Libration Point Orbits and Applications: Proceedings of the Conference Aiguablava, Spain 10 - 14 June 2002

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Proceedings of the Conference

Libration Point rbits and

Applications

This page intentionally left blank

edited by

Gerard Gomez EEC - Universitat de Barcelona, Spain

Martin W. Lo Jet Propulsion Laboratory, Caltech, USA

JosepJ. Masdemont EEC - Universitat Politecnica de Catalunya, Spain

Proceedings of the Conference

Libration Point ^•^

rbits and Applications Aiguablava, Spain

10-14 June 2002

world Scientific New Jersey • London • Singapore • Hong Kong

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

LIBRATION POINT ORBITS AND APPLICATIONS F'roceedings of the Conference Copyright 0 2003 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-363-8

printed in Singapore.

PREFACE

The International Conference on Libration Point Orbits and Applications was held at the Parador d’aiguablava on the Costa Brava of Girona, Spain, on June 10th through 14th in 2002. This conference is the latest in a series of meetings under the name of “Libration Point Mission Design Workshop” that were organized at JPL in the past six years. The purpose for these earlier informal meetings was to bring a few of us together who were interested in the applications of dynamical systems theory to space mission design. This was the first formal meeting in this series we have organized on libration missions and the first to be held in Europe. Our goal for this meeting was slightly different from that of the previous meetings. We wanted to use this opportunity to provide a historical perspective, capture the stateof-the-art, and survey some of the future directions in this rapidly changing field. We hope that this resulting book of proceedings will provide a good starting point for researchers and students of this field. In these past six years, the application of tools comming from the general field of dynamical systems has gone from a mathematical curiosity in the space community to become a serious methodology for the design and operation of real space missions. In 2001, first MAP then the Genesis Mission were successfully launched. Both missions were designed using dynamical systems theory. MAP is designed and operated by NASA’s Goddard Space Flight Center; Genesis is designed and operated by NASA’s Jet Propulsion Laboratory. Moreover, future missions such as Herschel, Plank and Gaia of the European Space Agency, the Terrestrial Planet Finder, the Next Generation Space Telescope, the Lunar Sample Return Missions, and many others, are all using dynamical systems concepts for their mission design. These examples provide a measure of the success of dynamical systems applications as a mission design tool within the space community.

vi

Preface

The highly successful and well attended Aiguablava Conferece being documented by this Proceedings (almost 700 pages) is another indication of the growing interest and importance of dynamical systems interacting and colaborating with traditional methods within the space community. The organization of this Conference and the result of these proceedings reflects the interdisciplinary nature of this field from pure mathematics to software engineering and from space mission design to dynamical astronomy. Although each of these topics is of interest in itself, the focus of the Conference was the application of these different elements to space mission design. In this regard, this reflects the modern trend in engineering where both theoretical and computational mathematics and scientific disciplines work together to produce new solutions to real life engineering problems. The sharp boundaries between academic disciplines established in medieval universities appear artificial in this new context. What makes binds these different elements together and make them all work together is the final software and the computer system. This suggests that we may need to reexamine the organization of our institutions and their operational philosophies that they support and facilitate this new and fruitful1 approach to solving engineering problems. This also underscores the necessity for active research and development in the field of mission design. The list of papers that have been submitted for these proceedings have been organized in sections as they appeared in the conference. The fist section deals with Mission Analysis and Operations, the second one is devoted t o Dynamics around the Libration Points, the third one t o Software Tools and the last one to Solar System Dynamics and Applications. We thank all the authors the time they spent and the care they had in the preparation and submission of the papers, making possible this book. We wish also to thank the European Space Agency -and very specially Dr. RodriguezCanabal from ESOC- the Spanish Ministery of Science and Technology, the Catalan goverment, the Universities of Barcelona and Polytechnical of Catalonia, and the Terrestrial Planet Finder Project and the Navigator Program at JPL for helping and providing the funding for this conference. Finally, we thank also each of the participants of this conference whose contributions and enthusiasm made this an informative, exciting, and delightful conference. Gerard G6mez Martin Lo Josep Masdemont

LIST OF PARTICIPANTS

Miquel Angel Andreu

Esther Barrab6s

[email protected]

[email protected]

Dpt. Matematica Aplicada i Anhlisi Universitat de Barcelona Av. Gran Via 585, 08007 Barcelona, Spain

Dpt. Informitica i Matematica Aplicadi Universitat de Girona Campus Montilivi, Edifici P4, 17071 Girona, Spain

Juan Carlos Bastante

Mark Beckman

[email protected]

[email protected]

DEIMOS SPACE SL Sector Oficios 34, 1, 28760 Tres Cantos, Madrid, Spain

Goddard Space Flight Center Code 572, Greenbelt, MD20771, USA

Miguel Bell6-Mora

Antonio F. Bertachini

[email protected]

[email protected]

DEIMOS SPACE SL Sector Oficios 34, 1, 28760 Tres Cantos, Madrid, Spain

INPE-DMC Av. dos Astronautas 1758, Sao Jose dos Campos-SP 12227-10, Bra

Andrew D. Burbanks

Elisabet Canalias

[email protected]

elisabet.cana1iasB.upc.e.s

School of Mathematics University of Bristol University Walk, Bristol BS8 l T W , United Kingdom

Facultat de Matematiques Universitat Politixnica de Catalunya Diagonal 647, 08028 Barcelona, Spain

John Carrico

Jordi Cobos

[email protected]

jordi.cobosBesa.int

Analytical Graphics, Inc. 40 General Warren Blvd. Malvern, PA 19355, USA

ESA/ESOC Robert-Bosch Str. 5, 64293 Darmstadt, Germany

vii

...

viii

Last of Participants

Gerald L. Condon

Iharka Csillik

[email protected]

[email protected]

NASA Johnson Space Center 2101 NASA Road One / EG5, Houston, Texas, TX 77058, USA

Astronomical Observatory 3400 Cluj-Napoca, str. Ciresilorp. 19, Romania

Donald J. Dichmann

Eusebius Doedel

[email protected]

[email protected]

Astrodynamics Consultant 20821 Amie Ave #120, Torrance, CA 90503, USA

Department of Computer Science Concordia University 1455 blrd. de Maisonneuve West, Montreal, Quebec H3GlM8, Canada

David Dunham

Pere Durbh

[email protected]

pdurba(Qindra.es

Applied Physics Laboratory Johns Hopkins University Mail Stop 2-155, 11100 Johns Hopkins Road, USA

Indra Espacio S.A Avd. Diagonal 218-188, 08018 Barcelona, Spain

Natan Eismont

Emmet Fletcher

[email protected]

[email protected]

Space Research Institute 117997, Profsoyuznaya street 84/32, Moscow, Russia

Analytical Graphics, Inc Paseo de la Castellana 141, 8 planta, 28046 Madrid, Spain

F'rederic Gabern

Iwona Gacka

[email protected]

gackaQastro.uni.wroc.pl

Dpt. Matemgtica Aplicada i Analisi Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain

Instytut Astronomiczny Uniwersytetu Wroclawskiego ul. Kopernika 11, 51-622 Wroclaw, Poland

Gerard Gbmez

Martin Hechler

[email protected]

[email protected]

Dpt. Matematica Aplicada i Analisi Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain

European Space Operations Centre Robert-Bosch-Str. 5, 64293 Darmstadt, Germany

Jacques Henrard

Martin B. Houghton

[email protected]

[email protected]

Departement de Mathematique FUNDP 8, Rempart de la Vierge, B.5000, Namur, Belgique

NASA Goddard Space Flight Center 571-Bldg. 11/Rm. E109, Greenbelt, MD 20771, USA

List of Participants

ix

Kathleen C. Howell houellQecn.purdue.edu School of Aeronautics and Astronautics Purdue Universityi, X1281 Grissom Hall, West Lafayette, IN 47907, USA

Charles Jaffh cjarf eQuvu.edu Department of Chemistry West Virginia University Morgantown WV 26506, USA

Angel Jorba [email protected] Dpt. Matemitica Aplicada i Anilisi Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain

Justyna Kaczmarek

Jean A. Kechichian

Wang-Sang Koon [email protected] Control and Dynamical Systems California Institute of Technology MC 107-81, Pasadena, CA 91125, USA

Jean.A.KechichianQaero.org

The Aerospace Corporation MS M4/947 P.O. Box 92957, Los Angeles, California 90009, USA

[email protected] Obserwatorium Astronomiczne UAM ul. Sloneczna 36, 60-286 Poznan, Poland

Martin W. L o mulQjpl.nasa.gov MS 301/142 Jet Propulsion Laboratory 4800 Oak Grove Drive, Pasadena, CA 91109-8099, USA

Manuel Marcote [email protected] Dpt. Matemitica Aplicada i Anilisi Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain

Regina Martinez [email protected] Department de Matemitiques Universitat Autbnoma de Barcelona 08193 Bellatena, Barcelona, Spain

Josep J. Masdemont josspQbarquins.upc.es Dpt. Matemitica Aplicada I, ETSEIB Universitat Politknica de Catalunya Diagonal 647, 08028 Barcelona, Spain

Jose Maria Mondelo [email protected] Dpt. Matemitica Aplicada I, ETSEIB Universitat Polit&cnica de Catalunya Diagonal 647, 08028 Barcelona, Spain

Cesar Ocampo Dpt. Aerospace Engineering The University of Texas at Austin Room 412B, Mail Code C0600, Austin, TX 78712-1085 USA

Estrella Olmedo [email protected] Dpt. Matemitica Aplicada i Anilisi Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain

Nadege Pie nadegepieQhotmail.com Univesity of Texas at Austin 63 rue de l'Abondance, 690003 Lyon, France

[email protected]

x

List of Participants

[email protected] Goddard Space Flight Center Code 453.2, Greenbelt, Maryland 20771, USA

Jose Rodriguez-Canabal [email protected] European Space Operations Centre Robert-Bosch-Str. 5, P.O. Box 406, 64293 Darmstadt, Germany

Merch Romero

Aexey E. Rosaev

[email protected] Facultat de MatemBtiques Universitat Polithcnica de Catalunya Diagonal 647, 08028 Barcelona, Spain

rosaev@nedra. ru FGUP NPC NEDRA Svobody, 8/38, Yaroslavl, 150000, Russia

Craig E. Roberts

Shane Ross

Anna Samh

[email protected] Control and Dynamical Systems California Institute of Technology MC 107-81, Pasadena, CA 91125, USA

[email protected] Departament de Matemhtiques Universitat Autbnoma de Barcelona 8193 Bellaterra, Barcelona, Spain

Daniel J. Scheeres

Carles Sim6

[email protected] Dpt . of Aerospace Engineering The University of Michigan 1320 Beal Ave., 3048 FXB Building, Ann Arbor, MI 48109-2140, USA

[email protected] Dpt. Matemhtica Aplicada i AnBlisi Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain

Charalampos Skokos

Alexander Sukhanov

[email protected] Department of Mathematics (CRANS) University of Patras, GR-26500, Patras, Greece Research Center for Astronomy Academy of Athens, GR-10673, Athens, Greece

[email protected] Space Research Institute (IKI) 117997, 84/32 Profsoyuznaya Str, MOSCOW,Russia

Natalia N. Titova

Robert Tolson

[email protected] A.A. Dorodnitsyn Computing Center Russian Academy of Sciences Vavilov Str. 40, 117967 Moscow, Russia

[email protected] School of Engineering and Applied Sciei George Washington University 6529 Koula Drive, Diamondhead, MS 39525-3821, USA

List of Participants

Turgay User tuzercPgonzo.physics.gatech.edu School of Physics Georgia Institute of Technology Atlanta, Georgia 30332-0430, USA

Roby S. Wilson roby.wilsoncPjpl.nasa.gov NASA - Jet Propulsion Laboratory 4800 Oak Grove Drive, Pasadena, CA 91109-8099, USA

xi

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CONTENTS

Preface List of Participants

V

vii

SECTION I: Mission Analysis and Operations 1.

Orbit Determination Issues for Libration Point Orbits M. Beckman

2.

Performance Requirements and Kickstage Disposal Options for a Cislunar Gateway Station Transfer Vehicle G.L. Condon, S. Wilson and C.L. Ranieri

3.

Libration Point Missions, 1978-2002 D. W. Dunham and R. W. Farquhar

1

19

45

4. Technical Constraints Impact on Mission Design to the Collinear Sun-Earth Libration Points N. Ezsmont, A. Sukhanov and V. Khrapchenkov 5.

Libration Orbit Mission Design: Applications of Numerical and Dynamical Methods D. Folta and M. Beckman

75

85

6.

Herschel, Planck and GAIA Orbit Design M. Hechler and J. Cobos

115

7.

Getting to L1 the Hard Way: Triana’s Launch Options M. B. Houghton

137

...

Xlll

xiv

8.

9.

10.

Contents

Solar Surveillance Zone Population Strategies with Picosatellites Using Halo and Distant Retrograde Orbits J.A. Kechichian, E. T. Campbell, M.F. Werner and E. Y. Robinson

153

The SOH0 Mission L1 Halo Orbit Recovery from the Attitude Control Anomalies of 1998 C.E. Roberts

171

Possible Orbits for the First Russian/Brazilian Space Mission A . A . Sukhanov

219

SECTION 11: Dynamics Around the Libration Points 11.

12.

13.

New Results on Computation of Translunar Halo Orbits of the Real Earth-Moon System M.A. Andreu

225

Impulsive Transfers to/from the Lagrangian Points in the Earth-Sun System A . F. Bertachini de Almeida Prado

239

Astrodynamical Applications of Invariant Manifolds Associated with Collinear Lissajous Libration Orbits J. Cobos and J. J. Masdemont

253

269

14.

Halo Orbits in the Sun-Mars System I. Gacka

15.

Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits G. Gdmez, M, Marcote and J. J. Masdemont

287

Libration Point Orbits: A Survey from the Dynamical Point of View G. Gdmez, J. J. Masdemont and J.M. Mondelo

311

16.

Contents xv

17.

Dynamical Substitutes of the Libration Points for Simplified Solar System Models G. Gdmez, J.J. Masdemont and J.M. Mondelo

18.

Navigation of Spacecraft in Unstable Orbital Environments D. J. Scheeres

19.

Low Thrust Transfer to Sun-Earth L1 and Constraint on the Thrust Direction A . A . Sukhanov and N . A . Eismont

L2

373

399

Points with a 439

SECTION 111: Software Tools 20.

Satellites Formation Transfer to Libration Points J. C. Bastante, L. Pefiin, A . Caramagno, M. Bello-Mora and J. Rodn’guez- Canabal

21.

Software Architecture and Use of Satellite Tool Kit’s Astrogator Module for Libration Point Orbit Missions J. C a m t o and E. Fletcher

22.

23.

The Computation of Periodic Solutions of the 3-Body Problem Using the Numerical Continuation Software AUTO D. J. Dichmann, E. J. Doedel and R. C. Paffenroth An Architecture for a Generalized Spacecraft Trajectory Design and Optimization System C. Ocampo

455

471

489

529

SECTION IV: Solar System Dynamics and Applications 24.

25.

Restricted Four and Five Body Problems in the Solar System F. Gabern and A. Jorba Invariant Manifolds, the Spatial Three-Body Problem and Petit Grand Tour of Jovian Moons G. Gdmez, W.S. Koon, M . W. Lo, J.E. Marsden, J. J. Masdemont and S.D. Ross

573

587

xvi

26.

27.

28.

29.

30.

31.

Contents

Perturbing Action of the Earth’s Third-Degree Harmonics on Periodic Orbits Around Geostationary Equilibria J. Kaczmarek, I. Wytrzyszczak and I. Gacka

603

One Kind of Collision Orbits Related to Lagrangian Libration Points A.E. Rosaev

613

The Investigation of Stationary Points in Central Configuration Dynamics A.E. Rosaev

623

Statistical Theory of Interior-Exterior Transition and Collision Probabilities S. Ross

637

Smaller Alignment Index (SALI): Determining the Ordered or Chaotic Nature of Orbits in Conservative Dynamical Systems Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N . Vrahatis

653

Locating Periodic Orbits by Topological Degree Theory C. Polymilis, G. Servizi, Ch. Skokos, G. Turchetti and M. N . Vrahatis

665

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

ORBIT DETERMINATION ISSUES FOR LIBRATION POINT ORBITS M. BECKMAN Guidance, Navigation & Control Center’s Flight Dynamics Analysis Branch (Code 572) NASA’s Goddard Space Flight Center, Greenbelt, MD 20771, USA

Libration point mission designers require knowledge of orbital accuracy for a variety of analyses including station keeping control strategies, transfer trajectory design, and formation and constellation control. Past publications have detailed orbit determination (OD) results from individual libration point missions. This paper collects both published and unpublished results from four previous libration point missions (ISEE3, SOHO, ACE and MAP) supported by Goddard Space Flight Center’s Guidance, Navigation & Control Center. The results of those missions are presented along with OD issues specific to each mission. All past missions have been limited t o ground based tracking through NASA ground sites using standard range and Doppler measurement types. Advanced technology is enabling other OD options including onboard navigation using onboard attitude sensors and the use of the Very Long Baseline Interferometry (VLBI) measurement Delta Differenced One-way Range (DDOR). Both options potentially enable missions to reduce coherent dedicated tracking passes while maintaining orbital accuracy. With the increased projected loading of the DSN, missions must find alternatives to the standard OD scenario.

1. INTRODUCTION Orbit determination for libration point orbits (LPOs) is quite unique. The regime is far from the Low Earth Orbits (LEO) typically supported by the Goddard Space Flight Center (GSFC) and far from the interplanetary orbits supported by the Jet Propulsion Laboratory (JPL). The regime offers very little dynamics thus requiring an extensive amount of time and tracking data in order to attain a solution. NASA’s GSFC has designed and supported every libration point misI

2

M. Beckman

sion except the recent Genesis mission out of JPL. This paper presents a summary of the analyses and orbit determination results of four previous libration point missions supported by GSFC’s Guidance, Navigation & Control Center (GNCC). ISEE-3 was the first libration point mission in 1978. ISEE-3 stayed in a halo orbit for only three years before departing on the next phase of its mission. It was 17 years later before a second mission, SOHO, was flown to a libration point. SOHO was the first mission designed to remain in the vicinity of a libration point for the mission duration. SOHO was quickly followed by ACE, the first mission to follow the quasi-periodic Lissajous orbit pattern. The latest mission, MAP, the first mission to the Earth-Sun L2 point, is also presented here with as yet unpublished results. The majority of these missions have used the Deep Space Network (DSN) assets to support tracking services. However, with the projected loading on the DSN in future years, this option is becoming far less feasible. Alternatives include commercial tracking assets such as the Universal Space Network (USN), which is scheduled to support the future Triana mission, and the use of advanced technology to reduce the required tracking services from the DSN. One such technology is the use of Celestial Navigator (CelNav). CelNav is an onboard Kalman filter that processes one-way forward Doppler measurements and onboard attitude sensor data. CelNav analysis results are presented here. Another alternative is the use of the Very Long Baseline Interferometry (VLBI) measurement called Delta Differenced One-way Range (DDOR). DDOR is actually an angular measurement from a nearby quasar to the spacecraft. DDOR is being implemented at some of the DSN sites and its applicability to LPOs is presented here. 2. PAST MISSIONS

2.1. International Sun-Earth Explorer-3 (ISEE-3) The first libration point mission was ISEE-3 (Figure 1). Launched as part of an international cooperative agreement between ESA and NASA on August 12, 1978, ISEE-3 entered a large halo orbit about the Earth-Sun L1 point on November 20, 1978. The spacecraft remained in the halo orbit for 3 1 / 2 years before departing on June 10, 1982 for the second phase of its mission. ISEE-3 flew in a large halo orbit about L1 of approximately 600,000 km in Y-amplitude (in ecliptic plane perpendicular to Earth-Sun line). The

Orbit Determination Issues for Libration Point Orbits 3

Fig. 1. ISEE-3 Spacecraft

dynamics in the vicinity of the libration point are not significantly different for different size halo or Lissajous orbits. All have an approximate period of 6 months. The class, phase and Z-amplitude (out of the ecliptic) of the halo or Lissajous orbit all have an effect on the orbital accuracy. However, these effects are small and, given the number of variables affecting orbital accuracy, it is usually measured only to order-of-magnitude. ISEE-3 was ground tracked by NASA S-band Tracking Data Network (STDN) sites during the halo orbit phase. The tracking schedule was irregular but generally consisted of multiple short passes (5 minutes) at acquisition-of-signal (AOS), maximum elevation, and near loss-of-signal (LOS) from each station (Joyce '). Covariance analysis was performed pre-mission to assess the expected orbital uncertainties. Covariance analysis indicated an optimum batch tracking arc length of 21 days. Stationkeeping maneuvers were performed every 45 days enabling two completely independent orbital solutions between each maneuver. OD was performed every other week giving a 7-day overlap period. The covariance analysis was comparable to comparisons between consecutive definitive solutions obtained during the actual mission. The definitive overlap comparisons are obtained by differencing the trajectories obtained by the two overlapping solutions. Table 1 (Joyce 7, details the covariance analysis and definitive

4

M.Beckman

overlap comparisons. Table 1. ISEE3 Comparisons of Overlap Differences and Covariance Analysis. Period

Overlap Compare (km)

A B C

8.1 9.0

3.6

Covariance Analysis (km) 6.0 5.5 5.4

Corresponding velocity uncertainties were 0.3 to 2.0 cm/sec from the definitive overlap differences. Definitive overlap comparisons are not a direct measure of absolute orbital accuracy. However, without an independent tracking source, they are the best available measure. 2.2. Solar

tY Heliospheric Observatory ( S O H O )

SOHO was launched on December 2, 1995 as a joint ESA and NASA mission. SOHO performed a direct insertion into a large Earth-Sun L1 halo orbit with a Y-amplitude of approximately 670,000 km (see Figure 2).

Fig. 2. SOHO Trajectory in Solar Rotating Coordinates.

SOHO tracking is performed by the DSN, primarily the 26-m antennas, but some 34-m and 70-m MARK IVA antennas are also used. The MARK

Orbit Determinotion Issues for Libmtion Point Orbits 5

IVA SRA ranging system is generally slightly more accurate. The nominal tracking schedule for SOHO is 5 hours per day from alternating DSN sites. This schedule is extremely inconsistent for SOHO however. Covariance analysis was performed pre-mission in order to assess orbital accuracy and to determine the batch arc length. The covariance analysis used a conservative tracking schedule of only 1 hour per day. This analysis indicated that an optimum arc length of 21 days would give orbital accuracy to less than 9 km. Table 2 (Jordan ') details these results. Table 2. SOHO Covariance Analysis Results.

Data Span ( 4 14 21 14 21

Maximum Total Error 60d Pred Vel POS Vel (cm/s) (km) (cm/s) 0.79 0.42 17.4 0.34 0.19 8.4 20.1 1.05 0.39 0.06 0.26 11.3

Def Period Axis

Y Y Z Z

Pos

(W 10.1 8.6 6.6 5.7

SOHO performs station-keeping maneuvers every 8 to 12 weeks. Additionally, attitude maneuvers are performed much more frequently with the use of spacecraft thrusters. While the attitude maneuvers are designed for zero delta-V, thruster performance and misalignments contribute about a 5% error. The batch definitive arcs are broken at all maneuver points instead of attempting to model these maneuvers. Modeling would add an additional error source into the solution and would require a detailed engine model in the OD software. Data arcs were generally kept at the standard 21 days when possible, but were often shorter. As part of the solution process, the solar radiation pressure coefficient ( C p )was estimated along with range biases for each pass from the MARK IVA antennas (averaging about 6 per solution). SOHO's definitive overlap requirements were 50 km and 3 cm/sec. During long periods free of spacecraft perturbations, overlap comparisons were obtained. Actual definitive overlap comparisons average about 7 km. That uncertainty is primarily in the cross-track direction (plane-of-the-sky perpendicular to the projection of the velocity vector into that plane). Radial uncertainties are generally less than 1 km. Table 3 details the position and velocity definitive overlap comparisons. These overlap compares were obtained during long periods without spacecraft perturbations. The routine OD for SOHO was not typically this

6

M. Beckrnan

Table 3. SOHO Definitive Overlap Comparisons.

Pos (km) Vel (mmis)

RSS 7 0.4

Radial 1 0.1

Along-track 2 0.2

Cross-track 7 0.3

accurate due to the use of much shorter data arcs. The predicted orbital uncertainty requirement after a 44-day propagation is 100 km and 10 cm/sec. Definitive solutions were compared to predicted solutions after 44 days of propagation to obtain a predictive overlap comparison. The SOHO predictive overlaps were generally around 14 km. Table 4 details the predictive overlap comparisons. Note that the radial component is no longer constrained by the measurement data and grows significantly. Table 4. SOHO Predictive Overlap Comparisons.

Pos (km) Vel (mrnls)

RSS 14 0.7

Radial 9 0.2

Along-track 2 0.3

Cross-track 11 0.6

An additional study was performed for the MAP mission using real SOHO tracking data. This analysis was performed to show the effects of reducing the 5 hours per day of SOHO tracking data to only 37 minutes per day for MAP. The SOHO definitive ephemeris using all available tracking data was used as the truth ephemeris. Table 5 (Nicholson 'O)shows the comparisons for the reduced tracking data solutions. The results are somewhat erratic but generally show a degradation of accuracy of less than 2 km. Table 5. SOHO Reduced Tracking Data Results. Epoch

Editing

C,

980111 980111 980111 980321 980321 980405 980405 980417 980417

None 37 min/ day 37 min twice/day None 37 min/day None 37 min/day None 37 minldav

1.399 1.394 1.396 1.384 1.278 1.371 1.360 1.389 1.392

PosRSS (km) NA 8.31 0.71 NA 19.6 NA 0.65 NA 1.21

VelRSS (cm/sec) NA 0.236 0.081 NA 3.82 NA 0.142 NA 0.411

Orbit Determination Issues for Libmtion Point Orbits 7

2.3. Advanced Composition Ezplored (ACE) ACE was launched on August 25, 1997 as a NASA Explorer program mission. ACE performed a direct insertion into an Earth-Sun L1 small Lissajous orbit with a Y-amplitude of about 150,000 km (see Figure 3). ACE was the first spacecraft to fly in the quasi-periodic Lissajous pattern. The periodic halo orbits do not exist at the smaller amplitudes.

Fig. 3. ACE Trajectory in Solar Rotating Coordinates.

The ACE spacecraft is spin-stabilized at 5 rpm with the spin axis of the spacecraft required to point within 20 degrees of the Sun at all times. In addition, the High Gain Antenna (HGA) is required to point Earthward within 4.5 degrees. These two constraints require ACE to perform reorientation maneuvers as frequently as every 5 days. These maneuvers are performed with thrusters and therefore force the analysts to break the arc around these maneuvers to obtain clean data arcs free of spacecraft perturbations. Thus, ACE uses data arcs of 4 to 14 days, which are clearly not optimal for OD accuracy (Colombe 3). For the longer data arcs, C, and pass dependent range biases from 70-m sites are estimated. ACE gets approximately one 3.5-hour pass per day from the DSN with an additional 2 or 3 one-hour passes per week. The DSN data is primarily from the 26-m and 34-m sites. Because ACE extends the batch data arc as long as possible between attitude maneuvers, there are no definitive overlap comparisons available. Single point overlaps are obtained by differencing consecutive definitive solutions at the time of the attitude maneuvers. Those overlaps indicate a

8

M. Beckman

mean position difference of 10 km and a velocity difference of 1.2 cm/sec. Table 6 details the ACE overlap comparisons. Table 6. ACE Overlap Comparisons.

Pos (km) Vel (mm/s)

Pos (km) Vel (mm/s)

RSS Radial Along-track Definitive Point Overlap 10 4 5 0.1 1.2 0.9 2-Week Predictive Overlap 23 8 6 0.1 1.2 0.9

Cross-track 8 0.9 21 0.9

ACE never attempted to model the spacecraft maneuvers in order to obtain longer tracking arcs. Analysis was done for this particular scenario for a future libration point mission, Constellation-X. At the time, the current Constellation-X design called for momentum unloads using spacecraft thrusters every other day. Since a two-day arc was clearly not sufficient for OD, modeling of the maneuvers would be required. Covariance analysis was performed for a scenario with 4 mmlsec deltaVs applied every other day with a thruster performance uncertainty of either 3% or 5%. The delta-Vs were applied toward the Sun in hopes absorbing some of the error in the estimated C,.. The tracking schedule used was 10 minutes of range and Doppler tracking data every day from a single station with a 21-day tracking data arc. Estimating the spacecraft maneuvers was not possible due to the sheer number. The definitive OD position and velocity errors for the 3% and 5% deltaV error cases as shown in Table 7 (Marr ’). Table 7. Constellation-X Covariance Analysis Assuming Multiple Spacecraft Maneuvers. Delta-V Error 3% 5%

Pos Error (km) 12-47 16-78

Vel Error (cm/sec) 3.5-4.0 5.8-6.5

Note that the errors seen in the Constellation-X analysis are considerably highly than that seen for ACE using shorter data arcs. In addition, the larger velocity errors would require much more frequent station-keeping maneuvers, and a higher station-keeping delta-V budget, in order to maintain the Lissajous orbit.

Orbit Determination Issues for Libration Point Orbits 9

2.4. Microware Anisotropy Probe ( M A P )

MAP is the latest libration point mission. MAP was launched on June 30, 2001 and used a lunar swingby to insert into a small Lissajous orbit about the Earth-Sun La point (see Figure 4). MAP is the first mission to remain in the vicinity of the L2 point for an extended period of time.

Fig. 4.

AP Trajectory in Solar Rotating Coordinates.

Tracking services for MAP are provided by the DSN. MAP receives a minimum of one 45-minute pass per day from the DSN 34-m or 70-m sites. Because MAP receives exclusively MARK IVA tracking data, MAP has the highest quality measurement data set of any previous mission. However, MAP does not possess an equivalent quantity of measurement data than earlier missions. The MAP spacecraft is spin stabilized about an axis that precesses once per hour about a 22.5-degree half-angle cone about the Sun-MAP line. Because of the unique attitude requirements for MAP, the cross-sectional area for solar radiation pressure forces is nearly constant. This greatly reduces attitude dependent errors on solar radiation force modeling which is typically a large error source. Most missions estimate the C, but current GSFC software limits the solar radiation force calculation to a fixed cross-sectional area and a single constant estimated C, over the entire data arc. Thus, the estimated C, normally soaks up changing forces due to attitude changes and solar events. For MAP, this estimated C, is extremely consistent ( z t

10

M. Beckman

0.005) and varies only in response to solar events (Fink ‘). This improves overall OD accuracy for MAP. Because of the C, consistency, MAP is able to use longer data arcs than other missions. MAP uses a minimum of 14-day arcs after maneuvers up to a current maximum of 72 days of spacecraft unperturbed motion. In addition, because MAP receives a large amount of 70-m tracking data, they have been able to calibrate the range biases from various stations and are able to apply these biases to future solutions. This eliminates numerous parameters from the estimated state vector (Fink 4). Since OD data arcs are extended to much longer lengths for MAP, overlap differences do not exist. However, post-processed solutions using two consecutive 5-week arcs do give adequate comparisons. Over the short prediction span of 5 weeks, the overlap differences were 2.0 km and 0.83 mm/s. The overlaps increase when the prediction span is increased to 9 weeks: 6.7 km and 3.9 mm/s. Table 8 (Fink ‘) details the MAP results. These MAP results are airly optimistic as they are taken during a period of relative solar inactivity and continuous science mode operation. Results are significantly worse with irregularly high solar winds or when the science mode attitude is changed. Table 8. MAP Overlap Comparisons.

Pos (km) Vel (mm/s\ , , \

Pos fkm) vel (mrnis)

RSS Radial Along-track 5-Week Predictive Overlap 2.0 0.3 1.4 0.83 0.36 0.40 9-Week Predictive Overlap 6.7 6.2 2.3 3.9 3.8 0.4

Cross-track 2.0 0.79 1.8 0.6

3. ADVANCED TECHNOLOGY

3.1. Celestial Navigator (CelNav) CelNav is a part of the Goddard Enhanced Onboard Navigation System (GEONS) software package developed by GSFC’s GNCC. CelNav uses standard spacecraft attitude sensors and communication components to provide autonomous navigation. Analysis to date indicates that real-time autonomous navigation accuracies to 10 km RMS for LPO missions are achievable using high-accuracy attitude sensors and one-way Doppler mea-

Orbit Determination Issues for Libration Point Orbits 11

surements (Folta CelNav uses directional measurements from standard attitude sensors (eg. Earth and Sun sensors) and one-way forward-link Doppler measurements from a ground station communications receiver augmented with a Doppler extraction capability (see Figure 5 for a schematic). The one-way forward Doppler is obtained from the spacecraft communication link, thus eliminating the need for dedicated tracking services. The directional measurements are the angles of the line-of-sight unit vector from the sensor to the celestial object, measured with respect to the sensor frame of reference. Simulated analyses using realistic and optimistic levels for the measurement noise and biases and the Doppler tracking frequency have been performed. Directional measurement noise standard deviations were selected to be consistent with the current digital sun sensor technology of 1 arcminute and an onboard attitude determination accuracy of 1 arc-minute (achievable using star trackers). The one-way Doppler measurement accuracy is primarily dependent on the noise and stability characteristics of the onboard oscillator that provides the frequency reference used in the Doppler extraction process. The optimistic reference frequency quality was modeled based on expected performance of a typical ultra-stable oscillator (USO) (Folta 5 ) .

Autonomous Navigation Scenario

1

$9 T;";

_--/--4-

SIC to Sun directtonal measurement

measurement/

/

measurement

: Earth

Fig. 5.

CelNav Measurement Sources.

For the optimistic case using unbiased Earth and Sun directional mea-

12 M. Beckman

surements with noise consistent with current digital sun sensor technology and Doppler measurements referenced to an USO, orbital error was 7 km and 2 mm/sec. Various tracking scenarios are shown in Table 9 (Folta 5 ) . When Doppler tracking was eliminated, orbital errors increased significantly. The addition of more realistic parameters including a noisy US0 (10 times the noise sigma), reduced Doppler tracking data, directional measurement biases, and the elimination of Earth directional measurements all degraded solution accuracy to a range of 14 to 22 km. Table 9. CelNav Solution Accuracy. Tracking Scenario Nominal Eliminated Doppler tracking Increased Doppler measurement noise from 0.001 Hz to 0.01 Hz Reduced Doppler tracking from 2 to 14-hr pass per day Added directional measurement bias of 0.1 arc-minute Eliminated Earth directional measurements

Pos Error (km) 7

Vel Error (cm/sec) 2

62

30

22

3

17

NA

22

4

14

NA

Steady-state accuracy was not found to be very sensitive to elimination of Sun directional measurements or a 4-fold increase in the directional measurement noise to 6 arc minutes (consistent with existing Earth sensor technology and 0.1 degree accurate attitude determination). Figures 6 and 7 compare the steady-state position and velocity performance for the optimistic case with a realistic case starting at the least favorable tracking geometry and including 0.1 arc-minute directional measurement biases and Doppler measurements from a noisy US0 with Doppler tracking reduced to one 2-hour contact every other day (Realistic with Sun, Earth, and Doppler) and a realistic case identical to above but without Doppler tracking (Realistic with Sun and Earth). As a comparison, analysis using the realistic sensor parameters gave rather good results for a highly elliptical (1.8 by 9 Re) orbit. Attitude sensors alone gave a position RMS of 15 km, while the addition of Doppler

Orbit Determination Issues for Libmtion Point Orbits 13

........... tOO 80

00 40

.............

............................................................. ..................

............................... A

Fig. 6. CelNav Position Errors Based on Various Sensor ACCUI .acies.

data dropped that error to 1.5 km (Long *). 100

............................ ............................. ................................................

................................................

Fig. 7.

CelNav Velocity Errors Based on Various Sensor Accuracies.

3.2. Delta Diflerenced One- W a y Range ( D D O R )

All previous LPO missions have used ground based tracking using range and Doppler measurement types. Both of these measurements give information only along the spacecraft line-of-sight. Information perpendicular to this line is inferred only from time-varying changes in these measurements and the dynamical model used. Thus, the radial component of the orbital uncertainty is considerably more accurate than the plane-of-the-sky

14 M. Beckman

components. DDOR is a true VLBI measurement type that is being implemented at the DSN 34-m and 70-m (X-band only) sites as a nominal measurement type by May 2003 (Cangahuala 2 ) . DDOR is obtained by double differencing simultaneous observations of the spacecraft from two widely separated ground sites followed immediately by observations from an angularly nearby quasar (see Figure 8 for a schematic). The differential range to both the spacecraft and the quasar is determined from the observations. These measurements are then algebraically differenced to provide a precise determination of the angular position offset between the two sources as common measurement errors tend to cancel. With multiple baselines, the 2D angular component can be determined. This information provides previously unavailable plane-of-thesky knowledge. Potentially, the use of this measurement type could reduce plane-of-the-sky orbital uncertainty t o the current radial levels.

Fig. 8.

DDOR Measurement Type.

Each tracking station simultaneously views the spacecraft and records radio tones being broadcast. The antennas then simultaneously slew off the spacecraft and record the signals from a reference quasar which is located angularly near the spacecraft. The calculation of the angular separation between the spacecraft and the quasar is then,

a=

(ATS/C

- ATQ)c 7

B where a is projection of the angular separation between the spacecraft and quasar onto the baseline between the two stations, A T S ~ is C the time de-

Orbit Determination Issues for Libmtion Point Orbits

15

lay between when a radio signal from the spacecraft is received at the first station and when it is received at the second station, ATQ is the same for the quasar signal, c is the speed of light, and B is the baseline length (Pollmeier 'I). Intercontinental baselines between DSN stations range from 8,000 to 10,500 km. Accuracies are typically expressed as a distance measurement (numerator of above equation) since the baseline lengths vary. DDOR has been used operationally before on interplanetary missions such as Voyager, Galileo, and Magellan. It has also been tested on Mars Global Surveyor and Mars Odyssey. Accuracies of 21 to 50 cm were seen for Voyager measurements (30 cm equals 37.5 nrad at 8000 km baseline) [Border 821. The DDOR requirement for Galileo was 50 nrad [Pollmeier 921. Current DDOR implementation states accuracies of 7.5 nrad with telemetry subcarriers and 5 nrad with Differenced One-way Range tones (Cangahuala 2). Figure 9 (Cangahuala 2, shows the DDOR error budget for the current implementation.

Fig. 9. DDOR Error Budget.

Covariance analysis was performed by GNCC at the request of NASA HQ to assess the use of DDOR for LPOs. Analysis indicates improved orbital accuracy can be obtained while reducing tracking times by 80%. Table 10 details some of the analysis for a SOH0 orbit using 14 hours per week of DSN tracking. The use of DDOR measurements could reduce that tracking to 2.5 hours per week and improve total position uncertainty by more than 25%. The use of DDOR has many advantages. DDOR is one-way data type (downlink only). There is no need to calibrate the spacecraft uplink for

16 M. Beckman

refraction which simplifies ground station operation. Spacecraft angular position, or plane-of-the-sky position components, is more accurately determined by DDOR; fivefold improvements are possible. However, there are drawbacks to the use of DDOR. For each acquisition during Voyager, lo9 bits were reduced to obtain one measurement. This extensive post-processing typically took up to 24 hrs (Border ’). Thus far, DDOR has been used as a supplemental measurement source with independent solutions obtained from standard measurement types used as references. An increased reliance on DDOR and large reductions in standard tracking would reduce the quality assurance of DDOR. Table 10. Covariance Analysis Results Using DDOR for LPOs. R&D Sch (*)

DDOR

2 hrs/day 3 hrs/3 days 3 hrs/3 days

None None Oncejday

Baseline

Def Pos Acc (km) 3.8 6.5 2.8

Tot DSN Trk Time (hrs/wk) 14 2.5 2.5

NA NA 50% Gds-Mad 12% Can-Mad 38% Gds-Can (*) Rotating stations each day including both northern/southern hemisphere.

4. CONCLUSIONS

Previous LPO missions have obtained OD accurate to 2 to 10 km. The best accuracy has been achieved by MAP and is due in part to the favorable attitude and consistent Cr estimates. The worst accuracy has been achieved by ACE and is due primarily to the shortening of the batch data arc due to frequent spacecraft perturbations. The amount of tracking data received for each mission is not highly correlated with the OD accuracy achieved. This suggests that other issues such as spacecraft perturbations, spacecraft attitude, and use of MARK IVA data are more important than quantity of tracking data. The use of DDOR measurements can increase the accuracy of a standard range and Doppler tracking scenario by 25% while reducing the total amount of tracking time by 80%. DDOR data greatly improves the planeof-the-sky position error components. While DDOR has not yet been used for any LPO missions, it has been used operationally on interplanetary missions. The use of CelNav would eliminate the need for all coherent dedicated

Orbit Determination Issues for Libmtion Point Orbits 17

tracking passes. The performance of CelNav using realistic sensor performance indicates that autonomous navigation using directional and Doppler measurements can meet onboard navigation requirements on the order of 30 km. Higher accuracy is achievable by reducing measurement noise and increasing the Doppler tracking frequency. Autonomous navigation using only directional measurements can provide a lower-cost navigation method for missions with less stringent onboard navigation requirements, i.e. greater than 50 km. References 1. Border, J. S., et al.: Determining Spacecraft Angular Position with Delta VLBI: The Voyager Demonstration, AIAA-82-1471, AIAAIAAS Astrodynamics Conference, San Diego, CA, Aug 9-11, 1982. 2. Cangahuala, L. A.: Navigation Measurements: Overview, Performance and f i t u r e Plans, Navigation Tracking Requirements Peer Review slides, June 26, 2001. 3. Colombe, B.: SOHO orbit analyst, Personal communication on SOHO OD strategies, April 2002. 4. Fink, D.: M A P L 2 OD Accuracies Report, Internal memorandum dated Apr 17 2002. 5. Folta, D., et al.: Autonomous Navigation Using Celestial Objects, AAS 99439,1999. 6. Jordan, P., et al.: Solar and Heliospheric Observatory (SOHO) Mission Description and Flight Dynamics Analysis Reports, Revision 2, CSC/TM91/6030ROUDO, Sep 1993. 7. Joyce, J. B., et al.: Dajectory Determination Support and Analysis for ISEE3 from Halo Orbit to Escape from the Earth/Moon System, AIAA-84-1980, AIAAIAAS Astrodynamics Conference, Seattle, WA, Aug 20-22, 1984. 8. Long, A., et al.: Autonomous Navigation of High-Earth Satellites Using Celestial Objects and Doppler Measurements, AIAA 2000-3937, AIAAIAAS Astrodynamics Specialist Conference, Denver, CO, Aug 14-17, 2000. 9. Marr, G.: Constellation-X L 2 Orbit Determination OD Error Analysis with Impulsive Momentum Unloading Maneuvers, memorandum to ConstellationX project dated June 5, 2000. 10. Nicholson, A.: M A P Backing Assessment for Reduced Volume of Data, Computer Sciences Corporation memorandum dated Sept 10, 1999. 11. Pollmeier, V. M. and Kallemeyn, P. H.: Galileo Orbit Determination from Launch through the First Earth Flyby, The Institute of Navigation, 47th Annual Meeting, 1992.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

PERFORMANCE REQUIREMENTS AND KICKSTAGE DISPOSAL OPTIONS FOR A CISLUNAR GATEWAY STATION TRANSFER VEHICLE

G.L.CONDON Senior Engineer, NASA Johnson Space Center (JSC), Houston, Texas

S. WILSON Elgin Software, Inc., Columbia, Missouri C.L. RANIERI Co-operative Education Student, NASA JSC, Houston, Texas

The notion of human missions to libration points has been proposed for more than a generation l-'. A human-tended Gateway Station at the cislunar ( L 1 ) libration point could support an infrastructure expanding human presence beyond low Earth orbit and serve as a staging location for human missions to the lunar surface, Mars, asteroids, and other libration points. Human occupation of the Gateway Station requires a transfer system in the form of a Libration point Transfer Vehicle (LTV) designed to ferry the crew between low Earth orbit and the Gateway Station. Assuming the LTV uses an expendable kickstage for the Earth orbit departure maneuver, a key problem in the design of such a system is the economical and safe disposal of that kickstage. After investigating the basic performance requirements for delivering the crew vehicle to Li , several options for kickstage disposal are explored. These include: return to Earth (ocean impact), lunar surface impact, lunar swing-by into heliocentric orbit, and insertion into a long-lifetime geocentric parking orbit. If there is no radioactive or comparably hazardous material in the kickstage, results indicate that the best option from the standpoint of public safety, aesthetics, and economy is direct return to one or the other of two mid-ocean lines on the Earth surface. Because L1 is an unstable libration point, another important problem involves potential unplanned return of the Gateway Station (or associated support spacecraft parked at L l ) , brought on by failure to perform required orbit correction maneuvers. Accordingly, another part of the study determined orbit lifetimes for vehicles at L1 having velocities of varying magnitude and direction relative to the libration point.

19

20

G.L.Condon, S. Wilson and C.L.Ranieri

1. Introduction After addressing the performance requirements for transferring a spacecraft from low Earth orbit (LEO) to a Gateway Station at the L1 (cislunar) libration point ', this paper looks into options for disposing of an expendable kickstage used t o supply the required Earth orbit departure (EOD) velocity increment. This particular inquiry is restricted to human spacecraft for which - because of operational considerations and mass penalties involved with providing life support to the flight crew during transit - there are advantages in choosing flight times on the order of 3.5 days, as opposed to the much longer ones that are required to achieve absolute minima in the propulsive AV requirement. In addition, this paper examines the orbit lifetime for the Gateway Station - or a compromised vehicle in its near vicinity - when, for some reason, it is not possible to perform required station-keeping maneuvers. Orbit lifetimes are also briefly examined for vehicles at L1 that experience impulsive velocity increments having particular magnitudes and directions. 2. Nominal Earth-to-Gateway Crew Transfer Sequence The nominal Earth-to-Gateway crew transfer sequence begins with the EOD maneuver that places the LTV crew module/kickstage on a transfer orbit targeted to the L1 point (Figure 1). The optimum flight time depends on the geocentric distance of the Moon at libration point arrival (LPA) time. In this study, the flight time between EOD and libration point arrival (LPA) was fixed at 3.5 days. This yields, for any Earth-Moon distance, velocity increments very near those obtained at the first local minimum in the curve of transfer AV versus flight time. At some point on the transfer orbit, prior to arrival at L1, the LTV crew module and kickstage separate (figure 2). The crew module continues on to L1 where it performs an LPA maneuver to achieve rendezvous with the Gateway station. Sometime after separation, when the required AV reaches a minimum, the kickstage performs a stage deflection maneuver that puts it on a path to the disposal destination. A priori assumptions for this study include departure from an International Space Station (ISS) orbit having an altitude of 407 km and an inclination of 51.6'. Over an 18-year period, the inclination of the Moon's orbit to the Earth equator varies between 18.3' and 28.6O (Figure 3). Assuming coplanar departure from the ISS orbit (which is always very nearly

Requirements ond Options for o Cislunar Gotewoy Station %nsfer

/

Vehicle 21

Moon

Fig. 1. The Lunar L1 Gateway Station provides an evolutionary infrastructure for future exploration missions. It represents a transportation waypoint and safe haven for crews as well as a staging point for future human and robotic missions of exploration and development of space.

LTV Craw ModulaIICiokstage lnjaction Toward L l

"-

LTV Crew Module I Lclckatage Ssparation

LTV Crew Module

, . I Continues t to L l

a

LW IQCkshga Dlvartad to DIsposaI Dastlnntlon

Fig. 2. In the nominal Earth-to-Gateway crew transfer sequence, the EOD maneuver is followed by a coast and then separation of the LTV crew module from the kickstage. The crew module continues on to L1 and, at an appropriate time, the kickstage performs a deflection maneuver that sends it to a desired disposal destination.

the case when the transfer AV is minimized), when the Moon's orbit inclination is 28.6" the LPA plane-change requirement may be its small as 23" or as great as 80.2", depending on whether the ascending node of the ISS (at EOD time) is aligned with the ascending or the descending node of the Moon. The EOD AV for a local-optimum transfer exhibits a slight dependency on lunar distance, but is virtually unaffected by the wedge angle between the plane of the transfer orbit and that of the Moon. However, the magnitude of the LPA velocity increment depends strongly on wedge angle, and to a

22

G.L. Condon, S. Wilson and C.L. Ranien'

1

I

\

Fig. 3. The 18-year lunar inclination cycle shows a variation of the Moon's orbit inclination, with respect to the Earth's equator, of 1 8 . 3 O to 28.6". This results in a possible libration point arrival (LPA) plane change range of 2 3 O to 80.2O.

lesser extent on the geocentric distance of the Moon at LPA time. It follows from these facts, and from those described in the preceding paragraph, that one needs to look at transfer trajectories in a month when the lunar orbit inclination is at or very near 28.6' to determine maximum performance requirements for the LTV.

2.1. LPA Time Frame for Evaluation of Performance Requirements

A two-week period in October of 2006 was chosen for an evaluation timeframe. As indicated in Table I, libration point arrivals in that period exhibit a near-maximum variation in LPA plane change requirement (Xfr Orbit iEMP). This period begins with the Moon simultaneously very near perigee and its ascending node on Earth's equator, and ends with the Moon very near apogee and its descending node. It can be seen that an aggregate of 22 launch opportunities were examined. Libration point arrivals at perigee with were combined with minimum and maximum LPA plane change angles, and likewise for arrivals at apogee, with a variety of combinations between these extremes. As illustrated in Figure 4, in a real-world situation where the ascending node of the ISS orbit is precessing under the natural influence of Earth

Requirements and Options for a Cislunar Gateway Station lhnsfer Vehicle 23

Table 1. Earth to L1 transfer and upper stage disposal data. Transfers involve coplanar departure from circular Earth parking orbit having an altitude of 407 km and an inclination of 51.6 deg. RA: right ascension.

# 1 2 3 4 5 6 7 8 9 10 11

Arr Time (Nominal) 2006 Oct l0/06/06 04:OO 10/07/06 04:OO l0/08/06 04:OO 10/09/06 08:OO 10/11/06 0O:OO 10/12/06 08:OO 10/13/06 18:OO 10/15/06 18:OO 10/17/06 04:OO 10/18/06 12:OO 10/19/06 18:OO

Lunar L1 RA deg -1.0 12.4 26.2 42.9 68.1 88.5 109.2 135.4 151.8 166.2 179.2

Earth Decl deg -0.1 7.2 14.0 20.7 27.1 28.7 27.2 20.7 14.0 6.9 -0.1

Dist. 1000 km 304 304 305 309 317 324 332 339 343 344 345

Park Orbit RAN Epoch 2006 Oct 10/02/06 16:OO 10/03/06 16:OO 10/04/06 16:OO 10/05/06 20:OO 10/07/06 12:OO l0/08/06 20:OO 10/10/06 06:OO 10/12/06 06:OO 10/13/06 16:OO 10/15/06 0O:OO 10/16/06 06:OO

oblateness, at most a couple of launch opportunities can occur during any two-week interval. In this study, for the purpose of sampling all combinations of lunar distance and LPA plane change angle, the ascending node location for the ISS orbit (the columns labeled ”Park Orbit RANo” in Table I) was treated as an arbitrary parameter that could be changed at will. 2.2. Nominal Earth-to-Gateway

AV

Cost

Figures 5 and 6 show the nominal AV performance cost (excluding upper stage deflection to a disposal locale) - in terms of EOD AV, LPA AV, and their sum - for L1 arrivals over the chosen two-week period. The greatest range of AV variation (approximately 30%) occurs for LPA. The EOD maneuver cost remains fairly constant (within 0.5%), resulting in a variation for the total transfer AV of up to approximately 6%. This variation results from the combined effects of variations in LPA plane change and LPA maneuver altitude. Some of the curves shown in Figures 5 and 6 (and many others to follow) are distinctively different for northerly and southerly LPA pre-maneuver geocentric velocity azimuths. The data come from the calculation of EarthL1 transfer trajectories which arrive at the libration point over a two-week period beginning when the Moon was very near perigee as it crossed the Earth’s equatorial plane heading northward (Oct 6, 2006). Had the time period begun when the Moon was near perigee as it crossed the equator

24

G.L. Condon, S. Wilson and C.L. Ranieri

Table 1. (Cont.) Earth to L1 transfer and upper stage disposal data. GO, DROA, LVI, HO and SROA maneuver times selected to minimize AV for stage disposal. RAN: Right ascension of ascending node; RANo: right ascension of ascending node at RAN epoch; iEMP: Inclination of Xfr orbit with Earth-Moon plane; EOD: Earth orbit departure to L1 lunar libration point; LPA: Libration point arrival (3.5 days after EOD); GO: Upper stage disposal ‘in “safe” geocentric orbit (6600 km perigee alt ,300000-370000km apogee alt); DROA: Upper stage disposal in remote ocean area (direct 20 deg atmospheric entry angle, 240 deg longitude spread); LVI: Upper stage disposal on lunar surface (vertical impact); HO: Upper stage disposal in heliocentric orbit (via lunar swingby); SROA: Upper stage disposal in remote ocean area (via lunar swingby); OC: Overlapped conic trajectory; MC: Multiconic trajectory.

Park Xfr Orbit Orbit # RAN0 iEMP 1 -1.0 23.7 2 6.7 24.0 3 14.7 24.3 4 25.0 28.7 5 43.0 35.0 6 61.2 44.0 7 84.0 55.2 8 117.6 69.2 140.3 75.4 9 10 160.7 78.7 11 179.3 80.1 Park Xfr Orbit Orbit # RAN0 iEMP 178.9 81.0 1 2 198.1 79.8 3 217.7 75.9 4 240.9 68.2 273.2 54.4 5 295.8 44.3 6 7 314.5 35.9 333.3 28.0 8 343.4 24.9 9 10 351.8 23.3 11 359.2 23.3

Northerly L1 Arrival Azimut Maneuver AV, m/s EOD LPA GO DROA LVI MC MC OC MC MC MC 87 88 52 50 3061 782 45 87 88 3059 784 59 87 88 42 3060 781 61 3060 781 65 43 93 94 53 101 101 3063 776 63 3063 787 62 59 110 109 3066 810 59 61 115 115 58 117 118 3071 851 61 3072 875 63 53 116 117 115 117 3074 890 65 51 49 114 117 3074 900 66 Southerly L1 Arrival Azimut Maneuver AV, m/s EOD LPA GO DROA LVI MC MC OC MC MC MC 3060 984 55 104 106 91 55 105 106 91 3061 980 3059 960 90 55 106 106 3059 916 87 55 107 108 3064 838 77 58 109 109 3063 786 61 62 110 109 109 109 3066 748 33 69 83 107 107 7 3070 726 5 89 105 106 3072 724 10 92 104 105 3073 727 93 104 106 11 3073 733

HO SROA OC OC 66 106 66 106 65 111 71 117 78 126 86 132 92 135 96 134 95 132 95 132 94 131

HO SROA OC OC 87 124 88 126 87 128 87 131 86 132 87 132 83 129 83 124 82 120 82 120 81 121

heading southward (e.g., Oct 24,2003), the shapes and trends of the curves would have been similar, but the adjectives ”norther!y” and ”southerly” would have to be interchanged in the captions that identify the direction

Requirements and Options for a Cislunar Gateway Station lhnsfer Vehicle

0

m

m

40

m

im

im

25

ia

ErmEOITh~,d8ys~gD.Jm~@

Fig. 4. Outbound Earth to L1 and inbound L1 to Earth opportunities for selected transfer times occur with a frequency of about every 10 days.

Transfer Deita-V vs. Libration Pdnt Arrival Time TOW t f a r f s r AV L €00 AV + LPA AV

3.5 Day Trip Tlme

Fig. 5. The nominal AV performance cost shows the EOD, LPA and total (EOD LPA) AV for L1 arrival during the two-week period October 6-20, 2006.

of the arrival velocity azimuth.

+

26

G.L. Condon, S. Wilson and C.L. Ranieri

-

-

Total Transfer Delta-V M. Libration Point Arrival Time EOD AV + LPA AV: M-i LPA P b m c h m o . 23PM.2'

Total banshrAV

Fig. 6. The nominal AV performance cost shows a more detailed view of the total (EOD + LPA) AV for L1 arrival during the two-week period October 6-20, 2006.

3. Kickstage Disposal Options After execution of the EOD maneuver, the LTV kickstage and crew module share a trajectory having a perigee altitude near that of the pre-departure orbit, an apogee altitude equal to that of the libration point and, significantly, an orbit orientation and energy comparable to that required for reaching the near vicinity of the Moon. These circumstances immediately bring to mind the trajectory design problems solved in the Apollo program, wherein the trans-lunar injection (TLI) maneuver routinely put the command service module/lunar module (CSM/LM) on or very near a trajectory that provided a free return to the Earth, and the spent Saturn S-IVB stage was variously diverted (after TLI) onto trajectories that ended in heliocentric orbit or with lunar impact. Accordingly, the following options for upper stage disposal were selected for evaluation: (1) Lunar Swing-by to Heliocentric Orbit (HO) (2) Lunar Vertical Impact (LVI) (3) Direct Return to Remote Ocean Area (DROA) (4) Lunar Swingby to Remote Ocean Area (SROA) (5) Transfer to Long Lifetime Geocentric Orbit (GO)

Requirements and Options for o Cislunar Gotewoy Station lhnsfer Vehicle 27

3.1. Methodology The time of execution for the stage deflection maneuver (SDM) in every case was selected to minimize the AV for kickstage disposal, subject to imposed constraints. The various disposal trajectory solutions are considered to be a practical attempt to minimize these maneuver AVs (e.g., coplanar kickstage deflection maneuver assumed optimal for some disposal options) and not rigorous global optimizations analysis. This study employed a trajectory scan tool called Earth Orbit to Lunar Libration point (EOLL), which is based on a four-body model (Earth, Moon, Sun, spacecraft) and uses Jean M e e d s analytic lunar and solar ephemeredes. EOLL uses an overlapped conic approach to solving the split boundary value problem, and individually calibrates each solution to multi-conic accuracy 9. Trajectory solutions were verified by feeding EOLL post-maneuver state vectors into the Astrogator module of the Satellite Tool Kit, and propagating them to see if they went where they were predicted to go.

3.2. Option 1 . Lunar Swing-by t o Heliocentric Orbit ( H O ) This would seem to be the ideal choice from the standpoint of aesthetics and public safety since (at least in the short term) it does not clutter the Moon, the Earth, or geocentric space. In addition, escape from the EarthMoon system can be achieved at relatively low cost by diverting the stage onto a posigrade low-altitude encounter with the Moon (figure 7). However, there is a long-term possibility for an Earth impact. Absent maneuvers in heliocentric space to raise perihelion, the upper stage will of necessity follow a path, which recurrently crosses that of the Earth, resulting in close encounters at intervals measured in decades. The disposal trajectories chosen were those that required the minimum SDM AV, subject to a constraint that the post-encounter geocentric v-infinity vector magnitude had to be greater than or equal to 800 m/s (i.e., C3 > 0.64 km2/s2). The deflection AV cost for the HO transfer lies in the 65-96 m/s range with the entire variation occurring with the northerly lunar libration point azimuth arrival (figure 8). The northerly arrival deflection AV shows a strong dependence on the transfer orbit inclination (with respect to the Earth-Moon plane). The deflection AV for the southerly azimuth arrival ranges only from 81-88 m/s.

28

G.L. Condon, S. Wilson and C.L. Ranieri

Fig. 7. Following nominal insertion of the LTV crew module and kickstage onto an L1-bound trajectory, the kickstage performs a maneuver targeting it to pass behind the limb of the Moon, achieving escape from Earth-Moon gravity into heliocentric space.

E"

m

u 0 I M M O M lW@M M M b M *HuIbM - O M

* M I b I W O W - M O M

L M o r r n a n u w m n ~ m

m)

Fig. 8. The deflection AV cost of a heliocentric orbit transfer of the spent LTV kickstage ranges from 65-96 m/s with the kickstage, over a specified two-week period in October 2006, achieving a geocentric V-infinity > 800 m/s (C3 > 0.64 km2/s2).

3.3. Option 2. Lunar Vertical Impact (LVI)

In terms of trajectory characteristics (figure 9) everywhere except very near the Moon, lunar vertical impact typifies impact at any accessible lunar im-

Requirements and Options for a Cislunar Gateway Station Zhnsfer Vehicle 29

pact site. The magnitude of the SDM velocity increment is roughly the mean of those required for impact at any accessible lunar site, the variation amounting to something like plus or minus 15 m/s. Since it precludes Earth impact, this option would maximize public safety. However, for aesthetic and perhaps scientific reasons, it seems undesirable to institute a long-term program of deliberately cluttering the lunar landscape. Such objections might be ameliorated by confining impacts to some chosen lunar surface area of appropriate size. The cost in terms of AV probably would not be great. Depending on the chosen impact site, it would be somewhere between 5 and 30 m/s greater than heliocentric orbit disposal. But midcourse corrections (requiring extended system lifetime) would almost certainly be needed for precise control of the impact location. The deflection AV cost for LVI disposal lies in the 88-118 m/s range with the entire variation occurring with the northerly lunar libration point azimuth arrival (figure 10). Like the HO disposal option, the northerly arrival deflection AV shows a strong dependence on the transfer orbit inclination (with respect to the Earth-Moon plane). The deflection AV for the southerly azimuth arrival ranges only from 105-109 m/s.

Fig. 9. Following nominal insertion of the LTV crew module and kickstage onto an L1-bound trajectory, the kickstage performs a maneuver targeting it to impact the lunar surface.

30

G.L. Condon, S. Wilson and C.L. Ranieri

Fig. 10. The deflection AV cost of a lunar vertical impact transfer of the spent LTV kickstage ranges from 88-118 m/s with the kickstage, over a specified two-week period in October 2006.

3.4. Option 3. Direct Return to Remote Ocean Area

(DROA) The disposal trajectories for this option (figure 11) were constrained to have an entry angle of 20°, yielding a short atmospheric flight segment and a small footprint for wreckage from stage breakup in the atmosphere. The SDM AV budget for this option includes an allowance for varying the entry time over a span of 16 hours, yielding 240° of impact longitude control. The latter is sufficient to guarantee the capability for nominal impact on one or the other of two mid-ocean lines (one in the Pacific and one in the Atlantic, figure 12), no matter what the date of Earth orbit departure might be. The impact coordinates for the trajectory requiring the minimum SDM AV (subject to the entry angle constraint) depend on the coordinates of the Moon when the LTV spacecraft arrives at L I . The impact latitude lies within bounds of something like plus and minus 50°, but the longitude is essentially random over the long term. Since 240° is the maximum longitude difference between the two mid-ocean lines, sizing the SDM AV budget to provide that amount of longitude control guarantees a capability to reach one or the other of the mid-ocean lines, no matter where the minimum-AV trajectory lands.

Requirements and Options for a Cislunar Gateway Station Thnsfer Vehicle

6. C m module a

@

n h

at LI

31

Moon

//

il

1. Lunarmnsbr VOhkk (LW cmw modulewl(hKloLstage in hltW 407 x 407 km Wrklng OrMt

0

6.

Kkkstagereiumsb Esrlh Rr OC(M improt

.Jeulsoned lackshqo pecnm

narwrto achlave 2(p 8tmospherk mby an@eand mtd-oceanImpact 2. K k b l a g e hJectscmw module 3. Coast phase; 6 lackstageonto b a s b r K k b w IMmn ImJeCtoIy-Id Ll

Fig. 11. For the DROA case, following nominal insertion of the LTV crew module and kickstage onto an L1-bound trajectory, the kickstage performs a small maneuver targeting it to impact the Earth surface in a remote ocean area.

In light of these facts, unless the upper stage contains radioactive material (which seems very unlikely, given that it has a short operating life and the velocity increment required of it is only about 3 km/s) or something comparably hazardous, this option should be acceptable from the standpoint of public safety, and aesthetics as well. With regard to the latter, it should be borne in mind that the total increase in wreckage on the sea floor - produced by a century of any kind of Earth-to-LI shuttle operation sustainable by the global economy - would be almost infinitesimal compared to the residue of two world wars. The remoteness of the mid-ocean lines from any major land mass, combined with the small wreckage footprint resulting from the steep atmospheric entry angle, means that precise trajectory control (which would extend the operating lifetime of the kickstage) should not be necessary. As can be seen in Figure 13, the deflection AV cost for the DROA transfer ranges from 42-93 m/s with the southerly lunar libration point arrival azimuth showing the greatest AV cost. The deflection AV range for the southerly arrival is 55-93 m/s while the AV cost range for the northerly arrival is 42-61 m/s. Generally, there are two local optima for the location of the kickstage maneuver point in the Earth-to-LI transfer trajectory, of which the better one was always chosen. An advantage to this approach is that, assuming kickstage disposal is not allowed to constrain the primary

32

G.L. Condon, S. Wilson and C.L. Ranieri

mission, this option is one of three (HO,DROA,GO) requiring the lowest AV budget that could be found (less than 100 m/s in all three cases). In addition, avoidance of close lunar encounter, combined with steep entry over wide areas of empty ocean minimizes criticality of navigation and maneuver execution errors. This approach may not be appropriate if the kickstage contains radioactive or other hazardous material.

Fig. 12. For the DROA case, the kickstage is targeted to an Earth impact in a remote ocean area. The white shaded area shows a 240° longitude control capability allowed for in the LTV kickstage AV budget.

4. Lunar Swingby Return to Remote Ocean Area (SROA)

3.5. Option

The only constraints applied to the disposal trajectories for this option (figure 14) were that the kickstage enter the Earth atmosphere at an angle of 20° or more, after having experienced a retrograde close encounter with the Moon. For any particular case, the minimum SDM AV magnitude satisfying those criteria represents a lower bound for that which would be required to satisfy the more stringent constraints applied in Option 3 (DROA). Since the lower bounds thus determined (figure 15) are greater than the AV requirements for DROA, there is no good reason to consider SROA any further.

Requirements and Options for a Cislunar Gateway Station Tknsfer Vehicle 33

imoso

imam

mmm010 w ~ z m m i-wo

1p11(u1omia~mmomiommom

U m ( l o n P o k t A M T h . (ImMSW b h m

Fig. 13. The deflection AV cost of a DROA transfer of the spent LTV kickstage ranges from 42-93 m/s with the kickstage, over a specified two-week period in October 2006, achieving a direct impact in a remote area of the Earth’s oceans.

2.KWrdq.ktchcrnrpdrh *Uck8awol*Db.n.*r

‘s.catw; K*-

-(RnrdL*

Fig. 14. For the SROA case, following nominal insertion of the LTV crew module and kickstage onto an L1-bound trajectory, the kickstage performs a small maneuver targeting it to a lunar swing-by to impact in a remote ocean area.

3.6. Option 5. h n s f e r to a Long Lifetime Geocentric Orbit ( G O ) This option requires that the kickstage be transferred to a geocentric orbit having an adequate orbit lifetime to minimize concern over possible future

34

G.L. Condon, S. Wilson and C.L. Ranien’

Lmamtrohtclk.l’lb.ll*dnr Man1

Fig. 15. These lower bounds on deflection AV cost of a SROA transfer of the spent LTV kickstage range from 106-135 m/s with the kickstage, over a specified two-week period in October 2006, achieving a lunar swing-by and subsequent entry into the Earth atmosphere at an angle of 20° or more.

impact on the Earth. Although some insight was gained (see “Earth Moon L1 Libration Point Orbit Lifetime Analysis” section), the geocentric orbit characteristics required to guarantee an adequate orbit lifetime were not determined in this study. To establish a lower bound on performance requirements for the GO disposal mode, minimum AV was determined for raising perigee of the postEOD upper-stage trajectory to an arbitrary altitude of 6,600 km, without altering the apogee altitude nor the plane of the orbit. For less than 30% of the totality of launch opportunities were these optimistic bounds lower than the AV necessary for guaranteed direct return to a mid-ocean line (Option 3, DROA). Since the unaltered apogee altitude ranges between 300,000 and 370,000 km, one or both of the unexplored orbit modifications almost certainly would be required - at considerable cost in AV - to preclude lunar perturbations that could make the upper stage strike the Earth after a comparatively short interval of time. As shown in Figure 17, the lower bound on deflection AV cost for the GO transfer lies in the 5-91 m/s range with the entire variation occurring with the southerly lunar libration point arrival azimuth. The deflection AV

Requirements and Options for a Cislunar Gateway Station lltansfer Vehicle 35

b.(.day(ar.ld

Ll

Fig. 16. For the geocentric orbit (GO) case, following nominal insertion of the LTV crew module and kickstage onto an L1-bound trajectory, the kickstage performs a maneuver targeting it to a geocentric parking orbit.

cost shows a correlation with inclination of the transfer orbit relative to the Earth-Moon plane. The deflection AV cost for the northerly azimuth arrival remains relatively confined, ranging only from 52-66 m/s. The GO disposal option would seem preferable to DROA or SROA if the kickstage carries hazardous material. Taking an optimistic viewpoint that acceptable geocentric orbit lifetime could somehow be attained with the bounding values on deflection AV shown in Figure 17, a case might be made for its preference over DROA even if the kickstage contains no hazardous material. In 4 of the 22 cases studied, the lower bound on AV for GO disposal (into an orbit having a perigee altitude of 6600 km and an apogee altitude in the range of 300000 - 370000 km) was less than 12 m/s. Assuming the 22 cases represent an unbiased sample of all possible transfers between Earth orbit and L I ,this implies that a 12 m/s budget would suffice if it were permissible to forgo all but about 20% of the otherwise-available transfer opportunities. However, the GO disposal option carries a disadvantage of promising to deposit more orbital debris in geocentric space. In addition, the 12 m/s budget described above would increase the average interval between usable transfers to a frequency of approximately one every 50 days. With no constraint imposed by the AV budget, there exists an opportunity approx-

36

G.L. Condon, S. Wilson and C.L. Ranieri

imately once every 10 days. To achieve acceptable orbit lifetime, lunar and solar perturbations may necessitate a higher perigee and/or lower apogees, either of which will require SDM AV magnitudes greater than those shown in Figure 17.

ionmom iwm090 lomnm mnmoem ununom m o m -om

iotmoom

Ul.anpatAnk.lTanm(nwmlyy Unn)

Fig. 17. The deflection AV cost of a GO transfer of the spent LTV kickstage ranges from 5-91m/s over a specified two-week period in October 2006 with the kickstage achieving a geocentric parking orbit.

3.7. LT V Kickstage Disposal Options - Conclusion

In terms of propulsive AV, it was found that sending the upper stage to any Earth atmosphere entry point via close encounter with the Moon (SROA) is the most expensive of all the disposal modes studied (figure 18). On the other hand, guaranteed direct return to a mid-ocean line (DROA) is cheaper than lunar impact (LVI) or heliocentric orbit (HO) disposal, and probably cheaper than disposal in any geocentric orbit (GO) having an adequate lifetime to satisfy public safety concerns. A brief and cursory look at geocentric orbit lifetime is found in the "Earth Moon L1 Libration Point Orbit Lifetime Analysis" section. The DROA disposal option appears to offer the best suite of desirable features. It provides for controlled Earth contact using a relatively small disposal AV. It also avoids a close encounter with the Moon, which would result in not only a greater AV cost, but also a near certainty of requiring

Requirements and Options for a Cislunar Gateway Station Transfer Vehicle 37

midcourse correction maneuvers before and after perisel passage. In addition, DROA avoids littering the lunar surface and geocentric space with debris. However, this approach would not serve well for cases where the LTV kickstage contains hazardous (e.g., radioactive) materials. In that case the ”next best” disposal option (HO) avoids Earth or lunar disposal issues (e.g., impact location, debris footprint, litter) by taking the kickstage around the Moon and into heliocentric space. This approach also carries a relatively low AV cost. Further study would be required for this disposal option to determine the probability of subsequent re-contact with the Earth, and the cost of precluding such an event.

Fig. 18. A summary of LTV disposal maneuver deflection AVs shows the lunar swingby disposal option (SROA) to be the most costly while the direct Earth remote ocean area (DROA) return provides a cheaper return than a disposal t o lunar impact (LVI), heliocentric orbit via lunar swing-by (HO) and, in most cases, a geocentric parking orbit (GO). Note that the GO data in this plot does not reflect a ”safe” orbit lifetime as reflected in the ”Earth Moon L1 Libration Point Orbit Lifetime Analysis” section.

4. Earth Moon L1 Libration Point Lifetime Analysis

This section examines the orbit lifetime of spacecraft using the Earth Moon L1 Libration Point (EM L1). The GO disposal option for the LTV kickstage would require a long lifetime parking orbit. This study assumes a 100-year orbit lifetime as a long lifetime parking orbit. An unplanned or uncontrolled

38

G.L. Gondon, S. Wd30n and C.L. Ranieri

return of a spacecraft to Earth could be a potential threat to public safety. This threat is aggravated if the spacecraft contains radioactive or otherwise hazardous material. Whether or not the LTV kickstage contains any hazardous material, mission planners should provide safe disposal guidelines for such spacecraft. With this in mind, three cursory studies were performed to gauge the safety of a GO disposal method. The purpose of these studies was to gain a better understanding of the nature of parking orbit lifetime for a spacecraft under multi-body influence (i.e., Earth, Sun, Moon). These studies also endeavor to provide a better understanding of the fate of a spacecraft at EM I51 that either loses station-keeping control or suffers an unplanned jet firing without recovery capability. 4.1. Methodology

Each study examined the fate of a spacecraft following GO disposal for parking orbit propagation periods of up to a 100 years. Five possible outcomes for these "long life" orbits include: Earth impact, lunar impact, lunar fly-by to a heliocentric orbit, lunar fly-by to a heliocentric orbit with a subsequent Earth return and impact, or a continuous 100-year period in an Earth parking orbit. The studies were performed with Satellite Tool Kit (STK) Astrogator using a high fidelity propagation model that included gravity from multiple bodies of the solar system including: the Earth (as the central gravitation source), the Moon, the Sun, Jupiter, and Mars. For all three of the studies, the spacecraft was placed in the same orbital plane of the Earth and the Moon. This provided an anticipated worst-case scenario as the spacecraft experiences the largest lunar perturbations in the same orbital plane as the Moon due to longer durations in proximity to lunar gravitational perturbations as compared with planes out of the Earth-Moon plane. 4.2. Study One: Spacecmft left on a high eccentricity

parking orbit The first study focuses on a spacecraft (e.g., LTV kickstage) left on a high eccentricity parking orbit with perigee and apogee in the region of LEO and the EM L1 point, respectively. Propagation of the kickstage was initiated at the apogee of the orbit. The orbits examined include LEO to EM L1 transfer orbits as well as orbits with higher perigees resulting from post EM I51 arrival apogee AV maneuvers. The orbit perigee radii ranged from 6,600 km to 20,000 km. The apogee radius range of 300,000 to 371,000 km

Requirements and Options for a Cislunar Gateway Station Transfer Vehicle 39

contained the location of the EM L1 point. Forty- five different orbits were examined in these apogee and perigee ranges. The lunar perturbations for a spacecraft left in these high apogee orbits were very large and caused significant changes in the size and shape of these orbits. The results of 45 orbit propagations indicated that a spacecraft in an orbit in the abovementioned apogee and perigee ranges would impact the Earth in less than 10 years in 56% of the cases. The other 44% of the orbit propagations saw the spacecraft either achieve heliocentric orbit or impact the Moon. While the 45 case run matrix was not exhaustive, it was sufficient to demonstrate that a spacecraft left in a parking orbit with apogee and perigee ranges identified above would likely impact the Earth in a much shorter period of time than that considered a long lifetime orbit (i.e., > 100 years). This preliminary survey was intended to provide a cursory understanding of possible outcomes of selected orbit propagations. The 3-D contour plot in figure 19 shows the results of 45 completed orbit propagations for the range of apogee and perigee radii mentioned above. Positive values on this plot represent the time in orbit before Earth impact while negative values represent orbits that transition to either a heliocentric orbit or a lunar impact. This plot shows a fairly random distribution of orbit lifetime results and also reflects particular orbits that have not yet been examined. Orbits with apogees around the 350,000 to 360,000 km range appear to have no Earth impacts. However, a more exhaustive run matrix would better determine if this set of orbits would serve as a safe disposal region as well as the associated AV cost required to achieve such parking orbits.

4.3. Study Two: Spacecraft Left at EM L1 W i t h No Station Keeping Capability This study examined the orbit lifetime for a spacecraft that has already performed the maneuver to place itself at the EM L1 point. This implies that the spacecraft’s initial condition has zero position and velocity relative to the EM L1 point. Since the L1 point is not a stable equilibrium point, with no station keeping maneuvers, the spacecraft will drift from the EM L1 due to the Moon’s eccentric orbit and other perturbations such as solar radiation and gravity. Once the spacecraft leaves the EM L1 point, the Earth’s and Moon’s gravity affect its orbit until the spacecraft moves further and further away from its initial equilibrium position. For this study, an orbit lifetime of 100 years was examined for a spacecraft with an initial condition at the EM L1 position and velocity. A scan of orbit propagations

40

G.L. Condon, S. Wilson and C.L. Ranieri

Fig. 19. This plot reflects the propagated orbit lifetime for a spacecraft placed in a parking orbit with indicated apogee radius and perigee radius. Note: a negative value indicates the propagation resulted in either a transfer to heliocentric space or a lunar impact.

was performed using different propagation epochs to express the changing initial position and velocity of the EM L1 due to the Moon’s eccentric orbit. Note that this scan was performed over a little more than one complete lunar orbit period and does address the longer term effect of the orientation of the lunar orbit plane as it changes with successive lunar orbit periods. The goal of this cursory study was to determine the final destination of a free-drifting spacecraft with an initial position and velocity matching that of the EM L1 point and the effect of variations in the relative positions of the Earth and Moon at the beginning of the propagation. Results of this study show that the spacecraft that is initially at the EM L1 position and velocity does not impact the Earth, independent of the EM L1 position (e.g., for the Moon at apogee or perigee or somewhere between the two). Figure 20 shows the orbit lifetime results for this study. The x-axis indicates the days in a lunar cycle while the y- axes show the orbit lifetime (left axis) and L1 radius from Earth during a lunar cycle (right axis). This plot indicates the orbit lifetime for a free drifting spacecraft initially matching the L1 position and velocity. Orbit lifetimes of 100 years indicate that the spacecraft achieved a heliocentric orbit (departed the Earth-Moon system) and, in this case, did not subsequently return to

Requirements and Options for a Cislunar Gateway Station Transfer Vehicle 41

Earth. Values less than 100 years indicate the spacecraft orbit lifetime prior to a lunar impact. For this study, no trajectories impacted the Earth. This plots the EM L1 position and shows how the orbit lifetime results seem to be independent of the EM L1 initial position. This plot does not appear to show a distinct correlation between L1 radius and possible outcomes for a propagated free-drifting spacecraft.

Fig. 20. The propagated trajectory of a free-drifting spacecraft, initially at the EM L1 position and velocity, results in an orbit lifetime > 100 years (transition to heliocentric trajectory) or significantly less than 100 years (lunar impact). No trajectories, evaluated in this study, impacted the Earth.

4.4. Study Three: Spacecmft Left A t EM L1 With No

Station Keeping Capability And A n Initial Impulsive AV This final look at orbit lifetimes for spacecraft at the EM L1 point has a similar set up as the previous study. This study again starts with a spacecraft with its initial position and velocity matching that of the EM L1 point. However, this study examines the effects that a range of imparted AV perturbations (magnitudes and directions) have on the spacecraft orbit lifetime and final destination. These AV impulses were evenly spaced O from Oo-360’ from the spacecraft’s velocity vector in 45’ increments. The magnitude of the AVs ranged from 1 to 500 m/s. These AV impulses reflected possible

42

G.L. Condon, S. Wilson and C.L. Ranien'

anomalous situations such as uncontrolled spacecraft thrusting followed by inability to null out or recover from the undesired impulse. The AVs were all confined to the lunar orbital plane to maximize the duration and effects of the lunar perturbations. The trajectory propagation began immediately after an imparted AV to a spacecraft initially possessing the position and velocity of the EM L1. Three examined epochs for commencement of the orbit propagations corresponded to the EM L1 apogee, perigee, and midpoint locations. The results of this study indicated that spacecraft at or near the EM L1 position and velocity are not very likely to impact the Earth. However, these orbits are not 100% safe. Figure 21 shows the ultimate fate of 161 different cases of propagated spacecraft trajectories. It shows that 51% of the orbits impacted the Moon, 44% of the orbits transitioned to heliocentric space and only 2% directly impacted the Earth. Additionally, 2% achieved a heliocentric orbit, but eventually returned to Earth-Moon space and impacted the Earth. A small percentage of the cases (- 1%)resulted in the spacecraft remaining in a high Earth orbit for more than 100 years without impacting either the Earth or the Moon or transitioning to a heliocentric orbit. In total, only 7 out of 161 propagated orbit cases resulted in an Earth impact. However, Earth impact did result for a AV as small as 10 m/s and very close Earth encounters resulted for AVs as small as 1 m/s. This indicates that while leaving a spacecraft at or near the EM L1 point position and velocity is relatively safe, an in depth study with the exact initial conditions must be performed before considering the spacecraft in a safe orbit for no Earth impacts. These results suggest that, while leaving a quiescent spacecraft at or near EM L1 may not result in an Earth impact, imparting a AVimpulse to the spacecraft could result in approximately a 2% chance of a direct Earth impact and a 2% chance for Earth impact following initial transition to heliocentric space.

Acknowledgments The authors acknowledge Daniel M. Delwood for providing trajectory analysis and Richard Ramsell €orproviding a conceptual graphic model of a LTV kickstage.

References 1. Farquhar, R.W.: Future Missions for Libration-point Satellites, Astronautics & Aeronautics, pp. 52-56, May 1969.

Requirements ond Options for a Cislunar Gotewoy Stotion Thnsfer Vehicle

43

Fig. 21. The pie chart represents destination results for 161 orbit propagations of a spacecraft initially possessing the same position and velocity as the EM L1 and after a perturbing AV impulse in the range of 1 to 500 m/s in a range of directions from Oo to 360° in 45O increments. For these orbit propagations, the spacecraft impacted the Moon 52% of the time, transitioned to heliocentric space (without returning in i 100 years) 44% of the time, impacted the Earth directly 2% of the time, impacted the Earth after time in heliocentric space, and spend > 100 years in a geocentric orbit 1% of the time.

2. Farquhar, R.W.: The Utilization of Halo Orbits in Advanced Lunar Operations, NASA TN D-6365, July 1971. 3. D’Amario, L.A.: Minimum Impulse Three-Body lhjectories, Massachusetts Institute of Technology T-593, June 1973. 4. Farquhar, R.W. and Kamel, Ahmed, A.: Quasi-Periodic Orbits About the nanslunar Libration Point, Celestial Mechanics 7, pp. 458-473, 1973. 5. Farquhar, R.W., and Dunham, D.W.: Use of Libration-Point Orbits for Space Observatories, Observatories in Earth Orbit and Beyond, Kluwer Academic Publishers, pp. 391-395, 1990. 6. Bond, V.R., et.al.: Cislunar Libration Point as a nansportation Node for Lunar Exploration, AASIAIAA Spaceflight Mechanics Meeting, February 1991. 7. Farquhar, R. W.: The Role of the Sun-Earth Collinear Libration Points in Future Space Exploration, SPACE TIMES, pp. 10-12, November-December 2000. 8. Condon, G. L., Pearson, D. P.: The Role Of Humans I n Libration Point Missions With Specific Application To A n Earth-Moon Libration Point Gateway Station, AAS 01-307, AASIAIAA Astrodynamics Specialist Conference, Quebec City, Canada, July, 2001. 9. D’ Amario, L. A., Edelbaum, T. N.: Error Analysis of Multi-Conic Techniques, AAS 73-217, AASIAIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

LIBRATION POINT MISSIONS, 1978 - 2002 D.W. DUNHAM and R.W. FARQUHAR Johns Hopkins University, Applied Physics Laboratory Mail Stop 2- 155, 1 1 100 Johns Hopkins Road Laurel, Maryland 20723-6099, USA

This paper summarizes the six missions to the vicinity of libration points that have been flown up to the time of this conference in June 2002. The first libration-point mission, the third International Sun-Earth Explorer (ISEE-J), is emphasized because it laid the groundwork for so many later missions, most of which are covered more thoroughly in other papers given at this conference. First, the authors present some basic properties of libration-point orbits, and some history of their development for early missions. Only brief information is given here; details can be found in the references.

1. Introduction

In 1772, the French mathematician, J. L. Lagrange, showed that there are five positions of equilibrium in a rotating two-body gravity field. Three of these “libration points”, or Lagrangian points, are situated on a line joining the two attracting bodies, and the other two form equilateral triangles with these bodies. All five libration points lie in the orbit plane of the two primary bodies. Fig. 1 shows a sketch of the libration points near the Earth. Their proximity makes them most attractive for possible space mission applications. Fig. 2 shows the basic linearized equations of motion near the Sun-Earth L1 libration point. The out-of-plane frequency is slightly different from the in-plane frequency, resulting in motion that describes a Lissajous figure as viewed from the Earth. It is well-known that the collinear libration points are unstable and that the equilateral points Lq and L5 are only quasi-stable in the Earth-Moon system. Therefore, some form of station keeping control 45

46

D.W. Dunham and R.W. Farquhar

Fig. 1. Libration Points near the Earth.

is needed to maintain a spacecraft at or near a libration point, especially the collinear ones. G . Columbo first showed that the AV cost for maintaining a satellite near a collinear point was mainly a function of the accuracy of its orbit determination and could be as small as 10 m/sec per year '.

Fig. 2. Equations of Motion near the Sun-Earth L1 Libration Point.

2. History and Use

The libration points in the Earth-Moon system were known by the early pioneers of space flight, but it was not until 1950 that Arthur C. Clark suggested that the La point of the Earth-Moon would be an ideal site to

Libmtion Point Missions, 1978-2002

47

broadcast radio and TV signals to colonies on the back side of the Moon 2, 3. But at lunar La, the comsat would be invisible from Earth. In 1966, Farquhar proposed a Lissajous path around lunar La to keep stationkeeping costs under about 10 m/sec per year and allowing visibility from Earth most of the time 4. But to allow continuous communication with Earth, periodic out-of-plane maneuvers are needed to effectively change the period of the out-of-plane motion to match that of the in-plane motion, allowing what was coined a “halo” orbit to be flown. A data-relay satellite that would fly such a path was considered for Apollo 17 when that mission would have been the first manned landing on the far side of the Moon; see Fig. 3. But that idea was dropped when the Apollo program was shortened and Apollo 17 was changed to a near-side mission. In 1973, Farquhar and Kame1 discovered that when the in-plane oscillation is greater than 32,379 km, there is a corresponding value of the out-of-plane amplitude that has the same period, producing a “natural” halo orbit 5 . More information about the early development of Iibrationpoint orbits, and attempts to use them for space missions such as Apollo, is given in a recent paper by Farquhar 6 .

Fig. 3.

Lunar Farside Communications Link.

With the end of the Apollo program, interest in lunar missions waned. As early as 1964, Farquhar recognized that the Sun-Earth L1 point would be an ideal location to continuously monitor the interplanetary environment

48

D . W . Dunham and R . W.Farquhar

upstream from the Earth '. From then until 1970, several attempts were made to convince scientists of the value of an L1 monitor to measure the solar wind before it reached the Earth. But there seemed to be a fear of doing something this new so that little interest was expressed for this proposal; see Fig. 4.

Fig. 4. Fear of Libration-Point Missions.

3. ISEE-3

In 1971, the fears subsided when N. F. Ness, a renowned space physicist at NASA's Goddard Space Flight Center, was impressed with the utility of a spacecraft near the Sun-Earth L1 point to monitor the upsteam solar wind. In 1972, it was decided to include such a satellite in a proposed three-spacecraft program that became known as the International Sun-Earth Explorer (ISEE) Program *. I S E E l and ISEE-2 would stay in a highly elliptical Earth orbit with an apogee distance of about 24 Earth radii. The separation of the two spacecraft would be controlled to measure the magnetosphere's fine structure. ISEES would be located in a halo orbit about the Sun-Earth L1 point to monitor the solar wind about one hour before it reached the magnetosphere, and I S E E l and 2. A relatively small-amplitude Lissajous path was ruled out due to frequent crossings of the solar radio interference exclusion zone, a 3" radius

Libmtion Point Missions, 1978-2002

49

centered on the Sun (as seen from the Earth) where S-band communication would be difficult or impossible; see Fig. 5.

Fig. 5. Lissajous Path crossing Solar Radio Interference Zone.

Instead, a halo orbit was desired that would avoid the exclusion zone. ISEE-3 was designed to be a spinning spacecraft with a pancake-beam antenna on its axis, which would be kept perpendicular to the ecliptic plane within &lo. Since the pancake beam antenna would work over a range of 12' centered on the spacecraft's "equator", this imposed additional constraints on the orbit shown in Fig. 6. The Z-amplitude selected for ISEE-3's orbit was 120,000 km, subtending 4.5" at the 1.5-million-km distance of the Sun-Earth L1 point, so the trajectory missed all of the constaint zones by the maximum amount, 1.5". The corresponding Y-amplitude is 666,670 km. An isometric view of this periodic halo orbit is shown in Fig. 7. Figure 8 shows the ISEE3 spacecraft in its flight configuration with its booms and antennas deployed. The drum-shaped spacecraft is spin stabilized with a nominal spin rate of 20 rpm. A pair of Sun sensors, accuracy about 0.1", determine attitude. A hydrazine propulsion system is used for attitude and AV maneuvers. There are 12 thrusters, four radial, four spinchange, two upper-axial, and two lower-axial. Eight conospherical tanks held 89 kg of hydrazine at launch, providing a total AV capacity of about 430 m/sec. Since a libration-point mission had never been flown before, this large capacity provided margin in case the actual station-keeping costs

50

D . W. Dunham and R . W. Farquhar

i I

ZONE

\

OF SOLAR INTERFERENCE [DIAMETER

-6Of

Fig. 6. ISEE-3 Halo Orbit Constaints (view from Earth).

ro SUN

A W

n

l

I

’% Y

Fig. 7. ISEE-3 Halo Orbit Around Sun-Earth L1 (isometric view).

were higher than theoretical models predicted. Detailed descriptions of the 13 science experiments are given in Ref. 9. ISEE-3 was launched by Delta rocket #144 on August 12, 1978. The spacecraft was built and operated by NASA’s Goddard Space Flight Center (GSFC). ISEE-3’s 100-day transfer trajectory is shown in Fig. 9, which like most of the following orbit plots is a rotating ecliptic-plane view with the Sun-Earth line fixed (horizontal). Three AV maneuvers totaling 57 m/sec removed launch injection errors and inserted into the desired halo orbit. These costs were not optimized due to several operational considerations lo.

Libmtion Point Missions, 1978-2002

51

MAGNETOMETER

Fig. 8. ISEE-3 Spacecraft, Flight Configuration.

Fig. 9. ISEE-3 Transfer Trajectory to Halo Orbit.

Before launch, a contingency study was made to see how large the transfer trajectory insertion underperformance could be and still be able to reach the halo orbit with ISEE-3’s AV capacity. It was assumed that a first midcourse correction (MCC) maneuver could be performed no earlier than 18 hours after launch. The MCC and halo orbit insertion (HOI) maneuvers were then optimized using the full-force model Goddard Mission Analysis System (GMAS) software to calculate the direct transfer costs shown in the

52

D.W. Dunham and R.W. Farquhar

right-hand column of Table 1. Table 1. Direct Transfers for V-ncr (in m/sec) Injection Errors (Launch August 12, 1978).

AV2xy 92.4 123.8 155.5 187.4

n 3 4 5 6

Z2

11.0 14.8 18.7 22.7

AV2 93.1 124.7 156.6 188.8

AVIXY 32.4 30.5 28.6 26.8

ZIN 2.8 -3.8 -4.8 -5.9

CVIN 35.2 34.3 33.4 32.7

TCV 128.3 159.0 190.0 221.5

A one-sigma ( a ) error by the Delta upper stage amounts to about 5 m/sec at injection, but since the velocity decreases rapidly with time from injection, the MCC needed to correct the error must be about six times larger according to the variant of the vis-viva equation: P 2VAV = -Aa. a2

Since about 200 m/sec capacity was wanted after arriving at the halo orbit, Table 1 shows that direct transfers could be used to correct injection errors as large as 6a. However, for large errors, a better solution to the problem was found. By allowing the spacecraft to complete one orbit rather than performing MCC right away, a maneuver is performed instead at the perigee following the injection. Then the injection error can be corrected for approximately its size without the factor of 6 penalty that results when MCC is performed 18h after injection. Another maneuver is performed a day or two after the perigee maneuver. The trajectory for the V-30 case with this strategy is shown with a solid curve in Fig. 10. Compared with the nominal transfer, shown as a dashed line, the new transfer costs more at P1 Id and at HOI so that the total cost is a little larger than for the direct strategy listed in Table 1. But with larger errors, the situation with the new strategy improves because the period of the first orbit is shorter. Consequently, the transfer following the P1 maneuver is closer to the nominal transfer (less rotation due to less time in the rotating frame), so the HOI cost decreases with larger errors. At 5a, the new strategy costs less total AV than the direct one. The trajectory for V-6a is shown in Fig. 11, which also shows the trajectory in case of 5% execution errors of the perigee maneuver. Like the nominal transfer, the trajectory is sensitive to initial velocity errors, but these can easily be removed a day or so after the perigee. In some cases, a maneuver

+

Libration Point Missions, 1978-2002

53

Fig. 10. V-3a Contingency Case.

at the first apogee, A1, is needed to raise perigee to prevent atmospheric reentry. Injection errors as large as -2Ou could have been corrected with this strategy ll. But fortunately this contingency plan was not needed because the actual injection error was only about 112-0 . ISEE-3 was injected into its planned halo orbit on November 20, 1978. During the four years it remained in the halo orbit, less than ten m/sec of AV were needed each year to maintain the orbit. This was perhaps twice the amount needed to maintain a quasi-periodic “balanced” orbit, removing only the unstable part of the motion, considering the orbit determination errors. ISEE-3 used what would be considered a slightly inefficient strict control strategy, always targeting back to minimize the residuals from the nominal path rather than a loose strategy such as the “energy balancing” maneuvers used by some of the later missions 12. With a large fuel supply, there was little incentive to maintain the halo orbit in a very optimum way; even with the strict control strategy, ISEE-3 could have been maintained in the halo orbit for about 30 years. But Farquhar and several scientists had other ideas for ISEE-3’s future. 4. Double Lunar Swingby Orbits

Some scientists were concerned about ISEE-3’s measurements being made from a relatively fixed distance from the Earth. What they really wanted was t o explore the geomagnetic tail of the Earth, swept back by the solar

54 D . W. Dunham and R . W. Farquhar

I

i

i

\

Fig. 11. V-6a Contingency Case with AVper errors.

wind, and to sample it at different distances from the Earth, everywhere from the Moon’s orbit to near L2, at 1.5 million kilometers, about four times the lunar distance. They wanted to take ISEES out of the halo orbit to make these new measurements in the opposite direction. In 1981 and 1982, funds for space science were limited and turning off operating spacecraft that had fulfilled their planned mission was being considered. There were strong motives to do something new with ISEE-3 at the end of its planned 4-year mission. At the same time, scientists were considering ISEE follow-on missions called Origins of Plasmas in the Earth’s Neighborhood (OPEN) that later evolved into the International Solar-Terrestrial Physics (ISTP) Program. The geomagnetic tail was a high priority for OPEN. Some sort of highly elliptical orbit would be needed to study the “geotail” at the desired different distances less than the L2 distance, but how could this be done ? Fig. 12 shows that a highly elliptical orbit generally maintains its orientation in inertial space so that the apogee is in the tail for only about one month of the year; the rest of the time, the spacecraft would spend most of its time outside the magnetosphere. What was needed was some way to rotate the line of apsides at the rate that the Earth moves around the Sun, about lo per day. In that way, the apogees could be kept in the geotail, as shown in Fig. 13. Geotail phenomena could be measured at different distances from the Earth. But how could the line of apsides be rotated at the needed rate ? It would cost about

Libmtion Point Missions, 1978-2002

55

Fig. 12. Uncontrolled Argument of Perigee (Line of Apsides Fixed).

400 m/sec per month to do this with AV maneuvers, clearly prohibitive. It was realized that lunar swingby maneuvers must hold the key to solve this problem, but how could it be done ? Astrodynamicists around the world worked on the problem, but the first ideas resulted in orbits that passed too close to the Moon, actually a little under the surface when solar perturbations and the approximately 0.05 eccentricity of the Moon’s orbit were taken into account. Farquhar discovered a good solution to the problem in 1979 13. The trajectory in an inertial frame is shown in Fig. 14. Starting at apogee A l , the spacecraft completes about 314 an orbit, encountering the Moon at 5’1. This trailing-edge swingby sends the spacecraft into a higher orbit, past Az, taking 33 days to reach the Moon’s orbit again (the dots in Fig. 14 are at l-day intervals). By then, the Moon will have completed a little more than one revolution so that it is in position for a leading-edge swingby that lowers the orbit into one like the initial orbit, but with the line of apsides rotated through an angle Aw. The process can then be repeated; two cycles of this double-lunar swingby trajectory are shown in the figure. It is called a one-month double-lunar swingby orbit because the duration of the outer loop is just over one month. The perigee and apogee distances of the inner orbit, and the lunar swingby distance, can be varied until Aw divided by the (in this case 2-month) cycle time equals the Earth’s mean motion around the Sun. These double-lunar swingby orbits are doubly periodic because they are

56

D.W. Dunhom and R.W. Forquhor

Fig. 13. Controlled Argument of Perigee (Rotating Line of Apsides).

Fig. 14. One-Month Double Lunar Swingby Orbit, Inertial Frame.

periodic in both lunar and solar rotating frames. The trajectory in the lunar rotating frame is shown in Fig. 15. Roger Broucke claims that he found this orbit in a comprehensive study of periodic orbits in the circular restricted three-body problem with the Earth-Moon mass ratio undertaken in the 1970’s, but he did not publish it, and did not realize its utility. The utility of double-lunar swingby orbits is seen best when they are portrayed in a rotating coordinate system with the Sun-Earth line fixed.

Libration Point Missions, 1978-2002 57

Fig. 15. One-Month Double Lunar Swingby Orbit, Earth-Moon Rotating Frame.

The “one-month” orbit is shown in the solar rotating frame at the top of Fig. 16. Since the geotail points approximately in the anti-Sun direction, it can be seen that the trajectory spends most of its time in the geotail, and traverses different distances along it. By decreasing the lunar swingby distance, it is possible to achieve other double-lunar swingby orbits with higher outer-loop apogees, to dwell in the geotail even longer and measure it over greater distances. Orbits with outer loops just over two months and three months are shown in the middle and bottom of Fig. 16, respectively. It is also possible to complete multiple revolutions in the inner orbit, increasing the time between the S1 and S2 swingbys to just under two months or even just under three or more months; in these cases, the distance between 5’1 and S2 increases. Keep in mind that the trajectories in Figures 14-16 were computed with patched conics and circular orbits for the Sun and Moon. How are double-lunar swingby orbits related to libration-point orbits?a Both are high-altitude orbits that maintain a fixed orientation aAfter the presentation of this paper at the conference in June 2002,M. Hechler claimed that he independently discovered these double lunar swingby orbits in 1979. This claim is not supported by the facts. Although Hechler may have independently calculated these orbits, he acknowledges in a July 1979 ESOC internal document where he first presented his work on them that “Another orbit-type was suggested by Farquhar for the OPEN project which unfortunately was overlooked in the previous studies” (p. 26) 32. In a letter t o Farquhar dated July 6,1979,Hechler’s colleague J. Cornelisse requested a preprint of Ref. 13 after learning about the new double lunar swingby technique from Farquhar at

58

D. W. Dunham and R. W. Farquhar

in the Sun-Earth rotating frame, so both are of interest for space physics studies. Libration-point orbits are higher; in fact, in the real solar system with full perturbations, double-lunar swingby orbits with five-month and longer outer loops pass near, or even around, the L1 or Lz libration point. If the timing of the Moon is right (and that can be designed), it is very easy to transfer between these two types of orbits.

Fig. 16. Double Lunar Swingby Orbits, Sun-Earth Rotating Frame.

the Astrodynamics Specialist Conference in Provincetown, Massachusetts in late June, 1979. Earlier in 1979, Hechler and Cornelisse developed and published a different, less practical lunar swingby technique that required AV maneuvers to avoid lunar impact 33.

Librntion Point Missions, 1978-2002

59

5. The ISEE-3/ICE Extended Mission In March 1981, Fred Scarf, the principal investigator for ISEE-3’s plasma wave experiment, wanted to use the spacecraft to explore the distant geomagnetic tail or perhaps even to fly through the tail of a comet. He contacted Farquhar about these possibilities; Farquhar realized that ISEE-3, orbiting the LI libration point, could easily leave the halo orbit (on an unstable manifold, in current terminology) to travel to a wide variety of locations, perhaps with the help of lunar swingbys. At first, the low telemetry rates of ISEE-3’s antenna seemed to preclude a comet option. But in July 1981, Joel Smith and Warren Martin at JPL noted that, with upgrades that had recently been made to the Deep Space Network (DSN) antennas, it would be possible to support a data rate of 1000 bits/sec from ISEE-3 at a distance of 0.5 A.U. Momentum built for an extended mission to a comet, preferably also including a geomagnetic tail excursion, especially after plans for a separate dedicated U.S. mission to Halley’s comet were abandoned in September 1981. For various reasons, mainly the shorter communication distance, an encounter with comet Giacobini-Zinner (GZ) was selected 6 . The energies were right, but at first it was not clear how to reach GZ. It was a difficult two-point boundary value problem, with ISEE-3 flying in a fixed halo orbit and the encounter with GZ having to occur on September 11, 1985 when the comet crossed the ecliptic plane. As explained in Ref. 6, double lunar swingby orbits provided the key. The best way to initiate a lunar swingby sequence was found, illustrated in Fig. 17. A 4 m/sec retro AV caused ISEE-3 to slowly leave the halo orbit and fall towards the Earth and the lunar orbit. Solar perturbations robbed energy from the orbit during late 1982, setting up the good geomagnetic tail passage during a “3month” loop in early 1983. Since ISEE-3’s trajectory was not in the lunar orbit plane, a 34 m/sec out-of-plane maneuver was needed near apogee on February 8 to target the first lunar swingby, 4 ,on March 30, 1983. Backwards integrations from GZ also showed how the end of the trajectory must be, from the comet to S4 in Fig. 21. The backwards integrations even found S3, approximately. But how could Sl in Fig. 17 be matched with Ss in Fig. 21 ? Many possible combinations of lunar swingbys were investigated 14. The best solution turned out to be one of the simplest, a long five-month outer loop that spent a few months near LZ shown in Fig. 18. In 1982 when calculating these trajectories, Dunham discovered an interesting trajectory by decreasing the lunar swingby distance in Fig. 18.

60 D . W. Dunham and R . W. Farquhar

Fig. 17.

ISEE-3 Transfer from Halo Orbit to Geomagnetic Tail.

1 Dav

-

f l az

__y_LI

Sun-EanhUne

Geoma&tic Tail

s1: 330-83 s2: 4-23-63

$3: 9-27-83

Fig. 18. ISEE-3 Five-Month Geotail Excursion.

Earlier, he added a subroutine to produce a printer plot of the final trajectory calculated during a run of the program, but unfortunately the plot he produced was lost. Due to its possible current interest, he regenerated the trajectory. First, he started with an actually determined state vector for ISEE-3 just after S1 provided by Craig Roberts, and used the Swingby program to rather closely duplicate Fig. 18; the result is in Fig. 19. Dunham added very small retro AV’s at the Pz perigee on April 2, 1983, decreasing the lunar swingby distance to achieve longer outer loops, as he did in 1982. By decreasing the velocity by only 4 mm/sec, the lunar swingby distance was decreased by 49 km, resulting in the trajectory to a small-amplitude Lissajous orbit about Lz shown in Fig. 20. The trajectory

Libmtion Point Missions, 1978-2002

61

Rorating Ecliptic-PI&e View, Fixed Sun-Earth Line

S beyond lunar orbit

Fig. 19. Recent Reconstruction of ISEE-3’s Five-Month Geotail Excursion.

did not satisfy the comet goals of ISEE-3 at the time, so unfortunately Dunham did not publish it. But it proved the concept of using a lunar swingby to achieve a small-amplitude LZ Lissajous orbit for very little (only statistical) AV several years before such trajectories were planned for the Relict-2 and MAP missions.

Rotating EcliptiePlane View, Fied Slm-Earth t i e

Sr lunar swingby

Fig. 20. Possible ISEE-3 Trajectory to 152 Lissajous Orbit (1982).

62

D. W. Dunham and R . W . Farquhar

ISEE-3 missed its chance to become the first Sun-Earth L2 satellite (that honor goes to the Microwave Anisotropy Probe, or MAP, spacecraft launched in 2001, as noted below), but instead it became the first spacecraft to make in-situ measurements of a comet, a more important distinction to most. Continuing from Fig. 18, ISEE-3’s escape trajectory is shown in Fig. 21. A close-up view of the S5 lunar swingby that made the spacecraft’s trajectory hyperbolic relative to the Earth is shown in Fig. 22. Just after that swingby, NASA re-named the spacecraft the International Cometary Explorer (ICE). The first 3.4 years of ICE’S heliocentric orbit is shown in a much larger ecliptic-plane view, rotating with the Sun-Earth line fixed, in Fig. 23. Three AV maneuvers totaling 42 m/sec were performed in 1985 to target ICE to fly through the tail axis of Comet GZ about 8000 km from the nucleus. More details of the highly successful encounter are given elsewhere 6 . In 2014, ICE will pass near the Earth, and an in-plane AV of 1.5 m/sec and an out-of-plane AV of 39 m/sec were performed on February 27 and April 7, 1986, respectively, to target a lunar swingby on August 10, 2014. That swingby plus some small maneuvers could capture ICE back into an Earth orbit, perhaps even returning it to a libration-point orbit 1 5 . But another possibility was found in 1998. With a AV of about 25 m/sec performed on January 10,2010, ICE could swing by the Earth at a distance of about 36 Earth radii and encounter GZ a second time on September 19, 2018 6 . ISEE-3/ICE may be known to most for its comet “first”, but in astronautics it is most famous for pioneering the use of both libration-point and double-lunar-swingby orbits. 6. Relict-2, First Plans for an Lz Astronomical Satellite

ISEE-3 proved the utility of an orbit about the Sun-Earth L1 point for space physics (especially upstream solar wind) measurements. Orbits about the Sun-Earth L2 point could be used to memure the geomagnetic tail, but already ISEE-3 showed that double-lunar swingby orbits were better for that purpose. However, in the late 1980’s, many mission planners learned the value of orbits near the Sun-Earth L2 point for astronomical observations 16. A satellite there would have an unobstructed view of well over half of the sky with no interference from either the Sun, the Earth, or the Moon, all of which would remain within about 15” of the direction to the Sun. Especially observations in the infrared would benefit since the geometry and construction of the spacecraft would allow passive cooling to very low temperatures; the solar cell panels pointing towards the Sun could shade the

Libration Point Missions, 1978-2002

63

As

IAV

* 6.5 mkecl

ISEE-ITm,eclw Rdanve to Fixed SUrrEanh Llna

Escape Trajectory

Fig. 21. ISEE-3 Escape Trajectory.

-

GMT 1753 18 19

1859 1845 18 47

Fig. 22.

Close-up of ISEE-3’s 5th Lunar Swingby, December 22, 1983.

scientific instruments. A small-amplitude Lissajous orbit about Lz would be better than the large-amplitude one that would be required by a periodic halo orbit. A dish antenna to send data back to Earth would not have to swivel as far with a small-amplitude orbit. Like for ISEE-3, there would be a central “exclusion zone” both for receiving commands from Earth and for possible long eclipses. The apex of the Earth’s shadow almost reaches the mean L2 distance so that total eclipses are rare, but deep partial eclipses

64

D . W. Dunham and R. W. Farquhar

Fig. 23.

Initial Heliocentric Orbit of ICE with Comet GZ Flyby in 1985.

could damage the spacecraft. Maneuvers to avoid the exclusion zone would be similar to those needed for station keeping, to remove the unstable component of the motion. A Russian microwave astronomy satellite called Relict-2 was the first one proposed to use a Sun-Earth L2 orbit in about 1990 17. Since the spacecraft would have only a limited AV capacity, a lunar swingby would be used to achieve the desired small-amplitude orbit. A possible trajectory for Relict-2 published in Ref. 17 is shown in Fig. 24, the usual rotating ecliptic-plane view with fixed horizontal Sun-Earth line. Unfortunately, the mission has yet to be funded due to financial problems with the Russian space program following the collapse of the Soviet Union. There is still interest in the mission. The 271 lunar swingby shown in Fig. 24 needs to be performed on just one day each month when the Sun-Earth-Moon angle is about 135” between new and full moon. In order to have a reasonable launch window, the spacecraft would be launched into an elliptical “phasing orbit” with apogee just beyond the Moon’s orbit where it would stay for several weeks before the lunar swingby. This allows a dozen or more launch opportunities each month

Libmtion Point Missions, 1978-2002

65

c-

Fig. 24. Possible Relict-2 Trajectory to Sun-Earth Lz using Lunar Swingby.

rather than the single one that a direct launch to the Moon would entail. Phasing orbits to target a lunar swingby were first used by the Japanese Hiten double-lunar swingby mission in 1990, the second one after ISEE-3 to fly a double-lunar swingby orbit 1 8 . The cartoons in Fig. 25, drawn at the Japanese Institute of Space and Astronautical Sciences to explain the phasing orbits used by Hiten, show the advantages and disadvantages of different numbers of phasing orbits.

