thecomponents of the four-vectors a and b.

3.6.2 Gauge freedom in KS space

In the following lineswe extend the concept of the gauge freedomto the natural elements in KS space. As discussed in the previoussection in absence of perturbations Eq. (3.39) reduces to a har-monic oscillator that admits the trivial solution

u(φ) = a cosφ

2+ b sin

φ

2≡ f(φ; a,b) (3.44)

with a and b constant. In addition, the velocity in U4 takes theform

u∗(φ) = −a2

sinφ

2+

b2

cosφ

2≡ g(φ; a,b)

The symmetry of these equations yields a simple form of the La-grange and Poisson brackets. The only nonzero terms are[ai, bi

]= −[bi, ai

]= 1/2

ai, bi= −bi, ai

= 2

in which ai and bi are the components of a and b, and i =1, 2, 3, 4. Following the procedure presented in Sect. 2.7 we de-fine the gauge as

Φ =∂f∂a da

dφ+∂f∂b db

dφ=

dadφ

cosφ

2+

dbdφ

sinφ

2(3.45)

Its full derivative takes the form

dΦdφ=∂Φ

∂φ+∂Φ

∂a da

dφ+∂Φ

∂b db

dφ(3.46)

Equation (2.44) translates to KS language as∑j

[oen, oe j

]doe j

dφ= −

(Q +

dΦdφ

)· ∂f∂oen

− ∂g∂oen

·Φ

with

Q =1

16ω2

2V u + r

[∂V

∂u− 2L⊤(u) p

]+

1ω

dωdφ

dudφ

It renders the gauge-generalized form of Eq. (19,52) from Stiefeland Scheifele (1971, p. 89):

dadφ= +2

(Q +

dΦdφ

)sin

φ

2+Φ cos

φ

2(3.47)

dbdφ= −2

(Q +

dΦdφ

)cos

φ

2+Φ sin

φ

2(3.48)

Equation (3.40) defined the evolution of the osculating total en-ergy. Because of the gauge invariance of this variable this equationstill holds and defines the evolution of the energy as:

dωdφ= − r

8ω2

∂V

∂ t− 1

2ω[u∗ · L⊤(u)p] (3.49)

The decomposition of the time in a secular part and a time ele-ment is G -independent so Eq. (3.43) still holds.

Equations (3.50–3.51) include the full derivative of thegauge. But this term involves the derivatives of the elements, asshown by Eq. (3.46). It is then convenient to rearrange theseequations so that the derivatives of the corresponding elementsare all grouped in the left-hand side:

M(φ)dadφ= +2

(Q +

∂Φ

∂φ

)sin

φ

2+

(I cos

φ

2+ 2

∂Φ

∂b

)Φ (3.50)

M(φ)dbdφ= −2

(Q +

∂Φ

∂φ

)cos

φ

2+

(I sin

φ

2− 2

∂Φ

∂a

)Φ (3.51)

This is achieved by introducing the matrix

M(φ) = I − 2∂Φ

∂asin

φ

2+ 2

∂Φ

∂bcos

φ

2

with I the identity matrix of dimension four.

40 3 The Kustaanheimo-Stiefel space and the Hopf fibration

In sum, the gauge-generalized equations of motion in KSspace are:

dτdφ=

1ω3 (µ − 2rV ) − r

16ω3

u ·

[∂V

∂u− 2L⊤(u) p

]− 2ω3

dωdφ

(u · u∗) (3.52)

dωdφ= − r

8ω2

∂V

∂ t− 1

2ω[u∗ · L⊤(u)p] (3.53)

dadφ= M−1

[+2

(Q +

∂Φ

∂φ

)sin

φ

2+

(I cos

φ

2+ 2

∂Φ

∂b

)Φ

](3.54)

dbdφ= M−1

[−2

(Q +

∂Φ

∂φ

)cos

φ

2+

(I sin

φ

2− 2

∂Φ

∂a

)Φ

](3.55)

The equations are completed with the initial conditions:

φ0 = E0 : τ = 0, ω =

√µ − r0V0

2(r0 + ||u′0||2), a = a0, b = b0

The non-osculating elements a0 and b0 are solved from the oscu-lating elements a0 and b0 given the equations

f(φ0; a0, b0) = f(φ0; a0,b0) (3.56)g(φ0; a0, b0) = g(φ0; a0,b0) +Φ(φ0; a0, b0) (3.57)

Recall that f = a cos(φ/2) + b sin(φ/2) and g = ∂f/∂φ. Thechoice of the gauge must make the matrixM(φ) regular.

3.7 Orthogonal bases

3.7.1 Basis attached to the fiber

In Section 3.2.2 it is shown that fibers are circles in U4. Let uand w = R(ϕ; u) be two vectors attached to a fiber F. Theyspan a plane containing the fiber. This plane is not a plane of theLevi-Civita type because ℓ(u,w) , 0. Since trajectories intersectfibers at right angles this subspace is transversal to the flow. Anorthogonal basis can be attached to the resulting plane, allowingprojections on the transversal subspace. Although arbitrary or-thonormal bases can be constructed via the Gram-Schmidt pro-cedure (Nayfeh and Balachandran, 2004, pp. 529–530), the basisdescribed in this section appears naturally in the formulation.

Associated to every vector u there is a KS matrix L(u). Thecolumns of the matrix define a vector basisB = u1,u2,u3,u4,with u1 ≡ u. The basisB is orthogonal,

ui · u j = rδi j

Here δi j denotes Kronecker’s delta. Assuming x ≥ 0 every pointin the fiber generated by w = R(ϕ; u) lies in the plane spannedby u1 and u4, i.e. w · u2 = w · u3 = 0 for all ϕ. Conversely, forx < 0 the fiber is confined to the u2u3 plane. The basis B is anorthogonal basis attached to the fiber at u. In addition,

u4 · u′ = ℓ(u, u′) = 0

meaning thatu′ is perpendicular tou4. In fact,u4 = −t, as shownby Eq. (3.17).

The vectors arising from the products L(u)ui, i = 1, 2, 3 havea vanishing fourth component. They are equivalent to vectors inE3. However, the fourth component of the product L(u)u4 is notzero. The three vectors obtained by these transformations corre-spond to the position vector r, and a pair of orthogonal vectorsspanning the plane tangential to the two-sphere at r inE3. Thesevectors are the columns of the associated Cailey matrix.

3.7.2 Cross product

Stiefel and Scheifele (1971, pp. 277–281) sought a definition ofcross product in the parametric space U4 when discussing theorthogonality conditions of vectors and Levi-Civita planes. Al-though the cross product of two vectors inR3 is intuitive, its gen-eralization to higher dimensions is not straightforward. Indepen-dent proofs from different authors (see for example Brown andGray, 1967) show that the cross product of two vectors only existsin dimensions 1, 3, 7; for n dimensions the cross product involvesn − 1 vectors. Stiefel and Scheifele (1971) defined the productp = u × v as

p = L(u) v4

where v4 = [v4,−v3, v2,−v1]⊤ is the fourth column of L(v).The properties of this construction motivated the authors to call(p1, p2, p3) the cross product of u × v, with p4 = u · v. In thefollowing lines we analyze in more detail this construction andconnect with alternative definitions provided by Vivarelli (1987)and Deprit et al. (1994).

Let e1, e2, . . . , en be an orthogonal basis inRn. The Grass-mann exterior product gives rise to the bivectors ei∧e j, trivectorsei∧ e j∧ ek, and successive blades of grade m ≤ n (Flanders, 1989,§II). They constitute the subspaces

∧m Rn of the exterior algebra:

∧Rn = R ⊕ Rn ⊕

2∧Rn ⊕ . . . ⊕

n∧Rn

noting that∧0 Rn = R and

∧1 Rn = Rn. Without being ex-haustive we simply recall that such exterior algebra is associativewith unity, satisfying ei ∧ e j = −e j ∧ ei and ei ∧ ei = 0. Theexterior product of two parallel vectors vanishes. We shall writeei j...k = ei ∧ e j ∧ . . . ∧ ek for brevity.

In Section 3.7.1 an orthogonal basis attached to u was de-fined, where two of its vectors are KS-transformed to vectorsspanning the plane tangent to the two-sphere in E3. Identifyingui =

√r ei, the exterior product of vectors u1,u2,u3,u4 gen-

erates the oriented hypervolume

u1 ∧ u2 ∧ u3 ∧ u4 = −r2e1234

provided that det(L(u)) = −r2. In three dimensions the exteriorproduct is equivalent to the cross product, given e1 × e2 = e3,e1 × e3 = −e2 and e2 × e3 = e1. Applying the cross product tothe first three elements ofB provides

u1 × u2 × u3 = ru4

This result confirms thatB is, indeed, an orthogonal basis.

§3.8 Conclusions 41

Vivarelli (1987) and Deprit et al. (1994) worked in the moregeneral Clifford algebra Cl3. Introducing the Clifford product oftwo vectors

ab = a · b + a ∧ b

the exterior algebra over R3 can be identified with the Cliffordalgebra Cl3: bivectors and trivectors become ei ∧ e j → eie j ande1∧e2∧e3 → e1e2e3. Note that ba = a ·b−a∧b , ab, so theClifford product is not commutative. Thus, the even subalgebraCl+3 = R⊕

∧2 R3, isomorphic to the quaternion algebraH, is notcommutative either.

The algebra H ≃ Cl+3 is endowed with the multiplicationrules e2

1 = e22 = e2

3 = −1, and the product of vectors is anti-commutative, eie j = −e jei. Identifying these bivectors with thequaternion basis elements

e2e3 = i, e3e1 = j, e1e2 = k

the product of two quaternions u and v is established. Vivarelli(1987) rewrote Stiefel and Scheifele’s form of the cross product interms of the quaternion product

u × v = 12

(ukv∗ − vku∗) (3.58)

where ∗ denotes the involution u∗ = u1 + u2i + u3j − u4k. Disre-garding the arrangement of the components, Deprit et al. (1994)defined the cross product of two quaternions as

u × v = 12

(vu† − uv†)

Here † denotes the quaternion conjugate. The difference be-tween these quaternionic definitions and the original one fromStiefel and Scheifele is the fact that u × v is a pure quaternion,i.e. ℜ(u × v) = 0, whereas the fourth component of Stiefel andScheifele’s product u × v is u · v.

3.8 Conclusions

The topology of the KS transformation has important conse-quences in the stability and accuracy of the solutions in KS space.There are two key aspects to consider when studying the stabil-ity of the motion. First, the presence of a fictitious time that re-places the physical time as the independent variable. Second, thedimension-raising nature of the Hopf fibration.

Classical theories of stability are based on the separation be-tween nearby trajectories. Having introduced a fictitious time,the question on how to synchronize the trajectories arises. Thenumerical stabilization of the equations of motion by KS reg-ularization relates to solutions synchronized in fictitious time.But a theory of stability synchronized in physical time allows thetranslation of concepts such as attractive sets, Lyapunov stability,Poincaré maps, etc. to KS language.

The additional dimension provides a degree of freedom tothe solution in parametric space. In general, the free parametercan be fixed arbitrarily, with little or no impact on the result-ing trajectory in Cartesian space. However, as strong perturba-tions destabilize the system, different values of the free parameter

may result in completely different solutions in time. This phe-nomenon is caused by numerical errors and destroys the topolog-ical structure ofKS transformation: points in a fiber are no longertransformed into one single point. By monitoring the topologi-cal stability of the integration it is possible to estimate an indicatorsimilar to the Lyapunov time.

The geometry of theorbitalmotion inKS space attains a sym-metric form. The natural vectorial elements introduced by Stiefeland Scheifele (1971, §19) can be given a geometrical interpreta-tion, thanks to having characterized orthogonal bases inKS space.From this construction we derived, for the first time, the gauge-generalized equations of orbital motion in KS space.

—TheDromos at Luxor

4The Dromo formulation

T he Space Dynamics Group at the Technical Univer-sity of Madrid (SDG-UPM) was born as “Grupo deDinámica deTethers” (GDT) in 1995. Founded by Prof.

Jesús Peláez, it focused on the analysis, design, and simulationof space tethers. The group established active collaborationswith institutions like the European Space Agency, the Harvard-SmithsonianCenter forAstrophysics, or theUniversity ofMichi-gan Ann-Arbor, among others.

The bead model is a widely used technique for simulatingthe dynamics of space tethers. The tether is discretized in punc-tual masses (or beads) and their constrained dynamics are mod-eled individually. As the precision required by the simulationincreases the number of beads grows significantly. The compu-tational load may become intractable for long simulations. Ef-ficient integration techniques are required in this context. Moti-vated by this problem, Peláez et al. (2007) developed a new specialperturbation method, calledDromo.* The formulation relies onthe concept of Hansen ideal frames to decouple the in-plane dy-namics from the motion of the orbital plane, following the workby Deprit (1975) and Palacios and Calvo (1996).

The Dromo elements are the components of the eccentricityvector in an ideal frame, the inverse of the angular momentum,and the four components of a quaternion that defines the orbitalplane with respect to an inertial reference. Replacing the tradi-tional Euler angles by the Euler parameters avoids potential sin-gularities and transforms trigonometric relations intopolynomialequations, at the cost of increasing the dimension of the system.Possibly the first application of this technique to perturbed or-

*The name derives from the ancient Greek word δρoµoς (dromos), whichmeans racetrack. It also refers to the ceremonial pathways to temples and burialsites. The propagator was given this name because of its speed.

bital motion was due to Broucke et al. (1971) and Deprit (1975).Dromo features a second order Sundman transformation that in-troduces the ideal anomaly, which behaves as the independentvariable. Later studies defined a time element to improve the in-tegration of the time transformation (Baù et al., 2014; Baù andBombardelli, 2014). The formulation has been progressively im-proved and the latest version is that by Urrutxua et al. (2015b).A complete and very instructive review can be found in the the-sis of Urrutxua (2015, Chap. 4). In a series of works Baù et al.(2015) presented elegant modifications of the method based on adifferent time transformation and the introduction of a perturb-ing potential. Roa and Peláez (2015d) recently derived a reformu-lation of Dromo that is specifically conceived for hyperbolic or-bits. They formulated the evolution of the eccentricity vector inMinkowski space-time to take advantage of the hyperbolic natureof thismetric. Chapter 5 is devoted to this particular formulation.

Dromonot only presents an efficient integration scheme, butit has provenuseful formany analytical studies. Bombardelli et al.(2011) published an asymptotic solution to the dynamics of aspacecraft perturbed by a constant tangential acceleration, whileGonzalo and Bombardelli (2014) and Urrutxua et al. (2015a) fo-cused on the radial case. Bombardelli (2014) arrived to an ana-lytical description of collision maneuvers, a study that resultedin an interesting strategy for finding optimal collision avoidancemaneuvers (Bombardelli and Hernando-Ayuso, 2015). Amatoand Bombardelli (2014) and Amato et al. (2016) studied the po-tential of Dromo for propagating resonant trajectories of Poten-tially Hazardous Asteroids focusing on the error growth aftersuccessive close encounters. The integration of Dromo with thesemi-analytic propagation theory developed at the Draper Labo-ratory at theMassachusetts Institute ofTechnology also providedpromising results. Dromouses a gyroscopic descriptionof the dy-

43

44 4 The Dromo formulation

namics of the orbital plane. It is based on the evolution of the an-gularmomentum vector following the typical approach formod-eling rigid body dynamics. In fact, Urrutxua and Peláez (2014)took advantage of this connection to derive a roto-translationalpropagator based onDromo. Roa et al. (2015b) detected a singu-larity in the equations of Dromo: when the angular momentumvanishes the orbital plane becomes undetermined and the gyro-scopic definition is no longer valid.

In this chapterwe first introduce the concept ofHansen idealreference frames. The Dromo formulation is built upon thismathematical contrivance in Sect. 4.2. The numerical advantagesof Dromo are discussed in Sect. 4.3 over a few examples. So far,these sections revisit concepts that can be found in the litera-ture. Next, Sect. 4.4 provides the first formal discussion aboutthe noncanonical nature of the Dromo formulation. In Sect. 4.5we present, for the first time, the gauge-generalized equations ofDromo following the procedure described in Sect. 2.7, togetherwith the Lagrange and Poisson brackets related to the Dromo el-ements. The main original contribution of this chapter is char-acterizing the singularity in Dromo (see Sect. 4.6 and Roa et al.,2015b). Alternative versions of Dromo are presented in Sect. 4.7,in an attempt to weaken the resulting numerical issues. Finally,numerical examples assessing the behavior ofDromo in the vicin-ity of the singularity can be found in Sect. 4.8.

4.1 Hansen ideal frames

Let I = iI, jI,kI be the inertial frame, and A = iA, jA,kAan arbitrary rotating reference frame. The velocity of a particle,defined by the time derivative of the radius vector, can be com-puted from theperspective of the inertial or the rotating referenceframe. The absolute and relative velocities relate through

drdt

∣∣∣∣∣I

=drdt

∣∣∣∣∣A

+ ωAI × r

whereωAI is the angular velocity vector (or the Darboux vector)of frame A with respect to the inertial space. The absolute ve-locity decomposes in the velocity relative to themoving referenceplus the evolution of the rotating reference itself. If the movingframe A is attached to the orbital plane, then the angular veloc-ityωAI governs the dynamics of this plane. Thus, in the unper-turbed problem it isωAI = 0 (the orbital plane is fixed)meaningthat

drdt

∣∣∣∣∣I

=drdt

∣∣∣∣∣A

(4.1)

That is, the time derivatives of r in frames I andA coincide. Thisphenomenonmotivated Peter A.Hansen to propose the conceptof ideal coordinates, back in 1857. The original definition, takenfrom Hansen (1857, p. 66), reads

“I call ideal coordinates of a planet, comet, or satellitethose having the property that not only them but alsotheir derivatives with respect to time have the same formin the perturbed and unperturbed motion.”

AlthoughHansen did not work with vectors, in modern applica-tions the concept of Hansen ideal coordinates has been extended

to define Hansen ideal frames. Let us recover the rotating frameA, for which ωAI = 0 when there are no perturbations. In theKeplerian case Eq. (4.1) holds because the orbital plane is fixed.FrameA is said to be Hansen ideal if the velocity relative toA inthe perturbed problem (withωAI , 0) is still given by Eq. (4.1).This yields the condition

drdt

∣∣∣∣∣I

=drdt

∣∣∣∣∣A

+ ωAI × r =drdt

∣∣∣∣∣A

=⇒ ωAI × r = 0

The angular velocity vectorωAI must be parallel to the radial di-rection for frame A to be ideal, a phenomenon that was alreadynoted by Hansen (1857, p. 67). Thus, the angular velocity ofa Hansen ideal frame follows the direction of the radius vector.More details and some new results regardingHansen ideal frameswere recently presented by Jochim (2012).

The main advantage of introducing ideal frames is that it al-lows a separate treatment of the in-plane motion and the mo-tion of the orbital plane itself. Herget (1962) explored this con-cept when trying to eliminate the singularity related to the defi-nition of the apses when e → 0. Broucke et al. (1971) combinedHansen’s results with replacing the Euler angles by the Euler pa-rameters, an approach that was later refined by Deprit (1975)when he published a special perturbation method based on idealelements. In a sequel, he characterized ideal frames from an alge-braic perspective (Deprit, 1976).

4.2 Dromo

Under a normalization such that µ = 1, the governing equationsof motion for the two-body problem reduce to

d2rdt2 +

rr3 = ap, r = ||r||

The initial value of the semimajor axis is taken as the unit oflength, meaning that the mean motion reduces to unity. Let Lbe the orbital frame, with L = i, j,k. Such basis is defined bythe expressions:

i =rr, k =

hh, j = k × i (4.2)

where h ∈ R3 is the angular momentum vector of the particle.The dynamics of the orbital plane are modeled as if it were a

rigid body. A reference frame, equivalent to a body-fixed frame, isattached to it and themotionof this framedescribes the evolutionof the orbital plane. The orbital plane can be defined geometri-cally by infinitely many frames sharing their z-axis and rotatingwith different angular velocities. This yields different definitionsof the perifocal frames P = iP, jP,kP: for example, the onedefined by the osculating values of the eccentricity and angularmomentum vectors is the osculating perifocal frame. However,we shall look for an alternative definition of the perifocal frame: adefinition that makes the perifocal frame ideal in Hansen’s sense.Describing the motion of the orbital plane with a Hansen idealframe will yield the dynamical decomposition discussed in theprevious section. This frame, referred to as the perifocal depar-ture frame (a term coined by Deprit, 1975) is given a dynamicaldefinition derived in the following lines.

§4.2 Dromo 45

The evolution of the orbital frame is governed by the differ-ential equations

didt

∣∣∣∣∣I

= ωLI × i =ddt

(rr

)=

1r

(v − dr

dti)

dkdt

∣∣∣∣∣I

= ωLI × k =ddt

(hh

)=

1h

(dhdt− dh

dtk)

LetωLI = ωx i+ωy j+ωz kbe the angular velocity of the orbitalframe with respect to the inertial frame. Projecting the previousequations ontoL yields:

ωx = −1h

(dhdt· j

)= −1

h(r × ap) · j = r

h(ap · k)

ωy = 0

ωz =1r

(v · j) = −1r

v · (i × k) =1r2 (k · h) =

hr2

meaning that

ωLI =hr2 k +

rh

(ap · k) i = ωLP + ωPI (4.3)

The orbital frame is defined by the rotation of an intermediateframe along the radial direction i, plus a rotation along the k di-rection. Motivated by this result, Peláez et al. (2007) defined theperifocal departure frame as the one that rotates with an angularvelocity

ωPI =rh

(ap · k) i (4.4)

It is ωPI ∥ r and therefore P is a Hansel ideal frame. The ori-entation of this frame at departure can be fixed at convenience,but the obvious choice is to make it coincide with the osculatingperifocal frame at the departure epoch:

iP =e0

e0, kP =

h0

h0, jP = kP × iP

The relative motion between the orbital and the perifocal depar-ture frame abides by

ωLP =hr2 k

The independent variable in Dromo is the ideal anomaly,σ,defined in terms of the time transformation

dtdσ=

r2

h(4.5)

The ideal anomaly decomposes in

σ = β + ϑ

Here β is the angle between the osculating eccentricity vector ande0, and ϑ is the true anomaly. Figure 4.1 shows the geometry ofthe different reference frames that have been introduced. The re-lation between the ideal anomaly and the true anomaly can beobserved in this figure.

Figure 4.1: Geometrical definition of the problem.

4.2.1 The eccentricity vector

The velocity of the particle is given by

v =drdt=

drdt

i +hr

j

Thus, the eccentricity vector e admits the following definition

e =v × hµ− r

r=

(h2

r− 1

)i − h

drdt

j (4.6)

This expression is projected on the departure perifocal frameP toprovide:

e = +[(

h2

r− 1

)cosσ + h

drdt

sinσ]

iP

+

[(h2

r− 1

)sinσ − h

drdt

cosσ]

jP = ζ1 iP + ζ2 jP

That is, the pair (ζ1, ζ2) defines the components of the eccentric-ity vector in the departure perifocal frameP. Recall that the angleβ describes the evolution of the osculating eccentricity vector, i.e.tan β = ζ2/ζ1. The projections ζ1 and ζ2 admit two alternativedefinitions:

ζ1 =

(h2

r− 1

)cosσ + h

drdt

sinσ ≡ e cos β

ζ2 =

(h2

r− 1

)sinσ − h

drdt

cosσ ≡ e sin β

Introducing the inverse of the angular momentum,

ζ3 =1h

the triple (ζ1, ζ2, ζ3) defines the first three Dromo elements. Inthe unperturbed case it is β = 0, and therefore ζ1 = e, ζ2 = 0and ζ3 = 1/h0 for all t.

4.2.2 The orbital plane

Let rI, rP ∈ H be two pure quaternions associated with the com-ponents of vector r in the inertial and in the departure perifocal

46 4 The Dromo formulation

reference frames, I and P respectively. Both quaternions relatethrough the rotation action

rI = n rP n†, n ∈ H (4.7)

Quaternion n† is the conjugate of n. See Appendix A for detailson the algebra of quaternions.

The unit quaternion n = η4 + i η1 + j η2 + k η3 relates to theEuler angles by means of successive applications of Euler’s theo-rem:

η1 = sini2

cosΩ − ω

2, η2 = sin

i2

sinΩ − ω

2

η3 = cosi2

sinΩ + ω

2, η4 = cos

i2

cosΩ + ω

2

with i the osculating inclination, Ω the osculating right ascen-sion of the ascending node, and ω the departure argument ofthe pericenter. It differs from the osculating argument of thepericenter by the angle β = arctan ζ2/ζ1. The Euler parameters(η1, η2, η3, η4) are the last four Dromo elements.

4.2.3 Evolution equations

The Dromo elements form the set (ζ1, ζ2, ζ3, η1, η2, η3, η4). Inthis section we will derive the equations governing the evolutionof the elements.

The radial distance in Dromo variables reads

r =h2

1 + e cosϑ=

1ζ2

3 (1 + e cosϑ)=

1ζ2

3 s

with s = 1 + e cosϑ = 1 + ζ1 cosσ + ζ2 sinσ. The evolutionof the elements with σ is integrated together with the Sundmantransformation (4.5), in order to propagate the physical time.The time behaves as a dependent variable. This differential equa-tion is rewritten in Dromo elements to provide:

dtdσ=

1ζ3

3 s2(4.8)

The equation of the hodograph is obtained from the crossproduct of k and the eccentricity vector, given by Eq. (4.6),

v = ζ3(j + k × e)

Considering that e = ζ1 iP + ζ2 jP, the hodograph equationtransforms into

ζ1 jP − ζ2 iP = −j +vζ3

This equation is differentiated with respect to ideal anomaly σand projected along iP and jP to provide

dζ1

dσ= −s sinσ(ap · i) +

[ζ1 + (1 + s) cosσ

](ap · j)

dζ2

dσ= −s cosσ(ap · i) +

[ζ2 + (1 + s) sinσ

](ap · j)

where the term ap stands for ap = ap/(ζ43 s3). It is the normalized

perturbing acceleration.

The angular momentum vector changes according to the ex-pression

dhdt= r × ap = r(i × ap)

where it is h = h k. Considering that

dhdt=

dhdt

k + hdkdt=

dhdt

k + h(ωLI × k) =dhdt

k − r(ap · k) j

the evolution equation for h is projected along k to provide

dhdt= r(ap · j) =⇒

dhdσ=

r3

h(ap · j)

The derivative of ζ3 results in

dζ3

dσ= −ζ3(ap · j)

Note that the angular momentum of the particle is only affectedby the projection of the perturbing acceleration in the along-trackdirection.

LetwPI be the pure quaternion defined by the projections ofvectorωPI/2 in frame P. This quaternion relates to quaternionn by means of

2wPI = 2n†dndt

This equation is inverted to provide the time evolution of quater-nion n,

dndt=

12n(2wPI)

Expanding the quaternion product and taking into account thedefinition ofωPI given in Eq. (4.4) provides the evolution equa-tions for the components of n. The full system of eight ordinarydifferential equations becomes

dtdσ=

1ζ3

3 s2(4.9)

dζ1

dσ= −s sinσ(ap · i) +

[ζ1 + (1 + s) cosσ

](ap · j) (4.10)

dζ2

dσ= −s cosσ(ap · i) +

[ζ2 + (1 + s) sinσ

](ap · j) (4.11)

dζ3

dσ= −ζ3(ap · j) (4.12)

dη1

dσ=

(ap · k)2

(−η4 cosσ − η3 sinσ) (4.13)

dη2

dσ=

(ap · k)2

(−η3 cosσ + η4 sinσ) (4.14)

dη3

dσ=

(ap · k)2

(−η2 cosσ + η1 sinσ) (4.15)

dη4

dσ=

(ap · k)2

(−η1 cosσ − η2 sinσ) (4.16)

§4.3 Improved performance 47

Equations (4.9–4.16) need to be integrated from the initial con-ditions

σ = σ0 : t = t0, ζ1 = e0, ζ2 = 0, ζ3 = 1/h0, n = n0

Recall that ap = ap/(ζ43 s3).

The position and velocity vectorswritten in the orbital frameL read

r =1ζ2

3 si and v = ζ3(u i + s j)

with

s = 1 + ζ1 cosσ + ζ2 sinσ, u = ζ1 sinσ − ζ2 cosσ

The unit vectors i and j can be referred to the departure perifocalframe thanks to

i = cosσ iP + sinσ jP, j = − sinσ iP + cosσ jP

Finally, the transformation from the components in the perifocaldeparture frame to the inertial frame is given by the quaternionicproduct in Eq. (4.7).

4.3 Improved performance

The motivation behind the development of Dromo was toimprove the numerical performance of the propagation com-pared to the straightforward integration inCartesian coordinates.Many comparisons can be found in the literature (Urrutxua et al.,2011; Urrutxua and Peláez, 2012; Baù et al., 2014; Roa et al., 2014;Amato and Bombardelli, 2014). For completeness, this sectionpresents two examples of application to evaluate the nominal per-formance of Dromo compared to Cowell’s method. The inter-ested reader may findmore detailed numerical analyses in the ref-erences mentioned above, as well as in Urrutxua (2015, Chap. 8).

The first example (Problem 1) is the test problem 2b pro-posed by Stiefel and Scheifele (1971, p. 122). It is a highly ellip-tical orbit perturbed by the gravitational attraction of the Moonand the Earth’s J2. Both forces are computed from the simplifiedmodel defined in Stiefel and Scheifele (1971, p. 119). The orbit isintegrated for about 50 revolutions, and the osculating elementsat departure are defined in Table 4.1.

Table 4.1: Initial conditions of the test problems.

Units Problem 1 Problem 2

MJD0 (ET) − N/A 51544.0000

t f days 288.1277 20.0000

a km 136000.4185 10000.0000

e − 0.9500 0.3000

i deg 30.0000 50.0000

ω deg -90.0000 180.0000

Ω deg 0.0000 150.0000

M0 deg 0.0000 162.3605

The performances of Dromo and Cowell are compared inFig. 4.2. The figure shows the relation between the final error

Figure 4.2: Performance of the integration of Problem 1.

in position and the number of function calls. An efficient for-mulation requires less function calls for the same level of accu-racy. The performance curves are generated by changing the in-tegration tolerance from ε = 10−6 to 10−14 and the error is mea-suredwith respect to an accurate solution integrated inquadrupleprecision floating-point arithmetic with different formulations,making sure that the first sixteen digits coincide. In this examplethe orbits are integrated using a RKF5(4) scheme and Dromo re-duces the required number of function calls by almost an orderof magnitude when compared to the integration in Cartesian co-ordinates.

The second example (Problem 2) is a realistic orbit aroundthe Earth, with the initial conditions defined in Table 4.1 (givenin the ICRF/J2000 reference system with the reference plane de-fined by the Earth mean equator and equinox at epoch). TheEarth’s gravity field is modeled with a 40 × 40 grid using theharmonics from the GGM03S solution. The orientation of theEarth is given by the IAU 2006/2000A, CIO based standard(using X, Y series). In order to model the effects of the atmo-spheric drag, the area-to-mass ratio of the spacecraft is assignedthe value A/m = 0.0075 m2/kg, and the drag coefficient is cD =

2.0. Space weather data with the three-hour values of the atmo-spheric parameters, as well as the Earth Orientation Parameters,are taken from Celestrak. The atmospheric density is given bythe MSISE90 model. The acceleration due to the solar radiationpressure is computed assuming a reflectivity coefficient cR = 1.2,a nominal solar flux of Φ = 1367 W/m2 with periodic seasonalvariations, and a conic shadow model. The perturbations fromthe Sun, the Moon, and all the planets in the Solar System areconstructed from the DE430 ephemeris. The propagator, PER-FORM, is explained in detail in Chap. 6.

The orbit is integrated with LSODAR, a variable order, vari-able step size and multistep Adams-based method implementedin the library ODEPACK (Hindmarsh, 1983). In addition, Cowell’smethod is also integrated with an eighth-order Störmer-Cowellscheme. Figure 4.3 displays the performance of each integration.Although the Störmer-Cowell integration is faster than Dromofor low-accuracy integrations, its accuracy does notmatch the oneprovided by LSODAR. With this integrator Dromo proves fasterthan Cowell for a given level of accuracy. In addition, the mini-mum error reached byDromo is two orders ofmagnitude smallerthan the minimum error reached by the integration in Carte-

48 4 The Dromo formulation

Figure 4.3: Performance of the integration of Problem 2.

sian variables. The improvements in performance come from thecombination of two factors. First, a more efficient step-size con-trol thanks to the second-order Sundman transformation. Sec-ond, a smoother evolution of the right-hand side of the differen-tial equations to be integrated because of using elements in lieuof coordinates. The advantage of this technique has been widelydiscussed in Sect. 2.5.

4.4 Variational equations and the non-canonicity of Dromo

The present section is devoted to deriving the variational equa-tions of Dromo. The Lagrange and Poisson brackets are pre-sented here for the first time. The variational equations and theLagrange brackets will be recovered in the next section for deriv-ing the gauge-generalized Dromo equations, and in Chap. 8 forsolving the relative dynamics in Dromo variables (see Roa andPeláez, 2015a, for a generic method for propagating the varia-tional equations numerically). With these results at hand, somecomments on the noncanonicity of Dromo will be included.

In Sect. 2.6 we proved that the canonical equations based ona homogeneous Hamiltonian can be easily referred to a fictitioustime. The physical time becomes a dependent variable associatedto an additional coordinate, q0 ≡ t. The corresponding canonicalequation referred to the ideal anomaly renders

dq0

dσ=∂Hh

∂p0=

r2

h

which turns out to be the Sundman transformation. Followingthe procedure in Sect. 2.6 Dromo can be understood as a trans-formation

D : (σ; q0, . . . , q3, p0, . . . , p3) 7→ (σ; q0, . . . , q3, p0, . . . , p3)

where q0 = q0, qi = ζi (with i = 1, 2, 3), p0 = p0, and pi arethree of the components of quaternion n. The fourth compo-nent is not independent from the others; in order to ensure thatthe Jacobian of the transformation is regular the constraint:

η21+η

22+η

23+η

24 = 1 =⇒ η1 δη1+η2 δη2+η3 δη3+η4 δη4 = 0

needs to be satisfied. A similar discussion can also be found in thework of Broucke et al. (1971) or Gurfil (2005a), among others.

The conditions for canonicity depend on the form of the Ja-cobian of D, denoted A. This matrix is built from the partialderivatives of the position and velocity vectors, r = f(σ; oe) andv = g(σ; oe):

∂f∂ζ1= − cosσ

ζ23 s2

i,∂f∂ζ2= − sinσ

ζ23 s2

i,∂f∂ζ3= − 2

ζ33 s

i

∂f∂η1= − 2

ζ23 sη4

[N+1324 j − (N+1144 sinσ + N−3412 cosσ)kP

]∂f∂η2= +

2ζ2

3 sη4

[N−1423 j − (N+2244 cosσ − N+1234 sinσ)kP

]∂f∂η3= − 2

ζ23 sη4

[N+3344 j + (N+1423 cosσ − N−1324 sinσ)kP

]and∂g∂ζ1= +ζ3 jP,

∂g∂ζ2= −ζ3 iP

∂g∂ζ3= −(ζ2 + sinσ) iP + (ζ1 + cosσ) jP

∂g∂η1= −2ζ3

η4

N+1324[(ζ1 + cosσ)iP + (ζ2 + sinσ)jP]

− [N−1234(ζ2 + sinσ) + N+1144(ζ1 + cosσ)]kP

∂g∂η2= +

2ζ3

η4

N−1423[(ζ1 + cosσ)iP + (ζ2 + sinσ)jP]

+ [N+2244(ζ2 + sinσ) + N+1234(ζ1 + cosσ)]kP

∂g∂η3= −2ζ3

η4

N2

3344[(ζ1 + cosσ)iP + (ζ2 + sinσ)jP]

− [N+1423(ζ2 + sinσ) + N−1324(ζ1 + cosσ)]kP

They are written in terms of the coefficient N±i jkℓ = ηiη j ± ηkηℓ.In this case η4 has been chosen as the dependent component.

Having identified [q1, q2, q3]⊤ ≡ r, [p1, p2, p3]⊤ ≡ v, qi =

ζi and pi = ηi (with i = 1, 2, 3), the nonzero Lagrange bracketsread[ζ1, ζ2

]=

1ζ3s2 ,

[ζ1, ζ3

]=

2s sinφ − u cosφζ2

3 s2[ζ2, ζ3

]= −ζ1 + (1 + s) cosφ

ζ23 s2

,[ζ3, η1

]=

2η4ζ

23

N+1324

[ζ3, η2

]=

2η4ζ

23

N−2314,[ζ3, η3

]=

2η4ζ

23

N+3344

[η1, η2

]=η4

η1

[η1, η3

]=η4

η2[η2, η3]=

4ζ3

For completeness, the Poisson brackets read:ζ1, η1

=−η2

η1

ζ1, η2

=η2

η4

ζ1, η3

=−ζ3η2

2[ζ1 + (1 + s) cosσ]

ζ2, η1

= −η2

η1

ζ2, η2

=η2

η4

ζ2, η3

=ζ3η2

2(u cosσ − 2s sinσ)

ζ3, η1

= −η2

η1

ζ3, η2

=η2

η4

ζ3, η3

=ζ2

3η2

2ζ1, ζ2

= ζ3s2,

η2, η3

=ζ3

4N+1324

η1, η2=ζ3

4N+3344,

η1, η3

=ζ3

4N−1423

§4.6 Singularities 49

The condition (2.36) is violated meaning that Dromo is notcanonical. This is equivalent to noting that the Jacobian matrixis not symplectic.

4.5 Gauge-generalized Dromo formula-tion

Equations (4.9–4.16) canbe generalizedby introducing anontriv-ial gauge, following the procedure presented in Sect. 2.7. Thanksto having defined the Lagrange brackets explicitly Eq. (2.44) leadsto the general form of the gauge-generalized Dromo equationswhen considering the time transformation from Eq. (4.8):

∑j

[oen, oe j

]doe j

dσ=

(∆Fζ3

3 s2− dΦ

dσ

)· ∂f∂oen

− ∂g∂oen

·Φ

In this case the gauge functionΦ relates to the elements bymeansof

Φ =∑

j

∂f∂oe j

doe j

dσ

Let Φ = Φ/(ζ3s) be the adjusted gauge function. Denoting(ap,x, ap,y, ap,z) and (Φx, Φy, Φz) the components of ap and Φ inthe orbital frameL, the gauge-generalized version of Eqs. (4.10–4.16) is

dζ1

dσ= +s sinσ (ap,x − Φ′x) + [ζ1 + (1 + s) cosσ] (ap,y − Φ′y)

+ (ζ1 + cosσ)Φx + (2ζ2 + ℓ2 sinσ/s)Φy

dζ2

dσ= −s cosσ (ap,x − Φ′x) + [ζ2 + (1 + s) sinσ] (ap,y − Φ′y)

+ (ζ2 + sinσ)Φx − (2ζ1 + ℓ2 cosσ/s)Φy

dζ3

dσ= −ζ3

s[s(ap,y − Φ′y) + sΦx − uΦy

]dη1

dσ=

12

(+η4 cosσ − η3 sinσ)(ap,z − Φ′z)

+12s

sη2 Φy + [η3(ζ1 + cosσ) + η4(ζ2 + sinσ)]Φz

dη2

dσ=

12

(+η3 cosσ + η4 sinσ)(ap,z − Φ′z)

− 12s

sη1 Φy + [η4(ζ1 + cosσ) − η3(ζ2 + sinσ)]Φz

dη3

dσ=

12

(−η2 cosσ + η1 sinσ)(ap,z − Φ′z)

+12s

sη4 Φy − [η1(ζ1 + cosσ) + η2(ζ2 + sinσ)]Φz

dη4

dσ=

12

(−η1 cosσ − η2 sinσ)(ap,z − Φ′z)

− 12s

sη3 Φy − [η2(ζ1 + cosσ) − η1(ζ2 + sinσ)]Φz

In these equations Φ′ denotes the full derivative of the gauge. Itis important to note that Eq. (4.9) needs to be integrated togetherwith this system of equations. The fact the time transformationdepends on ζ3 apart from r causes that the physical time is no

longer invariant to the selection of the gauge. Thus, the timetransformation should be defined as

dtdσ= ζ3r2 =

1ζ3

3 s2

in terms of the osculating value of ζ3. The transformation be-tween the osculating and the nonosculating elements is given inEqs. (2.40–2.40).

The referred equation involves the full derivative of the gaugefunction, given by

dΦdσ=∂Φ

∂σ+

∑j

∂Φ

∂oe j

doe j

dσ

When the convective terms do not vanish the evolution of theDromo elements is given explicitly by

∑j

([oen, oe j

] − ∂Φ

∂oe j· ∂f∂oen

)doe j

dσ

=

(∆Fζ3

3 s2− ∂Φ∂σ

)· ∂f∂oen

− ∂g∂oen

·Φ

The terms multiplying the derivatives of the elements in the left-hand side of this equation form amatrix that needs to be invertedin order to define the evolution equations. The final form of theequations depends on the selection of the gauge function.

4.6 Singularities

There are two specific situations that make the integration of theDromo equations problematic:

Asymptotes: Equations (4.9–4.16) depend explicitly on theperturbing term ap = ap/(ζ4

3 s3) and become singular for(ζ4

3 s3)→ 0. The term s = 1+e cosϑ vanishes in two cases:(i) in the case of a parabolic orbit (e = 1), when the parti-cle is sufficiently far from the attractive center, as ϑ → π.(ii) in the case of a hyperbolic orbit (e > 1), when the par-ticle approaches the asymptote of the osculating hyperbolaϑ→ arccos(−1/e). In addition, if (ζ3

3 s2) vanishes Eq. (4.9)becomes singular. Even for Keplerian orbits with constantnonzero angular momentum s vanishes when approachinginfinity along an asymptote.

Angular momentum: If the position and velocity vectors areparallel, r×v = 0, the angularmomentum vanishes and thedeparture perifocal frameP is undetermined. This problemmay appear at departure if r0 ∥ v0, or during the integrationprocess due to external perturbations. Strong perturbationsin the along-track direction have an important effect on theangular momentum of the particle, and may make it zero.

Both situations are intimately related with flyby trajectories. Thefirst problem is a geometric issue that appears only in hyperbolicand parabolic orbits. The second problem relates to the magni-tude and the effective direction of the perturbing term. In prac-tice both phenomena are usually coupled.

50 4 The Dromo formulation

The angular momentum appears recursively in the deriva-tion of the equations of Dromo. The orbital plane is defined bymeans of the angular momentum vector. The angular velocity ofthe perifocal departure frame P, given in Eq. (4.4), becomes sin-gular when the angular momentum becomes zero. Indeed, if theposition and velocity vectors are parallel they do not define anyplane, but a straight line. The direction of this line is given bythe radius vector of the particle, which follows a rectilinear orbitalong this line. The trajectory of the particle can be considered adegenerated conic section.

The description of the orbital plane adopted in Dromo isbased on the dynamics of the rigid-body. Relying on the angularmomentum of the particle a gyroscopic description of the orbitalplane is provided. The quaternion n is introduced to overcomethe singularities associated to the usual Euler angles. But the gyro-scopic approach is no longer validwhen the equivalent rigid-bodyis no longer spinning, i.e. when the angularmomentumvanishes.

Equation (4.12) shows the dependency of the angular mo-mentum with the perturbing terms. The angular momentum isaffected by the component of the perturbing term along the di-rection of j. Numerical instabilities may be encountered whenthis particular projection of the perturbing acceleration grows.This is the typical case when the particle approaches a massiveplanet (such as Jupiter) along an incoming trajectory that leadsto a deep flyby.

4.7 Modified Dromo equations

Two alternative formulations (I and II) of Dromo are presentedin this section. Each formulation introduces a new set of Dromoelements, denoted oeI and oeII, respectively. The problem is re-formulated in an attempt to weaken the singularity. Note, how-ever, that Dromo includes a Sundman transformation of ordern = 2 and according to Thm. 1 it is not possible to achieve thefull regularization of the dynamics. In the next chapter Dromois redefined to derive a new propagator for hyperbolic orbits, notaffected by any singularity.

The first formulation eliminates all denominators from theright-hand side of the system of differential equations. To do so,an alternative definition of the fictitious time is provided. Thisintroduces a slower time scale in the problem that may improvethe accuracy of the integration. In addition, it may help to con-trol the integrator step size more efficiently. The transforma-tion from the elements oeI to the state vector in Cartesian coor-dinates becomes singular for s → 0 and the physical time be-comes constant, even though there are no singularities in the evo-lution equations. This formulationmay outperform the originalDromo numerically, but does not fully overcome the singulari-ties.

The second formulation eliminates the singularities in thestrict sense: neither s nor the angular momentum appear in anydenominator. However, as s → 0 the right-hand side of the dif-ferential equation for the time goes to zero too. That is, althoughthe formulation is not singular for s→ 0, the time becomes con-stant and the integration freezes in the limit case. It may behavebetter than Dromo as s→ 0 or h→ 0.

4.7.1 Formulation I

Consider the new set of variables ψ1, ψ2, ψ3 defined as

ψ1 = 1 + ζ1 cosσ + ζ2 sinσ ≡ s

ψ2 = ζ1 sinσ − ζ2 cosσ ≡ u

ψ3 =1ζ2

3ℓ≡ a

with the auxiliary term

ℓ = 1 − ζ21 − ζ2

2 = 2ψ1 − ψ21 − ψ2

2

Thenew set of variables isoeI = [ψ1, ψ2, ψ3, η1, η2, η3, η4]⊤. Thevariables ψ1 and ψ2 are proportional to the radial and circum-ferential components of the velocity. They are not elements, be-cause they are not constant in the Keplerian case. They have beendefined in terms of the original Dromo elements oe, and the in-verse map oeI 7→ oe reads

ζ1 = − cosσ + ψ1 cosσ + ψ2 sinσζ2 = − sinσ + ψ1 sinσ − ψ2 cosσ

ζ3 =1√ψ3ℓ

Under this new formulation the position and velocity vectors ofthe particle result in

r =ψ3ℓ

ψ1i

v =1ψ3ℓ

(ψ2 i + ψ1 j)

The derivative of the energy with respect to ideal anomalyσ,

dE

dσ=

12ψ2

3

dψ3

dσ=

drdσ

(ap · i) + r(ap · j)

provides the evolution of ψ3:

dψ3

dσ=

2ψ33ℓ

ψ21

[ψ2(ap · i) + ψ1(ap · j)

]Note that the problem is still singular for ψ1 = 0. A completeregularization of the equations requires not only the dependentvariables to be changed, but the independent variable. The or-bital plane is parameterized using the same quaternion than theoriginal Dromo formulation, defined by (η1, η2, η3, η4).

Let χ be the new independent variable, related to the idealanomaly through a higher order Sundman transformation

dσdχ= ψ3

1 (4.17)

§4.8 Numerical experiments 51

The new form of the Dromo equations is:

dtdχ= (ψ3ℓ)3/2ψ1 (4.18)

dψ1

dχ= −ψ2ψ

31 + 2(ψ3ℓ)2ψ1(ap · j) (4.19)

dψ2

dχ= (ψ1 − 1)ψ3

1 + (ψ3ℓ)2[ψ1(ap · i) + ψ2(ap · j)]

(4.20)

dψ3

dχ= 2ψ3

3ψ1ℓ[ψ2(ap · i) + ψ1(ap · j)

](4.21)

dη1

dχ=

(ψ1ℓ)2

2(−η4 cosσ − η3 sinσ)(ap · k) (4.22)

dη2

dχ=

(ψ1ℓ)2

2(−η3 cosσ + η4 sinσ)(ap · k) (4.23)

dη3

dχ=

(ψ1ℓ)2

2(−η2 cosσ + η1 sinσ)(ap · k) (4.24)

dη4

dχ=

(ψ1ℓ)2

2(−η1 cosσ − η2 sinσ)(ap · k). (4.25)

Equations (4.17–4.25) define the new system of differential equa-tions that govern the evolution of the Dromo elements. Thenumber of equations to be integrated has been increased fromeight to nine, because the ideal anomaly σ is no longer the inde-pendent variable and is integrated from Eq. (4.17).

The initial conditions are defined by the value of the param-eters at epoch χ = 0 (σ = σ0), corresponding to

χ = 0 : σ = σ0, t = 0, ψ1 = 1+ e0 cosσ0, ψ2 = e0 sinσ0

This formulation is free of singularities in the strict sense.However, as ψ1 → 0 the right-hand side of the equations van-ishes and the integration freezes. In addition, the velocity mayencounter a singularity as the term ψ3ℓ vanishes: note that it isℓ = 1 − e2, which goes to zero as e→ 1.

4.7.2 Formulation II

Let (φ1, φ2, φ3) be the triple

φ1 = ζ1, φ2 = ζ2, φ3 =r2

2=

12 ζ4

3 s2

The first two terms are equivalent to the first Dromo elements,whereas the third term is proportional to the distance to the cen-ter of attraction, and it is not an element.

The new set of variables is oeII = [φ1, φ2, φ3, η1, η2, η3, η4]⊤.In this case the map oe 7→ oeII is already established by the defini-tion of oeII. The inverse mapping only requires

ζ3 =1

4√

2s2φ3

Let Υ be the new independent variable, related to the idealanomaly through

dσdΥ=

sr2 =

s2φ3

= ζ43 s3 (4.26)

Under this new formulation, the evolution equations become

dtdΥ=

√s√2φ3

(4.27)

dφ1

dΥ= − s sinσ(ap · i) +

[φ1 + (1 + s) cosσ

](ap · j) (4.28)

dφ2

dΥ= −s cosσ(ap · i) +

[φ2 + (1 + s) sinσ

](ap · j) (4.29)

dφ3

dΥ= φ1 sinσ − φ2 sinσ (4.30)

dη1

dΥ=

(ap · k)2

(−η4 cosσ − η3 sinσ) (4.31)

dη2

dΥ=

(ap · k)2

(−η3 cosσ + η4 sinσ) (4.32)

dη3

dΥ=

(ap · k)2

(−η2 cosσ + η1 sinσ) (4.33)

dη4

dΥ=

(ap · k)2

(−η1 cosσ − η2 sinσ) (4.34)

These equations may become singular as φ3 → 0, i.e. r → 0.The equations are not fully regularized, but the singularity fors→ 0 is overcome. The state vector is defined by

r =√

2φ3 i

v =

√1

s√

2φ3

[− (φ2 + sinσ) iP + (φ1 + cosσ) jP]

4.8 Numerical experiments

The singularity inDromo appears under very specific conditions.Among the different scenarios in which s → 0, deep flybys willbe considered. When the particle approaches the sphere of influ-ence of the attracting body it may enter the sphere close to theasymptote along a hyperbolic orbit. The incoming velocity vec-tor would be quasi-parallel to the radius vector, leading to thediscussed singularities. As the minimum flyby distance decreasesthe perturbation of the third body increases, with a direct im-pact on the angular momentum. A number of theoretical flybysabout the Earth and Jupiter are designed to test the performanceof Dromo and the new versions Dromo-I and Dromo-II.

The designed experiment consists in propagating the helio-centric orbit of a virtual asteroid subject to the gravitational at-traction of a certain planet. The model is reduced to the circularrestricted three-body problem (CRTBP) in order to gain full con-trol on the perturbations and dynamics of the encounter. The or-bit of the asteroid intersects the orbit of the planet and performsa planetocentric flyby. There are two possible approaches to theproblem. On the one hand, the perturbed heliocentric orbit ofthe asteroid can be integrated directly. On the other hand, theorbit can be divided in three arcs: (i) the initial heliocentric arcuntil the asteroid reaches the sphere of influence of the planet,(ii) the planetocentric flyby inside the sphere of influence of theplanet (perturbed by the Sun), (iii) the final heliocentric arc oncethe asteroid has left the sphere of influence of the planet. Thissecond approach requires to switch the primary attracting bodywhen entering and leaving the sphere of influence. Figure 4.4 de-picts the geometry of the problem.

52 4 The Dromo formulation

Table 4.2: Definition of the numerical test cases.

Planetocentric Phase Heliocentric Phase Planetary Constants

dper [km] e∞ ϑ∞ [deg] ϑasy [deg] αenc [deg] ∆v [km s-1] e0 µ [km3s-2] a [AU]

Geocentric Case 1 −21565 19.404 −194.18 ±196.10 179.34 −3.236 0.63.99×105 1.00

Flyby Case 2 −11010 13.221 −107.45 ±108.08 180.63 −3.234 0.6

Jovicentric Case 3 −53170 11.000 −179.08 ±179.64 115.34 −0.091 0.11.27×108 5.20

Flyby Case 4 246438 11.007 −166.17 ±173.30 113.26 −1.920 0.1

Figure 4.4: Geometry of the three-body problem. The primary body

is C , S denotes the secondary body,O denotes the orbiting particle

andB is the barycenter of the system.

Let C denote the primary attractive center, let S denote thesecondary and let O be the asteroid. Neglecting themass of the as-teroid (mO ≪ mC ,mS ) the restricted three-body problem is gov-erned by the equation

d2rCO

dt2 + µC

rCO

r3CO

= −µS

(rSO

r3SO

− rSC

r3SC

)= ap

where vector ri j reads ri j = r j − ri for i, j = C , S ,O and ri j =

||ri j||. The perturbing acceleration is then written:

ap = −µS

(rSO

r3SO

− rSC

r3SC

)= −µS

rSO

r3SC

(r3

SC

r3SO

− 1)− rOC

r3SC

When rCS ∼ rSO , it is r3

SC/r3SO ∼ 1 and the subtraction may result

in important losses of accuracy. Introducing the auxiliary terms

f (Λ) = Λ3 + 3Λ + Λ2

(1 + Λ)3/2 + 1and Λ =

rOC · (rOC + 2rSO)rSO · rSO

the perturbation from the third body becomes

ap = −µS

rSO

r3SC

f (Λ) − rOC

r3SC

and involves no sensitive subtractions.

In the CRTBP the secondary body is assumed to follow a cir-cular orbit about the primary body. Under this assumption thesynodic rate ωs = [(µC + µS )/r3

SC ]1/2 is constant. Equations arenormalized so as µC = 1 and rSC = 1. The dynamical equationsin the CRTBP admit a first integral of motion, namely the Jacobi

integral. If v denotes the velocity of the particle in the synodicframe and V is the potential, the Jacobi integral reads

v2 = −J − 2V

with J the Jacobi constantLet (xs, ys, zs) be the barycentric coordinates of a particle in

the synodic rotating reference frame. The Jacobi constant is givenby

J = x2s + y

2s − (x2

s + y2s + z2

s) +2ϱrSO

+2(1 − ϱ)

rCO

where ϱ = mS/(mC + mS ) is the mass ratio. How the Jacobiconstant is preserved during the integration indicates the stabilityof the formulation; it is a necessary but not sufficient conditionfor accuracy. An enlightening discussion about this subtle phe-nomenon can be found in Quinlan and Tremaine (1992). Theorbit of the asteroid is propagated for one complete revolutionabout the Sun: the final value of the Jacobi constant, J f , is thencompared to the initial value, J0, and the relative error in preserv-ing the Jacobi constant is computed

Error =|J f − J0|

J0=∆J

J0

Four different flyby scenarios are designed, as summarizedin Table 4.2. In the first two cases the asteroid encounters theEarth, whereas in the third and fourth cases the perturbing planetis Jupiter. Each flyby is defined by the pericenter height (dper)and the eccentricity of the planetocentric orbit when enteringthe sphere of influence of the planet (e∞). The direction of theasymptote (ϑasy) can be compared to the initial true anomaly onthe planetocentric orbit (ϑ∞). Note that Case 3 represents a di-rect impact against Jupiter along a quasi-parabolic orbit. Case 4is a moderate version of Case 3, where the asteroid is initially veryclose to the asymptote. Both cases capture the discussed singular-ities of Dromo. Table 4.2 also presents the effective ∆v achievedby the asteroid during the flyby. The heliocentric orbit is initiallydetermined by the initial eccentricity e0. The orbital encounteroccurs at α = αenc (see Figure 4.4), being the encounter the in-stant of the closest approach to the planet.

The performance of the original Dromo scheme is com-pared against the new versions Dromo-I and Dromo-II. Cowell’smethod, the solution from theKustaanheimo-Stiefel transforma-tion (Kustaanheimo and Stiefel, 1965), and the Sperling-Burdetregularization (Burdet, 1968) are integrated as reference solutions.These formulations are comparednot only in termsof the error in

§4.9 Conclusions 53

preserving the Jacobi constant, but also in terms of the number offunction calls. The problem is integrated using a Runge-Kutta-Fehlberg 8(7) integration scheme, in double precision. Both thedirect integration of the problem (where the Sun is always the pri-mary body) and the solutionwhere the primary is switchedwhenthe asteroid is inside the sphere of influence of the planet (theprimary body is now the planet, and the Sun is the secondary)are computed. Figure 4.6 displays the number of function callsand the relative error in preserving the Jacobi constant for each ofthe four cases defined in Table 4.2. Each point in the figures cor-responds to a different integration tolerance, which varies from10−5 to 10−15.

Dromo-II introduces a high-order time transformationwhich refines the analytic step-size adaption. The result is a moreefficient step-size control that describes the close encounterwith ahigher resolutionwhen integrating theproblemdirectly (withoutswitching the primaries). This phenomenon is easily observed forCase 1 andCase 2, which are qualitatively similar and far from thesingularity. When the primaries are switched the advantages ofan improved step-size management are not so significant. In gen-eral, the direct integration requires less function calls for the sameaccuracy. This is due to the fact that switching the primaries re-quires additional root-finding operations related to entering andleaving the sphere of influence. The performance of Cowell’smethod, the original Dromo, Dromo-I, the KS and the SB reg-ularization is comparable in these cases.

Case 3 is a limit case specifically conceived for reaching thesingularity in Dromo. The direct integration of the problemfails when the formulation becomes singular. The numerical is-sues encountered by Dromo can be analyzed in Fig. 4.5. Asthe asteroid approaches Jupiter the term ζ3 grows rapidly, sincethe angular momentum goes to zero. Despite the growth of ζ3the denominator ζ4

3 s3 that defines the term ap = ap/(ζ43 s3) in

Eqs. (4.9–4.16) becomes zero. During the integration the prod-uct ζ3

3 s2 goes to zero too. Hence, the transformation in Eq. (4.8)becomes singular at that point. This is caused by the fact thats → 0 compensates the growth rate of ζ3. This phenomenonmakes the equations ofDromo singular. The singularity is foundat Tcrit = 1100.72 days. It appears when the position and ve-locity vectors become parallel due to the perturbation from thethird body. The fact that ζ4

3 s3 goes to zero has additional conse-quences. In Dromo-II the derivative of the ideal anomaly withrespect to the new independent variableΥ—Eq. (4.17)— is pre-cisely ζ4

3 s3. Hence, the ideal anomaly becomes constant and theformulation freezes. The same issue is detected when integratingDromo-I.

When propagating Case 3 the direct integration of the prob-lem using any version of Dromo leads to an extremely highnumber of function calls, and for finer tolerances the formula-tion freezes. The KS transformation slightly outperforms Cow-ell’s method and the SB formulation. When the primaries areswitched, version II is the only Dromo method capable of com-puting the entire trajectory.

Case 4 is still a deep flyby, but it is possible to integrate theproblem with Dromo and Dromo-II. When considering the di-rect integration of the problem Dromo-II, but not Dromo, isable to match the performance of the KS and SB formulations.Dromo-I is problematic to integrate in this case, the resulting

Figure 4.5: Evolution of the terms ζ43 s3 and ζ3 in Case 3.

computational times are completely out of scale and the resultsare not included in the figure. However, when the primaries areswitched the performance of the formulations is similar. Dromo-II provides the most accurate solution, although requires morefunction calls.

4.9 Conclusions

The power of Hansen ideal frames resides in the effective decou-pling between the in-plane dynamics and the motion of the or-bital plane itself. Once the perturbation is projected in the orbitalframe the normal component only affects the dynamics of the or-bital plane. Dromo exploits all the potential of this decomposi-tion by introducing a quaternion to model the evolution of theorbital plane. In addition, the dynamical decomposition yieldsan intuitive interpretation of the effects of each component ofthe perturbing acceleration.

Dromo combines different regularization and stabilizationtechniques (see Chap. 2). The use of the Sundman transforma-tion enhances the step-size regulation. Moreover, the degree twoof the transformation increases the benefits of the time transfor-mation: the new independent variable is equivalent to the trueanomaly, which captures in more detail the dynamics around pe-riapsis. The integration of elements instead of coordinates yieldsa smoother integration scheme, specially when facing weak per-turbations.

Roa et al. (2015b) discovered a singularity in the equations ofDromo: when the angular momentum vanishes the gyroscopicdescription of the orbital plane is no longer valid and it becomesundetermined. This singularity is predicted by Thm. 1. The be-havior in the vicinity of the singularity can be improved by in-troducing a higher-order time transformation. The analytic step-size adaption is more efficient and it is possible to integrate orbitsreaching smaller values of the angular momentum. Recall thatthe fundamental theorem presented in page 12 shows that a sec-ond order Sundman transformation cannot achieve a full regu-larization of the dynamics. However, the singularity in this casecomes fromhow the orbital plane is given a gyroscopic definition.

54 4 The Dromo formulation

Figure 4.6: Function calls vs accuracy of the integration, measured as the relative error in the preservation of the Jacobi constant. The figures

on the left correspond to the direct integration, and the figures on the right show the performance of the switching technique.

“It must be nice to ride these fields on the cob of mathematics proper,while the likes of us must trudge along on foot.”

—A. Einstein to T. Levi-Civita. Lucerne, August 2, 1917

5Orbit propagation in Minkowskian geometry

T he numerical integration of perturbed hyperbolic or-bits is challenging due to the rapid changes in the rela-tive velocity during pericenter passage, which may lead

to considerable uncertainties (Hajduková Jr, 2008). Aproper for-mulation should also exhibit numerical stability during the inte-gration. Cowell’s method provides a simple integration schemefor arbitrary orbits. Due to the Lyapunov instability of orbitalmotion, numerical errors in Cowell’s method grow rapidly (seeChapter 2). The instability of the motion is governed by theeigenvalues shown in Eq. (2.2): the exponential behavior due tothe eigenvalue (2µ/r3)1/2 is responsible for the strong divergenceof the integral flow during close flybys.

Hyperbolic orbits are closely related to the dynamics and evo-lution mechanisms of bodies and particles leaving the solar sys-tem. Comets are a relevant example. Long-period comets ejectedfrom the Oort cloud due to the effect of passing stars typicallyenter the inner solar system with quasi-parabolic orbits (Hills,1981; Królikowska, 2001). Gravitational and non-gravitationaleffects may cause the comets to leave the solar system follow-ing hyperbolic orbits (Królikowska, 2006). Special attention hasbeenpaid to thenumerical propagationof suchorbits (Yabushita,1979; Thomas andMorbidelli, 1996), since they can be integratedbackward in time in order to analyze the origin and evolution ofcomets (Everhart, 1976; Ipatov, 1999). Another example is therole of hyperbolic orbits in the dynamics of dust particles, in par-ticular the flux of β-meteoroids incoming from the inner solarsystem following hyperbolic orbits and detected by the Ulyssesmission (Wehry and Mann, 1999).

In asteroid-planet close approaches the relative orbit of theasteroid with respect to the planet is typically hyperbolic. Such

phenomena are important because impacts may occur. Nowa-days, onemajor concern is evaluating the collision risk of theNearEarth Asteroids (Milani and Knežević, 1990; Muinonen et al.,2001). The dynamics of the encounter are complex and orbitalresonances are particularly critical. Planetary perturbations maylead to orbital resonances that produce high risk close encoun-ters (Kozai, 1985; Ferraz-Mello et al., 1998). Milani et al. (1989)explored the behavior of resonant asteroids through massive nu-merical simulations. Valsecchi et al. (2003) developed an analyt-ical theory for modeling resonant returns. If sufficient observa-tional data is available, the asteroid mass and gravitational distri-butionmaybemodeledwhen considering its dynamics (Yeomanset al., 1997; Miller et al., 2002). Roa and Handmer (2015) stud-ied the likely fate of asteroid fragments after disruption in a lu-nar distant-retrograde orbit using various campaigns of numeri-cal simulations, in which flybys around the Moon have a strongscattering effect. An accurate description of hyperbolic orbits in-creases the scientific value of the results.

Many exploration missions involve hyperbolic orbits. Intheir way to interstellar space, both Voyager 1 and 2 spacecraftfollowed gravity-assist swingby trajectories about Jupiter and Sat-urn. Voyager 2 also visited Uranus and Neptune, and it is cur-rently traveling through the heliosheath toward interstellar space(seeKohlhase andPenzo, 1977, for the original design, andFig. 1.3for its orbit). The Cassini-Huygens mission is another exampleof gravity-assist trajectories; to reach Saturn, the spacecraft per-formed three planetary flybys about Venus, the Earth and Jupiter(Burton et al., 2001, and Fig. 1.4). During the mission remark-able advances on several fields transpired. From detecting in-terstellar dust (Altobelli et al., 2003) to general relativity exper-iments (Bertotti et al., 2003), including Titan’s atmosphere char-

55

56 5 Orbit propagation in Minkowskian geometry

acterization (Waite Jr et al., 2005) and Jupiter observations (Porcoet al., 2003). The Europa mission, the next one in NASA’s Flag-ship Program, will perform 45 flybys of Europa looking for suit-able conditions for life. The unprecedented scientific results ex-pected from themission require extremely accurate propagationsfor modeling the dynamics of the probe.

In Chapter 4 we introduced the concept of Hansen idealframes and discussed its potential for numerical propagation.The works byDeprit (1975), Palacios et al. (1992) and Peláez et al.(2007) exploited this idea to derive a set of regular elements. Theunperturbed radial motion in hyperbolic orbits is well-known interms of hyperbolic trigonometry (Battin, 1999, p. 167):

r = a(1 − e cosh H)

Roa and Peláez (2015d,e) derived a special perturbation methodusing the variation of parameters technique: for coherence withthe radial solution the motion of the eccentricity vector is de-scribed using hyperbolic angles and rotations, and not the tra-ditional Euclidean definitions. They formulated the dynam-ics using the mathematical constructions underlying Minkowskispace-time. In the field of special relativity,HermannMinkowski(1923) created a four-dimensional description of time and spaceto simplify the formulation of Einstein’s theory. The geome-try on the two-dimensional Minkowski space-time (namely theMinkowski planeR2

1) naturally introduces hyperbolic trigonom-etry (Saloom and Tari, 2012). Some authors refer to this plane asthe Lorentz plane (Sobczyk, 1995).

Hypercomplex numbers provide useful geometrical repre-sentations. The components of a two-dimensional hypercom-plex number can be understood as the coordinates of a point ona certain plane. Depending on the metrics, hypercomplex num-bers define different geometries. For instance, the modulus of acomplex number is equivalent to the Euclidean distance from theorigin to the point its components define. Similarly, the modu-lus of a hyperbolic number leads to the definition of distance onthe Minkowski plane (Boccaletti et al., 2008, chap. 4). Complexnumbers and hyperbolic numbers are special cases of hypercom-plex numbers. Hyperbolic numbers are also known as dual num-bers (Kantor et al., 1989) or perplex numbers: the latterwas intro-duced by Fjelstad (1986), who described the hyperbolic construc-tions in special relativity through them. Ulrych (2005) extendedhyperbolic numbers to quantum physics and Motter and Rosa(1998) stated their formal inclusion inClifford algebras. Cariñenaet al. (1991) explored Minkowskian metrics for representing or-bits in the configuration and in the velocity spaces. Multicomplexnumbers, a special subset of hypercomplex numbers, generalizedcomplex numbers to higher dimensions (Price, 1991). Lantoineet al. (2012) applied these numbers to the computation of high-order derivatives.

The properties of Minkowskian geometry were exploited byRoa and Peláez (2015d,e) to formulate hyperbolic motion. Rota-tions and metrics in the Minkowski planeR2

1 are described usinghyperbolic numbers. The ultimate goal is to provide amore accu-rate and stable description of such orbits, in which the propaga-tion error is not affected byperiapsis passage. The new set of eightelements, according to the definition of element by Stiefel andScheifele (1971, chap. 3), derives from the theory of hypercomplexnumbers. The first element is a time element. The second and

third elements are the components of the eccentricity vector onR2

1. The fourth element is the semimajor axis. The componentsof the eccentricity vector are referred to a certain reference frame,attached to the orbital plane. The motion of the orbital plane isuniquely determined by the motion of this frame. It is definedon the inertial reference by means of a quaternion. The compo-nents of the quaternion are the last four elements. Quaternionsare treated as particular instances of hypercomplex numbers.

This chapter is organized as follows. Section 5.1 defines theorbital problem. The evolution of the eccentricity vector on theMinkowski plane is determined. This section closes with the def-inition of the bijection between the components of the eccentric-ity vector on the Euclidean and the Minkowski plane. The varia-tionofparameters technique is applied in Sect. 5.3, where the evo-lution equations for the orbit geometry are obtained. Section 5.4describes themotion of the orbital plane. The time element is de-fined in Sect. 5.5. A brief summary of the equations is presented.Finally, Sect. 5.6 analyzes the performance of themethod througha set of numerical experiments. Hypercomplex numbers are de-scribed in Appendix A. General definitions and properties of theassociated algebras are stated.

5.1 Orbital motion

Let r ∈ R3 denote the position vector of a particle in an inertialreference frame I. The perturbedmotion in a central gravity fieldis governed by the equation:

d2rdt2 = −

rr3 + ap (5.1)

with r = ||r|| and a normalized gravitational parameter µ = 1.The orbital frame L = i, j,k rotates with the particle and

is given the usual definition:

i =rr, k =

hh, j = k × i

The angularmomentum vector is h, and h = ||h||. The departureperifocal frame, P, is described by the basis iP, jP, kP,

iP =e0

e0, kP =

h0

h0, jP = kP × iP

just like in Chap. 4.LetA denote an intermediate frame, withA = iA, jA,kA.

The xAyA-plane is contained in the orbital plane, i.e. kA ∥ kP.At t = t0 this frame coincides with frame P but there exists a rel-ative angular velocity between both frames, which is normal tothe orbital plane. There is a degree of freedom in the definitionof frameA: the angle between the xA-axis and the xP-axis. Thereason for introducing the intermediate frame is that the eccen-tricity vector can be written

e = ρ1iA + ρ2jA

with (ρ1, ρ2) ∈ R × R to be defined at convenience. Once thecomponents of the eccentricity vector are defined (Sect. 5.1.1), anexplicit expression for the relative velocityA/P will be provided

§5.1 Orbital motion 57

(Sect. 5.4). Initially it is ρ1(0) = e0 and ρ2(0) = 0. Figure 5.1depicts the described reference frames.

Figure 5.1: Schematic view of the reference frames to be used. The

true anomaly is denoted byϑ, and ν defines the angle between versoriA and the radius vector, that is ν = α + ϑ.

5.1.1 The eccentricity vector

The evolution of the eccentricity vector is defined in the interme-diate referenceA in terms of

e = ρ1iA + ρ2jA

Let ze ∈ C be the representation of e on theGauss-Argand plane.This number is written

ze = ρ1 + iρ2

with i =√−1 the imaginary unit.

The Euclidean norm of the eccentricity vector is given by

||e||2E2 = zez

†e = ρ

21 + ρ

22 ≡ |ze|2C (5.2)

where z†e is the complex conjugate of ze.Alternatively, an equivalent representation of e can be found

on the Minkowski plane, namely

we = λ1 + jλ2, we ∈ D, (λ1, λ2) ∈ R × R

HereD denotes the field of hyperbolic numbers,

D = x + jy | (x, y) ∈ R × R, j2 = +1

and j is the hyperbolic imaginary unit. Note that j =√+1 < R.

The formal definition of hyperbolic numbers and its connectionwith Minkowskian geometry can be found in Appendix A.

The norm is defined by the metrics in R21,

||e||2R2

1= wew

†e = λ

21 − λ2

2 ≡ |we|2D (5.3)

The norm of the eccentricity vector is the osculating eccentric-ity, e. The different definitions of norm in the Euclidean and theMinkowskian vector space provide two different expressions for

the osculating eccentricity e, given in Eqs. (5.2) and (5.3). But theeccentricity of the orbit has a clear physical meaning and it is pos-sible to assume that both definitions of normyield the same phys-ical quantity (Roa andPeláez, 2015d,e). Thus, equating Eqs. (5.2)and (5.3) establishes a relation between the Euclidean and theMinkowskian components of e:

e2 = ρ21 + ρ

22 = λ

21 − λ2

2

The evolution of the eccentricity vector is defined in terms ofa circular angle α on the Gauss-Argand plane, and in terms of ahyperbolic angleγ on theMinkowski plane. This is easily inferredfrom the polar form of ze andwe,

ze = e e iα = e(cosα + i sinα)

we = e e jγ = e(cosh γ + j sinh γ)

That is, vector e is determined by the circular angle α or, equiv-alently, by the hyperbolic angle γ, together with the osculatingeccentricity e. The numbers ze andwe satisfy

ze = e(cosα + i sinα) = ρ1 + iρ2

we = e(cosh γ + j sinh γ) = λ1 + jλ2

Given the previous expressions it is natural to write

ρ1 = e cosα, λ1 = e cosh γρ2 = e sinα, λ2 = e sinh γ

Provided that ze and we represent the same physical entity—the eccentricity vector e— there exists a bijective map betweenthe Gauss-Argand plane and the Minkowski plane, (ρ1, ρ2) ↔(λ1, λ2). This bijection can be derived geometrically when ana-lyzing the components of e on both planes. Figure 5.2 shows thegeometrical construction of the problemwhen superposing bothplanes. Whenα changes the osculating eccentricity vector definesa circumference of radius e, ce, on the Gauss-Argand plane. In asuch case, it defines a rectangular (or equilateral) hyperbola on theMinkowski plane with semimajor axis equal to the osculating ec-centricity, he. Hyperbolic rotations are described in Appendix A.

The geometry shown in Fig. 5.2 leads to the relation:

α = gd (γ)

where gd (γ) refers to the Gudermannian function (Battin, 1999,p. 168). The explicit expression of the Gudermannian is

gd (γ) =∫ γ

0sech t dt = arctan sinh γ = 2 arctan eγ − π

2(5.4)

This function verifies

cos gd (γ) = sech γ, sin gd (γ) = tanh γ

From these properties of the Gudermannian the mapping(λ1, λ2) 7→ (ρ1, ρ2) is established,

ρ1 = λ1

(1 −

λ22

λ21

), ρ2 =

λ2

λ1

√λ2

1 − λ22 (5.5)

and the inverse mapping (ρ1, ρ2) 7→ (λ1, λ2) reads

λ1 = ρ1

(1 +

ρ22

ρ21

), λ2 =

ρ2

ρ1

√ρ2

1 + ρ22

58 5 Orbit propagation in Minkowskian geometry

Figure 5.2: Geometrical representation of the eccentricity vector

superposing the Gauss-Argand plane and theMinkowski plane.

5.1.2 The hyperbolic anomaly

The hyperbolic anomaly H is the hyperbolic angle between theosculating apse line and the direction from the center of the hy-perbola, O, to the particle, P . It is the hyperbolic equivalent tothe true anomaly, ϑ (the circular angle between the osculatingapse line and the radius vector). The direction of the apse lineis given either by the circular angle α or by the hyperbolic angleγ. The axes on theMinkowski plane are oriented as frameA, andcentered at O.

In the perifocal reference frame the eccentricity vector pointstoward positive abscissa. The orbit corresponds to the left branchof the hyperbola. Let ve ∈ D be a hyperbolic number that fixesthe direction of the apse line with respect to the center of the hy-perbola, O, in referenceA. On h1

+ it is v+e = e jγ, whereas on h1− it

is v−e = −e jγ, for compliance with the sign criterion. Hyperbolicrotations will then be defined on h1. Recall that this hyperbola isdifferent from the osculating orbit.

Let v−r ∈ D be a hyperbolic number that provides the direc-tion from O to the particle, with v−r ⊂ h1

−. From the definition ofthe hyperbolic anomaly it follows

v−r = v

−e e−jH = −e jγ e−jH = −e−j(H−γ) = −e−ju (5.6)

That is, the direction to the particle can be directly referred to thexA-axis using thehyperbolic angleu = H−γ. This angle becomesthe new independent variable. Figure 5.3 shows the geometry ofthe problem.

The hyperbolic anomaly satisfies the relations:

cosh H = cosh γ cosh u + sinh γ sinh u (5.7)

sinh H = sinh γ cosh u + cosh γ sinh u (5.8)

If there are no perturbations it is γ = 0 and H ≡ u. The trueanomaly can be obtained from the hyperbolic anomaly (Vallado,1997, pp. 221–222),

cosϑ =e − cosh H

e cosh H − 1, cosh H =

e + cosϑe cosϑ + 1

sinϑ =ℓ sinh H

e cosh H − 1, sinh H =

ℓ sinϑe cosϑ + 1

Figure 5.3: Geometrical interpretation of the hyperbolic anomaly and

the independent variable u on theMinkowski plane. The focus of the

hyperbola is denoted byF .

with ℓ =√

e2 − 1. Initially it is γ = 0 and u0 = H0, so the apseline coincides with the xA-axis.

5.2 Hyperbolic rotations and the Lorentzgroup

The eccentricity vector can be written e = e ue, in which ue fol-lows the direction of the line of apses. According to Eq. (5.6) theeccentricity vector can be projected in A by means of the linearmapping

R(γ) =[

cosh γ sinh γsinh γ cosh γ

]with det(R) = 1.

Let T denote the metric tensor attached to the coordinatespace. In Euclidean space E2 the signature of T is ⟨+,+⟩. Al-though the determinant of R(γ) is unity, in E2 the operator R(γ)is not orthogonal and does not define a rotation. Indeed, consid-ering that R(γ) = R⊤(γ) it is easy to verify that the orthogonalitycondition

R(γ) T R⊤(γ) = T

is not satisfied, because

R(γ) T R⊤(γ) ≡ R(γ) R(γ) =[

2 cosh2 γ − 1 2 cosh γ sinh γ2 cosh γ sinh γ 2 cosh2 γ − 1

]Since the operator is not orthogonal it does not represent arotation in E2. But if one migrates to the two-dimensionalMinkowski space-time M the metric tensor is endowed with asignature ⟨+,−⟩ (see for example Penrose, 2004, chap. 18). Themodification of the signature is equivalent to introducing a time-like component, and in this case the condition

R(γ) T R⊤(γ) = T

holds. The operator R(γ) is orthogonal inM and det R(γ) = 1,so it can be seen as a rotation action.

This construction is simply a subgroup of the more generalLorentz groupO(3, 1). If one identifies β = v/c and the Lorentz

§5.3 Variation of Parameters 59

factor g = 1/√

1 − β2 with g = cosh γ and −βg = sinh γ, thetwo-dimensional Lorentz boost[

g −βg−βg g

]reduces tomatrix R. The Lorentz group leaves the space-time ori-gin invariant.

5.3 Variation of Parameters

The solution to the generic perturbed problem is found using thevariation of parameters procedure. First, the solution rk = rk(t)to the Keplerian problem is obtained in terms of different con-stant parameters. Seeking a solution of the same form, the per-turbed problem is solved considering that the parameters are nolonger constant but functions of u.