7. SOHO The Solar Heliospheric Observatory (SOHO), the 2nd ISTP mission, was the 2nd libration-point mission, a sophisticated ESA solar observatory launched with an Atlas from Cape Canaveral on December 2, 1995 and operated by GSFC. It entered the L1 halo orbit on February 14, 1996. The continuous detailed solar observations available to all via the Web have set a new standard for solar observation. Many dozens of small “sun-grazing” comets have been discovered with SOHO’s coronagraph. But its orbit is rather unremarkable, a periodic halo orbit with Z-amplitude 120,000 km being a virtual carbon copy of ISEE-3’s orbit; see the ecliptic-plane view in Fig. 26 and References 12, 19, and 20. Communication with SOHO was lost for 6 weeks in mid-1998 due to an attitude maneuver mishap that temporarily crippled the spacecraft. Recovery of the mission, and the heroic efforts to work around the loss of all of SOHO’s gyros, make an interesting story 21.

66

D. W. Dunham and R . W. Farquhar

1) U R E V .

* CORRECrWN OF m

N ERROR NEUIS

LllRoEDELTA-V. A

8. ACE

The Advanced Composition Explorer (ACE) was the 3rd libration-point mission, launched with a Delta from Cape Canaveral on August 25, 1997. ACE, a particles and fields spacecraft, was built at the Applied Physics Laboratory and is operated by GSFC. Like ISEE-3 and SOHO, ACE was placed into orbit about the Sun-Earth L1 point. ACE started its Lissajous orbit, with an X-amplitude of 81,755 km and a Z-amplitude of 157,406 km, on December 13,1997. Its in-plane motion is shown in Fig. 26. Fig. 27 shows the out-of-plane motion, and locations of the first several stationkeeping AV maneuvers, for both ACE and SOHO. ACE was the first spacecraft to fly a Lissajous orbit, including “Z-axis control” maneuvers to avoid the solar exclusion zone 12722123.

Libmtion Point Missions, 1978-2002

67

t Earth's&

Fig. 26. jection.

SOHO Halo and ACE Lissajous L1 Orbits, Solar Rotating Ecliptic-Plane Pro-

Fig. 27. SOHO and ACE Orbits with Stationkeeping AV Locations, View Looking Towards the Sun.

9. WIND

The WIND spacecraft, a space physics spacecraft that like SOHO is part of the ISTP program, was funded by NASA and launched with a Delta from Cape Canaveral on November 1, 1994. The spacecraft, operated by GSFC, used four phasing orbits before its initial lunar swingby. For the first time, WIND used a Sunward-pointing double lunar swingby orbit to repeatedly cross and measure the forward bow shock region of the magne-

68 D.W. Dunham and R.W. Farquhar

tosphere. But following the first lunar swingby, WIND made a large loop around the Sun-Earth L1 point, from February to June, 1995, during its initial 7-month outer loop, qualifying it as the 4th libration-point mission. The next several outer loops were below L1, as shown in Fig. 28. From November 1997 to June 1998, near the end of its nominal mission, WIND flew an &month outer loop, again passing around L 1 . During its extendedmission phase, WIND has continued its pioneering orbital acrobatics, including the first extensive out-of-plane measurements of the Earth’s magnetosphere and the use of a two-week “back-flip” (coined by C. Uphoff) trajectory using two close lunar swingbys connected by an out-of-plane loop to change its sunward-pointing double lunar swingby trajectory to an anti-sunward-pointing one 24,25y26,27.

d...

To SUn

Fig. 28. WIND Trajectory, November 1994 V September 1997, Solar Rotating Ecliptic Plane Projection.

10. MAP

The Microwave Anisotropy Probe (MAP), an astronomical satellite designed primarily to measure the “Big-Bang” background radiation, was funded by NASA and launched with a Delta from Cape Canaveral on June 30, 2001. Following Relict-2’s design l7 (compare Fig. 24 with Fig. 29 below), MAP used phasing orbits and a lunar swingby to achieve a smallamplitude Lissajous orbit about the Sun-Earth L2 point on October 1, 2001, becoming the 5th libration-point mission. It was the 2nd (after ISEE3) mission to obtain measurements near L2 and the first mission dedicated to this purpose, the first “observatory” to use L2 as proposed in 1990 1 6 . The need to avoid even shallow lunar partial eclipses complicated the orbit design for MAP 23t28.

Libmtion Point Missions, 1978-2002

Fig. 29.

69

MAP Trajectory, Solar Rotating Frame, Isometric View.

11. Genesis

The Genesis spacecraft, designed to collect samples of the solar wind and return them to Earth for detailed analysis, was funded by NASA and launched with a Delta from Cape Canaveral on August 8,2001. Following a trajectory very similar to ISEE-3’s (compare Figures 9 and 17 with Fig. 30 below), Genesis launched into a transfer orbit towards a large-amplitude Sun-Earth 151 Lissajous orbit with a Z-amplitude of about 120,000 km, very similar to ISEE-3’s halo orbit. The Lissajous orbit insertion occurred on November 16, 2001 to become the 6th libration-point mission. The spacecraft collectors were deployed to start capturing solar wind particles a few days later; this phase will last 29 months. Like ISEE-3, Genesis will use lunisolar perturbations to shape its return trajectory, but this time the target is a large Air Force test range in Utah. The spacecraft will use aerobraking during its descent into the atmosphere during a morning in August 2004. During that time of day and year, the weather in Utah is normally very favorable for the return capsule recovery operations, but if necessary, the spacecraft can complete another orbit to return about 20 days later 29930.

70

D.W. Dunhom and R . W. Foquhor

Fig. 30.

Genesis Trajectory, Solar Rotating Ecliptic Plane View.

12. Future Libration-Point Missions

For 18 years after ISEE-3’s launch, there were no further libration-point missions. But in the six-year period starting in December 1995, five libration-point missions were successfully launched and operated, gathering important new scientific results in a new cost-effective way, as described above. But these missions only scratch the surface of the potential returns that libration-point orbits can deliver. Table 2 lists the six missions that have now made the flight of these orbits almost routine, and also lists eight planned missions during the next dozen years, all observatory missions to Sun-Earth Lz orbits. Besides these, the TRIANA spacecraft, designed to image the Earth continuously from near the Sun-Earth L1 point, has already been built, but it is not clear now when or if the spacecraft will be launched 31. Nevertheless, a rich future of libration-point missions is assured, building on the pioneering work of ISEE-3 and the five other libration-point missions that are still operating. For more information, and the latest developments, the Halo Orbit and Lunar Swingby Missions section of the following Web site provides links to the Web sites of most of the missions listed in Table 2: http://highorbits.jhuapl.edu.

Libration Point Missions, 1978-2002

71

Table 2. Flown and Planned Libration-Point Missions. *Acronyms: ISEE (International Sun-Earth Explorer; SOHO (Solar Heliosphere Observatory); ACE (Advanced Composition Explorer); MAP (Microwave Anisotropy Probe); GAIA (Global Astrometric Interferometer for Astrophysics); NGST (Next Generation Space Telescope); TPF (Terrestrial Planet Finder). Sun-Earth Date of Orbit Lib. Point Insertion Mission Purpose L1, L2 1978, 1983 Solar wind, cosmic rays, plasma studies 1996 Solar observatory SOHO (ESA/NASA) L1 1997 Solar wind, energetic L1 ACE (NASA) particles L1 1995 Solar-wind monitor WIND (NASA) 2001 Cosmic microwave LZ MAP (NASA) background L1 2001 Solar-wind composition Genesis (NASA) L2 2007 Far infrared telescope Herschel (ESA) 2007 Cosmic microwave L2 Plank (ESA) background L2 2008 Stellar observations Eddington (ESA) L2 2010 Deep space observatory NGST (NASA) L2 2011 X-ray astronomy Constellation-X (NASA) Galactic structure, GAIA (ESA) L2 2012 Astrometry 2012 Detection of distant L2 TPF (NASA) planets 2014 Detection of Earth-like L2 DARWIN (ESA) planets

Mission* I S E S 3 (NASA)

References 1. Columbo, G.:The Stabilization of an Artificial Satellite at the Inferior Conjunction Point of the Earth- Moon System, Smithsonian Astrophysical Observatory Special Report No. 80, November 1961. 2. Clarke, A. C.:Interplanetary Flight, Temple Press Books Ltd., London, 1950, pp. 111-112. 3. Clarke, A. C.:The Making of a Moon, Harper & Brothers, New York, 1957, pp. 192-198. 4. Farquhar, R. W.:Station-Keeping in the Vicinity of Collinear Libration Points with an Application to a Lunar Communications Problem, AAS Science and Technology Series: Space Flight Mechanics Specialist Symposium Vol. 11, pp. 519-535 (presented at the ASAS Space Flight Mechanics Specialist Conference, Denver, Colorado, July 1966). 5. Farquhar, R. W. and Kamel, A. A.:Quasi-Periodic Orbits About the 'Ikanslunar Libration Point, Celestial Mechanics, Vol. 7, No. 4, June 1973, pp. 458-

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473. 6. Farquhar, R. W.:The Flight of ISE&3/ICE: Origins, Mission History, and a Legacy, J. Astronautical Sciences, Vol. 49, No. 1, January 2001, pp. 23-73 and presented at the AIAA f AAS Astrodynamics Conference, Boston, Massachusetts, August 11, 1998. 7. Farquhar, R. W.:Linear Control System for a Satellite at the Sun-Earth Collinear Libration Points, Lockheed Interdepartmental Communication FM-52-20-274, March 9, 1964. 8. Ogilvie, K. W. et al.:International Sun-Earth Explorer: A Three-Spacecraft Program, Science, Vol. 198, No. 4313, October 14, 1977, pp. 131-138. 9. Special Issue on Instrumentation for the International Sun-Earth Explorer Spacecraft, IEEE Transactions on Geoscience Electronics, Vol. GE-16, July 1978. 10. Farquhar, R. W. et al.:Tkajectories and Orbital Maneuvers for the First Libration-Point Satellite, J. Guidance and Control, Vol. 3, No. 6, November 1980, pp. 549-554. 11. Dunham, D.:Contingency Plans for the ISEE-3Libration-Point Mission, AAS Paper 79-129 presented at the AASf AIAA Astrodynamics Specidist Conference, Provincetown, Massachusetts, June 25-27, 1979. 12. Dunham, D. W. and Roberts, C. E.:Stationkeeping Techniquesfor LibrationPoint Satellites, J. Astronautical Sciences, Vol. 49, No. 1, January 2001, pp. 127-144 and presented at the AIAAIAAS Astrodynamics Conference, Boston, Massachusetts, August 11, 1998. 13. Farquhar, R. W. and Dunham, D. W.:A New Trajectory Concept for Exploring the Earth’s Geomagnetic Tail, J. Guidance and Control, Vol. 4, No. 2, March 1981, pp. 192-196, presented as AIAA Paper 80-0112 at the AIAA Aerospace Sciences Meeting, Pasadena, California, January 14-16, 1980. 14. Muhonen, D. P. et al.:Alternative Gravity Assist Sequences for the ISEE-3 Escape Tkajectory, J. Astronautical Sciences, Vol. 33, No. 3, July 1985, pp. 255-288. 15. Roberts, C. E. et aZ.:The International Cometary Explorer Comet Encounter and Earth Return Trajectory, Advances in the Astronautical Sciences, Vol. 69, 1989, pp. 709-725. 16. Farquhar, R. W. and Dunham, D. W.: Use of Libration-Point Orbits for Space Observatories, Observatories in Earth Orbit and Beyond, Kluwer Academic Publishers, 1990, pp. 391-395. 17. Eismont, N., et al.:Lunar Swingby as a Tool for Halo-Orbit Optimization in Relict-2 Project, ESA SP- 326, December 1991, pp. 435-439. 18. Uesugi, K.:Space Odyssey of an Angel Summary of the Hiten’s Three Year Mission, Advances in the Astronautical Sciences, Vol. 84, 1993, pp. 607-621. 19. Dunham, D. W., et al.:Tkansfer Trajectory Design for the SOHO LibrationPoint Mission, IAF Paper 92-0066, September 1992. 20. Domingo, V., et al.:The SOHO Mission: An Overview, Solar Physics, Vol. 162, No. 1-2, December 1995, pp. 1-37. 21. Roberts, C. E.:The SOHO Mission Halo Orbit Recovery from the Attitude Control Anomalies of 1998, presented at the International Conference on

Libmtion Point Missions, 1978-2002 73

Libration Point Orbits and Applications, Aiguablava, Spain, June 10, 2002. 22. Stone, E. C., et al.:The Advanced Composition Explorer, Space Science Reviews, 1998. 23. Beckman, M.:Orbit Determination Issues for Libration Point Orbits, presented at the International Conference on Libration Point Orbits and Applications, Aiguablava, Spain, June 13, 2002. 24. Dunham, D. W. et al.:Double Lunar-Swingby Trajectories for the Spacecraft of the International Solar-Terrestrial Physics Program, Advances in the Astronautical Sciences, Vol. 69, 1989, pp. 285-301. 25. Acuiia, M. H. e t al.:The Global Geospace Science Program and Its Investigations, Space Science Reviews, Vol. 71, 1995, pp. 5-21. 26. Franz, H., e t al.: WIND Nominal Mission Performance and Extended Mission Design, J. Astronautical Sciences, Vol. 49, No. 1, January 2001, pp. 145167 and presented at the AIAA/AAS Astrodynamics Conference, Boston, Massachusetts, August 11, 1998. 27. Uphoff, C. W.:The Art and Science of Lunar Gravity Assist, Advances in the Astronautical Sciences, Vol. 69, 1989, pp. 333-346. 28. Cuevas, 0. e t al.:An Overview of Trajectory Design Operations for the Microwave Anisotropy Probe Mission, AIAA Paper 2002-4425, presented at the AIAA/AAS Astrodynamics Specialist Conference, Monterey, California, August 5, 2002. 29. Lo, M. e t al.:Genesis Mission Design, J. Astronautical Sciences, Vol. 49, No. 1, January 2001, pp. 169-184 and presented at the AIAAIAAS Astrodynamics Conference, Boston, Massachusetts, August 11, 1998. 30. Wilson, R. S.:The Genesis Mission: Mission Design and Operations, presented at the International Conference on Libration Point Orbits and Applications, Aiguablava, Spain, June 10, 2002. 31. Houghton, M. B.:Getting to L1 the Hard Way: TRIANA’s Launch Options, presented at the International Conference on Libration Point Orbits and Applications, Aiguablava, Spain, June 10, 2002. 32. Hechler, M.:On the Orbit Selection for a GEOS 3 Magnetotail Mission, European Space Operations Centre, Mission Analysis Division Working Paper No. 102, July 1979. 33. Hechler, M.:GEOS 3 Magnetotail Mission: On Maneouvre Assisted Multiple Lunar Swingby Orbits, European Space Operations Centre, Mission Analysis Division Working Paper No. 87, February 1979.

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Libration Point Orbits and Applications G . Gbmez, M. W. Lo and J . J . Masdemont (eds.) @ 2003 World Scientific Publishing Company

TECHNICAL CONSTRAINTS IMPACT ON MISSION DESIGN TO THE COLLINEAR SUN-EARTH LIBRATION POINTS N. EISMONT, A. SUKHANOV and V. KHRAPCHENKOV Space Research Institute, Russian Academy of Sciences 11 7997,Profsoyuznaya street 84/32, Moscow, Russia

For the practical realization of the mission t o the collinear Sun-Earth libration points technical constraints play a significant role. In the paper the influence of the constraints generated by the use of piggi-back mode of the delivering spacecraft to the vicinity of libration points are studied. High elliptical parking orbit of Molniya is taken as initial orbit for start to the 151, I52 libration points. The parameters of this orbit are supposed to be fixed and determined by the main payload demands. The duration of the passenger payload keeping on the mentioned 12 hours period orbit is limited for the case when launcher upper stage is used for the velocity impulse applying to put spacecraft onto transfer orbit to the libration point. The possibility to use one axis attitude control of the spacecraft for the executing correction maneuvers is investigated, supposing that spacecraft is spin stabilized with the spin axis directed to the Sun and maneuver impulse goes along this axis. The cost of constraints is presented in terms of characteristic velocity and time of transfer to the libration point vicinity. The goal of the paper is to understand the possibility of using regular launches of Molniya communication satellite by Soyuz-Fregat launch vehicle for sending low cost scientific spacecraft to Sun-Earth libration points.

1. Introduction

The mission to the vicinity of Sun-Earth collinear libration points are fulfilled and planned for the scientific experiments gaining big advantages from use of this region of space for optimal measurement conditions. Some of these experiments demand to keep spacecraft comparatively close to the libration points, for example the ones intended for microwave background and infrared radiation studies. The other experiments such as ones for solar wind exploration allow high values of amplitude in spacecraft motion relative to the libration points. 75

N. Easmont, A. Sukhanov and V. Khrapchenkov

76

So it is a matter of scientific interest to explore feasibility to put spacecraft into orbits around collinear libration points supposing that deviation of the spacecraft from these points in the limits inside 1400 thousands km in transversal towards Sun-Earth line direction is acceptable. Such approach is dictated by the impact of technical constraints on the possibilities to put s/c onto orbit around libration points. These constraints are generated by the necessity to decrease the cost of launch and mission at large. 2. List and nature of technical constraints

In our further consideration we accept assumption that possible libration points missions are restricted by possibilities of passenger launches by Russian launch vehicles, which are used for putting payload onto high elliptical orbits. Now in use are the following launch vehicles for these purposes:

- Proton with DM on Breeze-M upper stage; - Molniya; - Soyuz with fiegat upper stage. Proton is used for the launches onto geostationary orbit, Molniya and Soyuz launch Molniya communication satellites and Oko military SIConto high elliptical 12 hours period orbit. During launch onto geostationary orbit s/c is put onto geostationary transfer high elliptical orbit (GTO). It is important to underline that parameters of the orbit for the mentioned payload are fully determined by the payload demands including date and time of launch. The initial parameters of these payloads are close the following l : Molniya: 0 0

0 0 0

--

period 0.5 star days, perigee height 640 km, inclination 63', perigee argument 288', ascending node longitude is determined by demands of constellation configuration.

Oko: the similar parameters besides perigee argument what is about 320'. For Proton GTO is quite typical with inclination about 47.5', perigee height 200 km and apogee height 35920 km, perigee argument 0' '.

Constraints Impact on Mission Design 77

In case of Proton use for passenger payload launch the main obstacle is necessity to modify upper stage in order to mount additional payload. Quite obviously this payload is to be mounted between upper stage and main spacecraft to be launched on GTO what is not so easy to do taking into account a broad variety of possible s/c to be put at the GTO to GSO by Proton. In case if upper stage to be used for maneuver to further transfer to Halo-orbit time delay for the appropriate additional engine burn is sufficient technical constraint (now the upper limit for this delay is several hours). It means that the s/c for the Halo-orbit mission is to be equipped by its own engine or Proton upper stage is to be significantly modified. Use of Molniya launch vehicle excludes fully additional burn of the upper stage for our purposes because it can be started only once. The most convenient option is the use of Soyuz-Fregat launch vehicle because F’regat upper stage equipped by multistarted engine. But even in this case there are restrictions on the time interval between start of the launcher and last engine burn of the upper stage engine. Given above can be summarized as constraint on the time interval between launch of the main s/c and the last maneuver to put onto Halo-orbit passenger payload. In addition in many cases it means that nominal transfer from initial high elliptical orbit to the Halo-orbit is to be executed by one impulse maneuver. To decrease the cost of the s/c the simplest attitude control is to be considered such as the case of spin stabilization. Example of such stabilization is Russian Prognoz series s/c with the spin axis periodically targeted towards Sun. One of the s/c of this series was planned to be used in so-called “Relict2” project with the main goal of investigation of microwave background radiation. Due to possibility to target the spin axis only in Sun direction the correction maneuvers of this s/c were to be fulfilled in Sun (or opposite to the Sun) direction, what is to be added to the list of possible technical constraints. The other example of the attitude constraint is related to the libration point mission with the use of Solar Electric Propulsion (SEP) ’. As it is well known SEP demands rather high electrical power. In optimal case solar panels are to be kept in position orthogonal to the Sun direction and thruster axes are to be directed along velocity vector, what can be achieved only with rotating panels or thrusters. To avoid this difficulty and to simplify attitude control it was proposed to use spin stabilization with

78 N . Eismont, A . Sukhanov and V . Khmpchenkov

spin axis being orthogonal towards Sun direction and being in orbit plane. Thrust is applied along spin axis. Solar panels form cylinder surface with the axis along spin axis. Thrusters are on when the angle between velocity vector and spin axis is less then 60". So the technical constraints listed above look rather strong to explore very possibility of mission to collinear libration points. 3. Launch date impact on halo-orbit if ascending node is fixed

Figures 1, 2 present trajectories to the vicinity of LZ libration point and around it supposing one impulse transfer from GTO having inclination 62" and argument of perigee 0" with respect to equator and ascending node longitude 0'. The trajectories are given in solar-ecliptic coordinate system with tics interval equal 4 days. Four launch dates are checked: 15.01.98, 15.02.98, 27.02.98 and 22.03.98. One can see from these figures that for dates from 15.02 to 22.03 the resulting Halo-orbit are rather similar with Y-axis amplitude within 800 000 km, X-coordinate changing within limits from 1 150 000 km to 1 750 000 km and Z-axis amplitude 150 000 km. For the launch date 15.01 Yaxis amplitude is increased to 1 300 000 km with similar rise at motion amplitudes along other axes. But in any case Figures 1, 2 illustrate feasibility of broad enough launch window to put s/c onto Halo-orbit, more then two months, at least for the initial orbits with perigee argument close to zero. 4. Ascending node longitude influence on halo-orbit characteristics

Most regular launches onto high elliptical orbit are planned for Molniya communication satellites. The main source of concern for using these launchers for putting passenger s/c onto Halo-orbit is value of the argument of perigee (288') what means too big angle between ecliptic and line of apsides. Figures 3,4 illustrate the feasibility to put s/c onto Halo-orbit under so strong constraints. Three trajectories are presented for different values of ascending node longitude of initial (parking) orbit: 0 deg., 90 deg. and 180 deg. First case corresponds to minimum angle between apogee-perigee line and elliptic plane and the last one - maximum angle. Consequently in the

Constraints Impact on Mission Design

1500

Launch date impact on the one impulse Halo-orblt

Launch date -15 0 1 98 0 -15 02 98 * -27 0 2 98 0 -22 03 98

1000

2

79

500

P

9 2

O

5 ’-500

-1000

-l500!,

,,,,,,, ,,,

-2000

I , r I I

I

-1500

-1000

I_r__l,,,,,,,,,,, -500

0

500

X, thousands krn

Fig. 1.

-300 -1500

-1000

-500

500

1000

1500

Y, thousands krn

Fig. 2.

first case Halo-orbit looks the most “n0rma1’~and the last case presents the trajectory with quite visible deviations from the libration point: up to 1 850 000 km in Y direction, and with approaching to the Earth in X-axis direction to 340 000 km. The trajectories presented on the Figures 3,4 are related to the different

80

N. Eismont, A . Sukhanov and V. Khmpchenkov

launch dates, depending on ascending node longitude: 0 deg. - 12.12.98, 90 deg. - 22.02.98, 180 deg. - 10.07.98. The values of osculating eccentricity e and semimajor axis a are the following: e = 0.98, 0.99, 0.995; a = 652 154 km, 915 990 km, 1 525 700 km.

One impulse Halo-orbit with initial pengee argument 290 deg , inclination 62 deg and ascending node longitude 0, 90 and 180 deg 1500

1000

* - 0 deg A-

4

90 deg

- 180 deg

500

3 4

0

2

'

-500

i -1000

-1500 ~

-2000 -1800 -1600 -1400 -1200 -1000 -800

-600

-400

-200

0

200

X. thousands k m

Fig. 3.

Similar calculations have been done for perigee argument 320" (case of Oko s/c as a main payload) which confirmed the feasibility to put s/c onto trajectory in vicinity of L2 point using mentioned orbit as initial one (parking orbit). 5. Parking orbits

Despite broad window of dates for launching s/c onto Halo-orbit it may happen that the waiting time on the orbit of the main payload will be more than 8 months. It leads to the necessity to optimize parking orbit parameters. In the case when upper stage allows to put passenger payload onto higher orbit than GTO or Molniya orbit, the necessary propellent mass

Constraints Impact on Mission Design 81

One impulse Halo-orblt with lnltlal perigee argument 290 deg inclination 62 deg and ascendmg node longltude 0, 90 and 180 'deg

Y, thousands km

Fig. 4.

onboard s/c itself may be significantly reduced. But higher orbits are more influenced by Moon and Sun perturbation. The most critical here is perigee height evolution. To analyze evolution of the parking orbit the following formulae can be used 4:

-

ba = 0,

(1 - e2

+ 5e2 sin2w) ,

(5 cos2 i sin2 w

+ (1 - e2>(2 - 5 sin2 w)>,

(1) given secular evolution of the osculating parameters per orbit. Here a, e, Q, i, w, hp are osculating semimajor axis, eccentricity, ascending node longitude, perigee argument, height of perigee with respect to the

82

N. Eismont, A . Sukhanov and V. Khrapchenkou

perturbing body orbit plane, in our case ecliptic plane (approximately), po is Earth gravitational constant, ,ul is perturbing body gravitational constant (Moon and Sun in our case). From these formulas one can see that perigee height increases if perigee lies in the second or fourth quarter of the orbit counted from ascending node on ecliptic and rate (per day) of perigee height change is proportional a5f2.

It means that parking orbit with perigee argument coinciding with Molniya orbit one for some positions of ascending node values (with respect to equator) does not satisfy this requirement and perigee height will decrease for these values of node longitude. So to exclude additional constraint on initial orbit parameters the parking orbit is proposed to be chosen with Molniya orbit parameters as a baseline option. For this option for transfer onto libration point orbit delta-V impulse about 700 m/s is necessary. Taking into account possible nominal and correction maneuvers, additional 400m/s delta-V capacity is to be reserved onboard s/c (about 300 m/s in case if amplitude of Halo- orbit is to be decreased from maximum to zero 5, and 100 m/s for correction maneuvers). As to the case of Oko as a main payload (argument of perigee 320O)perigee height rises for any longitude of ascending node, so it is safe to transfer our s/c to the orbit with higher apogee using upper stage engine, for example onto orbit with 100 000 km semimajor axis applying -540 m/s delta-V with corresponding decreasing propellant onboard s/c. 6. Feasibility of maneuvers with direction of thrust constraints

As it was mentioned above constraints in engines thrust direction may be imposed as a low cost attitude control concept. It was shown in that for correction maneuvers in order to keep s/c on Halo-orbit, Sun-directed delta-V impulses are enough for solving this problem. The propellant loses in this case do not exceed 18 percent. It gives the possibility to use for Halo-orbit mission spin stabilized s/c with spin axis periodically targeted to Sun. Mission to libration point with the use of SEP and spin stabilized s / c was analyzed in 3 . It was supposed that thrust is applied when the angle between thrust and velocity vectors is inside 60' limits. Results of this work have confirmed the possibility of such approach.

Constraints Impact on Mission Design 83

Propellant consumption in this case will increase by 17 percent and time of transfer will be longer with factor 1.7 time comparing with case when thrust is directed along velocity vector.

7. Conclusions Technical constraints influence on feasibility and principal characteristics of Halo-orbit missions have been studied. The main source of these constraints are demands to decrease the cost of mission. It was shown that under constraints generated by requirements to launch s/c as a passenger together with the most regular mission such as Molniya, Oko and putting satellites onto geostationary transfer orbit, the mission to the collinear libration points are feasible. In the worst case spacecraft is to be equipped by engine unit with delta-V capacity up to 1100 m/s. Also it was concluded that use of spin-stabilized s/c with engine thrust along spin axis is possible. It is true for the case when spin axis is targeted to Sun and also for the solar electric propulsion when spin axis is orthogonal to the Sun direction and lies in orbit plane.

References 1. Another Molniya-3. Novosti Kosmonavtiki, #12 (227), 2001, pp. 45-46. 2. Raduga-I: Military Comsat System Replenished. Novosti Kosmonavtiki, #12 (227), 2001, pp. 37-38. 3. A.A. Sukhanov, N.A. Eismont:Low Thrust nansfer to Sun-Earth L1 and L2 Points with a Constraint of the Thrust Direction. Paper presented to this Conference. 4. Introduction to the Theory of Flight of Artifical Earth Satellites (in Russian), p. 491. Moscow, Nauka, 1965. 5. N. Eismont, D. Dunham, S.-C. Jen and R. FarquharLunar Swingby its a Tool for H a l e Orbit Optimization in Relict-2 Project. Proceedings of the ESA Symposium on Spacecraft Flight Dynamics, Germany, 30.-4 October, 1991 (ESA SP-326, December 1991), pp.435-439. 6. P. Eliasbeng, T. Timokhova:Orbital Correction of Spacecraft in Vicinity of Collinear Center of Libration (in Russian). Space Research Institute Preprint 1003, Moscow, 1985.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

LIBRATION ORBIT MISSION DESIGN: APPLICATIONS OF NUMERICAL AND DYNAMICAL METHODS D. FOLTA and M. BECKMAN Guidance, Navigation d Control Center’s Flight Dynamics Analysis Branch (Code 572) NASA’s Goddard Space Flight Center, Greenbelt, MD 20771, USA

Sun-Earth libration point orbits serve as excellent locations for scientific investigations. These orbits are often selected to minimize environmental disturbances and maximize observing efficiency. Trajectory design in support of libration orbits is ever more challenging as more complex missions are envisioned in the next decade. Trajectory design software must be further enabled to incorporate better understanding of the libration orbit solution space and thus improve the efficiency and expand the capabilities of current approaches. The Goddard Space Flight Center (GSFC) is currently supporting multiple libration missions. This end-to-end support consists of mission operations, trajectory design, and control. It also includes algorithm and software development. The recently launched Microwave Anisotropy Probe (MAP) and upcoming James Webb Space Telescope (JWST) and Constellation-X missions are examples of the use of improved numerical methods for attaining constrained orbital parameters and controlling their dynamical evolution at the collinear libration points. This paper presents a history of libration point missions, a brief description of the numerical and dynamical design techniques including software used, and a sample of future GSFC mission designs.

1. Introduction

Sun-Earth libration point orbits serve as excellent locations for scientific investigations of stellar and galactic physics. These orbits are often selected to minimize environmental impacts and disturbances and to maximize observing efficiency. Trajectory design in support of such missions is challenging as more complex mission designs are envisioned. To meet these challenges, trajectory design software must be further enhanced to incorporate better understanding of the libration orbit solution space and to encompass new 85

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optimal methods. Thus the support community needs to improve the efficiency and expand the capabilities of current approaches. For example, invariant manifolds, derived from dynamical systems theory, have been applied to trajectory design over the past few years. The manifold approach offers new insights into the natural dynamics associated with the multibody problem.1$2Overall it allows a more rapid and robust methodology to libration orbit and transfer orbit design when used in combination with numerical techniques. Trajectory design approaches should also include improved numerical targeting methods that allow optimization and a dynamical view of the state space allowing the user rapid intuitive feedback.

1.1. A n Overview of N A S A Themes Involving Libmtion Orbits The NASA Enterprises of the Space Sciences (SSE) and Earth Sciences (ESE) are a combination of several programs and themes that will benefit from the applications of improved numerical and dynamical approaches to meet mission trajectory design^.^ The Space Sciences Enterprise includes themes such as Sun-Earth-Connections (SEC), Origins, the Structure and Evolution of the Universe (SEU), and Exploration of the Solar System (ESS). Each of these themes call for missions in libration point orbits. The attainment and maintenance of the particular orbits will be a challenge for the mission designer. For example, the desire to obtain a specific libration orbit and eliminate shadows and minimize fuel requirements while meeting specific payload needs will be a significant technology payoff. SEC missions will use orbits that provide unique coverage for solar observation and Earth’s environmental regions. Recent SEC missions included the L1 libration point mission SOH0 and the LI/L2 WIND mission; both missions used conceptual manifold implementations. The Living With a Star (LWS) theme of the SEC may require mission design of trajectories that place spacecraft into heliocentric orbits and libration orbits at the L1 and L3 Sun-Earth libration points. Other Space Science challenges include the Structure and Evolution of the Universe (SEU) program. Currently the Micro Arcsecond X-ray Imaging Mission (MAXIM) and Constellation-X missions of the SEU are libration point orbiters, with each mission a formation-flying concept. As found on the Origins web site, “The Origins Program has embarked on a series of closely linked missions that build on prior achievements. As each Origins mission makes radical advances in technology, innovations will

Libration Orbit Mission Design 87

be fed forward from one generation of missions to the next. By the end of the decade, we will have combined the very best imaging, formation flying, and other visionary technologies,giving us the power of enormous telescopes at a fraction of the cost.” A major goal of the Space Sciences Origins Program is the launch of the James Webb Space Telescope (JWST) and The Terrestrial Planet Finder (TPF). Each mission is to the L2 libration point. The mission design of JWST is currently employing the use of invariant manifolds to seed numerical targeting schemes. The Triana mission of the Earth Science Enterprise (ESE) is the lone ESE mission not orbiting the Earth. It is a mission that has relied solely on manifolds for computation of its mission baseline to L1 . While this mission is a significant deviation from traditional low Earth orbiters, it represents the possibilities of other Earth observing mission at unique vantage points. A major challenge to many of the above missions is the use of interferometry to form a virtual telescope. By placing telescope components on individual formation flying spacecraft, they would form a constellation or formation that would provide a powerful single telescope. Spacecraft carrying such instruments would have to fly in a precise formation, one that would provide us with the greatest possible information. Not only will spacecraft be separated across small to very large distances, they’ll constantly be turning and pointing at different stars, expanding and contracting the distance between them. We’ll need improved numerical and dynamical system applications and optimal control methods to monitor and maintain less than centimeter-sized changes in position in order to make the individual systems act as one large spacecraft.

1.2. Historical Missions Even though libration orbits have become more mainstream and several missions to the Sun-Earth collinear libration points are now proposed, current NASA libration missions have been few in number totaling only seven with five true orbiters, International Sun-Earth Explorer (ISEE-3), Solar Heliospheric Observatory (SOHO), Advanced Composition Explorer (ACE), Microwave Anisotropy Probe (MAP), and Genesis and two that stayed briefly in a libration orbit, WIND and Geotail. The Flight Dynamics Analysis Branch (FDAB) of the Goddard Space Flight Center (GSFC) has designed and supported all but one of these missions. While libration orbits share many dynamic properties, their diversity is revealed by how mission constraints are met. In a Sun-Earth rotating co-

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ordinate frame centered at the libration point, their geometric orientations and amplitudes may vary significantly. Also, the mathematical approach for representation of the reference orbit may differ. For ISEE-3 the complexity of mission design was handled through a combination of analytical and numerical methods to predetermine the reference libration orbit, along with proven operational numerical techniques for targeting and optimization.*y5 The International Solar Terrestrial Program (ISTP) SOHO mission was the next true libration orbiter, with orbit amplitudes equal to that of the ISEE-3 m i ~ s i o n .While ~ > ~ the SOHO transfer and mission orbit is similar to ISEE-3, the stationkeeping control method does not follow the ISEE-3 method of re-targeting back to a predetermined reference path. SOHO stationkeeping is performed to ensure that the orbit completes another revolution which has an added benefit of minimizing the AV required. The WIND spacecraft of the ISTP Program was originally planned as a libration orbiter, its trajectory was designed using multiple lunar gravity assist for rotation of the line of apsides to coincide with the Sun-Earth line before insertion into the L1 lissajous.* ACE was designed differently in that a direct transfer orbit was adjusted to allow a capture into a small L1 Lissajous orbit.g The most recent GSFC mission was MAP. MAP included a lunar gravity assist to obtain a small amplitude lissajous orbit.1° The Genesis mission is similar to SOHO but utilizes invariant manifolds. 1.3. Future Mission Challanges

At GSFC, there is currently one mission awaiting launch, Triana, and two L2 missions in design or formulation, the James Webb Space Telescope (JWST) l 1 and Constellation-X. Future missions of formations such as Maxim and Stellar Imager are in their conceptual stage. These and other new missions such as the Terrestrial Planet Finder or European Space Agencys Darwin drive designs of constrained transfer trajectories and mission orbits l2>l3.These missions are designed to meet orbit goals for specific Lissajous orbits, to minimize fuel or operational requirements, and to provide formation or constellation options. Traditionally, libration orbit design has been analyzed with a baseline trajectory concept set in place by project requirements or analytical boundary methods; that is, a trajectory had been baselined so that science requirements are met. Future mission design requires a more generalized approach as operational considerations require the launch window, gravity assist, transfer trajectories, final orbit geometry and orientation, and the number of spacecraft to be as flexible as possible

Libmtion Orbit Mission Design 89

to optimize science return while minimizing operational and launch requirements. Upcoming missions also bring new challenges that individually may easily be met, but in combination they become problematic. These may include 0 0

0

0 0 0

0 0 0 0

0 0 0 0

Biased Orbits when using large sun shades. Reorientation to different planes and classes. Minimal Fuel. Earth-Moon libration orbits. Constrained communications. Low thrust transfers. Shadow restrictions. Solar sail applications. Very small amplitudes. Continuous control to reference trajectories. Limited AV directions. Human exploration. Use of external libration orbits to Lq, Lg. Quasi-stationary orbits. L3 Co-linear orbits. Servicing of resources in libration orbits.

1.4. A Brief History of Trajectory Design and Capabilities

GSFC libration point mission design capabilities have significantly improved over the last decade. The success of GSFC support is based on an accurate numerical computational regime. Before 1990, mainframe computers were the only resource to compute high fidelity trajectories for libration orbits. The software of choice at that time was the Goddard Mission Analysis System (GMAS). This software had complete optimization functionality as well as the capability to model all the required perturbing forces. The software was unique at the time since it allowed object modules to be linked into the run sequence as a way to allow the user access to data for trajectory analysis. During the early 199O’s, the GSFC operational PC program called Swingby was deve10ped.l~Swingby was developed as a replacement for GMAS with an interactive graphical user interface to provide instantaneous feedback of the trajectory design in multiple coordinate systems. It was designed to be a generic tool to support a variety of missions including, lunar, planetary, libration, and deep space and of course gravity assisted trajectory designs. Swingby provides complete mission analysis and opera-

90 D. Folta and M. Beckman

tions for the WIND, SOHO, ACE, and is currently being used for Triana analysis and as an independent check for MAP. Additionally, the lunar orbiter missions of Lunar Prospector and Clementine also used Swingby for mission design and maneuver planning. With the unprecedented success of Swingby, GSFC invested in a commercial program called Astrogator, produced by Analytical Graphics Inc. that is based on Swingby design and mathematical specification^'^. Table 1 presents both historic and future planned and conceptual missions. 2. Numerical and Dynamical Targeting Methods

It is important that libration trajectories be modeled accurately. The software must integrate spacecraft trajectories very precisely and model both impulsive and finite maneuvers. Swingby and Astrogator allow this by incorporating various high order variable or fixed step numerical integrators (Runge Kutta, Cowell, and Bulirsch-Stoer). Precise force modeling includes up to lOOx 100 Earth and lunar gravity potentials, solar radiation pressure, multiple 3rd-body perturbation effects and an atmospheric drag model. Varying user-selected parameters to achieve the required goals performs trajectory targeting and optimization. A differential corrector (DC) is routinely used as the method of choice for targeting. Both programs use B-plane and libration coordinate targets. These software tools are also excellent for prelaunch analysis including error analysis, launch window calculations, finite engine modeling, and ephemeris generation.

2.1. Numerical Shooting Methods Any trajectory design for libration orbit transfers and stationkeeping can be computed using GSFCs Swingby or Anaytical Graphics (AGI) Astrogator software. Currently, both of these programs use a direct shooting approach (forward or backward) for targeting and meeting mission goals. A shooting method using a differential corrector (DC) is widely used to achieve orbit goals in these programs although both provide the user with a limited Quasi-Newton / Steepest Descent method. All three methods use numerical partial derivatives to calculate the direction for convergence. The DC in Swingby uses the first derivative information. The partial derivatives are calculated by numerically propagating to the stopping condition, changing the independent variable with a small perturbation and re-propagating. The change in the goals divided by the change in the variables are used to

Libmtion Orbit Mission Design 91

compute the partials. The usual sequence of a forward shooting method is to vary the initial conditions though predefined perturbations. The initial conditions include the orbital initial conditions, an applied AV, or spacecraft design parameter to meet goals that include orbital parameters such as period, position, velocity, amplitude, etc. Table 1. Libration Orbit Missions. Amplitudes Launch Total DV Transfer (Ax, Ay, Az) Year Allocation Type (mb) ISEE-3 c L1Halo/Lz/Comet 175000, 660670, 1978 430 Direct 1st mission 120000 WIND+ c L1-Lissajous 10000, 350000, 1994 685 Multiple 250000 Lunar Gravity Assist S OH0 c L1-Lissajous 206448, 666672, 1995 275 Direct 120000 ACE c L1-Lissajous 81775, 264071, 1997 590 Direct 1st small amplitude 157406 (Constrained) MAP 2001 127 c Lz-Lissajous n/a, 264000, Single Lunar 1st Lz Mission 264000 Gravity Assist Genesis c L1-Lissajous 25000, 800000, 2001 540 Direct 250000 Triana f L1-Lissajous 81000, 264000, Direct # 620 Launch Constrained 148000 JWST* f Lz-Lissajous 290000, 800000, Direct # 150 131000 SPECS Direct f Lz-Lissajous 290000, 800000, # Tbd Tethered Formation 131000 MAXIM Direct f Li-Lissajous Large Lissajous # # Formation Constellation-X f Lz-Lissajous Large Lissjaous 2010 150-250 Single Lunar Loose Formation Gravity Assist Darwin f L1-Lissajous 300000, 800000, 2014 # # Large Lissajous 350000 Stellar Imager f Lz-Lissajous Large Lissajous 2015 # Direct -30 S/C Formation TPF f Ln-Lissajous Lissajous # # # Formation Notes: c represents current missions, f represents future mission concepts. * This information represents concept only, # =unknown at this time, + WIND originally had a L1 Lissajous orbit as part of baseline trajectory. Mission

Location Type

The general procedures used in developing a baseline L2 direct transfer trajectory are: 0

Target a trajectory that yields an escape trajectory towards a libration

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0

0

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point with the Moon at the appropriate geometry Target the anti-Sun right ascension and declinations at the appropriate launch epoch Target the solar rotating coordinate system velocity of the Sun- Earth rotating coordinate x - z plane crossing condition to achieve a quasilibration orbit, Lz x-axis velocity 0 Target a second x - z plane crossing velocity which yields a subsequent x - z plane crossing, then target to a one period revolution at L2 In all above conditions, vary the launch injection C3 and parking orbital parameters ( w , 0,parking orbit coast and inclination) Incorporate conditions to achieve the correct orientation of the Lissajous pattern N

0

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0

While the basic Swingby DC targeting procedures used in developing a baseline lunar gravity assist trajectory for L2 were: 0

0

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0

Target the Moon at the appropriate encounter epoch to achieve an anti-Sun outgoing asymptote vector Target the lunar B-Plane condition to achieve gravity assist parameters and a perpendicular Sun-Earth rotating coordinate x - z plane crossing Target z - z plane crossing velocities which yields a second x - z plane crossing and target to a one period revolution at L2 Re-target lunar B-plane conditions to achieve the correct orientation of the Lissajous pattern with respect to the ecliptic plane

In both procedures, target goals may include Time (epoch, durations, flight time), B-Plane conditions (B.T B.R angle, B magnitude, outgoing asymptote vector and energy), Libration sun-Earth line crossing conditions (position, velocity, angle, energy, or a mathematical computation (eigenvectors), or other parameters at intermediate locations. Targets may be single event string, nested, or branched to allow repeatable targeting. Maneuvers can be inserted were appropriate. These procedures are duplicated for significant changes in launch date and to include phasing loop strategies. The phasing loop strategy allows time between launch and a lunar encounter, thus providing a longer Iaunch window since the phasing loop periods can be adjusted by maneuvers to arrive at the chosen epoch and lunar phase angle with respect to the SunEarth line. Targeting to an opening Lissajous pattern assures that the spacecraft will not pass through the shadow for at least 3 years (assuming control of the unstable mode). Retarget conditions via addition of deter-

Libmtion Orbit Mission Design 93

ministic AVs can be used to achieve the correct orientation and Lissajous pattern size with respect to the ecliptic plane. This procedure is duplicated for significant changes in launch date or to include lunar phasing loop strategies. Targeting to an opening Lissajous pattern assures that the spacecraft will not pass through the shadow for multiple revolutions assuming control of the unstable mode.

Fig. 1.

Sample Windows and DOS versions of Swingby.

While this procedure will achieve the required orbit, it is not robust for rapidly changing requirements. In order to decrease the difficulty in meeting mission orbit parameters and constraints in a direct targeting approach, the application of a dynamical system approach is investigated and incorporated into the overall trajectory design technique. This procedure can also be used for backward targeting, that of using a predefined libration orbit and targeting backward in time to the launch / parking orbit conditions. This procedure also involves the use of a DC to provide maneuvers to attain the mission orbit and parking orbit constraints. Using parametric scans, DC, and multiple targets, a more robust design can be achieved. Considerations are being given to new strategies that incorporate optimization routines into this scheme to ensure minimal fuel or time requirements can be met. Figure 1 presents sample output of the Windows and DOS versions of Swingby used to support GSFC missions.

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2.2. Dynamical Systems Approach

As mission concepts become more ambitious, increasing innovation is necessary in the design of the trajectory. Design capabilities for libration point missions have significantly improved in recent years. The success of Swingby for construction of trajectories in this regime is evidence of the improvement in computational capabilities. Nevertheless, conventional tools do not currently incorporate any theoretical understanding of the multi-body problem and do not fully exploit dynamical relationships. An in depth discussion of the versatility of dynamical systems as they apply to libration trajectory design were previously presented and is summarized below with permission.2 2.3. Invariant Manifolds

Nonlinear dynamical systems theory (DST) offers insights in multi-body regimes, where qualitative information is necessary concerning sets of solutions and their evolution.2 DST is, of course, a broad subject area with applications to many fields. For application to spacecraft trajectory design, it is helpful to first consider special solutions and invariant manifolds, since this aspect of DST offers immediate insights. Under a GSFC grant, Purdue University investigated various dynamical systems methodologies that now are included in software called Generator. In Generator, different types of solution arcs, some based on dynamical systems theory, are input to a process that differentially corrects the trajectory segments to produce a complete path in a complex dynamical mpdel. A two level iteration scheme is utilized whenever differential corrections are required. This approach produces position continuity and then a velocity continuity for a given trajectory. An understanding of the solution space then forms a basis for computation of a preliminary libration and transfer orbit solution and the end-to-end approximation can then be transferred to a direct targeting methods like Swingby for final adjustments for launch window, launch vehicle error analysis, maneuver planning, or higher order modeling. Our current goal is to blend dynamical systems theory, which employs the dynamical relationships to construct the solution arcs into Swingby or Astrogator with strength in numerical analysis. The geometrical theory of dynamical systems is based in phase space and begins with special solutions that include equilibrium points, periodic orbits, and quasi-periodic motions. Differential manifolds are introduced as the geometrical model for the phase space of dependent variables. An invariant manifold is defined as an n-dimensional surface such that an orbit

Libration Orbit Mission Design 95

starting on the surface remains on the surface throughout its dynamical evolution. So, an invariant manifold is a set of orbits that form a surface. Invariant manifolds, in particular stable, unstable, and center manifolds, are key components in the analysis of the phase space. Bounded motions which include periodic orbits such as halo orbits exist in the center manifold, as well as transitions from one type of bounded motion to another. Sets of orbits that approach or depart an invariant manifold asymptotically are also invariant manifolds (under certain conditions) and these are the stable and unstable manifolds, respectively, associated with the linear stable and unstable modes. The periodic halo orbits, as defined in the circular restricted problem, are used as a reference solution for investigating the phase space in this analysis. It is possible to exploit the hyperbolic nature of these orbits by using the associated stable and unstable manifolds to generate transfer trajectories as well as general trajectory arcs in this LZ region of space. 2.4. L i s s a j o u s - M a n i f o l d - ~ n s f e rGeneration

The computation process of the stable and unstable manifolds, shown in Table 2, is associated with particular halo orbit design parameters and is accomplished numerically in a straightforward manner. The procedure is based on the availability of the monodromy matrix (the variational or state transition matrix after one period of motion) associated with the lissajous orbit. A similar state transition matrix of this sort can be computed using the state equations of motion based on circular three-body restricted motion. This matrix essentially serves to define a discrete linear map of a fixed point in some arbitrary Poincare section. As with any discrete mapping of a fixed point, the characteristics of the local geometry of the phase space can be determined from the eigenvalues and eigenvectors of the monodromy matrix. These are characteristics not only of the fixed point, but also of the lissajous orbit. The local approximation of the stable and unstable manifolds involves calculating the eigenvectors of the monodromy matrix that are associated with the stable and unstable eigenvalues. This approximation can be propagated to any point along the halo orbit using the state transition matrix. The first step is to generate the lissajous orbit of interest. This is indicated in Table 2 by “Lissajous”. With this information, the monodromy matrix can then be computed (assuming periodic motion). Also, in the “Monodromy” block, the eigenvalues/eigenvectors associated with the nominal

96 D . Folta and M . Beckman Table 2. Utility Phase (Generic Orbit) Lissajous Monodromy (Periodic Orbit) Manifold Transfer

Dynamical System Approach Segments. Input User Data Universe and User Data Universe and Lissa.jous Output

Universe and Monodromy Output Universe, User Selected Patch, Points, Manifold Output

output Control Angles for Lissajous Patch Point and Lissajous Orbit Fixed Points and and Stable and Unstable Manifold Approximations 1-Dimensional Manifold Transfer Trajectory from Earth to L1 or Lz

orbit are computed and near the fixed point, the half-manifold is determined to first order, by the stable eigenvector. The next step is then to globalize the stable manifold. This can be accomplished by numerically integrating backwards in time. It also requires an initial state that is near but not necessarily on the halo orbit. A linear approximation is utilized to get this initial state displaced along the stable eigenvector. Higher order expressions are available but not necessary. A displacement is selected that avoids violating the linear estimate, yet the displacement is not so small that the time of flight becomes too large due to the asymptotic nature of the stable manifold. Note that a similar procedure can be used to approximate and generate the unstable manifold. The stable and unstable manifolds for any fixed point along a halo orbit are onedimensional and this fact implies that the stable/unstable manifolds for the entire halo orbit are two-dimensional. This is an important concept when considering design options. With the manifold as an initial guess, one can then perform differential corrections in the Transfer block that meet all the trajectory constraints while achieving an Earth access region. This final step provides the necessary conditions that are used in the numerical shooting process. Figure 2 presents some of the menus for the generation of invariant manifolds. This information is then transferred to the numerical operational GSFC tools for further refinement of the trajectory using the highest fidelity models available.

Libration Orbit Mission Design 97

Fig. 2.

Generator menu samples and manifold output.

3. Applications of Numerical and Dynamical Methods

We now investigate the use of the above numerical and dynamical system approaches as they are applied to the JWST, Constellation-X, and two conceptual missions. These examples demonstrate the design of the libration orbit and the transfer orbit.

3.1. J W S T Trajectory Design: Libration Orbit

The design of the JWST libration point trajectory begins with the generator dynamical system approach.16The required y-axis amplitude parameter of 800,000 km is input into the generation of a lissajous orbit. The resulting output as shown in Figure 3 is a result of the lissajous segment. This orbit reflects the use of multiple bodies, semi-analytical elliptical approximation of the orbit, and solar radiation pressure (SRP). The algorithms include parameters of a Richardson-Cary model as a first guess to obtain a differentially corrected orbit using a full planetary ephemeris. The orbit as shown meets all the JWST requirements because this is the starting point versus the end conditions of a shooting method. Figure 3 shows the JWST orbit in an Solar Rotating Coordinate (SRC) frame. It is a class I orbit that has an opening z-axis component. Figure 3 shows the complement of the Sun-Earth-Vehicle (SEV) angle. A maximum of 30" and minimum of 4' is achieved to meet all lighting constraints.

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Fig. 3.

M. Beckman ,

,

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W" ULYI

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,

,

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,

I

Lissajous pattern and Sun-Earth-angles for 800,000 km orbit.

3.2. J WST Trajectory Design: Direct Transfer

Given a libration orbit with the above JWST requirements, a transfer trajectory is sought that will also minimize fuel requirements and incorporate possible JWST constraints. While a trajectory design approach similar to that used for SOH0 or ACE can be pursued, the application of a dynamical system approach is investigated and is incorporated into the overall trajectory design technique. Using invariant manifolds and the JWST orbit parameters, libration orbits and transfer paths can be computed; a surface is projected onto configuration space and the three-dimensional plots appears in Figures 4 upside-down to show detail. This particular section of the surface is associated with the "Earth Access region" along the Lz libration point orbit. An interesting observation is apparent as motion proceeds along the center of the surface. The smoothness of the surface is interrupted because a few of the trajectories pass close to the Moon upon Earth departure. Lunar gravity assists were not incorporated into the approximation for the manifolds, but no special consideration was involved to avoid the Moon either. Using information available in Figure 4 the one trajectory that passes closest to the Earth is identified and used as the initial guess for the transfer path. The larger size of the Lissajous orbit reduces the Earth passage distance and minimizes any insertion AV. Given the initial guess, the transfer is differentially corrected to meet the requirements of achieving both the lissajous orbit and an Earth parking orbit. Fkom this point, the solution is input directly into numerical tools and appears in Figure 5. Swingby/Astrogator and other tools are used for further visualization, analysis of launch vehicle and maneuver errors, midcourse corrections, and other design considerations.

Libration Orbit Mission Design 99

Fig. 4. JWST 800 km Y-axis amplitude.

Fig. 5.

Numerical targeting.

3.3. Continuous Low Thrust Options

Alternative JWST trajectory options have been investigated recently. They include a low thrust propulsion system and possible L2 servicing options. Low thrust trajectory solutions exist for the collinear libration points and have been ana1y~ed.l~ The trajectory generally consists of spiraling out to lunar orbit with periods of thrusting and coasting and targeting the postlunar leg to insert into the periodic orbit by varying coast times. The thrust can be along the velocity vector or at an angle to it to achieve maximum efficiency. The problem with most low thrust trajectory designs however is the extensive time-of-flight. This is amplified by the mass of JWST which in this analysis is in the vicinity of 10,000 kg. Most low thrust engines would

100 D. Folta and M. Beckman

take many years to raise the orbit to lunar distances. A recent investigation included the use of nuclear powered electric propulsion. The proposed system would provide 1.2 N of thrust at an Isp of 4800 sec. At this thrust level, an JWST trajectory design was completed with 510 days of continuous thrust followed by an 85-day coast into the nominal LZ orbit. Figure 6 shows the ecliptic plane view of the transfer. A small insertion maneuver is also required.

Fig. 6. JWST trajectory design: Low thrust transfer.

The design of this trajectory was accomplished using the numerical process of Swingby to model the continuous propulsion system, maintain the proper attitude profile, and perform a shooting method to achieve the libration orbit goals. While the DC method was chosen for this analysis, other utilities can be used to optimize transfer time (coast and finite burn sequence) or minimize fuel for the trajectory. Also generator can be used to setup initial conditions for the parking orbit and final targets. 3.4. Earth Return

/ Servicing

missions

The possibility of Lz servicing brings up numerous scenarios. One such scenario studied recently is to return JWST to LEO to be serviced at the 1%. Unstable manifolds from the nominal LZorbit that pass near the Earth are used as initial estimates. The manifold is targeted to meet inclination and dynamic pressure constraints. A large drag apparatus would be used to aerocapture at the Earth. After first perigee at the Earth, an apogee maneuver would be required to retarget perigee to the original 107 km

Libration Orbit Mission Design 101

altitude. After three perigees, perigee altitude remains constant and the spacecraft is aerocaptured within 4 days. Figures 7 and 8 show the transfer to and from Lz in the rotating frame and in the inertial frame near the Earth.

Fig. 7. JWST trajectory design: Servicing return with aerobraking transfer.

Fig. 8. JWST trajectory design: Servicing return with aerobraking transfer.

3.5. Constellation-X Constellation-X is more challenging. It involves a scientific desire to have four spacecraft in relative close proximity to one another while the transfer design requires a lunar gravity assist The mission separations are not determined as of this date, but initial goals indicate separations of greater than 50km but less than 50,000km. The spacecraft must maintain this separation throughout the mission. Thus part of the trajectory design 18919.

102 D. Folto and M. Beckmon

challenge is to launch two spacecraft from one launch vehicle and perform a lunar gravity assist to attain a libration orbit that meets a relative formation requirement. The mission orbit was chosen primarily to meet the following sky coverage requirement, the mission orbit and attitude constraints must be such that 90% of the sky is accessible at least twice per year, with viewing windows not shorter than 2 weeks in duration; and 100% of the sky is available at least once a year with a minimum viewing window of one week. The spacecraft will be inserted into the Lissajous orbit via a lunar swingby. The lunar swingby is necessary in order to reduce the amount of onboard AV and the C3 (launch energy) needed from the launch vehicle. Smaller (more negative) values of C3 yield a larger payload capability. In order to increase the number of launch opportunities, a number of phasing loops will be performed prior to the lunar swingby. Figure 9 shows the Constellation-X transfer trajectory and the characteristics of this approach assuming 3 1/2 phasing loops. Different numbers of loops could be considered for various launch days to increase the number of launch opportunities. Under a current concept, two Constellation-X spacecraft will be placed in a highly eccentric injection orbit by the launch vehicle. Maneuvers will be performed roughly centered on the phasing loop apses, using the spacecraft propulsion system, to properly time the spacecraft’s encounter with the moon. The timing and geometry of the lunar encounter will be chosen to allow the spacecraft to be inserted into the L2 Lissajous orbit with little or no maneuver required and still meet separation requirements. The trajectory design was computed using Swingby. No requirement has been specified regarding the depth or duration of acceptable Earth or lunar shadows. In designing the nominal trajectory, an effort will be made to minimize shadows without significantly increasing total AV during all phases of the mission. However, some shadows may be unavoidable during design, or others may crop up during flight due to contingencies. Phasing loop maneuver adjustment will be used to mitigate cruise phase (after lunar swingby but prior to lissajous insertion) shadows post-launch. Unacceptable shadows in the lissajous orbit may be avoided through propulsive maneuvers, however no fuel has been budgeted for that purpose. The total AV required in this particular example is approximately 160 m/sec per spacecraft. This includes correcting for launch vehicle errors, targeting the lunar swingby, mid-course correction maneuvers, Lissajous orbit insertion and station-keeping maneuvers. Table 3 details the AV budget

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for each spacecraft with a launch that assumes a C3 of -2.60 km2/s2.

Fig. 9. Constellation-X transfer trajectory.

3.6. Conceptual Missions GSFCs Flight Dynamics Analysis Branch performs many future mission studies in the conceptual phase. One recent analysis included a mission to study the distant geotail between the Earth-Sun L2 point. The investigator wanted to obtain coincident measurements of the geotail over a wide spatial region from L2 to about twice L2. The resulting trajectory design uses the periodic orbits about Lz to initiate slightly perturbed trajectory arcs that all cross into the geotail within a week of each other as shown in figure 10. A mother ship would release 16 small spacecraft into perturbed orbits, each cumulatively 10 m/s in along-track AV off the periodic orbit. The mother ship would spend 333 m/sec in AV during the release phase over four days including returning to the periodic orbit. The small spacecraft would not have any propulsion system. 4. Libration Formation Flying

In addition to the need for improved numerical and dynamical system approaches to libration trajectory design, capabilities are required to meet new multiple spacecraft mission goals of interferometer and optical measurements and need to include new methods for operational application to support the trajectory design. As seen in Table 1, there are at least five missions that require formations. In combination with dynamical and improved direct methods, algorithms are being developed for complete “system” control of formation flying spacecraft. These methods employ linear

104 D . Folta and M. Beckman Table 3.

Sample Constellation-X Delta-V Estimated Budget.

LV Error Correction

Spacecraft-1 Spacecraft-:! AV (m/s) AV (m/s) 20 20

Deterministic Phasing Loops A1 AV P1 AV A2 AV P2 AV T3 AV P 3 AV Midcourse Corrections Libration Orbit Insertion

35 0 25 5 0 5 0 4 2

39 0 25 5 3 6 0 4 2

Stat ion-keeping

40

40

Other Subtotal Finite Burn Losses, Momentum Dumping, ACS Subtotal Contingency (Launch window, etc.) Subtotal Cosine Losses

10 111 22

10 115 23

133 13

138 14

146 4

152 5

Total

151

157

Assumptions/Comments Performed between TTI+8hrs to P2 Lunar swingby timing

Post-lunar swingby Assumes Midcourse Correction successfully remove LO1 targeting errors Estimated at 4 man/yr for 10 years Calibration burns, etc. ~20% of above

-10% of above

Assumes all thrusters canted loo

and non-linear feedback control systems that can be managed to analyze cooperative spacecraft. Currently, formation flying spacecraft control is being extensively researched and has been demonstrated autonomously for in low Earth orbit. A growing interest in formation flying satellites demands development and analysis of control and estimation algorithms for stationkeeping and formation maneuvering. This development of controllers, such as discrete linear-quadratic-regulator control or non-Iinear approaches for formations in the vicinity of the co-linear sun-Earth libration point will be necessary. This development may include an appropriate Kalman filter as well. Formation flying control can be performed in three ways-centralized, decentralized, or in combinations. With centralized control, one spacecraft or processor calculates and commands the motion of the entire formation. With decentralized control, each spacecraft, with input from the

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Fig. 10. Conceptual trajectory design to twice La distance.

rest of the formation, processes its own control requirements. Speyer first introduced a decentralized linear-quadratic-Gaussian control method.20 Folta and Carpenter applied this work to formation flying satellites, and further expanded it to deal with both time-invariant and time-varying systems.21 Speyer’s method produces identical results to the centralized linear-quadratic-Gaussian control method, and it also minimizes data transmission. NASA has several distributed spacecraft libration missions planned for the next decade and beyond. The Stellar Imager22 , Constellation-X, and MAXIM will image stars and black holes while TPF will look for planets. They also rely on the capability of correctly modeling the dynamics of the libration region and the inclusion of the formation control method into the overall picture. For example in a recent libration formation flying research using the Stellar Imager mission that uses 31 spacecraft in close formation at L2 as a baseline, the dynamics of the libration region were incorporated into the control state space to ensure accurate modeling and therefore more accurate control results. The modeling of the dynamics creates much more realistic Lissajous orbits than those derived from the circular restricted three-body problem. Also one can numerically compute and output the dynamics matrix, for a single/multiple satellite at each epoch. This matrix is not computed from the pseudopotential but is a monodromy matrix which incorporates the full perturbations and third body ephemeris data.

106 D. Folta and M. Beckman

4.1. Stellar Imager Ezample

Stellar Imager (SI) is a concept for a space-based, UV-optical interferometer. The purpose of the mission is to view many stars with a sparse aperture telescope in an attempt to better understand the various effects of stars’ magnetic fields, the dynamos that generate them, and the internal structures and dynamics of stars. The leading concept for SI is a 500-meter diameter Fizeau-type interferometer composed of 30 small drone satellites that reflect incoming light to a hub satellite. The hub will recombine, process, and transmit the information back to Earth. As Figure 11 shows, in this concept, the hub satellite lies halfway between the surface of a sphere containing the drones and the sphere origin. Focal lengths of both 0.5 km and 4 km are being considered. This would make the radius of the sphere either 1 km or 8 km. The type of orbit and location in space is an important part of mission design. The best orbit choice for the formation after consideration of gravity gradients, scattered and stray light, and element replacement is a Lissajous orbit around the Sun-Earth Lp point. The y-amplitude of the Lissajous orbit will be about 600,000 km, but is not critical to the mission. With this orbit, SI will be able to cover the entire sky every half year while maintaining an aim perpendicular to the sun. For useful imaging, SI must aim within 10 degrees of perpendicular from the sun. To function properly, SI will need to accommodate a wide range of control functions. In addition to maintaining its desired trajectory around Lp, the formation must slew about the sky requiring movement of a few kilometers and attitude adjustments of up to 180 degrees. While imaging, though, the drones must maintain position within 3 nanometers of accuracy in the direction radial from the hub and within 0.5 millimeters of accuracy along the sphere surface. The accuracy required for attitude control while imaging is 5 milli-arcseconds tip and tilt (rotations out of the surface of the sphere). The rotation about the axis radial from the hub (rotation within the sphere) is a much less stringent 10 degrees. 4.2. SI Formation Flying Results

A common approximation in research of this type models the dynamics of a satellite in the vicinity of the sun-Earth Lp point using the circular restricted three-body assumptions. These assumptions only account for gravitational forces from the sun and Earth. The moon is also included, but not as an independent body. The masses of the earth and moon are com-

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Drrmts

S p h Origin

500 m

Fig. 11. Stellar imager geometry.

bined and assumed to be at the earth-moon barycenter. The motion of the sun and the earth-moon barycenter is also assumed to be circular around the system barycenter. There is extensive literature available on the application of linear quadratic regulator (LQR) control to formation flight in the two-body problem (2BP) and also a few examples of its application in the circular-restricted three-body problem (CR3BP). The numerical implementation of this type of control is often difficult because of the assumptions in its original development are based on two-body dynamics, general optimal control theory and calculus of variations. The most significant of these assumptions is that the nonlinear system dynamics are linearized relative to a constant equilibrium solution. This results in a linear system that assumes constant matrices. In spite of this critical assumption, the available literature includes numerous examples in which this result is extended to time-varying systems, where the system matrices are actually time varying. Although the controller may appear to work on a case-by-case basis, most of the available research on formation flight to date provides no sound mathematical justification for this extension. firthermore, since the most essential assumption in the development of this controller is violated, it is safe to say that the resulting controller is not truly optimal. To properly apply LQR control to formation flight in the three-dimensional CR3BP, particularly in the vicinity of periodic orbits near L1 and L2, it is necessary to account for the time-varying nature of the linearized dynamics; for a halo orbit, the system matrix is periodic. This SI analysis uses high fidelity dynamics based on a dynamical systems simulation. This creates much more realistic Lissajous orbits than those derived from the circular restricted three-body problem. Using ephemeris files, we take into account the effects of eccentricity, an independent moon, the other planets of the solar system, and solar radiation

108 D. Folta and M. Beckman

pressure. The resulting Lissajous orbit can then be used as a more accurate reference orbit. In addition to providing the reference positions and velocities, the dynamics matrix is also numerically computed at each epoch. The SI reference orbit is shown in Figure 12 with the earth as the origin. The X coordinate connects the two primary bodies, the 2 coordinate is parallel to their angular velocity of the system, w, and the Y coordinate completes a right-handed system. Three different scenarios make up the SI formation control problemmaintaining the Lissajous orbit, slewing the formation to aim at another star, and reconfiguring the formation to take another snapshot of a star when necessary. Following the Lissajous orbit is not a problem of formation control, but rather a problem of maintaining an orbit. Therefore, only the hub satellite needs simulation to determine the amount of control and fuel needed to maintain a Lissajous orbit. The results can be extended similarly to other satellites in the formation. x 106 1

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............

:.

..... t i*:. ......... ...... ...........

:

o Y

..........-...... - 1 0

..;

'

2

xIoB

Fig. 12. Generator reference orbit.

4.3. Formation Maintenance Results

Averaging the determined AV from a dozen simulations, the AV required to keep a satellite in a Lissajous orbit about LZ for two orbital periods is approximately 0.38 meters per second. A key part of the SI mission is to image many stars. Following a Lissajous orbit around L z , SI could view

Libration Orbit Mission Design 109

the entire sky approximately every half-year while slewing about just the radial (x) axis. This maintenance method will also maintain the aiming angle perpendicular to the sun. The formation slewing simulation follows a similar algorithm as the Lissajous orbit simulation. Figure 13 provides an image of the entire SI formation slewing 90 degrees, with a 0.5 km focal length. The black dots represent drones at the beginning of the simulation, and the red circles represent drones at the end of the simulation. The hub is the black asterisk at the origin. The upper-right plot illustrates the Golomb15 rectangle formation projected into the x-z plane. The lower-left plot clearly shows the drones slewing 90 degrees about the hub-centered x axis. The formation slewing simulation runs for one day, with one maneuver per minute (1440 maneuvers), whereas the Lissajous orbit simulation runs for 359 days with one maneuver per day. Table 4 shows the average AV’s for a dozen simulations for the various scenarios. The larger the angle the formation slews through, the more AV is needed. Also, the larger the focal length, the more AV required.

4

(I

Drone s/c at beginning

Fig. 13. SI slewing configuration.

5

110 D . Folta and M. Beckman Table 4. Average formation slewing AV’s. Focal Length Slew Angle Hub Drone 2 Drone 31 (km) (deg) AV (m/s) AV (m/s) A V ( m / s ) 0.5 30 1.0705 0.8271 0.8307 0.5 90 1.1355 0.9395 0.9587 1.2688 1.1189 4 30 1.1315 4 90 1.8570 2.1907 2.1932

5. Future Missions Designs Needs 5.1. Improved Tools

If one includes search methods and optimization in numerical and dynamical approaches, a full system architecture can be made for design for libration missions that include single and multiple spacecraft, and Human Exploration and Development of Space (HEDS) missions. New search methods, such as genetic and simulated annealing algorithms, combined with indirect and direct optimization techniques can be applied to best meet scientific and HEDS libration formation flying requirements. These tools would provide the best-case scenario for the formation or orbit type, fuel cost, and transfers to minimize overall system cost. While the roadmap is still being worked, the outlook is very promising. Figure 14 gives a possible path to a level that incorporates all the best capabilities of numerical and dynamical methods. The idea is to merge the best of current targeting, optimizing, and control applications. Additionally, these tools must be able to interface with one another.

Fig. 14. Roadmap to the future.

Libmtion Orbit Mission Design

111

5.2. Innovative Trajectory Design Concepts and

Visualization As numerical and dynamical systems are improved to incorporate high fidelity modeling of dynamics of libration points, new optimal targeting schemes that include direct and indirect methods or stochastic approaches, a new class of missions and capabilities will emerge just as over the past 25 years. There are several notions on the horizon that represent challenges. These incorporate research on quasi-stationary orbits23, the use of weak stability boundary dynamics24, heteroclinic/hornoclinic t r a j e c t o r i e ~and ~~, improved numerical propagation schemes for formations of spacecraft, and stochastic optimization. Also, as computer capabilities improve, the analysis and design of libration orbits should become more ordinary, just as the thought of designing an Earth orbiter was once viewed with awe. These improvements can be seen already in applications using 3-dimensional graphics which can be rotated to give the analyst a more intuitive approach to meeting mission requirements and goals. The use of intelligent systems that can be used in automation and multiple constraint checking is starting to find its way into everyday analysis, which will help the analyst. 6. Conclusions

Trajectory design in support of libration missions is increasingly challenging as more constrained mission orbits are envisioned in the next few decades. Software tools for trajectory design in this regime must be further developed to incorporate better understanding of the solution space, improving the efficiency, and expand the capabilities of current approaches. Improved numerical and dynamical systems offers new insights into the natural dynamics associated with the multi-body problem and provide to methods to use this information in trajectory design. The goal of this effort is the blending of analysis from dynamical systems theory with the well-established NASA Goddard software programs such as Swingby to enhance and expand the capabilities for mission design and to make trajectories more operationally efficient.

References 1. K. Howell, B. Barden, and M. Lo, “Application of Dynamical System Theory to Trajectory Design for a Libration Point Mission”, Journal of the Astro-

112

D . Folta and M. Beckman

nautical Sciences, Vol. 45 No. 2, 1997, pp. 161-178. 2. J.J. Guzman, D. S. Cooley, K.C. Howell, and D.C. Folta, “Trajectory Design Strategies that Incorporate Invariant Manifolds and Swingby”, AAS 98-349 AIAA/AAS Astrodynamics Conference, Boston, August 10-12, 1998. 3. NASA WebSites, http://sec.gsfc.nasa.gov, http://cossc.gsfc.nasa. gov, http://uw.nasa.gov, 2002 4. R.W. Farquhar, D. Muhonen, and D. Richardson, “Mission Design for a Halo Orbiter of the Earth”, Journal of Spacecraft and Rockets, Vol 14, No. 3 , 1977, pp. 170-177. 5. R. W. Farquhar, “The Flight of ISEE-J/ICE:Origins, Mission History, and a Legacy”, AIAA/AAS Astrodynamics Conference, Boston, AIAA-98-4464, August 10-12, 1998. 6. D. Dunham, S. Jen, C. Roberts, A.Seacord, PShearer, D. Folta, and D. Muhonen, “Transfer Trajectory Design for the SOHO Libration Point Mission”, 43rd Congress of the International Astronautical Federation, Washington D.C. 1992. 7. S. Stalos, D. Folta, B. Short, J. Jen, and A. Seacord, “Optimum Transfer to a Large-Amplitude Halo Orbit for the Solar and Heliospheric Observatory (SOHO) Spacecraft”, AAS 93-294, Flight Mechanics Symposium, GSFC, 1993. 8. D. Folta and P. Sharer,“Multiple Lunar Flyby Targeting for the WIND Mission”, AAS 96-104, AAS/AIAA Space Flight Mechanics Meeting, February 1996, Austin, Tx 9. P. Sharer and T. Harrington, “Trajectory Optimization for the ACE Halo Orbit Mission”, AIAA/AAS Astrodynamics Specialist Conference, July 1996, San Diego, CA. 10. M. Mesarch, Andrews, “The Maneuver Planning Process For The Microwave Anisotropy Probe (Map)”, AIAA/AAS Astrodynamics Specialist Conference, Monterey, CA August 5-8, 2002 11. JWST Website, http://JWST.gsfc.nasa.gov/, Goddard Space Flight Center 12. T P F Website, http: //tpf .jpl .nasa.gov/library/tpf-book, 2002 13. ESA Darwin Website, http://scsi.esa.int/, 2002 14. J. Carrico, C. Schiff, L.Roszman, H.Hooper, D. Folta, and K. Richon, “An Interactive Tool for Design and Support of Lunar, Gravity Assist, and Libration Point Trajectories” , AIAA 93-1126, AIAA/AHS/ASEE Aerospace Design Conference, CA., 1993. 15. Analytical Graphics Incorporated, STK / Astrogator, commercial software package 2002 16. D. Folta, S Cooley, K Howell, “Trajectory Design Strategies For The NGST L2 Libration Point Mission” , AAS 01-205, AAS Astrodynamcis Conference, CA., 2001. 17. D. Folta et al., “Servicing And Deployment Of National Resources In SunEarth Libration Point Orbits”, 53rd International Astronautical Congress, The World Space Congress - 2002, 10-19 Oct 2002 / Houston, Texas 18. M. Houghton,“Getting to L1 the Hard Way: Triana’s Launch Options”, Li-

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bration Point Orbits and Applications June 10-14, 2002, Girona, Spain 19. Constellation-X web site, http: //constellation.gsfc.nasa.gov ,2002 20. L. Newman, GSFC Memo, Constelllation-X Trajectory Design and Navigation Report 21. J. L. Speyer, “Computation and Transmission Requirements for a Decentralized Linear-Quadratic-Gaussian Control Problem,” IEEE Transactions on Automatic Control, Vol. AC-24, No. 2, April 1979, pp. 266-269. 22. D. Folta, R. Carpenter, and C. Wagner, “Formation Flying with Decentralized Control in Libration Point Orbits”, International Space Symposium, Biarritz, France. June 2000 23. N. Hamilton, D. Folta, and R. Carpenter, “Formation Flying Satellite Control Around The L2 Sun-Earth Libration Point”, To be presented at the AIAAf AAS Astrodynamics Specialist Conference, Monterey, Ca, August, 2002 24. E. Belbruno, “ The Dynamical Mechanism of Ballistic Lunar Capture Transfers in the Four Body Problem from the Perspective on Invariant Manifolds and Hill’s Region”, Centre De Recerca Maematica, Institut D’Estudis Catalans, No. 270, November 1994 25. E. Belbruno, “Analytic Estimation of Weak Stability Boundaries and Low Energy Transfers”, Contemporary Mathematics, Vol. 292, 2002 26. M. Lo et al. “New Dynamical Systems Application in Mission Design”, Institute of Geophysics and Planetary Physics, Nov., 1999

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

HERSCHEL, PLANCK AND GAIA ORBIT DESIGN M. HECHLER Mission Analysis Section, ESA/ESOC, Robert-Bosch-Str. 5, D-64.293 Darmstadt, Germany J. COBOS T E R M A , Flight Dynamics Division, ESA/ESOC, Robert-Bosch-Str. 5, 64293 Darmstadt, Germany

HERSCHEL/PLANCK (double launch in 2007 on ARIANE) and GAIA (launch in 2010 on SOYUZ/FREGAT) are Astronomy missions in the ESA Scientific Program with different objectives but with quite common requirements of a highly stable thermal environment and sky viewing conditions unobstructed by Earth and sun. A class of orbits near the La libration point (outside Earth) in the sun-Earth system has been selected for these projects. Not differentiating in the conventional way between Halo or Lissajous orbits, a family of non-escape orbits around La has been classified solely by their property of neither falling towards the Earth nor to the sun within the limits of numerical precision of the initial conditions. The stable manifolds of some of the Lissajous orbits in this family (generally with large amplitudes) touch perigee conditions which can be naturally injected into by a launcher, e.g. a low perigee altitude, and specifically for ARIANE a line of apsides near the equator plane. Thus a 'free' transfer t o some of these orbits which requires no manoeuvres after perigee, except stochastic orbit corrections, exists. Starting from the free transfers to the large amplitude non-escape orbits, transfers to small amplitude (e.g. maximum sun-spacecraft-earth angle below 10') have been constructed combining the linear theory for orbits in the restricted circular three body problem with the numerical algorithm. The linear theory defines directions of escape (e+xt-term) and non-escape in the velocity subspace. Manoeuvres along the non-escape direction which are optimum in the linear theory to change the orbit amplitude, are numerically corrected along the escape direction. By the same approach also optimum eclipse avoidance strategies could be derived, which guarantee a mission of at least 6 years without eclipse with a manoeuvre of typically 15 m/s. The eclipse avoidance manoeuvres are perfor115

116 M. Hechler and J . Cobos med during the last revolution before the eclipse occurs near one of the maximum amplitudes, and essentially revert the track of the motion in the plane orthogonal to the sun Earth line. Herschel and Planck will share a launcher, either an ARIANE 5 ESV (upper stage with delayed ignition after coast arc) or an ARIANE 5 ECA (cryogenic upper stage) launched from Kourou. Herschel will remain on the large amplitude orbit to which the launcher naturally delivers, whereas Planck will perform an amplitude reduction manoeuvre. GAIA will be launched by a Soyuz/Fkegat launch from Baikonur, with or without lunar gravity assists and the necessary phasing orbits. In all cases the combination of the orbit amplitudes (in ecliptic and orthogonal to it) for a given size (maximum sun-spacecraft-Earth angle) and the initial phase in the orbit will depend on the launch date. Launch windows will be given. Finally the navigation and orbit maintenance aspects of the mission have been studied for the transfer and for the phase in the Lissajous orbit comparing several maintenance strategies. The preferred strategy is using a constant manoeuvre direction (along the escape line in the linear theory) which allows a dedicated spacecraft design.

1. Introduction

The orbits around L2 or L1 in the Earth-sun system have become of particular interest for astronomy missions. ESA mission or missions with ESA participation in this category are listed in Table 1. Table 1. ESA missions at Libration point.

Project SOH0

Launch 1995

Objective Sun observations

Smart 2

2006

Herschel

2007

Technology demonstration for LISA and Darwin Far infrared astronomy

Planck

2007

Cosmic background

Eddington

2008

Star seismology

NGST

2009+

GAIA

2010+

Next Generation Space Telescope Astrometry

Orbit, Remarks Halo at L1. Still in operations Cluster of S/C near L1. Drag free, drift away Large amp. Lp Lissajous. Stable manifold transfer from ARIANE launch 15' Lissajous at L z . Double launch with Herschel on ARIANE Lissajous at L2. Herschel/Planck s/c bus reuse Lissajous at L p . NASA 15' Lissajous at Lz Orbit control with FEEPS

From a spacecraft near L2 uninterrupted sky observations are possible as

Herschel, Planck and GAIA Orbit Design

117

Earth and sun remain more or less close together seen from the spacecraft. A major advantage for modern to be cooled detectors is that with the proper spacecraft design a very stable thermal environment can be achieved near L2. The drawbacks of these orbits are the long transfer duration, and the necessity of orbit maintenance maneuvers about once per month to counteract the instability. The large communications distance is less important todays with higher frequency systems, and use of directivity in the special geometry.

2. Lissajous Orbits Around L z 2.1. Escape Direction i n the Linear Problem ’Quasi periodic orbits’ around L2 have first been proposed for space missions in reference 3. Analytic theories for their construction under the full dynamics of the solar system have been developed under ESA contracts in the ’80s and are documented in Most of the basic properties of these orbits are inherited from those of the linearised circular restricted threebody problem. Therefore a few basic equations of the linear theory will be repeated below, as the rather powerful numerical orbit construction method developed thereafter will be based on these properties alone. With the ecliptic plane as the xy-plane with the x-axis from the sun to the Earth-moon baricentre in the rotating frame, and z out of the ecliptic, and introducing the usual coordinate transformations with the distance unit as sun-Earth distance and the time unit as 1 sidereal year / 27r, in the well known linear approximation for the circular restricted three-body problem, the differential equations of the relative motion around L2 can be written as

’.

?-2Y-(1+2K) x = o y 2 j.- (1 - K ) y = 0

+

%+Kz=O

}

(1)

With K = 3.940522 for L2. The complete solution of this homogeneous system can be written as z = e’=gt

. A1 + e-’=ut . A2 + cos (wzyt + +),

y = c1 eXzyt . A1 - c1 e-’=st z = cos (w,t +*) * A ,

+

. A,

. A2 - c2 sin ( w z y t + + ),

.A,

}

(2)

where c1, c2, w,,, A,, and w, are constants depending on K. The integration constants A l , A2 are linear functions of the initial conditions

M. Hechler and J . Cobos

118

with

dl = c 1 L y

+

c2wxy

d2 = ClWxy - C2Xxy The X-Y part of state vector ( x ~ , y o , k : ~ ,isy on ~ ) ~a Lissajous orbit if A1 = A2 = 0. A1 = 0 gives the stable manifold. The motion in z is an uncoupled oscillation at a different frequency. It can be seen that when starting from a state vector which satisfies A1 = 0, then a velocity increment A V = (Ako, A ~ owith ) ~

(-2,d> (ti;)= o

(3)

will not lead to an escape from that family of "orbits around L2". From this we define the escape direction of AV components creating an unstable motion by the line (see figure 1)

and the non-escape direction orthogonal to u by

( +, 2 )

A

fsT= f

(5)

The escape line is +28.6" from the x-axis (=sun to Earth axis) and the non-escape line is -61.4" from the x-axis. It can be observed that: 0

0 0

In the linear problem these directions do not depend on the point in the orbit (homogeneous) Velocity increment components along f u control the stability Velocity increment components in the plane spanned by s and the zdirection will only change amplitude or phase of a non-escape orbit

From these properties a simple but effective method has been derived for the numerical construction of the Lissajous (or better non-escape) orbits in

Herschel, Planck and GAIA Orbat Design

119

X

A

Fig. 1. Escape and non-escape directions.

the full nonlinear problem with any type of perturbations. And also orbit maintenance strategies, methods to reduce the orbit amplitude and eclipse avoidance manoeuvres can be constructed using the same basic principle. 2.2. Numerical Construction of Non-escape Orbits

We start at any point on a Lissajous orbit, or on the stable manifold of it, either obtained from any analytic approximation or from the end point of the transfer orbit, or as continuation of any preceding orbit construction, e.g. stopping at the x-x-plane crossing. A bisection of velocity increments in the escape direction, combined with forward integration will then automatically find the non escape orbit through this initial position. The bisection process is continued until after a forward integration over e.g. 450 days the orbit does neither escape to the sun nor fall to the Earth. The z-component is not controlled. Table 2 shows the bisection process at the first x-z-plane crossing for the reference orbit of Herschel. AV is the velocity increment along the unstable direction, Tfin days and Rf in km give time and radius from Earth when the integration stops either above 2x106 km or below 0 . 5 lo6 ~ km. 2.3. Numerical Construction of Transfer Orbit A specific feature of an ARIANE launch is introduced by the launch site near the equator. The orbits around the libration points lie near the ecliptic plane. So orbits into which AFUANE can deliver large payloads may not always be suited to start a transfer to the Lz region. Therefore for

120 M. Hechler and J . Cobos Table 2. Numerical orbit generation by bisection.

Rf

no. 1 2

0.0000000000 2.5000000000

341.391 148.000

194782.7 2503830.5

8 9 10

0.0390625000 0.0195312500 0.0097656250

248.000 262.000 280.000

2537365.8 2519552.8 2559225.2

............

........

.........

16 17

0.0007629395 0.0006866455

410.000 452.379

2515283.6 44090.8

... ...

AV

............

Tf

........

Fig. 2. Numerical construction of Herschel orbit. The Transfer construction method leads to selection of Lissajous orbit (Ay, A,, I&) around L2 such that its stable manifold touches the best ARIANE launch conditions. Repeated numerical corrections at each crossing of the x-z plane, each time numerically generate a non escape orbit over 450 days.

Herschel/Planck, rather than prescribing the target orbit, orbits around L2 were searched which can be reached from maximum mass ARIANE launch conditions. This led to the class of large size Lissajous orbits. The stable

Herschel, Planck and GAIA Orbit Design 121

manifold of these orbits ”touches” the conditions reached by an ARIANE launch for a particular launch date and hour. The ”escape direction” on this stable manifold] and at the perigee point, has been assumed to be along the perigee velocity direction, independent of above discussion. This directly leads to an orbit construction and orbit selection method for Herschel. The amplitude reduction for Planck will be discussed below. Before starting the bisection the “fuzzy boundary” must be localised. Table 3 and Figure 3 display the scan in perigee velocity which defines the zone between escape to the sun and a closed orbit at Earth to the step size of the scan. The velocity increment AV, (m/s) at perigee is stated relative to that of a geostationary transfer orbit. Table 3. Scan in perigee velocity for fuzzy boundary. no. 9 10 11 12 13 14 15

AV, 741 738 735 732 729 726 723

Tf 64.3 107.1 77.8 55.8 45.1 38.2 33.2

Rf 2525932.4 2546203.3 90498.8 27342.1 12583.2 9398.2 7589.2

Fig. 3. Scan for fuzzy boundary.

Once the fuzzy boundary has been localised a bisection in the pericentre velocity will do the rest as shown in table 4 and Figure 4. At the end (in-

122

M. Hechler and J . Cobos

tegration does not reach any stop conditions within 450 days) it converges to an orbit which remains captured in the L2 region. And this is the orbit searched for the space project. Table 4.

Bisection in perigee velocity.

Tf

Rf

107.157 115.362

2546203.3 451451.3

no. 1 2

AVP 738.0000000000 736.5000000000

..............

.......

......

20 21

736.8734264374 736.8734292984

407.022 453.494

16355.0 194775.5

1 e+w

500000

-500000

-let06

-a

Fig. 4. Bisection in perigee velocity.

2.4. Transfer between Orbits of Different Size

Both spacecraft, Herschel and Planck, will be delivered by ARIANE into the stable manifold of a Lissajous orbit with a large size

S,, = 2/A$ + A : in the yz-plane as shown in figure 2. S,, is equivalent to the maximum sunspacecraft-Earth angle. Planck will have to be manoeuvred from there to an orbit with a smaller size. From the linear theory such amplitude reduction manoeuvres will be in the plane spanned by the non escape direction in the x-y-plane and by z. Since the motion in the xy-plane and the motion in z are decoupled, the problem of finding a minimum AV transfer between

Herschel, Planck and GAIA Orbit Design

123

two Lissajous orbits to reduce the size can first also be decoupled into two problems. It can then be analysed in which cases the two possible manoeuvres may combine by vector addition. In the xy-plane, an obvious approach is to enter at some point into the stable manifold of a Lissajous orbit with a different amplitude (coupled A,,A,). This will be done with a velocity increment AV along the nonescape direction (A1 = 0), written as

The A,-amplitude (remember A, = c2As ) after the manoeuvre can be expressed as

Inserting (6) into (7) renders a quadratic equation for a as function of (x, y). The solution of this equation with the smaller modulus (minimum AV) satisfies

E),

with p =< (2, &7z >, and the 2-amplitudes A:) and A:) of the initial and final orbit. From this it can be concluded that the minimum of a as function of (x, y) is obtained if (x, $) is aligned to (cp, -c1) which (0)

implies p = A, . From this it follows that there are two solutions for the time on the initial orbit at which this condition is satisfied each 4.7883 days before the {y = 0)-plane crossing. This time will be fixed when constructing the numerical solutions. Finally the value of the AV can be expressed as function of the ampIitude change (in km) aIone as

AV = AA, * 3.648001 x 10-7~-1.

(9)

Similar to this, a manoeuvre changing the z-amplitude can be represented in a closed form and it can be concluded that the minimum AV, is reached for z = 0, and

A& = AA, a3.952326 x 10-7s-1

(10)

The discussion on how to combine the two independent components for a given size reduction at minimum AV, and in particular on how to choose

124

M. Hechler and J. Cobos

the two target Amplitudes A , , A , is a little more complicated. Using a point ( a ,b) according to (f)

( a ,b) = Sye . (0.6783,0.7348)

the following algorithm can be derived

(11)

lo:

The numerical implementation uses the time and direction information for the manoeuvres as derived from linear theory, but regenerates non-escape trajectories after the manoeuvre by a bisection along the escape direction, until escape terms are suppressed. An example of this is shown in table 2.4 for the Planck orbit insertion manoeuvre. Table 5. no. 1

Bisection for Planck insertion manoeuvre. Avcorr

0.0000000

Tf 89.4

..................

7 8 9

21.9531250 21.9140625 21.9335937

17 18 19 20

21.9484710 21.9484329 .21.9484519 21.9484615

288.0 241.9 260.2

.................. 430.0 414.8 446.9 450.0

Rf 121548.0

Avtot 170.9

...................... 2516405.7 210446.5 112360.6

172.4 172.4 172.4

2556265.1 217892.5 102305.4 2241871.7

172.4 172.4 172.4 172.4

......................

I can be seen that the correction along the escape direction is as much as 22 m/s, but the change by this component of the amplitude reduction manoeuvre is less than 2 m/s. However the direction of the manoeuvre is changed from 118.6O according to the linear approximation to 124.9’. Figure 5 contains the amplitude reduction manoeuvre together with a numerical generation of the Planck orbit over 5 revolutions.

Herschel, Planck and GAIA Orbit Design

--t

-

I

125

1

x (W)

Fig. 5. Numerical construction of Planck orbit. Double launch on ARIANE 5 with Herschel, separation at launch, same transfer orbit, amplitude reduction manoeuvre 4.6 days before x-z-plane crossing.

2.5. Eclipse Avoidance

Because the two oscillation periods in y and z are different for a Lissajous orbit around Lz (see figure 6 for a corresponding case with a small amplitude) a spacecraft on such an orbit will enter into eclipse some day depending on the initial phase angle &. The time span from eclipse to eclipse can be proven to be about 6 years, so either the initial phase may be chosen to reach 6 years without eclipse for a space mission or a manoeuvre strategy may be introduced to avoid an eclipse. Using the derivations of the preceding section it can be demonstrated l1 that an orbit with exactly the same amplitude (in linear theory) will be reached by inverting the z-velocity at the point with maximum lyl position component (~ = 0). It is easy to see that this corresponds to go exactly "time back" not only in the z-component but also in y, due to the symmetry of y relative to the maximum. This has the effect that if an orbit was to go

126

M. Hechler

and . I . Cobos

Fig. 6. Lissajous Orbit projection into yz-plane as seen from Earth, starting tangential at Earth shadow, high order analytic propagation in restricted circular 3-body problem for 6.8 years until next eclipse.

into eclipse at the next occasion after this maximum ( y (passage, after such a z reversal manoeuvre it will move for another 6 years without eclipse. A rough estimate of the size of the z-velocity reversal manoeuvre comes out to be

N

1

- s4.4 x lo9 m2/s (for 15'0rbit)

A, r o is the radius of an exclusion zone, e.g. the penumbra of the Earth at L2 distance. A similar strategy can also be constructed for a manoeuvre in the xy-plane along the non-escape direction. The size of such a manoeuvre will depend on the z-amplitude as follows.

21

1 Az

- .4.0 x lo9 m2/s (for 15"orbit)

The manoeuvre is executed at a point with maximum lyl-component (j, = 0 ) . This point is 4.7883 days before the maximum in IzI is reached. With above rough estimations, it is easy to see that the criterion in order 5 -= 0.923, then to choose one or the other strategy would be: if

2

Herschel, Planck and GAIA Orbit Design

127

a t-manoeuvre is cheaper. The most expensive case is for A,/A, = 0.923.

Thus a global bound depending on S,, = l*vl

I 2\/wz+ 21

(

5.8397258 c2

only is defined by.

)2zJ%x

15.1 m/s (for 15"orbit)

In the numerical implementation, the possibility of "going closer to the shadow" on the way back has been implemented, and both strategies are then explicitly compared rather than taking a decision on above linear theory criterion. The feasibility of such a manoeuvre and the global bound has been confirmed after numerical correction of the nonlinearity effects for a large amount of cases over the yearly launch windows of the different projects. In most cases in the launch window the z-reversal strategy comes out to be preferable. This allows conclusions on the preferred sun aspect angle of the manoeuvre.

g N

Fig. 7.

1

0 .............................

I _

"

4

' .. _..........<............. .."

L*:::::.::. :.:: -__ :.&.-.-

~I ................ : ........... ,..

Eclipse avoidance manoeuvre (2-reversal).

Figure 7 shows the y-z projection of the orbit through such a reversal of the z-velocity at maximum IyI. The motion before the eclipse (dashed line) starts at the x-z-plane crossing (start point marked by a manoeuvre square). Without the eclipse avoidance manoeuvre at minimum y, the eclipse would be entered one revolution later (dashed line). The backward motion after

128 M. Hechler and J . Cobos

the manoeuvre i shown by the continuous line. After the eclipse avoidance manoeuvre ther will be always at least another 6 years without eclipse. 2.6. h n s i t i o n to Nonlinear Theory

With a single spacecraft e.g. launched by a Soyuz launcher from Baikonur (Eddington, GAIA missions) there is no reason 'a priori' to inject with the launcher onto the stable manifold of a Lissajous orbit. To demonstrate that, also for that case, the chosen strategy is not far from optimum, a much more complicated approach had to be taken. Differently from previous studies with a prescribed target orbit the amplitudes were to be made part of the optimisation (with the sun-spacecraft-Earth angle condition) and also an injection manoeuvre onto the stable manifold of such an orbit rather than onto the orbit itself was to be allowed. For this a new theory of representing the stable manifold of Lissajous orbit in the nonlinear problem has been developed l2 expanding the nonlinear terms in the equation of motion into Legendre polynomials and deriving a formal series solution according to the Lindstedt-Poincar6 method. The result of this study was that an improvement of only less than about 10 m/s could be obtained when injecting to the stable manifold of a given amplitudes Lissajous orbit rather than to the orbit itself. 3. Project Applications

3.1. Launch Windows For Herschel/Planck The derivations of the previous sections have been extensively used to construct transfer orbits from ARIANE launch conditions, amplitude reduction manoeuvres and forward propagations of orbits for Herschel and Planck, including the eclipse avoidance strategy as outlined. this has been done on a grid of launch date over a whole year and launch hours over the range of times in which the direct transfer to L2 is feasible. On each grid point the orbit insertion manoeuvre size for Planck and additional conditions are evaluated. The most important such conditions are:

0

the sun aspect angle during the ARIANE powered ascent duration of eclipse in the transfer

The launch window will be defined by the possible combinations of lift off times an launch dates for which several conditions are satisfied. Figure 8

Herschel, Planck and GAIA Orbit Design

129

presents these conditions for an ARIANE 5 / ESV launch in the form of level lines as function of launch date and launch hour. It can be seen that the dominating condition in this case is the tank size limit on Planck, namely the AV limit, consisting of the insertion AV (amplitude reduction) plus the eclipse correction. All stochastic allocations are accounted separately in the propellant budget. The condition that there shall be no eclipse during the transfer (battery dimensioning) takes out a triangular region near the equinoxes and a line before. The sun aspect angle condition during the ARIANE ascent (worst angle at upper stage ignition for ARIANE 5 / ESV) is not actually active for this case.

."".

LRUNCH DATE AV (4s)

total orbit AV z 180 m/s s u n aspect at EPS ignition c 6 0 deg

total orbit

eclipse during transfer > 75 rnin

time to eclipse 4.6.8 hours (in days)

Fig. 8. ARIANE5/ESV Planck launch window for 15' sun-spacecraft-Earthangle, with eclipse avoidance (2007).

Figure 9 shows that for an ARIANE 5 / ESC launch (cryogenic upper stage without possibility to introduce a coast arc by delayed ignition of the upper stage) the sun aspect angel condition (in this case at fairing separation) becomes very important. In fact a sub-optimum ascent trajectory will be used to move the line of apsides of the reached orbit relative to the Earth, only for the purpose of improving this condition slightly.

130 M. Hechler and J. Cobos

totol orbit AV > 325 m/s Isun aspect a t foiring separation < 60 deg eclipse during tronsfer > 75 mm

LRUNCH D A T E total orbit AV (m/s) time to eclipse 4.6.8 hours (tn days)

Fig. 9. ARIANE5/ECA Planck launch window for 15' sun-spacecraft-Earth angle, with eclipse avoidance (2007).

3.2. Navigation and Orbit Maintenance

Independent of the deterministic transfer strategy outlined above, a strategy to handle deviations from the planned trajectory has to be derived. To quickly remove the launcher dispersion is particularly important for the nearly parabolic transfers to libration point regions, as for this type of orbits errors quickly amplify and possibly the target cannot be reached any more. Also when the Lissajous orbit has been reached regular orbit correction manoeuvres are necessary to counteract the instability. A navigation process which combines orbit determination to detect the deviation from the desired trajectory and the optimisation of re-targeting manoeuvres will have to be implemented. Mission analysis studies will assess the statistics of these stochastic manoeuvres to derive a propellant estimate. The first orbit correction during the transfer of Herschel and Planck has been studied using the same thinking as for the orbit construction itself. For each random point of a Monte Carlo simulation of the launcher error

Herschel, Planck and GAIA Orbit Design

131

using the launcher dispersion matrix, a manoeuvre 2 days from launch is calculated along the velocity direction such that a non escape orbit is reached for Herschel. It is not evident that on day 2 from perigee the orbital velocity is close to the escape direction. The escape directions on the stable manifold of a Lissajous orbit remain to be derived. For each case the Planck insertion manoeuvre is recalculated. This procedure has been repeated on a reduced grid of launch times and a propellant estimate (a 99-percentile) for the first orbit correction on day 2 (50 m/s for the percentile) and the stochastic increment in the orbit insertion of Planck has been sampled. Another pair of maneuvers of about 3 m/s each will be necessary for further targeting about 10 days from the first correction and 10 days before Lissajous orbit injection. These manoeuvres have been assessed using conventional methods of interplanetary navigation. The navigation in the operational orbit has been studied in detail 13. A large number of simulations has been performed. These simulations first demonstrate that 0.3 mm/s Doppler and 10 m range tracking (la assumed accuracies) from 1 ground station are sufficient to reach a position accuracy of 10 km in the plane of sky throughout the mission. In the viewing direction and also away from zero declination the position accuracy is in the order of 1 km. From the tracking data the unstable deviations from the desired motion around Lz will be observed. Deviations will have to be removed without too much time delay, the spacecraft has to be targeted back to a quasi-periodic motion around L2. Several orbit maintenance strategies to achieve this have been tested: 0 0

0 0

Classical interplanetary navigation with a shifting target position. Linear Quadratic Control as suggested in the phase A for SOH0 4 . Removal of the velocity component along the escape direction. Re-computation of future periodic trajectory at each manoeuvre time.

Independent of the method it could be demonstrated that an allocation of not more than 1 m/s per year is sufficient to maintain the orbit, provided there are no major perturbations as un-symmetric wheel off-loading etc.. A maneuver will be necessary at least once per month. The very simple strategy of removing the unstable motion (along the linear escape direction)

132 M. Hechler and J . Cobos

seems to perform well and has a property from which the spacecraft design could take advantage of. All manoeuvres will be nearly aligned or opposite to a direction 28.6" from the sun to Earth x-axis.

3.3. Launch window for GAIA Soyuz will inject its payload from Baikonur first into a circular orbit (about 190 km altitude) at 51.8" inclination. A second burn of the F'regat upper stage at optimum time in this orbit will then inject the spacecraft into its transfer orbit to Ls. Consequently, for a Soyuz launch, there is one more degree of freedom than for ARIANE, to optimise the transfers. The argument of perigee can be arbitrarily chosen. Also the actual lift-off time can be exactly prescribed for a Soyuz launch, a daily slot is not required from the launcher side. This additional degrees of freedom open the possibility to include the condition that the orbit should not have an eclipse over 5 year as a constraint in the minimisation of the size of the insertion manoeuvre. Figure 10 shows this AV and the optimum lift-off time as function of the launch date. The argument of perigee has been optimised as well. The target orbit is free of eclipse over 5 years. It can be seen that a similar propellant allocation as for Planck will be necessary, e.g. 180 m/s for a 6 months launch window, however the launch window will rather consist of one slot in winter and spring rather than two slots near the equinoxes. This comes from the variation of the orbit inclination relative to the ecliptic over the year. It has

145

,

1W

,

,

0

,

IW I

I

,

2w

, ,

,

3w

,,

hi

.

I

Fig. 10. GAIA - Soyuz/Fregat launch window for 15' sun-spacecraft-Earth angle.

been assumed that the second Fregat burn can only be scheduled near the

Herschel, Planck and GAIA Orbit Design

133

ascending node of the orbit after about one full revolution, due to ground st at ion coverage requirements. The orbit maintenance can be shown to be feasible for GAIA, but eclipse avoidance manouevres, as conceived for Herschel and Planck, will not be possible for GAIA because of the extreme low thrust level of the electric propulsion system. However as discussed above, for the Spyuz launch there will always be a choice of the lift-off time and the argument of perigee such that the transfer as constructed ends into an orbit without eclipse. An error analysis for this has not yet been done.

It has also been demonstrated l 3 that for a Soyuz launch from Baikonur transfer strategies with a lunar gravity assist will always be possible, with a proper phasing strategy before the lunar fly-by. This will safe about 50 m/s propellant. For the time being a direct transfer has been taken as baseline.

4. Conclusions

Lissajous orbits around Lz have been sel6cted for the ESA Astronomy missions Herschel, Planck, Eddington and GAIA. Rather than Halo orbits, different types of Lissajous orbits will be flown. Starting from the dynamic features of the orbits around Lz as seen in the linear theory a rather simple but quite general and systematic approach a has been derived for all orbit manoeuvres. In all cases the linear theory is applied to calculate a first guess of a manoeuvre in the linear approximation of the non-escape direction. A bisection method along the (also guessed) escape direction is then used to re-generate the motion around L2. This is applied to 0 0 0

0 0

generate the transfer orbit, (bisection at perigee) calculate the orbit insertion manoeuvre of Planck (amplitude reduction) generate the Lissajous orbits (small corrections at inferior x-z-plane crossings) calculate eclipse avoidance manoeuvres assess the orbit maintenance manoeuvres.

A basic difference to previous work is that the target orbits are never pre-

134

M. Hechler

and J . Cob08

scribed. The sole criteria of an orbit to be suited for a particular space mission are those induced by the mission requirements themselves. These are:

0

remain within a certain sun-spacecraft-Earth angle no eclipse

An essential contribution to introduce the above orbit design is the possibility to easily avoid eclipses (or more general e.g. for L2 an exclusion zone in some angle) by manoeuvres. The derived numerical methods are rather fast and robust. They could therefore be used to derive the launch window for Herschel/Planck and GAIA with a Soyuz launch. An interesting application was a transfer to an orbit around L2 ’via’ L1 for an ARIANE double launch near midnight g.

References 1. R. W. Farquhar, Preliminary Considerations for the Establishment of a Satellite in the Neighbourhood of Centres of Libration, M.S. Thesis, UCLA, Dec. 1960. 2. G. Colombo, The Stabilisation of an Artificial Satellite at the Inferior Conjunction Point of the Earth-Moon-System, Smithsonian Astrophysics Observatory Special Report No. 80, Nov. 1961. 3. R. Farquhar, The Control and Use of Libration Point Satellites, Stanford University Report SUDAAR-350, July 1968. 4. J. Rodriguez Canabal, M. Hechler, Orbital Aspects of the SOH0 Mission Design, AAS 89-171. 5. R. Farquhar, Halo Orbits and Lunar Swingby Missions of the 199O’s, Acta Astronautica, Vol 24, 1991, pp. 227-234. 6. G. G6mez, A. Jorba, J. Masdemont, C. Sim6, Study Refinement of Semi-Analytical Halo Orbit Theory, Final Report ESOC Contract 8625/89/D/MD(SC), Barcelona, April 1991. 7. G. G6mez, J. Llibre, R. Martinez, C. Sim6, Dynamics and Mission Design Near Libration Point, Vols 1to 4,World Scientific Monograph Series in Mathematics, Singapore 2001. 8. E. Belbruno, G.B. Amata, Low Energy aansfer to Mars and the Moon Using Fuzzy Boundary Theory, ALENIA doc. SD-RP-AI-0202, Final Report of ESA Contract, August 1996. 9. M. Hechler, GAIA/FIRST Mission Analysis: ARIANE and the Orbits around L2, MAS W P 393, ESOC February 1997.

Herschel, Planck and GAIA Orbit Design

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10. M. Hechler, J. Cobos, FIRST Mission Analysis: Transfers to Small Lissajous Orbits around L s , MAS WP 398, ESOC July 1997. 11. M. Hechler, J. Cobos, FIRST/PLANCK and GAIA Mission Analysis: Launch Windows with Eclipse Avoidance Manoeuvres, MAS WP 402, ESOC December 1997. 12. J. Cobos, M. Hechler, FIRST/PLANCK Mission Analysis: Transfer to Lissajous Orbit Using the Stable Manifold, MAS W P 412, ESOC December 1998. 13. M. Be116 Mora, F. Blesa Moreno, Study on Navigation for Earth Libration Points, Final Report ESA Contract No. 12571/97/D/IM(SC), 1999. 14. M. Hechler, Herschel/Planck Consolidated Report on Mission Analysis, Issue 2.0, ESOC June 2002.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont ( 4 s . ) @ 2003 World Scientific Publishing Company

GETTING TO

L1

THE HARD WAY: TRIANA’S LAUNCH OPTIONS M.B. HOUGHTON

NASA’s Goddard Space Flight Center, Greenbelt, Maryland 20771, USA

Over the past four years, NASA’s Goddard Space Flight Center has built and tested the Triana observatory, which will be the first Earth observing science satellite to take advantage of the unique perspective offered by a Lissajous orbit about the first Earth-Sun Lagrange Point ( L 1 ) . Triana was originally meant t o fly on the U.S.Space Transportation System (a.k.a. the Space Shuttle), but complications with the shuttle manifest have forced Triana into a “wait and see” attitude. The observatory is currently being stored a t NASA’s Goddard Space Flight Center, where it waits for an appropriate launch opportunity to materialize. To that end, several possible alternatives have been considered, including variations on the nominal shuttle deployment scenario, a high inclination Delta-type launch from Vandenberg Air Force Base, a Tsyklon class vehicle launched from Baikonur, Kazakhstan, and a ride on a French Ariane vehicle out of French Guiana into a somewhat arbitrary geostationary transfer orbit (GTO). This paper chronicles and outlines the pros and cons of how each of these opportunities could be used to send Triana on its way to L1.

1. Introduction The Triana observatory, built by NASA’s Goddard Space Flight Center and shown in Figure 1, will be the first Earth science mission to take advantage of the unique perspective offered by a Lissajous orbit about the first EarthSun Lagrange Point (Ll). From this vantage point, Triana will, for the first time ever, send back images of a nearly fully lit Earth disk (see Figure 2), in 10 different wavelengths (UV, visible, and near IR), 24 hours a day.

137

138 M.B. Houghton

In addition to its Earth viewing instruments, Triana has a suite of solar wind (i.e. space weather) instruments. Collectively, the data from Triana will be used to drive investigations in various research areas and to set the foundation for various education and outreach programs '. MmAR

FarsdevcttP

""&,!

EPIC T Pa % : l Y n r B j

Fig. 1.

ILArtist'srendering" of fully deployed Wana observatory.

Fig. 2.

Image representative of those to be taken by Triana at L1.

2. Baseline Trajectory & Orbit

Triana was originally meant to fly on the US. Space Shuttle. Its baseline trajectory is a direct transfer to a 15 x 4 deg (sun-earth-observatory angle) Lissajous orbit about L1. This trajectory flows directly from a circular

Getting to L1 the Hard Way: Zkiana’s Launch Options

139

low Earth orbit (nominally a Space Shuttle orbit) and has an outgoing asymptote that is essentially right along the Earth-sun line (Figure 3). It was chosen to simplify the mission design and to minimize manifesting constraints (i.e. no lunar timing issues) with the Space Shuttle. Triana’s baseline mission orbit at L1 (a Lissajous) was designed so that the sun-earth-observatory angle never exceeds 1 5 O , or goes below 4O. This was done to maximize the value of the science data and to avoid any communications issues, respectively. It was designed to accommodate a 2 year minimum mission with a 5 year goal (Figure 4). Ecliptic Plane E$rth-Sun Line

Lissajous Orbit -’*--

Transfer Trajectory 2.2 days out

From Earth Fig. 3.

Views of Triana’s baseline direct trajectory to L1 orbit.

3. Triana as a Shuttle Payload

Triana was originally meant to be flown on the U.S. Space Shuttle. As such, NASA’s Goddard Space Flight Center built and tested the Gyroscopic Upper Stage (GUS) to deliver the final thrust needed to get Triana from

140 M.B.Houghton

‘.

FroinEarth Fig. 4.

./,

\--* ’

Triana’s baseline 15 x 4 deg mission orbit (2-5 year life).

the Shuttle’s low earth orbit (LEO) to an L1 transfer trajectory and to interface Triana with the Italian Research Interim Stage (IRIS) airborne support equipment ’. The arrangement is illustrated in Figures 5 & 6. The procedure for actually flying Triana on the Space Shuttle is long and arduous, but begins with the Shuttle Program Office specifying an inclination and altitude for a given manifested mission and the Guidance, Navigation, and Control Center (GNCC) at Goddard generating firstcut LEO state vectors that match the specified conditions and are appropriate for use in getting to L1. Goddard uses a software tool called Generator, which was developed by Purdue University in West Lafayette, Indiana. Generator uses Dynamical Systems Theory (DST) to work “backwards” (using an ephemeris model) from a given Lissajous orbit to a specific LEO state. It uses invariant manifolds (Figure 7) to get initial trajectories which it then passes through a differential corrector to match the LEO conditions. The final solutions are verified by way of “proven” software at Goddard, such as the PC-based Swingby program (which uses high-fidelity force models), and sets of Transfer Trajectory Insertion (TTI) points (and associated A V s ) are then produced for use by the Shuttle Program in planning the given Shuttle mission (Table 1 and 2) 3 . The Shuttle Program Office works with a given set of TTI points to determine a mission timeline in which they are able to deliver Triana to a specified TTI point at the specified time to within a specified tolerance. In

Getting to L1 the Hard Way: W a n a ’ s Launch Options

141

Fig. 5. Triana atop its upper stage in proximity to its IRIS “cradle”

Fig. 6. Triana’s originally manifested STS-107 accommodations.

terms of cross-track and along-track errors, the agreed upon toler- ances boil down to 3.8” in right ascension and 1 minute in release (along-track) timing. Typical Shuttle orbits regress at rates as high as 0.47” per orbit, SO the right ascension constraint gives the Shuttle Program a 16 orbit window to work with - 8 on either side of the TTI point (Figure 8). Once Triana is released from the Shuttle and the Shuttle is given enough time to maneuver to a “safe distance”, the GUS is ignited. The GUS provides a fixed AV that is set, by way of ballast weights, prior to launch, in order to match the required AV, as computed from the TTI table for

142 M.B.Houghton

the given mission scenario (altitude and inclination). Figure 9 shows how little the actual value changes with TTI date. The GUS is set to match the average value for a given launch period. Triana's hydrazine system is later used to make-up for any release and GUS AV errors.

Fig. 7.

Sample manifold used in Triana's trajectory design process.

Fig. 8. Triana's release window from Shuttle mission perspective.

I

,

nl ''W

'%.

II -I

LlWl

-3 "5-

Fig. 9.

7 m .

nm%

>.-

%(,%a*

Representative TTI AV variation over the course of 1 year.

Getting to L1 the Hard Way: lkiana’s Launch Options 143 Table 1. Sample set of ’Pransfer Trajectory Insertion (TTI) points and velocity information for use in planning Triana’s egression from the Shuttle. For each T T I Date and Time, the Pre-TTI Positon Vector (ECI MJ2000) (km), the Pre-TTI Velocity Vector (km/s) and the Post-TTI Velocity Vector (ECI MJ2000) (km/s) are given. TTI Date & Time (UTC)

X-Pos (pre) X-Vel (pre) X-Vel (post) 03/12/200204~54~55.246 -6496.295660 0.545300198 0.768524656 03/13/2002 07:29:08.686 -6533.324861 0.823688739 1.161076325 03/14/2002 11:58:51.406 -6509.307254 1.102960140 1.554967693 03/15/2002 13:43:58.606 -6483.153613 1.295358237 1.826309911 03/16/2002 14:19:15.406 -6457.246812 1.453538958 2.049361716 03/17/2002 14:43:35.566 -6431.384405 1.593303689 2.246415876 03/18/2002 14:38:07.246 -6405.365815 1.721233509 2.426759620 03/19/2002 14:29:37.486 -6379.117024 1.840557349 2.594951562 03/20/2002 14:20:15.886 -6352.544855 1.953424733 2.754027949 03/21/2002 14:11:54.766 -6325.505068 2.061569344 2.906439171 03/22/2002 14:07:35.566 -6297.772497 2.166611243 3.054476240 03/23/2002 14:00:23.566 -6268.927544 2.270522812 3.200928512 03/24/2002 13:53:02.926 -6238.485220 2.375112945 3.348355520

Y-Pos (pre) Y-Vel (pre) Y-Vel (post) -1031.834389 -6.882538272 -9.699978797 - 1117.128821 -6.788818011 -9.569556432 -1313.011839 -6.732747449 -9.491915786 -1459.700794 -6.691632520 -9.434451754 -1583.433335 -6.654868669 -9.382777809 -1694.035083 -6.619817798 -9.333351763 -1795.941048 -6.585488814 -9.284851969 -1891.370362 -6.551457919 -9.236721675 -1981.857707 -6.517431534 -9.188574452 -2068.667391 -6.483115177 -9.140017506 -2152.998693 -6.448146761 -9.090560728 -2236.342517 -6.411941921 -9.039401677 -2320.072543 -6.373859016 -8.985655214

Z-Pos (pre) Z-Vel (pre) Z-Vel (post) 1017.328778 -3.498582205 -4.930764184 607.607440 -3.624969011 -5.109776909 454.330635 -3.655202927 -5.153153097 373.483822 -3.667561448 -5.170850526 313.368529 -3.675165133 -5.181658658 262.749165 -3.680525835 -5.189212655 217.673800 -3.684501314 -5.194762340 176.299456 -3.687490786 -5.198892597 137.518166 -3.689721093 -5.201938340 100.489825 -3.691335137 -5.204113586 64.463595 -3.692422921 -5.205556890 28.582285 -3.693033680 -5.206350159 -7.931316 -3.693171205 -5.206510374

144 M.B. Houghton

Table 2. Sample set of Transfer Trajectory Insertion (TTI) points and velocity information for use in planning Triana’s egression from the Shuttle. For each TTI Date and Time, the Pre-TTI State Vector (ADBARV MJ2000) and the TTI Thrust Unit Vector (ECI MJ2000) are given. TTI Date & Time (UTC) 03/12/2002 04:54:55.246

03/13/2002 07:29:08.686

03/14/2002 11:58:51.406

03/15/2002 13:43:58.606

03/16/2002 14:19:15.406

03/17/2002 14:43:35.566

03/18/2002 14:38:07.246

03/19/2002 14:29:37.486

03/20/2002 14:20:15.886

03/21/2002 14:11:54.766

03/22/2002 14:07:35.566

03/23/2002 14:00:23.566

03/24/2002 13:53:02.926

RA (deg) A (deg) x (unit) 189.02514096 117.21885841 0.070452714 189.70312839 118.05433633 0.106420344 191.40427361 118.25230998 0.142502122 192.68872125 118.33302626 0.167359915 193.77808485 118.38262822 0.187796842 194.75660548 118.41757169 0.205854436 195.66247199 118.44347175 0.222382968 196.51477783 118.46294053 0.237799614 197.32680159 118.47746143 0.252382092 198.10960415 118.48796871 0.266354394 198.87384987 118.49505001 0.279925830 199.63306922 118.49902699 0.293351231 200.40000554 118.49992508 0.306864305

Dec (deg) R (km) Y (init) 8.79185282 6655.937 -0.889223044 5.23771139 6655.937 -0.877113296 3.91402220 6655.937 -0.869868965 3.21672100 6655.937 -0.864557015 2.69854325 6655.937 -0.859807240 2.26239085 6655.937 -0.855278794 1.87411844 6655.937 -0.850843620 1.51780227 6655.937 -0.846446957 1.18387106 6655.937 -0.842050873 0.86507153 6655.937 -0.837617333 0.55492557 6655.937 -0.833099543 0.24604340 6655.937 -0.828422002 -0.06827454 6655.937 -0.823501813

B (ded V (km/s) z-(uditj 90.0000 7.739945918 -0.452016363 90.0000 7.739955649 -0.468344933 90.0000 7.739955925 -0.472251129 90.0000 7.739955150 -0.473847894 90.0000 7.739954216 -0.474830345 90.0000 7.739953149 -0.475523012 90.0000 7.739952045 -0.476036711 90.0000 7.739950938 -0.476423018 90.0000 7.739949854 -0.476711240 90.0000 7.739948685 -0.476919846 90.0000 7.739947542 -0.477060458 90.0000 7.739946434 -0.477139437 90.0000 7.739945337 -0.477157272

4. Optional Shuttle Scenarios Several alternate Shuttle mission scenarios have been considered in an attempt to make Triana as compatible as possible with the Shuttle manifest.

Getting to

L1

the Hard Way: Triana’s Launch Options 145

Fig. 10. M A N plot showing 12 opportunities on joint mission (Cam et aL3.

Triana was originally meant to fly on STS-107, at 150 nautical miles with an inclination of 39O,but that opportunity evaporated due to circumstances outside the control of Goddard, or anyone involved in the Triana project. I’riana has re-baselined for a generic research flight at 150 nautical miles with an inclination of 28.5O. This is considered to be the most likely (Shuttle) scenario. The inclination change causes no real problems. Generator is able to find solutions over a very wide range of inclinations, as evidenced below, without changing the shape of the final Lissajous. The feasibility of taking Triana along on a retrieval (e.g. UARS) or a rendezvous (e.g. Space Station) mission has been explored. These scenarios pose a problem only in that the right ascension of the ascending node (RAAN) of an object in low earth orbit regresses at a rate of several degrees per day (5 for Space Station, for example), due to the Earth’s oblateness, and the RA of Triana’s TTI point precesses at roughly 1°/day, due to the Earth’s motion about the sun. This creates a situation in which the two coincide (a requirement, if the Shuttle is going to do both on the same flight) only 6 times per year. This number can be doubled by recognizing the fact that their exists one set of TTI points for use near a LEO ascending node and one set for use near a LEO descending node. Combining these sets gives 12 opportunities per year (Figure 10).

A “random release” joint Space Station mission has also been considered. “Random release” refers to the right ascension (RA) constraint on

146 M.B.Houghton

Triana’s release point. In this scenario, the Shuttle is free to release Triana on any orbit, of any mission, as long as the Triana Project is allowed to still specify where in the orbit the observatory is released. This way, the Triana Project can constrain Triana’s RA at release to lie within a given hemisphere, and, consequently, its apogee (post GUS burn) to lie within a given hemisphere. The GUS ballast, in this case, is set such that Triana is injected into a highly elliptical orbit (HEO) with a period of 14 days, twice the moon’s orbital period, so as to minimize the moon’s impact on Figure 11: Illustration of “random release” Shuttle mission scenario the orbit. This way, Triana can loiter while the Earth-sun line precesses around (at roughly 1’ per day) to line up with Triana’s apogee (Figure 11).Triana’s hydrazine system is then used to send Triana the rest of the way to LI.Since the release and GUS AV corrections, as well as the final TTI maneuver, can all be done at perigees, this scenario does not differ, considerably, from the baseline scenario, in terms of total AV costs.

Triana loiters in HE0 for up to 7 months Fig. 11. Illustration of “random release” Shuttle mission scenario.

Getting to L1 the Hard Way: W a n a ’ s Launch Options

147

5. Looking for an ELV

In light of the lack of success in finding a spot on any near-term Shuttle flight, the Triana Project has explored various expendable launch vehicle (ELV) options, as well, in hopes that an opportunity might ultimately present itself. To that end, Triana’s energy requirement can be summarized in one of two ways. Either the observatory, itself, needs to be boosted to a C3 of -0.7 km2/s2, or the Triana/GUS stack needs to be placed in a sufficiently high-energy LEO such that the AV provided by the GUS is enough to put Triana on the proper trajectory. These relationships are shown in Figures 12 and 13, with the difference being the weight and the required velocities. Domestically, Delta I1 ELVs have the capability to boost Triana directly onto its transfer trajectory ‘. The Triana Project considered them at length. Even a flight out of Vandenberg Air Force Base, on the U.S. West Coast, where the resulting inclination would have been in excess of 60°, was considered. Generator was able to find solutions at inclinations as high as 80”. A Delta I1 appears to be a very viable option, but has never come to fruition, primarily due to a lack of such funding on the Triana Project. Triana’s Shuttle launch was meant to be “free”. 11.2 11.0

T Y

i

iB E.

=

10.0 9.8 9.6

0

ZOO

400

600

BOO 1wO 1200 1400 1600 1800 2000 ~ - w n l

Fig. 12. Energy requirement for Triana observatory (C3 of -0.7).

Discussions with United Start, a Delaware based corporation owned

148 M.B. Houghton

by the American company Assured Space Access, Inc. and the Russian company ZAO Puskovie Uslugi (Launch Service Provider) 5 , have led t o the conclusion that a 3-stage version of the Ukrainian Tsyklon vehicle, being proposed by United Start, would be capable of putting the Triana/GUS stack into an orbit with a perigee of 120 km and an apogee of 200 km at an inclination of 51.6O. Figure 14 shows that, as long as the orbit’s perigee is within roughly 60” of Triana’s TTI point, the Tsyklon/ GUS combination is enough to get Triana to L1.It is assumed that the Tsyklon vehicle places Triana/GUS in an orbit whose RAAN is specified by the Triana Project.

6. Triangular Transfer Trajectories

The final and possibly the most difficult to accommodate ELV considered for use by the Triana Project is the French Ariane 5. What makes this opportunity difficult is the fact that Triana would need to be co-manifested with some other payload, and the Ariane’s standard product is a rather specific geostationary transfer orbit (GTO). The Ariane “constraints” would put Triana into an orbit who’s line of apsides is as much as 45 minutes (11.25’ in RA) off the Earth-sun line and 23.4’ off the ecliptic 6 . Compared to the Shuttle tolerance of 3.8’ in RA, 11.25” is excessive. What makes this scenario possible is a “triangular” transfer trajectory first proposed by Purdue University as a possible Shuttle contingency option ’. These trajectories have departing asymptotes that are much farther off the Earth-sun line than the baseline trajectory (which is right along the Earth-sun line), easily encompassing the 11.25O (Figure 15). Putting Triana into an orbit with a 7.5 day period (roughly) would allow the triangular transfer trajectories to be achieved, ultimately, from anywhere within the line of apsides RA range (Figure 16). It would just be a matter of letting the Earth-sun line precess to the proper spot, while waiting in up to 2 phasing loops. As for the inclination issue, it’s possible to enter into a Lissajous at L1 from any of the given ecliptic inclinations, but doing so determines the z-amplitude of the resulting Lissajous (Figure 17). A AV would be required to lower (or raise) the z-amplitude to match the mission orbit. The cost of doing so was shown to be manageable (10’s of m/s). There is an additional issue with particularly low inclinations in that there will be

Getting to L1 the H a d Way: i’kiana’s Launch Options

149

a communications loss as the observatory passes in front of the sun. All of these nuances are ultimately manageable *. 80-

I

I

6 4

n

200

400

600

am

tono

1200

1400 1600

isno

2000

Fig. 13. Requirement placed on LEO state for use by Triana/GUS.

-180

-135

-90

4.5

M 4 . 1Anwno*

0

45

90

135

180

(r = a7.7 minnin)

Fig. 14. Tsyklon/GUS “energy balance” for getting Triana to L1.

7. Conclusion There are several ways of getting a satellite, such as Triana, to L1, and they’re not all equal. Clearly the most sensible solution is to have a dedicated launch vehicle for any satellite that is going to L1, or any Lagrange

150

M.B. Houghton

Lwfing ”Lhm’n” into the EciQlic Plane

Fig. 15.

“Triangular” transfer trajectories for use in getting to L1

Fig. 16. Line of apsides with respect to the Earth-sun line (Ariane).

point for that matter. From the standpoint of targeting, Lagrange point missions are essentially no different than interplanetary missions. They’re relatively hard to hit. Anything that makes that task more difficult than it already is should be avoided. The sensible approach not withstanding, Goddard Space Flight Center has shown that there are several options for getting Triana to L1, apart from the baseline Shuttle deployment scenario. It was shown that there are feasible ways of flying Triana on a rendezvous/retrieval type Shuttle mission and that the TTI point RA constraint can, if necessary, be lifted, through the use of phasing loops, allowing for even greater flexibility. The inclination of the LEO orbit from which Triana departs was also shown to be relatively unimportant. This was evident in the Delta I1 study, which looked at inclinations as high as 80”. Other considered options include a version of the Ukrainian Tsyklon vehicle and a co-manifested flight on an

Getting to

Fig. 17.

Tri angi

L1

the Hard Way: Ih’ana’s Launch Options 151

:ent inclinations.

Ariane 5 launch vehicle. The latter option would involve the use of phasing loops and a “triangular” transfer trajectory. Such flexibilities in mission planning approaches make a plethora of options ultimately available. References 1. J. Watzin; “The Triana Mission Implementation - A Unique Mission with a Unique Approach”; 2001. 2. C. Tooley, M. Houghton, et al.; “The IRIS-GUS Shuttle Borne Upper Stage System” AIAA 2002-3761. 3. G. Marr, S. Cooley, et al.; “Triana Trajectory Design Peer Review” (Package); September 26, 2001. 4. Delta I1 Payload Planners Guide - yuu.boeing .corn. 5. United Start Corporation - yuu.unitedstart. corn. 6. Ariane 5 User’s Manual - yww. arianespace .corn. 7. K. Howell, J. Anderson; “Triana Contingency Analysis Progress Report” ; March 2001, IOM AAE-0140-001. 8. M. Beckman; Personal communications and analysis.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and 3. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

SOLAR SURVEILLANCE ZONE POPULATION STRATEGIES WITH PICOSATELLITES USING HALO AND DISTANT RETROGRADE ORBITS J.A. KECHICHIAN, E.T. CAMPBELL, M.F. WERNER a n d E.Y. ROBINSON Astrodynamics Department, The Aerospace Corporation, EL Segundo, CA 90245-4691, USA

Solar surveillance missions that provide ample warning time of impending solar storms to satellite users in earth orbit are naturally designed t o allow the weather forecasting spacecraft to wander as far away from the earth as possible, in the direction of the sun. Vehicles positioned on the L1 halo orbits do not provide more than an hour of warning time due to the fact that the L1 equilibrium point of the sun-earth system is located at some 1.5 million km from the earth. In order to increase this warning time by a factor of two or more, use is made in this paper of distant retrograde orbits which allow compact sensorcraft to remain in the vicinity of the earth but at substantially larger distances from it than the L1 point. However, the sensorcraft transit periodically and only for a limited time inside the surveillance zone centered on the sun-earth axis, such that a certain number of such sensorcraft is needed for continuous surveillance capability. An alternative scheme produces a finite number of passes through the surveillance zone with some transits reaching much further out toward the sun, over several years. Another scheme utilizes a series of miniature probes released from a multi-probe carrier vehicle from a parking halo orbit, t o travel on that halo orbit invariant unstable manifold in successive single passes through the surveillance zone, requiring continuing replenishment. This paper shows several sensorcraft and sensor probes release strategies and their associated trajectories for both the distant retrograde orbit and the halo orbit cases. It also provides an estimate of the minimum number of probes needed for continuous coverage for a given time span.

1. Introduction

It is well know that high velocity plasma structures called coronal mass ejections (CMEs) emanating from the sun can interact with the magnetic field of the earth and trigger geomagnetic storms that can affect adversely power grids, communications and spacecraft in earth orbit. Initial studies 153

154 J . A . Kechichian, E.T. Campbell, M.F. Werner and E.Y. Robinson

focused on the use of halo orbits around the sun-earth L1 inner libration point using micro satellites equipped with a wide angle imager to track the coronal ejections, and a magnetometer to determine the magnetic polarity of the solar wind. This polarity must have the potential to interact with the magnetic field of the earth to trigger an alarm and alert government and utilities agencies of an impending geomagnetic storm. Although the small size and low cost of these small satellites enable affordable periodic replenishment and continuous presence at L1, the warning time is limited to about 45 minutes due to the L1 point being distant by some 1.5 million km from the earth, and the solar wind velocity of about 500 km/s. In order to increase this warning time by a factor of two or more, and provide ample time for the confirmation of the impending threat and study the CME characteristics, the spacecraft must wander beyond the LI point with at least one satellite transiting through the surveillance zone centered on the sun-earth line of sight at a given time for continuous coverage and monitoring of the solar weather. To this end, two different architectures are examined and analyzed in this paper. The first architecture makes use of the distant retrograde orbits (DROs) which stay in the vicinity of the earth while crossing the earth-sun line periodically at larger distances than the L1 point. The second architecture uses halo orbits from which a string of satellites is released at regular intervals of time such that they are allowed to travel on the halo orbit unstable manifold which branches out of the L1 location towards the sun in a threedimensional twisting path that resides for a certain time within the solar surveillance zone before wandering out of the zone in its heliocentric orbit. In the first architecture, a series of satellites is released from a GTO-type orbit at regular times, after some maneuvering, on a path that mimics a multi-period DRO such that these satellites transit through the zone of interest in succession for a continuous monitoring of the CMEs. Section 2 discusses this architecture. Drawing on the pioneering work of Farquhar and Richardson 2 , regularized nonsingular variables are used to describe the DRO type trajectories and the transfer trajectories that originate from earth orbit. The libration point orbit dynamics have benefited from the contributions in Refs. 8,9J0 especially as related to the geometry of the stable and unstable manifolds which enable zero cost transfers between certain orbits. The asymptotic departure from halo orbits with zero cost is adopted by the second architecture deployment strategy shown in Section 3. The investigation and use of the DRO and halo orbits are depicted in Refs. in a thorough way. 37475

697

11&%13>14715

Solar Surveillance Zone Population Strutegies with Picosatellites 155

2. Populating a Certain Distant Retrograde Orbit (DRO) with Small Satellites

Drawing on the considerable contributions of HQnon 1 2 7 1 3 , Ocampo and Ocampo and Rosborough l 4 have suggested the use of Distant Retrograde Orbits for a constellation of satellites to monitor the interplanetary medium at distances larger than the 1.5-106km of the L1 point in order to increase the warning times of impending geomagnetic storms. The restricted circular three-body model as well as Hill's approximation or simplified model are used 7%14 t o study via the PoincarC map the characteristics of the DRO orbits in the sun-earth-moon system where the earth and the moon are considered as a single body with mass m2 such that p = mz/(ml m2) with ml the sun's mass. In a rotating reference system centered at m2 with x along the ml - m2 direction, y in the ecliptic plane, and z along the out-of-ecliptic direction, and the usual normalizations which set to unity the sun-earth distance a, the system mean motion n, and the sum ml m2, the differential equations of the circular model are given by

+

+

+

+ +

+

where r1 = ((x 1)2 y2 + z2)'l2,and 7-2 = (x2 y2 z 2 ) ' / 2 are the ml-s/c and m2-s/c distances respectively. Hill's model is generated by rescaling the distance unit by p 1 / 3 and taking the limit p + 0 617

X

X - 2y = 3~ - -, r3

(2)

y+2i=-- Y r3

z =

z

(3)

'

--.

(4)

T3

The Jacobi constant C for the restricted circular three-body model and the Hill model respectively are given by i2+j,' i 2= 2R - C and i2 G2 +i2= 2RH where fi = (l/2)((x 1 - p)2 y2) (1 - , % ) / T I p / r 2 and RH = (3/2)x2 1 / with ~ r = ( x 2 + y2 z 2 ) l I 2 .Various classes of DRO orbits can be investigated by first generating Poincar6 sections in the form of the ( x , i ) plane for the planar problem of motion in the ecliptic plane (y = 0), for various values of the energy constant C. Considering the i < 0

c,

+

+

+

+

+

+

+

+

156 J.A. Kechichian, E.T. Campbell, M.F. Werner and E.Y. Robinson

crossings through the PoincarC section C, retrograde orbits about the origin, or the earth, are generated using Hill’s model. It is known that a periodic orbit exists at the center of the bounded curves of the PoincarC section plots. Thus a family of single periodic DRO’s, stable in the linear sense are obtained as well as n-period unstable retrograde orbits which do not feature any close encounters with the earth 7 J 4 . Unstable periodic orbits with repeated passes close to the earth are discussed in Ref. and examples of simple period as well as 2 and 4 period orbits are shown. These close encounter orbits are used to inject a spacecraft from low earth orbit, and an insertion maneuver will later capture a DRO-type orbit to carry out its mission.

In the restricted circular three-body model, both stable DRO’s as well as unstable ERO’s (Earth Return Orbits) can be generated The stable simple period DRO’s are shown to be linearly stable up to x values of 10 million km from the earth. The simple period ERO’s belong to the family of the planar Lyapunov orbits about either the L1 or L 2 collinear libration points which are continued and extended towards the earth until a close approach is achieved. Multiple period or n-cycle ERO’s are also shown in Refs. However these multi-period ERO’s tend to get less wide along the x direction than their simple period counterpart for the same closest distance to the earth. An example using xo = 100,000 km at closest approach is used in Ref. ’, and a simple period ERO with a period of roughly 400 days, as well as a period 2 and period 4 ERO of period 565 and 974 days respectively, are generated. The simple period ERO wanders beyond the 3 million km mark along x only once in its orbit but the 2 and 4period ones wander at a maximum x value of 2.106 km twice, and at 1.5.106 km and 2.5.106 km twice each respectively, albeit along longer period orbits. However, the multi-period ERO’s provide a higher frequency of entry into the surveillance zone centered along the x axis, per time unit than their simpleperiod counterpart, and for this reason they are more interesting as candidate trajectories for inexpensive throw-away small satellites which would be on station more than once in their mission lifetime. 7114.

7314.

In order to search for distant retrograde orbits with close passage near the earth a computer program developed in Ref. is used. The software uses the restricted circular model but its rotating axes are centered at the L1 libration point with x pointing towards the earth, and y along earth’s velocity vector in its heliocentric orbit. The z axis is along the normal to

Solar Surveillance Zone Population Strategies with Piwsatellites

157

the ecliptic plane. The barycentric equations of motion are given by

+

+ +

+

+ +

where Fl = (z p)E yjj zd, F2 = (z- 1 p)E yjj zd are the sun-s/c and earth-s/c vectors with the hatted quantities standing for unit vectors along the three rotating axes. The L1-centered equations are given by

where once again p and (1-p) are the dimensionless masses of the earth and the sun and where Y L is the L1-earth distance. 1-1 and r 2 are now written as TI = ((1 - Y L z)' y2 z2)lI2and 7-2 = ((z- 7 ~ y2 ) ~z2)lI2.The coordinates z,y, z are dimensionless and the derivatives are with respect to non-dimensional time. A regularized form of the L1-centered equations is used for accurate numerical integration

+ + +

+ +

158 J.A. Kechichian, E.T. Campbell, M.F. Werner and E.Y. Robinson

nE = 1.990986606-10-7 rad/s is the sun-earth angular rate, T is a fictitious time related to the physical time t by dt = rzdr, and the quantity is given by

+ +

The Jacobi constant CJ is given by CJ = 2U* - (i2 y2 .i2) with the pseudopotential U* written as U* = (1/2)((x+l-p-y~)~+y~)+(l-p)/~1+ p / r g . The u-variables are related to the Cartesian coordinates through x* = u: - ui - ui ui,y = 2 2 ~ 1 2 ~-2 2 ~ 3 ~ z4=, 2 ~ 1 ~ 23~ 2 ~with 4 , x* = x - y~ snd the forcing functions given by

+

+

,

The physical time is obtained from the integration of d t / d r = uy

+ ui +

ug +ti;. Given initial conditions in the form of X O ,yo, zo, io,yo, i o , the corresponding eight quantities ul(O),u2(0),u3(0),u4(0),ui( 0 ) ,ui(O),ui(0),uk(0)

are determined and the simultaneous integration of the equations (11)(15) as well as d t / d r carried out. Backwards integration is carried out with respect to r’ = -r after changing the sign of the second term in each of Eqs. (12)-(15) and after dropping the n E and n$ factors in F;, F;, F; and to render all the quantities at hand unitless. Starting from a point in space, the backwards integration is stopped at closest approach to the earth by also iterating on the transfer time t f such that the closest approach point takes place at time zero 5 . Restricting the motion to the ecliptic plane, small velocity changes AX and Ay at the initial point are guessed and the backwards integration carried out, and these two search velocity changes searched on until a desired earth flyby distance hT and argument of perigee WT are matched All the necessary transformations to the earth-centered equatorial system are

‘.

Solar Surveillance Zone Population Strategies with Picosatellites

159

shown in detail in Ref. from which the osculating orbit elements are readily obtained.

Fig. 1.

Conical surveillance zone and trajectory geometry.

The geometry of the trajectory and its intersection with the conical surveillance zone is shown in Figure 1. As a function of x, the radius of the base of the 15 deg half angle cone is given by rmas = (R1 - x) tan 15" where R1 is the L1-earth distance. The spacecraft is located from L1 and the earth center by r L and R,, while its distance from the x axis is depicted by r l = (y2+z')~/~.A numerical search is next carried out o the Cartesian coordinates starting at point xo = -3.5. lo6 km, yo = 0, zo = 0 , 20 = 0, yo = 350 m/s, 20 = 0 and searching on the A i l Ay quantities such that the backwards iterated trajectory achieves a close approach of the earth. A solution using AX = 0.614847 km/s Ay = 1.617502 km/s achieving an h~=137.616km is found and shown traced forward in time in Figure 2 from the earth t o the x = -3.5. lo6 km point which in effect is equivalent to a distance of 5 million km from the earth itself. Figure 3 shows the evolution of the various quantities, namely x, y, r,,,, r I and Rg as a function of time t during the first 300 days of this trajectory. The portions of these type of trajectories that are within the surveillance zone must have x negative and r l r < r,, which is clearly the case just before the 300 day mark for a duration of about 20 days. If this trajectory is integrated further in time, it is seen in Figure 4 that another passage in the surveillance zone takes place before the s/c moves farther out

160 J . A . Kechichian, E.T. Campbell, M.F. Werner and E.Y. Robinson

0

-2woooo -4000000 -6000000 E -8000000 Y

r: -10000000 -12000000

-14000000 -16000000

-18000000 0

60000

4oooO

20000

00

00

00

20000 00

40000 00

60000 00

x, km

Fig. 2.

z - y trace of a backwards generated trajectory intersecting the earth.

0 Fig. 3. Evolution of z,y, rmar, r l and Rg distances during first 300 days from earth launch.

from the vicinity of the earth. It will eventually return near the earth after moving completely once around the sun in this rotating frame depiction. This numerical search is pursued further by selecting other WT target values leading to a path that closely resembles a multiple-cycle DRO as shown in Figure 5 . Here five passages in the surveillance zone are possible before the spacecraft departs farther away from the vicinity of the earth. The initial conditions at departure from the earth at time zero are given

Solar Surveillance Zone Population Strategies with Picosatellites

161

2000MN)o

10000000 0

-10000000 -20000000

d

-30000000

>;

-40000000 -50000000 -60000000 -70000000 -80000000 0 15000 000

10000 000

50000 00

50000 00

10000

15000

000

000

x, km

Fig. 4.

Repeated passages of spacecraft through surveillance zone: x - y trace.

5.OE+7 4.OE+7 3.OE+7 E 2.0E+7 x

x

1.OE+7 O.OE+O

-1 .OE+7 -2.OE+7 -1.5E+7

-1 .OE+7

-5.OE+6 O.OE+O x, km

5.OE+6

1 .OE+7

Fig. 5. Distant retrograde orbit-type trajectory emanating from earth.

by the following elements, namely a0 = -129966.259 km, eo = 1.043418887, io = 23.44 deg, Ro = 0 deg, wo = 319.283459 deg, , 0; = 0 deg with 0; standing for the true anomaly. This trajectory is effectively planar and it is contained entirely in the ecliptic plane. Its perigee radius rp at departure from the earth at time zero is 5642.990 km which is less than the radius of the earth Re. which is considered here as a point mass. Minor adjustments in a0 and eo will provide an r p > Re as desired while still keeping the overall geometry of Figure 5. In fact changing only a0 to -139966.259 km will increase rp to 6077.17 km without affecting substantially the geometry of the first four passages through the surveillance zone.

162

J.A. Kechichian, E.T. Campbell, M.F. Werner and E. Y . Robinson

E

Y

0

200

400

600

800

1000 1200 1400 1600 1800 2

oc

t, days

In Figure 6, the x and y components of the trajectory of Figure 5 are shown as a function of time. In this same figure, the time evolution of T , , ~ , T I and R, is also displayed. The five entries into the surveillance zone are clearly visible in Figure 6. If succeeding satellites trace this path, then a fleet of about twenty such vehicles will suffice to ensure that at least one of them will be transiting through the surveillance zone at a given time for a continuous monitoring of the solar weather upstream of the L1 point. The spacecraft distance and velocity relative to L1 are shown in Figures 7 and 8 with the first two as well as the last entry into the zone taking place at or near the 5 million km mark. Further shaping of the trajectory may lead to further entry periods, and if maneuverable vehicles are used instead, to effective n-cycle periodic DRO’s more extended useful lifetimes.

2.1. A Possible Deployment Stnztegy When maneuverable vehicles are considered, it is advantageous to deploy several vehicles on a single launcher into an intermediate orbit such as a GTO type of orbit before injecting each individual spacecraft onto the deep space trajectory.

A GTO type orbit inclined at 23.44 deg of say a = 24446 km, e = 0.724 will experience a nodal regression of about 0.45 degf day and an advance of its perigee of about 0.66 degfday due to the earth gravity field. If we consider a launch of say five satellites in a cluster onto this GTO orbit, then

Solar Surveillance Zone Population Strategies with Picosatellites

Fig. 7.

163

Evolution of L1-relative distance of DRO-type transfer trajectory.

0

200 400 600 800 10001200 1400 1600 18002000 1. days

Fig. 8. Evolution of L1-relative velocity of DRO-type transfer trajectory.

releasing or injecting each vehicle from this GTO at a 20 day interval, will result in the string of spacecraft spaced at 20 days as desired to repeat the pattern of Figure 6 for continuous presence in the surveillance zone. However each vehicle must adjust its orbit argument of perigee and node prior to injection. In a crude analysis, the GTO itself could be biased in both w and R in such a way that the first and last vehicles would perform

164 J.A. Kechichian, E.T. Campbell, M.F. Werner and E.Y. Robinson

equally demanding maneuvers prior to injection, with the in-between vehicles requiring much less AV to perform smaller adjustments in w and R.

Table 1. Five-Satellite Maneuvering Requirements.

Aw s/c

1 2 3 4 5

(ded 26 13 0 13 26

(d%) 18 9 0 9 18

AVN (m/s) 167 84 0 84 167

Avh (m/s) 504 252 0 252 504

Av total (m/s) 1763 1428 1092 1428 1763

Table 1 shows an initial bias in w of 26 deg. and in R of 18 deg such that with w and R drifting in time, the subsequent second and third spacecraft require smaller changes, with the fifth vehicle requiring a similarly larger change as vehicle 1. The AV calculations can be carried out through the use of the vacant focus theory which shows that the in-plane orbit elements obey the following perturbation laws

1 = 2(e + c p ) -f T V

r fN -, a V r- c p ) fN a V '

- -sg.

eLj = 2.99.-f T + ( 2 , + V fT and f N are the tangential and normal components of the perturbation acceleration vector such that the velocity changes AVT and AVN are given by fTAT and fNAT respectively. If a change Aw is desired while constraining Ae = 0, then from Eq. (18)

or

AVT =

r s p AVN 2a(cg* e) .

+

This expression is now used in Eq. (19) written as

2Av~ eAw = -SO' V

( 2 , + ace.) r +V

,

Solar Surveillance Zone Population Strategies with Picosatellites

yielding for @*= 180° and corresponding V = m d ( 1 - e)/(l

Av, = e Aw

1+3e

a

165

+ e),

l+e'

At apogee, AVT = 0 from Eq. (20) such that the Aa change from Eq. (17) is also equal to zero 2a2 P

Aa = -VAVT. Therefore, AV, is the total AV needed to make the Aw change using a single impulse at the apogee of the GTO orbit. This maneuver is more conservative than an optimal two-impulse maneuver but it is sufficient for our first order analysis. For our GTO orbit, Eq. (21) yields a value at AVN of about 6.43 m/s per degree of Aw rotation of the line of apsides. The corresponding values are shown in Table 1 for the five vehicles. It is however more expensive to adjust the R values prior to the injection of each vehicle from the GTO drifting orbit. From the general perturbation equation

where 6 = w + @*,and fhAT = Avh represents the out-of-plane velocity change, we have

which for w = 0 for convenience, has a maximum at ce* = -e resulting in an impulse location at T = a. For our GTO orbit, @*= 136.385 deg and

Avh = 1.606266801AR,

(25)

or roughly 28 m/s per degree of R rotation. Because the perigee velocity of the GTO orbit is V, = 10.092 km/s, and because the example of Figure 5 requires a perigee velocity at departure of V:= 11.184 km/s, then an injection AV from the GTO orbit at perigee of 1.092 km/s is also needed besides the AVN and Avh maneuvers of Table 1. Adding all three AV's to the GTO orbit results in the totals also shown in Table 1.

166 J . A . Kechichian, E.T. Campbell, M.F. Werner and E.Y. Robinson

3. Deployment from an L1-halo Orbit onto Invariant Unstable Manifolds

In this scenario, only early CME warnings that are at least twice the warning time afforded by a spacecraft in an L1 halo orbit are considered. Further, useful warnings are assumed possible only when the spacecraft resides within a tube having a radius of 4 sun radii that is centered on the earth-sun line. Using this early warning criterion, what performance can be gained from a “garage”, holding many small spacecraft, that resides in an L1 halo? One possible solution that minimizes the propulsion needs of the small spacecraft in the “garage”, is to boost them out of the “garage” onto invariant unstable manifolds at intervals that will provide earth with continuous early warning.

To begin, a circular restricted three-body problem L1 halo having a maximum z excursion, away from the ecliptic plane, of 400,000 km was chosen for the nominal “garage” orbit. Figure 9 shows the three projections of the orbit in the usual earth centered rotating frame. (Note: Similar to the previous section of 12 this paper, the mass of the earth and the moon have been combined to compose a single fictitious planet, referred to here as the earth/moon, that is the second primary in this three-body model.) The size of the halo has not been optimized, but has been used rather as a representative case; it is large enough to allow uninterrupted communication with earth, but small enough to give reasonable on-station time. The six eigenvalues of the monodromy matrix associated with this nominal, periodic, LI halo, consist of one stable (< l),one unstable (> l),and 4 unity magnitude (= 1) eigenvalues. Accordingly, a twodimensional invariant stable manifold, in six-dimensional state space, asymptotically approaches the periodic solution as time increases in a positive direction. Similarly, by reversing the flow of time, a twodimensional invariant unstable manifold, in six-dimensional state space, asymptotically approaches the periodic solution. The unstable invariant manifold can be approximated at any given point on the halo, thus reducing the dimensionality of the manifold to one, by the eigenvector associated with the unstable eigenvalue of the appropriate monodromy matrix (this particular monodromy matrix is found by integrating the state transition matrix from the point of interest over one halo period). This unstable eigenvector is guaranteed to be tangent to the

Solar Surveillance Zone Population Strategies with Pacosatellites

167

Fig. 9. Three planar projections of a 400,000 km L1 halo orbit as seen in the earth centered rotating reference frame. The black asterisks represent points on the halo that will be used to approximately locate the nearby invariant unstable manifold.

unstable invariant manifold only at the halo orbit. However, a small perturbation, on the order of a couple of hundred kilometers in position, in the direction of the eigenvector will remain close enough to the unstable manifold that integration backwards in time from this perturbed state will still asymptotically approach the halo orbit. Further, this perturbation is large enough to allow one to quickly follow the unstable invariant manifold away from the vicinity of the halo. It is the behavior of the invariant unstable manifold, away from the halo, that will be leveraged in the design of this early CME warning system. Figure 10 shows the 400,000 km L1 halo with three representative unstable manifolds emanating from three points on the halo, each depicted with a black asterisk (these same halo departure points are also denoted by black asterisks in Figure 9). The manifolds, themselves, initially spiral about the sun-earth line while progressing ahead of the earth due to a larger effective mean motion. These three manifolds, as seen in Figure 11, are closer to the sun than the “garage” and stay within the prescribed early warning sunearth tube for almost 290 days. However, they remain at least twice as far from the earth as the L1 point and in the coverage tube for only 38 days. Thus, to guarantee continuous warning times that

168 J.A. Kechichian, E.T. Campbell, M.F. Werner and E.Y. Robinson

are two times better than is possible from a L1 halo, a new probe must be launched from the “garage” roughly every 38 days. Launching 10 probes per year would only be feasible for very small inexpensive probes. To this end, one advantage to using the unstable invariant manifold trajectories is that the “garage” could provide the boost necessary to enter the trajectory, thus reducing the propulsion needs (i.e., size, complexity, and cost) of the probes.

5 2

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(W

4 5

0

10’

5

0

2

rtw

4

6

x

Id

Fig. 10. Three planar projections of three invariant unstable manifolds emanating from a 400,000 km L1 halo. The black asterisk signifies where on the halo the manifolds originate.

Future considerations are focused at increasing the on-station time from 38 days per spacecraft. This may be accomplished by optimizing the size of the nominal halo and by performing critically placed maneuvers. Of course, there will be a trade-off between requiring fewer, but more sophisticated, spacecraft that can stay on-station longer versus needing more inexpensive small spacecraft to meet the mission objective of providing the earth with continuous early CME warning.

Solar Surveillance Zone Population Strategies with Picosatellites

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lhrlda9

Fig. 11. The three manifolds integrated from the 400,000 km halo stay within the 4 Sun Radii constrainted cylinder radius for nearly 290 days, but only give better than twice the warning time of a spacecraft in a L1 halo for 38 days.

4. Conclusion

Two solar surveillance zone population strategies using a series of small satellites that transit through the earth-sun line at distances larger than the L1 libration point have been discussed. The first architecture consists of using an intermediate GTO-type orbit from where the vehicles are injected into certain distant retrograde orbits at regular intervals of time flying in the ecliptic plane and crossing the earth-sun line periodically at distances larger than the L1 point for increased warning time of impending geomagnetic storms. Because a cluster of vehicles must be flown on a single booster to the GTO orbit, and due to the precession of that parking orbit between successive releases onto the DRO-type orbits, it is shown that AV requirements for maneuvering prior to injection onto the ecliptic plane are not excessive in view of the small mass of each individual spacecraft. A second architecture that releases each spacecraft at regular intervals of time from a parking halo orbit onto its corresponding unstable manifold with essentially zero cost in AV has also been discussed and the three dimensional flight paths depicted after assuming a proper dispersion of the spacecraft from the carrier vehicle in halo orbit prior to individual release. For both architectures, the minimum number of vehicles is determined for continuous solar weather monitoring over a given time span.

170 J.A. Kechichian, E.T. Campbell, M.F. Werner and E. Y. Robinson

References 1. Farquhar, R. W., “The Control and Use of Libration-Point Satellites,” NASA TR R-346, 1970. 2. Richardson, D. L., “Analytic Construction of Periodic Orbits About the Collinear Points,” Celestial Mechanics, Vol. 22, 1980, pp. 241-253. 3. Stiefel, E. L., and Scheifele, G. Linear and Regular Celestial Mechanics, Springer-Verlag, NY, 1971. 4. Howell, K.C., Mains, D.L., and Barden, B.T., “Transfer Trajectories from Earth Parking Orbits to Sun-Earth Halo Orbits,” Paper No. 94-160, AAS/AIAA Space Flight Mechanics Conference, Cocoa Beach, Florida, February 1994. 5 . Kechichian, J. A., “Computational Aspects of Transfer Trajectories to Halo Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 4, JulyAugust 2001, pp. 796-804. 6. Szebehely, V., Theory of Orbits, Academic Press, New York, 1967. 7. Ocampo, C., “Trajectory Optimization for Distant Earth Satellites and Satellite Constellations,” PhD Thesis, University of Colorado, Dept. of Aerospace Engineering Sciences, 1996. 8. Howell, K.C., Barden, B.T., and Lo, M. W., “Application of Dynamical Systems Theory to Trajectory Design for a Libration Point Mission,” The Journal of the Astronautical Sciences, Vol. 45, No. 2, April-June 1997, pp. 161-178. 9. Koon, W.S., et al, “Shoot the Moon,” AAS 00-166, AAS/AIAA Space Flight Mechanics Meeting, Clearwater, FL, 23-27 January 2000. 10. G6mez, G. and Mastemont, J., “Some Zero Cost Transfers Between Libration Point Orbits,” AAS 00- 177, AAS/AIAA Space Flight Mechanics Meeting, Clearwater, FL, 23-27 January 2000. 11. Franz, H., “Design of Earth Return Orbits for the Wind Mission,” AAS 02170, AAS/AIAA Space Flight Mechanics Meeting, San Antonio, Texas, 27-30 January 2002. 12. HBnon, M., “Numerical Exploration of the Restricted Problem. V. Hill’s Case: Periodic Orbits and Their Stability,” Astronomy and Astrophysics, (l),1969. 13. Hknon, M., “Numerical Exploration of the Restricted Problem. VI. Hill’s Case: Non-Periodic Orbits,” Astronomy and Astrophysics, (9), 1970. 14. Ocampo, C. A., and Rosborough, G . W., “Transfer Trajectories for Distant Retrograde Orbiters of the Earth,” AAS 93-180, AAS/AIAA Spaceflight Mechanics Meeting, Pasadena, CA, 22-24 February 1993. 15. G6mez, G., Llibre, J., Martinez, R., and Sim6, C., “Dynamics and Mission Design Near Libration Points, Vol. I Fundamentals: The Case of Collinear Libration Points,” World Scientific Monograph Series in Mathematics, Vol. 2, World Scientific Publishing Co., 2001.

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

THE SOHO MISSION L1 HALO ORBIT RECOVERY FROM THE ATTITUDE CONTROL ANOMALIES OF 1998 C.E. ROBERTS Computer Sciences Corporation, NASA Goddard Space Flight Center, Greenbelt, M D 20771, USA The joint European Space Agency (ESA) and National Aeronautics and Space Agency (NASA) mission called the Solar and Heliospheric Observatory (SOHO) is historically the second of five deepspace missions to be operated at one of the Sun-Earth collinear libration points by the NASA Goddard Space Flight Center (GSFC). Launched in December 1995 with a goal of revolutionizing solar science, SOHO has flown a halo-type libration point orbit (LPO) around the Sunward L1 point since March 1996. The billion-dollar SOHO mission was intended to have a two-year minimum lifetime, followed by an extended mission phase of at least four years. However in 1998 SOHO’s life was nearly cut short twice by separate, very different onboard anomalies of the most threatening natures. The first mishap, which occurred on June 25, 1998, saw SOHO lose 3-axis attitude control, with the resulting tumble severing communications with Earth. The loss of communications lasted until early August when radar signals rediscovered the practically powerless, frozen spacecraft. By that time, by chance, the slowly spinning spacecraft’s non-nominal orientation was such that the solar panels were receiving some sunlight every spin period, thus enabling intermittent communications. This good fortune provided the opportunity for a rescue, and a strategy was developed for slowly, carefully thawing the spacecraft and re-charging the batteries. Delicate, extremely intensive rescue efforts conducted over the next several weeks were ultimately successful, and by the end of September 1998 the spacecraft itself was functioning nearly normally (though not yet the science instruments). Recovery operations -including a series of delta-V maneuvers to restore the decaying halo orbit- continued during the autumn of 1998, and optimism about SOHO’s chances was high until the last of the gyroscopes failed just before Christmas. Though this second anomaly did not lead to another loss of the Sun-pointing attitude control as in the June event, it did lead to an autonomous fail-over of attitude control from the momentum wheels to thrusters. But the lack of gyroscopes presented a huge problem. Without them there was no way to effect the transition of attitude control back to the momentum wheels and thereby halt the attitude thrusting-thrusting that was unbalanced in terms of delta-V and gradually increasing the energy of the orbit.

171

172 C.E. Roberts The continual thrusting posed a dual threat. Not only would SOHO eventually run out of fuel, but the cumulative delta-V imparted to the halo orbit -at times as much as 0.65 m/sec per day- threatened to push SOHO away from the L1 region and into an independent and useless solar orbit. The mission appeared doomed, but after a weeks-long struggle to rescue it a second time, SOHO was once again saved. Thus two of history’s most extraordinary spacecraft recoveries succeeded just half a year apart. The aspects of the rescues that this paper will address primarily concern the halo orbit. Not only are halo orbits extraordinarily sensitive to perturbations, but by their nature the delta-V costs t o correct the orbit grow exponentially with the time elapsed from experiencing the perturbation. Due to the perturbations involved, both of the SOHO accidents threatened escape from the L1 region. Hence, the mission could still have been lost despite all other efforts to re-establish control of the spacecraft’s attitude and on-board functions, had personnel at GSFC’s Flight Dynamics Facility (FDF) not found ways to restore the orbit while contending with numerous adverse circumstances arising from the anomalies. Among the concomitant problems faced was an inoperative closed-loop maneuver control system, and orbit determination results degraded to the point of uselessness. In response, a number of critical improvisations were devised. Of the two most important improvisations, developed following the December mishap, the first was to model the trajectory with continuous low-thrust applied. That was the only way for updating and assessing the orbit throughout the 40-day contingency. The second was a strategy for correcting the orbital energy via a time- staggered series of delta-V maneuvers designed to gradually counteract the effects of variable attitude thrusting. In sum, these were feats never before performed with a Libration Point Orbit mission.

1. Introduction

Libration point orbits (LPOs) -whether halo or Lissajous orbits- are inherently unstable and must be maintained via occasional propulsive maneuvers, as has been done for the Solar Heliospheric Observatory (SOHO) since arriving at L1 in early 1996.l These maneuvers are widely referred to as “stationkeeping” maneuvers. In the absence of stationkeeping, LPO perturbations lead to exponential decay away from the libration point region. If the net effect of all perturbations -especially those due to the errors from propulsive maneuvers- imparts an excess of orbital energy over that needed to keep the orbit “in balance,” the result will be eventual escape from the Earth-Moon system into an independent orbit around the Sun. Alternatively, if the net effect is a deficit of energy needed for balance, then the halo will decay leading to a trajectory coming back toward Earth. This latter case has itself two types of outcome depending on the particular orbital energy of the return trajectory. For a fairly narrow range of the highest

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return energies, there may not be a capture by the Earth-Moon system, or at least not a strong one, and luni-solar perturbations will eventually cause escape into solar orbit. The other type of return consists of lower energy trajectories that do result in capture. These capture orbits generally display quite chaotic behavior within the Earth-Moon system, and this behavior is very sensitive to the prevailing gravitational conditions (positions of the various bodies) at the time of escape from the LPO. Luni-solar perturbations, particularly those due to the inevitable close lunar encounters, have profound effects on the future course of these capture orbits, with escape some months or years later often the outcome. Both of the SOHO anomalies of 1998 threatened the survival of the halo orbit, and each time the outcome could have been one of the types discussed above. The problem for SOHO was that if an orbit recovery effort failed, then the result would be an uncontrolled, decaying trajectory that would either escape into heliocentric orbit or “fall” back toward Earth. During the first anomaly, the possible outcomes seemed virtually infinite in number, because we did not know how much thrusting SOHO performed following the loss of contact, and therefore we were in the blind as to its orbital energy state. During the second anomaly, the threat posed by the virtually continuous thrust situation was escape into solar orbit. For both anomalies, full recovery of both the spacecraft and the halo orbit were absolutely essential to continuing the mission. A partial or temporary recovery, particularly any recovery ending with exhaustion of fuel or other shortcoming, could still have resulted in any of the mission-ending trajectory scenarios discussed above. But it is the stories of these two anomalies and subsequent successful recoveries -not the infinitude of possible decay scenarios- that are the main subjects of this paper. Some points should be made about the scope and organization of the paper. The paper concentrates on the halo orbit and maneuver aspects of the recovery stories. Given the enormity and complexity of each anomaly and recovery in its entirety, many facets of both are simply beyond scope. For instance, the issues involved with attitude determination and control, as well as other spacecraft engineering areas, were extremely involved and could in no way be treated adequately here (these aspects deserve their own papers). Just enough is presented on these non-orbit-specific topics to provide background for, and their relationship to, the halo orbit recovery stories. On another note, a significant portion of the intended audience for

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this paper consists of current and future SOHO mission team members, especially those who may become the trajectory analyst successors to the present author. So, it is anticipated that not all portions of the paper may be of equal interest to all readers. Therefore, an attempt was made to organize the paper in such a way that different sections may be read fairly independently, hopefully without too much loss of continuity. The paper is organized into five main sections. The first section provides an overview of the SOHO spacecraft and a discussion of its L1 halo orbit and propulsion system. In the second section, the focus is on SOHO stationkeeping methodology and a brief history of the mission’s stationkeeping maneuvers prior to the anomalies of 1998. In the third section, the story of the first major anomaly of June 25th, 1998, and the subsequent recovery is presented. Similarly, the fourth section covers the second anomaly of December 21, 1998, and its recovery. The final section, an epilogue, treats the aftermath.

2. The SOHO Spacecraft and its Mission

2.1. The Spacecmft Historically, the Solar and Heliospheric Observatory -a 1.2 billion dollar joint ESA and NASA deep space mission to study the Sun- is the second spacecraft to fly a Lagrange-point ~ r b i t . Launched ~?~ December 2, 1995, SOHO is at the time of this writing in a quasiperiodic halo orbit around the Sun-Earth collinear point L1 -an orbit almost identical to the one flown by the pioneering International Sun-Earth Explorer-3 (ISEE-3) over two decades ago.4 It carries a suite of 12 scientific instruments including imaging sensors to study phenomena relating to the solar surface and atmosphere, solar dynamics, and the solar corona and solar wind. The Sun-Earth L1 point region is the perfect location from which to conduct continuous, direct observation of the Sun while simultaneously making in situ measurements of the solar wind upwind of the Earth. This 3-axis stabilized spacecraft maintains one axis fixed upon the Sun’s center at all times. This attitude keeps the suite of scientific instruments always trained upon the Sun with an accuracy as good as one arc-second for around-the-clock data gathering. The spacecraft was designed for attitude stabilization and pointing control via a closed-loop system. This

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system originally employed an inertial reference unit consisting of 3 roll gyroscopes (used mainly when attitude control is via thrusters), a fourwheel reaction wheel assembly for momentum management, a fixed-head star tracker, and 2 fine sun sensors -all under the control of an onboard computer (OBC). The onboard control system is supported by a variety of ground-based attitude determination and control functions including momentum management.5 The relevant environmental torque is that due to solar radiation pressure.

A large spacecraft, SOHO had a total mass of 1,863 kg at launch -of which 251 kg was fuel. Electrical power is generated primarily by two large solar cell array panels yielding approximately 1,150 W. The solar panels, taken together with the main bus (Service Module) and payload (Instrument Module), give SOHO a deployed cross-sectional area of 21.9 m2. The Service and Payload modules together give SOHO a boxy, rectangular look (see Figure A-1 of Appendix A), and it is parallel to the long axis that the body X-axis -the X,, or roll, axis- is defined. The +XB direction is that along which the instrument boresights are intended to point at the Sun. The ZB axis, or yaw axis, is orthogonal to XB,and it is positive in an “up” sense that is intended to be parallel with the Sun’s positive spin axis direction, i.e., its North Pole. The Zg axis is also the axis of maximum inertia. The YB axis, or pitch axis, completes the right-handed triad. Angular rotations about these body axes are defined consistent with the usual right-hand rule. In its nominal operations mode, SOHO communicates with the ground via an aftmounted, gimbaled, Earth-pointing parabolic High Gain Antenna (HGA). SOHO S-band tracking and telecommunications are performed via NASA’s Deep Space Network (DSN), with the tracking data and telemetry data forwarded to NASA’s Goddard Space Flight Center (GSFC), where the ground operations are conducted by a joint ESA/NASA Flight Operations Team (FOT). The GSFC Flight Dynamics Facility (FDF) conducts a variety of flight dynamics support functions, including attitude analysis, trajectory design, and control, maneuver planning,6 tracking data evaluation and orbit determination (OD), and DSN station acquisition data generation.

176 C.E. Roberts

2.2. The SOHO Mission Halo Orbit

The SOHO mission orbit is a Class 2-type halo orbit; meaning that its sense of revolution about L1 -as it appears from Earth- is counterclockwise. Libration point orbit characteristics are often discussed with reference to a well-known non-inertial coordinate system called the Rotating Libration Point (RLP) frame. This is an L1-centered frame where the X-axis points from L1 to the Earth-Moon barycenter, the Z-axis points up toward the North Ecliptic Pole (NEP), and the Y-axis completes the right-handed frame, pointing approximately along the direction of Earth’s velocity vector. The primary mission constraints placed on the halo orbit were: 1) that the minimum Sun-Earth-Vehicle (SEV, and Vehicle = SOHO) angle never be less than 4.5 degrees and, 2) that the maximum SEV angle never be greater than 32 degrees. The minimum SEV angle constraint satisfies SOHO’s Solar Exclusion Zone (SEZ) requirement (the spacecraft must skirt the SEZ for the sake of avoiding solar interference with communications). The maximum SEV constraint derives from the gimbal angle limits of the HGA dish, and was germane to the design of the Earth to L1 transfer traject~ry.~?~ The halo orbit selected for SOHO during the prelaunch mission design phase satisfies the mission constraints quite easily. The orbit comes close to in SEV angle of 4.5 degrees during the nearer, Earth-side crossing of the RLP XZ-plane, and is at about 5.0 degrees at the farther, Sun-side crossing. At the RLP extreme Y-axis points of the halo, the SEV angle is never more than approximately 25.5 degrees. The RLP-frame oscillation amplitudes corresponding to this 4.5-by25.5 degree orbit -as specified during the pre-launch mission design phasewere as follows 6,7:

Ax Ay Az

= = =

206,448 km 666,672 km 120,000km

These amplitudes are virtually identical to those specified for the first LPO halo mission, ISEE-3, and hence SOHO has the same 178-day period as ISEE-3 had.8*9SOHO’s orbit is shown projected in the solar rotating-

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frame XY-plane, XZ-plane, and YZ-plane, in Figures 1 through 3, respectively. (The Advanced Composition Explorer (ACE) mission'S 6-by-10 degree Lissajous orbit appears in Figures 1 and 2 for contrast.")

Lunarm Solar Ecllptic Rotating Frame XY Plane PmlecUon

Fig. 1. SOHO Halo Orbit XY-Plane Projection.

SOHO L1 Halo Orbil and ACE L1 Lissajjous OrMt

Solar Ecliptic Rotating Frame XZ Plane Projection

Fig. 2.

SOHO Halo Orbit XZ-Plane Projection.

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Fig. 3.

SOHO Halo Orbit YZ-Plane Projection.

2.3. Overview of the SOHO Propulsion System

SOHO maneuvering capability is provided by a blow-down, monopropellant hydrazine propulsion system.'>'' The fuel -251 kg at launch- is carried by a single tank of prolate spheroidal shape which keeps the hydrazine separated from its gaseous helium pressurant by a diaphragm. The total delta-V capability at launch was approximately 318 m/sec (neglecting canting and plume impingement losses), though only 275 m/sec of that was allocated to the overall delta-V budget. The remainder was reserved for attitude control including momentum management maneuvers. Sixteen thrusters arranged into two fully redundant branches of 8 thrusters each provide both delta-V and attitude control as needed. The primary branch (A-branch) is used for commanded maneuvers, while the redundant branch (B-branch) is used mainly in a hard-wired, emergency attitude control mode (more on this in later sections). Each thruster carried a beginning-of-life (BOL) thrust rating of 4.2 Newtons (N) and a specific impulse of 220 seconds (at 22.4 bars of pressure). End-of-life (EOL) thrust is expected t o be approximately 2.2 N (at 6.6 bars). All nominal thrust components are in the body X Z plane. The primary thruster designations and their functions are given in Table 1.

The SOHO M i s s i o n L1 Halo Orbit Recovery 179 Table 1. SOHO Thrusters (data applies to both A- and B-branches). Thruster Pair 1 and 2 (1 is canted 30° down, 2 is 30° up) 3 and 4 5 and 6 7 and 8

Thruster Locations

Fore (Sunward)

Delta-V Direction (body frame)

Primary Attitude Control

Thrust XB-cosine

Thrust ZB-cosine

-XB

Pitch

-0.866

1 : (+0.5) 2 : (-0.5)

Aft (Earthward) Top (+ZB) Bottom ( - Z B )

+XB -zB

+ZB

Yaw Roll Roll

$1 0

0

0 -1 +1

When in a certain closed-loop control mode, SOHO is capable of autonomously stabilizing its Sun-pointing attitude using thrusters. Delta-V maneuvers, however, are always planned and computed on the ground by a GSFC FDF trajectory analyst. Then, maneuver command loads specifying ignition time, selected delta-V thrusters, and total burn duration are constructed in the SOHO Mission Operations Center (MOC) at GSFC for eventual uplink via the DSN. Prior to December 1998, the SOHO OBC would govern orbit maneuvers by firing the delta-V thrusters at an average duty cycle of 75% and by on-pulsing appropriate attitude control thrusters as necessary. However, this picture changed substantially after the last of the two anomalies to be described in sections 3 and 4 below.

3. Stationkeeping of the SOHO Halo Orbit

3.1. Geneml Approach The basic SOHO stationkeeping strategy is an orbital energy balancing technique, also referred to as “single-axis” control, which means the entire delta-V is applied along one specified axis For SOHO, this axis is the spacecraft-to-Sun line. Since SOHO is a Sun-pointer, its body-frame X B axis is always aligned with the spacecraft-to-Sun line, which incidentally is always nearly parallel with the Earth-to-Sun line. A delta-V toward the Sun will increase orbital energy (thereby preventing orbit decay Earthward), and a delta-V in the opposite direction will decrease orbital energy (preventing escape into solar orbit). (A history and explanation of the development of this energy-balancing technique can be found in Dunham et a l l . ) The delta-V can be applied at any desired time or arbitrary position ‘ 7 l 2 .

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within the halo orbit. Given that the preference is to maneuver before the error growth has reached one meter/sec, the energy adjustments are typikm2/sec2. It should cally quite small -in terms of C3, on the order of be noted that prior to launch a Project-level requirement was placed on the mission that all stationkeeping delta-Vs be parallel (anti-parallel) to the SOHO-Sun line. Given SOHO’s solar-pointing attitude and thruster configurations, this requirement dovetails perfectly with the single-axis control technique, and so it might seem that the requirement derived from spacecraft design. However, that was not quite the case, as this requirement actually replaced its antithesis, i.e., an earlier requirement that specified that stationkeeping delta-Vs be normal to the SOHO-Sun line. Another significant Project preference was that stationkeeping (SK) maneuvers be scheduled together with the momentum management maneuvers (MMMs). This stipulation has definite impact on the frequency, spacing, and modelling of SK burns, and not only because of the independent momentum management requirements and schedule. One very important impact is that there is always some small but non-negligible, net delta-V left over at the conclusion of a MMM, and given the sensitivity of halo orbits, this residual delta-V must be dealt with. Typical MMMs, prior to June 1998, imparted a residual delta-V of about 2 cm/sec toward the Sun. After the recoveries of 1998-99 (Sections 3 and 4 below), residual delta-Vs are now more typically in the 6 to 8 cm/sec range. These residual delta-Vs are typically Sunward (though in some unusual circumstances they can be anti-Sunward). The MMM thrusting is modeled as an independent perturbation to the trajectory, the effect of which is accounted for within the SK maneuver targeting cycles. Thus the MMM delta-Vs are negated, or as happens sometimes, the delta-Vs are helpful if their direction is the same as that needed for the SK burn. In general they change the magnitude of the SK delta-V by an amount equal to the MMM residual delta-V. All SOH0 SK burns have coincided with one or more MMMs concluded within hours, or at most a day or two, of the SK burn itself. Generally, the length of the intervals between LPO stationkeeping maneuvers may depend on a number of factors -especially those arising from operational considerations. Just how much time should be allowed to elapse between SK burns depends very much on the particular mission and its requirements, constraints, fuel budget, attitude stabilization modes, propulsion performance, and ground support details including station coverage

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schedules Generally, it is best to maneuver before the halo correction cost has grown “too large,” but each LPO mission must judge for itself what “too large” is depending on its operational realities. For SOHO, it was originally preferred to do a burn before the delta-V grew to 1.50 m/sec, but on occasion it was allowed go somewhat higher than that to accommodate schedule realities. Therefore, in the early days of the mission, when forecasting an upcoming SK maneuver, that future epoch was sought where the halo correction delta-V would be no larger than about 1.50 m/sec. But whatever the case, when forecasting SK maneuvers, certain behaviors should be recognized. 178.

First, burn performance errors tend to be the dominant perturbations affecting the course of the post-burn orbit, much more so than orbit knowledge errors (in-plane orbit determination velocity knowledge is generally good to about 1 mm/sec). Second, halo correction delta-V costs increase exponentially with time elapsed since the last maneuver, and, as is well known, the doubling time constant for this exponential behavior is approximately 16 days. Hence -if maximizing the interval between SK maneuvers is the goal (and for SOHO it is)- it is obvious why one should strive to account for, minimize, and if possible offset, all maneuver error sources during the planning and delta-V targeting process. Changes in targeting results from one maneuver planning iteration t o the next are generally only marginal -especially during the final stages- yet where LPOs are concerned every little bit counts. Even a onemillimeter per second error can matter. It is also plain to see why doing small burns is preferable. Given that propulsion system performance dispersions could be expected to be within some (hopefully) small range (typically within a few percent for most missions), then the absolute burn errors will be the smaller, the smaller the burn is itself.

3.2. Use of the Swingby l h j e c t o r y Design Program for SOHO Stationkeeping

All SOHO trajectory design and control support is performed with the GSFC Mission Design and Analysis TOO^,^^-'^ a PC-based, interactive, menu-driven graphics software program more widely and popularly known as “Swingby”. With Swingby it is possible to construct long-duration deepspace trajectories in great detail via its spacecraft and mission event models, its high precision, full-force model numerical propagators and its trajectory

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targeting capabilities. For the specific support of SOHO, a number of essential coordinate frames, models and parameters must be described to Swingby via its menu systems. Among these are the delta-V and attitude coordinate frames, the finite burn propulsion model, and the nominal alignment of the SOHO body coordinate frame relative to the mission attitude frame.

3.3. Swingby Program Coordinate Frame Constructions

The Swingby program provides the capability to define coordinate frames of convenience to the user via onscreen menu inputs. As SOHO is a 3axis stabilized, solar pointing spacecraft, we have defined in Swingby a special delta-V coordinate frame that is a local (i.e. spacecraft-centered) Sun-pointing frame. This frame is referred to by the designation “Delsun”. The construction of this frame (from vectors expressed in Earth-centered inertial (ECI) mean-of-J2000 coordinates) is as follows l 3:

XD ZD

= =

YD

=

Spacecraft-to-Sun vector orthogonal to XD,up towards the direction of the North Ecliptic Pole (NEP) ompletes the right-handed triad.

The Delsun frame is convenient for SOHO maneuver work for two reasons. First, it provides delta-Vs relative to the Sun along a single axis. Second, it is a natural for the control of LPOs because its axes are aligned quite closely with the RLP and related solar rotating frames. (The convenience is of course for the analyst using Swingby; the program transforms Delsun delta-V vectors to ECI 52000 coordinates for its internal calculations.) Next, Swingby defines a SOHO attitudemode reference frame as follows: the +ZA (yaw axis) is parallel to the NEP, + X A (roll axis) is towards the Sun, and +YA (pitch axis) completes the right-handed triad. Swingby makes the further assumption that the body axes are by default co-aligned with the attitude frame’s axes, i.e., +XB = + X A , +YB = +YA, and +ZB = +ZA.

The SOHO Mission

L1

Halo Orbit Recovery 183

Thus, the body axes are assumed to be at their default zero points in the attitude frame. In the current operational version of Swingby, there is no provision for non-zero yaw and pitch angles. Thus, the XB-axis is assumed always pointed precisely at the solar center. However, Swingby does take a fixed roll angle attitude (though not a roll rate) as input, so that the inertial direction of thrust for the ZB-axis thrusters may be accurately modeled when the spacecraft will be in a fixed, non-zero roll attitude. (Normally, SOHO’s onboard roll control laws have the ZB-axis follow the Sun’s polar axis, so that in observing mode the SOHO roll angle only varies within a range of 0.0f7.5degrees over the course of a year.) 3.4. Swingby Progmm Propulsion System Model

A SOHO-specific propulsion model is defined in Swingby via menu system inputs, giving it the capability to target and model SOHO finite burns. The eight thrusters of each branch are defined in terms of their individual body-frame thrust direction cosines and their thrust and specific impulse performance polynomials. (The thrust and specific impulse are functions of the tank pressure.) Duty cycles and thrust scaling factors (TSF) can be specified for each thruster via user input. Additional inputs include the dry spacecraft mass, remaining fuel mass, fuel tank pressure and temperature, hydrazine density, and roll angle. The user selects which thrusters are to be used for a given burn. The “Delsun” system described above supplies the fundamental attitude orientation of the spacecraft. The attitude is assumed Sun-fixed and roll-fixed for the duration of the burn. Swingby can take the above-mentioned information for selected thrusters plus an impulsive delta-V and make a good first guess of overall burn duration using fundamental rocket formulas. Burn duration can then be refined further during the finite burn trajectory targeting process described below. Swingby finite burn model outputs include the delta-V vector in a choice of reference frames, maneuver duration, fuel usage, final tank pressure, and ignition and burnout state vectors and associated orbit parameters. 3.5. SOHO SK Targeting Technique using the Swingby

Progmm Using Swingby, the SK maneuvers are initially designed as impulsive deltaV maneuvers. To start, a given state (e.g., an up-to-date OD solution)

184 C.E. Roberts

is propagated up to a future candidate maneuver epoch. At that epoch, the Swingby numerical targeting sub-system applies an initial small velocity perturbation (usually 5 km/sec) in a direction either parallel or anti-parallel to the S/C-Sun line, i.e., a Delsun Xo-axis component of delta-V. This delta-V component becomes the independent variable in the differential-corrections (DC) targeting scheme. In the DC scheme, perturbed trajectory states constructed by Swingby are propagated forward in time toward a targeting goal (dependent variable) described at a future crossing of the RLP XZ-plane (i.e. YRLP = 0 km). This goal -evaluated at the propagation stop for the specified XZ-planecrossing- is that the RLP X-component of velocity be equal to zero (i.e., the RLP-frame VX = 0 km/sec, within a tolerance no greater than km/sec). (For halo orbits, driving the VX component to nearly zero at the plane crossing constrains the trajectory to penetrate the plane perpendicularly.) The delta-V ultimately achieving the goal at the second future XZ-plane crossing supplies the orbital energy change sufficient to maintain the halo orbit for at least a full revolution. The SK maneuver delta-V can then be refined further by targeting the third and fourth XZ-plane crossing. The impulsive delta-V thus obtained becomes input for the subsequent SK finite maneuver targeting process. Targeting involving the finite burn model begins with making a first guess of the burn duration based on the impulsive delta-V magnitude and fuel remaining (Swingby does this via rocket formulas upon user request). Other essential user inputs include an appropriate selection of thrusters and the prevailing spacecraft propulsion system conditions. Then, burn duration replaces delta-V as the independent variable in the DC targeting process while the dependent goal remains the same, as does the rest of the targeting process. The Delsun-frame assumption and a roll angle value specify the Sun-pointing orientation of SOH0 -and hence all of its thrusters- in inertial space (the necessary transformations are automatically performed internally within Swingby). During targeting, the trajectories are integrated with a full force model including, of course, the thrust. As the targeting problem is basically solved at the impulsive delta-V stage, the finite burn targeting convergence is usually rapid. The end result is a finite maneuver plan defining all thrusters to be used, ignition epoch, and individual thruster burn times, as well as delta-V vector and fuel usage prediction.

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3.6. Brief History of SOHO S K Maneuvers prior to the 1998 Anomalies Between the halo orbit insertion of March 1996 and the June 25, 1998, anomaly, there occurred eight SK maneuvers. The first, designated SK01, occurred on May 23, 1996, while the last (SK-08) of that period was performed on April 17,1998. The basic history of that period is summarized in Table 2, first published in Dunham et e l 1 . Table 2. SOHO Stationkeeping Maneuver History through April 17, 1998. Orbit Man. Event SK-01 SK-02 SK-03 SK-04 SK-05 SK-06 SK-07 SK-08 Notes:

Date (m/d/y) . . . .

Days since Jets Planned Achieved Delta-V Last Orbit Used Delta-V Delta-V Error Burn Event (m/sec) (m/sec) 5/23/96 63.0 1,2 0.3067 0.3089 +0.714 9/11/96 112.0 1,2 0.4541 0.4578 +0.808 1/14/97 124.7 1,2 0.0432 0.0411 -4.861 4/11/97 87.2 1,2 0.1887 0.1892 +0.235 9/04/97 146.06 1,2 1.8876 1.8972 +0.506 11/29/97 85.9 3,4 0.0396 0.0408 $2.84 12/19/97 20.0 1,2 0.3984 0.3956 -0.703 4/17/98 118.9 1,2 1.4375 1.4350 -0.179 On 4/17/98, fuel = 205.87 kg; total SOHO mass = 1818.57 kg.

Fuel Used (kg) 0.3353 0.4925 0.0490 0.2064 2.0258 0.0345 0.4263 1.5441

As can be seen from the table, the SK maneuvers after SK-01 were generally infrequent and typically modest in magnitude -usually well under 1 m/sec. Six of the eight burns were significantly under 0.5 m/sec, and SK-03 and SK-06 were both less than 0.05 m/sec. In those cases where the interval between burns was not at least 100 days, it is generally due to having to keep schedule with the MMMs. A few maneuvers were affected by spacecraft attitude control emergencies, which necessitated attitude recovery thrusting that adversely impacted the orbit (causing the subsequent SK burn to occur much earlier than otherwise). The period of the mission covered by the table represents roughly four halo revolutions, so that the SK frequency averaged approximately two burns per revolution, or four burns per year. In total, about 4.75 m/sec was expended, representing an average of 1.2 m/sec per revolution, or 2.4 m/sec per year. Thrusters 1 and 2 were required for all burns except SK-06; this was probably due to systematic Sunward delta-V errors from the MMMs. Lastly, it should be emphasized that up to June 1998, SOHO was a well-behaved spacecraft during SK maneuvers. Attitude control was steady, precise, predictable, and reliable, so that delta-V directional errors were absent or at least indiscernible. On top

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of that, overall the thrusters displayed good repeatability.

4. The SOHO Attitude Control Anomaly of June 25, 1998

Going into June 1998, a ninth stationkeeping (SK) maneuver -one that was going to be very small at 5.7 cm/sec (neglecting the residual (MMM) contribution)- was planned for the 28th of that month. But that SK maneuver was never performed due to an anomalous loss of attitude control and communication during the night of June 24th - 25th. The series of events began when SOHO unexpectedly triggered two briefly separated Emergency Sun Reacquisition (ESR) events following the completion of a scheduled MMM. An ESR can be triggered by a variety of anomalies or upsets to the spacecraft, usually having to do with the attitude sensors or the Attitude Control Unit (ACU). SOHO had a history of several of these ESR events prior to June 24, 1998. 4 . 1 . Emergency Sun Reacquisition ( ES R ) Events

ESRs are problematic for a number of reasons, including the elaborate and complicated procedures that the FOT must step through to restore the craft to a normal operating mode. But from an orbit perspective, their worst feature is that Sun-pointing attitude stabilization is maintained via a hardwired control loop that relies on the continual pulsing of the B-branch thrusters lB, 2B, 3B, and 4B. Thrusters 1B and 2B provide the pitch control, while the 3B and 4B pair provide the yaw control. The spacecraft is free to roll, though slowly. The thrusting would not necessarily be bad if no net delta-V resulted. Unfortunately, the 3B/4B pair dominates the 1B/2B pair, pulsing as much as 5 times as often. This imbalance imparts a net delta-V in the Sunward direction, and at a rate that has been observed to be as high as 0.65 m/sec over a day (though on average more like 0.4 to 0.5 m/sec per day). Worse, in ESR mode the thruster counts are not present in the telemetry data, precluding direct knowledge of the actual thruster on-times. Nor is there a ground-based spacecraft simulator capable of providing a useful picture of the thruster behavior during ESR mode. Fortunately, khe delta-V (specifically, the radial component of deltaV projected along the station-to-spacecraft line-of-sight) can be measured on the ground via Doppler tracking data. (The delta-V (AV) is

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related to its radial component ( A V ~ ~ d ivia ~ l )the SEV angle, as AV = A V ~ ~ d icos(SEVan,gZe). ~l/ Since SOHO is never more than 25.5 degrees from the Earth-Sun line, approximately 90% or more of any SOHO-Sunline delta-V is observable in this way.) Thus we at least know how much deltaV any given ESR is responsible for. Later, after the ESR thrusting has ended, a recovery maneuver can be designed to offset the ESR’s negative effects on the orbit. These recovery maneuvers resemble SK maneuvers in all respects, except that they are generally based not on post- burn orbit determination, but rather on a predicted post-ESR trajectory obtained as a result of modeling the ESR delta-V into the pre-ESR trajectory. 4.2. The Accident

But now back t o June 25th. The loss of communication occurred because of a faulty recovery attempt from the second of the two ESRs, which had a disastrous result 2 . In brief, SOHO’s roll rate began increasing dangerously, and then attitude control failed completely as SOHO rolled into a tumble while still thrusting. In short order, there was a loss of high gain antenna (HGA) coverage of the Earth and, worse, loss of solar power as well. In the early aftermath, repeated attempts to communicate with SOHO failed, and the situation appeared truly grim. In hopes of finding some way to recover the spacecraft, the various SOHO support teams regrouped by specialization to attempt to find some answers. Though the results of some of these broader efforts and recovery sub-plots will be briefly touched upon, they are largely beyond the scope of this paper, which shall concentrate mainly on the orbit aspects of the anomalies and the recoveries. From the trajectory analyst’s perspective, we knew that at the time of the loss, SOHO had as much as 206 kg of fuel remaining, representing an average delta-V capability of approximately 225 m/sec. That amount was vastly more than needed to complete the remaining four years of SOHO’s planned 6-year mission (contingencies aside). So there was hope that if contact could ever be reestablished, with luck the 206 kilograms would be enough fuel to repair whatever damage the halo orbit sustained. 4.3. Early Post-anomaly Assessment

From the orbit perspective, orbit determination updates would no longer be possible for lack of tracking data. Therefore the only hope for achieving

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post-accident orbit knowledge was through analysis of the delta-V from thrusting as measured via Doppler during the accident, and the subsequent modeling of that delta-V into the known pre-accident orbit. Before the accident, it was known that a SK maneuver -if it were to be performed on June 24th- would have required a AV of about -5.7 cm/sec (i.e., in an anti-Sunward direction), the MMM residual delta-V not included. What was measured via Doppler leading up to the tumble were AVs from the MMM, the two ESRs, and the ESR recovery attempts (some attitude control thrusting continues during control mode transitional states). Once the tumble commenced, however, it was not long before the tracking data ceased (the drop-out occurred June 25th at 04:38:22 UTC). But from the data available, these AVs were as shown in Table 3. Table 3. Delta-V Sequence during the Anomaly of June 25, 1998. Thrusting Event Delta-V (cm/sec) Momentum Burn +1.787 Post-MMM +0.179 ESR#1 +0.017 Post-ESR#l +0.680 +0.032 ESR #2 Post-ESRs2 -1.273 Net Delta-V +1.422 Note: +/- indicates Sunward/anti-Sunward.

The problem was, we did not (and still do not) know exactly what the nature of the thrusting that may have occurred following loss of contact was like. For all we knew at the time, SOHO might have continued thrusting until fuel exhaustion. Had that happened, the mission surely would have had no hope of recovery. What could be said, however, was that if thrusting had ceased, or if at least no net delta-V resulted from the tumble, following loss of contact, then the cost of halo re-balancing on June 25th would have been approximately -5.7 cm/sec plus +1.4 cm/sec, for a net -4.3 cm/sec. Without this correction, SOHO would inevitably escape into solar orbit. 4.4. Post-Anomaly Trajectory Prediction

A SOHO “best estimated trajectory,” or BET, was constructed based on the observed thrusting described above, applying the 1.4 cm/sec delta-V that was reconstructed from the data. Of course, just what the real situation

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was, we did not know. A first order of business was to gain a sense of how much time we might have before it would become too difficult for DSN search efforts to succeed. This required a quick trajectory dispersion study. To model possible dispersion trajectories, one of which the spacecraft might be on, eight impulsive delta-Vs ranging over 2 orders of magnitude (1 cm/sec to 100 cm/sec) were imparted to the BET at the epoch of dropout. The delta-Vs were applied either parallel (+) or anti-parallel (-) with the SOHO-to-Sun line direction, respectively increasing or decreasing orbital energy. Table 4 below gives the epochs where the dispersion trajectories diverged from the BET by an angular separation of 0.135 degree as seen from the Earth. Accordingly, these were the epochs when a DSN 34-meter station beam-width could no longer simultaneously cover both the BET and a diverging hypothetical dispersion trajectory, assuming the tracking was centered on the BET sky path. For a given dispersion delta-V magnitude, the epochs where the angular separation exceeded 0.135 degree proved to be identical for both -t and - cases. Table 4. June 25, 1998, Anomaly Delta-V Dispersion Study. Dispersion Delta-V (cm/sec) f 100 f 10 f 5

fl

Epoch of 0.135 deg Angular Separation 10 Aug 1998 19 Sept 1998 13 Oct 1998 26 Dec 1998

Days from June 25th, 1998 46 86 110 185

From these results alone it was clear that time was of the essence, and that search efforts had to be organized and ramped up quickly. There was not a lot of time available to locate SOHO if in fact the post-loss delta-V actually had been large, e.g., one meter per second or larger. It was assumed at the time that if further post-accident analysis or simulations indicated a better estimate of the overall delta-V occurring after the loss of contact, then we would produce an updated BET ephemeris using the improved data. However, such results were never forthcoming, so the initial postaccident BET was the one used over the ensuing several weeks to support operations, in particular for the generation of predicted DSN site views and station antenna pointing vectors. 4.5. Decay h j e c t o r y Sensitivities

The problem for SOHO was that if recontact and recovery could not be achieved, it would be flying an uncontrolled, decaying trajectory that

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eventually would either escape into heliocentric orbit or come back toward Earth. For all practical purposes, the possible outcomes were virtually infinite because we did not know how much actual thrusting SOHO performed following the loss of contact, and therefore we could not know its actual energy state. To gain insight into the sensitivities involved in the problem, some trajectory simulations were performed involving application of a narrow range of delta-V dispersions to the BET. This quick-study was never meant to be a thorough-going investigation of LPO escape trajectories, but rather was designed t o provide the SOHO Project with an overview of the complexity and severity of the orbit problem. The point was that it was possible for even a seemingly small difference in departure delta-V to result in hugely and dramatically different trajectory outcomes. As will be made plain below, it followed that there could be no single orbit recovery scenario or strategy that would apply to all escape cases. The orbit recovery strategy would need to be tailored to the specific escape trajectory. But that would be impossible to do without knowing what the actual departure delta-V was, or without resuming contact with, and control of, the spacecraft. Further, given the fact of LPO exponential decay, the specific recovery strategy would also greatly depend on how much time passed before both contact and control were resumed. But the first step was to see what the BET itself would do. Propagating from the epoch of telemetry loss, it was seen that the BET itself would diverge only very slowly from the halo orbit. It retained quite well its halo characteristics up to about mid-November 1998, after which time its divergence would accelerate with the result being escape into heliocentric orbit. This result is indicated in Figures B-1 and B-2 of Appendix B. (Appendix B contains eight archival, annotated, screen-snap images generated with the Swingby program during the early days of the crisis and presented at SOHO situation meetings.) Figure B-1 depicts an escape case called the “Telemetry Dropout Case”, or “TDC”, along with the results of five dispersion trajectories created in 1 cm/sec increments from +1 to +5 cm/sec (delta-Vs applied parallel to the SOHO-Sun line toward the Sun). (The TDC was a virtually identical, early precursor trajectory to the eventual BET.) All six trajectories drift away from the halo orbit at varying energies, going directly into solar orbit. Figure B-2 depicts in a solar rotating-frame what becomes of the TDC, i.e., it escapes into a heliocen-

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tric orbit having a 341-day period. In this inferior orbit having somewhat greater orbital speed than Earth’s, SOHO would eventually “lap” Earth, returning to the Earth’s vicinity after 11.6 years. Other TDC dispersion cases were developed as well. All trajectories shown in the eight figures of Appendix B are uncontrolled, that is, there are no delta-Vs introduced after the dispersion delta-V, so the trajectories are free to evolve under natural gravitational influences only. Figure B-3 shows results for delta-Vs of -1 to -6 cm/sec at 1 cm/sec increments, all escape trajectories of slightly lesser energies than those of Figure B-1. The -7 cm/sec case of Figure B-4 is the first where a return to Earth is seen, but a stable capture is not achieved and escape to solar orbit is the result. Figures B-5 and B-6, for the -8 cm/sec and -9 cm/sec cases respectively, show two different Earth return trajectories that do not quite come within the orbit of the Moon, and so are before long whisked into heliocentric orbit. Figure B-7 shows the first fairly solid Earth capture case, corresponding to a dispersion of -10 cm/sec. Figure B-8 shows the last Earth-capture case looked at -for -11 cm/sec- and it is very different from that of Figure B-7. Both trajectories -propagated for about 2.5 years- are quite chaotic and greatly influenced by lunar encounters. It should be emphasized that these cases were not studied extensively; they were quick-looks only. Of course it was not known whether SOHO was truly on a solar escape trajectory or on an Earth-return trajectory. However, it was apparent at the time that if SOHO were on some Earthreturn trajectory, the recovery strategy would have had to be adapted to that trajectory’s particular characteristics. Once on its way back to Earth, there would be no stopping it. The trick would have been to see whether the anticipated Earth and Moon encounters -occurring weeks or months in the future- could be exploited somehow to set up an eventual lunar flyby gravitational-assist trajectory back toward L1 with a minimum of propulsive delta-V. (This is an area where the GSFC FDF has a wealth of experience, with the NASA GSFC GGS/Wind mission being a prime example.16) Assuming a workable recovery solution would exist at all, some maneuvers -and perhaps some large ones- would doubtless have been necessary to set up useful gravitational-assist encounters. In many scenarios, such a task would probably have been a herculean challenge for a potentially damaged SOHO, a spacecraft that was never especially nimble for deep space flight in the first place.

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4.6. Presumed Condition of the Spacecmfl

Meanwhile, post-accident attitude analysis indicated that SOHO was likely to have settled into a slow spin around its major axis of inertia (ZB-axis) 17. However it was not known what inertial orientation or spin sense the ZBaxis itself might have attained. In any case, the solar panels could be receiving only partial sunlight at best. In a worst case, the ZB-axis orientation might be such that the angle between the Sun direction and the solar panel normals (which were in the spin plane) was too great for useful power generation. However, starting from the assumption (and the hope) that SOHO might still be receiving some solar power, numerous attempts were carried out during July 1998 to prod SOHO into emitting bursts of its carrier signal from its Low Gain Antenna. As it turned out, many such carrier “spikes” were received.17 This was the initial indication that not only was SOHO managing to generate at least briefly intermittent power from sunlight, but that it was also still close to the path of the BET ephemeris. This was both immensely fortunate and encouraging.

4.7. Detection of the Spacecmfl

Another excellent indication of the continuing validity of the BET ephemeris came on July 23, when bistatic radar tests conducted with the Arecibo radio telescope in Puerto Rico and DSN Goldstone successfully bounced echoes off SOHO (at a range of 1,473,000 km!). Analysis of the radar data by a group at Cornell University17 revealed a radar cross section similar to SOHO’s dimensions, and that the craft had a spin period of about 54 seconds. Additionally, after some initial confusion concerning the Doppler analysis, we were told that a radial velocity difference of -12 cm/sec was observed as compared to the BET ephemeris. This was not a huge discrepancy for a period of 29 days, and was further proof that the net delta-V of the accident had not been large, as feared, but rather that it must have been quite modest, even small. As 29 days represented 29/16 = 1.8125 doubling periods, the -12 cm/sec radial velocity difference suggested that there was a delta-V of approximately -12/21.8125= -3.4 cm/sec that had not been observed after loss of contact on June 25th. If true, then a halo correction delta-V of approximately -4.3 (-3.4) = -7.7 cm/sec would have been needed right after the accident.

+

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Thus, while we were viewing the radar test results skeptically yet hopefully, it seemed clear that the post-accident orbit was not diverging all that rapidly from the pre-accident halo. This suggested that there might still be ample time for recovery before halo correction costs grew to uncomfortable levels. Finally, a brief link was established between Goldstone and SOHO on August 3, 1998, and the threshold for the enormous task of recovery operations was reached.

4.8. The Recovery

from the June 1998 Anomaly

The ensuing recovery of SOHO was a long and complicated story. Only a brief account of it can be given here. As gradually longer contacts with SOHO were achieved during early August, it was confirmed from Sun Sensor data that SOHO was slowly spinning about its +ZB-axis. This axis was then nearly parallel with the ecliptic plane, pointing some 37 degrees to the West of the Sun (as of August 11th). This situation was truly fortunate enough for recovery prospects, because the Eastward motion of the Sun over the coming weeks would decrease the angle between itself and SOHO's spin plane. Since the solar panel normals were in the spin plane, there would then be a substantial fraction of every spin period during which the panels would receive useful ~un1ight.l~ In mid-August, the FOT began gradual, carefully paced and monitored cycles of battery recharging, and cycles of heating for the frozen propulsion system.17 By mid- September, with the propulsion system thawed, the batteries restored, and improved telemetry coverage established, the time was right for using thrusters to reestablish the Sun-pointing, 3-axis stabilized attitude, to be followed by return of control to the wheels. Not long after the success of this delicate, weeklong operation, the use of the HGA was restored as well. Then, with high-rate telemetry reestablished, preliminary system and instrument checkouts showed that everything had come through the accident in good working order with the exception of two of the three roll control gyros, which were now useless. (Almost needless to say, bringing SOHO back to life was a delicate, arduous, tense job that involved an enormous amount of work from a large number of people on both sides of the Atlantic. But it is too long and involved a story to be told in full here, so we return to the orbit recovery story.) Meanwhile, over what remained of August, new tracking data were grad-

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ually accumulated, and numerous attempts to redetermine the orbit with useful accuracy were made. However, due to coverage spottiness and other problems with the tracking data, the orbit determination solutions did not achieve a useful quality until near the end of August. (Though solutions were available earlier that demonstrated that the halo orbit had not deteriorated too seriously, they were not yet good enough for accurate halo orbit retargeting.) By September lst, a solution of sufficient quality was used to show that if a halo correction maneuver were performed that day, the delta-V cost would be -0.967 m/sec. As 68 days -or 68/16 = 4.25 doubling periods- had elapsed since the accident, this delta-V implied that the halo correction cost early on June 25th would have been approximately 0.967/24.25= -0.051 m/sec. This amazing result suggested that an effective delta-V of only about one cm/sec had gone unaccounted-for at the time of the accident. Given such a small delta-V, the most likely accident scenario appeared to be that the thrusters ceased firing very soon after contact was lost. Hence it seems also that very little fuel was lost during the accident.

4.9. The Recovew Maneuvers

Finally, by September 25th, it was possible to attempt the first orbit recovery maneuver (RM-01). Halo orbit retargeting with the Swingby program showed that an anti-sunward AV of 6.21 m/sec was required, meaning that thrusters 1A and 2A were needed. In excess of 3 m/sec of this deltaV was needed to counter the Sunward perturbations imparted from the continual attitude recovery thrusting over the period of September 16th to the 23rd. That attitude recovery maneuver was performed in an ESR mode, having the typical ESR attributes described previously. (Though a Sun-pointing attitude was reestablished fairly early on during that week, SOHO remained in the ESR mode while preparations for a return to 3axis stabilization with the momentum wheels were completed.) During the ESR maneuver, the cumulative delta-V (some 3.16 m/sec as revealed by the Doppler tracking data) was modeled into the trajectory using Swingby. It was known that by September 16th (when the attitude recovery maneuver began) the halo correction cost had grown to just 1.76 m/sec. Had there not been the intervening ESR, it would have been only 2.52 m/sec by September 25th, not bad given all SOHO had been through. Thus it was owing to the Sunward-thrusting ESR event lasting over 6 days that the halo correction cost had more than doubled.

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There were a number of uncertainties going into RM-01 following the attitude reorientation ESR. These uncertainties included: 1) pre-ESR orbit determination that was still somewhat rough around the edges; 2) how much ESR delta-V may have been normal to the station line-of-sight and so not measurable; 3) the accuracy of our methods for modeling a weeklong series of ESR perturbations into the trajectory; 4) whether SOHO’s propulsion and closed-loop systems would respond normally to commands; 5) and lastly, the fact that there was not a sufficiently long arc of tracking data obtainable between September 23rd and the 25th to compute a valid postESR OD solution. Nevertheless, the maneuver was carried out successfully in two segments. The first segment was a brief 2-minute burn, deliberately kept short to guard against any propulsion system problems. None occurred, and so with the thrusters commanded for the nominal 75% duty cycle, the second segment was completed in 47.5 minutes, again without problems. The second segment had been designed deliberately to undershoot the targeted delta-V by 5% because we wanted to guard against possible “hot” performance by 1A and 2A that might force us to use the under-calibrated jets 3A and 4A for the follow-on recovery maneuver. By all appearances the September 25th burn seemed quite successful, achieving the desired delta-V exactly. But we had deliberately biased it low by approximately 31 cm/sec. So, given that and the considerable uncertainty in the pre-burn orbit as well, we realized we would need to do another correction burn within a few weeks at most. Nevertheless, we were feeling as though we were virtually “out of the woods” at last. But as it turned out, there was some rockiness left to the recovery where the orbit was concerned.

4.10. Developments Complicating the Recovery

By early October, while flight controllers and spacecraft engineers were busy with a continuing series of systems checkouts and science instrument recommissioning, the FDF flight dynamics team began seeing peculiar new orbit determination (OD) results. In brief, the mystery centered on strangely inflated values for the solved-for coefficient of reflectivity (CR) that began appearing in the OD solutions in the weeks following the September 25th correction burn. The CR -which should have a value between 1.0 (perfect

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absorber) and 2.0 (perfect reflector), and for SOHO was normally about 1.40- began showing values over 2.0, and eventually reaching over 9.0. This can happen with the orbit determination software in situations where there is an extraneous, unaccounted-for acceleration on the spacecraft. In fact, we calculated that the CR values we were seeing could correspond to an acceleration away from the Sun on the order of m/sec2, in turn indicating an applied force on the order of N. These numbers were not incompatible with the possibility of a fuel leak, and so that became the initial leading hypothesis for explaining the OD results. While the effort to either establish or disprove the leak hypothesis proceeded, a major distraction came along in the form of the upcoming November 17th Leonids meteor shower -possibly a “storm”- and its perceived dangers. Comet Temple-Tuttle -parent comet of the annual Leonids showerhad returned to perihelion in February 1998, and some astronomers were pointing out the possibility of a major Leonid “storm” of magnitude perhaps great enough to pose a threat to Earth’s spacecraft fleet. In particular, the densest portion of the great cometary stream of material was predicted to pass close to L1,which meant that SOHO might be facing greater danger than most. In any case, there was a NASA imperative to address the issue and come up with a risk amelioration plan. Thus, at an inopportune time, SOHO flight dynamics team resources were diverted to assess the threat, educate the flight teams and various levels of management about comets, meteors, and the risks, and to analyze and propose possible courses of action. Against this backdrop, the October recovery maneuver was planned. The halo targeting results for the October maneuver varied depending on whether the anomalous “acceleration7’was assumed to be real or not. While no corroborating proof could be adduced for the fuel leak hypothesis -in particular, there seemed to be no discernable response by the momentum wheels to any anomalous torque- various orbit maneuver options were considered with respect to it. Various finite burn cases considering the acceleration to be real, not real, and for cutting the difference were developed, including computing the effects on a possible follow-on maneuver of the chosen assumption being wrong. Finally, it was decided to assume the acceleration was not real (i.e., trajectory propagations did not include it), and a maneuver (RM-02) of 2.0 m/sec (it would have been 2.84 m/sec if we had assumed the acceleration was real) was planned for October 16th. Though we were somewhat surprised by the size of this delta-V, we attributed it

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mainly to the likely large orbit uncertainties surrounding the September series of maneuvers. And despite the conservatism surrounding the RM-01 planning, the orbit determination results insisted that the delta-V needed to be Sunward, requiring jets 3A and 4A after all. As it happened, SOHO turned in a somewhat cold performance for RM-02, achieving only 1.92 m/sec. (This was not too surprising, as the thrusters 3A and 4A were not all that definitively calibrated due to rare prior use.) Post-burn OD showed that the anomalous “acceleration” was still present, but events were at this point dominated by preparations for the Leonids. (The course of action chosen was to do a special roll maneuver to turn the sole remaining gyro and the star trackers away from the Leonid radiant.) But the discomfiting development was that the halo re-targeting results now showed that a new correction burn of a magnitude comparable to the one just completed (and again in the Sunward direction) would be needed just a month later. This time it was decided to assume the anomalous acceleration was real, and a maneuver (RM-03) of 2.29 m/sec using jets 3A and 4A was performed on November 13th, 1998. Simultaneously, a renewed push to solve the “anomalous acceleration” problem was made, and shortly after the Leonids special operations campaign was past, and with a number of hypothetical causes rejected, the suggestion came up that a tracking problem called a “range ambiguity” might be to blame. This was quickly shown to be the case, and following the completion of requested reconfigurations at the DSN tracking stations the problem was solved. (Though the technical explanation of a “range ambiguity” lies beyond the scope of this paper, it effectively amounts to a mis-measurement of the radial position of the spacecraft. In SOHO’s case, the range discrepancy had amounted to approximately 1200 km.) While a study of why the range ambiguity came about was never made, it seems probable that it developed during the last half of September 1998, which was characterized by a long period of variable thrusting episodes and substantial orbit solution uncertainties both prior to and after the burns.

...

4.11. Finally Out of the Woods or So W e Thought

Though the range ambiguity problem had caused considerable consternation for the orbit recovery process, by the last half of November the situation for SOHO was rapidly improving. With new and correct OD results now in hand, the SOHO halo orbit seemed to have “snapped back”. The follow-

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on orbit correction delta-V was predicted to be only about 10 cm/sec by December 21, 1998, and it was planned for that soon only because a long, maneuver-free science observing campaign was scheduled to begin early in 1999. Although the autumn 1998 orbit maneuvers were designed and performed in the same way as “stationkeeping” burns always had been, given their circumstances we are classifying all three as a special type, that is, a 5-ecovery maneuver” (RM).

As December 1998 arrived, all of the SOHO flight support teams were looking forward t o a return to some semblance of normalcy. However, it was not to be.

5. The Anomaly of December 21, 1998, and the “Great ESR” On December 21,1998, as the SOHO Flight Operations Team was preparing the spacecraft for a round of momentum burns and a small SK burn, the lone remaining gyroscope failed irretrievably. As this happened, we once again found ourselves in ESR mode with the B-branch thrusters back in action. But this time there was a difference. There was no way to return to a normal mode of attitude control without at least one operational gyro. Thus the one thing we had feared above all else had happened. Early on during the recovery effort from the June 1998 anomaly, planning was underway to LLre-invent”SOHO as a gyroless spacecraft, but the massive changes required for the onboard flight software and the ground procedures were not expected t o be ready until the end of 1999. Hence, the mission was once again in grave trouble. SOHO was now akin to a continuous low-thrust spacecraft. Recall that for ESR mode the aft-end, yaw-control thrusters 3B and 4B dominate the pitch control thrusters 1B and 2B located on the Sunward side of SOHO. Via the Doppler data, we detected early on that an average of around +0.45 m/sec (Earth-SOH0 radial component) was being imparted to the orbit per day (that implied an average acceleration of approximately 5 m/sec2 away from Earth). Given that SOHO was near its western- most elongation at the time, the Doppler results meant that the total delta-V directed Sunward was averaging around 0.5 m/sec per day. (As the crisis wore on, this running average varied considerably.) Doppler results for the

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first 15 days of the crisis are shown in Figures C-1 and C-2 of Appendix C, which contains a selection of archival plots from the second anomaly. Figure C-1 gives the daily delta-V imparted by the ESR, while C-2 shows cumulative ESR delta-V relative t o the preanomaly orbit.

As previously noted, perturbations of such magnitudes spell trouble for LPOs. In this particular case, with orbital energy continually increasing, the threat was that SOHO would eventually escape into solar orbit. (An example of the escape trajectory caused by a continuing -or “indefinite”ESR event is shown in Figure C-3. Shown for comparison, and overlying the “indefinite ESR” escape trajectory, are the nominal halo orbit and a one-day ESR event trajectory. The latter also escapes though not nearly as quickly.) However the ESR delta-V was not imparted at a strictly constant rate, but varied somewhat day to day. Nevertheless, the cumulative harm being done was mounting daily, as was the delta-V cost of the required corrective action. So, the issues that were obviously paramount were the damage being done to the halo orbit, what it would take to fix it, and the rate of fuel usage. All three issues were addressed during the early days of the crisis.

5.1. Modeling the ESR Using the Swingby Progmm

With the version of the Swingby program in use at the time, poweredflight segments of space trajectories could only be modeled in terms of what Swingby defines as a “Maneuver Event” 13. That was the only way to apply the thrust term in the trajectory integrations. In normal usage, a “Manuever Event” was a planned, finite-duration maneuver. It had not previously been used in such a way as to model a long-duration (or openended), continuous thrusting situation. Fortunately though, Swingby had the inherent flexibility to do just that. It had four key features that were advantageous for our problem. It had the built-in, menu-driven functionality to: 1) model arbitrarily long propulsive maneuvers; 2) model a large number of separate maneuvers in sequence with each other and/or with free-flight propagation segments; 3) easily incorporate or re- configure different propulsion models; and 4) employ unique thruster configurations and attributes for each separate maneuver. As events unfolded, these capabilities were exploited fully. Fortunately, though the ESR’s acceleration of SOHO tended to vary,

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there were often long stretches of time where it was at least approximately constant. Thus the ESR behavior could be modeled within Swingby in a piece-wise fashion using sequences of separate, constant duty-cycle maneuvers. The duty cycle of each such maneuver, or "ESR piece", was adjusted until the achieved radial delta-V (the component of the delta- V parallel to the Earth-SOH0 line) matched the observed Doppler over the time period modeled. It was not necessary to micro-model the daily ESR activity. Taking averages, it was generally adequate to model a given day's activity in one or two pieces, though occasionally it required more to do the job accurately. 5.2. Estimating the ESR-mode thruster duty cycles

A crucial part of the puzzle consisted of estimating the duty cycles of the thruster pairs lB, 2B and 3B, 4B so that accurate ESR burn modeling and continuous-thrust trajectory propagations could be made. The duty cycling was not some known value fixed by flight software, nor could it be commanded from the ground or determined from the telemetry (recall that the ESR-mode telemetry lacked thruster firing data). Through clues including results of spacecraft simulations performed on the ground, it was estimated that the 3B, 4B pair was pulsing roughly 5 times as often as the lB, 2B pair. But this was relative performance, and we needed to determine absolute. Since there were time spans where the thrust level was essentially fixed (or could be averaged) as seen from Doppler, both a delta-V (AV) and a burn duration (AT) were thus provided as measurements. With such measurements, and with the 3B/4B-to-lB/2B pulsing ratio estimated to be 5:1, then the duty cycles of each pair could be estimated beginning with the formula F = m(AV/AT). Since the spacecraft mass, m, was known, and both the AV and the burn time span AT, were measurable from the Doppler, then the net thrust force F could be calculated directly. Further, F could be resolved into the four individual thruster components and exploited to find an average duty cycle -denoted by x- over a given ESR span, for thruster set 1B and 2B, by writing an equation with a single unknown as follows:

This simple expression assumes:

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Individual thruster forces are known, computed from the propulsion blowdown polynomials (functions of the fuel tank pressure, which is known from telemetry) The forces for lB, 2B are negative because they are anti-Sunward; these thrusters are also canted by 30 degrees to the X B axis, and hence also to the SOHO-Sun line Thruster calibration scale factors are all equal to 1.0 (the B-branch thrusters are not subject to calibration due to lack of the appropriate telemetry data types) Duty cycling is equivalent to scaling the forces.

To demonstrate, the hydrazine tank conditions on December 21,1998 -186 kg of fuel at a pressure of 15.84 bars- meant that at the outset of the ESR, FIB= 2.94 N, F ~ B = 2.95 N, F ~ B = 3.00 N, and F ~ B = 3.02 N. Assuming an observed AV of 0.45 m/sec over a 24 hour period (AT = 86,400 sec), and given that the SOHO mass was 1799 kg, the force on the spacecraft was calculated to be F = 9.37(10-3) N. Inserting these values into the above expression for F and solving for the unknown duty cycle, 2, gives 2 = 3.748(10-4). So, the duty cycle for the l B , 2B pair was 0.03748%,and for the 3B, 4B pair, it was 5 times greater, or 0.18741%. This approach enabled computation of a good first guess for the duty cycling of each of the two thruster pairs. Differential adjustment was used to fine tune the duty cycle values input to the Swingby program, so that the software-computed radial delta-V ( A V ~ ~ d imatched ~l) that observed from the Doppler data for a given time span. With the fundamental duty cycle determined, it was possible to configure precisely the modeling of each ESR piece, or span, on a daily basis. As the ESR delta-V rate would change, the duty cycles could be recomputed, or differentially adjusted, to keep pace. Thus, we had in place a method for modeling the extended, on-going ESR event as a variable, continuous thrust, long duration “maneuver”, and in that way we were able to keep our orbit knowledge current. This was a fortunate thing because it was the only way to keep our orbit knowledge updated. The normal operational orbit determination process (a batch least-squares processing of range and rangerate tracking data types) was not designed to obtain solutions with the presence of the effectively continu-

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ous thrust perturbations. So, the ESR model within Swingby was extended daily as new Doppler observations came in. In addition to serving a a substitute orbit determination process, the Swingby-based approach naturally provided fuel usage results as well.

5.3. Early Assessment of the Problem Initially, relative calm prevailed amongst the various teams involved. But concern grew daily as the crisis lengthened and the author’s predictions for the fate of the halo orbit grew to be fully appreciated. The author personally spent the 1998 Christmas holidays thinking intensely on how to prevent SOHO from escaping the L1 region. Clearly, it was going to be no mean feat. Apart from the modeling challenges the ESR posed, it was not certain that the crippled spacecraft would even be capable of performing the delta-V maneuvers that would be needed to save it. The spacecraft had been designed to perform delta-V maneuvers of commanded duration via closed-loop OBC control that required gyroscopes in the loop. Now we were faced with devising an approach to doing all delta-V maneuvers (and eventually momentum maneuvers as well) with open loop commanding. This was a challenge that was going to be tricky for a number of reasons, with the need for numerous flight software and procedural workarounds prominent on the list. But the first order of business was to assess what we were facing. Implementing the detailed ESR modeling (described above) in the Swingby program was absolutely critical to developing an accurate picture of what was happening to the halo orbit, and to designing the eventual solution to the threat. It was certain that we needed to take energy out of the orbit to prevent escape from &. But there were some big initial questions that needed answers. Such as, how many maneuvers would it ultimately take to do the job ? How soon did we need to get the maneuvers started ? And, lastly, could we possibly wait until spacecraft engineers found a way of taking SOHO out of ESR-mode before starting ? Following an early estimate from the spacecraft engineers that it could take one to two months to bring about an end to the ESR, some quick-anddirty orbit recovery scenarios were constructed to get a feel for the problem. For example, two sample continuous thrust trajectories were propagated up to candidate dates of January 20th and February 19th, respectively,

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from which dates a single impulsive, anti-Sunward delta-V was targeted simply to hold SOHO to the L1 region (though not re-establish the mission halo). The delta-Vs obtained, however, were both impracticably large: 30 m/sec for the January case and a far worse 158 m/sec for the February case. (See Figure C-3 for the locations of these hypothetical burns on the “indefinite ESR”-escape trajectory.) From these results it was obvious that to prevent rapid escape in a practical, affordable way, we needed to begin to do counteractive delta-V maneuvers just as soon as possible. During the final week of 1998, engineers concluded that we could reasonably attempt a first-ever open-loop delta-V maneuver by January 7th, and so that date was chosen to plan toward. Around the same time, an estimate came in from the spacecraft engineers that taking SOHO out of ESR-mode might be possible by February 1st. So that became another target date; one we all hoped was achievable. If it was achievable, then we could also hope for a first orbit determination solution by February 1l t h (a minimum 10-day arc of freeflight tracking data was considered essential for a reasonably accurate solution). February 11th would then become a candidate date for another correction maneuver. Having these dates specified provided the sort of LLboundaryevents” needed to facilitate construction of various orbit control and recovery scenarios. One early scenario envisioned doing the bulk correction delta-V on January 7th with the remainder left to February l l t h , which would follow an end of the ESR on February 1st. However, the delta-Vs computed for those two dates were nearly 20 m/sec and 4 m/sec respectively -the first of them considered two times too large for SOHO to safely attempt. Yet another two-burn scenario considered doing a fixed 10 m/sec on January 7th and a final corrective delta-V on February 11th. However, the February 11th delta-V for this case was an even more impermissible 48 m/sec. Thus, in general the 2-burn scenario did not look promising. Next, an alternative conception involved a series of fixed duration burns designed to gradually counteract the ESR’s Sunward accelerations and bring orbital energy back to a rough halo value by February 1st. Then, a final clean-up recovery burn could be performed by mid-February for around 5 m/sec or less. Historically, what was first proposed was a series of four burns -each under 10 m/sec- to be executed at one-week intervals beginning on January 7th. The details of this straw-man scheme are provided in Table 5.