5.3.1 The new independent variable

The new independent variable u is related to the physical timethrough the Sundman transformation

dtdu= r√

a (5.9)

and defines a fictitious time. Baù et al. (2014,2015) explored asimilar transformation for the elliptic case. In the following thesemimajor axis is assumed to be positive, a > 0. So is the energy,E > 0. The radial velocity is

drdt=

1r√

adrdu

Derivation with respect to physical time yields

d2rdt2 =

1r2a

d2rdu2 −

1r3a

(drdu

)2

− 12a2r2

drdu

dadu

This expression relates the radial acceleration with the fictitiousradial acceleration. In particular,

d2rdu2 = r2a

d2rdt2 +

1r

(drdu

)2

+12a

drdu

dadu

(5.10)

Equation (5.10) governs the radialmotion, r(u). To complete thisequation, explicit expressions for the terms in the right-hand sideare required, namely the radial acceleration, the fictitious radialvelocity, and the evolution of the semimajor axis. The three fol-lowing propositions derive these expressions.

Proposition 1: The radial acceleration is given by

d2rdt2 = −

1r2 +

h2

r3 + (ap · i) (5.11)

where h is the angular momentum of the particle.

Proof: Let γ ∈ R3 denote the acceleration vector. Equation (5.1)is rewritten as

γLI = −1r2 i + ap

This expression provides the acceleration of the particle relativeto the inertial reference I. The in-plane motion is governed bythe acceleration γLP, considering the inertia terms

γLP = −1r2 i + ap − γPI − 2ωPI × vLP

The angular velocities that appear in the problem are

ωLP =hr2 k, ωPI =

rh

(ap · k) i (5.12)

The definition of these angular velocities by Peláez et al. (2007)can be found in Eq. (4.3). The acceleration γLP is expanded,

γLP = −1r2 i +

[I − (k ⊗ k)

] ap (5.13)

where I is the secondorder unit tensor, representedby the identitymatrix, (k ⊗ k) denotes the dyadic product, and () refers to thecontracted product. The tensorial term yields[I − (k ⊗ k)

] ap = ap − (ap · k) k = (ap · i) i + (ap · j) j ≡ a∥,p

Sincek is normal to the orbital plane a∥,p defines the in-plane per-turbing acceleration.

The in-plane acceleration can alternatively be determinedthrough its kinematic definition in polar coordinates. That is,

γLP =

(d2rdt2 −

h2

r3

)i +

1rdhdt

j (5.14)

By equating Eqs. (5.13) and (5.14) the following relation is ob-tained:

− 1r2 i + a∥,p =

(d2rdt2 −

h2

r3

)i +

1rdhdt

j

If projected along the direction of i, this expression leads to theradial acceleration,

d2rdt2 = −

1r2 +

h2

r3 + (ap · i)

affected by the perturbing term (ap · i) = (a∥,p · i).

Proposition 2: The evolution of the semimajor axis in terms ofthe fictitious time u abides by

dadu= −2a2

[drdu

(ap · i) + h√

a(ap · j)]

(5.15)

Proof: The Keplerian energy of the system is of the form:

E =v2

2− 1

r=

12a, with a > 0, E > 0

The time evolution of the energy is obtained in terms of thepower performed by the perturbing forces. That is,

dE

dt= ap · v (5.16)

60 5 Orbit propagation in Minkowskian geometry

This equation is a particular case of a more general result: con-sider a generic constant in Kepler’s problem, κ. Its time evolutiondue to external perturbations ap isdκdt=∂κ

∂v· ap

Since the partial derivative of the energy with respect to the ve-locity vector is the velocity vector itself, for κ = E the previousequation reduces to Eq. (5.16).

The velocity is projected in the orbital frame as

v =drdt

i +hr

j (5.17)

and the power becomesdE

dt=

drdt

(ap · i) +hr

(ap · j)

This equation provides a simple expression for the time evolutionof the energy.

The time evolution of the semimajor axis is obtained fromdadt= −2a2 dE

dtand, considering the aforementioneddefinitionof thepower, thisbecomesdadt= −2a2

[drdt

(ap · i) +hr

(ap · j)]

(5.18)

Equation (5.18) is transformed into the derivative with respect tou by applying the transformation given in Eq. (5.9),

dadu= −2a2

[drdu

(ap · i) + h√

a(ap · j)]

Proposition 3: The fictitious radial velocity squared is(drdu

)2

= r2 − h2a + 2ra

Proof: Considering Eq. (5.17) the definition given to the energycan be expanded,

2E =

(drdt

)2

+h2

r2 −2r=

1a

The term (1/a) reads

1a=

(drdt

)2

+h2

r2 −2r

(5.19)

From Eq. (5.19) and the definition of the fictitious time it followsthat(drdu

)2

=

(drdt

)2 (dtdu

)2

= r2 − h2a + 2ra

Considering Props. 1 and 3, Eq. (5.10) becomes

d2rdu2 − a − r = r2a(ap · i) +

12a

drdu

dadu

(5.20)

The last term is intentionally left unchanged.

Solution to the unperturbed case

In absence of perturbations (ap = 0) the semimajor axis remainsconstant and Eq. (5.20) reduces to

d2rdu2 − a − r = 0

Integration of this equation yields

r(u) = a(B1 cosh u + B2 sinh u − 1) (5.21)

where a > 0 is constant, and B1 and B2 are two integration con-stants to be determined. InKeplerianhyperbolic orbits the radiusvector is a well known function of the form:

r = a(e cosh H − 1). (5.22)

Equating Eqs. (5.21) and (5.22) leads to

B1 = e, B2 = 0, u ≡ H

The fictitious radial velocity of the particle is simply

drdu= a(B1 sinh u + B2 cosh u) (5.23)

which reduces todrdu= ae sinh H

in the unperturbed case.

Extension to the perturbed case

The solution to the general perturbed problem is constructedthrough the variation of parameters procedure. The equa-tions developed in this section define the evolution of the tripleλ1, λ2, λ3with respect to fictitious time u, which is the new in-dependent variable. From Eq. (5.21) a more general solution issuggested,

r(u) = λ3(u)[λ1(u) cosh u + λ2(u) sinh u − 1

](5.24)

where λ1, λ2 and λ3 ≡ a are no longer constants because of theperturbing forces. This expression can be obtained directly fromEq. (5.22),

r(u) = a(u)[e(u) cosh H(u) − 1

]and introducing Eq. (5.7):

r(u) = a(u)[e(u)(cosh γ cosh u + sinh γ sinh u) − 1

]= a(u)

[λ1(u) cosh u + λ2(u) sinh u − 1

]In sum, the solution r = r(u) is written

r(u) = λ3(u)[λ1(u) cosh u+λ2(u) sinh u−1

]= λ3(u)r(u) (5.25)

with r(u) = λ1(u) cosh u + λ2(u) sinh u − 1.To compute the fictitious radial velocity Eq. (5.25) is derived

to provide:

drdu=dλ3

dur + λ3

(dλ1

ducosh u +

dλ2

dusinh u

)+ λ3(λ1 sinh u + λ2 cosh u)

§5.4 Orbital plane dynamics 61

We assume a null gauge,

Φ ≡ dλ3

dur + λ3

(dλ1

ducosh u +

dλ2

dusinh u

)= 0 (5.26)

Imposing this condition guarantees that the fictitious velocityon the Keplerian orbit defined by the instantaneous values ofλ1, λ2, λ3 coincides with that of the true orbit. Hence, such aKeplerian orbit is in fact the osculating orbit. The derivative of rreduces todrdu= λ3(λ1 sinh u + λ2 cosh u) (5.27)

5.3.2 Evolution equations

The second derivative of Eq. (5.27) yields

d2rdu2 =

dλ3

duq + λ3

(dλ1

dusinh u +

dλ2

ducosh u

)+ λ3(r + 1)

with q = λ1 sinh u + λ2 cosh u. Using this result Eq. (5.20) be-comesdλ1

dusinh u +

dλ2

ducosh u = λ2

3r2(ap · i) −q

2λ3

dλ3

du(5.28)

The derivative of the semimajor axis, defined in Eq. (5.15), is com-pleted when considering the form of r = r(u),

dλ3

du= −2λ3

3[q(ap · i) + ℓ(ap · j)

]where

ℓ =√λ2

1 − λ22 − 1 =

√e2 − 1

Equations (5.26) and (5.28) form a system of equations fromwhich the derivatives of the variables λ1 and λ2 can be solved.They provide the governing equations, Eqs. (5.29–5.31),

dλ1

du= λ2

3

(−ℓ2 sinh u + 2λ2r)(ap · i)

+ ℓ[λ1 + (r − 1) cosh u](ap · j)

(5.29)

dλ2

du= λ2

3

(−ℓ2 cosh u + 2λ1r)(ap · i)

+ ℓ[λ2 − (r − 1) sinh u](ap · j)

(5.30)

dλ3

du= −2λ3

3

[q(ap · i) + ℓ(ap · j)

](5.31)

These equations are equivalent to the Gauss’ form of the La-grange planetary equations, in terms of the new set of variables.

The bijection between the Gauss-Argand plane and theMinkowski plane, defined in Eq. (5.5), relates the variables(ρ1, ρ2) and (λ1, λ2). These expressions may be derived to pro-vide

dρ1

du=

(2 − e2

λ21

)dλ1

du− 2

λ2

λ1

dλ2

du

dρ2

du=

1e

λ32

λ21

dλ1

du+

1λ1

(λ1 − 2λ2

2

) dλ2

du

Replacing the value of the derivatives of (λ1, λ2), given in Eqs.(5.29–5.31), leads to the evolution equations for variables ρ1 andρ2:

dρ1

du=λ2

3

λ21

[ℓ2e2 sinh u + 2λ2(ℓ2 − r)

](ap · i)

+ ℓ[(ℓ2 + 1)(1 − r) cosh u + λ1(2r2 + ℓ2 − 1)

](ap · j)

(5.32)

dρ2

du=

λ23

eλ21

− [ℓ2e2λ1 cosh u + (1 − r)ℓ4

+ ℓ2(1 − 3r − (1 + r)λ21) − 2r

](ap · i)

+ ℓ[e2(1 − r)q + λ1λ2(r2 + ℓ2)

](ap · j)

(5.33)

Equations (5.29–5.30) provide the evolution of the eccen-tricity vector on the Minkowski plane, whereas Eqs. (5.32–5.33)provide the evolution of the eccentricity vector on the Cartesianplane with e > 1.

5.4 Orbital plane dynamics

The orbital frame L rotates around the normal to the orbitalplane with angular velocity ωLA. The motion of frame L re-ferred to the inertial reference is composed of the in-planemotionand the motion of the orbital plane itself. That is,

ωLI = ωLA + ωAI

The term ωLA refers to the in-plane motion, whereas ωAI ac-counts for the motion of the orbital plane with respect to the in-ertial reference I. The angular velocity of the orbital plane is thengiven by

ωAI = ωLI − ωLA (5.34)

The second term corresponds to the time evolution of angle ν,

ωLA =dνdt

k

Recall that Eq. (5.12) provided explicit expressions for the dynam-ics of the departure perifocal frame P, namelyωLP andωPI:

ωLI = ωLP + ωPI

In sum, Eq. (5.34) can be written

ωAI = ωLP + ωPI − ωLA

which results in

ωAI =rh

(ap · k) i +(

hr2 −

dνdt

)k = ω1i + ω3k (5.35)

The pair (ω1, ω3) defines the components of ωAI in frame L.Projected onto frameA the angular velocity reads

ωAI = paiA + qajA + rak

62 5 Orbit propagation in Minkowskian geometry

The angular velocity referred to the fictitious time becomes

ΩAI = paiA + qajA + rak, with ΩAI = ωAI

dtdu

Let rA, rI be two pure quaternions associated to the com-ponents of r in frames A and I, respectively. The connectionof quaternions with the set of four-dimensional hypercomplexnumbers H4 is discussed in Appendix A. Let q ∈ H4 be the unitquaternion that defines the position of frame A on the inertialframe. Quaternion q is written in terms of its components asq = χ4 + i χ1 + jχ2 + kχ3. It defines the rotation rA 7→ rI,

rI = q rA q†

The derivative of quaternion qwith respect to fictitious time u isgiven by

dqdu=

12q(2mAI)

with 2mAI = 0 + i pa + j qa + k ra. Computing the quaternionproduct leads to the equations:

dχ1

du=

12

(−χ4 pa − χ3qa + χ2ra) (5.36)

dχ2

du=

12

(−χ3 pa + χ4qa − χ1ra) (5.37)

dχ3

du=

12

(−χ2 pa + χ1qa + χ4ra) (5.38)

dχ4

du=

12

(−χ1 pa − χ2qa − χ3ra) (5.39)

The components (pa, qa, ra) of the fictitious angular velocity inframeA are obtained from

pa = ω1 cos ν, qa = ω1 sin ν, ra = ω3

withΩAI = ω1i + ω3k. The first component ω1 of the angularvelocity is obtained directly from Eq. (5.35), when referred to thefictitious time,

ω1 = λ23

r2

ℓ(ap · k)

The second component, ω3, is

ω3 =λ2

3

e2λ1

[ℓλ1(r − ℓ2) + e

(ℓ2(1 − r) − 2r

)](ap · i)

+ q[λ1(ℓ2 + r) + ℓe(r − 1)

](ap · j)

(5.40)

5.5 Time element

Equations (5.29–5.31) provide the geometry of the orbit, andEqs. (5.36–5.39) define the motion of the orbital plane. Theright-hand side of these equations vanisheswhen there are noper-turbations. However, the evolutionof the time givenbyEq. (5.9),does not vanish in the unperturbed case, since λ3/2

3 r is not zero.As discussed in Sect. 2.5 introducing time elements typically im-proves the numerical performance of the integration. The time

element is obtained as the correction to the unperturbed timedueto perturbations. That is,

t = tnp + tte (5.41)

where tnp refers to the term that does not include perturbations,and tte is the time element, satisfying tte = 0 if ap = 0. The termtnp abides by

tnp − tnp0 = a3/2(e sinh H − H)

with tnp0 = −a3/20 (e0 sinh H0 − H0). Introducing the new set of

variables the previous equation becomes

tnp − tnp0 = λ3/23 (q − u)

Deriving the definition given in Eq. (5.41) provides the evo-lution of the time element,

dttedu=

dtdu−

dtnpdu

resulting in

dttedu= λ7/2

3[(

r2 + 2(2r − ℓ2) − 3qu)(ap · i) + ℓ(2q − 3u)(ap · j)

](5.42)

Summary

The final system of equations is:

dttedu= λ7/2

3[r2 + 2(2r − ℓ2) − 3qu](ap · i)

+ ℓ(2q − 3u)(ap · j)

(5.43)dλ1

du= λ2

3(−ℓ2 sinh u + 2λ2r)(ap · i)

+ ℓ[λ1 + (r − 1) cosh u](ap · j)

(5.44)dλ2

du= λ2

3(−ℓ2 cosh u + 2λ1r)(ap · i)

+ ℓ[λ2 − (r − 1) sinh u](ap · j)

(5.45)dλ3

du= −2λ3

3[q(ap · i) + ℓ(ap · j)

](5.46)

dχ1

du=

12

[ω1(−χ4 cos ν − χ3 sin ν) + ω3χ2

](5.47)

dχ2

du=

12

[ω1(−χ3 cos ν + χ4 sin ν) − ω3χ1

](5.48)

dχ3

du=

12

[ω1(−χ2 cos ν + χ1 sin ν) + ω3χ4

](5.49)

dχ4

du=

12

[ω1(−χ1 cos ν − χ2 sin ν) − ω3χ3

](5.50)

Equations (5.43–5.50) need to be integrated from the initial con-ditions:

t = t0 : u0 = H0, λ1 = e0, λ2 = 0, λ3 = a0, tte = 0

When the time element is not included, Eq. (5.43) is replaced byEq. (5.9),

dtdu= λ3/2

3 r

§5.6 Numerical evaluation 63

The reader willing to implement this algorithm is encouraged tofollow the sequence:

1. Given the hyperbolic angle u and the pair (λ1, λ2) computethe auxiliary terms:

r = λ1 cosh u + λ2 sinh u − 1q = λ1 sinh u + λ2 cosh u

e =√λ2

1 − λ22 and ℓ =

√λ2

1 − λ22 − 1

2. The state vector is obtained from

r = λ3 r i, v =1

r√λ3

(q i + ℓ j)

Equation (5.51) provides the transformation to the compo-nents in the inertial reference. The physical time reads

t = λ3/23 (q − u) − a3/2

0 (e0 sinh H0 − H0) + tte

The perturbing term ap can now be obtained.

3. Compute the angle α and the hyperbolic angle γ usingEq. (5.4),

tanh γ =λ2

λ1, α = gd (γ) = arctan sinh γ

4. Solve the hyperbolic anomaly, H = u + γ.

5. Obtain the true anomaly,

ϑ = atan2(ℓ sinh H, e − cosh H

)and the angle ν, that provides the evolution of frameLwithrespect to frameA, ν = α + ϑ.

6. Compute the components of the angular velocity in A, ω1and ω3:

ω1 = λ23

r2

ℓ(ap · k)

ω3 =λ2

3

e2λ1

[ℓλ1(r − ℓ2) + e

(ℓ2(1 − r) − 2r

)](ap · i)

+ q[λ1(ℓ2 + r) + ℓe(r − 1)

](ap · j)

7. Integrate Eqs. (5.43–5.50) to advance one step, and return

to 1.

The Cartesian coordinates of the particle in the inertial refer-ence are obtained from

rI = s rL s†, and vI = s vL s

† (5.51)

where quaternions rL and vL are rL = 0 + i xL + j 0 + k 0 andvL = 0 + i xL + j yL + k 0. The components of quaternions = s4 + i s1 + j s2 + k s3 are

s1 = χ1 cosν

2+ χ2 sin

ν

2, s2 = χ2 cos

ν

2− χ1 sin

ν

2

s3 = χ3 cosν

2+ χ4 sin

ν

2, s4 = χ4 cos

ν

2− χ3 sin

ν

2

The following expressions provide the orbital elements from thenew set of elements:

a = λ3

e =(λ2

1 − λ22

)1/2

i = atan2(χ2

1 + χ22, χ

23 + χ

24

)ω = atan2

(C−1 cosα +C+2 sinα, C+2 cosα −C−1 sinα

)Ω = atan2

(C+1 ,C

−2)

with C±1 = (χ1χ3 ± χ2χ4) and C±2 = (χ1χ4 ± χ2χ3).

5.6 Numerical evaluation

Theperformance of the proposed formulation is evaluated in thissection. The orbits of four hyperbolic comets (Table 5.1), and thegeocentric flybys of NEAR, Cassini, and Rosetta (Table 5.3) areintegrated using different formulations. They are compared interms of accuracy and runtime. The new formulation is com-pared against Cowell’s method, the Kustaanheimo-Stiefel (KS)transformation, the Sperling-Burdet (SB) regularization, and theDromo formulation. The simulations are conducted with PER-FORM (see Chap. 6 for details), in particular with the LSODARintegrator.

In order to determine the accuracy of the propagation a refer-ence solution is constructed. First, the orbits are integrated withCowell’s method in quadruple precision floating-point arith-metic and setting the integrator tolerance to εtol = 10−22. Adense output is generated providing the position of the bodieswith high accuracy. The results are now truncated to the firstsixteen digits, so the solution is reduced to double precision.The numerical test cases will be integrated in double precisionfloating-point arithmetic and compared to these reference valuesthat are considered the exact solution to the problem.

The error ϵ is defined as the mean value of the error in posi-tion across all the points in the dense output. Denoting by ϵ j theerror at a certain date j, the error in the solution is

ϵ =1N

N∑j=1

ϵ j, with ϵ j = ||r j − rrefj || (5.52)

Each point from the reference solution is denoted by rrefj , and N

is the total number of points in the dense output.The runtime refers to the time since the integration starts un-

til the final point is reached. It includes the time for computingthe perturbations at each integration step. The runtime is mea-sured as the mean runtime after forty consecutive integrations.This procedure averages the effect of possible instabilities in theintegration. Computations are performed on an Intel Core i7-4770@3.40GHz machine.

5.6.1 Hyperbolic comets

Table 5.1 summarizes the orbital parameters defining the orbits offour hyperbolic comets. Their heliocentric orbits are integratedfor four years: two years before periapsis passage, and two yearsafter.

64 5 Orbit propagation in Minkowskian geometry

Figure 5.4: Integration runtime vs. accuracy of the solution for the integration of the hyperbolic comets. The curves correspond to Cowell's for-

mulation, Dromo, the new solution without (``New'') andwith the time element (``New-TE''), the Kustaanheimo-Stiefel transformation (``KS''),

and the Sperling-Burdet regularization (``SB''). The curves are parameterized in terms of the tolerance, εtol.

Table 5.1: Orbits of the four selected hyperbolic comets.

Comet q [AU] e [−] i [deg] Ω [deg] ω [deg] Osculation

C/1999U4Catalina-Skiff 4.92 1.0076 51.93 32.29 77.52 2002-Jan-03

C/2002 B3 Linear 6.06 1.0075 73.69 289.36 123.21 2002-Feb-11

C/2008 J4McNaught 0.45 1.0279 87.37 289.69 92.18 2008-May-17

C/2011 K1 Schwartz-Holvorcem 3.38 1.0053 122.61 70.74 167.09 2011-Jun-03

Note: The orbits of the comets are defined by means of the perihelion distance, q, eccentricity, e, inclination, i, right ascension of the ascending node,Ω, andargument of perigee,ω. Angles are referred to the ICRF/J2000.0 reference frame, and the reference plane is the EarthMean Equator and Equinox of reference

epoch. The values in this table correspond to the epoch of osculation. (Source: JPL/Small-Body Database)

Table 5.2:Mean value (µϵ) and standard deviation (σϵ) of themost accurate error profile from each formulation.

C/1999U4Catalina-Skiff C/2002 B3 Linear C/2008 J4McNaught C/2011 K1 Sch.-Holvor.

µϵ [m] σϵ [m] µϵ [m] σϵ [m] µϵ [m] σϵ [m] µϵ [m] σϵ [m]

Cowell 14.43 15.97 13.88 14.54 17.25 23.68 5.67 11.51

Dromo 3.61 1.48 7.92 3.09 7.49 5.06 1.24 1.40

New 6.82 3.39 3.50 2.45 6.73 8.28 1.69 2.07

New-TE 0.46 0.26 0.45 0.19 0.40 0.50 0.76 0.94

KS 3.69 2.49 5.64 3.73 3.82 4.35 3.44 4.18

SB 1.44 1.04 1.69 1.32 3.29 1.59 1.37 1.57

§5.6 Numerical evaluation 65

The propagator includes three sources of perturbations:third-body perturbations, relativistic corrections, and the Sunoblateness. Theperturbations from themajor planets, Pluto, andthe Moon, as well as the four major asteroids (Ceres, Vesta, Pal-las, and Hygiea) are computed from the DE431 ephemeris. Rel-ativistic corrections account for the effects of the spherical cen-tral body, Lense-Thirring, oblateness, and rotational energy. Ad-ditional non-gravitational accelerations are not considered in themodel.

Figure 5.4 displays the machine runtime for different levelsof accuracy in the solution. The error in position displayed inthis figure corresponds to the definition given in Eq. (5.52). Eachpoint is obtained by varying the integration tolerance down tothe machine precision. The machine zero for the selected com-piler is 2.2204 × 10−16, so the tolerance spans from εtol = 10−5

to εtol = 10−15. The smallest tolerance corresponds to the lastpoints to the left of the curves. The new formulation with thetime element (New-TE) is the most accurate in all four cases. Noother formulation is able to reduce the error below one meter,not even with the finest tolerance. Moreover, improvements inaccuracy beyond one order of magnitude with respect to Cow-ell’s method are observed when integrating the orbits of the fourcomets. To achieve these results the integration of the New-TErequires more time, because it involves more algebraic operationsthan the rest. But the excess in runtime depends on the case:for comets C/1999 U4 Catalina-Skiff, C/2002 B3 Linear, andC/2011 K1 Schwartz-Holvorcem the total runtime of the New-TE is about twice the runtime for Cowell’s method, which is thefastest. The New-TE exhibits its best performance when propa-gating the orbit of C/2008 J4McNaught: it is faster than KS andSB, and the accuracy is improved by about one order of magni-tude even with respect to these formulations. The new formu-lation without the time element does not exhibit clear improve-ments in performance nor accuracy.

The time evolution of the error is plotted in Fig. 5.5. Thisfigure shows the error in position at date j, ϵ j. The values corre-spond to the tolerance for which each formulation is more accu-rate. The New-TE not only yields the smallest mean error, butit also exhibits the most stable behavior. It is remarkable howthe formulation is not affected by the perihelion passage: whilethe error of the reference formulations grows around the perihe-lion (specially for comets C/1999 U4 Catalina-Skiff, and C/2008J4 McNaught) the error curve of the New-TE remains flat. Forcomet C/2011 K1 Schwartz-Holvorcem this formulation is stillthe most stable during the integration, although the error growsslightly toward the end. The SB is the second formulation interms of accuracy and stability of the error, but error peaks about

the perihelion of comet C/2008 J4McNaught still appear. In thisfigure it is easily observed how the introduction of the time ele-ment improves the performance of the proposed formulation.

Table 5.2 shows the mean value and the standard deviationof the error distribution for the cases displayed in Fig. 5.5. Thesevalues correspond to the last point to the left (best accuracy) inFig. 5.4. The numerical values in this table confirm that the er-ror profile for the New-TE is the most stable in the proposed testcases, and also the most accurate. Note that the mean and stan-dard deviation of the error is below one meter only for the New-TE. For the rest of formulations the mean value and standard de-viation of the error remains over one meter in all four cases.

5.6.2 Geocentric flybys

This section deals with the geocentric flybys of NEAR, Cassini,and Rosetta spacecraft. The integration starts approximatelywhen the spacecraft enters the sphere of influence of the Earth,and stops when it leaves the sphere. This corresponds to a timespan of 4 days for NEAR, 2 days for Cassini, and 6 days forRosetta. Table 5.3 defines the reference orbits.

The propagator now includes different sources of perturba-tions: third-body perturbations, relativistic corrections, and theEarth gravity field. The perturbations from themajor planets andthe Moon, as well as the four major asteroids (Ceres, Vesta, Pal-las, andHygiea) are computed from the DE431 ephemeris. Rela-tivistic corrections account for the effects of the spherical centralbody, Lense-Thirring, oblateness, and rotational energy. A 100×100 gravity field is implemented, based on the GGM03S model.The Earth rotation model corresponds to the IAU 2006/2000A,CIO based (X-Y series) standard. The Earth Orientation Param-eters from Vallado and KelsoVallado and Kelso (2013) are con-sidered. Additional non-gravitational accelerations are not com-puted in the model.

Figure 5.6 displays the machine runtime for different levelsof accuracy in the solution. The error in position displayed inthis figure corresponds to the definition given in Eq. (5.52). It isobserved that the new formulation with the time element (New-TE) achieves a level of accuracy that cannot be reached by the restof formulations. This solution is the most accurate in the threetest cases. For the case of Rosetta, for example, the second mostaccurate formulation is the KSmethod, and its error doubles thatof the New-TE. In addition, the New-TE is as fast or even fasterthan the KS and SB methods. For Cassini it is slightly faster.

The time evolution of the error is plotted in Fig. 5.7. Thisfigure shows the error in position at date j, ϵ j. The values cor-

Table 5.3: Definition of the geocentric flybys of NEAR, Cassini and Rosetta.

Spacecraft a [km] hper [km] e [−] i [deg] Ω [deg] ω [deg] Closest approach

NEAR 8496 533 1.8134 108.0 88.3 145.1 1998-Jan-23 07:23:00

Cassini 1555 1172 5.8565 25.4 3.2 248.5 1999-Aug-18 03:30:00

Rosetta 26702 1954 1.3121 144.9 170.2 143.1 2005-Mar-04 22:11:00

Note: The orbits of the spacecraft are defined by means of the semimajor axis, a, the perigee height, hper , eccentricity, e, inclination, i, right ascension of theascending node,Ω, argument of perigee,ω, and time of closest approach. Angles are referred to the ICRF/J2000.0 reference frame, and the reference plane is

theEarthMeanEquator andEquinoxof reference epoch. The values in this table correspond to theosculating values at closest approach. (Source: JPL/Horizons)

66 5 Orbit propagation in Minkowskian geometry

Figure 5.5: Time evolution of the error in position at epoch, ϵ j, for the four hyperbolic comets.

respond to the smallest tolerance limited by the machine zero,εtol = 10−15. The New-TE solution exhibits the most stable be-havior, and it is also the most accurate. Note that this formula-tion is not affected significantly by the perigee passage, while therest of formulations show certain error peaks about the perigee.In particular, the integration error of the orbits of NEAR andRosetta show significant jumps at periapsis when propagated inKS variables.

Table 5.4 shows the mean value and the standard deviationof the error distribution for the cases displayed in Fig. 5.7. Thesevalues are obtained with the smallest admissible tolerance, corre-sponding to the last point to the left in Fig. 5.6. These numericalvalues show that the New-TE is, indeed, the most accurate andstable of the compared formulations. Reductions in the meanerror of about one or two orders of magnitude are encountered,depending on the formulation. The new formulation withoutthe time element matches the performance of the KS and SBmethods, and may outperform these methods in particular cases(Cassini).

Table 5.4:Mean value (µϵ) and standard deviation (σϵ) of the posi-tion error for the smallest admissible tolerance, εtol = 10−15.

NEAR Cassini Rosetta

µϵ [m] σϵ [m] µϵ [m] σϵ [m] µϵ [m] σϵ [m]

Cowell 3.75 3.79 34.25 32.51 1.96 2.24

Dromo 0.52 0.46 1.84 1.16 1.90 0.96

New 2.21 1.19 1.05 0.91 0.34 0.27

NewTE 0.11 0.09 0.28 0.21 0.06 0.04

KS 2.32 1.04 2.00 1.25 1.34 1.16

SB 0.30 0.22 2.38 0.88 0.16 0.15

5.7 Conclusions

Describing hyperbolic orbits is intimately related to hyperbolicgeometry. The formulation presented in this chapter, developedbyRoa and Peláez (2015d), fills the gap between hyperbolic prob-

§5.7 Conclusions 67

Figure 5.6: Integration runtime vs. accuracy of the solution for the integration of the geocentric flybys of NEAR, Cassini and Rosseta.

The curves correspond to Cowell's formulation, Dromo, the new solution without (``New'') andwith the time element (``New-TE''), the

Kustaanheimo-Stiefel transformation (``KS''), and the Sperling-Burdet regularization (``SB''). The curves are parameterized in terms of the

tolerance, εtol. The last point to the left corresponds to εtol = 10−15.

lems in astrodynamics and the mathematical foundations of hy-perbolic geometry. The advances in special relativity, in partic-ular the algebraic and geometric characterization of Minkowskispace-time, provide a number of useful mathematical tools. Thegeometry inMinkowski space-time is successfully extended to ce-lestial mechanics.

Three main conclusions are drawn from this analysis: (i) De-spite being more intuitive, Euclidean geometry is not the onlypossible choice for defining vectors and rotations. They can bedefined equivalently in the two-dimensional Minkowski space-time. This geometrical consideration ismotivated by the fact thatthe unperturbed motion of a particle along a hyperbolic orbit istypically parameterized using hyperbolic trigonometry. The evo-lution of the eccentricity vector on the Minkowski plane is for-mulated in terms of hyperbolic functions, that comply with thesolution on the osculating orbit. This leads to an adapted versionof the special perturbation method Dromo (Peláez et al., 2007),conceived for hyperbolic orbits. (ii) The bijection between theMinkowski plane and the Euclidean plane has been establishedrigorously. This bijection may apply to interdisciplinary studies,whereMinkowskian geometry appears. (iii)The resulting formu-lation improves the stability of the numerical integration. Thishas a direct impact on the accuracy of the solution, having re-duced the propagation error in the proposed test cases.

The strongest point of this new formulation is that it the er-ror evolution is not affected by periapsis passage, a typical down-side of Cowell’s method. In addition, it reaches levels of accuracythat cannot be achieved with the compared formulations. This

extra level of accuracy may be important in some critical scenar-ios. The complexity of the equations might increase the compu-tational time with respect to similar methods, although the even-tual increase in CPU time is quite affordable.

68 5 Orbit propagation in Minkowskian geometry

Figure 5.7: Time evolution of the error in position at epoch, ϵ j, for NEAR, Cassini and Rosseta. The orbit is sampled every 40min for NEAR,

20min for Cassini and, 1 h for Rosetta.

“That fondness for science, that affability and condescension whichGod shows to the learned, […], has encouragedme to compose a shortwork on calculating by al-Jabr and al-Muqabala.”

—Al-Khwārizmī, father of Algebra البر)الوارزمات

1

, al-Jabr) andAlgorithmics (

البرالوارزمات

1

, al-Khwarizmia)

6PERFORM: Performance Evaluation of

Regularized Formulations of Orbital Motion

M any different formulations and numerical schemesfor propagating the equations of orbital motion canbe found in the literature. Almost always, the numer-

ical experiments following the derivation of a new formulationshow the apparent superiority of the new method with respectto the existing ones. The point of view of Zare and Szebehely(1975)maybe toopessimistic, as they claim that “ always favorableclaims give […] an overwhelming feeling of nonscience and dis-honesty,” but it is undoubtedly true that standardization and re-producibility will enhance the scientific value of any experiment.This idea motivated the development of PERFORM (PerformanceEvaluation of Regularized Formulations of ORbital Motion), adedicated software for orbit propagation including different for-mulations and integrators. The ultimate goal is to have a way tocompare the numerical performance of the formulations whenrunning on the same platform, with the same force models, etc.

In its current version, PERFORM includes thirty-six differentformulations and seven integrators (see Table 6.2 for a list of for-mulations and Appendix B for details).* Apart from a collectionof formulations, the software includes high-fidelity force modelsin order to test the performance of the formulations in a widevariety of practical scenarios, not limited to simplified academicexamples.

This chapter makes no claims about the performance of spe-cific formulations. It simply shows how tools like PERFORMshall be used to decide which combination of orbital formula-tion and numerical integrator yields the most efficient solutionfor a given problem. One should not expect one formulation to

*Hodei Urrutxua and Giulio Baù made relevant contributions by providingcertain numerical integrators and formulations.

outperform the rest in every single scenario. This kind of anal-ysis is important when thousands of similar propagations needto be carried out. Relevant applications are trajectory optimiza-tion and planetary protection studies. When optimizing a trajec-tory, the solver will call the cost function many times. Evaluat-ing the cost function and its gradient typically requires propagat-ing the trajectory. Algorithms for local optimization will restrictthe search to bounded neighborhoods around the nominal tra-jectory. Thus, small differences in performance between trajecto-ries in the search space should be expected, and finding the mostefficient strategy for integrating the nominal orbit will improvethe overall exploration. Most interplanetary mission designs in-clude planetary protection studies. They reduce to campaignsof Monte-Carlo simulations (or equivalent techniques) in whichthe nominal trajectory is perturbed slightly to check the proba-bility of crashing with celestial bodies without a sufficiently thickatmosphere in which the spacecraft could disintegrate. One in-teresting example is the Cassini spacecraft, which is powered bya plutonium-filled radioisotope thermoelectric generator. Theplanetary protection teams had to make sure that Cassini wouldnot hit the Saturnianmoons. Placing a radioactive heat source onan icy moon like Enceladus might have important consequences.In particular, if life were ever found in the moon, it would be un-clear whether it was originated by Cassini’s heat source, or by nat-ural processes.

Section 6.1 explains how PERFORM is implemented. De-tails on the orbital formulations, numerical integrators, and forcemodels are provided. The propagator in tested in different realexamples of application, presented in Sect. 6.2. Finally, Sect. 6.3shows how PERFORM can be used for evaluating the performanceof different formulations when integrating a given problem. The

69

70 6 PERFORM: Performance Evaluation of Regularized Formulations of Orbital Motion

concept of the sequential performance diagram (SPD) is pre-sented here for the first time.

6.1 Implementation

The core of PERFORM is implemented in Fortran, with a numberof auxiliary Python scripts to manage the input and output filesand some legacy code in Pascal (the Brown-Newcomb lunisolarephemeris). The program can be compiled and run both in dou-ble and quadruple precision floating-point arithmetic.

6.1.1 Force models

Gravitational perturbations from the planets in the Solar Systemare computed according to the JPLDE series of ephemeris. PER-FORM currently admits DE405, DE430, DE430t, DE431, DE432,and DE432t. In addition, it allows the user to select the Brown-Newcomb semianalytic ephemeris for retrieving the states of theMoon and the Sun. The series solution is implemented in Pas-cal following Montenbruck and Pfleger (1994, p. 161). The statesof Ceres, Pallas, Vesta and Hygiea are retrieved by interpolatingunformatted binary files generated from JPL/Horizons data.

Solar radiation pressure is computed accounting for sea-sonal variations of the solar flux and a conic shade model. Theatmospheric density models implemented in PERFORM are theMSISE90 (Hedin, 1991), the Jacchia 70 and77 (Jacchia, 1977), anda simple exponential model. Space weather data (including thethree-hour variations) are retrieved from Celestrack. The grav-ity field of the Earth, the Moon and Mars are modeled with theGrace Gravity Model (GGM03S), Grail’s lunar model, and theMRO95A model from the Mars Reconnaissance Orbiter. Theperturbations are gravitational harmonics are computed follow-ing the technique by Cunningham (1970). PERFORM has been in-tegrated with the SOFA package provided by the IAU. The orien-tation of the Earth can be determined using different standards:

a) IAU 2000A, CIO based, using (X,Y) series.

b) IAU 2000B, CIO based, using (X,Y) series.

c) IAU 2000A, equinox based, using classical angles.

d) IAU 2006/2000A, CIO based, using classical angles.

e) IAU 2006/2000A, CIO based, using (X,Y) series.

The Earth Orientation Parameters (EOP) providing the correc-tions to the position of the pole and the length of day are retrievedfrom either the International Earth Rotation and Reference Sys-tem Service (IERS) or the US Naval Observatory (USNO). Thesimplified model without nutation or precession by Archinalet al. (2011) is also available in PERFORM. Table 6.1 shows the dif-ferences between the different conventions and EOP. Given anequatorial vector defined in the GCRS it is then transformed tothe ITRS using two different models. Then, the angle betweenthe resulting vectors provides the offset between the conventions.The first row corresponds to the difference in the rotationdefinedby model a) and the rest of models. The second row corresponds

to the rotation difference between the same model but when us-ing the IERS and the USNOEOP. Finally, the last row shows thetotal runtime for computing the GCRS to ITRS rotation matrix5,000 consecutive times. Using the B series over the A series re-duces the runtime one order of magnitude. The difference withrespect to the rotation using the A series is about 62.5 µas. Theapproximate model from Archinal et al. (2011) yields an angulardifference of 237′′, but the total runtime is reduced by a factorof 500. The difference in using one source of EOP or the other is140 mas in all five cases.

Relativistic corrections follow froma simplificationof thePa-rameterized Post-Newtonian (PPN)model, including a term dueto the gravitational and rotational energy of the central body, itsoblateness, and the Lense-Thirring effect. For heliocentric prop-agations the contribution from the Sun J2 is accounted for. Spe-cific perturbations like the continuous thrust from the enginesor the non-gravitational actions upon comets (following Kró-likowska, 2006) have also been implemented.

6.1.2 Formulations

An orbit propagator is defined by two major elements. First,the formulation chosen to model the dynamics of the system.Second, the integrator used to propagate numerically the cor-responding equations of motion. Both elements are importantwhen assessing the performance of a given integration.

Chapter 2 presented a plethora of methods and techniquesfor improving the numerical performance of the propagation, re-sulting in different formulations of the equations of orbital mo-tion. The formulations implemented in PERFORM are listed inTable 6.2, and explained in Appendix B. The first property of agiven formulation is the dimension of the system, i.e. the num-ber of equations to be integrated (Neq). Next, the number of el-ements Nel is important because it defines the number of equa-tions whose right-hand side will be zero (or at least constant) dur-ing the propagation of Keplerian orbits. For Neq = Nel all thevariables are elements, and all the benefits presented in Sect. 2.5may be expected when integrating weakly perturbed problems.The table also shows the time transformation featured in the for-mulation. The time transformation is closely related to the useof time elements, which may be constant (vanishing derivative inthe Keplerian case) or linear (constant derivative in the Kepleriancase). The ID assigned to each formulation is the internal identi-fication used by PERFORM. The formulations are divided in fourclasses:

Class 1: element-based formulations (i.e., Nel ≥ Neq/2) includ-ing a Sundman transformation of order one.

Class 2: element-based formulations including a Sundmantransformation of order two.

Class 3: element-based formulations with no time transforma-tion (the physical time is the independent variable).

Class 4: formulations based on coordinates including a Sund-man transformation of order one.

§6.1 Implementation 71

Table 6.1: Performance of the different GCRS to ITRS transformations at JD(TT)=2457066.50.

Units Model a) Model b) Model c) Model d) Model e) Archinal et al. (2011)

Offset w.r.t. model a) (IERS) mas − 0.0625 3.10× 10-6 0.0777 0.0776 2.37× 105IERS vs USNO mas 140.70 140.870 140.68 140.70 140.70 −Runtime s 6.651 0.578 6.460 6.549 4.945 0.015

Table 6.2: List of propagationmethods implemented in PERFORM (see Appendix B for a detailed description of each formulation).

Neq Nel dt/ds Time Elem. ID Class Reference

Cowell 6 0 − − COW − Cowell and Crommelin (1908)

Stab. Cowell 8 2 r Linear SCW 4 Janin (1974)

Dromo 8 7 r2/h − DRO 2 Peláez et al. (2007)

Time-Dromo 8 7 − − TDR 3 −Minkowskian + T.E. 8 8 r/

√2E Const. HDT 1 Roa and Peláez (2015d)

Minkowskian 8 7 r/√

2E − HDR 1 Roa and Peláez (2015d)

KS-a 9 0 r − KS_ 4 Kustaanheimo and Stiefel (1965)

KS-b 9 1 r Linear KST 4 Kustaanheimo and Stiefel (1965)

KS-c 10 1 r − KSR 4 Kustaanheimo and Stiefel (1965)

KS-d 10 2 r Linear KRT 4 Kustaanheimo and Stiefel (1965)

Sperling-a 7 0 r − SP_ 4 Sperling (1961)

Sperling-b 7 1 r Linear SPT 4 Sperling (1961)

Sperling-c 13 4 r − SPR 4 Sperling (1961)

Sperling-d 13 4 r Linear SRT 4 Sperling (1961)

Deprit∗ 8 7 − − DEP 3 Deprit (1975)

Palacios∗ 8 5 r2/h − PAL 2 Palacios et al. (1992)

Milankovitch† 7 6 − − MIL 3 Rosengren and Scheeres (2014)

Stiefel-Scheifele 10 10 r/√−2E Linear SSc 1 Stiefel and Scheifele (1971, p. 91)

Equinoctial 6 5 − − EQU 3 Walker et al. (1985)

Classical 6 5 − − CLA 3 −Burdet elem. (BG14) 11 11 r Const. BCP 1 Burdet (1969)

Unified StateModel 7 3 − − USM − Vittaldev et al. (2012)

EDromo-0‡ 8 7 r/√−2E − ED0 1 Baù et al. (2015)

EDromo-1‡ 8 8 r/√−2E Linear ED1 1 Baù et al. (2015)

EDromo-2‡ 8 8 r/√−2E Const. ED2 1 Baù et al. (2015)

BF-0H‡ 10 1 r2/h − B0H 2 Baù et al. (2015)/Ferrándiz (1988)

BF-1H‡ 10 1 r2/h Linear B1H 2 Baù et al. (2015)/Ferrándiz (1988)

BF-2H‡ 10 2 r2/h Const. B2H 2 Baù et al. (2015)/Ferrándiz (1988)

BF-3H‡ 11 1 r2/h − B3H 2 Baù et al. (2015)/Ferrándiz (1988)

BF-0C‡ 10 1 r2/h − B0C 2 Baù et al. (2015)/Ferrándiz (1988)

BF-1C‡ 10 1 r2/h Linear B1C 2 Baù et al. (2015)/Ferrándiz (1988)

BF-2C‡ 10 2 r2/h Const. B2C 2 Baù et al. (2015)/Ferrándiz (1988)

BF-3C‡ 11 1 r2/h − B3C 2 Baù et al. (2015)/Ferrándiz (1988)

Stiefel-1‡ 10 9 r − ST0 1 Baù et al. (2015)

Stiefel-2‡ 10 10 r Linear ST1 1 Baù et al. (2015)

Stiefel-3‡ 10 10 r Const. ST2 1 Baù et al. (2015)

∗Code provided byHodei Urrutxua†Aaron Rosengren greatly contributed to the implementation.‡Code provided by Giulio Baù

72 6 PERFORM: Performance Evaluation of Regularized Formulations of Orbital Motion

6.1.3 Numerical integration

A sophisticated numerical integrator can improve the efficiencyof the propagation in Cartesian coordinates, just like regularizedformulations reduce the error growth rate and improve the stabil-ity compared to Cowell’s method. An adequate combination offormulation/integrator might maximize the potential of both el-ements. For this reason, PERFORM includes numerical integrationschemes of different nature. A comprehensive survey of availablemethods for propagating ordinary differential equations can befound in Urrutxua (2015, chap. 7). For details on the methods,we refer to the classic book by Hairer et al. (1991).

The baseline integrator is LSODE, the Livermore Solver forOrdinary Differential Equations implemented by Hindmarsh(1983). This algorithm is distributed together withmany variantsin the library ODEPACK. It is based on an Adams-Moulton mul-tistep scheme, and switches automatically to implicit backwarddifferentiation when the integrator detects a stiff problem. Thisalgorithm has been in use for over 30 years, meaning that it hasbeen deeply tested and optimized.

The secondorder equations ofmotion canbe integratedwitha Störmer-Cowell integration scheme. The implementation inPERFORM follows the guidelines in Berry (2004). This is a sec-ond order multistepmethod, which is particularly well suited forintegrating the equations of motion in Cartesian coordinates.

Hodei Urrutxua kindly provided a family of explicit Runge-Kutta methods with variable step size. More in particular, PER-FORM includes the following Runge-Kutta-Fehlberg methods:RKF5(4), RKF6(5), RKF7(6), and RKF8(7). The DoPri853scheme is also incorporated for completeness (Dormand andPrince, 1980).

6.1.4 Variational equations

PERFORM can be used to propagate the variational equations oforbital motion numerically. Denoting ν the generic independentvariable, and y = y(ν) the state vector, propagating the orbitmeans solving the initial value problem

dydν= f(ν; y), with y(ν0) = y0 (6.1)

Given a nominal orbit y = y(ν), in many problems it is interest-ing tomonitor the evolution of nearby solutions, defined initiallyas y′0 = y0 + δy0. If the initial separation δy0 is small, the relativedynamics can be described using a linear model,

δy(ν) = Ξ(ν; y0) δy0

in which Ξ(ν; y0) represents the state-transition matrix. In prac-tice this matrix is obtained numerically because it depends on theperturbations and the state vector at each integration step. Itsevolution is governed by the initial value problem

∂Ξ

∂ν=∂f∂y

∣∣∣∣∣νΞ(ν; y0), with Ξ(ν0; y0) = I

These equations are integrated together with Eq. (6.1), and thecomponents of the state-transitionmatrix are attached to the statevector. The Jacobian matrix ∂f/∂y|ν is defined by the partial

derivatives of the right-hand side of Eq. (6.1) with respect to thestate vector, keeping constant ν. It is computed numerically ateach integration step using finite central differences.

6.2 PERFORM as a high-fidelity propagator

6.2.1 The orbit of WT1190F

OnNovember 13, 2015, a man-made space object burnt in the at-mosphere during reentry 100 km south of the coast of Sri Lanka.The object was designated WT1190F. Observations date back to2013 and link this piece of debriswithUDA34A3 andUW8551D,observed in February and November 2013, respectively. The ex-periment in this section evaluates the sensitivity of the problem tomodel uncertainties and different parameterizations of the forcemodel.

The origin of WT1190F is uncertain due to the difficultiesin propagating its orbit backward in time. Observations prior toimpact show that the object was in a strongly perturbed, highlyelliptical orbit beyond theMoon. Due to its large area-to-mass ra-tio the solar radiation pressure perturbs its motion significantly.There are different hypotheses about its origin: the fact that itsorbit reaches the Moon suggests that it might be a piece of hard-ware from the Apollo era, or any other lunar mission. A closeencounter with the Moon could provide enough energy to de-part from the Earth-Moon system, to be injected into a helio-centric orbit and then return after several decades. On the otherhand, WT1190F has been linked to another Earth-orbiting ob-ject, 9U01FF6. It was observed in 2009 and 2010, meaning that itwould have already been in the Earth-Moon system for six yearsbefore impact.*

Initial conditions

The initial conditions on 03-Oct-2015 provided by CNES aresummarized in Table 6.3. The object is orbiting the Earth in ahighly eccentric orbit that crosses that of the Moon.

Table 6.3: Osculating elements at epoch JD 2457298.5 with respect

to the ICRF/J2000 referred to the Earthmean equator and equinox

of reference date.

Units Value

a km 338296.386469173

e − 0.937268501114443

i deg 3.1967

Ω deg 311.55613

ω deg 314.04406

M0 deg 6.19095

The physical parameters defining the object are given in Ta-ble 6.4. They are the area to mass ratio, A/m, the drag coefficientCD, and the reflectivity coefficient CR.

*http://projectpluto.com/temp/wt1190f.htm

§6.2 PERFORM as a high-fidelity propagator 73

20−Mar−2012 06−Feb−2013 26−Dec−2013 14−Nov−2014 03−Oct−2015−1

−0.5

0

0.5

1

Spec

ific

ener

gy [

km

2/s

2]

J

2J

2+J

3J

2+J

3+J

4

10−Jul−2013

Figure 6.1: Evolution of the energy for different degrees of the zonal

gravity field.

Table 6.4: Physical properties.

Units Value

A/m m2/kg 0.00375

CD − 2.2

CR − 1.0

The orbit is propagated backward to determine the possibleorigin of the object. The likely fate of the orbit is to impact theEarth, the Moon, or to gain enough energy to leave the Earth-Moon system and to orbit around the Sun.

Analytic ephemeris

The first propagations of the orbit are integratedwith the follow-ing force model:

• Luni-solar perturbations computed from the Brown-Newcomb analytic ephemeris.

• Zonal harmonics from the Earth gravity field (GGM03Smodel).

• Solar radiation pressure with a conic shadow model.

• Atmospheric drag including an exponential density model.

The transformations between the GCRF and the ITRF framesare computed with the simplified model from Archinal et al.(2011) not including precesion nor nutation terms. The Brown-Newcomb ephemeris are implemented in Pascal, together withthe solar model, the transformations from ecliptic to equatorialcoordinates and between equinoxes. Analytic ephemeris are usedin this first integration to show the sensitivity of the problemwhen compared to the solution with accurate ephemeris.

The orbit is propagated using the Dromo formulation. It isintegrated with the variable step size and variable order multistepLSODAR scheme in double precision floating-point arithmetic.

Simulations show thatWT1190Fwas ejected from the Earth-Moon system between 2012 and 2013, depending on the tuningof the force model. Three different (backward) propagations are

compared in Fig. 6.1. In the first case the Earth’s gravity field isreduced to the J2 coefficient. The second and third cases also in-clude J3 and J3 + J4. The three solutions coincide until a closelunar flyby on 10-Jul-2013. Small differences in the entry condi-tions (due to the different force models) yield important differ-ences after the flyby. The ejection from the Earth-Moon systemoccurs in June 2012, April and May 2013.

The strong effect of the lunar flyby proves that the selec-tion of the lunar ephemeris is critical. Figure 6.3 shows the dif-ference between the solution provided by the analytic Brown-Newcomb luni-solar ephemeris and the preciseDE431 ephemerisdistributed by JPL. The difference is of about 1′′. For the case ofthe Moon it reaches a maximum of 4.′′58 on 28-Jul-2013, just 18days prior to the lunar flyby discussed above. The introductionof amore precise solution for the orbit of theMoonmay improvethe description of the ejection/capture mechanism.

Precise ephemeris

A high-fidelity force model is considered in this section. It in-cludes:

• Perturbations from the Sun, all the planets in the Solar Sys-tem and theMoon (DE431 ephemeris). Themajor asteroidsCeres, Pallas, Vesta and Hygea are also included.

• A 40 × 20 Earth gravity field (GGM03S model).

• Solar radiation pressure with a conic shadow model.

• Atmospheric drag computed with the MSISE90 densitymodel.

The Earth rotation model now includes the IAU2000/2006Astandards for the precesion and nutation, and the position of thepole is solved from the CIO based solution using (X,Y) series.The daily Earth Orientation Parameters are retrieved from theObservatoire de Paris, and TAI-UTC data comes from the USNaval Observatory. The space weather solution is provided byNOAA. The MSISE90 model uses the three-hourly values of thecoefficients.

First, the orbit is propagated forward to reproduce the im-pact. On November 13, 2015 at 06:18:40.0 the orbit reaches analtitude of approximately 20 km. The coordinates of the impactpoint are 536′16.′′5N 8148′27.′′0 E, about 100 km southeast ofHambantota, Sri Lanka. Figure 6.2 displays the groundtrack dur-ing the entry phase computed with PERFORM.

Next, the orbit is propagated backward in time. By changingthe lunar solution the close approaches provide less energy andthere are no escapes between March 2012 and October 2015. Infact, the object reenters the Earth atmosphere on 22-Sep-2012.

Sensitivity: physical parameters and numerical er-ror

Having fixed the force model we analyze the sensitivity of the so-lution to the values of A/m, CR and CD. Modifications on thevalues of these parameters can yield close approaches to the Earthat higher altitudes. The atmospheric drag might not be capable

74 6 PERFORM: Performance Evaluation of Regularized Formulations of Orbital Motion

01:21:33.5

06:18:40.0

06:09:45.205:34:06.0

Figure 6.2: Groundtrack onNovember 13, 2015, computedwith the high-fidelity forcemodel. The green dot represents the impact point.

20−Mar−2012 06−Feb−2013 26−Dec−2013 14−Nov−2014 03−Oct−20150

1

2

3

4

5

Dif

fere

nce

[ar

csec

]

Moon Sun

Figure 6.3: Difference between the analytic Brown-Newcomb luni-

solar ephemeris and the DE431 solution.

of forcing the reentry of the object and it will remain orbiting theEarth-Moon system. Table 6.5 presents a number of cases withmodified values of the physical parameters.

Table 6.5: Different configurations ofWT1190F.

A/m [m2/kg] CR CD

Case 1 0.00610 1.3 2.0

Case 2 0.00150 1.2 2.2

Case 3 0.00750 1.2 2.0

The strong perturbations make the problem extremely sen-sitive not only to the definition of the initial conditions, but tothe accumulation of numerical error. A backward integration of

50 years is a long time span compared to the time scales of pertur-bations such as the Moon attraction or the Earth gravity field.

Case 1Larger values of the the area-to-mass ratio and the reflectivity co-efficient increase the effect of the solar radiation pressure. Theorbit no longer reenters the Earth atmosphere and it escapes theEarth-Moon system by June 2008. Figure 6.4 (top figure) showsthe distance to the Earth during a backward propagation from03-Oct-2015 to 02-Oct-1965, corresponding to propagations indouble and quadruple precision floating-point arithmetic (withintegration tolerances ε = 10−12 and ε = 10−19, respectively).The object escapes from the Earth-Moon system in June 2008 butwith slightly different states. There is a fundamental difference inthe results: the second close-approach to the Earth-Moon systemin the 1970’s varies, resulting in a capture only when integrated inquadruple precision. Thenumerical error in the double precisioncasemakes the orbit tomiss the capture. According to the preciseintegrationWT1190F would have been orbiting the Earth-Moonsystem until January 1979.

Case 2The object remains in the Earth-Moon system until escape inApril 2004 (integration in double precision) or in January 2005(quadruple precision), as shown in Fig. 6.4 (central figure). Qual-itatively both solutions describe the same behavior, although theperiod of the relative motion is different: in double precision theclose approaches occur in April 1969 and October 1986, whereasin quadruple precision they occur in October 1971 and 1988.

Case 3

§6.2 PERFORM as a high-fidelity propagator 75

Figure 6.4: Distance to the Earth for a 50 year propagation.

In this case the escape/capture by the Earth-Moon system variessignificantly depending on the numerical precision (see Fig 6.4,bottom figure): in quadruple precision the object is captured inJanuary 2003, whereas in double precision it was orbiting theEarth since October 1999. The solution computed in quadrupleprecision exhibits a shorter period for the relative motion, result-ing in two close approaches in March 1978 and September 1990.In double precision, however, only one close approach is found(August 1981).

Comments about the orbit solution

The orbit of WT1190F goes beyond the Moon and the gravita-tional attraction from the Earth’s satellite has a strong impact onthe dynamics. In addition, due to the low perigee height a high-order model for the gravity field of the Earth, and a precise at-mospheric density model are required. The strong perturbationsmake the orbit extremely sensitive to changes in the initial condi-tions or the physical characterization of the object. Small vari-ations at departure will grow rapidly in time, specially becauseof the divergence of close trajectories after a number of flybys.

The perturbation due to the solar radiation pressure proves im-portant; a more detailed model for the shape and attitude of theobject should be used.

In order to predict the origin ofWT1190F the pure propaga-tion should be contrasted against actual observations. By linkingthe propagation with orbit solutions for UDA34A3, UW8551D,and even 9U01FF6 the uncertainties in the definition of the prob-lem could be assessed. It will be hard, if possible at all, to linkbackward-propagated trajectories returning to the Earth with ac-tual Apollo-era launches without a model for the orbital maneu-vers. For how long the propagation can be trusted remains anopen question.

6.2.2 Impacts of Near Earth Asteroids

The Dromo formulation was presented in Chap. 4, and the po-tential singularities related to deep flybys and collision courseswere discussed. An important question is how realistic these sin-gular situations truly are. In order to test the formulation in realscenarios, the terminal orbits of two near-Earth asteroids that im-pacted the Earthwill be integrated. The first asteroid is 2008TC3,which impacted the Earth in October 7, 2008. The second aster-oid is 2014 AA, which impacted the Earth in January 2, 2014. Inaddition, asteroid 2014 LY21 passed at roughly 3,000 km over thesurface of the Earth in June 3, 2014, and its orbit will be includedas a third example.

Table 6.6 summarizes the physical parameters that are rel-evant for the example. The minimum angular momentumreached during the propagation (hmin) is included in the table.This value is interesting because the singularities in Dromo areexpected for h→ 0. In dimensionless variables it is h ∼ 1, whichis far from the singularity.

Table 6.6: Characterization of the selected asteroids.

Asteroid Closest Approach hmin [km2/s] Diameter [m]

2008 TC3 2008-Oct-07 9.867×103 4.1±0.312014 AA 2014-Jan-02 1.602×104 2.5±0.522014 LY21 2014-Jun-03 5.065×103 6.0±2.03

1Jenniskens et al. (2009); 2Chesley et al. (2014); 3MPEC 2014-L48: 2014LY21

Consequently, even for impact trajectories the propagation willnot reach the singularity. Dromo will not encounter any numer-ical issue when integrating these orbits. The accuracy of the inte-gration is evaluated by comparing the results with the trajectorysolution provided by JPL/Horizons. The initial date is two daysprior to the closest approach. The final errors in position are ofthe order of the radii of the asteroids. The integration of the or-bits of 2014 AA and 2014 LY21 is particularly accurate, as the er-rors are smaller than the uncertainty in the measurement of theradii. The error in the propagation of 2008 TC3 grows linearly,but the final error remains of the order of the radius of the aster-oid. These results not only show that the singularity detected inDromo may be hard to reach in practice, but they provide an ad-ditional validation of the forcemodels implemented in PERFORM.

76 6 PERFORM: Performance Evaluation of Regularized Formulations of Orbital Motion

Figure 6.5: Validation of Dromo formulation in propagating asteroid

close approaches.

6.3 Evaluating the performance

This section presents two examples in which PERFORM is usedto evaluate the performance of different formulations. The firstexample is the test problem already introduced in Table 4.1. Ahighly eccentric geocentric orbit is perturbed by the Moon andthe oblateness of the Earth, as defined by Stiefel and Scheifele(1971, p. 122). The second example simulates a low-thrust trans-fer from LEO to GEO. The communications satellite ABS-3A (aBoeing 702SP spacecraft) is taken as a representative example.

For convenience, the initial conditions and the reference so-lutions of the two problems are summarized in Table 6.7. Thereference solution is constructed by integrating the problem inquadruple precision floating-point arithmetic with the finest tol-erance using different formulations. Then, only the commondigits are retained. Because the numerical experiments will becarried out in double precision, the solution can be consideredexact as long as the first sixteen digits coincide (the machine zerois 2.2204 × 10-16).

Table 6.7: Initial conditions and reference solutions of the numerical

test cases.

Units Problem 1 Problem 2

MJD0 − N/A 57083

t f days 288.12768941 100.00000000

x0 km 0.000000000000000E+00 1.115741121014817E+04

y0 km -5.888972700000000E+03 1.019260166000129E+03

z0 km -3.400000000000000E+03 -2.268170222538688E+03

vx,0 km/s 1.069133800000000E+01 7.309103529900698E-01

vy,0 km/s 0.000000000000000E+00 3.662867091587677E+00

vz,0 km/s 0.000000000000000E+00 3.991045017757724E+00

x f km -2.421905011593605E+04 -1.814081886073619E+04

y f km -2.279621063730220E+05 1.145173099550784E+04

z f km -1.297534424000825E+05 -5.027819073450440E+02

vx, f km/s -3.072444684207348E-01 -1.256074877885349E+00

vy, f km/s 1.539502056878936E-01 -2.613579325742426E+00

vz, f km/s 7.809786648117930E-02 -3.413479931546313E+00

6.3.1 Problem 1

Problem 1 is integrated with all the formulations listed in Ta-ble 6.2 except for the Minkowskian propagator (see Chap. 5),which is only valid for hyperbolic orbits. The orbit is propa-gated with two different integrators as representative examples:RKF5(4) and LSODAR. The performance of the integration isevaluated with the traditional plots that present the error in thepropagation and the number of function calls (it is equivalent tothe total runtime). The performance curves are built by changingthe integration tolerance down to themachineprecision. Inorderto simplify the visualization of the results, the data is presented inthe form of sequential performance diagrams (SPD).

Figure 6.6 is the SPD corresponding to the integration withthe RKF5(4) scheme. The central figure shows the performancecurves of the most efficient formulation of each class, plus the so-lutions usingCowell’smethod and the unified statemodel, whichdo not fit in any of the four classes. The criterion for choosing themost efficient formulation is based on a trade-off between speedand accuracy. The selection is still done manually, although pro-cedures for the automatic classification of the solutions are be-ing developed (Urrutxua et al., 2016). The second level of perfor-mance plots in the SPD corresponds to the performance of eachformulation within a given class. The curves have been arrangedso that the most efficient formulation (the one selected to be rep-resented in the central figure), is plotted first. Certain formula-tions have different versions. In the second level of figures onlythe most efficient version of each formulation is presented. Theperformance of all the versions of a given formulation can be ana-lyzed in the third level of figures. The introduction of a Sundmantransformation determines the performance of the methods: thecentral figure shows that using the time as the independent vari-able (Class 3 formulations, COW, and USM) renders performancecurves with the same slope. Indeed, the use of a fictitious timebehaves as an analytic step-size adaption (see Sect. 2.2), and hasa direct effect on the error-control routines. The KS transforma-tionwith redundant equations and a time element (KRT) exhibitsthe best compromise betweenmaximumaccuracy and speed. It isinteresting to note that the SPD shows that the integration of theredundant equations (both for the KS and the SB formulations)has a significant effect on the performance, much more signifi-cant than introducing a time element.

The SPD for LSODAR is depicted in Fig. 6.7, which followsthe same structure than Fig. 6.6. The irregularities in the curvesare due to the changes in the order of the integrator, a task thatis performed automatically in ODEPACK. This particular integra-tor includes a large collection of features that make the algorithmmore robust and efficient. For the same level of accuracy it is oneorder of magnitude faster than the RKF5(4). However, the min-imum error that can be reached is about two orders ofmagnitudelarger than the one obtained with the RKF5(4). The limit comesfrom theminimum admissible tolerance, which is constrained bythe machine zero. Disregarding the discontinuities related to thechanges in the order of the integrator, LSODAR reduces the dif-ferences in performance between formulations in the same fam-ily. This is because the integrator is highly flexible and adaptable.In particular, for the KS and SB families integrating redundantequations does not improve the overall performance. The SPD

§6.3 Evaluating the performance 77

Figure 6.6: Sequential performance diagram (SPD) for the

Stiefel-Scheifele problem integratedwith a RKF5(4) scheme.

78 6 PERFORM: Performance Evaluation of Regularized Formulations of Orbital Motion

Figure 6.7: SPD for the Stiefel-Scheifele problem integrated

with LSODAR.

§6.4 Conclusions 79

Figure 6.8: Performance of Cowell's method using different integra-

tors.

shows that Dromo (DRO) and the KS regularization (KS_) yieldthe best performance, followed closely by the use of the Stiefel-Scheifele elements (SSc). There is a clear separation between themethods involving a time transformation and those using thephysical time as the independent variable (COW,USM, andClass 3formulations).

In some scenarios the integration of the equations in Carte-sian coordinates is preferred. For instance, when the high-ordervariational equations are propagated together with the state vec-tor. Finding the most adequate integrator for a given applicationcan have the same effect than using a regularized formulation in-stead of Cowell’s method. In fact, many research groups focustheir attention on the integrator part, rather than on the formula-tionpart.* Figure 6.8 depicts the performance ofCowell’smethodwhen using different integrators, including LSODAR, RKF5(4),RKF8(7), DoPri853, and Störmer-Cowell of orders 8 and 12. Theperformance curves presented in this figure reveal that significantdifferences should be expected depending on the choice of theintegration scheme. More in particular, Störmer-Cowell is thefastest, and the accuracy is comparable to that of LSODAR. De-spite being slower, the methods of the Runge-Kutta family aremore accurate in this particular application.

6.3.2 Problem 2

The spacecraft mass is 1954 kg and the area-to-mass ratio is as-signed the value A/m = 0.0767 m2/kg. The drag coefficientand the reflectivity are cD = 2.0 and cR = 1.2, respectively.The atmospheric density is retrieved from the MSISE90 model.The Earth gravity field is modeled with a 10× 10 expansion, andthe orientation of the Earth is computed neglecting the preces-sion and nutation effects. The perturbations from the Sun andthe Moon are defined using the DE430 ephemeris. The space-craft is provided with three ion engines, each generating 165 mNof thrust. The thrust is assumed constant and directed along thevelocity vector. The changes in the mass of the spacecraft are ne-glected. The orbit is propagated for 100 days, which corresponds

*An example is the Mission-analysis, Operations, and Navigation ToolkitEnvironment (MONTE) used at the Jet Propulsion Laboratory. This tool includesaccurate physical models, it is highly flexible, and the use of Cartesian variablesgreatly simplifies the implementation. In order tomaximize the performance, adedicated integrator was developed: DIVA (Krogh, 1974).

to roughly 870 revolutions of the spiral. The initial semimajoraxis is 10,000 km, and the eccentricity at departure is 0.15.

The performance of the different formulations integratedwith the RKF5(4) scheme can be studied in the SPD in Fig. 6.9.Because of the perturbations being small, the use of elements ver-sus coordinates drives the classification of the methods in termsof their performance. Cowell’s method is entirely based on co-ordinates, and exhibits the worst performance. Next, the uni-fied state model (USM) and the KS formulation with redundantequations (KSR) show similar performances. These methods in-clude both coordinates and elements. Finally, the Milankovitchelements (MIL), the Stiefel-Scheifele elements (SSc), and Dromo(DRO) show similar performances, as they are all based on ele-ments. Being the eccentricity small, the difference between usinga first order or a second order time transformation is not impor-tant. In this particular example the performance curves of B1H,B2H, B1C, and B2C are out of scale: the time element in this ver-sion of the Burdet-Ferrándiz formulation is not well suited fornumerical integration with a RKF5(4) scheme.

The SPDusingLSODAR is shown in Fig. 6.10. Like in Prob-lem 1, the adaptability of the integrator makes the performanceof formulations within the same class very similar. In addition,the overall speed of the propagations is improved with respectto the RKF5(4) integator, with little or no penalties in the accu-racy. In this particular case the use of elements is as important asthe time transformation. Indeed, the performance of Class 3 (el-ements with no time transformation) and Class 4methods (coor-dinates with time transformation) are comparable, and similar tothe performance of formulations using both elements and a ficti-tious time. Cowell’s method is the least efficient in this example,and its performance improves significantly when embedding theenergy and using a time element (SCW). The unified state model(USM) models the orbit with three elements, a technique that re-sults in significant improvements in the computational efficiency.

6.4 Conclusions

There are many problems in astrodynamics and orbital mechan-ics that require the integration of millions of trajectories. Typ-ically, the reach of such numerical explorations is limited by thecomputational resources. Before launchingmassive campaigns ofsimulations, it is worth finding which is the best combination ofnumerical integrator and orbital formulation to solve the prob-lem. In this way, the runtime may be reduced by several ordersof magnitude while retaining the same level of accuracy. Even ifthe computational time is not the driving factor, the right choiceof the formulation and integrator may significantly improve theaccuracy of the propagations.

Unfortunately, each formulationwill perform differently de-pending on the problem to be integrated. Claims about the ad-vantages of a given formulation should always be supported bydetails about the integration setup, and an explanation of the or-bital regime. For this reason, tools like PERFORM are necessary.This software combines high-fidelity force models with the im-plementation of a large collection of propagation methods andnumerical integrators. This allows the user to compare differentoptions on the same platform and for the problem of interest.

80 6 PERFORM: Performance Evaluation of Regularized Formulations of Orbital Motion

Figure 6.9: SPD for Problem 2 integratedwith RKF5(4).

§6.4 Conclusions 81

Figure 6.10: SPD for Problem 2 integratedwith LSODAR.

Part II

Applications

83

“Henceforth, space by itself, and time by itself, are doomed to fadeaway into mere shadows, and only a kind of union of the two willpreserve an independent reality.”

—Hermann Minkowski

7The theory of asynchronous relative motion

L inear solutions to relativemotion are preferred formanypractical applications like, for example, their integrationin navigation algorithms or their implementation in on-

board computers. In their pioneering work, Clohessy and Wilt-shire (1960) arrived to the exact solution to the linear equationsthat govern the circular rendezvous problem. These equationswere originally posed by Laplace (1799, p. 152) and further de-veloped by Hill (1878). They linearized the equations of motionassuming that the relative distance is small compared to the ra-dius of the reference orbit. This hypothesis has been adopted inmost of the subsequent studies. Tschauner and Hempel (1964,1965) addressed the elliptic case and formulated the governingequations of motion. In deriving his theory of the primer vector,Lawden (1954) had published a set of equations of motion thatare equivalent to those of the elliptic rendezvous. He provideda semianalytical solution to these equations in his book, whichrequires the numerical evaluation of an integral (Lawden, 1963,p. 85). Carter (1990, 1998) paid special attention toLawden’s inte-gral and proposed different methods to simplify it. Lawden andCarter’s work motivated Yamanaka and Ankersen (2002) to de-rive a state-transition matrix for the elliptic case that representsthe exact solution to the linearized problem. De Vries (1963) for-mulated the same problemmaking extensive use of the elliptic el-ements. He provided an approximate series solution bymeans ofan expansion in powers of the eccentricity. His solution iswrittenin terms of the true anomaly.

Alfriend et al. (2000) and Broucke (2003) solved the linearproblem through a geometric construction. The relative orbit isdetermined by applying a set of small differences to the orbitalelements defining the reference orbit. Alfriend et al. (2000) pio-neered this technique, and referred to it as the geometric method.

It allows to solve the problem using different formulations anddifferent sets of elements. The variational form of the solutionhelps to interpret the dynamics of the problem. Schaub (2004)presented a geometric method based entirely on the use of dif-ferences on the classical elements of the leader’s orbit, account-ing for the effect of the Earth oblateness. Given this solution, thesecular drift due to differences on the semimajor axes can be eas-ily analyzed. Gim and Alfriend (2003, 2005) posed the problemusing the set of nonsingular equinoctial elements and publisheda state-transition matrix including the J2 perturbation. Casotto(1993) conducted an interesting study of the transformations be-tween differences on the initial state vector and differences in theorbital elements. D’Amico andMontenbruck (2006) introducedthe concept of the differential eccentricity/inclination vector. Byparameterizing the relative dynamics in terms of the differences intheprojections of the eccentricity vector and a combinationof therelative inclination with the relative ascending node, they arrivedto a compact formulation. The versatility of the formulation wasproven by accounting for the effects of the differential drag andJ2 effects. We refer to the work by Lee et al. (2007) for a formalproof of the validity of the solutions obtained through the vari-ational form of the equations of motion. Melton (2000) foundan alternative state-transition matrix by means of an expansionin powers of the eccentricity that is written with the time as theindependent variable.

The solution to the linear equations of motion includes sec-ular terms. Such terms may be canceled if the initial conditionssatisfy the so called energy-matching condition (Alfriend et al.,2009, chap. 4). This condition implies coorbitalmotion of leaderand follower. Special attention has been paid to how the initialconditions relate to the relative drift between the spacecraft (Gur-

85

86 7 The theory of asynchronous relative motion

fil, 2005b; Gurfil and Lara, 2013). In the general case, the linearsolution is depreciated as time advances and the separation be-tween the spacecraft grows. For sufficiently large relative distancesnonlinear effects are no longer negligible and affect the accuracyof the solution. Different approaches to accounting for nonlineareffects can be found in the literature. Gurfil and Kholshevnikov(2006) presented an elegant characterization of the manifolds ofnonlinear relativemotion. Gim andAlfriend (2003) proposed aningenious interpretation of the problem to reduce the effects ofnonlinearities. They introduced a set of curvilinear coordinatesthat relate naturally to the geometry of the problem, a techniquethat was later recovered by Bombardelli et al. (2015). The lin-ear solution remains valid as long as the difference between thesemimajor axes is small. Vallado and Alfano (2014) analyzed indetail the transformation between cartesian and curvilinear coor-dinates, assessing their accuracy. Bombardelli et al. (2016) deriveda compact approximate solution for relative motion between or-bits with moderate eccentricity and inclination. Some authorshave studied the second-order equations of the circular problem(London, 1963; Karlgaard and Lutze, 2003), deriving solutionsthrough different methods. Vaddi et al. (2003) tackled the ec-centric case relying on a number of asymptotic expansions. Theyprovided the required corrections on the initial conditions thatyield periodic solutions when considering higher order terms.Sengupta et al. (2007) developed a tensorial representation of thesecond-order state-transition matrix for the elliptic case, coupledwith differential gravitational perturbations.

The exact solution to the unperturbed fully nonlinear prob-lem can be obtained by subtracting the state vectors of the leaderand follower spacecraft. Gurfil and Kasdin (2004) proposed amethod for obtaining solutions of arbitrary orders, relying on anexpansion in Fourier series. In a collection of papers, Conduracheand Martinuşi (2007b,c, 2009) wrote the exact solution to theproblem by referring the state of the follower spacecraft to theorbital elements defining the reference orbit. A more compactrepresentation in quaternionic form has been presented, which isindependent from any geometric assumptions (Condurache andMartinuşi, 2010). This method was later recovered by Martin-uşi and Gurfil (2011) to incorporate different terms of the gravityfield of the central body. In a different context, Gurfil and Kas-din (2003) derived the set of epicyclic orbital elements. They arisefrom the canonicalmodelling of the problem of coorbital relativemotion and are suitable for accounting for the mutual gravita-tional attraction between bodies.

The relative state vector is computed at a given time. It rep-resents the difference in position and velocity between two space-craft, measured at that particular time. This renders the time-synchronous solution. Since regularization deals with time trans-formations, a newdefinition of time is introduced. And the ques-tion about the synchronism of relative motion arises. Should therelative state vector be synchronized in time, or in fictitious time?This question has been widely discussed in Chap. 2 when explor-ing the Lyapunov instability and the structural stability of orbitalmotion. Solutions synchronized in time suffer from Lyapunovinstability, whereas solutions synchronized in fictitious time en-joy orbital stability. Thus, we will explore an alternative defini-tion of relative motion, in which the states are computed for thesame fictitious time. A time delay will appear, and it will be cor-

rected a posteriori to recover the physical meaning of the solu-tion. The ultimate goal is to improve the accuracy of the dynam-ical model.

In this chapter we formulate the key concepts of the noveltheory of asynchronous relative motion. The theory has several ad-vantages. First, it provides the variational equations for any for-mulation relying on an independent variable different from time.It greatly simplifies the derivation of the partial derivatives of theorbital elements in the unperturbed case. In the literature thereare very few studies about how the variational equations of regu-larized formulations should be propagated, and the present the-ory fills that gap. Second, it allows to introduce nonlinear effectseasily in the linear solution to relative motion, leading to impor-tant improvements in accuracy. Thenonlinear correction is basedon the dynamics of the problem and it is computed a posteriori.Third, the proposed theory is a physical concept: it can be ap-plied to any of the existing linear solutions, and it admits the in-troduction of arbitrary perturbations. It is not restricted to theKeplerian case. The correction depends on the time delay in theasynchronous solution. A general expression for the time delay isprovided in this chapter by means of the equinoctial orbital ele-ments. Using this set of elements guarantees that the problem isfree of singularities. As representative examples, the method willbe applied to theClohessy-Wiltshire (CW) solution in the circularcase, and to the solution from Yamanaka-Ankersen’s (YA) state-transition matrix in the elliptic case. The new form of these solu-tions providesmore accurate results both in position and velocity.

First, the concept of synchronism is introduced and the non-linear correction is derived from the main definitions. A simpleexample is included to showhow theproposedmethod is applied.Section 7.3 generalizes the theory to any formulation. The ex-plicit form of the time delay is given in terms of the equinoctialorbital elements. The connection with the solution provided byGim and Alfriend (2005) is discussed. The circular case is ad-dressed in Sect. 7.4 and the corrected form of the CW solution ispresented. Special attention is paid to how the nonlinear correc-tion relates to high-order solutions. Section 7.5 includes a num-ber of numerical test cases, both perturbed and unperturbed,where the performance of the corrected solutions is comparedwith the original methods. The inverse transformations can befound in Appendix D.

It is worthwhile to emphasize that the applications of thistheory go beyond spacecraft relative motion. This chapter fo-cuses on this particular topic, but the concepts presented herecan be applied to any problem involving the partial derivatives ofthe states. This includes orbit determination, optimization, thesearch for periodic orbits, and many others.

7.1 Definition of the problem

The problem of relative motion reduces to describing the rela-tive dynamics of a follower spacecraft, f , with respect to a leaderspacecraft, ℓ. Let L = i, j,k be the Euler-Hill reference frame,centered at the leader spacecraft and defined by the basis:

i =rℓrℓ, k =

hℓhℓ, j = k × i

§7.2 Synchronism in relative motion 87

where r is the radius vector and h denotes the angular momen-tum. The subscript ℓ refers to the leader spacecraft, while the sub-script f refers to the follower spacecraft. Frame L rotates aboutthe inertial reference I = iI, jI,kIwith angular velocity:

ωLI =hℓr2ℓ

k

The relative position is defined by

δr = x i + y j + z k

The relative velocity referred to the fixed reference is denoted byδv, whereas the relative velocity from the perspective of the rotat-ing frame is δr. They relate through the inertia terms

δr = δv − ωLI × δr

and their components in the rotating reference are δv = vxi +vyj + vzk and δr = x i + y j + z k, respectively. Figure 7.1 depictsthe geometry of the problem.

Figure 7.1: Geometrical definition of the problem.

7.2 Synchronism in relative motion

Let r ∈ R3 be the solution to the two-body problem

d2rdt2 =

µ

r3 r + ap (7.1)

inwhich r = ||r|| is the distance to the origin, µ is the gravitationalparameter, and ap is the perturbing acceleration. Introducing acharacteristic length ℓc and a characteristic time tc = (ℓ3

c/µ)1/2,variables are normalized according to

r =rℓc, v =

vℓc/tc

, t =ttc, ap =

ap

ℓc/t2c

Here v denotes the velocity. Although ℓc can be chosen at con-venience, in this chapter we shall make it equal to the semimajoraxis of the reference orbit, ℓc ≡ aℓ, in order the mean motion tosimplify to unity, nℓ = 1. To alleviate the notation all variablesare assumed to be normalized, and the tilde will be droppedfrom now on.

Equation (7.1) needs to be integrated from the initial condi-tions

at t = t0 : r(t0) = r0, v(t0) = v0 (7.2)

in order to obtain the state vector x = [r⊤, v⊤]⊤, formed by theposition and velocity vectors,

x = f(t, oe0) (7.3)

in normalized variables. The vector oe0 denotes the initial valuesof the set of parameters used to describe the solution, i.e. the sixintegration constants. If the equations of motion are propagatedin Cartesian coordinates oe0 reduces to the initial conditions inEq. (7.2). If the orbit is modeled with a set of orbital elements,then oe0 are the osculating values of the elements at departure. Acommon choice is to take oe = [a, e, i, ω, Ω, ϑ0]⊤, correspondingto the classical elements of the orbit. Alternative sets of elementshave been discussed in Sect. 2.5. For Keplerian orbits (ap = 0)the elements oe are constant and they are solved from the initialconditions (7.2); for perturbed problems, however, they are func-tions of time t.

In the following lines wewill first derive the usual solution tolinear relativemotion,whatwe shall call the time-synchronous so-lution. Then, the asynchronous approach will be presented, andwe will show how the asynchronous solution can be transformedinto the time-synchronous one via a first-order correction of thetime delay. Section 7.2.3 contains a simple example of applica-tion. Finally, Sect. 7.2.4 discusses how to take advantage of thisconstruction to define a high-order correction that improves thelinear solution easily. This last section includes a diagram sketch-ing how the solution is built.

7.2.1 Time-synchronous approach

Let xℓ and x f be the inertial state vectors of the leader and fol-lower spacecraft, respectively. Both states relate by means of

x f (t) = xℓ(t) + δx(t) (7.4)

where δx is the relative state vector. Similarly, if oeℓ denotes theelements of the leader, then the elements of the follower read

oe f (t) = oeℓ(t) + δoe(t) (7.5)

and initially it is

oe f ,0 = oeℓ,0 + δoe0

Vector δoedenotes the differential elements. In absence of pertur-bations the elements will remain constant, to wit, δoe(t) ≡ δoe0.

If the relative separation and velocity are small compared tothe absolute states, the differential elements will be small com-pared to the absolute elements too. Under this assumption andconsidering the solution (7.3), the state vector of the follower canbe obtained from the series expansion

x f = f(t, oeℓ,0+δoe0) ≈ f(t, oeℓ,0)+∂fℓ∂oe0

∣∣∣∣∣tδoe0 = xℓ+

∂fℓ∂oe0

∣∣∣∣∣tδoe0

Identifying this result with Eq. (7.4) proves that the variations onthe leader position and velocity vectors yield the relative state vec-tor, δx:

δx = Jt δoe0, where Jt =∂fℓ∂oe0

∣∣∣∣∣t

(7.6)

88 7 The theory of asynchronous relative motion

Here Jt represents the Jacobian matrix of the orbit of the leaderspacecraft. The problem of relative motion reduces to comput-ing this Jacobian matrix, defined by the partial derivatives of theleader state vector xℓ = f(t, oeℓ,0) with respect to oe0.

Since the leader and follower states need to be computedat the same time, t is kept constant when deriving the Jaco-bian matrix, which justifies the subscript in Jt. This is the time-synchronous (or simply synchronous) Jacobian matrix. In thisway, the relative orbit can be constructed geometrically by apply-ing a set of small differences δoe0 to the reference orbit. Whenperturbations are present Jt is typically propagated numerically,following the usual procedure summarized in Sect. 6.1.4.

7.2.2 Asynchronous approach

The solution to linear relative motion presented in the previoussection is based on the partial derivatives of the reference orbit,computed with constant time. Although this provides the physi-cal solution to the problem, it is subject to the known Lyapunovinstability of orbital motion. This section explores an alternative(asynchronous) approach, in which the partial derivatives are ob-tained keeping constant an independent variable different fromtime, often equivalent to an angle. The time is no longer con-stant, and the resulting time delay needs to be corrected in orderto obtain exactly the solution from the previous section. As a re-sult, thanks to the Poincaré stability of motion, all secular termsare confined to the time delay.

The problem can be formulated using any set of parametersoe, and the physical time can be replaced by an alternative inde-pendent variable, ϕ, which behaves as the fictitious time. The so-lution can be expressed like

x = g(ϕ, oe0) (7.7)t = T (ϕ, oe0) (7.8)

The physical time becomes a dependent variable defined byEq. (7.8), which is equivalent to Kepler’s equation in the corre-sponding variables. In any case, Eq. (7.8) is an invertible bijectivefunction in R, whose inverse provides

ϕ = Φ(t, oe0)

The bijection defines the mapping T : ϕ 7→ t, andΦ : t 7→ ϕ.Now the inertial state vector of the follower at ϕ can be writ-

ten

x f (ϕ) = xℓ(ϕ) + δxasyn(ϕ). (7.9)

The relative state vector in the present asynchronous approach,δxasyn, is different from the one appearing in Eq. (7.4), δx, as wewill prove in what remains of this section.

Given Eq. (7.7), the state vector of the follower abides by

x f = g(ϕ, oeℓ,0 + δoe0) ≈ g(ϕ, oeℓ,0) +∂gℓ∂oe0

∣∣∣∣∣ϕδoe0

= xℓ +∂gℓ∂oe0

∣∣∣∣∣ϕδoe0 (7.10)

By virtue of Eq. (7.9) the variations on the leader position andvelocity vectors result in the relative state vector, δxasyn, given by:

δxasyn = Jϕ δoe0, where Jϕ =∂gℓ∂oe0

∣∣∣∣∣ϕ

(7.11)

Matrix Jϕ is the Jacobian matrix of (7.7), particularized along theorbit of the leader. Defining the Jacobianmatrix requires the par-tial derivatives of the leader state vector xℓ = g(ϕ, oeℓ,0) with re-spect to oe0. When perturbations are considered, the Jacobianis computed numerically (see Sect. 6.1.4 for details on the imple-mentation). The states of the leader and follower are computedat the same value of independent variable. Thus, ϕ is kept con-stant when computing the partials leading to the Jacobian ma-trix. This provides the ϕ-synchronous (or asynchronous) Jacobianmatrix, Jϕ. Comparing Eqs. (7.6) and (7.11) clearly shows the dif-ference between the synchronous, δx, and the asynchronous so-lutions, δxasyn, which comes from the different definitions of theJacobian:

δx = Jt δoe0, and δxasyn = Jϕ δoe0

In the asynchronous approach, the state vector of the leaderxℓ and the state vector of the follower x f are defined at differenttimes, tℓ and t f , which come from Eq. (7.8):

t f = T (ϕ, oe f ,0) and tℓ = T (ϕ, oeℓ,0)

Following the sameprocedure that yieldedEq. (7.10), it is possibleto relate these two different times thanks to

t f = T (ϕ, oeℓ,0 + δoe0) ≈ T (ϕ, oeℓ,0) +∂Tℓ∂oe0

∣∣∣∣∣ϕ· δoe0

= tℓ +∂Tℓ∂oe0

∣∣∣∣∣ϕ· δoe0

in which a time delay δt appears, defined as

δt = t f − tℓ =∂Tℓ∂oe0

∣∣∣∣∣ϕ· δoe0 (7.12)

This means that the relative state vector δxasyn determines wherethe follower will be in a time δt in the future, or where it was−δtago. In general, the partials in Eq. (7.12) will be solved numer-ically, although in the Keplerian case they can be computed ex-plicitly. It is worth noticing that if δoe0 is small, the time delaywill also be small. Figure 7.2 depicts the configuration of the so-lution. Three dots represent the position of the leader spacecraftat t0, t1, and t2. Similar dots show the position of the followerspacecraft at those same times. The relative states between thesepairs of points furnish the synchronous solution to the problemδx (in gray). The positions of the leader spacecraft at t1 and t2correspond to the values ϕ1 and ϕ2. The stars mark the pointsin the follower orbit corresponding to these same values of ϕ1and ϕ2. The resulting state vectors define the asynchronous so-lution δxasyn, and the time delay connects the synchronous andasynchronous solutions (points and stars, respectively).

Now that the asynchronous solution and the time delay areknown, the question on how to recover the synchronous solu-tion arises. That is, how can we refer the solution x f (t f ) to thereference time of the leader spacecraft, x f (tℓ)? The answer comesfrom the series expansion

x f (tℓ) = x f (t f − δt) = x f ,asyn −∂f f

∂t

∣∣∣∣∣asynδt + O(δt2) (7.13)

§7.2 Synchronism in relative motion 89

Figure 7.2: Schematic representation of the synchronous and asyn-

chronous solutions, together with the time delay.

The derivatives appearing in the expansion are

∂r f

∂t

∣∣∣∣∣asyn= vℓ + δvasyn

∂v f

∂t

∣∣∣∣∣asyn= −

r f

r3f

∣∣∣∣∣asyn+ ap, f

= −rℓr3ℓ

+ ap,ℓ +1r3ℓ

[I − 3(i ⊗ i)

]δrasyn + δap

(7.14)

The term between square brackets results in a three-by-three ma-trix, defined by the identity matrix I and the dyadic product ⊗.The unit vector i, attached to the orbital frame, reads i = rℓ/rℓ.

The perturbations in the leader orbit are ap,ℓ, the perturba-tions affecting the follower orbit will be written ap, f , and δap =

ap, f − ap,ℓ denotes the differential perturbations. When the sep-aration is small, ||δx||/||xℓ || ∼ ε ≪ 1, the time delay and the dif-ferential perturbations will be small too, δt/tℓ ∼ ||δap||/||ap,ℓ || ∼ε ≪ 1. Subtracting the leader state vector xℓ(tℓ) from the expan-sion and retaining only first-order terms in ε results in

δx = δxasyn −∂fℓ∂t

δt (7.15)

Equation (7.15) can be separated to define the relative positionand velocity vectors,

δr = δrasyn − vℓ δt (7.16)

δv = δvasyn +

(rℓr3ℓ

− ap,ℓ

)δt (7.17)

The asynchronous solution can be transformed into the true,synchronous solution by applying an intuitive linear correction,which is equivalent to assuming locally rectilinear motion.

We will now derive the relation between the synchronousand the asynchronous Jacobian matrices, Jt and Jϕ, respectively.Equations (7.3) and (7.7) are two alternative solutions to the sameproblem, meaning that the following relations hold:

f(t, oe0) = g(Φ(t, oe0), oe0) (7.18)g(ϕ, oe0) = f(T (ϕ, oe0), oe0) (7.19)

The partial derivatives of Eq. (7.18), particularized along theleader orbit, render

∂fℓ∂oe0

∣∣∣∣∣t=∂gℓ∂oe0

∣∣∣∣∣ϕ+∂gℓ∂ϕ

∂Φ

∂oe0

∣∣∣∣∣t=⇒ Jt = Jϕ +

∂gℓ∂ϕ

∂Φ

∂oe0

∣∣∣∣∣t

(7.20)

The first summand appearing in the definition of Jt, Jϕ, is theasynchronous Jacobianmatrix. The second summand is a correc-tion that accounts for the variation on the independent variableϕdue to the differences δoe0, keeping the time constant. It recoversthe time-synchronism. The first term is usually straight-forwardto compute. But the second term is cumbersome since Kepler’sequation (or equivalent) needs to be differentiated. It is worthnoticing that this is the approach found in the literature (see forexample Broucke, 2003), although no references to the synchro-nism of the solutions are made.

The partial derivatives of Eq. (7.19) furnish the relation

∂gℓ∂oe0

∣∣∣∣∣ϕ=

∂fℓ∂oe0

∣∣∣∣∣t+∂fℓ∂t

∂T∂oe0

∣∣∣∣∣ϕ=⇒ Jϕ = Jt +

∂fℓ∂t

∂T∂oe0

∣∣∣∣∣ϕ

(7.21)

Subtracting Eqs. (7.20) and (7.21) yields the following identity re-lating the partials of t and ϕ:

∂gℓ∂ϕ

∂Φ

∂oe0

∣∣∣∣∣t= −∂fℓ

∂t∂T∂oe0

∣∣∣∣∣ϕ

(7.22)

Indeed, recovering Eq. (7.6) andmaking use of Eq. (7.20) and theidentity (7.22) leads to

δx = Jt δoe0 =

(Jϕ +

∂gℓ∂ϕ

∂ϕ

∂oe0

∣∣∣∣∣t

)δoe0

=

(Jϕ −

∂fℓ∂t

∂T∂oe0

∣∣∣∣∣t

)δoe0 = δxasyn −

∂fℓ∂t

δt

which is none other than Eq. (7.15). This means that Eq. (7.15)is the exact solution to the linear equations of relative motion.Moreover, the synchronous Jacobian reads

Jt = Jϕ −∂fℓ∂t

∂T∂oe0

∣∣∣∣∣ϕ

(7.23)

and this form of the Jacobian is easier to compute than the one inEq. (7.21).

The corrected form of the relative velocity referred to the ro-tating frameL is

δr = δrasyn + δt[( 1

rℓ− v2

ℓ

)I + (vℓ ⊗ vℓ)

]rℓr2ℓ

− ap,ℓ δt

The asynchronous solution is corrected with the instantaneousvelocity and acceleration of the leader spacecraft, multiplied bythe time delay.

7.2.3 A simple example

This example will demonstrate how to compute the asyn-chronous solution and the time delay, and how the first-ordercorrection of the time delay transforms the asynchronous solu-tion into the synchronous one. We will compute the radial sep-aration between the leader and follower spacecraft, δr, due to a

90 7 The theory of asynchronous relative motion

small difference in the semimajor axis, δa. The spacecraft sharethe same values of the rest of orbital elements. No perturbationsare considered in this example.

The radial distance is defined in time by the functional rela-tion

r = f (t, a) (7.24)

Following Eq. (7.6), the solution to the radial motion when onlythe semimajor changes is

δr = Jt δa ≡∂ f∂a

∣∣∣∣∣tδa (7.25)

In practice, the radial motion is easily written in terms of the trueanomaly ϑ,

r = g(ϑ, a) ≡ a(1 − e2)1 + e cosϑ

The true anomaly relates to physical time bymeans of the inverseKepler equation

ϑ = Φ(t, a)

In order to differentiate Eq. (7.24) keeping constant the time weapply the chain rule

∂ f∂a

∣∣∣∣∣t=∂g

∂a

∣∣∣∣∣ϑ+∂g

∂ϑ

∂Φ

∂a

∣∣∣∣∣t

(7.26)

as shown in Eq. (7.20). The first term reduces to:

∂g

∂a

∣∣∣∣∣ϑ=

ra

and the first factor in the second term is

∂g

∂ϑ=

r2e sinϑa(1 − e2)

The last term requires the derivatives of Kepler’s equation withconstant time.

Differentiating Kepler’s equation,

n(t − tp) = E − e sin E

(inwhich E is the eccentric anomaly and tp is the time of periapsispassage) with respect to a yields

−3n2a

(t − tp) =∂E∂a

∣∣∣∣∣t(1 − e cos E) (7.27)

The eccentric anomaly relates to the true anomaly by means of:

sin E =

√1 − e2 sinϑ

1 + e cosϑ, and cos E =

e + cosϑ1 + e cosϑ

These expressions are differentiated to provide

∂E∂a

∣∣∣∣∣t=

√1 − e2

1 + e cosϑ∂Φ

∂a

∣∣∣∣∣t

Introducing this result in Eq. (7.27) furnishes the relation

∂Φ

∂a

∣∣∣∣∣t= −3n

2a(1 + e cosϑ)2

(1 − e2)3/2 (t − tp) = −32

na√

1 − e2

r2 (t − tp)

which completes the second term from Eq. (7.26),

∂g

∂ϑ

∂Φ

∂a

∣∣∣∣∣t= −3

2ne sinϑ√

1 − e2(t − tp) (7.28)

Finally, Eq. (7.25) leads to the time-synchronous solution to theproblem:

δr =[

ra− 3

2ne sinϑ√

1 − e2(t − tp)

]δa (7.29)

This result can alsobe found in, for example, theworkbyBroucke(2003), Gim and Alfriend (2003) or Schaub (2004).

Now the same result will be derived applying the theory ofasynchronous relative motion, according to Eq. (7.15):

δr = δrasyn −∂ f∂tδt (7.30)

in which

δrasyn =∂g

∂a

∣∣∣∣∣ϑδa =

raδa, and

∂ f∂t≡ r =

µ

he sinϑ

are the asynchronous solution and the radial velocity, respectively.The time delay is solved from its definition, given in Eq. (7.12),

δt =∂T∂a

∣∣∣∣∣ϑδa

The function T (ϑ, a) is Kepler’s equation:

t = T (ϑ, a) ≡ tp + [E(ϑ) − e sin E(ϑ)](a3/µ)1/2

Differentiating with respect to a, keeping constant the anomalyϑ, leads to

∂T∂a

∣∣∣∣∣ϑ=

32a

(t − tp)

and the time delay takes the form

δt =32a

(t − tp) δa

Consequently, Eq. (7.30) becomes

δr =raδa−3

2ne sinϑ√

1 − e2(t−tp) δa =

[ra− 3

2ne sinϑ√

1 − e2(t − tp)

]δa

This is the same exact result provided by Eq. (7.29), although thisresult has been obtained from the first-order correction of thetime delay applied to the asynchronous solution. Note also that

−∂ f∂t∂T∂a

∣∣∣∣∣∣ϑ

= −32

ne sinϑ√

1 − e2(t − tp)

equals the expression in Eq. (7.28), which proves the iden-tity (7.22).

§7.3 Generalizing the transformation 91

7.2.4 Improving the accuracy with second-order correctionsof the time delay

The time delay has been corrected thanks to the series expansionin Eq. (7.13). Retaining only first-order terms provided the ex-act solution to the linear equations of relative motion, Eq. (7.15).If higher order terms are considered, the accuracy of the correc-tion will improve because of introducing nonlinearities in the so-lution. Extending Eq. (7.13) yields

x f (tℓ) = x f ,asyn −∂f f

∂t

∣∣∣∣∣asynδt +

12∂2f f

∂t2

∣∣∣∣∣asynδt2 + O(δt3) (7.31)

Apart from the terms in Eq. (7.14), this expansion also involves

∂2v f

∂t2

∣∣∣∣∣asyn= −vℓ

r3ℓ

+ 3rℓrℓr4ℓ

+ O(ε) ≈ − 1r3ℓ

[I − 3(i ⊗ i)

]vℓ

The correction is an approximation of the solution, which triesto retain accuracy without complicating excessively the compu-tations. A rigorous definition of the second-order correction ofthe time delay requires the explicit value of the differential per-turbations δap. Although it can be solved by evaluating the par-tial derivatives of the perturbing acceleration, this is not a usualfeature of orbit propagators, and important modifications in thecode would be required. For this reason and for the sake of sim-plicity, we neglect the contribution of the differential perturba-tions to the second-order correction of the time delay. It is wor-thy emphasizing thatwe are onlyneglecting δap when computingthis high order correction, but not in the propagation of the lin-ear solution. The partials leading to the linear solution and thetime delay are integrated considering all the perturbations.

As a result, retaining terms up to second order in ε inEq. (7.31), the relative position and velocity corrected to secondorder (denoted with a star ⋆) read

δr⋆ = δrasyn − (vℓ + δvasyn)δt −(

rℓr3ℓ

− ap,ℓ

)δt2

2(7.32)

δv⋆ = δvasyn +

(rℓr3ℓ

− ap,ℓ

)δt +

δt2r3

ℓ

[I − 3(i ⊗ i)

](2δrasyn − vℓδt)

(7.33)

This technique is a generic concept, not restricted to any spe-cific formulation or perturbation model. In fact, combiningEqs. (7.16–7.17) and (7.32–7.33) shows how the linear solution(δr and δv) can be improved by introducing nonlinear terms:

δr⋆ = δr − δv δt + rℓ2r3

ℓ

δt2 (7.34)

δv⋆ = δv +δt

2r3ℓ

[I − 3(i ⊗ i)] (2δr + vℓδt) (7.35)

The only extra variable that we need, apart from the linear so-lution and the state of the leader, is the time delay. Recall thati = rℓ/rℓ. In a series of papers, Roa and Peláez (2015b,f, 2016d)andRoa et al. (2015a) arrived to different expressions for the timedelay using regularized variables, which will be discussed in thenext chapter. In the next section we will derive a simple explicitsolution for the time delay using the set of equinoctial elements

Figure 7.3: Diagram showing the construction of the solution.

in the unperturbed case. When perturbations are present, thetime delay needs to be propagated numerically. Figure 7.3 depictsthe flowchart of the solution. The linear solution to relative mo-tion (7.6) can be obtained with both the usual time-synchronousapproach (in line with the references in the introduction) or withthe asynchronous approach. In any case, from this solution andwhen the time delay is known, a corrected, more accurate solu-tion can be built. See Sect. 7.5 for examples of the improvementsin accuracy.

7.3 Generalizing the transformation

Consider that the linear, time-synchronous solution to the prob-lem (δr and δv) has been obtained using a certain formulation.The formulations discussed in the introduction are good exam-ples. We will now see how the accuracy can be improved thanksto the second-order correction of the time delay. In what remainsof the chapter the relative velocity is defined in the rotating refer-enceL, i.e. δx = [x, y, z, x, y, z]⊤. Recall thatδv = δr+ωLI×δr,with δr = x i + y j + z k. Combining Eqs. (7.34–7.35) yields thecorrected velocity in the rotating frame,

δr⋆ = δr +δtr4ℓ

[(rℓ − h2

ℓ )I + h2ℓ (k ⊗ k)

]δr

− rℓ(i ⊗ i)(3δr + vℓδt) + r2ℓhℓ × δr

As summarized in Fig. 7.3, the linear solution can be im-

proved without the need for computing the asynchronous solu-tion (only the time delay is required). Therefore, this section isdevoted to deriving the generic definition of the time delay, so itcan be applied to any existing solution directly. No perturbationsare considered in this section (ap = 0, meaning that δoe ≡ δoe0).

92 7 The theory of asynchronous relative motion

The resulting expression is parameterizedwith the equinoctial or-bital elements, in order to avoid singularities.

7.3.1 The time delay using equinoctial orbital elements

In order to derive a generic nonsingular expression for the timedelay, the variational solution to relativemotion is obtained usingthe equinoctial orbital elements. This set of elements, introducedby Brouwer and Clemence (1961), overcomes the singularities ofthe classical elements for equatorial and circular orbits. Gim andAlfriend (2005) obtained the time-synchronous solution to rela-tive motion in curvilinear coordinates using these elements. Thetransformation proposed in this section is derived in the usualrectangular coordinates. All variables refer to the leader space-craft.

Let q ∈ R7 denote the equinoctial elements, ordered as q =[a, q1, q2, k1, k2, λ0, tp]⊤, where:

q1 = e cos(ω + Ω), k1 = tani2

cosΩ

q2 = e sin(ω + Ω), k2 = tani2

sinΩ

and tp is the time of pericenter passage. The true anomaly ϑ isreplaced by the true longitude

λ = ω + Ω + ϑ

Introducing the auxiliary variables η2 = 1 − q21 − q2

2 (equivalentto the angular momentum squared) and

s = 1 + q1 cos λ + q2 sin λ, u = q1 sin λ − q2 cos λ

the position and velocity vectors read:

r =η2

si, and v =

1η

(u i + s j)

Recall that both the normalized semimajor axis and themeanmo-tion are equal to one, so that h ≡ η. The equinoctial referenceframe Q (Broucke and Cefola, 1972) is defined in the inertial ref-erence I bymeans of: (i) a rotation about the z-axis of magnitudeΩ, which defines the node, (ii) a rotation about the rotated x-axisofmagnitude i, (iii) a rotationofmagnitude−Ω about the rotated

z-axis. The equinoctial frame is shown in Fig.

z-axis. The equinoctial frame is shown in Fig. 7.4.

Figure 7.4: Equinoctial frame.

The rotation Q 7→ I is represented by the matrix

P =1ℓ2

1 + k2

1 − k22, +2k1k2, +2k2

+2k1k2, 1 − k21 + k2

2, −2k1

−2k2, +2k1, 1 − k21 − k2

2

with ℓ2 = 1 + k2

1 + k22. It defines the transformation

rI = P rQ

which maps the coordinates in the equinoctial frame, rQ, to thecoordinates in the inertial frame, rI. The orbital frame is ob-tained through a rotation of magnitude λ about the zQ-axis.

The dimension of the relative state vector is extended toinclude the time delay as the seventh component, resulting inδx′asyn = [x, y, z, x, y, z, δt]⊤. For computing the asynchronoussolution it is more convenient to use the angular displacement intrue longitude as the independent variable:

γ = λ − λ0

so the initial condition λ0 contained in q can be modified. Ac-cording to thenotation in Sect. 7.2, it isϕ ≡ γ. The asynchronoussolution is

δx′asyn(γ) = Jγ(γ) δq

with δoe ≡ δq = [δa, δq1, δq2, δk1, δk2, δλ0, δtp]⊤. In this sec-tion δq ≡ δq0. The Jacobian is

Jγ =

r, −2q1 + r cos λs

, −2q2 + r sin λs

, 0, 0,η2us2 , 0

0, 0, 0,2rk2

ℓ2 , −2rk1

ℓ2 , r, 0

0, 0, 0,2r sin λℓ2 , −2r cos λ

ℓ2 , 0, 0

− u2η,

uq1 + η2 sin λ

η3 ,uq2 − η2 cos λ

η3 , 0, 0,s − 1η

, 0

− 3s2η,

3q1 s + 2η2 cos λη3 ,

3q2 s + 2η2 sin λη3 ,

2uk2

ηℓ2 , −2uk1

ηℓ2 , −uη, 0

0, 0, 0,2ηℓ2 (q1 + cos λ),

2ηℓ2 (q2 + sin λ), 0, 0

32

(t − tp),η

s2e2 [η2q2 − q1u(s + 1)], − η

s2e2 [η2q1 + q2u(s + 1)], 0, 0,η3

s2 , 1

§7.3 Generalizing the transformation 93

Its determinant is simply:

det(Jγ

)= 2η3/(s2ℓ4)

so the matrix is regular even in the circular and equatorial cases.However, the time delay (given by the last row of the matrix) be-comes singular for e → 0. This singularity can be overcome byreplacing the second and third columns by a linear combination

with the seventh column, which removes e2 from the denomina-tor. The only terms affected by this combination are those cor-responding to the time delay. If coli denotes the i-th column ofmatrix Jγ, this matrix is transformed according to

col2 7→ col2 −ηq2

e2 col7, and col3 7→ col3 +ηq1

e2 col7

Matrix Jγ becomes:

Jγ =

r, −2q1 + r cos λs

, −2q2 + r sin λs

, 0, 0,η2us2 , 0

0, 0, 0,2rk2

ℓ2 , −2rk1

ℓ2 , r, 0

0, 0, 0,2r sin λℓ2 , −2r cos λ

ℓ2 , 0, 0

− u2η,

uq1 + η2 sin λ

η3 ,uq2 − η2 cos λ

η3 , 0, 0,s − 1η

, 0

− 3s2η,

3q1 s + 2η2 cos λη3 ,

3q2 s + 2η2 sin λη3 ,

2uk2

ηℓ2 , −2uk1

ηℓ2 , −uη, 0

0, 0, 0,2ηℓ2 (q1 + cos λ),

2ηℓ2 (q2 + sin λ), 0, 0

32

(t − tp), − ηs2 [q2 + (s + 1) sin λ],

η

s2 [q1 + (s + 1) cos λ], 0, 0,η3

s2 , 1

(7.36)

The normalized semimajor axis, a = 1, has been omitted forbrevity. The determinant of the matrix is not affected by the lin-ear combinationof the columns. The timedelay is given explicitlyby

δt = d · δq (7.37)

where vector d is the last row of matrix Jγ. That is,

δt =32

(t − tp)δa − η

s2 (C2δq1 −C1δq2) +η3

s2 δλ0 + δtp (7.38)

with C1 = q1 + (s + 1) cos λ and C2 = q2 + (s + 1) sin λ. Thetime delay introduces a secular term in the solution. This termappears due to a variation on the semimajor axis. In the case ofcoorbital motion between leader and follower the semimajor axescoincide and δa = 0. The solution becomes periodic and therelative orbits are closed.

The linear transformation Jγ : δq 7→ δx′asyn maps a set of dif-ferences on the equinoctial elements to the asynchronous relativestate vector and the time delay. According to Fig. 7.3, the asyn-chronous solution can be corrected to define the synchronousone—Eqs. (7.16–7.17)— and then improved to account for non-linearities—Eqs. (7.34–7.35)—. Equations (7.32) and (7.33) canbe solved directly to go from the asynchronous solution to theimproved one including nonlinearities.

No perturbations are considered in this section, so δq ≡ δq0is constant. This set of differences can be determined initially bymeans of the inverse transformation:

δq = J−1γ (0) δx′0

It is important to note that the initial value of the time delay isalways zero, δt0 = 0. The inverse matrix J−1

γ can be found in

Appendix D. From this expression, the state-transition matrix

Ξ(γ) = Jγ(γ) J−1γ (0) (7.39)

can be constructed, providing the asynchronous relative state vec-tor:

δx′asyn(γ) = Ξ(γ) δx′0The last element of this matrix product, which involves the state-transition matrix, defines the time delay in terms of the relativeinitial conditions, and δx′0 = [x0, y0, z0, x0, y0, z0, 0]⊤. It reads:

δt = δtxx0 + δtyy0 + δtx x0 + δtyy0 (7.40)

Note that the out-of-plane motion does not affect the time delay.The contribution of each term is solved from the matrix productin Eq. (7.39):

δtx =3r2

0

(s0 + 1)(t − t0) +u0

η+

s0

ηs2

[C1(2C20 − s0 sin λ0)

−C2(2C10 − s0 cos λ0)]

δty = −3u0

r20

(t − t0) +1

s2η

[C1(s2

0 cos λ0 −C30)

+C2(s20 sin λ0 +C40) + s0(η2 − s2)

]δtx =

3u0

η(t − t0) − r

s[(s + 1) cos γ + (s0 − 1)

]+ 2r0

δty =3ηr0

(t − t0) +r0

s2 (C1C20 −C2C10)

with

C1 = q1 + (s + 1) cos λ, C2 = q2 + (s + 1) sin λC3 = uq2 + cos λ + q1, C4 = uq1 − sin λ − q2

94 7 The theory of asynchronous relative motion

These expressions provide the evolution of the time delay froma set of initial conditions. Once the time delay is obtained —evaluating Eq. (7.40)— it can be applied to the asynchronoussolution to obtain the synchronous one. In addition, a second-order correction can be computed to improve the accuracy of thelinear solution, according to Fig. 7.3.

7.3.2 Connection with the time-synchronous solution

The time-synchronous Jacobian matrix written in terms of theequinoctial elements, already provided by Gim and Alfriend(2003, 2005), can be recovered from matrix Jγ. First, let us calld the first six elements of the last row of Jγ. Second, we ignorethe last row and last column of Jγ so it reduces to a 6 × 6 matrix.

Then, considering the identity in Eq. (7.23),

Jt = Jγ −∂fℓ∂t⊗ d (7.41)

(recall that vector d has already been introduced in Eq. (7.37) andhere ap = 0) the partials ∂f/∂t take the form

∂fℓ∂t=

vℓ(

1rℓ− v2

ℓ

)I + (vℓ ⊗ vℓ)

rℓr2ℓ

Evaluating Eq. (7.41) renders the synchronous Jacobian Jt,

which is precisely the one derived by Gim and Alfriend (2003),although they interpreted the solution in curvilinear coordinates.The synchronous Jacobian matrix results in:

Jt =

−3u2η

(t − tp) +η2

s, − cos λ, − sin λ, 0, 0, 0

− 3s2η

(t − tp),1s

[q2 + (s + 1) sin λ], −1s

[q1 + (s + 1) cos λ],2η2k2

ℓ2 s, −2η2k1

ℓ2 s,

2η2

s

0, 0, 0,2η2 sin λℓ2 s

, −2η2 cos λℓ2 s

, 0

3s2

2η4 (1 − s)(t − tp) − u2η,

s2 sin λη4 , − s2 cos λ

η3 , 0, 0, 0

3s2u2η4 (t − tp) − 3s

2η,

sq1 + (η2 + s2) cos λη3 ,

sq2 + (η2 + s2) sin λη3 ,

2k2uℓ2η

, −2k1uℓ2η

,2uη

0, 0, 0,2(q1 + cos λ)

ℓ2η,

2(q2 + sin λ)ℓ2η

, 0

(7.42)

The determinant

det(Jt)= 4η/ℓ4

confirms that there are no singularities in the formulation. Theinverse matrix is given in Appendix D. An important differencebetweenmatrices Jγ and Jt is that the only secular terms inmatrixJγ appear in the time delay. However, inmatrix Jt there are secularterms in the radial and along-track components of the solution.This is a consequence of the stability properties of orbitalmotion.The asynchronous solution is based on the geometry of the orbitsand, consequently, exploits the Poincaré stability of motion. The

effect of Lyapunov instability appears only when considering thetime within each orbit, and affects only the time delay.

7.4 The circular case

The simplest case of relativemotion is themotion about a circularreference orbit. In this case, the orbital radius of the leader space-craft rℓ is constant and equal to the semimajor axis. The eccen-tricity is zero and the γ-synchronous Jacobian matrix is denotedby Jγ,c. The state-transition matrix is obtained from the productΞc(γ) = Jγ,c(γ) J−1

γ,c(0), resulting in:

Ξc(γ) =

4 − 3 cos γ, 0, 0, sin γ, 2(1 − cos γ), 0, 0

0, 1, 0, 0, 0, 0, 0

0, 0, cos γ, 0, 0, sin γ, 0

3 sin γ, 0, 0, cos γ, 2 sin γ, 0, 0

6(cos γ − 1), 0, 0, −2 sin γ, 4 cos γ − 3, 0, 0

0, 0, − sin γ, 0, 0, cos γ, 0

6(γ − sin γ), 0, 0, 2(1 − cos γ), 3γ − 4 sin γ, 0, 1

§7.4 The circular case 95

The angular displacement γ relates to the normalized time tthrough the linear equation:

γ = t − t0

because the normalized mean motion equals unity.From the state-transition matrix it is possible to derive

in closed form the asynchronous relative state vector, namelyδxasyn(γ) = Ξc(γ) δx0:

xasyn(γ) = 2(2x0 + y0) − (3x0 + 2y0) cos γ + x0 sin γ (7.43)yasyn(γ) = y0 (7.44)zasyn(γ) = z0 cos γ + z0 sin γ (7.45)

xasyn(γ) = x0 cos γ + (3x0 + 2y0) sin γ (7.46)yasyn(γ) = −3(2x0 + y0) + 2(3x0 + 2y0) cos γ − 2x0 sin γ

(7.47)zasyn(γ) = z0 cos γ − z0 sin γ (7.48)

This system of equations is the asynchronous equivalent to theCW solution. The time delay becomes:

δt = 2x0 + 3(2x0 + y0)γ − 2x0 cos γ − 2(3x0 + 2y0) sin γ

The secular term vanishes when y0 = −2x0 (recall that variablesare nondimensional). This is in fact the condition for coorbitalmotion from the CW solution (Alfriend et al., 2009, p. 86).

Following the diagram in Fig. 7.3, a first-order correction ofthe time delay will transform Eqs. (7.43–7.48) into the true CWsolution. In the circular case, the linear corrections to the positionand velocity, given in Eqs. (7.16–7.17), reduce to

δr = δrasyn−δt j, and δr = δrasyn+δt i−k×(−δt j) = δrasyn

The term k× (−δt j) is the correction to the inertia terms becauseof defining the velocity in the rotating frame. This factor cancelsthe first term in the correction of the time delay. Hence, the asyn-chronous velocity coincides with the synchronous velocity. Theonly component of the relative position vector that is affected bythe correction is the along-track component. The equations cor-rected up to first order become:

x(γ) = 2(2x0 + y0) − (3x0 + 2y0) cos γ + x0 sin γy(γ) = (y0 − 2x0) − 3(2x0 + y0)γ + 2x0 cos γ

+ 2(3x0 + 2y0) sin γz(γ) = z0 cos γ + z0 sin γ

x(γ) = x0 cos γ + (3x0 + 2y0) sin γy(γ) = −3(2x0 + y0) + 2(3x0 + 2y0) cos γ − 2x0 sin γz(γ) = z0 cos γ − z0 sin γ

These equations are, indeed, the solution to relative motionabout a circular reference orbit already provided by Clohessy andWiltshire (1960).

7.4.1 Second-order correction

The CW solution can be improved using the second-order cor-rection of the time delay introduced in Sect. 7.2.4. According to

Eqs. (7.34–7.35), the corrected state vector becomes:

δr⋆ = δr − δt2(2δrasyn + 2k × δrasyn − δt i

)δr⋆ = δr + δt

[(k ⊗ k) − 3(i ⊗ i)

]δrasyn + k × δrasyn

Expanding these expressions yields the explicit form of the im-proved solution. The in-plane relativemotion innormalized vari-ables is given by:

x⋆(γ) = 2A1 +12

A4A5 + (x0 − A4A2) sin γ − (A2 + A4 x0) cos γ(7.49)

y⋆(γ) = A8A5 + y0 + (x0A5 − 2A2A8) sin γ + A−6 sin 2γ− (2x0A8 + A2A5) cos γ + 2x0A2 cos 2γ (7.50)

x⋆(γ) = −3A1A5 + (A2A7 − x0A5) sin γ + (x0A7 + A2A5) cos γ− A−6 sin 2γ − 2x0A2 cos 2γ (7.51)

y⋆(γ) = −A+6 − 3A1 + (A2A5 − 2x0) sin γ + (2A2 + x0A5) cos γ− 2x0A2 sin 2γ + A−6 cos 2γ (7.52)

which has been written in terms of the following coefficients:

A1 = (2x0 + y0), A2 = (3x0 + 2y0), A3 = (y0 − x0)

A4 = (2A3 − 3A1γ), A5 = (2x0 + 3A1γ), A±6 = (A22 ± x2

0)A7 = 1 + 6A1, A8 = A1 − 1

The +/− sign is chosen according to the sign that appears inEqs. (7.49–7.52).

The out-of-plane motion is governed by the equations:

z⋆(γ) = −B−4 + B1 sin γ − B2 cos γ + B−3 sin 2γ + B+4 cos 2γ(7.53)

z⋆(γ) = −B+3 + B2 sin γ + B1 cos γ − B+4 sin 2γ + B−3 cos 2γ(7.54)

where

B1 = (z0A5 + z0), B2 = (z0A5 − z0)B±3 = (z0A2 ± z0 x0), B±4 = (z0A2 ± x0z0)

The time delay is the only source of secular terms. The conditionfor closing the relative orbits when using the corrected solution isstill the same condition that applies to the original CW solution,y0 = −2x0. For A1 = 0 the corrected solution is periodic, justlike the CW solution. The out-of-plane motion is now coupledwith the in-plane motion.

The expressions provided in this section introduce nonlinearterms to the CW solution in a simple way, which does not involvethe second derivatives of the relative state vector. Nevertheless, itshould be noted that Eqs. (7.49–7.54) are neither a solution tothe second or third-order equations of circular relative motion.The third-order equations are (see, for example, Richardson andMitchell, 2003):

x − 3x − 2y +32

(2x2 − y2 − z2) − 2x(2x2 − 3y2 − 3z2) = 0

y + 2x − 3xy +3y2

(4x2 − y2 − z2) = 0

z + z − 3xz +3z2

(4x2 − y2 − z2) = 0

96 7 The theory of asynchronous relative motion

(a) a = 6900 km, ||δr0 || = 137.5m, ||δr0 || = 1.1m/s

(b) a = 8500 km, ||δr0 || = 65.6m, ||δr0 || = 0.1m/s

Figure 7.5: Error comparison between the improved CW solution

(``CW(⋆)''), and the exact solution to the first (``1st-Ord''), the second

(``2nd-Ord''), and the third-order (``3rd-Ord'') equations of circular

relativemotion.

Figure 7.5 compares the accuracy of the improved CW so-lution —Eqs. (7.49–7.54)— against the numerical integrationof the first, second, and third-order equations of circular relativemotion. Note that the solution to the first-order equations is sim-ply the CW solution. The error is measured as the norm of thedifference with respect to the integration of the exact nonlinearequations of relative motion. It is observed that the solutions tothe second and third-order equations are more accurate at first,but they diverge from the exact solution after several revolutions.The theory of asynchronous relative motion will eventually pro-vide a more accurate description of the dynamics after sufficientrevolutions. No divergence in the corrected solution has been ob-served. The initial conditions are small to ensure that the linearapproach is valid.

7.5 Numerical evaluation

This section evaluates the performance of the new theory of asyn-chronous relativemotionwith a series of examples. First, we con-sider a circular and an elliptic Keplerian orbit. Next, we pro-ceed to the perturbed case and propagate the relative dynamicsof spacecraft orbiting the Earth and Jupiter.

The error in determining the relative position at time j, ϵ j, ismeasured as:

ϵ j = ||δr j − δrref||

whereδrref is the reference solution. To compute this solution theorbits of the leader and the follower are solved separately, thenprojected in the rotating Euler-Hill frame and subtracted. Thesame definition of error applies to the relative velocity, δr.

7.5.1 Keplerian motion

The reference orbits for the two examples of Keplerian orbits aredefined inTable 7.1. They are propagated for 15 revolutions. Thefirst example refers to a circular reference orbit, whereas the sec-ond corresponds to elliptic relative motion. The initial relativestate vector is defined in the rotating reference.

Table 7.1: Definition of the reference orbits and relative initial condi-

tions in theL frame.

Id. a [km] e i [deg] ω [deg] Ω [deg] ϑ0 [deg]

#1 8500 0.0 0 10 20 0

#2 30000 0.6 70 20 60 50

x [m] y [m] z [m] x [m/s] y [m/s] z [m/s]

#1 -30 -30 50 0.0 0.1 0.0

#2 -60 -100 1000 0.2 0.2 -1.0

The linear solution to relative motion is computed using theClohessy-Wiltshire solution in the circular case, the Yamanaka-Ankersen state-transition matrix in the elliptic case, and the vari-ational solution using equinoctial elements (see Sect. 7.3.2) is ap-plied to both cases. All these approaches yield the same exactresults. Then, the accuracy of these linear solutions will be im-proved using the second-order correction of the time delay de-scribed in Sect. 7.2.4. Finally, the solution in curvilinear coordi-nates provided byGim andAlfriend (2005) is also included in thecomparison.

Figure 7.6 presents the evolution of the propagation errorsboth in position and velocity. Figure 7.6(a) corresponds to thecircular case. The nonlinear correction improves the accuracy inthe relative position by more than one order of magnitude after15 revolutions, and in the relative velocity by almost three ordersof magnitude. It is interesting to note that the curvilinear ap-proach exhibits similar advantages when propagating the relativeposition, but when dealing with the velocity it is clearly outper-formed by the new theory. In the elliptic case (Fig. 7.6(b)) theuse of curvilinear coordinates does not improve the accuracy ofthe linear solution noticeably, whereas applying the new nonlin-ear corrections reduces the error by over one order of magnitudeboth in position and velocity. The exact values of the final errorscan be compared in Table 7.2. The numbers in the table corre-spond to the errors divided by themagnitude of the relative sepa-ration and relative velocity, computed with the exact solution tothe problem. In this way, it is easy to evaluate the impact of theerror in each case and, more importantly, the significance of theimprovements in accuracy achievedwith the nonlinear correctionof the solutions. All linear approaches yield the same result.

§7.5 Numerical evaluation 97

Table 7.2: Final error in position and velocity relative to themagni-

tude of the relative separation and velocity between the spacecraft.

Cases 1 and 2. The star⋆ denotes the improved solutions.

Case 1 Case 2

ϵ f /||δr f || ϵ f /||δr f || ϵ f /||δr f || ϵ f /||δr f ||

CW 1.07× 10-3 1.04× 10-3 - -

CW (⋆) 2.85× 10-5 2.28× 10-6 - -

YA 1.07× 10-3 1.04× 10-3 4.12× 10-2 6.80× 10-2YA (⋆) 2.85× 10-5 2.28× 10-6 1.63× 10-3 5.02× 10-3

Eq. 1.07× 10-3 1.04× 10-3 4.12× 10-2 6.80× 10-2Eq. (⋆) 2.85× 10-5 2.28× 10-6 1.63× 10-3 5.02× 10-3

Curv. 3.08× 10-5 2.85× 10-3 3.96× 10-2 7.22× 10-2

7.5.2 Perturbed motion

Having discussed the accuracy of the method in the Kepleriancase, we now consider the effect of generic perturbations on therelative dynamics. The synchronous and the asynchronous Jaco-bian matrices are propagated numerically together with the timedelay: the synchronous solution is computed by direct propaga-tion of the variational equations of orbital motion in Cartesiancoordinates; next, integrating the variational equations attachedto the equinoctial elements, using γ = λ− λ0 as the independentvariable, renders the asynchronous solution and the time delay(see Chap. 6 for details on the implementation). In these exam-ples, we will focus on the difference between the purely linear so-lution and the one improved with the second-order correction ofthe time delay. All computations are carried out with PERFORM.

The first example (Case 3) corresponds to the motion of a

spacecraft around the Earth integrated for 12 h (roughly sevenrevolutions). The model includes luni-solar perturbations, anonuniform Earth gravity field, atmospheric drag, and solar ra-diation pressure. The gravitational attraction from the Sun andthe Moon are computed using the DE431 ephemeris solution.The Earth gravity field is reduced to a 4 × 4 grid using theGGM03S model. For such a short propagation the nutationand precesion of the Earth is ignored and its orientation is com-puted with the simplified model fromArchinal et al. (2011). Theleader and follower spacecraft are assigned an area-to-mass ratioof 1/500 m2/kg, a drag coefficient cD = 2.0, and the atmo-spheric density is modeled with the MSISE90 standard. The re-flectivity coefficient for the two spacecraft is the same and equalto cR = 1.2. The values of the osculating elements at the initialepoch (MJD 55198 ET) can be found in Table 7.3.

Table 7.3: Definition of the reference orbits and relative initial

conditions in theL frame. The reference orbits are defined in the

ICRF/J2000 system, using the Earthmean equator and equinox as

the reference plane.

Id. a [km] e i [deg] ω [deg] Ω [deg] ϑ0 [deg]

#3 7400 0.100 30.00 90.00 0.00 0.00

#4 673048 0.222 25.55 114.53 -0.03 0.60

x [m] y [m] z [m] x [m/s] y [m/s] z [m/s]

#3 0 0 0 1 1 1

#4 500000 -10000 0 0 10 0

Figure 7.7(a) displays the error in the propagation of the rel-ative position and velocity in Case 3. The nonlinear correctionreduces the error-growth rate and, as a result, the error in the fi-

(a) Case 1 (circular) (b) Case 2 (elliptic)

Figure 7.6: Error in the relative position and velocity using the linear solutions (``CW-YA/Eq.''), the improved version of these formulations

(``CW-YA(⋆)/Eq.(⋆)''), and the solution using curvilinear coordinates (``Curv'').

98 7 The theory of asynchronous relative motion

(a) Case 3 (geocentric orbit) (b) Case 4 (Jupiter-Europa system)

Figure 7.7: Propagation error for Cases 3 and 4.

nal position is reduced from 1,561 m down to 181 m. The oscilla-tions in the error profile come from the orbital frequency, as themagnitude of the relative perturbations is small compared to thedifferential gravity. The error in velocity is also reduced by oneorder of magnitude, although the improvements are smaller (inrelative terms) than in the relative position. The reason is that forthe second-order correction in velocity we neglected the contri-bution of the differential perturbations.

The second example (Case 4) corresponds to a formationaround Jupiter propagated for 13.45 days (approximately fourrevolutions). More in particular, the leader spacecraft describesa distant retrograde orbit around Europa. The initial conditionsfor this orbit were first computed in the circular restricted three-body problem, and they were then refined using a Levenberg-Marquardt algorithm in order to minimize the separation inphase space between the initial crossing of the Poincaré sectionand the crossing after four revolutions. The initial values (MJD55255 ET) of the osculating elements with respect to Jupiter arepresented in Table 7.3. The model accounts for the gravitationalperturbations from the Sun, Io, Europa,Ganymede, andCallisto,using their true ephemeris. The accuracy of the propagation isshown in Fig. 7.7(b). The improvements in accuracy introducedby the nonlinear correction are of about one order of magnitudeafter four revolutions. In this case the relative initial conditionsare larger than the ones considered in Case 3, as the relative sep-aration is close to 0.1% of the radius of the reference orbit. Thisexplains the large errors in position observed in this case. Like inCase 3, the improvements in accuracy when propagating the rel-ative velocities are more subtle, because of omitting the differen-tial perturbations in the correction. Table 7.4 presents the exactvalue of the final propagation errors, divided by the magnitudeof the relative separation and velocity at that time. The error inposition for Case 3 with the linear solution is about 10% of therelative separation, and the improved solution reduces the errorto 1.2%. Similarly, in Case 4 the error in position is reduced from13% down to 2.5%.

Table 7.4: Final error in position and velocity relative to themagni-

tude of the relative separation and velocity between the spacecraft.

Cases 3 and 4.

Case 3 Case 4

ϵ f /||δr f || ϵ f /||δr f || ϵ f /||δr f || ϵ f /||δr f ||

Linear 1.02× 10-2 2.17× 10-2 1.33× 10-1 2.46× 10-1Corr. 1.17× 10-3 2.97× 10-3 2.52× 10-2 7.53× 10-2

7.6 Conclusions

The concept of asynchronous relative motion has been definedformally. This theory has two main applications. First, it pro-vides a systematic technique for propagating the partial deriva-tives of any formulation using independent variables differentfrom time. Second, it is easy to introduce nonlinear terms in thelinear solution to improve its accuracy. As a result, the error in thepropagation of the relative orbits between spacecraft is reducedsignificantly.

The method presented in this chapter is not restricted to anyparticular formulation nor perturbation model. The proposedmethod can be applied eventually to any of the existing solutionsto the linear equations of relative motion, and to the motion un-der any perturbation. For example, the corrected form of theClohessy-Wiltshire solution has been provided explicitly. Thecorrected solution reduces the propagation error by several ordersof magnitude. Such improvements may have a positive impacton the design of GNC and filtering algorithms. The fact thatone only needs to propagate the reference trajectory makes themethod appealing for modeling the dynamics of swarms includ-ing hundreds or thousands of spacecraft.

Regularizationmethods introduce time transformationsmo-tivated by the structural stability of orbital motion, trying to es-

§7.6 Conclusions 99

cape from the known Lyapunov instability. Similarly, the theoryof asynchronous relativemotion shows how the secular terms canbe confined to the definition of a time delay, without affecting therest of variables in the so-called asynchronous solution. The timedelay plays a key role in the variational solution and, because ofits physical meaning, the mechanism for introducing nonlineari-ties is simple and does not require the computation of the secondpartial derivatives of the state vector, nor deriving multiple scalesexpansions. The improved equations of motion are not the exactsolution to the second-order equations of motion. The solutionmay be, in fact, more accurate than the second and third-ordersolutions after sufficient revolutions, because it is not affected bythe divergence of these equations.

This research has implications beyond spacecraft relativemo-tion. On the one hand, it shows how to propagate the state-transition matrix (which maps the initial value of the separationin position and velocity up to a given time) when using formu-lations different from the Cartesian coordinates and, more im-portantly, when the physical time is no longer the independentvariable. This result is useful when the variational equations ofmotion need to be integrated numerically. Thanks to this theoryit is now possible to extend the use of regularized formulations,which behave very well for numerical integration, to the problemof propagating the partial derivatives. On the other hand, we pre-sented a generic method for increasing the accuracy of the prop-agation of the variational equations through simple correctionsapplied a posteriori.

“And here there dawned on me the notion that we must admit, insome sense, a fourth dimension of space for the purpose of calculatingwith triples…An electric circuit seemed to close, and a spark flashedforth.”

—Sir William Rowan Hamilton

8Regularization in relative motion

B y definition, the relative motion between two particles isthe difference between their respective absolute motions.When their relative separation is small this difference is

linearized and the relative state vector is computed directly with-out the need for solving the dynamics of the two particles inde-pendently. The outcome of the linearization process depends onthe parameterization of the dynamics, i.e. on the variables chosento write the analytic solution to the problem.

Cartesian coordinates are the most intuitive variables. Thisform of the equations of relative motion dates back to Laplace(1799, book II, chap. II, §14). But the problem can be formu-lated using any set of variables or elements. Finding the mostadequate representation of the equations of motion for a givenapplication may provide a deep insight into the dynamics of theproblem, simplify its analysis, and even allow to obtain solutionsthat would be intractable otherwise. In Chap. 7 we have alreadypresented a more detailed survey of existing solutions.

Carter (1990) found the connectionbetween the equations inLawden’s primer vector theory and the linearization by De Vries.He recovered the integral (Lawden, 1963, pag. 85, eq. 5.51)

I(ϑ) =∫ ϑ

ϑ0

dχ(1 + e cos χ)2 sin2 χ

(8.1)

and succeeded in solving it in closed-form by replacing the trueanomaly ϑ by the eccentric anomaly. Yamanaka and Ankersen(2002) advanced on Carter’s work and arrived to a simplifiedstate-transition matrix for solving the elliptic rendezvous prob-lem. The resulting state-transition matrix is explicit both in timeand true anomaly. Because of how compact the solution is, andthe fact that it is valid for moderate eccentricities, this method

has been applied in many practical scenarios. Alfriend et al.(2009, chap. 5) presented a detailed overview of a number ofstate-transition matrices that can be found in the literature, andCasotto (2014) referred to an interesting classification of them.

The formulations discussed in the previous lines and inChap. 7 become singular as the eccentricity goes to one. Not onlyare they not valid for describing relative motion about parabolicor hyperbolic orbits, but the performance may be affected whenthe elliptic orbit is quasi-parabolic. Little attention has been paidto this particular issue. Carter (1990) introduced the eccentricanomaly for solving Lawden’s integral in the elliptic case, mean-ing that the result will only be valid for elliptic orbits. He ad-vanced on this result and was able to find the solution to Eq. (8.1)for the case of open orbits, although I(ϑ) takes different valuesdepending on the eccentricity of the orbit.

There is a renewed interest in relative motion about hyper-bolic orbits. Missions designed to fly by a certain asteroid, comet,or planet and to deploy a landing probe may fall in this category.The concept of the Aldrin cycler (Byrnes et al., 1993) is a goodexample. Landau and Longuski (2007) proposed a solution tohyperbolic rendezvous based on impulsive maneuvers and geo-metric constructions. Carter (1996) analyzed optimal impulsivestrategies to rendezvous with highly eccentric orbits.

An important part of the theory of regularization deals pre-cisely with finding suitable parameterizations of orbital motion.Based on this theory, we aim for a fully regular and universal so-lution to the relative dynamics. First, regular means that thereare no singularities; in this scenario the typical singularity corre-sponds to e → 1, and not so much to r → 0. Second, a solu-tion is said to be universal if there is a unique form of the equa-

101

102 8 Regularization in relative motion

tions that is valid for any type of reference orbit (circular, elliptic,parabolic, and hyperbolic). Section 2.4.3 already presented someuniversal solutions to Kepler’s problem using regularization. Ev-erhart and Pitkin (1983) explained in detail how introducing theStumpff functions leads to universal solutions to the two-bodyproblem, nomatter the eccentricity of the reference orbit. Danby(1987) discussed exhaustively the role of the Stumpff functions insolving Kepler’s equation. Folta and Quinn (1998) investigatedthe use of universal variables in the problem of relative motion.

In the present chapter we will solve the problem of linearKeplerian relative motion using the Dromo formulation, theKustaanheimo-Stiefel (KS) transformation, and Sperling-Burdet(SB) regularization. The goal is to provide formulations that arecompletely free of singularities, and valid for any type of refer-ence orbit. Roa and Peláez (2014, 2015b, 2016d) already exploredthe potential of regularization in relativemotion. The KS and SBregularizations use the same time transformation, and the solu-tion to relativemotion admits a compact tensorial representation.In order to simplify the derivations, the solution is built usingthe theory of asynchronous relativemotion (see Chap. 7 andRoaet al., 2015a). Thanks to this theory, the linear solution can alsobe corrected to account for nonlinear terms. This correction de-pends strongly on the form of the time delay, and consequentlyon the definition of the time transformation. Thus, it is inter-esting to compare the performance of the Dromo-based solution(based on dt/ds = r2/h) with the performance of the KS and SBsolutions (with dt/ds = r).

Section 8.1 solves the problem of relative motion using theDromo elements. Next, Sects. 8.2 and 8.3 present the solutionusing the SB and KS variables. The orbits are assumed Keple-rian. Practical comments about the fictitious time can be found

in Sect. 8.4. Numerical examples showing the accuracy of themethods and the impact of the nonlinear correction appear inSect. 8.5. An additional result obtained by applying the theoryof asynchronous relative motion is a generic method for propa-gating numerically the variational equations of formulations us-ing an independent variable different from time. This method ispresented in Sect. 8.6, and admits any kind of orbital perturba-tions.

8.1 Relative motion in Dromo variables

Let q = [ζ1, ζ2, ζ3, η1, η2, η3, η4, σ0]⊤ denote the set of Dromoelements. The orbit of the follower spacecraft can be constructedin terms of a set of differences on the elements defining the leaderorbit, q f = qℓ + δq. Assuming that the difference between theelements is small (or equivalently that the relative separation issmall), the first-order asynchronous solution is given by the linearequation

δxasyn = Dasyn(σ) δq (8.2)

Here δx = [δx, δy, δz, δvx, δvy, δvz]⊤ is the relative state vectorfrom the perspective of the inertial frame. In this chapter no per-turbations are considered, meaning that vector δq is constant.The velocity in the rotating Euler-Hill frameL, δr, follows from

δr = δv − ωLI × δr

The matrix Dasyn(σ) contains the partial derivatives of the statevector with respect to Dromo variables, and reduces to (seeSect. 4.4):

Dasyn =

− cosσζ2

3 s2− sinσζ2

3 s2− 2ζ3

3 s0 0 0 0

uζ2

3 s2

0 0 0 +2η2

ζ23 s

−2η1

ζ23 s

+2η4

ζ23 s

−2η3

ζ23 s

1ζ2

3 s

0 0 0 +2N+34

ζ23 s

−2N−43

ζ23 s

−2N+12

ζ23 s

+2N−21

ζ23 s

0

ζ3 sinσ −ζ3 cosσ u −2ζ3sη2 +2ζ3sη1 −2ζ3sη4 +2ζ3sη3 −ζ3

ζ3 cosσ +ζ3 sinσ s +2ζ3uη2 −2ζ3uη1 +2ζ3uη4 −2ζ3uη3 0

0 0 0 2ζ3M+34 −2ζ3M−43 −2ζ3M+12 2ζ3M−21 0

We have introduced the auxiliary terms M±i j = uN±i j ± sN∓ji andN±i j = ηi cosσ ± η j sinσ to simplify the notation. Recall thats = 1 + ζ1 cosσ + ζ2 sinσ and u = ζ1 sinσ − ζ2 cosσ.

The elements (η1, η2, η3, η4) are the components of a unitquaternion, |n| = 1. Thus, the set of differences δηi are not inde-pendent as they must satisfy the constraint (see Sect. 4.5)

η1 δη1 + η2 δη2 + η3 δη3 + η4 δη4 = 0 (8.3)

This means that one of the four components δηi cannot be de-

cided at convenience. For example, choosing η4 as the dependentvariable δη4 takes the form

δη4 = −1η4

(η1 δη1 + η2 δη2 + η3 δη3)

If η4 = 0, then another component should be chosen as the de-pendent variable.

The physical time is obtained by integrating the Sundman

§8.2 Relative motion in Sperling-Burdet variables 103

transformation in Eq. (4.8),

t = T (σ; q) ≡ t0 +∫ σ

σ0

1ζ3

3 (1 + ζ1 cos χ + ζ2 sin χ)2dχ

Differentiating this equation with respect to each Dromo ele-ment while keepingσ constant yields the time delay

δt = I1 δζ1 + I2 δζ2 + I3 δζ3 + I4 δσ0

defined in terms of

I1 =∂T∂ζ1

∣∣∣∣∣∣σ

=

∫ σ

σ0

2 cos χζ3

3 (1 + ζ1 cosχ + ζ2 sinχ)3dχ

I2 =∂T∂ζ2

∣∣∣∣∣∣σ

=

∫ σ

σ0

2 sin χζ3

3 (1 + ζ1 cosχ + ζ2 sinχ)3dχ

I3 =∂T∂ζ3

∣∣∣∣∣∣σ

=

∫ σ

σ0

3ζ4

3 (1 + ζ1 cosχ + ζ2 sinχ)2dχ

I4 =∂T∂σ0

∣∣∣∣∣∣σ

=s2

0 − s2

ζ33 s2

0s2

In the Keplerian case the eccentricity vector is constant so ζ1 ≡ e,ζ2 = 0, and δζ2 = 0. Therefore

δt = I1 δζ1 + I3 δζ3 + I4 δσ0 (8.4)

Matrix D is valid for any type of orbit and it is regular, except forthe case when the angular momentum vanishes or there is a col-lision. A dedicated analysis of the singularities in Dromo can befound in Sect. 4.6. Although these limitations exist, in practicethe formation flight scenarios are far from the singularities. Theformulation is universal in formalthough the explicit solutions tothe integrals Ii are different depending on the type of orbit. It isworth noticing that a similar difficultywas encountered byCarter(1990) when solving Lawden’s integral. In fact, in the elliptic andhyperbolic cases it is

I1 = −3 ζ1(t − t0)

1 − ζ21

+1

ζ33 (1 − ζ2

1 )

(1 + s

s2 sinσ − 1 + s0

s20

sinσ0

)I3 =

3ζ3

(t − t0)

whereas in the parabolic case the integrals reduce to

I1 =1

10ζ33

[5(tan

σ

2− tan

σ0

2

)− tan5 σ

2+ tan5 σ0

2

]I3 =

12ζ4

3

[3(tan

σ

2− tan

σ0

2

)+ tan3 σ

2− tan3 σ0

2

]under the assumption ζ2 = 0.

The transformation from the relativeDromo elements to therelative state vector needs to be inverted at departure in order tocompute δq from the initial conditions δx0, defining the linearmap

δq = Q(σ0) δx0

CombiningmatrixQ andmatrixDasyn provides the asynchronousstate transition matrix

δx(σ) = Ξasyn(σ,σ0) δx0

withΞasyn(σ,σ0) = Dasyn(σ)Q(σ0). Details on the inverse trans-formation can be found in Appendix D. The orbit is defined inDromo variables in terms of the eccentricity vector. When theorbit is circular the eccentricity vector vanishes and the Dromodifferential elements δq cannot be initialized properly. This sin-gularity in the transformation translates into matrix Q becomingsingular for circular reference orbits.

Summary

When using the Dromo formulation the solution is obtained ac-cording to the following steps:

1. Use matrix Q (defined in Appendix D) to transform theinitial relative state vector to the relative Dromo elements,δq = Q δx0. The velocity needs to be referred to the inertialframe, δv0 = δr0 + ωLI,0 × δr0, withωLI,0 = h/r2

0 .

2. Propagate the state vector using Eq. (8.2) and matrix Dasyn.

3. Solve the time delay from Eq. (8.4).

4. Recover the synchronism through the first-order correctiondefined in Eqs. (7.32–7.33).

5. (Optional) If more accuracy is needed, the linear solutioncan be refined by introducing nonlinear terms according toEqs. (7.34–7.35). They give the improved solution δr⋆ andδv⋆.

8.2 Relative motion in Sperling-Burdetvariables

The explicit solution to Kepler’s problem via the SB regulariza-tion is given in Eq. (2.19). Assuming µ = 1, introducing the aux-iliary vector

d = −(ω2r0 + e) (8.5)

and considering the modified argument z = ω2s2, the dynamicsof the leader spacecraft are governed by the equations

rℓ(s) = rℓ,0 + s rℓ,0C1(z)vℓ,0 + s2C2(z) dℓ (8.6)

r′ℓ(s) = rℓ,0C0(z)vℓ,0 + sC1(z) dℓ (8.7)

Recall that vℓ = r′ℓ/rℓ. They are defined in terms of the initialconditions rℓ,0 and vℓ,0. The radial distance evolves like

r(s) = r0C0(z) + r′0sC1(z) + s2C2(z) (8.8)

Similarly, Eq. (2.20) defines the universal form of Kepler’s equa-tion:

t = T (t; r, r′) ≡ t0 + sr0C1 + s2r′0C2 + s3C3 (8.9)

104 8 Regularization in relative motion

The solution to the problem of relative motion is obtainedby computing how vectors rℓ and vℓ change given a set of differ-ences in the initial conditions δr0 and δv0. The relative velocitywill be solved from the fictitious velocity r′ℓ. In what remains ofthe chapter all variables are referred to the leader spacecraft; toalleviate the notation the subscript ℓ will be omitted. The asyn-chronous solution reads

δxasyn = Ξasyn(s) δx0

Here Ξasyn(s) denotes the asynchronous state transition matrix.It can be written in blocks as:

Ξasyn(s) =

∇r0

∣∣∣s r, ∇v0

∣∣∣s r

∇r0

∣∣∣s v, ∇v0

∣∣∣s v

(8.10)

The blocks correspond to the partial derivatives of the positionand velocity vectors with respect to the initial conditions, keep-ing constant the fictitious time. Each block is computed as thegradient of the corresponding vector field. The resulting rank-two tensors are:

∇r0

∣∣∣sr = I + sr0

(v0 ⊗ ∇r0

∣∣∣sC1

)+ s2[(d ⊗ ∇r0

∣∣∣sC2

)+ C2∇r0

∣∣∣sd

](8.11)

∇v0

∣∣∣sr = sr0

[C1I +

(v0 ⊗ ∇v0

∣∣∣sC1

)]+ s2[(d ⊗ ∇v0

∣∣∣sC2

)+ C2∇v0

∣∣∣sd

](8.12)

∇r0

∣∣∣sv =

1r2

[r∇r0

∣∣∣sr′ − (r′ ⊗ ∇r0

∣∣∣sr)

](8.13)

∇v0

∣∣∣sv =

1r2

[r∇v0

∣∣∣sr′ − (r′ ⊗ ∇v0

∣∣∣sr)

](8.14)

The gradients of the Stumpff functions are obtained attending tothe derivation rules (C.3–C.4) established in Appendix C:*

∇r0

∣∣∣sCk =

(∂ωCk

)∇r0

∣∣∣sω =

s2

r30

C∗k+2r0

∇v0

∣∣∣sCk =

(∂ωCk

)∇v0

∣∣∣sω = s2

C∗k+2v0

Equations (8.11–8.12) then become:

∇r0

∣∣∣sr = I +

s3

r20

C∗3(v0 ⊗ r0) +

s4

r30

C∗4(d ⊗ r0)

+ s2C2∇r0

∣∣∣sd (8.15)

∇v0

∣∣∣sr = sr0C1I + s3r0C

∗3(v0 ⊗ v0) + s4

C∗4(d ⊗ v0)

+ s2C2∇v0

∣∣∣sd (8.16)

The gradients of vector d are required. Differentiating Eq. (8.5)renders

∇r0

∣∣∣sd = −ω

2I +2r3

0

(r0 ⊗ r0) − ∇r0

∣∣∣se

∇v0

∣∣∣sd = 2(r0 ⊗ v0) − ∇v0

∣∣∣se

*We abbreviate the notation of partial derivatives according to

∂

∂y≡ ∂y

in order to simplify the notation.

The gradients of the eccentricity vector are obtaineddirectly fromits definition:

∇r0

∣∣∣se =

(v2

0 −1r0

)I +

1r3

0

(r0 ⊗ r0) − v0 ⊗ v0

∇v0

∣∣∣se = 2(r0 ⊗ v0) − v0 ⊗ r0 − r′0I

Recall that r′0 = (r0 ·v0). At this point the solution is completelyreferred to the initial conditions r0 and v0. The gradients of vec-tor d reduce to:

∇r0

∣∣∣sd = −

1r3

0

[r2

0I− (r0⊗r0)]+v0⊗v0, ∇v0

∣∣∣sd = r′0I+v0⊗r0

Considering the dyadic products

d ⊗ r0 =

(r′0v0 −

r0

r0

)⊗ r0 and d ⊗ v0 =

(r′0v0 −

r0

r0

)⊗ v0

Equations (8.15–8.16) transform into:

∇r0

∣∣∣sr = I +

sr4

0

[r2

0(r0C1 + s2C∗3)v0 + r0sC2r0

+ s3C∗4(r0r′0v0 − r0)

] ⊗ r0 − C2s2

r0[I − r0(v0 ⊗ v0)]

∇v0

∣∣∣sr = sr0C1I + s2

C2[r′0I + (v0 ⊗ r0)

]− s4

r0C∗4(r0 − r0r′0v0) ⊗ v0 + s3r0C

∗3(v0 ⊗ v0)

The first two blocks of the state transition matrix Masyn(s) aregiven by the initial conditions and the fictitious time.

The gradients of the velocity vector require the gradients ofboth the fictitious velocity vector r′ and the radial distance r.From the solution to the radial distance r(s) —Eq. (8.8)—, it fol-lows:

∇r0

∣∣∣sr = sC1v0 +

1r3

0

(r2

0C0 + s2r0C1 + s3r′0C∗3 + s4

C∗4)r0

∇v0

∣∣∣sr = sC1r0 + s2(r0C1 + r′0sC∗3 + s2

C∗4)v0

These equations are already referred to r0 and v0 and involve noadditional terms. Computing the partial derivatives of Eq. (8.7)renders

∇r0

∣∣∣sr′ =

1r4

0

[sr0C1r0 + r2

0(r0C0 + s2C1)v0

− s3C∗3(r0 − r0r′0v0)

] ⊗ r0 +sr0C1

[r0(v0 ⊗ v0) − I

]∇v0

∣∣∣sr′ = r0C0I + sC1

[r′0I + v0 ⊗ (r0 + r0sv0)

]− s3

r0C∗3(r0 − r0r′0v0) ⊗ v0

These results complete the gradients of the velocity. The four

§8.3 Relative motion in Kustaanheimo-Stiefel variables 105

blocks in matrixMasyn, given in Eqs. (8.11–8.14), take the form:

∇r0

∣∣∣sr = I +

sr4

0

[r0(r0a+13 + s3r′0C

∗4)v0 + sa−24r0

] ⊗ r0

− C2s2

r0[I − r0(v0 ⊗ v0)] (8.17)

∇v0

∣∣∣sr = sb+12I + s2

C2(v0 ⊗ r0) − s4

r0C∗4(r0 − r0r′0v0) ⊗ v0

+ s3r0C∗3(v0 ⊗ v0) (8.18)

r2∇r0

∣∣∣sv = −

srr0C1I +

s2

r0

(C

21r0 + r0sC1C2v0

) ⊗ v0

= +sr4

0

[r0b+01C1 + s2a+14C1 + a+02a−13

]r0

+ r0s[r2

0C0C2 + s2(b+13C2 − b+01C∗4)]v0

⊗ r0 (8.19)

r2∇v0

∣∣∣sv = rb+01I +

s2

r0(C2

1r0 + r0sC1C2v0) ⊗ r0 +s3

r0

[(a+14C1

− a+02C∗3)r0 + r0s(b+13C2 − b+01C

∗4)v0

] ⊗ v0 (8.20)

having introduced the auxiliary terms:

a±i j = r0Ci ± s2C∗j and b±i j = r0Ci ± sr′0C

∗j

Recall that r, r′, r and v refer to the leader spacecraft.The physical time is given explicitly by Eq. (8.9), the general-

ized formofKepler’s equation. The gradients of the timewith re-spect to the initial state vector are obtained by deriving this equa-tion, and result in

∇r0

∣∣∣sT =

sr3

0

(r0a+13 + s3r′0C

∗4 + s4

C∗5)r0 + s2

C2v0

∇v0

∣∣∣sT = s2

C2r0 + s3(r0C∗3 + sC∗4r′0 + s2

C∗5)v0

The time delay δt reduces to

δt = (∇r0

∣∣∣sT · δr0) + (∇v0

∣∣∣sT · δv0) (8.21)

and it is completely determined by the equations given above.Having derived the asynchronous state transition matrix and thetime delay, the solution is complete.

Summary

The method presented in this section is summarized as follows:

1. Given the initial relative state vector δr0 and δv0, computethe gradients of the state vector, defined in Eqs. (8.17–8.20).The velocity needs to be referred to the inertial frame, δv0 =

δr0 + ωLI,0 × δr0, withωLI,0 = h/r20 .

2. Build the asynchronous state transition matrix Ξasyn(s) de-fined in Eq. (8.10) and find the asynchronous relative statevector, δxasyn = Ξasyn(s)δx0.

3. Solve the time delay from Eq. (8.21).

4. Recover the synchronism through the first-order correctiondefined in Eqs. (7.32–7.33).

5. (Optional) If more accuracy is needed, the linear solutioncan be refined by introducing nonlinear terms according toEqs. (7.34–7.35). They give the improved solution δr⋆ andδv⋆.

8.3 Relative motion in Kustaanheimo-Stiefel variables

In KS language the position and velocity vectors are written interms of the coordinates u and u′ in R4. Combining [u⊤,u′⊤]into vector y, the relative state vector δx can be defined in termsof δy thanks to

δxasyn(s) = Masyn(s) δy(s) (8.22)

The relative state vector in KS space δy evolves in fictitious time,meaning that

δy(s) = Nasyn(s) δy0 (8.23)

Finally, the vector δy0 relates to the initial conditions bymeans of

δy0 = T(x0) δx0

Thematrix T(x0) is solved from the linear formof theHopf fibra-tion in Appendix D. It connects the relative state vector in Carte-sian space with the relative state vector in KS space.

Differentiating Eq. (3.6), r = L(u) u, provides the relativeposition vector:*

δr = δL(u) u + L(u) δu = L(δu) u + L(u) δu = 2L(u) δu (8.24)

The fact that the fourth component of δr is zero proves that thebilinear relation ℓ(u, δu) = 0 holds, whichmotivates the last sim-plification. Equation (8.24) defines only the relative position vec-tor. The relative velocity results in

δv =2r[L(u′) δu + L(u) δu′

] − 4r2 (u · δu) L(u) u′ (8.25)

Equations (8.24) and (8.25) canbewritten inmatrix form inorderto defineMasyn,

Masyn(s) =

2L(u), 04

2rL(u′),

2r2

rL(u) − 2

[L(u) u′ ⊗ u]

(8.26)

which has already been introduced in Eq. (8.22). Matrix 04 is thezero-matrix of dimension four.

Next, we focus on matrix Nasyn(s) —Eq. (8.23)—. In the Ke-plerian case Eq. (3.31) can be solved explicitly to provide

u(s) = u0D0(z)+ s u′0D1(z) (8.27)

u′(s) = u′0D0(z) − ψ2s u0D1(z) (8.28)

with ψ2 = −E/2 and having introduced the modified Stumpfffunctions Dk(z) = Ck(z/4) ≡ Dk. This is the solution to Ke-pler’s problem in KS variables. The Ck and Dk functions relate

*We have relaxed the notation by denoting r and v the extensions to R4 ofthe position and velocity vectors, [x, y, z, 0]⊤ and [vx, vy, vz, 0]⊤ respectively.

106 8 Regularization in relative motion

through the half-angle formulas, given in Appendix C. Comput-ing the partial derivatives of these equations yields

Nasyn(s) =

∇u0

∣∣∣su, ∇u′0

∣∣∣su

∇u0

∣∣∣su′, ∇u′0

∣∣∣su′

(8.29)

Kriz (1978) also relied on the partial derivatives of u and u′with respect to u0 and u′0 when solving the perturbed two-pointboundary value problemusing theKS transformation. In the fol-lowing lines a new derivation is presented in tensorial form. SeeShefer (2007) and the references therein for an overview of someworks on this topic that appeared in the Russian literature.

Applying the gradients ∇u0 ,∇u′0 : R4 → R4 × R4 to Eqs.(8.27–8.28) yields:

∇u0

∣∣∣su = −ψ

2sD0 I4 + u0 ⊗ ∇u0

∣∣∣sD0 + ψ

2s(u′0 ⊗ ∇u0

∣∣∣sD1)

∇u′0

∣∣∣su = −ψ

2sD1 I4 + u0 ⊗ ∇u′0

∣∣∣sD0 + ψ

2s(u′0 ⊗ ∇u′0

∣∣∣sD1)

∇u0

∣∣∣su′ = −ψ2sD1 I4 + u′0 ⊗ ∇u0

∣∣∣sD0 − ψ2s(u0 ⊗ ∇u0

∣∣∣sD1)

∇u′0

∣∣∣su′ = −ψ2sD0 I4 + u′0 ⊗ ∇u′0

∣∣∣sD0 − ψ2s(u0 ⊗ ∇u′0

∣∣∣sD1)

From the properties of the Stumpff functions it follows that:

∇u0

∣∣∣sD0 = ∂ψD0 ∇u0

∣∣∣sψ = s2 ψ

r0D1 u0,

∇u0

∣∣∣sD1 = s3 ψ

r0D∗3 u0

∇u′0

∣∣∣sD0 = ∂ψD0 ∇u′0

∣∣∣sψ =

s2

r0D1 u′0

∇u′0

∣∣∣sD1 =

s3

r0D∗3 u′0

The gradients of vectors u and u′ then become

∇u0

∣∣∣su = D0 I4 + ψ

2 s2

r0

(D1u0 + sD∗3u′0

) ⊗ u0 (8.30)

∇u′0

∣∣∣su = sD1 I4 +

s2

r0

(D1u0 + sD∗3u′0

) ⊗ u′0 (8.31)

∇u0

∣∣∣su′ = +ψ2 s

r0

[(2D1 − ψ2s2

D∗3)u0 + sD1u′0

] ⊗ u0

− ψ2sD1 I4 (8.32)

∇u′0

∣∣∣su′ =

sr0

[(2D1 − ψ2s2

D∗3)u0 + sD1u′0

] ⊗ u′0

+D0 I4 (8.33)

These expressions complete matrix Nasyn according to its defini-tion given in Eq. (8.29).