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Delta-V

Fuel Use

Duration

7.43 7.37 46 Jan 14 7.24 7.16 46 Jan 21 7.08 6.96 46 Jan 28 Totals 29.38 29.1’ 184 *Total fuel usage over the period was 52 kg when ESR continuous thrust included.

This scheme -proposed on Dec. 31, 1998- assumed a constant average duty cycle ESR up to February 1st. Although it was clear as early as Dec. 31st that the constant ESR pulsing assumption was not strictly valid, there existed no other way to approach the problem of propagating the continuous thrust trajectory for long periods (i.e., on the order of weeks or months). This was primarily because the variability of the pulsing rate could not be predicted in advance. No simulator existed that could adequately account for and duplicate the variability seen up to that point, let alone predict its future course over weeks or months. It was obvious that the delta-V imparted would decrease over time due to fuel blowdown (note the declining delta-V and fuel consumption over the series of fixed-duration maneuvers shown in Table 5 -an effect of the declining fuel pressure). But it was not known how much the pulsing, or the yaw/pitch pulsing ratio, would change in response to the declining thrust forces. Plausibly, the pulsing might have increased, implying greater fuel consumption over time and bringing an earlier end to the mission. As it was, though, the technique was adequate to make a fair estimate of when fuel would be exhausted. Scenarios showed that if the series of deltaVs shown in Table 5 were executed -but the effort to take SOH0 out of ESR mode in February subsequently failed- then another maneuver could be done in early February to prevent escape for another several months. This February maneuver (it would be on the order of 10 m/sec) would work by reducing orbital energy by just the right decrement that the ESR thrusting would restore it gradually, so as to maintain a halo-like orbit of virtually nominal dimensions. At the time, the author referred to this result as the “guided continuous thrust trajectory”, or GCT trajectory (see Figure C4, in Appendix C). In principle, the GCT trajectory could be maintained through at least the point where fuel was exhausted, which propagation

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scenarios showed could happen around October 1, 1999. Fuel exhaustion would mean the end for the ESR thrusting and, for that matter, attitude control. SOHO would then be free to gradually escape the L1 region, but that outcome would have been of no consequence for a dead spacecraft. The existence of the GCT solution indicated that rescue efforts could continue well into 1999, with the hope that success would come soon enough that some fuel would still be available for restoring the halo orbit, thus continuing the mission for at least a while longer. Though a thorough-going study of the issue was never made, a brief investigation showed that for a GCT trajectory the cost to restore a free-flight halo orbit would exceed the needed fuel supply by early August 1999 (the cost was nearing 20 m/sec by August 1). This result suggested that ideally we really needed to escape ESR at least a couple of months earlier than August, given the long-range ESR fuel consumption rate uncertainties. (Though the ESR consumption rate was varying considerably day-to-day, it was estimated to be mostly within a 0.5 to 0.7 kg per day rangeover the first couple of weeks of the crisis.) This analysis also effectively killed some early speculation on the part of the Project that we might be able to stay in ESR mode until the end of 1999, when the fully updated gyroless flight software and procedures were expected to be ready.

So, in summary, it became clear that we needed to begin maneuvers just as soon as possible. There could be no long wait for the spacecraft to be extricated from ESR mode first. It was all just too uncertain as to whether or when that might succeed. The halo orbit would simply decay too fast in the meantime, with the recovery delta-V costs compounding continuously all the while. It was also apparent that the recovery would have to be conducted gradually with a series of maneuvers. Resolving the orbit decay problem with just one or two large burns would be beyond the crippled spacecraft’s safety and capability margins. As grim as the overall situation was, SOHO nevertheless possessed the luxury of being fuel-rich (about 186 kg) when the crisis began. That fact afforded us substantial breathing room, despite the on-going fuel drain.

5.4. The Struggle

After New Year’s, rescue activities escalated as the urgency of the crisis grew ever more apparent. Situation and planning meetings -bringing to-

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gether personnel from the SOHO Project, Flight Operations Team, Flight Dynamics Facility, SOHO experts from the European Space Agency (ESA) and Matra Marconi Space, and various NASA consultants- were convened daily, and the effort began to take on qualities akin to fighting a war. Meanwhile, planning for the first burn of the series -set for January 7thcontinued. Since repeated attempts to determine the orbit via normal tracking data processing were unsuccessful, the effort to model the ESR within Swingby (as described earlier) in as accurate and detailed a way as possible was crucial. That was the only way to update our knowledge of the orbit state. Then, the maneuver series discussed above was repeatedly retargeted and re-planned in cycle with updates of the ESR behavior. The January 7th recovery burn occurred as originally scheduled -the only maneuver of the series to do so. At this point some 17 days into the crisis, the ESR had already imparted about 9 m/sec of delta-V (Doppler component) to the orbit. This burn -designated RM-04- with a nominal delta-V of 7.44 m/sec was planned in three segments, with the first segment made short to guard against problems. Caution was certainly necessary, and not only because this was the first-ver attempt to do a planned delta-V maneuver (with A-branch thrusters) while an active ESR was in progress (on B- branch thrusters) -something the spacecraft was never intended to do. Without the gyros, the maneuver had to be commanded open-loop, which meant that the firing details for both delta-V thrusters and A-branch attitude control thrusters basically had to be spoon fed to the spacecraft. This also was new and untried. Unsurprisingly, with all the new techniques and procedures being used, there was one significant hitch, but luckily only one. The problem was effectively a duty cycling problem, which was later found to be due to onboard software behavior. It caused delta-V shortfalls in all three segments. A fourth segment was added toward the end to make up the deficit. (All maneuvers were monitored via real-time Doppler data, which enabled evaluation of the maneuver in near real-time. Thus the final segment of a maneuver could be adjusted -or another segment added- as needed.) But on balance RM-04 was successful. In the days that followed, the schedule for the remaining maneuvers shifted numerous times due to a variety of reasons, ranging from changes in the DSN support schedule to the fluid readiness of all the different players generally. This factor coupled with the variable ESR behavior made maneuver planning and fuel consumption forecasting extremely difficult. Scenarios

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and plans were updated very frequently, seemingly with no end. Another important development that had impact on maneuver planning was the introduction during the middle of January of “yaw braking”. Conceived by the spacecraft engineers, yaw braking was the occasional, commanded pulsing of A- branch thrusters 1A and 2A to null, or at least slow, the yawing rate (though primarily responsible for pitch, 1A and 2A possess yaw components as well). After some experience was gained with this novel (for SOHO) technique, it came to greatly reduce the pulsing of the B-branch yaw thrusters, causing the average ESR yaw-to-pitch firing ratio to become more like 1:l. (It should also be noted that occasional, commanded roll control thruster pulsing -to prevent the roll rate from exceeding the spacecraft’s roll stability limit- had already been going on since early in the crisis. Also, due to SOHO’s crippled condition, the roll rate could only be monitored via Doppler data.) The yaw braking had the twin benefit of reducing both the Sunward delta-V and the fuel consumption. Toward the end of the crisis, yaw braking was reducing daily net delta-V to nearly the zero-level. Though it complicated the maneuver planning efforts significantly, yaw braking was nevertheless a welcome development, ultimately reducing the delta-V needed for the remaining recovery burns.

No attempt will be made here to describe all the twists and turns leading up to the remaining maneuvers of the series. A brief recounting of the main results will suffice. Several scenarios were developed for the second burn of the series, with its date, magnitude, and the follow-on burns changing significantly with each. The recovery burn RM-05 -intended to be 8.69 m / s e was finally executed in three segments on January 19th. The commanding details and thruster duty cycles had once again been modified, including rectification of the problem that had caused the January 7th burn to be 17% “cold”. The final segment of RM-05 was extended slightly to make up for modestly cold performance over the first two segments. As a result, the achieved delta-V was very close to planned. The final two burns of the series likewise were re-planned several times, and were significantly affected by the growing efficacy of the yaw braking efforts. (The yaw braking effects were beneficial in terms of reducing the overall recovery delta-V needed, yet the day-to-day unpredictability of their results complicated the maneuver planning process.) So the third burn (RM-06) was performed on January 26th, again in three segments. Like RM-05, the third and final segment of RM-06 was extended to make up for

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the slightly cold performance of the preceding two. That the final result was about 1%“hot” was of little consequence. By the time we were executing this burn, the end of the crisis was just around the corner. 5.5. Success

On January 31, 1999, spacecraft engineers successfully took SOH0 out of ESR mode, as had been planned for days. Attitude control was finally returned to the momentum wheels and the B-branch attitude thrusting was over. There was a collective sigh of relief, of course, but there was one small recovery burn left to do, and that had been pegged for the following day, February 1st. Though it was small -intended to be just 0.32 m/secthere remained plenty of reason for anxiety, as we were implementing yet another major change in the way delta-V burns were commanded. While we had been in ESR-mode, a way was devised to fire the delta-V thrusters at a 100% duty cycle (continuous burning), as they were for burns RM-05 and RM-06. Now, with the momentum wheels back in service, we needed to use a new “pulse train” approach to firing the thrusters. In this approach, pulse duration and the interval between pulses for each delta-V thruster and attitude thruster -as well as total numbers of pulses for each- were specified in detail in the command load to be up-linked for on-board execution. The pulse sequences were computed so as to minimize wheel torques and preserve attitude stability during the burn. The intervals between delta-V pulses were quite long relative to their duration, making for a very small duty cycle of just 5%. (The new 5% duty cycle recommendation had recently come from Matra, SOHO’s prime manufacturer.) So, this new pulse train approach increased the execution time of delta-V maneuvers of given size by a factor of 15 as compared to formerly, when the OBC had governed 75% duty cycle pulsing autonomously. But as things turned out, RM-07 -split into an initial short segment followed by a wrap-up segment- performed smoothly and accurately.

6. Epilogue to the 40-Day Mad Scramble

At last we could relax a little, after what had been a 40-day long, high profile ordeal. Though the spacecraft engineers still had issues to attend to, from a trajectory point of view the crisis was over. Finally, we could do orbit determination in the normal, routine way again. Within a couple

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of weeks, post-ESR orbit determination and trajectory analysis showed that the combination of both the ESR modeling efforts and the recovery maneuvers had been highly effective and accurate. There was no more than approximately 18 cm/sec of residual delta-V remaining to correct. This residual was in the anti-Sunward direction, meaning that all told we had overshot just a little with the recovery burns. Still, given what we had been up against, it was a very small residual indeed, and a maneuver to clean it up was planned for early March. However, one more scare occurred on Valentine’s Day when SOHO endured yet another ESR event that lasted for four days. Though disappointing, at least this time the techniques and procedures were in hand for a relatively quick and confident extrication. The end result was not so bad from the orbit point of view, however, as a planned March 5th Sunward trim burn (RM48) was reduced to just 0.13 m/sec from an anticipated 0.76 m/sec, thanks to the ESR coincidentally imparting delta-V in the needed direction. Following the success of RM-08 -the last of the recovery burns (see Table 6, where the first eight SK burns are retained for reference)- both the trajectory situation and science operations at last returned to normalcy. But the massive effort by the spacecraft engineers to rewrite and revise much of the flight software, spacecraft commands, and ground procedures to support a new era of gyroless operations continued for months more. Although the magnitudes of the February 1st and March 5th burns were comparable to those of typical stationkeepings, we have classified them as the last of the series of recovery burns. The first burn after March 5, 1999, that we termed a routine stationkeeping burn was SK-09, performed on June 17, 1999 (see Table 6). SK-09 performed so accurately that SOHO could have coasted until late November or early December 1999 before another stationkeeping was needed. However the very next burn was scheduled for September of that year for purposes of helping to test and commission new gyroless operations flight software.

It is worth reiterating that ever since the recovery, delta-V maneuvers must be performed with A-branch duty cycles no greater than 5%. Formerly, maneuvers were performed with 75% duty cycles, as all of the first eight SK burns were. The impact of that change has been that the wall time needed to execute a maneuver of a given size is now 15 times longer than

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C.E. Roberts Table 6. SOHO Recovery Maneuver History. Orbit Date Days since Jets last orbit used maneuver (m/d/y) event burn event SK-01 5/23/96 63 1,2 SK-02 9/11/96 SK-03 1/14/97 SK-04 4/11/97 SK-05 9/04/97 SK-06 11/29/97 SK-07 12/19/97 SK-08 4/17/98 RM-01 9/25/98 RM-02 10/16/98 RM-03 11/13/98 RM-04 1/07/99 RM-05 1/19/99 RM-06 1/26/99 RM-07 2/01/99 RM-08 3/05/99 SK-09 6/17/99 104 3,4 RM = Recover?, Maneuver. *large AV erro; due to onboard software

Planned AV (m/sec) 0.3067 0.4541 0.0432 0.1887 1.8876 0.0396 0.3984 1.4375 6.21 2.0 2.285 9.77 8.69 4.06 0.32 0.125 0.4652

Achieved AV AV error (m/sec) (%I 0.3089 +0.714 +0.808 0.4578 0.0411 -4.861 0.1892 +0.235 1.8972 +0.506 0.0408 +2.84 0.3956 -0.703 1.4350 -0.179 6.21 0.0 -3.78 1.924 +0.350 2.293 -17.28’ 8.082 -0.576 8.64 -2.62 3.95 +0.77 0.323 -2.35 0.122 0.4596 -1.21

Fuel used (kg) 0.3353 0.4925 0.0490 0.2064 2.0258 0.0345 0.4263 1.5441 6.666 1.687 1.960 8.60 9.20 4.27 0.267 0.130 0.3835

problem related to ESR mode

before. Therefore, we now try to plan to do SK maneuvers before the halo correction delta-V exceeds 0.75 m/sec, because of execution time considerations. For instance, as of this writing it takes about 110 minutes to impart 0.75 m/sec using the canted thrusters 1A and 2A. Thus the practicality of large delta-V maneuvers has become seriously eroded. Despite all the troubles, SOHO still had perhaps as much as 145 kg of fuel remaining at the end of the crisis, representing an estimated AV capability of around 170 m/sec. This amount was much more than was needed to complete the remaining three years of its nominal extended mission, which SOHO has since achieved as of March 2002. (SOHO, as of this writing, is in an “extended, extended” mission phase.) Though SOHO still suffers from occasional ESR events (2 to 3 per year on average), the recoveries of both spacecraft and halo orbit are generally swift and assured. The gyroless operations paradigm has proven on the whole to be very successful, and the solar science achievements of the mission have been, and continue to be, immense.

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7. Conclusion

Certainly the stories of SOHO’s two mission-threatening anomalies and subsequent rescues rank with the most spectacular in the history of space flight to date. The first anomaly involved the complete loss of communications with, and control of, a deepspace mission flying a Sun-Earth L1 halo orbit. SOHO was out of contact for several weeks, and three months passed before SOHO was rehabilitated enough to perform maneuvers. Given the halo orbit’s extreme sensitivity to perturbations and exponential decay rate, this was an extremely dangerous situation. The second anomaly involved hardware failures (loss of gyroscopes) coupled with loss of normal attitude control. That resulted in the onset of autonomous attitude thruster activity that was unbalanced in terms of delta-V, threatening SOHO with a mission-ending escape to solar orbit. Considering the dire nature of these emergencies it is amazing that total disaster was averted, let alone that the mission was ultimately restored to full working condition both times. No one could have reasonably expected as much. The gratifying outcomes are a tribute to the dedication and quality professionalism of all the SOHO teams involved in the rescue efforts. Given that these were the first anomalies of their type for a LPO mission, there are important lessons to be drawn from SOHO’s experience. (Surely, these lessons are indispensable should the same or similar contingencies occur again for SOHO itself.) First, intervention on behalf of the LPO that is as prompt as possible is critically important. By this is meant beginning recovery maneuvers to counter the delta-V imparted by an anomaly before the recovery delta-Vs have grown to magnitudes that are formidable, or impossible to do anything about. Second, in the event of an accident involving thruster activity, or some runaway situation involving on-going autonomous thruster activity, monitoring the radial delta-V imparted to the orbit as revealed by the Doppler tracking data is likewise critical. That could provide the best means, if not the only means, to reconstruct accurately what happened, or is happening, to the orbit. However the third lesson is the realization that trajectory design software with flexible thrust modeling functionality would be required to ta$e full advantage of the results from the Doppler. The software would have to provide the means to model and propagate an effectively continuous thrust trajectory simultaneously with the capability to model and to target finite delta-V burns -potentially a series of discrete burns- for any desired epoch, or lo-

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cation, within the halo orbit. Lastly, SOHO was the first LPO spacecraft to fly (unintentionally) an effectively continuous low-thrust trajectory, and did so for 40 days straight. The technique developed during the SOHO crisis to control this situation and prevent escape was to use a counteractive delta-V maneuver -or series of maneuvers- to essentially “guide” the continuous thrust trajectory such that it followed an LPO-like path around L1 . This worked by designing stationkeeping-style maneuvers where the thrust direction was used as the independent targeting variable to force the trajectory to follow a path very close to the nominal halo orbit. So close, in fact, that by the end of recovery operations the orbit was practically indistinguishable from the pre-anomaly halo orbit. In SOHO’s case, the maneuvers had to take energy out of the orbit to adjust for the energy that the autonomous continuous thrust was gradually adding. However, in cases requiring it, the reverse would work as well, i.e., add energy with the targeting variable (planned maneuver) that anomalous thrust is taking out of the orbit.

Acknowledgments The author would like to acknowledge four fellow GSFC Flight Dynamics Facility colleagues for their collaboration and crucial contributions in their areas of specialty during the SOHO contingencies. They are Steve Hendry (Honeywell, tracking systems engineering and orbit determination), John Rowe (Computer Sciences Corporation (CSC), attitude determination and control), and Richard Arquilla (CSC) and Steve Thorpe (CSC) (attitude and orbit determination operations). This work was supported by the National Aeronautics and Space Administration (NASA) under contract NAS 9-98100.

References 1. Dunham, D. W. and Roberts, C. E., Stationkeeping Techniques for LibrationPoint Satellites, The Journal of the Astronautical Sciences, Vol. 49, No. 1, January- March 2001, pp. 127-144. 2. Website http: //sohow.nascom.nasa. gov for more information.

The SOHO Mission L1 Halo Orbit Recovery 213 3. Domingo, V., et al., The SOHO Mission: An Overview, Solar Physics, Vol. 162, No. 1-2, December 1995, pp. 1-37. 4. Farquhar, R. W., The Flight of ISEE-3/ICE: Origins, Mission History, and a Legacy, Journal of the Astronautical Sciences, Vol. 49, No. 1, January-Mach 2001, pp. 23-73. 5. Becher, T., et al., Solar and Heliospheric Observatory (SOHO) Mission Flight Dynamics Support System (FDSS) Mathematical Background, CSC/TR92/6100ROUDO, Computer Sciences Corporation (CSC), January 1993. 6. Jordan, P., et al., Solar and Heliospheric Observatory (SOHO) Mission Description and Flight Dynamics Analysis Reports, Revision 2, CSC/TM91/6030R2UDO, Computer Sciences Corporation, September 1993. 7. Dunham, D. W., et al., Transfer Trajectory Design for the SOHO LibrationPoint Mission, IAF Paper 92-0066, September 1992. 8. Farquhar, Robert W., Muhonen, Daniel P. and Richardson, David L., Mission Design for a Halo Orbiter of the Earth, Journal of Spacecrafi and Rockets, Vol. 14, NO. 3, March 1977, pp. 170-177. 9. Richardson, D. L., Halo-Orbit Formulation for the ISEE-3 Mission, Journal of Guidance and Control, Vol. 3, No. 6 , November-December 1980, pp. 543548. 10. Roberts, Craig E., Long Duration Lissajous Orbit Control for the ACE SunEarth L1 Libration Point Mission, AAS 01-204, presented at the AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, California, February 2001. 11. Jordan, P., Hametz, M., McGiffin, D., and Roberts, C. E., SOHO Maneuver Operations Handbook, draft version, Computer Sciences Corporation, November 1995. 12. Roberts, C. E., SOHO Halo Orbit Maintenance Study, Memorandum 5682204, Computer Sciences Corporation, April 17, 1996. 13. Yom, S., DiRosario, R., Gates, D., and Jose, R., Mission Analysis and Design Tool (Swingby) User’s Guide, Revision 3, CSC/SD-93/6071R3UDO, Computer Sciences Corporation, September 1995. 14. McGiffin, D., et al., Mission Analysis and Design Tool (Swingby) Mathematical Principles, Revision 1,draft version, CSC/TR-92/6091R1UDO1 Computer Sciences Corporation, September 1995. 15. Carrico, J., et al., An Interactive Tool for Design and Support of Lunar, Gravity Assisted, and Libration Point Trajectories, AIAA Paper 93-1126, presented at the AIAA/AHS/ASEE Aerospace Design Conference, Irvine, California, February 1993. 16. Franz, Heather, A Wind Trajectory Design Incorporating Multiple Transfers Between Libration Points, AIAA-2002-4525, presented at the AIAA/AAS Astrodynamics Specialist Conference, Monterey, California, August 2002. 17. Olive, Jean-Philippe, SOHO Recovery, NASA/CP-1999-209235, presented at the 1999 Flight Mechanics Symposium, NASA Goddard Space Flight Center, Greenbelt, Maryland, May 1999.

214

C.E. Roberts

Appendix A.

I

Posirive Roll about x: y ==> 2 ==> I positive Pitch about Y : 2: x =a1 Positive Yaw about

Fig. A-1. SOHO Spacecraft and Body Axes (Source: SOHO Project.)

The SOHO Mission L1 Halo Orbit Recovery

215

Appendix B. Possible Dispersion Trajectories Pertaining to the June 25, 1998 SOHO Anomaly

Fig. B-1. Disp. Traj. +1 to +5 cm/s

Fig. B-2. TDC Solar Orbit I

Fig. B-3. Disp. Traj. -1 to -6 cm/s

Fig. B-4. Disp. Traj. -7 cm/s

\ =u

Fig. B-5. Disp. Traj. -8 cm/s

Fig. B-6. Disp. Traj. -9 cm/s

E . &

Fig. B-7. Disp. Tkaj. -10 m/sec

TLtr. -6 C4.m

Fig. B-8. Disp. Traj. -11 m/sec

216

C.E.Roberts

Appendix C. The S O H 0 Anomaly of December 21,1998

Figure C-2. ESR cumulative delta-V relative to No-burn Orbit

The SOH0 Mission L1 Halo Orbit Recovery 217

Figure C-4. Overlay of Indefinite ESR Escape Trajectory with Guided Continuous Thrust Trajectory through Burnout and Fall-back to Earth

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

POSSIBLE ORBITS FOR THE FIRST RUSSIAN/BRAZILIAN SPACE MISSION A.A. SUKHANOV Space Recearch Institute (IKI) of the Russian Academy of Sciences 84/32 Profsoyuznaya St., Moscow 11 7997, Russia

A Brazilian satellite is to be designed for participation in the first Rus sian/Brazilian space mission. The mission goal is obtaining data in different areas of solar-wind and solar-terrestrial physics. Possible orbits of the Brazilian satellite are considered in the paper. All the orbits approach the Sun-Earth L1 point or both 151 and L2 points; some of the orbits include L1 or L2 halo orbit. Both direct and lunar gravity- assist trajectories are considered.

1. Introduction

A first Russian/Brazilian space mission is being started now. The main mission goals are space weather monitoring and prediction, study of the solar wind interaction with the Earth magnetosphere, possibly experiments in the geomagnetic tail etc. Brazil intends to participate in the mission with its own satellite of about 300-kg mass to be designed as a universal platform for this and future missions. The satellite is to perform experiments jointly with Russian and Ukrainian satellites. An important goal for the Brazilian satellite is also space engineering experiments for testing new technologies and satellite systems. The satellite orbit is not defined yet; ideally it should satisfy different, sometimes contradictory, demands of the Brazilian and Russian scientists. The demands imply the following satellite locations: 219

220

A.A. Sukhanov

In a vicinity of the L1 point for the solar-wind physics and magnetic storm prediction; - In a vicinity of the L2 point for the geomagnetic tail exploration; - Crossing the magnetosphere bound for investigation of the solar wind interaction with magnetosphere. -

One of the mission options having been considered was the following: The Russian satellite is inserted into a highly eccentric Earth orbit with apogee of about 400,000 km and apsid line lying in the ecliptic plane; - The Brazilian satellite is piggybacked with the Russian one and then launched from the Russian satellite orbit to the Sun-Earth L1 point and then possibly to the Lapoint. -

This option has been used for the preliminary mission analysis; several possible orbits for the Brazilian satellite more or less satisfying the demands listed above were found. There are both direct transfers and ones using lunar swingby among the orbits. The orbits can be conditionally divided into the following groups: (1) (2) (3) (4)

Quasi-periodic orbits approaching L1 point and returning to the Earth. L1 halo orbits. Quasi-periodic orbits approaching L1 and L2 points in turn. Flying by the L1 point and subsequent transfer to an L2 halo orbit.

2. Description of the orbits

The Brazilian satellite orbits shown and described below start from the parking orbit; the parking orbit has a 400,000-kmapogee and apsid line lying in the ecliptic plane. Launch from the parking orbit can be performed using a delta-V maneuver in the parking orbit perigee (direct transfer) or using lunar swingby. The orbits have substantial out-of-plane components, although only projections of the orbits onto the ecliptic plane are shown in Figures 1-12. Group 1: Quasi-periodic orbits approaching L1 point and returning to the Earth. -

Orbit 1: This orbit is shown in Fig. 1; orbital period is about five months. A lunar swingby is used for the insertion into the orbit.

Possible Orbits for the First Russian/Bmzilian Space Mission 221 -

Group 2: L1 halo orbits. - Orbit 2. A very short transfer to a halo orbit is presented in Fig. 2 (the transfer duration is 1.5 month); although the transfer has a high AV cost: total AV = 432 m/sec. This is a direct transfer without a lunar swingby. - Orbit 3. The orbit shown in Fig. 3 actually belongs both to the first and second groups: the spacecraft approaches L1 vicinity, then returns to the Earth and then is placed into L1 halo orbit. The transfer does not include any lunar swingby. - Orbit 4. The transfer to an L1 halo orbit shown in Fig. 4 is performed using lunar gravity assist maneuver. Group 3: Quai-periodic orbist approaching L1 and L2 points in turn. Orbit 5. As is seen in Fig. 5 this orbit approaches the L1 and L2 points in a rather long distance, returns closely to the Earth and goes round the lunar orbit. Insertion into Orbit 5 is direct, without a lunar gravity assist. - Orbit 6. This quasi-periodic orbit shown in Fig. 6 passes by the L1 and L2 points very closely and returns to the Earth in a short distance. The insertion is direct without a lunar swingby. - Orbit 7. This orbit presented in Fig. 7 is a modification of the Orbit 1 (see Fig. 1). A lunar swinby is used for insertion into Orbit 7. - Orbit 8. This orbit resembling an animal’s face (see Fig. 8) uses a lunar gravity assist.

-

Group 4: Flying by the L1 point and subsequent transfer to an L2 halo orbit. Orbit 9. This orbit shown in Fig. 9 is a continuation of the transfer given in Fig. 2 (Orbit 2) if the spacecraft is not inserted into the L1 halo orbit but continues its flight to the L2 point. The transfer does not use any lunar swingby. - Orbit 10. This orbit presented in Fig. 10 is a continuation of the Orbit 3 shown in Fig. 2 if the spacecraft is not inserted into the L1 halo orbit but continues its flight to the L2 point. This is a direct transfer without a lunar swingby. - Orbit 11. This orbit shown in Fig. 11 is a continuation of the Orbit 4 (see Fig. 4) if the spacecraft does not perform an insertion maneuver into the L1 halo orbit. Actually Orbit 11 can be related both to the Group 2 and 4 because the spacecraft stays in the L1 halo orbit for almost six months without any AV maneuver. A lunar gravity assist

-

222

A . A . Sukhonov

-

maneuver is needed for this orbit. Orbit 12. This orbit is shown in Fig. 12; a lunar swingby is used.

3. Summary of the transfer parameters

Some of the transfer parameters are given in Table 1 for all 12 orbits described above. The flight duration is given in the Table for the orbit parts shown in Figures 1-12. If the spacecraft is inserted into a halo orbit then the flight duration is given for the transfer till the insertion instance.

i

7

I"'

-7

"1

Fig. 1. Orbit 1.

i

Fig. 2. Orbit 2.

i

i

Fig. 3. Orbit 3.

Fig. 4. Orbit 4.

Y

Y

Fig. 5. Orbit 3.

Fig. 6. Orbit 6.

Possible Orbits for the First Russian/Bmxilian Space Mission 223 Table 1. Transfer parameters. Orbit no.

Lunar swingby

1 2 3 4 5 6 7

Yes No No Yes No No Yes Yes No No Yes Yes

a 9 10 11 12

AV of launch from the parking orbit (m/sec) 0 62 58

AV of insertion into a halo orbit (m/sec) 370

0

9 -

57 62 0 0 62 58 0 0

150

300 176 95 167

Total AV (m/sec)

Flight duration (Ym)

0 432 208 9 57 62 0 0 362 234 95 167

i

i

Fig. 7. Orbit 3.

Fig. 8. Orbit 8.

?

YI

Fig. 9. Orbit 3.

Fig. 10. Orbit 10.

Y

Y

Fig. 11. Orbit 11.

Fig. 12. Orbit 12.

2 0.13

1.1 0.3 1.5

1 2 1 0.5

1.5 2 0.7

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

NEW RESULTS ON COMPUTATION OF TRANSLUNAR HALO ORBITS OF THE REAL EARTH-MOON SYSTEM M.A. ANDREU Dept. Matembtica Aplicada i Anblisi, Univ. Barcelona, Gmn Via 585, 08007 Barcelona, Spain.

The computation of translunar Halo orbits of the real Earth-Moon system (REMS) has been an open problem for a long time, but now, it is possible t o compute Halo orbits of the REMS in a systematic way from quasi-periodic Halo orbits of the Quasi-bicircular Problem (QBCP). The QBCP is a model for the dynamics of a spacecraft in the Earth-Moon-Sun system. This model is a restricted four body problem where Earth, Moon and Sun are moving in a self-consistent motion close t o bicircular. The dynamics of a spacecraft under the influence of the other three bodies are described by a Hamiltonian system with three degrees of freedom and depending periodically on time. The similarity between the Halo orbits of REMS and the ones of the QBCP show that the QBCP is a good model of reference t o study the dynamics around L2 in the REMS.

1. Introduction

In the last years, an increase of the application of dynamical systems techniques in the design of trajectories in astrodynamics has been observed. The knowledge of the dynamics of a problem helps to get orbits which probably would not be obtained from a single numerical approach. Moreover, the dynamical systems approach provides more insight of the problem studied and it can offer families of solutions depending on one or more parameters, providing more flexibility to the mission design. The use of four-body problems as models of reference for the design of some space missions has also started in the last years. The Genesis mission, 225

226

M.A. Andreu

launched in August 2001, is an example of that, see Howell et al. l o and Koon et al. 14. In Genesis mission, the Bicircular Problem (BCP) was used for the computation of the nominal orbit. The BCP is a restricted four body problem which considers the Earth and the Moon moving in circular orbits around their barycenter, and the barycenter moving also in a circular orbit around the Sun, see Huang l l . In spite the BCP is not a self-consistent model, for the fact that the motion of primaries is not a solution of the Three Body Problem, it has been useful to study the dynamics around the triangular libration points of the Earth-Moon system, see Wiesel 17, G6mez et al. and Jorba 1 2 . In the beginning of 1970s, Farquhar proposed the use of translunar Halo orbits to establish a continuous communication link between the Earth and the far side of the Moon, see Farquhar 6 . However, as far as we know, it seems that nobody was able to produce Halo-like orbits for the REMS revolving around the Earth-Moon axis for a prolonged time interval. The main reason of this difficulty comes from the fact that the RTBP and the BCP are not good models of the dynamics around of the L2 point. The RTBP is not a good model because the existence of the Sun highly perturbs the orbit of the Moon and so the effect of the Sun must be included in the model in some way. On the other hand, the BCP is not a good model because of its lack of consistency, which is especially relevant because of the existence of the 1:2 resonance between the inner frequency of the Halo family and the synodical frequency of the Moon. The Quasi-bicircular problem (QBCP), which was introduced by the author in l , is a model similar to BCP but it is self-consistent. The BCP and the QBCP are Hamiltonian systems with three degrees of freedom which depend periodically on time. This means that the use of the QBCP, instead of the BCP, does not introduce any relevant increase of the complexity of the model. The QBCP is a model sufficiently simple to be studied and sufficiently complex to keep the main features of the REMS around L2. In this paper, we summarize a systematic method to compute Halo orbits of the REMS, see Andreu 3 . The main tools to obtain this orbits have been the QBCP as model of reference, a method for the computation of invariant tori and a relaxed Newton’s method to get orbits of the REMS from orbits of the QBCP.

Computation of Zhnslunar Halo Orbits of the Real Earth-Moon System 227

2. The Quasi-bicircular Problem

The QBCP is a restricted four body problem where three masses revolve in a quasi-bicircular motion, the fourth mass being small and not influencing the motion of the primaries, see Andreu In this work, the primaries are Earth, Moon and Sun, so the parameters of the model are: the Earth-Moon mass parameter p, the mean distance between the Earth-Moon barycenter and the Sun a,, and the synodic frequency of the Moon w,. These parameters have been set up as

'.

p = 0.0121505816, a, = 388.81114, w, = 0.925195985520347.

The mass of the Sun is taken as rn, = ( 1 - w , ) ~u: - 1 N 328900.54, from the third Kepler's law. To express the Hamiltonian of the fourth particle under the influence of the primaries, it is convenient to use a rotating pulsating system of reference, centered at the Earth-Moon barycenter and in such a way that the Earth and the Moon are located at fixed points ( p ,0,O)and ( p - 1,0,0), respectively (see figure 1). In this frame of reference, the Sun moves in a T-periodic orbit, where T = 2n/w,.

Y

Fig. 1. Geometry of the QBCP in a synodical system of reference.

228

M.A. Andreu

The Hamiltonian of the QBCP is 1 H = - a l ( p ~ + p ~ + P ~ ) + f f 2 ( p z z + p , Y + P , z ) + f f 3 ~-P,z)+ zY 2

where

+ y2 4-z2 , qim = (z- p + 1)2 + y2 + z2 , Qis= (z- + (y - + z2 ,

qie = (z- p)2

and f f k are T-periodic functions known, whose Fourier coefficients decay quickly, and therefore it is necessary to keep only a reduced number of Coefficients in order to achieve high accuracy. A set of coefficients of the f f k functions can be found in Andreu l . Around the L2 of the QBCP, there is a family of Halo orbits similar to the one of the RTBP, but in the QBCP, each Halo orbits lies on a 2 D torus and the family is not continuous, it is a Cantor set, see Jorba and Villanueva 13. Using the reduction of the Hamiltonian to the center manifold, a full description of the dynamics in a large neighbourhood around L2 can be obtained, see Andreu 2 . However, Halo orbits are beyond the agreement zone where the reduced Hamiltonian can be used. So, to study the family of Halo orbits, it is necessary to use other numerical methods for the computation of invariant tori. 3. The Halo family of the QBCP

To compute Halo orbits of the QBCP, the Jorba’s method for the computation of invariant tori has been used, see Castella and Jorba and Jorba 12. This method reduces the problem of obtaining a 2D invariant torus to the computation of the Fourier coefficients of a section of the torus. Figure 2 shows a picture to explain the method. We denote by g the time T map defined by the flow of the equations of the QBCP. The objective is to obtain an invariant curve y of the map g, with a given rotation number p, so y should satisfy the equation g(r(f3))= + 2TP) 9 vf3 E [O,2x1 . (2) To determine y with a given precision, the Fourier coefficients up to a certain order are computed by discretizing (2) and solving the corresponding nonlinear system of equations by means of Newton’s method.

Computation of Thnslunar Halo Orbits of the Real Earth-Moon System 229

To compute highly unstable tori, it is more convenient to use several sections 7 0 , . .. ,ym-l, instead of only one. Then, the Fourier coefficients of the m curves are computed from the invariance equations gi(ri(e))

.

=ri+l(e)

gm-l(”lm-l(@))=^lo(e

i = 0 , . .. , m - 2,

+ 2.rrP) ,

1

ve E [ 0 , 2 ~ 1 ,

where gj is the map defined by the flow of the QBCP from time t = iT/m to t = (i l ) T / m ,for i = 0,. ..,m - 1.

+

....... _.-..

....! i

y i !:......................

.’

.......... ....... ........... ......

.......

! ._

t=O

t=T

Fig. 2. Jorba’s method for the computation of an invariant torus.

In our case, Halo 2D tori of the QBCP have been computed using four sections, yi,i = 0,1,2,3. The rotation number p , which is defined modulus one, has been taken as p=P W - 2 , ws where wp is the inner frequency of the torus and w, is the synodic frequency. In this way, the rotation number is close to zero because all the orbits considered in this work have wp N 2w,.

4. Translunar Halo orbits of the REMS

The motivation for the introduction of the QBCP was the interest in the study of the orbits around LZ of the REMS and in particular, the desire

230 M.A. Andreu

of obtaining translunar Halo orbits of the REMS for a long time interval (for example, 20 years). From the early 1970s, some people had work on this topic but, as far as we know, nobody was able to produce such kind of orbits. In 1997, the first Halo-like orbits of the REMS for long time span were presented in the NATO AS1 which was held in Maratea (Italy), see Andreu 4. Our goal is to get a Halo orbit of a differential equation as

2 = F(t,x), (3) where x E R6 (position and velocity), and F is the vector field of the REMS, which is modeled accurately by the JPL ephemeris. One of the difficulties to get Halo orbits of the REMS for a long time interval is that they are highly unstable, but this is easily overcome using a parallel shooting method, see Stoer and Bulirsch 16. In this way, it is necessary to compute a set of points to,^^), (tl,xl),. .. , (tn,xn) , belonging to the same orbit, so they must satisfy the matching equations 4(tk,tk+l,xk) =xk+l , k=O,**.,n-1. (4) where +(t,t',x) is the flow of the equation (3) going from time t to t' with initial condition x.

The main difficulty to obtain Halo orbits of the REMS for a long time span is to have a good initial appoximation of the orbit desired. If the initial orbit is not close to the orbit that we want to get, the iterative process of refinement probably would not be convergent. Halo orbits of the QBCP has been used successfully to obtain Halo orbits of the REMS. The linearization of (4)directs to

A1 -I A2

-I A3

-I

... ... An

-

")

,

(5)

bn

where Ak are 6 x 6 matrices coming from the integration of the variational equations of (3); the independent term is formed by the vectors bk = Xk $J(tk-l,tk,Zk-l) and the quantities Axk are the corrections to the states Xk.

Computation of 'hnslunar Halo Orbits of the Real Earth-Moon System 231

The linear system (5) is infradetermined, having 6n equations and 6n+6 unknowns. Then, it is possible to add some minimising condition in order to get an orbit close to the initial orbit. For instance, the minimising function can be taken as P j=O

where only the states multiples of a certain m are involved. To find a solution of the matching equations, a relaxed Newton's method has been used. This means that the sequence of states {zp)}jis computed as

where Azp) is the solution of (5) minimising (6) and constant, T E (0,l).

T

is a relaxation

The previous method has been used to compute more than thirty Halo orbits of the REMS for a time interval longer than 41 years. Halo orbits of the QBCP have been taken as initial guess for the iterative refinement process. Due to the instability of the orbits in the vicinity of L2, it has been necessary to keep about two points per day in the parallel shooting method. The minimising condition (6) has been used with m = 16 and the relaxation constant has been taken usually as T = 0.2. In this way, the convergence is very slow, but the sequence orbits tends to an orbit of the REMS with a normal component rather small, except for the Halo orbits with smallest z-amplitude. Figures 3a-d show several projections of a Halo orbit of the QBCP. Figures 3e-h show a Halo orbit of the REMS computed from the previous one using the methodology explained above. We recall that Halo orbits have an inner period of about fifteen days, so figures 3d,h show about one thousand oscillations of z . In both figures, the profile formed by the maximum z of each revolution around L2 shows a long period wave of about 4900 days. From now on, time is measured in days starting from January 1st 2000. In principle, a Halo orbit of the REMS can have all the frequencies of the Solar System, but most of them with an amplitude certainly very small. Apart from its main inner frequency, a Halo orbit can have also some

232 M.A. Andreu

0.1 -

0.1 z

L

0-

0-

4.1 -

-0.1 -

.

1

.

-0.2

.

.

.

.

.

0.1

0

4.1

.

I

I

0.2

4.6

I

-0.2

-0.4

0.6,

.

I

0.4

0.6

(4

.

,

0.2

0 Y

X

,

,

,

,

0.2 I

I

,

,

,

.

.

,

,

.

.

I

,

,

.

.

0.4 0.2

Y

01

. z

0 -

0

-0.2 -0.4

.

-0.6 -

-0 1

-0.2

0

-0.1

0.1

I

0.2

.

8000

4004

0

12OW

I 16000

t (days)

X

0.2

,

(f

01 -

0.1 *

z

z

0 -

0~

-0.1 -

-0 1

.

I

.

.

.

.

0

-0.1

-0.2

.

.

.

0.1

-

I

I

0.2

-0.6

4.4

-0.2

x

0

0.2

6

0.4

Y

(g> 0.6,

I

,

'

'

,

,

'

'

I

.

'

'

I

,

'

'

I

0.4 -

0.2 Y

0 -0.2 -0.4 -

-0.6l

-0.2

-0.1

0 X

0.1

0.2

I 0

.

. 4000

.

.

8000 1 (days)

.

. 12000

. 16000

Fig. 3. (a)-(d) Halo orbit of the QBCP labelled as HOQ1. (e)-(h) Halo orbit of the REMS labelled as HORl and computed from the previous one.

Computation of Thnslunar Halo Orbits of the Real Earth-Moon System 233

component in the normal frequency to the Halo family. In such case, to be precise, this orbit should be called quasi-halo, see Gdmez et al. We want to remark that there is no reason at all to assume that the orbit obtained with the Newton’s method has only one inner frequency. Figure 4 show the (time,z ) projection of a quasi-halo orbit. This orbit was computed from a Halo orbit of the QBCP, using two initial iterates of the relaxed Newton’s method with T = 0.2 and after these, 12 iterates more without relaxation. In Andreu l , some quasi-halo orbits were shown, some of them exhibiting a large normal component.

0.1

z 0

-0.1

0

4000

8000

12000

16000

t (days)

Fig. 4.

Graph of z(t) of a quasi-halo orbit, labelled as HOR3.

In order to study the main frequencies involved in the Halo family, Fourier analysis of the computed Halo orbits was performed by means of Mondelo’s software, which uses a slight modification of the Laskar’s method to detect relative maxima of the Fourier Transform, see Laskar et al. l5 and Mondelo g. After the Fourier analysis, it is convenient to express every frequency as a linear combination of some frequencies having physical meaning. The frequency analysis shows that, apart from the inner frequency of the orbit, the second frequency that is important is 2w,. The importance of this frequency for a wide range of Halo orbits comes from the proximity of the 1:2 resonance. The following frequency in importance is the anomalistic frequency, which is w, = 0.99154522 using the units of the QBCP and its corresponding period is 27.55 days.

234

M.A. Andreu

Coming back again to figures 3d,h and 4, the frequency analysis also shows that the long period wave formed by the relative maxima and minima of I has the period of 2 r / ( w p - 2w,). This is about 4900 days for the orbits HOQl and HORl, and about 4500 days for the orbit HOR3. In the profile of the orbit of figure 4, another superimposed wave with a shorter period is seen. It has been checked that this wave is directly related to the normal component of the orbit, having the period of 2 r / w n . In this orbit, the normal frequency has been estimated as W n = 0.0334138, so the corresponding period is about 820 days. In order to display the Halo family of the QBCP, it is necessary to take some point or quantity as a representative of each torus. We recall that 70is the section of a torus for t = 0. Then, the intersection of 70 with the plane y = 0 is considered. In the case of Halo orbits, this intersection usually has two points. We define the representing point of a torus as the point with maximum z coordinate in the intersection of 70with y = 0. In the case of the REMS, we can proceed in a similar way with the purpose of comparing the orbits of the REMS with their corresponding orbits of the QBCP. Let ro be the set of points of a Halo orbit of the REMS for t = to k T , k E Z, where to is the epoch of the first full moon of the year 2000. In this case, ro does not fill a closed curve as before. Figures 5a,b show (2,y) and (y, z ) projections of 70 of the orbit HOQl and the ro of the orbit HOR1. In ro, several small loops are seen. The frequency analysis has shown that these loops are produced by some frequencies related to the anomalistic frequency.

+

In principle, all the frequencies of the Solar System might appear in To. Nevertheless, I'o seems to be close to a 2D torus, having the amplitude of one of the frequencies much larger than the other one. For this reason, the 2D torus can be approximated by its core, which is a closed curve, and from this curve, a representing point can be taken in the same way as in the QBCP. This approximation allow to establish a preliminary comparison between the Halo family of the QBCP and the computed Halo orbits of the REMS, which can be found in figures 6a,b. In spite of the fact that the Halo family of the QBCP is a Cantor set, we want to point out that it is possible to follow the family of tori as if it were a continuous family, except when big gaps appear. Some of these big gaps in the family are seen for p < -0.0065, which is called the R zone. We

Computation of lhnslunar Halo Orbits of the Real Earth-Moon System 235

0.6 -

Y

0.4

.

0.2

.

0.2

,

(b) .....

....................

0.1 -

z

0 -

0 -0.2 -0.4 -

-0.1 -

-0.6'

I

"

-0.2

"

"

0

-0.1

"

0.1

I

0.2

.... ......

... ..--

HORl

~

H001

-0.6

-0.4

-0.2

X

0 Y

0.2

0.4

0.6

+

Fig. 5. Section for time t = to kT,k E Z,of the orbits HOQl and HORl. In the case of HOQl, to = 0. For HOR1, to corresponds to the first full moon of the year 2000.

0.008

QBCP REMS 0 0.004 -

aa

0 2

P

4.004 4.008 0'

-0.012

.

'

'

4.008

"

"

0

4.004 P

'

"

0.004

'

0.008

-

4.012 0.13

0.135

0.14

0.145

0.15

0.155

X

Fig. 6. Comparison between Halo families of the QBCP and the REMS. (a) ( p , z ) projection. (b) ( 2 ,p ) projection.

conjecture that the big gaps of the R zone are associated with resonances involving the normal frequency, but this fact has not been checked yet. The nice agreement observed between both families of Halo orbits is a clear evidence of the convenience of using the QBCP as model of reference for the study of the dynamics around the L:!of the REMS. We want to stress that the dominant feature of the Halo family of the REMS comes from the 1:2 resonance, as in the QBCP. It is also true, that some discrepancies have been found in the R zone. There, some of the computed orbits of the REMS exhibit an important normal component. It seems that the Halo family of the REMS also has some gaps as the one of the QBCP, but these gaps are shifted about 0.001 units in the rotation number, from one model to the other.

236

M.A. Andreu

Acknowledgements

I would like to express my gratitude to C. Simb, G. Gbmez, J. Masdemont, A. Jorba, E. Castellii and J.M. Mondelo for their fruitful contribution, comments and suggestions in differents parts of this work. The author also wants to acknowledge the support of the Catalan grants CIRIT 1998SGR00042 and 2000SGR-00027 and the Spanish grant DGICYT BFM2000-805.

References 1. Andreu, M.A.: 1998, The Quasi-bicircular Problem, Thesis, Dept. Matemitica Aplicada i Andisi, Universitat de Barcelona. 2. Andreu, M.A.: 2002, Dynamics in the center manifold around La in the Quasi-bicircular Problem, Celestial Mechanics and Dynamical Astronomy 84(2), 105-133. 3. Andreu, M.A. : 2002, Preliminary study on the translunar Halo orbits of the real Earth-Moon system, Celestial Mechanics and Dynamical Astronomy, (in press). 4. Andreu, M.A. and Sim6, C.: 1999, 'Translunar Halo orbits in the Quasibicircular Problem', B.A. Steves and A.E. Roy editors, The Dynamics of Small Bodies in the Solar System, NATO AS1 (1997, Maratea, Italy), pp. 309-3 14. 5. Castell&.,E. and Jorba, A.: 2000, O n the vertical families of two-dimensional tori near the triangular points of the Bicircular problem, Celestial Mech. 76(1), 35-54. 6. Farquhar, R.W.: 1970, The Control and Use of Libration-Point Satellites, NASA TR R346. 7. G6mez, G., Llibre, J., Martinez, R. and Sim6, C.: 1987, Study on orbits near the triangular libration points in the perturbed restricted three body problem, ESOC contract 6139/84/D/JS(SC), Final Report. 8. G6mez, G., Masdemont, J. and Sim6, C.: 1998, QvasihaZo orbits associated with libration points, J. of the Astronautical Sci., 46. 9. Gbmez, G., Mondelo, J.M. and Sim6, C.: 2001, Refined Fourier analysis: procedures, error estimates and applications, (preprint). 10. Howell, K.C., Barden, B.T., Wilson, R.S.,Lo, M.W.: 1997, Rajectory Design Using a Dynamical Systems Approach with Application to Genesis, AAS/AIAA Astrodymics Specialist Conference, AAS Paper 97-709. 11. Huang, S.S.: 1960, Very restricted four body problem, NASA Technical Note D-501. 12. Jorba A.: 1999, O n practical stability regions for the motion of a small particle close t o the equilateral points of the real Earth-Moon system, (preprint). 13. Jorba, A., Villanueva, J.: 1997, O n the persistence of lower dimensional in-

Computation of lhnslunar Halo Orbits of the Real Earth-Moon System 237

14. 15.

16. 17.

variant tori under quasi-periodic perturbations, Journal of Nonlinear Science, 7, 427-473. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: 1999, The Genesis Tbajectory and Heteroclinic Connections, AASjAIAA Astrodynamics Specialist Conference, Girdwood, Alaska, AAS99-451. L a s h , J., Froeschlb, C., Celletti, A.: 1992, The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping, Physica D, 56, 253-269. Stoer, J., Bulirsch, R.: 1980, Introduction to numerical analysis, SpringerVerlag. Wiesel, W.: 1984, The restricted Earth-Sun-Moon problem I: Dynamics and libration point orbits, Dept. Aeron. and Astron., Air Force Institute of Technology Wright-Patterson AFB, Ohio.

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Libration Point Orbits and Applications G. G6mez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

IMPULSIVE TRANSFERS TO/FROM THE LAGRANGIAN POINTS IN THE EARTH-SUN SYSTEM A.F. BERTACHINI DE ALMEIDA PRADO Instituto Nacional de Pesquisas Espaciais - INPE - Brazil S6o Jose‘ dos Campos - SP - 12227-010 - Bmzil

This paper is concerned with trajectories to transfer a spacecraft between the Lagrangian points of the Sun-Earth system and the primaries of that system. The Lagrangian points have important applications in astronautics, since they are equilibrium points of the equation of motion and very good candidates to locate a satellite or a space station. The planar circular restricted three-body problem in two dimensions is used as the model for the Sun-Earth system, and Lemaitre regularization is used to avoid singularities during the numerical integration required t o solve the Lambert’s three-body problem. The results show families of transfer orbits, parameterized by the transfer time.

1. Introduction The well-known Lagrangian points that appear in the planar restricted three-body problem (Szebehely 6 , are very important for astronautical applications. They are five points of equilibrium in the equations of motion, what means that a particle located at one of those points with zero velocity will remain there indefinitely. Their locations are shown in Figure 1. The collinear points (L1, L2 and L3) are always unstable and the triangular points (L4 and L5) are stable in the present case studied (Sun-Earth system). They are all very good points to locate a space-station, since they require a small amount of AV (and fuel) for station-keeping. The triangular points are specially good for this purpose, since they are stable equilibrium 239

240

A.F. Bertachini de Almeida Prado

points. In this paper, the planar restricted three-body problem is regularized (using Lemaitre regularization) and combined with numerical integration and gradient methods to solve the two point boundary value problem (the Lambert’s three-body problem). This combination is applied to the search of families of transfer orbits between the Lagrangian points and the primaries, in the Sun-Earth system, with the minimum possible energy. This paper is a continuation of two previous papers that studied transfers in the Earth-Moon system: Broucke l , that studied transfer orbits between the Lagrangian points and the Moon and Prado 4 , that studied transfer orbits between the Lagrangian points and the Earth.

2. The planar circular restricted three-body problem

The model used in all phases of this paper is the well-known planar circular restricted three-body problem. This model assumes that two main bodies ( M I and M2) are orbiting their common center of mass in circular Keplerian orbits and a third body (A&), with negligible mass, is orbiting these two primaries. The motion of M3 is supposed to stay in the plane of the motion of M I and M2 and it is affected by both primaries, but it does not affect their motion (Szebehely 6). The canonical system of units is used, and it implies that: i) The unit of distance (1) is the distance between M I and M2; ii) The angular velocity ( w ) of the motion of M I and M2 is assumed to be one; iii) The mass of the smaller primary ( M 2 ) is given by p = m2/(m1 m2), (where ml and m2 are the real masses of M I and M2, respectively) and the mass of M2 is (1 - p ) , so the total mass of the system is one; iv) The unit of time is defined such that the period of the motion of the primaries is 27r;v) The gravitational constant is one. Table 1 shows the values for these parameters for the Sun(Ml)-Earth(Mz) system, that is the case considered in this paper. Then, the equations of motion are:

+

Table 1. Canonical system of units. Unit of distance Unit of time Unit of velocity

149,596,000km 58.13 days 29.8 km/s

Impulsive Zhnsfers to/’m

the Lagmngian Points in the Earth-Sun System

241

where R is the pseudo-potential given by:

R = -(x2 1 2

+ y2) + l-p+!i. 9-2 9-1

This system of equations has no analytical solutions, and numerical integration is required to solve the problem. This system has an invariant called Jacobi integral. There are many ways to define the Jacobi integral (see Szebehely pg. 449). In this paper the definition shown in Broucke is used. Under this version, the Jacobi integral is given by: 1 E = -(s2 + y2) - R(x,y) = Const. 2 The three-body problem has also two important properties:

(1) The existence of five equilibrium points (called the Lagrangian points L1, L2, L3, L4 and Lg, as seen in Figure l),that are the points where dR/ax = OR/ay = 0. It corresponds to say that a particle in this point with zero initial velocity remains in its position indefinitely. The first three (L1, L2, L3) are called collinear points (because they are in the “x” axis) and they are unstable for any value of p. The last two (L4 and L5) are called triangular points (because they form an equilateral triangle with MI and M2) and they are stable for p < 0.03852, what means that they are stable in the present case (Sun-Earth), since p = 0.0000030359. Their locations and potential energy are shown in Table 2. Table 2. Position and potential energy of the five Lagrangian points.

L1 L2 L3

L4 Lg

X

Y

-0.9899909 -1.0100702 1.0000013 -0.4999969 -0.4999969

0 0 0 $0.8660254 -0.8660254

Eo -1.5004485 -1.5001481 -1.5000015 -1.4999984 -1.4999984

(2) The curves of zero velocity, that are curves given by the expression:

Eo = - R ( X , Y ) ,

(3)

242

A . F . Bertachini de Almeida Pmdo

f’

Fig. 1. Geometry of the problem of three bodies.

that is the equivalent of equation (2) in the special case where x = = 0. They are important, because they determine forbidden and allowed regions of motion for M3 based in its initial conditions. More details about these and other properties of the three-body problem can be found in Szebehely and Broucke l.

3. Lemaitre regularization The equations of motion given by equation (1)are not suitable for numerical integration in trajectories passing near one of the primaries. The reason is that the positions of both primaries are singularities in the potential V (since T I or T Z goes to zero, or near zero) and the accuracy of the numerical integration is affected every time this situation occurs. The solution for this problem is the use of regularization, that consists in a substitution of the variables for position (x-y)and time ( t ) by another set of variables (WI,W Z ,T), such that the singularities are eliminated in these new variables. Several transformations with this goal are available in the literature (Szebehely 6 , chapter 3), like Thiele-Burrau, Lemaitre and BirkhofT. They are called “global regularization”, to emphasize that both singularities are eliminated at the same time. The case where only one singularity is eliminated at a time is called “local regularization”. For the present research the Lemaitre’s regularization is used. To perform the transformation it is necessary first to define a new complex variable q = q1 +iqz (i is the imaginary

Impulsive lhnsfers to/fmm the Lagrangion Points in the Earth-Sun System 243

unit), with q1 and

92

given by: q1

=x+1/2-p,

92

=y.

(4)

Now, in terms of q, the transformation involved in Lemaitre regularization is given by:

( ):

g=f(w) = - w2+4

,

(5)

for the old variables for position (x-y) and:

where f'(w) denotes d f l d w , for the time. In the new variables the equation of motion of the system is:

w"

+ 24 f'(w)I2w'

= grad,R*,

(7)

where w = w1 + iw2 is the new complex variable for positions, w' and w" denotes first and second derivatives of w with respect to the regularized time I-, grad,R* represents dR*/dwl+idR*/dw2 and R* is the transformed pseudo-potential given by:

where C = p ( 1 - p ) - 2 E . Equations (7) in complex variables can be separated in two second order equations in the real variables w1 and 202 and organized in the standard first order form, that is more suitable for numerical integration. The final form, after defining the regularized velocity components w3 and w4 as wi = w3 and wi = w4 , is:

Another necessary set of equations is the one to map velocity components from one set of variables to another. They are:

244

A.F. Bertachini de Almeida Pmdo

4. The mirror image theorem

The mirror image theorem (Miele 3, is an important and helpful property of the planar circular restricted three-body problem. It says that: “In the rotating coordinate system, for each trajectory defined by (z(t),y ( t ) , 2 ( t ) ,$ ( t ) that is found, there is a symmetric (in relation to the “z” axis) trajectory defined by (z(-t), -y(-t), - 2 ( - t ) , - ~ ( - t ) ” .The proof is omitted in this paper, but the reader can verify its veracity by substituting those two solutions in the equations of motion. With this property, there is no need to calculate the returning trajectories from the primaries to the Lagrangian points. For the collinear points, the symmetric of the trajectory that goes from the Lagrangian point to the Earth is the trajectory that goes from the Earth to the Lagrangian point. For the triangular points the situation is a little more complex, since these points are not in the “z” axis. The symmetric of the trajectory that goes from Lq to the Earth is the trajectory that goes from the Earth to L5 and the symmetric of the trajectory that goes from L5 to the Earth is the trajectory that goes from the Earth to L4.

5. The Lambert’s three-body problem The problem that is considered in the present paper is the problem of finding trajectories to travel between the Lagrangian points and the primaries. Since the rotating coordinate system is used and all the primaries and the Lagrangian points are in fixed known positions, this problem can be formulated as: “Find an orbit (in the three-body problem context) that makes a spacecraft to leave a given point A and goes to another given point B”. That is the famous TPBVP (two point boundary value problem). There are many orbits that satisfy this requirement, and the way used in this paper to find families of solutions is to specify a time of flight for the transfer. Then, the problem becomes the Lambert’s three-body problem, that can be formulated as: “Find an orbit (in the three-body problem context) that makes a spacecraft to leave a given point A and goes to another given point B, arriving there after a specified time of flight”. Then, by varying the specified time of flight it is possible to find a whole family of transfer orbits and study them in terms of the AV required, energy, initial flight path angle, etc.

Impulsive Bansfers to/from the Lagmngian Points in the Earth-Sun System 245

6. The solution of the TPBVP The restricted three-body problem is a problem with no analytical solutions, so numerical integration is the only possible approach to solve it. To solve the TPBVP in the regularized variables the following steps are used: (1) Guess a initial velocity v’i, so together with the initial prescribed position the complete initial state is known; (2) Guess a final regularized time ~f and integrate the regularized equations of motion from TO = 0 until ~ j ; (3) Check the final position Ff obtained from the numerical integration with the prescribed final position and the final real time with the specified time of flight. If there is an agreement (difference less than a specified error allowed) the solution is found and the process can stop here. If there is no agreement, an increment in the initial guessed velocity Gi and in the guessed final regularized time is made and the process goes back to step i).

<

The method used to find the increment in the guessed variables is the standard gradient method, as described in Press et. al. The routines available in this reference are also used in this research with minor modificat ions.

‘.

7. Numerical results

The Lambert’s three-body problem between the primaries and the Lagrangian points is solved for several values of the time of flight. Since the regularized system is used to solve this problem, there is no need to specify the final position of M3 as lying in an primary’s parking orbit (to avoid the singularity). Then, to make a comparison with previous papers (Broucke and Prado 4, the center of the primary is used as the final position for M3. The results are organized in plots of the energy (as given by equation (2)) and the initial flight path angle in the rotating frame against the time of flight. The definition of the angle is such that the zero is in the “x” axis, (pointing to the positive direction) and it increases in the counter-clockwise sense. Plots of the trajectory in the rotating system are also included. This problem, as well as the Lambert’s original version, has two solutions for a given transfer time: one in the counter-clock-wise direction and one

A.F. Bertachini de Almeida Pmdo

246

in the clock-wise direction in the inertial frame. In this paper, emphasis is given in finding the families with the smallest possible energy (and velocity at the Lagrangian points, as a consequence of equation (2)), although many other families do exist.

7.1. Trajectories f m m L1 In the nomenclature used in this paper, L1 is the collinear Lagrangian point that exists between the Sun and the Earth. It is located about 1,496,867km from the Earth. Figure 2 shows the results for the least expensive family of transfers to the Sun that was found in this research and Figure 3 shows the transfers to the Earth. The local minimum for a transfer to the Sun occurs for a time of flight close to 64 days, requires an energy E = -1.0101 and has a initial flight path angle of 90 deg. In terms of velocity increment (AV) it means an impulse of 29.5 km/s (0.9903 canonical units) applied at L1. For a transfer to the Earth, the minimum occurs for a time of flight close to 35 days, requires an energy E = -1.5004 and has a initial flight path angle of 248 deg. In terms of velocity increment (AV) it means an impulse of 0.298km/s (0.01canonical units) applied at L I . 05

03

040

Y 02

060

01

40 = 20 01 10

08

06

x

04

02

00

120

0

20 40 €4 80 100 120 140 la Transfer Time [days)

0

20 40 60 80 100 120 140 Transler Time [days]

0

Fig. 2. Transfers from L1 to the Sun.

7.2. Trajectories fmm L2 In the nomenclature used in this paper, Lz is the collinear Lagrangian point that exists behind the Earth. It is located about 1,506,915km from the Earth. Figure 4 shows the results for the least expensive family of transfers to the Sun that was found in this research and Figure 5 shows the transfers

Impulsive h n s f e r s to/from the Lagmngian Points in the Earth-Sun System 247 0 001

, I $

-1 4975

2 .1 4980 ;-1 4985 .1 4990 220

-1.5000 -1 5005 -1 5010

-0 064-1 000 .0998 .O 996 -0934 -0992 ,0390

x

Fig. 3.

= 200

o

180 10 20 30 40 50 60 70 a0 go100 Transfer Time [days)

o

10 20 30 40 50 60 70 a0 9 o i o o Transfer Time [daysl

Transfers from L1 to the Earth.

to the Earth. The local minimum for a transfer to the Sun occurs for a time of flight close to 64 days, requires an energy E = -0.9896 and has a initial flight path angle of 88 deg. In terms of velocity increment (AV) it means an impulse of 30.1 km/s (1.0104 canonical units) applied at L z . For a transfer to the Earth, the minimum occurs for a time of flight close to 35 days, requires an energy E = -1.5004 and has a initial flight path angle of 67 deg. In terms of velocity increment (AV) it means an impulse of m/s (canonical units) applied at Lz. 0

°

1

7

4

0

1

P

9- 100-

I80$n

60-

.

40-

-

= 20-

Sun

-10.1 2 .1 0

- 0

-0.8 -0.6 a.4 4 2 X

o

Tiansler Time [days)

Fig. 4.

20

40 03 a0 100 120 Transfer Time [days!

10

Transfers from Lp to the Sun.

7.3. Trajectories from L3 In the nomenclature used in this paper, L3 is the collinear Lagrangian point that exists on the opposite side of the Sun (when compared to the position of the Earth). It is located about 149,595,740 km from the Sun, what means that it is almost at the same distance that the Earth is, but in the opposite direction. Figure 6 shows the results for the least expensive

248

A.F. Bertachini de Almeida Pmdo

0 0025

0 0020 Y

00015

OW05

o m

1 5w5

lOl2

1004

looB

1030

0

X

20

40

60

80 100 1

Ttamfer Tnne [&ys)

Fig. 5. Transfers from Lz to the Earth.

family of transfers to the Sun that was found in this research and Figure 7 shows the transfers to the Earth. The local minimum for a transfer to the Sun occurs for a time of flight close to 70 days, requires an energy E = -0.9965 and has a initial flight path angle of 275 deg. In terms of velocity increment (AV) it means an impulse of 29.9 km/s (1.0035 canonical units) applied at Ls. For a transfer to the Earth, the minimum occurs for a time of flight close to 169 days, requires an energy E = -1.4390 and has a initial flight path angle of 253 deg. In terms of velocity increment (AV) it means an impulse of 10.4 km/s (0.3493 canonical units) applied at Ls. 01

Y4 4a132

04

-

3

Wlon

3

080 075

? 250 f 240

05 00

m

1GU

06

02 04 06

08 10

0

12

X

o

$260

i085

20 40 60 80 1M120140160

0 20 40 60 Bo lMl2J140160

T r m f e i Tm ldaysl

Trmder Tme Id&

Fig. 6. Transfers from L3 to the Sun.

7.4. Trajectories from

L4

L4 is one of the triangular Lagrangian points. Its location is the third vertice of the equilateral triangle formed with the Sun and the Earth, in the semiplane of positive y. In the present case under study (Sun-Earth system) it is a stable equilibrium point. It is a very important point, because it is

Impulsive Thnsfers to/fmm the Lagmngian Points in the Earth-Sun System 249

Fig. 7. Transfers from L3 to the Earth.

an excellent location for a space station. Its stability property makes the fuel required for station-keeping almost zero. Figure 8 shows the results for the least expensive family of transfers to the Sun that was found in this research and Figure 9 shows the transfers to the Earth. The local minimum for a transfer to the Sun occurs for a time of flight close to 64 days, requires an energy E = -0.9999 and has a initial flight path angle of 29 deg. In terms of velocity increment (AV) it means an impulse of 29.8 km/s (1.0001 canonical units) applied at Lq. For a transfer to the Earth, the minimum occurs for a time of flight close to 174 days, requires an energy E = -1.4790 and has a initial flight path angle of 313 deg. In terms of velocity increment (AV) it means an impulse of 6.1 km/s (0.2049 canonical units) applied at L4.

YO5

04

03

02 01

00 05 0 4

03 0 2 0 1 00 0 1

0

X

Fig. 8. Transfers from L4 to the Sun.

a

A . F . Bertachini de Almeida P m d o

250

Y 04

290

..

02

p -1.2

00

-1.4

270

-1.6

o

-1.0 -0.3 -0.8 -07 .0.6 .0.5 -0.4

30

X

60 30 1 2 ~150 180 Tiansfer Time [days]

40

60

m 100 120 140 160

1 0

Transfer Time (days]

Fig. 9. Transfers from L4 to the Earth.

7.5. Trajectories j h m

L5

L5 is the other triangular Lagrangian point. Its location is the point symmetric to L4 (in relation to the “x” axis),the third vertice of the equilateral triangle formed with the Earth and the Moon, in the semi-plane of negative y. It is also stable and a very important point, for the same reasons that L4 is an important point. Figure 10 shows the results for the least expensive family of transfers to the Sun that was found in this research and Figure 11 shows the transfers to the Earth. The local minimum for a transfer to the Sun occurs for a time of flight close to 64 days, requires an energy E = -0.9999 and has a initial flight path angle of 149 deg. In terms of velocity increment (AV) it means an impulse of 29.8 km/s (1.0001 canonical units) applied at L5.For a transfer to the Earth, the minimum occurs for a time of flight close to 291 days, requires an energy E = -1.4910 and has a initial flight path angle of 252.40 deg. In terms of velocity increment (AV) it means an impulse of 4.0 km/s (0.1342 canonical units) applied at L5. 02

$ 140

Tmi?

Y 04

06

08

05

$130

2 07

= 110 90

10

11 06

-:la,

09

L5 08

5 120

04

02

x

00

02

o

10

m

m

30 40 50 60 80 90 Transfm Tmc [days)

ion

80

o

Fig. 10. Transfers from L5 to the Sun.

10

m

m

30 40 50 w 80 90 Transler T!me [da*Sl

n

Impulsive lhnsfers to/from the Logmngian Points in the Earth-Sun System 251 02

02 Y 04

: *.

06

B

08

w

* * **.*

10 12

10

08

x

-96

04

-1 2 14 16

2 170 150

0

50 100 150 200 250 300 Tiansfer Time [days]

Fig. 11. Transfers from

L5

0

50 100 150 200 250 300 TiansferTime [day$]

to the Earth.

7.6. Comparison between the trajectories The results show that it is much more expensive (in terms of energy and AV) to go to the Sun from the Lagrangian points than to go to the Earth. For all points considered here there is a family of small AV transfers to the Earth and those families were not found in the case of transfers to the Sun. The physical explanation is that the Sun, the Earth and the Lagrangian points are fixed points in the rotating system, but they are not fixed in the Sidereal (inertial) system. In the sidereal system, all bodies and points are rotating with unit angular velocity and it means that their linear velocity is equal to their distance from the center of the system. The Earth and the Lagrangian points have distances from the center of the system in the same order of magnitude (the approximate values are: Earth = 0.9999969, L1 = 0.9899909, L2 = 1.0100702, L3 = 1.0000013, Lq = Ls = 0.9999984), so their linear velocities are similar and a small AV is enough to cause an approximation and the rendezvous desired. However, the Sun has a very small distance from the center (0.0000030359), and in consequence a very small linear velocity. As seen from the Sun, in the inertial coordinate system, to transfer a satellite from one of the Lagrangian points (or the Earth) to the Sun is equivalent to transfer a satellite from a high (about 1,500,000 km) circular orbit with a transverse velocity near 29.8 km/s (the circular velocity for this altitude) to the Sun. The best way to do it is to apply a AV of about 29.8 km/s in the opposite direction of the motion to reduce the velocity to a value in the order of magnitude of the velocity required by a Hohmann-type transfer (not the same value, because this is not a two-body problem, since the Moon is still acting in the system). The order of magnitude of this velocity is near zero, considering a two body Hohmann type transfer from the Lagrangian

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A.F. Bertachini de Almeida Pmdo

point to the center of the Earth. This means that a AV in the order of magnitude of 29.8 km/s is always required for those transfers, and there is no hope to reduce it by a large amount.

8. Conclusions In this paper, the Lemaitre regularization is applied to the planar restricted three-body problem to solve the Lambert’s three-body problem (TPBVP) and it gives families of transfer orbits between the primaries and all the five Lagrangian points that exist in the Sun-Earth system. Those trajectories are shown and a comparison is realized. This comparison showed that transfers with small fuel consumption, that exist in transfers to/from the Earth, does not exist to/from the Sun. The results shown here can help mission designers to find useful trajectories for real spacecrafts.

Acknowledgments The author is grateful to CNPq (National Council for Scientific and Technological Development) - Brazil for the contract 300221/95-9 and to FAPESP (Foundation to Support Research in S k Paul0 State) for the contract 2000/14769-4.

References 1. Broucke, R.: Traveling Between the Lagrange Points and the Moon, Journal of Guidance and Control, Vol. 2, N 4, July-Aug. 1979, pp. 257-263. 2. Faxquhar, R. W.: Future Missions for Libration-point Satellites, Astronautics and Aeronautics, May, 1969. 3. Miele, A.: Theorem of Image Trajectories in the Earth-Moon Space, XIth International Astronautical Congress, Stockholm, Sweden, Aug. 1960; Astronautica Acta, pp. 225-232. 4. Prado, A.F.B.A.: Traveling Between the Lagrangian Points and the Earth. Acta Astronautica, Vol. 39, No. 7 (October/96), pp. 483-486. 5. Press, W. H.; Flannery, B. P.; Teukolsky, S. A. and Vetterling, W. T.: Numerical Recapes, Cambridge University Press, NewYork, 1989. 6. Szebehely, V.: Theory of Orbits, Academic Press, New York, 1967.

Libration Point Orbits and Applications G. Gdmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

ASTRODYNAMICAL APPLICATIONS OF INVARIANT MANIFOLDS ASSOCIATED WITH COLLINEAR LISSAJOUS LIBRATION ORBITS J. COBOS TERMA, Flight Dynamics Division, ESA/ESOC, Robert-Bosch-Str. 5, 64293 Darmstadt, Germany J.J. MASDEMONT IEEC & Departament de Matemdtica Aplicada I , Uniuersitat Politdcnica de Catalunya, E. T.S.E. I. B., Diagonal 647, 08028 Barcelona, Spain

A method of transfer between two Lissajous around the same collinear equilibrium point is developed making use of the geometry of the phase space around these points. The proposed solution is actually expected t o be the optimal strategy for the problem. As one of the applications, optimal strategy for eclipse avoidance around L2 is also developed in this context. This gives for the mission FIRST/Planck of the European Space Agency (ESA), 6 years free of eclipse expending a delta-v of 15 m/s typically.

1. Introduction

Libration orbits in a vicinity of the collinear equilibrium points of the EarthSun system, L1 about 1.5 x lo6 km from the Earth towards the Sun and La at about the same distance away from the Sun, have been proved to be suitable for space missions since 1978 when ISEE-3 was the first spacecraft launched to a halo orbit around L1 (see 3, till the SOH0 spacecraft launched in 1995 (see lo) which is still a in a halo orbit around L1. Since the collinear libration points L1 and L2 provide the requirements of a highly stable thermal environmental and sky viewing conditions unob253

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Cobos and

J.J.

Masdemont

structed by Earth and Sun, recently many astronomy missions have been considered to be placed there. Just to cite a few of them, FIRST, Planck and GAIA are missions in the ESA Scientific Program for which a class of orbits near Lp have been selected. Orbits around Lp are also going to be used for future astronomy missions as likely the Next Generation Space Telescope. Again in the NASA Scientific Program missions like GENESIS, TPF or even locations for space stations have been considered (see 1 , 4 ) . Instability is a basic dynamical property of the collinear libration points (see l l ) . This property is inherit by all the orbits which exist around the libration points (see for the description of the neigbourhood of the libration point orbits). A spacecraft placed in a libration point orbit, will move away from it because of the small perturbations of the orbit that cannot be avoided. 59697

Moreover to place a spacecraft in the equilibrium point is not feasible for two reasons. The fist one is that the amount of propellant, Av, to stop the spacecraft at the libration point would be prohibitive, and the second one is that for the L 1 case, the equilibrium point appears as a spot in the solar disc. The radiation comming from the sun would make impossible the communication with the satellite and so, an exclusion zone around the solar disc, when the satellite is seen from Earth, has to be avoided. In a similar way, and exclusion zone around Lz has to be avoided to skip the Earth half-shadow as a requirement of the mission. Traditionally, the exclusion zone has been avoided selecting the so called halo orbits. Halo orbits are a family of periodic orbits which leave the equilibrium point inside the bounded component when they are looked from the Earth. The main disavantadge of halo orbits is that they appear very elongated with a big excursion in the direcction of the ecliptic plane with respect to the small excursion of the out of plane component. This fact difficults the communication with the satellite since it has to be keeped pointing towards the Earth at the end of the big excursions. There are also a family of orbits around the libration points which are well suited for space missions. They are the so-called Lissajous orbits. Lissajous orbits appear as quasi-periodic orbits filling any chosen rectangle around the libration point as seen from the Earth. Its drawback is that at certain times they cross the exclusion zone and so manoeuvres are required to avoid crossings. They have not used since the present beacuse of the big

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expenditures of the planned maneouvres. The strategy described in this paper gives a way of both transfering between lissajous orbits and avoiding the exclusion zone with a little cost. It was developed during the mission analysis of FIRST, Planck and GAIA of the European Space Agency and actually it is going to be applied for missions FIRST and Planck (see FIRST is the cornerstone project in the ESA Science Program dedicated to far infrared Astronomy. Planck, renamed from COBRASISAMBA, is to map the microwave background over the whole sky and is now combined with FIRST for a launch in 2007. Several possible options where considered during the orbit analysis work. The final one adopted was the so-called “Carrier”. Both spacecraft will be launched by the same Ariane 5, but will separate after launch. For this option, the optimum solution is a free transfer to a large amplitude Lissajous orbit. FIRST will remain in this orbit whereas Planck, of much less mass, will perform a size reduction manoeuvres.

2. Linear Approach

Let us perform de computations using the classical Restricted Three Body Problem (RTBP). The differential equations of motion in the sinodical reference frame are (see 1 1 ) , X - 2 Y = anlax Y 2 x = an/au 2 = an/az

+

+

+ %+ +

where O ( X ,Y,2 ) = $ ( X 2 Y 2 ) $p(l - p ) , and T I , r2 denote the distances from the spacecraft to the primaries. T: = ( X - p)2+Y2 Z 2 , and ri = (X 1- p)2 + Y 2+ Z 2 .

+

+

Following ’, Let us write the equations of motion (1) with origin a selected collinear libration point and scaling the longitudes such that the new distance from the origin (equilibrium point) to the closest primary be equal to one.

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J. Cobos and J.J. Masdemont

In this new reference system, the form of the linearized equations are,

x - 2 y - (1+2c2)

2 =0 y + 2 i+ (c2 - 1) y = 0

f+c2z=0 where, following the notation of masses.

6t2,

c2

}

(2)

is a constant depending only on the

Quasiperiodic solutions of the linear system (2) are characterized by an harmonic motion in the ecliptic xy plane (also known as “in plane component”) and an uncoupled oscillation in the z direction (also known as “out of plane component”) with a different period. The general solution is obtained adding the hyperbolic exponential parts, which due to the Hamiltonian character of the RTBP equations have both stable and unstable components with opposite exponents,

z ( t )= AleXt+ A2e-xt + A3 cos wt + A4 sin wt y ( t ) = cAleXt- cA2e-Xt - LA4 cos wt + LA3 sin wt z ( t ) = As cos vt + As sin ut

}

(3)

where Ai are arbitrary constants and c, 2, w , X and u are constants depending on c2 only:

It is also convenient to look at the oscillatory solution of the linear part as having an amplitude and a phase,

z(t)= AleXt + A2e-xt + A, cos (wt + 4) y ( t ) = cAleXt- c A ~ e - ’ + ~ LA, sin (wt + 4) z ( t ) = A, cos (vt + $)

}

(5)

where the relations are A3 = A, cos 4, A4 = -A, sin 4, As = A, cos II, and A6 = -A, sin $. We note that choosing, A1 = A2 = 0, we obtain a periodic motion in the x y components together with a periodic motion in z of a different period. This represents the Lissajous orbits in the linearized restricted circular three-body problem, A,, A, being the maximum in plane and out of plane

Astrodynamical Applications of Invariant Manifolds

257

amplitudes respectively. The first integrals A1 and A2 are directly related to the unstable and stable manifold of the linear Lissajous orbit. For instance, the relation A1 = 0, A2 # 0, defines a stable manifold . Any orbit orbit verifying this condition, will tend forward in time to the Lissajous (or periodic) orbit defined by A,,A, since the A2-component in (3) will die out. A similar fact happens when A1 # O,A2 = 0, but now backwards in time. Then, this later condition defines a unstable manifold. In our analysis of the transfer we want to avoid unstable motions forward in time, this means that we require the condition A1 = 0, which is equivalent to (see details in 2),

It is important to note that, given a position x, y of the spacecraft, we have an explicit formula for the set of possible velocities, x, y, for which escape is avoided. As a result, the non scape condition is kept by manoeuvres (Ak, AG) of the form,

where,

ICY[,

indicates the size of the manoeuvre.

Due to the decoupling between the in plane and out of plane motions in the linear theory, the problem of finding a minimum Av transfer between two Lissajous orbits of different size can also first be decoupled in two problems which we address in the next section.

3. Changing in Plane and Out of Plane Amplitudes

Let us assume that at a given time t , we perform a (Ax, Ay) manoeuvre. The central xy part (libration terms) will change from an initial in plane amplitude A): to a final one A ): given by (see details in 2), (i) 2

= A,

+ c2 4

+

1

d:

XC

- 2Ty(tm)Ax d2

+

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J. Cobos and J.J. Masdemont

where ?(t;) has to be understood as limt,tm,t
which tells us the final in plane amplitude that is reached when a manoeuvre (A?, A$) is done at time t , in the z(tm),y(t,) position corresponding to a Lissajous orbit. We note that up to this point we have not still required the no escape condition to the manoeuvre. Let us now assume that the manoeuvre at time t , is done in order to perform a transfer from a Lissajous orbit to another one with a different in plane amplitude, leaving the out of plane motion untouched. The natural approach is to insert into the stable manifold of the final Lissajous orbit (f) from the departure position when possible. The manoeuvre will select A, keeping A1 = 0. The new A2-term which appear as a consequence of the manoeuvre, since it is acompained by the exponential decay in (3), tends to vanish. The result is that we will reach the final orbit asymtotically with no more manoeuvres. For this strategy we have only to select the (A?, Ay) manoeuvre in the direction of the non escape condition according to formula (7). Then for the final in plane amplitude we have,

where the function p ( t ) is defined as,

p ( t ) = A:) sin ( w t

+ +i

- p),

and the angle /3 is given by the direction of the vector ( c , i). This means that, for a given time, the magnitude of the manoeuvre necessary to reach a target in-plane amplitude is given by the quadratic equation, a2 - 2p(t,)a

(f)2

- (A,

(i)2 )2,

- A,

Astrodynamical Applications of Invariant Manifolds

259

and so, the magnitude of the manoeuvre is given by,

Using this expression, we observe that if,

0 0

A"' > A(<)the transfer manoeuvre is possible at any time. - 6)' A , < A , , the transfer manoeuvre is possible only when the expression inside the square root is positive; more precisely, when t E [6, G - 61 U

rf)

A(f)

[G + 6, % - 61, where 6 = i(arccos (3) - di + p). Once the target A:) amplitude is selected, we note the two basic possibilities that we have when selecting the manoeuvre.

0

0

Select t , in such a way that the Av expended in changing the amplitude be a minimum. As we said this corresponds to the minimum of IaI. Select t , in such a way that you arrive at the target orbit with a selected phase.

We will comment these possibilities and its implications in the next section. In the case of an out of plane manoeuvre (z-manoeuvre) the analogous formula to (10) is (see 2 ) ,

where essentially we see that the formal role of (Y is played now by A i l u and the former angle, p, is now zero. The corresponding discussion about the possibility of transfer is the following. If,

A'f' > A:' the transfer manoeuvre is possible at any time. A!) A:) the transfer manoeuvre is possible only if the expression inside the square root is possitive; more precisely, when t E [E, - E] U

2

[F+E,$-E],

F

A(f)

where&= :(arccos(;;b)-$i).

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J. Cobos and J.J. Masdemont

3.1. Optimal Manoeuvres Let us assume that A:) # A:). Looking for the local minima of a(tm)we get that performing the manoeuvre when t , verifies, wt,

+ & - p = i,

or

wt,

+ g5i

-

/3 =

F,

(both mod 2 ~ ) , (12)

this is, when the angle, wt,+di, is orthogonal to p. Then the minimum fuel (f) (i) expenditure for the manoeuvre, Av = IaminI = JA, - A, J is obtained”. In a similar way, when possible, the manoeuvre to change the out of (f) plane amplitude from A:’ to A , is optimal when t , verifies, Vtm

+ $Ji = i,or

vt,

+ $Ji = %,

(both mod 2 ~ ) .

(13)

In this case, the minimal Aw = /Ail is given acording to, A i = ”(A:’ A,(f)), or A i = v(A,(f) - A:’), for the ~ / or 2 3 ~ / 2cases respectively.

+

Since the out of plane movement is governed by z ( t ) = A:’ cos (vt &), we note that the optimal times correspond to the z = 0 crossing. This is when the satellite crosses the ecliptical plane, which is natural if we think in terms of energy. We note also that the plane orthogonal t o z = 0 and crossing z = 0 following the direction of /3 has the same properties for the in plane maneouvres.

4. Changing Phases

At the time t , of the manoeuvre, we change the in plane amplitude from

A:)

=

Jm, J-, to A:) -

which will remain along (f)

the new trajectory. Depending on t m , the components A, and A:) can vary in the circle of fixed radius A:), giving as a result, to reach the target orbit at different phases. Assuming that the manoeuvre (AX, Ay) is done in the non escape direction (7), and using the definition of p, the relation between the amplitudes

Jn

< A!), because in this case the points of discontinuity verify, (i) > la,,,inl,when A (f) , < cos(wt++i - p ) = A , / A , , giving, la1 = aEven in the case A:)

(f)

A!’

Astrodynamical Applications of Invariant Manifolds

261

A3 and A4 are (see ’), (f)

A, = A, - a(t,) sin (wt, - p), (f) (4 a(tm)cos (wtm - p), A4 = A4 (a)

+

(14)

where in the case that the manoeuvre is done when the satellite is on a (f) Lissajous orbit, and the target amplitude, A, , can be reached, the value of a(tm)is given by (10). Proceeding in a similar way, we can obtain the components of the target z-amplitude, which give us the out of plane phase. The relations are,

4.1. In Plane Phase Change Manoeuvres Maintaining

Amplitudes

As a particular case of the change of phase, let us study the case where the in plane amplitude is maintained. This special case will be very usefull when avoiding the exclusion zones. Let us assume that we peform the manoeuvre in a Lissajous orbit with in-plane amplitude A, with the pourpose of arriving assimtotically to another one with in-plane amplitude A:) = A:), which in the remaining of this subsection will be simply called A,. We want to link q5f with q$ depending on t,. Using the non trivial manoeuvre of (10) and the equations (14), after some trigonometry we end up with, (f)

A3 = A, cos (2(wt, - p)

+ &),

(f)

A,

= A, sin (2(wt, - p) (f)

+ $i). (f)

Comparing these expressions with their alternative ones, A3 = A, cos + f , (f) and A:’ = -A, sinq5f, we see that the relation between the phases is, q5f - +i = -2(wt,

-p

+ c$i)

(mod 27r).

(15)

Also we can consider a manoeuvre in the z component to change the out of plane phase without changing the amplitude. Using the non trivial manoeuvre given by the equation (11)we obtain, Ai - = 2A, sin (vt, $i), (16) Y

+

262

J. Cobos and J.J. Masdemont

and the corresponding change of phase as a function oft, is, $f

- $i = -2(vt,

+ $i)

(mod 2n).

We observe in this case that the manoeuvre (16) corresponds to invert the z component of the velocity.

4.2. Effective Phases

Looking at the central part of (5) or equivalently, if the satellite is in a Lissajous orbit we have,

Let us define the Effective Phase @ E as all the epochs t and all the phases such that @ E = wt (mod 2n). In the same way we define the Effective Phase PSZE as all the epochs t and all the phases $ such that @ E = vt ?I, (mod 27r). Although effective phases are subsets in the space R x [0,2n],for convenience they will be identified by the numbers @ E and @ E in [O,2n].

+

++

+

From equations (17) and taking also into account the velocities we note that there is a biunivocal correspondence between a pair of effective phases ( @ E ,Q E ) and a state (z, y, z , k,jl, i )on a Lissajous orbit of amplitudes A, and A,. In fact, from a dynamical systems point of view this is a consequence that Lissajous orbits are 2D tori since @ E and @ E are identified (mod 2n). This is, we have the tori in action-angle coordinates. The convenience of using the effective phases becomes clear since in the space ( @ E , @ E ) a trajectory such as (17) with initial phases +i, $i, is a straight line of slope w/v, starting at the point ( + i , $ i ) , and followed with constant velocity components w and v respectively in the directions @ E and @ E . So, dymamics are much easier. As a first application of the effective phases, looking (12) we see that the optimal manoeuvre to change the A, amplitude have to be done when the trajectory in the space of effective phases crosses either the line @E = p+ f or @ E = p %f.Also according to (13) the optimal change of A, have to be done either when crossing @ E = f or @ E =

+

4.

Astrodynamical Applications of Invariant Manifolds 263

4.3. Eclipse Avoidance Stmtegy. L O E W E problem

Usually a technical requirement for libration point satellites is to avoid an exclusion zone. For orbits about L1 in the Sun-Earth system the exclusion zone is about the solar disk as seen from Earth (see 3 ) . For orbits about L2 in the Sun-Earth system sometimes the Earth half-shadow has to be avoided. In both cases, since Sun and Earth are located in the x axis, the exclusion zone is set as a disk in the yz plane centered at the origin. Traditionally, halo orbits have been used to avoid the exclusion zone (see 3, however they drawback is that the y excursions of the satellite are very big. This fact increases the complexity and cost of some hardware parts of the satellite. Lissajous orbits suit much better in most of the cases. However, if the duration of the mission is long enough, the satellite will irremediably cross the exclusion zone. The time to enter eclipse deppends on the initial phases r$i, $i, and in the best case the time span between eclipses is about 6 years for an orbit of moderate size (see ’). We have developed a completely new eclipse avoidance strategy based on the phase change explained in a previous section. This eclipse strategy is optimum for La, and it could be seen that it is optimum in general for exclusion zones of radious smaller than one half of the amplitudes of the Lissajous. So, it will be optimum for L1 as well if we consider a big Lissajous orbit.

As we will see, the idea is to perform the manoeuvre near the “corners” of the Lissajous figure corresponding to the yz projection, were velocities are small. We are near these corners just when we are near to enter to or exit from an eclipse. This will provide us maximum time without eclipse after performing the manoeuvre. The best way to represent the LOEWE (Lissajous Orbit Ever Without Eclipse) problem is using the space of effective phases. Let us assume that the satellite is in a Lissajous orbit (17) of amplitudes A , and A,. The exclusion zone is set in the yz plane as a disk of radius T , y2 + z2 < r 2 . Of course, T < A, = EA,, and T < A,. The border of the disk in the plane of effective phases satisfies the equation,

A: cos2 Q E

+ k A, sin2 @ E = r 2 , -2

2

(18) and are the ellipse likk plots represented in figure 1. When the lissajous trajectory represented by a line in the plane of effective phases cuts one of

264

J. Cobos and J.J.

Masdemont

these curves it means that the satellite is entering the exclusion zone (see figure 1).

Fig. 1. Exclusion zone in the plane of effective phases (left) and lissajous trajectory hitting an exclusion zone (right). In the right hand side figure the basic left figure is periodically extended for displaying better the trajectory.

Z manoeuvre strategy. If we perform a z-manoeuvre corresponding to the inversion of the z velocity component just in a point with maximum lyl-component (and so y = 0), it is easy to see that this corresponds to go exactly “time back” not only in the z component, but in the yz projection as well due to the symmetry of the y component with respect to the maximum. We can also change this t , a little bit, in order to go not exactly “time back” in the yz projection, but “returning” tangential to the exclusion zone (disc of radius T O ) . This is actually the optimum cost manoeuvre time, and the one that maximize time without eclipse as well.

Y manoeuvre strategy. If we introduce the expression

+f = -+i 2(wt, - ,8) in the part of the y component of (17), and set t = t,, we

obtain:

+ +

y ( t m ) = -A, sin (wtm +i - (T y(tm) = wAy cos ( ~ t m 4i - ( T

+ 2,8)), + 2P)).

So, there is an invertion of y plus a “shift in time” (and some small and exponentially decreasing terms).

265

Astrodynamical Applications of Invariant Manifolds

+

If we perform the manoeuvre at a time At = (5 p) before a maximum in IzI (approx. 4.7 days in the Sun-Earth L2 case), due to the symmetry of the z component with respect to this maximum, and considering the shift in time for the y (in fact, 2 and y) component, we return exactly “time back” in the projection yz for the linear equations. The same considerations as in the z-manoeuvre case can be done in order to return tangential to the exclusion zone, getting the real minimum.

Rough estimation of the cost of the manoeuvre

It is clear that, in order to maximize the time without eclipse and to minimize the cost, the manoeuvre have to be done near the last corresponding maximum in z or y before entering into eclipse. Then, we are near the corner of the Lissajous figure, where velocities are small. Let us compute a rough estimate of the cost for this method. For this pourpose, we approximate a “revolution” of the Lissajous by an ellipse (this is we take w N v, see the almost straight trajectory in figure l),and consider the trajectory tangent to the exclusion zone (we take a disc of radius T O ) . Since the equations of motion are autonomous we can assume that at t = 0 we have y = 0 and x > 0. Using (17) the projection of the Lissajous trajectory in the yz plane is, y ( t ) = -A, sinwt,

where A, = EA, and

+

~ ( t=)A , cos (vt $),

(19)

II, is going to be estimated.

Let us approximate the radius of the yz projection given by,

+ cos2wt + A: cos (2vt + 2 ~ ) ) )

r ( t ) 2= ~ ( t+ )~ ~( t= )A; ~sin2wt + A: cos2 (vt $) =

N

N

-

Ai+A: 2

Ai+AZ 2

Ag+A: 2

1 + -(-A; (we take w 2 v) 2 1 + s( (A: cos 2$ - A;) cos 2wt - A: sin 2$ sin 2wt) + -21 ~ ( A CZoS ~ I I-, A ; ) ~+ (A: sin 2$12 cos (2wt + 71, N

N

where y = arctan2(A: sin 2$, A: cos 2$ - A ; ) ) . Finally the condition of tangency of the ellipse to the disk of radius

TO,

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J . Cobos and J.J. Masdemont

+ z2. This is, A: + A$ - 2A;A: cos 2$,

is given by the minimum of r 2 = y2

AZ+A: - _1 2 2 which gives us the approximate value of $ that we have to consider for the Lissajous trajectory (19), ro2 N

cos2$

N

A:

+ A:

-

+

(A; + A2 - 27-3' - 2(A; A:)$ - 27-04 - 1. 2A;A: AEAZ

Once we have the approximation of the Lissajous trajectory tangent to the exclusion zone, let us estimate the cost of the z-maneouvre strategy and the cost of the xy-manoeuvre strategy, z-manoeuvre strategy, We need to apply a manoeuvre, Ai- = 2i-, at the time of maximum lyl, this is at t , = &.

]Ail = 1221

= 2A,vl

sin (ut,

+ $)I

N

2A,ul sin (wt,

+ $)I

=

With the current constraints for the FIRST/Planck Mission (Sunspacecraft- Earth angle 15", avoiding Earth half-shadow projection, ro "/ 14000 km), the last factors are practically constants in the family of orbits considered. We obtain,

]Ail 2~

4.4 x 109

m2/s. A, So, a z-manoeuvre is expensive for orbits with small A,. x-y manoeuvre strategy. In this case we have to apply a manoeuvre in the non-escape direction of magnitude Q = 2p(tm) at t m = t,,,, $ ( $ +p).

+

Q

= 2p(t,) = 2A, sin (wt, - p) = 2A, cos (wt, -

(z+ p)) n-

=2

Yzmaz

~

this is, Q is proportional to the y-velocity at the time of maximum IzI. Proceeding similarly to the z-manoeuvre case (it is pretty symmetric) we have, 4.0 x 109 I(Ai,AG)I = IQI 2 2wr0 A, m 2 / % kw A , 7 -

Jm-

,

Astrodynamical Applications of Invariant Manifolds 267

for the FIRST/Planck Mission requirements. So, a zy-manoeuvre is expensive for orbits with small A,.

Cost-estimate for FIRST/Plank. Using these rough estimations, it is easy to see that the criterion in order to choose one or the other strategy would be, if A* < 1 = 0.923, A, - kw 7

in the FIRST/Planck case, then a z-manoeuvre is cheaper, otherwise, a zy-manoeuvre is cheaper. The most expensive case is the one that verifies the equality. In this way we can get a global bound depending only on A,, = A: A::

+ $--

for the FIRST/Plank mission.

So, the cost of a manoeuvre, choosing the suitable strategy, should not exceed this order of magnitude. Anyway, for a concrete orbit, it will be better to compute both strategies and adopt the better one, instead of using this A,/A, criterion based on a rough estimation. After this manoeuvre, we always obtain, in the case of the current mission constraints, a time of at least 6 years without eclipse.

Acknowledgements The research of J.M. has been partially supported by grants DGICYT BFM2000-0805 (Spain) and CIRIT 2000SGR27 and 2001SGR-70 (Catalonia) .

References 1. The Terrestrial Planet Finder. A NASA Origins Program to search for Habitable Planets. May 1999, JPL Publication 99-003.

268

J. Cobos

and

J.J. Masdemont

2. J. Cobos, J. Masdemont. Transfers between Lissajous Libmtion Point Orbits. In preparation. 3. R.W. Farquhar, D.P. Muhonen, C. Newman, H.Heuberger. The First Libration Point Satellite. Mission Overview and Flight History. AAS/AIAA Astrodynamics Spec. Conf., 1979. 4. R.W. Farquhar. The Role of Sun-Earth Collinear Libration Points an Future Space Exploration. AAS Annual Meeting, 1999. 5. G. G6mez, A. Jorba, J. Masdemont, C. Sim6. Study Refinement of Semi-Analitycal Halo Orbit Theory, Final Report ESOC Contract 8625/89/D/MD(SC). Barcelona. April 1991. 6. G. G6mez, J. Masdemont, C. Sim6. Quasihalo Orbits Associated with Libmtion Points. The Journal of the Astronautical Sciences, Vol46, No 2, 135-176. 1998. 7. A. Jorba, J. Masdemont. Dynamics in the Center Manifold of the Collinear Points in the Restricted Three Body Problem. Physica D, 132, 189-213. 1999. 8. M. Hechler, 3. Cobos. FIRST/Planck and GAIA Mission Analysis: Launch Windows with Eclipse Avoidance Manoeuvers. MAS W P 402, ESOC December 1997. 9. D.L. Richardson. A Note on a Lagrangian Formulation for Motion about the Collinear Points. Celestial Mechanics 22, 231-236, 1980. 10. F. C. Vandenbussche, P. Temporelli. The Trip to the L1 Halo Orbit. ESA Bulletin 88, 1996. 11. V. Szebehely. Theory of orbits. Academic Press, 1967.

Libration Point Orbits and Applications G . Gbrnez, M. W. Lo and J. J. Masdemont ( 4 s . ) @ 2003 World Scientific Publishing Company

HALO ORBITS IN THE SUN-MARS SYSTEM I. GACKA Znstytut Astronomicmy, Uniwersytetu Wroctawskiego ul. Kopernika 11, 51-622 Wroctaw, Poland

This paper shows families of halo orbits in the Sun-Mars system. They have been computed numerically. In calculation of a periodic orbit around collinear points L1 and L2 the method of initial conditions numerical improvement has been used. Then the program follows the path of vector of initial conditions to get a family of these orbits. The set of differential equations of motion and variational equations has been integrated by using the Bulirsch-Stoer method. The stability of periodic orbits in the circular restricted threebody problem with respect to some perturbations has been studied. In this work there has been included the disturbance from the Earth as well as from Jupiter. The JPL ephemerides have been used, what has given the position of these planets. There are shown stable orbits there, for which parameters of the stability perform the condition /PI< 2 and I&[ < 2.

1. Introduction Lagrange showed that the three-body problem has five relative equilibrium configurations, at which the particle could be stationary in a reference frame rotating with the two other bodies. There are three collinear points L1, Lp, L3 and two triangular points L4, Ls. The first study, which was connected with motion resulting from particular initial conditions which produce periodic, doubly symmetrical orbits around L1 and L2 points, was issued by Bray and Goudas Early in the eighties Farquhar and Kame1 found a

'.

269

270 I. Gacka

family of three-dimensional orbits around the translunar libration point and called them 'halo orbits'. Later on, Breakwell and Brown extended the numerical calculations for the L2 family of periodic orbits and obtained a new class of linear orbits placed in the neighborhood of the Moon. The approach of different procedures to obtain a description of the dynamics around the libration points was suggested by G6mez et al. '. As is well known, orbits around the L1 and L2 points of the Sun-Earth and the Earth-Moon systems are of great interest for applications in astronautics. In the vicinity of the collinear point there are placed satellites there. This study is concerned with the Sun-Mars system. In Sec.2 the reference frame, the equation of motion and the method of determining a family of halo orbits are described. Sec.3 presents results of the survey of periodic orbits. In Sec.4 there are shown the stable orbits and the influence of the perturbation from the Earth and Jupiter.

2. Equations of motion

There is used the standard coordinate system for the restricted three-body problem (see Fig.1). It can be defined in the following way: this is a rectangular reference system, in which x - y plane is the one where Mars and the Sun move. This system has its origin at the Sun-Mars barycenter and rotates around normal to x - y plane with angular velocity n equal to mean motion of the planet. The Sun is located in ( ( p - 1)a,0, O)T and Mars in the ( p ~ , O , 0point, ) ~ where a is the semi major of Mars orbit. The equations of motion of the test particle have the form:

where -2ny and +2nx terms are Coriolis terms, which depend on the velocity of the particle in the rotating reference frame. U = U ( x ,y, z ) is given bY

where the first term is the centrifugal potential, the second term is the potential from the Sun and Mars and the third term is the disturbing function. In this equation rs, r M , pi and ri are the distance between particle and Sun,

Halo Orbits in the Sun-Mars System 271

Mars, planet and planet from the origin of reference frame, respectively.

+ y2 + 2 2 , ?-M = (z - ( p - 1)u)2 + y2 + z2, 2 p: = (Xi - z)2 + (yi - y y + (2i - 2) , 7-;

= (z - pa)2

2

?-:

= zp +yz2 +zp.

Fig. 1. The reference frame.

The coordinates of the perturbing planet have been taken from JPL DE406 ephemerides. The initial epoch was fixed at 2452275.5 JD. To have the coordinate of the planet in the reference system given above the transformation of the form

R=Cr+Rb has been taken, where R = ( & q ,C)T is the vector of position of the planet in the ecliptic system whose origin is at the Sun, Rb = (&, q,,Cb)T is the vector of position of barycenter Sun-Mars in the ecliptic system, r = (z,y, z ) is~ the vector of position of the planet in the rotating system whose origin is at the center of mass of the Sun-Mars system. The matrix of revolution C is given by coscp sincp 0

272

I. Gacka

where cp = n . t . The adopted units for computation are the same as the ones of the JPL ephemeris, and are the following: gravity potentials of the bodies in AU3/day2, the unit of distance in AU, the unit of time in day. The computation has been divided into four parts: the C model (the classical model), i.e. the restricted three-body problem, when the disurbing function is zero; the E model, i.e. the RTBP with including the perturbation from the Earth into the total potential; the J model, i.e. the RTBP with including the perturbation from Jupiter and the last model (the E+J model) containing simultaneously the perturbation from the Earth and Jupiter in equations of motion. For each model there has been used the following method of computation. To obtain the family of periodic halo orbits a numerical continuation method, including the double symmetry (see Bray and Goudas ', Breakwell and Brown 2 , Howell 5 ) , has been used. We consider the initial vector x (0 ) = ( ~ 0 , 0 , ~ 0 , O , y 0 ,at 0 )t~ = 0, which is perpendicular to the x - z plane. Using the Bulirsch-Stoer procedure equations are integrated until the y component attains zero. This moment is a half period (t = T / 2 ) and the test particle is located in the point x ( T / 2 ) = ( x , O , ~ , x , $ , iIn ) ~order . to have a periodic orbit it is sufficient that x = i = 0 at T / 2 . If this condition is true we can determine a new initial conditions, if not, we must improve initial conditions. To improve the initial conditions we have added the correction Ax(0) = (6x0,0,6~0,0,6$0,0), which can be calculated from

AX = G-' H , where a41

+ 8 4 @ 2 1 @43 + 84@23 @45 + p4@25 @63 + b6@23 @65 + b6@25 c 2

and

c 3

Halo Orbits in the Sun-Mars System 273

c, = -1,

8is the transition matrix and the differential equation

where @ ( t )= for this matrix is

where 0 and 1 are the zero and identity matrix, U,, is symmetric matrix of second derivatives of U again x,y and z. This correction process has to be repeated a number of times in order to reach the closest periodic orbit. The criterion for terminaling iterations is the size of the quantity (i; i 3 ) l j 2 which must be less then E (see Bray and Goudas ’).

+

If the orbit is periodic, the first order stability can be determined by using the transition matrix at the end of a complete cycle @(T). Two stability parameters have been defined

P = ;(a + [a2- 4(p - 2)]’i2), Q = $ ( a- [a2- 4(p - 2)l1j2), a = 2 - TT(@(T)), p = ;[a2 + 2 - Tr(@2(T))]. The orbit is stable, if the moduli of P and Q are less or equal 2.

IPI 1

2 i

IQI 5

2

If one of this conditions or all are not true the orbit is unstable (see Bray and Goudas ’). To calculate initial conditions for new halo orbit (XI= ( q , z l , y l ) * ) , there can be used continuation method ( G6mez et al. ). By integrating the followingsystem of equations along the s parameter, a new initial vector can be determined. dli -- A3 -dx= - A1 _ -dz_ -- A2 ds Ao’ ds Ao’ ds Ao’ where

A0 = (A: A1 = (f:.fi - fif;),

A2 =

+ A; +

-(f;.fi- fif;),

A3 =

-(fif; - fif:),

and f1 = 2ny

where

+ u,,

f2

= u,,

fi[t!Glare the partial differentials of fl(’) to z ( z , 6) components.

274

I. Gacka

3. The family of halo orbits

9

sa

-1.524

-1.521

-1.518

-1.51

-1.524

-1.521

-1.518

-1.51

x IAUI

0.003

I L

-0.0060.003 0 0.0030.006 Y [AUI

Fig. 2. The family of halo orbits around the L1 point. Two solid lines indicate the region of stable halo orbits.

By using the described above method there have been determined 220 halo orbits with step s = fO.OOO1. The computation has been made for the classical RTBP (the C model), then there has been included the perturbation from the Earth only (the E model), from Jupiter only (the J model) and the last model has contained the conjunct disturbance from the Earth and from Jupiter (the E+J model). All computations have been begun at the initial vector x(0) equal (-1.516422 AU, 0.005931AU, -0.000094 AUlday) for the L1 family and (-1.531328AU, 0.004556AU, -0.000084 AUlday) for the Lz family. By using the Lindstedt-Poincarh method (details in Richard-

Halo Orbits in the Sun-Mars System 275

s s

0.006

0.009

0.003

0.006

5 4

0

0.003

N

%

-0.003

0 ....

-0.006 -1.533

-1.53

-0.003

-1.527

-1.524

-1.533

-1.53

-1.527

-1.524

x “Jl

0.009/ 0.006

4

=,

-0.0064.003 0 Y

0.003 0.006

“Jl

Fig. 3. The family of halo orbits around the La point. Two solid lines indicate the region of stable halo orbits.

son ‘) these initial conditions have been calculated. Fig.2 and Fig.3 present members of the family of orbits around the L1 and L2 point, respectively. There are shown orbits for the classical model. Present families include both stable and unstable orbits. A stable region is drawn between two solid lines.

If we add the perturbation, initial conditions are changed. In the Fig.4 there is presented the path of initial conditions for each family on the x - z , x - y and z - y plain. On the x - z and x - y graph the right curve and the left curve refer to results for the L1 and L2 family, respectively. In turn the bottom curve and the top curve on the z - y plain are for

276 I. Gacka

0.008

0

0.006

9

s N

0.004 > 2.

0.002

-5e-05 -0.0001

n -1.532 -15 2 8 -1.524 -15 2 -15 16

I -0.00015 I -1.532 -1 5 2 8 -1.524 -15 2 -1.511

5e-05

a

I

F -0.00015 0

0.002 0.004 0.006 0.008

z [AUI

Fig. 4. The path of initial conditions for the L1 and L2 family of halo orbits. On the x - z and x - y plain the right curve and the left curve refer to results for the Li and Lz family, respectively. In turn the bottom curve and the top curve on the last figure is for the L1 and L Z point, respectively.

the L1 and L2 point, respectively. One symbol appoints to one orbit of the particular model. There are marked initial conditions there: for the C model as the cross, for the model with the Earth as the star, the square refers to the J model and the circle refers to the E+J model. It is invisible in the scale of this figure. All paths of the particular model are covered. The next figure (Fig.5) shows differences of initial conditions between the classical model and the model with the perturbation for the L1 family. The solid line indicates the influence of the Earth on the orbit, the dashed line is the difference between the C model and the J model, and the dotted line includes the disturbing function from the Earth and Jupiter. On the

Halo Orbits in the Sun-Mars System 277

0.1

le-07

50

150 number 100

.1

200

i

Fig. 5 . The difference of initial conditions for the L1 family between the C model and the E (cross), J (star), E+J (square) model.

abscissa axis there is the number of the halo orbit (no.1 refers to the orbit which is the closest to the L1 point and no.220 is for the orbit which is the closest to Mars) and on the ordinate axis there are differences of the 2,z and 6 components. After adding the disturbance the initial vector of position can move about several thousand kilometers, but it can move only several kilometers. The difference of the velocity can reach even five hundred, but can be far more. Between no.158 and no.168, where the difference of 2 and i component is the smallest, simultaneously, there are placed stable orbits there. The similar situation is for the Lz family. It is presented in Fig.6. Here stable orbits are between 110.168 and no.178. For both families the influence of the Earth is weaker than of Jupiter.

278

I. Gacka

1 :

.. ..

.

0.1

0.01

-I

..

;

,

1

0.001

0

z

0.0001

”?%

le-05

0

>”

1e-06 1e-07 50

100 150 number

200

Fig. 6. The difference of initial conditions for the L2 family between the C model and the E (cross), J (star), E+J (square) model.