The universal Kepler equation is written in terms of vectorsu0 and u′0 as

t = T (s; u0,u′0) ≡ s2

r0(1+C1)+2s2 (u0 ·u′0)C2+2s3(u′0 ·u′0)C3

Note that it involves the Stumpff functions Ck and notDk. Thetime delay is determined through

δt = ∇u0

∣∣∣sT · δu0 + ∇u′0

∣∣∣sT · δu

′0 (8.34)

where the gradients of time are

∇u0

∣∣∣sT =

8r0

s3ψ2[s(u0 · u′0)C∗4 + s2(u′0 · u′0)C∗5 +r0

4C∗3]u0

+ s(1 + C1

)u0 + 2s2

C2 u′0

∇u′0

∣∣∣sT =

2s2

r0

4s2[(u′0 · u′0)sC∗5 + (u0 · u′0)C∗4

]u′0

+ r0[u′0(C2 + C

∗3)s + u0C2

]Equation (8.34) is now defined in terms of the initial state vectorin KS variables and the solution is complete.

8.3.1 Summary

The following steps summarize the algorithm for computing therelative state vector from the KS transformation.

1. Transform the initial separation δx0 to the differences δy0using matrix T(x0) (see Appendix D). The velocity needs tobe referred to the inertial frame, δv0 = δr0 + ωLI,0 × δr0,withωLI,0 = h/r2

0 .

2. Use Eqs. (8.30–8.33) to compute the matrix Nasyn.

3. GetmatrixMasyn(s) fromEq. (8.26) (all variables are referredto the leader spacecraft).

4. The asynchronous solution to the problem is:

δxasyn = Masyn(s)Nasyn(s)T(x0) δx0

5. Compute the time delay δt from Eq. (8.34).

6. Recover the synchronism through the first-order correctiondefined in Eqs. (7.32–7.33).

7. (Optional) If more accuracy is needed, the linear solutioncan be refined by introducing nonlinear terms according toEqs. (7.34–7.35). They give the improved solution δr⋆ andδv⋆.

8.4 On the fictitious time

In practice, the solution needs to be propagated across a certaintime interval, t ∈ [t0, t f ]. The fictitious time is initially zero, span-ning across s ∈ [0, s f ]. The final value of the fictitious time, s f ,is solved from the universal Kepler equation:

t f − t0 = s f r0C1(z f ) + s2f r′0C2(z f ) + s3

fC3(z f ) (8.35)

Battin (1999, §4.5) wrote this equation in terms of the universalfunctionsUk(z), which relate to the Stumpff functions by meansof Uk(z) = skCk(z). The universal functions are also referred toas generalized conic functions (Everhart andPitkin, 1983). Specialattention has been paid to the numerical resolution of this formof Kepler’s equation (Burkardt and Danby, 1983; Danby, 1987).

Battin (1999, p. 179) creditedCharlesM.Newman fromMITwithderiving an explicit expression for the final valueof s in termsof

s f = ω2(t f − t0) + r′f − r′0, with r′ = (r · v)

§8.5 Numerical examples 107

This expression is useful when the relative state vector is to becomputed from the position and velocity of the leader spacecraft.Given the initial and final times, the final value of the fictitioustime is solved from Eq. (8.35), which is not affected by the eccen-tricity of the orbit.

The fictitious time can be related to the true anomaly. How-ever, since how the true anomaly relates to the time depends onthe type of orbit, the transformation from the true anomaly tothe fictitious time will also be eccentricity-dependent. But thisdoes not affect the universality of the solution. From the Sund-man transformation it follows

dsdϑ=

dsdt

dtdϑ=

rh

In the limit case e→ 1, this differential equation leads to

s f = h(tan

ϑ

2− tan

ϑ0

2

)Simple expressions can be obtained for the elliptic and hyperboliccases:

e < 1 : s f =√

a(E − E0)

e > 1 : s f =√|a|(H − H0)

in which E and H are the eccentric and hyperbolic anomaly, re-spectively.

8.5 Numerical examples

This section is devoted to testing the performance of the second-order correction of the time delay for the four different types ofreference orbits: circular, elliptic, parabolic, and hyperbolic. Theaccuracy of the proposed formulations is analyzed by comparingthem with the exact solution to the problem. The error at eachstep is measured as:

ε = ||δr − δrref||, ε = ||δr − δrref||

where δrref and δrref are the exact relative position and velocityvectors. The exact solution is constructed by solving the nonlin-ear two-body problem for the leader and the follower, and thensubtracting the absolute state vectors. Vector δr = [δx, δy, δz]⊤

denotes the relative velocity from theperspective of theEuler-Hillreference frame that rotates with the leader spacecraft.

Table 8.1 defines four test cases, giving both the reference or-bit and the initial relative conditions. In the circular and ellip-tic cases the solution is propagated for 15 complete revolutions.The Clohessy-Wiltshire (CW) and Yamanaka-Ankersen (YA) so-lutions will be included as references. In the parabolic and hyper-bolic cases the propagation spans from ϑ0 to −ϑ0. Examples ofequatorial-retrograde and polar orbits are selected to show thatthe KS and SB formulations are not affected by typical singulari-ties such as i = 0 or e = 0, and that the Dromo-based solution isvalid for any noncircular orbit.

Figure 8.1(a) displays the relative orbit and the error in po-sition and velocity for Case 1, an equatorial, retrograde, circularorbit. The solution is computed only with the KS and SB for-mulations, because the Dromo-based solution is singular in this

Table 8.1: Definition of the reference orbits and relative initial condi-

tions in theL frame.

Id. a [km] e i [deg] ω [deg] Ω [deg] ϑ0 [deg]

#1 7500 0.0 180 50 0 -45

#2 9000 0.7 90 100 50 10

#3 ∞ 1.0 20 250 10 -120

#4 -20000 1.4 20 100 60 -130

δx [m] δy [m] δz [m] δx [m/s] δy [m/s] δz [m/s]

#1 -150 50 -200 -0.6 -0.1 -0.1

#2 0 -50 150 0.1 0.1 0.5

#3 -170 160 -20 -1.7 -1.0 0.5

#4 -60 50 100 -0.2 0.2 -0.5

Note: the reference orbit for Case 3 is defined through the angular

momentum, which is h =60000 km2/s

case. The linear solution (“Linear”) obtained with the KS and SBmethods coincides exactly with the CW and YA results, becausethey all are the exact solution to the same problem (linearized rel-ative motion). The improved solution (“KS-SB (⋆)”) adequatelycaptures the nonlinear behavior of the dynamics and improvesthe accuracy of the propagation by one order of magnitude bothin position and velocity. This improved formulation introducesnonlinear terms using the time delay. Consequently, since the KSand SB schemes use the same time transformation, they yield thesame exact accuracy.

The results for Case 2 are presented in Figure 8.1(b). Thiscase is an example of a polar, highly eccentric reference orbit. It ispropagated with the YA solution, and the Dromo, SB, and KSformulations. The linear solution (with the first-order correc-tion) coincides in practice for the four methods. However, thesecond-order correction depends on the time transformation andthe results are slightly different. The Dromo formulation relieson a second-order Sundman transformation, whereas the timetransformation in the SB and KS schemes is of first order. Thediscretization of the orbit is different: the former is equivalent tothe true anomaly, whereas the latter corresponds to the eccentricanomaly. In this particular case, the solution corrected to secondorder in Dromo variables (“Dromo (⋆)”) is slightly more accuratethan the equivalent solution using theKSor the SB (“KS-SB (⋆)”),which coincide exactly. These differences will grow with the ec-centricity of the reference orbit. The second-order correction ofthe time delay improves the accuracy of the propagation by al-most two orders of magnitude in position and one order of mag-nitude in velocity after 15 revolutions.

Figures 8.2(a) and 8.2(b) correspond to parabolic and hyper-bolic reference orbits, respectively. The three formulations pre-sented in this chapter can be applied directly to these type oforbits, although the Dromo-based formulation requires chang-ing the form of the integrals Ii (see p. 103). The results in bothcases are qualitatively similar. It is observed that the error in ve-locity is maximum around perigee no matter the formulation.This is caused by the strong divergence of the dynamics aroundperiapsis. The nonlinear correction partially mitigates this phe-nomenon in the parabolic case, although no clear improvements

108 8 Regularization in relative motion

Table 8.2: Absolute and relative errors in position and velocity for the proposed formulations (final value).

Case 1 Case 2 Case 3 Case 4

ε f [m] ε f [mm/s] ε f [m] ε f [mm/s] ε f [m] ε f [mm/s] ε f [m] ε f [mm/s]

CW 875.46 12.68 − − − − − −YA 875.46 12.68 8224.9 5309.0 − − − −SB 875.46 12.68 8224.9 5309.0 5.22 0.07 231.6 5.28

SB (⋆) 103.78 4.29 221.7 1475.5 4.84 0.07 249.0 5.33

KS 875.46 12.68 8224.9 5309.0 5.22 0.07 231.6 5.28

KS (⋆) 103.78 4.29 221.7 1475.5 4.84 0.07 249.0 5.33

Dromo − − 8224.9 5309.0 5.22 0.07 231.6 5.28

Dromo (⋆) − − 188.9 1542.4 12.35 3.05 238.5 19.73

ε f /||δr f || ε f /||δr f || ε f /||δr f || ε f /||δr f || ε f /||δr f || ε f /||δr f || ε f /||δr f || ε f /||δr f ||

CW 7.68×10-3 2.03×10-2 − − − − − −YA 7.68×10-3 2.03×10-2 2.99×10-2 8.14×10-3 − − − −SB 7.68×10-3 2.03×10-2 2.99×10-2 8.14×10-3 3.43×10-4 4.70×10-4 9.93×10-4 8.89×10-4SB (⋆) 9.11×10-4 6.90×10-3 8.05×10-4 2.26×10-3 3.18×10-4 4.01×10-4 1.07×10-3 8.97×10-4

KS 7.68×10-3 2.03×10-2 2.99×10-2 8.14×10-3 3.43×10-4 4.70×10-4 9.93×10-4 8.89×10-4KS (⋆) 9.11×10-4 6.90×10-3 8.05×10-4 2.26×10-3 3.18×10-4 4.01×10-4 1.07×10-3 8.97×10-4

Dromo − − 2.99×10-2 8.14×10-3 3.43×10-4 4.70×10-4 9.93×10-4 8.89×10-4Dromo (⋆) − − 6.86×10-4 2.36×10-3 8.11×10-4 1.83×10-3 6.17×10-3 2.05×10-3

are observed in the hyperbolic case. In the parabolic case, thesecond-order time transformation used in the Dromo formula-tion exhibits small advantages in the propagation of the veloc-ity, whereas the propagation of the position is less accurate. TheDromo-based solution seems to be less accurate in the case ofopen orbits. The loss of accuracy is attributed to the issues relatedto parabolic and hyperbolic orbits reported in Sect. 4.6.

Table 8.2 summarizes the previous discussions on the accu-racy of the methods. The final errors in position and velocity arepresented, showing both their absolute magnitude and the valuerelative to the final relative separation and velocity. In this way,the real impact of the error on the solution can be quantified, andthe significance of the error reductions when introducing non-linearities is easier to appreciate. The improved solutions are de-noted with a star (⋆). This table clearly shows that the linear so-lutions coincide identically, as they are the exact solution to thesame system of equations. The nonlinear correction, conversely,depends on the time transformation and therefore the accuracyof the KS and SB methods coincides (both use dt/ds = r), but itis different from that of Dromo (which uses dt/ds = r2/h).

8.6 Generic propagation of the varia-tional equations

This section summarizes the procedure for propagating the state-transition matrix using an arbitrary formulation, subject to anysource of perturbations ap. The special set of variables oe relatesto the state vector x by means of an invertible transformation q :

x 7→ oe:

oe(ϕ) = q(t(ϕ); x), x(t) = q−1(ϕ(t); oe)

The linear form of these transformations is given in terms of thematrices Q and Q†:

δoe(ϕ) = Q(t(ϕ); x) δx, δx(t) = Q†(ϕ(t); oe) δoe

HereQ† is the pseudo-inverse ofQ, because this is not necessarilya square matrix. The physical time and the independent variableϕ relate by means of

ϕ = Φ(t; x), t = T (ϕ; oe)

with T ≡ Φ−1. Under this notation the generalized Sundmantransformation reads∂t∂ϕ= T ′(ϕ; oe) (8.36)

The evolution of the set oe(ϕ) obeys the differential equation

∂oe∂ϕ= g(ϕ; oe, ap) (8.37)

This equation is propagated together with Eq. (8.36), which pro-vides the physical time, and from the initial conditions

ϕ = 0 : oe(0) = oe0, t(0) = t0

The solution to this initial value problem determines the evolu-tion of the set oe, namely

oe(ϕ) = G(ϕ; oe0), with G(ϕ; oe0) =∫ ϕ

ϕ0

g(ϕ; oe0) dϕ

§8.6 Generic propagation of the variational equations 109

(a) Case 1: circular reference orbit

(b) Case 2: elliptic reference orbit

Figure 8.1: Relative orbit and propagation error for the circular and elliptic cases.

Following Eq. (7.15), a small separation δoe0 can be propa-gated in time according to

δoe(t) = δoeasyn(ϕ(t)) − ∂oe∂t

δt

This renders the synchronous variation on the set oe at time t.Using matrix notation it is

δoe(t) = M(t, t0) δoe0 (8.38)

inwhichmatrixM(t, t0) can be referred to the asynchronous tran-sition matrix

Masyn(ϕ, ϕ0) = ∇oe0

∣∣∣ϕG

by means of

M(t, t0) = Masyn(ϕ(t), ϕ0) − 1T ′

[g(ϕ; oe, ap) ⊗ ∇oe0 T ]

Considering the transformation between the set oe and thestate vector x the previous expressions can be extended to definethe state-transition matrix Ξ(t, t0). This matrix can be used topropagate the relative state vector

δx(t) = Ξ(t, t0) δx0

Indeed, from Eq. (8.38) it follows

δx(t) = Q†(ϕ(t), oe) δoe(t) =[Q†(ϕ(t), oe)M(t, t0)Q(t0, x0)

]δx0

This identity proves that the state-transitionmatrix is built from

Ξ(t, t0) = Q†Masyn(ϕ(t), ϕ0) − 1

T ′[g(ϕ; oe, ap) ⊗ ∇oe0 T ]

Q0

(8.39)

Matrices Q† and Q contain the partial derivatives of the state vec-tor with respect to oe, and vice-versa. These two matrices can

110 8 Regularization in relative motion

(a) Case 3: parabolic reference orbit

(b) Case 4: hyperbolic reference orbit

Figure 8.2: Relative orbit and propagation error for the parabolic and hyperbolic cases.

be computed analytically. The term T ′ is the right-hand side ofthe Sundman transformation (8.36), and g(ϕ; oe, ap) is the right-hand side of the evolution equations (8.37). If t is considered partof the setoe, as it is the usual practice, then∇oe0 T is simply the rowof Masyn corresponding to the physical time. Similarly, T ′ will beone of the components of g(ϕ; oe, ap). Matrix Masyn is the solu-tion to the initial value problem∂Ξasyn

∂ ϕ= ∇oeg(ϕ; oe, ap)Ξasyn

Ξasyn(0) = I

(8.40)

Here∇oeg(ϕ; oe, ap) is the Jacobianmatrix of the function (8.37).The initial value problem (8.40) is integrated together withEqs. (8.36) and (8.37). Thus, at every integration step the solu-tions oe(ϕ), Ξasyn(ϕ, ϕ0), and t(ϕ) are available. In addition, theright-hand sides g(ϕ; oe, ap) and T ′ are also known. Since ∇oe0 T

is a row ofΞasyn(ϕ, ϕ0), all the terms in Eq. (8.39) are known andthis equation furnishes the state-transition matrix. As shown inSect. 6.1.4, PERFORM can be used for propagating the variationalequations following this procedure.

When propagating the variational equations of a formula-tion different from Cowell’s method, two factors should be con-sidered. First, the initialization of the independent variable. Sec-ond, the correction of the time delay when using a time element.

8.6.1 Initializing the independent variable

The initial value of the new independent variable is given in a spe-cific way, and it may be ϕ0 , 0. For example, if the independentvariable is the true anomaly its initial value is solved from the pro-jections of the position vector onto the perifocal frame.

§8.6 Generic propagation of the variational equations 111

When the initial conditions change, as it is the case when thevariational formulation is integrated, then the initial value of theindependent variable will change too. The gradient of the set oedoes not account for the change inϕ0,meaning that in the currentsetup this informationwill be lost and the differentiationwill fail.To solve this issue, the formulation must be adapted so that theindependent variable is always zero at departure. This is achievedby introducing a modified independent variable, ϕ∗ = ϕ − ϕ0.There are two systematic ways of modifying the formulation ac-cordingly (Roa and Peláez, 2015a):

1. To attach ϕ0 to the vector of elements or coordinates oe. Itremains constant throughout the integration process, andthe original value of the independent variable is recoveredsimply by means of ϕ = ϕ∗ + ϕ0. This approach requireslittle modification of the equations ofmotion, but increasesthe size of system.

2. To reformulate the problem in terms of the modified inde-pendent variable ϕ∗. This approach preserves the dimen-sion of the system, at the cost of having to derive the modi-fied equations of motion.

Examples of formulations requiring this correction are Dromo(Chap. 4), the Minkowskian formulation for hyperbolic orbits(Chap. 5), the Stiefel-Scheifele method (Sect. 3.6), the equinoc-tial elements with the longitude as the independent variable(Sect. 7.3), etc.

8.6.2 The time element

Time elements separate the physical time t in a term that dependson the perturbations (the time element per se, tte) and a term notaffected by perturbations, tnp,

t(ϕ; oe) = tte(ϕ; oe) + tnp(ϕ; oe) (8.41)

The time element vanishes if there are no perturbations. Thederivative of the time element scales with the perturbation, andyields a smoother evolution in weakly perturbed problems.

The presence of a time element complicates slightly the defi-nition of the time delay: the gradient ∇oe0 t is no longer given ex-plicitly by the state-transitionmatrixΞasyn, since thismatrix prop-agates the time element and not the physical time. In fact, takingthe gradient ∇oe0 in Eq. (8.41) provides

∇oe0 t = ∇oe0 tte(ϕ; oe) + ∇oe0 tnp(ϕ; oe)

where∇oe0 tte(ϕ; oe) is the corresponding row of the matrixΞasyn.The second term is required for adjusting the physical time fromthe variation on the time element. For convenience, the followingsections present the explicit form of the partial derivatives of theformulations discussed in this thesis.

Time-element in Minkowskian variables

The time element improves the numerical performance ofthe method defined in Chap. 5, for which it is oe =

[t, λ1, λ2, λ3, χ1, χ2, χ3, χ4]⊤. The physical time decomposes in

t = tte + λ3/23 [(λ2

1 − λ22)1/2 sinh H −H]− λ3/2

30 (λ10 sinh H0 −H0)

The time delay is obtained from the gradient ∇oe0 t. Consideringthe function

d(u, x) =∂tte∂x+ λ3/2

3

(sinh u

∂λ1

∂ξ0+ cosh u

∂λ2

∂x

)+

32

√λ3(r + 1 − u)

∂λ3

∂x

the partial derivatives forming the gradient take the form:

∂ t∂λ10

= d(u, λ10) − λ3/230 sinh H0

∂ t∂λ30

= d(u, λ10) − 32

√λ30 (r0 + 1 − H0)

∂ t∂H0

= d(u,H0) − 32

√λ30 (λ10 cosh H0 − 1)

∂ t∂ξi,0

= d(u, ξi,0), ξ0 = [tte0, λ20, χ10, χ20, χ30, χ40]⊤

The partial derivatives with respect to the initial values of the el-ements are given by the integration procedure described in theprevious lines.

Time-element in KS variables

In KS variables it is oe⊤ = [t, u⊤,u′⊤]. The time delay is given by

δt = ∇u0 t · δu0 + ∇u′0 t · δu′0

The derivatives of time t with respect to the initial conditions u0and u′0 read

∇u0 t = ∇u0 tte −1

2rα(ru′ + 2r′u

) ∇u0 u

− 12rα2 (2r′u′ + rαu) ∇u0 u′

∇u′0 t = ∇u′0 tte −1

2rα(ru′ + 2r′u

) ∇u′0 u

− 12rα2 (2r′u′ + rαu) ∇u′0 u′

Typically, the energy α is integrated together with the time andthe coordinates u and u′. In such a case the gradients ∇u0α and∇u′0α are given by the corresponding row of the state-transitionmatrix. The previous equations then reduce to

∇u0 t = ∇u0 tte −1

2α(u ∇u0 u′ + u′ ∇u0 u

)+

(u · u′)2α2 ∇u0α

∇u′0 t = ∇u′0 tte −1

2α(u ∇u′0 u′ + u′ ∇u′0 u

)+

(u · u′)2α2 ∇u′0α

Finally, the last term required for solving the t-synchronousstate-transition matrix is t′:

t′ = t′te −1

2rα(ru′ + 2r′u

) · u′ − 12rα2 (2r′u′ + rαu) · u′′

Time-element in Sperling-Burdet variables

Both the Sperling-Burdet regularization and the stabilized Cow-ell method (Sect. 2.3) introduce a fictitious time ϕ ≡ s by

112 8 Regularization in relative motion

means of the Sundman transformation dt = r ds. Whenintroducing a time element, for the former method it isoe = [tte, r⊤, r′⊤, r, r′, α, µe⊤]⊤, and for the latter oe =

[tte, r⊤, r′⊤, α]⊤. Considering the auxiliary terms

r′ =r · r′

r, α =

1r

(2 − ||r

′||2r

)the gradient ∇oe0 t consists of

∇r0 t = ∇r0 tte +1

r4α2

[(r′||r′||2r − r3α r′

) ∇r0 r

− r2(2r′r′ + αr r) ∇r0 r′

]∇r′0 t = ∇r′0 tte +

1r4α2

[(r′||r′||2r − r3α r′

) ∇r′0 r

− r2(2r′r′ + αr r) ∇r′0 r′

]and the derivative with respect to fictitious time reads

t′ = t′te +1

r4α2

[(r′||r′||2r − r3α r′

) · r′ − r2(2r′r′ + αr r) · r′′]

8.7 Conclusions

The theory of asynchronous relative motion allows to solve therelative dynamics using regularized formulations of orbital mo-tion. The theory provides a systematic technique that can beapplied eventually to any formulation relying on an indepen-dent variable different from time. When the motion is formu-lated using fully regular schemes, like the Sperling-Burdet or theKustaanheimo-Stiefel methods, the result is a universal formula-tion that is valid for any kind of reference orbit: circular, elliptic,parabolic, andhyperbolic. The use ofDromo elements also yieldsa solution valid for any type of reference orbit, although there is asingularity arising from the indeterminacy of the eccentricity vec-tor for circular orbits.

The nonlinear correction of the time delay can be appliedeasily and the accuracy of the linear solution is increased signifi-cantly. Formoderately to highly elliptical orbits the discretizationof the reference orbit in terms of the true anomaly seemsmore ef-fective than the discretization in terms of the eccentric anomaly.This conclusion translates into the Dromo-based solution beingmore accurate than the KS and SB schemes in the elliptic case.However, following the discussion in Chap. 2, the full regulariza-tion of the problem will only be possible if the Sundman trans-formation is of order n < 3/2.

Another outcome of the theory is a general procedure forpropagating the variational equations of any formulation thatuses an independent variable different from time. The typical nu-merical procedure defines the transition matrix at every integra-tion step. But the corresponding partial derivatives are computedwith constant fictitious time (or a given angle), rather than withconstant time. This introduces an intrinsic time delay in the solu-tion. It can be corrected easily using the theory of asynchronousrelative motion.

“Eadem mutata resurgo [Although changed, I arise the same].”

—Jakob Bernoulli’s tombstone

9Generalized logarithmic spirals: a new analytic

solution with continuous thrust

E lement based formulations have proven efficient forpropagating weakly perturbed orbits (see Sect. 2.5). Theright-hand side of the equations of motion scales with the

magnitude of the perturbation, and evolves in a similar time scale.A problem of fundamental importance in astrodynamics that in-volves small perturbations is the use of low-thrust propulsion sys-tems. The low acceleration from the engines results in trajectoriesthat spiral away or toward the central body. When the spacecraftspirals around the primary for hundreds or thousands of revolu-tions the slow time scale of the thrust acceleration couples withdifferent frequencies coming from, for example, the high-ordergravity harmonics of the central body. This phenomenon slowsdown the integration significantly due to an inefficient step-sizecontrol. It might become an issue when many propagations arerequired, as it is the case for trajectory optimization. Finding asuitable set of elements for integrating such trajectories may im-prove the design process.

The evolution of the elements is typically formulated apply-ing the variation of parameters technique to an analytic solution.The key to deriving an efficient representation is finding an an-alytic solution to a problem that is sufficiently close to the per-turbed one. This is why special perturbation methods are con-structed by perturbing Keplerian orbits. Motivated by these con-siderations, Roa and Peláez (2015c); Roa et al. (2016a) became in-terested in analytic solutions with continuous thrust fromwhichspecial perturbation methods could be built. Their goal was tofind a set of elements that reduce the computational cost of in-tegrating such problems. In the field of low-thrust preliminarytrajectory design, many authors have devoted themselves to find-ing special trajectories that can be defined in closed form. Themost common approach is the so-called shape-based method: in-

stead of integrating the equations of motion given a thrust pro-file, the shape of the trajectory is assumed a priori and then thrustrequired to follow such curve is obtained.

The shape-based approach was conceived for providing suit-able initial guesses for more sophisticated algorithms for trajec-tory optimization. Bacon (1959) and Tsu (1959) explored theequations of motion for the case of logarithmic spirals, the lat-ter considering the use of solar sails. However, the flight-pathangle is constant along a logarithmic spiral so they turn out notto be practical for trajectory design. Petropoulos and Longuski(2004) proposed a more sophisticated curve, the exponential si-nusoid, which describes the trajectory bymeans of four constantsand resulted in a celebrated mission design technique. They de-rived analytic expressions for the required tangential accelerationand for the angular rate. Practical applications of this solution tooptimization problems have also been discussed by McConaghyet al. (2003), Vasile et al. (2006), and Schütze et al. (2009). Izzo(2006) made a significant contribution by formulating and solv-ingLambert’s problemwith exponential sinusoids, andpresenteda systematic treatment of the boundary conditions. Wall andConway (2009) suggested generalized forms of the exponential si-nusoid together with inverse polynomial functions. Alternativefamilies of solutions havebeendiscussedbyPetropoulos andSims(2002). Pinkham (1962) published a particular representation ofthe logarithmic spiral and improved its versatility by introducinga scaling factor of the semilatus-rectum, whereas Lawden (1963)established the connection between spirals and optimality condi-tions. Perkins (1959) obtained interesting analytic results whenstudying the motion of a spacecraft leaving a circular orbit undera constant thrust acceleration. We refer to the work by McInnes(2004, chap. 5) for a detailed review of solutions arising from the

113

114 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

design of solar sails. The design method presented by De Pascaleand Vasile (2006), which was later improved byNovak and Vasile(2011), relies on the use of pseudo-equinoctial elements for shap-ing the trajectory. A dedicated analysis of the optimality of themethod canbe found inVasile et al. (2007). Taheri andAbdelkha-lik (2012, 2015a,b) followed a different approach bymodeling thetrajectory using Fourier series. The method proves flexible fordifferent preliminary design scenarios. Gondelach and Noomen(2015) published an ingenious technique based onworking in thehodographplane andmodeling the trajectory using the velocities.

A different problem is that of solving the equations of mo-tion given the thrust profile. Analytic solutions are only availablefor very specific cases, but due to their theoretical and practicalimportance it is worth exploring the conditions that yield closed-form solutions. Possibly the first comprehensive approach to thedynamics of a spacecraft under a constant radial and circumfer-ential acceleration is due to Tsien (1953). He solved explicitly theradial case and proposed an approximate solution for the circum-ferential part. The explicit solution to the constant radial thrustproblem is found in terms of elliptic integrals (Battin, 1999, pp.408–418), and alternative solutions involving the Weierstrass el-liptic functions (Izzo and Biscani, 2015), approximate methods(Quarta and Mengali, 2012) and asymptotic expansions (Gon-zalo and Bombardelli, 2014) have been proposed. Urrutxua et al.(2015b) solved the Tsien problem in closed form relying on theDromo formulation (see Chap. 4).

Benney (1958) analyzed for the first time the escape trajec-tories of a spacecraft subject to a constant tangential thrust andprovided approximate solutions for the motion, while Lawden(1955) had already found the optimal direction of the thrust tominimize propellant consumption. Boltz (1992) advanced onthese pioneering works by assuming that the ratio between thethrust and the local gravity is constant, insteadof assumingpurelyconstant thrust. He derived approximate solutions for both thehigh and low-thrust cases. Exact solutions to the constant tan-gential thrust problem have eluded researchers but explicit so-lutions to certain variables can be found. For instance, the ex-pressions defining the escape conditions or the amplitude of theboundedmotion have been provided by different authors (Pruss-ing and Coverstone-Carroll, 1998; Mengali and Quarta, 2009).Bombardelli et al. (2011) derived an alternative asymptotic solu-tion using regular sets of elements. A more didactic approachto perturbation methods applied to the motion of a continu-ously accelerated spacecraft is presented by Kevorkian and Cole(1981), who took the atmospheric drag as the perturbing acceler-ation. In a series of works, Zuiani et al. (2012) and Zuiani andVasile (2015) exploited the advantages of approximate analyticalsolutions based on equinoctial elements. Colombo et al. (2009)adopted a thrust profile that decreases with the square of the ra-dial distance and arrived to a semi-analytical solution. They ap-plied the method to the design of asteroid deflection missions.

The existence of integrals of motion enriches the dynamicsof any system. The analysis of the constant radial thrust prob-lem benefits from the fact that the force is conservative. Theperturbing potential determines the total energy, which is con-served. The angular momentum is also conserved. Prussing andCoverstone-Carroll (1998) and Akella and Broucke (2002) ap-proached the problem from the energy perspective and discussed

the corresponding integrals of motion. General considerationson the integrability of the system can be found in the work bySan-Juan et al. (2012). For the tangential thrust problem Battin(1999, p. 418) arrived to an equation with separate variables thatis integrated to define a first integral of the motion.

This chapter is organized as follows. In Sect. 9.1 we recoverthe tangential thrust profile that Bacon (1959) related to the log-arithmic spiral and formulate the dynamics. The first reason forrecovering the logarithmic spiral is how simple the thrust profileis. It decreases with the square of the radial distance, which seemsconvenient for solar sails or solar electric propulsion. The secondreason is the number of mathematical properties of the logarith-mic spirals.* The equations of motion are then solved with noprior assumptions about the shape of the trajectory. Two first in-tegrals appear naturally. If the equations of motion under suchacceleration are solved rigorously one finds that the logarithmicspiral is not the only solution, but a particular case of an entirefamily of curves that we call generalized logarithmic spirals, S .The integrals ofmotion derived in this section yield a natural clas-sification of the trajectories in elliptic, parabolic, and hyperbolicspirals. Sections 9.2, 9.3 and9.4 present the fully analytic solutionfor each type of spiral. Closed-form expressions for the trajectory,velocity, and time of flight are provided. They are summarizedin Sect. 9.5. Section 9.6 analyzes the evolution of the orbital ele-ments. Properties related to the osculating orbits are highlighted.The departure conditions for spirals emanating from aKeplerianorbit are analyzed in Sect. 9.7. Practical comments can be foundin Sect. 9.8, together with a numerical example comparing thenew spirals with a solar sail trajectory. The continuity of the so-lution is proved in Sect. 9.9.

9.1 The Equations of Motion. First Inte-grals

Let r ∈ R3 denote the position vector of a particle. The dynam-ics under the action of a central gravitational acceleration and aperturbation ap abide by

d2rdt2 +

µ

r3 r = ap, with r = ||r||

Shape-based approaches first assume that the trajectory of theparticle can be described by a certain curve, and then the requiredacceleration to generate such trajectory is obtained. In particular,Bacon (1959) proved that the thrust that renders a logarithmicspiral can be represented by the perturbing acceleration

ap =µ

2r2 cosψ t (9.1)

whereψ is the flight direction angle (complementary of the flight-path angle) and t denotes the vector tangent to the trajectory. Ba-con considered ψ constant, but we shall make no assumptions

*This curve captivated Jakob Bernoulli in the 17th century, so much thathe wanted a logarithmic spiral to be engraved on his tombstone as his coat ofarms. His epitaph reads “Eadem mutata resurgo”, which translates to “althoughchanged, I arise the same” and refers to the self-similarity properties of this curve.More details on its construction can be found in Lockwood (1967). Anecdotally,the craftsman made a mistake and carved an Archimedean spiral instead.

§9.1 The Equations of Motion. First Integrals 115

and let ψ change. In what follows the problem is normalized sothat µ = 1. By definition it is

t =vv, and (t · r) = r cosψ

Here v ∈ R3 is the velocity vector and v = ||v||. If the perturbingterm is confined to the orbital plane the problem is planar and canbe described by the polar coordinates (r, θ). Figure 9.1 depicts thegeometry of the problem.

Figure 9.1: Geometry of the problem. The velocity vector v follows

the direction of t.

The orbital frameL = i, j, k is given the usual definition

i =rr, k =

hh, j = k × i

The angular momentum vector is h = r × v. The velocity of theparticle projected inL reads

v = r i + rθ j

where denotes derivatives with respect to time. The problemcan be formulated using intrinsic coordinates in terms of the tan-gent and normal vectors,

t =1v

(r i + rθ j

), and n =

1v

(r j − rθ i

)(9.2)

which define the intrinsic frame T = t,n,b with b ≡ k. Thetangent vector can be alternatively defined in terms of the flightdirection angle ψ:

t = cosψ i + sinψ j

Recall that ψ is not necessarily constant. Comparing this expres-sion with Eq. (9.2a) it follows

cosψ =rv, sinψ =

rθv, tanψ = r

dθdr

(9.3)

The specific forces acting on the particle are the gravitationalattraction of the central body, ag, and the perturbing accelerationdefined in Eq. (9.1), ap. These accelerations are written in T as

ag = −1r2 (cosψ t − sinψ n)

ap =cosψ2r2 t

The intrinsic equations of motion turn out to be

dvdt= − cosψ

2r2 (9.4)

v2k = +sinψ

r2 (9.5)

where k is the curvature of the trajectory, namely

k =dds

(ψ + θ) (9.6)

Here s denotes an arclength. It relates to the velocity bymeans of

v =dsdt

(9.7)

Combining this identity with Eqs. (9.4–9.5) and the geometricrelations in Eq. (9.3) yields the final form of the equations of mo-tion:

dvdt= − cosψ

2r2 (9.8)

vddt

(ψ + θ) = +sinψ

r2 (9.9)

cosψ =rv

(9.10)

sinψ =rθv

(9.11)

and they must be integrated from the initial conditions

t = 0 : v = v0, ψ = ψ0, r = r0, θ = θ0

An additional equation relates the polar angle and the radius,

tanψ = rdθdr

(9.12)

If the inertial reference frame I = iI, jI,kI is defined sothat the angular momentum is h = h kI, then θ > 0 and fromEq. (9.11) it follows sinψ > 0 ⇒ ψ ∈ (0,π). This assumptionforces the angular velocity to be θ > 0. The retrograde case couldbe derived in an analogous way, but it is obviated for the sake ofclarity. The limits ψ = 0 and ψ = π are also omitted since theyyield rectilinear orbits crossing the origin.

9.1.1 The equation of the energy

Let Ek denote the specific Keplerian energy of the system, definedby the vis-viva equation

Ek =v2

2− 1

r

Its time evolution is given by the power performed by the per-turbing acceleration,

dEk

d t= ap · v

From Eq. (9.1) it follows

dEk

d t=

cosψ2r2 (t · v) =

v cosψ2r2

116 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

The geometric relation between the radial velocity and cosψ—Eq. (9.10)— yields an equation with separate variables,

dEk

d t=

12r2

drdt=⇒ dEk

d r=

12r2

which is integrated to provide a first integral of motion:

Ek = −12r+

K1

2=⇒ v2 − 1

r= K1 (9.13)

Here K1 is a constant of integration, determined by

K1 = v20 −

1r0

(9.14)

in termsof the initial values of the radius and velocity, r0 and v0. Itcan take positive or negative values. The integral ofmotion (9.13)is the generalized integral of the energy.

For the velocity to be real it must be

K1 ≥ −1r=⇒ 1 + K1r ≥ 0 (9.15)

which sets a lower bound to the values that K1 can take.Combining the integral of motion (9.13) with the equation

of the Keplerian energy furnishes a local property that holds forany value of r:

2v2 − v2k = 2(K1 − Ek)

This expression relates the velocity at r of the spiral trajectorywiththe velocity of the Keplerian orbit with energy Ek.

9.1.2 The equation of the angular momentum

In this sectionweprove the existence of an additional first integralrelated to the angularmomentum. Dividing Eq. (9.8) by Eq. (9.9)and introducing Eq. (9.12) it transforms into an equation withseparate variables,

dvv= − cotψ

2(dψ + dθ) = −dr

2r− 1

2cotψ dψ

which is integrated easily,

2 ln v = − ln r − ln(sinψ) + ln K2 =⇒ v2r sinψ = K2 (9.16)

considering that sinψ > 0. This result defines an integral of mo-tion written in terms of the constant K2, which is determinedfrom the initial values r0, v0, ψ0:

K2 = v20r0 sinψ0 = v0r2

0 θ0 (9.17)

The integral of motion (9.16) is the generalization of the angularmomentum equation. It relates to the angularmomentumof theKeplerian orbit at r, hk, by means of

vk sinψk

v2 sinψ=

hk

K2

If the orbit is prograde, (h · kI) > 0, and non-degenerate,h , 0, then K2 > 0. The condition for the orbit to be prograde

can be satisfied by an adequate choice of the inertial reference sothat θ > 0. The limit K2 → 0 yields degenerate rectilinear orbitsfor which θ = θ0 and ψ = 0 or ψ = π.

The form of Eq. (9.17) yields an interesting property of thegeneralized spirals: since sinψ0 is symmetric with respect toψ0 =

π/2, two different trajectories that emanate from the same radiusand with the same value of K1 and K2 can be found. The firsttrajectory will depart in lowering regime (ψ+0 = π/2 + δ) andthe second in raising regime (ψ−0 = π/2 − δ), while sharing thesamevalueof K2. These two trajectories relate to the twodifferentregimes that characterize the generalized spirals considered in thischapter. The radius of the orbit will increase/decrease dependingon the sign of r. Since ψ ∈ (0,π), Eq. (9.10) shows that there aretwo possibilities:

• Raising regime: corresponding to cosψ > 0 (r > 0).

• Lowering regime: corresponding to cosψ < 0 (r < 0).

In raising regime the perturbing acceleration ap moves the parti-cle away from the attracting body (the origin). In lowering regimethe opposite happens. These two regimes are separated by theclassical circular Keplerian orbit. In this analysis circular Keple-rian orbits appear whenψ = π/2 and the perturbing accelerationap vanishes. When ψ crosses π/2 the direction of the thrust vec-tor changes although it remains tangential to the velocity. Hence,it is important to understand the evolution of the flight directionangle.

9.1.3 The flight direction angle ψ

The value of sinψ can be solved from Eq. (9.16) and the equationof the energy:

sinψ =K2

r v2 =K2

1 + K1r(9.18)

This relation is the solution to the differential equation

dψdr= − K1

rv2 tanψ ordψdt= −K1

rvsinψ (9.19)

that governs the evolution of the flight direction angle ψ. Notethat for K1 = 0 the angleψwill be constant. The trajectory in thiscase reduces to a logarithmic spiral. Logarithmic spirals are not ofmuch interest in mission design for this reason. Nonzero valuesof the constant K1 yield an entire family of generalized spirals. IfK1 < 0 the flight direction angle will always grow in time, and forK1 > 0 it will decrease. Note that, since 0 < sinψ ≤ 1, it mustbe K2 ≤ |1+K1r| = 1+K1r. The last simplification comes fromEq. (9.15), because K1 ≥ −1/r.

The term cosψ can be defined from Eq. (9.18), resulting in

cosψ = ±

√1 −

K22

(1 + K1r)2 = ±

√(1 + K1r)2 − K2

2

1 + K1r(9.20)

The condition from Eq. (9.15) is required for the last simplifica-tion. The +/− sign of the cosine relates to the raising/loweringregime of the trajectory. In what remains of the chapter the firstsign will always correspond to raising regime, and the second to

§9.1 The Equations of Motion. First Integrals 117

lowering regime. Considering Eqs. (9.18) and (9.20) tanψ can bewritten as

tanψ = ± K2√(1 + K1r)2 − K2

2

(9.21)

9.1.4 The fundamental theorem of curves

Let k (s) and τ(s) be two singled-valued functions of the ar-clength. The fundamental theoremof curves states that any curvewith torsion τ(s) and nonzero curvature k (s) is uniquely deter-mined except for a Euclidean transformation (rotation, transla-tion and/or reflection). The arclength, curvature, and torsion ofthe curve are invariant under such actions.

Equation (9.7) defines the velocity along the trajectory asthe time derivative of the arclength. Combining this result withEq. (9.10) it follows

ds = v dt = secψ dr (9.22)

The arclength of the spiral connecting r1 and r2 is obtained fromthe integral of the previous expression, considering Eq. (9.20). Ityields

s =1|K1|

∣∣∣∣∣ √(1 + K1r2)2 − K22 −

√(1 + K1r1)2 − K2

2

∣∣∣∣∣=

K2

|K1|(|cotψ2| − |cotψ1|

)If there are transitions between raising and lowering regimes thearclengthhas to be computed in separate arcs. Once the trajectoryis solved for each type of spiral the arclength will be given as afunction of the polar angle, s = s(θ).

The curvature k (s) has already been introduced by Eq. (9.5),and reduces to

k =sinψ(rv)2 =

K2

r (1 + K1r)2

The curvature depends only on the radial distance and the con-stants K1 and K2 by means of Eqs. (9.13) and (9.18). For K2 =

0 the curvature vanishes and the trajectory becomes rectilinear.This case will not be considered in this chapter.

An important result in differential geometry is that the tor-sion of any planar curve is zero. Hence, two spirals with the samevalues of K1 and K2 have the same curvature and torsion. By thefundamental theorem of curves the resulting curves are equal ex-cept perhaps for a Euclidean transformation. This result suggeststhe concept of generating spiral:

Definition 1: (Generating spiral Γ) The generalized logarith-mic spiral defined by K1 and K2 that spans from θ → −∞to θ → +∞ with an arbitrary orientation is called the gen-erating spiral, and it is written Γ-spiral. It contains all thepossible solutions to Eqs. (9.8–9.11) given K1 and K2, except per-haps for a rotation or reflection depending on the initial conditions.

Two spirals that originate from the same Γ-spiral have the sameK1 and K2. If they are initially in the same regime but defined by

(a) AT-symmetric spiral (b) A pair of C-symmetric spirals

Figure 9.2: Graphical representation of theT- and C-symmetries.

different initial positions (r0, θ0), then they relate through a sin-gle rotation. On the contrary, if the initial positions are the samebut the departure regime changes, then the trajectories relate bya reflection R. This last reflecting transformation is particularlyinteresting because of its inherent symmetry:

Definition 2: (C-symmetry) Two curves are said to be C-symmetric if they originate from the same Γ-spiral (they share thesame values of K1 and K2) and relate by a reflection about a polaraxis, R : S 7→ S†, where S† is the C-symmetric of S . The spiralsare reflected about the axis defined by θ0. One spiral is initially inraising regime, whereas the other is in lowering regime.

The reflection R simply transforms a spiral which is initially inraising regime into another one initially in lowering regime (orvice-versa), while keeping constant K1, K2 and (r0, θ0). The re-flection R transforms the trajectory but, since the orbit is as-sumed to be prograde, the initial velocity vectors are not reflectedabout θ0. They are reflected about the perpendicular to the initialradius vector. It is natural to consider the following concept:

Definition 3: (R-invariance) A spiral S is R-invariant if itcoincides with the C-symmetric spiral, i.e. S ≡ S†.

The concept of C-symmetry involves two spirals that origi-nate from the same Γ-spiral. It should not be confused with animportant intrinsic property of a single spiral, which is

Definition 4: (T-symmetry) A spiral r = r(θ) is said to beT-symmetric if r(θm + δ) = r(θm − δ), with δ ≥ 0. The polarangle θm is the axis ofT-symmetry.

The concept ofT-symmetry yields two important propertiesof the transformation R. First, if a spiral is T-symmetric thenthe reflection R can be represented by a rotation. Second, if aspiral is R-invariant then the spiral is T-symmetric. The oppo-site is not true, in general. This statement follows from the factthat the definition of a reflection with respect to a polar axis θaxis R : r(θax + δ) 7→ r(θax − δ). If the Γ-spiral is R-invariantthen r(θax + δ) = r(θax − δ), which is exactly the definition ofT-symmetry for θm ≡ θax. On the other hand, a T-symmetricspiral is not R-invariant unless the reflection is applied about theaxis of symmetry, θax ≡ θm. Figure 9.2 helps in visualizing theT-symmetry of a spiral and a pair of C-symmetric spirals.

118 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

9.1.5 Families of solutions

Like Ek in the Keplerian case, different values of the constant ofthe generalized energy K1 yield different types of solutions. Theresulting families are now introduced briefly, andwill be analyzedin detail in Secs. 9.2, 9.3 and 9.4, respectively:

Elliptic spirals: (K1 < 0) There is a physical limit to the ra-dius, rℓ, set by the condition v2 ≥ 0 in Eq. (9.14):

rℓ = −1

K1, with K1 < 0 (9.23)

Elliptic spirals are dynamically bounded and symmetric.In Section 9.2 an additional, more restrictive limit to theradius is found. Since ψ grows continuously in time —seeEq. (9.19b)— the sign of r changes when crossing the valueψ = π/2 and, in such a case, the trajectory evolves fromraising to lowering regime.

Parabolic spirals: (K1 = 0) In this case it is v2 = 1/r mean-ing that the velocity at every point of the spiral matches thelocal circular velocity. A particle in raising regime will reachinfinity with v → 0. The angle ψ remains constant and thetrajectory reduces to a logarithmic spiral.

Hyperbolic spirals: (K1 > 0) A particle in raising regimewill spiral away and reach infinity with nonzero finite veloc-ity, v → v∞ ≡

√K1 as r → ∞. The constant K1 equals

the characteristic energy C3. There are two subfamilies ofhyperbolic spirals:

• Spirals of Type I, defined by K2 < 1. They only haveone asymptote.

• Spirals of Type II, defined by K2 > 1. They have twosymmetric asymptotes.

Both subfamilies are separated by an asymptotic case, K2 =

1, as it will be proved later on.

The discussed families of solutions can be distin-guished in Fig. 9.3. The figure depicts the regionscorresponding to each type of spiral on the (K1,K2)plane. The spirals are elliptic for K1 < 0, parabolic forK1 = 0, and hyperbolic for K1 > 0. It has alreadybeen stated that two subfamilies of hyperbolic spirals exist: TypeI (K2 < 1) and Type II (K2 > 1). The transition corresponds tothe limit case K2 = 1. Elliptic spirals exist only for K2 ∈ (0, 1).

Figure 9.4 depicts the three different families of generalizedspirals, with the elliptic and hyperbolic spirals bifurcating fromthe parabolic (logarithmic) one. Although all trajectories are inraising regime (r > 0), the angleψ only grows for the case of ellip-tic spirals. For parabolic (logarithmic) spirals it remains constant,and decreases in the hyperbolic case.

Figure 9.3: Types of spirals in the parametric space (K1,K2).

Figure 9.4: Families of generalized spirals: elliptic (K1 < 0),parabolic (K1 = 0) and hyperbolic (K1 > 0).

In the following sections Eqs. (9.8–9.11) are solved in closedform to determine the trajectory and time of flight. The solutiontakes different forms depending on the type of spiral. The conti-nuity of the solution is proved in Sect. 9.9.

It is worth mentioning that the solutions can be unified us-ing the Weierstrass elliptic functions. In this formalism there isno need to distinguish the different types of orbits. The mostgeneral form of the referred universal solution will be derived inSect. 12.8, after the properties of each family have been character-ized.

9.1.6 Form of the solution

The problem has been reduced to the planar case. Thus, there arefour constants ofmotion defining the shape of the trajectory, andrelating the position in the orbit with the time. In order to inte-grate the equations of motion (9.8–9.11) we shall first obtain therelation between the polar angle and the radial distance, θ(r). In-verting this equation yields the trajectory, r(θ). Finally, the timeevolution needs to be accounted for. Typically one seeks a rela-tion θ(t) that yields r(θ(t)), usually obtained as the inverse of t(θ).In some cases, like Keplerian orbits, the equation t(θ) cannot be

§9.2 Elliptic Spirals (K1 < 0) 119

inverted explicitly and θ(t) needs to be solved numerically. Thisis none other than Kepler’s equation. In the present problem weadopt a slightly different approach by writing t(r). The explicitexpressions for t(r) derived in the following sections are the spiralform of Kepler’s equation, keeping in mind that t(r) ≡ t(r(θ)).Like Kepler’s equation, the equation for the time of flight cannotbe inverted analytically.

9.2 Elliptic Spirals (K1 < 0)

The family of elliptic spirals is defined by K1 < 0, meaning thatthe velocity is bounded by the circular velocity

v2 <1r

As v2 → 1/r or K1 → 0 elliptic spirals converge to parabolic(logarithmic) spirals.

Since sinψ, r, v > 0, and K1 < 0, Eq. (9.19b) shows that thetime derivative of ψ is always positive. That is, the angle ψ alwaysgrows in an elliptic spiral. For this kind of orbits the constant K2always belongs to the open interval K2 ∈ (0, 1). Indeed, fromEq. (9.18)

K2 = sinψ0(1 + K1 r0) < sinψ0 ≤ 1 =⇒ K2 ∈ (0, 1)

If the orbit is initially in lowering regime, (ψ0 > π/2 and cosψ0 <0), the orbit will reach the origin r = 0 which is a singular pointin this formulation. Therefore, the final fate for an orbit that isinitially in the lowering regime is r = 0 and ψ = π − arcsin(K2).

If the orbit is initially in raising regime (ψ < π/2 and cosψ >0) it will eventually reachψ = π/2 so cosψwill become negative,changing the sign of the radial velocity. The radius reaches amax-imum when ψ = π/2. By Eq. (9.18) the apoapsis radius reads

rmax =1 − K2

−K1(9.24)

Once the maximum radius is reached cosψ changes its sign andthe spiral enters lowering regime, r < 0. Thus, elliptic spiralstransition from raising regime to lowering regime naturally. Theopposite is not true, i.e. the trajectorywill never change from low-ering to raising regime. The limit rmax will always be smaller thanrℓ = −1/K1.

Elliptic spirals cannot escape to infinity. Once they transitionto lowering regime (cosψ < 0) they will remain in that regimeand fall toward the origin. The spiral can only be in raising regime(cosψ > 0) if it is initially in raising regime, i.e. cosψ0 > 0. Ifinitially it is r0 = rmax then the spiral immediately transitions tolowering regime and falls toward the origin.

When r = rmax the velocity vector is normal to the radiusvector (ψ = π/2) and from Eq. (9.13) the velocity will be

vm =

√K1 +

1rmax

=

√−K1K2

1 − K2=

√K2

rmax(9.25)

The velocity in an elliptic spiral isminimumat apoapsis (ψ = π/2and r = rmax), i.e. min(v) = vm.

9.2.1 The trajectory

Inwhat remains of the section the equations ofmotion are solvedfor the elliptic case. When substituting Eq. (9.21) in Eq. (9.12) ityields

dθ = ± K2 dr

r√

(1 + K1r)2 − K22

(9.26)

Recall that the first/second sign corresponds to raising/loweringregime. This expression can be integrated to provide the evolu-tion of the polar angle,

θ − θm = ∓K2

ℓ

∣∣∣∣∣∣arccosh[rmax

r

(1 +

1K2

)− 1

K2

]∣∣∣∣∣∣= ∓K2

ℓ

∣∣∣∣∣∣arccosh[− 1

K2

(1 +

ℓ2

K1r

)]∣∣∣∣∣∣ (9.27)

with ℓ = (1 − K22 )1/2. Here θm defines the orientation of the

apoapsis, θm = θ(rmax), which is equivalent to the apse-line.When implementing this method the value of θm is solved ini-tially fromEq. (9.27) particularized at (r0, θ0). Depending on theinitial regime of the spiral a pair of C-symmetric elliptic spiralsappear, defined by

θm = θ0 +K2

ℓ

∣∣∣∣∣∣arccosh[− 1

K2

(1 +

ℓ2

K1r0

)]∣∣∣∣∣∣ (9.28)

θ†m = θ0 −K2

ℓ

∣∣∣∣∣∣arccosh[− 1

K2

(1 +

ℓ2

K1r0

)]∣∣∣∣∣∣ (9.29)

Equation (9.28) applies to spirals initially in raising regime, andEq. (9.29) corresponds to spirals initially in lowering regime. Thepair of resulting spirals are C-symmetric and relate simply by areflection about θ0.

A pair of C-symmetric spirals generated by the same Γ-spiralis depicted in Fig. 9.5. One curve is initially in lowering regime,whereas the other is initially in raising regime. The angular mo-mentum is positive, so θ > 0. Being θ0 = 0, Rdefines a reflectionwith respect to the horizontal axis. The direction of the initial ve-locity vector determines which trajectory is the one followed bythe particle.

Figure 9.5: A pair of C-symmetric elliptic spirals departing from (),propagated forward and backward.

120 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

The equation for the trajectory is obtained upon inversion ofEq. (9.27) and yields:

r(θ)rmax

=1 + K2

1 + K2 cosh β(9.30)

where β denotes the spiral anomaly:

β =ℓ

K2(θ − θm), β† =

ℓ

K2(θ − θ†m)

The first argument, β, corresponds to a spiral initially in raisingregime. The second, β†, defines the trajectory of a spiral initiallyin lowering regime.

Elliptic spirals are T-symmetric with respect to the axis de-fined by θm, i.e. r(θm + δ) = r(θm − δ) with δ ≥ 0. This prop-erty follows from the equation of the trajectory, Eq. (9.30). Thus,the reflection R can alternatively be represented by a rotation* ofmagnitude ϑ, with

ϑ =2K2

ℓ

∣∣∣∣∣∣arccosh[− 1

K2

(1 +

ℓ2

K1r0

)]∣∣∣∣∣∣An elliptic spiral in raising regime will always intersect itself atleast once when propagated for sufficient revolutions and withadequate initial conditions. Intersections occur on the axis ofsymmetry, θ = θm.

From Eq. (9.30) and the symmetry properties of the trajec-tory it can be verified that limθ→+∞ r(θ) = limθ→−∞ r(θ) = 0,which means that all elliptic spirals fall toward the origin for suf-ficiently large times, and also when propagated backward. TheΓ-spirals depart form the origin, reach rmax, and fall to the originagain.

The arclength is measured from θm so it takes positive valuesfor θ > θm and negative values for θ < θm. It can be solved interms of the polar angle by introducing Eq. (9.30) into Eq. (9.22),and then integrating from θm to θ thanks to Eq. (9.10). It resultsin

s(θ) =ℓK2 sinh β

(−K1)(1 + K2 cosh β)

For spirals initially in raising regime, r0 > 0, s(θ) relates to thepolar angle through the spiral anomaly β. Conversely, if r0 < 0then β is replaced by β†. The total arclength of the Γ-spiral, sΓ,reduces to

sΓ =2ℓ

(−K1)

It only depends on the constants K1 and K2, equivalent to theenergy and angular momentum. Given K1 < 0 and K2 the corre-sponding Γ-spiral contains all the possible elliptic spirals definedby K1 and K2. In addition, sΓ is the maximum arclength that theparticle can travel along the spiral.

*The trajectory being T-symmetric is a necessary condition for R-invariance. An elliptic spiral will be R-invariant if both the axes ofT-symmetryand C-symmetry coincide. This is equivalent to θ0 = θm + nπ, with n =0, 1, 2, . . .. From Eq. (9.30) it follows

R-invariance ⇐⇒ r0,inv

rmax=

1 + K2

1 + K2 cosh(

nπℓK2

)If initially r0 ≡ r0,inv then the axis of symmetry θm coincides with the initialdirection given by θ0. The axis ofT-symmetry is R-invariant.

9.2.2 The time of flight

In order to solve completely the equations of motion —Eqs.(9.8–9.11)— a relation between the position in the orbit and thetime is required. The radial velocity, defined in Eq. (9.10), can beinverted to provide

dtdr=

secψv= ±

√r(1 + K1r)

(1 + K1r)2 − K22

(9.31)

Integrating this expression from rmin to r yields the time of flightas a function of the radial distance and referred to the time ofapoapsis passage, tm:

t(r) − tm = ±(

rvK1

√1 − sinψ1 + sinψ

+K2∆ E−k′2∆Π

vmK1k

)(9.32)

Here ∆ E and ∆Π are the difference between the incomplete andthe complete elliptic integrals of the second and third kinds,

∆ E = E(ϕ, k) − E(k), ∆Π = Π(n; ϕ, k) − Π(n; k)

Their argument, modulus, and parameter are

sin ϕ =vm

v

√2

1 + sinψ, k =

√1 − K2

2, n =

K2 − 12K2

and k′ denotes the complementary modulus, which relates to themodulus k by means of k2 + k′2 = 1 (see Appendix E for a dis-cussion about the notation and required properties). This formof the equation for the time of flight resembles Kepler’s equation,provided that tm can be seen as the spiral equivalent to the timeof periapsis passage. Its value is solved initially from Eq. (9.32)particularized at t(r0) = t0,

tm = t0 ∓(

r0v0

K1

√1 − sinψ0

1 + sinψ0+

K2∆ E0 −k′2∆Π0

vmK1k

)(9.33)

The sign is chosen according to the initial regime of the spiral.The time of flight to describe the generating spiral, tΓ, is

tΓ =2[K2 E(k) − k′2Π(n; k)

]vmK1k

This expression depends only on K1 and K2. It is the maximumtime that any elliptic spiral generated by Γwill require to connecttwo points.

9.3 Parabolic Spirals (K1 = 0)

Parabolic spirals are equivalent to the well-known logarithmicspirals. The velocity coincides with the local circular velocity,

v =1√

r

and shows that parabolic (logarithmic) spirals reach infinity withzero velocity, limr→∞ v = 0, just like parabolic orbits in theKeple-rian case. The evolution of the angle ψ is governed by Eq. (9.19).

§9.4 Hyperbolic Spirals (K1 > 0) 121

For K1 = 0 it vanishes and proves that the angle ψ is constantalong a parabolic spiral.

The definition of K2 provided in Eq. (9.16) becomes

sinψ = K2, and cosψ = ±√

1 − K22 = ±ℓ

Beingψ constant, the regime is defined initially by the value ofψ0.In the limit case ψ = π/2 (K2 = 1) parabolic spirals degenerateinto circular Keplerian orbits. The thrust vanishes and, since ψis constant, it will remain zero. The constant K2 is restricted toK2 ∈ (0, 1]. Figure 9.6 shows a pair of C-symmetric parabolicspirals. The corresponding flight direction angles make sinψ =sinψ† and therefore the C-symmetry.

Figure 9.6: Pair of C-symmetric parabolic spirals, withψ = 88 andψ† = 92, respectively.

9.3.1 The trajectory

This section proves that the solution to the system of equations(9.8–9.11) when K1 = 0 is indeed a logarithmic spiral. Equa-tion (9.12) is integrated to provide

θ − θ0 = ±K2

ℓln(r/r0) (9.34)

The fact that limr→∞ θ(r) = ∞ shows that the particle follows aspiral branch when reaching infinity.

Equation (9.34) is inverted to define the trajectory of the par-ticle,

r(θ) = r0 e(θ−θ0) cotψ (9.35)

which is none other than the equation of a logarithmic spiral.If measured from the initial position θ0 the arclength* takes

the form

s(θ) =r0

cosψ[e(θ−θ0) cotψ − 1

]This definition yields positive values of the arclength for θ > θ0,and negative values if propagated backward (θ < θ0).

*The original expression for the arclength of the logarithmic spiral is at-tributed to Evangelista Torricelli back in the 17th century. He derived the firstrectification of a transcendental curve and was able to integrate the length of thespiral by evaluating the sum of small values of dθ.

9.3.2 The time of flight

The time of flight is solved from the inverse of the radial velocityand results in:

t − t0 = ±23ℓ

(r3/2 − r3/2

0)

(9.36)

It takes an infinite time to reach r → ∞, as shown by the limitlimr→∞ t(r) = ∞. On the contrary, parabolic spirals in loweringregime reach the attractive center in finite time:

limr→0

t(r) =2r3/2

0

3ℓ

Of course forψ = π/2 (ℓ = 0) the spiral never reaches the origin,since the spiral degenerates into a circular Keplerian orbit.

9.4 Hyperbolic Spirals (K1 > 0)

Hyperbolic spirals are generalized logarithmic spirals with posi-tive constant of the generalized energy. When in raising regimespirals of this family will reach infinity with a finite, nonzerovelocity v∞. The hyperbolic excess velocity can be solved fromEq. (9.14):

v∞ = limr→∞

√K1 +

1r=

√K1 =⇒ K1 ≡ C3 (9.37)

This result is analogous to the Keplerian case, where the hyper-bolic excess velocity is defined by the characteristic energy, C3 =

v2∞. The time derivative ofψ, given by Eq. (9.19b), proves that the

angle ψ always decreases along hyperbolic spirals.While parabolic spirals reach infinity along a spiral branch,

hyperbolic spirals reach infinity along an asymptotic branch. Inthe limit case r → ∞ the polar angle converges to a finite valueθas. Sinceψdecreases in time the position and velocity vectorswillbecome parallel at infinity. The impact parameter of the result-ing asymptotes, c, can be solved from the equation of the angularmomentum, Eq. (9.16),

h = v∞c =⇒ c =K2

K1(9.38)

which defines the minimum distance from the asymptote to thecenter of attraction.

Equation (9.18) is valid for all types of spirals. It relates theconstants K1 and K2 with the flight direction angle ψ, and ofcourse sinψ ≤ 1. For K2 < 1 this condition is satisfied natu-rally, provided that K1r > 0. However, in the case K2 > 1 thiscondition holds only if

r ≥ rmin =K2 − 1

K1

A minimum radius exists for hyperbolic spirals with K2 > 1, sothey will never reach the center of attraction. This radius is re-ferred to as the periapsis of the spiral. The expression for rmin isequal to that for rmax (for elliptic spirals), the only difference be-ing the values that K1 and K2 can take.

122 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

There are two different types of hyperbolic spirals: thosewith K2 < 1 (Type I) and those with K2 > 1 (Type II). Fig-ure 9.7 shows the evolution of the radius and the angle ψ for thetwo types of hyperbolic spirals. Type II spirals in lowering regimereach rmin, then transition to raising regime and escape. Type Ispirals in lowering regime simply converge to the origin, and es-cape to infinity if in raising regime. Also, cosψ can change its signonly for Type II spirals, which means that the trajectory transi-tions from lowering to raising regime. The two types of hyper-bolic spirals are separated by the asymptotic limit K2 → 1.

Figure 9.7: Evolution of the radius and angleψ along hyperbolic spi-

rals of Types I and II initially in lowering regime.

This figure also unveils a fundamental difference between hyper-bolic spirals of Types I and II: a hyperbolic spiral of Type I onlyhas one asymptote, whereas spirals of Type II have two asymp-totes.

9.4.1 Type I hyperbolic spirals

Spirals of this type are defined by (K1,K2) ∈ (0,∞) × (0, 1).There are no limitations to the values the radius can take, since thecondition sinψ ≤ 1 holds naturally. If the trajectory is initiallyin lowering regime (cosψ0 < 0) then the particle falls toward theorigin. The dynamics of the particle when reaching the origin canbe understood by taking the limit r → 0 in Eq. (9.18),

limr→0

ψ(r) = π − arcsin K2 (9.39)

which only depends on the angular momentum. Since K2 < 1 itis arcsin K2 < π/2, meaning that ψ will never cross ψ = π/2as shown in Fig. 9.7. That is, the trajectory remains in lower-ing regime. If initially it is cosψ0 > 0 then the spiral will bein raising regime forever. The asymptotic escape is defined bylimr→∞ ψ(r) = 0 so the position and velocity vectors becomepar-allel. In this case ψ0 < π/2 and, since the flight direction anglealways decreases, it is not possible for the trajectory to transitionto lowering regime.

A particle in lowering regime will always fall to the origin,whereas particles in raising regime will always escape to infinity.

Natural transitions between regimes are not possible since cosψcan never change its sign.

The trajectory

When assuming K2 < 1 the integration of Eq. (9.26) yields

θ − θ0 = ±K2

ℓln

[r sinψ

r0 sinψ0

(1 − K2 sinψ0 + ℓ |cosψ0|1 − K2 sinψ + ℓ |cosψ|

)](9.40)

There are two possible C-symmetric solutions, given by thechoice of the sign. Having obtained the solution for the polar an-gle, taking the limit r → ∞ in Eq. (9.40) provides the directionto the asymptote, θas = limr→∞ θ(r):

θas = θ0 +K2

ℓln

[K2(ℓ |cosψ0| + 1 − K2 sinψ0)

(K2 − sinψ0)(1 + ℓ)

](9.41)

This equation is valid for spirals initially in raising regime. Theasymptote of the C-symmetric spiral, initially in lowering regime,follows a different direction given by

θ†as = θ0 −K2

ℓln

[K2(ℓ |cosψ0| + 1 − K2 sinψ0)

(K2 − sinψ0)(1 + ℓ)

](9.42)

Once K1 and K2 are fixed, two different possible trajectories areobtained: one departs in raising regime, and the other in loweringregime. They are C-symmetric and relate by a reflection R aboutθ0. Type I hyperbolic spirals can never be R-invariant, since theyare notT-symmetric.

The equation for the trajectory is obtained upon inversion ofEq. (9.40) and introducing the direction of the asymptote. Aftersome simplifications it takes the form

r(θ) =ℓ2/K1

2 sinh β2

(sinh β

2 + ℓ coshβ2

) (9.43)

with

β =ℓ

K2(θas − θ), β† = − ℓ

K2(θ†as − θ)

The argument β applies to spirals initially in raising regime, andβ† defines the C-symmetric spiral initially in lowering regime.The direction of the initial velocity vector determines which oneto choose.

Figure 9.8 shows two C-symmetric hyperbolic spirals ofType I. The corresponding asymptotes are denoted by θas and θ†as.They intersect on the axis of C-symmetry, at a distance d fromthe origin, which follows from the impact parameter defined inEq. (9.38):

d =K2

K1 |sin(θas − θ0)| =K2

K1 |sin(θ†as − θ0)|

This type of spirals have only one asymptote, so the Γ-spiral con-nects the origin with infinity.

§9.4 Hyperbolic Spirals (K1 > 0) 123

Figure 9.8: Two C-symmetric hyperbolic spirals of Type I.

The arclengthmeasured from the direction of the asymptoteθas is obtained by integrating Eq. (9.22) and results in

s(θ) =

[(1 − 2ℓ2) cosh β − ℓ sinh β − 1

]csch2 β

2

4K1(1 + ℓ coth β2 )

This expression yields positive values of the arclength. If the spiralis in lowering regime, then the argument β is replaced by β† andthe arclength takes negative values.

The time of flight

Integration of Eq. (9.31) for K2 < 1 requires the use of the in-complete elliptic integrals of the second, E = E(ϕ, k), and thirdkinds,Π = Π(n; ϕ, k):

t(r) = K4 ±(

rvK1

√1 + sinψ1 − sinψ

− K2 E+k′2ΠK1√

K1K2/2

)(9.44)

with

sin ϕ =

√2rK1 sinψ

(1 + K2)(1 − sinψ), k =

√1 + K2

2, n =

1 + K2

2K2

The complementary modulus is k′2 = (1− K2)/2. The constantof integration K4 is obtained from the initial conditions:

K4 = t0 ∓(

r0v0

K1

√1 + sinψ0

1 − sinψ0− K2 E0 +k′2Π0

K1√

K1K2/2

)(9.45)

The time to reach the origin derives from the limitlimr→0 t(r) = K4. It is finite and depends on the initial condi-tions by means of Eq. (9.45).

9.4.2 Type II hyperbolic spirals

A particle moving along a Type II hyperbolic spiral in loweringregime (cosψ < 0) will reach a limit radius rmin with ψ = π/2 infinite time. Then, it enters raising regime and escapes to infinity.It can never reach the origin. Once the particle is in raising regime(cosψ > 0) it will stay in that regime forever and escape to infin-ity. This behavior follows from the fact that ψ decreases in time.

It can only be in lowering regime (cosψ < 0) if it is initially inlowering regime, i.e. cosψ0 < 0. If initially it is r0 = rmin thenthe spiral immediately transitions to raising regime and escapes toinfinity.

The velocity of the particle when reaching the periapsis is ob-tained from Eq. (9.14), and reads

vm =

√K1K2

K2 − 1=

√K2

rmin(9.46)

This expression coincides with the one obtained for the ellipticcase in Eq. (9.25). The velocity is maximum at periapsis.

The trajectory

The evolution of the polar angle for Type II hyperbolic spiralscomes from solving Eq. (9.26) assuming that K2 > 1:

θ − θm = ±K2

ℓ

π2 + arctan

1 + K1r − K22

ℓ√

(1 + K1r)2 − K22

(9.47)

and with ℓ = (K22 −1)1/2. The value of θm is solved initially from

θm = θ0 −K2

ℓ

π2 + arctan

1 + K1r0 − K22

ℓ√

(1 + K1r0)2 − K22

(9.48)

θ†m = θ0 +K2

ℓ

π2 + arctan

1 + K1r0 − K22

ℓ√

(1 + K1r0)2 − K22

(9.49)

Like in the case of elliptic spirals these expressions define twodifferent trajectories, depending on the initial regime. Equa-tion (9.48) is valid when r0 > 0, and it is replaced by Eq. (9.49)if r0 < 0. The adequate choice is given by the direction of theinitial velocity vector.

From Eq. (9.47) it is possible to solve for the radial distance,

r(θ)rmin=

1 + K2

1 + K2 cos β(9.50)

where the spiral anomaly reads

β =ℓ

K2(θ − θm), β† =

ℓ

K2(θ − θ†m)

When the spiral is initially in raising regime the spiral anomalyis β, and if initially in lowering regime it is replaced by β†. Thesolution takes the samevalues for θ > θm and for θ < θm,meaningthat hyperbolic spirals of Type II areT-symmetric with respect tothe axis defined by θm, i.e. r(θm + δ) = r(θm − δ) with δ ≥ 0.

The reflection R connecting the C-symmetric spirals S andS† can alternatively be represented by a rotation* of magnitude

*Since this type of spirals is T-symmetric the spirals can be R-invariant.The condition for R-invariance reduces to θ0 = θm + nπ, or

R-invariance ⇐⇒ r0,inv

rmin=

1 + K2

1 + K2 cos(

nπℓK2

)If r0 = r0,inv then the axis of T-symmetry coincides with the axis of C-symmetry.

124 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

ϑ, with

ϑ =2K2

ℓ

π2 + arctan

1 + K1r0 − K22

ℓ√

(1 + K1r0)2 − K22

Type II hyperbolic spirals initially in lowering regime will al-ways intersect themselves at least once when propagated for suffi-cient revolutions and with adequate initial conditions. Intersec-tions occur on the axis of symmetry, θ = θm. Initially in loweringregime the spiral reaches rmin and then enters raising regime, fol-lowing aT-symmetric path.

How the spiral reaches infinity is studied from the limitr → ∞. The existence of two asymptotes can be proved fromEq. (9.50): the denominator cancels for

β = ± arccos(− 1

K2

)=⇒ θ±as = θm ±

K2

ℓarccos

(− 1

K2

)This equation defines the orientation of the two asymptotes, thatare symmetric with respect to θm. The C-symmetric spiral is de-fined by θ†m, given in Eq. (9.49). Its two asymptotes correspondto

θ±as†= θ†m ±

K2

ℓarccos

(− 1

K2

)The difference between the asymptotes θ±as and θ±as

† comes fromthe fact that θm , θ

†m. The geometry of the spirals and the asymp-

totes can be analyzed in Fig. 9.9. Each spiral isT-symmetric withrespect to its corresponding θm. In addition, the spirals are C-symmetric with respect to the initial polar angle, θ0 = 0.

Figure 9.9: Pair of C-symmetric hyperbolic spirals of Type II, with

their corresponding asymptotes and axes ofT-symmetry.

Like in the elliptic case, the arclength ismeasured from θm, sothat s(θ) > 0 for θ > θm and s(θ) < 0 when θ < θm. ConsideringEq. (9.50) the arclength takes the form

s(θ) =ℓK2 sin β

K1(1 + K2 cos β)

Depending on the initial regime of the spiral the argument βmight be replaced by β†. The arclength of the Γ-spiral is infinite,as deduced from the limit limθ→±∞ sm = ±∞.

The time of flight

Integrating Eq. (9.31) and introducing the time of periapsis pas-sage tm (following the same procedure applied to the elliptic spi-rals) defines the time of flight. It reads

t(r) − tm = ±1

K3/21

√z(z2 − K2

2 )(z − 1)

− arcsinh

√(z − 1)(z − K2)z + K2(2z − 1)

+

√2 [k′2(Π + K2 F) − K2 E]

k√

K2

(9.51)

with z = 1 + K1r. In this case the arguments of the elliptic inte-grals are

sin ϕ =

√n (z − K2)k2(z − 1)

, k =

√2

1 + K2, n =

1K2

The time of flight to rmin is solved from Eq. (9.51) particular-ized at r = r0:

tm = t0 ∓1

K3/21

√z0(z2

0 − K22 )

(z0 − 1)− arcsinh

√(z0 − 1)(z0 − K2)z0 + K2(2z0 − 1)

+

√2 [k′2(Π0 + K2 F0) − K2 E0]

k√

K2

(9.52)

and it is defined in terms of the constants K1 and K2.

9.4.3 Transition between Type I and Type II spirals

The limit K2 → 1 defines the transition regime between Type Iand Type II hyperbolic spirals. The continuity of the transitionis discussed in Sect. 9.9. As shown in the previous, the flight di-rection angle for Type I spirals in lowering regime reaches a limitvalue given in Eq. (9.39). For K2 = 1 it is limr→0 ψ(r) = π/2.This result, together with the fact that limK2→1 rmin = 0 showsthat the limit case K2 = 1 can be understood as a Type II hy-perbolic spiral with a periapsis radius equal to zero. Finally, fromEq. (9.47) it is limK2→1 θm = ±∞. The spiral can never reach theaxis of symmetry and transitions from raising lowering regime toraising regime are not possible. The spiral approaches the originalong a spiral branch. These spirals are notT-symmetric.

The trajectory

Imposing K2 = 1 in Eq. (9.26) and integrating the result yields

θ = K3 ∓√

1 +2

K1r, with K3 = θ0 ±

√1 +

2K1r0

(9.53)

The inverse of this equation defines the trajectory for K2 = 1,

r(θ) =2

K1[(θ − K3)2 − 1

] (9.54)

The definition of K3 in Eq. (9.53b) contains the information onthe initial regime of the spiral.

§9.5 Summary 125

The asymptote in this case is given by the simple relation:

θas = limr→∞

θ(r) = K3 − 1 = θ0 −1 − √

1 +2

K1r0

(9.55)

if the spiral is initially in raising regime. In the opposite case it is

θ†as = K†3 + 1 = θ0 +

1 − √1 +

2K1r0

(9.56)

These results provide a more compact description of the trajec-tory

r(θ) =2/K1

β (β + 2)or r†(θ) =

2/K1

β†(β† − 2)(9.57)

with β = (θas − θ) and β† = (θ†as − θ). The first equation definesthe trajectory of a spiral initially in raising regime, and the secondcorresponds to a spiral initially in lowering regime. The asymp-tote is defined by Eqs. (9.55) and (9.56), depending on the regimeof the spiral.

Transitions from lowering to raising regimes are not possible,so the spirals of this type have one asymptote. Figure 9.10 showsan example of two C-symmetric spirals with K2 = 1.

Figure 9.10: Pair of C-symmetric hyperbolic spirals in the limit case

K2 = 1.

When in raising regime, the length of the curve measuredfrom the direction of the aymptote, θas takes the form

s(θ) =2(β + 1)

K1β (β + 2), β = (θas − θ)

whereas for a spiral in lowering regime s†(θ) is obtained by re-placing β with β† = (θ†as − θ) in the previous. The sign criterionhas been chosen so that s(θ) > 0 and s†(θ) < 0. Due to theC-symmetry of the spirals, when |θ − θas| = |θ − θ†as| then it iss(θ) = −s†(θ).

The time of flight

For K2 = 1 the integration of Eq. (9.31) reduces to

t − t0 = ±1

2K3/21

2(Ξ − Ξ0) − ln

[1 + 2(z + Ξ)

1 + 2(z0 + Ξ0)

](9.58)

where z = 1 + K1r and Ξ =√

z(1 + z). The time for reachingthe origin is solved from the limit

limr→0

t(r) =1

2K3/21

2(Ξ0 −

√2) + ln

[3 + 2

√2

1 + 2(z0 + Ξ0)

]Although the number of revolutions is infinite, it is compensatedby a decreasing period that yields a finite time to reach the origin.

9.5 Summary

Table 9.1 summarizes the different types of solutions that can befounddepending on the values of the parameters K1 and K2. Theequations defining the trajectory and the time of flight are refer-enced in this table for convenience.

Table 9.1: Equations describing the different families of generalized

logarithmic spirals.

Type of spiral K1 K2 Trajectory TOF

Elliptic < 0 ∈ (0, 1) Eq. (9.30) Eq. (9.32)Parabolic = 0 ∈ (0, 1] Eq. (9.35) Eq. (9.36)Hyperb. Type I > 0 ∈ (0, 1) Eq. (9.43) Eq. (9.44)Hyperb. Transition > 0 = 1 Eq. (9.57) Eq. (9.58)Hyperb. Type II > 0 > 1 Eq. (9.50) Eq. (9.51)

Taking the limit K1 → 0 in the equations of the trajectory forelliptic and hyperbolic spirals yields the equation of the trajectoryfor parabolic (logarithmic) spirals. Similarly, the limit K2 → 1for hyperbolic spirals ofTypes I and II is continuous and the tran-sition between both types is given by the solution for K2 = 1.

9.6 Osculating Elements

The orbital elements defining the osculating orbit can be relatedto the constants of motion K1 and K2. The eccentricity of theosculating orbit is obtained from the definition,

e = v × h − rr= (K2 sinψ − 1) i − K2 cosψ j (9.59)

which provides

e(r) =

√1 − K2

2

(1 − K1r1 + K1r

)(9.60)

It has been proven that elliptic spirals (K1 < 0) are boundedby a maximum radius, rmax, defined in Eq. (9.24). Similarly, forhyperbolic spirals of Type II (K1 > 0 and K2 > 1) a minimumradius rmin exists. The eccentricity of the osculating orbit at thosepoints is

em = 1 − K2, for K1 < 0 (9.61)em = K2 − 1, for K1 > 0,K2 > 1 (9.62)

These expressions show that for K2 → 1 the final orbit is quasi-circular, and that the eccentricity at r = rmax and r = rmin

126 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

only depends on the angular momentum K2. The osculating or-bit will never be perfectly circular except for the case of degen-erate parabolic spirals. In fact such orbits turn out to be circu-lar Keplerian orbits. All elliptic spirals, together with parabolicand Type I hyperbolic spirals in lowering regime fall toward theorigin after sufficient time. The osculating eccentricity becomeslimr→0 e(r) = (1 − K2

2 )1/2.The eccentricity of the osculating orbit can be rewritten in

terms of the flight direction angle:

e =√

K22 + 1 − 2K2 sinψ (9.63)

It only depends on the constant K2 and the angle ψ. This expres-sion, together with Eqs. (9.61–9.62), shows that the eccentricityof the osculating orbit reaches a minimum value at r = rmin orr = rmax.

The eccentricity always decreases along an elliptic spiral inraising regime. It grows in lowering regime. Conversely, the ec-centricity grows for all hyperbolic spirals in raising regime, anddecreases in lowering regime. The eccentricity of the osculatingorbit at infinity is defined by

e∞ = limr→∞

e(r) =√

1 + K22 (9.64)

so the osculating orbit is always hyperbolic. The eccentricity ofthe osculating orbit of a parabolic (logarithmic) spiral is constantand equals e = (1 − K2

2 )1/2. Equation (9.62) relates the eccen-tricity at r = rmin with the value of K2. A simple analysis ofthis expression shows that hyperbolic spirals with K2 = 2 yieldparabolic osculating orbits at r = rmin. If K2 > 2, then the oscu-lating orbit is hyperbolic for every r.

The angular momentum reduces to

h = r × v = rv sinψ k =K2

vk = K2

√r

1 + K1rk (9.65)

The semimajor axis can be solved from Eqs. (9.60) and (9.65):

a(r) =h2

1 − e2 =r

1 − K1r(9.66)

The semimajor axis grows in raising regime and decreases in low-ering regime, nomatter the type of spiral. It only depends on theenergy and the radial distance. This property leads to a relationbetween the osculating semimajor axes at two different points ofthe spiral, r1 and r2:

1r2− 1

r1=

1a2− 1

a1(9.67)

If two isoenergetic spirals intersect, the corresponding osculatingorbits have the same semimajor axis at the intersection point. Thesemimajor axis of the osculating orbit at r = rmax and r = rmin isgiven by a single expression:

am =K2 − 1

K1(2 − K2)

It defines the maximum value the semimajor axis can reach. Onthe other hand, for spirals reaching infinity (K1 ≥ 0) it is

a∞ = limr→∞

a(r) = − 1K1

(9.68)

and this result confirms a beautiful connection with Keplerianorbits: the osculating orbit at r → ∞ is parabolic if the spiral isparabolic, and hyperbolic if the spirals is hyperbolic.

Note that the denominator in Eq. (9.66) vanishes for r =1/K1 and only when K1 > 0. Equation (9.60) shows that thissingularity corresponds to the point where the osculating orbitbecomes parabolic, e = 1. That is, when the particle escapes fromthe gravitational field of the central body,

r∗ = 1/K1 =⇒ e(r∗) = e∗ = 1

In this case it is v∗ = (2K1)1/2 and ψ∗ = arcsin(K2/2). It is in-teresting to note that the flight direction angle at escape, ψ∗, onlydepends on the generalized angular momentum K2. In the limitcase K2 = 1, which separates Type I and Type II hyperbolic spi-rals, it is ψ∗ = π/6. Hyperbolic spirals of Type I (K2 < 1) aredefined by ψ∗ < π/6, whereas those of Type II (K2 > 1) satisfyψ∗ > π/6.

Combining the limits from Eqs. (9.64) and (9.68) it fol-lows the expression for the impact parameter c, already given byEq. (9.38),

c = |a∞|√

e2∞ − 1 =

K2

K1

The argument of periapsis, ω, can be solved from the pro-jection of the eccentricity vector in the inertial frame I. FromEq. (9.59) it is

ω = arctan[sin θ + K2 cos(θ + ψ)cos θ − K2 sin(θ + ψ)

](9.69)

Equation (9.69) determines the evolution of the apse line. Theargument of periapsis ω always increases, no matter the type ofspiral. The apse line rotates counterclockwise in the inertial ref-erence.

9.7 In-Orbit Departure Point

The constant K1 is defined initially by r0 and v0, and K2 alsorequires ψ0. Although an initial ∆v can be applied to providethe adequate energy and angular momentum, it is interestingto study the spirals that emanate naturally from a Keplerian or-bit with no additional maneuvers. That is, the thrust given inEq. (9.1) has initially the direction of the velocity in the departureorbit. The eccentricity of this orbit is e0, a0 is the semimajor axis,and h0 is the angular momentum.

The velocity is expressed in terms of the orbital elements andthe true anomaly, ν, as

v =1h0

√1 + e2

0 + 2e0 cos ν

and the angle ψ is obtained from the expressions:

sinψ =1 + e0 cos ν√

1 + 2e0 cos ν + e20

, cosψ =e0 sin ν√

1 + 2e0 cos ν + e20

Since the eccentricity of the departure orbit is fixed, the angle ψonly depends on the departure point, ν0. From the definitions of

§9.7 In-Orbit Departure Point 127

K1 and K2, given in Eqs. (9.14) and (9.16), it follows

K1 =e0

h20

(e0 + cos ν) and K2 =

√1 + 2e0 cos ν + e2

0 (9.70)

The constants K1 and K2 define the type of spiral. The previ-ous equations corroborate that all generalized spirals which em-anate fromcircular orbits are degenerate parabolic spirals (K1 = 0and K2 = 1). Equation (9.70) proves that, once the initial orbitis given by a0 and e0, K1 and K2 only depend on the departurepoint ν(t0) = ν0. In addition, if the departure orbit is ellipticthe constant K2 is constrained to K2 ∈ (0, 2), as deduced fromEq. (9.70b). If K2 = 2 the departure orbit is parabolic, and forK2 > 2 it is hyperbolic.

It is clear from Eq. (9.70) that the eccentricity of the depar-ture orbit limits the values that K1 and K2 can take. The thrustprofile defined in Eq. (9.1) may not be adequate for departingfrom a circular orbit. If it is not possible to meet the designrequirements with that thrust, then an initial maneuver is re-quired. A solution to this problem without the need for impul-sivemaneuvers can be found in Sect. 10.2, and it is later applied inChap. 11 for the design of orbit transfers (Roa and Peláez, 2016a).Given the design values of K1 and K2, together with the energyof the departure orbit (a0), Eq. (9.70) can be inverted to definethe required eccentricity and departure point:

e0 =

√A0 − K2

2

A0and cos ν0 =

K22 (1 + K1a0) − A0√

A0(A0 − K22 )

with A0 = 1 + 2K1a0.Let φ1 = arccos(−e0) and φ2 = 2π − φ1, with e0 < 1.

The angles ν = φ1 and ν = φ2 correspond to the semiminoraxis of the ellipse, since r(φ1) = r(φ2) = a0. Parabolic (logarith-mic) spirals in raising regime originate from ν0 = φ1, and thosein lowering regime depart from ν0 = φ2. Elliptic spirals in raisingregime correspond to ν0 ∈ (φ1,π), and for lowering regime it isν0 ∈ (π, φ2). If the particle departs from apoapsis (ν0 = π) thenr0 = rmax and it immediately enters lowering regime. The spi-rals departing from ν0 ∈ (0, φ1) are hyperbolic spirals in raisingregime, whereas ν0 ∈ (φ2, 2π) corresponds to hyperbolic spiralsin lowering regime. For hyperbolic spirals departing from ν0 = 0it is r0 = rmin, so the spiral immediately enters raising regime.

Figure 9.11 depicts the different regions found in an ellipticKeplerian orbit. If the thrust starts between periapsis and apoap-sis then the spiral is in raising regime. Between apoapsis and peri-apsis the resulting spirals are in lowering regime.

The maximum radius that an elliptic spiral can reach, rmax,relates to the departure point by means of

rmax =h2

0(e0 + 2 cos ν0)

(e0 + cos ν0)(√

1 + 2e0 cos ν0 + e20 + 1

) (9.71)

with ν0 ∈ (φ1,π). In order to analyze how much the radius ofthe orbit can be raised from a certain departure orbit the limit ofEq. (9.71) shall be considered. In particular, for an elliptic spiralin raising regime the final radius maximizes in the limit case

rmax∣∣∣max = lim

ν0→φ−1rmax(ν0) = +∞

Figure 9.11: Diagram of departure points and spiral regimes.

Obviating practical limitations such as maximum admissiblethrust, there is no theoretical limit to the maximum radius an el-liptic spiral can reach. There is, however, a minimum value forthe maximum reachable radius:

rmax∣∣∣min = lim

ν0→πrmax(ν0) = a0(1 + e0) (9.72)

For a transfer to apoapsis, the apoapsis of the target orbit must behigher than that of the departure orbit.

For the hyperbolic case the type of the spiral (I or II) comesfrom the value of K2. From Eq. (9.70b) it follows that K2 = 1 forν0 = φ3 or ν0 = φ4, with φ3 = arccos(−e0/2) and φ4 = 2π−φ3.If ν0 ∈ (φ3, φ1) ∪ (φ2, φ4) then K2 < 1, and for K2 > 1 it isν0 ∈ [0, φ3)∪(φ4, 2π]. Figure 9.12 depicts the discussed intervals.

Figure 9.12: Hyperbolic spirals emanating from an elliptic orbit (e0 =

0.5).

The limit radius for a hyperbolic spiral with K2 > 1 is de-fined by an expression equivalent to that for the elliptic case —Eq. (9.71)—:

rmin =h2

0(e0 + 2 cos ν0)

(e0 + cos ν0)(√

1 + 2e0 cos ν0 + e20 + 1

) (9.73)

with ν0 ∈ (φ4, 2π). The minimum radius rm is reduced forK1 → 0 and K2 → 1. But the previous analysis, as summarizedin Fig. 9.12, shows that the interval where K2 > 1 is limited byφ4.The limit K1 → 1 (ν0 → φ2) cannot be reached with K2 > 1.Hence, the minimum possible value of rm to be reached by thehyperbolic spiral is nil. In analogy with the elliptic case there is

128 9 Generalized logarithmic spirals: a new analytic solution with continuous thrust

no theoretical limit to how much the orbit can be lowered by ahyperbolic spiral. But there exists an upper limit,

rmin∣∣∣max = lim

ν0→0rmin(ν0) = a0(1 − e0) (9.74)

for the limit radius. That is, for transfers to periapsis the periapsisof the target orbit must be lower than that of the departure orbit.

The upper limit for rmax and the lower limit for rmin are adirect consequence of the continuity of the solution. The limitν0 → φ−1 shows that the maximum radius grows unboundedly,and for ν0 = φ1 the spiral is parabolic. Parabolic spirals are in factelliptic spirals with infinite rmax. On the other hand, the limitν0 → φ−4 in the case of Type II hyperbolic spirals leads to rmin =

0, which defines the limit case K2 = 1 for ν0 = φ−4 .

9.8 Practical Considerations

The fact that the acceleration in Eq. (9.1) decreases with 1/r2 sug-gests that it might be similar to that originated by a solar sail. Fig-ure 9.13 shows an example of a spiral transfer between the Earthand Mars, assuming circular orbits with radii 1 and 1.524 AU re-spectively and neglecting the phasing of the planets. The acceler-ation from the solar sail reads

ap = βµ

r2 cos2 α ns

where β denotes the lightness number, ns is the unit vector nor-mal to the sail, and the pitch angle α is the angle between ns andr. Using a solar sail with constant pitch angle α = 76.95 andlightness number β = 0.35 the spacecraft reaches the orbit ofMars along a logarithmic spiral with ψ = 88 after 980.43 days(seeMcInnes, 2004, pp. 129–136 for details). The same exact tra-jectory can be obtained with a parabolic generalized logarithmicspiral, and the total time of flight is 978.75 days. The accelerationprofiles are not the same meaning that the velocities will be dif-ferent too, causing the difference observed in the times of flight.The figure also depicts a number of elliptic and hyperbolic spi-rals connecting the same departure and arrival points in differenttimes.

Figure 9.13: Spiral transfers from the Earth toMars.

The accelerationprofiles are plotted in Fig. 9.14. First, the fig-ure shows that the magnitude of the acceleration exerted by thesail coincides almost exactly with the acceleration required by aparabolic spiral. Changing the value of K1 the solutions to thetwo-point boundary-value problem are elliptic and hyperbolicspirals. The acceleration profile separates from that of the solarsail as K1 increases in magnitude. Elliptic spirals require higheracceleration levels at first but it decreases rapidly. On the con-trary, the acceleration along hyperbolic spirals grows at first andthen decreases after reaching amaximum. The times of flight varyaccordingly. Hyperbolic spirals are faster than the parabolic onefor a given transfer geometry, whereas elliptic spirals are slower.In practice, small values of K1 will lead to long transfer timeswith small accelerations, whereas high-energy transfers yield shorttimes and high acceleration levels. Parabolic spirals are very lim-ited because of having a constant flight direction angle. Thus, themission analyst seeking low-energy spirals should focus on theelliptic and hyperbolic cases with K1 small. Chapter 11 presentsa design strategy using generalized logarithmic spirals, satisfyingconstraints not only on the position and velocity vectors, but alsoon the time of flight. The technique is based on matching thevalues of the constants of motion, just like the patched conicsmethod in the Keplerian case.

Figure 9.14: Acceleration profile along the spiral transfers.

9.9 Continuity of the solution

Different expressions for the trajectory have been provided de-pending on the type of spiral. The solution is continuous, asproved in this section. Proofs are developed for the raising regimeto simplify the formulation. The definition of the family ofgeneralized spirals will be continuous as long as limK1→0− rell =limK1→0+ rhyp = rpar, where rell, rpar and rhyp denote the equa-tions for the trajectory in the elliptic, parabolic and hyperbolic(Type I) cases. They correspond to Eqs. (9.30), (9.35) and (9.43),respectively.

In addition, the continuity of the solution for the hyperboliccase is determined by the condition limK2→1− rI = limK2→1+ rII =rlim. Here rI and rII denote the trajectory for Type I and Type IIhyperbolic spirals, corresponding to Eqs. (9.43) and (9.50). Thetrajectory in the asymptotic limit K2 → 1, rlim, is defined inEq. (9.57).

§9.10 Conclusions 129

9.9.1 Elliptic to parabolic

The transition from elliptic to parabolic spirals is defined by thelimit K1 → 0− in Eq. (9.30). Considering the argument

β =ℓ

K2(θ − K3) + ln(−K1K2) = α + ln(−K1K2)

the limit K1 → 0− shows that

limK1→0−

2ℓ2

(−K1)(2 − K1K2

2 eα − e−α/K1) = 2ℓ2 eα (9.75)

The argument α = ℓ(θ − K3)/K2 depends on the value of K3,which is given by the definition of θm. The limit K1 → 0− yields

α =ℓ

K2(θ − θ0) − ln

(2ℓ2

r0

)With this result Eq. (9.75) readily becomes the equation for thetrajectory of parabolic (logarithmic) spirals, r = r0 eα′ with α′ =ℓ(θ − θ0)/K2.

9.9.2 Hyperbolic to parabolic

Hyperbolic spirals ofType I define the transition fromhyperbolicto parabolic spirals, so the constant K2 is compatible. Taking thelimit K1 → 0+ in Eq. (9.43) yields limK1→0+ r(θ) = 2ℓ2 eα, withα = ℓ(θ − K3)/K2. In the limit case the constant K3 reduces to

limK1→0+

K3 = θ0 +K2

ℓln

(2ℓ2

r0

)and, as proved for the elliptic case, the equation for the trajectorybecomes limK1→0+ r(θ) = r0 eα′ .

9.9.3 Asymptotic limit from Type I hyperbolic spirals

Let δ > 0 be a parameter such that K2 = 1 − δ. The limit K2 →1− is equivalent to δ → 0+. It can be obtained from the seriesexpansion of the numerator and denominator in Eq. (9.43) andretaining only first order terms, i.e.

limδ→0+

r(θ) =−4δ/K1 + O(δ3/2)

−2[θ − θ0 −

√1 + 2/(K1r0)

]2 − 1δ + O(δ3/2)

=2

K1[(θ −C)2 − 1

]where C = θ0 +

√1 + 2/(K1r0) is equivalent to K3 in the limit

case K2 → 1−, as defined in Eq. (9.53b). The result is indeed theequation of the trajectory for K2 = 1 given in Eq. (9.54).

9.9.4 Asymptotic limit from Type II hyperbolic spirals

Consider the expression defining the axis of symmetry, θm, givenin Eq. (9.48). The trajectory —Eq. (9.50)— can be rewritten as

r =ℓ2

K1

1 + K2 sin β′

K22 cos2 β′ − ℓ2

, β′ =ℓ

K2(θ − K3)

where K3 is defined initially by the initial conditions. Introducingδ > 0 so that K2 = 1+δ, the limit K2 → 1+ transforms into δ→0+. The numerator and denominator in Eq. (9.50) are expandedto provide

limδ→0+

r(θ) =−4δ + O(δ3/2)

−2K1[θ − θ0 −

√1 + 2/(K1r0)

]2 − 1δ + O(δ3/2)

=2

K1[(θ −C)2 − 1

]Again, C ≡ K3 as defined in Eq. (9.53b).

9.10 Conclusions

A priori solutions for shape-based approaches might only repre-sent a particular case of all the possible trajectories rendered by agiven thrust profile. A rigorous approach to the case of logarith-mic spirals shows that entire families of solutions were missing,and that the system admits two integrals of motion. One of thefirst integrals is a generalization of the equation of the energy andallows an energy-based analysis, analogous to the Keplerian case.Three different families of solutions are found: elliptic, parabolicand hyperbolic. Parabolic spirals turn out to be pure logarith-mic spirals. Transitions between the families of spirals have beenproven to be continuous. The existence of upper/lower limitradii yields bounded and symmetric trajectories. The second in-tegral of motion relates to the angular momentum.

The definitions here provided are purely dynamical. The pa-rameters defining the spirals are constants of motion with a clearphysical interpretation, and not shape parameters introduced ar-tificially. This is particularly useful for parametric analyses. Al-though mathematically interesting, the logarithmic spiral has re-ceived little attention for practical applications in the past due toits constant flight direction angle. If one thinks about its Keple-rian equivalent, the parabola, this is also the least practical conicsection from the mission design perspective. Just like Keplerianellipses and hyperbolas are of much more interest, the new ellip-tic and hyperbolic spirals prove more flexible than the parabolicspiral, and are still fully integrable.

“The discussion of the relations of two or more places of a heav-enly body in its orbit as well as in space, furnishes an abundance ofelegant propositions, such as might easily fill an entire volume.”

—Karl Friedrich Gauss

10Lambert’s problem with

generalized logarithmic spirals

L ambert’s problem is solved for the case of a spacecraft ac-celeratedby a continuous thrust. The solution is basedonthe family of generalized logarithmic spirals, which pro-

vides a fully analytic description of the dynamics including thetime of flight and involves two conservation laws (see Chap. 9).Before solving the boundary-value problem an extended versionof the generalized logarithmic spirals is presented, including acontrol parameter. The structure of the solution yields a collec-tion of properties that are closely related to those of theKepleriancase. A minimum-energy spiral is found, with pairs of conjugatespirals bifurcating from it. Thanks to the integral of motion re-lated to the energy the solutions are classified in elliptic, parabolicand hyperbolic. The maximum acceleration reached along thetransfer can be solved in closed form. The problem of design-ing a low-thrust trajectory between two bodies reduces to solv-ing two equations with two unknowns. Double-time opportu-nity transfers appear naturally thanks to the symmetry propertiesof the generalized logarithmic spirals. Comparing the Keplerianand spiral pork-chop plots in an Earth-Mars example shows thatthe spiral solution might increase the mass fraction delivered tothe final orbit, thanks to reducing the magnitude of the impul-sive maneuvers at departure and arrival.

10.1 Introduction to Lambert’s problem

Originally formulated by Leonhard Euler and Johann H. Lam-bert in the 18th century, the problem of finding the orbit thatconnects twoposition vectors in a certain timehas occupiedmanyauthors throughout the years. Due to Lambert’s pioneering con-tributions the problem is typically referred to as Lambert’s prob-

lem. EvenGauss, the Prince ofMathematicians, said that this par-ticular problem is “to be considered among the most importantin the theory of the motions of the heavenly bodies”. He pub-lished the first formal solution to the problem from the mathe-matical point of view in his treaty on the motion of celestial bod-ies in 1809 (Gauss, 1809, §3). He devoted the entire third sec-tion of his book to the problem of relating several places in orbit.Gauss rewrote Kepler’s equation by means of Cotes’ formulas,and established the connection with previous work from Euler.He then furnished a number of transformations to be solved byseries expansions represented by continuous fractions (seeGauss,1809, pp. 112–117). It is interesting to note that Bate et al. (1971)referred to Lambert’s problem as the Gauss’ problem instead, asa tribute to his contributions to orbit determination.

Battin was particularly captivated by this classical problemand is responsible for a large collection of properties of the so-lutions and reviews of existing methods (see Battin, 1999, §6 and§7). He greatly refined and tuned Gauss’ method for numericalimplementation. In fact, his formulation of Gauss’ method (Bat-tin, 1977)was implemented in the onboard guidance computer ofthe NASA space shuttle. Based on his previous discovery of theinvariance of the mean or normal point (Battin et al., 1978), Bat-tin and Vaughan (1984) derived an algorithm that successfully re-moved the singularity in Gauss’ method when the transfer angleequals π. A simplified version of Battin and Vaughan’s methodhas been proposed by Avanzini (2008). He recovered an inter-esting property of the eccentricity vector of the families of solu-tions: its component along the chord remains constant (Battin,1999, p. 256). This property motivated the author to parame-terize the problem in terms of the component of the eccentricityvector normal to the chord, which yields a monotonic evolution

131

132 10 Lambert’s problem with generalized logarithmic spirals

of the time. This algorithm competeswith the originalmethod inaccuracy and computational cost, but it is simpler to implement.

Finding auniversal descriptionof the problemoccupiedLan-caster and Blanchard (1968), Battin (1977) and Bate et al. (1971).Themethod fromLancaster andBlanchard (1968), originally pre-sented in a short note (Lancaster et al., 1966), was further im-proved by Gooding (1990), who focused on the computationalaspects. He relied on Halley’s cubic iteration process (Gander,1985) to solve for the universal variable and showed that typicallyonly three iterations are required for convergence. In the workof Peterson et al. (1991) the performance of different algorithmsis assessed. Sarnecki (1988) investigated the along-track compo-nent of the motion and found a physical interpretation for theuniversal variable introduced by Lancaster and Blanchard (1968),making use of the components of the velocity. Later improve-ments in Lancaster-Blanchard’s method have been proposed byIzzo (2014). He introduced a new Lambert-invariant variableto parameterize the problem. His main contribution is that thecurve of the time of flight exhibits two asymptotes, and it canbe inverted approximately to provide adequate initial guesses forthe iterative procedure. The overall performance of the methodmatches Gooding’s, although the algorithm is simpler. Similarly,Arora andRussell (2013) have recently recoveredBate’s algorithmin order to improve the computational efficiency by introducinga new iteration parameter, which relates to the cosine of the dif-ference in eccentric anomaly. The formulation has been extendedto account for the partial derivatives (Arora et al., 2015). Startingfrom Bate’s method Luo et al. (2011) tackled the quasi-Lambertproblem, where the constraint on the transfer angle is replacedby a constraint on the departure flight-direction angle. The pseu-dostate method was also exploited by Senent (2010) for findingabort return trajectories for manned missions to the Moon. Nonumerical optimizers are required and flyby return trajectoriesare also found by the algorithm. Sun (1981) followed a differ-ent approach by finding a universal differential equation to de-scribe the problem. He made extensive use of hypergeometricfunctions and analyzes carefully the differences with respect toBattin’s method.

The rise of interplanetary exploration has brought renewedinterest into Lambert’s problem for mission design applications.It is the cornerstone of the automated method for gravity-assisttrajectory designproposedbyLonguski andWilliams (1991). Thetransfer legs form a patched-conic solution and the algorithm ad-mits constraints in both the number of admissible revolutionsand the time of flight. Another relevant example is the Star algo-rithm developed at JPL by Landau and Longuski (2006). It hasbeen designed for solving multiple gravity assist problems in anautomatic way, being useful for broad searches and preliminaryinvestigations of candidate trajectories. Sims et al. (1997) pro-posed techniques for maximizing the energy gains from gravity-assist maneuvers. Vasile and De Pascale (2006) posed the prob-lemof designing interplanetary trajectories withmultiple gravity-assists as a global optimization problem. Izzo et al. (2007) fol-lowed a heuristic approach and published search space pruningmethod for finding optimal solutions.

Low thrust is a powerful technique for the design of in-terplanetary trajectories due to the high specific impulse of thepropulsive systems. For this reason, Izzo (2006) explored the po-

tential of exponential sinusoids for solving the accelerated Lam-bert problem. A different solution is that of Avanzini et al.(2015), who wrote the problem in equinoctial elements and de-rived a series solution. The present chapter formulates the spiralLambert problem using the generalized logarithmic spirals pro-posed presented in Chap. 9. The structure of the solution is sur-prisingly similar to that of the Keplerian case. The approach fol-lowed in this chapter is motivated by Battin’s ideas (1999, p. 237):

“[I] have been fascinated by this subject for many yearsand have collected (almost as a hobby as others would col-lect stamps) a number of delightful and often useful prop-erties of the two-body, two-point, boundary-value prob-lem.”

Special attention is paid to formally stating the properties of thesolution. The families of solutions are parameterized using theconstant appearing in the generalized equation of the energy. Aminimum-energy transfer is foundwith pairs of conjugate spiralsemanating from it. Alternative discussions can be found in Roaet al. (2016b); Roa and Peláez (2016b).

This chapter is organized as follows. Section 10.3 solves thetwo-point boundary-value problem with a continuous acceler-ation and free transfer time. The minimum-energy spiral, thefamilies of solutions, and their properties are discussed. Section10.4 introduces the constraint on the time of flight to tackle Lam-bert’s problem. The design of repetitive transfers is addressed inSect. 10.5. Finally, Sect. 10.6 analyzes the practical advantages ofusing the spiral solution in place of the Keplerian one when de-signing transfers to Mars. Additional properties and dynamicalconstraints are analyzed in Sects. 10.7 and 10.8.

10.2 Controlled generalized logarithmicspirals

It is worth recalling that generalized logarithmic spirals are the so-lution to the dynamics under the acceleration in Eq. (9.1):

ap =µ

2r2 cosψ t (10.1)

Mission analysts will notice the limitation of this thrust profile:there are no control parameters. Because there are no degrees offreedom the spirals are simply the analytic solution to an initialvalue problem. In addition, if the departure orbit is circular thethrust will always be zero unless an impulsivemaneuver is consid-ered.

In order to overcome this limitation we shall consider themore general perturbation

ap =µ

r2

[ξ cosψ t + (1 − 2ξ) sinψ n

](10.2)

involving both tangential and normal components of the thrustvector. Here ξ is a constant that behaves as a control parameter.Thenormalized gravitational parameterµ = 1will be omitted forbrevity. Note that Eq. (10.2) reduces to Eq. (10.1) when ξ = 1/2(Roa and Peláez, 2016a). The angle between the radial directionand the thrust vector will be denoted φ. Figure 10.1 sketches thegeometry of the problem.

§10.2 Controlled generalized logarithmic spirals 133

Figure 10.1: Geometry of the problem.

The choice of the thrust in Eq. (10.2) is not arbitrary. It isthe natural generalization of the acceleration (10.1), because bothproblems share the same exact properties.

Combining the thrust profile in Eq. (10.2) with the gravita-tional acceleration

ag = −1r2 (cosψ t − sinψ n)

yields the equations of motion projected in the intrinsic frame:

dvdt=ξ − 1

r2 cosψ (10.3)

vddt

(ψ + θ) =2(1 − ξ)

r2 sinψ (10.4)

drdt= v cosψ (10.5)

dθdt=v

rsinψ (10.6)

In the limit case ξ → 1 the thrust compensates the gravitationalattraction of the central body and the velocity remains constant.We consider this a natural limit to the control parameter, so ξ <1. The control parameter ξ can be bounded from limitations onthe ratio between the thrust acceleration and the local gravity.

The problem can be transformed by means of a linear appli-cation S of the form

S : (t, v, r, θ, ψ)→ (τ, v, r, θ, ψ)

involving the constant parameters α, β, and δ:

τ =tβ, r =

rα, v =

v

δ

The parameter α is simply a dilation of the solution, so it can beset to unity without loss of generality. In order the parameters tobe physically compatible it must be

β =

√α3

2(1 − ξ) and δ =α

β=

√2(1 − ξ)

α

Under the action of S Eqs. (10.3–10.6) transform into

dvdτ= − cosψ

2r2

vddτ

(ψ + θ) = +sinψ

r2

drdτ= v cosψ

dθdτ=v

rsinψ

These equations are none other than the equations of motion re-sulting from the acceleration in Eq. (10.1), Eqs. (9.8–9.11). Thatis, S is a similarity transformation that transforms the problemperturbed by the thrust acceleration in Eq. (10.2) to the simplifiedproblem governed by Eq. (10.1). The solution to the more gen-eral problem can be easily obtained bymapping the inverse trans-formation S−1 over the simplified solution r(θ) and τ(r), whichcorresponds to that in Chap. 9. There is no need to re-derive thesolution including the control parameter ξ. The time of flight,trajectory, and velocity of the general problem is obtained fromthe simplified one thanks to

t = β τ, r = α r, v = δ v

Due to the importance and reach of such an intriguing transfor-mation the entire Chap. 12 is devoted to analyzing this transfor-mation, and new families of spirals will be obtained. For prac-tical reasons in this chapter we only recover the basic propertiesrequired for designing orbit transfers. The complete reformula-tion of the solution with the control parameter can be found inAppendix F.

The constants ofmotion K1 and K2 relate to the transformedvalues, κ1 and κ2, by means of

K1 = δ2κ1 = v

2 − 2(1 − ξ)r

(10.7)

and also

K2 = αδ2κ2 = rv2 sinψ (10.8)

The fact that the integrals of motion (10.8) and (9.16) have thesame form justifies the choice of the acceleration (10.2).

The flight-direction angle is defined in terms of

sinψ =K2

2(1 − ξ) + K1r(10.9)

It is important to note that the constant κ2 is not affected by thevalue of the control parameter ξ. The apoapsis radius for ellipticspirals is

rmax =2(1 − ξ) − K2

(−K1)

and the periapsis of a hyperbolic spiral of Type II reads

rmin =K2 − 2(1 − ξ)

K1

Hyperbolic spirals ofType I correspond to K2 < 2(1−ξ), equiva-lent to κ2 < 1, and spirals of Type II are those with K2 > 2(1−ξ)or κ2 > 1.

134 10 Lambert’s problem with generalized logarithmic spirals

The tangent and normal vectors, t and n, can be referred tothe orbital frameL = i, j,k,

i =rr, k =

hh, j = k × i

by means of

t = cosψ i + sinψ j and n = − sinψ i + cosψ j

The thrust acceleration from Eq. (10.2) becomes

ap =1r2

[ξ cos2 ψ − (1 − 2ξ) sin2 ψ] i + (1 − ξ) sinψ cosψ j

The angle between the thrust direction and the radial direction,φ, is defined by

tanφ =(1 − ξ) sinψ cosψξ − (1 − ξ) sin2 ψ

=(1 − ξ) sin 2ψ

3ξ − 1 + (1 − ξ) cos 2ψ(10.10)

For ξ = 1/2 it follows φ = ψ, which is the case of purely tan-gential thrust discussed in Chap. 9. For ξ = 1/3 this equationreduces to φ = 2ψ. That is, the velocity lies on the bisection be-tween r and ap.

As discussed above, the original family of generalized loga-rithmic spirals (ξ = 1/2) requires an impulsive ∆v for departingfrom circular orbits. This comes from the combination of twofactors: First, the velocity on the circular orbit is v = r−1/2, whatmakes K1 = 0 and the trajectory is a pure logarithmic spiral. Inthis case the flight-direction angle ψ is constant. Second, on a cir-cular orbit it is always ψ = π/2 so the thrust vanishes. Since ψremains constant along the logarithmic spiral the thrust acceler-ation will always be zero and the particle describes a Kepleriancircular orbit. In the general case ξ , 1/2, the presence of a nor-mal accelerationovercomes this limitation, and impulsivemaneu-vers canbe avoided. When applying the similarity transformationS−1 it is

v(r) = δv =

√2(1 − ξ)

r

meaning that it is possible to describe the same trajectory withdifferent velocities. When ξ , 1/2 logarithmic spirals are suitablefor departing from circular orbits.

10.3 The two-point boundary-value prob-lem

Consider the problem of finding a generalized logarithmic spiralS that connects a departure point P1, defined by (r1, θ1), witha final point P2, defined by (r2, θ2), with θ2 > θ1. The posi-tion vectors for P1 and P2 are r1 and r2, respectively. In orderto account for the number of revolutions n, θ2 is decomposed inθ2 = θ2 + 2nπ, with n = 0, 1, 2 . . . and θ2 ∈ [0, 2π). Changingthe number of revolutions is equivalent to modifying the geom-etry of the boundary-value problem (BVP). No constraints areimposed on the time of flight in this section. In order the trajec-tory to depart from P1 and arrive to P2 the following equationmust be satisfied:

r2 = r(θ2; ξ,K1,K2, r1, θ1) (10.11)

The function r(θ; ξ,K1,K2, r1, θ1) is the equation of the trajec-tory and depends on the type of spiral. The polar angle θ is theindependent variable, the constants K1 and K2 define the shapeof the spiral, (r1, θ1) are the initial conditions at P1, and ξ is thecontrol parameter.

In the Keplerian case the minimum-energy ellipse plays a keyrole in the configuration of the transfers (Battin, 1999, pp. 240–241). The Keplerian energy (or equivalently the semimajor axisof the orbit) yields an intuitive parameterization of the solutions.An analogous parameterization can be introduced for the spiralcase in terms of the constant K1. With this technique the solu-tions are easily classified in elliptic, parabolic or hyperbolic spiraltransfers.

Given the boundary conditions (r1, θ1) and (r2, θ2), the con-trol parameter ξ, and a certain value of K1 the problem reducesto solving for K2 in Eq. (10.11). Note that once the value of K1 isfixed, the departure and terminal velocities are known:

v1 =

√K1 +

2(1 − ξ)r1

and v2 =

√K1 +

2(1 − ξ)r2

If the initial and final radii are the same, r1 = r2, then the depar-ture and terminal velocities in a generalized spiral trajectory arealso equal, v1 = v2.

Since K1, ξ and r1 are fixed Eq. (10.9) shows that solving forK2 is equivalent to solving for the initial flight-direction angle,ψ1. In fact, due to the C-symmetry of the spirals it is more con-venient to solve for the departure angleψ1 in order not to lose anysolution, and then compute the value of K2 from Eq. (10.9).

Elliptic spirals are those with the minimum value of the con-stant K1. In this case the boundary-value problem reduces tofinding the zeros of the function

f (ψ1; ξ,K1, r1, r2) ≡ r2

rmax− 2(1 − ξ) + K2

2(1 − ξ) + K2 cosh β2= 0 (10.12)

where β2 is the spiral anomaly at P2 and rmax is the apoapsis of thespiral, defined in Eq. (F.10).

Figure 10.2 shows the effect of the constant K1 on the zerosof the function f (ψ1) for an example transfer geometry. This fig-ure proves that a minimum value of K1 exists: for K1 > K1,minthere are two solutions, that collapse toonedouble solutionwhenK1 = K1,min. If K1 < K1,min the function has no roots: the BVPhas no solution.

Figure 10.2: Zeros of the function f (ψ1) for increasing values ofK1 < 0. In this example r2/r1 = 2, θ2 − θ1 = 2π/3, and ξ = 1/2.

§10.3 The two-point boundary-value problem 135

10.3.1 The minimum-energy spiral

LetSdenote the set of all generalized logarithmic spirals S that aresolutions to the BVP. The elliptic spiral for which K1 = K1,min isreferred to as the minimum-energy spiral, Sm ∈ S. The constantof the energy relates to the specific Keplerian energy Ek bymeansof

K1 = 2(Ek +

ξ

r

)(10.13)

Consider two spirals Si, S j ∈ S, with i , j. Thanks to Eq. (10.13)it is possible to compute the value of K1 for both spirals from thevalues of the energy of the osculating orbits at r1, Ek,i and Ek, j.If Si is the minimum-energy spiral (i ≡ m), then the osculatingKeplerian energy at r1 is also minimum. The osculating orbits atP1 and P2 for theminimum-energy spiral are thosewith themini-mumKeplerian energy and semimajor axis. When the thrust van-ishes the osculating orbits at P1 and P2 coincide, and correspondto the transfer orbit. If they are the minimum-energy orbits theywill reduce to theminimum-energyKeplerian transfer. Thus, theminimum-energy spiral can be seen as the generalization of theminimum-energy ellipse.

The value of K1,min can be obtained through the followingnumerical procedure. The function f (ψ1) defined in Eq. (10.12)reaches a minimum at ψ∗1, so that f ′(ψ∗1; ξ,K1, r1, r2) = 0. Theprime ′ denotes the derivative with respect to ψ1. The min-imum value of K1 for which a solution to the boundary-valueproblem exists corresponds to the value that yields only one rootof the function f (ψ1). That is, when the minimum of f (ψ1) isexactly zero. Given the control parameter and the boundary con-ditions r1 and r2 theminimum-energy transfer, defined by K1,minand ψ∗1, is the spiral that satisfies

f (ψ∗1; ξ,K1,min, r1, r2) = 0 (10.14)f ′(ψ∗1; ξ,K1,min, r1, r2) = 0 (10.15)

The first condition forces the minimum to be zero, so there isonly one root, and the second equation determines where theminimum is located. Note that K1,min is defined in terms ofθ2 = θ2 + 2nπ and takes different values depending on the num-ber of revolutions n, even if θ2 remains the same.

Figure 10.3 shows how the value of K1,min depends on the ge-ometry of the transfer. As the transfer angle increases the value ofK1,min decreases inmagnitude, meaning that increasing the num-ber of revolutions reduces the energy of the transfer. The evo-lution of the values of K1,min proves that limθ2→∞ K1,min = 0−;transfers with negative K1 always exist, and the magnitude of theconstant K2 can be arbitrarily small. This yields an important re-sult, which is that the BVP defined by r1 and r2 will always admitthree types of solutions: elliptic, parabolic and hyperbolic spiraltransfers.

10.3.2 Conjugate spirals

There are two solutions to Eq. (10.12) for each value of the con-stant of the energy and control parameter. This behavior is inti-mately related to the conjugateKeplerian orbits that appear in the

Figure 10.3:Minimum K1 for different tranfer geometries. In this

figure each curve corresponds to a different value of r2/r1.

unperturbed form of Lambert’s problem (Battin, 1999, p. 244):

Definition 5: (Conjugate spirals) Two generalized logarithmicspirals S , S ∈ S are said to be conjugate if they share the samevalue of the constant of the energy K1 and control parameter ξ.The conjugate of S is denoted S .

Figure 10.4 shows the spiral transfer corresponding to K1,minand two pairs of conjugate spirals, Si and S i. A spiral and its con-jugate are always separated by theminimum-energy spiral. As K1becomes more negative both trajectories come closer, and theycollapse into one single trajectory, Sm, for K1 = K1,min. In thisexample both the spiral Si and its conjugate S i reachP2 in the low-ering regime, after having crossed the maximum radius rmax.

Figure 10.4: Examples of pairs of conjugate spiral transfers and the

minimum-energy spiral.

Given a pair of conjugate solutions to the BVP, one will befaster than the minimum-energy transfer, whereas its conjugatewill be slower. Conjugate spirals can be characterized by a collec-tion of properties connecting the values that certain variables takeat departure and arrival, summarized in Sect. 10.7. For example,the conservation of Eq. (10.7) proves that the departure and ar-rival velocities are the same for two conjugate spirals, i.e. v1 = v1and v2 = v2.

136 10 Lambert’s problem with generalized logarithmic spirals

10.3.3 Families of solutions

Parameterizing the solutions of the BVP in terms of the con-stant K1 yields a natural classification of the orbits: starting fromK1,min < 0, the spirals are elliptic in the interval K1 ∈ [K1,min, 0);solutions with K1 = 0 are parabolic spirals, and for K1 > 0 thespirals are hyperbolic.

Figure 10.5 shows an illustrative example of the families of so-lutions. First, note that the minimum-energy spiral separates thefast and slow transfers along elliptic spirals. Reducing the value ofK1 makes the pairs of conjugate spirals converge to theminimum-energy spiral. On the contrary, increasing K1 reduces the size ofthe fast transfers and increases the size of the slow ones. The lim-its are set by the corresponding parabolic spirals (K1 = 0). Thesolution to the transfer along a parabolic (logarithmic) spiral canbe given in closed form. The constraint in Eq. (10.11) combinedwith Eq. (F.19) provides

r2 = r1 e(θ2−θ1) cotψ1 =⇒ ψ1 = arctan[θ2 − θ1

ln(r2/r1)

](10.16)

and defines the departure flight-direction angle. If the spiral is inthe lowering regime, r2 < r1, the solution to Eq. (10.16) is nega-tive and then the departure angle takes the value π + ψ1. Equiva-lently the value of K2 results in

K2 =b

√1 + b2

with b =θ2 − θ1

ln(r2/r1)(10.17)

This value of K2 is unique no matter the regime of the spiral,thanks to the C-symmetry of the trajectory. Note that Eq. (10.16)has only one solution: the second conjugate solution is not real,since it connects both points through infinity. That is equivalentto an elliptic spiral with rmax → ∞. If K1 increases it becomespositive and yields hyperbolic spirals. Pairs of conjugate solu-tions also exist in this case, one of them being a fictitious solutionthrough infinity. The fast transfers are below the parabolic trans-fer and connect the two points directly. In the limit K1 → ∞the hyperbolic spiral degenerates into a rectilinear orbit along thechord connecting r1 and r2. The conjugate solutions connect thetwo points by reaching infinity, so the solution does not exist inpractice. In the multi-revolution case the transfer for K1 → ∞decomposes into two rectilinear segments connecting P1 with theorigin and the origin with P2.

Figure 10.5: Families of solutions for r2/r1 = 2, θ2 − θ1 = 2π/3,and fixed ξ.

The existence of conjugate spirals and the discussed behaviorcan be understood from Fig. 10.6. There are two elliptic spiraltransfers with the same value of K1. For K1 = K1,min the pair ofconjugate spirals converge to theminimum-energy spiral, and forK1 → 0− the apoapsis of the slow transfer becomes infinite. Thebranch of fast transfers (ψ1 > ψ1,m) exists for both the parabolicandhyperbolic cases, but the slow transfers (ψ1 < ψ1,m) exist onlyfor the elliptic case. The branch of conjugate slow hyperbolic so-lutions corresponds to fictitious transfers through infinity, justlike Keplerian hyperbolas. Despite the fact that the minimum-energy spiral Sm always departs with ψ1 ∈ (0,π/2), this con-straint does not apply to generic fast transfers: arbitrary solutionswith ψ1 ≥ π/2 may be found.

Figure 10.6: Departure flight-direction angle as a function of the

constant of the energy K1 for fixed values of ξ. Different transfergeometries are considered, keeping constant the radii r2/r1 = 2.

The trajectory of a particle following a hyperbolic spiral(K1 > 0) takes different forms depending on the value of K2,the transition being K2 = 2(1 − ξ). It is possible to find a limitvalue of K1, K1,tr, such that if K1 < K1,tr then S ∈ S is hyperbolicof Type I, and of Type II for K1 > K1,tr. It reads

K1,tr =4(1 − ξ)[2 √

(r2 − r1)2 + r1r2∆θ2 − (r2 + r1)∆θ]

r1r2∆θ(4 − ∆θ2)(10.18)

with∆θ = θ2−θ1. When K1,tr → 0 all hyperbolic spirals in S areof Type II. It is interesting to note that for transfers where r2 = r1it is K1,tr = 0, which proves that for r1 = r2 all hyperbolic spiralsS ∈ S are of Type II.

For r1 = r2 the direction of the axis of T-symmetry θm re-duces to θm = (θ2 + θ1)/2. In his book, Battin (1999, pp. 250–256) discusses a number of properties involving the bisection ofthe transfer angle. In the spiral case we found that for transferswith r1 = r2 all spiral solutions areT-symmetric and the axis ofT-symmetry coincides with the bisection of the transfer angle.

The dynamics of the transfer depend on whether the spiraltransitions between regimes or not. For a formal treatment of thesolutions we introduce the following definition:

Definition 6: (Direct and indirect transfers) A spiral transferS ∈ S is direct if the evolution of the radius r from r1 to r2 is

§10.3 The two-point boundary-value problem 137

monotonic. On the contrary, if r reaches a minimum or maxi-mum value during the transfer, then the solution is an indirecttransfer. Indirect transfers can only be elliptic or hyperbolic spiralsof Type II, and the axis of T-symmetry lies between θ1 and θ2,θm ∈ [θ1, θ2].

To determine whether a solution S ∈ S is direct or indirectone should solve for θm in Eq.(F.24) and check if it lies between θ1and θ2. Similarly, these relations are useful when looking explic-itly for direct or indirect transfers. All spiral transfers connectingr1 = r2 are indirect. The case K1 = 0 yields a degenerate indirecttransfer, corresponding to a circular Keplerian orbit.

10.3.4 The thrust acceleration

The thrust acceleration profile is defined in Eq. (10.2), and de-pends on the radial distance and the flight-direction angle. Figure10.7 depicts the evolution of the acceleration due to the tangentialthrust (ξ = 1/2) for different types of spirals. Initially the accel-eration ismaximum for slow elliptic transfers. It decreases rapidlyand vanishes for r = rmax. When the spiral transitions to loweringregime the acceleration changes its sign. The figure shows howthe acceleration profile resembles the typical thrust-coast-thrustsequence. The minimum-energy transfer separates the slow andthe fast elliptic spiral transfers. The acceleration along hyperbolicspirals of Type II is zero for r = rmin, and also ap → 0 as r → ∞.That is, the magnitude of the thrust acceleration decreases alongthe asymptotes. The thrust along parabolic or Type I hyperbolicspirals never vanishes, because there are no transitions betweenregimes.

Figure 10.7: Dimensionless thrust acceleration along different types

of spiral transfers with ξ = 1/2.

In practice, knowing a priori the maximum value of thethrust greatly helps in the design process, since solutions that re-quire a propulsive acceleration over the admissiblemaximum canbe easily discarded. Themagnitude of the acceleration due to thethrust ap takes the form

ap(r) =1r2

√ξ2 cos2 ψ + (1 − 2ξ)2 sin2 ψ

=

√ξ2[2(1 − ξ) + K1r]2 + K2

2 (1 − 4ξ + 3ξ2)

r2[2(1 − ξ) + K1r]

The thrust dependsonK1, K2, the radial distance, and the controlparameter ξ. Once the transfer spiral is selected the problem offinding themaximumvalue of ap(r) reduces to finding the radiusr∗ that maximizes ap(r), so that

ap,max =

√ξ2[2(1 − ξ) + K1r∗]2 + K2

2 (1 − 4ξ + 3ξ2)

r∗2[2(1 − ξ) + K1r∗](10.19)

The maximum acceleration exerted by the thrust ap between P1to P2 occurs at r∗ = min(r1, r2), except for the case of hyperbolicspirals of Type II, where r∗ is given by:

ξ ∈ [ξ−, ξ+] :

min(r1, r2), rQ ≤ min(r1, r2), Dmax(r1, r2), rQ ≥ max(r1, r2), DrQ , rest

ξ < [ξ−, ξ+] :

rmin, Imin(r1, r2), rest

(10.20)

Here D and I denote direct and indirect transfers, respectively.The value of rQ is:

rQ =6K1K2(1 − ξ)

R1/3

[(1 − 3ξ)K2 cos

(ϑ + 2π

3

)− ξ

√2(1 − ξ)(3ξ − 1)

]where

R = 54K61 K3

2ξ3√

2(3ξ − 1)3(1 − ξ)3

ϑ = arctan

√

2[K22 (3ξ − 1) − 2ξ2(1 − ξ)]

2ξ√

1 − ξ

The limits ξ+ and ξ− are the solutions to the equation

|1 − 2ξ|r2min

=

√ξ2[2(1 − ξ) + K1rQ ]2 + K2

2 (1 − 4ξ + 3ξ2)

rQ2[2(1 − ξ) + K1rQ ]

and determine the values of ξ that make the acceleration at theperiapsis of the spiral to match the acceleration at rQ . The con-straint on K2 becomes:

K22 ≤

a2p,maxr

∗4[2(1 − ξ) + K1r∗]2 − ξ2[2(1 − ξ) + K1r∗]2

1 − 4ξ + 3ξ2

This expression leads to a criterion for bounding the search in thespace of solutions.

10.3.5 The ∆v due to the thrust

The total ∆v imparted to the particle by the continuous thrust isdefined by the integral

∆vthr =

∫ t2

t1||ap|| dt

which in general needs to be evaluated numerically. However, forthe special case ξ = 1/2 (purely tangential thrust) this integral

138 10 Lambert’s problem with generalized logarithmic spirals

can be solved in closed form, admitting two different solutionsdepending on whether the spiral transfer is direct or indirect. Forthe direct case and ξ = 1/2 it is

∆vthr =

∣∣∣∣∣∣√

K1 +2(1 − ξ)

r1−

√K1 +

2(1 − ξ)r2

∣∣∣∣∣∣ = |v1 − v2|

(10.21)

The minimum-energy spiral involves the maximum change be-tween the departure and arrival velocities. Thus, if theminimum-energy spiral defines a direct transfer, then the resulting ∆vthr isthe maximum among all the possible direct transfers with ξ =1/2. If the spiral transfer is indirect it reaches amaximumormin-imum radius rm during the transfer, and the value of ∆vthr is:

∆vthr = |v1 − vm| + |v2 − vm| = |v1 + v2 − 2vm| (10.22)

10.3.6 The locus of velocities

Figure 10.8 depicts the locus of velocity vectors projected onskewed axes (Battin, 1999, p. 244). The departure velocity vectorv1 decomposes in

v1 = vρ u1 + vc uc

where u1 = r1/r1 and uc is the unit vector along the chord con-necting P1 and P2. The figure shows that the locus of minimumdeparture velocities, vm = ||vm||, is tangent to the locus of solu-tions at the point where K1 = K1,min: the departure velocity forSm is the minimum among all possible transfers. Increasing thedeparture velocity yields two intersection points with the locusof solutions. When the velocity becomes equal to [2(1 − ξ)/r]1/2

for r = r1 the solutions are parabolic spirals, with

vp =

√2(1 − ξ)

r1(cosψ1 u1 + sinψ1 u2)

Here, vector u2 is defined by the inplane perpendicular to u1, incounter-clockwise direction, and ψ1 is given in Eq. (10.16). Thevelocity vk corresponds to the magnitude of the departure veloc-ity for theminimum-energy ellipse in the Keplerian case. It is ob-served that a spiral transfer allows to reduce the minimum veloc-ity required for the transfer, vm < vk. The required ∆v for leav-ing the departure orbit will be smaller than the one of the bal-listic case. This is an important improvement with respect to thelogarithmic spirals (Petropoulos et al. (1999) andMcInnes (2004)showed that logarithmic spirals typically require higher v∞ at de-parture than the ballistic case). Increasing the magnitude of thedeparture velocity yields hyperbolic spiral transfers. This branchof solutions converges asymptotically to the chord connecting P1and P2.

10.4 Fixing the time of flight

Generalized logarithmic spirals admit closed-form solutions forthe time of flight. They depend on the type of spiral. This sectionanalyzes the spiral transfers from r1 to r2 given a constraint on thetime of flight.

Figure 10.8: Locus of velocity vectors projected on skewed axes for

ξ = 1/2.

The simplest solution is the transfer along a parabolic spiral(K1 = 0), which corresponds to a logarithmic spiral. CombiningEqs. (F.20) and (10.16) the time of flight reduces to

t2 =2√

2(1 − ξ)[ln2(r2/r1) + (θ2 − θ1)2]

3 ln(r2/r1)(r3/2

2 − r3/21

)(10.23)

This expressiondepends only on the boundary conditions. In thelimit case r2 = r1 and ξ = 1/2 this expression provides

limr2→r1

t2 = limr2→r1

23

(θ2 − θ1)

r3/22 − r3/2

1

ln(r2/r1)

= (θ2 − θ1) r3/21

which is in fact the normalized time of flight corresponding to acircular Keplerian orbit of radius r1.

Figure 10.9 shows the influence of the constant of the energyin the time of flight from P1 to P2. As K1 → 0− the branch offast solutions converges to the fast parabolic transfer. It definesthe transition from elliptic to hyperbolic spirals. Note that K1 =

0 behaves as a vertical asymptote: the parabolic solution alongthe slow branch requires an infinite time to reach P2. This is dueto the fact that the slow parabolic transfer connects both pointsthrough infinity, as discussed from Fig. 10.5. Increasing K1 fromthat point and along the slow branch yields the fictitious set ofhyperbolic solutions that connect P1 and P2 through infinity.

Figure 10.9: Dimensionless time of flight parameterized in terms of

the constant of the energy, K1.

§10.5 Repetitive transfers 139

Given the time of flight, increasing the number of revolu-tions of the spiral transfer increases the value of K1 of the solu-tion. Consequently, increasing the number of revolutions of di-rect transfers always reduces the total ∆v due to the thrust.

The parameterization used to construct Fig. 10.9 shows thatthe time of flight does not have any minima. Given the timeof flight, the geometry of the transfer and the control parame-ter ξ there is only one value of K1 that satisfies the constraints.This statement holds for a given number of revolutions (definedas part of the geometry of the transfer) and for prograde orbits.There is an equivalent retrograde spiral transfer, ψ1 ∈ (π, 2π),with a different time of flight, unless the transfer is symmetricwith respect to θ1. Similarly, from Fig. 10.9 it follows that differ-ent solutions to Lambert’s problem can be obtained by changingthe number of revolutions, i.e. the transfer angle.

The constraint on the time of flight leaves one degree of free-dom associated to the control parameter ξ, apart from the num-ber of revolutions and the selection of prograde/retrograde mo-tion. The spiral Lambert problem translates into solving for K1and ψ1 in

r2 = r(θ2; ξ,K1, ψ1, r1, θ1) (10.24)t2 = t(r(θ2); ξ,K1, ψ1, r1, θ1) (10.25)

Recall that the transfer angle θ2 accounts for the number of rev-olutions n, so that θ2 = θ2 + 2nπ with θ2 ∈ [0, 2π). The retro-grade solutions can be found by solving a complementary prob-lemwhere θ2 is replaced by θ2,ret, defined as θ2,ret = 2(n+1)π−θ2.The solution to Eqs. (10.24) and (10.25) obtained for the comple-mentary problem, ψ1,ret, is modified so that it yields a retrogradeorbit, i.e. ψ1 = 2π − ψ1,ret. Figure 10.10 depicts a family of so-lutions to the spiral Lambert problem obtained by changing thevalue of the control parameter ξ. The time of flight is the samefor all the spirals in the figure.

Figure 10.10: Family of solutions to the spiral Lambert problem pa-

rameterized in terms of the control parameter ξ.

An initial guess (K1, ψ1) is required in order to initialize theiterative procedure. Thus, given the geometry of the transfer, thetime of flight, and the control parameter, the first step is to com-puteK1,min andψ∗ fromEqs. (10.14–10.15). For theparabolic caseψ1 has been solved in closed form in Eq. (10.16). The correspond-ing times of flight follow naturally and it is now possible to deter-mine the type of solution and provide a consistent initial guess.The next step is to solve for K1 andψ1 in Eqs. (10.24–10.25). Themethod of generating an initial guess can be obviated if a betterestimate is available.

10.5 Repetitive transfers

Assume that an elliptic spiral trajectory intersects the axis of sym-metry in the raising regime. Due to the T-symmetry of the tra-jectory it will always intersect again the axis at that exact point inthe lowering regime after sufficient time. Hence, if the axis ofsymmetry is aligned with r2 and the spiral connects P1 and P2 inm revolutions, then it will always pass again through P2 after nrevolutions. We refer to this kind of transfers as m : n repetitivetransfers. The departure flight-direction angle can be selected sothat the integers m and n take prescribed values.

When defining this kind of transfers K1 is fixed and cannotbe decided by convenience: it depends on the geometry of thetransfer. The simplest way to formulate the problem is to solvefor K1 and ψ1 in the system:

r2 = r(θ2 + 2mπ; ξ,K1, ψ1, r1, θ1)r2 = r(θ2 + 2nπ; ξ,K1, ψ1, r1, θ1)

The solution depends on the configuration of the m :n sequence.Solutions only exist for elliptic spirals and hyperbolic spirals ofType II.

Figure 10.11 depicts two examples of different repetitivetransfers. The solutions are indirect, by definition. If m + n iseven then themaximum is oriented in the direction of θ2, whereasif m + n is odd then the maximum corresponds to θ2 + π. Themagnitude of the maximum radius depends on n−m and on thevalue of m, which indicates how soon the point P2 is reached forthe first time.

(a) m = 0, n = 1 (b) m = 1, n = 5

Figure 10.11: Examples ofm : n repetitive transfers for r2/r1 = 2and θ2 − θ1 = 2π/3.

During the transfer and along the spiral the values ofK1,K2, ξ are constant. This means that the only difference be-tween crossings is the regime of the spiral, and the velocities afterm and n revolutions (respectively v2 and v′2) are the same, v2 = v

′2.

Similarly, the conservation of K2 shows that the pair of solutionsare C-symmetric. It then follows ψ2 +ψ

′2 = π. Equation (10.28)

proves that the thrust acceleration atP2 only depends on K1,K2and r2. Thismeans that themagnitude of the thrust accelerationsatP2 for consecutive passes is the same, but one favors the velocitywhereas the other opposes it.

The search for repetitive transfers is an additional techniquefor controlling the time of flight. Although the configuration of

140 10 Lambert’s problem with generalized logarithmic spirals

the transfer is given by discrete values of m and n, it might be use-ful for different mission scenarios. Figure 10.12 shows the time offlight for the two repetitive passes, t2 and t′2, for different m : nsequences. It is interesting to note that increasing the number ofrevolutions between the two passes reduces t2, while the time offlight t′2 increases. For constant n increasing m yields slower trans-fers. Similarly, for constant m larger values of n increase the timeof flight. When m → ∞ the difference in the time of flight forthe two passes decreases, and the time of flight approaches a limitvalue. The existence of a finite limit relates to the fact that increas-ing the number of revolutionsmakes K1,min smaller inmagnitudeuntil Sm becomes parabolic in the theoretical limit (see Fig. 10.3).

Figure 10.12: Dimensionless time of flight for differentm : n repeti-

tive configurations. The size of themarkers scales with the values of

the index n > m. Diamonds represent the time to the first pass, and

circles correspond to the second pass.

10.6 Evaluating the performance

The simplest approach to the design of an orbit transfer betweentwo bodies is to connect them using a Keplerian arc. This meanssolving the ballistic Lambert problem given the state of the firstbody at departure, and the state of the second body at arrival.Two impulsive maneuvers are required (one at departure, andone at arrival) in order to meet the conditions on the velocity. Inthis chapter, we explored the potential of replacing the Keplerianarc with a continuous-thrust arc. This allows to reduce the mag-nitude of the impulsive maneuvers, which leads to more efficienttransfers (see Table 1.2 for a comparison of specific impulses).In this section, we will evaluate the performance of the new de-sign strategy by comparing it to the purely ballistic approach, us-ing the 2016–2018 Mars launch campaign as an example. Theyare compared in terms of the required propellant mass fraction,the departure characteristic energy C3, and the arrival v∞. Theephemeris of the Earth and Mars are retrieved from the DE430ephemeris, using SPICE.

The overall performance of the ballistic and accelerated trans-fers can be compared in Figs. 10.13(a) and. 10.13(d). The pork-chop plot in Fig. 10.13(a) shows the two launch windows cor-responding to the 2016 and 2018 period. The figures show thepropellant mass fraction required for the transfer. This bud-get includes the impulsive maneuvers required at launch andto rendezvous with Mars. In the accelerated case, the propel-

lant expenditures associated to the continuous-thrust phase arealso accounted for. The specific impulses for the impulsive andcontinuous-thrust maneuvers are Isp = 350 s and Isp = 3500 s,respectively. The control parameter is fixed to ξ = 1/2 (purelytangential thrust). Equations (10.24–10.25) are solved with aLevenberg-Marquardt algorithm. The spiral transfers are re-stricted to the zero-revolutions case (θ2 − θ1 < 2π). With theballistic solution more than 80% of the total mass budget will beoccupied with propellant. The spiral transfer is able to reducethis figure to about 50%, which brings a significant increment inthe payload mass delivered to the final orbit. Although the opti-mal spiral transfers require twice the time of flight compared tothe Keplerian ones, the launch windows are broader and, for thesame time of flight, the spiral solution still outperforms the bal-listic approach.

Figures 10.13(b) and 10.13(e) depict the departure C3 re-quired for the transfer. The spiral solution extends againthe launch opportunities, with transfers with C3 as low as0.03 km2/s2. This is a significant reduction compared to the7.73 km2/s2 expected for May, 2018.

Finally, Figs. 10.13(c) and 10.13(f) focus on the arrival v∞when reaching Mars. The main advantage of the accelerated ap-proach is its flexibility, as it extends the launch opportunities. Inaddition, the optimum solutions reduce the v∞ atMars by aboutone order of magnitude.

10.7 Additional properties

In this section, additional properties of the spiral two-pointboundary-value problem are presented. They have been groupedin this section to simplify the reading of this chapter.

Property 1: (Change in the semimajor axis) Given ξ, the changein the inverse of the semimajor axis between two points of the spiraldepends on r1 and r2 alone, and reduces to:

1a2− 1

a1= 2ξ

(1r2− 1

r1

)

Property 2: (Minimum velocity) The departure and terminalvelocities, v1 and v2, are minimum on the minimum-energy spiral.

Property 3: (Change in the velocity and semimajor axis) Thechange in the velocity from P1 to P2, ∆v = |v2 − v1|, is maximumalong the minimum-energy spiral. Similarly, the change inthe semimajor axis, |∆a| = |a2 − a1|, is minimum along theminimum-energy spiral.

In order to establish the properties of conjugate spirals it isimportant to note that:

Lemma 1: If a certain variable γ depends only on K1 and r,γ = γ(r; ξ,K1), then γ1 = γ1 and γ2 = γ2 for all pairs ofconjugate spirals. Here γ1 = γ(r1) and γ2 = γ(r2).

This Lemma is a powerful contrivance for finding variables that

§10.8 Additional dynamical constraints 141

(a) Propellant mass: Ballistic (b) LaunchC3 : Ballistic (c) Arrival v∞ : Ballistic

(d) Propellant mass: Accelerated (e) LaunchC3 : Accelerated (f) Arrival v∞ : Accelerated

Figure 10.13: Comparison of the ballistic and accelerated pork-chop plots for the Earth toMars transfers.

take the same values on a spiral and its conjugate, and provides

Property 4: (Departure and terminal velocities) Two conjugategeneralized logarithmic spirals S and S have the same departureand terminal velocity, i.e. v1 = v1 and v2 = v2.

Property 5: (Departure and terminal semimajor axes) Twoconjugate spirals S and S have the same departure and terminalosculating semimajor axes, i.e. a1 = a1 and a2 = a2.

Property 6: (Angular velocities) The ratio between the initialand terminal angular velocities is the same for two conjugatespirals, θ1/θ2 =

˜θ1/˜θ2.

Property 7: (Flight-direction angle) The ratio between theinitial and terminal values of sinψ is the same for two conjugatespirals, sinψ1/sinψ2 = sin ψ1/sin ψ2.

Property 8: (Circumferential velocities) The ratio between theinitial and terminal circumferential velocities vθ = v sinψ is thesame for two conjugate spirals, vθ1/vθ2 = vθ1/vθ2.

Property 9: (Angular momentum) The ratio between the

initial and terminal angular momenta (or equivalently theorbital parameter, p) is the same for two conjugate spirals,h1/h2 = h1/h2 and p1/p2 = p1/p2.

Property 10: (Eccentricity) The initial and terminal valuesof the eccentricity of two conjugate spirals satisfy the relation(1 − e2

1)/(1 − e22) = (1 − e2

1)/(1 − e22).

10.8 Additional dynamical constraints

Apart from the boundary conditions (r1, θ1), (r2, θ2) and fixingthe time of flight, additional limitations to the solutionsmight beimposed. They typically originate from technical limitations ormission requirements. The goal of this section is to provide theconditions that K1,K2 must satisfy so that the transfer meetscertain requirements. They are useful for rapidly rejecting solu-tions that violate the constraints, without having to integrate thetrajectory. The conditions are similar to those related to the max-imum thrust acceleration, presented in Sect. 10.3.

142 10 Lambert’s problem with generalized logarithmic spirals

10.8.1 Arrival conditions

Once the transfer is determined by means of K1,K2, ξ the ar-rival conditions can be solved explicitly. The velocity is given by

v2 =

√K1 +

2(1 − ξ)r2

(10.26)

Property 4 showed that the arrival velocity for a pair of conjugatespirals is the same. The flight-direction angle at r2 is solved fromthe definition of K2:

ψ2 = arcsin[

K2

2(1 − ξ) + K1r2

](10.27)

The two solutions to this equation are C-symmetric. When thespiral reaches P2 in the raising regime it is ψ2 ∈ [0,π/2]. Thiscorresponds to direct transfers where r2 > r1 or indirect hyper-bolic transfers of Type II. On the contrary, if the spiral is in thelowering regime at P2 it is ψ2 ∈ [π/2,π]. This occurs for directtransfers where r2 < r1 and for indirect elliptic transfers.

The magnitude of the thrust acceleration is

ap,2 = ±

√ξ2[2(1 − ξ) + K1r2]2 + K2

2 (1 − 4ξ + 3ξ2)

r22[2(1 − ξ) + K1r2]

(10.28)

The+/− is selected depending onwhether the spiral is in the rais-ing/lowering regime at P2.

10.8.2 Radius

Consider that the trajectory is boundedby aminimumand amax-imum radius, so that

rmin ≤ r(θ) ≤ rmax

where of course [rmin, rmax] ⊃ [r1, r2]. For direct transfers theprevious consideration makes the condition on the radii to holdnaturally. Only for the case of indirect transfers r(t) can takevalues outside the interval [r1, r2], when rmax > max(r1, r2) orrmin < min(r1, r2).

Indirect transfers can only be elliptic spirals or hyperbolic spi-rals of Type II. A parameterization in terms of the constant K1yields a straightforward classificationof the solutions. For the caseof elliptic spirals the condition r(θ) ≥ rmin is satisfied naturally,whereas r(θ) ≤ rmax requires

rm ≤ rmax =⇒ K2 ≥ 2(1 − ξ) + K1rmax

The solution S ∈ S will be a Type II hyperbolic spiral as long asK1 > K1,tr, where the limit value K1,tr is given in Eq. (10.18). Itthen follows

rm ≥ rmin =⇒ K2 ≥ 2(1 − ξ) + K1rmin

10.8.3 Eccentricity

Assume that the transfer orbit must satisfy a constraint of theform e(r) ∈ [emin, emax]. The eccentricity of the osculating orbit

is defined explicitly in Eq. (F.26). For elliptic spirals it always de-creases in the raising regime. It grows in the lowering regime. Theeccentricity at r = rmax is minimum, and denoted e(rmax) = em.The eccentricity of the osculating orbit of a parabolic (logarith-mic) spiral is constant. The eccentricity grows for all hyperbolicspirals in the raising regime. It decreases in the lowering regime.The eccentricity at r = rmin is maximum, and denoted e(rmin) =em.

The minimum andmaximum eccentricity for indirect trans-fers is given by the same expression, em = |1 − K2|, which de-pends only on the constant of the angular momentum K2. Forthe case of indirect elliptic transfers the spiral is initially in theraising regime, it reaches the maximum radius and then transi-tions to lowering regime. In this case the minimum eccentricityoccurs at r = rmax, and corresponds to em = 1−K2. A constrainton the minimum eccentricity translates into K2 ≤ 1 − emin. Themaximum eccentricity occurs at r1 or r2, whichever is smaller. Anupper limit to the osculating eccentricity translates into

K22 ≥ (1 − e2

max)Q(r∗) with r∗ = min(r1, r2)

and

Q(r) =2(1 − ξ) + K1r

2ξ − K1r

Note that in the case r1 = r2 it is indifferent which value tochoose. Transfers to the same radius are always indirect. In ad-dition, the axis of symmetry coincides with the bisection of thetransfer vectors due to theT-symmetry of the spiral.

For direct transfers there are no transitions between rais-ing/lowering regimes. If r2 < r1 the spiral is in the loweringregime, and if r2 > r1 the spiral is in the raising regime. Notethat the transfer r2 = r1 is not possible along direct elliptic spi-rals. To satisfy the constraints on the eccentricity it suffices thatK2 verifies

r2 > r1 : (1 − e2max)Q(r1) ≤ K2

2 ≤ (1 − e2min)Q(r2) (10.29)

r2 < r1 : (1 − e2max)Q(r2) ≤ K2

2 ≤ (1 − e2min)Q(r1) (10.30)

For the case of parabolic spirals (K1 = 0) the eccentricity re-mains constant, meaning that the constraints on the values of theeccentricity reduce to (1 − e2

max)1/2 ≤ K2 ≤ (1 − e2

min)1/2. Re-call that the required value of K2 for transfers along parabolic spi-rals admits a closed-form solution, given in Eq. (10.17). The con-straint on the eccentricity can be referred directly to the geometryof the problem

1 − e2max

e2max

≤[θ2 − θ1

ln(r2/r1)

]2

≤1 − e2

min

e2min

The maximum and minimum eccentricity of the osculatingorbit along Type I hyperbolic spirals occurs at r1 and r2: it is min-imum at the smallest radius, and maximum at the largest. Notethat solutions of this type are always direct. The constraint on K2is obtained by switching r1 and r2 in Eqs. (10.29–10.30).

The constraints onType II hyperbolic spirals require an anal-ysis of the orientation of the axis of T-symmetry. For the caseof indirect transfers the maximum eccentricity corresponds to

§10.9 Conclusions 143

r = rmin, and reads em = K2 − 1. The constraint on the max-imum eccentricity reduces to K2 ≤ 1 + emax. The eccentricitywill always be above the lowest bound as long as

K22 ≤ (1 − e2

min)Q(r∗)

For the case of direct transfers the conditions on K2 are the sameas for the case of hyperbolic spirals of Type I.

10.8.4 Semimajor axis

Consider that the semimajor axis of the osculating orbit is con-strained toa ∈ [amin, amax]. It grows in the raising regime andde-creases in the lowering regime, no matter the type of spiral. Thesemimajor axis will reach its minimum value when the radius isminimum, and its maximum value when the radius is maximum.

The semimajor axis only depends on the energy of the spi-ral, K1, and the control parameter, ξ. Provided that the familiesof solutions to the transfer have been parameterized in terms ofthe value of K1 the adequate selection of K1 ensures that the con-straints are satisfied:

2ξrmin− 1

amin≤ K1 ≤

2ξrmax− 1

amax

Here rmin and rmax denote theminimumandmaximumradii thatthe spiral reaches. For indirect elliptic transfers the maximum ra-dius correspond to rm, and for indirect hyperbolic spirals of TypeII the minimum radius is rm.

10.8.5 Periapsis and apoapsis radii

Consider the constraints on the radius at periapsis and apoapsis,rp ∈ [rp,min, rp,max] and ra ∈ [ra,min, ra,max]. They are definedexplicitly by combining Eqs. (F.26) and (F.28):

ra =r

2ξ − K1r

1 +√

1 −K2

2

Q(r)

rp =

r2ξ − K1r

1 −√

1 −K2

2

Q(r)

They grow in the raising regime, and diminish in the loweringregime. Therefore, for the case of direct transfers the constraintson the apoapsis radius reduce to

K22 ≤ Q(r∗)

1 −[ra,min

(2ξr∗− K1

)− 1

]2 , r∗ = min(r1, r2)

K22 ≥ Q(r∗)

1 −[ra,max

(2ξr∗− K1

)− 1

]2 , r∗ = max(r1, r2)

The constraint on theperiapsis radius takes the same form, simplyinverting the inequality signs, and changing ra by rp.

Differentiating the definition of rp and ra with respect to rshows that both the apoapsis and periapsis radii have an extremepoint which corresponds to rmax if K1 < 0, and rmin if K1 > 0

and K2 > 2(1 − ξ). For the case of elliptic spirals the osculatingorbit at r = rmax is defined by means of

rp =K2[2(1 − ξ) − K2]

(−K1)(2 − K2)and ra =

2(1 − ξ) − K2

(−K1)

These are the maximum values that rp and ra can take. They arereached only for the case of indirect transfers. In such a case theconstraints translate into

K22 − K2[2(1 − ξ) − K1rp,max] ≥ 2K1rp,max (10.31)

K2 ≥ 2(1 − ξ) + K1ra,max (10.32)

The conditions related to rp,min and ra,min are equal to those fordirect transfers. Similarly, for indirect hyperbolic transfers ofType II theminimumvalues of rp and ra are the opposite of thosefor the elliptic case: the constraints are obtained by replacing theminimum/maximum values for the maximum/minimum onesinEqs. (10.31–10.32), and inverting the inequality signs. The con-ditions for rp,max and ra,max are those for the direct transfers.

10.9 Conclusions

The connections with the Keplerian case found in the definitionof the family of generalized logarithmic spirals simplify the studyof Lambert’s problem under a continuous acceleration. The exis-tence of two integrals of motion provides a clear structure of thesolution and yields dynamical properties of theoretical and prac-tical interest. Classical geometric and dynamic properties of theKeplerian Lambert problemhave an equivalent expression in spi-ral form. There is a minimum-energy spiral transfer, with pairsof conjugate spirals bifurcating form it. Conjugate spirals are en-dowedwith a collectionofproperties similar to those of conjugateKeplerian orbits.

All the numerical computations reduce to solving a systemoftwo equations with two unknowns. Once the solution to the sys-tem is known, no other iterative processes are required, providedthat all the relevant variables have been defined analytically. Theresolution is simplified thanks to having located special cases forwhich the transfer can be solved analytically (parabolic spirals) orthere is a specific procedure to compute them (minimum-energyspiral). When dealing with an arbitrary transfer knowing a prioritwo points on the curve for the time of flight helps to determinethe range where the solution is to be found.

The generalized logarithmic spirals might improve the effi-ciency of the transfers computed with the Keplerian Lambertproblem. Depending on the mission requirements the launchwindows can be stretched, or the mass delivered to the final orbitmight increase because of reducing the magnitude of the initialand final impulsive maneuvers. The time of flight will typicallyincrease, although special caseswhere the total∆v is reducedwhilemaintaining the time of flight can be found. Thequestion of howrealistic the adopted thrust acceleration is requires further stud-ies, in particular due to the dependency with the flight-directionangle.

“We choose to go to the Moon in this decade and do the other things,not because they are easy, but because they are hard […]”

—John F. Kennedy

11Low-thrust trajectory design with

controlled generalized logarithmic spirals

C hapter 9 introduced the family of generalized loga-rithmic spirals, a new analytic solution with continuousthrust. One of the possible practical applications of this

kind of solutions is the preliminary design of low-thrust trajecto-ries. The pioneering work by Petropoulos (2001), and Petropou-los andLonguski (2004) showed the full potential of shape-basedmethods for exploring large spaces of solutions in mission de-sign. They exploited their own analytic solutionwith continuousthrust in order to derive low-thrust gravity-assist mission designtools.

In this chapter, we shall explore the potential of generalizedlogarithmic spirals for low-thrust mission design. Thanks to hav-ing introduced a control parameter in Sect. 10.2, there is nowade-gree of freedom that can be adjusted to meet design constraints.The new control law takes the form

ap =µ

r2

[ξ cosψ t + (1 − 2ξ) sinψ n

](11.1)

where ξ is the control parameter.The main advantage of using generalized logarithmic spirals

over existing shape-based methods is the fact that the solution isdynamically intuitive. The constants to adjust are not arbitraryshape parameters, but constants of motion with a clear physicalmeaning. The conservation laws and the properties of the newspirals are closely related to Keplerian orbits. As a result, workingin the spiral realm is similar to working with Keplerian solutions,and classical design techniques can be recovered and applied tothe accelerated problem. This might help the mission analyst inthe design process, leading to a “patched spirals” approach forlow-thrust mission design: the boundary conditions define thevalues that the constants of motion must take, and different spi-

ral or coast arcs can be defined easily. The continuity of the tra-jectory is guaranteed by matching the values of the constants ofmotion in the transition nodes.

Section 11.1 presents the design strategy. The first step is tocharacterize transfers between circular orbits. The second stepconsists in generalizing this method to the design of transfers be-tween arbitrary orbits. Special attention is paid to the existenceof solutions for a given transfer configuration. Periodic orbits aredesigned in Sect. 11.2. In Sect. 11.3 the flexibility of the designmethodology is improved by introducing multiple nodes in theshaping sequence. Up to this point the analysis focuses on theplanar case. Section 11.4 extends these techniques to the three-dimensional case. Finally, an example of application is presentedin Sect. 11.5, consisting in the design of a mission to Ceres, in-cluding a flyby around Mars. In Appendix G we will explore anovel approach to shape-based trajectory design. Seiffert’s spher-ical spirals are recovered in order to design missions for planetaryexploration.

11.1 Orbit transfers

The control parameter provides a degree of freedom that can beadjusted at convenience to meet certain requirements. In addi-tion, the transfer can be decomposed in a number of legs withdifferent values of the control parameter, or even considering Ke-plerian arcs. The transition between two spiral arcs with differentvalues of ξ is equivalent to changing the thrustmagnitude and di-rection. No adjustments in the orbital velocity are required.

The control parameter only affects the value of the constantK1. Changing the value of ξ adjusts the values of the constant of

145

146 11 Low-thrust trajectory design with extended generalized logarithmic spirals

the generalized energy, K1, but not the values of K2. If K2 needsto be changed along the transfer, coast arcs should be introduced.

11.1.1 Bitangent transfers

Consider the spiral transfer between two circular orbits of radiir0 and rF . Along a spiral arc the evolution of the flight-directionangle ψ is monotonic, so if it is initially ψ0 = π/2 (tangentialdeparture) it is not possible to haveψF = π/2 (tangential arrival)with one single spiral arc. Bitangent transfers are the compositionof at least two continuous spiral arcs with different values of ξ.Whenψ = π/2 the spiral is either a hyperbolic spiral at rmin or anelliptic spiral at rmax. Parabolic spirals are not considered becauseof having a constant flight-direction angle.