The Fig.7 shows the behavior of parameters of stability versus the period for the L1 family. In graphs each point refers to one orbit for the C model (cross), the E model (star), the J model (square) and the E+J model (circle). In this case almost all points are covered. For the increasing period the value of the P parameter is decreasing and keeps below a value of 2. The Q parameter, however, is increasing and reaches the value of a thousand. The most unstable orbits (orbits which break up after one period) have the period above a value of 280 days and they are lain the closest to the L1 point. The disturbance, of course, changes the period and parameters of the stability. It is seen in the Fig.8. On the ordinate axis there is the difference of the period and parameters of the stability between the C model and

Halo Orbits in the Sun-Mars System 279

4

3 0-2

0 800

I , I ;

1

160 200 240 280 320 360

Fig. 7. The P and Q parameter for the L1 family of halo orbits.

the model with the disturbing function including the Earth (cross), Jupiter (star) and the Earth+Jupiter (square). The period can be changed about one day for the most unstable orbit and less then 0.001 (one thousand) for the stable orbit. Changes of the P parameter are below 0.1 (one tenth) and are increasing, while changes of the Q parameter are decreasing from 100 to 0.001. Very similar situation is for the L2 family. The Fig.9 presents the P and Q parameter of the period. For the increasing period the value of the P parameter is decreasing and keeps below the value of 2. The Q parameter, however, is increasing and reaches the value of a thousand. Next figure (Fig.10) shows differences of the period, the P and Q parameters between the classic model and the model with the Earth (cross), with Jupiter (star), with the Earth and Jupiter (square). The variation of the period is below one day and can reach a value even 0.001 for stable orbits. The variation of the P parameter is increasing but still remains below 1. The difference of the Q parameter is decreasing.

4. The stable halo orbits

10 stable orbits around L1 and L2 point have been found. Tab.4 and Tab.4 contain initial conditions (ZO, Z O , yo), the period, and two parameters of the stability for each stable orbits for the C model. For remaining models stable orbits have the same number.

280

-g

-

f t

I. Gacka

10

L

1

7

0.1 0.01

-

;

0.001 -

0.0001 50

loo

150

50

200

number

100

100 150 number

200

r

10 1 0.1 0.01 0.001 0.0001 L 50

100 150 number

200

Fig. 8. The difference of the period and parameters of the stability for the L1 family between the C model and E (cross), J (star), E+J (square) model.

The study of the effect of the perturbation on the test particle has been realized by means of the stable orbit evolution study. The behaviour of the particle on the stable orbit in the time has been investigated. So, the orbit no.163 from the L1 family (see Tab.4) has been singled out as an example. The studies have been carried out for one period of particle, one and 50 orbital period of Mars. If we calculate the difference between the classical model and the model with the disturbing function, then it is seen how the additional planet influence on the behavior of the test body. This is shown in Fig.11. In the left graph there is presented the difference of the distance between the classical model ( r c )and the model with the disturbance ( r d ) . In turn, in the right one there is the difference of the velocity. The solid,

Halo Orbits in the Sun-Mars System 281

4

2000

I

1600

3

1200

0

a 2

800 1

400

0 160 200 240 280 320 360 T [day1

0 160 200 240 280 320 360

T [day1

Fig. 9. The P and Q parameter for the L2 family of halo orbits. Table 1. Initial conditions of stable orbits for the L1 point. NO.

158 159 160 161 162 163 164 165 166 167 168

zo [AU] -1.51895 -1.51900 -1.51905 -1.51910 -1.51915 -1.51920 -1.51925 -1.51929 -1.51934 -1.51939 - 1.51944

zo [AU] 0.008860 0.008868 0.008877 0.008884 0.008891 0.008897 0.008902 0.008907 0.008912 0.008916 0.008920

YO [AUldayl

-0.0000909 -0.0000903 -0.0000897 -0.0000891 -0.0000885 -0.0000879 -0.0000873 -0.0000866 -0.0000860 -0.0000853 -0.0000846

T [day] 249.486 247.804 246.134 244.472 242.820 241.176 239.536 237.902 236.272 234.644 233.018

P 1.017 1.022 1.045 1.091 1.161 1.258 1.378 1.516 1.667 1.824 1.983

Q 1.820 1.498 1.220 0.987 0.801 0.660 0.560 0.495 0.458 0.442 0.442

Table 2. Initial conditions of stable orbits for the L2 point.

NO. 168 169 170 171 172 173 174 175 176 177 178

GO [AU] -1.52834 -1.52829 -1.52825 - 1.52820 -1.52815 -1.52810 -1.52805 -1.52800 -1.52795 -1.52790 -1.52786

zo [AU] 0.008876 0.008884 0.008892 0.008899 0.008905 0.008910 0.008915 0.008920 0.008924 0.008927 0.008930

$0

[AU/day] 0.0000904 0.0000899 0.0000893 0.0000887 0.0000881 0.0000875 0.0000869 0.0000863 0.0000856 0.0000850 0.0000843

T [day] 249.976 248.284 246.606 244.960 243.282 241.634 239.994 238.360 236.732 235.108 233.486

P 1.035 1.037 1.055 1.096 1.161 1.251 1.366 1.499 1.646 1.800 1.956

Q 1.821 1.495 1.212 0.974 0.582 0.635 0.530 0.461 0.421 0.402 0.400

282

I. Gacka

'. ..'. I

0.001 -

0

0.0001

.

T

100

10

1 0.1 0.01 0.001 0.0001

50

100 150 number

200

Fig. 10. The difference of the period and parameters of the stability for the La family between the C model and the E (cross), J (star), E+J (square) model.

dashed and dotted lines indicate the influence of the Earth, Jupiter and the Earth+Jupiter, respectively. The hummock that has arisen in both figures is the result of the approach of the test body to Mars. Next figures'(Fig.12) show the chosen particle for one orbital period of Mars. The colour are the same as in the previous figure. This time the test particle has had three close passages to Mars (there are shown three hummocks there). The position and the velocity are decreasing but the test body is staying on the halo orbit. After about 22.5 orbital period of Mars the test body has been thrown away to the halo region by the Earth. The influence of Jupiter has caused that this particle has left the halo region after 3 orbital period of Mars and has been placed on the almost circular

Halo Orbits in the Sun-Mars System 283

F e E

Y

0.1

'

0.01 . le-07

0.001

0

50

100

150

200

50

0

100

150

200

t [day1

t [day1

Fig. 11. The difference between the classical model and model with the disturbing function including the Earth (solid), Jupiter (dashed) and the Earth+Jupiter (dotted). This is for the stable halo orbit for one period of a particle.

0.1 0.01 0.001

0

0.2

0.4

0.6

0.8

t [the orbital period of Mars]

1

le-06

7

le-07

-

0

'

0.2 0.4 0.6 0.8 1 t [the orbital period of Mars]

Fig. 12. The difference between the C model and model with the disturbing function including the Earth (solid), Jupiter (dashed) and the Earth+Jupiter (dotted). This is for the stable halo orbit for one orbital period of Mars.

orbit around the Sun-Mars barycenter (see Fig.13).

To find the moment of this throwing away two limits have been intercepted, bottom and upper, which appoint to the halo region, i.e. the region which the particle on the halo orbit occupies. For the L1 family the bottom and the upper limit have been accepted as the mean distance from the origin equal 1.518 AU and 1.524 AU, respectively. In turn, for the family

I. Gacka

284

le+10 I

E y 2 I

P -

I

-8

1e+06

10000 100

0.01

2 E i5 -

0.001

-

0.0001 le-05

.p

0.01

1

0.1

1e-06 0

10 20 30 40 50 t [the orbital period of Mars]

0

1 0 2 0 3 0 4 0 5 0 t [the orbital period of Mars]

Fig. 13. The difference between the classical model and model with the disturbing function including the Earth (solid), Jupiter (dashed) and the Earth+Jupiter (dotted). This is for the stable halo orbit for 50 orbital periods of Mars. Table 3. The moment of escape from the halo region of the L1 point. The time is given in the orbital period of Mars (day). No. 158 159 160 161 162 163 164 165 166 167 168

the E model 1.455 (999.5) 1.806 (1240.5) 2.460 (1751.5) 10.659 (7322.5) 15.893 (10918.0) 22.469 (15435.0) 9.748 (6696.5) 7.279 (5000.5) 8.938 (6140.0) 8.171 (5613.0) 25.528 (17543.5)

the J model 3.144 (2159.5) 3.827 (2629.0) 4.921 (3380.5) 5.444 (3739.5) 6.194 (4255.0) 3.080 (2115.5) 2.066 (1419.0) 2.030 (1394.5) 2.011 (1381.5) 2.002 (1375.5) 2.009 (1380.0)

the E - J model 3.550 (2439.0) 4.933 (3389.0) 4.500 (3366.0) 5.474 (3760.5) 3.795 (2607.0) 3.096 (2126.5) 2.093 (1438.0) 2.040 (1401.5) 2.014 (1383.5) 1.996 (1371.5) 1.985 (1363.5)

around the L2 point there are the 1.523 and 1.529 distances from the center of mass of the Sun-Mars system. When the particle has left this halo region the time of leaving and planet which has caused it have been noted. The results for the family around the L1 and La point are tabled in Tab.4 and Tab.4, respectively. The moment of the escape is given in the orbital period of Mars and in day in brackets. The behaviour of test particles for 50 orbital periods of Mars has been studied. The test particle on the halo orbit in the classical model has survived over at least the time quoted in the same shape in the determined region. In turn, if one includes other planets the particle leaves the halo region. For both families one observes that the moment of

Halo Orbits in the Sun-Mars System 285

Table 4. The moment of escape from the halo region of the Lz point. The time is given in the orbital period of Mars (day). No. 168 169 170 171 172 173 174 175 176 177 178

the E model 2.527 (1736.0) 2.526 (1735.0) 2.863 (1967.0) 2.888 (1984.0) 3.584 (2462.0) 16.041 (11019.5) 34.195 (23490.5) 23.108 (15874.5) 7.625 (5238.0) 7.942 (5456.0) 24.234 (16647.5)

the J model 1.087 (746.5) 1.096 (753.0) 1.450 (994.0) 1.809 (1242.5) 2.514 (1727.0) 2.148 (1475.5) 2.113 (1451.5) 1.753 (1204.5) 1.736 (1192.5) 1.727 (1186.5) 1.729 (1187.5)

the E - J model 1.817 (1248.0) 1.825 (1253.5) 2.162 (1485.0) 2.149 (1476.0) 2.131 (1464.0) 1.793 (1231.5) 1.765 (1212.5) 1.749 (1201.5) 1.737 (1193.0) 1.729 (1187.5) 1.726 (1186.0)

escape increases first and then decreases. The influence of the Earth on a particle for the L1 family is greater than for the L2. And in the opposite way it is for the perturbations from Jupiter, they are stronger for the L2 halo orbits than for the L1. The combined effect of the disturbances causes that particles on orbits around the L2 point leave earlier the halo region than for the second family. The last value in the second column is caused by the action of Mars, which is bigger than the influence of the Earth (see Tab.3 and Tab.4).

5. Conclusions

Halo orbits exist near the L1 and L2 point at the Sun-Mars system. Families of halo orbits are comparable in size and all orbits decrease in period as they approach Mars. The influence of Earth on initial conditions, period and parameters of the stability are weaker than the influence of Jupiter. Stable orbits exist nearer the planet than the libration point. If the disturbing function was zero, the particle on the stable orbit would survive integration even for 100 years and keep its halo shape. If one studies the evolution of the stable orbit, one can see that the influence of the Earth for the L2 family is weaker than the L1 family. In turn the effect of Jupiter on the L2 family is stronger than on the L1 family. Under the influence of the Earth and Jupiter, however, members of the family around the L2 point escape from the halo region faster than members of the L1 family.

286

I. Gacka

Acknowledgments

This work was supported by the KBN, project number 4612/PB/IA/01. I would like to thank Dr T.Kozar for help and critical reading of the paper and Mgr T.Kruk for support.

References 1. Bray, T.A. and Gouda, C.L., Doubly Symmetric Orbits about the Collinear Lagrangian Points, Astron. J., 72, 202-213, 1967 2. Breakwell, J.V. and Brown, J.V., The Halo Family of 3-Dimensional Periodic Orbits in the Earth-Moon Restricted 3-Body Problem, Celest. Mech., 20, 389404, 1979 3. Farquhar, R.W. and Kamel, A.A., Quasi-Periodic Orbits About the Tkanslunar Libration Point: Celest. Mech., 7, 458-473, 1973 4. Gbmez, G., Jorba, A., Masdemont, J., Sim6, C., Dynamics and Mission Design Near Libration Points, World Scientific, 2001 5. Howell, K.C., Three-Dimensional, Periodic, Halo Orbits, Celest. Mech., 32, 53-72, 1984 6. Richardson, D.L., Analytical Constructaon of Periodic Orbits About the Collinear Points, Celest. Mech., 22, 241-253, 1980

Libration Point Orbits and Applications G . G h e z , M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

TRAJECTORY CORRECTION MANOEUVRES IN THE TRANSFER TO LIBRATION POINT ORBITS G. GOMEZ, M. MARCOTE IEEC & Departament de Matemdtica Aplicada i Andlisi Universitat de Barcelona, Gran Via 545, 08007 Barcelona, Spain

J.J. MASDEMONT IEEC & Departament de Matemdtica Aplicada I Universitat Polittcnica de Catalunya, Diagonal 64 7, 08028 Barcelona, Spain

In this paper we study the manoeuvres to be done by a spacecraft in order to correct the error in the execution of the injection manoeuvre in the transfer trajectory. We will consider the case in which the nominal trajectory is a halo orbit around the collinear equilibrium point L1. The results can be easily extended to the L2 point and to other kinds of libration point orbits, such as Lissajous and quasi-halo orbits. For our study we use simple dynamical systems concepts related with the invariant manifolds of the target orbit, and we compare our results with those obtained by Serban et al. l4 using optimal control.

1. Introduction

This paper is devoted to the study of the so called Trajectory Correction Manoeuvres (TCM) problem, that deals with the manoeuvres to be done by a spacecraft in the transfer segment between the parking orbit and the target nominal one. The main purpose of the TCMs is to correct the error introduced by the injection manoeuvre in the transfer trajectory due to the 287

G. Gdmez, M . Marcote and J.J. Masdemont

288

inaccuracies of the launch vehicle. In connection with the Genesis mission (see l o ) , the TCM problem has been studied in For this mission a halo type orbit, around the L1 point of the Earth-Sun system, is used as nominal orbit. Since this orbit has a strong hyperbolic character, following the ideas introduced in it is possible to use its stable manifold for the transfer, avoiding the insertion manoeuvre into the halo orbit. This is what is know in the literature as the dynamical systems approach to the transfer problem. Other approaches use straightforward propagation from Earth launch conditions to find orbits between the Earth and the halo orbits, keeping some boundary conditions and constraints, at the same time that minimise the total fuel consumption during the transfer (see In any case, one of the conclusions of all these studies is that the insertion manoeuvre, from a parking orbit around the Earth to the transfer trajectory, is a large one, with a Av of the order of 3000 m/s. For the Genesis mission the error in its execution was expected to be about a 0.2 % of Aw (1 sigma value) and a key point to be studied is how large is the cost of the correction of this error when the execution of the first correction manoeuvre is delayed. 8914.

476,

3t799J1).

For the purpose of comparison, in the present study we will use for the main parameters the same values used in 14. More concretely, we will take as reference model for the simulations, the Restricted Three Body (RTBP) Problem with the same value of the mass ratio p = 0.3035910E - 05, so the gravitational effect of the Moon on the transfer trajectory will not be considered (see '). We will also use the same launch conditions near the Earth, which are given in table 1 (from Serban et d . 14). Table 1. Adimensional initial conditions for the reference transfer trajectory. porn 0

gym Znom

0

-1.0000355656083653 + 00 -1.2989505271354733 - 05 -1.6571725774653463 - 05

?$om

%Om

1.5475858756450793- 01 -3.1578000358609183 - 01 -1.1674380533701183 - 01

Since the target halo orbit is not explicitly given in 14, we have used one with approximately the same size as the one displayed in the Figures of the paper, this is a halo periodic orbit with normalised z-amplitude (see l2 for the definition) ,8 = 0.28 corresponding to initial conditions: ~ ( 0=) -0.9922709412937017, ~ ( 0=) 0 , ~ ( 0=) -0.002456251256325228, k(0) = 0 , ~ ( 0=) 0.01191138815471799, i ( 0 ) = 0. It must be noted that the value of

Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits 289

the Jacobi constant of the halo orbit, C = 3.000771793017166, and the one of the above initial conditions for the transfer, C = 3.000782265790755,do not agree. This means, in particular, that the reference transfer trajectory with the initial conditions given in table 1, is not an orbit in the stable manifold of the halo periodic orbit. Nevertheless, approximately 110 days after launch, the transfer orbit is very close to the halo one and, at that point, a manoeuvre of about 13.5 m/s inserts the spacecraft into the halo. Of course, this insertion manoeuvre could be skipped if the reference initial conditions would belong to the stable manifold but, unfortunately, the departure point rarely meets the constraints associated with actual launch conditions.

-1.m2

Fig. 1. hference transfer trajectory and nominal halo orbit, a s given in Serban et al., for the study of the TCM problem (adimensional units). The departure and arrival points are separated, approximately 110 days of time of flight.

In Figure 1 we have displayed the solution with the initial conditions given in table 1 as well as the nominal halo orbit. In Figure 2 we show the different coordinate projections of both the reference transfer trajectory and some “nearby” orbits of the stable manifold of the nominal halo orbit. In the paper by Serban e t al. 14, two different strategies are considered to solve the TCM problem: the Halo Orbit Insertion (HOI) technique and the Manifold Orbit Insertion (MOI) technique. For the HOI technique, an insertion point in the halo orbit is fixed, in this way at least two manoeuvres must be done: the first one (TCM1) a few days after the departure and

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G. Gdmez, M. Marute and J.J. Masdemont

a-1

0-

4-

9-

0

0-

0-

I

Fig. 2. Projections and 3D representation of the transfer trajectory used by Serban et al. and “nearby” orbits on the stable manifold of the nominal halo orbit (RTBP units).

the last one at the HOI point. It is numerically shown that, in practice, the optimal solution can be obtained with just two TCMs, so the TCM2 is performed at the HOI point. The time of flight is not fixed in the simulations and, for the optimal costs obtained, it is found that the cost behaves almost linearly with respect to both TCMl epoch and launch velocity error. The halo orbit insertion time is always close (with variations of the order of 20%) to that of the reference transfer trajectory (transfer trajectory with no insertion error). For the MOI problem, the last manoeuvre is an insertion on the stable manifold of the nominal halo orbit, so there is no manoeuvre of insertion onto the halo orbit. The numerical results obtained with this approach are very close to the ones corresponding to the HOI technique. The main technical tool used through the paper is, as in the classical approach to the transfer problem, an optimisation procedure: the software package COOPT, developed at the University of Santa Barbara 15. This software is used to do an optimisation of the cost function (total Aw) subject to the constraint of the equations of motion. In the same reference, a parametric study of the cost of the TCM is done changing and delaying the execution of the first impulse.

Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits

291

In the present paper we perform the same kind of parametric study as in l4 but without using any optimal control procedure. The quantitative results, concerning the optimal cost of the transfer and its behaviour as a function of the different free parameters, turn out the same. Additionally, we provide information on the cost of the transfer when the correction manoeuvres cannot be done at the optimal epochs. These results are qualitatively very close to those obtained in l3 for the cost of the transfer to a Lissajous orbit around Lz,when the time of flight between de departure and the injection in the stable manifold is fixed, but the target state (position and velocity) on the manifold is varied. For this problem it is found that the cost of the transfer can rise dramatically, as will be shown also later on.

2. The TCMl problem for halo orbits 2.1. Description of the method We use as reference departure state, the one given in table 1. We will always start from the fixed initial position (close to the Earth) given by this reference point. To simulate the injection error, and following 14, we modify the modulus of the velocity at this initial condition according to

where E is a parameter that it is allowed to vary between -6 m/s and +6 m/s, and .',", = (-4.612683390613825, 9.412034579485869, 3.479627336419212)T km/s, which, in adimensional units, correspond to the values given in table 1. As it has already been mentioned, the departure point ( X d e p ) constructed in this way, is not on the stable manifold of the nominal halo orbit selected, but rather close to it. The transfer path has three different legs, qualitatively represented in Figure 3: The first leg goes from the fixed departure point to the point where the TCMl is performed. Usually, this correction manoeuvre takes place few days after the departure.

292

0

0

G. Gdmez, M. Marcote and J.J. Masdemont

The second leg, between the two trajectory correction manoeuvres TCMl and TCMZ, is used to perform the injection in the stable manifold of the nominal orbit. The last path corresponds to a piece of trajectory on the stable manifold. Since both TCMl and TCM2 are assumed to be done without errors, the spacecraft will reach the nominal halo orbit without any additional impulse.

Due to the autonomous character of the RTBP, the origin of time can be arbitrarily chosen. We assume that at the departure t = 0. As it is explained later, we will select an "arrival point" to the halo. In this way, the TCM1, TCM2 and arrival epochs, will be denoted by t l , t 2 and t 3 , respectively. The values of the correction manoeuvres at tl and t 2 will be denoted by Avl and Avz, respectively.

Fig. 3.

The three legs used for the computation of the transfer solutions

When we say that we reach the nominal halo orbit, we mean that we are within a certain distance of a point of it, in the direction of the stable manifold. More precisely, this means that if we select a certain (short) distance, d, and an arrival point on the halo, X,", the point that in fact we reach is X , = X," d * V s ( X , " )where , Vs(X,h)is the linear approximation of the stable direction at the point X,". A value of d = 200 km gives good results as is shown in '. We remark that the stable manifold is a two dimensional manifold (a surface in the 6-dimensional space of positions and velocities) which can be parametrised in the following way. Once a displacement d has been selected, given a point X h on the halo orbit we

+

Tmjectory Correction Manoeuvres in the lPransfer to Libmtion Point Orbits 293

can get an initial condition on the stable manifold X h + d . V s ( X h ) .Following the flow backwards we get all the points in the manifold associated with X h . In this way X h can be thought as one of the parameters which generate the manifold. In what follows, we will call it the parameter along the orbit. The other one is the elapsed time, following the flow, from the initial condition X , = X t d . V S ( X t )to a certain point. We call this time interval the parameter along the Aow. We remark that this parametrisation depends on the choice of d, a small change in d produces an effect equivalent to a small change in the parameter along the orbit. This is: with a small change in d we can get the same orbits of the manifold as with a small change of X h and only a small shift in the parameter along the orbit will be observed. This is because the stable direction is transversal to the flow.

+

We denote by $ ( X , t ) the image, under the flow of the Restricted Three Body Problem, of the point X after t time units. Given the departure state, Xdep, and the time tl, we define X 1 = C#J(Xdep, t l ) .Then, the transfer condition is stated as

$(Xl + Awl, t2 - tl) + Av2 = $(Xa, t 2 - t3),

(2)

+

where, in this relation, a term like X1 Avl has to be understood as: to the state X I (position and velocity) we add Av1 to the velocity. Note that for a given insertion error E (which determines &ep) we have six equality constraints, corresponding to the position and velocity equations (2), and ten parameters: t l , t2, t3, Av1, Av2 and X , (given by the parameter along the orbit) which should be chosen in an optimal way within mission constraints. The sketch of the exploration procedure is the following. To start with, we consider 6 and tl fixed. Two types of explorations appear in a natural way: the fixed time of flight transfers, for which t3 is fixed, and the free time of flight transfers, where t3 is allowed to vary. In both cases, we start the exploration fixing an initial value for the parameter along the orbit, X,. In the case of fixed time of flight, the problem then reduces to seven parameters (t2, Avl, Av2) and the six constraints (2). Using Avl and AVZt o match the constraints (2), the cost of the transfer, llAvll = IlAv1II IlAw211, is seen as a function of t2. In the case of free time of flight, llAvll is seen as a function of t2 and t3, or equivalently, as a function of t2 and the parameter along the flow, t,, = t3 - t 2 .

+

Once we have explored the dependence of the transfer cost with respect

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to t2 and t 3 , we study the behaviour moving the parameter along the orbit, X,,and finally, the dependence with respect to the parameter E (which is determined by the launch vehicle) and tl (which, due to mission constraints, is enough to vary in a narrow and coarse range). We will see that we have some simple linear relations between them.

In order to solve equation (2) by some differential correction procedure, we need an initial guess. This is taken from the solution obtained when E = 0. For most of the simulations, as well as for the parametric study, we use a continuation procedure to get the initial approximation of the solution. It must also be noted, that due to the strong hyperbolic behaviour of the orbits under consideration, it can be necessary to solve equation (2) using some multiple shooting method (see 16). We could use a slight variation of the multiple shooting procedure to recover the MOI technique with more than two TCM used in 14, although this possibility has not been implemented.

As a first example, Figure 4 shows the results obtained when: E = -3m/s, the first manoeuvre is delayed 4 days after the departure (tl = 4), the total time of flight, t 3 , is taken equal to 173.25 days and the arrival point is the one given in table 2. In the next section we will come back to this Figure. Non-linear approximation of the stable manifold In the previous section, we have discussed how the linear approximation of the stable direction (obtained using the linearisation of the flow) can be used to globalise and parametrise the stable manifold of a periodic halo orbit. In a second approach we have used a non-linear approximation of the stable manifold. In the case of halo orbits and using the parameters mentioned in the preceding section, the results obtained with the linear approximation and the ones using the non-linear one are almost the same. Since the increase in computational cost doesn’t give any extra advantage, all the computations that we present have been done using the linear approximation of the manifolds. However this non-linear study is very useful when dealing with the study of the TCM problem for Lissajous libration point trajectories, specially with big amplitudes. Following 2 , in this section we summarise the procedure for the compu-

Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits

----__. 40

60

m

liim elapsed

/’‘

2,

120

100

295

---.-.-‘. 140

b e w m h e depamre and (he sscmd mmow-

160 (days)

D

Fig. 4. Cost of the trajectory correction manoeuvres when T C M l is delayed 4 days after departure and the total time of flight is fixed to 173.25 days. The arrival point on the nominal halo orbit is given in table 2. The curves labelled with (a) correspond to llAvll1, those with (b) to llAvzII and those with (c) to the total cost: llAvlII llAv2II.

+

tation of the non-linear approximation of the stable manifold for the Lissajous and halo orbits. Consider the linearised equations of the restricted three body problem around any collinear equilibrium point

x - 2?j

+ 2 4 z = 0,

- (1

+ 2s + (c2 - 1)y = 0 , 2 + c2z = 0 , where c:, is a parameter depending on the mass ratio and the equilibrium point considered (see 12). The solution of these equations is given by,

+ a2 e-’ot + a3 cos(w0t + dl), y = L3.21e’ot - &a1 e-’ot + &a3 sin(w0t + +I),

z = 01 e’ot

t = a4

cos(v0t

+

42),

where &, i 2 , WO, vo and A0 are constants which can be written in terms of c2. Finally, ai and q5i are free amplitudes and phases. Taking a1 = a2 = 0 we get libration solutions. In case that a1 # 0 or a 2 # 0 we get exponentially increasing or decreasing translations along privileged directions in the phase space. So, we can consider them as amplitudes in the unstable and stable directions respectively. In particular,

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G. Gdmez, M. Marcote and J.J. Masdemont

setting a1 = 0, we get initial conditions for orbits in the stable manifold of a certain linear Lissajous orbit corresponding to the linear equations. Using a Lindsted-Poincark procedure we can look for a formal series solution of the nonlinear equations in terms of the four amplitudes ai and the following three variables,

el = wt + +1,

e2 = ut + 42,

e3 = At.

The expansions of x, y and z are given by,

where 0 = @I1 + q82) and the summation is taken with respect to the integer index i, j , k, m, p and q in a suitable way. Also, according to the Lindstedt-PoincarB procedure, in order to avoid secular terms the frequencies w , u and X must be expanded in formal power series of the four amplitudes, w = x w i j k m ff!

fft

=

a?, &jkm

u=

uijkm af a; fft ff?,

af a: ffi a?,

being the independent terms, woooo = W O , uoooo = vo and XOOOO= XO. So the expansions truncated at first order reproduce the solution of the linear equations of motion. Moreover, if we skip the terms of the expansion related with i and j (this is i - j # 0) we have expansions for Lissajous orbits but not for their invariant manifolds which turn out be the same as the ones given in

‘.

In the halo periodic case the procedure must be slightly modified. The solution depends only on one frequency and this fact introduces a relation between the two central amplitudes a3 and a4.The formal series expansion are given by,

where again,

Trajectory Correction Manoeuvres in the Transfer to Libmtion Point Orbits 297

but now one must take into account a relation between amplitudes which is given by a series expansion of the type,

In all these expansions there are symmetries which make many of the coefficients zero. This fact saves storage and computing time. In the Lissajous expansions have been tested. Using order 25 (i.e. terms up to i j k m = 25), differences less than 100 km between the numerically integrated solution and the direct evaluation of the expansion are obtained for the orbits of the manifolds up to about a distance of 500000 km from the Lissajous orbit.

+ + +

2.2. Fixing the arrival point and the time of flight For the first study of the cost of the TCM, we have taken tl = 4 days and E = -3 m/s. For the time of flight we have used the values obtained in l4 for the optimal solution, this is t 3 = 173.25 days. Since the arrival point is not explicitly given in the above reference, we have used the following approximation (which corresponds to integrate the reference initial state during 173.25 days) Table 2. Approximation used for the dimensional coordinates of the arrival point, Xa,of the optimal solution. la

Yo za

-9.89856258326291093 - Of 4.1836615583455538E- 03 2.1771475345925264E- 03

Xa

ka ia

3.39137363195719843 - 03 -6.2057458211230666E - 03 4.1484980583161675E- 03

As it has already been said, with the values of these parameters fixed, we get a one dimensional set of possibilities, which are the ones displayed in Figure 4. In the Figure, we show the cost of the two Trajectory Correction Manoeuvres, as well as the total cost, in front of the epoch of execution of the second manoeuvre, t 2 . Several remarks should be done in connection with the Figure:

298 G. Gdmez, M. Marcote and 3 . J . Masdemont

0

The solutions of equation (2) are grouped along, at least, three curves. For t2 = 99.5 days there is a double point in the cost function, corresponding to two different possibilities. In Figure 5 we have represented both as well as the orbit of the stable manifold where we perform the injection. The qualitative behaviour of both solutions is rather different.

Fig. 5. Coordinate projections and 3D representation of the two solutions obtained for t 2 = 99.5 days (double point of the cost function). In the Figures we have represented also the orbit of the stable manifold of the nominal orbit where we perform the injection.

0

0

0

For t2 = 113 days we get the optimum solution in terms of fuel consumption: llAvlII llAv2II = 49.31 m/s. This value is very close to the one given in l4 for the MOI approach, which is 49.1817 m/s. The discrepancies can be attributed to slight differences between the two nominal orbits and the corresponding target points. When t2 is small or very close to the final time, t 3 , the total cost of the TCMs increases, as it should be expected. Around the values t2 = 92, 97 and 102 days, the total cost increases

+

Trajectory Correction Manoeuvres in the Transfer to Libmtion Point Orbits 299

Fig. 6. Using the values of the parameters to get Figure 4, here we represent the angle (in radians) between the two velocity vectors, the ones just prior and after the TCM2 epoch.

abruptly. This sudden grow is analogous t o the one described in in connection with the TCM problem for the Genesis mission. It is also similar to the behaviour found in l3 for the cost of the transfer to a Lissajous orbit around Lz, when the time of flight between the departure and the injection in the stable manifold is fixed. To explain this fact, we have computed the angle between the two velocity vectors at t = t 2 , this is when changing from the second to the third leg of the transfer path. This angle has been represented in Figure 6, and looking at it we see that it also increases sharply at the corresponding epochs. This seams to be the geometrical reason for the detected behaviour.

As a second step, we have done a first parametric study allowing variations in the epoch of the execution of TCM1, t l , and in E . Partial results are given in table 3. In the last column of this table we include the numerical results obtained by Serban e t al. (I4) for the MOI strategy (those corresponding to HOI trajectories are similar), which are very close to ours. From this table, it is clearly seen that the behaviour of the optimal cost with respect to c is linear. In Figure 7 we represent the results corresponding to a larger set of explorations, where we allow variations in the magnitude of the error, E , and in the epoch t l . From it, it is also clear a linear behaviour of the optimal cost with respect to t l . In the next step of our study we allow variations in the parameter along the orbit. Assuming the periodic halo orbit parametrised by time (the pe-

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G. Gdmez,

M. Marcote and J.J. Masdemont

Table 3. Numerical results of the parametric study of the TCM cost. The simulations have been done fixing the arrival point as in table 2 and the total time of flight t 3 = 173.25 days in order to compare the results with the ones obtained by Serban et al. which are displayed in the last column.

3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5

4 3 2 1 0 -1 -2 -3 -4 -5 5 4 3 2 1 0 -1 -2 -3 -4 -5 5 4 3 2 1 0 -1 -2 -3 -4 -5

41.2822 31.0891 20.8642 10.5610 0.0974 10.8051 21.7421 32.7661 43.8878 55.1127 59.0547 47.4145 35.7505 24.0530 12.2530 0.0987 12.7285 25.5010 38.3502 51.3480 64.5745 66.1938 53.1763 40.1315 27.0602 13.8820 0.1172 14.6289 29.1862 43.8275 58.8400 74.2527

14.7060 14.1944 13.7963 13.6390 14.0324 13.4634 12.8390 12.2004 11.5459 10.8796 15.2508 14.5247 13.8649 13.3523 13.1986 14.0261 12.9531 11.9398 10.9663 9.8576 8.9561 15.2117 14.3203 13.4972 12.8413 12.6547 14.0080 12.3175 10.8579 9.5245 8.1134 6.6145

55.9882 45.2835 34.6605 24.2000 14.1298 24.2685 34.5811 44.9665 55.4337 65.9923 74.3055 61.9392 49.6154 37.4053 25.4516 14.1248 25.6816 37.4408 49.3165 61.2056 73.5306 81.4055 67.4966 53.6287 39.9015 26.5367 14.1252 26.9464 40.0441 53.3520 66.9534 80.8672

45.1427 55.6387 65.9416

49.1817 61.5221 73.4862

53.9072 66.8668 81.1679

riod of the orbit is approximately equal to 180 days) we have taken a total number of “arrival points” equal to 36, evenly spaced in time. In Figure 8 we show the behaviour of the total cost of the trajectory correction manoeuvres when the parameter along the orbit is changed around the value corresponding to the optimal solution (which is also displayed in the Figure). In the left plot the displayed curves correspond to adding 5,lO and 15 days respectively, to the parameter along the orbit and the one in the right hand side to decrease this parameter in 5 and 10 days. We represent only

Tmjectory Correction Manoeuvres in the Transfer to Libmtion Point Orbits

Fig. 7. Behaviour of the optimal cost vs

8

301

for different values of t ]

+

TCM with a total llAvll = llAvlII IlAvzll smaller than 300 m/s. Increasing o decreasing the values of the parameter along the orbit out of the range of the ones represented in the figures, the total cost increases, and the results obtained are always over the threshold fixed for the representation. This is also the reason because one of the three pieces of the optimal solution has disappeared from the plots. In Figure 9 we plot the surface representing the cost when changing the parameter along the orbit (the value 0 of this parameter corresponds to the point X , given in table 2). Since the total time of flight has been fixed, we get only total TCM costs below 300 m/s within the ranges displayed in the figures.

Fig. 8. Total cost of the trajectory correction manoeuvres when the “arrival” point at the halo orbit is changed. Numerical results correspond to changing the arrival point by adding (left figure) or subtracting (right figure) 5, 10 and 15 days to the parameter along the orbit corresponding to the optimal solution which is represented by the lowest curve.

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Fig. 9. Total cost of the trajectory correction manoeuvres when the "arrival" point, represented by the parameter along the orbit, is moved around the point X , given in table 2, the total time of flight is fixed to 173.25 days and the first manoeuvre is delayed 4 days after the departure. We display the results for negative and positive variations of the parameter along the orbit on the left and right-hand side figures, respectively.

2.3. h e time of flight

To start with, we take tl = 4 days, E = -3 m/s and the arrival point of the preceding sections. With all these parameters fixed, the transfer condition (2) has a two dimensional set of solutions, which can be parametrised by t2 and the parameter along the flow, t,, = t3 - t2, which give the insertion point into the stable manifold. In Figure 10 we show some sections of this surface, for different values of the parameter along the flow, t,, ranging from 40 days (right curve) to 125 days (left curve) as well as the solution that we have obtained in the preceding section for t3 = 173.25 days. Several remarks should be done with respect to this figure: 0

There are values of t2 and t,, (for instance t2 = 108.125, t,, = 65 days) for which the total cost is less than the values we have obtain for t3 = 173.25 days. If we take into consideration that the curves that we have plotted in Figure 10 correspond to evenly spaced values o f t , , , it seems that the value of t2 that makes the cost optimal is a linear function of t,,, at least in the right hand side of the figure where we are close to the optimum values (t2 > 100; the curves in this region correspond to values of t,, equal to 70, 65, 60, 55, 50, 45 and 40 days). Assuming t2 = m(tws- t:,) t! the value of m is close to minus one, since the couple (t2,t,,) that makes minimum the cost verifies t,, t2 N 173.3 days. This fact justifies why the cost function we obtain for t3 = 173.25 days is very close to the optimal solution for t3 free.

+

+

Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits

303

Fig. 10. The curves appearing in the figure are slices of the surface representing the minimum total cost of the TCM, for different values of t w s . In the computations, the first manoeuvre has been delayed 4 days after the departure, the total time of flight is free and the arrival point is fixed. The lowest curve, which almost envelopes the different slices, is the cost function when the total time of flight is fixed to 173.25 days.

To study the influence of the variations in the parameter along the orbit, which is equivalent to change the arrival point, we have taken 12 arrival points evenly spaced in time, displayed in Figure 11. In Figure 12(a) we show the behaviour of the optimal cost for the first six values of the parameter along the orbit. After point number six, the cost function increases sharply and we have not represented the results associated to them. We see that in the region between the 4th and 5th point there is an optimal solution. Taking values of the parameter along the orbit between these two ones, in Figure 12(b) we show the curves of minimum cost as a function of tws. Each curve corresponds to a different value of the parameter along the orbit varying between 0.8 and 1.6 (with step 0.1). As before, it is interesting to observe that the values of t z and t,, that minimise the total cost behave linearly, with respect to the parameter along the orbit, when we are near to the optimal solution. This is shown in Figures 12(c) and 12(d). Using this fact we have obtained that the optimal solution corresponds to t h = 61.34 days with a total cost of 49.1861 m/s. The insertion manoeuvre takes place 111.14 days after the departure with a total time of flight of 172.27 days. This optimal solution is displayed in Figure 13.

304 G. Gdmez, M. Marcote and J.J. Masdemont

Fig. 11. 3D representation of the nominal halo orbit and the 12 “arrival points”, evenly spaced in time, that have been used in our simulation.

As a final exploration we allow variations in the size of the target halo orbit. We have done the computations using halo orbits with values of the z-amplitude p equal to 0.08 and 0.18 in addition to the value 0.28 used in the preceding simulations. In table 4 we give the results obtained using the same nominal departure point for all of them. We remark that when the amplitude of the nominal orbit decreases, the total cost of the optimal TCM increases as well as the value of the parameter along the flow ( t w s ) which corresponds to the optimal solution. Table 4. Optimal solution for different normalised z-amplitude (j3) halo orbits. The departure point has been taken as in table 1.

B t z (days) t,, (days) Cost

(m/s)

0.28 111.14 61.13 49.1861

0.18 146.61 111.55 97.0549

0.08 69.70 156.80 170.9650

3. Departing from the stable manifold

In this section we show the results corresponding to take the departure point on the stable manifold of the target orbit. Now, instead of using the

Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits

305

*I

1

\

1

UI

-.

h..

Fig. 12. (a) Minimum total cost of the trajectory correction manoeuvres when the first manoeuvre is delayed 4 days, E = -3 m/s, the total time of flight is free and the arrival point is varied. (b) Section of the cost surface for different values of the parameter along the orbit between the 4th and 5th arrival point. (c) Optimal time between the insertion into the stable manifold and the arrival when the arrival point is varied. (d) Optimal time of flight when the arrival point is varied.

departure conditions given in table 1,we take as initial position and velocity a point on the stable manifold, with the velocity components affected by some error. If the error is set equal to zero, then no TCM is needed to reach the target orbit. We use the same nominal halo orbit of the preceding sections, this is: a halo orbit around the L1 point of the Earth-Sun system, with normalised z-amplitude p=0.28. Taking the parameter along the orbit between [0,2n], in Figure 14 we represent the minimum distance to the Earth of the stable manifold of the nominal orbit at its first close passage following the parameter along the flow. As it can be seen, there are orbits which collide with the Earth (their minimum distance is below its equatorial radius). We have selected for the departure point one on the orbit associated to a value of the parameter along the orbit equal to t h = 3.66000 and at a distance from the centre of the Earth equal to 6578 km; the adimensional coordinates of this point are given in table 5 and, as it can be seen, are not too far from those given in table 1.

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G. Gdmez, M. Marcote and J.J. Masdemont

f

.........

........

Fig. 13. Projections and 3D representation of the optimal transfer trajectory with t2 = 111.14 days, tws = 61.13 days and total bv = 49.1861 m/s.

Fig. 14. Minimum distance to the Earth of the orbits of the stable manifold of the nominal halo orbit with j3 = 0.28. The distance is below 6578 km for the values of the parameter along the orbit (with values in [0,2n]) t h between 3.648511 and 4.207157 as can be seen in the magnification.

Now, adding E = 7 m/s to the three velocity components of the nominal point given in table 5 , we compute the departure point which will be used for the explorations (the parametric study varying the value of E gives results qualitatively analogous to the ones already described). As time of flight we

-

Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits 307 Table 5. Adimensional coordinates of the nominal departure point on the stable manifold. znom -1.000036453220198E 00 &:Om 1.618213815598005E - 01 0 yr"' -9.466006191933124E - 06 $;Om -2.526026481278061E - 01 zrom -1.128673413649424E - 05 igom -2.308399055627169E - 01

+

Fig. 15. Total cost of the TCM, as a function of tz, for different values of t i . The points with a cross on each curve correspond to the optimum cost

take the value t 3 = 217.28, which is the total time required by the orbit with the initial conditions given in table 5 to reach the arrival point, X,, at the halo orbit (always at a distance of 200 km, in the direction of the stable manifold). In Figure 15 we show the total cost (in m/s) of the TCM as a function of t 2 , for different values of tl between 1 and 7 days. On each curve we have marked with a cross the points corresponding to the minimum cost. From this figure one clearly sees that: (1) As tl increases, the cost of TCMs also does, and it behaves almost linearly with respect to tl in the selected range. (2) The cost of the TCMs is about a 20% less than the values given in table 3, when the departure point is not taken on the stable manifold. (3) The optimal values of t 2 move around t 2 = 58 days, and approximately after 82 days (t2=140 days) one finds also values for TCMl very close to the optimal ones.

As a final exploration we allow variations of the insertion point along the stable manifold. In particular if we fix tl = 1 day and t 2 = 57 days, we get a target insertion point on the stable manifold, which corresponds

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G. Gdmez, M. Marcote and J.J. Masdemont

Fig. 16. Total cost of the TCM for different values of the parameter along the orbit.

to a parameter along the orbit approximately equal to 3.66. Now, we have allowed values of the parameter along the orbit between 3.64 and 3.68 and we have studied the total cost of the TCM, keeping fixed the values of tl and t 2 , and taking as insertion point the one at minimum distance to the target point of insertion already described. The results are given in Figure 16, from which we see that the cost increases when we move away from the most "natural" trajectory.

4. Conclusions

(1) The TCM problem can be studied just by using simple Dynamical Systems concepts. (2) For the optimal TCM, the results obtained with this approach agree, qualitatively and quantitatively, with those obtained with the help of optimal control software. (3) For periodic halo orbits, the use of the linear approximation of the stable manifold gives the same results as the non linear one. (4) The developed procedure can be used for any kind of libration point orbit.

Acknowledgements The research of G.G. and J.M. has been partially supported by grants DGICYT BFM2000-805 (Spain) and CIRIT 2000 SGR-27 (Catalonia). M.M.

Trajectory Correction Manoeuvres in the Tmnsfer to Libmtion Point Orbits 309

wishes t o acknowledge t h e support of the doctoral research grant AP20013064 from the Spanish Ministerio de E d u c a c i h , Cultura y Deportes.

References 1. J. Cobos and M. Hechler. FIRST/PLANCK Mission Analysis: Transfer to a Lissajous Orbit Using the Stable Manifold. Technical Report MAS Working Paper No. 412, ESOC, 1998. 2. J. Cobos and J.J. Masdemont. Transfers between Lissajous Libration Point Orbits. Technical report, In preparation. 3. R.W. Farquhar. The Control and Use of Libration Point Satellites. Technical Report TR R346, Stanford University Report SUDAAR-350 (1968). Reprinted as NASA, 1970. 4. G. Gbrnez, A. Jorba, J. Masdemont, and C. Sim6. Study of the Transfer from the Earth to a Halo Orbit Around the Equilibrium Point L1. Celestial Mechanics, 56(4):541-562, 1993. 5. G. Gbmez, A. Jorba, J.J. Masdemont, and C. Sim6. Dynamics and Mission Design Near Libmtion Point Orbits - Volume 3: Advanced Methods for Collinear Points. World Scientific, 2001. 6. G. G6mez, J . Llibre, R. Martinez, and C. Sim6. Dynamics and Mission Design Near Libmtion Point Orbits - Volume 1: Fundamentals: The Case of Collinear Libmtion Points. World Scientific, 2001. 7. M. Hechler. SOH0 Mission Analysis L1 Transfer Trajectory. Technical Report MA0 Working Paper No. 202, ESA, 1984. 8. K.C. Howell and B.T. Barden. Brief Summary of Alternative Targeting Strategies for TCM1, TCM2 and TCM3. Private communication. Purdue University, 1999. 9. J.A. Kechichian. The Efficient Computation of Transfer Trajectories Between Earth Orbit and L1 Halo Orbit within the Framework of the Sun-Earth Restricted Three Body Problem. In A A S / A I A A Space Flight Mechanics Meeting, Clearwater, Florida, USA, AAS Paper 00-174, 2000. 10. M.W. Lo, B.G. Williams, W.E. Bollman, D. Han, Y. Hahn, J.L. Bell, E.A. Hirst, R.A. Corwin, P.E. Hong, K.C. Howell, B.T. Barden, and R.S. Wilson. Genesis Mission Design. In A I A A Space Flight Mechanics, Paper No. A I A A 98-4468, 1998. 11. D.L. Mains. Transfer Trajectories from Earth Parking Orbits to L1 Halo Orbits. Master’s thesis, Department of Aeronautics and Astronautics, Purdue University, Purdue, USA, 1993. 12. D.L. Richardson. Analytical Construction of Periodic Orbits About the Collinear Points. Celestial Mechanics, 22(3):241-253, 1980. 13. R.S. Wilson, K.C. Howell and M.W. Lo. Optimization of Insertion Cost Transfer Trajectories to Libration Point Orbits. Advances in the Astronautical Sciences, 103:1569-1586, 2000.

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14. R. Serban, W.S. Koon, M.W. Lo, J.E. Marsden, L.R. Petzold, S.D. Ross, and R.S. Wilson. Halo Orbit Mission Correction Maneuvers Using Optimal Control. Automatica, 38:571-583, 2002. 15. R. Serban and L.R. Petzold. COOPT - A Software Package for Optimal Control of Large-Scale Differential-Algebraic Equation Systems. Journal of Mathematics and Computers in Simulation, 56(2):187-203, 2001. 16. J . Stoer and R. Bulirsch. Introduction to Numerical Analysis. Springer Verlag, 1983.

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) 0 2003 World Scientific Publishing Company

LIBRATION POINT ORBITS: A SURVEY FROM THE DYNAMICAL POINT OF VIEW G. GOMEZ IEEC d Departament de Matemcitica Aplicada i Ancilisi Universitat de Barcelona, G m n Via 585, 08007 Barcelona, Spain J.J. MASDEMONT and J.M. MONDELO IEEC d Departament de Matemcitica Aplicada I, Universitat Polit6cnica de Catalunya, E. T.S.E.I.B., Diagonal 647, 08028 Barcelona, Spain

The aim of this paper is to provide the state of the art on libration point orbits. We will focus in the Dynamical Systems approach to the problem, since we believe that it provides the most global picture and, at the same time, allows to do the best choice of both strategy and parameters in several mission analysis aspects.

I. Dynamics and phase space around the Libration Points 1. Equations of motion and Libration Points

1.1. The Restricted Three Body Problem and its perturbations

It is well known that several very simple models, such as the Two Body Problem or the Restricted Three Body Problem (RTBP), are suitable for spacecraft mission design, since they give good insight of the dynamics in many real situations. In this section we will review some of the most 311

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G. Gdmez, J.J. Masdemont and J.M. Mondelo

relevant restricted models for the analysis of the motion in the vicinity of the libration points. Most of the well known restricted problems take as starting point the circular RTBP, that models the motion of a massless particle under the gravitational attraction of two punctual primaries revolving in circular orbits around their center of mass. In a suitable coordinate system and with adequate units, the Hamiltonian of the RTBP is (Szebehely 70)

being p = rnz/(ml+ m2), where ml > m2 are the masses of the primaries. In order to get closer to more realistic situations, or simplifications, this model is modified in different ways. For instance, (1) Hill’s problem. Is useful for the analysis of the motion around r n 2 . Can be obtained setting the origin at m2, rescaling coordinates by a factor p1l3and keeping only the dominant terms of the expanded Hamiltonian in powers of p1/3. The Hamiltonian function is 1

2

2

= S ( P , +P,

2 +PZ)

+ yp,

1 - zpy - (”2 + y2 + z2)1/2 -

2

1

+ S(Y

2

2

+ 1.

This Hamiltonian corresponds to a Kepler problem perturbed by the Coriolis force and the action of the Sun up to zeroth-order in p1l3.Hill’s model is the appropiate first approximation of the RTBP for studying the neighborhood of m2 which takes into consideration the action of the primary m l (Sim6 & Stuchi 6 8 ) . This model has a remarcable set of solutions known as the Variation Orbit Family. This is a family of 27rm-periodic solutions (rn is the parameter of the family) which serves as the first approximation in the modern theory of lunar motion. (2) Restricted Hill four body problem. This is a time-periodic model that contains two parameters: the mass ratio p of the RTBP and the period parameter rn of the Hill Variation Orbit. The RTBP is recovered as rn + 0, and the classical Hill model is recovered as p -+ 0, both in the proper reference frames (Scheeres ‘ O ) . (3) The elliptic RTBP. It is a non-autonomous time-periodic perturbation of the RTBP in which the primaries move in an elliptic orbit instead of a circular one (Szebehely 70).

Libration Point Orbits: A Survey from the Dynamical Point of View 313

(4) The Bicircular Restricted Problem. Is one of the simplest restricted problems of four bodies, obtained from the RTBP by adding a third primary. It can be also considered a periodic perturbation of the the RTBP in which one primary has been splitted in two that move around their common center of mass. This model is suitable to take into account the gravitational effect of the Sun in the Earth-Moon RTBP or the effect of the Moon in the Sun-Earth RTBP. In a coordinate system revolving with Earth and Moon, the Hamiltonian of this problem is (see Sim6 et al. 65)

-

1-P

+ y 2 + 22)1/2

((z - p ) 2

-

-

P

((z - p

+ 1)2 + y 2 + z2)1/2

mS

((z - a s c o s q 2

+ (y + a s s i n e ) 2 + z 2 ) 1 / 2

mS

-

4

- zcOse),

+

with B = wst 80, where ws is the mean angular velocity of the Sun, ms its mass and U S the distance from the Earth-Moon barycenter to the Sun. ( 5 ) Coherent models. They are restricted four body problems in which the three primaries move along a true solution of the three body problem. These models have been introduced for the study of the motion around the geometrically defined collinear and triangular equilibrium points of the Earth-Moon system (Andreu l, Howell et al. 40) and the Sun-Jupiter system perturbed by Saturn (Gabern and Jorba 19). The Hamiltonian of these problems can be written as 1

H = -adpa 2

+ P; + P2) + aZ(YPs - w y )

+a3(zpz

+ ypy + r p z ) +

Ql4z

+ a5y

+ ((z - a 7 ) 2 + (ymS- a s p + 22)1/2 ) ’ where the ai are time-periodic functions, with the same basic frequency as the Bicircular Problem. In a different approach, instead of taking as starting equations those of the RTBP, we can consider Newton’s equations for the motion of an in-

314

G . Gbmex, J.J. Masdemont and J.M. Mondelo

finitesimal body in the force field created by the bodies of the Solar System

Performing a suitable change of coordinates (see G6mez et al. the above equations can be written in Hamiltonian form with the following Hamiltonian function 26y32),

H = Pl(P2 + P;

+ P2) + P2(.Pz + YPy + Z P z ) + P3(YPz - ZPy) +

+ P5z2 + P6Y2 + P7z2 + P 8 z Z + + PlOPy + PllPz + P12z -!- P13Y + P14z +

+P4(zPy - YPz) +PSPz

where S* denotes the set of bodies of the Solar System except the two selected as primaries, and the pi are time dependent functions that can be computed in terms of the positions, velocities, accelerations and overaccelerations of the two primaries. Notice that this Hamiltonian is, formally, a perturbation of the RTBP one. Most of all the intermediate models that have been mentioned are particular cases of this one. Once two primaries have been selected, a Fourier analysis of the pi functions (G6mez et al. 32) allows the explicit construction of a graded set of models with an increasing number of frequencies, that can be considered between the RTBP and the true equations.

1.2. Libration Points and dynamical substitutes

As it is well known, the RTBP has five equilibrium points: three (L1, Lp, L3) are collinear with the primaries and the other two (Lq and L5) form an equilateral triangle with them. Although the models introduced in the preceding section are close to the RTBP, all of them, except Hill's model, are non autonomous, so they do not have any critical point. If the model is time-periodic, under very general non-resonance conditions between the natural modes around the equilibrium points and the perturbing frequency, the libration points can be continued to periodic orbits of the model. In the continuation process, the periodic orbit can go through bifurcations to end up in more than a single periodic orbit or reach a turning point

Libmtion Point Orbits: A Survey from the Dynamical Point of View 315

and disappear. These periodic orbits, which have the same period as the perturbation, are the dynamical substitutes of the equilibrium points. For models with a quasi-periodic perturbation the corresponding substitutes will be also quasi-periodic solutions (see Figure 1).

Fig. 1. Dynamical substitutes for the L1 point in the Earth-Moon system for a timeperiodic (left) and a quai-periodic (right) model.

Dynamical substitutes of the triangular points, for several of the models already mentioned, have been studied in G6mez et al. 24, Sim6 et al. 65 and Jorba et al. 44. For the collinear points of the Sun-Earth system, the dynamical substitutes of L2 for time-periodic models have been given by Farquhar 17, Howell 40. Andreu does a complete study of the substitutes of the collinear libration points for a coherent model close to the EarthMoon problem and compares some of the results obtained with the ones corresponding to a bicircular model. For models depending in more than one frequency one can find results in these Proceedings.

2. The phase space about the Libration Points In this section we will describe the dynamics near the collinear equilibrium points L1, L2, always in the framework of the RTBP. Since we are interested in the motion in the vicinity of a given libration point, following Richardson ", we set the origin of coordinates at a given libration point and scale variables in such a way that the distance from the smallest primary to the selected equilibrium point will be equal to one. Expanding T I = ((x- p)2 y2 z2)'l2 and 7-2 = ((z - p y2 + z2)1/2in power

+ +

+

+

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G. Gdmez, J.J. Masdemont and J.M. Mondelo

series, one gets

n12

+ +

where p2 = x2 y2 z 2 , the c, are constants depending on the equilibrium point and the mass ratio p , and P, is the Legendre polynomial of degree n. With a linear symplectic change of coordinates (G6mez et al. 23), the second order part of the Hamiltonian is set into its real normal form,

where, for simplicity, we have kept the same notation for the variables. Here, A, w p and w, are positive real numbers. From H2, it is clear that the linear behaviour near the collinear equilibrium points is of the type saddle x centre x centre. Hence, one can expect families of periodic orbits which in the limit have frequencies related to both centers: wp and w, (called planar and vertical frequencies, respectively). This is assured by the Lyapunov center theorem, unless one of the frequencies is an integer multiple of the other (which only happens for a countable set of values of the mass ratio (see Siege1 and Moser 62)). Near the libration points we can also expect 2D tori, with two basic frequencies which tend to wp and w, when the amplitudes tend to zero. The rigorous existence of these tori is more problematic. First, the basic frequencies at the collinear point can be too close to resonant. Furthermore, the frequencies change with their amplitudes and so, they go across resonances when the amplitudes are changed. This leads to a Cantor set of tori. The proof of the existence of these tori follows similar lines to the proof of the KAM theorem (see Jorba and Villanueva 46). Close to the L1 and L2 libration points, the dynamics is that of a strong unstable equilibrium, because of the saddle component of the linear approximation. This is the reason why is not feasible to perform a direct numerical simulation of the Poincar6 map in order to get an idea of the phase space. Due to the center x center part, and when we consider all the energy levels, there are 4D center manifolds around them (they are also called neutrally stable manifolds). On a given energy level this is just a 3D set where dynamics have a “neutral behavior”. On it there are periodic orbits and 2D invariant tori. The L3 point has the same linear behavior, however the instability is quite mild. Nevertheless, the long term effects associated to the unstable/stable manifolds of L3 or to the ones of the central manifold

Libmtion Point Orbits: A Survey from the Dynamical Point of View 317

around L3 are extremely important (see Gcimez e t al. In this section we will show results about the phase space in a large neighbourhood of the collinear libration points and will see how all the mentioned invariant sets (periodic orbits and tori) are organized. 24926).

2.1. Local (semi-analytical) approach The analysis of the dynamics in the center manifold for values of the energy close to the one of the equilibrium point can be done in a semi-analytical way using different strategies. One consists in performing a reduction of the Hamiltonian that decreases the number of degrees of freedom, removes the hyperbolic directions and allows the numerical study of the Poincarh map in a vicinity of the equilibrium points (see Gcimez et al. 23 and Jorba and Masdemont 45). This approach is usually known as the reduction to the center manifold. Note that, generically, the expansions required for these computations cannot be convergent in any open set, because of the crossing of resonances. Another procedure consists in the use of Lindstedt-Poincarh methods to explicitly compute the periodic orbits the invariant tori (see Richardson 57 and Gcimez et al. It looks for analytical expressions for them in terms of suitable amplitudes and phases. Both approaches are limited by the convergence of the expansions used for the changes of coordinates and the Hamiltonian, in the first case, and for the periodic orbits and invariant tori, in the second one, which is discussed in the mentioned papers. The general ideas and main results obtained with both procedures will be discussed in the next two sections. 23726933),

Reduction to the center manifold

The reduction to the center manifold is similar to a normal form computation. The objective is to remove not all the monomials in the expansion H (up to a given order) but to remove only some, in order to have an invariant manifold tangent to the elliptic directions of H2. This is done through a series of changes of variables which can be implemented by means of the Lie series method (Deprit 12). The Hamiltonian of the RTBP, with the second order terms in normal

318

G. Gdmez, J.J. Masdemont and J.M. Mondelo

form, can be written, in a suitable set of coordinates and momenta, as

where Hn denotes an homogeneous polynomial of degree n. To remove the instability associated with the hyperbolic character of H2, the instability is given by the term Xq3p3. Thus, in the linear approximation of the equations of motion, the central part is obtained setting 43 = p3 = 0. If we want the trajectory to remain tangent to this space when adding the nonlinear terms, this is, with q 3 ( t ) = p 3 ( t ) = 0 for all t > 0 once we set q3(0) = p 3 ( 0 ) = 0, we need to have 43(0) = p 3 ( 0 ) = 0. Then, because of the autonomous character of the system, we will obtain q 3 ( t ) = p 3 ( t ) = 0 for all t 2 0.

H , we first note that, in the second order part of the Hamiltonian,

Recalling the form of the Hamiltonian equations of motion, qi = H , , , pi = - H q i , one can get the required condition, q3(0) = p3(0) = 0 for q~(0= ) p 3 ( 0 ) = 0, when, in the series expansion of the Hamiltonian, H , all the monomials hijqip3 with i 3 j 3 = 1 have h, = 0 (i and j stand for ( i l , i 2 , i 3 ) and ( j I , j 2 , j 3 ) , respectively). This happens if there are no monomials with 23 j 3 = 1. Of course, other expansions could give us the same required tangency condition, but this is the one that needs to cancel less monomials in (2) and, in principle, it is better behaved both in terms of convergence and from a numerical point of view.

+

+

All the computations can be implemented using specific symbolic manipulators that can carry out the full procedure up to an arbitrary order (see Jorba 45). In this way, we end up with a Hamiltonian H ( q , p ) = H ~ ( q , p ) R N ( q , p ) , where H N ( q , p ) is a polynomial of degree N in ( q , p ) without terms with i 3 + j 3 = 1, and R N ( q , p ) is a remainder of order N 1 that is skipped in the computations.

+

p3

+

In order to reduce the number of degrees of freedom, after setting 43 = = 0 in the initial conditions we look only for orbits in the same energy

level; in this way only three free variables remain. A further reduction is obtained by looking not at the full orbits, but just at their crossings of a surface of section. Now, all the libration orbits with a fixed Hamiltonian value can be obtained just varying two variables in the initial conditions. For instance, the initial conditions can be chosen selecting arbitrary values for

Libmtion Point Orbits: A Survey from the Dynamical Point of View 319

Fig. 2. Poincar6 maps on the section z = 0 (in RTBP coordinates) of the orbits in the central manifold of L1 (two top figures) and L2 (two bottom figures) for the following values of the Jacobi constant: 3.00085,and 3.00078515837634.RTBP mass parameter of the Earth+Moon-Sun system, p = 3.040423398444176x

and p 2 , setting q1 = 0 (the surface of section), and finally computing p l in order to be in the selected level of Hamiltonian energy. The propagation of this initial condition, looking when and where it crosses the surface of section again and again, gives what is called the images of the Poincar6 map on the Poincar6 section q1 = 0. Alternatively] the plane z = 0 (in RTBP coordinates) can be used t o get a more familiar picture. Note that, due to the linear part of the RTBP equations of motion around the collinear equilibrium points (3), z = 0 is a surface of section for all the libration orbits in a neighbourhood of the equilibrium point except for the planar ones, which are contained in the z = 0 plane. q2

This is the procedure used to get Figure 2, where the libration orbits around L1 and La are displayed for two different values of the Jacobi constant] CJ, of the RTBP. From Figure 2, we note that for each level of CJ there is a bounded region in the Poincar6 section. The boundary of the region is the planar Lyapunov orbit of the selected energy (related to the planar frequency wp of H z ) , and is completely contained in the surface of

320 G . G b m e t , J.J. Masdemont and J.M. Mondelo

section. The fixed point, in the central part of the figures, corresponds to an almost vertical periodic orbit, related to the vertical frequency wv. Surrounding the central fixed point, we have invariant curves corresponding to Lissajous orbits. The motion in this region is essentially quasi-periodic (except for very small chaotic zones that cannot be seen in the pictures). Depending on the value of the Jacobi constant, there appear two additional fixed points close to the boundary. These points are associated to halo orbits of class I (north) and class I1 (south). Surrounding the fixed points corresponding to the halo orbits, we have again invariant curves related to quasi-periodic motions. These are Lissajous orbits around the halos that we call quasi-halo orbits (see G6mez et al. 3 3 ) . Finally, in the transition zone from central Lissajous to quasi-halo orbits there is an homoclinic connection of the planar Lyapunov orbit. We note that the homoclinic trajectory that goes out from the orbit and the one that goes in do not generally coincide; they intersect with a very small angle. This phenomenon is known as splitting of separatrices. We also note in this case, that the planar Lyapunov orbit is unstable even in the central manifold. Lindstedt-Poincark procedures: halo, quasi-halo, and Lissajous orbits The planar and vertical Lyapunov periodic orbits, as well as the Lissajous, halo, and quasi-halo orbits, can be computed using Lindstedt-Poincar6 procedures and ad-hoc algebraic manipulators. In this way one obtains their expansions, in RTBP coordinates, suitable to be used in a friendly way. In this section we will give the main ideas used for their computation. We will start with the computation of the Lissajous trajectories (2D tori) and halo orbits (1D tori or periodic orbits). The RTBP equations of motion can be written as

Libration Point Orbits: A Survey from the Dynamical Point of View 321

with c,, p and P, as in (1).The solution of the linear part of these equations is

where w p and w, are the planar and vertical frequencies and K. is a constant. The parameters a and ,B are the in-plane and out-of-plane amplitudes of the orbit and $1, $2 are the phases. These linear solutions are already Lissajous trajectories. When we consider the nonlinear terms, we look for formal series solutions in powers of the amplitudes a and 6 , of the type

+

+

where 81 = wt $1 and 82 = ut $1 . Due to the presence of nonlinear terms, the frequencies w and v cannot be kept equal to w p and w,, and they must be expanded in powers of the amplitudes

i,j=I

i,j=l

The goal is to compute the coefficients X i j k p n , Y i j k m , Z i j k m , w i j , and vij recurrently up to a finite order N = i j. Identifying the coefficients of the general solution (5) with the ones obtained from the solution of the linear part (4),we see that the non zero values are x 1 0 1 0 = 1, YIOIO = K., ~ 1 0 1 0= 1, WOO = w P and uoo = w,. Inserting the linear solution (4)in the equations of motion, we get a remainder for each equation, which is a series in a and ,B beginning with terms of order i j = 2. In order to get the coefficients of order two, this known order 2 terms must be equated to the unknown order 2 terms of the left hand side of the equations. The general step is similar. It assumes that the solution has been computed up to a certain order n - 1. Then it is substituted in the right hand side of the RTBP equations, producing terms of order n in Q and p. This known order n terms must be equated with the unknown terms of order n of the left hand side.

+

+

The procedure can be implemented up to high orders. In this way we get, close to the equilibrium point, a big set of KAM tori. In fact, between these tori there are very narrow stochastic zones (because the resonances are

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dense). Hence we will have divergence everywhere. However, small divisors will show up only at high orders (except the one due to the 1:l resonance), because at the origin wp/w, is close to 29/28. The high order resonances have a very small stochastic zone and the effect is only seen after a big time interval. Halo orbits are periodic orbits which bifurcate from the planar Lyapunov periodic orbits when the in plane and out of plane frequencies are equal. This is a 1:l resonance that appears as a consequence of the nonlinear terms of the equations and, in contrast with the Lissajous orbits, they do not appear as a solution of the linearized equations. Of course, we have to look for these 1-D invariant tori as series expansion with a single frequency. In order to apply the Lindstedt-Poincarb procedure, following Richardson 57, we modify the equations of motion (3) by adding to the third equation a term like A . z , where A is a frequency type series

c 00

A=

dijai@,

i,j=O

that must verify the condition A = 0. We start looking for the (non trivial) librating solutions with frequency wp

We note that after this step, halo orbits are determined up to order 1, and A = 0 is read as do0 = 0. Halo orbits depend only on one frequency or one amplitude since they are 1-D invariant tori, so we have not two independent amplitudes Q and p. The relation between Q and p is contained in the condition A = 0 which implicitly defines a = ~ ( p ) . When we consider the full equations, we look for formal expansions in powers of the amplitudes Q and ,B of the type

+

where 6 = wt # and, as in the case of 2-D invariant tori, the frequency w must be expanded as w = C&o~ij~iPj. The procedure for the computation of the unknown coefficients X i j k , & j k , Z i j k , wij and d i j is similar to the one described for the Lissajous trajectories.

Libration Point Orbits: A Survey from the Dynamical Point of View 323

Quasi-halo orbits are quasi-periodic orbits (depending on two basic frequencies) on two dimensional tori around a halo orbit. Given a halo orbit of frequency w,the series expansions for the coordinates of the quasi-halo orbits around it will be of the form

These expansions depend on two frequencies (w,v) and one amplitude, y (related to the size of the torus around the halo orbit). The frequency v is the second natural frequency of the torus, and it is close to the normal frequency around the base halo orbit. The amplitude, y, is related to the size of the torus around the “base” halo orbit which is taken as backbone. In order to apply the Lindstedt-Poincar6 method to compute the quasihalo orbits, it is convenient to perform a change of variables transforming the halo orbit to an equilibrium point of the equations of motion. Then, orbits librating around the equilibrium point in the new coordinates correspond to orbits librating around the halo orbit in the original ones. The details of the procedure for their computation can be found in G6mez et aZ. 33.

In Figure 2 we display a sample of the different kind of orbits computed using the Lindsted-Poincar6 procedure according to the previous explanations.

2.2. Numerical approach

In this section we will show how, with a numerical approach, the analysis of the phase space using semi-analytical methods, can be extended to a wider range of energy values, including several bifurcations and also to the L3 libration point. The approach is based in the computation of the families of periodic orbits and 2D invariant tori of the center manifolds of the three collinear libration points. Numerical methods have been widely used to compute fixed points and periodic orbits and we will nor enter into the details for their computation here. The reader can find an excellent exposition in the paper by Doedel et al. 13. There are not many papers dealing with the numerical computation

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Fig. 3. 3D representation of several types of orbits around L1. Upper left: vertical periodic orbit with a = 0.0 and p = 0.1 (obtained as a Lissajous orbit with a = 0). Upper right: Lissajous orbit with a = 0.05 and p = 0.15. Lower left: halo orbit with p = 0.1. Lower right: quasi-halo orbit with p = 0.2 and y = 0.067.

of invariant tori. For this purpose, there are mainly two different methods: one is based in looking for a torus as a fixed point of a power of the Poincark map, P", with x being a real number and where P" is obtained by interpolation. The details of the method, as well as some numerical examples, can be found in Sim6 63. The second procedure, introduced in Castelk and Jorba *, is based on looking for the Fourier series of the parametrization of an invariant curve on a torus, asking numerically for quasi-periodic motion. This has been the approach, combined with a multiple shooting procedure, that we have used to study the quasi-periodic motions in a neighbourhood of the collinear libration points (Mondelo 5 5 , G6mez et al. 34).

As a first step of the numerical approach, the study of the families of periodic orbits around the libration points and their normal behavior must be done. Normal behavior around a periodic orbit

Libration Point Orbits: A Survey from the Dynamical Point of View 325

Let cpt(x)be the flow of the RTBP. The normal behavior of a T-periodic orbit through xo is studied in terms of the time-T flow around XO,whose linear approximation is given by the monodromy matrix M = D V T ( X O ) of the periodic orbit. As the monodromy matrix M is symplectic, we have that SpecM = {1,1,XA,I '; ,

~ 2A;}'. ,

The stability parameters of the periodic orbit, that are defined as s j = X j AT' for j = 1 , 2 , can be of one of the following kinds:

+

0 0

0

Hyperbolic: s j E R, l s j l > 2. It is equivalent to X j E R\{-1,l). Elliptic: s j E R, I s j ( < 2. It is equivalent to X j = eiP with p E R (if lsjl = 2, then it is said to be parabolic). Complex unstable: s j E C\R. It is equivalent to X j E @\R, l X j l # 1.

If sj is complex unstable, then s3-j is also complex unstable and, in fact, = q.After the complex unstable bifurcation, following a Hamiltonian Hopf pattern, there appear invariant tori, as is shown in Pacha 5 6 . If s j is hyperbolic, then the periodic orbit has stable and unstable manifolds, whose of M are tangent to the sections at xo through the {Xj,X;l}-eigenplane { X j , Xi'}-eigenvectors at xo. If s j is elliptic, the { X j , Xi'}-eigenplane of M through xo is foliated (in the linear approximation) by invariant curves of the restriction to this eigenplane of the linearization of VT (that is, the map x 4 xo M ( x - m)), which have rotation number p. For the full system, some of these invariant curves subsist and give rise to 2D tori. s3-j

+

In what follows, we will say that a periodic orbit has central part if one of the stability parameters s1,s2 is elliptic. The tori of the central manifolds will be computed starting from the central part of such orbits. Numerical computation of invarian tori

We look for a parametrization of a 2-dimensional torus $J : T2 = R2/2nZ 4 R6,satisfying

+(e + w t ) = V t ( q ( e ) ) ,

ve E T ~ vt, E R,

(7)

where w = (w1,wa) E R2 are the frequencies of the torus and cpt(x)is the flow associated to the RTBP. Let us denote by Ti the period corresponding

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G. Gdmez, J.J. Masdemont and J.M. Mondelo

to the wi frequency, that is Ti = 2 ~ / w i ,and 8 = (<,q).In order to reduce the dimension of the problem, instead of looking for the parametrization of the whole torus, we can look for the parametrization of a curve { q = q o } (or {< = (0)) on the torus, which is invariant under ' p ~ namely ~ , 'PT2 ($(<, q o ) ) = $(<

+ w1T2,

Then, we look for a parametrization

'p

vo),

v< E T1*

(8)

: T1 + W" satisfying

+ PI = $6 ('p(J))

vc E T1,

(9) where 6 = T2 and p = 6wl. Note that p is the rotation number of the curve we are looking for. We assume for 'p a truncated Fourier series representation 'p(<

7

Nf

p(<)= Ao

+ x ( A k cos(k<) + Bk sin(k<)),

(10)

k=l

with Ak, B k E R".This representation of the geometrical torus {$J(O)}O,p is non unique for two reasons: (1) For each choice of r]o we have a different 'p in (lo), i.e., a different invariant curve on the torus. (2) Given the parametrization (lo), for each 50 E T1,'p(<-
+

<

The details of the computational aspects (implementation, computing effort, parallel strategies, etc.) of this procedure are given in G6mez et al. 34. As a sample of the tori that can be computed with this procedure, in Fig. 4 we display families around bifurcated halo-type orbits of L1 and Lz with central part. Invariant tori starting around vertical orbits In Fig. 5 we have displayed the region (in the energy-rotation number plane) covered by the 2-parametric family of tori computed starting from

Libration Point Orbits: A Survey from the Dynamical Point of View 327

L1, period duplication

L1,period triplication

-0.1 -0.2

0.4

L2, period duplication

L2, period triplication

0 -0.1

0.4

-1.1

Fig. 4. Tori around the bifurcated halo-type orbits. The two on the top are in the families around Li and have energy h = -1.501. The two on the bottom are in the families around Lz and have energy h = -1.507.

the vertical L1 Lyapunov families of periodic orbits with central part. The diagrams corresponding to Lz and L3 are similar (see G6mez et al. 34). The boundary has different pieces: 0

0

0

The lower left piece a (from vertex 1 to 2) is related to the the planar Lyapunov family. The orbits of this family represented in the curve are just the first piece of the family with central part. The horizontal coordinate is the energy level h of the curve and the vertical coordinate is p = ( 2 ~ ) ~ / ( 2 7-rv) - 27~,where 2cosv is the stability parameter of the orbit. The upper piece p (from vertex 2 to 3) is strictly related to the vertical Lyapunov family. The points on this curve are ( h ,p ) where h is the energy of the orbit and the rotation number p is such that the elliptic stability parameter of this orbit is 2 cosp. Note that this relation between p and v is different from the previous item, and this is so in order to have continuity of p along an isoenergetic family of tori. The bottom boundary y (from vertex 3 to l),that corresponds to p = 0,

328

G . G6rne.z. J.J. Masdemont and J.M. Mondelo

begins a t the value of the energy where the halo families are born. It is related to a separatrix between the tori around the vertical Lyapunov families and the halo ones.

L1 0.6

I,

-1.6

-1.58

-1.56

-1.54 Energy

-1.52

-1.5

Fig. 5. Region in the energy-rotation number plane, covered by the two-parametric family of tori computed starting at the vertical Lyapunov family of p.0. for L1. The number of harmonics used for the computation of Fourier representation of the tori (< 25, < 50, < 100 and > 100) is shown in the figure. Vertex 1 is at the value of the energy at which the halo family is born. Vertex 2 is at the value of the energy of the equilibrium point. Vertex 3 is at the value of the energy of the first bifurcation of the vertical Lyapunov family.

There are different ways of computing the tori within the region surrounded by the curves mentioned above. We always start from the pieces of boundary formed by periodic orbits. One possibility is then to perform the continuation procedure keeping fixed the value of the energy h. Another one is to allow variation of the energy but keeping fixed the rotation number p. In this last case, and in order to be as close as possible of conditions that guarantee the existence of tori, it is convenient to set the rotation number “as irrational as possible”. To this end, when we have used this second strategy, we have set the values of p such that 2 r l p is an integer plus the golden number. In both cases, and for all L1, Lz and L3 cases, we have always reached a region where the number of harmonics is larger than the maximum value allowed, which at most has been set equal to 100. Larger values of this parameter make computing time prohibitive. Just to have an idea of the computing effort, the constant rotation number family

Libration Point Orbits: A Survey from the Dynamical Point of View 329

with p = 0.176 requires about 3 days of CPU time of an Intel Pentium I11 at 500MHz.

A ((secondview” of the center manifold Using the periodic orbits and the tori computed using the afore mentioned strategies, we have been able to extend the Poincark map representation of the central manifolds around the collinear libration points. Figures 6 and 7 show the results for L1, and L3, respectively (the results for Lz are close to the ones obatined for L1).In all these figures we have represented the z-y coordinates at the intersections with z = 0, p , > 0. All the plots have a similar structure. The exterior curve in each plot is the Lyapunov planar orbit of the energy level corresponding to the plot. As this orbit is planar, it is completely included in the surface of section, and is the only orbit for which this happens. For the three equilibrium points, and for small energy values, the whole picture is formed by invariant curves surrounding the fixed point associated to the vertical orbit. They are related to the intersections of the Lissajous type trajectories around the vertical periodic orbit. The halo orbits appear at the energy levels corresponding to the first bifurcation of the Lyapunov planar family. This can be seen clearly in the Poincark map representations, since there appear two additional fixed points surrounded by invariant curves. Increasing the values of the energy, the L1 and Lz families of halo orbits have two relevant bifurcations, by period triplication and duplication (see Figure 4). Both bifurcations can be also detected on the Poincark representations. This additional structure has not been detected for the L3 case. Within the bifurcated families there are some with central part, which are surrounded by invariant tori. These tori give rise to the typical “island chain” structure of two-dimensional areapreserving maps. This behavior is more clearly seen in a magnification of the figures, as is shown in G6mez et al. 34. The region between the tori around the vertical Lyapunov orbit and the tori around the halo orbits is not empty, as it appears in the above figures, and should contain, at least, the traces, on the surface of section, of the invariant manifolds of the Lyapunov planar orbit. These manifolds act as separatrices between both kinds of motion. The same thing happens between the islands of the bifurcated halo-type orbits and the tori around the halo orbits. In this case, the region between both kinds of tori is filled with the traces of the invariant manifolds of the bifurcated hyperbolic halo-type orbits. In all these boundary regions, the motion should have a chaotic behavior. With the current tools we have not

330

G . Gbmez, J.J. Masdemont and J.M. Mondelo

Energy -1.59

Enemy -1.575

-0.855 -0.85 -0.845-0.84 -0.835 -0.83 -0.825

-0.87 -0.86 -0.85 -0.84 -0.83 -0.82 -0.81

Energy -1.529

0.25

Energy -1.4991

0.3 0.2 0.1 0

-0.05 -0.1 -0.15 -0.2 -0.25 -0.96 -0.93 -0.9 -0.87 -0.84 -0.81 -0.78

-0.1 -0.2 .n-.-3 -1

Fig. 6. Energy slices of the section z = 0, p ,

-0.95

-0.9

-0.85

-0.8

-0.75

> 0 of the invariant tori around L1

been able to compute these separatrices. Energy -1 50

Fig. 7.

Energy slices of the section z = 0, p ,

Energy -1.06

> 0 of the invariant tori around L3.

3. Computations in very accurate models of motion The purpose of this section to show procedures to get solutions, close to the ones previously obtained for the RTBP, of more realistic equations of motion as Newton's equations using JPL ephemeris for the motion of the

Libration Point Orbits: A Survey from the Dynamical Point of View 331

bodies of the solar system, or some of the intermediate models mentioned in previous sections. For these more realistic models there no complete study of the phase space around the libration points (or their dynamical substitutes) like the one that exists for the RTBP. Since the solutions will be computed numerically and the equations of motion are time dependent, an initial epoch and a fixed time span is selected and the orbit is computed for this period of time. In the following section we describe a multiple shooting procedure similar to the one used for the numerical solution of boundary-value problems (see Stoer and Bulirsch sg).

3.1. Multiple shooting

As in the standard multiple shooting method, the total time span is splitted into a number of shorter subintervals selecting, for instance, N equally spaced points t l , t z , ...,t N . (tl is the initial epoch and t N - t l the length of the time interval mentioned above). Different time intervals could have also been used. Let us denote At = ti+l - ti and by Qi

== ( t i ,xi,y i , ~ iki, , $ i , ii,

i = 1,2,

...,N

the points on a fixed orbit of the RTBP, equally spaced ( A t ) in time. This orbit can be, for instance, any of the ones that we have been able to compute their formal expansions using a Lindstedt-Poincark method. Let 4(Qi) be the image of the point Qi under the flow associated to the equations of motion in the solar system after an amount of time At. As, in this way, the epochs ti are fixed, we can write Qi = (xi,yi,zi,&, y i , ii,)T. I f all the points Qi were be on the same orbit of the new equations, we would have q5(Qi) = Qi+l for i = 1,...,N - 1. Since this is not the case, a change of the starting values is needed in order to fulfill the matching conditions. In this way, one must solve a set of N - 1 nonlinear equations, which can be written as

F QN

QN

QN-1

QN

Newton's method is used to solve the above system. If

Q(j)

=

332

G. Gbmez, J.J. Masdemont and J.M. Mondelo

( Q (8 1 ,Q2(j), ..., Qg’) T , denotes the j-th iterate of the procedure, Newton’s equations can be written as

D F ( Q ( ~ .) ()~ ( j + l )- ~

( $ 1= - F ( Q ( ~ ) ) ,

where the differential of the function F has the following structure

with D@ = diag(A1, A2, AN- AN-^). As each of the transition matrices, Ai, that appear in D @ are 6 x 6 , at each step of the method we have to solve a system of ( N - 1) x 6 equations with 6 x N unknowns, so some additional conditions must be added. This is the only difference with the standard multiple shooting method and is due to the fact that our problem is not a real boundary-value one. As additional equations we could fix some initial and final conditions at t = t o and t = t N . In this case one must take care with the choice because the problem can be ill conditioned from the numerical point of view. This is because the matrix D F ( Q ) can have a very large condition number. To avoid this bad conditioning, we can choose a small value for At, but in this case, as the number of points Qi increases (if we want to cover the same time span), the instability is transfered to the procedure for solving the linear system. Also, the extra boundary conditions can force the solution in a non-natural way giving convergence problems when we try to compute the orbit for a long time interval. To avoid these problems, we can apply Newton’s method directly. As the system has more unknowns than equations, we have (in general) an hyperplane of solutions. h o m this set of solutions we try to select the one closer to the initial orbit used to start the procedure. This is done by requiring the correction to be minimum with respect to some norm (i.e. the euclidean norm). The use of the normal equations must be avoided because they are usually ill conditioned too. More precisely, denoting by A Q ( j )

and requiring llAQ(j)112 to be minimum, one gets the Lagrange function L ( A Q ,p ) with (vector) multiplier p

L ( A Q ,P ) = AQT AQ + p T . ( F ( Q )+ D F ( Q ) . AQ).

Libration Point Orbits: A Survey from the Dynamical Point of View 333

By solving the corresponding Lagrange multipliers problem, we obtain

which gives the value of AQ(j) explicitly. However, since the matrix DF(Q(j)) is usually very big, a special factorization in blocks is suitable to get the solution 11 in a computationally and efficient way. See G6mez et al. 33 for the implementation and the properties of the algorithm. In order to illustrate the procedure we reproduce the details of some iterations of the computation of a particular solution using JPL ephemeris DE403. The algorithm is started using as initial nodes, Qi, that is, the components of Q(O),points on a quasihalo orbit of the Sun-Earth+Moon system around the L1 point with /3 = 0.20 and y = 0.08. The initial epoch is fixed to be January 1 2000, and 40 nodes are used with a time step, At, between them of 180 days. This covers a total time span of 19.7 years. So, the total number of revolutions “around” the equilibrium point L1 is approximately 39 and, in order to perform the multiple shooting, approximately one point per revolution has been taken. In Figure 8 we show the (z,y) projection of the orbit after different iterations of the procedure. All the figures are represented in normalized coordinates centered at the L1 point. The first plot corresponds to the orbit from where the points Qi were taken, which was computed with the analytical expansions. It is an approximate solution (due to the truncation and asymptotic character of the series) of the RTBP equations of motion. The next two plots, showing large discontinuities at some points, are the results obtained after the first two iterations. The different pieces that constitute the orbit do not match at the nodes in these first steps because the initial conditions were taken from a solution of the RTBP and now we are integrating these initial conditions in a model including all the bodies of the solar system with its real motion. These discontinuities are so large because of the highly unstable character of the solution and because of the small number of nodes per revolution that have been taken. The last plot corresponds to the orbit computed after 8 iterations. The discontinuities that appear in the first iterations are reduced to “zero” by the method. In the first step, adding the corrections applied at all ...) is the nodes, the total correction in position (lAQti,312 lAQ$912 of 319600.6 km and of 9360.6 km/day in velocity, which means an average value for the corrections at each point of 8000 km and 235 km/day. After

+

+

G. Gdmez, J . J. Masdemont and J.M. Mondelo

334

eight iterations the total amount of the corrections has been reduced to 37 mm and less than 1 mm/day, for positions and velocities, respectively. Taking shorter time intervals between consecutive nodes, the norm of the function F is much smaller at the first steps and the number of Newton iterations decreases. For the Sun-Earth+Moon system, a value of At equal to 7 days requires no more than 4 or 5 iterations to get a final solution with discontinuities at the nodes smaller than tracking errors. For the EarthMoon system, computations must be done more carefully and a time step of half a day usually gives good results.

0.8 I

,

,

,

,

,

,

,

0.8

2.

0.4

.

0.4

.

0.2

.

0.2

.

.

0 -

2.

.

4.2

4.4

-

4.4

.

4.8

.

4.6

.

'

4.2

' 4.1

'

0

x

'

'

'

0.1

02

0.3

0.8

.

0.4

.

1

0.4

.

43

0.8

I

0.8

4.1

0

0.1

0.2

0.3

,

,

,

' 0.1

'

'

0.2

0.3

0.4

I .

.

.

.

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,

04.

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.

0 -

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I

0.6 .

08.

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4 8 .

'

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'

4 2

4.1

0

.

x

0.1

'

0.2

' 0.3

I

0.4

-0.81 4.3

4.2

x

I

0.4

Fig. 8. (I,y) projections of the orbits obtained with the multiple shooting procedure at different steps. The figure on the left upper corner is the orbit of the RTBP, computed from the expansions, from which the initial points Qi are taken. The orbits with large jumps discontinuities are the ones obtained after the first two iterations. The figure on the right lower corner is the orbit computed after 8 iterations. The initial orbit is a quasihalo orbit with p = 0.2 and 7 = 0.08.

Libration Point Orbits: A Survey from the Dynamical Point of View 335

3.2. Resonances The L2 point in the Earth-Moon system is quite close to a resonance with solar effects. Because of that, in some sense, the “distance” between the RTBP model and the real one is too large to extend easily the solutions of the RTBP to the real problem. In order to deal with this situation Andreu has introduced a timedependent restricted four body model that, within a certain degree of simplicity, captures some of the most basic dynamical properties of the true motion around the libration points. The model is time-periodic, since it depends in just one frequency: the difference between the mean sinodical frequencies of the Sun and the Moon. This makes the computation of the most relevant invariant objects of the dynamics simpler. The main success of the model is that it has allowed the computation of halo-type orbits around L2 for very large time spans, covering at least a Saros period. When the analytical techniques are applied to get the center manifold around the dynamical substitutes of the libration points, the results allow only the exploration of energy levels very close to the one of he dynamical substitute, so the information obtained is poor.

11. Applications to spacecraft missions 4. Transfer to libration point orbits 4.1. Transfer using invariant manifolds Libration point orbits have, for values of the energy not too far from the ones corresponding to the libration points, a strong hyperbolic character. It is thus possible to use their stable manifolds in order to obtain a transfer. This is what is known in the literature as the dynamical systems approach to the transfer problem. Other ways to obtain transfer trajectories from the Earth to a libration point orbit use optimization procedures. These methods look for orbits between the Earth and the libration orbit maintaining some boundary conditions, subject to some technical constraints, and minimizing the total amount of fuel to be spent in manoeuvres during the transfer

336

G. G6mez, J.J. Masdemont and J.M. Mondelo

(see Hechler 35). According to Masdemont approach one can proceed as follows:

54*21,

in the dynamical systems

(1) Take a local approximation of the stable manifold at a certain point of the nominal orbit. This dermines a line in the phase space based on a point of the nominal orbit and formed by initial conditions on the stable manifold. (2) Propagate, backwards in time, the points in the line of the local approximation of the stable manifold until one or several close close approaches to the Earth are found (or up to a maximum time span is reached). In this way some globalization of the stable manifold is obtained. (3) Look at possible intersections (in configuration space) between the parking orbit of the spacecraft and the stable manifold. At each of these intersections, the velocities in the stable manifold and in the parking orbit have different values, v, and up. A perfect manoeuvre with Av = v, - up would move the spacecraft from the parking orbit to an orbit in the stable manifold that will reach the nominal orbit without any additional manoeuvre. (4) Then lAvl can be minimized by changing the base point of the nominal orbit at which the stable manifold has been computed (or any equivalent parameter).

Note that, depending on the nominal orbit and on the parking orbit, the intersection described in the third item can be empty, or the optimal solution found in this way can be too expensive. To overcome these difficulties, several strategies can be adopted. One possibility is to perform a transfer to an orbit different from the nominal one and then, with some additional manoeuvres, move to the desired orbits. In a next section we will show how these last kind of transfers can be done. Another possiblility is to allow for some intermediate manoeuvres in the path from the vicinity of the Earth to the final orbit. In the case in which the nominal orbit is a quasihalo or Lissajous orbit and any phase can be accepted for the additional angular variable, the stable manifold has dimension 3. This produces, on one side, a heavier computational task than in the case of halo orbits, but, on the other side, it gives additional possibilities for the transfer. One should think that the stable manifold of the full center manifold of the collinear libration points

Libration Point Orbits: A Survey from the Dynamical Point of View 337

(for a fixed value o f t ) have dimension 5, which offers a lot of possibilities.

4.2. T h e TCM problem

The Trajectory Correction Manoeuvres (TCM) problem deals with the manoeuvres to be done by a spacecraft in the transfer segment between the parking orbit and the target nominal one. The purpose of the TCMs is to correct the error introduced by the inaccuracies of the injection manoeuvre. In connection with the Genesis mission (see Lo et al. 53), the TCM problem has been studied in Howell 37 and Serban et al. " . For this mission, a halo type orbit around the L1 point of the Earth-Sun system is used as nominal orbit. The insertion manoeuvre from the parking orbit around the Earth to the transfer trajectory is a large one, with a Av of the order of 3000 m/s; for the Genesis mission, the error in its execution was expected to be about a 0.2 % of Av (1 sigma value), and a key point to be studied is how large is the cost of the correction of this error when the execution of the first correction manoeuvre is delayed. In the paper by Serban et al. 6 1 , two different strategies are considered to solve this problem, both of them using an optimization procedure and producing very close results. It is numerically shown that, in practice, the optimal solution can be obtained with just two TCMs and that the cost behaves almost linearly with respect t o both the TCMl epoch and the launch velocity error. The same results can be obtained without using any optimal control procedure. This is what is done in G6mez et al. 28. The quantitative results, concerning the optimal cost of the transfer and its behaviour as a function of the different free parameters, turn out the same as in Serban e t al. 61. Additionally, we provide information on the cost of the transfer when the correction manoeuvres cannot be done at the optimal epochs. These results are qualitatively very close to those obtained in Wison et al. 59 for the cost of the transfer to a Lissajous orbit around L z , when the time of flight between the departure and the injection in the stable manifold is fixed, but the target state (position and velocity) on the manifold is varied. For this problem, it is found that the cost of the transfer can rise dramatically. In our approach the transfer path is divided in three different legs:

338

0

0

0

G. G d m e z , J.J. M a s d e m o n t and J.M. M o n d e l o

The first leg goes from the fixed departure point to the point where the TCMl is performed. Usually, this correction manoeuvre takes place few days after the departure. The second leg, between the two trajectory correction manoeuvres TCMl and TCM2, is used to perform the injection in the stable manifold of the nominal orbit. The last path corresponds to a piece of trajectory on the stable manifold. Since both TCMl and TCM2 are assumed to be done without errors, the spacecraft will reach the nominal halo orbit without any additional impulse.

Let t l , t 2 and t 3 be the TCM1, TCM2 and arrival epochs, respectively, and Awl, A w 2 the values of the correction manoeuvres a t tl and t 2 . In this way, given the departure state, X d e p , and the time t l , we define X I = ( P t l ( X d e p ) , where y t ( X ) denotes the image under the flow of the point X after t time units. Then, the transfer condition is stated as (Ptz-tl

( X l + Awl) +

= (Ptz-ts ( X a ) ,

(12)

where X , represents the arrival state to the target orbit, which is chosen as X a = X," d . V5(X,")in the linear approximation of the stable manifold based at the point X,". In (12), the term X I Awl has to be understood as: t o the state X 1 (position and velocity) we add Awl to the velocity. Note that, for a given insertion error E (which determines X d e p ) , we have six equality constraints, corresponding to the position and velocity equations (12), and ten parameters: t l , t 2 , t 3 , Awl, A w 2 and X , (given by the parameter along the orbit), which should be chosen in an optimal way within mission constraints.

+

+

The sketch of the exploration procedure is the following. To start with, we consider the error of the injection manoeuvre and tl fixed. Two types of explorations appear in a natural way: the fixed time of flight transfers, for which t 3 is fixed, and the free time of flight transfers, where t 3 is allowed to vary. In both cases, we start the exploration fixing an initial value for the parameter along the orbit, X,. In the case of fixed time of flight, the problem then reduces t o seven parameters ( t 2 , Awl, A w 2 ) and the six constraints (12). Using Awl and A w 2 t o match the constraints (12), the cost of the transfer, llAwll = IIAw,II IlAw211, is seen as a function of t 2 . In the case of free time of flight, llAwll is seen as a function of t 2 and t 3 , or equivalently, as a function of t 2 and the parameter along the flow, t,, = t 3 - t 2 .

+

Libmtion Point Orbits: A Survey from the Dynamical Point of View

339

Once we have explored the dependence of the transfer cost with respect to t 2 and t 3 , we study the behavior moving the parameter along the orbit, X u , and finally, the dependence with respect to the magnitude of the error (which is determined by the launch vehicle) and tl (which, due to mission constraints, is enough to vary in a narrow and coarse range).

As an example, Figure 9 shows the results obtained when: the magnitude of the error in the injection manoeuvre is -3 m/s, the first manoeuvre is delayed 4 days after the departure (tl = 4), and the total time of flight, t 3 , is taken equal to 173.25 days.

40

Bo tM 120 140 160 lime dspssd beweenUw d e m r e and the m u n d manoeuvre Wvsl

60

m

Fig. 9. Cost of the trajectory correction manoeuvres when TCMl is delayed 4 days after departure and the total time of flight is fixed t o 173.25 days. The curves labelled with (a) correspond to llAwlII, those with (b) to llAw2II and those with (c) to the total cost: IlAviII IlAvzII.

+

Several remarks should be done in connection with the Figure:

0

The solutions of equation (12) are grouped along, at least, three curves. For t2 = 99.5 days there is a double point in the cost function, corresponding to two different possibilities. For t2 = 113 days we get the optimum solution in terms of fuel consumption: IlAqII IlAv2ll = 49.31 m/s. This value is very close to the one given in Serban et al. for the MOI approach, which is 49.1817 m/s. The discrepancies can be attributed to slight differences between the two nominal orbits and the corresponding target points.

+

340

0

0

G. Gdmez, J.J. Masdemont and J.M. Mondelo

When t 2 is small or very close to the final time, t 3 , the total cost of the TCMs increases, as it should be expected. Around the values t 2 = 92, 97 and 102 days, the total cost increases abruptly. This sudden grow is analogous to the one described in Howell and Barden 37 in connection with the TCM problem for the Genesis mission. It is also similar to the behaviour found in Wilson e t al. 59 for the cost of the transfer to a Lissajous orbit around La, when the time of flight between the departure and the injection in the stable manifold is fixed. This fact can be explained in terms of the angle between the two velocity vectors at t = t 2 , this is when changing from the second to the third leg of the transfer path. This angle also increases sharply at the corresponding epochs.

5. Transfers between libration point orbits

In this section we will show a method for performing transfers between libration point orbits around the same equilibrium point. The interest on this problem was initially motivated by the study of the transfer from the vicinity of the Earth to a halo orbit around the equilibrium point L 1 of the Earth-Sun system (G6mez et al. Masdemont 54). There, it was shown that the invariant stable manifolds of halo orbits can be used efficiently for the transfer from the Earth, if we are able to inject the spacecraft into that manifolds. This can be achieved easily when the orbits of the manifold come close to the Earth. But this is true only when the halo orbit is large enough, or when the effect of the Moon, bending some orbits of the manifold, is big enough to take these orbits near the Earth. For small halo orbits, if a swingby with the Moon is used, there are launch possibilities only during two or three days per month (Eismont 14, G6mez et al. 2 2 * 2 1 , Masdemont 54). These launching possibilities can be longer for halo orbits with larger z-amplitude. This is because they have a stable manifold coming closer to the Earth. After the transfer from the Earth to a large halo orbit has been done, we must be able to go from it to a smaller one in a not very expensive way, in terms of the Av consumption and time. Although this rule also applies for the transfer to Lissajous orbits, the study of the transfer between Lissajous orbits was first motivated by the missions FIRST, Plank and GAIA of the European Space Agency Scientific Program. FIRST is the cornerstone project in the ESA Science Program dedicated 23122,

Libration Point Orbits: A Swwey from the Dynamical Point of View

341

to far infrared Astronomy. Planck, renamed from COBRAS/SAMBA, is to map the microwave background over the whole sky and is now combined with FIRST for a launch in 2007. Several possible options where considered during the orbit analysis work. The final one adopted was the so-called “Carrier”, where both spacecrafts will be launched by the same Ariane 5, but will separate after launch. For this option, the optimum solution is a free transfer to a large amplitude Lissajous orbit. FIRST will remain in this orbit, whereas Planck, of much less mass, will perform size reduction manoeuvres. In what follows we will consider the problem of the transfer between both halo type and Lissajous orbits, always around the same libration point.

5.1. Transfers between halo orbits The method that we present is based on the local study of the motion around halo orbits, and uses of the geometry of the problem in the neighbourhood of an orbit of this kind (Masdemont 54, G6mez et al. 29). The approach is different from the procedure developed by Hiday and Howell 36,41 for the same problem. In their approach, they select departure and arrival states on two arbitrary halo orbits, and take a portion of a Lissajous trajectory as a path connecting these states. At the patch points there are discontinuities in the velocity which must be minimized. The primer vector theory (developed by Lawden 51 for the two body problem) is extended to the RTBP and applied to establish the optimal transfers. In our approach, we first study the transfer between two halo orbits which are assumed to be very close in the family of halo orbits. With this hypothesis, the linear approximation of the flow in the neighbourhood of the halo orbits, given by the variational equations, is good enough to have a better understanding of the transfer. Assume that at a given epoch, tl, we are on a halo orbit, H I , and that at this point a manoeuvre, Ad’), is performed to go away from the actual orbit. At t = t 2 > tl, a second manoeuvre, Ad2),is executed in order to get into the stable manifold of a nearby halo orbit H2. Denoting by AD the difference between the 2-amplitudes of these two orbits, the purpose of an optimal transfer is to perform both manoeuvres in such a way that the

342

G. Gdmez, J.J. Masdemont and J.M. Mondelo

performance function

will be maximum. Let cp be, as usual, the flow associated to the differential equations of the RTBP and cpT(y) the image of a point y E R6 at t = 7, so we can write C~T(Yf h) = %(Y)

f &~(y)h

+ o(l h 12)

= ~ T ( Y f)

A(7)h f o(l h 12)-

Let x p be the initial point on a halo orbit, H I , with z-amplitude p, and let us denote &(P) = cp,(xp). The corresponding points in the phase space at t = t l ,t2, neglecting higher order terms and assuming that the time required to execute the manoeuvre can be also neglected, will be, respectively,

At t = t2, the insertion manoeuvre, Ad2), into the stable manifold of the halo orbit of z-amplitude p A@ is done. So, denoting by At,,t, = A ( t ~ ) A ( t l ) - l we must have

+

4tz

(P + Ap)

72e2,p+Ap(t2)f 73e3,p+Ap(t2),

where ez,p+ap(tz) and es,p+~p(tz) are the eigenvectors related to the stable direction and to the tangent to the orbit direction, respectively, of the orbit of amplitude ,B Ap at t = t2. The first term in the right hand side of the above equation can be written as

+

We want to compute the cost of the transfer per unit of z-amplitude, so we set Ap = 1 and the equation to be solved is, after expanding ei,p+Ap by Taylor around P and neglecting higher order terms,

from which we can isolate Ad'), Ad2) getting

Av(') = u10

+

+

~ 2 ~ 1 73u13, 2

Libmtion Point Orbits: A Survey from the Dynamical Point of View 343

for suitable uij. All the magnitudes that appear in these two equations, except the scalars 7 2 and 7 3 , are three-dimensional vectors. As Ap = 1 has been fixed, the maxima of the performance function corresponds to the minima of llA~(~)112llAd2)112.Computing the derivatives of this function with respect to 7 2 and 7 3 and equating them to zero, we get a system of two polynomial equations of degree four in the two variables 7 2 and 7 3 , that must be solved for each couple of values t l , t 2 (which are the only free parameters).

+

The results of the numerical computations show that for a fixed value of t l , there are, usually, two values of t 2 at which the performance function has a local maximum (for values of tl close to 90" and 240" there are three and four maxima). The difference between these two values of t 2 is almost constant and equal to 180". That is, after the first manoeuvre has been done, the two optimal possibilities appear separated by a difference of 1/2 of revolution. The cost of the transfer using the optimal t 2 is almost constant and the variation around the mean value do not exceeds 4%. As an example, if the z-amplitude of the departure orbit is ,B =0.1, the optimum value is reached using the first maximum for t l = 102" and t 2 = 197". For these particular values, the cost of the transfer, per unit of Ap, is of 696 m/s. The cost increases with p: for p =0.15, the optimal value is Av = 742 m/s (tl = 102", t 2 = 193") and for ,B =0.2 Av = 785 m/s ( t l = 101", t 2 = 187"). It has been found that the variation with p of the optimal value of the cost is almost linear. The value of t l for the first manoeuvre is almost constant and equal to lOO", the corresponding point in the physical space being always very close to the z = 0 plane. For very small values of p, the second manoeuvre must be done after t 2 = 270°, but this value decreases quickly and for p E (0.1,0.3) it is of the order of t 2 = 190°, approximately. That is, one has to wait, typically, 1/4 of revolution after the first manoeuvre, to do the second one. The transfer computed with the above procedure is not optimal if the initial and final orbits are not close. This is because the solution given by the linear analysis is not good enough when the orbits have very different z-amplitudes. Several possibilities are discussed in G6mez et ~ 1 . ~ ' .As a final conclusion we can say that the cost of a unitary transfer is of 756 m/s

344 G . Gdmez, J.J. Masdemont and J.M. Mondelo

and the behaviour with the r-amplitude p is almost linear. In this way, the cost of the transfer between two halo orbits of amplitudes ,B =0.25 and 0.08 is (0.25 - 0 . 0 8 ) ~756 m/s = 128.5 m/s. In Figure 10 we show the three projections of a transfer trajectory that goes form ,f3 = 0.25 to p = 0.08

Fig. 10. Projections of the transfer trajectory starting at a departure orbit of zamplitude p = 0.25 and arriving at a final one with z-amplitude p = 0.08. The dotted points correspond to the epochs at which the manoeuvres have been done.

5.2. Transfers between Lissajozls orbits The method of this section is based in the dynamical study of the linearized RTBP equations of motion about a collinear equilibrium point. The development was iniciated during preliminary studies of the FIRST/Plank mission (see Cobos and Hechler 9, and is fully developed in Cobos and Masdemont lo ll. 3

Let us start with the solution of the linear part of the equations of

Libration Point Orbits: A Survey from the Dynamical Point of View 345

motion (3) which can be written as,

+

+

+

x(t) = AleXt A2e-Xt A3 cos w t A4 sin w t y(t) = cAlext - ~ A z e - ’ ~ LA4 cos w t iA3 sin w t z ( t ) = A5 cos vt A6 sin vt

+

where Ai are arbitrary constants and c, pending only on c2.

E,

+

w , X and v are constants de-

Introducing amplitudes and phases (13) can also be written as

+

+ +

+

x(t) = AleXt A2e-xt A, cos ( w t 4) y(t) = cAleXt- ~ A z e - ’ ~ KA, sin ( w t 4) ~ ( t=)A , cos (vt $)

+

+

}

(14)

where the relations are A3 = A, cos 4, A4 = -A, sin 4, A5 = A , cos $Jand A6 = -A, sin$. The key point is that, choosing A1 = A2 = 0, we obtain periodic motions in the xy components with a periodic motion in the z component of a different period. These are the Lissajous orbits in the linearized restricted circular three-body problem, A,, A, being the maximum in plane and out of plane amplitudes respectively. The first integrals A1 and A2 are directly related to the unstable and stable manifold of the linear Lissajous orbit. For instance, the relation A1 = 0, A2 # 0, defines a stable manifold . Any orbit orbit verifying this condition, will tend forward in time to the Lissajous (or periodic) orbit defined by A,, A,, since the A2-component in (13) will die out. A similar fact happens when A1 # O,A2 = 0, but now backwards in time. Then, this later condition defines a unstable manifold. The analysis proceeds by computing the manoeuvres that keep the A1 component equal to zero in order to prevent escape from the libration zone, and how do the amplitudes change when a manoeuvre is applied. We note that, for the linear problem, the motion in the z-component is uncoupled from the motion in the xy component, and z-manoeuvres only change the A, amplitude but do not introduce instability. Assume that the motion takes place in a Lissajous orbit with amplitude A:) and phase $i, and that the desired final z-amplitude is A $ f ) .The possible z-manoeuvres A i wich perform the transfer at time t , are given by,

A i = A, v

(i)

We note that

sin (vt,

+ &) f JArf)’

- A!’

2

cos2 (vt,

+ $i)

(15)

G . G d m e t , J.J. Masdemont and J.M. Mondelo

346

0 0

if A:’ 2 A:), the transfer manoeuvre is possible at any time, but if A:’ < A:), the transfer manoeuvre is possible only if the expression inside the square root is possitive; more precisely, when t E - E ] U ;[

+E, 5-

A(f)

where E = i(arccos ( f ~ -) $i). This con4‘ dition essentially says that it is not possible to reduce the amplitude with an impulsive manoeuvre in case that the position at time t , has (f) a z component bigger than A , . [E,

E],

The change in the in-plane amplitude is a little more tricky since one must keep the unstable component equal to zero. Assuming that the motion takes place in a Lissajous orbit with amplitude A t ) and phase 4i and the desired final in-plane amplitude is ALf), the possible manoeuvres at time tm are given by

where a , indicating the size of the manoeuvre can be 0

= A:) sin (wt,

+ 4i- p) f 4~~ (f)’ - A:)’

COS’

(wt,

+ 42 - P),

where P is a fixed angle given by the direction of the vector (c, k). Again we observe that 0 0

if A:) L A:), the transfer manoeuvre is possible at any time, but if A:) < A:), the transfer manoeuvre is possible only when the expression inside the square root is positive; more precisely, when

t

A(f)

E [ 6 , 3 - 61 U

[E + 6, % - 61, where 6 = : ( a r c c o s ( z )

- di

+ p).

We also note that the manoeuvre (16) is always in the same direction. This direction plays a similar role to the direction orthogonal to the z-plane in the case of the previously commented z-manoeuvres. Once the target amplitudes are selected, the epochs of the manoeuvres can be chosen essentially according to the following possibilities:

0

Select t , in order to minimize the Av expended in changing the amplitude. Select t , in order to reach the target orbit with a selected phase.

Libration Point Orbits: A Survey from the Dynamical Point of Vaew 347

Assuming that the amplitudes before and after the manoeuvres are different, in the first case the optimal t , for changing the in-plane amplitude verifies that the angle ut, q$ equals kx, k E Z. In this case (f) (i) the minimum fuel expenditure for the manoeuvre is ( A , - A, 1. In a similar way, the optimal t , for a change in the out-of-plane amplitude verifies vt,+$i = ;+k7r, k E Z, and the manoeuvre is given by A2 = v(Alf’-A:)).

+ +

+

In case that we decide to arrive at the selected Lissajous orbit with a certain phase, the analysis proceeds by considering the in-plane and out-ofplane amplitudes A, and A, written in term of its respective components A S ,Ad and As, As, and studying the angle which they define. A particular interesting case is the one in which the manoeuvres maintain the amplitudes (the non trivial possibilities of (15) and (16)). In this case, an in-plane manoeuvre (16) at time t , produces an in-plane change of phase given by

and an out-of-plane manoeuvre (15) produces an out-of-plane change of phase given by $1 - $i = -2(vt,

+ $i)

(mod 27r).

(18)

These manoeuvres give two strategies for the avoidance of the exclusion zone needed in many missions (see for instance Farquhar 16). Besides the well known z-strategy given by (18), we have another zy one given by (17), which for the FIRST/Plank mission implies a delta-v expenditure of only 15 m/s every six years (see Cobos and Hechler ’).

5.3. Homoclinic and heternclinic connections

In the preceding sections, we have shown how to use the local dynamics around a halo orbit for “local” purposes. In this one we will study the global behaviour of the invariant stable/unstable manifolds of the central manifolds of L1 and L2, in order to perform some acrobatic motions connecting libration orbits around these equilibrium points. In order to show some heteroclinic trajectories between libration orbits around L1 and L2, we have to match an orbit of the unstable manifold of a libration orbit around one point with another orbit, in the stable manifold of a libration orbit around the other point. This is, both orbits have to be

348

G . Gbmez, J.J. Masdemont and J.M. Mondelo

the same one. Since these orbits, when looked in the X coordinate of the RTBP system, have to go from one side of the Earth to the other one, the place where we look for the connection is the plane X = p - 1, this is, the plane orthogonal to the X axis that cuts it at the point where the center of the Earth is located. Although the technical details are much more complex, the main idea is similar to the computations introduced in G6mez et al. 27 for L 4 , 5 connections. Once a Jacobi constant is fixed, we take initial conditions in the linear approximation of the unstable manifold of all the libration orbits inside the level of energy. Since the energy is fixed, we have three free variables (usually 41, qz and p z ) . A scanning procedure in these variables is done. Since the selected orbits will leave the neighbourhood of the libration point, each initial condition in the variables ( q , p ) is translated into RTBP coordinates and then propagated forward in time until it crosses the plane X = p - 1. We do the same process for the orbits in the stable manifold, where all the propagation is done backward in time. We have to remark that, as usual, the unstable and stable manifolds have two branches. In the process we select only the branches that, at the initial steps of the propagations, approach the X = p - 1 plane. Since the Jacobi constant is fixed, the set of all RTBP values C = {(Y, Y ,2, 2))obtained, characterize the branch of the manifold of all the libration orbits around the selected equilibrium point for the particular section. Let us denote these sets like C+”j, where or - denote the branch of the s (stable) or u (unstable) manifold of the L j , j = 1 , 2 libration orbits at the i-th intersection with the X = p - 1 plane.

+

Looking at the above mentioned branches of the manifolds, the simplest heteroclinic orbits will be obtained from TI- = CFsl n CTu2 and TI+ = CFul n CFs2.Both sets give transfer orbits that cross once the plane X = p - 1. w e will denote by 1 , - (respectively &+) the set of heteroclinic trajectories from L2 to L1 (resp. from L1 to L2) that cross k times the plane X = p-1, following the above mentioned branches of the manifold. We note that, due to the symmetries of the RTBP equations, for any heteroclinic orbit from L1 to L2 we have a symmetrical one from L2 to L1 and so just one exploration must be done. Unhopefully, it has been found (see G6mez e t al.

30)

that TI+ is empty

Libmtion Point Orbits: A Survey from the Dynamical Point of View 349

and so one must look for connections crossing at least twice the plane X = p - 1. In this case many possibilities of connections appear. As an example, in Figure 11 a connection between a Lissajous orbit around L 2 and a quasi-halo orbit around L 2 is displayed. Both the 3-D representation of the homoclinic orbit and the intersections with the surface of section 2 = 0, around both equilibrium points, are given. As another kind of connection, the homoclinic orbit inside the central manifold that marks the transition from the central Lissajous orbits to the quasi-halo ones is computed in G6mez et al. 30. These kind of solutions are interesting because they perform a transition from a planar motion (close to a Lyapunov orbit) to an inclined orbit (close to the quasi-halo orbits) without any Av. Figure 12 shows one of these orbits in central manifold (q,p ) variables. Unfortunately, the transition is very slow but probably, with very small Av, it could be possible to accelerate the transition from planar to inclined motion.

6. Low energy transfers According to Sim6 64: “It seems feasible to produce accurate and enough complete descriptions of the dynamics on the center manifolds of the collinear libration points as well as large parts of the corresponding stable and unstable manifolds. Having these concepts in hand, the design of space missions B la carte, involving the vicinities of these points, could be done in an automatic way”. Although not all the theoretical and practical questions underlying the above idea, and required for its implementation, have been solved, some progress has been done and will be summarized in this section. The invariant manifold structures associated to the collinear libration points provide not only the framework for the computation of complex spacecraft mission trajectories, but also can be used to understand the geometrical mechanisms of the material transport in the solar system. This has been the approach that has been used recently for the design of low energy transfers from the Earth to the Moon (Koon et al. 49) and for a “Petit Grand Tour” of the Moons of Jupiter (Koon et G6mez et al. 2 5 ) . It has also been used to explain the behaviour of some captured Jupiter comets, see Howell et al. 4 2 , Koon et al. 50.

350

G. Gdmez, J.J. Masdemont and J.M. Mondelo

1 1 ,

,

.

,

,

,

.

I

om,

.

,

,

,

,

,

Fig. 11. Li-Lz heteroclinic connection between a Lissajous orbit around La and a quasi-halo orbit around L1. In the lower pictures the intersections of the orbits with the surface of section ( Z = 0) for Lz (left) and for L1 (right) are displayed with crosses.

6.1. Shoot the M o o n

The goal is to produce transfer orbits from the Earth ending at a lunar capture orbit, using less fuel than in a Hohmann transfer. This problem was first considered by Belbruno and Miller and applied to the Hiten mission in 1991. The present procedure, developed by Koon et al. 48, is based in the construction of trajectories with prescribed itineraries and has the following three key steps:

(1) Decouple the Sun-Earth-Moon-Spacecraft system (which is a restricted 4-body problem) in two restricted 3-body problems: the Sun-EarthSpacecraft and the Earth-Moon-Spacecraft systems. (2) Use the stable/unstable manifolds of the periodic orbits about the SunEarth system Lz libration points to provide a low energy transfer from

Libration Point Orbits: A Survey from the Dynamical Point of View 351

Fig. 12. Homoclinic connection between Lyapunov orbits inside the central manifold (in central manifold coordinates).

the Earth to the stable/unstable manifolds of periodic orbits around the Earth-Moon Lz libration point. The "low energy" required is needed because some manoeuvres must be done in order to depart slightly from the manifolds and also because the manifold intersection is not a true one, since they are related to different restricted problems. (3) Finally, use the unstable manifolds of periodic orbits around the EarthMoon libration points to provide a ballistic capture about the Moon. In fact, the procedure works as follows: first a suitable Sun-Earth Lz periodic orbit is computed, as well as their stable and unstable manifolds. Some orbits on the stable manifold come close to the Earth and, at the same time, points close to the unstable manifold propagated backwards in time come close to the stable manifold. So, with an small Av is possible t o go from the Earth to the unstable manifold of this periodic orbit. At the same time, when we consider the LZ point of the Earth-Moon system, is has periodic orbits whose stable manifold ''intersect" the unstable manifold that we have reached departing from the Earth and are temporarily cap-

352

G. Gdmez, J . J . Masdemont and J.M. Mondelo

tured by the Moon. With a second small Av, we can force the intersection to behave as a true one. The orbit computed in this way can be used as an intial guess to find a true solution, in the JPL ephemeris model, performing the prescribed acrobacies.

6.2. P e t i t Grand Tour The general idea for the “Petit Grand Tour” of the Moons of Jupiter is similar to the one of “Shoot the Moon”. In a first step, the Jovian moon nbody system is decoupled into several three-body systems. The tour starts close to the LZ point of an outer moon (for instance Ganymede). Thanks to an heteroclinic connection between periodic orbits around L1 and L2, we can go from the vicinity of L2 to the vicinity of L1 and, in between, perform one o several loops around Ganymede. Now, we can look for “intersections” between the unstable manifold of the p.0. arond the L1 point and the stable manifold of some p.0. around the La point of some inner moon (for instance Europa). By the same considerations, we can turn around Europa and leave its influence through the L1 point. Once the orbits have been obtained, they are easily refined to a more realistic model.

6.3. Solar s y t e m low energy transfers and astronomical applications

As Lo and Ross 52 suggested, the exploration of the phase space structure, as revealed by the homoclinic/heteroclinic structures and their association with mean motion resonances, may provide deeper conceptual insight into the evolution and structure of the asteroid belt (interior to Jupiter) anf the Kuiper belt (exterior to Neptune), plus the transport between these two belts and the terrestrial planet region. Potential Earth-impacting asteroids may utilize the dynamical channels as a pathway to Earth from nearby heliocentric orbits in resonance with the Earth. This phenomenon has been observed recently in the impact of comet Shoemaker-Levy 9 with Jupiter, which was in 2:3 resonance with Jupiter just before impact.

Libration Point Orbits: A Survey from the Dynamical Point of View 353

Numerical simulations of the orbital evolution of asteroidal dust particles show that the earth is embedded in a circumsolar ring of asteroidal dust, known as the zodiacal dust cloud. Both simulations and observations reveal that the zodiacal dust cloud has structure. When viewed in the Sun-Earth rotating frame, there are several high density clumps which are mostly evenly distributed throughout Earth’s orbit. The dust particles are belived to spiral towards the Sun from the asteroid belt, becoming trapped temporarily in exterior mean motion resonances with the Earth. It is suspected that the gross morphology of the ring is given by a simpler RTBP model involving the homoclinic and heteroclinic structures associated with the libration points.

7. Station keeping 7.1. The Target mode approach and the Floquet mode approach The problem of controlling a spacecraft moving near an inherently unstable libration point orbit is of current interest. In the late 1960’s, Farquhar l5 suggested several station-keeping strategies for nearly-periodic solutions near the collinear points. Later, in 1974, a station-keeping method for spacecraft moving on halo orbits in the vicinity of the Earth-Moon translunar libration point (Lz) was published by Breakwell, Kame1 and Ratner 7. These studies assumed that the control could be modeled as continuous. In contrast, specific mission requirements influenced the stationkeeping strategy for the first libration point mission. Launched in 1978, the International Sun-Earth Explorer-3 (ISEE-3) spacecraft remained in a near-halo orbit associated with the interior libration point (L1) of the Sun-Earth/Moon barycenter system for approximately three and one half years (Farquhar la).Impulsive manoeuvres at discrete time intervals (up to 90 days) were successfully implemented as a means of trajectory control. Since that time, more detailed investigations have resulted in various station-keeping strategies, including the two identified here as the Target Point and Floquet Mode approaches. The Target Point method (as presented by Howell and Pernicka 43, Howell and Gordon 39, and Keeter 47 ) computes correction manoeuvres by

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minimizing a weighted cost function. The cost function is defined in terms of a corrective manoeuvre, as well as position and velocity deviations from a nominal orbit at a number of specified future times ti. The non-final state vectors at each time ti are denoted as ”target points.” The target points are selected along the trajectory at discrete time intervals that are downstream of the manoeuvre. In contrast, the Floquet Mode approach, as developed by Sim6 et al. 66,67, incorporates invariant manifold theory and Floquet modes to compute the manoeuvres. Floquet modes associated with the monodromy matrix are used to determine the unstable component corresponding to the local error vector. The manoeuvre is then computed such that it eradicates the dominant unstable component of the error. It is noted that both approaches have been demonstrated in a complex model such as the Earth-Moon system.

Target Point Approach The goal of the Target Point station-keeping algorithm is to compute and implement manoeuvres to maintain a spacecraft ”close” to the nominal orbit, i.e., within a region that is locally approximated in terms of some specified radius centered about the reference path. To accomplish this task, a control procedure is derived from minimization of a cost function. The cost function, J , is defined by weighting both the control energy required to implement a station-keeping manoeuvre, Av, and a series of predicted deviations of the six-dimensional state from the nominal orbit at specified future times. The cost function includes several submatrices from the state transition matrix. For notational ease, the state transition matrix is partitioned into four 3 x 3 submatrices as

The controller, in this formulation, computes a Av in order to change the deviation of the spacecraft from the nominal path at some set of future times. The cost function to be minimized is written in general as

where superscript T denotes transpose. The variables in the cost function include the corrective manoeuvre, Av, at some time t,, and p l , p2 and p3 that are defined as 3 x 1column vectors representing linear approximations of the expected deviations of the actual spacecraft trajectory from the nominal

Libmtion Point Orbits: A Survey f r o m the Dynamical Point of View 355

path (if no corrective action is taken) at specified future times t l , t 2 and t3, respectively. Likewise, the 3 x 1vectors ~ 1 , 2 1 2and v3 represent deviations of the spacecraft velocity at the corresponding ti. The future times at which predictions of the position and velocity state of the vehicle are compared to the nominal path are denoted as target points. They are represented as At, such that ti = t o At,.The choice of identifying three future target points is arbitrary.

+

In Eqn. (20), Q, R, S, T , R,, S,, and T,, are 3 x 3 weighting matrices. The weighting matrix Q is symmetric positive definite; the other weighting matrices are positive semi-definite. The weighting matrices are generally treated as constants that must be specified as inputs. Selection of appropriate weighting matrix elements is a trial and error process that has proven to be time-consuming. A methodology has been developed that automatically selects and updates the weighting matrices for each manoeuvre. This "time-varying" weighting matrix algorithm is based solely on empirical observations. Determination of the Av corresponding to the relative minimum of this cost function allows a linear equation for the optimal control input, i.e., Av* =

+ B&SBzo + B&TB30 + DToRUDio+ D&SUDzo + D$TvD30] -' x [ (BgRBio + B&SBzo + B&T&o + DT&Dio + D & S ~ D Z + O D&TuD3o vo ) + (BToRAlo + B?oSAzo + B&TA3o + DToRUCio + D&SvCzo + D&TuC3o) P O ] , +

- [Q BLRBio

where vo,is the residual velocity (3x 1vector) and po is the residual position (3 x 1 vector) relative to the nominal path at the time to. The performance of the modified Target Point algorithm is not yet truly "optimal," though it has been demonstrated to successfully control the spacecraft at reasonable costs. This accomplishment alone provides the user with a quick and efficient way to obtain reasonable station-keeping results. Given some procedure to select the weighting matrices, the manoeuvre is computed from the above equation. The corrective manoeuvre (Av*) is a function of spacecraft drift (in both position and velocity with respect to the nominal orbit), the state transition matrix elements associated with the nominal orbit, and the weighting matrices. It is assumed here that there is no delay in implementation of the manoeuvre; the corrective manoeuvre occurs at the time t o , defined as the current time. Note that this general method could

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certainly accommodate inclusion of additional target points. Although the nominal orbit that is under consideration here is quasiperiodic, the methodology does not rely on periodicity; it should be applicable to any type of motion in this regime. In this application, three additional constraints are specified in the station-keeping procedure to restrict manoeuvre implementation. First, the time elapsed between successive manoeuvres must be greater than or equal to a specified minimum time interval, tmin. This constraint may be regulated by the orbit determination process, scientific payload requirements, and/or mission operations. Time intervals of one to three days are considered in the Earth-Moon system. The second constraint is a scalar distance (pmin) and specifies a minimum deviation from the nominal path (an isochronous correspondence) that must be exceeded prior to manoeuvre execution. For distances less than pmin, manoeuvre computations do not occur. Third, in the station-keeping simulation, the magnitude of POsition deviations are compared between successive tracking intervals. If the magnitude is decreasing, a manoeuvre is not calculated. For a corrective manoeuvre to be computed, all three criteria must be satisfied simultaneously. I

After a manoeuvre is calculated by the algorithm, an additional constraint is specified on the minimum allowable manoeuvre magnitude, Av,in. If the magnitude of the calculated - Av is less than Avmin then the recommended manoeuvre is cancelled. This constraint is useful in avoiding "small" manoeuvres that are approximately the same order of magnitude as the manoeuvre errors. It also serves to model actual hardware limitations. Floquet Mode Approach An alternate strategy for station-keeping is the Floquet Mode approach, a method that is significantly different from the Target Point approach. It

can be easily formulated in the circular restricted three-body problem. In this context, the nominal halo orbit is periodic. The variational equations for motion in the vicinity of the nominal trajectory are linear with periodic coefficients. Thus, in general, both qualitative and quantitative information can be obtained about the behavior of the nonlinear system from the monodromy matrix, M , which is defined as the state transition matrix (STM) after one revolution along the full halo orbit.

Libration Point Orbits: A Survey from the Dynamical Point of View 357

The knowledge of the dynamics of the flow around a halo orbit, OF any solution close to it, allows other possibilities in addition to the stationkeeping procedure described here, such as the computation of transfer orbits both between halo orbits and from the Earth to a halo orbit (G6mez et al. The behavior of the solutions in a neighborhood of the halo orbits is determined by the eigenvalues, Xi, i = 1,.. . , 6 and eigenvectors e i , i = 1,.. . , 6 of M. Gathering the eigenvalues by pairs, their geometrical meaning is the following: 21329).

a) The first pair (XI, A), with X1.X2 = 1 and X1 x 1500, is associated with the unstable character of the small and medium size halo orbits. The eigenvector, el(to), associated with the largest eigenvalue, XI, defines the most expanding direction, related to the unstable nature of the halo orbit. The image under the variational flow of the initial vector el(to), together with the vector tangent to the orbit, defines the linear approximation of the unstable manifold of the orbit. In a similar way, ez(t0) can be used to compute the linear approximation of the stable manifold. b) The second pair (As, X4) = (1, 1) is associated with neutral variables (i.e., unstable modes). However, there is only one eigenvector with eigenvalue equal to one. This vector, e 3 ( t o ) is the tangent vector to the orbit. The other eigenvalue, A4 = 1, is associated with variations of the energy (or the period) of the orbit through the family of halo orbits. Along the orbit, the vectors e3 and e4 span an invariant plane under the flow. c) The third couple, (As, Xe), is formed by two complex conjugated eigenvectors of modulus one. The restriction of the flow to the corresponding two-dimensional invariant subspace, is essentially a rotation. This behavior is related to the existence of quasiperiodic halo orbits around the halo orbit (see G6mez et a2. 3 3 ) .

When considering dynamical models of motion different from the restricted three-body problem, halo orbits are no longer periodic, and the monodromy matrix is not defined. Nevertheless, for quasiperiodic motions close to the halo orbit (and also for the Lissajous orbits around the equilibrium point) the unstable and stable manifolds subsist. The neutral behavior can be slightly modified including some instability which, from a practical point of view, is negligible when compared with the one associated with XI.

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Instead of the vectors e i ( t ) it is convenient to use the Floquet modes E i ( t ) which, for the periodic case, are defined as six periodic vectors from which the ei(t) can be easily recovered (see Wiesel 'l). For instance E , ( t ) is defined as e l ( t ) exp[-(t/T)logX~], where T is the period of the halo orbit. The control algorithm is developed to utilize this information for station-keeping purposes. The emphasis is placed on formulating a controller that will effectively eliminate the unstable component of the error vector, 6 ( t ) = (6x,Sy,6z, 6x,6y, 6.2) defined as the difference between the actual coordinates obtained by tracking and the nominal ones computed isochronously on the reference orbit. At any epoch, t , 6 can be expressed in terms of the Floquet modes 6

i=l

The controller objective is to add a manoeuvre such that the magnitude of the component of the error vector in the unstable direction, C Y ~ ,is reduced to zero. The five remaining components do not produce large departures from the reference orbit. By contrast, the component of the error vector along the unstable mode increases by a factor of A1 in each revolution. Denoting the impulsive manoeuvre as A = (O,O, 0, A,, A,, A,)T, to cancel the unitary unstable Floquet mode, it is required that 6

F'rom these equations A,, A,, A, can be obtained as a function of c5 and c6. These free parameters are determined by either imposing a constraint on the available directions of the control or by minimizing a suitable norm of A. For practical implementation it is useful to compute the so-called projection factor along the unstable direction. It is defined as the vector 'IT such that 6 . T = c q . Note that for the computation of T only the Floquet modes are required, so it can be computed and stored together with the nominal orbit. To annihilate the unstable projection, a1, with a manoeuvre, Av = (O,O, 0, A,, A,, A,)T, we ask (6 Av) . 'IT = 0. In this way,

+

+

A,'IT~ A,TS

+ Az'IT~+ a1 = 0

(23)

Libmtion Point Orbits: A Survey from the Dynamical Point of View 359

is obtained, where ~ 4 ~5, and 7r6 are the last three components of T . Choosing a two axis controller, with Az = 0, and minimizing the Euclidean norm of Av, the following expressions for Ax and Ay are obtained,

A x --_-

a1r4

T i + T,z

A -_y -

a1T5 T%+T,z'

(24)

In a similar way, a one or three axis controller can be formulated. Once the magnitude of the manoeuvre is known, an important consideration is the determination of the epoch at which it must be applied. The study of this question requires the introduction of the gain function, g ( t ) = JJAJJ-l, where A is the unitary impulsive manoeuvre. It measures the efficiency of the control manoeuvre along the orbit to cancel the unitary unstable component. This component is obtained using the projection factors and the error vector. As the projection factor changes along the orbit, the same error vector has different unstable components. It is natural then to consider a delay in the manoeuvre until reaching a better epoch with less cost. So, the function to be studied is

However, as is shown in Sim6 et al. 66, this function is always increasing, therefore, it is never good to wait for a manoeuvre except for operational reasons.

As it has been said, when the station keeping has to span for a long time, the satellite can tend to deviate far away from the nominal orbit. This could happen since the cancellation of the unstable component does not take care of the neutral components which might grow up to the limit of loosing controllability. In order to prevent large deviations of the satellite from the nominal orbit, it is advisable to perform manoeuvres of insertion in the stable manifold. The main idea of the strategy is to put the satellite in a state such that approaches the nominal orbit assimptotically in the future. This strategy is in principle much cheaper than to target to the nominal orbit itself since the latter case can be considered, from an implementation point of view, as a sub-case of targeting to the stable manifold. Moreover, even when the controllability using only unstable component cancellation manoeuvres (UCCM) is assured, it can be advantageous to perform insertion in the stable manifold since the control effect of this manoeuvres usually persist for a longer time span than UCCM. Moreover, subsequent

360 G. Gdmez. J.J. Masdemont and J.M. Mondelo

UCCM would be cheaper due to the fact that the satellite is closer to the nominal orbit, and consequently the projection of the deviation in the unstable component is smaller. Although the idea is simple, the implementation is not so easy since in first place, the target state in the stable manifold cannot be accomplished with a single manoeuvre as it happens with UCCM and, in second place, the actual state of the satellite is known but affected by tracking errors. Moreover the manoeuvres to be done will be noised by some errors too. We refer the interested reader to G6mez et al. 31 for the details of the implementation. As a final remark, several constraints that impact the manoeuvres must be specified in the procedure. The most relevant are the time interval between two consecutive tracking epochs (tracking interval), the minimum time interval between manoeuvres, and the minimum value of a1 that cannot be considered due solely to tracking errors. Special emphasis must be placed on the evolution of al. With no tracking errors, the evolution of this parameter is exponential with time (see Figures 13 and 14). When adding tracking errors, and in order to prevent a useless manoeuvre, this value must be greater than the minimum. So, the minimum value must be selected as a function of the accuracy in orbit determination. On the other hand, the value of al should not be too large, because this increases the value of the manoeuvre in an exponential way. Thus, a maximum value is chosen such that, if a1 is greater than the maximum, a control manoeuvre will be executed to cancel the unstable component. When a1 is between the minimum and maximum values, the error can be due to small oscillations around the nominal orbit. In this case, a manoeuvre is executed only if the error has been growing at an exponential rate in the previous time steps and the time span since the last manoeuvre agrees with the one selected. Also, if the magnitude of the calculated Av is less than Av,in, then the recommended manoeuvre is cancelled. Once these parameters have been fixed, there are no more free variables allowing any further minimization.

Libration Point Orbits: A Survey from the Dynamical Point of View

No orbit determination.No Ws insertion.No errors

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Fig. 13. From top to bottom, evolution with time of the unstable component, position deviations with respect t o the nominal trajectory, and velocity deviations. In all the figures, no orbit determination has been performed, because the simulations have been done with no errors for the tracking and the execution of the manoeuvres. There is only an error at the initial insertion epoch. In the left hand side figures there are no manoeuvres for the insertion in the stable manifold, while in the right hand side there are. These manoeuvres can be clearly seen, because after its execution the distance to the nominal orbit goes to zero both in position and in velocity. The discontinuities that appear in these two figures are associated to the execution of the manoeuvres. The points marked with a cross are those at which the tracking has been performed, and the ones marked with a star are those at which a manoeuvre has been executed.

7.2. Numerical results In Figures 13 and 14, we show some results of simulations done for a halo orbit around LZ in the Earth-Moon system. We display the evolution of

G. Gbmez, J.J. Masdemont and J.M. Mondelo

362

No orbit determination. Ws insertion. Errors 6e-05,

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0 800

300

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Fig. 14. From top to bottom, evolution with time of the unstable component, position deviations with respect t o the nominal trajectory, and velocity deviations. In the left figures no orbit determination if performed, while in the right ones it is done. In all the figures, the manoeuvres and the tracking are performed with errors. There is also an error at the initial insertion epoch. There are manoeuvres for the insertion in the stable manifold that can be clearly seen at the times at which the distance t o the nominal orbit decreases t o very small values (which are not equal t o zero because there is an error added to the manoeuvres). The points marked with a cross are those at which the tracking has been performed, and the ones marked with a star are those at which a manoeuvre has been executed.

the unstable component and the deviations from the nominal trajectory in position and velocity in different situations. In Figure 13 the manoeuvres are done without any error, while in Figure 14 they are performed with errors.

Libmtion Point Orbits: A Sumey from the Dynamical Point of View 363

In each Figure, we show the results of the station keeping strategy with and without stable manifold insertion manoeuvres. From them, it becomes clear both the exponential grow of a1 and the role of the insertion manifold manoeuvres. Finally, in Figure 15 the averaged Av used for the station keeping is displayed. As before, the simulations correspond to a halo orbit around LZ in the Earth-Moon system. If no insertion manifold manoeuvres are done, there appears an exponential grow of the Av, while if these manoeuvres are done, the station keeping cost per year remains constant. From this Figure, it is also clear that a good orbit determination procedure can be useful to reduce the total Av. 3500 3000

F$

-5 Fi

2500 2000

i 1 : ii :;

1500

lo00

/ ;

1..-. _ _ _ ................................................ _ _ ~ . . ..... ...... ~

500

-

500

__..__...-.. .:;

-

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Fig. 15. Averaged Av used for the station keeping, in cm/s/year, in different situations. The two curves with an exponential grow of the Av correspond to simulations with no insertions in the stable manifold. For the upper curve, there was no orbit determination. For the other two curves, for which there seems to be a finite limit for the Av, we have used insertion manoeuvres in the stable manifold. The one with lower cost uses orbit determination, whereas the other does not.

8. Application of libration point orbits to formation flight

The excellent observational properties of the L2 point of the Earth-Sun system have lead to consider this location for missions requiring a flying multiple spacecraft in a controlled formation. Darwing, LISA and TPF are three of the more challenging examples of such missions.

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Two basic orbital strategies have been analyzed for a formation flying mission at the libration points: a nominal orbit strategy and a base orbit strategy. In the nominal orbit strategy each spacecraft follows its own predefined orbit, while in the base orbit strategy each spacecraft follows an orbit relative to a predefined one, known as the base orbit. The base orbit may have no spacecraft on it. In the next two sections we will briefly discuss the results that have been obtained in both approaches.

8.1. T h e nominal orbit strategy Barden and Howell 3,4,38 have considered the possibility of using quasiperiodic solutions around the halo orbits and the libration points, as natural locations for a constellation. In their studies, a certain number of spacecraft are initially placed along a planar curve close to any of the two kinds of the above mentioned tori and, in a first step, they analyze their natural motion. In the quasi-halo case, the torus itself is related to an underlying periodic halo orbit. As the initial planar curve proceeds in time along the torus, in the direction of motion of the underlying halo orbit, there are certain aspects of the evolution of the curve that are of particular interest. The curve appears simply closed and nearly circular in configuration space. When the amplitude of the curve is small, i.e., less than 1000 km, the curve is considered to be planar. However, as the curve evolves, it changes size and shape. Identifying the plane containing the curve, one can view the constellation from a point along the normal to the plane. Although the plane will not persist, the deviations from a reference one are small: less than 1%of the distance between spacecraft when they are on opposite sides of the constellation. In addition to the variations in size and shape, there is a winding aspect of the motion due to the change of the relative locations of the points of the curve. This is because the torus is self-intersectiong in the configuration space at the two xz-plane crossings. This type of natural motion, as an option for formation flying, is very appealing from a dynamical perspective. From a practical standpoint, however, this formation will likely not meet the constraints and scientific requirements of a generic mission. The likely scenario is that some prespecified formation will be mandated.

Libration Point Orbits: A Survey from the Dynamical Point of View 365

As an example of non-natural formation, Barden and Howell consider six spacecraft evenly distributed on a circle of radius 100 km in a plane coincident with the rotating libration point coordinates y and z (parallel to the yz plane) and around a Lissajous type orbit. At each manoeuvre, the formation is enforced to be on the plane, but there will be out-off-plane excursions for each of the spacecrafts between the manoeuvres; the amplitude of the excursions will vary for each vehicle. In a first simulation, four manoeuvres per revolution in the xy plane (nearly equally spaced in time) are executed where all six spacecraft implement their respective manoeuvres simultaneously. The size of the manoeuvres ranges from 0.043 m/s to 0.12 m/s for a total cost of 2.93 m/s for a duration of 355 days (which is equivalent to two revolutions along the baseline Lissajous trajectoy in the xy plane. These manoeuvres are necessary to define a nominal path for each of the spacecraft; additional stationkeeping manoeuvres will also be required to accommodate errors and uncertainties. Even for the baseline motion, however, out-of-plane excursions between the manoeuvres reach a maximum value at any one time of approximately 20 km in this example. The only means of reducing this deviation is to increase the frequency of the manoeuvres. With manoeuvres every 11 days, instead of 44 days, the outof-plane deviations never exceed 1.8 km. The total cost is of 2.77 m/s which is smaller than the 2.93 m/s required. However, fixed planes can be specified where the total cost increases with the increased number of manoeuvres.

8 .2 . Formation flight in the vicinity of a libration point. TPF case

The TPF Mission (Terrestrial Planet Finder) is one of the center pieces of the NASA Origins Program. The goal of TPF is to identify terrestrial planets around stars nearby the Solar System (see Beichman e t al. 5 ) . For this purpose, a space-based infrared interferometer with a baseline of approximately 100 m is required. To achieve such a large baseline, a distributed system of five spacecraft flying in formation is an efficient approach. Since the TPF instruments needs a cold and stable environment, near Earth orbits are unsuitable. Two potential orbits have been identified: a SIRTF-like heliocentric orbit and a libration orbit near the L2 Lagrange point of the Earth-Sun system. There are several advantages to a libration orbit near Lz. Such orbits are easy and inexpensive to get to from Earth. Moreover, for missions with heat sensitive instruments (e.g. IR detectors), libration orbits

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provide a constant geometry for observation with half of the entire celestial sphere available at all times. The spacecraft geometry is nearly constant, with Sun, Earth, and Moon always behind the spacecraft, thereby providing a stable observation environment, making observation planning much simpler. In this section we present some of the results of G6mez et al. 20, which contain preliminary computations of the TPF mission. From the dynamical point of view, the TPF Mission can be broken into four scenarios: Launch and Transfer Phase

For the simulation, it is assumed that the spacecraft starts in a typical 200 km altitude parking orbit at 28.5 deg inclination, and a halo orbit is used as a target Baseline Orbit. At the appropriate time, the spacecraft performs a major manoeuvre of about 3200 m/s. This injects the spacecraft onto the stable manifold of the halo orbit to begin the Transfer Phase. The transfer trajectory is designed by using an orbit of the stable manifold with a suitable close approach to the Earth. Deployment Phase

It is assumed that all the spacecraft of the formation reach the Baseline Orbit in a single spacecraft. This begins the Deployment Phase. The five satellites are manoeuvreed to reach their initial positions on the different points of a 20-gon of 100m diameter at the same time. The Deployment Phase can last several hours and simulations between 1 and 10 hours have been done. Assuming that deployment is performed using two impulsive maneouvres, and that a selected satellite has to be put in the edge of an 20-gon of diameter D meters, and after that starts doing R revolutions per day, the following table summarizes the estimation of the deployment cost in cm/s as a function of the deployment time. Deployment Time

10 Hr 100 Hr

Libration Point Orbits: A Survey from the Dynamical Point of View 367

Pattern Maintenance Phase Once the initial configuration has been established, the spacecraft will manoeuvre to follow the edge of the 20-gon to provide a suitable spin rate for the formation. The nominal spin rate used for this simulation is 360 deg every 8 hours. The period where the pattern is maintained is called the Pattern Maintenance Phase. Assuming that a spacecraft is spining in a 20gon of diameter D meters and doing R revoulitions per day, it is obtained that: Formation maintenance cost per satellite in cm/s per Day = 0.0023 D R2. Reconfiguration Phase Once sufficient data has been acquired for one star system, the formation will be pointed to another star for observation. Repointings occur during the Reconfiguration Phase. The computations of the Reconfiguration Phase cost is similar to the Deployment Phase, except that the spacecraft do not depart from the same location (i.e. the Mother Ship). Estimation of TPF budget for a ten year’s mission A table presenting an estimation of the AV cost associated to satellites located in an N-gon of 50 and 100 m around a L2 base halo orbit, spining at the rate of 3 revolutions per day, for a 10 years mission is also given. Manoeuvres per S/C in m/s Halo Insertion Initial Deployment (10h) Formation Maintenance Station Keeping (2-Axis) Reconfiguration (est.) 10 Year DV Budget (m/s)

50m Diameter Case 5 0.009 O.l/Day 3/Yr O.O5/Day 585

lOOm Diameter Case 5 0.018 O.Z/Day 3/Yr O.l/Day 1135

Halo insertion cost, due to transfer from the Earth, and station keeping cost (including avoidance of the exclusion zone, that could be required in case of using an La Lissajous orbit) are also included. The usual station keeping can be assumed to be absorbed in the so often performed pattern maintenance manoeuvres. Manoeuvres are also considered to be done

368 G. Gdmez, J.J. Masdemont and J.M. Mondelo

without error, so control correction manoeuvres are not included. Finally, the paper ends with some issues related to the TPF simulations and the visualization tools suitable for the design of the mission.

Acknowledgments The work has been partially supported by DGICYT grant PB94-0215 and CIRIT grant 2000 SGR-27. The authors are indebted to their colleages K. Howell, A. Jorba, W.S. Koon, J. Llibre, M.W. Lo, J.E. Marsden, R. Martinez, S. Ross, and, very specially, to C. Sim6, with whom they have done most of the work contained in this review paper.

References 1. M.A. Andreu: The Quasibicircular Problem. PhD thesis, Dept. Matemhtica Aplicada i Anhlisi, Universitat de Barcelona, Barcelona, Spain, 1999. 2. M.A. Andreu: Dynamics in the Center Manifold Around 12 in the Quasi-Bicircular Problem. Celestial Mechanics and Dynamical Astronomy, 84(2) :105-133, 2002. 3. B.T. Barden and K.C. Howell Formation Flying in the Vicinity of Libration Point Orbits. In A A S Paper 98-169, Monterey, CA., 1998. 4. B.T. Barden and K.C. Howell: Dynamical Issues Associated with Relative Configurations of Multiple Spacecraft near the Sun-Earth/Moon 11 Point. In A A S Paper 99-450, Girdwood, Alaska., 1999. 5. C.A. Beichman, N.J. Wolf and C.A. Lindensmith: The Terrestrial Planet Finder (TPF). JPL Publications 99-003, 1999. (http://tpf.jpl.nasa.gov). 6. E. Belbruno and J. Miller: Sun-Perturbed Earth to Moon Transfers with Ballistic Capture. Journal of Guidance, Control and Dynamics, 16:77&775, 1993. 7. J.V. Breakwell, A.A. Kame1 and M.J. Ratner: Station-Keeping for a Translunar Communica!ions Station. Celestial Mechanics, 10(3):357-373, 1974. 8. E. Castellh and A. Jorba: On the Vertical Families of Two-Dimensional Tori Near the Triangular Points of the Bicircular Problem. Celestial Mechanics and Dynamical Astronomy, 76:35-54, 2000. 9. J. Cob- and M. Hechler: FIRST Mission Analysis: Transfer to Small Lissajous Orbits around L2. Technical Report MAS Working Paper No. 398, ESOC, 1997. 10, J. Cobos and J.J. Masdemont: Astrodynamical Applications of Invariant Manifolds Associated with Collinear Lissajous Libration Orbits. In Libration Point Orbits and Applications, 2003.

Libration Point Orbits: A Survey from the Dynamical Point of View 369

11. J. Cobos and J.J. Masdemont: Transfers Between Lissajous Libration Point Orbits. Technical report, In preparation. 12. A. Deprit: Canonical Transformations Depending on a Small Parameter. Celestial Mechanics, 1(1):12-30, 1969. 13. E.J. Doedel, R.P. Paffenroth, H.B. Keller, D.J. Dichmann, J. Galbn-Vioque, and A. Vanderbauwhede: Computation of Periodic Orbits of Conservative Systems with Applications to the 3-body Problem. Int. J. Bifurcation and Chaos, 2002. 14. N. Eismont, D. Dunham, J. Shc-Chiang and R.W. Farquhar: Lunar Swingby as a Tool for Halo-Orbit Optimization in Relict-2 Project. In Third International Symposium on Spacecmft Flight Dynamics, E S A SP-326. European Space Agency (Darmstadt, Germany), 1991. 15. R.W. Farquhar: Far Libration Point of Mercury. Astronautics & Aeronautics, 5(8):4, 1967. 16. R.W. Farquhar: The Control and Use of Libration Point Satellites. Technical Report TR R346, Stanford University Report SUDAAR-350 (1968). Reprinted as NASA, 1970. 17. R.W. Farquhar: The Moon’s Influence in the Location of the Sun-Earth Exterior Libration Point. Celestial Mechanics, 2(2):131-133, 1970. 18. R.W. Farquhar, D.P. Muhonen, C.R. Newman and H.S. Heuberger: TPrajectories and Orbital Maneuvers for the First Libration-Point Satellite. Journal of Guidance and Control, 3(6):549-554, 1980. 19. F. Gabern and A. Jorba: A Restricted Four-Body Model for the Dynamics near the Lagrangian Points of the Sun-Jupier System. Discrete and Continuous Dynamical Systems. Series B, 1(2):143-182, 2001. 20. G. G6mez, M.W. Lo, J.J. Masdemont and K. Museth: Simulation of formation flight near 12 for the tpf mission. In A S S / A I A A Space Flight Mechanics Conference. Paper A A S 01-305, 2001. 21. G. Gbmez, A. Jorba, J. Masdemont and C. Sim6: Study of the Transfer from the Earth to a Halo Orbit Around the Equilibrium Point L1. Celestial Mechanics, 56(4):541-562, 1993. 22. G. G6mez, Jorba, J.J. Masdemont and C. Sim6: Moon’s Influence on the Transfer from the Earth to a Halo Orbit. In A. E. Roy, editor, Predictability, Stability and Chaos in N-Body Dynamical Systems, pages 283-290. Plenum Press, 1991. 23. G. Gbmez, A. Jorba, J.J. Masdemont and C. Sim6: Dynamics and Mission Design Near Libration Point Orbits - Volume 3: Advanced Methods for Collinear Points. World Scientific, 2001. 24. G. G6mez, A. Jorba, J.J. Masdemont and C. Sim6: Dynamics and Mission Design Near Libration Point Orbits - Volume 4: Advanced Methods for Triangular Points. World Scientific, 2001. 25. G . G6mez, W.S. Koon, M.W. Lo, J.E. Marsden, J.J. Masdemont and S.D. Ross: Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design. Advances in The Astronautical Sciences, 109, 1:3-22, 2001. 26. G. G6mez, J. Llibre, R. Martinez and C. Sim6: Dynamics and Mission Design Near Libration Point Orbits - Volume 1: Fundamentals: The Case of

A.

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Collinear Libration Points. World Scientific, 2001. 27. G. G6mez, J. Llibre and J. Masdemont: Homoclinic and Heteroclinic Solutions in the Restricted Three-Body Problem. Celestial Mechanics, 44:239259, 1988. 28. G. G6mez, M. Marcote and J.J. Masdemont: Trajectory Correction Manoeuvres in the Transfer to Libration Point Orbits. In these Proceedings. 29. G. G6mez, J. Masdemont and C. Sim6: Study of the Transfer Between Halo Orbits. Acta Astronautica, 43(9-10):493-520, 1998. 30. G. G6mez and J.J. Masdemont: Some Zero Cost Transfers Between Halo Orbits. Advances in the Astronautical Sciences, 105(2):1199-1216, 2000. 31. G. G6mez and J.J. Masdemont: Refinements of a Station-Keeping Strategy for Libration Point Orbits. In preparation, 2002. 32. G. G6mez, J.J. Masdemont and J.M. Mondelo: Solar System Models with a Selected Set of Frequencies. Astronomy €4 Astrophysics, 390:733-749, 2002. 33. G. G6mez, J.J. Masdemont and C. Sim6: Quasihalo Orbits Associated with Libration Points. Journal of The Astronautical Sciences, 46(2):1-42, 1999. 34. G. G6mez and J.M. Mondelo: The Dynamics Around the Collinear Equilibrium Points of the RTBP. Physica D, 157(4):283-321, 2001. 35. M. Hechler: S O H 0 Mission Analysis L1 Transfer Trajectory. Technical Report M A 0 Working Paper No. 202, ESA, 1984. 36. L.A. Hiday and K.C. Howell: Transfers Between Libration-Point Orbits in the Elliptic Restricted Problem. In AAS/AIAA Spaceflight Mechanics Conference, Paper A A S 92-126., 1992. 37. K.C. Howell and B.T. Barden: Brief Summary of Alternative Targeting Strategies for TCMl, TCM2 and TCM3. Private communication. Purdue University, 1999. 38. K.C. Howell and B.T. Barden: Trajectory Design and Station Keeping for Multiple Spacecraft in Formation Near de Sun-Earth L1 Point. In General Conference of the International Astronautical Federation. Paper IAFN-A. 7.07, 1999. 39. K.C. Howell and S.C. Gordon: Orbit Determination Error Analysis and a Station-Keeping Strategy for Sun-Earth L1 Libration Point Orbits. Journal of the Astronautical Sciences, 42(2):207-228, April-June 1994. 40. K.C. Howell and J.J. Guzman: Spacecraft Trajectory Design in the Context of a Coherent Restricted Four-Body problem with Application to the MAP Mission. In Congress of the International Astronautical Federation, volume IAF Paper 00-A.5.06, 2000. 41. K.C. Howell and L.A. Hiday-Johnston: Time-Free Transfers Between Libration-Point Orbits in the Elliptic Restricted Problem. Acta Astronautics, 32:245-254, 1994. 42. K.C. Howell, B.G. Marchand and M.W. Lo: Temporary Satellite Capture of Short-Period Jupiter Family Comets from the Perspective of Dynamical Systems. In AAS/AIAA Space Flight Mechanics Meeting, A A S Paper 00-155, 2000. 43. K.C. Howell and H.J. Pernicka: Stationkeeping Method for Libration Point Trajectories. Journal of Guidance, Control and Dgnamics, 16(1):151-159,

Librution Point Orbits: A Survey from the Dynamical Point of View 371 1993.

44.

A. Jorba:

Quasiperiodic Perturbations of Ordinary Differential Equations. PhD thesis, Universitat de Barcelona, Barcelona, Spain, 1991. 45. A. Jorba and J.J. Masdemont: Dynamics in the Center Manifold of the Restricted Three-Body Problem. Physica D, 132:189-213, 1999. 46. A. Jorba and J. Villanueva: On the Normal Behaviour of Partially Elliptic Lower Dimensional Tori of Hamiltonian Systems. Nonlinearity, 10:783-822, 1997. 47. T.M. Keeter: Station-Keeping Strategies for Libration Point Orbits: Target Point and Floquet Mode Approaches. Master’s thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, 1994. 48. W.S. Koon, M.W. Lo, J.E. Marsden and S.D. Ross: Heteroclinic Connections

Between Periodic Prbits and Resonance Transitions in Celestial Mechanics. Chaos, 10(2):427-469, 2000. 49. W.S. Koon, M.W. Lo, J.E. Marsden and S.D. Ross: Low Energy Transfer to the Moon. Celestial Mechanics and Dynamical Astronomy, 81( 1):63-73, 2001. 50. W.S. Koon, M.W. Lo, J.E. Marsden and S.D. Ross: Resonance and capture of jupiter comets. Celestial Mechanics and Dynamical Astronomy, 81( 1):27-38, 2001. 51. D.F. Lawden: Optimal Trajectories for Space Navigation. Butterworths & Co. Publishers, London, 1963. 52. M.W. Lo and S. Ross: SURFing the Solar Sytem: Invariant Manifolds and the Dynamics of the SoIar System. Technical Report 312/97, 2-4, JPL IOM, 1997. 53. M.W. Lo, B.G. Williams, W.E. Bollman, D. Han, Y. Hahn, J.L. Bell, E.A.

54.

55.

56.

57. 58. 59.

Hirst, R.A. Corwin, P.E. Hong, K.C. Howell, B.T. Barden and R.S. Wilson: Genesis Mission Design. In AIAA Space Flight Mechanics, Paper No. AIAA 98-4468, 1998. J.J. Masdemont: Estudi i Utilitzacid de Varietats Invariants en Problemes de Mechnica Celeste. PhD thesis, Universitat Polit&cnica de Catalunya, Barcelona, Spain, 1991. J.M. Mondelo: Contribution to the Study of Fourier Methods for QuasiPeriodic Functions and the Vicinity of the Collinear Libration Points. PhD thesis, Dept. Matembtica Aplicada i Anblisi, Universitat de Barcelona, Barcelona, Spain, 2001. J.R. Pacha: On the Quasi-Periodc Hamiltonian Andronov-Hopf Bifurcation. PhD thesis, Dept. de Matemitica Aplicada I, Universitat Politkcnica de Catalunya, Barcelona, Spain, 2002. D.L. Richardson: Analytical Construction of Periodic Orbits About the Collinear Points. Celestial Mechanics, 22(3):241-253, 1980. D.L. Richardson: A note on the Lagrangian Formulation for Motion About the Collinear Points. Celestial Mechanics, 22(3):231-235,J1980. R.S. Wilson, K.C. Howell and M.W. Lo: Optimization of Insertion Cost Transfer Trajectories to Libration Point Orbits. Advances in the Astronautical Sciences, 103:1569-1586, 2000.

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60. D.J. Scheeres: The Restricted Hill Four-Body Problem with Applications to the Earth-Moon-Sun System. Celestial Mechanics and Dynamical Astronomy, 70(2):75-98, 1998. 61. R. Serban, W.S. Koon, M.W. Lo, J.E. Marsden, L.R. Petzold, S.D. Ross and R.S. Wilson: Halo Orbit Mission Correction Maneuvers Using Optimal Control. Autornatica, 38:571-583, 2002. 62. C.L. Siege1 and J.K. Moser: Lectures on Celestial Mechanics, SpringerVerlag, 1971. 63. C. Simb: Effective Computations in Hamiltonian Dynamics. In Societ6 Mathematique de Fkance, editor, Cent ans a p r h les Mithodes Nouvelles de H. Poincari, pages 1-23, 1996. 64. C. Sim6: Dynamical Systems Methods for Space Missions on a Vicinity of Collinear Libration Points. In C. Sim6, editor, Hamiltonian Systems with Three or More Degrees of Freedom, pages 223-241. Kluwer Academic Publishers, 1999. 65. C. Sim6, G. G6mez, A. Jorba and J. Masdemont: The Bicircular Model Near the Triangular Libration Points of the RTBP. In A. Roy and B. Steves, editors, From Newton t o Chaos, pages 343-370. Plenum Press, 1995. 66. C. Sim6, G . G6mez, J. Llibre and R. Martinez: Station Keeping of a Quasiperiodic Halo Orbit Using Invariant Manifolds. In Second International Symposium o n Spacecraft Flight Dynamics, pages 65-70. European Space Agency, Darmstadt, Germany, October 1986. 67. C. Sim6, G. G6mez, J. Llibre, R. Martinez and R. Rodriquez: On the Optimal Station Keeping Control of Halo Orbits. Acta Astronautica, 15(6):391-397, 1987. 68. C. Sim6 and T.J. Stuchi: Central Stable/Unstable Manifolds and the Destruction of KAM Tori in the Planar Hill Problem. Physica D , 140(1-2):l-32, 2000. 69. J. Stoer and R. Bulirsch: Introduction to Numerical Analysis. Springer Verlag, 1983. 70. V. Szebehely: Theory of Orbits. Academic Press, 1967. 71. W. Wiesel and W. Shelton: Modal Control of an Unstable Periodic Orbit. Journal of the Astronautical Sciences, 31(1):63-76, 1983.

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

DYNAMICAL SUBSTITUTES OF THE LIBRATION POINTS FOR SIMPLIFIED SOLAR SYSTEM MODELS G. GOMEZ

IEEC €4 Departament de Matemcitica Aplicada i Ancilisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

J.J. MASDEMONT and J.M. MONDELO IEEC €4 Departament de Matemcitica Aplicada I, Universitat Politdcnica de Catalunya E.T.S.E.I.B., Diagonal 647, 08028 Barcelona, Spain

The purpose of this paper is to develop a methodology to generate simplified models suitable for the analysis of the motion of a small particle, such as a spacecraft or an asteroid, in the Solar System. The procedure is based on applying refined Fourier analysis methods to the timedependent functions that appear in the differential equations of the problem. The equations of the models obtained are quai-periodic perturbations of the Restricted Three Body Problem that depend explicitly on natural frequencies of the Solar System. Some examples of these new models are given and compared with other ones found in the literature. For two of these new models, close to the Earth-Moon system, we have computed the dynamical substitutes of the collinear libration points.

1. Introduction

The main goal of this paper is the construction of quasi-periodic analytic models suitable for the study of the motion of a small particle in the Solar 373

374

G. Gdmez, J.J. Masdemont and J.M. Mondelo

System. Without any simplification, the equations of the general problem form a set of 60 first order differential equations difficult to analyze. It is well known that very simple models, such as the Two Body Problem or the Restricted Three Body Problem (RTBP), are suitable for many purposes, since they give a good insight of dynamics in large regions of the phase space of the problem. Some of these models are restricted, which means that the small particle does not have any influence in the motion of the remaining bodies. The models introduced in this paper will be also restricted but not so simple as the ones already mentioned. We will try to keep within them the behavior of the dynamics related to the resonances between natural and excitation frequencies. Most of the well known restricted problems take as starting point the RTBP. In our approach, instead of taking as starting equations those of the RTBP, we will consider Newton's equation for the motion of an infinitesimal body in the force field created by the bodies of the Solar System. Following the ideas of l , in Sect. 2 we introduce suitable reference systems and units such that, after selecting two bodies of the Solar System as primaries, the previously mentioned equations are set as a perturbation of the RTBP. In Sect. 3, and for two particular choices of primaries, we perform the Fourier analysis of all the time periodic functions that appear in the new equations. In this way we are able to introduce a graded set of models with an increasing number of frequencies, that can be considered between the RTBP and the true equations. This is done in Sect. 4. Finally, in the last section we compute the dynamical substitutes of the collinear equilibrium points for two of the intermediate models introduced, close to the Earth-Moon system. The computation of the dynamical substitutes of the libration points in models different from the ones developed here can be found in 233,435

Most of the results to be developed in this paper can be also found in '. This reference includes additional details about the development of models for the Earth-Moon system, which are not be given here.

Dynamical Substitutes of Libration Points 375

2. General equations of motion

Through the full paper, the set of bodies of the Solar System will be denoted by

S = {Pl,...,Pll}

(1)

where PI,. . . ,Pll are the nine planets, the Moon and the Sun, respectively. The mass of P E S will be denoted by mp. Let us consider, in an intertial reference frame, Newton's equations for the motion of an infinitesimal body in the force field created by the bodies of the Solar System,

were G is the gravitational constant, R is the position of the infinitesimal body, Ri is the position of the Solar System body i in the inertial system chosen and mi its mass. The associated Lagrangian is 1

+ i E S IIRGmi - Rill '

L(R,R',t*)= T(R',R')

where R = ( X ,Y,Z ) T is the position of Q, the prime denotes the derivative with respect to time, t*, (R',R') is the dot product between R' and R', G is the gravitational constant, Ri is the position of the body i E S and 11 . 11 denotes the Euclidean norm. In practice, it is convenient that the reference frame and units, both in space and time, are consistent with the ephemerides data files used for the determination of Ri. Since we are interested in writing the equations of motion for Q as a perturbation of the RTBP equations, we must select two bodies I , J E S with mI > m J , which will play the role of primaries. In this way, the mass parameter, p, is defined as p = mJ/(mI m J ) ,and so 1 - p = mI/(mr+mJ).Next, we must introduce the synodic reference frame. Recall that the origin of this system is set at the barycenter of I , J and that the positions of the primaries are fixed at ( p ,0,O) and ( p - 1,0,0) '.

+

Following G6mez e t a1.l , the transformation from synodical coordinates, r = (2,y,z ) ~to , sidereal ones, R, is defined by

R = B + IcCr,

(3)

376

G . Gdmez, J.J. Masdemont and J.M. Mondelo

where -

The translation, B, is given by

B=

-

-

mIRI + mJRJ mI+mJ ’

that clearly puts the barycenter of the primaries at the origin, The orthogonal matrix C = ( e l , e 2 , e 3 ) , sets the primaries on the zaxis and turns the instantaneous plane of motion of the primaries into the zy plane (by requiring that the relative velocity of one primary with respect to the other has its third component equal to zero). The columns of C are

being RJ I = RI - RJ . k = ~~RJIII is a scaling factor which makes the distance between the primaries to be constant and equal to 1.

It is important to remark that this change of variables is non-autonomous, since B, k and C depend on time through the components of RI and RJ. The change of coordinates given by Eq. (3) is checked to preserve the Lagrangian form of the equations and the new Lagrangian becomes 1 L ( r ,r’, t*)= -(B’, B‘) + k’(B’,s ) + k(B‘,s‘) 2 Gmr k [ ( z - p)2 y2 z2]1/2 A[(. - p

+

+ +

+

GmJ

+ 1)2 + y2 + 221112

where s = Cr, ri is the position of the body i in dimensionless coordinates and S* represents the set of Solar System bodies without the two primaries I , J. To get the above expression of L , we use that C defines an orthogonal transformation and, hence, it preserves the scalar product and the Euclidean norm. Finally, we want to use the same time units as those usual for the RTBP, where 27r time units correspond to one revolution of the primaries. If t* is

Dynamical Substitutes of Libration Points

377

some dynamical time and n is the mean motion of J with respect to I , then we perform the change of independent variable through

t = n(t* - tt;),

(4)

where t: is a fked epoch. From now on, t will be called dimensionless time. In Table 2 we give the values of n for the Earth-Moon and the Sun-(Earth+Moon) systems. In this second case, Earth+Moon means the Earth-Moon barycenter and, for this system, the Earth and the Moon are substituted in S by a fictitious body of mass mE mM behaving as their barycenter. The values in Table 2 are averaged values through the 6000 years covered by the JPL ephemerides file DE406 8 , and have been computed from this file. Using Kepler's third law, G ( m l + m J ) = n2u3, we can also define the mean semi-major axis of the orbit of one primary around the other; these values are also given in Table 2.

+

Table 1. Values for the mean motion and mean semi-major axis used in the Earth-Moon and Sun(Earth+Moon) cases. The units are (Julian Days)-' and km.

n a

Earth-Moon 0.22997154619514 384601.25606767

Sun-( Earth+Moon) 0.01720209883844 149598058.09228115

If we denote with a dot the derivative with respect t o t, remove those terms independent of r and r and multiply by the scaling factor u/(G(mr m J ) )= l/(n2u2), the new Lagrangian can be written as

+

1 L(r,i., t ) = - (k(B, s ) U2

1 + k ( B , 6 ) + -21P. ( r , r) + ~cic(s,k) + -P(s, 6)) 2 1-P

[(x - p)2 + y2 + z2]1/2

+ [(x - p + 1)2P+ y2 + z2]1/2

Since e1,e2,e3 form an orthogonal basis, we have that (ei,ej) = &j, (ei,ej) = -(ei,ej) and (e1,ei) = 0 for i , j = 1,2,3. It can be further shown that ( e 1 , e 2 ) = 0, (e2,e3) = 0 and (elle3) = 0. Recalling that r = ( ~ , y , z )writing ~, s = Cr = e1x e2y egz and using the previous

+

+

378

G. Gdmez, J.J. Masdemont and J.M. Mondelo

relations, we get

+ i 2 )+ az(x3i: + y~ + z i ) + a3(x3j - iy) +a4(yi - Gz) + a5z2 + a6g2 + a 7 Z 2 + a 8 Z z +a95 + a106 + a l l i + a 1 2 5 + a139 + a 1 4 2

L(r,r,t) = al(i2+ g2

1-P

+U15

([(z - p)2 + y2 + 22]1/2 +

c

i G *

+ [(z- /I + 1)2P+ + y2

Pi

[(z- X i ) 2

+ (y - yip + (2 - Zi)2]'/2

22]1/2

1

'

where the ai are time dependent functions that can be computed in terms of the positions, velocities, accelerations and over-accelerations of the two primaries. Using Lagrange equations ( d ( d L / b r ) / d t = dL/dr) we get the second-order differential equations

+ b8y + b g z + b 1 3 -dR , dX dR y = b2 - b.5k + b4G + b 6 i - bsz + bloy + bllz + b13-, 89 dR d = b3 - b6G + b 4 2 + bgz - blly + b 1 2 z + b13-, dz

2 = bi

+ b4X

b5G

b7x

being

-1 .. b2 = -(Bez),

k

b3

-1 .. = -(Be3)

b4

-2k =-

k

b7

= (elel) -

bs =

2k T(eie2)

k z, a3

+ (e1e2), b13 = -,k3

k

b5 = 2 ( e l e 2 ) ,

k blo = ( e 2 e 2 ) - -.

k

(5)

Dynamical Substitutes of Librotion Points 379

We note that setting bi = 0 for i # 5,7,10,13, b5 = 2, b, = blo = b13 = 1 and skipping the sum over S* in Eq. (6), the Eqs. (5) become the usual RTBP equations with mass parameter p. Therefore, we can see Eqs. (5) as a perturbation of the RTBP equations. Once the primaries have been fixed, we will get an idea of the order of magnitude of the perturbation, by looking at the first coefficient of the Fourier expansions of the b, functions. The Fourier analysis of this functions will be done in the next sections for two different systems.

If we we introduce momenta as

we have

It is known that, in this case, the Hamiltonian of the model is given in terms of the Lagrangian as H(r,p, t ) = Xp, #py i p , - L(r, r,t ) , where p = (p,,p,,pz)T and 5 , g, i can be written in terms of p,, p,, p, from (8). After expanding the previous expression of H , skipping terms that do not depend on r, p and collecting we obtain

+

+

380 G. Gdmez, J.J. Masdemont and J.M. Mondelo

where

a2 c1 = 2k2’ -k c2=-,

-1. c10 = -Be2

c3 = ele2,

-1. c11= -Be3,

CQ

k

-1

*

= -Bell k

k

,

k

(

c5=- 2a2 k2 (e1e2)2 - e 1 e 1 ) ) ,

-(e1e2)(e2e3)

- ele3

C15

-a =-

k

Note that setting c1 = 1/2, c3 = 1, C15 = 1,the rest of ci equal to zero, and skipping the sum over S*, we recover the Hamiltonian of the RTBP 7 . The Hamiltonian of the bicircular coherent models 2,4 can also be recovered by setting c1 = a1/2,c2 = 0 2 , c3 = a3, c12 = a4,~ 1 = 3 as, ~ 1 = 5 -a6, the rest of Ci equal to zero, and letting the sum over S*run only over the Sun, with xs = a7, ys = a8 and zs = 0.

3. Fourier analysis This section is devoted to the results of the Fourier analysis applied to all the time-dependent functions appearing in Eqs. (5), (6), this is: the bj functions and the coordinates, xi,yilZilof the bodies of the Solar System included in S*.The Fourier analysis follows the methodology developed in Gmez et aL9, which is a refined procedure that allows a very accurate determination of frequencies and amplitudes for analytic quasi-periodic functions. Here we will discuss the selection of the main parameters used in the method as well as the results obtained. Although the analysis can be done for any set of primaries, we have selected two different couples -the Earth and

Dynamic51 Substitutes of Libration Points

381

the Moon and the Sun and the Earth-Moon bxycenter- because of their relevance in many spacecraft mission analysis simulations.

3.1. Fourier analysis of the bi functions Using the algorithm described in 9, we have performed Fourier analysis of the {bi}i=1,...,13 functions, both for the Earth-Moon case and the Sun(Earth+Moon) case. This means that for each bi function we have obtained a set of frequencies and amplitudes that define its quasi-periodic approximation as a trigonometric polynomial, &ai ( t ) .As for any Fourier procedure, the most relevant parameters to be specified are the size, T, of the time (sampling) interval and the number, N , of equally spaced sampling points chosen in the interval. These parameters define, for instance, the Nyquist critical frequency, we = N/(2T), that fixes the window within we will find all the frequencies (true or aliased) of our time series. So, the first thing that we need is some criteria to choose properly T and N. Due to our implementation of the Fourier analysis procedure, the parameter N must range over powers of two. For consistency, the length, T, of the time-interval has also been chosen to range over a geometric progression, and the time-interval has always started at January lst, 2001. The smallest time-interval length, Tmin,has been taken of 95 years (34698.75 Julian days) and the greatest timeinterval length, T,,,, has been chosen as the maximum time-interval covered by the JPL DE406 ephemerides after Jan 1st 2001, which is 364938 Julian days (999.15 years). Therefore, we have let T range over the set {6nT,i,}~~o where 6 = (Tma/Tmin)l’lo- The time units used are revolutions of the secondary ( J ) around the primary (I)or, equivalently, dimensionless time divided by 27r. The reason for this is that, in this way, the frequency 1.0 corresponds to one revolution of J around I , which has a more intuitive meaning (one lunar month in the Earth-Moon case, one sidereal year in the Sun-(Earth+Moon) case) that will help in the elaboration of the intermediate models of motion. Moreover, in order to evaluate the trigonometric approximations of the bi functions, we only have to multiply the frequencies found by the dimensionless time, without the need of an additional 27r factor. The maximum number of samples Nma has been chosen to be 220, in order to allow for “comfortable” runs on machines with 64MB of memory

382

G. Gdmez, J.J. Masdemont and J.M. Mondelo

(or, equivalently, bi-processor machines with 128MB). For each value of T, the minimum number of samples has been chosen such that $ 2 1.5, in order to make the maximum detectable frequency to be at least 1.5. Assume that, for certain fixed values of T and N, we have performed Fourier analysis of a given function bi(t) obtaining the trigonometric polynomial Qbi(t). Then, we can easily compute the maximum difference between the analyzed function and its quasi-periodic approximation at the sampling points, that is,

where t l = 1 (TIN), 1 = 0,. . . ,N - 1 are the sampling epochs. In Figure 1 (Sun-(Earth+Moon) case, see for the Earth-Moon one) we have represented, for all the bi functions, the minimum of dm, with respect to the different values of N explored, when varying T according to the preceding discussion. To reduce the leakage effect, in all the computations we have multiplied our data by a Hanning function of order two

H%t) = 2 3 (1 - cos (2.;))2 The advantages of the Hanning function with respect to other well-known window functions lo are its simplicity and its degree of differentiability. For instance, HF(t) has degree 2n, whereas a general “triangle window function” T$(t) has degree just n. This higher degree of regularity implies a faster decay of the Fourier coefficients (see for more details). In order to control aliasing, two different strategies have been followed. The first one is based on time-domain, and consists in computing the maximum difference between the initial function and its quasi-periodic approximation, over a refinement of the initial grid of data used for the Fourier analysis. This difference will be denoted as CQ. If it increases significantly when increasing the number of points of the grid, then aliasing is very likely to occur. For this test, we have used a refinement with 16N equally spaced points in [O,T]. The second anti-aliasing strategy is based on frequency-domain. It consists in computing the number of rightmost consecutive harmonics of the residual Discrete Fourier Transform (DFT) that have modulus less than a

Dynamical Substitutes of Libmtion Points 383

b, ,Sun-Earth+Mwn

4 ,Sun-Earlh+Mwn

years (sidereal)

years (sidereal)

years (sidereal)

b, , Sun-Earth+Moon

9 ,Sun-Earth+Mwn

ba, Sun-Earlh+Mwn

le-06

0.0007

O.OOO35

0.W

0.00025 0.m2

0.W 0.0002 0.0001

0.00015

0.0001

0

3OOo 6ooo 9ooo 12000 years (sidereal)

59-07 0

b,Sun-Eatlh+Mwn

6ooo 9000 12000 years (sidereal)

3000

0

3000 6000 9OW 12000 years (sidereal)

b ,Sun-Earth+Mwn

b,Sun-Earlh+Mwn

years (sidereal)

years (sidereal)

years (sidereal)

blO, Sun-Earih+Mwn

b,, , Sun-Earth+Mwn

b,, , Sun-Earlh+Mwn

0.0008

8e-06 7e-06

0.0007 O.ooo6

69-06

O.WO5 O.ooo4 0.W 0.0002

5e-06

0.0001 00

3OOo

woo

so00

4e-06 -06 28-06 12000 19-06

O.ooo8

0.0002 0.0001

0 0

3OOo 6000 9OW 12000 years (sMereal)

0

3000

6ooo years (sidereal)

90W

0

3000 6000 9ooo 12000 years (sidereal)

b,, , Sun-Earlh+Mwn

0.W 0.0002 0.0001 0

0

6M)o 90W years (sidereal)

3000

12WO

Fig. 1. Error results of the Fourier analysis of the b; functions in the Sun-(Earth-tMoon) case. For each value of T explored, we have represented the minimum value of d,,, with respect to N . The values of T are given in sidereal years (revolutions of the Earth around the Sun).

384

G. Gdmez, J.J. Masdemont and J.M. Mondelo

fraction of the maximum modulus of the residual DFT. Then, we divide this number by N/2, the total number of harmonics, and this defines the parameter a2. That is, if Ei(t)= bi(t) - Qbi(t) is the error of the trigonometric approximation of bi(t) and C E ~ , T , N ( ~S )E, ~ , T , N ( ~j ) ,= 0,. . .,N/2, are the cosine and sine coefficients of its DFT, we compute 2 112 P E ; , T , N ( ~= ) ( ( c E ~ , T , N ( ~ ) ()s~E ~ , T , N ( ~ ) )

+

Pmax

=

a 2

=

m a

j=Q, ...,N / 2

,

PE;,T,N(~),

min{j :P E < , T , N ( ~_<) pmaX/25for 1 = j , . . . ,N / 2 }

N/2 Then, for instance, a value of 0.2 for a2 means that there are no frequencies greater than 0.8 = 0 . 8 . (N/2T), with amplitude greater than 1/25 times the modulus of the residual DFT, so we do not expect aliasing in the corresponding Fourier analysis. We are assuming here that amplitudes decrease as frequencies increase, which is ensured by the Cauchy estimates of the Fourier coefficients for an analytic quasi-periodic function.

As an example of aliasing and how the two previously-described strategies detect it, we have represented in Figure 2 the residual DFT of some of the Fourier analysis of the bl function in the Earth-Moon case. Some numerical values of these analysis are given in Table 3.1. In the left plot, we see that for N = 16384 there are frequencies of high amplitude near wmaX = 4.02903. As we increase N , the amplitude of the frequencies near urnax decrease and the values of d,, as well as the parameter a1 of the first anti-aliasing strategy become closer. N = 32768

N = 16384 3e-05 2.5e-05 2e-05 152-05 1e-05 58-06 0

3e-05 2.5e-05 2e-05 1.5e-05 1e-05 58-06 0 00.5 1 1.52 2.5 33.54 4.5

N = 65536 3e-05 2.58-05 2e-05 1.5e-05 1e-05 5e-06 0

0 1 2 3 4 5 6 7 8 9

0 2 4 6 81012141618

Fig. 2. Modulus of the residual DFT some of the Fourier analysis of the bl function in the Earth-Moon case. The values of the parameters of these analysis are given in Table 3.1.

According to this, for the results displayed in Figure 1 only those analyses with a1 < 1.2dmaXand a 2 0.2 have been taken into account.

>

Dynamical Substitutes of Libmtion Points

385

For the generation of simplified models for the Solar System, among all the analysis performed we have selected the best ones in terms of minimum d,,,. The corresponding parameters of these “best” analysis are given in Tables 3.1 (Earth-Moon) and 3.1 (Sun-(Earth+Moon)). Table 2. Parameters associated to the Fourier analysis of Fig. 2. From left t o right: dayo, initial Julian day since Jan lst, 2001; dayf, final Julian day (same maxiunits); T, time interval in J-revolutions; N, number of points used; umaxr maximum modulus of the residual DFT; dmax, mum detectable frequency; p,,, maximum difference between bl and its quasi-periodic approximation over the samples; 01, a2,values of the two anti-aliasing parameters. T N Wmax Pmax dmax ffl ffz dayo dayf 366 55917.4 2033.24 16384 4.02903 2.663-05 4.903-04 2.293-03 0.0007 366 55917.4 2033.24 32768 8.05806 2.663-05 5.303-04 5.673-04 0.1633 366 55917.4 2033.24 65536 16.1161 2.663-05 5.633-04 5.673-04 0.5816

Table 3. Values of the parameters for the best Fourier analyses of the bi functions for the Earth-Moon case. function bi

T (days) 55551.4 55551.4 55551.4 55551.4 43904.0 70288.7 55551.4 55551.4 70288.7 55551.4 70288.7 43904.0 55551.4

T (J-rev.) 2033.24 2033.24 2033.24 2033.24 1606.94 2572.64 2033.24 2033.24 2572.64 2033.24 2572.64 1606.94 2033.24

N 65536 65536 32768 65536 32768 32768 65536 524288 65536 65536 65536 32768 65536

Pmax

dmax

2.663-05 2.673-05 3.303-06 2.3 13-06 4.85606 3.923-08 3.5 13-06 1.963-07 1.973-08 3.513-06 1.67E-08 1.583-06 3.5 13-06

5.633-04 5.493-04 5.583-05 5.013-05 9.16E-05 1.133-06 7.81E-05 5.943-06 5.69607 7.833-05 5.053-07 3.293-05 7.99E-05

3.2. Fourier analysis of the positions of the planets

In order to complete the quasi-periodic approximation of all the timedependent part in the vector-field given by Eqs. (5), we have performed Fourier analysis of each coordinate xi, yi, zi of the Solar System bodies in dimensionless coordinates, using the same procedure as for the analysis of the bi functions. Plots of the minimum value of d,,, with respect to N , for fixed values of T , as well as tables with the values of the parameters for best analyses, can be found in

‘.

386

G. Gdmez, J.J. Masdemont and J.M. Mondelo Table 4. Values of the parameters for the best Fourier analyses of the b, functions for the Sun-(Earth+Moon) case. Note that, in this case, J-revolutions are sidereal years. function

bi

T (days) 142382.6 142382.6 112529.5 34698.8 34698.8 88935.7 34698.8 288422.1 88935.7 34698.8 70288.7 34698.8 34698.8

T (J-rev) 389.815 389.815 308.083 94.998 94.998 243.488 94.998 789.642 243.488 94.998 192.436 94.998 94.998

N 65536 65536 131072 4096 4096 262144 4096 524288 131072 4096 524288 4096 4096

Pmax

dmax

4.953-08 4.953-08 2.283-09 8.343-06 1.753-05 1.763-08 1.36605 9.653-08 9.713-09 1.363-05 2.353-08 3.92606 1.343-05

4.403-07 4.333-07 2.683-08 6.74605 1.26604 5.713-07 9.17E-05 1.673-06 3.193-07 9.173-05 2.38606 4.06605 9.473-05

4. Generation of simplified Solar System models

In this section we will generate several simplified Solar System models using the Fourier approximations computed according to the previous section. The models obtained will be compared with other ones through the computation of residual accelerations.

4.1. Adjustment using linear combinations of basic

frequencies In order to give a more physical meaning to the results obtained from the Fourier analysis, we will write the computed frequencies as linear combinations, with integer coefficients, of basic ones. These basic frequencies can be identified as “natural” frequencies of the planetary and lunar theories. The introduction in the Fourier expansions of the basic frequencies will be the key point for the construction of models of motion with increasing dynamical complexity. In principle, the basic frequencies will be extracted from the list of frequencies computed in the Fourier analysis and using the procedure explained below. Nevertheless, in some cases it can be convenient to introduce a fixed set of basic frequencies obtained by other means, for instance from an analytical lunar theory, and then write all the computed frequencies as

Dynamical Substitutes of Libmtion Points

387

linear combinations of the ones in this fixed set. Both approaches will be considered in what follows. In order to better describe the procedure, we need two definitions. Assume that w1, . ..,wn is a set of basic frequencies and that a frequency f can be written as f = k l w l + . . .+k,w, with k l , . . . ,k, integer numbers, then we say that f is a linear combination of w1 ,. . . ,wn of order k = 1 kl I . . 1 k, I. We say that f is a linear combination of w l , . . .,w , of order k within tolerance E > 0 if, for some kI, ...,kn such that k = Jkll . - . Iknl, we have

+. + + +

A simple approach for the determination of basic frequencies is: (1) Choose a maximum order of the linear combinations to be found. (2) Choose a tolerance for the adjustment of frequencies as linear combination of basic ones. (3) For each frequency, try out all the linear combinations of the current set of basic frequencies up the chosen maximum order. (4) If any of the linear combinations fulfills the tolerance requirements, add the current frequency to the set of basic ones.

This procedure may add extra basic frequencies (and thus end up with a rationally dependent set) in some cases, for instance, if the current frequency is an integer divisor of one of the basic frequencies. To avoid this, instead of trying to adjust the current frequency as linear combination of the basic ones, we will try to adjust zero as linear combination of the current frequency and the basic ones. If we succeed to do this and the current frequency gets a coefficient different from f l , it may be necessary to divide some basic frequencies by this coefficient.

4.2. Simplified models for the Sun-(Earth+Moon) system

For the models to be developed in this section, and leaving aside the two primaries -Sun and Earth-Moon barycenter--, we will not consider any perturbing body in S* As it will be shown, this provides rather accurate models and, at the same time, avoids the introduction of additional basic

388

G. Gdmex, J.J. Masdemont and J.M. Mondelo

frequencies. In this way, in the equations of motion (5), we will only use the Fourier expansions of b l , . .. ,b13, and its general expression for the equations of motion will be

being

The super-index (i) that we have used for the b y ) , j = 1, ..., 13, and functions will be used as a label for the different intermediate models, according to the number of basic frequencies retained in the Fourier expansions. The numerical data obtained (see l1 or contact the authors) suggests to take into consideration only the b 4 , b 5 , b7, b l o , b12 and b13 functions. In addition to this simplification, we will not consider any Solar System body in Eq. (6), since, just using the b i , we are already taking the Sun into account. Applying the procedure of Sect. 4.1 to the b13 function, with tolerance and maximum order 20, we get the following four basic frequencies: ul = 0.999992616, v2 = 0.6255242728, v3 = 0.9147445983, v4 = 1.8313395538. These four frequencies allow to adjust the frequencies of the Fourier analysis of the b4, b5, b7, blo and b12 functions. The results are given in Table 4.2. According to this, for i = 1 , . . ., 4 we define the model SSSMi as the one given by Eq. (ll),taking as b y ) the Fourier expansion of b j computed in Sect. 3.1, but keeping only the independent term and the frequencies that can be written as a linear combination of v1, . . .,vi. Once the different models have been produced, it is desirable to see if they are close or not the the “real” one, that is: the full equations of motion (Eqs. (5) and (6)) in which the time periodic functions, bi and x i , y i , zitare computed using the JPL ephemeris files. For these purposes we first select

Dynamical Substitutes of Libration Points Table 5. Fourier analysis results of the dominant bi functions in the Earth-Sun system. The frequencies have been adjusted as l i n e s combinations, C kiv,, of the four basic frequencies. The order of the linear combination, k, and the corresponding error are also displayed. Func b4

b5

b7

blo

b12

b13

Frequency 0.00000000000 0.99999261980 1.99998564390 1.25103997640 1.83134352170 0.91473091670 2.99997409570 0.00000000000 0.99999261700 1.99998563790 1.25103998380 0.91475203530 1.83134663800 2.99997541480 0.00000000000 0.99999261500 1.99998562010 0.91475953220 1.25103999430 2.99998010500 0.62552269280 0.00000000000 0.99999261500 1.99998562010 0.91475953220 1.25103999430 2.99998010500 0.62552269280 0.00000000000 0.99999262330 1.99998564990 1.83134558880 1.25103987210 2.99997235010 0.91470513360 0.00000000000 0.99999261640 1.99998562580 1.25104010020 0.91474459830 2.99997729050 1.83133955380

Amplitude 1.300003-09 3.337203-02 8.352803-04 3.938003-05 3.400503-05 2.849203-05 1.971603-05 2.000003+00 6.674903-02 1.392303-03 6.695503-05 6.124803-05 4.856903-05 3.016903-05 1.000423+00 5.008003-02 1.253503-03 4.823703-05 4.224403-05 3.080403-05 2.719003-05 1.00042E+00 5.008003-02 1.253503-03 4.823703-05 4.224403-05 3.080403-05 2.719003-05 -1.393003-04 1.669303-02 6.962303-04 3.110503-05 2.465503-05 2.260703-05 1.304503-05 1.000423+00 5.008003-02 1.253403-03 4.711803-05 4.674403-05 3.077603-05 2.812303-05

Error 0.000003+00 3.388003-09 4.110703-07 -8.56920606 3.967903-06 -1.368203-05 -3.753503-06 0.000003+00 5.51530610 4.050903-07 -8.561803-06 7.437003-06 7.084203-06 -2.434403-06 0.000003+00 -1.416603-09 3.872703-07 1.493403-05 -8.551303-06 2.255803-06 -1.580003-06 0.000003+00 -1.416503-09 3.8727OE-07 1.49340605 -8.551303-06 2.255803-06 -1.580003-06 0.000003+00 6.875503-09 4.171103-07 6.035003-06 -8.673503-06 -5.499103-06 -3.946503-05 0.000003+00 5.352903-12 3.930303-07 -8.445403-06 -4.825403-11 -5.587003-07 -9.859903-12

Icl

0 1 2 0 0 0 3 0 1

2 0 0 0 3 0 1 2 0 0 3 0 0 1 2 0 0 3 0 0 1 2 0 0 3 0 0 1 2 0 0 3 0

k2

0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 1 0 0 0 0 2 0 1 0 0 0 0 2 0 0 0 0 0 2 0 0 0

k3

0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0

k4

k

0 0 0 0

0 1 2 2 1 1 0 1 0 3 0 0

0 1

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1

2 2 1 1 3 0 1 2 1 2 3 1 0 1 2 1 2 3 1

0 1 2 1 2 3 1 0 1 2 2 1 3 1

389

390 G. Gdmez, J.J. Masdemont and J.M. Mondelo

a set of trajectories, Tz:

P

+

t

+ (r(t),+(t)),

P 6

along which the position, r(t), and velocity, i(t), are known. We have done two kinds of selections. In the first one we have chosen for T~ a family of periodic halo orbits with different z-amplitudes; these orbits are true solutions of the RTBP (see for their computation) and cover a large set of solutions with very different sizes. Then, given two models to be compared, with differential equations r = f(rji,t ) and r = g(r, i,t ) , respectively, and given a trajectory, yz, which does not need to be a true solution of any of the two models, we compute the “mean relative residual acceleration over y” as

where t is a fixed epoch (in dimensionless units) and L is the length of the trajectory yz (in configuration space).

For the second test the computations are similar except that we have taken instead of y z ( t ) a set of points uniformly distributed around a large neighborhood of the equilibrium points. We have also required to their energy (Jacobi constant) to be in a certain interval around the value associated to the equilibrium points. The results obtained are analogous to the ones obtained for the halo orbits, and will not be given here. In Table 4.2, we compare the models RTBP, SSSMl and SSSMl with the real Solar System using the residual acceleration test introduced above. We note that the SSSM4 model gives worse results than SSSMl. This is not a contradiction. Looking closer to Table 4.2 we can see that the maximum amplitude of the frequencies of b4, b5, b7, blo and b12 that are not multiple of v1 is 6.6953-05. Because of that, adding frequencies does not improve significantly the approximation of the bi functions, and in this way the structure of Eqs. (5) takes over the fact that the bi terms of SSSM4 are closer to the ones of the real Solar System than the corresponding terms of SSSMl. Therefore, for the Sun-(Earth+Moon) case, we will give SSSMl as simplified Solar System model. Note that this is a model with very few frequencies (just one) that significantly improves the RTBP.

Dynamical Substitutes of Libmtion Points

391

Table 6. Mean relative residual accelerations between several models and the real Solar System over selected halo orbits of the RTBP around L2 in the Sun-(Earth+Moon) case. z-a. 0.020000 0.024838 0.030846 0.038308 0.047575 0.059084 0.073376 0.091126 0.113169 0.140545 0.174543 0.216766 0.300000

RTBP 3.4464973-02 3.4111843-02 3.366579602 3.3135803-02 3.2547893-02 3.1943553-02 3.1373813-02 3.0890823-02 3.0537703-02 3.033772E-02 3.0285163-02 3.0341153-02 3.0475773-02

SSSMl 9.9015263-05 9.779360605 9.6169133-05 9.4163273-05 9.1751343-05 8.8956103-05 8.5828413-05 8.2361833-05 7.8599793-05 7.4502523-05 7.0207143-05 6.5794923-05 5.8980803-05

SSSM4 8.9054543-04 8.7686703-04 8.5895003-04 8.3641663-04 8.0925273-04 7.7768133-04 7.4204443-04 7.0264213-04 6.5972433-04 6.1356383-04 5.6438853-04 5.1270313-04 4.3238593-04

4.3. Simplified models for the Sun-(Earth+Moon) system

For the Earth-Moon models to be developed in this section, and leaving aside the two primaries -Earth and Moon- the Sun will be the only perturbing body in S *. As it will be shown, this provides rather accurate models and, at the same time, avoids the introduction of additional basic frequencies. In a rather accurate theory for the lunar motion, as the simplified Brown theory given in 12, the fundamental parameters can be expressed in terms of five basic frequencies. In terms of cycles per lunar revolution, their numerical values are

0 0

0

0

The mean longitude of the Moon, which is set equal to w1 = 1.0. The mean elongation of the Moon from the Sun, w2 = 0.925195997455093. This is the frequency of the time-dependent part of the Bicircular Problem (BCP) and the Quasi-Bicircular Problem (QBCP) mentioned in the Introduction. The mean longitude of the lunar perigee, w3 = 8.45477852931292-10-3. The longitude of the mean ascending node of the lunar orbit on the ecliptic, w4 = 4.01883841204748. The Sun’s mean longitude of perigee, w5 = 3.57408131981537.

392

G. Gdmez, J.J. Masdemont and J.M. Mondelo

The value of w5 is close to the lower frequencies computed in our Fourier expansions and, at the same time, is close to the precision we can expect in the determination of frequencies with the data used By these reasons and in order to have also a set of basic frequencies with astronomical meaning, we have adopted for the Earth-Moon models these frequencies as the basic set, instead of the ones provided by the procedure of Sect. 4.1.

’.

This basic set of frequencies needs to be modified in order to give a sequence of models that successively improve the RTBP. The Fourier analysis results suggest to consider a new frequency basis v l , . . . ,v5, defined as: 0

0

0 0

0

which is the main frequency of b l , b 2 , xs and ys, so it can be considered the “main planar frequency”, which is coherent with the fact that that it is also the frequency of the BCP and QBCP models (see 1 3 J ) . v2 = w1 - w3, which allows to complete a first approximation of the largest functions among the bi and xs, ys, zs. v3 = w1 - w2 w4, which is the main frequency of b3. v4 = w1 - w5, which is the first frequency of xs which cannot be expressed in terms of v l , v2, and v 5 = w5 -w2, which is the first frequency of bs that cannot be expressed in terms of v1, v2, v3, v4. v1

=

w2,

+

In order to check all the above statements, the full Fourier expansions can be found in l 1 or provided by the authors. The new basis verifies

0-1

0 0

1

Since the matrix in the above transformation is unimodular, a valid basic set of frequencies.

{vi}i=1,...,5is

The results for the residual accelerations are given in Table 4.3, using as test paths several halo orbits around the collinear equilibrium point L2. As it has already been mentioned, the results with other trajectories, or other equilibrium points, give the same qualitative information. From this table, it becomes clear that the best onefrequency models that we can be used,

Dynamical Substitutes of Libmtion Points

393

Table 7. Mean residual accelerations between several models and the real Solar System over selected halo orbits of the RTBP around L2 in the Earth-Moon case. The first column displays the z-amplitude of the halo orbit used as test orbit. The remaining columns show the mean residual acceleration between the corresponding model and the real Solar System over the test orbit. z-a. 0.020 0.025 0.031 0.038 0.048 0.059 0.073 0.091 0.113 0.141 0.175 0.217 0.269 0.300

RTBP 0.140126 0.136603 0.132882 0.129087 0.125352 0.121813 0.118614 0.115905 0.113823 0.112471 0.111872 0.111928 0.112400 0.112678

BCP 0.146459 0.142856 0.139025 0.135080 0.131141 0.127324 0.123757 0.120571 0.117895 0.115836 0.114443 0.113663 0.113311 0.113200

QBCP 0.138580 0.135174 0.131578 0.127914 0.124312 0.120905 0.117835 0.115249 0.113283 0.112037 0.111533 0.111672 0.112201 0.112492

SSSMl 0.365299 0.353302 0.340305 0.326550 0.312235 0.297505 0.282462 0.267173 0.251690 0.236056 0.220325 0.204551 0.188782 0.180899

SSSM2 0.095769 0.093293 0.090590 0.087699 0.084643 0.081429 0.078045 0.074461 0.070634 0.066510 0.062042 0.057196 0.051978 0.049240

SSSM3 0.010674 0.010388 0.010076 0.009744 0.009393 0.009024 0.008637 0.008229 0.007796 0.007331 0.006831 0.006292 0.005716 0.005417

SSSM4 0.001374 0.001346 0.001315 0.001282 0.001247 0.001210 0.001171 0.001128 0.001081 0.001030 0.000973 0.000910 0.000840 0.000802

SSSM5 0.000727 0.000720 0.000711 0.000702 0.000691 0.000678 0.000664 0.000646 0.000625 0.000598 0.000566 0.000526 0.000481 0.000456

according to the residual acceleration criteria, are the BCP and the QBCP. But, when we allow two or more frequencies, the models we get fit the JPL one much better. As it has been said, only the Sun has been taken into account in all the intermediate models. By adding additional Solar System bodies, the residual accelerations are of the same order of magnitude than the ones obtained just using the Sun. It is also clear that, from this point of view, there is not a significant improvement between the RTBP and the non-autonomous Bicircular and Quasi Bicircular models.

5. Dynamical substitutes of the collinear libration points As it is well known, the RTBP has five equilibrium points: three of them (L1, L s , L3) are collinear with the primaries and the other two (L4 and L5) form an equilateral triangle with them. Although the intermediate models introduced in the preceding section are close to the RTBP, they are non autonomous, so they do not have any critical point. If we consider the SSSMl model, since it depends on only one frequency, it can be seen as a periodic perturbation of the RTBP so, under very general non-resonance conditions between the natural modes around the equilibrium points and the perturbing frequency, the libration points can be continued to periodic

394

G. Gdmez, J.J. Masdemont and J.M. Mondelo

orbits of the model. These periodic orbits, which have the same period as the perturbation, are the dynamical substitutes of the equilibrium points. In this section we will show these substitutes for the three collinear equilibrium points for SSSMl, in the Earth-Moon system. For the other models, SSSM2,. . .,SSSM5, as the perturbation is quasi-periodic, the corresponding substitutes will be also quasi-periodic solutions. The methodology for their efficient computation, as well as the results obtained, will appear elsewhere. The dynamical substitutes of the triangular points in the Earth-Moon system, for models close to the ones of this paper, have been studied in 14,l3 and l5 and will not be considered here.

4.0807 42807

3.6a-07

0.02

0.2

0.

Fig. 3. Dynamical substitutes for the SSSMl model of the three collinear equilibrium points.

The numerical computation of the periodic orbits of SSSMl that substitute L1 and L3 has no problem and the results obtained are shown in Figure 3. We can see that L1 is replaced by a very small size periodic orbit and that the substitute of L3 is also almost planar but rather large in the (z,y)-plane. The computation of the substitute of L z , also displayed in Figure 3, requires more care. Mainly, we need to introduce a continuation parameter between the RTBP and SSSMl, so we consider the 1-parameter

Dynamical Substitutes of Libmtion Points 395

family of vector-fields which can be formally written as

+

(1 - &)RTBP ESSSMl.

If E = 0 we get the RTBP and when SSSMl .

E

= 1 we get the desired final model

The dynamical substitutes of L1,2,3 in SSSM2 are two-dimensional invariant tori. They can be computed as follows the SSSM2 model can be written as 5316:

x = f(z,ut) where u = (u1, u2) and f is 2n-periodic in ut. We do not actually compute a 2D invariant torus but an invariant curve inside it. For that, we solve numerically for cp the equation 2n cp(S + = (cp(l)) , VS E [O,2x1

G) &$,

where &(z) is the flow from time 0 to time t of

x = f(z,e + ut) (which is not SSSM2 if 8 # 0). The geometrical torus is then (o,w ,Yl

( 4 8 21)10, ,L%

E[0,27r)

(see Fig. 4).

Fig. 4.

Computation of twdimensional invariant tori.

Using continuation techniques ”, we can reach the substitutes of the collinear libration points in SSSM2 along the homothopy of models (1 E)SSSM1 ESSSM~.As an example, we display in Figure 5 the results for the L1 point of the Earth-Moon case.

+

396

G . Gdmez, J . J . Masdemont and J.M. Mondelo

E-

0.61184

2-1

z

Z

Fig. 5. Continuation from the dynamical substitute of L1 in SSSMl (a periodic orbit) t o the dynamical substitute of L1 in SSSMz (a 2-dimensional torus), in the Earth-Moon case. The continuation is performed with respect t o the parameter E along the family of models (1 - E)SSSM~ ESSSM~.

+

References 1. G. G6mez, J. Llibre, R. Martinez, and C. Sim6. Dynamics and Mission Design Near Libration Point Orbits - Volume 1: Fundamentals: The Case of Collinear Libration Points. World Scientific, 2001. 2. M.A. Andreu. The Quasibicircular Problem. PhD thesis, Dept. Matemhtica Aplicada i Andisi, Universitat de Barcelona, Barcelona, Spain, 1998. Available at http://vuv.maia.ub.es/dsg/. 3. K.C. Howell and J.J. G u z m h . Spacecraft trajectory design in the context

of a coherent restricted four-body problem with application to the MAP mission. In 51st International Astronautical Congress, Rio de Janeiro, 2000. 4. F. Gabern and A. Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete Contin. Dyn. Syst. Ser. B, 1(2):143-182, 2001. 5. E. Castelk and A. Jorba. On the vertical families of two-dimensional tori near the triangular points of the bicircular problem. Celestial Mech. Dynam. Astronom., 76(1):35-54, 2000. 6. G. Gbmez, J. J. Masdemont, and J. M. Mondelo. Solar system models with a

Dynamical Substitutes of Libration Points 397 selected set of frequencies. Astronomy d Astrophysics, 390(2):733-749, 2002. 7. V. Szebehely. Theory of Orbits. Academic Press, 1967. 8. E.M. Standish. JPL Planetary and Lunar Ephemerides, de405/le405. Technical Report JPL IOM 314.10-127, NASA-Jet Propulsion Laboratory, 1998. 9. G. G6mez, J.M. Mondelo, and C. Sim6. Refined Fourier Analysis Procedures. Preprint, 2001. 10. E.O. Brigham. The Fast Fourier Transfom. Prentice Hall, 1974. 11. J.M. Mondelo. Contribution to the Study of Fourier Methods for QuasiPeriodic Fvnctions and the Vicinity of the Collinear Libration Points. PhD thesis, Dept. Matemitica Aplicada i Anilisi, Universitat de Barcelona, Barcelona, Spain, 2001. Available at http://vvu.maia.ub.es/dsg/. 12. P.R. Escobal. Methods of Astrodynamics. J. Wiley & Sons, 1968. 13. C. Sim6, G. G6mez, A. Jorba, and J. Masdemont. The Bicircular Model Near the Triangular Libration Points of the RTBP. In A. Roy and B. Steves, editors, f i o m Newton to Chaos, pages 343-370. Plenum Press, 1995. 14. G. G6mez, A. Jorba, J. Masdemont, and C. Sim6. Dynamics and Massion Design Near Libration Point Orbits - Volume 4: Advanced Methods for !l’riangular Points. World Scientific, 2001. 15. A. Jorba. A Numerical Study on the Existence of Stable Motions near the Triangular Points of the Real Earth-Moon System. Astron. Astrophys., 364(1):327-338, 2000. 16. A. Jorba. Private communication. 17. G. G6mez and J. M. Mondelo. The dynamics around the collinear equilibrium points of the RTBP. Phys. D, 157(4):283-321, 2001.

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Libration Point Orbits and Applications G. Gdmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

NAVIGATION OF SPACECRAFT IN UNSTABLE ORBITAL ENVIRONMENTS D.J. SCHEERES Department of Aerospace Engineering The University of Michigan Ann Arbor, MI 48109-2140, USA

The novelty of libration point orbits is their hyperbolic instability. It is this basic property that allows them to serve as connections between disparate regions of space, and gives them their many practical uses. This same property also makes the navigation of spacecraft in libration point orbits a fascinating subject, one which exposes new questions in orbit determination and control. The problem of spacecraft navigation and control is essentially concerned with the statistical distribution of orbits in phase space, and how best these s t a t i s tical distributions can be sensed and controlled. When placed into an unstable orbital environment, the dynamics of these distributions become quite interesting, with volume preserving stretching and contraction of the phase flow occuring over times on the order of the Lyapunov Characteristic Time of the nominal orbit. The interplay between dynamics and spacecraft navigation in such an environment creates new opportunities for research and understanding. In this paper the basic theory of orbit determination and control will be presented in the context of libration point orbits.

1. Introduction This paper addresses the application of spacecraft navigation theory to missions with unstable orbital environments. By navigation we mean the prediction of spacecraft uncertainties in the future, the processing of orbit determination measurements to reduce this uncertainty, and the placement of correction maneuvers along the trajectory. Spacecraft navigation the399

400

D.J. Scheeres

ory is derived largely from linearized analysis about a nominal, non-linear trajectory. It concerns itself primarily with the prediction of uncertainty distributions (representing the spacecraft and associated models) and the sensing and control of these uncertainty distributions. See Battin, Chapter 14 and Montenbruck and Gill l9 for comprehensive introductions to the main elements of spacecraft navigation theory. The vast majority of spacecraft have generally been flown in nearKeplerian, or stable, orbital environments. The main hallmark of these environments is that neighboring trajectories in an uncertainty dispersion generally only deviate from each other at a constant rate in time (see Fig. 1). This particular dynamic has led to a number of useful “rule-of-thumb” relations in spacecraft navigation, mostly concerned with error propagaion along the down-track direction, which is the direction in which this linear drift primarily acts. In more challenging missions involving multiple gravity assists, such as planetary flybys or satellite orbital tours, the nominal trajectory can no longer be viewed as stable in a traditional sense as orbit uncertainties now get expanded hyperbolically through each planetary or satellite flyby. However, since these flybys occur at well-spaced intervals and the orbit is stable between flybys, the navigation design process can still be modeled following standard approaches. In these situations the measurements must be concentrated at specific times relative to the flybys in order to ensure that the trajectory uncertainties and their mappings are properly constrained. The orbit determination measurements are chosen, in part, to ensure that the uncertainty mappings never get beyond the linear regime, meaning that the phase volume is never allowed to become significanly stretched in phase space. When orbital missions are considered in continuously unstable dynamical systems the situation changes. Examples of such environments would be a Sun-Earth halo orbiter, an Earth-Moon halo orbiter, a high inclination Europa Orbiter, or other environments where trajectories are continuously subject to unstable dynamics that cause neighboring trajectories to diverge from each other hyperbolically over all time spans. There has been a history of great success in navigating spacecraft in such environments about However, as shown in Renault and the Earth-Sun libration points Scheeres 20, the instability about the Earth-Sun libration points is weak enough to allow the application of standard navigation practice. The same is not true for navigation of a spacecraft about the Earth-Moon libration 5,619,3.

Navigation of Spacecmfi in Unstable Orbital Environments

401

Libration Point

Fig. 1. Graphical description of the different types of dynamical environments spacecraft are navigated in.

points, and indeed in those situations the navigation process is strongly influenced by the instabilities. The control of spacecraft in these environments was first considered by Farquhar 4 , and control costs were estimated without considering the statistical orbit determination errors. It is the incorporation of statistical orbit determination errors into the analysis of this problem, in fact, that motivates the current paper and methods. In unstable environments the traditional interpretations of trajectory navigation begins to show their limitations. It is important to note that these limitations are not necessarily due to non-linearities acting on the system, and can largely be traced back to the fundamentals of linearized navigation analysis. The difference is that the linear systems that describe the uncertainty distributions are no longer stable or degenerate, but exhibit hyperbolic expansions and contractions, which impinge directly on the traditional interpretations. This paper first reviews navigation fundamentals from a Hamiltonian systems perspective. Next, for motivation, we introduce some model problems of unstable orbital environments, along with a simple one-degree of freedom (1-DOF) model that can be used to motivate our analytical evaluation. Then we develop analysis methods for the navigation of spacecraft in such unstable environments and apply them to our simple 1-DOF model. Our intent in this chapter is to develop an appropriate understanding of

402

D.J. Scheeres

navigation in unstable orbital environments to supplement the conventional understanding of navigation in more traditional environments.

2. Navigation Fundamentals

First we review some fundamental results for spacecraft navigation. We take a general approach to the problem in what follows, and assume that the dynamical system can be expressed in a Hamilton canonical form. That this is possible is trivially true, since the standard Newton's equations of motion expressed in an inertial frame are already in such a form.

2.1. Dynamical System

Our dynamical state is defined as a 6-dimensional vector x, consisting of three coordinates (9) and three momenta or velocities (p), arranged as x = [q;p]. A solution of the dynamical system is designated as x(t) = 4(t,to,xo,p)where x, is the spacecraft state at an epoch to and p is a vector of force parameters that influence the dynamics of the system. In this paper we do not consider the effect of force parameters on spacecraft navigation, although these are items of essential concern. The state satisfies a differential equation x(t) = F(x(t),t) where the force function F is in general a function of both time and the state. Since this is a Hamiltonian system, F = JdH/dx, where H(x, t ) is the Hamiltonian of the system. Note that we do not assume that the Hamiltonian is constant in our discussions, thus allowing for a time-varying system. The matrix J is:

'1

J = [ -I 0 where 0 and I are three-by-three zero and identity matrices, respectively. Corresponding to any region of phase space B,, there exists a corresponding region in which the flow of the system is defined, denoted as B ( t ) = +(t,to,23,). If we represent the initial set as xo Sx, E B, and assume that the size of Sx, is relatively small, we find an explicit solution for B ( t ) from the linear dynamics; $(t,t o ,B,) = 4(t,t o ,x,) +@(t, t,)Sx,+. . . = $(t,to,x,) Sx(t). Ignoring higher order terms, we find that Sx obeys a

+

+

Navigation of Spacecraft i n Unstable Orbital Environments

403

linear dynamics law: 8 2H S X = J-SX 8x2

where the matrix A = J S is evaluated along the nominal solution of the differential equation x ( t ) . Solving this linear dynamical equation from an initial state x, to a final state x(t) results in the general solution:

6x = @(t,to,X,)SX, where a is a 6-by-6 matrix (the state transition matrix) with unity determinant (due to Liouville’s theorem), Sx, is the initial deviation from the state x, and Sx is the computed linear deviation from the nominal trajectory. More generally, the matrix @ satisfies:

&(t,t o ) = A ( t ) @ ( tt,o ) ,

@(to,t o ) = I .

While only approximate, this is a powerful result as it provides a general (linear) solution to the dynamical equations in the vicinity of any nominal trajectory. The state transition matrix @ can be used directly to determine whether a given trajectory is stable or unstable. To do so we must compute the Lyapunov Characteristic Exponent, defined as:

where I] - 11 denotes the 2-norm of the matrix. If this limit exists and is finite, the trajectory is unstable and x is the characteristic exponent of the instability. Generally speaking, the value 1/x is the characteristic time and can be used as a measure of the swiftness with which the exponential effects will be detected. If, on the other hand, x + 0, then the trajectory is not exponentially unstable but has, at most, a polynomial growth in time 18.

2.2. State Measurements and Orbit Determination

The second building block of a navigation system, following proper specification of the dynamics, is the specification of the orbit determination measurements. These can generally be denoted as scalar functions of the state, time, and measurement parameters as h(x,t , p ) , where p is the vector of measurement parameters. The quantity h represents some measurable

404

D.J. Scheeres

component of the spacecraft state, or some combination of these components. Usual quantities are a line-of-sight velocity (Doppler shift) range (light-time), or an angle relative to some landmark (optical or VLBI-type measurements), where the actual observations are denoted as h. For a spacecraft moving along a trajectory we denote a series of measurements, each taken at a different time ti, by the sequence hi and the corresponding predicted values for an assumed trajectory x ( t i ) = + ( t i 7 t o , x , )as hi = h ( x ( t i ) , t i , p )We . note that the observables hi are equal to the observable function evaluated at the “true” state x * , denoted by hf = h(x:,t i , p ) , plus a measurement noise wi which is usually assumed to be uncorrelated in time (white) and to follow Gaussian statistics with a zero-mean and a variance u?. Then the orbit determination problem can be solved by finding the ini-

xEl

tial state x , such that the functional L = wi where the wi are “weights” that will be defined later. The necessary conditions for the minimum of L to exist are a L / a x , = 0, or N

It;.

where @ ( t i to, , x,) = dx/dxoltiand hxi = These necessary conditions are non-linear, since the initial state x , is present implicitly in hi, hxi and in @i. In practice, one assumes that a nominal trajectory is defined which is relatively “close” to the true trajectory in phase space, and that a small correction to the initial nominal state can satisfy the necessary conditions. Specifically, we wish to increment the nominal solution to x , + bx, and solve for the linear correction. Taking the transpose of the above expression, substituting x , Sx, for x,, and performing the expansion in Sx, we find the new necessary conditions:

+

N

N

i=l

i=l

where zi = hi - hi is ideally equal to the data noise if the nominal orbit equals the true orbit, meaning that we can never recover the exact conditions due to the noise terms. Ignoring higher order terms, we can im-

Navigation of Spacecraft in Unstable Orbital Environments

405

mediately solve for the correction bx, to satisfy the necessary conditions: N

bx, = A-’

C~iiPihz;Zi, T

i= 1

N

A =Cwi@~hxih~i@i,

(2)

a= 1

where A is referred to as the information matrix at the epoch t o , and will be invertible if the observations taken together span the full initial state. Since this is only an approximate solution, the procedure must be iterated to solve the non-linear conditions (Eq. 1). In general, if the nominal solution is close enough to the true solution, this iteration procedure will converge on the so-called least-squares orbit determination solution. With every measurement, or series of measurements, we can assign an information matrix, denoted here as bA, which adds to the current information matrix at epoch. From Eq. 2 we see that, in general, bA = [email protected] the measurements will occur at discrete times, we do not consider continuous formulations of measurement updates and instead represent the effect of a measurement at some time t as: A’ = h + 6 A , where A’ is the new information matrix, all evaluated at epoch. Computationally, the measurement information updates are usually defined as a Householder transformation operating on a square root information matrix 2 J .

2.3. Distributions of Orbit Uncertainty

Since the orbit determination procedure outlined above contains uncertain data measurement terms, which can be described statistically, the resulting solutions for the orbit must also be, in some sense, uncertain and describable using statistical concepts. We formalize these statements in the following.

2.3.1. Statistical Description of Orbits First, under the assumption that the measurement noise has zero mean (which means that there are no unmodeled biases in the measurement function h), is uncorrelated in time, and has a Gaussian distribution at each time step, it can be shown that the true solution equals the mean solution of the distribution. Furthermore, if the data weights are chosen such that wi = l / a f , then the information matrix A is the inverse of the state covariance matrix, P . Finally, it can be shown that the probability density

406

D.J. Scheeres

function (PDF) of the initial conditions can be described as:

1

f(x;x,P)=

(2x1N’2

e- ;6xTA6x

m

where 6x = x - Sr and N is the dimension of x. Using the PDF we can define the mean of the solution,

x=

1,

xf(x;x,P)dx

the covariance of the solution,

[x- x][x - XIT f(x;Z, P)dx and the probability that a spacecraft resides in some region space:

P(x E a) =

s,

f(x;%,P)dx

B

of phase

(3)

The integral J, is taken over the entire phase space, while the probability integral is only taken over the phase volume contained in the region B. While the description of the solution mean, covariance, and probability at the initial epoch to is useful, we would like to generalize this result to an arbitary time. The covariance and information matrices can both be viewed as dynamical quantities that vary in time, satisfying the equations:

P = A P + PA^,

A = - A ~ A - AA,

(4)

where A = JHxx has been defined previously. This allows us to specify the covariance and information matrices as functions of time: A(t, to,Ao, x,) and P(t,t,, Po,xo) where we have noted the explicit dependance of these dynamical quantities on the initial state and initial distributions. The specific solution to these equations can be formulated in terms of the state transition matrix:

A(t) = 9-T(t,to)Ao9-1(t,to). P(t)= 9(t,t,)P,9T(t,to), Thus, the PDF and the probability distribution can be defined as general functions of time:

3, P(t>> =

f(x;

P(x E B ( t ) ) =

Navigation of Spacecmft in Unstable Orbital Environments 407

The above equations neglect the effect of model and measurement parameter uncertainties, which can be brought into the dynamical equations for the covariance and information matrices 25.

2.3.2. Probability measure as an integral invariant First, consider some region of phase space B, defined at an initial epoch t o . As mentioned above, we can compute the probability that the spacecraft can be found within this phase volume as P ( x E B0).There are two quantities of interest that can be attached to this idea, the first is the evolution of the phase volume 6, as a function of time, and the second is the probability of finding the spacecraft within this evolving volume. The first consideration can be understood, in a non-statistical sense, as the evolution of the phase volume:

V(t>=

L(t)

dx =

J

dx

4(t,to,Bo)

where the integral occurs over the 6-dimensional region B ( t ) mapped in time. Since we have assumed a Hamiltonian structure to our dynamics, we can immediately apply Liouville’s Theorem lo and note that the volume is conserved. This is an instance of an absolute integral invariant, stating that an integrated quantity defined over an arbitrary region of phase space is constant in time. Formally, a phase space integral of maximal order can be stated as:

I =

L

M(x,t)dx

(5)

where we assume that the state follows the dynamics equation x = F ( x ,t ) . A necessary condition for I to be an integral invariant is that the scalar quantity M satisfies the condition lo:

dM dt

- + MTrace

(a,> aF

=0

For the case where M = 1, we see that I = V, the phase volume. In this case, d M / d t = 0 and the condition reduces to Trace = 0. Now recall that we are dealing with Hamiltonian dynamical systems, so F = J a H / a x . Allowing the state x to be split into vectors of coordinates, q, and momenta, p, we have x = [q,p]and we find the general result that qi = a H / a p i and

(E)

408

Pi

D.J. Scheeres

= -aH/dqi. This leads to:

establishing Liouville's Theorem.

It should be noted that the application given below assumes that the force parameters of the system are fixed, and have no range of uncertainties associated with them. This is a reasonable restriction on the system, but one that cannot always be applied in spacecraft navigation. Now note that the probability measure defined previously in Eq. 3 is in the proper form to be an integral invariant. Thus we can check to see if the probability measure is invariant under the dynamics of the system, where the PDF function f is identified with the M function in Eq. 5 . In the following we will assume that the region over which we integrate to find the probability of our system is relatively small compared to the actual state components, allowing us to use the linearized flow to describe motion. Let us restate the PDF, now set equal to the M functional, as

where 6x = x - Z. Assume that force model uncertainties are not included in the information matrix (although measurement parameter uncertainties can be included without affecting the following). From above, we already see that the second factor in Eq. 6 is satisfied, as we assume a Hamiltonian dynamical system. Thus, we only need to establish that d M / d t = 0, or:

First consider the time derivative of IAl. It can be shown that

!!dt

lo

= -2IAITrace(A) n

[d2H/dqidpi - d2H/dpidqi]

Trace(A) = i=l

which was the same condition as for Liouville's Theorem, and thus the determinant of A (and also P ) is a constant. It is important to note that this is no longer true if force model parameter uncertainties are included,

Navigation of Spacecmft in Unstable Orbital Environments 409

as then the information content will have a uniform decrease in time. Next consider the time derivative of the exponential function. Now we will invoke a linearization assumption to assume that b x = Abx. The condition then becomes:

6xT [ATA + A A + A] bx = 0 which is trivially satisfied, given Eq. 4,if no uncertainty in the force parameters are assumed. Thus we see that the probability of finding a spacecraft within a given region is an integral invariant if there are no uncertainties in the force model, meaning that this probability does not change its value over time.

A different, and equivalent, statement is that the PDF itself is a constant of the flow, or that:

A partial differential equation relation that holds everywhere in phase space. These may seem like obvious results, but we note that if force parameter uncertainties are included into the PDF they are no longer true and that the probability of finding a spacecraft within one evolving region of phase fluid is not constant in time. What occurs in this case is that the uncertainties in the dynamics allow possible trajectories to leave the nominally defined phase fluid volume. An interesting question is whether a suitably generalized description of the dynamics would allow the integral invariance to hold again. A deeper understanding of what occurs in these cases is still needed. In the following we will ignore the case of uncertain force parameters, focusing instead on the simpler case.