The value of the constant K2 corresponding to the first arc issolved initially from

K2,1 = r0v20 sinψ0 = 1

Since the control parameter does not affect the value of K2 bothspiral arcs share the same values of K2 ≡ K2,1 = K2,2 = 1. Theconstant of the generalized energy on the first spiral arc takes theform

K1,1 = v20 −

2(1 − ξ1)r0

=2ξ1 − 1

r0

because v20 = 1/r0. For transfers with r0 < rF the first arc is a

hyperbolic spiral of Type II (with r0 = rmin) and the second arcis an elliptic spiral (with rmax = rF ). The condition r0 = rmin issatisfied naturally by having imposed K2 = 1:

rmin =2ξ1 − 1

K1,1=⇒ rmin = r0

The radial distance at the transition point A is solved from theequation of the trajectory, Eq. (F.25):

rA

r0=

3 − 2ξ1

2(1 − ξ1) + cos ℓ1θA

(11.2)

Similarly, the arrival spiral arc is defined by

K1,2 = v2F −

2(1 − ξ2)rF

=2ξ2 − 1

rF

and K2,2 = 1. From this equation it immediately follows thatrmax = rF . The conditions at the transition point are obtainedfrom the equation of the trajectory, Eq. (F.16):

rA

rF

=3 − 2ξ2

2(1 − ξ2) + coshℓ2[θA − (2n + 1)π] (11.3)

The maximum radius occurs at θm = (2n + 1)π, where n is thenumber of revolutions. Dividing Eqs. (11.2) and (11.3) provides arelation between ξ1 and ξ2,

rF

r0=

3 − 2ξ1

3 − 2ξ2

2(1 − ξ2) + coshℓ2[θA − (2n + 1)π]

2(1 − ξ1) + cos ℓ1θA

(11.4)

In order the two spiral arcs to be compatible it must be

K1,2 =2ξ2 − 1

rF

= vA2 − 2(1 − ξ2)

rA

The velocity-matching condition at A , vA1 = vA2, follows fromthe integral of the generalized energy,

2ξ2 − 1rF

=2ξ1 − 1

r0+

2(ξ2 − ξ1)rA

This condition yields an expression of ξ2 as a function of ξ1 andthe boundary conditions,

ξ2 =[(1 − 2ξ1)rF − r0]rA + 2ξ1r0rF

2r0(rF − rA )(11.5)

The problem of designing a bitangent transfer then reduces tosolving for ξ1 in Eq. (11.4). The value of the control parameteron the second arc, ξ2, is given by Eq. (11.5). There is one degree offreedom in the solution. Under this formulation it correspondsto the angular position of the node defining the transition point,θA . In addition, the number of revolutions can be adjusted bychanging the values of n.

In a similar fashion, when r0 > rF the equation to be solvedfor ξ1 transforms into

rF

r0=

3 − 2ξ1

3 − 2ξ2

2(1 − ξ2) + cosℓ2[θA − (2n + 1)π]

2(1 − ξ1) + cosh ℓ1θA

(11.6)

Recall that ℓi = |1 − 4(1 − ξi)2|1/2. Equation (11.5) still holds.Figure 11.1 shows two examples of bitangent transfers with

zero and one revolutions. The problem is based on an Earth toMars transfer, where rF /r0 = 1.527. The transition point is se-lected arbitrarily. The departure spiral arc corresponds to a hy-perbolic spiral ofType II in raising regime, whereas the second arccorresponds to an elliptic spiral in lowering regime. Their mini-mum and maximum radii are r0 and rF , respectively.

Figure 11.1: Examples of bitangent transfers with 0 and 1 revolu-

tions. The marks the transition point, fixed to θA = 100 deg. The

spirals depart from and arrive to×.

Low-thrust transfers are considered because they typically in-crease the mass delivered to the final orbit, by decreasing the pro-pellant mass spent along the transfer. Figure 11.2 compares thefraction of mass that can be inserted into the final orbit usingthe Hohmann and the spiral transfers. The values of the spe-cific impulse for the electric propulsion system and for the impul-sive maneuvers are 2500 s and 250 s, respectively. In this example

§11.1 Orbit transfers 147

the zero-revolutions spiral transfer delivers over 50% of the initialmass, and the one-revolution case increases this percentage up to75%. For the case of the Hohmann transfer the fraction of deliv-ered mass falls to roughly 10% of the initial mass.

Figure 11.2: Fraction of mass delivered to the final circular orbit.

The time of flight for the different types of transfers is plottedin Fig. 11.3. The zero-revolutions spiral transfer is geometricallyequivalent to the Hohmann transfer and the times of flight arecomparable. Depending on the transition point spiral transfersthat are either slower or faster than theHohmann transfer can befound. Increasing the number of revolutions can reduce the pro-pellant consumption, but the time of flight grows significantly.In this example the bitangent spiral transfer with one revolutionincreases the time of flight by a factor of three when comparedwith the Hohmann transfer, and for two revolutions the time offlight increases approximately by a factor six.

Figure 11.3: Time of flight for Earth toMars bitangent transfers.

Figure 11.2 suggests that there is a particular position of thetransition point θA that maximizes the mass delivered into or-bit. The zero-revolutions case is geometrically equivalent to theHohmann transfer and can be compared directly in terms ofmassdelivered into orbit and time of flight. When considering the op-timal transition point the spiral transfer delivers intoMartian or-bit 52.74% of the launch mass, whereas the Hohmann transfercan only deliver 10.12%. The time of flight for the Hohmanntransfer is 259.38 d. The spiral transfer turns out to be faster, re-ducing the time of flight to 257.05 d. That means that the spiral

transfer is more efficient both in terms of the mass delivered intoorbit and in flight time.

Theoptimumspiral transfer is foundby choosing adequatelythe transition point θA . Under the assumption that the specificimpulse Isp remains constant, the mass fraction is given by Tsi-olkovsy’s equation

mF

m0= exp

(−∆vthrg0Isp

)Here ∆vthr stands for the time integral

∆vthr =

∫ tA

t0ap,1 dt +

∫ tF

tA

ap,2 dt

The integral of the thrust acceleration is decomposed in two arcs:the acceleration along the departure arc, ap,1, is defined by ξ1,while the acceleration along the second arc, ap,2, is defined byξ2. The change in the thrust profile at θA is the correction thatthe initial arc requires in order to meet the boundary conditionsat arrival. Hence, the optimal solution is the one that requiresthe smallest correction, i.e. the transfer on which the difference|ap,1 − ap,2| at the transition point is minimum. The optimumtransition point for maximizing the mass delivered into orbit isthe one that minimizes the function

f (θA ) = |ap,1 − ap,2|

The magnitude of the thrust acceleration at θA can be written

ap,i =1

rA2

√ξ2

i cos2 ψA + (1 − 2ξi)2 sin2 ψA

How the variables rA and ψA relate to θA depends on the type ofspiral.

11.1.2 Transfers between arbitrary orbits. Introducing coastarcs

Changing the control parameter ξ only adjusts the value of K1.The first integral (9.16) is not affected by the control parameter.If the values of K2 on the departure and arrival spiral arcs are thesame no further corrections are required. However, in a generalcase K2,1 , K2,2 and therefore the constant K2 needs to be ad-justed. A coast arc is introduced in order to connect the depar-ture and arrival spiral arcs with the adequate values of K2. Thefirst spiral arc connects the departure point with a node A . A Ke-plerian orbit connects A and B , and the final thrust leg connectsB and the final point, F . If the difference between K2,1 and K2,2is small the coast arc will be short too.

The transfer from an initial position and velocity vectors, r0and v0, to a given state rF and vF imposes three constraints onthe solution at θF . These are r = rF , v = vF and ψ = ψF . Thecontrol parameters on both spiral arcs ξ1 and ξ2 can be adjustedto solve the transfer, together with the orientation of the nodesA and B , defined by θA and θB . Having four variables for onlythree constraints there is a degree of freedom in the solution. Thepreferred choice is touse thepositionof the first nodeA as the freeparameter and then solve for ξ1, ξ2, and the position of node B .

148 11 Low-thrust trajectory design with extended generalized logarithmic spirals

The initial conditions define the values of the constants K1and K2 on the first spiral arc:

K1,1 = v20 −

2(1 − ξ1)r0

and K2,1 = r0v20 sinψ0

just like the arrival conditions define K1 and K2 on the final arc:

K1,2 = v2F −

2(1 − ξ2)rF

and K2,2 = rF v2F sinψF

The first arc can be propagated to A using the correspondingequation for the trajectory and for the flight-direction angle

rA = r(θA ; K1,1,K2,1), and sinψA =K2,1

2(1 − ξ1) + K1,1rA

The conditions at A provide the eccentricity, semimajor axis, andargument of periapsis of the Keplerian orbit:

e =√

K22,1 + 1 − 2K2,1 sinψA (11.7)

a =rA

2ξ1 − K1,1rA

(11.8)

ω = atan2(− sin θA − K2,1 cos(θA + ψA ),K2,1 cos(θA + ψA ) − sin θA ) (11.9)

The value of K1 that a spiral will take if the thrust is switchedon again at some point of the Keplerian orbit behaves as

K1 =2aξ − r

ra

The velocity-matching condition at B then yields

2aξ2 − rB

rB a= v2

F −2(1 − ξ2)

rF

Recall that the semimajor axis and eccentricity have already beensolved from the conditions at A . The state at node B is solved an-alytically from Kepler’s problem. The remaining two constraintsare set on the values of the radius and flight-direction angle at F .In sum, the variables ξ1, ξ2, θB are solved from the system ofnonlinear equations:

2aξ2 − rB

rB a+

2(1 − ξ2)rF

− v2F = 0

rF − r(θF ; K1,2,K2,2, rB , θB , ξ2) = 0

ψF − ψ(θF ; K1,2,K2,2, rB , θB , ξ2) = 0

(11.10)

The conditions at B have already been obtained by propagatingthe Keplerian arc.

Similarly, the evolution of K2 on the coast leg renders

K2 =√

1 + 2e cosϑ + e2 (11.11)

where ϑ denotes the true anomaly. In Fig. 11.4 it is possible tostudy the evolution of the values of K2 and the flight-directionangle depending on the position inside a Keplerian orbit. Spiralsarriving/departing in raising regime correspond to points in theKeplerian orbit that are between periapsis and apoasis, whereas

spirals in lowering regime arrive to/depart from points betweenapoapsis and periapsis.

Figure 11.4: Evolution of K2 andψ on a Keplerian orbit (e = 0.3).

In the interval ϑ ∈ (0,π) K2 decreases toward apoasis. Con-versely, for ϑ ∈ (−π, 0) K2 increases. This simple rule shows thatif the first spiral arc arrives at A in raising regime and the secondarc departs from B in raising regime too, then for K2,2 > K2,1 theparticle will travel ∆ϑ > π along the Keplerian arc. This yields along coast arc. The same discussion applies to the case where thespirals are in lowering regime: when K2,2 < K2,1 the transfer in-cludes a long coast arc, and a short coast arc otherwise. By virtueof Eq. (11.11) the condition K2,2 ≶ K2,1 translates into:

cosϑF ≶K2

2,1 − (1 + e2F )

2eF

This defines the points on the arrival orbit for which the constantK2,2 is below/above K2,1. In general, short transfers will be pre-ferred because of involving shorter times of flight.

At B the true anomaly is simply

ϑB = θB − ω

The position of the node B can be solved from the compatibilityequation on K2, which yields the relation

cosϑB =K2

2,2 − (1 + e2)

2e(11.12)

There are two possible solutions to this equation depending onwhich quadrant the solution is in. As shown in Fig. 11.4 if ϑB ∈(0,π) the second spiral arc will be in raising regime at B . If ϑB

then the second spiral departs in lowering regime. The simplestcriterion for selecting the adequate quadrant is:

rB < rF =⇒ ϑB ∈ (0,π)rB > rF =⇒ ϑB ∈ (−π, 0)

It is valid for direct transfers from B to F . Inmost practical appli-cations the spirals are direct. If the spiral transfer is indirect thenthe previous criterion is inverted.

§11.1 Orbit transfers 149

It is worth to emphasize that transfers to circular orbits willalways be possible as long as the intermediate Keplerian orbit isclosed. The target value of K2,2 in this case is K2,2 = 1, which canbe achieved by any Keplerian orbit.

The velocity-matching condition provides the relation

ξ2 =[2a − rF (1 + av2

F )]rB

2a(rB − rF )(11.13)

Thanks to the previous expressions the system of equations de-fined in Eq. (11.10) reduces to one single transcendental equation,

rF − r(θF ; K1,2,K2,2, rB , θB , ξ2) = 0 (11.14)

to be solved for ξ1. On each iteration the values of ξ2, θB aresolved fromEqs. (11.13) and (11.12), respectively. Recall that θB =

ω + ϑB .Figure 11.5 shows two examples of generic spiral transfers in-

cluding a coast arc. The first example is the result of circulariz-ing an elliptic orbit considering zero-revolutions spiral arcs. Theswitch point A can be adjusted at convenience. Two different so-lutions to the same transfer problem are presented. These are ex-amples of long coast arcs, because the particle crosses the periap-sis of the Keplerian orbit. The second example corresponds to anEarth to Ceres transfer. In this case the correction required on K2is small and therefore the coast arc is almost negligible. Differ-ent solutions are displayed corresponding to different positionsof the transition point A . In this example the coast arc is so smallthat it cannot be distinguished.

(a) Long coast arcs (b) Short coast arcs

Figure 11.5: Examples of transfers between arbitrary orbits. The

departure point is, the pointsA andB are denoted by , and thearrival point is×. The gray lines represent the departure and arrivalorbits. The coast arcs are plotted using dashed lines.

11.1.3 Existence of solutions. The admissible region

The goal of this section is to find the set of values K2,2 ∈ I forwhich transfers from an initial state vector are possible. We referto I as the admissible region and K2,2 ∈ I is a sufficient conditionfor existence of solutions. We anticipate that solutions in an in-terval K2,2 ∈ I∗ outside the admissible region (I∗ ∩ I = ∅) existunder some very specific conditions. Therefore K2,2 ∈ (I ∪ I∗) isa necessary condition for solutions to exist.

Figure 11.6: Compatibility conditions in terms of the control parame-

ter for an example transfer.

The admissible region I is constructed by considering howmuch K2 can change on the coast arc. If the value of K2,2 is largerthan themaximumvalue that K2 can reach on theKeplerian orbitthen the transfer is unfeasible.

On a Keplerian orbit K2 changes according to

K2 =√

1 + e (e + 2 cosϑ)

The maximum and minimum values of K2 occur respectively atperiapsis (ϑ = 0) and at apoapsis (ϑ = π), and are

K2,max = 1 + e and K2,min = 1 − e (11.15)

This means that the values that K2 will take on the coast leg areconfined to [1 − e, 1 + e]. The eccentricity of the orbit dependson the selection of the control parameter ξ1 and the transitionpoint A . Figure 11.6 builds the admissible region for an exampleset of initial conditions and one given value of θA . It is param-eterized in terms of the control parameter on the first spiral arc,ξ1. The upper branch is K2,max = 1 + e, and the lower branch isK2,min = 1− e. The eccentricity is solved from Eq. (11.7). Noticethat the number of revolutions along the first arc does have aneffect on the admissible region, since it yields different values ofsinψA and modifies e. Two conclusions can be derived from thisfigure: first, given a target value of K2,2 if it falls in the forbiddenregion (K2,2 < I) the transfer is not feasible. Second, if K2,2 ∈ I itis possible to bound the range of values of ξ1 where the solutionis. The parameter ξp reads

ξp = 1 −r0v

20

2

and it defines the value of ξ1 that makes the first spiral arcparabolic. That is, if ξ1 < ξp the spiral is elliptic, and for ξ1 > ξp

the spiral is hyperbolic. The value of the control parameter thatmakes K2,max = 2 (or equivalently that makes e = 1) is denotedby ξ∗1. There are two pairs of relevant points in the definition ofthe admissible region:

a± = 1 ± |K2,1 − 1|, b± = 1 ±√

K22,1 + 1 − 2K2,1 sinψ0

A

150 11 Low-thrust trajectory design with extended generalized logarithmic spirals

whereψ0A denotes the flight-direction angle at A propagatedwith

ξ1 = 0. The highest point in the region I∗ reads

c = 2 + K2,1

The difference between I and I∗ is that for K2,2 ∈ I the Keplerianorbit is elliptic, whereas for K2,2 ∈ I∗ theKeplerian orbit is hyper-bolic. In this case the true anomaly is restricted by the asymptotesof the hyperbola, i.e. cosϑ < 1/e. In order to find solutions in I∗two additional constraints need to be satisfied:

• The coast arc must be a short arc, as defined from Fig. 11.4.

• Equation (11.12) must yield cosϑB < 1/e.

We emphasize that K2,2 ∈ I is a sufficient condition for thetransfer to be feasible. If K2,2 ∈ [b−, a−] ∪ [a+, b+] the admissi-ble region defines the intervals where adequate initial guesses inξ1 can be found. The discussion about the existence of solutionsdoes not replace the procedure for computing the transfer; Equa-tion (11.14) still needs to be solved for ξ1.

In practice, the non-existence of solutions relates to too shorttransfers that involve noticeable changes in the eccentricity. In-creasing the number of revolutions changes the admissible regionand can make the transfer possible.

11.1.4 Controlling the time of flight

In the previous sections it has been shown that there is a degree offreedom in the solution to the problem of finding a transfer orbitconnecting two given state vectors. It can be adjusted in order tomeet an additional constraint regarding the time of flight, t = tF .The degree of freedom is controlled by the position of the switchpoint A , and its adequate value is solved from:

tF = t1 + tk + t2

Here t1 and t2 are the times of flight for the first and second spiralarcs, respectively. The time spent on the Keplerian orbit is de-noted tk.

The number of revolutions of the first and second spiral arcsis an alternative way of modifying the geometry of the transfer.Consequently, it also affects the time of flight and introduces ad-ditional degrees of freedom in the solution.

It is important to note that themethods presented in the pre-vious sections, which consider only two spiral arcs, are the sim-plest but not the only solutions. By introducing more spiral arcs(or even coast arcs) the degrees of freedom of the system can beincreased at convenience. Introducing control nodes in the prob-lem might provide a useful method for trajectory optimization.

Section 10.3.4 analyzed carefully the properties of the acceler-ation in Eq. (11.1). Closed-form expressions were given for locat-ing the maximum acceleration reached along the transfer. Withthese results at hand, one can identify easily the transfers that vi-olate limitations on the maximum admissible thrust.

11.2 Periodic orbits

The procedure for designing bitangent transfers can be extendedto generate arbitrary periodic orbits. Due to theT-symmetry of

elliptic and hyperbolic spirals of Type II, any bitangent transfercan be closed to define a periodic orbit. When the spiral inter-sects a certain axis with ψ = π/2 the trajectory can be extendedsymmetrically to close the orbit. This procedure is then general-ized to generate periodic orbits of different shapes.

11.2.1 Coaxial solutions

A symmetric extension of the bitangent transfers defines a peri-odic orbit. Having imposed a maximum and minimum radius,Eq. (11.4) or (11.17) can be solved to provide the generating bi-tangent spiral. The number of revolutions n can be replaced byn = m/2, where m controls the number of intersections with thehorizontal axis before reaching the maximum/minimum radius.Periodic orbits require only two spiral arcs that are repeated basedon the symmetry properties of the trajectory. The first, from θ0 toθA , is propagated with ξ = ξ1. For the second arc, spanning fromθA to 2mπ−θA , it is ξ = ξ2. The final arc that closes the trajectoryis defined again by ξ1 and is propagated from 2(m+ 1)π− θA upto 2(m + 1)π.

Examples of coaxial periodic transfers are presented inFig. 11.7. We emphasize that there is a degree of freedom in theconstruction process. It relates to the orientation of the transi-tion point A . Because of the symmetry of the spiral there are twosymmetric transition points.

(a) m = 0 (b) m = 1

Figure 11.7: Coaxial periodic orbits. The denotes the nodes wherethe control parameter ξ changes.

11.2.2 Generic periodic orbits

Relying on the symmetry properties of the generalized logarith-mic spirals arbitrary periodic orbits can be generated. Numeri-cally, the problem reduces to solving a bitangent spiral transferwhere the orientation of the arrival point is adjusted dependingon the periodicity conditions. Periodic orbits are constructed bycombining elliptic and Type II hyperbolic spiral arcs; the exteriorradius of the orbit corresponds to rmax, whereas the interior ra-dius defines rmin. Only two arcs are required to build a periodicorbit; they are repeated sequentially according to the symmetriesof the spirals.

Consider a periodic orbit with s = 3, 4, 5, . . . lobes. Ifthe spiral departs from the maximum radius (θ = 0) then theminimum radius will be reached at θ = πΛ(s). The param-eter Λ(s) depends on the number of lobes and determines the

§11.3 Multinode transfers 151

configuration of the periodic orbit. Typical values of Λ(s) areΛ(s) = (s − j)/s, with j = 1, 2, . . . , s − 1. Given a transitionpoint θA the method derived for solving coaxial bitangent trans-fers can be extended compute the values of ξ1 and ξ2 defining ageneric periodic orbit. When the periodic orbit intersects itself atleast once it is called an interior periodic orbit ( j < s−1), and ex-terior otherwise ( j = s − 1). Figure 11.8 depicts the constructionof an interior periodic orbit with s = 3 andΛ = (s−1)/s = 2/3.

Figure 11.8: Construction of an interior periodic orbit with s = 3.

The symmetry properties of the spirals guarantee that if thesecond spiral arc reaches aminimum rmin = rin at θm = (s−1)π/sthen the resulting trajectory is a periodic orbit with s lobes. Ifr0 = rin the equation to be solved for ξ1 is

rexrin=

3 − 2ξ1

3 − 2ξ2

2(1 − ξ2) + cosh[ℓ2(θA − Λπ)]

2(1 − ξ1) + cos ℓ1θA

(11.16)

The value of ξ2 is solved from the velocity-matching condition,Eq. (11.5). When the spiral is initially at the exterior radius it isr0 = rex and the equation becomes

rinrex=

3 − 2ξ1

3 − 2ξ2

2(1 − ξ2) + cos[ℓ2(θA − Λπ)]

2(1 − ξ1) + cosh ℓ1θA

(11.17)

Examples of interior periodic orbits with Λ = (s − 1)/s arepresented in Fig. 11.9. The circumference of radius rex is the exte-rior envelope of the orbits, and the circumference with r = rin isthe interior envelope. Alternative configurations can be found bychangingΛ(s). Having fixed the number of lobes s there are fourdegrees of freedom in the definition of the periodic orbits: the in-terior and exterior radii, the transition point θA , and the integerparameter j in the definition ofΛ(s).

(a) s = 3 (b) s = 4 (c) s = 5 (d) s = 6

Figure 11.9: Examples of periodic orbits with j = 1, θA = 90 deg,

and different number of lobes. The denotes the nodes where thecontrol parameter ξ changes.

Figure 11.10: Examples of periodic orbits for θA = 20 deg and differ-

ent combinations of s and j.

Figure 11.10 displays a number of example interior and exte-rior periodic orbits generated by combining two spiral arcs. Theentire orbit is generated by symmetrically extending a bitangenttransfer defined by two arcs.

11.3 Multinode transfers

We have shown that transfers between circular orbits require atleast one transition node, whereas transfers between arbitrarystate vectors require a coast arc. An arbitrary number of nodescan be introduced in order to increase the flexibility of the solu-tion.

Consider the problem of finding transfers between circularorbits introducing N ≥ 1 nodes. Let θn denote the orientationof the n-th node. There are 2N − 1 degrees of freedom in thesolution: the position of the nodes (θ1, θ2, θ3, . . . , θN) and thecontrol parameters of the interior arcs (ξ2, ξ3, . . . , ξN−1). The arcconnecting nodes n − 1 and n is labeled the n-th arc. With this,the initial arc connecting θ0 and θ1 is labeled “1”. The final arc,connecting θN and θF , is denoted by F . The problem consists insolving for ξ1 given the position of the nodes and the values ofthe control parameter for the intermediate arcs.

The trajectory is propagated sequentially through the nodes

rn = r(θn; ξn), n = 1, 2, . . . ,N

considering the initial conditions (rn−1, vn−1, θn−1, ψn−1) definedby the previous node. The final arc needs to satisfy two condi-tions: first, the velocity-matching condition

v2N = K1,F +

2(1 − ξF )rN

=⇒ ξF =2rF − rN[2 + rF (v2

N − v2F )]

2(rF − rN)

Second, the boundary condition

rF = r(θF ; ξF )

Figure 11.11 shows two examples of multinode transfers betweentwo circular orbits. Each figure presents two spiral transfers thatdepart/arrive from/to the same exact state vectors. The difference

152 11 Low-thrust trajectory design with extended generalized logarithmic spirals

(a) 1 revolution (b) 2 revolutions

Figure 11.11: Examples of bitangent transfers with N = 7. Thenodes are denoted by •, is the departure point and× is the arrivalpoint.

between the black and gray transfer is the position of the nodesand the values of the control parameters. It is worth to emphasizethat the transitions at the nodes are continuous (no additional∆v’s are required) and no correcting maneuvers are required atdeparture nor arrival.

Introducing multiple nodes in transfers between arbitrarystate vectors is a simple procedure: the problem reduces to solv-ing a thrust-coast-thrust transfer, but the boundary conditionsare given by the forward/backward propagation of the additionalspiral arcs. Consider that there are N1 spiral arcs until the thrustis switched off, and N2 arcs between the point where the thrustis switched on again and the arrival point. The initial conditionsto be considered for solving Eq. (11.14) are defined by the finalstate after propagating the spirals 1, 2, . . . ,N1 − 1. The arrivalconditions are obtained from the backward propagation of theN2,N2−1, . . . , 2 arcs in the second leg. Figure 11.12 shows exam-ple of transfers between two state vectors. The two trajectoriesplotted in each figure arrive to and depart from the same exactstate vectors, and the only difference between them is the defini-tion of the intermediate arcs. This technique can be generalizedevenmore by introducing an arbitrary sequence of spiral and Ke-plerian arcs.

Figure 11.12: Examples of multinode transfers between two arbi-

trary state vectors.

11.4 Three-dimensional motion

Let us now consider themore general case in which the transfer isno longer planar. The out-of-plane component of the motion, z,is different from zero. To prevent confusion the norm of vector rin the three-dimensional casewill bewrittenR, withR2 = r2+z2.The normalized (µ = 1) and perturbed two-body problem obeys

d2rdt2 +

rR3 = ap, R = ||r|| (11.18)

Vector r can be represented by the set of cylindrical coordi-nates (r, θ, z). The climb angle is λ. Figure 11.13 depicts the ge-ometry of the three-dimensional trajectory. The position of thesubpoint P ′ is given by r∥, so that

r(θ) = r∥(θ) + z(θ) kI

This means that r∥ describes the projectedmotion of the particle.Themain hypothesis we will adopt is that r∥(θ) defines a general-ized logarithmic spiral. It is called the base spiral (Roa and Peláez,2016e). Under this notation it is ||r∥|| = r.

Figure 11.13: Geometry of a 3D generalized logarithmic spiral.

Let F = P ; t,n,b denote the Frenet-Serret frame, where tis the unit vector tangent to the trajectory, i.e.

t =drds=

vv

Here s denotes an arclength along the curve. The unit vectors nand b are the normal and binormal unit vectors, an relate to eachother and to t thanks to the Frenet-Serret formulas. In particular,the first of the Frenet-Serret formulas provides

n =1k

dtds

where k is the nonzero curvature of the trajectory, defined explic-itly as

k =

∣∣∣∣∣∣∣∣∣∣ dtds

∣∣∣∣∣∣∣∣∣∣

§11.4 Three-dimensional motion 153

The binormal vector completes an orthonormal dextral referenceframe,

b = t × n

Figure 11.14 shows the configuration of the Frenet-Serret frame.The auxiliary frameF∥ = P ; t∥,n∥,b∥ is defined so that (t∥ ·t) =cos λ and (t∥ · kI) = 0. It relates to the Frenet-Serret frame F bymeans of a rotation of magnitude λ about the normal vector.

Figure 11.14: The Frenet-Serret frame.

The velocity, v ∈ R3, can be decomposed in

v = v∥ + vz

where v∥ is the in-plane component of the velocity (v∥ · kI = 0)and the out-of-plane component isvz = z kI. From the geometryof the problem it follows

zv∥= tan λ =⇒ dz

dt= v∥ tan λ (11.19)

Taking the time derivative of the out-of-plane velocity defines thetransversal accelerationd2zdt2 = tan λ

dv∥dt+ v∥ sec2 λ

dλdt

(11.20)

The time evolution of λ(t) and its derivative are supposed to beknown and depend on the shape assumed for the trajectory.

Under the hypothesis that r∥ defines a generalized logarith-mic spiral, the equations of motion that govern the in-plane dy-namics are those found in Sect. 11.1. When combined with Eqs.(11.19–11.20) it follows the system

dv∥dt=ξ − 1

r2 cosψ (11.21)

drdt= +v∥ cosψ (11.22)

dθdt= +

v∥r

sinψ (11.23)

dψdt=

sinψr2v∥

[2(1 − ξ) − rv2∥ ] (11.24)

dzdt= v∥ tan λ (11.25)

d2zdt2 =

ξ − 1r2 cosψ tan λ + v∥(1 + tan2 λ)

dλdt

(11.26)

which needs to be integrated from the initial conditions:

t = 0 : v∥(t0) = v∥,0, r(t0) = r0, θ(t0) = θ0

ψ(t0) = ψ0, z(t0) = z0, z(t0) = z0

In these equationsψ is the in-plane flight-direction angle, definedas the angle between the projections of the position and velocityvectors onto the x − y plane, i.e.

cosψ =(r∥ · v∥)

r v∥(11.27)

We devote the following lines to finding a similar expression forthe absolute flight-direction angle,Ψ, between vectors r and v.

Let i, j,k denote the orbital frame L. The unit vectorsdefining this frame are

i =rR, k =

r × v||r × v|| , j = k × i

Similarly, consider the orbital frame attached to the subpoint P ′,defined in terms of

i∥ =r∥r, k∥ =

r∥ × v∥||r∥ × v∥||

, j∥ = k∥ × i∥

and denoted L′. The Frenet-Serret frame referred to the pro-jected spiral relates to frameL′ by means of

t∥ = cosψ i∥ + sinψ j∥, n∥ = − sinψ i∥ + cosψ j∥

The vectors defining the Frenet-Serret frame attached to the tra-jectory admit an equivalent projection ontoL:

t = cosΨ i + sinΨ j, n = − sinΨ i + cosΨ j

whereΨ is the flight-direction angle defined by vectors r and v,

cosΨ =(r · v)

Rv(11.28)

We emphasize the difference between this equation andEq. (11.27). It then follows

t∥ = cosΨ cos λ i + sinΨ cos λ j − sin λkn∥ = − sinΨ i + cosΨ jb∥ = cosΨ sin λ i + sinΨ sin λ j + cos λk

The total velocity of the particle reads

v = v∥ cosψ i∥ + v∥ sinψ j∥ + z k∥

Provided that r = r i∥ + z k∥, Eq. (11.28) renders

cosΨ =rv∥ cosψ + zz

Rv

11.4.1 The thrust acceleration

The equations of motion (11.21–11.26) show that the total accel-eration governing the dynamics is

d2rdt2 =

ξ − 1r2 [cosψ (t∥ + tan λkI) − 2 sinψn] + v∥sec2 λ

dλdt

kI

154 11 Low-thrust trajectory design with extended generalized logarithmic spirals

The gravitational acceleration is known to be

ag = −r

R3 = −1

R3

[r(cosψ t∥ − sinψ n) + z kI

]The acceleration due to the thrust decomposes inap = ap,∥+ap,z.The in-plane component takes the form

ap,∥ =

(ξ − 1

r3 +1

R3

)r cosψ t∥ +

[2(1 − ξ)

r3 − 1R3

]r sinψn

whereas the out-of-plane component of the acceleration reducesto

ap,z =

[ξ − 1

r2 cosψ tan λ + v∥ddt

(tan λ) +z

R3

]kI (11.29)

The climb-angle λ and its rate of change are known, given theshape of the trajectory. The previous expression is directly re-ferred to tan λ in order to simplify the introduction of a steeringlaw.

When the motion is confined to the plane it is z = 0 and λ =0. This condition makes r ≡ R, and the perturbing accelerationreduces to

ap =1r2 [ξ cosψ t + (1 − 2ξ) sinψ n] (z = 0, λ = 0)

and coincides with the acceleration defined in Eq. (11.1).

11.4.2 Helices

The simplest law for the evolution of the climb-angle λ is

λ = const.

and of course its time derivative is null. The radial velocity hasbeen defined in Eq. (11.22) as the projection of the velocity alongthe direction of r∥. Combining this expression with Eq. (11.19)yields

dzdr=

tan λcosψ

(11.30)

The term cosψ is given explicitly in Eq. (F.12). It is interesting tosee that the sign of this expression depends on the regime of thespiral through the sign of cosψ. It becomes

dzdr= ± [2(1 − ξ) + K1r] tan λ√

[2(1 − ξ) + K1r]2 − K22

(11.31)

This equation shows that the out-of-plane motion can be solvedin closed-form when the angle λ is constant, being:

z(r) − z0 =K2

K1tan λ(cotψ − cotψ0) (11.32)

Recall that

tanψ = ± K2√[2(1 − ξ) + K1r]2 − K2

2

The constant K1 appears in the denominator of the solution inEq. (11.32). In the limit case K1 → 0 this expression will becomesingular. But the singularity is avoidable as shown by the limit

limK1→0

z(r) = z0 + 2(r − r0)(1 − ξ) tan λ secψ0

When K1 = 0 the spiral is parabolic (logarithmic) and the flight-direction angle remains constant, ψ = ψ0.

Figure 11.15 shows examples of different helicoidal trajecto-ries depending on the type of the corresponding base spiral. Byextension, we call elliptic helices those helices whose base spiralis elliptic. The same applies to parabolic and hyperbolic helices.Elliptic helices resemble spherical spirals, the poles correspondingto r → 0 in the base spiral. Parabolic helices evolve on the surfaceof a cone.

(a) Elliptic (b) Parabolic (c) Hyperbolic TI (d) Hyperbolic TII

Figure 11.15: Helicoidal trajectories depending on the type of base

spiral.

The geometry of the problem is such that

(t · kI) = sin λ

Taking the derivative with respect to the arclength, s, provides(dtds· kI

)= 0 (11.33)

The derivative of t with respect to the arclength relates to n bymeans of the first of the Frenet-Serret formulas,

dtds= k n

Therefore, Eq. (11.33) translates into

k (n · kI) = 0

so vector n is confined to the plane defined by kI, provided thatthe helix has nonzero curvature. This proves that the normal vec-tor of a generalized logarithmic helix n is always parallel to thex − y plane.

From a geometric perspective it is interesting to prove thatthe resulting curves satisfy the condition for curves being helices,given by Lancret’s theorem. It states that a curve is called a helixif the ratio between the curvature, k , and the torsion, τ, is con-stant.* The axis of the helix is defined by the unit vector kI, andcan be referred to the Frenet-Serret frame by virtue of

kI = sin λ t + cos λb*This theoremwas first presented by the engineerMichel A. Lancret inMé-

moire sur les courbes à double courbure, on April 26, 1802. But the formal proof

§11.4 Three-dimensional motion 155

Differentiating with respect to the arclength s yields

0 = sin λ(dtds

)+ cos λ

(dbds

)The Frenet-Serret formulas transform this expression into

k (s) sin λn − τ(s) cos λ n = 0 =⇒ k (s)τ(s)

= cot λ

so that the ratio k /τ is indeed constant when λ = const.The connection between Eq. (11.32) and the initial condi-

tions (z0, ψ0) comes from solving for the constant of integrationgiven the departure point. Without losing generality the constantof integration can be obtained from the conditions at apoapsis orperiapsis when the spiral is either elliptic or hyperbolic of Type II.This provides a new form of the equation of the trajectory:

z(r) − zm =K2 tan λK1 tanψ

(11.34)

Here zm = z(rmax) or zm = z(rmin), depending on whether thehelix is elliptic or hyperbolic of Type II.

This suggests a new concept of symmetry: a generalized helixis said to beZ-symmetric if

z(θm + ∆) = −z(θm − ∆)

where ∆ ≥ 0 denotes an arbitrarily large angular displacement.The component z depends on the polar angle through the re-lations r = r(θ) and ψ = ψ(θ). Z-symmetric trajectories areinvariant with respect to rotations of magnitude ϑ = nπ (withn = 1, 2, . . .) about the axis θ = θm located in the z = zm plane.Clearly aZ-symmetric trajectory isT-symmetric, because a max-imum or minimum radius exists. Similarly, allT-symmetric tra-jectories are Z-symmetric because Eq. (11.34) will always changeits sign when ψ reaches π/2.

Elliptic spirals are known to be bounded, so that r → 0whenθ → ±∞. Similarly, Eq. (11.34) proves that the out-of-plane mo-tion is also bounded:

limr→0

z(r) = zm ±ℓ

K1tan λ

This result clearly shows theZ-symmetry of the helix.On the contrary, for the case of parabolic and hyperbolic spi-

rals the particle escapes to infinity. This means limr→∞ z(r) =±∞ depending on the type and regime of the helix. For hyper-bolic helices of Type II there are two asymptotes that providez→ ±∞.

of the theorem is attributed to Saint-Venant, published in his Mémoire sur leslignes courbes non planes in 1845. He worked with the concept of the rectifyingplane (spanned by vectors t and b): by evaluating the relative inclination of theline of intersection between consecutive rectifying planes he proved that for ahelix such lines are parallel. Moreover, they describe a cylinder. He then reducedthe condition for the rectifying lines to be parallel to d(k /τ) = 0. We refer toBarros (1997) for a generalization of Lancret’s theorem.

11.4.3 Shaping the transversal motion

Polynomial shape

Following the shape-based approach the out-of-plane compo-nent of the motion can be modeled as

z(θ) =N∑

n=0

cnθn

The shape coefficients cn are stored in the vector cN ∈ RN+1. Thetransversal velocity follows from the relation

dzdt=

dzdθ

dθdt=v∥r

sinψN∑

n=1

ncnθn−1 (11.35)

having introduced Eq. (11.23).The steering law λ(θ) can be solved by equating Eqs. (11.35)

and (11.19):

tan λ =sinψ

r

N∑n=1

ncnθn−1

The transversal component of the thrust acceleration, given inEq. (11.29), involves the time derivative of tan λ:

ddt

(tan λ) =ddt

( sinψr

) N∑n=1

ncnθn−1+

v∥ sin2 ψ

r2

N∑n=2

n(n−1)cnθn−2

This solution requires an expression for d(sinψ/r)/dt, whichtakes the formddt

( sinψr

)= − sinψ cosψ

r2v∥(K1 + v

2∥ ) (11.36)

Fourier series

Consider an alternative shape of the transversal component of themotion, modeled by a Fourier series:

z(θ) = a0 +

N∑n=1

[an cos(nθ) + bn sin(nθ)]

The out-of-plane velocity abides by

dzdt= − v∥

rsinψ

N∑n=1

n[an sin(nθ) − bn cos(nθ)]

Similarly, the steering law becomes

tan λ = − sinψr

N∑n=1

n[an sin(nθ) − bn cos(nθ)]

Its time derivative, required for computing the out-of-plane ac-celeration, is

ddt

(tan λ) = − ddt

( sinψr

) N∑n=1

n[an sin(nθ) − bn cos(nθ)]

− v∥ sin

2 ψ

r2

N∑n=1

n2[an cos(nθ) + bn sin(nθ)]

The derivative that appears in the first term is given in Eq. (11.36).

156 11 Low-thrust trajectory design with extended generalized logarithmic spirals

11.4.4 Transfers between arbitrary orbits

Themethod for designing transfers between arbitrary orbits con-sists in two steps:

1. The target orbit is projected on the orbital plane of the de-parture orbit. The transfer between the departure orbit andthe projectionof the target orbit is solvedusing a generalizedlogarithmic spiral. This defines the in-plane components ofthe motion, the time of flight, the control parameter ξ, andthe constants of motion K1 and K2.

2. The out-of-plane motion is designed to satisfy the bound-ary conditions at departure and arrival. If the polynomiallaw from the previous section is used, the problem reducesto adjusting the values of the coefficients cn. A similar ap-proach is followed when implementing the Fourier shape,by solving for an and bn.

Given the initial conditions (z0, z0) two of the shape coeffi-cients need to be adjusted in order the trajectory to satisfy z(r0) =z0 and z(r0) = z0. These equations are linear in the coefficients ci.Let (z f , z f ) be the arrival conditions for the out-of-planemotion.The boundary conditions provide the relation z f = z(r f ), wherer f is the value of the cylindrical radius at the arrival point and z(r)is the assumed law for the evolution of z. The conditions on thevelocity render z f = z(r f ). The problem imposes a total of fourconstraints to the system: the trajectory is forced to depart from(z0, z0) and to arrive to (z f , z f ). At least four shape coefficientsare required.

Polynomial shape

One simple approach is to solve for the first four coefficientsn = 0, . . . , 3, stored in c3. But different strategies can be adopteddepending on the number of coefficients. The linear system ofequations can be written in matrix form as

z = M c3 + N c4:N (11.37)

Here vector z = [z0, z f , z0, z f ]⊤ contains the boundary con-ditions. Vector c4:N refers to the remaining coefficients not in-cluded in c3, and the matrices M ∈ R4×4 and N ∈ R4×(N−4) de-pend on the shape function. DenotingMn the (n+ 1)-th columnofmatrixM, andNm the (m+1)-th columnofmatrixN, it follows:

Mn =

θ n0

θ nf

nθ n−10 C0

nθ n−1f C f

and Nm =

θm0

θmf

mθm−10 C0

mθm−1f C f

with n = 0, . . . , 3 and m = 4, . . . ,N. For simplicity the coeffi-cient C = v∥ sinψ/r has been introduced.

The matrix equation (11.37) can be solved for c3 to provide

c3 = M−1(z − N c4:N)

The determinant of matrixM is

∆ = −C0C f (θ f − θ0)4

It will only vanish if θ0 = θ f or C j = 0. The former is not pos-sible since θ always grows in time. The latter only occurs whenψ → 0 (degenerate rectilinear orbit) or r → ∞ (escape); bothsituations are limit cases that are omitted for consistency. Underthese hypotheses matrixM is regular.

Fourier series

When considering the Fourier series representation the boundaryconditions can be written

z = M c + d

Vector c groups the coefficients [a0, a1, b1, a2]⊤ and matrix M

reads

M =

1 cos θ0 sin θ0 cos 2θ01 cos θ f sin θ f cos 2θ f

0 −C0 sin θ0 C0 cos θ0 −2C0 sin 2θ00 −C f sin θ f C f cos θ f −2C f sin 2θ f

The last vector, d, refers to the remaining terms of the series andtakes the form

d =

b2 sin 2θ0 +∑N

n=3[an cos(nθ0) + bn sin(nθ0)]

b2 sin 2θ f +∑N

n=3[an cos(nθ f ) + bn sin(nθ f )]

2C0b2 cos 2θ0 −C0∑N

n=3 n[an sin(nθ0) − bn cos(nθ0)]

2C f b2 cos 2θ f −C f∑N

n=3 n[an sin(nθ f ) − bn cos(nθ f )]

In this case the determinant ofM is

∆ = −C0C f

2[ sin(3θ0 − θ f ) + 6 sin(θ0 + θ f ) − sin(θ0 − 3θ f )

− 4 sin(2θ f ) − 4 sin(2θ0)]

meaning that the matrix is regular for C0 , 0 and C f , 0.

11.5 Applications

This section shows how themethod presented in this chapter canbe applied to the design of an interplanetary mission. In orderto have a reference solution to compare with, we recovered an ex-ample that appears recursively in the literature (Sauer, 1997; Mc-Conaghy et al., 2003; Petropoulos and Longuski, 2004): a low-thrust transfer to rendezvouswithCeres including a flyby atMars.

The problem is modeled in two different ways. First, a sim-plified ephemeris model is considered, following the examplesin the literature. In this case the orbits of the Earth, Mars, andCeres are assumed Keplerian and their inclination relative to theecliptic plane is neglected. The second model considers the realephemeris of the planets and Ceres. The design tool interactswith the SPICE library (Acton, 1996) to retrieve the states at date.

11.5.1 Simplified model

The goal of this simplified example is to show how the general-ized logarithmic spirals (GLS) perform when compared to othermethods such as the exponential sinusoids, by adopting the same

§11.5 Applications 157

exact transfer configuration. The orbits are assumed to be con-tained in the ecliptic plane, as defined in Table 11.1. The specificimpulse is assumed to be constant and equal to 3000 s. The space-craft departs from the Earth onMay 6, 2003 with v∞ = 1.6 km/sat launch. The mass injected into orbit is 568 kg.

The Earth-Mars (EM) transfer leg decomposes in two spiralarcs connected by a coast arc. In the EM leg both spirals are hyper-bolic of Type II (H-II). The spiral arcs are described in Table 11.2.

Table 11.1: Reference orbits (MJD 52765).

Units Earth Mars Ceres

a AU 1.0000 1.5237 2.7656

e − 0.0162 0.0936 0.0794

ω + Ω deg 103.93 336.06 154.60

ϑ0 deg 121.10 289.08 258.35

Table 11.2: Parameters defining the spiral segments.

Type ξ K1 K2

EM-1 H-II 0.4936 0.0885 1.1021

EM-2 H-II 0.5181 0.0111 0.9684

MC-1 H-II 0.4983 0.0912 1.1371

MC-2 Ellip 0.4938 -0.0279 0.9291

In the Mars-Ceres (MC) leg the first spiral is also hyperbolic ofType II, whereas the second spiral is elliptic.

The performance of the GLS is compared with the solu-tion provided by Petropoulos and Longuski (2004) and Sauer(1997) in Table 11.3. We also include the optimization usingGALLOP (Sims and Flanagan, 1999; McConaghy et al., 2003) thatPetropoulos andLonguski computed as a reference for their com-parison. The launch v∞ is equal in all cases. The configura-tion of the Mars flyby differs slightly with that predicted bySTOUR-LTGA (the low-thrust version of the design tool createdby Longuski, 1983), defining a faster and deeper flyby. The MCleg is solved in order the arrival velocity tomatch the orbital veloc-ity of Ceres, which yields a null v∞ at arrival. The times of flightare taken from the solution byPetropoulos andLonguski (2004):

271 and 862 days for the EM and MC transfer legs, respectively.The solution foundwith theGLSyields the samepropellantmassfraction asSTOUR-LTGA, but requires no additional impulsive∆vfor the final rendezvous with Ceres. Given the relatively low spe-cific impulses of liquid propellants this final maneuver may in-crease significantly the propellant mass fraction. Note that Sauerused a more sophisticated model for the thruster and propellantexpenditures, and he selected June 8, 2003, for the launch date.This is responsible for the higher mass fraction of his solution.

The trajectory is plotted in Fig. 11.16. Both the EM and theMC legs include coast arcs to gain control over the final state.Right after launch there is a short thrust arc in order to increasethe energy of the intermediate Keplerian orbit. The thrust isswitched on again when the values of K1 and K2 are compatiblewith a feasible flyby aboutMars. The flyby is designed so it yieldsadequate values of the constants ofmotion. TheMC leg is solvedfrom Eq. (11.14) having selected the switch point that minimizesthe propellant mass fraction. The configuration of the flyby isobtained with a genetic optimization algorithm.

Figure 11.16: Earth-Mars-Ceres spiral transfer. Solid lines corre-

spond to the thrust arcs, whereas dashed lines are coast arcs. The

black dots • represent the switch points.

Table 11.3: Comparison of the GLS solution with STOUR-LTGA, GALLOP and Sauer's solution.

Units GLS STOUR-LTGA ∗ GALLOP ∗ Sauer∗

Launch v∞ km/s 1.600 1.600 1.600 1.600

Mars flyby v∞ km/s 1.590 1.435 1.920 −Flyby altitude km 562 5432 200 −Propellant mass fraction − 0.256 0.256 0.234 0.275

Arrival v∞ km/s 0.000 0.237 0.000 0.000

TOF Earth-Mars days 271 271 271 250

TOFMars-Ceres days 862 862 862 (739) 845

TOF total days 1133 1133 1133 (1010) 1095

∗Data taken from Petropoulos and Longuski (2004).

158 11 Low-thrust trajectory design with extended generalized logarithmic spirals

Figure 11.17: Thrust profile for the Earth-Mars-Ceres transfer.

Figure 11.18: Thrust angle compared to the flight-direction angleψ.Whenφ = ψ the thrust vector is directed along the velocity.

The magnitude of the thrust required for the mission isplotted in Fig. 11.17. It is worth noticing that the requiredthrust exhibits some peaks that exceed the limit thrust adopted inPetropoulos and Longuski (2004); they considered that the avail-able thrust is 95 mN at 1 AU and it decreases with the power law1/r2. Despite the peaks of high thrust and due to the coast arcsthe final result is a propellant mass fraction that is comparable tothe optimized solution, as shown in Table 11.3.

The orientation of the thrust depends on the values of thecontrol parameters (see Table 11.2) and on the evolution of theflight-direction angle. Figure 11.18 shows how the thrust angleevolves compared to the flight-direction angle. The orientation(and magnitude) of the thrust vector is adjusted in order to meetthe constraints on the time of flight. In addition, in the MC legthe values of the control parameter are defined so that the finalvelocity matches that of Ceres. Along the final spiral arc the mag-nitude of the thrust decreases and the thrust angle separates fromthe flight-direction angle for matching the velocities.

11.5.2 Real ephemeris

In the case ofMars the low relative inclination (< 2 deg) will havelittle impact on the solution. However, the inclination of the or-bit of Ceres is approximately 10 deg and the out-of-planemotionis more relevant.

The configuration of the planar transfer is obtained througha global optimization procedure using a genetic algorithm. Thestate vector is composed by the date of departure, the date of ar-

Figure 11.19: Projection of the out-of-plane component of themo-

tion.

rival at Mars, the date of arrival at Ceres, the launch v∞, and theconfiguration of the flyby. The performance of the new solutioncan be analyzed inTable 11.4. The departure date changes toMay21, 2003, and the time of flight for each leg is reduced. The resultis a propellant mass fraction for the in-plane motion of 23.42%.When accounting for the out-of-plane component of the thrustthis figure raises to 28.94%.

Table 11.4: Definition of the three-dimensional tranfer.

Units Value

Launch v∞ km/s 1.995

Mars flyby v∞ km/s 1.272

Flyby altitude km 2058

Prop. mass fraction − 0.289

Prop. mass fraction (in-plane) − 0.234

Arrival v∞ km/s 0.000

TOF Earth-Mars days 246

TOFMars-Ceres days 846

TOF total days 1092

The shape of the trajectory is depicted in Fig. 11.19 and it isdefined in the ecliptic ICRF/J2000 reference system. The orbitis projected in the xz-plane to show in more detail the transversalcomponent of the motion (the z-axis is exaggerated). Gray linesrepresent the orbits of the Earth, Mars, and Ceres. The arrivalvelocity at Ceres also matches the transversal component of thevelocity of the minor planet.

The transversal component of the motion obeys a polyno-mial shaping law. The design procedure is based on the methoddescribed in Sect. 11.4.4, using only the first four coefficients (i.e.c4:N = 0). The evolution of the out-of-plane component of thethrust vector is plotted in Fig. 11.20. It is interesting to note that

§11.6 Conclusions 159

Figure 11.21: Visualization of the transfer in Cosmographia.

the first spiral arc and the coast arcs are coplanar (the transversalcomponent of the thrust is only switched on in the second spi-ral arc). This is the simplest decomposition of the transfer, inwhich the correction of the out-of-plane motion occurs only inthe last leg. Alternative strategies might be considered, like intro-ducing an intermediate plane in which the Keplerian leg will beconfined. The propellant expenditures associated to the out-of-plane dynamics are 18% of the total budget.

Figure 11.20: Transversal component of the thrust.

Figure 11.21 is a representation of the trajectory (red line) inthe Solar SystemusingCosmographia. The visualization softwareis integrated with SPICE and, provided that the design tool usesthe ephemeris model in SPICE, it provides an accurate represen-tation of the departure from the Earth, the Mars flyby, and therendezvous with Ceres.

11.6 Conclusions

Thanks to the introduction of a control parameter the versatil-ity of the family of generalized logarithmic spirals is improved.

As a result, these new curves define a novel transfer strategy be-tween arbitrary orbits. The existence of constants of motion ofdynamical nature yields an intuitive design methodology, basedon matching the values of the constants of motion along the spi-ral and coast arcs.

The feasibility conditions and design constraints reduce tosimple compatibility equations. This not only simplifies the de-signproblem to solving systems of algebraic equations, but allowsto define the conditions that ensure the existence of solutions.Additional degrees of freedom can be considered by introducingcontrol nodes. The trajectory is decomposed in consecutive spi-rals or Keplerian arcs. The simple solutions obtained with singlespiral arcs can then be generalized at convenience, or even be han-dled by an optimizer. The distribution of the control nodes andthe values of the control parameter on each spiral arc will mightadjusted in order to minimize a certain cost function.

A fully three-dimensional transfer strategy is defined basedon the family of planar spirals. It is based on a complete decou-pling of the transversal dynamics, and the design procedure re-duces to solving four linear equations. When applied to actualdesign problems the solution provided by thismethod is in agree-ment with well-known optimized solutions.

“Ergo ellipſis eſt Planetæ iter.”

—Johannes Kepler

12Nonconservative extension of Keplerian

integrals and new families of orbits

T he invariance of the Lagrangian to time translationsand rotations in Kepler’s problem yields the conserva-tion laws related to the energy and angular momentum.

Noether’s theorem reveals that these same symmetries furnishgeneralized formsof the first integrals in a special nonconservativecase, which is a generalization of the acceleration introduced inChap. 9. The system is perturbed by a biparametric accelerationwith components along the tangential and normal directions. Asimilarity transformation reduces thebiparametric disturbance toa simpler uniparametric forcing along the velocity vector. Thesolvability conditions of this new problem are discussed in thepresent chapter, and closed form solutions for the integrable casesare provided. Keplerian orbits and generalized logarithmic spi-rals appear naturally from this construction. After characteriz-ing the orbits independently, a unified form of the solution isbuilt based on the Weierstrass elliptic functions. Two new fami-lies of orbits are found in this chapter: the generalized cardioids,and the generalized sinusoidal spirals. The names come from thefact that in the parabolic case the orbits reduce to pure cardioidsand sinusoidal spirals. These orbits approximate some instancesof Schwarzschild geodesics. Finally, the connection with otherknown integrable systems in celestial mechanics (Roa, 2016).

12.1 The role of first integrals

Finding first integrals is fundamental for characterizing a dynam-ical system. The motion is confined to submanifolds of lower di-mensions on which the orbits evolve, providing an intuitive in-terpretation of the dynamics and reducing the complexity of thesystem. In addition, conserved quantities are good candidates

when applying the secondmethod of Lyapunov for stability anal-ysis. Conservative systems related to central forces are typical ex-amples of (Liouville) integrability, and provide useful analyticresults. Hamiltonian systems have been widely analyzed in theclassical and modern literature to determine adequate integrabil-ity conditions. The existence of first integrals under the actionof small perturbations occupied Poincaré (1892, Chap. V) backin the nineteenth century. Emmy Noether (1918) established inher celebrated theorem that conservation laws can be understoodas the system exhibiting dynamical symmetries. In a more gen-eral framework, Yoshida (1983a,b) analyzed the conditions thatyield algebraic first integrals of generic systems. He relied on theKowalevski exponents for characterizing the singularity of the so-lutions and derived the necessary conditions for existence of firstintegrals exploiting similarity transformations.

Conservation laws are sensitive to perturbations and theirgeneralization is not straightforward. For example, the Jacobi in-tegral no longer holds when transforming the circular restrictedthree-body problem to the elliptic case (Xia, 1993). Nevertheless,Contopoulos (1967) was able to find approximate conservationlaws for orbits of small eccentricities. Szebehely and Giacaglia(1964) benefited from the similarities of the elliptic and the cir-cular problems in order to define transformations between them.Hénon andHeiles (1964) deepened in the nature of conservationlaws and reviewed the concepts of isolating and non-isolating in-tegrals. Their study introduced a similarity transformation thatembeds one of the constants of motion and transforms the orig-inal problem into a simplified one, reducing the degrees of free-dom (Arnold et al., 2007, §3.2). Carpintero (2008) proposed anumerical method for finding the dimension of the manifold inwhich orbits evolve, i.e. the number of isolating integrals that the

161

162 12 Nonconservative extension of Keplerian integrals and new families of orbits

system admits.The conditions for existence of integrals of motion under

nonconservative perturbations received important attention inthe past due to their profound implications. Djukic and Vu-janovic (1975) advanced onNoether’s theorem and included non-conservative forces in the derivation. Relying onHamilton’s vari-ational principle they not only extended Noether’s theorem, butits inverse form and the Noether-Bessel-Hagen and Killing equa-tions. Later studies by Djukic and Sutela (1984) sought integrat-ing factors that yield conservation laws upon integration. Exam-ples of application of Noether’s theorem to constrained noncon-servative systems can be found in the work of Bahar and Kwatny(1987). Honein et al. (1991) arrived to a compact formulation us-ing what was later called the neutral action method. Remarkableapplications exploiting Noether’s symmetries span from cosmol-ogy (Capozziello et al., 2009; Basilakos et al., 2011) to string the-ory (Beisert et al., 2008), field theory (Halpern et al., 1977), andfluid models (Narayan et al., 1987). In the book by Olver (2000,Chaps. 4 and 5) an exhaustive review of the connection betweensymmetries and conservation laws is provided within the frame-work of Lie algebras. We refer to Arnold et al. (2007, Chap. 3)for a formal derivation of Noether’s theorem and a discussion onthe connection between conservation laws and dynamical sym-metries.

Integrals of motion are often useful for finding analytic orsemi-analytic solutions to a given problem. The acclaimed solu-tion to the satellite main problem by Brouwer (1959) is a clearexample of the decisive role of conserved quantities in deriv-ing solutions in closed form. By perturbing the Delaunay ele-ments Brouwer andHori (1961) solved the dynamics of a satellitesubject to atmospheric drag and the oblateness of the primary.They proved the usefulness of canonical transformations evenin the context of nonconservative problems. Whittaker (1917,pp. 81–82) approached the problem of a central force depend-ing on powers of the radial distance, rn, and found that thereare only fourteen values of n for which the problem can be in-tegrated in closed form using elementary functions or elliptic in-tegrals. Later, he discussed the solvability conditions for equa-tions involving square roots of polynomials (Whittaker andWat-son, 1927, p. 512). Broucke (1980) advanced on Whittaker’s re-sults and found six potentials that are a generalization of the in-tegrable central forces discussed by the latter. These potentialsinclude the referred fourteen values of n as particular cases. Nu-merical techniques for shaping the potential given the orbit so-lution were published by Carpintero and Aguilar (1998). Clas-sical studies on the integrability of systems governed by centralforces are based strongly on Newton’s theorem of revolving or-bits.* The problem of the orbital precession caused by centralforces was recently recovered by Adkins and McDonnell (2007),who considered potentials involving both powers and logarithms

*Section IX, Book I, of Newton’s Principia is devoted to themotion of bod-ies in moveable orbits (De Motu Corporum in Orbibus mobilibus, deq; motu Ap-sidum, in the original latin version). In particular, Thm. XIV states that “Thedifference of the forces, by which two bodies may be made to move equally, onein a quiescent, the other in the same orbit revolving, is in a triplicate ratio oftheir common altitudes inversely”. Newton proved this theorem relying on ele-gant geometric constructions. The motivation behind this result was the devel-opment of a theory for explaining the precession of the orbit of theMoon. A de-tailed discussion about this theorem can be found in the book byChandrasekhar(1995, pp. 184–201)

of the radial distance, and the special case of the Yukawa poten-tial (Yukawa, 1935). Chashchina and Silagadze (2008) relied onHamilton’s vector to simplify the analytic solutions foundbyAd-kins and McDonnell (2007). More elaborated potentials havebeen explored for modeling the perihelion precession (Schmidt,2008). The dynamics of a particle in Schwarzschild space-timecan be regarded as orbital motion perturbed by an effective po-tential depending on inverse powers of the radial distance (Chan-drasekhar, 1983, p. 102).

Potentials depending linearly on the radial distance appear re-cursively in the literature because they render constant radial ac-celerations, relevant for the design of spacecraft trajectories pro-pelled by continuous-thrust systems. The pioneering work byTsien (1953) provided the explicit solution to the problem interms of elliptic integrals, as predicted byWhittaker (1917, p. 81).By means of a special change of variables Izzo and Biscani (2015)arrived to an elegant solution in terms of the Weierstrass ellip-tic functions. These functions were also exploited by MacMillan(1908) when he solved the dynamics of a particle attracted by acentral force decreasing with r−5. Urrutxua et al. (2015a) solvedthe Tsien problem using the Dromo formulation, which mod-els orbital motion with a regular set of elements (see Chap. 4).The case of a constant radial force was approached by Akella andBroucke (2002) from an energy-driven perspective. They studiedin detail the roots of the polynomial appearing in the denomina-tor of the equation to integrate and connected their nature withthe form of the solution. We refer again to San-Juan et al. (2012)for a detailed discussion about the integrability of the system.

Another relevant example of an integrable system in celestialmechanics is the Stark problem, governed by a constant accelera-tion fixed in the inertial frame. Lantoine and Russell (2011) pro-vided the complete solution to the motion relying extensively onelliptic integrals and Jacobi elliptic functions. A compact formof the solution involving the Weierstrass elliptic functions waslater presented byBiscani and Izzo (2014), who also exploited thisformalism for building a secular theory for the post-Newtonianmodel (Biscani andCarloni, 2012). The Stark problemprovides asimplified model of radiation pressure. In general, the dynamicssubject to this perturbation cannot be solved in closed form. Anintuitive simplification that makes the problem integrable con-sists in assuming that the force due to the solar radiation pressurefollows the direction of the Sun vector. The dynamics are equiv-alent to those governed by a Keplerian potential with a modifiedgravitational parameter.

InChap. 9 it was proven that the problemdefined by the per-turbation

ap =µ

2r2 cosψ t

can be solved in closed form, and that there are two conservationlaws. This acceleration was later extended in Chap. 10 to includea control parameter, while preserving the properties of the sys-tem. In the present chapter we approach this problem from ageneric perspective, considering an even more general accelera-tion including two freeparameters. Following this newapproach,the problem is formulated in Sect. 12.2, where the biparametricacceleration is reduced to a uniparametric forcing thanks to a sim-ilarity transformation. The conservation laws hold in the moregeneral case and characterize this type of system. The first inte-

§12.2 Dynamics 163

grals are obtained by exploiting known symmetries of Kepler’sproblem. Before solving the dynamics explicitly we prove thatthere are four cases that can be solved in closed form using ele-mentary or elliptic functions. The first two families are conic sec-tions and generalized logarithmic spirals. Sections 12.5 and 12.6introduce the two new families of orbits. Section 12.7 is a sum-mary of the solutions, which are unified in Sect. 12.8 introduc-ing the Weierstrass elliptic functions. Finally, Sect. 12.9 discussesthe connection with known solutions to other problems, and theconnection with Schwarzschild geodesics.

12.2 Dynamics

The two previous chapters dealt with the two-body problem

d2rdt2 +

µ

r3 r = ap (12.1)

perturbed by the continuous acceleration defined in Eq. (10.2).This forcing term was defined as

ap =µ

r2

[ξ cosψ t + (1 − 2ξ) sinψ n

]The coefficient (1 − 2ξ) was chosen so that the integral of mo-tion (9.16) still held. Replacing this term by a generic parameterη renders

ap =µ

r2 (ξ cosψ t + η sinψn) (12.2)

This new form of the perturbation includes the Keplerian caseas a particular instance, because making ξ = η = 0 cancels theperturbation. In addition, generalized logarithmic spirals are thesolution to η = (1 − 2ξ). The forcing parameter ξ controls thepower exerted by the perturbation,

dEk

dt= ap · v =

ξµ

r2 v cosψ

with Ek the Keplerian energy of the system. The second param-eter ηmodulates the contribution of forces that do not performany work.

For convenience we shall replace η by the parameter

γ =1 + η1 − ξ

In this way, when the perturbation is added to the Keplerian ac-celeration it takes the simplified form

d2rdt2 = −

µ

r2 (1 − ξ)(cosψ t − γ sinψ n) (12.3)

The term µ(1 − ξ) can be regarded as a modified gravitationalparameter µ∗. Consequently, when γ = 1 Eq. (12.3) reduces toKepler’s problem written in terms of µ∗ instead of µ.

12.2.1 Similarity transformation

In Sect. 10.2 we discussed a similarity transformation that con-nects the solution to the original thrust (9.1) and the one includ-ing the control parameter ξ, Eq. (10.2). In the present section we

will introduce a more general similarity transformation and dis-cuss its properties in detail.

Consider a linear transformation S of the form

S : (t, r, θ, r, θ) 7→ (τ, ρ, θ, ρ′, θ′)

defined explicitly by the positive constants α, β, and δ = α/β:

τ =tβ, ρ =

rα, ρ′ =

rδ, θ′ = β θ

The constantsα, β, and δ have units of length, time, and velocity,respectively. The symbol′ denotes derivatives with respect to τ,whereas is reserved for derivatives with respect to t. The scalingfactor α can be seen as the ratio of a homothetic transformationthat simply dilates or contracts the orbit. Similarly, β representsa time dilation or contraction. The velocity of the particle trans-forms into

v =v

δ

In addition, β and δ are defined in terms of α by virtue of

β =

√α3

µγ(1 − ξ) and δ =α

β=

√µγ(1 − ξ)

α

We assume γ > 0 and ξ < 1 for consistency.Equation (12.3) then becomes

d2ρ

dτ2 = −1ρ2

(1γ

cosψ t − sinψn), (12.4)

which is equivalent to the normalized two-body problem

d2ρ

dτ2 = −ρ

ρ3 + ap (12.5)

This result shows thatS establishes, indeed, a similarity transfor-mation between the two-body problem perturbed by the acceler-ation (12.2) and the simpler problem in Eq. (12.5), perturbed bythe purely tangential acceleration:

ap =γ − 1γρ2 cosψ t (12.6)

That is, S−1 transforms the solution to Eq. (12.4) into the solu-tion to Eq. (12.3). It is worth mentioning that γ = 2 reduces theacceleration to

ap =cosψ2ρ2 t

which is none other than (9.1).Using q = [ρ, θ]⊤ and q′ = [ρ′, θ′]⊤ as the generalized coor-

dinates and velocities, respectively, the dynamics of the problemabide by the Euler-Lagrange equations

ddτ

(∂L∂q′i

)− ∂L∂qi= Qi

The Lagrangian of the transformed system takes the form

L =12

(ρ′2 + ρ2θ′2) +1ρ

(12.7)

and the generalized forces Qi read

Qρ =(γ − 1)γ

(ρ′

ρv

)2

and Qθ =(γ − 1)γ

(ρ′θ′

v2

)(12.8)

164 12 Nonconservative extension of Keplerian integrals and new families of orbits

12.2.2 Integrals of motion and dynamical symmetries

Let us introduce an infinitesimal transformation R:

τ→ τ∗ = τ + ε f (τ; qi, q′i)qi → q∗i = qi + εFi(τ; qi, q′i)

(12.9)

defined in terms of a small parameter ε ≪ 1 and the generatorsFi and f . For transformations that leave the action unchangedup to an exact differential,

L(τ∗; q∗i ,

dq∗idτ∗

)dτ∗ −L

(τ; qi,

dqi

dτ

)dτ = ε dΨ(τ; qi, q′i)

with Ψ(τ; qi, q′i) a given gauge, Noether’s theorem establishesthat∑

i

(∂L∂q′i

)Fi + f

[L −

(∂L∂q′i

)q′i

]−Ψ(τ; qi, q′i) = Λ (12.10)

is a first integral of the problem. Here Λ is a certain constant ofmotion. We refer the reader to the work of Efroimsky and Gol-dreich (2004) for a refreshing look into the role of gauge func-tions in celestial mechanics (see Sect. 2.7 for a summary).

Since the perturbation in Eq. (12.6) is not conservative weshall focus on the extension of Noether’s theorem to noncon-servative systems by Djukic and Vujanovic (1975). It must beFi − q′i f , 0 for the conservation law to hold (Vujanovic et al.,1986). For the case of nonconservative systems the generators Fi,f , and the gaugeΨ need to satisfy the following relation:

∑i

(∂L∂qi

)Fi +

(∂L∂q′i

)(F′i − q′i f ′) + Qi(Fi − q′i f )

+ f ′L + f

∂L∂τ= Ψ′ (12.11)

This equation and the condition Fi−q′i f , 0 furnish the general-ized Noether-Bessel-Hagen (NBH) equations (Trautman, 1967;Djukic and Vujanovic, 1975; Vujanovic et al., 1986). The NBHequations involve the full derivative of the gauge function and thegenerators with respect to τ, meaning that Eq. (12.11) depends onthe partial derivatives ofΨ, Fi, and f with respect to time, the co-ordinates, and the velocities. By expanding the convective termstheNBHequationsdecompose in the systemofKilling equations(Djukic and Vujanovic, 1975):

L∂ f∂q′j+

∑i

∂L∂q′i

∂Fi

∂q′j− q′i

∂ f∂q′j

= ∂Ψ

∂q′j

∂

∂τ( fL − Ψ) +

∑i

∂L∂qi

Fi +L∂ f∂qi

q′i + Qi(Fi − q′i f )

+∂L∂q′i

∂Fi

∂τ− q′i

∂ f∂τ+

∑j

(∂Fi

∂q jq′j − q′iq

′j∂ f∂q j

) − ∂Ψ∂qiq′i

= 0

(12.12)

The system (12.12) is composed of three equations that can besolved for the generators Fρ, Fθ, and f given a certain gauge. Ifthe transformation defined in Eq. (12.9) satisfies the NBH equa-tions, then the system admits the integral of motion (12.10).

Generalized equation of the energy

The Lagrangian in Eq. (12.7) is time-independent. Thus, the ac-tion is not affected by arbitrary time transformations. In the Ke-plerian case a simple time translation reveals the conservation ofthe energy. Motivated by this fact, we explore the generators

f = 1, Fρ = 0, and Fθ = 0

Clearly Fi − q′i f , 0. Solving Killing equations (12.12) with theabove generators it follows the gauge function

Ψ =γ − 1γr

Provided that the NBH equations hold the system admits the in-tegral of motion

v2

2− 1γρ= −Λ ≡ κ1

2(12.13)

written in terms of the constant κ1 = −2Λ. This term can besolved from the initial conditions

κ1 = v20 −

2γρ0

When γ = 1 the perturbation (12.6) vanishes and Eq. (12.13)reduces to the normalized equation of the Keplerian energy. Infact, in this case κ1 becomes twice the Keplerian energy of the sys-tem, κ1 = 2Ek. Moreover, the gauge vanishes and Eq. (12.10)furnishes the Hamiltonian of Kepler’s problem. The integral ofmotion (12.13) is a generalization of the equation of the energy.Similarly, for γ = 2 Eq. (12.13) reduces to the generalized equa-tion of the energy presented in Eq. (9.13).

Generalized equation of the angular momentum

In theunperturbedproblem θ is an ignorable coordinate. Indeed,a simple translation in θ (a rotation) with f = Fρ = Ψ = 0 andFθ = 1 yields the conservation of the angular momentum. In or-der to extend this first integral to the perturbed case we considerthe same generator Fθ = 1. However, solving for the gauge andthe remaining generators in Killing equations yields the nontriv-ial functions

Fρ =ρ′

θ′(1 − vγ−1)

f =1 − vγ−1

θ′+ (1 − γ)ρ2θ′vγ−3

Ψ =1

2ρθ′v2

[ρ − (3 − γ)ρvγ−1

]+ 2 − vγ−1[2γ − (3 − γ)ρ′2ρ]

+ vγ−3[ρ(ρ4θ′4 − ρ′4) − 2(1 − γ)ρ′2]

They satisfy Fi−q′i , 0. Noether’s theoremholds and Eq. (12.10)furnishes the integral of motion

ρ2vγ−1θ′ = Λ ≡ κ2 (12.14)

This first integral is the generalized form of the conservation ofthe angular momentum. Indeed, making γ = 1 Eq. (12.14) re-duces to

ρ2θ′ = κ2

§12.2 Dynamics 165

where κ2 coincides with the angular momentum of the particle.In addition, the generators Fρ and f , and the gauge vanish whenγ equals unity.

The integral of motion (12.14) can be written:

ρvγ sinψ = κ2

in terms of the coordinates intrinsic to the trajectory. It is eas-ily verified that γ = 2 makes this equation to coincide with theconservation law in Eq. (9.16). The fact that sinψ ≤ 1 forces

κ2 ≤ vγρ =⇒ κ22 ≤ v2γρ2

The second step is possible because all variables are positive.Combining this expression with Eq. (12.13) yields

κ22 ≤

(κ1 +

2γρ

)γρ2 (12.15)

Depending on the values of γ, Eq. (12.15) may define upper orlower limits to the values that the radius ρ can reach. In general,this condition can be resorted to provide the polynomial con-straint

Pnat(ρ) ≥ 0

where Pnat(ρ) is a polynomial of degree γ in ρ whose roots dic-tate the nature of the solutions. This inequality will be useful fordefining the different families of solutions.

12.2.3 Properties of the similarity transformation

The main property of the similarity transformation S is that itdoes not change the type of the solution, i.e. the sign of κ1 is notaltered. Applying the inverse similarity transformation S−1 toEq. (12.13) yields

κ1 =v2

δ2 −2αγr=

v2α

µγ(1 − ξ)−2αγr=

α

µγ(1 − ξ)

[v2 − 2µ

r(1 − ξ)

]and results in

K1 = δ2 κ1 = v

2 − 2µr

(1 − ξ) = 2(v2

2− µ

∗

r

)Since δ2 > 0 no matter the values of ξ or γ, the sign of K1 isnot affected by the transformation S. If the solution to the orig-inal problem (12.1) is elliptic, the solution to the reduced prob-lem (12.5) will be elliptic too, and vice-versa. The transformationreduces to a series of scaling factors affecting each variable inde-pendently.

The integral of the angular momentum transforms into

κ2 =r2vγ−1θ

αδγ=

K2

αδγ=⇒ K2 = αδ

γκ2

The constant κ2 remains positive, although the scaling factorαδγcan modify its value significantly.

The transformation is defined in terms of three parameters:α, ξ and γ. For ξ = ξγ, with

ξγ = 1 − αγ

S reduces to the identitymap. Choosingα = 1 the special valuesof ξ that yield trivial transformations for γ = 1, 2, 3 and 4 are,respectively, ξγ = 0, 1/2, 2/3 and 3/4. The similarity transfor-mation can be understood from a different approach: solving thesimplified problem is equivalent to solving the full problem butsetting ξ = ξγ.

The polar angle relates to the radius by the differential equa-tiondθdρ=

tanψρ

(12.16)

The right-hand side of this equation can be written as a functionof ρ alone thanks to

tanψ =nκ2√

(vγρ)2 − κ22

The parameter n is n = +1 for orbits in a raising regime (r > 0),and n = −1 for a lowering regime (r < 0). The velocity is solvedfrom Eq. (12.13). Integrating Eq. (12.16) furnishes the solutionθ(ρ), which can then be inverted to define the trajectory ρ(θ).

12.2.4 Solvability

The trajectory of the particle is obtained upon integration andinversion of Eq. (12.16). This equation can be written

dθdρ=

nκ2√Psol(ρ)

(12.17)

where Psol(ρ) is a polynomial in ρ, in particular

Psol(ρ) = ρ2Pnat(ρ)

The roots of Psol(ρ) determine the form of the solution and co-incide with those of Pnat(ρ) (obviating trivial solutions). The in-tegration of Eq. (12.17) depends on the factorization of Psol(ρ).This polynomial expression can be expanded thanks to the bino-mial theorem:

Psol(ρ) =γ∑

k=0

(γ

k

)2k

γk ρ4−kκ

γ−k1 − ρ2κ2

withγ , 0 an integer. Forγ ≤ 4 the polynomial is of degree four:when γ = 1 or γ = 2 there are two trivial roots and Eq. (12.17)can be integrated using elementary functions; when γ = 3 orγ = 4 it yields elliptic integrals. Negative values of γ or positivevalues greater than four lead to a polynomial Psol(ρ) with degreefive or above. The solution can no longer be reduced to elemen-tary functions nor elliptic integrals (Whittaker andWatson, 1927,p. 512), for it is given by hyperelliptic integrals. This special classofAbelian integrals can only be inverted in very specific situations(see Byrd andFriedman, 1954, pp. 252–271). Thus, we shall focuson the solutions to the cases γ = 1, 2, 3 and 4.

The following sections 12.3–12.6 present the correspondingfamilies of orbits. For γ = 1 the solutions to the reduced prob-lem are Keplerian orbits. For the case γ = 2 the solutions aregeneralized logarithmic spirals. The cases γ = 3 and γ = 4 yieldtwo new families of orbits, generalized cardioids and generalizedsinusoidal spirals, respectively (for κ1 = 0 the orbits are cardioidsand sinusoidal spirals).

166 12 Nonconservative extension of Keplerian integrals and new families of orbits

12.3 Case γ = 1: conic sections

For γ = 1 the integrals of motion (12.13) and (12.14) reduce tothe equations of the energy and angularmomentum, respectively.The condition on the radius given by Eq. (12.15) becomes

Pnat(ρ) ≡ κ1ρ2 + 2ρ − κ2

2 ≥ 0 (12.18)

For the case κ1 < 0 (elliptic solution) this translates into ρ ∈[ρmin, ρmax], where

ρmin =1

(−κ1)

(1 −

√1 + κ1κ

22

)ρmax =

1(−κ1)

(1 +

√1 + κ1κ

22

)These limits are none other than the periapsis and apoapsis radii,provided that κ1 and κ2 relate to the semimajor axis and eccentric-ity by means of:

1(−κ1)

= a and√

1 + κ1κ22 = e

For κ1 = 0 (parabolic case) the semimajor axis becomes infinite,and Eq. (12.18) has only one root corresponding to ρ = κ2

2/2.Note that it must be κ2

2 > −1/κ1. Similarly, for hyperbolic orbits(κ1 > 0) it is ρ ≥ ρmin as ρmax becomes negative.

Thus, the solution is simply a conic section:

ρ(θ) =h2

1 + e cos(θ − θm)=

κ22

1 +√

1 + κ1κ22 cos(θ − θm)

(12.19)

The angle θm = Ω + ω defines the direction of the line of apsesin the inertial frame, meaning that θ − θm is the true anomaly. Ifκ1 < 0 then ρ(θm) = ρmin, and ρ(θm + π) = ρmax. The velocity vfollows from the integral of the energy:

v =√κ1 + 2/ρ

It is minimum at apoapsis and maximum at periapsis.Applying the similarity transformation S−1 to the previous

solution leads to the extended integral

K1

2= δ2 κ1

2=v2

2− µ

r(1 − ξ) = v2

2− µ

∗

r

The factor µ∗ is the modified gravitational parameter. This kindof solutions arise from, for example, the effect of the solar radia-tion pressure directed along the Sun-line on a particle following aheliocentric orbit (McInnes, 2004, p. 121).

12.4 Case γ = 2: generalized logarithmicspirals

The case γ = 2 yields the family of generalized logarithmic spi-rals. These orbits have been discussed extensively in Chaps. 9–11.The corresponding trajectories, properties, and applications canbe found in the referred chapters.

12.5 Case γ = 3: generalized cardioids

The condition in Eq. (12.15), yields the polynomial inequality:

Pnat(ρ) ≡ κ31 ρ

3 + 2κ21 ρ

2 +

(4κ1

3− κ2

2

)ρ +

827≥ 0 (12.20)

The discriminant∆ of the polynomial Pnat(ρ) predicts the natureof the roots. It is

∆ = −4κ31κ

42(3κ1 − κ2

2) (12.21)

The intermediate value theorem shows that there is at least onereal root. For the elliptic case (κ1 < 0) it is ∆ < 0, meaning thatthe other two roots are complex conjugates. In the hyperboliccase (κ1 > 0) the sign of the discriminant depends on the valuesof κ2: if κ2

2 > 3κ1 it is∆ > 0, and for κ22 < 3κ1 the discriminant is

negative. This behavior yields two types of hyperbolic solutions.