2.3.3. Pro bability Computation As discussed above, the region over which we compute the probability of finding a spacecraft is arbitrary. In practice it is common to restrict this region to a generalized ellipsoid that uses the information matrix as a generator. The reasons for this restriction are two-fold, first it turns out that the probability computation over this region can be evaluated in closed form and is directly related to X2-probability distributions. Second, the probability ellipsoids are themselves invariants of the flow, and map into each

410 D.J. Scheeres

other. Should we consider some other region of phase space over which the probability computation would be carried out, we would not have these two properties, even though the probability measure would still be constant. Specifically, let us consider the probability of finding the spacecraft within a region defined by the ellipsoid:

SxTAGx 5 r2 where r is an arbitrary number, and bx and A can be considered to be evaluated at epoch. Then, B = {Sx 1 r2 - 6xTA6x 2 O } . Thus, the probability integral can be stated as:

By a suitable change of variables, this can be reduced to the form (for general N) :

P=

2 2N/2F(N/2)

+

1'

UN-l

e - l5u 2 du

(7)

where r(n) = ( n - l)! and r ( n 1/2) = 47(271)!/2~~/n!,where n is an integer. The coefficient of the integral in Eq. 7 represents the integral over the surface of a sphere in N dimensions divided by ( 2 1 r ) ~ /and ~ , the remaining integral represents the integral over the radius of that sphere. For the general case of spacecraft motion, N = 6 and the coefficient of the integral is 1/8. This integral can be rewritten in a standard form:

which is in the classic X2-probability integral form, for which tables exist. In usual navigation practice, the state of the spacecraft is only desired on some lower-dimensional surface. A classic example is the computation of probability of the spacecraft when projected onto the plane perpendicular to the approach trajectory to a target planet. This represents a computation of probability on a 2-dimensional surface, only involving the position components. h r t h e r simplifications occur if we map into a 1-dimensional subspace, which will happen when we consider the statistics of AV consumption to control an orbit. When computing the probability distribution in these sub-spaces, it is necessary to first compute the relevant PDF for that projection, meaning that the mean and covariance of the new variables

Navigation of Spacecmft in Unstable Orbital Environments 411

must be calculated. Two situations will occur for this case, in general. The first is that the projection is a simple linear combination of the state, and can be represented as v = Qx,the second is that the projection is the norm of a linear combination, and can be represented as w = ~ ~ Q xthe ~ ~2-norm 2, of a vector. In both cases the matrix Q is of order m x n, m 5 n. For the first case we find that the new mean and covariance are simply related to the mean and covariance of the state x:

v = QZ P,, = QPQT where K and P are the mean and covariance of the original state x. Then the computation of the probability can proceed using the PDF:

where we note that the covariance matrix P,, will be non-singular in general as m <_ n and P is non-singular.

For the second case, we see that the computations are simplified in that we must only compute the mean and variance of a scalar, but that, the integrations we must perform become more complicated as they are non-linear in the state x. Specifically, we find: -

v=

Irn

I)Qxllzf(x)dx

~5 =

Irn

[IIQxlla - V]' f(x)dx

(8)

The computation for the mean in Eq. 8 can be bounded. First note that the information matrix in the PDF, A, can arbitrarily be split into the product of the square-root information matrix, R, as A = RTR '. Then a change of variables to the vector u = Rx is performed, where the Jacobian lax/&^( = 1/IRI = resulting in the elimination of the covariance determinant from the integral. This is the transformation used to reduce the probability integral to the form found in Eq. 7. Thus, anticipating this transformation we can rewrite v = IIQR-'uIJ2<_ I(QR-1J1211ul12 yielding

m,

412 D.J. Scheeres

yielding the inequality:

A similar bound can’t be derived for the variance due to the subtraction, still this simplified formula allows for an estimate of the mean without carrying out a detailed integration. It is important to point out that these integrals can also be carried out in closed form in some situations. In Renault and Scheeres 2o these integrations are carried out for motion relative to an equilibrium point in a two degree of freedom dynamical system. For this case they can be reduced to elliptic integrals of the first and second kind. Carrying out these integrations for higher-dimensional or time-varying systems becomes more difficult. 2.4. Statistical maneuver design

To plan for the navigation of a spacecraft, it is necessary to develop a statistical model for the amount of fuel necessary to keep the spacecraft on course. With the above results and definitions, such an analysis can be performed. In its most general form, the problem can be stated as follows, using the notation q = r and p = v. Given a statistical distribution in position and velocity relative to the nominal trajectory at time to of the form 6r,, 6vo,what is the mean and variance in the cost of the maneuvers to reduce the system back to 6r = 6v = 0 at some future time. Generally, the errors in position and velocity arise from the previous maneuver and can be thought of as errors in knowledge of the spacecraft state. Practically, maneuver execution errors must also be incorporated, but we can ignore these for the generalized discussion provided here. For a general trajectory, a minimum of two maneuvers are required to get back on track. One maneuver to target back to the trajectory at some future time, and a second maneuver to reduce the relative velocity to zero at that crossing. Of course, at the time when the trajectory crossing occurs, errors from the epoch of the last maneuver manifest themselves in a new set of dispersions, which must themselves be corrected. By considering the new dispersions to be uncorrelated with the initial dispersions (a conservative assumption in general), we can isolate these effects from each other and perform an analysis on the two maneuvers alone (see Fig. 4). For the

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design of these maneuvers, we have two free parameters, the time at which we perform the first correction maneuver, t l , and the time at which the trajectory crossing will occur, t2, both relative to the initial epoch to where the Sr, and 6vo are defined. In some instances, it is possible to choose t2 - tl >> tl - to. Then the second correction maneuvers may occur after several time intervals of tl - to have passed. Practically speaking, in these situations, the originally designed second maneuvers will never occur, as each correction maneuver will retarget the second maneuver based on new measurements. The approach outlined above is not the only approach to orbit control, and in fact a number of different possible solutions to this problem can be formulated. G6mez et al. and Dunham and Roberts review the different approaches to this problem as applied to halo orbits. These alternate approaches can be summarized as targeting to minimize the future deviation from the trajectory and targeting to kill the unstable component of motion. In Renault and Scheeres 2o an explicit comparison of these different techniques is made for control near an unstable equilibrium point in the 3-body problem. This being said, in the current document we focus on the more classical approach to retargeting to the nominal trajectory as this can yield definitive results, and can be used as a general measure of statistical maneuver costs in orbit control problems. We can express the maneuver strategy explicitly. Assume the state transition matrix is partitioned as:

Then, given a set of initial errors at epoch to, Sr, and Svo,the state of the system at a later time tl is:

6rl = 47-T (tl, t o ) S r o + 49.u (tl ,t0)SVO SVl

= 4 z ) T (tl, t 0 ) S r o

+ 4 u u (tl ,tovvo

A targeting maneuver is designed to null the spacecraft position error at some future time t2, essentially targeting the spacecraft to cross the nominal trajectory:

6r2 = 0 = 4 T T (t27 tl P

l

+

4TU

(t2, tl)SV’,

The first maneuver is then computed as AV1 = Svi - Svl, and is explicitly

414 D.J. Scheeres

equal to:

A K = - [ + , - , 1 ( t 2 , t l ) + r r ( t 2 , t I ) + r r ( t l t t o ) + +vr(tl,to>]bro - [+,-,1(t2,tl)+rr(t2,tl)+ru(tl,to)

+ +vu(tl,to>]6vo

(9)

The second maneuver is then performed at time t 2 and nulls the relative speed of the spacecraft to the nominal trajectory, AV2 = -6v(t2), explicitly leading to:

Av2 = [+vv(t2,tl)+,-,l(t2,tl)+rr(t2,tl)+rr(tl,to)-+vr(t2,tl)+rr(tl,to)]

6ro + [ + v v ( ~ 2 , ~ l ) ~ , - , l ( t 2 , t l ) ~ ~ r ( t 2 , ~ ~l )0+) r- u~ (u~r l( ,~ 2 , ~ l ) + r u ( t l ,6vo ~o)] (10)

Thus the general correction maneuver is a formula of the form:

AK = I Q \ E ~ ~ x ~ where Q, is a matrix in general. To compute the statistical cost of these maneuvers requires us to compute the mean and variance:

-

AV = 2

=

JwAV f (xo)dxo

1,

(AV -

q2 f(xo)dx0 = (AV)2 - EV2

If we implement a series of M such maneuvers, each with the same assumed statistical and dynamical representation, the total mean maneuver cost is M D and the total variance is M o i V . Thus, if we wish to estimate the statistical cost of performing this sequence of maneuvers to within an nsigma probability value (l-D Gaussian), we find:

Instead, if we wish to bound the mean AV we find:

where N is the dimension of the phase space being considered and R is the square-root information matrix.

3. Model Problems of Unstable Orbital Environments

Having reviewed the basics of spacecraft navigation, we are interested in defining appropriate models that contain unstable orbital environments.

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415

Thus, to motivate our discussion, we present some of the classical models that do so.

3.1. Geneml models for unstable orbital environments For our generic model problem we take the following Lagrangian equation of motion in a rotating coordinate frame:

z.. = -

av

az

This can be shifted into canonical Hamiltonian form by defining q = r and p = r 5 x r. This leads to a Hamiltonian of general form:

+

Starting from this basic form we can define several possible models by specifying V, all of which contain unstable motion.

Restricted 3-body Problem

v = -21 (2+ y2) +

1-P J(X

-4

P

- 1+ p12 + y2 + 22 +

where p is the fraction of total mass that the second, smaller body has. The restricted 3-body problem can be considered as the simplest model of spacecraft motion in the Earth-Moon system, and contains a host of interesting unstable motions that spacecraft could be flown in. Recent interest has focused on maintaining halo orbits, or lissajous orbits, in the vicinity of the L1 libration point (between the Earth and the Moon) as a way-station for transfers into the solar system and into the Earth-Sun halo orbits. A variety of different uses of the Earth-Moon libration points (and associated regions) have been proposed, originally by Farquhar4. One challenge to the use of these orbits is that motion in the vicinity of the Earth-Moon libration points are dominated by a hyperbolic instability with characteristic time on the order of 2 days. Thus, we expect dynamical instabilities to play an

416 D.J. Scheeres

important role in controlling the implementation of spacecraft navigation for such systems.

The Hill Restricted 3-body Problem 1

=

dm

1 +

(3s2 - 22)

The Hill restricted 3-body problem can be considered to be the simplest model of motion in the Earth-Sun system, and contains analogous dynamics to the restricted 3-body problem in the vicinity of the secondary. The characteristic instability of halo orbits in the Earth-Sun problem is on the order of 25 days, much slower than for the Earth-Moon system, but still significant. While there have been several missions to these unstable regions, it is still considered to be a challenging environment for trajectory design and navigation. The Hill restricted 3-body problem has another unstable motion of practical interest contained within it. For orbital situations such as that found at Europa (the moon of Jupiter), which are well modeled using the Hill problem, it is found that low altitude, circular orbits with inclinations between 39 - 141' are unstable and will impact on the surface within a few weeks if uncontrolled 23. The discussions given below will also apply to this situation.

-

Of interest, combinations of the Earth-Moon and Earth-Sun dynamical systems have been carried out. The earliest attempts occuring in the 1960's A recent generalization of this model has been proposed 27, combining the Hill and Restricted 3-body problem models in a dynamically consistent fashion. 14115,21.

Asteroid Orbiter Problem

where 6 and X are the body-fixed latitude and longitude of the test spacecraft, C 2 0 and C22 are the gravity coefficients, and P 2 0 and P22 are the generalized Legendre functions. There has been extensive analysis performed for the asteroid orbiter problem 26,8,16 . One general result is that an unstable orbital environment usually exists within a few mean resonance radii of

Navigation of Spacecraft in Unstable Orbital Environments 417

the body.

3.2. Simplified model for analytical study

In this paper we will also consider a simplified model for illustrative purposes and to provide a better understanding of the differences between navigation in traditional and unstable orbital environments. Consider the 1-DOF dynamical system: i.+a?-=O,

[;]

=*(t)

[;I]

Three different types of motion are possible depending on the parameter a. If a > 0, motion is oscillatory, if a = 0, it is rectilinear (degenerate), and if a < 0 it is hyperbolic. In particular, we can find explicit solutions for each of these cases. Oscillatory Motion: 2

*(t)=

a=w>o,

cos(wt) sin(wt) -w sin(&) cos(wt)

Degenerate Motion:

a=o,

*(t)=

[;;I.

Hyperbolic Motion:

a = -A2 < 0,

cosh(At) sinh(At) @(t)= Asinh(Xt) cosh(At)

1

'

Each of these cases can be used as simplified models for different regimes of motion. The oscillatory case corresponds to completely stable motion, such as might be found for the out-of-plane motion of a satellite in geosynchronous orbit. The degenerate case corresponds to motion in an integrable, or near-integrable, environment and can be viewed as analogous to the constant downtrack drift a spacecraft experiences relative to its nominal trajectory. Finally, the hyperbolic case can be viewed as analogous to motion about an unstable libration point.

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4. Analyzing Navigation in Unstable Environments In the following we consider the three fundamental tasks of navigation: prediction of orbit uncertainties, estimation of orbits (which can be viewed as the sensing and reduction of orbit uncertainty distributions), and the control of orbits due to their statistical distributions. In developing each of these ideas we will first introduce a general analysis approach to the question, use our simple 1-DOF model to motivate our discussion, and highlight differences between the three dynamical regimes.

4.1. Propagation of Orbit Uncertainty Distributions

We first consider the geometry of an orbit distribution mapped in time. We are specifically interested in the direction and extent of the maximum and minimum uncertainty of our distribution. This relates specifically to the prediction problem in navigation.

4.1.1. Distributions in Tame First, consider the extremal extensions that a linear deviation from a trajectory can have, given a constrained initial state. Specifically, we wish to maximize or minimize the 2-norm of dx, or equivalently the function 6xT6x, subject to the constraint 6x;fAo6xo- 1 = 0, meaning that the initial distribution lies on an ellipsoid of constant probability. In other words, given an initial probability distribution, what is the maximum or minimum state that can result in the future, and what initial conditions lead to this extremal state. To answer this question we form the augmented Lagrangian:

L = 6xT6x - c2 (SxTA,Gx,

- 1)

s x = q t , to)6xo where c2 is the Lagrange multiplier, and compute the extremal conditions a L / d x o = 0, leading to:

Navigation of Spacecraft in Unstable Orbital Environments 419

From this we see that the extremal values of all the trajectories in the flow, given the initial constraint, is:

6xT6x = ff 2 where cr2 is an eigenvalue of Eq. 12. Thus, the maximum or minimum extent that a linear trajectory can have, starting from a constrained state distribution, equals the square root of the maximum or minimum eigenvalue of A;1@T9. We note that the eigenvector ax, is not an eigenvector of the matrix 9 in general, and that the resulting direction of this largest deviation, 6x = @Sxo,is not necessarily related to the eigenvectors of @ either. For time-invariant systems, or for time-periodic systems, these two will approach each other when the time grows arbitrarily long. For generalized motion, however, there is not necessarily a relationship between these quantities. A complementary question is what the extremal value of a distribution is at a given time. We restrict ourselves to trajectory dispersion distributions that initially lie within an ellipsoidal region, and hence will always remain within an ellipsoidal region: 6xTA(t,to)6x5 1. Now we recall that the information matrix can be mapped in time using the state transition matrix (in the absence of stochastic acceleration noise) A ( t , t , ) = 9(t,t , ) - T A o @ ( t , t o ) - l and we wish to maximize the value of 6 ~ ( t ) ~ 6 x subject (t) to lying on the surface of the current probability ellipsoid, leading to the Lagrangian:

L = 6xT6x - o2 (6xTA(t,to)6x- 1) and the extremal condition dLld6x = 0:

[$1 - A ( t ,

1

to)

6x = 0

Again, the maximum or minimum value of all trajectories given the final constraint is:

6xT6x = ff2 where l/a2 is the minimum or maximum eigenvalue, respectively, of the information matrix A ( t ) . Now note that the problem can be transformed by multiplying by the covariance matrix P ( t , t , ) = A(t,to)-l and u2 to find:

[P(t,t o ) - a21]6x = 0

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D.J. Scheeres

showing the equivalence of evaluating the distribution in terms of covariance or in terms of the information matrix. Now, to close the circle, let us evaluate the trajectory along the initial condition in Eq. 12 found to be an extremal of the resulting trajectory flow to see if this satisfies the extremal condition for a distribution constrained by probability at a given epoch t , Eq. 13:

cPT9bx0 = cr2Aodxo, aTSx = cr2A06xo, Sx = cr2@-TAoS~o, SX = a29-TAo9-1dx. But by definition,

= A(t, t o ) and we get

SX = cr2A(t, to)6x equivalent to Eq. 13. We can note here that the eigenvectors of A, do not map into the eigenvectors of A(t), meaning that the direction of maximum or minimum elongations map, in the future and in the past, into some other general direction on the ellipsoid surface. Thus, at a current time, the state that is furthest from the nominal trajectory will not retain this distinction, and will not satisfy the extremal conditions at an earlier or later time. The above discussion is in the context of extremal values of the state, which include maximum, minimum, and all intermediate axes on the ellipsoid. Evaluating the eigenvalues and eigenvectors of the covariance matrix immediately provides the values and directions of all these extreme values, where we note that the eigenvectors are all mutually orthogonal since the covariance matrix is symmetric in general. If the linear system is time invariant (motion about an equilibrium point) or corresponds to a time-periodic system (motion about a periodic orbit), then the limiting uncertainty directions can be defined as time grows large, and are identified as the unstable manifolds of the object. In this case, there is a correspondance between the limiting direction of uncertainty distribution and the eigenvectors and eigenvalues of the state transition matrix 9. In Scheeres et al. 24 a direct comparison between the maximum eigenvectors of 9 along an unstable halo orbit and the direction of maximum uncertainty were made. It was found that these objects align with each other within 2 or 3 characteristic times of the instability. It is significant to note that this is shorter than the period of the halo orbit, indicating that the eigenvectors of the state transition matrix start to converge on the manifolds of the periodic orbit well before the first return of the periodic orbit, defining the Poincark map.

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421

For a general trajectory not associated with an equilibrium point or periodic orbit, the limiting direction of uncertainty is not well defined over finite periods of time. Even though the characteristic exponent of the system is defined as t + 00, the evolution of this direction over time periods shorter than this may exhibit any number of transient phenomenon. Situations of interest here, that have yet to be studied, include orbit uncertainty distributions along quasi-periodic orbits (the Lissajous trajectories in the restricted three body problem) and distributions along stable and unstable manifolds (such as the Genesis transfer trajectory towards and away from the halo orbit). The implication of these results for orbit uncertainty distributions relate to the probability of finding a spacecraft within some region of phase space projected into the measurement space in which the spacecraft is sensed and tracked. As the direction of uncertainty becomes pulled along the unstable manifolds of an orbit, the projected area in position and velocity space where the spacecraft can be found can become large, even though the total volume of the phase distribution is constant. This is an important point. Indeed, even though the total phase volume of an uncertainty distribution is conserved as it is mapped forwards in time, projections of this distribution are not conserved in general. At the simplest level, if we consider the surface area of the 6-dimensional uncertainty ellipsoid associated with a generic unstable trajectory, we see that the total surface area of this distribution will be unbounded as t + 00, even though the total volume is conserved. Similarly, the projection of such a distribution in either position or velocity space may also lead to a distribution that is unbounded, even though the total volume in the 6-dimensional phase space is constant. Some of these issues are discussed in more detail in 25. Additionally, as certain regions of the probability distribution evolve far away from the nominal orbit for an unstable trajectory, the effect of non-linearities on the orbit distribution can begin to become important.

4.1.2. Application to the 1-DOF System

Given an initial uncertainty distribution defined by A,, what are the asymptotic values of the maximum and minimum uncertainty distributions?

[aT@- a2A,] u = 0

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D.J. Scheeres

For our 1-DOF system the essential question is, what is what direction does it lie along?

(T

for t >> 1, and

For the oscillatory case, for simplicity, we use a A,:

The eigenvalues are (T = 1,w, which map entirely into the T and v directions, respectively. Thus all deviations are bounded, as expected. Also, for this case, we can compute the “boundary” of the distribution, here being the circumference of an ellipse of semi-major axes 1 and w. An approximate formula for the circumference C of an ellipse of semi-axes a and b is C 2 7 r , / m , leading to a constant circumference for this example. This implies that the orbit uncertainty distribution for such a dynamical system will always be bounded, and definite constraints on where the orbit can be found for all future time can be derived.

-

For the degenerate case we assume:

-

When t >> 1 this results in eigenvalues (T t, l / t , and the deviations grow linearly in time along the position direction and contract inversely with time along the velocity direction. Thus, as a given uncertainty distribution is mapped forward in time, it becomes well determined in the velocity direction, but poorly determined in the position direction. This again agrees with the basic results found for uncertainty propagation of near-Keplerian (n/fi)t, and is unorbits. The circumference of this ellipse is now C bounded with time, but only linearly. Thus, as time grows large, we expect to find the distribution spread over a larger and larger region of space (position for our 1-DOF system). Specifically, the projection of the orbit distribution into the position coordinate is unbounded in time, while the projection of the orbit distribution into the velocity direction is bounded.

-

Finally, for the hyperbolic case we will treat our simplified problem in more detail, using the general analysis outlined above. First, since our system is time-invariant, we have an unstable manifold found by analyzing the eigenvalues and eigenvectors of the matrix a:

Navigation of Spacecmft in Unstable Orbital Environments 423

Realizing that the eigenvalue X can be interpreted as the tangent of an angle ym, defining the direction of the stable and unstable manifold in phase space, we re-write these as (Fig. 2):

Fig. 2.

Mapping of an uncertainty distribution in an unstable 1-DOF dynamical system.

Now, let us consider the separate issues of the direction of the maximal distribution and the initial condition direction for the maximum distribution. In each case we will consider an initial distribution Po = A. = I for simplicity. Then the maximal distribution direction at a given time t is found as the maximum eigenvalue and corresponding eigenvector of the equation1

[ q t ,t , ) V ( t ,t o )- 0211 u = 0 and the initial condition which gives the maximal distribution at a given time t is found as the maximum eigenvalue and corresponding eigenvector of the equation:

[aT@, to)@(t, t o )- u21]u, = 0

(15)

We note that these equations will have the same characteristic equation, as can be inferred given the correspondence between the eigenvectors: u = @ ( t to)uo. , For our simple case this results in a characteristic equation:

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D.J. Scheeres

with roots

02 = 1+ -12 sinh2(Xt) fsinh(Xt) (A

+

i)

{1+ ; 1isinh2(Xt)

The direction of the maximum uncertainty in phase space at a given time is controlled by the eigenvector of Eq. 14, and is defined by the angle TP : 1 tanyp = - tanh(Xt) 2 1 tan(2yp) = t anh(At) tan(%,)

nr 1 yp = - - arctan 2 2 where n even corresponds to u+ > 1 and n odd to u- < 1. Thus, as expected, we see that as t + 00, tanh(Xt) + 1 and yp + 7,. In general, we have the direction for the maximal distribution:

+

.=[.

]

+

[

cos yP s1n yP TOfind the initial condition that yields this maximal distribution, we can solve for the eigenvector of Eq. 15 or compute u, = W 1 ( t ,t,)u = @(-t)u, which would be simpler for this system, yielding tanyp - tanh(Xt) tany, tan ypo = tan ym tanyp tanh(Xt) - tany, which can be reduced to: 1 tan 2yp0 = tan(r - 27,) tanh(At) nr 1 tan(r - 27,) ypo = - - arctan tanh(At) 2 2 with the same rules for n even and odd, again. Now as At + 00 we see that YP, + r / 2 - ym.

1

Thus, the optimal initial condition to fall under the influence of the unstable manifold does not lie on the unstable manifold, but is shifted

Navigation of Spacecmft in Unstable Orbital Environments 425

off of it. This can be understood when one considers the optimal initial condition to minimize the state magnitude. This direction is orthogonal to the above direction, and thus is equal to -ym, which does lie on the stable manifold. That this minimizing direction must lie on the stable manifold can be understood in that any deviation from the stable manifold will carry the trajectory onto the unstable manifold, and will increase its length. Next, we note that the extrema of the state are eigenvectors of the matrix aT9, which is symmetric and which thus implies that all eigenvectors will be mutually orthogonal. Thus, since the stable and unstable manifolds are not mutually orthogonal in general, but are separated by an angle 2ym, the optimal initial condition to increase the state cannot lie on the unstable manifold.

-

Finally, if we consider the circumference of the resulting ellipse, we find (7r/fi)ext. Thus now the distribution grows exponentially and, in general, projects into both position and velocity spaces (Fig. 2). Indeed, for our 1-DOF system, projecting the distribution into either position or velocity yields a distribution that grows exponentially with time. Thus, in this case, even though we have a finite volume of phase space in which to find the orbit, in time this space can be distributed over all possible positions and velocities. The key to removing the apparent contradiction is that the position and velocity are highly correlated with each other. Indeed, if the position is determined at some point in the future, the uncertainty in the orbit’s velocity immediately collapses due to the strong correlation between these two.

C

4.2. Optimal Measurement Strategies

A formal presentation of optimizing orbit determination measurements and measurement schedules is presented in Battin (pp 687-693). Scheeres et al. 2001 24 considered this question in light of an unstable orbital environment, and established that orbit determination sensitivity can be related to the unstable and stable manifolds of an orbit. Thus, the idea of optimizing orbit determination measurements can be closely associated with the linear dynamics structure of the nominal trajectory. Following from this, we can concieve of taking orbit determination measurements at optimal times relative to the orbit manifolds that will maximally increase the information content of the orbit (i.e., decrease covariance), and of avoiding measure-

426

D.J. Scheeres

ments during times when we expect the increase in information content to be small, such as occurs when measurements are taken along the stable manifold (see Fig. 3). Along these lines, in the following we introduce some basic ideas of how orbit determination measurements can be viewed in terms of dynamical systems theory.

A+6A = A’ 6A = h, hxT

Fig. 3. Illustration of the idea behind optimal measurement epochs, based on an uncertainty distribution.

4.2.1. Characterizing measurements in phase space As the spacecraft moves along its trajectory it is occasionally tracked from an Earth station. In the following we assume that the tracking stations take Doppler data during each track. As is well known l1 the estimate that can be extracted from a pass of Doppler data is the spacecraft unit position vector and it’s line-of-sight velocity. Thus, in phase space a pass of Doppler data is approximated by the measurements:

where i is the unit position vector of the spacecraft and v is its velocity vector. From the classic Hamilton and Melbourne analysis l1 we note that the determination of angular direction can be singular in declination if the spacecraft lies in the equatorial plane of the Earth during the observations. For our current analysis we are interested in the information content of these measurements with respect to the state. This is computed by taking

Novigotion of Spocecmft in Unstable Orbital Environments

427

the partials of the measurements with respect to r and with respect to v: ah, 1 ah, - - 0, - = -u,,, dr T dV

u,,

= [ I - ii],

where U,, is a dyad operator that removes the vector component parallel to the unit vector f . Thus, i. .U,, = 0, and if i.1 .i= 0, then i.1. U,, = 21. In terms of the position component, the measurement h, has a null space for orbit uncertainty distributions along i., meaning that it cannot directly detect this component of an uncertainty distribution. The line-ofsight velocity measurement has it’s null space along this same direction, i, and also has a null space along the direction defined by i x v, and thus has a twedimensional space where orbit uncertainty distributions can “hide”. For the velocity component of the partial, the h, measurement has no information content and the h, measurement has a null space orthogonal to i. When we combine this with our realization that our orbit distribution will have a characteristic direction for maximum and minimum uncertainy, we see that there may be phase space geometries at which a measurement can have an optimal impact on the uncertainty distribution (when the maximum uncertainty falls into the sensed direction) and other geometries where measurements prove to be ineffective (when it lies in the null space of dhfdx).This issue has been studied for a spacecraft in an Earth-Sun halo orbit, tracked from the Earth, where it was found that the orientation of uncertainty was controlled to some extent by both the local unstable dynamics and by the phase space geometry 24. Specifically, in the absence of tracking the orbit uncertainty distribution is entrained along the unstable manifold of the orbit, as expected from our previous analysis. However, in the presence of measurements the unstable manifolds become better determined, due to the sensitivity of errors in this direction. The axis of maximum uncertainty for these cases was oriented perpendicular to the local unstable manifold, in general, indicating that these directions were preferentially determined. When the phase space geometry was aligned so that the unstable manifold fell into the unobservable direction of the measurements, however, this situation was reversed and the direction of maximum uncertainty and

428

D.J. Scheeres

the local unstable manifold became aligned again. This indicates that a systematic approach to choosing optimal measurement times may be available, based on the local dynamics of the trajectory. In particular, if there exist certain periods of time where a fortuitous alignment between the sensing geometry and the distribution direction exist, we would expect a commensurate increase in our ability to determine the orbit with a single measurement. Should periods of poor alignment exist, we may defer measurement, or schedule an increased number of measurements should this occur during a crucial period of the mission. This is, indeed, entirely analogous to the “zero declination” singularity in Doppler tracking, but instead of arising purely from geometry, it arises due to the local dynamics along the spacecraft trajectory. When considering future systems designed to track spacecraft in specific environments, such as the Earth-Moon LZ libration point, this approach can be used to maximize the efficiency of the orbit determination system used for control. Following Battin l, we can pose the question of optimizing an observation in terms of finding the measurement partial that yields the maximum decrease in the trace of the covariance matrix. First we note that:

assuming that h is a scalar measurement. As noted in Battin, if we assume a perfect measurement (a, + 0) then the h, that minimizes the trace will be along the maximum eigenvector of the covariance matrix. For an unstable orbital system, this effect can become quite strong, as the orbit distribution can be pulled exponentially along the unstable manifold. It is important to recall, however, that the direction of maximum uncertainty is not equal to the unstable manifold direction over finite time spans, although it is asymptotic to it over a few characteristic times. Another important item to note, and characterized in 24, is that repeated measurements along the unstable manifold causes the orbit uncertainty distribution to become prefferentially determined along this direction, ultimately leading to this direction becoming relatively insensitive to improvement. Arguing along similar lines, we also find that measurements along the stable manifold of a trajectory, where the orbit uncertainty is naturally smaller, are also relatively insensitive. In fact, given frequent measurements of an unstable trajectory we find that the hyperbolic stable and unstable directions become relatively well determined and the center manifold of the trajectory is the

Navigation of Spacecmft in Unstable Orbital Environments 429

most poorly determined direction.

4.2.2. Application to a simple 1-DOF system

Consider an arbitrary measurement partial h,, what should its elements be to minimize the trace of the covariance matrix? Given our previous results this question can be answered easily by direct correlation to the distribution dynamics. For the oscillatory case, we see that there is not a strong dependance on the orientation of h,. Thus, any measurement will contribute to the information matrix and there is no strongly preferred direction for an optimal measurement. For the degenerate case the situation changes since the orbit uncertainty becomes spread out over time. We find that as time grows large measurements along the position direction are optimal, and will collapse the uncertainty. Furthermore, since the uncertainty along the velocity direction reduces as l / t , velocity measurements contribute essentially nothing to the information and leave the covariance unchanged. For the hyperbolic case the situation is modified again. Now we find that as time grows large measurements along the unstable manifold are optimal, and measurements perpendicular to this direction (note, not along the unstable manifold) contribute nothing in the asymptotic case. For a hyperbolic motion the position and velocity directions are strongly coupled together. This leads to the interesting result that some position-velocity combinations of measurements can be ineffective, while other combinations can be extremely efficient.

4.3. Control of Unstable Trajectories

The mathematical instability of halo orbits is a positive characteristic when it comes to control, as it enables very small and infrequent maneuvers in the absence of navigation errors (as low as 5 cm/s/year, 4 maneuvers/year for an Earth-Sun libration point orbiter). See 5,6 for a description of the pioneering work on the control of unstable trajectories for ISEE3. When

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D.J. Scheeres

orbital errors do build up, it is typically more expensive to correct to a nominal trajectory than to correct to a near-by halo orbit. But the near-by halo orbit must be designed to compensate for the errors as well as fulfilling the mission requirements and constraints. Due to the infrequency of halo orbit maneuvers for the Earth-Sun case, navigation teams have been able to iterate this labor-intensive process between the maneuver analysts and the trajectory designers t o accomplish the error correction. The question we consider is more ideal and, ultimately, of most concern to systems that exhibit a rapid instability (such as the Earth-Moon system).

4.3.1. Optimal Statastical Maneuver Placement The fundamental metric of navigation performance is the fuel expended in controlling the trajectory, and the frequency of maneuvers needed to control the trajectory. As described previously, the statistical characterization of these costs can be defined for any general trajectory and probability distribution. Thus, a decrease in the cost of these maneuvers can be acheived in one of two ways, either reduce the overall uncertainty or choose the timings of the maneuvers to reduce the statistical cost. Reduction in uncertainty includes optimally tracking the spacecraft as well as reducing control errors (which are not explicitly discussed here). The issue of optimal tracking was already described above. Thus, here we consider the optimal placement of maneuvers. In general, we can use the mean AV to constrain the magnitude of the statistical maneuvers, allowing us to use the limit on derived previously:

m

which means that to bound the mean of any statistical maneuver will only require that the eigenvalues of R-TQTQR-l be evaluated, the coefficient of the inequality being:

Thus, the bound involves the linear solution of motion about the trajectory and the uncertainty distribution at the initial epoch. Considering the uncertainty distribution to be controlled by the measurement sequence,

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431

discussed previously, the only remaining degree of freedom is to choose the maneuver times tl and t 2 to minimize the mean maneuver size. For a given trajectory, and a given set of generic initial uncertainties, this is a relatively simple computation to make for a single maneuver. Of more interest is the design of a sequence of such maneuvers to find the optimal maneuver frequency. Recall Eq. 11, which states the total cost of a sequence of M statistical control maneuvers.

To generate a steady-state rate at which fuel is being consumed, assume that the start of each maneuver sequence is spaced a time T from the previous one (Fig. 4),thus after M maneuver sequences are initiated a total time of M T must have passed. This allows us to define the mean rate at which fuel for statistical correction maneuvers is being used:

If the total number of maneuvers considered is large, thanthe UAV term contributes a negligible amount and the rate is essentially w AV/T. Clearly, if the mean maneuvers are constants, then as T grows large this cost rate drops to zero. Also, if maneuvers are performed very frequently, then T is small and the cost rate increases.

If the spacecraft is being controlled about an equilibrium point, then the dynamics matrix ! I ! will be invariant from maneuver to maneuver, and a general formula can be easily developed for For the control of a trajectory along a periodic orbit, or along a more general non-periodic motion, the dynamics matrix will become a function of location along the orbit, leading to a more complicated analysis that relies on numerical solutions to a time-varying linear differential equation.

m.

Application of this approach to the spacing of statistical control maneuvers about equilibrium points in the R3BP and Hill 3-body problem were analyzed by Renault and Scheeres 2o using analytical integrations for the statistical costs. Resulting from that analysis, it was shown that the optimal spacing of maneuvers is apparently controlled by the characteristic time of the instability. The details of the computation are more protracted, but the bottom line is that an optimal maneuver frequency exists and is

432

D.J. Scheeres

AV

Fig. 4.

Illustration of a sequence of control maneuvers.

equal to the characteristic time of the instability. Table 1 summarizes results from this analysis. In particular, it shows that the statistical control cost of a spacecraft in the Earth-Moon system can be quite large, while the similar control cost of a spacecraft in the Earth-Sun system will be extremely small in general. Both results are due solely to the characteristic time of the instability in these two different systems. Table 1. Targeting the equilibrium in the Hill and CR3BP, ur = lOkm, u, = lcm/s. Main bodies Earth-Sun (period = 1 year) Earth-Moon (period = 1 month)

Problem Hill 3-Body 3-Body Hill 3-Body 3-Body

= 1.01 xo = 0.99 20

= 1.156 xo = 0.837

20

cost m/s/period .470 ,465 .475 3.36 2.66 4.32

optimal spacing 22.9 davs 23.1 days 22.7 days 38.2 hours 44.4 hours 32.8 hours

4.3.2. Application to a 1-DOF system Again, let us first apply these approaches to our oscillatory dynamical system. If the initial errors are acceptable, no maneuvers are needed since the trajectory will not disperse any further from its initial set. Otherwise, the

Navigation of Spacecraft in Unstable Orbital Environments

433

statistical cost rate of the system can be shown to be

av

J -

- N

5 +2xN

T

where N is an arbitrary integer. Thus, we see that taking N + 00 drives the statistical cost rate to zero. The interpretation of this is that it only requires a finite number of maneuvers to control the trajectory for all future time, thus the statistical rate approaches zero as these maneuvers can be spaced arbitrarily in time. Things are more interesting for the degenerate case. Going through the computation (with to = 0) yields:

We can take t2

+ 00

to find:

-

-AV +$ T

Then T is arbitrary but finite, as we need tz-tl + 00. Taking T large allows us to make the statistical cost rate as small as we like. This indicates that statistical control on a degenerate manifold is precision limited, namely that it is the fixed uncertainties in the maneuvers themselves that limit and define the total fuel cost to perform statistical control. Given perfect control, we can drive the statistical cost rate to zero. Finally, let us consider our unstable 1-DOF system again in more detail. Applying Eqs. 9 and 10 we find: coshX(t2 - to) sinhX(t2 - t o ) AVI = -A sinh X(t2 - tl) Sr, - sinhX(t2 - tl) SVO cash X(t1 - to) sinh X(t1 - to) AV2 = -A sinh X ( t 2 - tl) Sr, sinh X(t2 - tl) SVO

+

In this case choose the correction sequence as application of maneuver AVl after a time T from the previous AV1 maneuver, and specify the second maneuver to lie at t2 + 00, which drives AV2 + 0, meaning that the controlled trajectory will asymptotically approach the origin. Naturally, the second maneuver is never performed, and the first maneuver is repeated

434

D.J. Scheeres

after every time T . For our system, the cost of each maneuver, given a position and velocity error of Sr, and Sv, at time to is:

A& = eXT (X6r + Sv( Thus, the longer that the mmeuver is delayed, the larger it becomes. Carrying out the detailed integrations yields (assuming urv = 0 initially):

giving us explicit formulae for the statistical cost of maneuvers. Let us also develop the bound on the mean AV, independant of the probability integral outlined above. For this we need to properly identify the matrix €!J = eXT [A, 11 and the inverse of the square-root information matrix, R-l = [u,.,0 : 0, a,].Combining these together yields !PR-’ = e X T [ u r A ,u,], which has a 2-norm eXT,/-. Now note that N = 2 for our case, and thus that &%‘(3/2)/I?(l) = @, yielding the inequality:

EV 5 P e2’ T d which, using Eq. 17, reduces to the inequality

fi5 @.

We note immediately that the cost of maneuvers (both computed and bounded) will always be proportional to eXT. We can use this result to compute the optimal maneuver frequency along an unstable trajectory. Assume we wish to control a trajectory over an extended period of time r , and that we wish to perform a maneuver after every time T , resulting in a total of M = r/T maneuvers. Then the total statistical cost of this sequence of maneuvers is:

Thus we see that the total cost is proportional to the term:

where x = AT is the variable and AT is a free parameter. For simplicity, assume r + 00, meaning that the statistical cost of the maneuvers is

Navigation of Spacecmft in Unstable Orbital Environments

435

controlled by the mean value of each maneuver, and yielding the simpler proportionality factor ez fx. Taking the partial of this with respect to x, setting it equal to zero, and solving for x yields the simple optimum x = 1, which then gives us our optimal maneuver frequency as: 1 T=-

x

or one maneuver after every characteristic time of the unstable system. This is significant as it directly links the local characterization of the trajectory to the appropriate control strategy. Figure 5 shows Eq. 18 for several values of AT, showing that the optimum spacing does not vary much from AT = 1.

0

1

2

3

4

5

L.mbd.T

Fig. 5. Scaling factor for the statistical cost of stationkeeping manuevers as a function of frequency of stationkeeping maneuvers for different values of total time AT.

This simple result can be used as a design principle in developing a control strategy for an unstable trajectory, using the local characteristic time of the trajectory as the nominal correction maneuver time. When applying this result to full 3-DOF systems targeted to time varying trajectories, the resultant equations are not as simple. For example, it is not possible t o place the trajectory directly onto the stable manifold for a 2-DOF or higher system, as the stable and unstable manifolds will occupy different locations in configuration space, unlike the 1-DOF problem where these manifolds overlap in configuration space. Still, generalizations of this result to multiple maneuver correction strategies still result in optimal maneuver spacings on the order of one characteristic time20.

436

D.J. Scheeres

Finally, consider what occurs when X AV, yields:

AV, =

&.

<< 1, expanding the formula for

[1+ AT

+ .. .]

and we see that the statistical cost is similar to the degenerate case. This provides a strong indication that more “traditional” trajectory control approaches are sufficient for Earth-Sun libration point orbiters, where the characteristic time is relatively long. It also shows, explicitly, that these same approaches may not be valid for an Earth-Moon libration point, due to the short characteristic times of these orbits.

5. Conclusions This paper presents some basic ideas and facts associated with the navigation of spacecraft in unstable orbital environments. First, a review of spacecraft navigation is given utilizing the terminology of Hamiltonian systems theory. Following this, a brief discussion of model unstable dynamical systems in astrodynamics is given. Then we focus on how an unstable orbital environment influences the propagation of uncertainty distributions, how it affects the choice of optimal orbit determination measurements, and how it constrains the design of statistical control maneuvers. Throughout, a simple 1-DOF dynamical model is used to illustrate the difference that instability can make for these issues of navigation concern.

Acknowledgments The research described in this paper was sponsored by the IPN Technology Program by a grant from the Jet Propulsion Laboratory, California Institute of Technology which is under contract with the National Aeronautics and Space Administration.

References 1. Battin, R.H., “An Introduction to the Mathematics and Methods of Astrodynamics,” AIAA Education Series, American Institute of Aeronautics and

Navigation of Spacecraft in Unstable Orbital Environments 437 Astronautics, 1999. 2. Bierman, G.J., “Factorization Methods for Discrete Sequential Estimation”, Academic Press, 1977. 3. Dunham, D.W. and Roberts, C.E., “Stationkeeping Techniques for LibrationPoint Satellites,” Journal of the Astronautical Sciences, Vol. 49(1), 2001. 4. Farquhar, R.W., “The Utilization of Halo Orbits in Advanced Lunar Operations,” NASA Technical Note TN D-6365,July 1971. 5. Farquhar, R., Muhonen, D.P., Newman, C.R., and Heuberger, H.S., “Trajectories and Orbital Maneuvers for the First Libration-Point Satellite,” Journal of Guidance and Control, Vol.3,No.6,November- December 1980,No.80-4123. 6. Farquhar, R. W., Muhonen, D., and Church, L., “Trajectories and Orbital Maneuvers for the ISEE-3fICE Comet Mission,” Journal of the Astronautical Sciences, Vol. 33, July-September 1985, pp. 235-254. 7. Froeschlk, C., “The Lyapunov Characteristic Exponents - Applications to Celestial Mechanics,” Celestial Mechanics, Vol 34, 1984, pp. 95-115. 8. Hu, W.,“Orbital Motion in Uniformly Rotating Second Degree and Order Gravity Fields,” PhD. Thesis, The University of Michigan, 2002. 9. Gomez, G., Llibre, J., Martinez, R., Simo, C., “Dynamics and Mission Design Near Libration Points,” World Scientific Monograph Series in Mathematics, VOl. 2, 2001. 10. Greenwood, D.T., “Classical Dynamics”, Dover, 1997,pp. 182-183. 11. Hamilton, T.W., Melbourne, W.G., “Information Content of a Single Pass of Doppler Data from a Distant spacecraft”, JPL Space programs Summary, No 37-39,Vol 111, March-April 1966, pp 18-23. 12. Howell, K. C. and Pernicka, H. J., ”Stationkeeping Method for Libration Point Trajectories,” Journal of Guidance and Control, Vol. 16, No. 1, 1993, pp. 151-159. 13. Howell, K. C., Barden, B. T., Wilson, R. S., and Lo, M. W., ”Trajectory Design Using a Dynamical Systems Approach with Application to Genesi,” AAS Paper 97-709,AASfAIAA Astrodynamics Specialists Conference, Sun Valley, Idaho, August 4-7,1997. 14. Kamel, A. A. and Breakwell, J. V., “Stability of Motion Near Sun-Perturbed Earth-Moon Triangular Libration Points,” Periodic Orbits, Stability and Resonances, 82-90,1970. 15. Kolenkiewicz, R.and Carpenter, L., “Stable Periodic Orbits about the SunPerturbed Earth-Moon Triangular Points,” AIAA J. 6(7), 1301-1304,1968. 16. Lara, M., and Scheeres, D.J., “Stability bounds for three-dimensional motion close to asteroids,” paper presented at the 2002 Space Flight Mechanics Meeting, San Antonio, Texas, January 2002. 17. Lo, M., Williams, B., Bollman, W., Han, D., Hahn, Y., Bell, J., Hirst, E., Corwin, R., Hong, P., Howell, K., Barden, B., and Wilson, R., ”Genesis Mission Design,” AIAA 98-4468,AIAAfAAS Astrodynamics Conference, Boston, Massachusetts, August 10-12,1998. 18. Marchal, C., The Three Body Problem, Elsevier, 1990. 19. Montenbruck, 0.and Gill, E., Satellite Orbits: Models, Methods, Applications, Springer 2000.

438 D.J. Scheeres

20. Renault, C. and Scheeres, D.J., “Optimal placement of statistical maneuvers in an unstable orbital environment,” submitted to Journal of Guidance, Control, and Dynamics. 21. Schechter, H. B., “Three-Dimensional Nonlinear Stability Analysis of the Sun-Perturbed Earth-Moon Equilateral Points,” AIAA J. 6(7), 1223-1228, 1968. 22. Scheeres, D.J., Hsiao, F.-Y., and Vinh, N.X., “Stabilizing motion relative to an unstable orbit: Applications to spacecraft formation flight”, accepted for publication in the Journal of Guidance, Control, and Dynamics. 23. Scheeres, D.J. , Guman, M.D. and Villac, B., “Stability Analysis of Planetary Satellite Orbiters: Application to the Europa Orbiter,” Journal of Guidance, Control, and Dynamics 24(4): 778-787,2001. 24. Scheeres, D.J., Han, D., Hou, Y., “The Influence of Unstable Manifolds on Orbit Uncertainty”, Journal of Guidance, Control, and Dynamics, 24(3), 573585,2001. 25. Scheeres, D.J., “Characterizing the orbit uncertainty dynamics along an unstable orbit”, presented at the 2001 AASIAIAA Astrodynamics Specialist Meeting, Quebec City, Canada, August 2001. AAS Paper 01-302. 26. Scheeres, D.J., Williams, B.G., and Miller, J.K., “Evaluation of the Dynamic Environment of an Asteroid: Applications to 433 Eros,” Journal of Guidance, Control and Dynamics 23:466-475,2000. 27. Scheeres, D.J., “The Restricted Hill Four-Body Problem with Applications to the Earth- Moon-Sun System,” Celestial Mechanics and Dynamical Astronomy 70:75-98,1998. 28. Williams, K., Barden, B. T., Howell, K. C., Wilson, Lo, M. W. and R. S., “Genesis Halo Orbit Station Keeping Design,”, ISSFD Conference, Biarritz, France, June, 2000.

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

LOW THRUST TRANSFER TO SUN-EARTH L1 AND Ls POINTS WITH A CONSTRAINT ON THE THRUST DIRECTION A.A. SUKHANOV and N.A. EISMONT Space Recearch Institute (IKI) of the Russian Academy of Sciences 84/32 Profsoyuznaya St., Moscow 117997,Russia

Low-thrust transfers from a low Earth circular orbit (LEO) to the Sun-Earth L1 and then to L2 points are analyzed. A spin-stabilized spacecraft with the spin axis orthogonal to the Sun direction is considered. The thrusters provide jet acceleration along the spin axis in both directions. Thus, the thrust is always orthogonal to the Sun direction. The spiral spacecraft ascent from the LEO is considered first. Each orbit of the spiral has two thrust arcs and two coast ones. Then the spacecraft is inserted into an Li halo orbit in the ecliptic plane. After the operations in the halo orbit are completed the spacecraft is transferred to an L2 halo orbit. The transfers containing zero, one, or two complete orbits around the Earth are considered.

1. Introduction

A low-thrust mission to one or two of the Sun-Earth collinear libration points was being developed in Russia a few years ago. Main mission goals were solar wind exploration and magnetic storm prediction; however, perhaps most important goal was testing of new technologies. The mission concept was the following:

-

A light spacecraft equipped with solar electric propulsion (SEP) was to 439

440

A . A . Sukhanov and N . A . Eismont

be assembled at the International Space Stationa; - The spacecraft was to ascend from LEO in a spiral orbit, to be transferred to the L1 point, and to be inserted into a halo orbit using SEP; - After the operations in the L1 halo orbit are completed the spacecraft could transfer to an Lz halo orbit. The mission has not been adopted, nevertheless some of its features can be of a certain interest. For instance, the following: a spin-stabilized spacecraft was considered with immovable solar arrays and thrusters what simplified the spacecraft design and control and lowered the mission cost. The spin axis is orthogonal to the Sun direction and the thrusters provide the thrust along the spin axis in both directions. Thus, the thrust is always orthogonal to the Sun direction and all the transfers and maneuvers were to be performed under this constraint on the thrust direction. This paper presents main results of the mission analysis, namely the following:

- Earth to L1 halo transfer including the spiral ascent near Earth; - L1 halo to Lz halo transfers with different numbers of complete orbits around the Earth. Some of the important problems of the mission (such as communication, parameters of the film solar arrays, a prolonged being in the van Allen belts etc.) are not discussed in the paper because they were outside authors’ competence.

2. The spacecraft concept

The electrically propelled spacecraft design is subject to the following requirements:

- the continuous thrust direction must be close to the spacecraft velocity vector for a long time of the thrust run to provide maximum efficiency of the thrust; - the solar panels must have a big area and be directed to Sun for all time of the SEP run to provide the power-consuming thrusters with maximum electric power. aMir space station was considered for this purpose at an early phase of the project.

Low Thrust lhnsfer to Sun-Earth Li and LZ Points

441

These requirements often contradict each other, especially in the spiral orbit where the spacecraft performs hundreds of orbits and its thrust must follow the velocity vector in each of them. This would lead to complicated both the spacecraft construction and control.

A simple and elegant solution of the problem has been proposed for the considered mission. The spin-stabilized spacecraft reminds a bicycle’s wheel with the spacecraft body in the middle and the solar arrays along the rim (see Fig. 1).

I

Fig. 1. Overall view of the spacecraft.

The one-sided arrays form a cylindrical surface of 9-m diameter and 2-m height. Thus, the total area of the arrays is about 55 m2. Assuming 85% coverage of the arrays by photocells, maximum effective area of the arrays (i.e. cross-sectional area of the photocells orthogonal to the solar radiation flux) is about 15 m2. Film solar arrays providing about 1.5 kW of the electric power assumed to be used. Wet initial mass of the spacecraft was estimated as of about 290 kg. The spacecraft has 8 thrusters D-38 developed in the Energia Rocket and Space Corporation. Four of the thrusters are installed on one side of the spacecraft and another four on the opposite one; thus, their thrust was to be directed along the spacecraft spin axis in two opposite directions. However, only two of four co-directed thrusters run simultaneously; another pair is auxiliary. Parameters of the D-38 thruster are given in Table 1.

442

A . A . Sukhanov and N.A. Eismont Table 1. D-38 thruster parameters. Parameter name Power, (W) Specific impulse, ( s ) Efficiency (including losses in PPU) Thrust force, (N) Mass flow rate, (kg/s) Resource, (hours) Propellant

Parameter value 750 2200 0.5 0.035 1.6.10-6 3000 xenon

The spacecraft spin axis is orthogonal to the Sun direction during the SEP run. Thus, the thrust always lies in the plane orthogonal to the Sun direction. It is assumed that any thrust direction in the plane can be chosen. The spacecraft contains 85 kg of xenon what provides about 7.5 km/s of the spacecraft characteristic velocity.

3. Spacecraft ascent from LEO 3.1. Spiml ascent stmtegy Typical space station orbital parameters were taken for the initial spacecraft orbit: a circular orbit of the 400-km altitude and 51.6-degree inclination. After the separation from the space station and starting its ascent from

LEO by means of SEP the spacecraft moves in an expanding spiral orbit. While the spacecraft jet acceleration is much lower than the gravitational one the instantaneous spacecraft orbit remains very close to the circular one of the growing radius. Optimal thrust direction in this case is always close to the spacecraft orbital velocity vector. However this optimal direction cannot be provided for the spacecraft described above (except two points in each orbit where the spacecraft velocity is orthogonal to the Sun direction). The following strategy of the SEP control taking into account the spacecraft concept has been selected for the mission: the SEP runs dong two 120degree arcs, f 6 0 degree from the projection of the Sun direction onto the orbit plane; two thrusters providing proper thrust direction run in each of the arcs (see Fig. 2). This strategy leads to a loss of 17 percent of the SEP effectiveness (and respectively to a higher propellant consumption) and to longer with factor 1.7 time of flight comparing to the permanent tangential thrust. This is the

Low Thrust lfvrnsfer to Sun-Earth L1 and La Points 443

Fig. 2. The SEP runs in the Earth vicinity.

payment for the simplified spacecraft design and control. The 120-degree arc has been chosen as a compromise: a shorter arc would provide higher effectiveness of the SEP but the flight time would increase; a longer thrust arc would lead to a less effectiveness of the propulsion.

3.2. Shadowing The thrust arc behind the Earth (with respect to Sun) can be entirely or partly shadowed by the Earth for a long time (see Fig. 2). It is impossible to avoid the shadowing completely; the only way to diminish it is an appropriate selection of both the Sun position and the longitude of the ascending node of the spacecraft orbit at the launch time. Analysis shows that the spacecraft launch in June-July or December-January with the longitude of the ascending node of about 280 to 300 degrees minimizes the average shadowing down to 7.5 percent (i.e. in average about 7.5 percent of the whole thrust arc per one orbit is shadowed). However this optimal solution would lead to a high (higher than 50 degrees) final inclination of the spacecraft orbit to the ecliptic plane what is not good for the further insertion into the halo orbit. Therefore a compromise has been selected: launch in May or November with the longitude of the ascending node around 260 degrees. This gives the average shadowing of about 8.5 percent and the final inclination to the ecliptic plane of about 35 degrees (it is clear that the minimal possible inclination is about 28 degrees if the inclination to the equator is 51.6 degrees). The thrust arc shadowing and inclination to the ecliptic versus time are shown in Fig. 3; as is seen in the Figure one of the thrust arcs is completely shadowed at the beginning of flight.

444

A . A . Sukhanov and N . A . Eismont

im

0

Fig. 3.

200

lime dflight, days

Ja

Shadowed arc and inclination to the ecliptic versus time.

3.3. Pammeters of the spiml orbit

Radius of the spiral orbit versus time is given in Fig. 4; Table 2 presents parameters of the spiral orbit in the Earth vicinity.

im 200 TlmeofflgM, days

0

Fig. 4.

333

Orbit radius versus time.

Notice that the final inclination to the ecliptic is sensitive to the initial longitude of the ascending node: 10-degree variation of the longitude changes the inclination in 3 degree. Since the ascending node precession is about 5 degrees per day for the space station, the launch window providing

Low Thrust mansfer to Sun-Earth L1 and

L2 Points

445

Table 2. Parameters of the spiral orbit. Parameter name Time of flight, (days) Number of orbits Consumed characteristic velocity, (m/s) Propellant consumption, (kg) Spacecraft mass, (kg)

Parameter value 280 1330 6850 78.9 211.1

necessary inclination to the ecliptic plane is narrow. Therefore the spacecraft should be separated in advance in order to start operations exactly at a proper time.

4. Flight to L1 and insertion into halo orbit

Rather small halo orbit with amplitude A , M 60 thousand km has been selected for the mission. Fig. 5 gives two projections of the spacecraft transfer trajectory: projection onto the ecliptic plane (zy) and the orthogonal one (xz);the spiral shown in Fig. 5 begins from 50,000-km radius. The transfer trajectory shown in Fig. 5 corresponds to launch in November; for the May launch the xy projection will not change and the xz one will be mirrored with respect to the z axis. The trajectory includes the spiral part, the flight to L1, and the halo orbit. Bold arc at the end of the spiral orbit is the last thrust arc injecting the spacecraft into the transfer trajectory to L1. This arc is shorter than the typical 120-degree arcs and asymmetric; this is to provide necessary halo amplitude and z component close to zero during the insertion into the halo. The thrust arc lasts 4.5 days and consumes 268 m/s of the spacecraft characteristic velocity (corresponding 2.6 kg of xenon have been included in the spiral orbit propellant consumption). The bold arc near L1 shows the break maneuver inserting the spacecraft into the halo orbit. The axes ticks correspond to 200,000km distance, crosses on the curves mark 10-day time intervals after the injection into the transfer trajectory. Parameters of the transfer to and insertion into the halo orbit are given in Table 3.

As is seen in Fig. 5 a planar halo orbit has been chosen for the mission analysis. It is not necessary for the mission purposes; however the cost of

446

A.A. Sukhanou and N.A. Eismont Y

J

Fig. 5. The spacecraft trajectory to L1.

Table 3. Parameters of the flight to halo orbit. Parameter name Time of flight (after the spiral), (days) Characteristic velocity of the insertion into halo, (m/s) Propellant consumption, (kg) Spacecraft mass in the halo, (kg) Amplitude A, of the halo orbit, (km)

Parameter value 140 290 2.8 208.3 62.103

this insertion is just a little higher than of the insertion into a 3D halo with a reasonable A, amplitude. So the cost, 290 m/s (see Table 3), is an upper limit for the insertion into a halo with 60,000-km A, amplitude.

A tiny variation of the AV transferring the spacecraft to the libration point can dramatically change the halo orbit amplitude. The approximate dependencies are the following: the AV increment in 5 cm/s increases the Ay amplitude in 100 thousand km and reduces the characteristic velocity and the propellant consumption of the insertion into halo in 20 m/s and 0.2 kg respectively.

Low Thrust "hnsfer to Sun-Earth L1 and Lz Points 447

5. On the M o o n gravity assist Moon gravity assist for the transfer to L1 has been analyzed only for the spacecraft with 3-axis stabilization. The main advantage of this maneuver is that it can put the spacecraft trajectory very close to the ecliptic plane and hence lower delta-V of the insertion into the halo almost in 200 m/s (2 kg of xenon). However the Moon gravity assist may require waiting in a parking orbit for providing the Moon encounter conditions what can increase the total flight time in a few weeks.

6. Transfer from L1 to Lz 6.1. Introduction

Planar transfers from the L1 halo orbit to an L2 one are considered in this section. This part of the mission was considered rather as a mission extension, its profile was completely uncertain. In particular parameters of the L2 halo orbit were not defined. Therefore different options have been considered for the transfer trajectory design. There is a great amount of possible halo-to-halo transfers. Even if both of the halos are given the transfers differ by the number and location of the active maneuvers, their values, number of complete orbits around the Earth, transfer duration, use of the Moon gravity assist etc. Some of the transfers are described below.

6.2. Models and methods used

Since the specific impulse is given (see Table l), we have the constant exhaust velocity (CEV) case. In this case the thrust value is given by

where mp is consumed propellant mass, c is exhaust velocity, m is the spacecraft mass. Hill model was used for the motion analysis. The model is given by the

448

A . A . Sukhanov and N . A . Eismont

equations i. = v , P +a, 6 = n 2 N r + 2 n M v - -r r3

where p is the Earth gravity constant, r , v are the spacecraft position and velocity, T = Irl, n is the Earth mean motion, a is the thrust vector, 3

0

0

0

1

0

Thrust arcs in the transfer are short comparing to the whole transfer time. This is why at the first step the arcs can be approximated by the junction points (i.e. points of the impulsive thrust application). In order to obtain the junction points positions a combination of the Pontryagin’s maximum principle and Lawden’s primer vector was used l*z. The Hamiltonian is

where P O , p,, p , are costate variables ( p , is the Lawden’s primer vector), superscript T means transposition. Vectors p,., p , satisfy the adjoint variational equations

(2) T

P, = -

= -(G

dH

+n2N)p,,

= -p,-nM

(4) T

P,,

where

I is the unit matrix of 3rd order. The constraint on the thrust direction described above can be written as follows: xOTa = 0 , where zo is unit vector of the

2

axis. Constraint on the thrust value is

(5)

Low Thrust %nsfer

to Sun-Earth L1 and La Points

449

where a is given by (1). The optimal control a should provide maximum of function (3) under the constraints ( 5 , S ) , i.e. the following function is to be maximized:

L =H where XI,

A2

A2 + A1zOTa + -(aTa - a2), 2

are indeterminate multipliers. Then

Vector a can be easily found from (7, 5 , 6 ) as follows: PA a = a-, PA

where

is a projection of the p , vector onto the yz plane, P A = JpAI.Also it can be shown that the switching function K. = n(t) in this case is given by

.

K.=-

PAC

m

The junction points correspond to the maximum values of P A which can be found by means of numerical integration of equations (2,4) and using (8); transfer between the junction points is ballistic, i.e. a = 0 in (2) when integrating. After the junction points were found three-thrust-arc transfers were analyzed. The thrust arcs were the following: launch from the initial L1 halo; a midcourse maneuver applied at a junction point; insertion into the L2 halo orbit. Since planar haleto-halo transfers were considered, the thrust was always directed along the y axis. Thus, it was sufficient to find only three values of the three maneuvers (i.e. propellant consumptions or characteristic velocities or the spacecraft velocity changes). The characteristic velocity values Avl , Av2, AVScorresponding to the thrust arcs were varied in order to perform the transfer and to minimize the sum Av = Av1 + Av2 + A v ~ . The condition Av 5 Av, was also used where Av, is the rest of the available characteristic velocity after the insertion into the L1 halo has been performed.

450

A . A . Sukhanov and N . A . Eismont

6.3. Zero complete orbits around Earth The transfer trajectory for this case is shown in Fig. 6. The thrust is directed toward +y axis for Avl,S maneuvers and -y for Av2 one. Y

Fig. 6. Transfer with zero complete orbits

The transfer parameters are given in Table 4. Table 4. Transfer with zero complete orbits. Parameter name Consumed characteristic velocity, (m/s) Avi Avz Av3 Time between Avl and Avz, (days) The transfer duration, (days) Propellant consumption, (kg) Final spacecraft mass, (kg) Ay amplitude of the La halo, (km)

Parameter value 306 50 195.6 60.5 70 181 2.9 205.4 800,000

Duration of the transfer is relatively short in this case, just 6 months. However the available propellant allows only this large halo orbit around

6.4. One complete orbit around the Earth

Fig. 7 shows one of the possible transfers with final A, = 300 thousand km. The thrust is directed toward +y axis for all three AV maneuvers in this case.

Low Thrust Thnsfer to Sun-Earth L1 and Lz Points 451 V

Fig. 7. Transfer with one complete orbit.

Parameters of the transfer are given in Table 5. Table 5. Transfer with one complete orbit. Parameter name Consumed characteristic velocity, (m/s) Avi AVZ Av3 Time between Avl and Av2, (days) The transfer duration, (days) Propellant consumption, (kg) Final spacecraft mass, (kg) Ay amplitude of the L2 halo, (km)

Parameter value 224 65 18.1 141 82 259 2.2 206.1 300,000

This transfer has a longer duration (8.6 months) than the zero-orbit one, but can provide lower amplitude of the final halo orbit for lower propellant consumption. The available propellant could permit even lower amplitude than one indicated in Table 5.

6.5. Two complete orbits around the Earth

One of the options for the transfer with two complete orbits around the Earth is shown in Fig. 8; the thrust is also directed toward +y axis for all maneuvers. Table 6 gives the transfer parameters. This is the longest transfer (10.5 months), but it allows any amplitude of

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Y

Fig. 8. Transfer with two complete orbits.

Table 6. Transfer with two complete orbits. Parameter name Consumed characteristic velocity, (m/s) Avi Av2 Av3 Time between Avl and Av2, (days) The transfer duration, (days) Propellant consumption, (kg) Final spacecraft mass, (kg) Ay amplitude of the L2 halo, (km)

Parameter value 70 35 1.6 33.1 70 319 0.7 207.6 150,000

the L2 halo orbit for a very low cost. Note that in the case of two complete orbits a two-thrust-arc transfer is also possible. A symmetric two-thrustarc transfer is shown in Fig. 9; here AVI = Av2 = 43.3 m/s (0.8 kg of the propellant for both) and the transfer duration is 307 days. Y

1

i

Fig. 9.

Symmetric two-impulse transfer.

Low Thrust l h n s f e r to Sun-Earth L1 and LZ Points 453

6.6. Common notes

In all three transfer cases considered both the L2 halo amplitude and the transfer duration can be varied by means of changing the AV maneuvers values and positions. Lowering the amplitude in 30 thousand km costs about 10 m/s of the spacecraft characteristic velocity (-0.1 kg of xenon); lowering the transfer time in 10 days takes 5-7 m/s. The Moon gravity assist can be easily performed in the considered planar (or near-planar) transfer. Phasing of the spacecraft trajectory necessary to provide the encounter with Moon can be obtained by a very small variation of the launch maneuver in the L1 halo orbit. The gravity assist certainly could either lower the xenon consumption for the transfer or lower the L2 halo amplitude or the transfer duration. However this maneuver has not been analyzed yet.

7. Conclusion Table 6 summarizes characteristics of all the spacecraft movements; the duration of its stay in the L1 halo orbit is excluded from the total flight duration because it is still undefined. Table 7. Summary of the spacecraft transfers. Operation Launch Ascent in the spiral orbit Transfer to and insertion into L1 halo Transfer to and insertion into La halo Rest for the correction maneuvers

Flight time, (month) 0 9.3

Total Av (km/s) 0 6.85

Total xenon consumption (kg) 0 78.9

14.0

7.14

81.7

208.3

20-24.5

7.21-7.45

82.4-84.6

205.4-207.6

-

0.05-0.29

0.4-2.6

-

S/C mass

(kd 290 211.1

The run time of each thruster is within 2000 hours what is covered by the thruster resource time (see Table 1). Nevertheless an installation of a pair of spare thrusters is also possible. Thus, the spacecraft concept having been taken for the mission provides

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fulfillment of all operations necessary for the transfer to the L1 halo orbit and then to the La one for reasonable time and propellant consumption. This is mainly due to the fact that the thrust orthogonal to the Sun direction is very effective for changing the orbital parameters in the libration points neighborhood. Although this is also true for planetary missions 3, so this concept can be applied to them as well.

References 1. L. S. Pontryagin et al., Mathematical Theory of Optimal Processes, (in Russian). Moscow, Nauka, 1969. 2. D. F.Lawden. Optimal Tkajectories for Space Navigation, Butterworths, London, 1963. 3. Williams, S.N.; Coverstone-Caroll, V., “Benefits of Solar Electric Propulsion for the Next Generation of Planetary Exploration Missions”. Journal of Astronaut. Sci., 45 No. 2, 143-159 (1997).

Libration Point Orbits and Applications G. G h e z , M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

SATELLITES F O R M A T I O N T R A N S F E R T O LIBRATION POINTS J.C. BASTANTE, L. PEfiiN, A. CARAMAGNO, M. BELLO-MORA DEIMOS Space S.L., Ronda de Poniente 19, Edijicio Fiteni VI, 2-2- 28760 Tres Cantos, Madrid, Spain.

J. RODR~GUEZ-CANABAL

ESA/ESOC/MAS, Robert Bosch Str. 5, 64293 Darmstadt, Germany

This paper treats the transfer of a satellites formation to the collinear Libration points of the Sun-Earth/Moon system. The focus is set on relative dynamics and GNC issues (motion of the satellites with respect t o the any of the other satellites in the formation), analysed taking as a reference an optimised transfer trajectory to the L2 Libration point. The analysis here presented can be thus considered as a step stone t o allow facing the global problem of optimising multiple transfer trajectories for satellites in formation.

1. Introduction Several future European space missions whose launch and operational life is envisaged for the next decade base the achievement of their objectives on their privileged orbital position around Libration points of the SunEarth/Moon system. These environments present very well known advantages in terms of thermal stability, observation and communication geometries stability, and minimum levels of dynamic perturbations, thus requiring low frequencies and budgets for on-orbit maintenance manoeuvres. 455

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On the other side, the concept of satellites formation has definitely emerged as a very promising technology to enable the interplanetary exploration, achieving the missions goals by making an effective resources use and hence maximising the scientific return. Among the most noticeable advantages, the satellites formations offer an increased instrument resolution and the possibility of on-orbit mission re-configuration and overall system robustness. On the other hand, mission cost and complexity are issues to be carefully addressed. This paper focuses on the problematic associated to the transfer of a collection of satellites to a collinear Libration point of the Sun-Earth/Moon system. This FF scenario imposes a treatment combining trajectory optimisation concepts with those related to the guidance, estimation and control of the relative state vector between the satellites composing the formation. This coupling is clearly detected after considering, for instance, the consequences of an absolute trajectory selection on the relative evolution of the formation satellites; or taking into account the limitations that a desired relative evolution can impose on the selection of an optimum absolute trajectory. Typical questions arising when considering this coupling are:

0

How different is a trajectory optimised for one satellite from that optimised taking into account cost functions and parameters related to the relative formation state vectors? What is the impact of a reference absolute trajectory selection on the relative state vectors maintenance functions?

It can be stated that when considering a trajectory optimisation problem for a group of satellites requiring a coordinated maintenance of their relative positions it is necessary to give a coupled treatment to the problem, merging concepts from the trajectory optimisation methods and the relative GNC functions. A brief presentation of the methodology here adopted is hereafter presented: (1) Firstly, it is necessary to perform an analysis devoted to mission characterisation, focusing on the major perturbation inducing the relative dynamics that represents the main source for the GNC requirements. In this sense, each different environment has its own particularities that have to be studied in detail.

Satellites Formation lhnsfer t o Libmtion Points 457

(2) The dispersion and open loop analyses, regarding the formation e v e lution as a collection of satellites, are addressed on the basis of the absolute trajectory and dynamic environment. (3) A qualitative and quantitative characterisation of the problem allows then the GNC design. The closed loop analyses provide thus an estimate of the relative formation maintenance cost and operational needs. (4) Finally, it is then possible to select the parameters and variables to be considered for an optimisation process, thus providing a new absolute reference trajectory.

This paper focuses on the analysis of the relative motion for a transfer to a collinear Libration point, and as such, it can be considered as a study case application of almost the whole analysis cycle.

2. Transfer to a libration point: Problem description

The objective of the formation transfer to a Libration point can be formulated in a manifold way: 0

0

0

Firstly, to explore the characteristics of the relative formation dynamics during the transfer to one of the collinear Libration points in a three bodies problem; Afterwards, to estimate the needs of a relative control strategy to maintain a given relative configuration during the cruise; Finally, to estimate the formation maintenance performance and AV budget, with the objective of using these analyses in the search of an optimum formation topology during the transfer.

As study case we consider here the transfer of 2 SC to the L2 point of the Sun-Earth/Moon system. The assumptions taken for this analysis are: 0

0

Sun and Earth are taken as punctual masses, with circular orbits around their barycentre; A direct transfer trajectory to the Librtation point has been selected as reference absolute trajectory. As a consequence, no .v after escape manoeuvre for absolute orbit control purposes is provided. In other words,

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satellites arrive naturally to the L2 point after the escape impulse; Scenario simulation begins after separation from the launcher, and at a distance from Earth corresponding to a typical correction manoeuvre after escaping from the Earth (in the order of one or two hundreds thousand kilometres); No manoeuvre for correction or injection into Halo orbit is commanded; only the cruise phase is studied; Relative trajectory is controlled by continuous low thrusting (i.e. FEEPs);

With these assumptions, the approach followed in the analysis is hereafter presented: 0

0

0

Assuming as initial condition the escape velocity from the Earth, we search for an absolute transfer trajectory, which effectively injects the satellites into a stable Halo orbit around L2; In a second step, a dispersion analysis allows to address the problem of the formation separation during transfer, and to quantitatively estimate the amount of this divergence for each one of the causes generating the dispersion; A closed loop control scheme is designed and implemented. This study permits to obtain a first guess of the fuel budget needed for relative state control during a transfer to one of the Libration points of the Sun/Earth-Moon system.

3. Reference Absolute trajectory: Transfer to LZ orbit

An iterative method has been adopted with the objective of obtaining an absolute transfer trajectory to the L2 point of the Sun-Earth/Moon system. The process has provided as final condition a stable final orbit around the Libration point. For this search, it has been considered a typical Arianne 5,200~200 km, 5.2 inclination, parking orbit around the Earth as the starting point for the transfer. A convention to command the manoeuvre at the ascending node was adopted; the resulting orbit is shown in Figure 1, for which the reference frame is the one shown in Figure 2.

i

Satellites Formation lhnsfer to Libration Points 459

:

x 10’

-05 -1

-2

0

2

4

6

8 XaXl6km

10

12

14

16

18 x 10s

Fig. 1. Transfer orbit to a Lz orbit of the Sun-Earth/Moon system.

The escape AV computed for this trajectory is of 3.194 km/s, very close to the 3.197 km/s stated in the ”IRSI-Darwin Concept and Feasibility Study Report, ESA-SCI(2000)12”, July 2000, leading to a natural injection in a stable orbit around Lz.

4. Dispersion analysis

With the objective of assessing the need of a relative GNC function, a simulation campaign has been run to obtain a quantitative estimation of the dispersion of the formation during transfer. This is obtained by propagating small initial differences commanded on the relative positions and velocities, measuring the effect, in terms of relative state vector, at the injection point on the Halo orbit. The injection point is taken as the first cross of the y=O plane, in the synodical reference frame attached to the Sun-Earth system, with the x-axis containing both bodies and the z axis orthogonal to the motion plane. The reference frame considered in this analysis is shown in Figure 2. Simulation conditions and additional assumptions considered for the test cases are:

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Masfer safellke

Fig. 2. 0

0

0

Reference frame considered for the analysis.

Simulation starts after 4 days of transfer trajectory, some 500.000 km from the Earth. The absolute trajectory provides the state vector (position and velocity) of the reference satellite at this point, which constitutes the reference state taken for adding the increments that generate the dispersion; Dispersion is measured as differences in position and velocity at the injection point in the Halo orbit; Separations between satellites have been commanded in position and velocity, in the three coordinate directions;

Some of the results obtained with the dispersion analysis are shown in Figure 3 and Figure 4. In these plots, horizontal axes mean deltas commanded at the initial transfer point, meanwhile the vertical axes are dispersion values in position, velocity or time, at the injection point in the L2 orbit.

-16

-10

-5

,,a&,,

I

10

16

'$6

-10

-6

Dn

6

10

15

Fig. 3. Time, position and velocity dispersion results obtained for deltas in the initial position.

Satellites Formation l h n s f e r to Libmtion Points

461

Fig. 4. Time, position and velocity dispersion results obtained for deltas in the initial velocity.

Figure 5 shows the dispersion induced by the effect of Solar Radiation Pressure on satellites with differences in the ballistic coefficient.

\I

dmo

--0

C A a & . k w m a a , m dfi.nnahhWk-Wr4m

Fig. 5. Position dispersion results obtained for differences in the satellites ballistic coefficients.

The main conclusions derived from these analyses are: Formation evolution is highly sensitive to initial separations in the SunEarth direction (x axis in Figure 2), both for position and velocity. For instance, initial separations of 5 km in the x direction generate distances between the satellites at the Halo injection point of around 1.000 km. On the other hand, initial separations of 0.5 m/sec in the x direction generate a dispersion of around 10.000 km at the Halo injection point. Separations in initial position and velocity in the other two directions produce also important separations at the Halo injection point, but

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at least one order of magnitude smaller than those obtained for equal separations along the x-axis. Finally, differences in the satellites ballistic coefficients also produce important divergences in the relative vector at the Halo injection point: differences of 10% produce dispersions of around 1.000 km.

Though the deltas in the initial conditions are intentionally generated, it seems a direct conclusion the need for a strategy to maintain a given reference relative state. To this end, next paragraphs face the problem of implementing a closed loop algorithm for the relative motion.

5. Relative motion in the RTBP (linear time-variant system) The simplified equations of the relative motion in a three bodies problem are linear but contain time varying parameters. They are hereafter outlined:

where: (z, y, z ) is the relative position vector in the synodical reference frame of Figure 2; (fz,f,, fi)is the relative acceleration, other than the gravitational one induced by the Sun and the Earth; w is the angular velocity of the Earth around the Sun (constant in this analysis, since it is being considered a circular heliocentric orbit); nm, and n,, are angular orbital velocities related to the distances between the reference satellite, the Sun and the Earth, respectively:

As can be seen, these two variables depend on the distances between the reference satellite and the Earth, for the nme; and between the reference

Satellites Formation rrclnsfer to Libmtion Points 463

satellite and the Sun, for the nms; as a consequence, they are time dependent. For the selected transfer trajectory selected, these variables follow the evolution shown in Figure 6.

8 s 10

Fig. 6. n,,

1s

20

25

30

and rime and parameters evolution.

From these figures it follows that: nms can be considered as constant, since its range of variation goes rad/sec and 1.99 x rad/sec; from 1.965 x However, n,, changes considerably (up to two orders of magnitude) from the beginning to the end of the transfer phase;

As a consequence, the system has to be dealt as "Linear Time Variant" for the design of the relative state controller. This task is faced in the following paragraphs.

6. Controller design through analytical closed loop poles placement

The objective of this paragraph is to outline the process followed in the design of a closed loop controller for the relative state vector, based on a pole placement technique. We select the desired behaviour in closed loop

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by imposing the closed loop poles of the system. The closed loop control scheme is shown in Figure 7.

- c o r w

Fig. 7. IFF control scheme.

The open loop poles are shown in Figure 8, left. It is clearly shown the instability is analogous to the one of the orbits near the collinear Libration points: in addition the time varying behaviour of the system during the transfer is represented by the poles displacement. Figure 8 right shows what might be the desired positions of the closed loop poles:

a

A fixed, time constant position of the closed loop poles is imposed; With respect to the position of the open loop poles, the imposed changes are: - The time variant, real negative open loop pole is transformed into a

real fixed and negative closed loop pole ( p l ) ; - The time variant, real positive open loop pole is transformed into a

real fixed and negative closed loop pole (pz); - The pair of time variant, pure imaginary conjugated open loop poles

are transformed into a pair of complex conjugated fixed closed loop poles (psr i p 3 i and psr - i p s i ) , but with a negative real part, with the objective of limiting their oscillatory nature, and adding an exponential decreasing component in the step response.

+

Adopting 51s state vector the relative position and velocity, and the simplified formulation for the relative motion in Equation 1, the following set of equations provide the state space realisation of the system for the

Satellites Formation mansfer to Libration Points 465

*In,

--ale

Open Loop

'1

poles evolution aLLw

I

p.+l'p.

poles desired &I@

ace

--

I

&I@-

x

Closed Loop

-

m

+I , L1P"

Ux;e

b

PI Y

....

R

't ale

Wp.

x

Fig. 8. Poles of the relative motion dynamical system in open loop (left) and desired position for the closed loop (right).

design of linear state regulator.

i = Az+Bu, u = -Kx where:

Considering the closed loop poles assigned as shown in Figure 8, right, the gain matrix for the controller has to be time dependent. The control equation can be written as:

u = -K(nme, nms,pl ,p2,p 3 ~p3i, ) Pzr,pzi)z, (4) where it is highlighted the dependency of the K matrix with time and with the closed loop poles. Though it could seem that the control gain matrix K contains many terms to be assigned, the nature of the problem allows the adoption of several simplifications: 0

Firstly, and as can be seen from Equation 1, the z relative motion is uncoupled (with the linear assumption) from x and y motion. As a consequence, all terms relating the z component and any other (zor y) component can be taken as null for the linear regulator.

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On the other hand, and again from Equation 1,a position delta in the x direction is only (linearly) related to y velocities deltas, and vice-versa. This allows taking as null the cross K x y and K y x terms. As a consequence the K matrix can be greatly simplified:

An analytical formulation has been developed to solve this problem, providing the remaining K gain matrix terms as a non linear function of the referred parameters, as follows:

with N = nLe introduced:

+ nks and where two additional assumptions have been

On one side, it is assumed a dependency between the Kxx and the K y y terms through a design parameter, C:

Kxx= CKyy. On the other side, and based on the Equation 1, it has been considered that the cross terms in the derivative sub-matrix are of opposite sign:

K x y. --- K yl. Next paragraph shows the closed loop results obtained with this controller.

Satellites Formation Thnsfer to Libmtion Points

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7. Closed loop performances and budgets

Figure 9 to Figure 13a show the results obtained in terms of the following figures of merit:

Figure 9 shows a case in which an initial delta in y axis direction (1 and 10 kilometres) has been commanded. Left plot shows the 3D evolution, meanwhile the one on the right gives the Av budget evolution; Figure 10 shows the same figures of merit, but for different control stiffness, in such a way that it has been differentiated between soft, medium and hard control, depending on the closed loop poles positions; Figure 11 shows results for different deltas in the Solar Radiation Pressure ballistic coefficients; Figure 12 compares the behaviour of the controller when counteracting deltas in z and y directions; Finally, Figure 13 compares a time dependent K matrix and a constant one.

Fig. 9. Comparison of control for different initial separations in Y axis (1 and 10 km).

Y3ampling time for all the test cases shown in this paragraph was set to 20 seconds. This value has to be taken into account in order to correctly understand the "continuous" nature of the thrusting commands.

468 J.C.Bastante et al.

Fig. 10. Different formation stiffness: different positions for the adopted closed loop poles.

lSl

,

,

,

,

,

,

,

,

,

,

.;

'"

Fig. 11. Comparison for different values of delta in SRP ballistic coefficient.

8. Summary and conclusions

Following points provide a summary of the analysis presented, along with the most noticeable conclusions:

0

0

0

The relative dynamics in the RTBP has been modelled as a linear timevariant dynamical system. Dispersion analyses indicate a high sensitivity of the formation natural evolution to initial relative state and SC layout differences. Controller design, based on analytical closed loop pole placement, leads to a time-variant linear regulator. Sensitivity analyses have been performed for the derivation of the rela-

Satellites Formation Thnsfer to Libmtion Points 469

Fig. 12. Comparison for nominal state vector relative pos in z and y axes (10km).

Fig. 13. Comparison for time variable gain matrix K and constant matrix K (computed at the beginning or at the end of the trajectory and kept constant during all the transfer).

0

tionship among closed loop poles, formation maintenance performance and delta-V budget. System performances and budgets mainly depend on:

- Initial relative state vector, - Stiffness of the formation: time response and allowed variation of

spacecraft distance with respect to nominal state vector. (SRP ballistic coefficient).

- Differencesbetween satellites layouts 0

Quantitative effect of each factor is (see Figure 9 to Figure 13):

AT/ budget and formation dispersion are linearly proportional to nominal separation along the y axis; - Soft control (smaller absolute value of the closed loop poles) is more fuel demanding (more dispersion). Maximum thrust level w 10 mN;

-

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A difference in SRP ballistic coefficient of 10% implies increments in accumulated AV in the order of 50% (from 2.5 m/sec to 3.5 m/sec); - Initial or nominal separations in the Sun-Earth direction are detrimental for formation maintenance purposes. It is, on the other hand, more convenient to maintain a safe SC distance through separation along the y-axis. -

N

0

a

N

The closed loop performances of the gain matrix K for a stiff formation configuration show a low sensitivity to the time-varying parameters of the model. In any case, this time dependency is more or less important as a function of the closed loop poles of the system. As a rule of thumb, it can be concluded that formation maintenance during 100 days of transfer needs about 3 m/s for a flyer SC. N

References 1. Darwin, the Infrared Space Interferometer, Concept and Feasibility Study Report, ESA-SCI(2000)12, 2000. 2. R. Jehn and F. Hechler: I R Interferometer Corner Stone Mission Mission Analylsis, MAS Working Paper n 396, ESA/ESOC 1997, Germany. 3. P. Bainum and A. Strong and Z. Tan: Control of Formation Flying Satellites.

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

SOFTWARE ARCHITECTURE AND USE OF SATELLITE TOOL KIT'S ASTROGATOR MODULE FOR LIBRATION POINT ORBIT MISSIONS J. CARRICO and E. FLETCHER Analytical Graphics, Inc.

40 General Warren Blvd. Malvern, PA 19355, USA

The Satellite Tool Kit (STK) Astrogator software module is the third and most recent version of a program originally developed by NASA Goddard Space Flight Center (GSFC). This software lineage - Swingby, Navigator, Astrogator - started in 1989 and has since been used to design and operate many missions, including the non-low Earth orbit missions Clementine, Wind, SOHO, ACE, Lunar Prospector, the AsiaSat 3 rescue, and MAP. This paper describes the history of the software program and reasons behind the numerical methods employed. The authors also discuss the software design methodology and goals that led to this mature software product. Limitations encountered during analysis and operations use are described, as well as subsequent architecture changes made to alleviate them, reduce risk, and support automation.

1. Introduction

Astrogator is the maneuver planning and trajectory design module of Satellite Tool Kit (STK), a completely commercial off-the-shelf (COTS) software program developed by Analytical Graphics, Incorporated (AGI) '. Astrogator is fully integrated within STK and, among other things, can be used to generate the orbit ephemeris and attitude history of spacecraft. These data are then available for subsequent analysis and processing by other modules in STK, such as calculating station acquisitions, lighting times, communication links, coverage effectiveness, and Sensor obscurations. 471

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2. History and Use

The ISEE-3/ICE mission was designed and operated by NASA GSFC in Greenbelt, Maryland, USA. Launched on 12 August 1978, the spacecraft was transferred to the Sun-Earth L1 (interior colinear) libration point, and became the first spacecraft stationed in a libration point orbit. The software used for trajectory design and operations was the Goddard Mission Analysis System, GMAS 4 , and an early variation on the General MANeuver program, GMAN . (This special version was known as ICEMan.) In the late 1980's, the NASA GSFC Flight Dynamics Facility (FDF) initiated an effort to get rid of its mainframe computers, and because both these programs ran on the mainframe, alternatives were explored. One initiative, started in 1989, was to create a new trajectory design program based on a personal computer (PC). This task was given to Computer Sciences Corporation (CSC). In addition to meeting the requirements of the previous software, the new software was to incorporate graphics as an aid to tranamed jectory design and analysis. This program was called Swingby because the first mission it was designated to support was the double-lunar swingby (DLS) mission Wind 'l1OJ1. In 1992, however, the FDF was tasked with supporting trajectory design and maneuver operations for the Deep Space Program Science Experiment, DSPSE, also called Clementine 12913. Under the direction of the NASA GSFC FDF, working with the Naval Research Laboratory in Washington D.C., Swingby was enhanced to support Lunar orbit, asteroid rendezvous, and operational maneuver planning tasks. Clementine launched on 25 January, 1994 and performed a successful 2 month lunar orbit. (Unfortunately, afterwards the spacecraft suffered an on-board computer problem and was lost after it left the Moon on the way to the asteroid Geographos.) Subsequent to Clementine, Swingby was used in the NASA GSFC FDF to support Wind The Solar and Heliospheric Observatory, SOH0 the Advanced Composition Explorer, ACE17; and Lunar Prospector". 213

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NASA GSFC also distributed Swingby to several educational and government organizations within the United States. These included the United States Air Force Academy; University of Colorado, Boulder; University of Texas, Austin; NASA Ames18, and the Jet Propulsion Laboratorylg. Swingby earned Computer Science Corporation's Corporate Technical Excellence Award. In 1994, CSC commercialized Swingby and starting sell-

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ing the software with the name Navigator, in cooperation with AGI. On 25 December 1997, Hughes Aerospace Corporation was supporting the launch of a geostationary satellite, AsiaSat 3. However, the 4th stage only performed 1second of an intended 110-second burn, and the spacecraft was stranded in a useless transfer orbit inclined at 51.6 degrees. The spacecraft engineers calculated that they needed 2424 meters/second in order to perform a Hohman transfer to geostationary orbit, but the spacecraft only had 2020 meters/second on board. Using Navigator, however, the engineers designed a trajectory that used Lunar gravity assists. They started their rescue maneuvers on 10 April 1998 and brought the spacecraft to a useful geostationary orbit with extra fuel available for stationkeeping20. The engineers used Navigator because their routine maneuver planning software was not designed for such multi-body trajectories. In 1996 AGI bought Navigator from CSC, and obtained the rights to commercialize Swingby. At the request of the NASA GSFC Flight Dynamics Analysis Branch (FDAB), AGI incorporated Swingby into the STK product line, enhanced the capabilities, and started selling the software under the name STK/Astrogator. The details of the design process follow, but the first major goal was to support the analysis, launch window calculations, transfer trajectory operations, and stationkeeping of the Microwave Anisotropy Probe (MAP). The Astrogator software was ready on time for the pre-launch analysis tasks, such as fuel budget determination, calculation of the launch windows, and contingency planning. STK/Astrogator has successfully supported operations since launch on 30 June 2001 21,22,23724325.

STK/Astrogator was also used operationally at the Applied Physics Laboratory in Laurel, Maryland, USA, for the Comet Nucleus Tour mission (CONTOUR). Astrogator was used for planning and analysis of the cislunar phasing loop maneuvers. After the unfortunate loss of contact of the spacecraft after the solid motor firing, STK/Astrogator was used to generate possible burn and no-burn trajectories, two-line element sets, and antenna pointing angles to support the search efforts 2 6 .

A summary of the libration point and deep-space missions that employed STK/Astrogator for operations are shown in Table 1. In addition to being used operationally, Astrogator has been used by many government, educational, and commercial organizations to analyze

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Table 1. Operational Lunar, Libration Point, and Multi-Body Missions Supported with Swingby, Navigator, and STK/Astrogator. Mission Clementine Wind SOH0 ACE AsiaSa 3 rescue Lunar Prospector MAP CONTOUR

Launch 25 January, " , 1994 1 November 1994 2 December 1995 25 August 1997 25 Dec 1997 7 January, 1998 30 June 2001 3 July 2002

Regime Lunar Orbit I Asteroid DLS Sun-Earth L1 Sun-Earth L1 Lunar Gravity assist Lunar Orbit Sun-Earth LZ Cislunar phasing loop/Comet tour

and design future missions. In particular, Astrogator has been used for the Triana Sun-Earth L1 mission, and the NGST at LZ proposals. It has also been used for some Sun-Mars libration point studies In cislunar space it has been used to analyze some Earth-Moon libration-point missions as well as many lunar orbiting and landing missions, including several missions using the weak stability boundary (WSB) transfer 29. 27t28.

For these types of multi-body missions, Astrogator has been used for trajectory design, launch-window calculations, fuel estimates, developing station keeping strategies, and evaluating other mission requirements such as shadow analysis and communications link studies. The majority of Astrogator users have employed it for analysis and operations of many LEO and GEO missions. This, of course, reflects the population of spacecraft missions. This work has included orbit ascent, stationkeeping, ground track control, rendezvous, low-thrust, formation flying, and constellation missions. In addition, Astrogator has supported the analysis of several heliocentric missions, including solar sail missions, and those to the Sun, Mercury, Venus, Mars, Jupiter, Saturn, and Pluto 30.

It should be noted that STK and STK/Astrogator are being used Worldwide, most notably in the USA, but with a growing presence in prominent organisations in Europe and Asia.

3. Design Methodology and Goals

STK/Astrogator was developed after a requirements definition and design study. This section describes the process that was followed, the resulting

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requirements, and the specifications and methods used to meet those requirements .

3.1. Process

For three months in 1997, AGI technical personnel met several times in small, facilitated design groups with mission analysts from the GSFC FDAB, and from one of its subcontractors, CSC. The design groups listed past mission tasks and future mission requirements. They detailed their current work processes and developed work flow diagrams. These processes were studied, and the team developed suggestions for process improvement. In addition to the processes, the successes and the difficulties of the existing software were discussed. From these meetings requirements were derived. AGI personnel developed and presented mockup and then prototype graphic user interface (GUI) designs to analysts in the NASA GSFC FDAB. AGI then delivered beta software for analysts to use, and supplied training. After gathering feedback, AGI developed and delivered software to the Goddard FDAB every few months. During this development phase, the MAP mission required some of the newly implemented features. Consequently, one of the first tasks was to reproduce the existing Swingby setups with Astrogator. After AGI did this, the MAP trajectory design team used Astrogator in parallel with Swingby for a short while. The Goddard FDAB gave a contract to CSC to independently validate and verify Astrogator. When the MAP mission requirements were met, the MAP team switched to using Astrogator exclusively. Many of the requirements developed in the design groups were standard accuracy and numerical method requirements for a trajectory program. In addition, there were five major areas that the design group developed: analysis through operations support; multiple mission support; seamless operation with STK; automation support; and user extensibility. The reasoning behind these requirements are described in the following sections.

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3.2. Requirements on Numerical Methods

The design groups developed a set of requirements that would be standard for any operational trajectory software. The requirements can be simplified by stating that all forces must be modeled with sufficient accuracy to match the GSFC FDAB operational orbit determination software, the “Goddard Trajectory Determination System” (GTDS) 31. These forces include multiple and selectable gravitational bodies, the gravitational effect of a non-spherical central body, atmospheric drag, and solar radiation pressure. Additionally, Astrogator was required to model impulsive and finite maneuvers with accuracy as good as or better than the existing Swingby and the General MANeuver program (GMAN). To meet these requirements, a high-fidelity trajectory propagation system was specified, based on the legacy software. The details of STK’s numerical methods are found in the on-line help system and at AGI’s web site, www.stk.com. The algorithms themselves are well known in the industry and the literature: Numerical integration using Cowell’s formulation as well as Variation of Parameters for orbit propagation; differential correction and homotopy continuation for targeting; bounded search algorithms for event detection; published atmospheric density models for the Earth and Mars; standardized coordinate and time transformations; planetary ephemerides from the Jet Propulsion Laboratory; et cetera. The requirements that drove these numerical methods derive from the high-level requirement to support operations. Therefore, as mentioned above, the trajectory propagation was required to match that used for orbit determination. This caused subsequent requirements on the content and accuracy of the force models. Furthermore, because STK is required to calculate predicted antenna pointing data and timing, requirements on Earth shape models, Earth orientation (rotation, precession, nutation, pole wander), and station mask modeling were imposed. In addition, possible mobile vehicles with satellite tracking or communication equipment imposed an additional requirement on STK that was met with the capability to create station masks from surrounding terrain data.

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STK/Astrogator was also required to calculate maneuver plans with thruster timing and attitude data accurate enough for operations. This lead to derived requirements on finite- burn propulsion modeling for pressure regulated, blow-down, and ion-propulsion systems. Furthermore, Astrogator was required to support post-maneuver engine calibration, which led to requirements on the accuracy and consistency of orbit propagation during the maneuver compared to the orbit determination systems.

3.3. Analysis Through Operations Support The design groups desired that the new software, Astrogator, would support a mission from early conceptual phases, through the rigorous pre-launch analysis, and throughout all phases of operations. This created a contradiction: the workflow analysis determined that during the pre-launch and analysis phases of missions, analysts benefit from easy-to- use interactive software, using modern GUI controls, with the ability to quickly set any and all input parameters. Furthermore, sometimes using reduced fidelity (less accurate) force models enable quick studies to be done efficiently. This contrasts greatly with the necessity in operations for high accuracy and strict configuration control of all relevant data and setups. The design groups noted that when Swingby was first developed, its interactive GUI and graphics streamlined some trajectory design tasks by enabling the analyst to quickly try out many different ideas. The immediate graphical feedback helped analysts develop useful intuitions, especially in the 3- and 4-body dynamics of the missions the GSFC FDF was supporting at that time. However, when the Swingby software was used for simulations and then operations, several difficulties arose. These centered on the fact that to quality-check that the proper data were being used in the system, the analyst must display the user interface panel. When the panel displayed the data, it was possible that the analyst could accidentally adjust and therefore corrupt a numerical value. In fact, even if the analyst did not adjust the data, sometimes the GIU panel would slightly modify a number because the internal binary representation was converted to text for display, and then converted back to binary if the analyst hit the OK button on the panel. This caused a lot of extra work in operations to quality check maneuver plans before delivery.

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To get around this problem, STK/Astrogator was architected to store individually configured items in separate XML files. These items represented the objects that make up the numerical simulation. Force model, numerical integrator, and central-body parameters are some examples. During analysis tasks, the analysts can set the parameters in these objects easily through STK’s GUI. As launch approaches, and the setups start being used for simulations and finally operations, these individual object files can be flagged as ’read-only’ using standard operating system commands. STK/Astrogator honors this setting, and will display the data of these objects in its GUI, but will not allow the analyst to re-set the parameters. Because these files are XML, they can also easily be displayed in human readable form and can be differenced with baseline object files for quality check purposes. These objects are managed by implementation in C++ using a prototype pattern 32. This is a software design pattern that allows objects to be instantiated, copied, and otherwise controlled in a convenient way. A prototype of at least one of every object is available for the analyst to use. If a modification is needed, then the analyst copies a prototype, modifies it, and uses the copy instead. The design groups also noted that during spacecraft contingency and emergency operations, the tasks performed by analysts were similar to prelaunch activities: quick flexibility was needed. Therefore, AGI designed and built a system that allows the analyst to copy any configuration-controlled item to a new object, modify it’s parameters, and use the new object instead of the old one. Then, when the contingency is over, the new item can be placed under configuration control by setting its file permissions.

3.4. Multiple Mission Support

Because the GSFC FDAB studies and supports such varied topics, Astrogator was required to support missions ranging from low Earth to deep space, libration point, and missions to and around other planets, comets, asteroids, the Moon, and the Sun. In order to meet this requirement, AGI employed a strategy design pattern 32. This software pattern establishes well-defined interfaces for classes of objects, but allows the implementation to be different according to the need.

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For example, all objects of the engine class must calculate thrust and exhaust velocity data. So all engine objects in STK have this interface. However, the polynomial engine model calculates these data in a completely different way than the solar-electric propulsion engine. During numerical integration the force model calls the engine model, selected by the analyst, and it simply asks for certain data. The force model has no knowledge of how the data are calculated. This strategy pattern was employed for many objects, including central bodies, orbital elements, power supplies, vectors and coordinate systems. The strategy pattern, combined with the prototype pattern mentioned above, allows the analyst to configure the objects comprising the simulation to meet the specific mission needs. This reduces the number of options with which the analyst is presented. For instance, this eliminates the need to ’hard-code’ a libration-point coordinate system for every set of appropriate bodies in the solar system, which would be quite a long list. Instead, STK/Astrogator comes installed with just a few configured libration-point coordinate systems. If the analyst wants to study, for example, the SunNeptune system, then they duplicate the Sun-Earth system, change the central body, and select which libration point should be the origin. In this way the analyst has easily customized the software to a specific need, without the development team having to guess all possible combinations ahead of time.

3.5. Seamless Operation with STK In the support of various missions, designing the trajectory is only the first step. Previous to Astrogator, after the trajectory was designed, the ephemeris was saved to a file and post-processed with a variety of other software, including STK. Often during trajectory design the acceptance of a trajectory can only be verified by monitoring other parameters, such as ground station communication link coverage or solar lighting geometry. Therefore it was required that the trajectory design system seamlessly interchanges orbit and attitude data with STK. This requirement was met by specifying that Astrogator acted as an ephemeris and attitude source from within STK. An example of integration with STK that has helped libration-point

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missions is the ability to calculate shadows from the Earth and the Moon. The MAP mission, for instance, has such strict no-shadow requirements -because of the thermal sensitivities of its scientific payload- that even annular eclipses from the Moon had to be avoided. Because Astrogator is integrated within STK, it became a simple process to run a shadow report after each candidate trajectory was developed. Other examples include analyzing contact times to ground stations; communications link budget and interference analysis; radiation dosage studies, and generic figure-of-merit coverage analysis around the Earth, Mars, or other planets. The seamless integration with STK also benefits operations tasks by enabling the analyst to create a wide variety of data products for other groups. The data products can be customized and automated using STK’s ability to change units and precision; automatically call post-processing scripts for formatting and merging; and by using the STK/Connect module to automate common tasks. Another benefit of Astrogator’s integration with STK is realized with the STK Visualization Option (STK/VO). This is described later in the section “Visualization.”

3.6. Automation support

The ability to automate tasks such as parametric studies, Monte Carlo analyses, and customized search methods was critical for reducing the workload of the analyst users. In addition, operations personnel had relied on the ability to script routine operations to reduce both the cost of staff as well as reduce the risk of making mistakes. To support this, AGI required Astrogator to allow complete symbolic access (i.e., through a text name) to all input parameters. Symbolic access to each and every variable is useful, of course, only if the analyst can write control logic to change these parameters. AGI developed a custom script language, and this was demonstrated to several users as a prototype. The feedback, however, indicated that the majority of potential users were not willing to commit the time to learning a proprietary language without knowing that it would be successful. As a result, an interface became required that could be run from a variety of standard languages. A generic system was specified to include languages such as ’C’, C++’, PERL, Python, Java, VBScript, and MATLAB, to name

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a few. Once these standard languages were supported, the analysts readily started using automation in trajectory design analysis and operations.

One of the more recent uses of the scripting capability has been to take data created with other programs and import them into STK. For example, to support the Triana mission, the company Space Exploration Engineering, Inc. (SEE) 33 was tasked by the GSFC FDAB to write a MATLAB tool to take trajectories designed using the Generator software (developed at Purdue University 34) and import them into Astrogator as targeting constraints and first guesses at the maneuvers. Several other organizations have written similar scripts that integrate STK within other systems. Some have even gone as far as running STK completely automatically in a “lights out” operations center 35. Scripting, however, was not sufficient to automate all tasks. In studying past mission analyses, three additional requirements arose. The first can be understood with an example. Take the problem of modeling a trajectory that includes performing a 20 meter/second maneuver at each periapsis until the radius of apogee is halfway to the Moon. This type of problem yielded the requirement that trajectory related actions (such as a maneuver) could be triggered by events (such as periapsis). A generic system was implemented that can trigger when any orbit parameter crossed any user-defined value. In addition, a system was specified that allowed any trajectory sequence of arbitrary complexity, including targeting, to be triggered by any one of these events. The second additional requirement arose from missions that require more than one maneuver. During analysis and operations, after the initial conditions are changed and a maneuver is modeled, all the subsequent maneuvers must be re-planned. Therefore,the requirement arose that the maneuver targeting capability must allow automatic re- planning, without analyst intervention. This is especially important if the trajectory model is wrapped withh the loop of a parametric or Monte Car10 script. This requirement was met by specifying a robust targeting algorithm with specific features to enable convergence even with very non-linear problems. In particular, a differential corrector with normalized parameters, a search step-size control algorithm, and a homotopy continuation method were employed. The third additional requirement arose after studying how analysts and

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operators targeted trajectories; the ability to first achieve ’coarse’ intermediate goals and then continue with further refinement was required. The analyst must be able to set up one targeting problem, and then, within the same section of the trajectory, target different, refined goals with either the same or, perhaps, different controls constraints. This requirement resulted in the specification of “Targeting Profiles” which enable the analyst to pair up a set of controls with a set of desired constraints, and then name this profile. Furthermore, it was specified that the analyst could create any number of these named profiles, and run them automatically in sequence; as soon as one profile was run successfully, the next in the list must run automatically. This has been quite useful in targeting non-linear problems.

3.7. User Extensibility Before Swingby was built, the Goddard Mission Analysis System (GMAS) was used for trajectory design. GMAS ran on mainframe computers, and was written mostly in FORTRAN. One major feature that was used heavily in the design of complex trajectories was the ability for a programmer/user to create a new FORTRAN subroutine and have it available as a new module within GMAS. The analysts often used this to create new coordinate systems, new parameters for targeting or reporting, and to augment the propagator force models. The first specification to meet this requirement called for the ability for a programmer/user to write a ’C’ function with a specified function signature and compile it as a dynamic link library (DLL) file. Then, by placing the DLL file in the proper system folder, the user could “plug-in” a new function that would be available within Astrogator just as if it was built in from the beginning. This was successfully implemented, and the NASA GSFC FDD built several example plug-in DLL functions specific to deep space missions. However, during testing, it became clear that the typical trajectory analyst was not very proficient in the ’C’ language, nor had much desire to learn. After about a year, this DLL feature was no longer used, and was taken out before the commercial release of Astrogator. To meet the requirement, a new specification was derived with the lessons learned in mind. Specifically, it was desired to allow an analyst/user to augment the functionality of Astrogator without help from a programmer

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or from technical support. Therefore the new specification called for functions to be in a simple, standard script language. Initially three languages were chosen: MATLAB, PERL, and VBScript. (PERL and VBScript were chosen in particular because they are free to the analyst.) Examples were created with which the analysts can start, as well as documentation of the syntax. The specification further called for a simple text equation capability (not requiring source files) for the most common and simple task of defining a new generic orbit parameter as a function of exiting parameters. These enhancements were made and put into the commercial version of the software in early 2002. Specifically, AGI added plug-in points in Astrogator to allow an analyst to augment it with their own engine models; additional forces for orbit integration; vectors (and therefore axes and coordinate systems); and Astrogator calculation objects (scalar values used for stopping conditions, graphing, reporting, and targeting constraints). Additionally, plug-in points were added to STK for custom access and communication constraints and to model attitude dynamics and control.

4. Visualization

The stories are told that during the Apollo missions, trajectory analysts strung lights in darkened rooms in order to visualize and understand the 3body transfer trajectories. Later, the aforementioned GMAS program was able to create simple plots of trajectories using text characters on green & white computer paper. There were a few other attempts within the Goddard organizations at using graphics software, and by the time Swingby was being designed, it became a requirement to display the trajectory graphically, as it was being calculated. Once implemented, this yielded several helpful results. Primarily, when an analyst is trying to develop a trajectory, and attempting to target maneuvers using a shooting method, it is immediately apparent if the trajectories start to diverge. Watching the trajectories target also gives the analyst insight to the cause of divergence, and makes it easier to rectify the problem. Another benefit of the visualization arises from the fact that there are no easily understood metrics to describe the many trajectories in the 4-Body problem (Sun, Earth, Moon, and spacecraft). Much of the work of a trajectory analyst involves understanding the authority of the control maneuvers to cause a trajectory to meet mission requirements. The deviation and re-

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lationship of two trajectories with similar initial conditions in the 4- Body problem can be more easily understood graphically than numerically. Once the analyst gains this understanding, he or she can devise metrics to quantitatively control the trajectory. In addition, the animation of the dynamics helps give insight into trajectory controls that can affect the evolution of a trajectory. These insights help tremendously to determine regimes where linear approximations are appropriate, and where they fail. With the combination of STK/Astrogator and STK/VO, the analyst can interactively rotate the trajectory with the mouse. This has been quite helpful in understanding the effects of the controls, especially for transfers to the libration points, and even more so for those transfer trajectories involving lunar gravity assists such as MAP. Additionally, the 3-D STK/VO views were helpful in explaining the complex geometries to the reviewers, to the non-trajectory members, and to project management of the MAP team. In particular, the design of MAP’S phasing loop trajectories kept the spacecraft out of Earth shadow, but small (annular) Lunar shadows were sometimes problematic. The discussions in design reviews were greatly added by the 3-D images and animations. The 3-D visualization was also a great help in developing a forward targeting algorithm for Weak Stability Boundary trajectories 29. While developing a methodology to correct the trajectory due to launch uncertainties and delays, the 3-D views showed clearly where changes behaved linearly, and where they were highly sensitive. This then led to the choice of stopping conditions and targeting constraints to correct the trajectory to meet mission requirements.

5. Summary

The software linage of Swingby, Navigator, and STK/Astrogator has been used for the analysis and operations of the majority of Libration point trajectories, as well as several Cislunar, Lunar, asteroid, and interplanetary missions, since it’s first use for the Clementine (DSPSE) mission. The feedback from use in early missions has caused significant enhancements for the most recent incarnation, STK/Astrogator. The driving requirement to have a single software tool support pre-launch mission analysis, operations, and contingencies has been a major factor in selection of algorithms and meth-

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ods, and several innovative solutions have been implemented to accomplish this task. The close working relationship between software industry and government experts has been instrumental in the development of a commercial software product with a mature feature set able to support Libration point and other multi-body missions. Finally the integration of the trajectory software package Astrogator with the other mission analysis capabilities of STK has proven to be valuable for many analysis and operational tasks: The ’normal’ post-trajectorydesign numerical tasks have been streamlined and automated, ensuring consistency, reducing risks, and allowing studies that would otherwise be time and cost prohibitive. Additionally, the interactive and advanced computer graphics capabilities have become an invaluable aid to understanding the complex trajectories that exist in the multi-body problem.

References 1. Analytical Graphics, Inc. web site at www.stk.com. 2. R.W. Farquhar: The Flight of ISEE-3/ICE: Origins, Mission History, and a Legacy, AIAA 98-4464,AIAA f AAS Astrodynamics Specialist Conference, Boston, Massachusetts, USA, August 1998. 3. R.W. Farquhar, et al.: Trajectories and Orbital Maneuvers f o r the First Libration-Point Satellite, J. Guidance and Control, Vol. 3, No. 6,Nov-Dec 1980,pp. 549-554. 4. G M A S System Description, from the NASA Goddard Space Flight Center, Flight Dynamics Analysis Branch. 5. G M A N Mathematical Specification, from the NASA Goddard Space Flight Center, Flight Dynamics Analysis Branch. 6. J.P. Carrico, H.L. Hooper, L Roszman, C. Gramling: Rapid Design of Gravity Assist Trajectories, Proceedings of the ESA Symposium on Spacecraft Flight Dynamics, Darmstadt, Germany, September/October 1991. 7. J.P. Carrico, C. Schiff, L. Roszman, H.L. Hooper, D. Folta, K.V. Richon: An Interactive Tool f o r Design and Support of Lunar, Gravity Assist, and Libration Point Trajectories, AIAA 93-1126,AIAA/AHS f ASEE Aerospace Design Conference, Irvine, California, USA, February 1993. 8. J.P. Carrico, D. Conway, D. Ginn, D. Folta, K.V. Richon: Operational Use of Swingby-an Interactive Rajectory Design and Maneuver Planning Tool-for Missions to the Moon and Beyond, AAS 95-323,AASf AIAA Astrodynamics Specialist Conference, Halifax, Nova Scotia, Canada, August 1995.