12.5.1 Elliptic motion

The nature of elliptic motion is determined by the polynomialconstraint in Eq. (12.20). The real root is given by

ρ1 =Λ(Λ + 2κ2

1) + 3κ31κ

22

3(−κ1)3Λ(12.22)

with

Λ = (−κ1)3(−κ1)κ2

2

[√−3κ1(κ2

2 − 3κ1) + 3κ1

]1/3(12.23)

Equation (12.20) reduces to ρ − ρ1 ≤ 0 and we shall write

ρmax ≡ ρ1

Thus, elliptic generalized cardioids never escape to infinity be-cause they are bounded by ρmax. When in raising regime theyreach the apoapsis radius ρmax, then transition to lowering regimeand fall toward the origin. The velocity at apoapsis,

vm =√κ1 + 2/(3ρmax)

is the minimum velocity in the cardioid.Equation (12.16) can be integrated from the initial radius r0

to the apoapsis of the cardioid, and the result provides the orien-tation of the line of apses:

θm = θ0 +nκ2√

AB(−κ1)3/2[2 K(k) − F(ϕ0, k)]

where K(k) and F(ϕ0, k) are the complete and incomplete ellipticintegrals of the first kind, respectively. Introducing the auxiliaryterm

λi j = ρi − ρ j

their argument and modulus read

ϕ0 = arccos[

Bλ10 − Aρ0

Bλ10 + Aρ0

], k =

√ρ2max − (A − B)2

4AB

§12.5 Case γ = 3: generalized cardioids 167

(a) Elliptic (b) Parabolic (c) Hyperbolic Type I (d) Hyperbolic Type II (e) Hyperbolic transition

Figure 12.1: Examples of generalized cardioids (γ = 3).

The previous definitions involve the auxiliary parameters:

A =√

(ρmax − b1)2 + a21, B =

√b2

1 + a21

and

b1 =Λ(Λ − 4κ2

1) + 3κ31κ

22

6κ31Λ

, a21 =

(Λ2 − 3κ31κ

22)2

12κ61Λ

2

Recall the definition ofΛ in Eq. (12.23).The equation of the trajectory is obtained by inverting the

function θ(ρ), and results in:

ρ(θ)ρmax

=

1 +

AB

[1 − cn(ν, k)1 + cn(ν, k)

]−1

(12.24)

It is defined in terms of the Jacobi elliptic function cn(ν, k). Theanomaly ν reads

ν(θ) =(−κ1)3/2

κ2

√AB (θ − θm)

Equation (12.24) is symmetricwith respect to the apse line,ρ(θm+

∆θ) = ρ(θm − ∆θ), as shown in Fig. 12.1(a).

12.5.2 Parabolic motion: the cardioid

When the constant of the generalized energy κ1 vanishes the con-dition in Eq. (12.15) translates into ρ < ρmax, where the maxi-mum radius ρmax takes the form

ρmax =8

27κ22

Parabolic generalized cardioids, unlike logarithmic spirals orKep-lerian parabolas, are bounded (they never escape the gravitationalattraction of the central body).

The line of apses is defined by:

θm = θ0 + n[π

2+ arcsin

(1 − 27

4κ2

2ρ0

)]The equation of the trajectory reveals that the orbit is in fact a

pure cardioid:*

ρ(θ)ρmax

=12

[1 + cos(θ − θm)] (12.25)

This curve is symmetric with respect to θm. Figure 12.1(b) depictsthe geometry of the solution.

12.5.3 Hyperbolic motion

The inequality in Eq. (12.15) determines the structure of the so-lutions and the sign of the discriminant (12.21) governs the na-ture of its roots. There are two types of hyperbolic generalizedcardioids: for κ2 <

√3κ1 the cardioids are of Type I, and for

κ2 >√

3κ1 the cardioids are of Type II.

Hyperbolic cardioids of Type I

For hyperbolic cardioids of Type I there is only one real root. Theother two are complex conjugates. The real root is

ρ3 = −Λκ2

1(2 + Λκ1) + 3κ22

3κ31Λ

(12.26)

having introduced the auxiliary parameter

Λ =

3κ22

κ9/21

[3√κ1 +

√3(3κ1 − κ2

2)]

1/3

Provided that Λ > 0 Eq. (12.26) shows that ρ3 < 0. Therefore,there are no limits to the values that ρ can take. As a consequence,the cardioid never transitions between regimes. If it is initiallyψ0 < π/2 it will always escape to infinity, and fall toward theorigin for ψ0 > π/2.

The equation of the trajectory for hyperbolic generalized car-dioids of Type I is

ρ(θ)ρ3A=

(A + B) sn2(ν, k) + 2B[cn(ν, k) − 1](A + B)2 sn2(ν, k) − 4AB

(12.27)

*The cardioid is a particular case of the limaçons, curves first studied by theamateurmathematician Étienne Pascal in the 17th century. The general form ofthe limaçon in polar coordinates is r(θ) = a + b cos θ. Depending on the valuesof the coefficients the curve might reach the origin and form loops. It is worthnoticing that the inverse of a limaçon, r(θ) = (a + b cos θ)−1, results in a conicsection. Limaçons with a = b are considered part of the family of sinusoidalspirals.

168 12 Nonconservative extension of Keplerian integrals and new families of orbits

It is defined in terms of

A =√

b21 + a2

1, B =√

(ρ3 − b1)2 + a21

which require

b1 =Λκ2

1(Λκ1 − 4) + 3κ22

6κ31Λ

, a21 =

(Λ2κ31 − 3κ2

2)2

6Λ2κ61

There are no axes of symmetry. Therefore, the anomaly is referreddirectly to the initial conditions:

ν(θ) =κ3/2

1

κ2

√AB (θ − θ0) + F(ϕ0, k)

The moduli of both the Jacobi elliptic functions and the ellipticintegral, and the argument of the latter, are

k =

√(A + B)2 − ρ2

3

4AB, ϕ0 = arccos

[Aλ03 − Bρ0

Aλ03 + Bρ0

]The cardioid approaches infinity along an asymptotic branch,with v → v∞ as ρ → ∞. The orientation of the asymptote fol-lows from the limit

θas = limρ→∞

θ(ρ) = θ0 +nκ2

κ3/21

√AB

[F(ϕ∞, k) − F(ϕ0, k)

]Here, the value of ϕ∞,

ϕ∞ = arccos(A − B

A + B

)is defined as ϕ∞ = limρ→∞ ϕ. An example of a hyperbolic car-dioid of Type I with its corresponding asymptote is presented inFig. 12.1(c).

Hyperbolic cardioids of Type II

For hyperbolic cardioids of Type II the polynomial in Eq. (12.15)admits three distinct real roots, ρ1, ρ2, ρ3, given by

ρk+1 =2κ2√3κ3

1

cos[π

3(2k + 1) − 1

3arccos

( √3κ1

κ2

)]− 2

3κ1

with k = 0, 1, 2. The roots are then sorted so that ρ1 > ρ2 > ρ3.Since

ρ1ρ2ρ3 = −8

27κ31

< 0

then ρ3 < 0 and also ρ1 > ρ2 > 0 for physical coherence. Thepolynomial constraint reads

(ρ − ρ1)(ρ − ρ2)(ρ − ρ3) ≥ 0

and holds for both ρ > ρ1 and ρ < ρ2. The integral of mo-tion (12.13) shows that both situations are physically admissible,because v2(ρ1) > 0 and v2(ρ2) > 0. There are two families of so-lutions that lie outside the annulus ρ < (ρ2, ρ1). When ρ0 < ρ2the spirals are interior, whereas for ρ > ρ1 they are exterior spi-rals. The geometry of the forbidden region can be analyzed in

Fig. 12.1(d). The particle cannot enter the barred annulus, whoselimits coincide with the periapsis and apoapsis of the exterior andinterior orbits, respectively.

The axis of symmetry of interior spirals is given by

θm = θ0 +2nκ2[K(k) − F(ϕ0, k)]

κ3/21

√ρ1λ23

in terms of the arguments:

ϕ0 = arcsin

√ρ0λ23

ρ2λ03, k =

√ρ2λ13

ρ1λ23

The trajectory simplifies to:

ρ(θ)ρ3=

1

1 + (λ32/ρ2) dc2(ν, k)(12.28)

Here we made use of Glaisher’s notation for the Jacobi ellipticfunctions,* so dc(ν, k) = dn(ν, k)/ cn(ν, k). The spiral anomaly νtakes the form

ν(θ) =κ3/2

1

√ρ1λ23

2κ2(θ − θm)

The trajectory is symmetric with respect to the line of apses de-fined by θm.

For the case of exterior spirals the largest root ρ1 behaves asthe periapsis. A cardioid initially in lowering regime will reachρ1, then it will transition to raising regime and escape to infinity.Equation (12.16) is integrated from the initial radius to the peri-apsis to provide the orientation of the line of apses:

θm = θ0 −2nκ2 F(ϕ0, k)

κ3/21

√ρ1λ23

(12.29)

with

ϕ0 = arcsin√λ23λ01

λ13λ02, k =

√ρ2λ13

ρ1λ23

the argument and modulus of the elliptic integral.The trajectory of exterior spirals is obtained upon inversion

of the equation for the polar angle,

ρ(θ) =ρ2λ13 sn2(ν, k) − ρ1λ23

λ13 sn2(ν, k) − λ23(12.30)

The anomaly is redefined as

ν(θ) =κ3/2

1

√ρ1λ23

2κ2(θ − θm)

This variable is referred to the line of apses, given in Eq. (12.29).The form of the solution shows that hyperbolic generalized car-dioids of Type II are symmetric with respect to θm.

*Glaisher’s notation establishes that if p,q,r are any of the four letters s,c,d,n,then:

pq(ν, k) =pr(ν, k)qr(ν, k)

=1

qp(ν, k)

Under this notation repeated letters yield unity. See Appendix E for details.

§12.6 Case γ = 4: generalized sinusoidal spirals 169

Due to the symmetry of Eq. (12.30) the trajectory exhibitstwo symmetric asymptotes, defined by

θas = θm ±2κ2 F(ϕ∞, k)

κ3/21

√ρ1λ23

The argument ϕ∞ reads

ϕ∞ = arcsin√λ23

λ13

TransitionbetweenType I andType II hyperboliccar-dioids

The limit case κ2 =√

3κ1 defines the transition between hyper-bolic cardioids ofTypes I and II.Thediscriminant∆ vanishes: theroots are all real and one is a multiple root, ρlim ≡ ρ1 = ρ2. Theregion of forbidden motion degenerates into a circumference ofradius ρlim. The roots take the form:

ρ3 = −8

3κ1< 0 and ρlim ≡ ρ1 = ρ2 =

13κ1

(12.31)

The condition (ρ − ρ3)(ρ − ρlim)2 ≥ 0 holds naturally forρ > 0. When the cardioid reaches ρlim the velocity becomesvlim =

√3κ1 = ρ

−1/2lim . It coincides with the local circular velocity.

Moreover, from the integral of motion (12.14) one has

κ2 = ρlimv3lim sinψlim =⇒ sinψlim = 1

meaning that the orbit becomes circular as theparticle approachesρlim. When ψlim = π/2 the perturbing acceleration in Eq. (12.6)vanishes. As a result, the orbit degenerates into a circular Keple-rian orbit. A cardioid with ρ0 < ρlim and in raising regime willreach ρlim and degenerate into a circular orbit with radius ρlim.This phenomenon also appears in cardioids with ρ0 > ρlim andin lowering regime.

The trajectory reduces to

ρ(θ)ρlim=

cosh ν − 1cosh ν + 5/4

(12.32)

which is written in terms of the anomaly

ν(θ) =θ − θ0√

3+ 2mn arctanh

(3√

ρ0

8ρlim + ρ0

)The integer m = sign(1−ρ0/ρlim) determines whether the parti-cle is initially below (m = +1) or above (m = −1) the limit radiusρlim. The limit limθ→∞ ρ = ρlim = 1/(3κ1) shows that the radiusconverges toρlim. This limit only applies to the casesm = n = +1and m = n = −1. When the particle is initially below ρlim andin lowering regime, m = +1, n = −1, it falls toward the origin.In the opposite case, m = −1, n = +1, the cardioid approachesinfinity along an asymptotic branch with

θas = θ0 − 2√

3[mn arctanh

(3√

ρ0

8ρlim + ρ0

)+ arctanh(3)

]Two example trajectories with n = +1 are plotted in Fig. 12.1(e).The dashed line corresponds to m = +1 and the solid line to m =−1. The trajectories terminate/emanate from a circular orbit ofradius ρlim.

12.6 Case γ = 4: generalized sinusoidal spi-rals

Settingγ = 4 in Eq. (12.15) gives rise to the polynomial inequality

[4(κ2

1ρ + κ1 + κ2)ρ + 1] [

4(κ21ρ + κ1 − κ2)ρ + 1

]≥ 0 (12.33)

which governs the subfamilies of solutions to the problem. Thefour roots of the polynomial are

ρ1,2 = +κ2 − κ1 ±

√κ2(κ2 − 2κ1)

2κ21

ρ3,4 = −κ1 + κ2 ∓

√κ2(κ2 + 2κ1)

2κ21

and the discriminant of Pnat(ρ) is

∆ =κ6

2

κ201

(κ22 − 4κ2

1) (12.34)

The sign of the discriminant determines the nature of the fourroots.

12.6.1 Elliptic motion

When κ1 < 0 there are two subfamilies of elliptic sinusoidal spi-rals: of Type I, with κ2 > −2κ1 (∆ > 0), and of Type II, withκ2 < −2κ1 (∆ < 0). Both types are separated by the limit caseκ2 = −2κ1 that makes ∆ = 0.

Elliptic sinusoidal spirals of Type I

For the case κ2 > −2κ1 the discriminant is positive and the fourroots are real, with ρ1 > ρ2 > ρ3 > ρ4. Since ρ3,4 < 0 Eq. (12.33)reduces to

(ρ − ρ1)(ρ − ρ2) ≥ 0, (12.35)

meaning that it must be either ρ > ρ1 or ρ < ρ2. The integral ofmotion (12.13) reveals that only the latter case is physically possi-ble, because v2(ρ1) < 0. Thus, ρ2 is the apoapsis of the spiral:

ρmax ≡ ρ2 =κ2 − κ1 −

√κ2(κ2 − 2κ1)

2κ21

andρ ≤ ρmax. Equation (12.17) is then integrated fromρ0 to ρmaxto define the orientation of the apoapsis,

θm = θ0 −2nκ2[F(ϕ0, k) − K(k)]

κ21

√λ13λ24

The arguments of the elliptic integrals are

ϕ0 = arcsin√λ24λ03

λ23λ04, k =

√λ23λ14

λ13λ24

Recall that λi j = ρi − ρ j.Elliptic sinusoidal spirals of Type I are defined by

ρ(θ) =ρ4λ23 cd2(ν, k) − ρ3λ24

λ23 cd2(ν, k) − λ24(12.36)

170 12 Nonconservative extension of Keplerian integrals and new families of orbits

The spiral anomaly reads

ν(θ) =nκ2

1

2κ2(θ − θm)

√λ13λ24

The trajectory is symmetric with respect to θm, which corre-sponds to the line of apses.

Elliptic sinusoidal spirals of Type II

When κ2 < −2κ1 two roots are real and the other two are com-plex conjugate. In this case the real roots are ρ1,2 and the inequal-ity (12.33) reduces again to Eq. (12.35), meaning that ρ ≤ ρ2 ≡ρmax. However, the form of the solution is different from the tra-jectory described by Eq. (12.36), being

ρ(θ) =ρ1B − ρ2A − (ρ2A + ρ1B) cn(ν, k)

B − A − (A + B) cn(ν, k)(12.37)

The spiral anomaly is

ν(θ) =nκ2

1

κ2

√AB (θ − θm)

It is referred to the orientation of the apse line, θm. This variableis given by

θm = θ0 +nκ2

κ21

√AB

[2 K(k) − F(ϕ0, k)]

considering the arguments:

ϕ0 = arccos[

Aλ02 + Bλ10

Aλ02 − Bλ10

], k =

√(A + B)2 − λ2

12

4AB

The coefficients A and B are defined in terms of

a21 = −

κ2(κ2 + 2κ1)4κ4

1

and b1 = −κ1 + κ2

2κ21

namely

A =√

(ρ1 − b1)2 + a21, B =

√(ρ2 − b1)2 + a2

1

The fact that the Jacobi function cn(ν, k) is symmetric proves thatelliptic sinusoidal spirals of Type II are symmetric.

Transition between spirals of Types I and II

In this particular case of elliptic motion, −2κ1 = κ2, the roots ofpolynomial Pnat(ρ) are

ρ1 =3 + 2

√2

κ2, ρ2 ≡ ρmax =

3 − 2√

2κ2

, ρ3,4 = −1κ2

These results simplify the definition of the line of apses to

θm = θ0 +√

2 n ln √2(1 − ρ0κ2) +

√(3 − ρ0κ2)2 − 8

1 + ρ0κ2

Introducing the spiral anomaly ν(θ),

ν(θ) =

√2

2(θ − θm)

the equation of the trajectory takes the form

ρ(θ)ρmax

=5 − 4

√2 cosh ν + cosh(2ν)

(3 − 2√

2)[3 − cosh(2ν)](12.38)

The trajectory is symmetric with respect to the line of apses θm

thanks to the symmetry of the hyperbolic cosine.Figure 12.2(a) shows the three types of elliptic spirals. It is im-

portant to note that in all three cases the condition in Eq. (12.33)transforms into Eq. (12.35), equivalent to ρ < ρmax. As a result,there are no differences in their nature although the equations forthe trajectory are different.

12.6.2 Parabolic motion: sinusoidal spiral (off-center circle)

Making κ1 = 0 the condition in Eq. (12.33) simplifies to

ρ ≤ ρmax =1

4κ2

meaning that the spiral is bounded by amaximum radius ρmax. Itis equivalent to the apoapsis of the spiral. Its orientation is givenby

θm = θ0 + n[π

2− arcsin

(ρ0

ρmax

)]

The trajectory reduces to a sinusoidal spiral,* and its defini-tion can be directly related to θm:

ρ(θ)ρmax

= cos(θ − θm) (12.39)

The spiral defined in Eq. (12.39) is symmetric with respect tothe line of apses. The resulting orbit is a circle centered at(ρmax/2, θm) (Fig. 12.2(b)). Circles are indeed a special case of si-nusoidal spirals.

12.6.3 Hyperbolic motion

Given the discriminant in Eq. (12.34), for the case κ1 > 0 thevalues of the constant κ2 define two different types of hyperbolicsinusoidal spirals: spirals of Type I (κ2 < 2κ1), and spirals of TypeII (κ2 > 2κ1).

Hyperbolic sinusoidal spirals of Type I

If κ2 < 2κ1 then ρ1,2 are complex conjugates and ρ3,4 are bothreal but negative. Therefore, the condition in Eq. (12.33) holdsnaturally for any radius and there are no limitations to the values

*It was the Scottish mathematician Colin Maclaurin the first to study sinu-soidal spirals. In his “Tractatus de Curvarum Constructione & Mensura”, pub-lished in Philosophical Transactions in 1717, he constructed this family of curvesrelying on the epicycloid. Their general form is rn = cos(nθ) and different valuesof n render different types of curves; n = −2 correspond to hyperbolas, n = −1to straight lines, n = −1/2 to parabolas, n = −1/3 to Tschirnhausen cubics,n = 1/3 to Cayley’s sextics, n = 1/2 to cardioids, n = 1 to circles, and n = 2 tolemniscates.

§12.6 Case γ = 4: generalized sinusoidal spirals 171

(a) Elliptic (b) Parabolic (c) Hyperbolic Type I (d) Hyperbolic Type II (e) Hyperbolic transition

Figure 12.2: Examples of generalized sinusoidal spirals (γ = 4).

of ρ. The particle can either fall to the origin or escape to infin-ity along an asymptotic branch. The equation of the trajectory isgiven by:

ρ(θ) =ρ3B − ρ4A + (ρ3B + ρ4A) cn(ν, k)

B − A + (A + B) cn(ν, k)(12.40)

where the spiral anomaly canbe referreddirectly to the initial con-ditions:

ν(θ) =nκ2

1

κ2

√AB (θ − θ0) + F(ϕ0, k)

This definition involves an elliptic integral of the first kind withargument and parameter:

ϕ0 = arccos[

Aλ04 + Bλ30

Aλ04 − Bλ30

], k =

√(A + B)2 − λ2

34

4AB

The coefficients A and B require the terms:

b1 = −κ1 + κ2

2κ21

, a21 = −

κ2(κ2 + 2κ1)4κ4

1

being

A =√

(ρ3 − b1)2 + a21, B =

√(ρ4 − b1)2 + a2

1

The direction of the asymptote is defined by

θas = θ0 +nκ2

κ21

√AB

[F(ϕ∞, k) − F(ϕ0, k)

]This definition involves the argument

ϕ∞ = arccos(A − B

A + B

)The velocity of the particle when reaching infinity is v∞ =

√κ1.

Hyperbolic sinusoidal spirals of Type I are similar to the hyper-bolic solutions of Type I with γ = 2 and γ = 3. Figure 12.2(d)depicts and example trajectory and the asymptote defined by θas.

Hyperbolic sinusoidal spirals of Type II

In this case the four roots are real and distinct, with ρ3,4 < 0. Thetwo positive roots ρ1,2 are physically valid, i.e. v2(ρ1) > 0 andv2(ρ2) > 0. This yields two situations in which the condition

(ρ − ρ1)(ρ − ρ2) ≥ 0 is satisfied: ρ > ρ1 (exterior spirals) andρ < ρ2 (interior spirals).

Interior spirals take the form

ρ(θ) =ρ2λ13 − ρ1λ23 sn2(ν, k)λ13 − λ23 sn2(ν, k)

(12.41)

The spiral anomaly is

ν(θ) =κ2

1

2κ2

√λ13λ24 (θ − θm)

The orientation of the line of apses is solved from

θm = θ0 +2nκ2 F(ϕ0, k)κ2

1

√λ13λ24

(12.42)

with

ϕ0 = arcsin√λ13λ20

λ23λ10, k =

√λ23λ14

λ13λ24

Interior hyperbolic spirals are bounded and their shape is similarto that of a limaçon.

The line of apses of an exterior spiral is defined by

θm = θ0 −2nκ2 F(ϕ0, k)κ2

1

√λ13λ24

The modulus and the argument of the elliptic integral are

ϕ0 = arcsin√λ24λ01

λ14λ02, k =

√λ23λ14

λ13λ24

The trajectory becomes

ρ(θ) =ρ1λ24 + ρ2λ41 sn2(ν, k)λ24 + λ41 sn2(ν, k)

(12.43)

and it is symmetric with respect to θm. The geometry of thesolution is similar to that of hyperbolic cardioids of Type II,mainly because of the existence of the forbidden region plottedin Fig. 12.2(d). The asymptotes follow the direction of

θas = θm ±2κ2 F(ϕ∞, k)κ2

1

√λ13λ24

, with ϕ∞ = arcsin√λ24

λ14

172 12 Nonconservative extension of Keplerian integrals and new families of orbits

Transition between Type I and Type II spirals

When κ2 = 2κ1 the radii ρ1 and ρ2 coincide, ρ1 = ρ2 ≡ ρlim, andbecome equal to the limit radius

ρlim =1

2κ1

The equation of the trajectory is a particular case of Eq. (12.43),obtained with κ2 → 2κ1:

ρ(θ)ρlim=

[1 − 8

4 + m(sinh ν − 3 cosh ν)

]−1

(12.44)

The spiral anomaly can be referred to the initial conditions,

ν(θ) =nm√

2(θ−θ0)+ln

2(1 + 2κ1ρ0) +√

2 + 8κ1ρ0(3 + κ1ρ0)m(1 − 2κ1ρ0)

avoiding additional parameters. Here m = sign(1 − ρ0/ρlim) de-termines whether the spiral is below or over the limit radius ρlim.The asymptote follows from

θas = θ0 − n√

2 log(1 −√

2/2)

Like in the case of hyperbolic cardioids, for m = n = −1and m = n = +1 spirals of this type approach the circular or-bit of radius ρlim asymptotically, i.e. limθ→∞ ρ = ρlim. When

approaching a circular orbit the perturbing acceleration vanishesand the spiral converges to a Keplerian orbit. See Fig. 12.2(e) forexamples of hyperbolic sinusoidal spirals with m = +1, n = +1(dashed) and m = −1, n = +1 (solid).

12.7 Summary

The solutions presented in the previous sections are summarizedin Table 12.1, organized in terms of the values of γ. Each familyis then divided in elliptic, parabolic, and hyperbolic orbits. Thetable includes references to the corresponding equations of thetrajectories for convenience. The orbits are said to be bounded ifthe particle can never reach infinity, because r < rlim.

12.8 Unified solution in Weierstrassianformalism

The orbits can be unified introducing the Weierstrass ellipticfunctions. Indeed, Eq. (12.17) furnishes the integral expression

θ(r) − θ0 =

∫ r

r0

k ds√f (s)

(12.45)

Table 12.1: Summary of the families of solutions.

Family Type γ κ1 κ2 Bounded Trajectory

Elliptic 1 < 0 >√−1/κ1 Y Eq. (12.19)

Conic sections Parabolic 1 = 0 − N Eq. (12.19)Hyperbolic 1 > 0 − N Eq. (12.19)

Elliptic 2 < 0 < 1 Y Eq. (9.30)Generalized Parabolic 2 = 0 ≤ 1 N Eq. (9.35)∗logarithmic Hyperbolic T-I 2 > 0 < 1 N Eq. (9.43)spirals Hyperbolic T-II 2 > 0 > 1 N Eq. (9.50)

Hyperbolic trans. 2 > 0 = 1 N Eq. (9.57)

Elliptic 3 < 0 < 1 Y Eq. (12.24)Parabolic 3 = 0 ≤ 1 Y Eq. (12.25)†

Generalized Hyperbolic T-I 3 > 0 <√

3κ1 N Eq. (12.27)cardioids Hyperbolic T-II (int) 3 > 0 >

√3κ1 Y Eq. (12.28)

Hyperbolic T-II (ext) 3 > 0 >√

3κ1 N Eq. (12.30)Hyperbolic trans. 3 > 0 =

√3κ1 Y/N Eq. (12.32)

Elliptic T-I 4 < 0 > −2κ1 Y Eq. (12.36)Elliptic T-II 4 < 0 < −2κ1 Y Eq. (12.37)

Generalized Elliptic trans. 4 < 0 = −2κ1 Y Eq. (12.38)sinusoidal Parabolic 4 = 0 − Y Eq. (12.39)‡spirals Hyperbolic T-I 4 > 0 < 2κ1 N Eq. (12.40)

Hyperbolic T-II (int) 4 > 0 > 2κ1 Y Eq. (12.41)Hyperbolic T-II (ext) 4 > 0 > 2κ1 N Eq. (12.43)Hyperbolic trans. 4 > 0 = 2κ1 Y/N Eq. (12.44)

∗ Logarithmic spiral.† Cardioid.‡ Sinusoidal spiral (off-center circle).

§12.9 Physical discussion of the solutions 173

with

f (s) ≡ Psol(s) = a0s4 + 4a1s3 + 6a2s2 + 4a3s + a4 (12.46)

and a0,1 , 0. Introducing the auxiliary parameters

ϑ = f ′(r0)/4 and φ = f ′′(r0)/24

Eq. (12.45) can be inverted to provide the equation of the trajec-tory (see Appendix E and Whittaker and Watson, 1927, p. 454,for details),

r(θ) − r0 =1

2[℘(z) − φ]2 − f (r0) f (iv)(r0)/48

×[℘(z) − φ]2ϑ − ℘′(z)

√f (r0) + f (r0) f ′′′(r0)/24

(12.47)

The solution is written in terms of the Weierstrass elliptic func-tion

℘(z) ≡ ℘(z; g2, g3)

and its derivative℘′(z), where z = (θ−θ0)/k is the argument andthe invariant lattices g2 and g3 read

g2 = a0a4 − 4a1a3 + 3a22

g3 = a0a2a4 + 2a1a2a3 − a32 − a0a2

3 − a21a4

The coefficients ai and k are obtained by identifying Eq. (12.45)with Eq. (12.17) for different values of γ. They can be found inTable 12.2.

Table 12.2: Coefficients ai of the polynomial f (s) and factor k.

γ a0 a1 a2 a3 a4 k

1 κ1 1/2 −κ22/6 0 0 nκ2

2 κ21 K1/2 (1 − κ2

2)/6 0 0 nκ2

3 27κ31 27κ2

1/2 6κ1 − 9κ22/2 2 0

√27 nκ2

4 16κ41 8κ3

1 4κ21 − 8κ2

2/3 2κ1 1 4nκ2

Symmetric spirals reach a minimum or maximum radius rm,which is a root of f (r). Thus, Eq. (12.47) can be simplified if re-ferred to rm instead of r0:

r(θ) − rm =f ′(rm)/4

℘(zm) − f ′′(zm)/24(12.48)

This is the unified solution for all symmetric solutions, with zm =

(θ − θm)/k. Practical comments on the implementation of theWeierstrass elliptic functions can be found in Biscani and Izzo(2014). Although ℘(z) = ℘(−z) the derivative ℘′(z) is an oddfunction in z, ℘′(−z) = −℘′(z). Therefore, the integer n needsto be adjusted according to the regime of the spiral when solvingEq. (12.47): n = 1 for raising regime, and n = −1 for loweringregime.

12.9 Physical discussion of the solutions

Each family of solutions involves a fundamental curve: the caseγ = 1 relates to conic sections, γ = 2 to logarithmic spirals, γ =3 to cardioids, and γ = 4 to sinusoidal spirals. This section isdevoted to analyzing the geometrical and dynamical connectionsbetween the solutions and other integrable systems.

12.9.1 Connection with Schwarzschild geodesics

The Schwarzschild metric is a solution to the Einstein field equa-tions of the form

(ds)2 =

(1 − 2M

r

)(dt)2− (dr)2

1 − 2M/r−r2(dϕ)2−r2 sin2 ϕ (dθ)2

written in natural units so that the speed of light and the grav-itational constant equal unity, and the Schwarzschild radius re-duces to 2M. In this equation M is the mass of the central body,ϕ = π/2 is the colatitude, and θ is the longitude.

The time-like geodesics are governed by the differential equa-tion(drdθ

)2

=1L2 (E2 − 1) r4 +

2ML2 r3 − r2 + 2Mr

where L is the angular momentum and E is a constant of motionrelated to the energy and defined by

E =(1 − 2M

r

)dts

dtp

in terms of the proper time of the particle tp and theSchwarzschild time ts. Its solution is given by

θ(r) − θ0 =

∫ r

r0

nL ds√f (s)

and involves the integer n = ±1 that gives the raising/loweringregime of the solution. Here f (s) is a quartic function definedlike in Eq. (12.46). The previous equation is formally equivalentto Eq. (12.45). As a result, the Schwarzschild geodesics abide byEqs. (12.47) or (12.48) depending on the selection of the roots,and are written in terms of the Weierstrass elliptic functions andidentifying the coefficients

a0 = E2 − 1, a1 = M/2, a2 = −L2/6,

a3 = ML2/2, a4 = 0, k = nL

Compact forms of the Schwarzschild geodesics using the Weier-strass elliptic functions can be found in the literature (see for ex-ampleHagihara, 1930, §4). We refer to classical books like the oneby Chandrasekhar (1983, Chap. 3) for an analysis of the structureof the solutions in Schwarzschild metric.

The analytic solution to Schwarzschild geodesics involves el-liptic functions, like generalized cardioids (γ = 3) and generalizedsinusoidal spirals (γ = 4). If the values of κ1 and κ2 are adjustedin order the roots of Psol to coincide with the roots of f (r) in

174 12 Nonconservative extension of Keplerian integrals and new families of orbits

(a) γ = 3 (b) γ = 4

Figure 12.3: Schwarzschild geodesic (solid line) compared to general-

ized cardioids and generalized sinusoidal spirals (dashed line).

Schwarzschildmetric the solutionswill be comparable. For exam-ple, when the polynomial f (r) has three real roots and a repeatedone the geodesics spiral toward or away from the limit radius

rlim =1

2M− L2 + 4M2

2ML(2L +√

L2 − 12M2)

If we make this radius coincide with the limit radius of a hyper-bolic cardioid with κ2 =

√3κ1 [Eq. (12.31)] it follows

κ1 =1

3rlimand κ2 =

1√

rlim

Figure 12.3(a) compares hyperbolic generalized cardioids andSchwarzschild geodesics with the same limit radius. The inte-rior solutions coincide almost exactly, whereas exterior solutionsseparate slightly. Similarly, when f (r) admits three real and dis-tinct roots the geodesics are comparable to interior and exteriorhyperbolic solutions of Type II. For example, Fig. 12.3(b) showsinterior and exterior hyperbolic sinusoidal spirals of Type II withthe inner and outer radii equal to the limit radii of Schwarzschildgeodesics. Like in the previous example the interior solutionscoincide, while the exterior solutions separate in time. In or-der for two completely different forces to yield a similar trajec-tory it suffices that the differential equation dr/dθ takes the sameform. Even if the trajectory is similar the integrals of motionmight not be comparable. As a result, the velocity along the or-bit and the time of flight between two points will be different.Equation (12.16) shows that the radial motion is governed by theevolution of the flight-direction angle. For γ = 3 and γ = 4the acceleration in Eq. (12.6) makes the radius to evolve with thepolar angle just like in the Schwarzschild metric. Consequently,the orbits may be comparable. However, since the integrals ofmotion (12.13) and (12.14) do not hold along the geodesics, thevelocities do not necessarily match.

12.9.2 Newton’s theorem of revolving orbits

Newton found that if the angular velocity of a particle followinga Keplerian orbit is multiplied by a constant factor k, it is thenpossible to describe the dynamics by superposing a central forcedepending on an inverse cubic power of the radius. The addi-tional perturbing terms depend only on the angular momentumof the original orbit and the value of k.

Consider a Keplerian orbit defined as

ρ(θ) =h2

1 + e cosϑ(12.49)

Here ϑ = θ − θm denotes the true anomaly. The radial motionof hyperbolic spirals of Type II—Eq. (9.50)— resembles a Keple-rian hyperbola. The difference between both orbits comes fromthe angular motion, because they revolve with different angularvelocities. Indeed, recovering Eq. (9.50), identifying κ2 = e andρmin(1 + κ2) = h2, and calling the spiral anomaly β = k(θ − θm),it is

ρ(θ) =h2

1 + e cos β

When equated to Eq. (12.49), it follows a relation between thespiral (β) and the true (ϑ) anomalies:

β = kϑ

The factor k reads

k =κ2

ℓ=

κ2√κ2

2 − 1

Replacing ϑ by β/k in Eq. (12.49) and introducing the result inthe equations of motion in polar coordinates gives rise to the ra-dial acceleration that renders a hyperbolic generalized logarithmicspiral of Type II, namely

ar,2 = −1ρ2 +

h2

ρ3 (1 − k2) = ar,1 +h2

ρ3 (1 − k2)

This is in fact the same result predicted by Newton’s theorem ofrevolving orbits.

The radial acceleration ar,2 yields a hyperbolic generalizedlogarithmic spiral of Type II. This is a central force which pre-serves the angular momentum, but not the integral of motionin Eq. (12.14). Thus, a particle accelerated by ar,2 describes thesame trajectory as a particle accelerated by the perturbation inEq. (12.6), but with different velocities. As a consequence, thetimes between two given points are also different. The accelera-tion derives from the specific potential

V (ρ) = Vk(ρ) + ∆V (ρ)

where Vk(ρ) denotes the Keplerian potential, and ∆V (ρ) is theperturbing potential:

∆V (ρ) =h2

2ρ2 (1 − k2)

12.9.3 Geometrical and physical relations

The inverse of a generic conic section r(θ) = a+b cos θ using oneof its foci as the center of inversion defines a limaçon. In particu-lar, the inverse of a parabola results in a cardioid. Let us recoverthe equation of the trajectory of a generalized parabolic cardioid(a true cardioid) from Eq. (12.25),

ρ(θ) =ρmax

2[1 + cos(θ − θm)]

§12.10 Conclusions 175

Taking the cusp as the inversion center defines the inverse curve:

1ρ=

2ρmax

[1 + cos(θ − θm)]−1

Identifying the terms in this equation with the elements of aparabola it follows that the inverse of a generalized parabolic car-dioid with apoapsis ρmax is a Keplerian parabola with periapsis1/ρmax. The axis of symmetry remains invariant under inversion;the lines of apses coincide.

The subfamily of elliptic generalized logarithmic spirals is ageneralized form of Cotes’s spirals, more specifically of Poinsot’sspirals:

1ρ= a + b cosh ν

Cotes’s spirals are known to be the solution to the motion im-merse in a potential V (r) = −µ/(2r2) (Danby, 1992, p. 69).

The radial motion of interior Type II hyperbolic and TypeI elliptic sinusoidal spirals has the same form, and is also equalto that of Type II hyperbolic generalized cardioids, except for thesorting of the terms.

It is worth noticing that the dynamics along hyperbolic sinu-soidal spirals with κ2 = 2κ1 are qualitatively similar to the mo-tion under a central force decreasing with r−5. Indeed, the orbitsshown in Fig. 12.2(e) behave as the limit case γ = 1 discussedby MacMillan (1908, Fig. 4). On the other hand, parabolic sinu-soidal spirals (off-center circles) are also the solution to the mo-tion under a central force proportional to r−5.

12.10 Conclusions

The dynamical symmetries in Kepler’s problem hold under a spe-cial nonconservative perturbation, which is a generalization ofthe thrust that rendered generalized logarithmic spirals. Thereare two integrals of motion that are extended forms of the equa-tion of the energy and angularmomentum. A similarity transfor-mation reduces the original problem to a system perturbed by atangential uniparametric forcing. It simplifies the dynamics sig-nificantly, for the integrability of the system is evaluated in termsof one unique parameter. The algebraic properties of the equa-tions of motion dictate what values of the free parameter makethe problem integrable in closed form.

The extended integrals ofmotion include the Keplerian onesas particular cases. The new conservation laws can be seen as gen-eralizations of the original integrals. The new families of solu-tions are defined by fundamental curves in the zero-energy case,and there are geometric transformations that relate different or-bits. The orbits can be unified by introducing the Weierstrass el-liptic functions. This approach simplifies themodeling of the sys-tem.

The solutions derived in this chapter are closely related to dif-ferent physical problems. The fact that the magnitude of the ac-celeration decreases with 1/r2 makes it comparable with the per-turbation due to the solar radiation pressure. Moreover, the in-verse similarity transformation converts Keplerian orbits into theconic sections obtained when the solar radiation pressure is di-rected along the radial direction. The structure of the solutions,

governed by the roots of a polynomial, is similar in nature to theSchwarzschild geodesics. This is because under the consideredperturbation and in Schwarzschildmetric the evolution of the ra-dial distance takes the same form. Some of the solutions are com-parable to the orbits deriving from potentials depending on dif-ferent powers of the radial distance. Although the trajectory maytake the same form, the velocity will be different, in general.

“The more I learn, the more I realize how much I don’t know.”

—Albert Einstein

13Conclusions to the thesis

M ostproblems in astrodynamics and celestialmechan-ics are based on the same physical principles. Every ef-fort toward advancing our understanding of the dy-

namical laws that govern the motion in space will help findingsolutions to particular problems. In this sense, the present thesishas shown how the improved description of the dynamics pro-vided by the theory of regularization yields new results in differ-ent areas. These innovative techniques have been applied system-atically to three challenging problems in modern space engineer-ing: spacecraft relative motion, low-thrust mission design, andhigh-performance numerical propagation of orbits. The meth-ods and techniques presented here might be extended to otherareas of mission design or astronomy.

Special attention has been paid to the historical backgroundof regularization. First of all, in order to pay tribute to the greatminds who set the foundations of astrodynamics and, conse-quently, of spaceflightmechanics. But, hopefully, this reviewwillalso convince the reader of the power of regularization, by show-ing how different were the motivations that made astronomersand engineers derive new formulations, and how the same col-lection of techniques were useful in many scenarios. Computershave revolutionized science and engineering, enabling extremelycomplex numerical experiments that simulate physical processeswith unprecedented accuracy. As a result, many disciplines havebecome eminently numerical. And regularization in celestial me-chanics is one of these disciplines. Technical reports that Bond,Gottlieb, Stiefel, Velez, Janin, Baumgarte, etc. completed forboth NASA and ESA (ESRO at the time of writing of many ofthem) in the sixties, seventies, and eighties, were focused on thevirtues of regularization for numerical integration. Regulariza-tion is undoubtedly well suited for numerical propagation of or-

bits, but there is also much more.

13.1 Outlook and advances

Regularization techniques can be applied to virtually any orbitalproblem. Conceptually, the main change is the use of differenttime variables. This is not only a matter of representation: thephysical meaning of the variational equations, partial derivatives,relative states, etc. change because of using an alternative param-eterization. This new idea led to the development of the theoryof asynchronous relative motion, which provides a different ap-proach to the concept of variational formulation. The theory hasbeen presented in this thesis for the first time. Its application tospacecraft relative motion constitutes a more efficient and accu-rate method for propagating the dynamics. Because it is a genericconcept, the formulation admits any source of external perturba-tion. Other properties of regularized formulations can be lever-aged too. For example, the universal solution toKepler’s problemcanbe extrapolated toderive theuniversal solution to relativemo-tion.

The theory of asynchronous relative motion can be appliedto virtually any problem that can be solved using variationalmethods. This extends the reach of the theory to other problemssuch as orbit determination, trajectory optimization, periodic or-bit search, etc. The present dissertation has derived the first sys-tematic formulation of the variational equations of orbital mo-tion using regularization. Given the virtues of regularized tech-niques for numerical propagation and the interest of the varia-tional equations in many problems, allowing the user to explorethe vicinity of the nominal trajectory with efficient methods isadvantageous. The software tool PERFORM has specifically been

177

178 13 Conclusions to the thesis

adapted to thepropagationof the partial derivatives of a given tra-jectory, in order to study its sensitivity to the initial conditions.

Orbital elements prove useful when integrating weakly per-turbed problems. And low-thrust trajectories are good candi-dates, as even the name suggests. While seeking special sets of ele-ments that would improve the numerical integration of this kindof orbits, an entirely new family of analytic solutions with con-tinuous thrust was found during the present doctoral research,called generalized logarithmic spirals. Looking for elements isclosely related to finding conservation laws, and this was the keyto deriving the new solution. The properties of generalized log-arithmic spirals are surprisingly similar to Keplerian orbits- Eventhe spiral Lambert problemwith these neworbits is almost equiv-alent to the classical ballistic problem. Further studies have re-vealed that this new family of orbits is, in fact, a subclass of amoregeneral type of integrable system.

The new analytic solution obtained with regularized meth-ods is not simply a mathematical curiosity. Based on it, a newstrategy for low-thrust preliminary mission design has been de-rived. The advantages of the new methodology come from theunique dynamical properties of the solution. This is yet anotherproof of the importance of exploringnewmodels, properties, andsolutions when approaching a certain problem.

New formulations can be derived for specific applications. Itis clear that today’s challenges differ from those faced in the 1960s.But the principles are the same, and the basic concepts of regular-ization can still be applied to satisfy the present and future needs.Asmissions incorporatemore flybys, maintaining the accuracy ofthe propagations becomes an issue. In this context, a geomet-rical analysis of the problem showed how the propagation canbe improved by relying on unconventional constructions. Thisdissertation presents a new propagator specifically conceived forintegrating hyperbolic orbits. It recovers the geometrical foun-dations ofMinkowski space-time, and proves that hyperbolic ge-ometry is more convenient than Euclidean for these kind of tra-jectories. The main advantage of the method is that the accuracyof the propagation is not affected by periapsis passage, which is acritical point during the integration phase.

Regularization might also lead to more profound theoreti-cal conclusions. The analysis of the topology underlying a specialperturbationmethod yielded a new theory of stability. Althoughthe theory is entirely constructed in an alternative phase space,not intuitive, it simplifies the characterization of complex systemsthat evolve into chaotic regimes. An indirect outcome is the dis-covery of a new Lyapunov-like indicator, which can be used tomonitor the dynamical regimes of the orbits.

The orbit propagation tool PERFORM has been developed inan attempt to standardize the numerical experiments for evalu-ating the performance of orbital formulations. The software in-cludes accurate force models that allow the user to integrate real-life problems and determinewhich combination of numerical in-tegrator and dynamical formulation is best suited for the prob-lem of interest. With the aid of sequential performance diagrams(SPD) the selection of the best combinations is simplified. Inpractice, investing some time in finding an efficient propagationstrategy may save hours of computation when long simulationcampaigns need to be conducted.

For centuries, osculation has been a dogma in celestial me-chanics. The equations of evolution of the elements, obtainedvia the variation of parameters procedure, have traditionally beenderived under the assumption that the orbit defined by the or-bital elements has to be the osculating orbit. Recent discoveriesby Efroimsky (2002) showed that this constraint can be relaxedin practice, opening a whole new world of possibilities. This the-sis has extended this revolutionary concept to some of the mostcommon regularized schemes, and has explained how this con-cept can be extended to any other formulation relying on timetransformations.

13.2 Future work

With fast and accurate models at hand, the problems of develop-ing control strategies, optimizing the mission architecture, andoperating spacecraft formations will be simplified. In the future,it will be worth exploring the combination of regularized formu-lations and Taylor differential algebra (Berz, 1988; Di Lizia et al.,2008) for solvingproblems relyingon the variational equations ofmotion. This is a method that propagates high-order expansionsof the integral flow, and serves as an automatic derivation algo-rithm. Originally developed in the realm of particle physics, itsrecent extension to orbital mechanics showed promising results.The combination of the two may lead to highly efficient propa-gators.

The results in relative motion have great potential for thedevelopment of dynamical models for spacecraft swarms. Thisconcept is receiving more attention every year, and poses variouschallenges. First, the propagation of the relative orbits might beproblematic when tens of thousands of spacecraft are to be con-sidered. When perturbations are present, numerical simulationsare required, and the method needs to be as efficient as possible.In this sense, the combination of regularization with the theoryof asynchronous relative motion presents the perfect frameworkfor simulating large swarms.

The novel strategy for propagating the variational equationscan be applied to many problems beyond formation flying. Typ-ical tools for finding periodic orbits use differential correctionalgorithms. The monodromy matrix is propagated numericallyat each iteration, and the solver uses the information about theneighborhood to converge to the periodic orbit. With a methodthat ismore accurate the convergence ratewill improve, and fastersimulations will reduce significantly the number of function callsand, consequently, the runtime. Improvements in convergencewill possibly lead to the discovery of new families of orbits in sen-sitive domains.

The current version of the algorithm for low-thrust prelimi-narymission design utilizes a rather rigid design strategy, inwhichthe thrust-coast-thrust is imposed. The flexibility of the designstrategywill be improved if this sequence could be defined at con-venience. In this way, more transfer options will be available. Asthe flexibility increases the number of degrees of freedom growstoo. Adjusting all the parameters becomes a problem, and themethod should be combined with a global optimizer. Some ex-periments have been conducted using genetic optimizers. Alter-native strategies should be developed.

§13.2 Future work 179

The idea behind PERFORM is to have a unified way to testformulations under the same conditions. Thus, the software ismeant to be distributed and made accessible to other interestedresearchers. Adding new formulations and integrators should bean easy task, in order to motivate new users to implement theirownmethods. This will not only enrich the software itself, but itwill establish some standards for this kind of experiments.

Different formulations have been extended in order to intro-duce the gauge freedom in the elements. This technique increasesthe flexibility of the formulation, and allows the user to choosethe definition of elements that best fits certain needs. The use ofnonosculating elements is a promising technique that may haveapplications in mission design and orbit propagation, and theyare worth exploring in the future.

Appendices

181

AHypercomplex numbers

L et an denote an n-dimensional hypercomplex number,an ∈ Hn. The field of hypercomplex numbers is denotedHn. The hypercomplex number an can written

an = a0u0 + a1u1 + a2u2 + . . . + an−1un−1

where the coefficients ai ∈ R are the components of an, andui (with i = 0, . . . , n − 1) are the generalized imaginary units,u0 ≡ 1 ∈ R and ui < R for i , 0. By analogy with vector algebrathey are also referred to as versors, defining the n-dimensional ba-sis 1, u1, . . . , un−1 (Boccaletti et al., 2008, p. 5). The main rulesinHn are:

Equality: an = bn ⇐⇒ ai = bi, ∀ i

Addition: cn = an + bn ⇐⇒ ci = ai + bi, ∀ i

Null element w.r.t. sum: an + zn = an ⇐⇒ zi = 0, ∀ i

Distributive w.r.t. sum: (an + bn) + cn = an + (bn + cn)

Multiplication of two hypercomplex numbers an =∑

a ju j

and bn =∑

bkuk is defined in terms of the product (u j uk),

an bn =

(∑j

a ju j

)(∑k

bkuk

)=

∑j

∑k

a jbk(u j uk)

with j, k integers spanning from 0 to n − 1. Provided that Hn

is closed under multiplication (Kantor et al., 1989, chap. 5), theproduct (u j uk) is of the form

u j uk = ψ0j,k + ψ

1j,ku1 + ψ

2j,ku2 + . . . + ψ

n−1j,k un−1 (A.1)

withψℓj,k ∈ R, ∀ j, k, ℓ. That is, (u j uk) ∈ Hn. The values for the

coefficientsψℓj,k, given in the form ofmultiplication tables, definethe nature of hypercomplex numbers.

Theorem 3: Multiplication is commutative in Hn if and only ifψℓj,k = ψ

ℓk, j for all possible values of j, k, ℓ.

Proof: Consider two hypercomplex numbers an =∑

a ju j andbn =

∑bkuk. The product (an bn) is commutative if (u j uk) is

commutative, provided that a jbk = bka j. Multiplication yields

u j uk =∑ℓ

ψℓj,kuℓ, uk u j =∑ℓ

ψℓk, juℓ

When equating term by term: u j uk = uk u j ⇐⇒ ψℓj,k = ψℓk, j.

Theorem 4: Multiplication is associative in Hn,(ui u j) uk = ui(u j uk), if and only if ∑ℓ(ψℓi, jψ

mℓ,k − ψℓj,kψm

i,ℓ) = 0for all possible values of the coefficients i, j, k,m.

Proof: Consider the products

(ui u j) uk =∑ℓ

ψℓi, juℓ uk =∑

m

∑ℓ

ψℓi, jψmℓ,kum

ui(u j uk) =∑ℓ

ψℓj,kui uℓ =∑

m

∑ℓ

ψℓj,kψmi,ℓum

Both expressions are equated term by term to provide∑ℓ ψ

ℓi, jψ

mℓ,k =

∑ℓ ψ

ℓj,kψ

mi,ℓ for all values of the coefficients

i, j, k,m.

Multiplication inHn is equipped with additional properties:

183

184 A Hypercomplex numbers

• Let r ∈ R and an ∈ Hn. The product isran = r

∑aiui =

∑raiui = anr.

• Let p, q ∈ R and an, bn ∈ Hn. Then, it is(pq)(anbn) = (pan)(qbn).

• cn(an + bn) = cn an + cn bn and (an + bn)cn = an cn + bn cn.

Definition 1 A number pn ∈ Hn is said to be a divisor of zero ofsn ∈ Hn if the equation sn pn = 0 is satisfied for pn, sn , 0 .

The equation sn pn = 0 is expanded to provide,

sn pn =∑

i

∑j

si p juiu j =∑

i

∑j

∑k

si p jψki, juk = 0

Each component of sn pn in the basis 1, u1, . . . , un−1must can-cel to comply with this result. This leads to a homogeneous sys-tem of n linear equations:∑

i

∑j

si p jψki, j =

∑i

∑j

siψki, j p j = 0 , with k = 0, . . . , n−1

There exists an equivalent expression in matrix form,

Ap = 0 , with Ai j =

n−1∑ℓ=0

sℓψi−1ℓ, j−1 and p = [p0, . . . , pn−1]⊤

(A.2)

which provides the components of pn. Such components are in-cluded in the kernel of A. Note that if matrix A is singular, i.e.|A| = 0, there are infinitely many nontrivial solutions to the sys-tem. In that case pn is a divisor of zero of sn. The hypercomplexnumbers pn, sn then verify

pn =0

sn, 0

The condition for pn being a divisor of zero deals both with itscomponents and with the coefficients ψℓj,k from the multiplica-tion table.

A.1 Complex and hyperbolic numbers

Two-dimensional hypercomplex numbers of the form h = x +y u1 are particularly interesting. These numbers correspond tothe components (x, y) ∈ R × R in the basis 1, u1. From nowon the generalized imaginary unit u1 is simply written u. Multi-plication reduces to

u j uk = ψ0j,k + uψ1

j,k

These coefficients define the nature of the resulting hypercom-plex numbers. In particular, the table

ψ000 = 1, ψ1

00 = 0, ψ010 = −0, ψ1

10 = 1

ψ001 = 0, ψ1

01 = 1, ψ011 = −1, ψ1

11 = 0(A.3)

defines the field of complex numbers C,

C = x + iy | (x, y) ∈ R × R, i2 = −1

where i ≡ u is referred to as the imaginary unit. Since ψℓj,k =ψℓk, j, Thm. 3 shows that multiplication is commutative in C. Lets2 ∈ C be s2 = x + iy. Matrix A from Eq. (A.2) is such that||A|| = x2 + y2, that is nonzero for s2 , 0. The kernel of Areduces to the null vector, and there are no divisors of zero inC.

Now consider that ψ011 = −1 is replaced by ψ0

11 = 1 inEq. (A.3); this defines the field of hyperbolic numbersD,

D = x + jy | (x, y) ∈ R × R, j2 = +1

where j ≡ u is the hyperbolic imaginary unit, and j =√+1 <

R. Like in C, multiplication inD is commutative. Letw2 be thehyperbolic number w2 = x + jy. The determinant ||A|| is ||A|| =x2−y2, that cancels for y = ±x. Thus, hyperbolic numbers admitdivisors of zero.

A.1.1 The modulus

Given a complex number z ∈ C, there is a conjugate element inthe field of complex numbers, z =† x − iy, with z ∈† C. Themodulus of z is obtained from the definition,

|z|2C = z z =† (x + iy)(x − iy) = x2 − i2y2 = x2 + y2

The locus |z|C = const. results in a circumference on the (x, y)plane. This circumference is denoted by cr, with radius r = |z|C.

In a similar fashion, given a hyperbolic number w ∈ D, itsconjugate is a hyperbolic numberw† = x − jy. The definition ofmodulus inD yields

|w|2D = ww† = (x + jy)(x − jy) = x2 − j2y2 = x2 − y2

Provided this result, x2−y2 = const. defines the locus of invariantmodulus on the (x, y) plane. Geometrically, this invariant sub-space is a rectangular (or equilateral) hyperbola written ha, wherea is the semimajor axis. The right branch is ha

+, and ha− denotes

the left branch. This hyperbola should not be confused with theosculating orbit.

A.1.2 The geometry of two-dimensional hypercomplexnumbers

Two-dimensional hypercomplex numbers have been defined interms of a particular basis 1, u. Given a pair of components(x, y), the two-dimensional hypercomplex number is written h =x + u y. This interpretation is closely related to the definition ofvectors: vector v = xi + yj is the compact expression of the pair(x, y) in the basis i, j.

The (x, y) plane in C, with x = R(z) and y = I(z), is theGauss-Argand plane. The norm of vector v = xi + yj in two-dimensional Euclidean geometryE2 is equivalent to themodulusof z = x + iy,

||v||2E2 = x2 + y2 ≡ |z|2C

The Gauss-Argand plane is conceived like the Cartesian plane,where Euclidean geometry applies. This is one of themajor prop-erties of complex numbers.

§A.2 Quaternions 185

Just like complex numbers comply with Euclidean E2 ge-ometry, hyperbolic numbers relate to Minkowskian geome-try (Boccaletti et al., 2008, chap. 4). The four-dimensionalMinkowski space-time M is equipped with a metric tensor ofsignature ⟨+,−,−,−⟩. A detailed description of the metrics inMinkowskian space-time is given by Penrose (2004, chap. 18).TheMinkowski plane,R2

1, is the two-dimensional representationofM and it is endowedwith the inner product ⟨v, v⟩R2

1= x2−y2,

with v = xi+ yj inR21 (see Saloom and Tari, 2012; Yaglom, 1979,

pp. 242–257).Consider w = x + jy with w ∈ D. Minkowskian geometry

relates to the hyperbolic numbers in terms of the pseudo-norm:

||v||2R2

1= x2 − y2 ≡ |w|2D

The geometrical representation of hyperbolic numbers on theMinkowski plane simplifies the description of hyperbolic geome-tries. Considering the metrics on the Minkowski plane the dis-tance is preserved along rectangular hyperbolas. Hyperbolic an-gles and rotations appear naturally from this property, as ex-plained in Sect. 5.1. The formulation of the orbital motion ofa particle along a hyperbolic orbit benefits from this contrivance,due to a straight-forward definition of the hyperbolic anomaly(see Sect. 5.1.2).

A.1.3 Angles and rotations

Acomplexnumber z = x+iy admits an alternative representationon the Gauss-Argand plane. Instead of the Cartesian form, thepolar form is introduced,

z = r eiϑ, with r = |z|C, and tanϑ =y

x

where r is the modulus and ϑ is the argument of z. Recall thatunder the mapping z 7→ z eiϑ2 the modulus is invariant, |z eiϑ2 | =|z| |eiϑ2 | = |z|. This application defines a rotation along the cir-cumference cr of magnitude ϑ2.

The polar form is related to the Cartesian form through Eu-ler’s identity,

z = r eiϑ = r(cosϑ + i sinϑ) = x + iy

Its formal proof can be found in Ahlfors (1966, pp. 43–44).The polar form of hyperbolic numbers on the Minkowski

plane reads

w = s e jγ

where s is themodulus andγ the hyperbolic argument. The formof the exponential function inD leads to the hyperbolic form ofEuler’s identity (Catoni et al., 2011, p. 16),

e jγ = cosh γ + j sinh γ.

Considering this identity the hyperbolic components ofw on theMinkowski plane read

x = R(w) = s cosh γ, y = H(w) = s sinh γ

In this case, the modulus is invariant under the mappingw 7→ w e jγ2 . This application represents a hyperbolic rotation

of magnitude γ2 along h s. That is, circular rotations preservethemodulus in Euclidean geometrywhereas hyperbolic rotationspreserve the modulus in two-dimensional Minkowskian geome-try. The sign criterion defines positive hyperbolic rotations on h1

+

from bottom to top, and from top to bottom on h1−.

A.2 Quaternions

Quaternions are a particular instance of hypercomplex numbers.It is q ∈ H4, with q a quaternion

q = q0 + q1u1 + q2u2 + q3u3

Quaternion algebra is endowed with the multiplication table:

u21 = −1, u1 u2 = u3 = −u2 u1

u22 = −1, u2 u3 = u1 = −u3 u2

u23 = −1, u3 u1 = u2 = −u1 u3

The definition of multiplication leads to the properties:

• The product of quaternions is not commutative. Directproof is obtained from themultiplication table, noting thatu j uk = −uk u j and then it isψℓj,k = −ψℓk, j. This violates thecondition from Thm. 1.

• The product of quaternions is associative. Note that(u1 u2) u3 = u3 u3 = −1, equal to u1(u2 u3) = u1 u1 = −1.

• Matrix A is regular, so there are no divisors of zero.

Quaternions appear recursively in many texts in mathematics,physics, and engineering, and their representation varies acrossdisciplines. For instance, they are usually regarded as Pauli ma-trices, or as spinors.

A.2.1 Rotations in R3

Let D ⊂ H4 be the subset of all pure quaternions, D = q ∈H4

∣∣∣ q0 = 0. Any vector r ∈ R3 admits an equivalent quater-nionic representation in terms of r ∈ D, given by the compo-nents of r. A unit quaternion q defines a rotation action r 7→ r1by means of

r1 = q r q†, with q q† = 1 (A.4)

As a rotation, this transformation preserves the norm, i.e. |r1| =|r|. This transformation is invertible,

r = q† r1 q

A.2.2 Quaternion dynamics

Theorem5: Let r, r1 ∈ D be two pure quaternions defined by thecomponents of vector r in a rotating and a fixed reference frame,respectively. The angular velocity of the rotating frame is ω. Ifquaternion q defines the rotation r 7→ r1, the time evolution of qis

q =12q (2w)

186 A Hypercomplex numbers

where 2w ∈ D is defined by the components of ω in the rotatingframe itself.

Proof: If vector r denotes the position of a particle, the timederivative of Eq. (A.4) yields the absolute velocity projected onthe inertial reference, v1,

v1 = q r q† + q r q† = q q† r1 q q

† + q q† r1 q q† = q q† r1 + r1 q q

†

= w1 r1 + r1 w†1 (A.5)

wherew1 = q q† and the dot represents derivationwith respect

to time. From the condition q q† = 1 it follows

q q† + q q† = 0 =⇒ w1 = −w†1

That is,w1 ∈ D. It is then possible to define an equivalent vectorω1/2. The expression for the velocity then transforms into

v1 = w1 r1 − r1 w1 =⇒ v1 = 2(ω1

2× r1

)= ω1 × r1

The quaternionic product in Eq. (A.5) is equivalent to the crossproduct of the associated vector. Hence, quaternion 2w1 ∈ D

is equivalent to the angular velocity vector of the rotating frame,projected onto the fixed frame. The components of the angularvelocity in the rotating frame reduce to

2w = q† 2w1 q = 2q† q

The inverse equation yields the time evolution of quaternion q interms of the angular velocity 2w,

q =12q (2w)

BFormulations in PERFORM

C hapter 6 explained the details of the propagator PER-FORM. One of its main features is the fact that incorpo-rates a collection of orbital formulations. Table 6.2 listed

all the methods available in PERFORM. Each of the methods wasgiven a three-character identifier, and this appendix explains allthe formulations in terms of their identifier.

COW: Cowell’s method is the widely used propagator based onthe use of Cartesian coordinates (x, y, z).

SCW: the stabilized Cowell method is sometimes referred to ass-Cowell, because it replaces the physical time t by a ficti-tious time s. The general stabilization method is explainedin Sect. 2.3, arriving to the equation of motion (2.11). Themethod is implemented following Janin (1974, eqs. 3.10and 3.14).

DRO: the Dromo formulation has been explained in detail inChap. 4. The evolution equations to be integrated areEqs. (4.9–4.16).

TDR: TimeDromo is a modification of Dromo in which the in-dependent variable is the physical time, instead of the idealanomaly. Equations (4.10–4.16) are multiplied by ζ3

3 s2, andEq. (4.9) is no longer necessary. TimeDromo reduces the di-mension of the system by one, and root-finding algorithmsare no longer required for stopping the integration.

HDT: this formulation is theMinkowskian propagator defined inChap. 5 including the time element [Eqs. (5.43–5.50)].

HDR: this is the version of the Minkowskian propagator not us-ing the time element, Eqs. (5.44–5.50).

KS_: the transformation by Kustaanheimo and Stiefel (1965)leads to Eq. (3.19), which is integrated togther with theSundman transformation dt/ds = r to provide the physi-cal time and the vector u.

KST: this version of the KS transformation includes a time ele-ment, defined as

dτds=

12h

1 +

[4h

(u · u′) u′ + r u]· (L⊤p)

with

h =(1 − 2||u′||2)

r

The physical time is retrieved by means of the equation

t = τ − (u · u′)h

KSR: instead of computing the value of the constant of the en-ergy h at each integration step, it is possible to integrate thefollowing equation

dhds= −2(L⊤p) · u′

and treat this variable as an element. This is a redundantequation that is integrated together with (3.19) and theSundman transformation.

KRT: this version of the KS regularization includes both the timeelement and the redundant equation.

187

188 B Formulations in PERFORM

SB_: the formulation originally developed by Sperling (1961) waslater improved by Burdet (1967), and is given by Eq. (2.18).This formulation is called the Sperling-Burdet (SB) regular-ization in PERFORM.

SBT: given the form of the equations in the SB regularization, itis possible to use the same time element defined by Janin(1974, eq. 3.10).

SBR: instead of computing the values of the eccentricity vector,the energy, and the radial distance at each integration step,it is possible to introduce a set of redundant equations tointegrate these values directly. The equations can be foundin Bond and Allman (1996, table 9.1, p. 154).

SRT: the redundant equations defined in the previous formu-lation are complemented with the time element by Janin(1974, eq. 3.10).

DEP: this is an element based formulation which uses a quater-nion to model the orbital plane, and is conveniently sum-marized in Deprit (1975, table 1).

PAL: the method by Palacios and Calvo (1996) may be regardedas a version of the DEP.

MIL: the version of the Milankovitch elements implemented inPERFORM can be found in Rosengren and Scheeres (2014,proposition 3).

SSc: the entire Sect. 3.6 was devoted to the definition of KS ele-ments, and the evolution equationswere provided by Stiefeland Scheifele (1971, p. 91).

EQU: although defined originally byCefola (1972), PERFORM im-plements the version of the equinoctial elements by Walkeret al. (1985, eq. 9, taking into account the corresponding er-rata).

CLA: the traditionalGauss variational equations can be found in,for example, Battin (1999, p. 488, eq. 10.41).

BCP: the elements introduced byBurdet (1968, eqs. 306 and 307)are integrated following what he calls the companion proce-dure (eqs. 318–321 in the referred paper). Except for the factthat they embedded the Jacobi integral in the formulation,thismethod is equivalent in practice to the BG14method byBond and Gottlieb (1989).

USM: the unified state model is summarized in Vittaldev et al.(2012, §3.5).

ED0: the remaining formulations are different versions of themethods presented by Baù et al. (2015). In particular, ED0refers to the equations presented in Sect. 6.1 of the referredpaper, using the Sundman transformation instead of equa-tion (6.1) in the paper. That is, the time is modeled with atime variable and not a time element.

ED1: refers to the same system of equations, using equation (6.1)in the paper.

ED2: this formulation features a constant time element, definedin equation (4.5) instead of (6.1) in the paper.

B0H: Baù et al. (2015, §C2) presented different versions of an im-proved formof the Burder-Ferrándiz equations. The identi-fier B0H refers to the formulation inwhich the physical timeis a state variable, and it obeys a first order differential equa-tion.

B1H: the physical time is given by a time linear time element.

B2H: a constant time element replaces the linear time element.

B3H: the physical time is a state variable, and its evolution is gov-erned by a second order differential equation.

BXC: formulations B0C to B3C are the same than B0H to B3H;the only difference is the fact that the generalized angularmomentum replaces the total energy as an element.

ST0: defined in Baù et al. (2015, §C2), this is a modification ofthe Stiefel-Fukushima formulation. In this case the physicaltime is integrated as a state variable.

ST1: a linear time element is propagated instead of the physicaltime.

ST2: the linear time element is replaced by a constant time ele-ment.

CStumpff functions

K arl Stumpff (1947) introduced a special family of func-tions that appear recursively in astrodynamics. Theywere later called the Stumpff functions, and they are de-

fined in terms of the series:

Ck(z) =∞∑

i=0

(−z)i

(2i + k)!

The Stumpff functions are intimately related to universal vari-ables (Everhart and Pitkin, 1983; Battin, 1999, chap. 4). They al-low to generalize the solution to Keplerian motion so the formu-lation is unique no matter the eccentricity of the orbit. In thiswork the argument of the Stumpff functions is z = ω2s2, withω2 = −2E .

When the orbital energy vanishes (in the parabolic case) it fol-lows z = 0 and the Stumpff functions reduce to

Ck(0) =1k!

(C.1)

Theorem 6: The Stumpff functions converge absolutely.

Proof: Absolute convergence implies

∞∑i=0

∣∣∣∣∣ (−z)i

(2i + k)!

∣∣∣∣∣ < ℓ, ∀k ∈ N

where ℓ is a finite number. This series verifies∞∑

i=0

∣∣∣∣∣ (−z)i

(2i + k)!

∣∣∣∣∣ = ∞∑i=0

∣∣∣∣∣ zi

(2i + k)!

∣∣∣∣∣ < ∞∑i=0

∣∣∣∣∣ zi

(2i)!

∣∣∣∣∣, ∀k ∈ N

The problem is reduced to proving that∞∑

i=0

∣∣∣∣∣ zi

(2i)!

∣∣∣∣∣ < ℓIt is possible to find a bounding value of ℓ when considering thefollowing expression:∞∑

i=0

∣∣∣∣∣ zi

(2i)!

∣∣∣∣∣ < ∞∑i=0

∣∣∣∣∣ zi

i!

∣∣∣∣∣ = ∞∑i=0

|z|ii!= exp(|z|)

which is bounded for all finite z. An elegant discussion of thegrowth rate of each term in the series can be found in Spivak(1994, p. 308).

The first of the Stumpff functions admit simple closed formexpressions, as shown in Table C.1.

Table C.1: Explicit expressions for the first Stumpff functionsCk(z),with z = ω2s2.

ω2 > 0 ω2 < 0 ω2 = 0

k = 0 cos√

z cosh√−z 1

k = 1 (sin√

z)/√

z (sinh√−z)/√−z 1

k = 2 (1 − cos√

z)/z (1 − cosh√−z)/z 1/2

Increasing the degree k of the Stumpff functions yields:

Ck+1(z) =∞∑

i=0

(−z)i

(2i + k + 1)!, Ck+2(z) =

∞∑i=0

(−z)i

(2i + k + 2)!

189

190 C Stumpff functions

From the later it follows that

Ck+2(z) =1z

∞∑i=0

(−1)izi+1(2(i + 1) + k

)!=

1z

[1k!− Ck(z)

]

which provides the recurrence formula:

Ck(z) + zCk+2(z) =1k!

(C.2)

Techniques for computing the derivatives of the Stumpfffunctions can be found, for instance, in the book from Bond andAllman (1996, appx. E). Some useful relations are:

s[∂sCk(z)

]= Ck−1(z) − kCk(z), ∂sCk(z) = −ω2sC∗k+2(z)

(C.3)

ω[∂ωCk(z)

]= Ck−1(z) − kCk(z), ∂ωCk(z) = −s2ωC∗k+2(z)

(C.4)

where ∂s and ∂ω denote the partial derivatives with respect to sand ω, respectively. Note that Eqs. (C.3a) and (C.4b) are onlyvalid for k > 0. The auxiliary term C∗k+2(z) corresponds to:

C∗k+2(z) = Ck+1(z) − kCk+2(z)

Danby (1992, p. 171) discusses in detail the computational aspectsof handling the Stumpff functions (see Appendix A). It is moreconvenient to compute the highest-degree functions via the se-ries, and then to apply the recurrence formula —Eq. (C.2)—to obtain the remaining functions. If the high-degree functionswere to be computed from the low degree ones, the recurrencerelation may become singular for z. Usual applications of theStumpff functions are restricted to k ≤ 3, and do not considerhigher-degree functions (Danby, 1987; Sharaf and Sharaf, 1997).In this work Stumpff functions up to k = 5 appear, and the re-quired formulae are given explicitly in the following.

The numerical stability of the convergent series requires a de-tailed analysis. Let S i

k ∈ Rdenote the i-th termof the series defin-ing Ck(z):

S ik(z) =

(−z)i

(2i + k)!

Since the Stumpff functions converge absolutely the factorialterm compensates the power for i sufficiently large (Spivak, 1994,p. 308). FigureC.1 shows the evolution ofS i

4(z) andS i5(z) for dif-

ferent values of the argument z, and for increasing i. It is observedthat the terms S i

k experience changes of several orders of magni-tude. As the argument grows the amplitude of the this variationincreases.

This phenomenon may lead to important losses in accuracy,provided that the least significant digits of the series are lost whenadded or subtracted from large quantities. Performing the com-putations in quadruple precision delays the appearance of theseproblems since truncation errors are reduced. However, as thecomputation advances the loss of accuracy will eventually appearat some point. To fully overcome this issue the argument of theStumpff functions is reduced making use of the so called half-

Figure C.1: Analysis of the growth-rate of the terms S ik(z).

angle relations:

C0(4z) = 2[C0(z)

]2 − 1

C1(4z) = C0(z)C1(z)

C2(4z) =12[C1(z)

]2

C3(4z) =14[C2(z) + C0(z)C3(z)

]C4(4z) =

18[C2(z)

]2+ 2C4(z)

C5(4z) =

132

C3(z) + 2C5(z) + 2C1(z)C4(z)

Note that these expressions do not require the computation ofhigher-order terms. They can be applied repeatedly to reduce thevalue of z below a certain threshold zcrit. Danby (1992, p. 173)summarizes the algorithm, even though he only provides half-angle formulae up to k = 3.

Figure C.2 shows a simple experiment to illustrate the previ-ous discussion. The Stumpff functionC0(z) is computed in threedifferent ways and the results are compared to the exact solutionC0(z) = cos

√z. The function C0(z) is first computed by series.

Figure C.2: Error in computing the Stumpff functionC0(z) from the

definition by series, from the recurrence relation starting atC4(z),and from the recurrence relation starting atC10(z).

191

Then, it is computed by applying the recurrence relation in Eq.(C.2) starting at C4(z), and starting at C10(z). The goal of theexperiment is to show how the error grows with the argumentz, and how the recursive application of the half-angle formulaekeeps the error bounded (setting zcrit = 0.1). Computations areperformed both in double and quadruple precision, in order toevaluate the effect of truncation errors in the result. It is observedthat the error grows exponentially with increasing z. However,when the argument is reduced before computing the series andthe recurrence relations, the error exhibits no dependency on themagnitude of the argument. It remains under the tolerance setas the convergence criterion for computing the series. Perform-ing the computations in quadruple precision yields the same er-ror growth-rate than in double precision. Although it is possibleto obtain valid results up to higher values of z, numerical instabil-ities still appear. For small values of z there exists a noticeable dif-ference between the solution obtained by recurrence fromC10(z)and that from C4(z). As z grows this difference is diluted due totruncation errors. The difference disappears when the argumentis reduced below zcrit.

There are two possible ways of computing the Stumpff func-tions by series. First, the series can be truncated when a certainaccuracy has been reached. In this case each term is computed se-quentially. A possible way to compute the terms in the series is:

S ik = −

z(2i + k)(2i + k − 1)

S i−1k , with S 0

k =1k!.

Convergence will have been reached when |S ik | < εtol, where εtol

denotes the tolerance. Second, the series can be truncated a pri-ori and a nested expression for the Stumpff function can be con-structed. Different forms of nesting the terms in the Stumpfffunctions can be found in the cited works.

DInverse transformations

T he present appendix contains the linearized form of theinverse of the series of transformations applied to theproblem of relative motion in Chapters 7 and 8. Sec-

tionD.1 focuses on the transformations involving the equinoctialelements. Next, Sect. D.2 contains the linearized transformationfrom Cartesian coordinates to Dromo elements. Thanks to thisconstruction an arbitrary relative state vector can be easily trans-formed into the corresponding differences in the set of Dromoelements. Finally, the linearized form of the inverse KS transfor-mation can be found in Sect. D.3.

D.1 Inverse transformations in equinoc-tial variables

D.1.1 Asynchronous case

Writing A ≡ Jγ for short, where Jγ is defined in Eq. (7.36), thenonzero terms of the inverse matrix A−1 read:

A−11,1 =

2sη4

[2(u2 + η2) + 3s(s − 1)

]A−1

1,2 = −2us2

η4

A−11,4 =

usA−1

1,5 =2uη

A−12,2 =

[s(1 − s) − u2] sin λ − su cos λη2

A−12,3 = −

q2

q1A−1

3,3 =q2

η2 [k1(q1 + cos λ) + k2(q2 + sin λ)]

A−12,4 = − tan λ A−1

3,4 = η sin λ

A−12,5 =

η

s(u sin λ + 2s cos λ)

A−12,6 = −

q2

q1A−1

3,6 = −q2η

s(k1 sin λ − k2 cos λ)

A−13,1 =

3s2 sin λ − 2su cos λη2

A−13,2 =

[s(s − 1) + u2] cos λ − su sin λη2

A−13,5 =

η

s(2q2 + u cos λ + 2 sin λ)

A−14,3 =

ℓ2

2η2 (q2 + sin λ)

A−14,6 =

ℓ2η cos λ2s

A−15,3 = −

ℓ2

2η2 (q1 + cos λ)

A−15,6 =

ℓ2η sin λ2s

A−16,2 =

1r

A−16,3 = −

1η2 [k1(q1 + cos λ) + k2(q2 + sin λ)]

A−16,6 =

η

s[k1 sin λ − k2 cos λ]

A−17,1 = −

3s2

η4 (s + 1)(τ − τp) +uη

193

194 D Inverse transformations

A−17,2 =

3s2uη4 (τ − τ0) − s

η

A−17,3 =

1η

[k1(q1 + cos λ) + k2(q2 + sin λ)]

A−17,4 = −

3uη

(τ − τp) + 2r

A−17,5 = −

3sη

(τ − τp)

A−17,6 = −r(k1 sin λ − k2 cos λ)

A−17,7 = 1

D.1.2 Synchronous case

Like in theprevious sectionwewriteB = Jt, where Jt canbe foundin Eq. (7.42). The nonzero terms in matrix B−1 take the form:

B−11,1 =

2s2(s + 1)η4

B−11,2 = −

2s2uη4

B−11,4 =

2uη

B−11,5 =

2sη

B−12,1 =

3s2q2

η5 (s + 1)(τ − τp) +( s + 1η2 − 2

s

)C1

= + 3 cos λ +2u sin λ

s

B−12,2 = −

3q2s2uη5 (τ − τp) −

uC1

η2

B−12,4 = +

3q2uη2 (τ − τp) +

η3 sin λ + ηuC1

s2

B−12,5 = +

3q2sη2 (τ − τp) +

ηC1

s

B−13,1 = −

3s2q1

η5 (s + 1)(τ − τp) +( s + 1η2 − 2

s

)C2

= + 3 sin λ − 2u cos λs

B−13,2 = +

3q1s2uη5 (τ − τp) −

uC2

η2

B−13,4 = −

3q1uη2 (τ − τp) −

η3 cos λ − ηuC2

s2

B−13,5 = −

3q1sη2 (τ − τp) +

ηC2

s

B−14,3 =

ℓ2

2η2 (q2 + sin λ)

B−14,6 =

ℓ2η cos λ2s

B−15,3 = −

ℓ2

2η2 (q1 + cos λ)

B−15,6 =

ℓ2η sin λ2s

B−16,1 =

3s2

2η5 (s + 1)(τ − τp) −u

2η2

B−16,2 = −

3s2u2η5 (τ − τp) +

s2η2

B−16,3 = −

k1(q1 + cos λ) + k2(q2 + sin λ)2η2

B−16,4 =

3u2η2 (τ − τp) −

η

s

B−16,5 =

3s2η2 (τ − τp)

B−16,6 =

η(k1 sin λ − k2 cos λ)2s

D.2 Cartesian to Dromo

This section presents the linear form of the transformation fromCartesian coordinates to Dromo elements. It completes thederivation presented in Sect. 8.1. The relative state vector atdeparture δx⊤0 = [δr⊤0 , δv

⊤0 ] is transformed to the differential

Dromo elements δq = [δζ1, δζ2, δζ3, δη1, δη2, δη3, δη4, δσ0]⊤

by the linear map

δq = Q δx0

In order to computematrixQ there are two additional conditionsthat need to be accounted for. First, the condition nn† = 1 fromEq. (8.3). Second, the differential form of the angle β (betweenthe osculating eccentricity vector and the xP-axis of the departureperifocal frame):

δβ =−ζ2 δζ1 + ζ1 δζ2√

ζ21 + ζ

22

Incorporating these conditions the transformation is given explic-itly by

Q1,1 =ζ1

ζ2Q2,1 = ζ1sk, Q1,2 =

ζ1

ζ2Q2,2 = ζ1ℓ

2 f

Q1,4 =ζ1

ζ2Q2,4 =

ζ1 fζ3

3

, Q1,5 =ζ1

ζ2Q2,5 = ζ1 p

Q3,1 = −suQ3,2 = −ζ3

3 s Q3,5 = −1s

Q4,1 = −η2

η1Q5,1 =

η2

η4Q6,1 = −

η2

η3Q7,1 =

sη2 f2

Q4,2 = −η2

η1Q5,2 =

η2

η4Q6,2 = −

η2

η3Q7,2 = −

η2k2

Q4,3 =M−43

M+34Q5,3 = −

M−43

M−21Q6,3 = −

M−43

M+12Q7,3 = −

ζ23

2M−43

Q4,4 = −η2

η1Q5,4 =

η2

η4Q6,4 = −

η2

η3Q7,4 =

η2(1 − s)ζ3(1 − ℓ2)

Q4,5 = −η2

η1Q5,5 =

η2

η4Q6,5 = −

η2

η3Q7,5 =

η2u(1 + s)sζ3(1 − ℓ2)

§D.3 Linear form of the Hopf fibration 195

Q4,6 =N−43

N+34Q5,6 = −

N−43

N−21Q6,4 = −

N−43

N+12Q6,4 =

N−43

ζ3s

Q8,1 =guζ3

3 s, Q8,2 =

s2(1 − s) − u2(1 + s)ζ3

3 s2

Q8,4 =g(1 − s)

s2ζ63

, Q8,5 =gu(1 + s)

s3ζ63

with

f =ζ2

3 u

ζ21 + ζ

22

, k = (s − ℓ2)fu, p =

s2 − ℓ2

sζ3(ζ21 + ζ

22 )

N±i j = ηi cosσ ± η j sinσ, M±i j = uN±i j ± sN∓ji

Note that the transformation is singular for e2 = ζ21 + ζ

22 = 0,

because of the indeterminacy of theperifocal frame and the eccen-tricity vector. Similarly, the transformation from polar to Carte-sian coordinates becomes singular when the transformed vectorvanishes.

D.3 Linear form of the Hopf fibration

This section deals with the linearization of the inverse formof theHopf fibration. The inverse transformation has been defined intwo different ways depending on the sign of the component x0—Eq. (3.22)— (for more details see also Roa et al., 2016c). Thisgives rise to two alternative definitions of thematrix T : δy 7→ δx,

δy0 =

T+(x0) δx0, if x0 ≥ 0

T−(x0) δx0, if x0 < 0

The transformation is referred to δx⊤0 = [δr⊤0 , δr⊤0 ] meaning

that the vanishing fourth component of the position and veloc-ity vectors has been obviated. They do not contribute to δy⊤0 =[δu⊤0 , δu

′0⊤]. When the first component of the initial state vector,

x0, is positive the transformation is defined by the matrix T+(x0).Conversely, if the component x0 is negative then it is more con-venient to use the matrix T−(x0). These two matrices read:

T+(x0) = c0

0, 0, 0, 0, 0, 0

+2z0, +2y0z0

R, 2

z20

R− 4r0, 0, 0, 0

−2y0, 4r0 − 2y2

0

R, −2

y0z0

R, 0, 0, 0

−2R, −2y0 , −2z0, 0, 0, 0

−c1 , 2r0z0 − y0c1

R, −2r0y0 − z0

c1

R, 0, −2r0z0, 2r0y0

−Rz0 − z0 x0, −y0

(z0 +

z0 x0

R

), 2x0r0 − z0z0 −

z20 x0

R, 2r0z0, 0, −2r0R

Ry0 + y0 x0, y0y0 − 2x0r0 +y2

0 x0

R, z0

(y0 +

y0 x0

R

), −2r0y0, 2r0R, 0

c2 − x0R, y0c2

R− y0 x0 − 2y0r0, z0

c2

R− z0 x0 − 2z0r0, −2r0R, −2r0y0, −2r0z0

T−(x0) = c0

−2z0, 2y0z0

R, −4r0 + 2

z20

R, 0, 0, 0

0, 0, 0, 0, 0, 0

2R, −2y0, −2z0, 0, 0, 0

2y0, 4r0 − 2y2

0

R, −2

y0z0

R, 0, 0, 0

Rz0 − z0 x0, −y0

(z0 −

z0 x0

R

),

z20 x0

R− 2r0 x0 − z0z0, −2r0z0, 0, −2r0R

c1 , 2z0r0 − y0c1

R, −2y0r0 − z0

c1

R, 0, −2r0z0, +2r0y0

−Rx0 − c2, y0 x0 − 2y0r0 + y0c2

R, z0 x0 − 2z0r0 + z0

c2

R, 2r0R, −2r0y0, −2r0z0

y0 x0 − Ry0, y0y0 + 2x0r0 −y2

0 x0

R, z0

(y0 −

y0 x0

R

), 2r0y0, 2r0R, 0

having introduced the auxiliary variables:

c0 =

√2

81

r0√

R, c1 = y0z0 − z0y0

c2 = y0y0 + z0z0, R = r0 + |x0|

EElliptic integrals and elliptic functions

E lliptic integrals are oneof themany examples of geomet-rical problems leading to important advances in calculus.The English mathematician JohnWallis is well known for

introducing the symbol “∞” for denoting infinity, and he waspossibly the first to present a systematic treatment of the problemof rectifying curves (i.e., computing their arc length) in hisArith-metica infinitorum (Wallis, 1656). In the 17th century many au-thors dealt with the problem of computing the arc length of var-ious transcendental curves. The ellipse was deeply studied, andWallis and Newton himself derived series solutions to rectify thisconic section.

Hendrik van Heuraet in 1659 and Pierre de Fermat in 1660showed independently that the problem of finding the arc lengthof a certain curve can be reformulated into the problem of com-puting an area. This means solving an integral. Following thissame line of thought, JakobBernoulli published in 1694 the equa-tion of the arc length of his lemniscate, which is given by the inte-gral expression∫ s

s0

dt√

1 − t4(E.1)

Giulio Fagnano was deeply interested in this type of integral ex-pressions arising from transcendental curves, and tried to extendthemto the ellipse. BasedonFagnano’swork, itwasEulerwhode-rived awhole theory for the elliptic integrals. Later, Adrien-MarieLegendre made significant contributions to the theory and sug-gested alternative definitions and notation. Nowadays the termelliptic integral refers to any function that admits the definition∫

R(s, t) dt

in which s2(t) is a polynomial of degree three or four in t, andR(s, t) is a rational function including at least an odd power of s.According to this general definition, Eq. (E.1) can be regarded asan elliptic integral.

Legendre culminated Euler’s work and defined three funda-mental functions:

F(ϕ, k) =

∫ ϕ

0

dθ√

1 − k2 sin2 θ=

∫ sin ϕ

0

dt√

1 − t2√

1 − k2t2

E(ϕ, k) =

∫ ϕ

0

√1 − k2 sin2 θ dθ =

∫ sin ϕ

0

√1 − k2t2√

1 − t2dt

Π(n; ϕ, k) =∫ ϕ

0

dθ

(1 − n sin2 θ)√

1 − k2 sin2 θ

=

∫ sin ϕ

0

dt

(1 − nt2)√

1 − t2√

1 − k2t2

They are the incomplete elliptic integrals of the first, second, andthird kinds, respectively. Under this notation ϕ is the argument,k is the modulus, and n is the parameter of the elliptic integrals.When the argument takes the value ϕ = π/2 these expressionsreduce to the complete elliptic integrals of the first, second, andthird kinds:

K(k) = F(π/2, k)E(k) = E(π/2, k)Π(n; k) = Π(n;π/2, k)

197

198 E Elliptic integrals and elliptic functions

E.1 Properties and practical relations

Elliptic integrals appear recursively in the definition of the gener-alized logarithmic spirals introduced in Chap. 9. In this sectionwe shall discuss some properties of practical nature. These prop-erties are required for arriving to the solutions included in the re-ferred chapter. More details on the properties can be found inOlver (2010, §19).

A vanishing argument makes the elliptic integrals zero, i.e.

F(0, k) = E(0, k) = Π(n; 0, k) = 0

Similarly, taking k = 0 simplifies the elliptic integrals to

F(ϕ, 0) = E(ϕ, 0) = Π(0; ϕ, 0) = ϕ and Π(1; ϕ, 0) = tan ϕ

The incomplete elliptic integrals of the first and second kinds alsosatisfy the following identities

F(π/2, 1) = ∞, E(π/2, 1) = 1, E(ϕ, 1) = sin ϕ

E.1.1 Reciprocal-modulus transformation

The reciprocal-modulus transformation allows the user to con-fine the modulus of the elliptic integral to the interval k < 1.Given the modulus k > 1, the elliptic integrals can be referred tothe reciprocal modulus k1 = 1/k < 1 by means of the formulas:

F(ϕ, k1) = k F(β, k)

E(ϕ, k1) =1k

[E(β, k) − k′2 F(β, k)]

Π(n; ϕ, k1) = kΠ(nk2; β, k)

The arguments ϕ and β relate by means of sin β = k1 sin ϕ.

E.1.2 Imaginary-argument transformation

When working in the complex plane it is sometimes useful to ap-ply the following relations:

F(iϕ, k) = i F(ψ, k′)

E(iϕ, k) = i[F(ψ, k′) − E(ψ, k′) + tanψ

√1 − k′2 sin2 ψ

]Π(n; iϕ, k) =

i1 − n

[F(ψ, k′) − nΠ(1 − n;ψ, k′)

]These formulas involve the change of argument sinh ϕ = tanψand the modulus k is replaced by the complementary modulus,which satisfies

k2 + k′2 = 1

E.1.3 Imaginary-modulus transformation

Elliptic integrals involving a complexmodulus verify the formulas

F(ϕ, ik) = κ′ F(θ, κ)

E(ϕ, ik) =1κ′

[E(θ, κ) − κ2 sin θ cos θ

√1 − κ2 sin2 θ

]Π(n; ϕ, ik) =

κ′

n1

[κ2 F(θ, κ) + nκ′2Π(n1; θ, κ)

]

which involve the auxiliary variables

κ =k

√1 + k2

, κ′ =1

√1 + k2

, n1 =n + k2

1 + k2

and also the arguments

sin θ =√

1 + k2 sin ϕ√1 + k2 sin2 ϕ

E.2 Implementations

Most computing environments come with built-in implementa-tions of elliptic integrals. However, the notation differs from oneto another and migrating between them may not be straightfor-ward. The following examples show how elliptic integrals are im-plemented in Matlab, Maple, and Mathematica.

In Abramowitz and Stegun (1964, p. 614, table 17.5), it isshown that

F(π/3, 0.8660254040) = 1.21259661

Note that the notation has been adapted to meet the standardsof this chapter. InMaple (setting Digits:=15), this same resultcan be obtained evaluating the function[> evalf( EllipticF(sin(Pi/3),0.8660254040) );

1.21259661525498

The argument of the function EllipticF is sin ϕ, instead of ϕ.InMatlab, the functions elliptic12 and elliptic3* provideefficient implementations of elliptic integrals. The present exam-ple can be solved with

>> elliptic12(pi/3,0.8660254040ˆ2)

ans =1.21259661537869

The function takes ϕ = π/3 as the argument, but the modulusneeds to be squared. The convention inMathematica is the same,with:

In[1]:= EllipticF[Pi/3, 0.8660254040ˆ2]

Out[1]:= 1.21259661537869

The differences between implementations of the incompleteelliptic integrals of the third kind reduce to the order inwhich theinputs are given. According to Abramowitz and Stegun (1964,p. 625, table 17.9):

Π(0.5;π/3, 0.8660254040) = 1.47906

The corresponding implementation in Maple yields[> evalf( EllipticPi(sin(Pi/3),0.5,0.8660254040) );

1.47906355878139

*https://goo.gl/nsJFT1 and https://goo.gl/NiYTme. A wrapperfunction elliptic123 is also available.

§E.4 Weierstrass elliptic functions 199

In Matlab it is

>> elliptic3(pi/3,0.8660254040ˆ2,0.5)

ans =1.47906355895353

and in Mathematica it is written

In[1]:= EllipticPi[0.5, Pi/3, 0.8660254040ˆ2]

Out[1]:= 1.479063558953542

E.3 Jacobi elliptic functions

Inverting elliptic integrals is not an easy task, and occupied manymathematicians. Gauss approached the problem by the end ofthe 18th century, although he did not publish anything on thesubject (Stillwell, 2010, p. 234). Later, Abel (1827) publishedsome preliminary results that motivated Gustav Jakob Jacobi tocomplete his theory, presented in his book Fundamenta novatheoriae functionum ellipticarum in 1829.

Let u denote the incomplete elliptic integral of the first kind

u = F(ϕ, k) =∫ ϕ

0

dθ√

1 − k2 sin2 θ

The Jacobi amplitude is defined in terms of the inverse of the el-liptic integral, and takes the form

ϕ = F−1(u, k) = am(u, k)

From this result, three important definitions follow:

sin ϕ = sin(am(u, k)) = sn(u, k)cos ϕ = cos(am(u, k)) = cn(u, k)√

1 − k2 sin2 ϕ =

√1 − k2 sin2(am(u, k)) = dn(u, k)

The functions sn, cn, and dn are the Jacobi elliptic functions.This notation is actually due to Gudermann (1838), who also in-troduced a fourth function that is missing in most references (cf.p. 12):

tn(u, k) =sn(u, k)cn(u, k)

as it was replaced by sc(u, k). Glaisher (1879) workedwith a largercollectionof functions, unifiedunder the followingnotation. Letp, q, r be any of the letters s, c, d, n. Then

pq(u, k) =pr(u, k)qr(u, k)

=1

qp(u, k)

The Jacobi elliptic functions satisfy two identities, similar innature to the trigonometrical identity:

sn2(u, k) + cn2(u, k) = 1

dn2(u, k) + k2 sn2(u, k) = 1

In fact, the elliptic functions relate to the trigonometric functionsby means of

sn(u, 0) = sin(u), cn(u, 0) = cos(u), dn(u, 0) = 1

For k = 1 they reduce to the hyperbolic functions

sn(u, 1) = tanh(u), cn(u, 1) = sech(u), dn(u, 1) = sech(u)

The Jacobi elliptic functions are periodic, with

sn(u, k) = sn(u + 4m K(k), k)cn(u, k) = cn(u + 4m K(k), k)dn(u, k) = dn(u + 2m K(k), k)

and m = 0, 1, 2, . . .. That is, sn and cn have period 4 K(k),whereas dn is 2 K(k)-periodic.

Another special function named after Jacobi is the Jacobi zetafunction, Z(ϕ, k). It is defined in therms of elliptic integrals offirst and second kinds as

Z(ϕ, k) = E(ϕ, k) − E(k) F(ϕ, k)K(k)

E.4 Weierstrass elliptic functions

Karl Weierstrass’ first studies were in economy and laws, follow-ing the advice of his father, who worked in the Prussian admin-istration. Motivated by the reading of Abel’s work, he decidedto abandon his studies after four years and to pursue a career inmathematics. He was given the chance to study at the Academyin Münster, where he enrolled in 1839. He was eager to entertheAcademybecauseChristophGudermannwas a lecturer there,and was teaching possibly the first course in elliptic functions.Gudermann discovered the mathematical talent of Weierstrassand motivated him to work in his own theories.

Most of Weierstrass’ theory of elliptic function was devel-oped in a series of unpublished works, collected as part of his lec-ture series. Reviewing the entire theory is out of the scope of thisthesis and, for practical reasons, we shall focus on the propertiesof the ℘ functions appearing in Chap. 12, and the connection ofthese functions with elliptic integrals.

According to Whittaker and Watson (1927, p. 454), it wasWeierstrass’ doctoral student Wilhelm Biermann who publishedthe following result, attributing it to his advisor. Consider theproblem of inverting the integral function

z =∫ x

a

dt√f (t)

(E.2)

where f (x) is a quartic polynomial with no repeated roots, anda is not necessarily a zero of f (x). This means solving for x =x(z, a). Thanks to the ℘ functions, Eq. (E.2) can be inverted interms of

x = a +48℘′(z)

√f (a) + b f ′(a) + 2 f (a) f ′′′(a)4b2 − f (a) f iv(a)

having introduced the auxiliary coefficient b = 24℘(z) − f ′′(a).In this equation f n(x) refers to the n-th derivative of f (x). It iseasy to verify that, if a = x0 is a root of f (x), then the inverse ofEq. (E.2) reduces to

x = x0 +f ′(x0)

4[24℘(z) − f ′′(x0)]

200 E Elliptic integrals and elliptic functions

Writing

f (x) = a0x4 + 4a1x3 + 6a2x2 + 4a3x + a4

the invariant lattices

g2 = a0a4 − 4a1a3 + 3a22

g3 = a0a2a4 + 2a1a2a3 − a32 − a0a2

3 − a21a4

define the Weierstrass elliptic function

℘(z) = ℘(z; g2, g3)

in the form that is expected bymost software packages. For exam-ple, in Maple the ℘ and ℘′ functions are evaluated with[> WeierstrassP(.5, .2, .3);

4.00317028652538[> WeierstrassPPrime(.5,.2,.3);

−15.9846346069412

In Matlab they can be evaluated using MUPAD, and in Mathemat-ica they are given by

In[1]:= WeierstrassP[.5, .2, .3]

Out[1]:= 4.00317028652538 + 0. i

In[2]:= WeierstrassPPrime[.5, .2, .3]

Out[2]:= -15.9846346069412 + 0. i

FControlled generalized logarithmic spirals

T his appendix presents the complete reformulation ofthe generalized logarithmic spirals including the controlparameter ξ. Chapters 10–12 used this solution exten-

sively, taking advantage of the similarity transformation that con-nects the original and the extended solutions. In the followinglines the equations

dvdt=ξ − 1

r2 cosψ (F.1)

vddt

(ψ + θ) =2(1 − ξ)

r2 sinψ (F.2)

drdt= v cosψ (F.3)

dθdt=v

rsinψ (F.4)

are integrated explicitly. The solution to the motion of a particleaccelerated by

ap =1r2

[ξ cosψ t + (1 − 2ξ) sinψ n

](F.5)

is written here for convenience, instead of using the similaritytransformation introduced in Chap. 10 and further developed inChap. 12. This an extensionof the generalized logarithmic spirals.

Dividing Eqs. (F.1) and (F.3) providesdvdr=ξ − 1r2v

=⇒ v dv =ξ − 1

r2 dr

The resulting expression is an equation of separate variables thancan be integrated easily to define an integral of motion:

v2 − 2r

(1 − ξ) = K1 (F.6)

which introduces the control parameter ξ in the generalization ofthe equation of the energy provided in Chap. 9. The constant ofmotion K1 is solved from the initial conditions

K1 = v20 −

2r0

(1 − ξ) (F.7)

If ξ = 0 the integral of motion (F.6) reduces to the equation ofthe Keplerian energy Ek, with

K1(ξ = 0) ≡ 2Ek = v2 − 2

r

Dividing Eqs. (F.1) and (F.2) provides the relation

dvv=

12

cotψ (dψ + dθ)

By virtue of Eqs. (F.3–F.4) it follows an equation of separate vari-ables

2dvv+ cotψ dψ +

drr= 0

that can be integrated analytically and defines a first integral:

2 ln v + ln(sinψ) + ln r2 = lnC =⇒ rv2 sinψ = K2 (F.8)

This result proves the S-invariance of the integral of mo-tion (9.16).

K2 = r0vγ0 sinψ0 (F.9)

In the following sections the equations of motion are solved forthe elliptic, parabolic, and hyperbolic cases.

201

202 F Controlled generalized logarithmic spirals

F.1 Elliptic spirals

Elliptic spirals relate naturally to Keplerian ellipses; the particlenever escapes the potential well of the central body. When prop-agated forward and backwards in time the spiral falls towardsthe origin. The trajectory is T-symmetric and the axis of T-symmetry is equivalent to the apse line. There is a natural tran-sition from raising regime to lowering regime at the apoapsis ofthe spiral. They are defined by negative values of the constant K1.This determines the values of ξ that yield elliptic spirals:

K1 < 0 ⇐⇒ ξ < 1 −r0v

20

2

From the integral of motion (F.8) and the condition sinψ ≤1 it must be

K2

rvγ≤ 1 =⇒ r

[K1 +

2r

(1 − ξ)]γ/2≥ K2

Thanks to the assumption γ = 2 the previous expression simpli-fies to

K1r + 2(1 − ξ) ≥ K2

and shows that there is a maximum radius for the case of ellipticspirals (K1 < 0):

rmax =2(1 − ξ) − K2

(−K1)(F.10)

The condition on ξ that makes K1 < 0 ensures rmax > 0. Themaximum radius can be seen as the apoapsis of the spiral. Thevelocity at rmax is

vm =

√K2

rmax=

√−K1K2

2(1 − ξ) − K2

It is the minimum velocity that a particle can have on an ellipticspiral.

The flight-direction angle can be solved from the integral ofmotion (F.8) and results in

sinψ =K2

rvγ≡ K2

rv2 =K2

2(1 − ξ) + K1r(F.11)

Only positive values of sinψwill be considered. This restricts thesolution to the case of prograde motions. Retrograde motionscan be solved in an analogous way but are omitted for clarity.From Eq. (F.11) it follows

cosψ = ±

√[2(1 − ξ) + K1r]2 − K2

2

2(1 − ξ) + K1r(F.12)

The choice of the sign depends on the regime of the spiral. If thespiral is in raising regime it is ψ < π/2 and therefore cosψ > 0.If the spiral is in lowering regime it is cosψ < 0.

For K1 < 0 Eq. (F.11) shows that K2 is constrained to theopen interval

0 < K2 < 2(1 − ξ)

The tangent ofψ is obtained by combining Eqs. (F.11) and (F.12),

tanψ = ± K2√[2(1 − ξ) + K1r]2 − K2

2

and introducing this result in the quotient of Eqs. (F.4) and (F.3)it follows

dθdr=

tanψr=⇒ dθ = ± K2 dr

r√

[2(1 − ξ) + K1r]2 − K22

(F.13)

Integrating this equation determines the evolution of the polarangle θ as a function of the radial distance. This angle can be re-ferred to the axis of symmetry, defined by θ(rmax) = θm, intro-ducing the spiral anomaly:

β =ℓ

K2(θ − θm) (F.14)

The spiral anomaly evolves according to

β(θ) = ∓∣∣∣∣∣∣arccosh

rmax

r− 2(1 − ξ)

K2

(1 − rmax

r

)∣∣∣∣∣∣ (F.15)

having introduced the auxiliary parameter

ℓ =√

4(1 − ξ)2 − K22

The angle θm defines the orientation of the axis of symmetry, andcan be solved directly from the initial conditions. The first signcorresponds to raising regime and the second sign to loweringregime.

The equation for the trajectory r = r(θ) is solved in terms ofthe spiral anomaly by inverting Eq. (F.15),

r(θ)rmax

=2(1 − ξ) + K2

2(1 − ξ) + K2 cosh β(θ)(F.16)

TheT-symmetry of the trajectory is easily verified from this equa-tion. This proves that introducing the control parameter ξ doesnot break the symmetries of the original family of solutions.

F.1.1 The time of flight

The time relates to the radial distance by means of the equationfor the radial velocity, Eq. (F.3):

dtdr=

1v cosψ

= ±√

r[2(1 − ξ) + K1r]√ℓ2 + K1r[K1r + 4(1 − ξ)]

(F.17)

This equation is integrated to provide the time of flight:

t(r)−tm = ±rvK1

√1 − sinψ1 + sinψ

±2[2(1 − ξ)k′2∆Π − K2∆ E]

√1 − ξ

(−K1)3/2√

K2

(F.18)

that is referred to the time of passage through the apoapsis rmax,denoted tm. The solution is given in terms of the complete and

§F.3 Hyperbolic spirals 203

the incomplete elliptic integrals of the second, E(ϕ, k), and thirdkinds,Π(p; ϕ, k), namely:

∆ E = E(ϕ, k) − E(k), ∆Π = Π(p; ϕ, k) − Π(p; k)

Their argument, modulus and parameter are, respectively:

sin ϕ =vm

v

√2

1 + sinψ, k =

√−K1rmax

4(1 − ξ) , p =K1rmax

2K2

The complementary modulus k′ is defined as k′ =√

1 − k2.The time of apoapsis passage is solved from the initial condi-

tions:

tm − t0 = ∓r0v0

K1

√1 − sinψ0

1 + sinψ0

∓2[2(1 − ξ)k′2∆Π0 − K2∆ E0]

√1 − ξ

(−K1)3/2√

K2

The signs in the expressions for the time of flight follow the signcriterion that has already been discussed: the first sign corre-sponds to raising regime, and the second to lowering regime.

F.2 Parabolic spirals

Parabolic spirals are equivalent to logarithmic spirals. For arbi-trary values of ξ the velocity is no longer the local circular velocity,but a generalization in terms of the control parameter:

v(r) =

√2(1 − ξ)

r

This equation shows how changing the value of the control pa-rameter yields logarithmic spirals but with different velocity pro-files. This phenomenon was already discussed by Petropouloset al. (1999), who found a parameterization of the thrust accelera-tion required by logarithmic spirals in terms of the orientation ofthe thrust vector. The value of the control parameter that makesthe spiral logarithmic given a set of initial conditions is

ξ = 1 −r0v

20

2

Equation (F.11) governs the evolution of the flight-directionangle. Imposing K1 = 0 in this equation yields

sinψ =K2

2(1 − ξ)

so the flight-direction angle remains constant during the propaga-tion. The differential equation connecting the polar angle θwiththe radial distance is simply

dθdr=

tanψr= ±K2

rℓ

The equation of the trajectory takes the form

r(θ)r0= e±ℓ(θ−θ0)/K2 (F.19)

Observe that

± ℓ

K2= cotψ

which is constant, and therefore Eq. (F.19) is no other than theequation of a logarithmic spiral. The thrust acceleration in thiscase is not tangential to the trajectory, because the normal com-ponent defined in Eq. (F.5) is not zero in general. In fact, Eq.(10.10) shows that the thrust vector forms a constant angle withthe radial direction.

As long as K1 = 0 the trajectory is a logarithmic spiral. Ifthe flight-direction angle is fixed then changing the value of thecontrol parameter ξ changes themagnitude of the velocity on thespiral, although the trajectory remains the same.

F.2.1 The time of flight

Imposing the conditions K1 = 0 in Eq. (F.17) results in a simpleexpression that can be integrated to provide the time of flight

t(r) − t0 = ±2√

2(1 − ξ)3ℓ

(r3/2 − r3/2

0

)(F.20)

This expression is referred directly to the initial conditions of theproblem. The sign depends on the regime of the spiral like inthe previous cases. It takes an infinite time to escape to infinity,meaning that the parabolic spirals follow a spiral branch.

F.3 Hyperbolic spirals

The family of hyperbolic generalized logarithmic spirals is definedby K1 > 0. The integral of motion (F.6) sets no limits to thevalues that the radius can take. In fact, from this expression itfollows that a particle traveling along a hyperbolic spiral reachesinfinity with a finite, nonzero velocity:

limr→∞

v2 = K1

The constant K1 is equivalent to the characteristic energyC3 andreadily provides the hyperbolic excess velocity. Once the initialconditions are fixed hyperbolic spirals appear for values of thecontrol parameter satisfying:

ξ > 1 −r0v

20

2

Equation (F.11) sets a dynamical constraint that relates the ra-dius with K1 and K2, because sinψ ≤ 1. When K1 > 0 this ex-pression holds even for K2 > 2(1 − ξ), unlike for elliptic spirals,but the radius must then satisfy

r ≥ K2 − 2(1 − ξ)K1

This equation defines the minimum radius that the spiral canreach. This behavior suggests that there are two different sub-families of hyperbolic spirals, as it has already been anticipated.Hyperbolic spirals of Type I correspond to K2 < 2(1 − ξ). Theprevious constraint on the radius holds naturally so they escape

204 F Controlled generalized logarithmic spirals

to infinity if they are in raising regime, and fall to the origin if inlowering regime. Hyperbolic spirals of Type II (K2 > 2(1 − ξ))exhibit a minimum radius where the spiral transitions from low-ering regime to raising regime. These spirals describe two asymp-totes. The two types of spirals are separated by the limit caseK2 = 2(1 − ξ).

F.3.1 Hyperbolic spirals of Type I

This subfamily of spirals corresponds to K1 > 0 and K2 <2(1 − ξ). The polar angle is solved by integrating Eq. (F.13) andbecomes:

θ−θ0 = ±K2

ℓln

r sinψ

r0 sinψ0

[2(1 − ξ) − K2 sinψ0 + ℓ | cosψ0|2(1 − ξ) − K2 sinψ + ℓ | cosψ|

]The limit r → ∞ determines the orientation of the asymptote:

θas = θ0 ±K2

ℓln

[K2(ζ − ℓ − K2 sinψ0 + ℓ | cosψ0|)

r0K1ζ sinψ0

](F.21)

having considered the auxiliary variable ζ = 2(1 − ξ) + ℓ.This type of spiral is not T-symmetric so the definition of

the spiral anomaly given in Eq. (F.14) cannot be applied directly.Redefining the angular parameter β(θ) as

β(θ) = ± ℓ

K2(θas − θ)

the equation of the trajectory becomes

r(θ) =ζℓ2/K1

sinh β2

[4ζ(1 − ξ) sinh β

2 + (ζ2 − K22 ) cosh β

2

] (F.22)

The signofβ(θ)depends on the regimeof the spiral. So it does thevalue of the orientation of the asymptote given in Eq. (F.21). Theregime of the spiral is easily determined fromψ0. It is straightfor-ward to verify the existence of an asymptote for θ = θas (β = 0).

The time of flight

Inverting the equation of the radial velocity and integrating theresulting expression yields the time of flight,

t(r) = K4 ±

rvK1

√1 + sinψ1 − sinψ

−2 E−(1 − p)Π

√K2(1 − ξ)

K3/21

written in terms of a constant K4. This constant can be eas-ily solved from the previous equation particularized at the ini-tial time. The solution is given in terms of the incomplete el-liptic integrals of the second and third kinds, E = E(ϕ, k) andΠ = Π(p; ϕ, k), being its argument:

sin ϕ =

√K1r sinψ

pK2(1 − sinψ)

The modulus and the parameter of the elliptic integrals are:

k =12

√2(1 − ξ) + K2

1 − ξ and p =2(1 − ξ) + K2

2K2

F.3.2 Hyperbolic spirals of Type II

For the case K1 > 0 and K2 > 2(1 − ξ) the condition sinψ ≤ 1provides

rmin =K2 − 2(1 − ξ)

K1(F.23)

meaning that the spiral will never reach the origin. If the spiral isinitially in lowering regime it will reach rmin and at this point ittransitions to raising regime. In this case ℓ takes the form

ℓ =√

K22 − 4(1 − ξ)2

The polar angle is solved from Eq. (F.13) and yields:

θ − θm = ±K2

ℓ

π2 + arctan

2(1 − ξ)[2(1 − ξ) + K1r] − K22

ℓ√

[2(1 − ξ) + K1r]2 − K22

= ±K2

ℓ

π

2+ arctan

[2(1 − ξ) − K2 sinψ

ℓ | cosψ|

](F.24)

The angle θm defines the direction of the axis ofT-symmetry. Itcan be solved from the initial conditions and the previous equa-tion. Note that its definition depends on the regime of the spiral,because there are two possible C-symmetric trajectories.

This equation can be inverted to provide the equation of thetrajectory

r(θ)rmin=

2(1 − ξ) + K2

2(1 − ξ) + K2 cos β(F.25)

having introduced the spiral anomaly:

β(θ) =ℓ

K2(θ − θm)

Since the equation for the trajectory depends on the spiralanomaly by means of cos β the trajectory is T-symmetric. Thefact that there are two different values of β that cancel the de-nominator proves the existence of two distinct asymptotes. Theasymptotes can be solved from the limit r → ∞ in the equationfor the polar angle and correspond to

θas = θm ±K2

ℓ

π

2+ arctan

[2(1 − ξ)

ℓ

]The two asymptotes are given by the two different signs that ap-pear in this equation. They are symmetricwith respect to the apseline θm.

The time of flight

The time of flight is obtained following the same technique ap-plied to elliptic, parabolic and hyperbolic spirals of Type I. Inte-grating the inverse of the radial distance renders

t(r) − tm = ∓

[K2 + 2(1 − ξ)]K2 E−K1rmin[K2 F+2(1 − ξ)Π]

K1√

K1K2[K2 + 2(1 − ξ)]

+2(1 − ξ)

K3/21

arcsinh √

2K1r(rv2 − K2)

2√

K2rv2 + (rv2 − K2)(1 − ξ)

− v

K21

√r2v4 − K2

2

§F.4 Osculating elements 205

The solution is given in terms of the incomplete elliptic integralsof the first, F ≡ F(ϕ, k), second, E ≡ E(ϕ, k), and third kinds,Π ≡ Π(p; ϕ, k). The argument of the elliptic integrals is given by

sin ϕ =√

2(1 − ξ)(1 − sinψ)

k√

K2 − 2(1 − ξ) sinψ

and the modulus and parameter read:

k =2√

1 − ξ√K2 + 2(1 − ξ)

and p =2(1 − ξ)

K2

The time of flight is referred to tm. It denotes the time of periapsispassage and is solved from the initial conditions:

tm−t0=±

[K2 + 2(1 − ξ)]K2 E0 −K1rmin[K2 F0 +2(1 − ξ)Π0]

K1√

K1K2[K2 + 2(1 − ξ)]

+2(1 − ξ)

K3/21

arcsinh

√

2K1r0(r0v20 − K2)

2√

K2r0v20 + (r0v

20 − K2)(1 − ξ)

− v0

K21

√r2

0v40 − K2

2

The regime of the spiral needs to be accounted for when solvingthis equation.

F.3.3 Limit case K2 = 2(1 − ξ)

Hyperbolic spirals ofType I are defined by K2 < 2(1−ξ)whereashyperbolic spirals of Type II correspond to K2 > 2(1− ξ). In thelimit case K2 → 2(1 − ξ) the minimum radius from Eq. (F.23)vanishes. This means that a spiral in lowering regime will reachthe origin but will not be able to transition to raising regime andescape.

The equation for the evolution of the polar angle reduces to:

θ(r) − θ0 = ∓√1 +

4(1 − ξ)K1r

−√

1 +4(1 − ξ)

K1r0

There is only one asymptotewhose definition depends on the ini-tial regime of the spiral:

θas − θ0 = ∓1 − √

1 +4(1 − ξ)

K1r0

The spiral anomaly in this case is defined with respect to the di-rection of the asymptote,

β = θ − θas

Inverting the equation for the polar angle defines the trajectory:

r(θ) =4(1 − ξ)

K1β(β ∓ 2), with β = θ − θas

Recall that the first sign corresponds to raising regime and the sec-ond to lowering regime.

The time of flight

The time of flight can be directly related to the initial conditions,namely

t(r) − t0 = ±1

K3/21

Ξ − Ξ0 + (1 − ξ) ln r0v

20 + 1 − ξ + Ξ0

rv2 + 1 − ξ + Ξ

The auxiliary parameter Ξ = Ξ(r) reads

Ξ(r) = v√

r[rv2 + 2(1 − ξ)]

F.4 Osculating elements

The orbital elements defining the osculating orbit can be referredto the constants defining the spirals and the radial distance. Inparticular, the eccentricity reads

e(r) =√

K22 + 1 − 2K2 sinψ (F.26)

The angularmomentumrelates to the constant K2: Roa et al.(2016a) connect Eq. (10.8) with the torque from the perturbedforces, and from that derivation it follows

h(r) = K2

√r

2(1 − ξ) + K1r(F.27)

The semimajor axis is obtained by combining Eqs. (F.26) and(F.27), and results in

a(r) =h2

1 − e2 =r

2ξ − K1r(F.28)

The semimajor axis always grows in the raising regime and de-creases in the lowering regime, no matter the type of spiral.

GDynamics in Seiffert’s spherical spirals

S ome seventy years after Carl G. J. Jacobi (1829) publishedhis celebrated theory of elliptic functions, Alfred Seiffert(1896) equipped them with a beautiful geometric inter-

pretation. He showed that the Jacobi elliptic functions rendera three-dimensional curve confined to a spherical surface. Thepolar angle grows linearly with the arclength parameter and thecurve connects the poles of the sphere. This defines what waslater called a Seiffert spiral. More recently, Erdös (2000) recoveredSeiffert’s spirals to describe the motion of a plane circumnavigat-ing the Earth with constant velocity. Chapters 9–11 proved thepotential of using known curves for mission design. Followingthis line of thought, this chapter recovers Seiffer’s spirals in orderto characterize its dynamics from an orbital perspective.

G.1 Dynamics

Seiffert’s spherical spiral is confined to a sphere of constant radius.In normalized variables that make the radius of the sphere equalunity, the spiral can be defined in cylindrical coordinates as:

r = sn(u, k) (G.1)θ = θr + ku (G.2)z = cn(u, k) (G.3)

It is defined in terms of the Jacobi elliptic functions sn(u, k) andcn(u, k). The definition of these functions and some useful prop-erties can be found in Appendix E. Here k is their modulus andu is the argument. In order to avoid singularities at the poles weadopt the sign criterion from Erdös (2000): the cylindrical radiusr takes values in the interval r ∈ [−1, 1] and the negative values

of the radius ensure a smooth transition through the poles. Herecovered these spirals and used them as an excuse to explain theJacobi elliptic functions from a rather academic perspective.

The angular variable u takes the form

u = nt + ϕ0

and shows that the angular velocity in Seiffert’s spiral is constant.The coefficient n denotes the mean motion on the spiral, ϕ0 con-trols the phasing and θr determines the orientation of the spiralin the inertial space. The parameters n, k, ϕ0, θr define the tra-jectory.

LetL = ur,uθ,uz be a rotating frame defined by the radial,circumferential and normal unit vectors. The radius vector to theparticle r is given by

r = r ur + z uz

Figure G.1 depicts the construction of the problem referred to aninertial frame I.

Geometrically, the spiral is confined to a spherical surface ofradius R2 = r2 + z2 = 1. The (positive) modulus of the Jacobielliptic functions k controls the shape of the spiral. For k > 1 theout-of-plane component of the motion, z, never vanishes. Thetrajectory is confined to one of the hemispheres, the northern orthe southern. For k < 1 the spiral is able to transition from thenorthern hemisphere to the southern hemisphere, and vice-versa.The limit case k = 1 is the asymptotic limit for which z → 0 ast → ∞.

Figure G.2 shows examples of orbits around Mars for differ-ent values of the modulus of the Jacobi elliptic functions. The

207

208 G Dynamics in Seiffert’s spherical spirals

Figure G.1: Geometry of the problem.

(a) k < 1 (b) k = 1 (c) k > 1

Figure G.2: Seiffert's spherical spiral trajectories aroundMars de-

fined by different values of themodulus k.

case k < 1 provides a full coverage of the planet with the orbitgoing through the poles. For k = 1 the spiral only reaches thepole once and then approaches the equator. For k > 1 the orbitis confined to one single hemisphere. The vertical axis zI coin-cides with the polar axis of the planet.

The trajectory inherits the periodicity of the Jacobi ellipticfunctions: the spiral is periodic,

r(t + Tsp) = r(t) and z(t + Tsp) = z(t)

with period

Tsp = 4 K(k)/n

whereK(k) denotes the complete elliptic integral of the first kind.Note that for k = 1 it is

limk→1

K(k) = ∞ =⇒ limk→1

Tsp = ∞

so it takes an infinite time to complete one revolution. A revolu-tion in this case is considered to be complete when r and z takethe same value again.

The velocity on the spiral can be obtained by derivingEqs. (G.1–G.3) with respect to time and yields

r = +n cn(u, k) dn(u, k) (G.4)θ = kn (G.5)z = −n sn(u, k) dn(u, k) (G.6)

The angular velocity is simply kn. The initial position relates tothe parameters of the spiral n, ϕ0, k, θr through

r0 = sn(ϕ0, k)θ0 = θr + kϕ0

z0 = cn(ϕ0, k)

and the initial velocity in cylindrical coordinates reads

r0 = +n cn(ϕ0, k) dn(ϕ0, k)θ0 = kn

z0 = −n sn(ϕ0, k) dn(ϕ0, k)

If the parameters n, k, ϕ0, θr are fixed the resulting Seiffert spi-ral is completely determined. The initial conditions must satisfythe previous relations. On the contrary, if the problem is initial-ized using the initial state vector the set of parameters defining thespiral need to be computed.

G.1.1 The accelerated two-body problem

The dynamics of a particle orbiting a central body and subject toa perturbing acceleration ap abide by

d2rdt2 +

µ

R3 r = ap

In what follows the gravitational parameter µ is normalized tounity. Consider that the trajectory of the particle describes a Seif-fert spherical spiral, so itsmotion is constrained to the surface of asphere of unit radius, ||r|| = 1. The circular velocity on the spherebecomes unity with this normalization.

Under these assumptions the required acceleration for theparticle to follow a Seiffert spherical spiral takes the form:

ap = ap,r ur + ap,θ uθ + ap,z uz

and the components in frameL are

ap,r = sn(u, k)1 − n2[1 + 2k2 cn2(u, k)]

(G.7)

ap,θ = 2n2k cn(u, k) dn(u, k) (G.8)

ap,z = cn(u, k)1 − n2[1 − 2k2 sn2(u, k)]

(G.9)

The magnitude of the acceleration due to the thrust reads

ap =

√1 − 2n2 + n4[1 + 4k2 cn2(u, k)

]It only depends on the time by means of the term cn2(u, k). Themaximum and minimum accelerations are, respectively

ap,max =√

1 − 2n2 + n4(1 + 4k2)

ap,min = |n2 − 1|

The maxima and minima of the acceleration occur at

tap,max =1n[2 j K(k) − ϕ0

](G.10)

tap,min =1n[(2 j − 1) K(k) − ϕ0

](G.11)

§G.1 Dynamics 209

Here j is an integer that accounts for the periodicity of the func-tion. The maximum acceleration is reached at the pole of thesphere.

When n = 1 the minimum acceleration vanishes. The min-imum acceleration is obtained for cn(u, k) = 0, which corre-sponds to the point where the trajectory crosses the equator (r =1 and z = 0). In the limit case k = 1 the trajectory evolves fromthe north pole of the sphere to the equator following an asymp-totic trajectory. Given the limit limk→1 K(k) = ∞, the time toreach the maximum acceleration, defined in Eq. (G.10), will onlybe finite for j = 0. The minimum acceleration is reached in infi-nite timewhenpropagated both forwards andbackwards in time.

For k = 0 the acceleration reduces to

ap,r = (1 − n2) sin u

ap,θ = 0

ap,z = (1 − n2) cos u

and it vanishes for n = 1. This means that Seiffert’s spiral withk = 0 and n = 1 is in fact a Keplerian orbit, as it will be discussedin more detail in the following section. Introducing a positiveparameter δ to quantify how much n separates from unity,

n = 1 ± δ

it follows that the low-thrust scenario corresponds to k ≪ 1 andδ ≪ 1. The magnitude of the thrust in the tangential directionwill be small compared to the radial acceleration. In addition, adirect consequence of k ≪ 1 is that the angular velocity on thespiral will be small, because θ = kn.

G.1.2 Integrals of motion

The Keplerian energy of the system Ek evolves in time due to thepower performed by the disturbing acceleration ap,

dEk

d t= ap · v

Considering the components of the velocity and the accelerationap, given respectively in Eqs. (G.4–G.6) and (G.7–G.9), it is

dEk

d t= 0

so the Keplerian energy of the system is conserved. Moreover,the velocity on the spiral can be solved from Eqs. (G.4–G.6) andreduces to

v = n

provided that the angular velocity is constant and the motion isconstrained to a spherical surface. In the definition of the Keple-rian energy

Ek =v2

2− 1

r

both terms are constant. So it is the semimajor axis,

a =1

2 − n2

The acceleration due to the thrust can be rewritten in termsof the components of the motion as

ap,r(r) = r1 + n2[2k2(r2 − 1) − 1]

(G.12)

ap,θ(θ) = 2n2k cn(θ − θr

k, k

)dn

(θ − θr

k, k

)(G.13)

ap,z(z) = z1 + n2[2k2(1 − z2) − 1]

(G.14)

The components of the acceleration derive from a perturbing po-tential

ap = −∇Vp

This shows that the problem is conservative. The disturbing po-tential Vp is separable,

Vp(r, θ, z) = Vr(r) + Vθ(θ) + Vz(z)

meaning that*

ap,r = −∂Vr

∂ r, ap,θ = −ns

(θ − θr

k, k

)∂Vθ

∂ θ, ap,z = −

∂Vz

∂ z

These three equations of separate variables can be integrated toprovide

Vr(r) = − r2

2

1 − n2

[1 − k2(r2 − 2)

]Vθ(θ) = −n2k2 sn2

(θ − θr

k

)Vz(z) = − z2

2

1 − n2

[1 + k2(z2 − 2)

]the total energy

E = Ek + Vp =12

(k2n2 − 1) (G.15)

is conserved and depends only on the angular velocity kn.

G.1.3 The osculating orbit

The angular momentum vector is solved from its definition h =r × v and can be projected onto the rotating frameL as

h = nk sn(u, k)[sn(u, k) uz − cn(u, k) ur] + dn(u, k) uθ

Its magnitude is simply

h = n

which is constant. The equation of the angular momentum con-stitutes a first integral of the motion.

The eccentricity vector takes the form

e = |n2 − 1| [ sn(u, k) ur + cn(u, k) uz]= |n2 − 1| r

*In order to find the equation for ap,θ we transform Eq. (G.1) by means ofGlaisher’s rule:1r=

1sn(u, k)

= ns(u, k)

and noting that u = (θ − θr)/k.

210 G Dynamics in Seiffert’s spherical spirals

Although the direction of the eccentricity vector changes in timeits magnitude,

e = |n2 − 1|

remains constant, meaning that the acceleration in Eqs. (G.7–G.9) does not affect the eccentricity of the osculating orbit. Inaddition, the apse line is directed along the radius vector r. Theparticle is always at periapsis of the osculating orbit. Recall that||r|| = 1.

The thrust profile ap conserves the Keplerian (and the total)energy of the system so the osculating orbit remains unchanged.It is only rotated so that the eccentricity vector always follows thedirection of the radius vector.

The inclination of the osculating orbit canbe solved from therelation

cos i = k sn2(u, k)

It oscillates with period 4 K(k)/n and the osculating orbit be-comes polar (cos i = 0) at

tpolar =1n[2 j K(k) − ϕ0

]with j = 0, 1, 2, . . . This time coincides with the time when thespiral crosses the pole.

G.2 The geometry of Seiffert’s sphericalspirals

The equations for the trajectory have been defined in terms ofthe Jacobi elliptic functions sn(u, k) and cn(u, k). The geometryof the resulting Seiffert spiral is therefore defined by the proper-ties of such functions. Different families of solutions are founddepending on the values of the modulus k.

First note that kn < 0 defines retrograde motions (θ < 0).This problem is dynamically equivalent to the prograde case andis omitted from this analysis for convenience. In the limit casek = 0 Eq. (G.5) shows that the polar angle θ remains constant,so the trajectory degenerates into a circular (non-Keplerian) polarorbit on the spherical surface. The particle moves according to

r = sin u and z = cos u

Themotion in the circular orbit is given as an explicit function ofthe time, provided that u = nt + ϕ0. Introducing k = 0 in thedefinition of the thrust acceleration will render

ap = (1 − n2)(sin u ur + cos u uz) = (1 − n2) r (G.16)

The acceleration opposes the gravitational attraction of the cen-tral body andmodifies the velocity of the resulting orbit. In a Ke-plerian circular orbit with the same radius the velocity is normal-ized to unity, whereas for degenerate Seiffert’s spirals it is v = n.When n = 1 the acceleration in Eq. (G.16) vanishes and the ve-locity matches the circular velocity of the Keplerian orbit.

The modulus of the Jacobi elliptic functions is typically as-sumed to be k ∈ [0, 1]. For k < 1 the functions sn(u, k) and

cn(u, k) oscillate between −1 and +1. Since r ∈ [−1, 1] by thesign convention that we adopted, the spiral crosses the poles. Inaddition, the fact that z changes its sign means that the particlecan travel from the northern hemisphere to the southern hemi-sphere, and vice-versa. The angular velocity takes nonzero valuesand the polar angle evolves linearly in time.

When the modulus becomes unity the Jacobi elliptic func-tions transform into:

r = sn(u, 1) ≡ tanh u and z = cn(u, 1) ≡ sechu

At u = 0 it is r = 0 and z = 1, which corresponds to the northpole of the sphere. But for t → ±∞ (with negative timesmeaningbackward propagation) it is

limt→±∞

r = ±1 and limt→±∞

z = 0

The spiral reaches the pole once but it then approaches the equa-tor asymptotically. The pole is reached at t = −ϕ0/n.

If the modulus lies in the interval k > 1 the Jacobi ellipticfunctions defining the trajectory transform into

sn(u, k) =1ksn(ku, 1/k)

cn(u, k) = dn(ku, 1/k)

The out-of-plane component of the motion is now defined bythe function dn(u, 1/k), with 1/k < 1. This function does notchange its sign. The resulting spiral is confined to onehemisphereand it cannot cross the equator.

It is remarkable that the two-body problem subject to theperturbing acceleration in Eqs. (G.7–G.9) admits an explicit so-lution in terms of the time. This is not the case even in the non-circular Kepler problem, where the time dependency is given bythe solution to Kepler’s equation. Tanguay (1960) and Petropou-los et al. (1999) found similar behaviors for special cases arisingfrom the logarithmic spiral.

In the previous section it was shown that the accelerationwillbe small for k ≪ 1 and δ ≪ 1, with n = 1 ± δ and δ > 0. Thecondition n ∼ 1 is equivalent to the velocity on Seiffert’s spiralbeing close to the local circular velocity. Since k < 1 the particlewill travel from the north pole to the south pole and back, andk ≪ 1 makes the angular velocity small.

G.3 Groundtracks

Let (λ, ψ) denote the latitude and longitude of the sub-satellitepoint on the surface of the central body, and let ω0 denote theangular velocity for the rotational motion of the central body.

Consider that θ at t = 0 defines the polar angle between theparticle and the prime meridian at departure, i.e.

λ0 = θr + kϕ0

The longitude is then given by

λ(t) = λ0 + (kn − ω0)t (G.17)

The latitude of the sub-satellite point is solved from

ψ(t) = arctanzr= arctan

[cn(u, k)sn(u, k)

]= arctan

[cs(u, k)

](G.18)

§G.4 Relative motion between Seiffert’s spirals 211

The last simplification follows from Glaisher’s notation for theJacobi elliptic functions (see Appendix E).

Equation (G.17) shows thatwhen kn = ω0 the angular veloc-ity of the spiral matches that of the central body, so the longituderemains constant and equal to λ0. The latitude vanishes for

u = 2 j K(k) =⇒ t =1n[2 j K(k) − ϕ0

]meaning that the osculating orbit is a polar orbit precisely whentheparticle crosses the equator. In the limit case k = 1 the latitudegiven in Eq. (G.18) transforms into

ψ(t) = arctan(csch u)

This expression shows that the latitude becomes ψ = π/2 at t =0, and the orbit becomes equatorial for t → ±∞. This behaviorwas described in the previous section when studying the natureof the solutions. For k > 1 the latitude is

ψ(t) = arctan[k ds(ku, 1/k)

]Provided that ds(ku, 1/k) > 0 ⇔ ku > 0, the motion is con-strained to the northern hemisphere.

Seiffert’s spirals can be used to design missions that observethe same longitude. Canceling the secular term in Eq. (G.17) willmake λ(t) = λ0. For this, the parameter k must satisfy

k = ω0/n

In order to control themaximumvalue of the accelerationwe findthat

ap,max =√

1 − 2n2 + n4[1 + 4(ω0/n)2]

is minimum for

nopt =

√1 − 2ω2

0

This is the optimal value of the orbital velocity, provided that itminimizes the thrust acceleration. In order to ensure that k < 1it must be nopt < ω0. This condition yields a maximum heightover the surface of the central body above which it is not possibleto obtain constant groundtracks.

Figure G.3 shows a hypothetical example of an observationmission around Enceladus. Porco et al. (2006) found that itssouth pole is active. This region exhibits the highest tempera-ture of the ice crust, and there are jets of icy particles emanatingfrom the moon. The plume feeds Saturn’s E-ring and, in Octo-ber 28, 2015, the Cassini spacecraft passed through the geyser inorder to analyze its composition. If an orbiter were placed in apolar orbit around Enceladus, it would go through the plume re-peatedly and throw light on the mysterious origin and composi-tion of the moon. Seiffert’s spirals provide a very simple strategyfor designing polar orbits and controling their groundtrack. Fig-ure G.3 shows a possible orbit, in which the probe is constantlyobserving the same parallel. This technique maximizes the ob-servational time of important features in the moon’s south pole.This kind of mission is also suitable for the use of cubesats de-ployed from the carrier, which serve as a communication relay fortransmitting data back to Earth.

South

to

North

North

to

South

Figure G.3: Groundtracks on the surface of Enceladus.

The spiral is polar, and the altitudeof theorbit is 148km. Thesouth-pole coverage of a 2.23-day orbit can be seen in Fig. G.4.

Figure G.4: Coverage of Enceladus' south pole using a Seiffert spiral.

The required control acceleration is shown in Fig. G.5. Thedownside of this particular design is the high acceleration levelsrequired for sustaining the spiral trajectory, although it serves asan illustrative example of the geometry of the spirals. The propel-lant mass fraction is approximately 6% of the initial mass.

0 0.5 1 1.5 2 2.50

5

10

15

Time [d]

Accele

ration [m

m/s

2]

Figure G.5: Control acceleration in Seiffert's spherical spiral.

G.4 Relativemotion between Seiffert’s spi-rals

The simple formof the solution to themotion along Seiffert’s spi-rals invites to model the relative dynamics of spacecraft followingtwo different spirals. Let δr denote the relative position of thefollower spacecraft with respect to the leader spacecraft, both de-scribing a Seiffert spherical spiral. Assuming that the relative sep-aration is small the first order solution to relative motion reduces

212 G Dynamics in Seiffert’s spherical spirals

to:

δr = δr ur + rδθ uθ + δz uz

δr = (δr − rθδθ) ur + (rδθ + δrθ + rδθ) uθ + δz uz

The orbit of the follower spacecraft can be constructed by consid-ering that the parameters defining the orbit of the follower space-craft read n + δn, k + δk, ϕ0 + δϕ0, θr + δθr, and the radius ofthe spiral is 1+ δR. The set of differences δ are small to complywith the hypothesis of linear relative motion.

The components of the relative position vector take the form

δr = rδR − δkk′2

[ rnkΛ−(u, k) − z2kr

]+

rn

(tδn + δϕ0)

δθ = δθr + u δk + k(t δn + δϕ0)

δz = zδR − δkk′2

[ znkΛ−(u, k) + zkr2

]+

zn

(t δn + δϕ0)

We have introduced the auxiliary function

Λ±(u, k) = Z(u, k) + u

[E(k)K(k)

± k′2], k′2 = 1 − k2

defined in terms of Jacobi’s zeta function Z(u, k) (see Ap-pendix E). It is worth noticing that there are secular terms, grow-ing linearly with t or u. The condition for canceling these termsand therefore defining closed orbits is

δk = δn = 0

Equation (G.15) shows that the previous condition translates intoboth orbits sharing the same value of the total energy. This isequivalent in practice to the well-known energy matching condi-tion found in the unperturbed case (Alfriend et al., 2009, pp. 59–60).

The relative velocity is solved from

δr = r δR + An δn + Ak δk + Aϕ0 δϕ0

δθ = n δk + k δn

δz = z δR + Bn δn + Bk δk + Bϕ0 δϕ0

given the coefficients

An =rn− rt

nz2 (r2 + k2n2z4)

Ak = −2kr2r

k′2+

r(r2 + k2n2z4)nz2k′2k

Λ−(u, k)

Aϕ0 = −r

nz2 (r2 + k2n2z4)

Bn =zn− zt

nz(z2 − k2n2r4)

Bk =kz(1 − 2r2)

k′2+

z(z2 − k2n2r4)nr2kk′2

Λ+(u, k)

Bϕ0 = −z

nr2 (z2 − k2n2r4)

The secular terms appear in the coefficients corresponding to δnand δk. The energy matching condition ensures that the velocityis bounded too.

G.5 The significance of Seiffert’s spirals

The orbital dynamics in Seiffert’s spherical spirals are endowedwith interesting properties. First, the problem is conservative.The perturbation that is required in order to describe this partic-ular curve derives from an ordinary potential. Second, the orbitalvelocity is constant. This makes Seiffert’s spirals similar to Keple-rian circular orbits. In fact, the spirals will degenerate into a per-fectly circular orbit under certain conditions. Third, the orbitalframe and the osculating perifocal frame coincide. This is a sur-prising property that comes from the form of the perturbation,which makes the eccentricity vector parallel to the radius vector.

Thanks to having a fully analytic solution at hand it is possi-ble to derive very simple strategies for planetary exploration. Al-though the thrust profile is not flexible andmay exceed the limitsof electric propulsion systems, the geometry and simplicity of thespirals is appealing.

List of Figures

1.1 Artistic view of the Terrestrial Planet Finder. Source: NASA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The LightSail spacecraft. Source: The Planetary Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The orbit of Voyager 2 from 08/21/1977 (launch) to 12/01/1992, represented in the ICRF/J2000 frame. . . . . . . . . 41.4 The orbit of Cassini in the ICRF/J2000 frame centered at Saturn (06/20/2004-12/01/2005). . . . . . . . . . . . . . . 5

2.1 Evolution of the propagation error of a Keplerian orbit. The orbit is integrated with a variable step-size Störmer-Cowellintegrator of order nine with a tolerance of 10-14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Discretization of a highly elliptical orbit (e = 0.8) using the physical and the fictitious times and the same number of points. 122.3 Error in position for the integration of the uncontrolled equations (Eq. (2.10), in black) compared to the controlled

equations (Eq. (2.11), in gray). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Geometric decomposition of Laplace’s linearization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Evolution of the propagation error of the Keplerian orbit discussed in Fig. 2.1. The black line corresponds to the inte-

gration of Eq. (2.1), and the gray line shows the accuracy of Eq. (2.14). The latter is integrated with a variable order andvariable step-size Adams scheme, setting the tolerance to 10-13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Propagation error along a Keplerian orbit in Cartesian coordinates (black) and using a set of elements (gray). . . . . . . 19

3.1 Hopf link connecting two different fibers in KS space. The Hopf fibration is visualized by means of the stereographicprojection to E3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Stereographic projection to E3 of the Hopf fibration corresponding to a set of initial positions on the three-dimensionalsphere of radius r. The black semi-torus consists of all the fibers that transform into the semi-circumference on the two-sphere on the bottom-left corner. One single fiber Fi is plotted in white, corresponding to ri. . . . . . . . . . . . . . . 30

3.3 Construction of the fundamental manifold. The mapping gt : x0 7→ x(t) denotes the integration of the trajectory fromt0 to t. Similarly, gs refers to the propagation using the fictitious time. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Solution to the Pythagorean three-body problem. The thick dots represent the initial configuration of the system. . . . 353.5 K-separation for the Pythagorean problem computed from a reference trajectory with θ = 0 and ϕ = 120 deg. . . . . . 353.6 K-separation for the four-body problem computed from a reference trajectory with θ = 90 deg and a second trajectory

with ϕ = 30 deg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.7 Two solutions to the four-body problem departing from the same fiber F0: the top figure corresponds to θ = 90 deg,

and the bottom figure has been generated with θ = 120 deg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.8 Relative change in the energy referred to its initial value, (E(t) − E0)/E0. . . . . . . . . . . . . . . . . . . . . . . . . . 36

213

214 List of Figures

3.9 K-separation for the four-body problem for different integration tolerances. The solutions for ε = 10-15 and 10-17 arecomputed in quadruple precision floating-point arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.10 Elliptic orbit in the L-plane spanned by (a, b) inU4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.11 Geometrical interpretation of the elements a and b in the L-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Geometrical definition of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Performance of the integration of Problem 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Performance of the integration of Problem 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Geometry of the three-body problem. The primary body is C , S denotes the secondary body, O denotes the orbiting

particle and B is the barycenter of the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 Evolution of the terms ζ4

3 s3 and ζ3 in Case 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.6 Function calls vs accuracy of the integration, measured as the relative error in the preservation of the Jacobi constant. The

figures on the left correspond to the direct integration, and the figures on the right show the performance of the switchingtechnique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1 Schematic view of the reference frames to be used. The true anomaly is denoted by ϑ, and ν defines the angle betweenversor iA and the radius vector, that is ν = α + ϑ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Geometrical representation of the eccentricity vector superposing the Gauss-Argand plane and the Minkowski plane. . . 585.3 Geometrical interpretation of the hyperbolic anomaly and the independent variable u on the Minkowski plane. The

focus of the hyperbola is denoted by F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Integration runtime vs. accuracy of the solution for the integration of the hyperbolic comets. . . . . . . . . . . . . . . 645.5 Time evolution of the error in position at epoch, ϵ j, for the four hyperbolic comets. . . . . . . . . . . . . . . . . . . . 665.6 Integration runtime vs. accuracy of the solution for the integration of the geocentric flybys of NEAR, Cassini and Rosseta. 675.7 Time evolution of the error in position at epoch, ϵ j, for NEAR, Cassini and Rosseta. The orbit is sampled every 40 min

for NEAR, 20 min for Cassini and, 1 h for Rosetta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1 Evolution of the energy for different degrees of the zonal gravity field. . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Groundtrack on November 13, 2015, computed with the high-fidelity force model. The green dot represents the impact

point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3 Difference between the analytic Brown-Newcomb luni-solar ephemeris and the DE431 solution. . . . . . . . . . . . . . 746.4 Distance to the Earth for a 50 year propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.5 Validation of Dromo formulation in propagating asteroid close approaches. . . . . . . . . . . . . . . . . . . . . . . . 766.6 Sequential performance diagram (SPD) for the Stiefel-Scheifele problem integrated with a RKF5(4) scheme. . . . . . . 776.7 SPD for the Stiefel-Scheifele problem integrated with LSODAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.8 Performance of Cowell’s method using different integrators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.9 SPD for Problem 2 integrated with RKF5(4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.10 SPD for Problem 2 integrated with LSODAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1 Geometrical definition of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Schematic representation of the synchronous and asynchronous solutions, together with the time delay. . . . . . . . . . 897.3 Diagram showing the construction of the solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.4 Equinoctial frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.5 Error comparison between the improvedCWsolution (“CW(⋆)”), and the exact solution to the first (“1st-Ord”), the second

(“2nd-Ord”), and the third-order (“3rd-Ord”) equations of circular relative motion. . . . . . . . . . . . . . . . . . . . . 967.6 Error in the relative position and velocity using the linear solutions (“CW-YA/Eq.”), the improved version of these formu-

lations (“CW-YA(⋆)/Eq.(⋆)”), and the solution using curvilinear coordinates (“Curv”). . . . . . . . . . . . . . . . . . . . 977.7 Propagation error for Cases 3 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.1 Relative orbit and propagation error for the circular and elliptic cases. . . . . . . . . . . . . . . . . . . . . . . . . . . 1098.2 Relative orbit and propagation error for the parabolic and hyperbolic cases. . . . . . . . . . . . . . . . . . . . . . . . 110

9.1 Geometry of the problem. The velocity vector v follows the direction of t. . . . . . . . . . . . . . . . . . . . . . . . . 1159.2 Graphical representation of theT- and C-symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179.3 Types of spirals in the parametric space (K1,K2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189.4 Families of generalized spirals: elliptic (K1 < 0), parabolic (K1 = 0) and hyperbolic (K1 > 0). . . . . . . . . . . . . . . 1189.5 A pair of C-symmetric elliptic spirals departing from (), propagated forward and backward. . . . . . . . . . . . . . . 1199.6 Pair of C-symmetric parabolic spirals, with ψ = 88 and ψ† = 92, respectively. . . . . . . . . . . . . . . . . . . . . . 1219.7 Evolution of the radius and angle ψ along hyperbolic spirals of Types I and II initially in lowering regime. . . . . . . . . 1229.8 Two C-symmetric hyperbolic spirals of Type I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.9 Pair of C-symmetric hyperbolic spirals of Type II, with their corresponding asymptotes and axes ofT-symmetry. . . . . 124

List of Figures 215

9.10 Pair of C-symmetric hyperbolic spirals in the limit case K2 = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.11 Diagram of departure points and spiral regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279.12 Hyperbolic spirals emanating from an elliptic orbit (e0 = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279.13 Spiral transfers from the Earth to Mars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289.14 Acceleration profile along the spiral transfers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

10.1 Geometry of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13310.2 Zeros of the function f (ψ1) for increasing values of K1 < 0. In this example r2/r1 = 2, θ2 − θ1 = 2π/3, and ξ = 1/2. . . 13410.3 Minimum K1 for different tranfer geometries. In this figure each curve corresponds to a different value of r2/r1. . . . . . 13510.4 Examples of pairs of conjugate spiral transfers and the minimum-energy spiral. . . . . . . . . . . . . . . . . . . . . . . 13510.5 Families of solutions for r2/r1 = 2, θ2 − θ1 = 2π/3, and fixed ξ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.6 Departure flight-direction angle as a function of the constant of the energy K1 for fixed values of ξ. Different transfer

geometries are considered, keeping constant the radii r2/r1 = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.7 Dimensionless thrust acceleration along different types of spiral transfers with ξ = 1/2. . . . . . . . . . . . . . . . . . 13710.8 Locus of velocity vectors projected on skewed axes for ξ = 1/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.9 Dimensionless time of flight parameterized in terms of the constant of the energy, K1. . . . . . . . . . . . . . . . . . . 13810.10 Family of solutions to the spiral Lambert problem parameterized in terms of the control parameter ξ. . . . . . . . . . . 13910.11 Examples of m :n repetitive transfers for r2/r1 = 2 and θ2 − θ1 = 2π/3. . . . . . . . . . . . . . . . . . . . . . . . . . 13910.12 Dimensionless time of flight for different m :n repetitive configurations. The size of the markers scales with the values of

the index n > m. Diamonds represent the time to the first pass, and circles correspond to the second pass. . . . . . . . . 14010.13 Comparison of the ballistic and accelerated pork-chop plots for the Earth to Mars transfers. . . . . . . . . . . . . . . . 141

11.1 Examples of bitangent transfers with 0 and 1 revolutions. The marks the transition point, fixed to θA = 100 deg. Thespirals depart from and arrive to ×. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

11.2 Fraction of mass delivered to the final circular orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14711.3 Time of flight for Earth to Mars bitangent transfers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14711.4 Evolution of K2 and ψ on a Keplerian orbit (e = 0.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14811.5 Examples of transfers between arbitrary orbits. The departure point is , the points A and B are denoted by , and the

arrival point is ×. The gray lines represent the departure and arrival orbits. The coast arcs are plotted using dashed lines. 14911.6 Compatibility conditions in terms of the control parameter for an example transfer. . . . . . . . . . . . . . . . . . . . 14911.7 Coaxial periodic orbits. The denotes the nodes where the control parameter ξ changes. . . . . . . . . . . . . . . . . . 15011.8 Construction of an interior periodic orbit with s = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15111.9 Examples of periodic orbits with j = 1, θA = 90 deg, and different number of lobes. The denotes the nodes where the

control parameter ξ changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15111.10 Examples of periodic orbits for θA = 20 deg and different combinations of s and j. . . . . . . . . . . . . . . . . . . . 15111.11 Examples of bitangent transfers with N = 7. The nodes are denoted by •, is the departure point and× is the arrival point. 15211.12 Examples of multinode transfers between two arbitrary state vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15211.13 Geometry of a 3D generalized logarithmic spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15211.14 The Frenet-Serret frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15311.15 Helicoidal trajectories depending on the type of base spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15411.16 Earth-Mars-Ceres spiral transfer. Solid lines correspond to the thrust arcs, whereas dashed lines are coast arcs. The black

dots • represent the switch points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15711.17 Thrust profile for the Earth-Mars-Ceres transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15811.18 Thrust angle compared to the flight-direction angle ψ. When φ = ψ the thrust vector is directed along the velocity. . . . 15811.19 Projection of the out-of-plane component of the motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15811.21 Visualization of the transfer in Cosmographia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15911.20 Transversal component of the thrust. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

12.1 Examples of generalized cardioids (γ = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16712.2 Examples of generalized sinusoidal spirals (γ = 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17112.3 Schwarzschild geodesic (solid line) compared to generalized cardioids and generalized sinusoidal spirals (dashed line). . . 174

C.1 Analysis of the growth-rate of the terms S ik(z). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

C.2 Error in computing the Stumpff function C0(z) from the definition by series, from the recurrence relation starting atC4(z), and from the recurrence relation starting at C10(z). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

G.1 Geometry of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208G.2 Seiffert’s spherical spiral trajectories around Mars defined by different values of the modulus k. . . . . . . . . . . . . . 208G.3 Groundtracks on the surface of Enceladus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211G.4 Coverage of Enceladus’ south pole using a Seiffert spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

216 List of Figures

G.5 Control acceleration in Seiffert’s spherical spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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Index

absolute convergence, 189action integral, 14al-

Jabr, 69Khwārizmī, 69

algebraexterior, 41Grassman exterior, 40Lie, 162normed, 27

anomalyflyby, 4generalized, 13ideal, 45intermediate, 12, 13partial inferior and superior, 12Pioneer, 4projective, 13

Archimedean spiral, 114Asteroid Redirect Mission, 1asynchronous relative motion, 88

basin of attraction, 31Baumgarte, Joachim, 10bead model, 43Bernoulli, Jakob, 197BG14, 188Biermann, Wilhelm, 199bilinear relation, 28, 37binary trees, 27Birkhoff, George David, 17bivectors, 41blade, 40

Brown-Newcomb ephemeris, 74Bulirsch-Stoer integrator, 34Burdet

-Ferrándiz transformation, 26Claude Alain, 17, 26

Callisto, 9, 98canonical, 20, 48, 162cardioid, 161, 167, 170Cassini mission, 5, 63, 69Cayley

-Dickson construction, 27sextics, 170

central method, 12Ceres, 70chaos, 5, 33, 178Clairaut, Alexis Claude, 19Clifford product, 41Comet Encke, 12Commercial Resupply Services, 1complex numbers, 27, 184conjugate spirals, 135Cosmographia, 159Cowell

Phillip Herbert, 9propagation method, 47, 63, 72, 110see also Störmer-Cowell integrator

Crommelin, Andrew Claude de la Cherois, 9cubesats, 2

d’Alembert, Jean le Rond, 19Darboux

-Sundman transformation, 11

229

230 INDEX

vector, 44Delaunay

Charles-Eugène, 19elements, 19, 162

Deprit, André, 10differential eccentricity and inclination vector, 85Dione, 5Diophantus of Alexandria, 26direct imaging, 3distant retrograde orbit, 98DIVA, 79Draper Laboratory, 44Dromo, 43, 63, 73, 102, 193

Earth Orientation Parameters (EOP), 47, 65, 70eight square theorem, 27Einstein

Albert, 55field equations, 173

element, definition, 18elliptic coordinates, 16elliptic integral, 114, 120, 162, 165, 197Enceladus, 5, 69, 211energy

characteristic, 140conservation, 35conservation law, 20generalized, 116, 132, 146, 164Keplerian, 31, 163, 189, 209kinetic, 13matching condition, 86, 212total, 13, 15, 36, 38, 209

epicycloid, 170equinoctial elements, 19, 91, 114, 193Euclidean transformation, 117Euler

angles, 19Leonhard, 22, 26, 131, 197polynomial identity, 27problem of three bodies, 17theorem, 46

Europamission, 3moon, 9, 98

exponential sinusoid, 113, 132

Fangano, Giulio, 197Fermat, Pierre, 197fiber bundle, 27fictitious time, 11, 48, 59, 106first integral, 116, 133, 161focal method, 12Frenet-Serret frame, 152fundamental theorem of curves, 117

Galilean moons, 9Ganymede, 9, 98gauge, 164

freedom, 39

generalized Dromo elements, 49generalized KS elements, 36transformation, 28

GaussKarl Friedrich, 19, 131, 199planetary equations, 18, 61

Gauss-Argand plane, 57, 184general relativity, 55generalized eccentric anomaly, 37genetic algorithm, 157, 158GGM03S, 47, 65, 70, 97Glaisher’s notation, 168, 199global optimization, 158Goddard Trajectory Determination System, 14Goldbach, Christian, 26Gram-Schmidt procedure, 40Grand Tour, 4Grassman

exterior algebra, 40exterior product, 40

gravitational scattering, 35, 55gravity-assist maneuver, 4, 56, 132, 145Gudermann

Christoph, 199function, 57

Halley’s comet, 9Hamilton, Sir William Rowan, 27, 101Hansen

ideal coordinates, 44ideal frame, 19, 44, 53, 56lunar theory, 19Peter Andreas, 9, 44

Heggie-Mikkola’s method, 34helix, 154van Heuraet, Hendrik, 197Hohmann transfers, 146Hopf

fibration, 27, 28, 105, 195Heinz, 25, 27link, 28

Hurwitz, Adolf, 27Hygiea, 70hyperbolic

anomaly, 58number, 56, 184rotation, 57

hypercomplex number, 56, 183hyperelliptic integrals, 165

Iapetus, 5ideal, seeHansen ideal frame and coordinatesimpulsive maneuvers, 140∞ symbol, 197InSight mission, 3involution, 41Io, 9, 98isochronous correspondence, 11, 31isolating integral, 162

INDEX 231

Jacchia 70 and 77, 70Jacobi

amplitude, 199elliptic function, 162, 167, 199elliptic functions, 199Gustav Jakob, 199zeta function, 199

James Webb Space Telescope, 2

K-separation, 33Kepler space telescope, 3Kowalevski

exponents, 161Sophie, 16

Kriz, Jiri, 26KS

elements, 36inverse transformation, 29matrix, 27, 28orthogonal base, 29, 40space, 105transformation, 28, 53, 63, 76, 105variables, 27

Kustaanheimo, Paul, 25

Lagrangebrackets, 19, 21, 39, 48constraint, 22Joseph Louis, 19planetary equations, 18

LambertJohann H., 131problem, 6, 113, 131, 140, 178

LancretMichel Ange, 155theorem, 155

Laplace, Pierre Simon, 19launch window, 140Legendre, Adrien-Marie, 197lemniscate, 170, 197Lense-Thirring, 65, 70Levenberg-Marquardt, 98, 140Levi-Civita

mapping, 37plane, 37Tullio, 9, 26, 55variables, 16, 26

limaçon, 167Liouville integrability, 161Lorentz

boost, 59group, 58, 59plane, 56

LSODE, 72Lyapunov

exponent, 34indicator, 5, 178second method, 161stability, 11, 23, 30–32, 55, 88, 99

Maclaurin, Colin, 170manifold

fundamental, 30, 32stable and unstable, 11

MarCO, 3Melotte, Philibert Jacques, 9metric, 32

signature, 58, 185tensor, 58see also Schwarzschild

Milankovitch elements, 19, 79minimum-energy spiral, 135Minkowski

Hermann, 56, 85plane, 56, 57, 185space-time, 5, 43, 58, 178, 185variables, 76, 111

Mittag-Leffler, Magnus Gustaf, 11MRO95A, 70MSISE90, 47, 70, 97

NEAR mission, 63Newman, Charles M., 107Newton’s theorem of revolving orbits, 162, 174Noether

-Bessel-Hagen equations, 164Emmy, 161theorem, 164

normed algebras, 27nutation, 70, 97

ObservatoryGreenwich, 9of Paris, 73Seeberg, 12US Naval, 70

octonions, 27Oort cloud, 55Orbital ATK, 1

Painlevé, Paul, 16Pallas, 70parametric space, 27Pascal, Étienne, 167Pasiphae, 9patched spirals, 145Pauli matrices, 185PERFORM, 47, 63, 69, 97, 110, 178perihelion precession, 162Picard-Lindelöf theorem, 22planetary protection, 69Poincaré

control term, 15elements, 19Henri, 162map, 32section, 11, 32, 98stability, 11, 30–32, 88

Poisson brackets, 19, 21, 39, 48

232 INDEX

pork-chop plot, 140PPN model, 70precession, 70, 97principle of least action, 14projective

decomposition, 19space, 13

propulsion systems, 3Pythagorean three-body problem, 34

quadratic form, 26quaternions, 19, 27, 41, 43, 45, 56, 62, 102, 185

radial intermediary, 13radioisotope thermoelectric generator, 69real numbers, 27reciprocity of KS-operators, 28rectify curve, 197rectifying plane, 155Rhea, 5Rosetta mission, 63Runge-Kutta methods, 15, 47, 53, 72, 79

Saint Venant, Adhémar Jean Claude Barré, 155Saturn’s E-ring, 211Scheifele, Gerhard, 26Schwarzschild

geodesics, 6, 173metric, 173space-time, 162

sedenions, 27Seiffert spiral, 145sequential performance diagram (SPD), 70shape-based method, 113, 145similarity transformation, 133, 161, 163singularity, 10, 23, 26, 49, 75, 93, 112SOFA, 70solar flux, 47, 70solar sail, 4, 114, 128space weather, 47, 70SpaceX, 1special relativity, 56Sperling

-Burdet regularization, 53, 63, 76, 103, 188Hans, 17

SPICE, 140, 156spinors, 27, 185spirals

Cotes’s, 175generalized logarithmic, 118, 132, 166generating, 117Poinsot’s, 175sinusoidal, 161, 170

stabilityasymptotical, 32orbital, 11topological, 33, 35see also Lyapunov, see also Poincaré

Stark problem, 162

state-transition matrix, 72, 85, 93, 109stellar binary, 35stereographic projection, 30Stiefel

-Scheifele’selements, 36, 111method, 111problem, 47, 76

Eduard, 10, 25Störmer-Cowell integrator, 10, 18, 48, 72, 79Strömgren, Bengt, 19Stumpff

functions, 17, 102, 103, 189Karl, 189

SundmanKarl Frithiof, 11transformation, 11, 26, 43, 59, 70, 103

SYMBA, 34symplectic

integrator, 34map, 21matrix, 21, 49

synchronism, 87

tethers, 43time delay, 88, 91, 103, 105, 106time element, 19, 39, 111Titan, 5, 56Torricelli, Evangelista, 121transit detection, 3Tschirnhausen cubics, 170Tsien problem, 114, 162Tsiolkovsky

equation, 147Konstantin, 4

two-point boundary-value problem, 134

unified state model, 71, 188

variation of parameters, 22, 113variational equations, 72, 86, 99Vesta, 70Vitins, Michael, 26Voyager mission, 4

Waldvogel, Jörg, 11, 26Wallis, John, 197Weierstrass

elliptic functions, 118, 161, 162, 172, 199Karl, 16, 199

WT1190F, 72

Yukawa potential, 162