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9. P. Sharer, H. Franz, D. Folta: W I N D Trajectory Design and Control, Paper No. MS95f032, CNES International Symposium on Space Dynamics, Toulouse, France, June 1995. 10. H. Franz: W I N D Nominal Mission Performance and Extended Mission Design, AIAA 98-4467, August 1998. 11. H. Franz: W I N D Lunar Backjlip and Distant Prograde Orbit Implementation, AAS 01-173, AASIAIAA Spaceflight Mechanics Meeting, Santa Barbara, California, USA, February 2001. 12. J. Carrico et al.: Maneuver Planning and Results for Clementine (the Deep Space Program Science Experiment), AAS 95-129, AASIAIAA Spaceflight Mechanics Meeting, Albuquerque, New Mexico, USA, February 1995. 13. D.W. Dunham, R.W. Farquhar: Trajectory Design for a Lunar Mapping and Near-Earth-Asteroid Flyby Mission, AAS 93-145, Spaceflight Mechanics 1993, Vol. 82, Advances in the Astronautical Sciences, ed. R. Melton, pp. 605-624. 14. D.W. Dunham, et al.: Tkansfer Tkajectory Design f o r the S O H 0 LibrationPoint Mission, IAF-92- 0066, 43rd Congress of the International Astronautical Federation, Washington, D.C., USA, AugustfSeptember, 1992. 15. D.W. Dunham: Handbook on Trajectories, Mission Design, and Operations for the Spacecraft of the Solar-Terrestrial Science Project, Vol. 3, prepared by the Working Group 3 of the IACG for Space Science, September 1994. 16. D.W. Dunham, C.E. Roberts: Stationkeeping Techniques for Libration-Point Satellites, AIAA 98- 4466, AIAAfAAS Astrodynamics Specialist Conference, Boston, Massachusetts, USA, August 1998. 17. C.E. Roberts: Long Duration Lissajous Orbit Control for the A C E Sun-Earth L1 Libration Point Mission, AAS 01-204, AASfAIAA Space Flight Mechanics Meeting, Santa Barbara, CA, USA, February, 2001. 18. D. Lozier, K. Galal, D. Folta, M. Beckman: Lunar Prospector Mission Design and Trajectory Support, AAS 98-323, pp. 297-311. 19. P.A. Penzo: A Survey and Recent Developments of Lunar Gravity Assist, Space Studies Inst., Princeton University, May 1998. 20. J. Salvatore, C. Ocampo: Mission Design and Orbit Operations for the First Lunar Flyby Rescue Mission, IAF-99-A.2.01, 50th International Astronautical Congress, Amsterdam, The Netherlands, October 1999. 21. K.V. Richon, M.W. Mathews: An Overview of the Microwave Anisotropy Probe ( M A P ) Trajectory Design, Advances in the Astronautical Sciences, Vol. 97, 1998, pp. 1979-1998. 22. M. Woodard, 0. Cuevas: M A P Trajectory Design, Satellite Tool Kit User's Conference, Georgetown, Washington D.C., USA, June, 2002. 23. M. Mesarch: ontingency Planning for the Microwave Anisotropy Probe Mission, AIAA 2002-4426, AIAAfAAS Astrodynamics Specialist Conference, Monterey, CA, USA. 24. M. Mesarch, S. Andrews: The Maneuver Planning Process for the Microwave Anisotropy Probe ( M A P ) Mission, AIAA 2002-4427, AIAAf AAS Astrodynamics Specialist Conference, Monterey, CA, USA. 25. D. Folta, K.V. Richon: Libration Orbit Mission Design at L2: a M A P and N G S T Perspective, AIAA 98-4469, August 1998.

Software Architecture and Use of Satellite Tool Kit’s Astrogator Module 487 26. Private communication with David Dunham, Dan Muhonen, and Peter Sharer of the Johns Hopkins Applied Physics Laboratory, Laurel, Maryland, July/August, 2002. 27. J.D. Strizzi, J. Kutrieb, P. Damphousse, J. Carrico: Sun-Mars Libration Points and Mars Mission Simulations, AAS 01-159, AASIAIAA Spaceflight Mechanics Meeting, Santa Barbara, California, USA, February 2001. 28. J. Carrico, J. Strizzi, J. Kutrieb, P. Damphousse: Trajectory Sensitivities for Sun-Mars Libration Point Missions, AAS 01-327, AASIAIAA Astrodynamics Specialist Conference, Quebec City, Quebec, Canada, July/August 2001. 29. E.A. Belbruno, J.P. Carrico: Calculation of Weak Stability Boundary Ballistic Lunar Ransfer Dajectories, AIAA 2000-4142, AIAAIAAS Astrodynamics Specialist Conference, Denver, Colorado, USA, August 2000. 30. Y . Gao, R. Farquhar: ew Horizons Mission for the Pluto-Kuiper Belt Mission, AIAA 2002-4722, AIAAIAAS Astrodynamics Specialist Conference, Monterey, CA, USA, August, 2002. 31. Goddard Trajectory Determination System (GTDS) Mathematical Reference, NASA Goddard Space Flight Center, Flight Dynamics Division. 32. E. Gamma, R. Helm, R. Johnson, J. Vlissides: Design Patterns,l995 AddisonWesley Longman, Inc. 33. Space Exploration Engineering web site at www.see.com. 34. K. C. Howell: Families of orbits in the vicinity of the collinear libration points, AIAA 98-4465 , AIAAIAAS Astrodynamics Specialist Conference, Boston, MA, USA, August, 1998. 35. AGI S T K InView article about the PROBA mission, Fall 2002.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

THE COMPUTATION OF PERIODIC SOLUTIONS OF THE 3-BODY PROBLEM USING THE NUMERICAL CONTINUATION SOFTWARE AUTO D.J. DICHMANN Astrodynamics Consultant, 20821 Amie Ave #l20, Torrance CA 90503, USA

E.J. DOEDEL Department of Computer Science, Concordia University, 1455 blvd. de Maisonneuve West, Montreal, Quebec, H3GlM8, Canada

R.C. PAFFENROTH Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, USA

AUTO2000 is the most recent version of AUTO, a software tool for continuation and bifurcation problems in ordinary differential equations. This paper discusses the continuation methods and bifurcation-detection algorithms that underlie the AUTO2000 software, and describes a method for computing families of periodic solutions of conservative system. This software package has a number of features that make it useful as a space mission analysis and design tool. Using scripts written in the object-oriented scripting language Python to drive AUT02000, we have computed several families of periodic solutions emanating from libration points in the Circular Restricted 3-Body Problem (CR3BP) for the Earth-Moon system. By following branches of solutions we were able to locate a variety of periodic orbits that may be useful for space missions. For some of these orbits we gain greater insight by viewing the orbits in both rotating and inertial coordinates. In particular we discuss the family of “Backflip” periodic orbits in which a series of Double Lunar Swingbys can be used to periodically send a spacecraft beyond the ecliptic plane.

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1. Introduction In space mission design, the availability of a wider range of orbits gives the mission analyst more flexibility in the choice of trajectory, and may lead to new opportunities for observations. In this paper we describe AUTO2000 13, the most recent version of AUTO, a software tool for continuation and bifurcation problems in ordinary differential equations. We show how AUTO2000 can be used to compute families of periodic solutions of conservative systems, i.e., systems having a first integral. The method is applied to the computation of periodic solutions of the Circular Restricted 3-Body Problem (CR3BP). The CR3BP is a valuable model for space mission design because it has many of the essential features of a more complex high-fidelity force model, yet it is simple enough to lend itself to analysis and computation of families of periodic orbits. A solution of the CR3BP then provides an excellent first approximation of a trajectory in the high-fidelity force model. The computation at the heart of many mathematical models of physical systems is the solution of a parameter-dependent system of nonlinear equations. Such problems generally have more than one solution, and it is often desirable to compute a family of solutions and search for specific solutions with certainly desirable properties. A bifurcation diagram is a schematic representation of such a solution set. The computation of such bifurcation diagrams and their singularities, e.g., folds, bifurcation points etc., is the domain of numerical continuation algorithms. In Section 2, we review some basic notions of numerical continuation. The basic idea of the method for continuing a family of periodic orbits in a conservative system was already used in Doedel et al. 9; a more extensive analysis of the ideas in this paper can be found in Doedel et al. 14,44,48. The problem of computing a periodic orbit can be phrased as a two-point boundary value problem. However, in a Hamiltonian system such as the CRSBP, the continuation methods implemented in AUTO2000 are not directly applicable, because the existence of a first integral implies non-uniqueness of solutions when the problem is phrased as a two-point boundary value problem in the canonical way. In this paper we describe how such boundary value problems may be modified to account for the Hamiltonian structure. In Section 3, we show how the CR3BP problem may be reformulated using the above ideas so that we may apply numerical continuation methods.

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Our focus in this paper is on orbits in the Earth-Moon system. To understand better the physical properties of these orbits, we present in Section 4 the unit conversions to transform from the nondimensional barycentric, rotating system of the CR3BP into physical units for the Earth-Moon system. In Section 5 we use the formulation in Section 3 to compute families of periodic solutions emanating from the L1 libration point. The numerical continuation methods in AUTO2000 are not restricted to a neighborhood of the libration point, do not depend on any symmetry properties of the solution, and can compute periodic orbits of arbitrary extent until some singularity is reached, such as a collision with a primary. AUTO2000 provides for detection and continuation of bifurcating branches of solutions, so a detailed map of periodic orbits may be computed for each of the libration points for any given value of the ratio of the masses of the primaries. We give graphical results that illustrate the solution structure of periodic orbits emanating from the L I . All of the branches of solutions shown in Section 5 were generated automatically using a single script written in the object-oriented scripting language Python to drive AUTO2000 47. These periodic orbits include the Lyapunov orbits of the planar CR3BP model. However, the three-space dimension model has a much richer periodic solution structure, containing many bifurcations that can be detected easily in our two-point boundary value problem continuation approach. The resulting bifurcation diagram contains some well-known solutions such as the Halo orbits, whose behavior and stability properties have been computed before. (See, for example, Howell 35 for a detailed numerical study.) Moreover, the approach used in this paper also generates families of periodic solutions that appear to be less well known, though some appear in articles such as Zagouras and Kazantzis 5 8 . It is convenient to compute solutions of the CR3BP in rotating coordinates because the equations of motion are autonomous in that frame. However, in Section 6 we show that in some cases one can gain further insight by also visualizing the orbits in Earth Centered Inertial (ECI) coordinates. Finally, in Section 7, we discuss possible applications of some of the families of periodic orbits to space missions, both for scientific and operational space weather observations. We pay particular attention to a family of “Backflip” periodic orbits, reminiscent of the Backflip maneuver described by Uphoff 5 6 , in which a periodic Double Lunar Swingby allows a spacecraft to explore the space both near and beyond the ecliptic plane.

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The AUTO2000 source code is publicly available via HTTP from the web site:

http://auto2000. sourceforge.net.

2. Continuation of solutions Most existing numerical continuation algorithms are for the computation of one-dimensional solution manifolds called solution branches, e.g., H. B. Keller 41, Allgower and Georg ', Doedel et al. 1 1 ~ 1 2and Rheinboldt 49. To review the basic notions of continuation, first consider the finite-dimensional equation

R" X R + R", (1) where the bold font indicates a vector, F is assumed to be sufficiently smooth for the theory to apply, y E Rn,and X E R One may also write Equation 1 as F(x) = 0 by setting x = (y,A), and we will use this notation F(Y,X) = 0 ,

F

:

when the distinction between the components y and X is not pertinent. This system has one more variable than it has equations. Given a solution (yo,A,), there generally exists a locally unique one-dimensional family of points, called a solution branch, that passes through (yo,XO). To compute another nearby point (y1,XI) on this branch, one can use a root-finding procedure, such as Newton's method, to solve the extended system

using as an initial guess (yo,XI). In general F(y0,XI) # 0, but if the step size AX is sufficiently small, and F is sufficiently smooth, then (yo, XI) lies within the basin of attraction of the given root-finding procedure, so the procedure converges to (y1,XI). Note, in this case, the initial guess (yo,XI) already satisfies Equation 2b, so in practice only Equation 2a need be solved for y. This algorithm is called natural parameter continuation and is shown schematically in Figure 1. One may now iterate the procedure described in the previous paragraph: xo = (y0,Xo) is used to compute x1 = (yl,Xl), which in turn in used to compute x2, etc. As shown in Figure 2, the core principle of numerical continuation is that, under certain conditions on F, the solution branch

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I

Natural parameter continuation with two different initial approximations: Solving Equation 2 using Newton’s method entails the selection of an initial approximation. For example, one can use the approximation x1# = (yo,XI). On the other hand, one might want to use a higher-order approximation. For example, given XO,the unit tangent to the solution curve at xo, one can use x; = xg+Xo As as a more accurate initial approximation, where As is chosen to have x1 lie on the line X = XI.

Fig. 1.

is surrounded by a basin of attraction for Newton’s method. If we have a starting value xo on the solution branch then, if the step size is sufficiently small, all steps stay within the basin of attraction and Newton’s method converges for all steps of the calculation. On the other hand, if one were to search directly for a solution where X = X d e s i r e d , and one did not have a good initial approximation, then there is no guarantee Newton’s method would converge. Natural parameter continuation captures the central ideas of numerical continuation, but it has some weaknesses. One significant problem with natural parameter continuation is that it fails at a fold, such as the one shown in Figure 3, because it does not account for the local shape of the solution curve. To formulate a method which can compute around folds, we begin by constructing an initial approximation to x1 as x; = xo XO As, for some step size As,where Xo is the unit tangent to the solution curve at XO. If we differentiate Equation (1) with respect to the arclength s at XO, we find F,(xo) XO = 0. Thus the vector XO is a null vector of the Jacobian matrix F,(xo), and can be computed at little cost. However the higherorder approximation does not, by itself, suffice to compute around a fold because there may be no solutions for the desired value of A. One must

+

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D.J. Dichmann, E.J. Doedel and R.C. Paffenroth

h

h

known

desired

Staying within the basin of attraction: The core principle of numerical continuation is that, under certain conditions on F , the solution branch is surrounded by a basin of attraction for Newton’s method. Here the basin of attraction is shown as the region bounded by dotted lines. If we have a starting value xo = (yo,Xo) on the solution branch, then step sizes can be chosen so that all steps stay within the basin of attraction and Newton’s method converges. On the other hand, if one were to directly search for a solution where X = Ade&&, and did not have a good initial approximation such as x;zLess, then there is no guarantee Newton’s method would converge. Fig. 2.

allow values of X to be determined by the geometry of the solution curve. Accordingly, we replace the system in Equation 2 with

where the superscript T denotes the transpose. The magnitude of the step size As is normally adapted along the branch, depending for example on the convergence rate of Newton’s method. The result is a well-known algorithm, called the pseudo-arclength continuation method 41, and is shown schematically in Figure 3. The name pseudo-arclength arises from the fact that the step size As approximates the arclength along the curve.

It can be shown that the above continuation method works if xo is a regular solution, i.e., if the null space of F,(xo) is one-dimensional. In this case the Jacobian of Equation 3 evaluated at xo, i.e., the n 1 by n 1 matrix

+

(FfT’)

7

+

(4)

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Fig. 3. Pseudo-arclength continuation and folds: Instead of fixing the value of

A, we add a constraint as shown in Equation 3b which implies the new solution x 1 we compute lies in the hyperplane perpendicular to the tangent vector XO.In this way, the value of X is allowed to vary, and is computed as one of the solution components using Newton's method. As this figure indicates, pseudo-arclength continuation is well-suited for numerical continuation around folds.

can easily be shown to be nonsingular. By the Implicit hnction Theorem, this guarantees the existence of a locally unique solution branch through XO. This branch can be parameterized locally by As. Moreover, for As sufficiently small and the initial approximation xy) = xo+XoAs, Newton's method for solving (3) can be shown to converge. In this paper we focus on the problem of computing a branch of periodic solutions of a dynamical system

x'(4 = f(x(t),A) >

f

:

P" X

P

+ P",

(5)

where X E P is a physical parameter. In this case the continuation step corresponding to Equation 3 takes the form of the constrained periodic boundary value problem

ad

Xi(.)

a21

Xl(0)

as) b)

J:

= Tlf(Xl(T),Xl), =Xl(l), x 1 ( T ) T x ~ ( T ) dT = 0,

J;(X~(T)

- X O ( T ) ) ~ X O ( TdT )

+ (TI - 2'0)ib + (XI - X O ) ~ O= As.

(6) These equations are to be solved for ( x l ( . ) , T l , X ~ ) , given a solution ( x g ( . ) , T o , Xo) and the tangent ( X O ( . ) , ~ , io). , Here 2'1 E P is the unknown

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period, and T is a scaled independent variable such that the periodic solution has period 1as a function of T . Equation 6a2 imposes unit periodicity. Equation 6a3 is a phase condition, which fixes the phase of the new orbit xl(.)relative to the given orbit XO(.). This integral phase condition has the desirable property of minimizing phase drift relative to XO(.). Equation 6b is the functional form of the pseudo-arclength constraint 3b. For full details on this formulation see Doedel et al. 12. Equation 6 can be solved by numerical boundary value solvers. In particular, AUTO louses piecewise polynomial collocation, similar to COLSYS 2 , with adaptive mesh selection as described in Russell and Christiansen 53. By using collocation, Equation 6 becomes a discrete system of the form of Equation 1,which can be solved using the continuation methods described in this section. For full details see Doedel et al. 12.

3. Periodic Solutions of the Circular Restricted 3-Body Problem

Section 2 described the mathematical formulation of problems that may be addressed by numerical continuation methods. In this section we show how the Circular Restricted 3-Body Problem (CR3BP) can be expressed in the form of Equation 5 . The CRSBP describes the dynamics of a body with negligible mass under the gravitational influence of two massive bodies, called the primaries, where the primaries move in circular orbits about their barycenter. Let (x,y,z) denote the position of the negligible-mass body in a rotating coordinate system with the origin at the barycenter where the x-axis points from the larger to the smaller primary; the z-axis points along the vector normal to the orbit plane of the primaries; and the y-axis completes the right-handed orthogonal triad. The parameter p represents the ratio of the mass of the smaller primary to the total mass. In this paper we consider the Earth-Moon system, for which p = 0.01215. The units are chosen so that the distance between the primaries, the sum of the masses of the primaries and the angular velocity of the primaries are all equal to one. Consequently the larger and smaller primaries are located at (-p, 0,O) and (1 - p, 0, 0),

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respectively. The equations of motion for the CR3BP are 2" y"

= 2y' = -22'

+ 2 - (1 - p ) ( z + p ) r r 3 - p(2 - 1 + p ) r Y 3 , + y - (1- p ) y q 3 - pyr,3,

= -(I - p ) -3 ~ ~- l~ z T ; ~ ,

(7)

where r1 = d ( z + p ) 2 + y 2 + z 2 ,

r2=J(a:-1+p)2+~2+22.

A derivation of these equations can be found in Danby '.

-

The dynamical system in Equation 7 has one integral of motion, namely the Jacobi integral

c = 2 U ( 2 ,y, z) - (w3 + w; + w;), 1

U ( 2 ,y, 2) = 2("2

+ y2) +

r1

+ 14. r2

It is well-known that for each value of p this system has five equilibria, called libration points (or Lagrange points) 5 4 , which lie in the orbit plane of the primaries. Three of the libration points, denoted L1, L2 and L3, are collinear with the primary bodies; one of them, L1, lies between the two primaries. Each of the other two points, Ld and L5, forms an equilateral triangle with the primaries. There exist two well-known families of periodic solutions near each of the collinear libration points: the Lyapunov orbits that lie in the x - y plane, and the so-called Vertical orbits that arise from the purely vertical solutions in the linearized dynamics 36. In this paper we use numerical continuation methods to explore families of three-dimensional periodic solutions of Equation 7 that emanate from L1. We make three modifications to Equation 7 to obtain a system to which AUTO2000 and numerical continuation may be applied. First, the three-dimensional second-order system in Equation 7 is rewritten as a sixdimensional first-order system in the standard way because AUTO2000 is set up to handle first-order systems. Second, we perform the transformation in Equation 6 where we add boundary conditions which impose unit periodicity, a phase constraint, and we introduce the unknown period T which we solve for as part of the numerical continuation procedure. The final modification is introduced to allow us to use the continuation method described in Section 2 to compute periodic solutions of the CR3BP.

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The Circular Restricted 3-Body Problem is a conservative system, for which there exists a first integral. Such systems have special structure, and a theorem that one can apply in such a context is the Cylinder Theorem 43: An elementary periodic orbit of a system with an integral I lies in a smooth cylinder of periodic solutions parameterized by I . The Cylinder Theorem implies there exists a branch of periodic orbits in a system without a parameter. However the theory and algorithms developed in Section 2 , and especially Equations 5 and 6, rely on the presence of a parameter. To take advantage of the power of numerical continuation methods we must rephrase the problem in a form where such methods can be applied directly. We accomplish this with the introduction of an “unfolding parameter” A, which is treated as an unknown but which will be zero upon solution. For a further background on this technique, see Doedel et al. 14, Muiioz-Almaraz et al. 44, and Paffenroth et al. 48. The resulting system of differential equations is

x‘ = Tw, + X dC/dx,

+ +

yl = Tw, X dC/dy, z’ = Tw, X dC/dz, w; = T[2v, x - (1- p)(x p)rT3 - p(x - 1 p)rF3 W$ = T [ - ~ w+, y - (1 - p)yrT3 - pyrF3 X dC/d~,],

.;

+

+

+

+

+ x dC/dw,],

+x dc/d~,],

= q-(i - p ) z r ; 3 - p ~ r ; 3

(8)

with separated boundary conditions

41) = X ( O ) , Y(1) = Y@), 41) = @), %(1)= % ( O ) , Wy(1) = Wy(O), vz(1) = .Z(O),

(9)

where T is the unknown period, and X is the unfolding parameter.

4. Physical Units

In order to relate the nondimensional units discussed in this paper to the physical world we need to be able to describe them in engineering units. We will use the superscript * to denote quantities in physical units. The gravitational constants of the Earth and the Moon are GMLarth= 398600.5 km3/sec2 and GMG,,, = 4902.794 km3/sec2, so that p = 0.01215. Let G M * = GMLaVth GM;,,, = 403503.3 km3/sec2. We take

+

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the Earth-Moon distance to be R* = 384401 km, the Earth’s radius to be 6378 km = 0.0165 R*, and the Moon’s radius to be 1738 km = 0.0045 R* 57

Using Kepler’s Third Law the rotational period of the system would be PgM = 2 7 r , / m = 27.2846 days. In fact, due to third-body effects, the period of the Earth-Moon system is PgM = 27.3217 days 5 7 . Because the angular velocity WEM is set to 1 in the nondimensional rotating coordinate system, the orbit period PEMis 27~in the nondimensional system. To transform a distance T from nondimensional to physical units we use = R*r. To transform a time t from nondimensional to physical units we use t* = PgM/PEM t = T* t , where T*= PgM/(2T) = 4.3225 days. The T*

Jacobi integral C has units of velocity squared, so to transform the Jacobi integral from nondimensional to physical units we use C* = (R* / T * )2 C where (R*/T*)’= 1.049694 km2/sec2. (If PsMwere derived from Kepler’s Law, then we would have (R * / T * )2= G M * / R * . ) The period of rotation for the Sun-Earth system in the inertial frame is P;E = 365.256 days. In terms of the nondimensional Earth-Moon orbit period, the Sun-Earth rotational period is PSE = PiE/T* = (PiE/PgM)PEM = 13.3687 P E M .

5. Tour of the Bifurcation Diagram In this section we present a tour of the families of periodic orbits emanating from L1 that we have computed using AUT02000. Previous work has mapped portions of the families of periodic orbits for various values of p ; cf. Howell 36 and references therein. Some researchers has investigated bifurcations of these families, including Ichtiaroglou and Michalodimitrakis 39, H h o n 34 and Howell and Campbell 38. G6mez and Mondelo 32 computed the families of orbits arising from L1, L2 and L3, as well as their bifurcating branches, for the Earth-Moon system AUTO2000 is designed to follow a branch of solutions, starting from a known or approximate solution, and to locate bifurcation points along the branch. Previous versions of AUTO had a good interface for expert users,

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Fig. 4. The bifurcation diagram for periodic orbits emanating from L1: The curves represent some of the branches of periodic solutions computed using AUT02000. The Earth and Moon are shown as two textured spheres, and the five libration points are shown as cubes. The branch labelled “L” represents the Lyapunov orbits (see Figure 6 ) and the branch labelled “V” represents the Vertical orbits (see Figure 9). Each bifurcation point is marked as a small white sphere if we discuss the bifurcating branch in this paper; otherwise the bifurcation point is colored dark. The rectangle lies in the x - y plane, and has the property that any solution branch which touches it has a planar solution at that point. For example, the entire branch of Lyapunov orbits lies in the plane, so all the orbits on that branch are planar. Planar orbits are trivially symmetric about the 2 - y plane. If a solution is symmetric about the x - y plane but nonplanar, then we have depicted it lying in the same plane as the gray rectangle but not touching the gray rectangle. On the branch of Lyapunov orbits there are two bifurcations, the first giving rise to the branch of Halo orbits, labelled by “H” (see Figure 7), and the second giving rise to a branch of “Axial” orbits that connect the Lyapunov and Vertical branches, labelled by “A”(see Figure 8). On the branch of Vertical orbits away from L1 there are three bifurcations. The second bifurcation point, labelled B(V,BF), yields the branches, labelled ‘‘BFl” and “BFz”,that respectively represent the Class 1 and Class 2 families of Backflip orbits (see Figures 10 and 11).

but this interface was somewhat difficult for beginners. Accordingly, the detection of families of periodic solutions near a libration point, and the continuation of these branches was facilitated by a Python script 47. Near a

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collinear libration point there are two such branches, the Lyapunov branch and the Vertical branch. The Python script also computes branches that arise from each of the bifurcation pointsa along the branches emanating from the libration point. Our presentation is organized around the bifurcation diagram shown in Figure 4,in which each curve represents a branch of periodic solutions. Figure 4 and later figures in this section show the Earth and Moon as two textured spheres. The libration points are visualized as cubes, and the branches of periodic orbits are drawn as curves. For example, the branch of Lyapunov orbits, marked with an “L” and the branch of vertical orbits, marked with a “V”, both emanate from the cube representing L I . Each bifurcation point is marked as small white sphere if we follow the bifurcating branch in this paper; otherwise the bifurcation point is colored dark. There are many branches of solutions for this problem, and in this paper we only treat a small subset. In addition, we include a gray rectangle that lies in the z - y plane. Any solution branch that touches this rectangle has a planar solution at that point. For example, the entire branch of Lyapunov orbits is planar, so the entire line which represents the Lyapunov orbits touches the gray rectangle. Planar orbits are trivially symmetric about the 2 - y plane. Solution B(V,BF) is symmetric about the 2 - y plane but nonplanar. As a visual cue we have depicted it lying in the same plane as the gray rectangle but not touching the gray rectangle. We emphasize that, even though the various visualization aids are in the proper physical position with respect to each other, the bifurcating branches themselves are only schematic. The relative positions of the various solution branches should not be interpreted as signifying any physical property of the solutions, other than those discussed above.

There are five families of periodic solutions illustrated in Figure 4. On the branch of Lyapunov orbits there are two bifurcations. The first bifurcation from the Lyapunov family gives rise to the branch of Halo orbits labelled with an “H” (see Figure 7). The second bifurcation from the Lyapunov family yields the branch of orbits labelled with an “A” (see Figures 8). We call the solutions on this branch the “Axial” orbits. On the curve of Vertical orbits, labelled by a “V”, (see Figure 9) there is a biaIn this paper we reserve the terms bifurcation point and bifurcation orbit, when not further qualified, for transcritical and pitchfork bifurcations, excluding period-doubling, torus. and subharmonic bifurcations.

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Fig. 5. A close-up view of the bifurcation diagram in Figure 4. Here we can see more clearly the bifurcation points B(V, BF) and B(V, 3) and the two legs of the branch of Backflip orbits. The bifurcation point B(V,BF) is placed outside the rectangle because the orbit is not planar. Instead we have drawn B(V,BF) in the plane of the rectangle to indicate that the orbit is symmetric with respect to the 2 - y plane. The two legs BF1 and BF2 correspond to the Class 1 and Class 2 families of Backflip orbits (See Figures 10 and 11.) Bifurcation point B(V,3) lies in the rectangle to indicate that the bifurcating orbit is planar. Specifically, B(V, 3) is a reverse period-doubling bifurcation.

furcation that also gives rise to the branch of orbits labelled by an “A”. Thus the family of Axial orbits forms a connection between the Lyapunov and Vertical families. The second bifurcation from the branch of Vertical orbits produces the branches labelled “BFl” and ‘‘BFz”that represent the “Backflip” orbits (see Figures 10 and 11).Figure 5 gives a closer view of this bifurcation point, labelled B(V,B F ) . In general, we denote the bifurcation point that connects a branch labelled X and a branch labelled Y by B ( X ,Y).We use the same notation whether we refer to a “bifurcation point” (when the bifurcation is shown on a bifurcation diagram) or a “bifurcation orbit” (when the bifurcation is shown as a physical solution). If we do not pursue a branch in this paper then it is labelled with a number instead of a letter, and the bifurcation point is colored dark. For example, there is a dark-colored bifurcation point labelled B(V,3) at the end of the branch of Vertical orbits, representing a reverse period-doubling bifurcation to a branch of planar solutions. (When approaching B(V,3) along this branch of planar solutions, which is not further discussed here, the point

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B(V,3) will appear as a regular period-doubling bifurcation.)

Fig. 6. The family of planar Lyapunov orbits. The orbits are depicted in the figures in the barycentric rotating frame. The gray disk lies in the z - y plane, centered at the barycenter with radius equal to the Moon’s orbit radius R. The orbits emanate from L1 on the z-axis. The figure on the upper left shows the two bifurcation orbits in this family. The small white ball at the z-axis represents the Moon, drawn to scale. The thick tube labelled B ( L , H ) represents the fist bifurcation orbit to the family of Halo orbits in Figures 7. The f i s t bifurcation orbit has a y amplitude of 0.0559548 R = 21509 km and has a period of 2.74298 = 0.436559 P E M .The thick tube labelled B ( L ,A ) represents the second bifurcation orbit to the family of Axial orbits shown in Figure 8. The second bifurcation orbit has a y amplitude of 0.250569R = 96319 km and a period of 3.95007 = 0.628673 PEM.The figure on the upper right shows the continuation of the family of Lyapunov orbits beyond the second bifurcation, terminating in collisions with the primaries.

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We now examine the shapes of the periodic orbits associated with the various branches, beginning with the planar Lyapunov orbits shown in Figure 6 . Here and in Figures 7-10 we show both a family of orbits and the bifurcation diagram, with an arrow indicating the relevant branch. The orbits are depicted in the barycentric rotating frame. The gray disk in these figures lies in the x - y plane, centered at the barycenter with radius equal to the Moon’s orbit radius R. As a visualization aid we show bifurcation orbits as thickened tubes. The first bifurcation orbit on the Lyapunov branch, labelled B ( L ,H ) gives rise to the well-known Halo orbits. Accordingly, the thick curve labelled B ( L ,H ) in the plane of the Lyapunov orbits is the orbit from which the Halos bifurcate. The second bifurcation orbit on this branch, shown as a thick tube labelled B(L,A ) , represents the bifurcation orbit to the family of Axial orbits shown in Figure 8. The family of Lyapunov orbits terminates in collisions with the primary bodies. Figure 7 shows a selection of Northern Halo orbits 5,21,35. To reduce clutter in the diagram, we have only plotted the Northern Halo orbits and not the Southern Halo orbits. The Northern and Southern families of Halo orbits are related through the symmetry z + -z. In Figure 7 the upper left diagram shows the Halo orbits up to a dark bifurcation orbit B ( H , l), whose bifurcating orbits we do not pursue here. (See Doedel et al. l4 and Paffenroth et al. 48 for details on this branch.) The upper right diagram in Figure 7 begins where the upper left diagram ends. It shows the northern Halo orbits from the bifurcation B ( H , 1) up to the second bifurcation B ( H ,2) which gives rise to a branch of planar solutions, also not shown here. As the bifurcation diagram indicates, the Halo branch can be continued past the bifurcation to the planar solutions, giving rise to the symmetry-related branch of southern Halos. Accordingly, the branch of Halo orbits in the bifurcation diagram is a loop. The Axial orbits are shown in Figure 8. We call this family of orbits the Axial orbits, because each orbit is axially symmetric about the x-axis under the transformation y + -y, z + -2, t + - t . The thick curve B ( L ,A) is the bifurcation orbit connecting the Lyapunov branch with the Axial branch. The thick curve B(V,A ) , symmetric across the 2 - z plane, is the bifurcation orbit connecting the Vertical branch and the Axial branch. There is a second symmetry-related family not shown here, and the whole branch of orbits forms a loop as shown in Figure 4. Some of the orbits on the “A” branch were plotted in Zagouras and Kazantzis 58 for p = 0.00095.

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Fig. 7. The family of Northern Halo orbits. The upper left figure shows the Halos near the planar bifurcation orbit. The thick curve labelled B ( L ,H ) is the Lyapunov orbit from which the Halo orbits bifurcate. The thick curve B ( H , 1) corresponds to the next bifurcation point on the branch of Halo orbits. The zamplitude of the B ( H , 1) orbit is 0.287 R = 110323 km. The orbits in the upper left figure correspond to the part of the bifurcation diagram labelled as “Hl”. The upper right figure represents the collection of Halo orbits from the bifurcation orbit B ( H , 1) up to the final bifurcation point B(H, 2). The curve B ( H ,2) that encompasses the Earth is a bifurcation orbit that gives rise to a branch of planar orbits not shown here. Once the family passes through B ( H , 2 ) , it becomes the symmetry-related branch of Southern Halo orbits. Hence, the branch of Halo orbits in Figure 4 is a loop. The orbits in the upper right figure are found in the part of the bifurcation diagram labelled as “H2”.

These orbits were also computed by Gdmez and Mondelo 32 for the EarthMoon system.

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D . J . Dichmann, E.J . Doedel and R. C. Paffenroth

A branch of Axial (axially symmetric) orbits connecting the Lyapunov and Vertical orbits. The thick curve B ( L ,A ) is the bifurcation orbit connecting the Lyapunov branch with the Axial branch. The thick curve B(V, A), symmetric across the x - z plane, is the bifurcation orbit connecting the Vertical branch and the Axial branch. The other curves are a representative collection of the orbits that connect these two bifurcation orbits. The periods of the Axial orbits in the Earth-Moon system lie between 0.629 P E M ,where the family connects with the Lyapunov orbits, and 0.647 P E M ,where the family connect with the Vertical orbits. There is a second symmetry-related branch not shown here which consists of these above orbits reflected across the x - z plane. Accordingly, the whole branch of orbits forms a loop as shown in Figure 4. The part of the bifurcation diagram where these orbits are found is labelled as “A”. Fig. 8.

We now turn our attention to the Vertical orbits, shown in Figure 9. In this family there are three bifurcation points beyond L1, indicated by the tubes labelled B(V,A ) , B(V,B F ) and B(V,3 ) . The tube B(V,A ) represents the bifurcation orbit to the family of Axial orbits. As can be seen, the Vertical orbits grow to encompass the Earth. The nearly-planar orbit B(V,B F ) corresponds to the bifurcation point B(V,B F ) in Figures 4-5 and represents the bifurcation to the family of Backflip orbits. The Vertical family terminates in a reverse period-doubling bifurcating orbit B(V,3 ) , corresponding to the point B(V,3) in Figure 5 , where the Vertical branch connects to a branch of planar solutions. The doubly-symmetric Vertical solutions are described in Bray and Goudas and Zagouras and Kazantzis 58

The second and third bifurcation points on the branch of Vertical solutions can be seen more clearly in Figure 5. Two legs of the branch of Backflip

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The family of Vertical orbits. The orbits at the three bifurcation points in the family beyond L1 are shown as thick curves. The upper left figure shows the bifurcating orbit B(V,A). The Axially Symmetric orbits shown in Figure 8 bifurcate from this orbit. The white sphere near the origin of this barycentric system represents the Earth and the white sphere on the positive z-axis represents the Moon, both drawn to scale. The orbits in the upper left figure correspond to the part of the bifurcation diagram labelled as “Vl”. The orbits in the upper right figure are found in the part of the bifurcation diagram labelled as V 2 ” . The upper right figure represents the collection of Vertical orbits from the bifurcation orbit B(V,A) up to the final bifurcation orbit B(V,3). The bifurcation orbit B(V,BF) encompasses the Earth and lies close to the z-y plane. The Backflip orbits shown in Figures 10 and 11 bifurcate from this orbit. The third bifurcating orbit B(V,3), corresponds to a reverse period-doubling. Fig. 9.

orbits, BFl and BF2, arise from the bifurcation orbit B(V,BF). The family of orbits corresponding to the BFl branch are shown in Figures 10 and 11. We refer t o the orbits on the BF1 and BF2 branches as “Backflip” orbits,

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named after the Backflip maneuvers described in Uphoff 5 6 . G6mez and Mondelo 32 also computed a family of backflip orbits for the Earth-Moon system, following a period-doubling bifurcation. Each Backflip orbit consists of two arcs, one above the Earth-Moon orbit plane and one below the orbit plane. The orbits on branch BF2 are reflections across the x - y plane of the orbits on branch BF1. We refer to the orbits on branch BFl as the Class 1 Backflip orbits, and those on branch BF2 as the Class 2 Backflip orbits. The two classes BF1 and BF2. are related through the symmetry z -+ -2. The Backflip orbits vary smoothly along the branch portions BF1 and BF2. However, it is useful to identify five “phases” along each of these branch portions. The first phase of the family of Class 1 Backflip orbits begins at the bifurcation from the Vertical family with both the Northern and Southern arcs near the Earth-Moon orbit plane. Thereafter, as the upper left figure in Figure 10 shows, the z-amplitude of the Northern arc increases to a maximum value close to R. Meanwhile the Southern arc remains near the Earth-Moon orbit plane and moves outward. In the second phase, depicted in the upper right figure of Figure 10, the z-amplitude decreases from the maximum to a local minimum value of 0.6465 R. In this second phase the Southern arc moves downward, away from the EarthMoon orbit plane. In the third phase, depicted in the upper left figure of Figure 11, the z-amplitude of the Northern arc again increases to a value close to R, while the z-amplitude of the Southern arc grows to 2.279 R. In the fourth phase, shown in the upper right figure of Figure 11, the Northern arc drops toward the Earth-Moon orbit plane, while the z-amplitude of the Southern arc decreases. In the final phase, shown in the bottom figure of Figure 11, the Northern arc remains near the x-y plane and extends further outward from the Earth, while the z-amplitude of the Southern arc again increases. The Backflip family terminates in a collision with the Moon. The transitions between the phases in the Backflip family correlates in part with changes in the Jacobi integral and the minimum distance to the Moon over the course of the branch. For example, near the juncture between phases 1 and 2, the minimum distance to the Moon reaches a local minimum value of 0.006 R = 2275.641 km. Near the juncture between phases 2 and 3, the Jacobi integral achieves a local maximum value of 2.53169. Near the juncture between phases 3 and 4, the Jacobi integral has an inflection point.

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Fig. 10. This figure and Figure 11 depict the changes in the Class 1 Backflip

orbits along branch BF1. The upper left figure shows the first phase of the Class 1 family. The family begins at the tube labelled B(V,BF),representing the bifurcation from the Vertical branch. Thereafter in phase 1 the northern arc of the Backflip orbits increases in amplitude in the positive z direction, until a maximum amplitude close to R is reached. At the same time the Southern arc remains nearly planar but increases in radius. The end of the first phase is shown by a thickened tube whose Northern and Southern arcs are labelled 1N and lS, respectively. However, this tube does not represent a bifurcation. The second phase of the family, shown in the upper right figure, begins where the fist phase in the upper left ends. As we follow the Backflip family further, the positive z amplitude of the Northern arc decreases to a value 0.6465 R. At the same time the Southern arc moves further downward, away from the orbit plane of the primaries. The thick tube in the upper right figure, with Northern and Southern arcs 2N and 2S, respectively, corresponds to the last orbit in the upper left figure.

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Fig. 11. Continuation of the branch of Class 1 Backflip orbits. In the upper left

figure, the third phase begins where the second phase in the upper right figure in Figure 10 ends. In the third phase the Northern arc increases in z-amplitude until a maximum amplitude close to R is achieved by the orbit with Northern and Southern arcs 3N and 3S, respectively. In this phase the Southern arcs extend southward until a negative z-amplitude of 2.279 R is reached at the cusp in arc 3s. In the fourth phase, the z-amplitude of the Northern arc decreases toward zero. Meanwhile the Southern arc remains far below the Earth-Moon orbit plane. The fourth phase ends with the orbit with Northern and Southern arcs labelled 4N and 4s. In the final phase, shown in the bottom figure of Figure 11, the Northern arc remains near the x - y plane and extends further outward from the Earth, while the z-amplitude of the Southern arc again increases. The Backflip family terminates in a collision with the Moon. The orbit with arcs 5N and 5 s is very close to a collision, passing within 0.56 km of the Moon.

Near the juncture of phases 4 and 5, the Jacobi integral achieves a local minimum value of -0.932750.

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The third bifurcation point from the branch of Vertical solutions, B(V,3), is a reverse period-doubling bifurcation. Because the bifurcation orbit B(V,3) is planar, we have drawn bifurcation point B(V,3) inside the gray rectangle. The families of periodic orbits we have described exhibit some interesting symmetries. For the N-body problem, the Mirror Theorem 5 2 states that an orbit is periodic if a mirror configuration occurs a t two distinct epochs. This is not a necessary condition for periodicity, but the Mirror Theorem has been a effective tool for computing periodic solutions near collinear libration points. Robin and Markellos 50 noted that in the CR3BP there are only two types of mirror configurations. In the “P” configuration, the orbit crosses the x - z plane orthogonally; in the “A” configuration, the orbit crosses the 5 axis orthogonally. Due to symmetries of the CR3BP, an orbit with a point in the P configuration is symmetric across the x - z plane, whereas an orbit with a point in the A configuration is axially symmetric. In Howell 35 and Howell and Campbell 38, periodic orbits were determined by a two-point boundary value problem where each boundary condition describes a P configuration. In the present study we do not exploit the Mirror Theorem in the definition of the boundary conditions. It happens that all of the solutions presented here possess points in the P configuration (the Halo and Backflip families), points in the A configuration (the Axial family) or both (the Vertical and Lyapunov families). However, as was shown in Doedel et al. l 4 and Paffenroth et al. 48, we can use AUTO2000 to detect families of periodic solutions that possess neither of these types of configurations.

6. Rotating and Inertial Coordinates

In Section 3 we defined the barycentric rotating coordinate system. It is convenient t o study the CR3BP in rotating coordinates because the equations of motion are time-independent in that frame, which simplifies the analysis. Moreover the rotating coordinate system is a convenient frame in which t o view orbits that remain near a libration point. However, for some trajectories in the three-body problem we can gain further insight by viewing the trajectory in an nonrotating frame as well as the rotating frame. We chose t o represent some orbits in an Earth Centered Inertial (ECI) frame in which the axes are aligned, at time t = 0, with the axes

512 D.J. Dichmann, E.J. Doedel and R.C. Paffenroth

of the barycentric rotating coordinate system. The angular velocity of the nondimensional rotating frame defined in Section 3 is 1, so the rotational period of the Earth-Moon system is PEM= 27~. An orbit that is periodic in rotating coordinates is, in general, not periodic in inertial coordinates. However, suppose the period P of a solution in rotating coordinates has an m : n resonance with the rotational period PEM;that is, PIPEM = m/n for some integers m and n. Then the trajectory in inertial coordinates is also periodic. (Cf. HQnon 34, Chapter 3.) Specifically, after n Earth-Moon orbit periods, the orbit will complete m periods in rotating coordinates and close in inertial coordinates. This is analogous to the observation that a periodic Keplerian orbit has a repeated ground track if the period in Earth-Centered Inertial coordinates is commensurate with the rotational period of the Earth.

7. Applications to Space Missions The families of periodic orbits described above offer a variety of possible applications to space missions. Due to the fixed positions of L1 and L2 along the line between the primaries in the rotating frame, the regions around these libration points provide excellent locations for scientific observation spacecraft and for communication relays "r2'. Since 1978, orbits near L1 and L2, especially Halo orbits, have been used for scientific missions and there are several future missions planned for these regions In this paper we focus instead on mission applications of orbits that travel far away from the line between the primaries. 24742751,

3,22723730731.

7.1. Lyapunov Orbits One of the important applications of three-body orbits is to explore geospace, i.e., the fields, plasmas and particles around the Earth Even within the limitations of planar orbits there are some important mission applications. 15716126155.

Farquhar and Dunham l8 examined the use of orbits with a series of lunar swingbys to observe the Earth's magnetic tail, whose axis is nearly along with the Earth-Sun line. The Earth-Sun line rotates with angular

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rate WSE = %IT/PSE in the inertial frame. When viewed in the EarthMoon rotating frame, the angular rate of the Earth-Sun line is WSEM = W E M - WSE = 0.9252, and the period of rotation is P , E M= ' h I T / W S E M = 1.0808 PEM = 6.7909. If an orbit has a period that is a multiple of PSEM, then the orbit is periodic in the Earth-Sun rotating frame. Using AUT02000, it is straightforward to set parameters that cause the software tool to produce output when it encounters a solution with a specified period. In the class of Lyapunov orbits emanating from L1, we found two Lyapunov orbits that have periods equal to PSEM.The smaller orbit extends only 0.18 lunar orbit radii beyond the Moon7sorbit, and so would be of limited value in exploring the geomagnetic tail. The larger orbit, depicted in Figure 12, extends well beyond the Moon's orbit. In inertial coordinates the orbit appears as a sequence of ellipses, where the line of apsides is rotated by the lunar swingbys at a rate equal to the rotational rate of the Earth-Sun line. If the initial orientation of line of apsides is chosen to lie along the Sun-Earth line, the orbit may be used to monitor the Earth's magnetic tail. Alternatively, if the line of apsides initially points toward the Sun then the orbit can be used to monitor the solar wind. This particular orbit is an Egorov class orbit 18. If the initial orientation of the line of apsides is chosen to lie along the Sun-Earth line then the orbit can be used to observed the Earth's geomagnetic tail or to observe the solar wind. Unfortunately, at closest approach to the Moon the orbit in Figure 12 would lie 68 km below the surface of the Moon, so the orbit is not useful for mission applications. In fact, Farquhar and Dunham considered the orbit illustrated in Figure 12, and showed that orbits with Double Lunar Swingbys (DLSs) are far more promising for the exploration of the geomagnetic tail 18. (See also Dunham and Davis l6 and Uesegi et al. 5 5 . ) Nevertheless, using AUTO2000 we were able to readily identify candidate orbits for observation of the geomagnetic tail. In this study we have only considered families of orbits emanating from the libration point L1 between the Earth and the Moon. If we were to perform continuation starting with another family of orbits, we should be able to find more periodic DLS orbits to follow the Sun-Earth line. Another area of application of three-body orbits is the observation of Coronal Mass Ejections (CMEs) from the Sun. SOH0 has been used for several years to observe CMEs. However, as we remarked above, Halo orbit missions have been restricted in size to simplify communications. As a

514 D.J. Dichmann, E.J. Doedel and R.C. Pafienroth

Fig. 12. An orbit that tracks the Sun. This Lyapunov orbit has a period equal the period of rotation of the Earth-Sun line in the rotating frame. In this orbit a series of lunar swingbys rotate the line of apsides of the orbit to follow the Sun-Earth line. In this, and in the following figures, we draw the orthogonal axes with length equal to the Moon’s orbit radius. We also draw a gray disk with radius equal to the Moon’s radius to indicate the Moon’s orbit plane. The orbit is displayed in rotating coordinates in the top figure and in ECI coordinates on the bottom left figure. In all figures drawn in inertial coordinates, we also draw the Moon’s orbit. The orbit has been propagated for three periods in the ECI frame to demonstrate the rotation of the line of apsides. At closest approach the orbit is 1670 km from the Moon’s center, or about 68 km beneath the Moon’s surface, making it useless for practical space missions. The figure at the bottom right shows the orbit in an Earth-centered frame that rotate with the Earth-Sun line, and the z-axis points toward the Moon at the initial time.

consequence, the Halo orbits currently in use remain far from the Sun. The Lyapunov orbits in the Sun-Earth system offer an alternative to the Halos as a location from which to observe the Sun-Earth interaction. As can be

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seen in Figure 6 , Lyapunov orbits can be found that pass much closer to the Sun, and extend a significant distance orthogonal to the Sun-Earth line in the x - y plane. However, to offer regular coverage near the Sun-Earth line it would be necessary to insert many spacecraft along the same Lyapunov orbit. Other, similar families of orbits that have been considered for this type of application are Distant Retrograde Orbits about the Earth and Earth Return Orbits emanating from L2 27928140,45,46.

7 . 2 . Vertical Orbits

For many space missions it is necessary to travel beyond the orbit plane of the primaries. For missions that observe fields, plasmas and particles, travelling beyond the ecliptic plane permits the spacecraft to sample a broader region of space. For telescopes that observe in the infrared range, it would be extremely valuable for the orbit to extend well beyond the ecliptic plane where the zodiacal dust is concentrated 33. The large Vertical orbits shown in Figure 9 may be useful for both of these purposes. Because orbits in the Vertical family provide a view of the poles of one or both primaries, they might be useful for polar science missions 25. For example, an orbit such as the one shown in Figure 13 curves over the Moon’s poles and so allows for extended observations of the Moon’s polar regions. An analogous orbit in the Earth-Sun system would allow for extended observations of the Earth’s poles.

A larger member of the Vertical family, such as the one shown in Figure 14, extends over the poles of the larger primary. From Figure 14 we see that this particular orbit, with a period equal to one Earth-Moon orbit period, is close to being a circular, polar orbit. An analogous orbit in the Sun-Earth system could be used to observe the Sun’s polar regions, and because its period equals the Earth-Moon orbit period it would return to the vicinity of the Earth once a year.

7.3. Axial Orbits The Axial orbits shown in Figure 8 form a connection between planar Lyapunov orbits and the Vertical orbits. Indeed each Axial orbit exhibits a combination of the characteristics of the Lyapunov and the Vertical fami-

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Fig. 13. An example of a Vertical orbit displayed in rotating coordinates. This particular orbit is the bifurcation point from which the Axial orbits in Figure 8 arise. This orbit has a period equal to 0.625 PEM.The z-amplitude of this orbit is 0.240 R = 92250 km. The figure on the right gives a close-up view of the orbit, where the small gray ball represents the Moon, drawn to scale. Because the orbit bends over the Moon’s poles, it offers opportunities to observe the lunar polar regions.

A second example of a vertical orbit displayed in rotating coordinates on the left and in ECI coordinates on the right. This orbit has a period equal to P E M , the Earth-Moon orbit period. The orbit in ECI coordinates is nearly planar and nearly polar.

Fig. 14.

lies. One could consider using an Axial orbit in the Sun-Earth system much as one might use a Lyapunov orbit to explore the region between the Sun and the Earth. However, the fact that the Axial orbits extend out of the

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orbit plane of the primaries adds an extra dimension to the observations a spacecraft could make. In this sense the Axial orbits are similar to the periodic Halo orbits and the quasiperiodic Lissajous orbits. However, the Axial orbits in the Sun-Earth system share a limitation as the Lyapunov orbits: They cross the z-axis periodically, so Sun interference would make communications between the spacecraft and Earth a challenge.

7.4. Backflip Orbits

The family of Backflip orbits shown in Figures 10 and 11 may offer special opportunities for the exploration of geospace. We call this family “Backflip” orbits because they are reminiscent of the Backflip maneuver described by Uphoff 5 6 .

Fig. 15. The Backflip bifurcation orbit. This is the same as the orbit B(V,BF) shown in Figure 10. The orbit is displayed here in rotating coordinates on the left and in ECI coordinates on the right. The orbit has been propagated for 4 Earth-Moon orbit periods in the ECI frame. This orbit is doubly symmetric in rotating coordinates. In ECI coordinates the orbit is very close to being a slightly inclined circular orbit whose radius is the same as the Moon’s orbit radius. The period of the orbit is 1.002 Earth-Moon orbit periods, so it nearly closes in the ECI frame.

The Backflip maneuver is an extension of the concept of a Double Lunar Swingby that Farquhar and Dunham described for planar orbits. A Backflip maneuver consists of a pair of lunar swingbys where the first swingby is designed to send a spacecraft, initially moving in the orbit plane of the

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primaries, far beyond that plane then back for a second encounter with the Moon. The second lunar swingby is designed to return the spacecraft trajectory to the orbit plane of the primaries. By using a Backflip maneuver, a spacecraft can sample the region beyond the primaries’ orbit plane and still remain in the vicinity of the Earth. The Wind mission employed several Backflip maneuvers for this purpose in the Earth-Moon system 27. A Backflip orbit then consists of a periodic sequence of Backflip maneuvers. In the Sun-Earth system, a Backflip orbit could also make it possible for a space telescope to reduce the effects of the zodiacal dust. The periods of the Backflip orbits range upward from 1.002 Earth-Moon orbit periods. The bifurcation orbit, where the family of Backflips arises from the Vertical family, is shown in Figure 15. We have extended the family up to its collision with the Moon’s surface where the period is 2.286 PEM. Figures 15- 22 show representative members of the Class 1 Backflip orbits from Figure 10 and 11 in both rotating and ECI coordinates. As we noted in Section 5, the branch of Class 1 Backflip orbits can be divided into five phases, where the z-amplitude of the Northern arc of the orbit first increases in phase 1 to a value close to the Moon’s orbit radius R,then decreases in phase 2 to a local minimum of 0.6465R,then increases in phase 3 to a value near R again, then decreases in phase 4 toward zero. In the fifth and final phase the Northern arc remains near the 2 - y plane and the orbits move outward from the Earth. The five phases are depicted in Figures 10 and 11. Figure 16 is a representative member of the first phase of Backflip orbits. Because the orbit in Figure 16 has a 4 : 3 resonance, the orbit is periodic in the inertial frame and closes after 3 Earth-Moon orbit periods. It is not clear that a resonant orbit has any practical advantages over nonresonant orbits. However, the simple, symmetric structure of this orbit in inertial coordinates makes it easier to comprehend. The orbit in Figure 17, which lies at the junction between phases 1 and 2, has some very interesting properties. This orbit has a 3 : 2 resonance with the Earth-Moon orbit period. When viewed in inertial coordinates, the orbit is nearly planar. One portion of the orbit is approximately a polar, semicircular arc with radius near the Moon’s orbit radius. As a consequence, that portion of the orbit is traced out in one-half Earth-Moon orbit period. The remainder of the orbit with 3 : 2 resonance consists of two segments

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Fig. 16. A Backflip orbit with 4 : 3 resonance. This orbit is part of the first phase of Class 1 Backflip orbits. The orbit is displayed in rotating coordinates in the top figure. The bottom figures show the orbit in ECI coordinates, with a skew view on the left and a projection into the z - y plane of the right. The orbit has been propagated for 3 Earth-Moon orbit periods in the ECI frame. This orbit has a period of 4/3 P E M . Therefore the orbit closes in the ECI frame after 3 Earth-Moon orbit periods. This orbit, like those shown in Figures 17- 22, lacks symmetry across the z - y plane.

beyond the Moon’s orbit that are close to being linear and radial. Each of these orbit segments is traced out in one-half Earth-Moon orbit period. The nearly circular segment is connected to each of the nearly linear segments by a lunar swingby. While the orbit has a period of 1.5 Earth-Moon orbit periods in the rotating frame, it actually requires 3 Earth-Moon orbit periods to complete one orbit in the inertial frame.

As we trace the family of Class 1 Backflips further (represented in Fig-

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Fig. 17. A Backflip orbit with 3 : 2 resonance. This orbit lies at the junction between the fist and second phases of the Class 1 Backflips, and corresponds to the orbit with Northern and Southern arcs 1 N and 1s shown in Figure 10. The orbit is displayed in rotating coordinates on the left and in ECI coordinates on the right. The orbit has been propagated for 2 Earth-Moon orbit periods in the ECI frame. This orbit has a period equal to 3/2 PEM,and closes after two Earth-Moon orbit periods. It is remarkable that this orbit is nearly planar.

ures 15- 22), the maximum z-amplitude initially decreases t o a value of 0.646R, the maximum radius in the plane of the primaries increases, and the minimum distance t o the Moon increases. As we trace the family further, the maximum z-amplitude increases again and the near-planar portion of the orbit becomes significantly nonplanar. An orbit such as the one in Figures 20-22 would allow a spacecraft t o explore space to a z-amplitude greater than 2R. As the family of Class 1 Backflips is followed further, the nonplanar portion of the orbit lies entirely below the orbit plane of the primaries. Figure 22 shows the shape of the Backflip orbits just before collision with the Moon’s surface.

8 . Conclusion

In this paper we have shown how the numerical continuation methods in AUTO2000 may be applied to the computation of periodic orbits in the Circular Restricted 3-Body Problem (CRSBP) . Pseudo-arclength continuation provides a robust method for computing periodic orbits which neither depends on any symmetry properties of the desired solutions nor upon any

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Fig. 18. A Backflip orbit with 7 : 4 resonance. This orbit lies in the second phase

of the Class 1 Backflips. The orbit is displayed in rotating coordinates in the top figure. The bottom figures show the orbit is ECI coordinates, with a skew view on the left and a projection into the z - y plane on the right.

expansions about the equilibria from which the solution arise. The general theory of numerical continuation does not immediately apply to conservative systems such as the CR3BP. However with the introduction of a unfolding parameter we can reformulate the problem to make the techniques in AUTO2000 applicable. Through the use of a Python script to drive AUTO2000 we are able to compute periodic orbits of arbitrary extent and follow all bifurcating branches. Our focus in this paper has been on the computation and mission applications of a subset of the branches that we have computed for the CR3BP. Other branches of the bifurcation diagram are explored in Doedel et al. l4 and Paffenroth et al. 48 and references cited therein, where it is shown that there is a rich variety of

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Fig. 19. A Backflip orbit with the positive z-amplitude at a local minimum. This

orbit lies at the junction between the second and third phases of the Class 1 Backflips, and corresponds to the orbits with Northern and Southern arcs 2N and 2 s in Figures 10-11. The orbit is displayed in rotating coordinates on the left and in ECI coordinates on the right. The orbit has been propagated for 3 EarthMoon orbit periods in the ECI frame. The period of the orbit is 1.902 P E M . Because the orbit does not have a simple resonance, it does not close in a few Earth-Moon orbit periods in the ECI frame.

periodic solutions emanating from the various libration points. Indeed, the families of periodic orbits form connections between the different libration points. Deprit and Henrard in particular described intricate connections between families of periodic orbits in the planar CRSBP. 8y’

It is common to compute periodic solutions of the CRSBP in a rotating coordinate system, where the dynamical system is autonomous. However to understand the nature and applicability of solutions that travel far from a libration point, it useful to represent the solutions in an Earth Centered Inertial frame. We have also used a feature in AUTO2000 that allows one to identify members of a family with particular periods. These tools have allowed us to explore the applicability to space missions of members of the Lyapunov, Vertical, Halo and Axially Symmetric branches from L1. We have paid particular attention to members of the family of “Backflip” orbits and their application to the exploration of geospace. In this study we have identified some physical quantities that correlate with the phase transitions. It would be enlightening to examine in greater detail the geometry of the close approach with the Moon, through an examination of the B-plane parameters l7 for example.

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A Backflip orbit where the positive z-amplitude reaches a maximum value very close to R. This orbit lies at the junction between the third and fourth phases of the Class 1 Backflip orbits, and corresponds to the orbits with Northern and Southern arcs 3N and 3s in Figure 11. The orbit is displayed in rotating coordinates on the left and in ECI coordinates on the right. The orbit has been propagated for 3 Earth-Moon orbit periods in the ECI frame. The period of the orbit is 1.992 P E M .The orbit nearly closes in the ECI frame after one EarthMoon orbit period.

Fig. 20.

We have not addressed stability issues in this paper, although AUTO2000 computes Floquet multipliers and eigenvectors. For example, we found that many Backflip orbits have at least one large Floquet multiplier, indicating instability. Because many Backflip orbits pass close to Moon, it is not surprising that they would be unstable. For a Backflip orbit to be viable for a spacecraft mission, it would be necessary to demonstrate that the orbit can be stabilized using an acceptable fuel budget. Floquet multipliers and eigenvectors can also be used in AUTO2000 to determine stable and unstable manifolds associated with an orbit. Recent work in libration point dynamics has focused on the invariant manifolds associated with the libration points and orbits about them (see, for example, Barden and Howell 3 , G6mez et al. 29 and Howell et al. 37). In future work it would be interesting to use these manifolds to determine efficient ways to insert a spacecraft into one of the orbits discussed in this work.

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Fig. 21. A Backflip orbit where the Northern arc lies approximately in the orbit

plane of the primaries. This orbit lies at the juncture of the fourth and fifth phases of Class 1 Backflip orbits, and corresponds to the orbits with Northern and Southern arcs 4N and 4s in Figure 11. The orbit is displayed in rotating coordinates on the left and in ECI coordinates on the right. The orbit has been propagated for 3 Earth-Moon orbit periods in the ECI frame. The Northern arc of the orbit lies very close to the Earth-Moon orbit plane, while the Southern arc extends far below that plane.

Fig. 22. A Backflip orbit near collision with the Moon. This orbit is part of the fifth phase of Class 1 Backflips, and corresponds to the orbits with Northern and Southern arcs 5N and 5s in Figure 11. The orbit is displayed in rotating coordinates on the left and in ECI coordinates on the right. The orbit has been propagated for 3 Earth-Moon orbit periods in the ECI frame. This orbit has a period equal to 2.286 PEM and passes within 0.56 km of the Moon’s surface.

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Acknowledgements DJD thanks Chauncey Uphoff, Peter Sharer and Heather Franz for introducing him to the principles and applications of the Backflip maneuver. This work was also inspired in part by the studies of “orbital acrobatics” by Dr. David Dunham and his colleagues. Thank you to Dr. Kent Bradford for his constructive comments. The work of EJD and RCP has been partially supported by NSF grant KDI/NCC SBR-9873173. EJD is also supported by NSERC Canada, Research Grant A4274. Our thanks to Christine Thorn for her invaluable assistance with the graphics.

References 1. E. L. Allgower and K. Georg. Numerical path following. In P. G. Ciarlet and 3. L. Lions, editors, Handbook of Numerical Analysis, volume 5. North Holland Publishing, 1996. 2. U. M. Ascher, J. Christiansen, and R. D. Russell. Collocation software for boundary value ODES. A C M Trans. Math. Software, 7:209-222, 1981. 3. B. T. Barden and K. C. Howell. Fundamental motions near collinear libration points and their transitions. J. Astronautical Sciences, 46(4):361-378, 1998. 4. T. A. Bray and C. L. Goudas. Doubly symmetric orbits about the collinear Lagrangian points. The Astronomical Journal, 72(2):202-213, 1967. 5. J. V. Breakwell and J. V. Brown. The “Halo” family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem. Celestial Mechanics, 20:389-404, 1979. 6. J. M. A. Danby. Fundamentals of Celestial Mechanics. Willmann-Bell, 1992. 7. A. Deprit and J. Henrard. A manifold of periodic orbits. In Z. Kopal, editor, Advances in Astronomy and Astrophysics, pages 1-124. Academic Press, 1968. 8. A. Deprit and J. Henrard. The Trojan manifold - Survey and conjectures. In G. E. 0. Giacaglia, editor, Periodic Orbits, Stability and Resonances, pages 1-18. D. Reidel Publishing Co., Dordrecht-Holland, 1970. 9. E. J. Doedel, D. G. Aronson, and H. G. Othmer. The dynamics of coupled current-biased Josephson junctions 11. Int. J. Bifurcation and Chaos, 1(1):5166, 1991. 10. E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. Sandstede, and X. J. Wang. AUT097: Continuation and bifurcation software for ordinary differential equations. Available via http://cmvl.cs.concordia.ca, 1997. 11. E. J. Doedel, H. B. Keller, and J. P. KernBvez. Numerical analysis and control of bifurcation problems: I. Int. J. Bifurcation and Chaos, 1(3):493-520, 1991.

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12. E. J. Doedel, H. B. Keller, and J. P. Kernbvez. Numerical analysis and control of bifurcation problems: 11. Int. J. Bifurcation and Chaos, 1(4):745-772, 1991. 13. E. J . Doedel, R. C. PafTenroth, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. E. Oldeman, B. Sandstede, and X. J. Wang. AUT02000: Continuation and bifurcation software for ordinary differential equations. Available via http://cmvl.cs.concordia.ca,2000. 14. E. J. Doedel, R. P. PafTenroth, H. B. Keller, D. J. Dichmann, J. GalinVioque, and A. Vanderbauwhede. Computation of periodic solutions of conservative systems with application to the 3-body problem. Int. J. Bifurcation and Chaos, 2002. Accepted. 15. D. W. Dunham and S. A. Davis. Catalog of Double Lunar Swingby orbits for exploring the earth’s geomagnetic tail. Technical report, October 1980. Computer Sciences Corporation, CSC/TM-80/6322. 16. D. W. Dunham and S. A. Davis. Optimization of a multiple lunarswingby trajectory sequence. J. Astronautical Sciences, 33(3):275-288, JulySeptember 1985. 17. L. Efron, D.K. Yeomans, and A.F. Schanzle. ISEE-3/ICE navigation analysis. J. Astronautical Sciences, 33(3):301-323, July-September 1985. 18. R. Farquhar and D. Dunham. A new trajectory concept for exploring the Earth’s geomagnetic tail. J. Guidance and Control, 4(2):192-196, MarchApril 1981. 19. R. W. Farquhar. The flight of ISEE-3/ICE: Origins, mission history and a legacy. J. Astronautical Sciences, 49(1):23-73, 2001. 20. R. W. Farquhar and D. W. Dunham. Use of libration points for space observatories. In Observatories in Earth Orbit and Beyond, pages 391-395. Kluwer Academic Publishers, 1990. 21. R. W. Farquhar and A. K. Kamel. Quasi-periodic orbits about the translunar libration point. Celestial Mechanics, 7:458-473, 1973. 22. F. Felici, M. Hechler, and F. Vanderbussche. The ESA astronomy missions at L2: FIRST and Planck. J. Astronautical Sciences, 49(1):185-196, 2001. 23. D. Folta, S. Cooley, and K. Howell. Trajectory design strategies for the NGST L2 libration point mission. In A A S / A I A A Space Flight Mechanics Meeting, 2001. AAS 01-205. 24. D. Folta and K. Richon. Libration orbit mission design at L2: A MAP and NGST perspective. In A I A A / A A S Astrodynamics Specialist Conference, 1998. AIAA 98-4469. 25. D. Folta, C. Young, and A. Ross. Unique non-Keplerian orbit vantage locations for Sun-Earth connections and Earth Science Vision roadmaps. In N A S A Goddard Flight Dynamics Symposium, 2001. 26. D. C. Folta and S. L. Sauer. ISEE-3 trajectory control utilizing multiple lunar swingbys. In A A S / A I A A Space Flight Mechanics Meeting, August 1984. AIAA 84-1979. 27. H. Franz. Wind lunar backflip and Distant Prograde Orbit implementation. In A A S / A I A A Space Flight Mechanics Meeting, 2001. AAS 01-173. 28. H. Franz. Design of Earth Return Orbits for the Wind mission. In A A S / A I A A Space Flight Mechanics Meeting, 2002. AAS 02-170.

The Computation of Periodic Solutions of the %Body Problem Using A U T O 527 29. G. Gbmez, K. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont, and S. D. Ross. Invariant manifolds, the spatial three-body problem and space mission design. In A A S / A I A A Astrodynamics Specialist Conference, 2001. AAS 01301. 30. G. Gbmez, M. W. Lo, J. Masdemont, and K. Museth. Simulation of formation flight near Lagrange points for the TPF mission. In A A S / A I A A Astrodynamics Specialist Conference, 2001. AAS 01-305. 31. G. Gbmez, J. Masdemont, and C. Simb. Lissajous orbits around halo orbits. In A A S / A I A A Space Flight Mechanics Meeting, 1997. AAS 97-106. 32. G. Gbmez and J.M. Mondelo. The dynamics around the collinear equilibrium points of the RTBP. Physica D, 157:283-321, 2001. 33. P. Gurfil and N.J. Kasdin. Optimal out-of-the-ecliptic trajectories for spaceborne observatories. In A A S / A I A A Space Flight Mechanics Meeting, 2001. AAS 01-162. 34. M. Hknon. Generating Families i n the Restricted Problem. Springer-Verlag, 1997. 35. K. C. Howell. Three-dimensional, periodic, 'Halo' orbits. Celestial Mechanics, 32:53-71, 1984. 36. K. C. Howell. Families of orbits in the vicinity of the collinear libration points. In A I A A / A A S Astrodynamics Specialist Conference, 1998. AAS 98-4465. 37. K. C. Howell, B. T. Barden, and M. W. Lo. Applications of dynamical systems theory to trajectory design for a libration point mission. J. Astronautical Sciences, 45(2):161-178, 1997. 38. K. C. Howell and E. T. Campbell. Three-dimensional periodic solutions that bifurcate from Halo families in the circular restricted three-body problem. In Spaceflight Mechanics, 1999. AAS 99-161. 39. S. Ichtiaroglou and M. Michalodimitrakis. Three-body problem: The existence of families of three-dimensional periodic orbits. Astronomy and Astrophysics, 81:30-32, 1980. 40. J. A. Kechichian, E. T. Campbell, M. F. Werner, and E. Y. Robinson. Solar surveillance zone population strategies with picosatellites using Halo and Distant Retrograde Orbits. In International Conference on Libration Point Orbits and Applications, Aiguablava, Spain, 2002. 41. H. B. Keller. Numerical solution of bifurcation and nonlinear eigenvalue problems. In P. H. Rabinowitz, editor, Applications of Bifurcation Theory, pages 359-384. Academic Press, 1977. 42. M. W. LO, B. Williams, W. Bollman, D. Han, Y. Hahn, J. Bell, E. Hirst, R. Corwin, P. Hong, K. Howell, B. Barden, and R. Wilson. Genesis mission design. J. Astronautical Sciences, 49(1):169-184, 2001. 43. K. R. Meyer. Periodic Solutions of the N-Body Problem. Springer Verlag, 1999. 44. F. J. Muiioz-Almaraz, E. Fkeire, E. J. Doedel, A. Vanderbauwhede, and J. GalBn. Continuation of periodic orbits in conservative and Hamiltonian systems. In preparation. 45. C. Ocampo. Trajectory Optimization for Distant Earth Satellites and Satellite Constellations. PhD thesis, University of Colorado, 1996.

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46. C. Ocampo and G. W. Rossborough. Transfer trajectories for Distant Retrograde Orbiters of the Earth. In A A S / A I A A Space Flight Mechanics Meeting, 1993. AAS 93-180. 47. R. C. Paf€enroth and E. J. Doedel. The AUTO2000 command line user interface. In Proceedings of the 9th International Python Conference, pages 233-241, March 2001. 48. R. C. PafTenroth, E, J. Doedel, and D. J. Dichmann. Continuation of periodic orbits around Lagrange points and AUT02000. In A A S / A I A A Astrodynamics Specialist Conference, 2001. AAS 01-303. 49. W. C. Rheinboldt. Numerical analysis of parametrized nonlinear equations. Wiley-Interscience, 1986. University of Arkansas Lecture Notes in the Mathematical Sciences. 50. I. A. Robin and V. V. Markellos. Numerical determination of the threedimensional periodic orbits generated from vertical self-resonant satellite orbits. Celestial Mechanics, 21:395-434, 1980. 51. J. Rodriguez-Canabal and M. Hechler. Orbital aspects of the SOH0 mission design. In Orbital Mechanics and Mission Design, volume 69 of Advances in the Astronautical Sciences, pages 347-356, 1989. AAS 89-171. 52. A. E. Roy. Orbital Motion. Adam Hilger, 1988. 53. R. D. Russell and J. Christiansen. Adaptive mesh selection strategies for solving boundary value problems. SIAM J. Numer. Anal., 15:59-80, 1978. 54. V. Szebehely. Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, 1967. 55. K. Uesegi, J. Kawaguchi, S. Ishii, N. Ishii, M. Kimura, and K. Tanaka. Design of Double Lunar Swingby orbits for MUSES-A and GEOTAIL. In A A S / A I A A Space Flight Mechanics Meeting, volume 69 of Advances in the Astronautical Sciences, August 1984. AAS 89-169. 56. C. W. Uphoff. The art and science of lunar gravity assist. In Orbital Mechanics and Mission Design, volume 69 of Advances in the Astronautical Sciences, pages 333-346, 1989. AAS 89-170. 57. J. Wertz, editor. Spacecraft Attitude Dynamics and Control. D. Reidel, 1981. 58. C. G. Zagouras and P. G. Kazantzis. Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points. Astrophysics and Space Science, 61:389-409, 1979.

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

AN ARCHITECTURE FOR A GENERALIZED SPACECRAFT TRAJECTORY DESIGN AND OPTIMIZATION SYSTEM C . OCAMPO Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas, USA 78712

The elements of a general high precision system for trajectory design and o p timization for single or multiple spacecraft using one or more distinct propulsion systems, and operating in any gravitational environment within the solar system are discussed. The system architecture attempts to consolidate most all spacecraft trajectory design and optimization problems by using a single framework that requires solutions to either a system of nonlinear equations or a parameter optimization problem with general equality and/or inequality constraints. The use of multiple reference frames that generally translate, rotate, and pulsate between two arbitrary celestial bodies facilitates the analysis of multiple celestial body force field trajectories such as those associated with libration point missions, cycling trajectories between any set of celestial bodies, or any other type of trajectory or mission requiring the use of multiple celestial bodies. A basic trajectory building block, referred t o as the basic segment, that can accommodate impulsive maneuvers, maneuver and non-maneuver based mass discontinuities, and finite burn or finite control acceleration maneuvers, is used to construct single or multiple spacecraft trajectories. The system architecture facilitates the modeling and optimization of a large range of problems ranging from single spacecraft trajectory design around a single celestial body to complex missions using multiple spacecraft, multiple propulsion systems, and operating in multiple celestial body force fields.

1. Introduction

Spacecraft trajectory optimization is a field that has received considerable attention over the last several decades. The field continues to evolve as 529

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a result of innovations in trajectory dynamics associated with spacecraft utilizing the simultaneous gravitational attraction of two or more celestial bodies where the contribution from any of the celestial bodies influences significantly the motion of the spacecraft Further advancements have been made in both the analytical and numerical based solution methods required to solve these types of problems. The most recent developments in purely analytical solutions to low thrust orbit transfer problems have been given by Azimov and Bishop and a recent numerical based solution method capable of solving complex low thrust multiple body gravity trajectories has recently been documented by Whiffen and Sims 3 .

’.

Fundamental results in trajectory optimization are based in part on original research due to Lawden 4 . Lawden introduced the Primer Vector and its use in the optimization of space trajectories is known as Primer Vector Theory. The Primer Vector and its applications form an integral part of the current system but the specific details associated with the optimization of both impulsive and finite burn maneuvers are not discussed in this article. Associated with the theory of optimal space trajectories, are the numerical methods required to solve spacecraft trajectory problems in force fields where even closed form solutions of the uncontrolled trajectories are not available. A comprehensive survey on the computational issues applied specifically to the spacecraft trajectory optimization problem is given by Betts 5 . A clear exposition on the conversion of optimal control problems into sub-optimal parameter optimization problems whose solutions require nonlinear programming is given by Hull 6 . The focus of the current article is to describe a general approach that draws upon pertinent aspects of trajectory design and optimization theory. The effort is an attempt to present a single framework in which one or more spacecraft operating in a force field environment under the mutual attraction of one or more celestial bodies, and using one or more propulsion systems can be analyzed and solved efficiently. The framework for this architecture is currently implemented in a prototype trajectory design and optimization system called COPERNICUS that is under development at the University of Texas. The term ‘architecture’ in the title of this article refers to the structure and methodology of the system. The architecture is defined by its basic

A n Architecture for Spacecraft Rajectory Design and Optimization System 531

components which include the force models, the coordinate frames, the numerical methods, and the method used to model the trajectories. The term ‘generalized’ implies that the system is designed to handle many classes of problems involving common gravitational force fields and acceleration models, different sets of boundary conditions, and various types of propulsion systems without resorting to specific procedures or algorithms to solve each type of problem separately with unique and distinct methods. Common in the spacecraft trajectory optimization literature is the description and implementation of specific methods and algorithms to solve specific problems. Though this is entirely valid, the current architecture attempts to unify the approach to these same problems under a single framework. The terms ‘trajectory design’ refer to the process of generating nominal and feasible solutions that satisfy a predetermined set of constraints and boundary conditions without considering the optimization of any aspect of such a solution. The term ‘optimization’ refers to the process of generating a trajectory design solution that extremizes some general scalar quantity of the solution, regardless of how complex it may be, provided that it can be determined or computed deterministically. In a spacecraft trajectory optimization problem, the cost functions typically considered include, minimization of the total impulse required, minimization of the total transfer times, minimization of propellent used, or maximization of the final spacecraft mass. However, other allowable cost functions may include the minimization of the hyperbolic excess velocity relative to an arrival or flyby celestial body or the minimization of the value of the Jacobi constant of a spacecraft in a circular restricted three body system. Important here is that the cost function should be allowed to take on any value that is of interest to a mission, provided that it can be uniquely computed from the variables and system parameters used to model the problem. The current prototype of the system uses explicit numerical integration for state propagation since the force fields encountered in these problems are in general nonlinear and non-autonomous. The solutions to the different classes of problems are obtained as solutions to either a system of nonlinear equations or to a nonlinear constrained parameter optimization problem. Efficient gradient based nonlinear root finding algorithms and sequential quadratic programming algorithms are used to obtain these solutions.

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All spacecraft trajectory problems can be formulated as either multipoint boundary value problems or as nonlinear programming problems that explicitly extremize single or multi-objective cost functions. In the case of a multi-point boundary value problem formulation, two cases exist. First, feasible trajectory solutions with no implicit optimization satisfy only the boundary conditions associated with the kinematics or the physical variables of the problem such as the conditions related to the physical state of the spacecraft or the physical parameters of the force model, A simple example of this includes the central body, orbital two point boundary value problem that satisfies Lambed’s theorem ’. Here, a nonlinear search is required for the three components of the velocity vector that connects two distinct position vectors in a given flight time. However, there exist other types of targeting involving more complex boundary conditions that are general functions of the state variables. An example of one of these is to find the launch injection conditions from any given Earth centered parking orbit so that the trajectory terminates in a captured orbit about the Moon. This is the classic ballistic lunar capture trajectory first examined by Belbruno ’. For this problem, a common parameterization of the launch conditions is given by the hyperbolic excess velocity magnitude, v,, and the right ascension and declination of the outgoing asymptote, a, and ,,S respectively. The flight time, At, is a free parameter, and one of the boundary conditions for a successful capture at the Moon is that the value of the Jacobi Constant of the spacecraft’s state measured with respect to the Earth-Moon system be between the Jacobi Constant values associated with the interior and exterior libration points that are near the Moon. The other boundary condition is that the radial distance between the spacecraft and the moon be less than the radius of the Hill sphere around the Moon at the time of the capture. The problem is further complicated by the fact the gravitational attraction of the Sun is required for this solution to exist. This problem has four unknowns (v,, a,, S,, At) but three boundary conditions given as inequalities. It is a targeting problem, but much more complex than the orbital two point boundary value problem, that can still be solved as a system of nonlinear equations. The second case of a multi-point boundary value problem formulation results if optimal control theory is used to formulate the multi-point boundary value conditions associated with a specific problem. Here the solutions produced satisfy both the kinematic boundary conditions and the natural boundary conditions, commonly referred to as the transversality conditions,

An Architecture for Spacecrafi "hjectory Design and Optimization System 533

that are part of the first order necessary conditions for a solution of the optimal control problem. This case implicitly extremizes a scalar cost function given in either the Mayer, Lagrange, or Bolza forms and is referred to as an indirect method. The nonlinear constrained parameter optimization problem directly extremizes a scalar cost function based on a variable parameter vector that can include parameters of the model and/or all other variables defining the states and dynamics of the spacecraft. A hybrid formulation is also possible where the search variables associated with the optimal control formulation of the multi-point boundary value problem augment the parameter vector of the parameter optimization problem. This eliminates the necessity of deriving and implementing the transversality conditions as constraint conditions associated with the optimal control problem. Having stated and described a class of solution methods that can solve these problems, significant importance is placed on the modeling of the problems, the proper choice of coordinate frames for state definition and targeting, and the proper identification of the independent and dependent variables. Though there are multiple ways in which a problem can be modelled, there are some that have better convergence properties than others for a given choice of solution method. A well designed system should facilitate, via experimentation if needed, the modelling of many types of problems and the generation of the solution procedures required to solve them. A solution procedure entails identifying a single stage or multi-stage procedure that drives an initial estimate of a solution to convergence.

It is noted that the trajectory problems solved by this system, whether optimal in some sense or not, are referred to as 'open loop' solutions in contrast to the guidance or stabilization problems encountered in many control problems. These solutions will eventually have to be controlled with closed loop feedback systems because in a real operational implementation of the solutions, unmodelled perturbations, uncertainties in the state of the spacecraft, and maneuver execution errors will be present. Fundamental to the general methodology presented here is the use of a basic element referred to as the basic segment that is composed of several basic entities. The basic segment is the basic building block from which all trajectories are constructed. Fundamentally, the basic segment is an arc that connects two endpoint nodes. The segment can accommodate velocity

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and mass discontinuities at either or both of the endpoint nodes. The velocity discontinuities result from either an impulsive maneuver produced by an engine or from an approximation to a gravity assist flyby about a given celestial body. However, whether the gravity assist is approximated by an impulse change in the velocity vector or whether it is a fully integrated trajectory arc about a celestial body is a choice allowed in the definition of the segment. The mass discontinuities at either node result from the mass depletion associated with an impulsive maneuver, or the mass changes that result when part of a spacecraft is discarded or added. This results when an engine stage is discarded or when a spacecraft is captured after a rendezvous. The arc connecting the end point nodes can be either a ballistic arc or a controlled arc if a finite burn engine or any other device (such as a solar sail) is used to produce an independent acceleration to the spacecraft. A ballistic arc is the same as a coast arc. Each segment can be altered to form several segment types. For example, a segment can be defined to be an impulse followed by a coast period, or a finite burn arc without the endpoint impulses, or simply a node point defined with an epoch, a position, and a velocity. Inherent to the system is the capability to work with disconnected segments. A segment is disconnected to any other segment if any of the state variables (position, velocity, or mass) at either node of one segment is discontinuous with respect to any of the state variables at the nodes of the other segment. Common sets of disconnected segments are those that have position discontinuities. This allows multiple spacecraft missions to be studied because additional spacecraft are modelled as segments that may be dynamically uncoupled from the segments used to model the trajectories of the other spacecraft. If some of the spacecraft are required to interact with other spacecraft, the node points of the segments that represent them are coupled via suitable boundary conditions to examine intercept, rendezvous, constrained formation flight, or any other type of problem that requires constraining the dynamic state of any spacecraft with respect to other spacecraft.

A similar method of using disconnected segments was used by Byrnes and Bright to examine complex impulsive maneuver based multiple body flyby trajectories to connect and optimize initially disconnected segments similar in concept to those described here. Though their definition of a segment and how it is propagated is different than the one presented here, the

A n Architecture for Spacecruj? %jectory

Design and Optimization System 535

concept is similar in that a solution is achieved by simultaneously minimizing the total sum of the impulsive maneuvers and achieving continuity in position, and optionally the velocity vectors, at initially disconnected node points that they referred to as ‘breakpoints’. Their method was able to robustly produce solutions to the Galileo tour of the Jovian system and the heliocentric multi-planetary flyby trajectory for the Cassini mission. The principal goal of current the system and architecture is to solve complex problems in a standard way without the need to develop specific models and algorithms for specific problems. If designed and implemented correctly, a general trajectory design and optimization system should be the model of choice for any specific problem, regardless of its complexity. Such a system should be capable of solving practical trajectory design and optimization problems using multiple propulsion systems, multiple spacecraft, and multiple celestial bodies and with any set of measurable perturbing accelerations in the force field. Examples of the type and scope of the problems that can be solved by the system include:

(1) Trajectories about a single celestial body for orbit transfers, rendezvous, intercepts, arrival and capture, departure and escape; (2) Transfer and return trajectories between any pair of celestial bodies that orbit each other or that are in orbit about another celestial body; (3) Trajectories associated with the libration points of any two celestial body system, including transfer trajectories to and between libration points or libration point orbits; or libration point orbit trajectory design; (4) Sample return missions including descent and ascent at the target celestial body including any necessary rendezvous maneuvers; (5) Ballistic, low energy, or low thrust cycler trajectories between any pair of celestial bodies; ( 6 ) Multiple body gravity assisted trajectories in the Solar System or any central body with one or more natural satellites using any combination of impulsive and/or low thrust maneuvers; (7) Ballistic or controlled low energy capture and escape from any celestial body using the direct influence of other celestial bodies.

A secondary goal of this system is to efficiently produce an accurate solution to a problem with minimal effort. Minimal effort is defined to

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be a level of effort required to generate a solution based on a predefined procedure to model and solve a particular problem. Producing a predefined procedure for a given problem, i.e., determining how many segments should be used, what are the independent and dependent variables, etc., can be a lengthy process. However, once this process is shown to achieve convergence from a wide range of initial estimates, then it becomes an automated process facilitating the solution to similar problems with different parameters. Complex problems may require a multi-stage approach, where sub-problems are solved independently and then combined in an overall solution. A well designed system should at least generate solutions to the subproblems efficiently and attempt to solve the complex problem.

A general approach to the problem facilitates the solution and optimization of trajectories for many types of missions in any force field encountered in most spacecraft trajectory problems. Though it is desired that this architecture solve all spacecraft trajectory problems foreseen in the next several decades, problems may be posed that are not solvable with the current architecture without further analysis, development, and generalization. Common to all of these problems is the requirement to have available an initial estimate of a solution that leads to convergence. This initial estimate becomes increasingly difficult to produce as the problem increases in complexity. This initial estimate has often been referred to as the ‘first guess’, however, this term will be avoided, because it should not be a ‘guess’. There is a lot of information available in a problem that can be used to construct a first estimate. Without going into details, the analytical solutions or approximations for simple force models and the dynamical systems based analysis currently being developed for more complex models serve as a basis for the construction of these estimates. However, convergence to a solution from even a well founded initial estimate is not guaranteed, given the complexity and scope of the problems that can be considered. Several unresolved issues remain that need to be addressed eventually if the system described here is to achieve some of the goals stated. First, it will be necessary to explore simplifications in the construction of initial estimates to most or all of the problems attempted. Secondly, if a comprehensive spacecraft dynamics system is to be produced, it will be necessary to incorporate a six degree of freedom spacecraft model to account for the attitude reorientation maneuvers required to properly align the spacecraft

An Architecture for Spacecmfi 'hjectory Design and Optimization System 537

to perform the needed maneuvers. The spacecraft model considered in the current system is still restricted to be a three degree of freedom model. Thirdly, if a detailed spacecraft operations system is to be produced, it will be necessary to include the observability and navigation accuracy of the trajectory solutions as part of the cost function. And fourthly, for close proximity operations between multiple spacecraft such as formation flight, it will be necessary to add a general relative motion frame model to examine multi-spacecraft trajectory problems about arbitrary trajectories, flying in any force field environment. Notation: All scalar quantities are typeset as italicized uppercase or lowercase, i.e., a and B are scalars. Vectors are typeset as bold lower case, i.e., a and b are vectors. Matrices are typeset as bold uppercase, i.e., A and B are matrices. The definition and dimensions of these variables or constants are context dependent and appear in the text where appropriate. Vectors are column vectors, so that if a is an n vector and b is an m vector and a is a function of b, then da/db is an n x m matrix. Dots above any quantity represent differentiation with respect to time. Superscript and subscript symbols are used to further distinguish the meaning of a given quantity; the definition of these are given in the text where appropriate.

2. Trajectory Design and Optimization Architecture

This section describes the basic elements of the system. This includes the formal definitions of the segments, trajectories, missions, force fields, and the coordinate systems.

2.1. Definition of the Basic Segment The system makes use of what is termed the basic segment. It is a trajectory arc that connects two node points. The arc connecting the two node points is a solution to the equations of motion which are propagated by whatever means necessary. The force field models are generally smooth, highly nonlinear, and time dependent, thus requiring numerical based solutions of the equations of motion. A trajectory is composed of one or more segments. A complete mission is composed of one or more trajectories associated with one or more spacecraft. A spacecraft is any object in the model that is not

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a celestial body and does not influence the motion of any other object in the system such as another spacecraft or a celestial body. It is the object whose state at single or multiple times is being determined or controlled. A celestial body is any object that can influence the motion of a spacecraft via its gravitational potential or emitted radiation in the equations of motion. Celestial bodies also influence the motion of other celestial bodies in the system. This influence is directly available from a pre-computed ephemeris of the celestial bodies in the model. It is assumed that the motion of the celestial bodies is known, either from a realistic ephemeris or from an analytical approximation to a real ephemeris; i.e., the system only propagates the motion of one or all of the spacecraft in a mission. The node points are tagged with an epoch that is referenced to a specified, but otherwise arbitrary, reference epoch, denoted as tepoch. The epoch of the initial node point of any segment i is th and the epoch of the final node point is t ) . The superscript ‘i’ denotes the segment number. A mission can have any number of segments so that i = 1,..,n. There is no restriction on the values that ti and t ) can have, i.e.,

ti = t )

or

t;l > t;

or

t;l < t;

However, the values of th and t ) can be constrained in any way necessary during the solution process, i.e., both quantities can be independent if desired or functionally dependent with respect to each other or the time epochs of the node endpoints of any other segment. For example, the time of flight for a segment i can be constrained to be less than or equal to a specified time of flight, or the time of flight of any other segment k; i.e., dti = t ) - td 5 dtk.Forward time or backward time propagations are handled in the same way and the temporal direction is only determined by the specific values of ti and t ). If to # t f , all segment propagations are from to to t f ,regardless of their relative values. No propagation is made if to = t f . The state of the spacecraft at either node point is given by its position, and mass, mti,f,. The subscript (0, f) on each of these quantities denotes that the quantity is referenced to either the to node or the t f node. The superscript ‘-’ on v;sf) and states that the value of the velocity and the mass, respectively, is given prior to any possible discontinuities in their values.

rkJ,f, velocity v&,,

There are two types of velocity impulses allowed at either node. The first type is a maneuver based impulse provided by an engine and the sec-

An Architecture for Spacecraft h j e c t o r y Design and Optimization System 539

ond type is a gravity assist impulse. The gravity assist impulse is used to approximate the change in velocity relative to a fixed external reference frame not attached to the celestial body providing the gravity assist. It is useful only when solving a problem where the flyby celestial body is treated as a zero-point mass. The components of the gravity assist impulse are constrained to satisfy the conservation of energy across the flyby and optionally, a minimum flyby radius relative to the central body. A fully integrated flyby of a celestial body without any associated discontinuity in the velocity is also allowed, but this flyby is modelled as a numerically integrated segment with a non-zero time of flight duration. Thus two ways to model a gravity assist are available, with one of them being an approximation. The approximation can be used for broad searches that may include multiple flybys and accuracy is not critical. The integrated flybys are used for more accurate trajectories. Both types of velocity impulses are treated the same way, except for the mandatory constraint imposed on the gravity assist impulse and the fact that the gravity assist impulse does not have an associated mass depletion. To model a ‘powered’ gravity assist flyby where in addition to the gravity assist impulse there is an additional maneuver based impulse, two segments are used. One of the segments can be a simple node with an impulse representing the gravity assist, and the other segment has an impulse representing the maneuver. If the epochs for both segments are the same, the order of the segments is not important. The impulse at either node can have zero magnitude. After the impulse, the velocity vector is

where the superscript

‘+’ specifies the value after the impulse.

The evolution of the mass value across a node depends on three allowable and distinct mass discontinuities. The first mass discontinuity is a nonmaneuver mass discontinuity that can result from either a mass drop off or a mass add on, so it can be positive or negative and is labeled as Am&. In other words, a spacecraft component can be discarded prior to the impulse. Or, if the spacecraft has performed a rendezvous with another spacecraft and has captured it, a positive mass discontinuity represents that additional mass associated with this capture. The mass value after this first non-

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maneuver mass discontinuity is

where Am& f ) is the value of the mass change. Proceeding with the impulse, the mass value after the maneuver is ,i+

-

- -i,

(0,f)-

+

(0,f)+ Amfo?f)

(3)

where Am!o,f, is the mass change that results from the instantaneous depletion of propellant that results from a maneuver based impulse. It is zero for a gravity assist impulse. The maneuver based mass change is directly , ~ ,the , exhaust velocrelated to the magnitude of the maneuver, A V ~ ~and ity of the engine used to provide this maneuver, cia,f). The maneuver mass discontinuity is obtained from a form of the rocket equation, - ,i-

A40,f) -

+(e-Av/c (0,f)

- 1)

(4)

where the superscripts and subscripts on Av and c have been omitted for notational simplicity. The exhaust velocity by definition is related to the specific impulse of the propellent used and the reference gravity acceleration value at the Earth's surface, c ! ~ , ~=) (geartj,). Following the impulse, another non-maneuver mass discontinuity with the same characteristics as the one prior to the impulsive maneuver is allowed. The mass value at the end of either node is then

where A m $ ,f) is the post impulse, non-maneuver mass discontinuity. Typically this mass change results when the engine stage used to produce the impulsive maneuver is discarded. However, allowance is made so that a mass add on can occur at this point again for reasons associated with a spacecraft capture. The reason non-maneuver mass discontinuities are allowed on either side of the impulse at either node is because in the case of a spacecraft rendezvous and capture, the maneuver performing the rendezvous could occur at the initial node point of a given segment, or at the final node point of a previous segment to which the given segment is connected to. Based on the evolution of the velocity and mass across a node point, the following distinct times labels that are equal in value but are used to

An Architecture for Spacecmft lkajectory Design and Optimization System 541

distinguish the values of both the velocity and the mass are node initial time, and time prior to any velocity or mass discontinuities t& : time after first non-maneuver mass discontinuity t&,n : time after the velocity impulse : node final time, and time after the second non-maneuver mass “>f) discontinuity

tG,f,

:

The arc that connects both node points can be either a ballistic arc with no independent control, or a controlled arc with thrust or acceleration controls. The control is provided by a thrust vector from an engine or a controlled acceleration that results from a non-mass depleting device such as a sail using radiation from a photon emitting celestial body such as the Sun or a star, or any other external momentum transfer device. Any system used to provide this control will be referred to as a propulsion system, though some may not use propellant such as in the case of a sail. The arc connecting the to node to the t f node of a segment i is defined by both the parameters of the propulsion system and the equations of motion. The control vector, I’(t),will in general have the following functional dependence,

where c is the exhaust velocity, T is the thrust, P i s the power, E is the efficiency of the propulsion system, u is the control direction unit vector, m is the instantaneous mass, and ap is a vector containing any other parameter that defines the propulsion system. For example, in the case of a sail, ap will contain parameters such as sail area, surface reflectivity, and other parameters that uniquely define it.

For any segment i, the dynamic state of the spacecraft along the arc for times between tt+ and t;- is defined as an augmented state vector comprised T

of its position, velocity, and mass ( ri(t)vi(t)m i ( t ) ) and satisfies the first order vector equation of motion, a

(i)= (

v(t)

g(r,v,m, t ,ag) + r ( t )

+

-T(t)/c(t)

m p

(7)

)Z

Here, g(r,v,m,t ,ag)is the acceleration per unit mass resulting from control independent terms such as the acceleration due to the gravitational

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potential from any celestial body, radiation pressure from energy emitting celestial bodies, or nonconservative forces such as atmospheric drag. The symbol ‘g’ commonly has been used to represent the gravitational acceleration due to gravitating celestial bodies with a dependence only on position and time. Here it is generalized to also include a dependence on velocity, v, mass, m, and an additional problem specific acceleration, as . The acceleration vector ag in the acceleration vector contains all of the constant or variable parameters of the force model such as the gravitational constant, the mass of the celestial bodies, the non-spherical gravitational potential field of the celestial body, the atmospheric parameters of the celestial body, the radiation parameters of any energy emitting celestial body, and any other term that may appear in the ballistic acceleration of the spacecraft. This acceleration may be time dependent. The time dependency of the force field results in part from the motion of the celestial bodies in the model or the rotation of a body about an internal axis if the non spherical is a gravitational potential of a celestial body is used. The mass rate, mi, result of fuel consumed during an engine burn and is related to the thrust, T, and exhaust velocity c, or any other arbitrary mass depletion that results from intentional or unintentional continuous venting of liquids or gasses, or mass accumulation during flight through resisting a medium such as dust clouds or atmospheres. The non-thrust contribution to mi is given by the general term h,. In summary, the spacecraft equations of motion between node points at tb and t ) are completely general and arbitrary, but known.

For any segment, the parameters for three propulsion systems need to be specified: one for the initial impulsive maneuver, one for the controlled arc, and one for the final maneuver. If the impulsive maneuvers are due to a gravity assist, then the propulsion system parameters for these are unimportant. The propulsion system parameters can all be defined for a single propulsion system, i.e., the exhaust velocity for the initial and final maneuver based impulses, and the controlled arc, assuming it is a constant exhaust velocity system, can be defined to be the same system by equating the exhaust velocity values of each segment node and the exhaust velocity of the controlled arc. But since these can be independent, multiple propulsion systems can be used in one individual segment. If the segment is a pure coast or ballistic arc then all propulsion system parameters are set to zero. If a segment is initiated with a high thrust booster, whose maneuver can be approximated with an impulsive maneuver, and controlled arc of the segment uses an independent low thrust propulsion system, then the

An Architecture for Spacecraft Tmjectory Design and Optimization System 543

arc requires the definition of the initial node impulsive maneuver and the parameters defining the low thrust engine. Another key entity of the basic segment is referred to as floating node point. It is similar to the two final endpoint nodes in that it has an associated time tag denoted as ti that is required to lie between t o and t f , to

I ti L tf

(8)

The main restriction of the floating node point is that it cannot have any velocity or mass discontinuities. It is used as a position, velocity, and mass reference measured along the segment. Since it is a node, the state vector associated with it can be constrained, or it can serve as a constraint for other segments. For example, a rendezvous between a spacecraft on segment i and a spacecraft on segment k can occur at the floating node point of segment k. The boundary conditions would be r3 = rf and v; = v f . Another example for the use of the floating endpoint is to find the location along a segment i that is at the periapsis point with respect to a celestial body. Here, ti is a free parameter constrained by Eq. 8 and its value must be such that f i ( t i )= 0 and Y i ( t i ) > 0, where f i ( t i )is the radial velocity with respect to the celestial body, and P i ( t i ) is the second time derivative of the position magnitude. Both of these conditions are sufficient for finding the location of periapsis. These types of constraints could be imposed without the need to define a floating node point because either endpoint nodes could be used to serve the same purpose. However, including the floating node point, simplifies the modelling process by removing the need to include an additional segment for some types of problems.

Control AccelerationVector

Fig. 1. The basic segment Building Block. Velocity impulses can exist at either node and the arc connecting the nodes can be either a ballistic or an controlled arc with a time dependent variable control vector. Mass discontinuities can exist at either node that result from an impulsive maneuver or non-maneuver mass changes. A floating node point lies between the endpoint nodes.

A sketch of the basic segment is given in Figure 1. Figure 2 shows the

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exploded time scale for the segment and the locations in time of the discontinuous and continuos states of the spacecraft. A subset of the parameters that define the basic segment are listed in Table 2.1. Within an individual segment, some of these quantities are independent and some are dependent. Outside of the segment, as in a trajectory or a mission, any of these quantities can be constrained and will thus be dependent in the solution process. I

!.

T

j

IM

t

M AccelecaionControlled Prc

t

Fig. 2. Exploded View Representation of the basic segment. Mass discontinuities can exist at each node point. These are either stage drop offs, mass additions, or impulsive maneuver mass discontinuities. The non maneuver mass discontinuities occur on either side of the velocity impulses. The velocity impulses are due to either an impulsive maneuver or an approximated gravitational assist. The velocity before an impulsive is v- , the velocity after an impulsive maneuver is v+. The maneuver based impulses have an associated mass discontinuity, Am. The non-maneuver mass discontinuities are Am- if it occurs before the impulse, and Am+ if it occurs after the impulse. At either node point, the beginning mass value is m - . The mass value prior to a velocity impulse is m- +. The mass value after a velocity impulse is m+ -. The mass value after the second non-maneuver mass discontinuity is m+. The controlled accelerated are is between and t; . All state variables vary continuously along this arc. The simplest type of segment is a node point at to with all parameters set to zero except for the node state variables, ro,v,,rn, which are required to be defined.

tt

The basic segment also includes a Lagrange multiplier vector, A, adjoined to the physical state variables, r,v,m at each time instant associated with each node and between to+and t7. For to++ t7, the evolution of Lagrange multiplier vector is governed by the Euler-Lagrange differential

An Architecture for Spacecmft Thjectory Design and Optimization System

545

equations associated with the optimal control problem. Other names given to this vector include the adjoint vector and the costate vector. This vector is composed of the position costate vector, A,, the velocity costate vector , A,, and the mass costate scalar, A., Additionally, because of the basic segment definition, the segments can be all independent and initially disconnected. The dependency a segment may have with respect to another segment is specified in the constraint definitions. Because of this, the architecture allows then the modeling of multiple spacecraft problems. It is not necessary to specify how many spacecraft are in the model. All that is required is to model as many segments as needed and that they be constrained in any way necessary to represent the number of spacecraft in the model. For example, a spacecraft rendezvous between two spacecraft requires a minimum of two segments. A possible constraint is that the final endpoint nodes of both spacecraft have the same position and velocity at the final solution. If both spacecraft have maneuvering capability, these maneuvers are then defined and allowed to be adjusted for each segment. If one spacecraft remains passive, then its segment definition has its propulsion parameters nulled out. Since the segments can be initially disconnected, the introduction of additional variables and possible constraints, though increases the number of variables, also decreases the sensitivity of the iterated solutions compared to strictly forward time or backward time propagated trajectories if the magnitude of the duration of a segment is longer than allowed for perturbations to be linearly valid in gradient based solution methods. However, if this is not an issue, any segment j can be forced apriori to be connected sequentially to any other segment i at any of the node points and distinct times of segment i. The idea of propagating disconnected segments that will eventually be connected or at least constrained in some way is a generalization of what is known as direct multiple shooting. The definition of the basic segment described here can be altered to include more parameters. The working model described has been sufficient to examine a broad class of trajectory problems. The definition of the basic segment, however, is dynamic so that yet unforeseen trajectory problems can be accommodated. The simplest mission can be modelled with only one segment. More complex missions will require more than one segment.

546

C. Ocampo Table 1. Basic set of Elements Defining the Basic Segment.

Symbol to TO

-

110

ISPi

P

Variable Type/Name Dim Dep Range or Constraint initial time 1 i -00 < to< 00 position 3 i 0 5 l r o l < 00 velocity mass mass discontinuity Lagrange multiplier velocity impulse specific impulse maneuver mass discont . mass discontinuity velocity mass Lagrange multiplier intermediate time thrust specific impulse power thrust direction unit vec. engine efficiency final time position velocity mass mass discontinuity Lagrange multiplier velocity impulse specific impulse maneuver mass discont . mass discontinuity velocity mass Lagrange multiplier

Units day km km/s

Segment Types The general definition of the segment allows for the modeling of different segment types. The type and number of segments used depends on the nature of the problem, the number of spacecraft in the model, the propulsion systems used in the model, and the complexity of the trajectory or mission. Based on the basic segment, the segment types that can be modelled, by

An Architecture for Spacecmft Rajectory Design and Optimization System 547

properly specifying the values that define it, are listed in Table 2.1. Figure 3 shows the different segment types beginning with the most basic segment type and proceeding down to the simplest segment models which are simple node points or node points with impulsive maneuvers. Each segment type in Figure 3 is labeled with an integer identifier that corresponds to those listed in Table 2.1.

L 9

d .I1

0

0

10

.I2

Fig. 3. Segment Types Based on the basic segment. These are treated as building blocks to construct many types of trajectories for one or more spacecraft. The large vector arrows represent velocity impulses; the small vector arrows represent a finite acceleration control arc. Dynamically there is no difference between segment types 9 and 11 or 10 and 12 except that the time epochs are different since segment types 9 and 10 are the to node and segment types 11 and 12 are the t f node, with and without a velocity impulse as shown.

Recall that a trajectory for a single spacecraft is comprised of one or more segments. Therefore, it is possible to model a trajectory for a single spacecraft that is equipped with one or more propulsion systems. Further, mass discontinuities resulting from either impulsive maneuvers, mass drop offs, and mass additions are accounted for by properly defining these quantities in the definition of the segment. A mission which is composed of one or more trajectories, which in turn is composed of one or more segments,

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C. Ommpo Table 2. Segment Types as Sub Types of the Basic Segment.

Segment Type Segment Description tonode, AVO,controlled arc, tfnode, Avf 1 2 tonode, AVO,controlled arc, tfnode tonode, controlled arc, tfnode, Avf 3 tonode, controlled arc, tfnode 4 tonode, AVO,ballistic arc, tfnode, Avf 5 tonode, Avo, ballistic arc, tfnode 6 tonode, ballistic arc, tfnode, Avf 7 tonode, ballistic arc, tfnode 8 tonode, Avo 9 tonode 10 tfnode, Avf 11 12 tfnode

will have as its basic building block any of the segment types shown in Figure 3 and tabulated in Table 2.1. The choice of which segments need to be connected to which other set of segments is problem dependent. In the case of multiple spacecraft problems, the connectivity of the segments is determined by the type of multiple-spacecraft mission being analyzed. For independent spacecraft that are not required to rendezvous or intercept, these may all be disconnected but constrained in any way necessary, such as maintaining their positions to satisfy some geometrical formation, for example. For intercept or rendezvous problems, clearly, some segments will need to be connected; i.e., for rendezvous problems, position and velocity are required to be continuous, or for intercept problems, only position needs to be continuous after the maneuver. A rendezvous between two spacecraft where one of the spacecraft assumes the additional mass of the other spacecraft will require the addition of a non-maneuver mass discontinuity which can be a fixed or variable quantity depending on whether mass of the captured spacecraft is fixed or variable. Impulsive maneuvers, which are constrained t o occur only at the endpoints, can be referenced to any allowable fixed-center frame or relative to the trajectory state associated with the respective node point allowing either velocity vector or radius vector referenced maneuvers. Finite acceleration maneuvers (finite engine burns, sails, etc.) can also be steered relative to the trajectory or remain fixed in any fixed-centered frame. In the case of finite burn maneuvers, both thrust constrained and power constrained thrust arcs can be modelled. Algorithmically, no distinction is made between finite burn, high thrust or low thrust systems. Both are treated

An Architecture for Spacecraft Zhjectory Design and Optimization System 549

equally and the specified propulsion system parameters further defines the type of thrust arc used. The variation of the thrust magnitude, the exhaust velocity, and the power are constant or variable depends on how each of these are constrained.

2.2. A Conceptual Modelling Example To illustrate the use of using the segment building blocks to construct a complex mission, consider the following hypothetical mission. It is desired to transfer a main spacecraft stationed at the vicinity of the interior libration point of the Earth-Moon system to the vicinity of the Jovian moon, Europa. A landing and ascent vehicle of fixed and known initial mass is attached to the main spacecraft. At Europa the lander separates from the main spacecraft, descends to the surface of Europa, performs the required objectives, and ascends to rendezvous with the main spacecraft. The main spacecraft returns to its starting point in the Earth-Moon system, but on its return route, it intercepts and flies by a comet. The main spacecraft is equipped with a nuclear powered continuous thrust, variable specific impulse engine. The lander/ascent vehicle is equipped with a high thrust constant specific impulse engine with fixed propellant mass. The mission performance objective is to minimize the initial mass of the main spacecraft while constraining the final mass (the dry mass) of the main spacecraft. The total mission duration is constrained to be no greater than a given value and the stay time for the lander at Europa is constrained to be no less than a given value. The mission can utilize any beneficial gravity assist maneuvers around celestial bodies that exist in the model. Clearly, there are many more details required to fully model this mission. However, the basic parameters stated are enough to conceptually illustrate the modeling of this type of mission. Figure 4 illustrates a possible set of segment blocks that can be used to construct the first iterate. The short dashed lines indicate the node points that need to be connected and required to be continuous at least in position and possibly velocity. In this sketch, the first assumption is that the mission can be modelled with ten segments. Other parts of the mission, such as the landing and ascent phases can be decomposed into further segments if necessary To correctly design and optimize such a mission, it will be necessary to include the gravitational attraction from at least the principle celestial bodies involved which in this case include the Sun, Earth, Moon, Jupiter, and Europa. The comet can be assumed to be a non-gravitating celestial

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C. Ocampo

body and can be modelled as an independent segment if its state vector is known at some epoch, or it can be treated as a gravitating celestial body if its ephemeris is known. Further, it will be necessary to specify the initial state or orbit of the main spacecraft with respect to a reference frame in which the location of the interior libration point of the Earth-Moon system is known. At departure, the trajectory may take advantage of the complex force field that is dominated by the simultaneous attraction from the Sun, Earth, and Moon. At Jupiter, the trajectory may also exhibit a complex behavior since the force field will be dominated by a restricted four body model that includes the Sun, Jupiter, and Europa. Though a solution to this mission is not presented here, it is one example of a type of mission that can be modelled and solved with the system described here.

2.3. The Equations of Motion and the Propagation Reference Frame For a general system, the choice of reference frame in which to model the spacecraft’s three degree of freedom point dynamics and the associated equations of motions is critical from several viewpoints. First, it is desired to have the simplest reference frame model allowable without compromising the validity of a solution and its accuracy. Second, in a comprehensive model, that may include multiple spacecraft, multiple propulsion systems, and multiple celestial bodies, it is desirable to work with a coordinate frame for state propagation in which transformations between the frames in which the states, targets, and maneuvers are referenced to are simple and computed efficiently. If the dynamics are modelled correctly, any reference frame used must yield the same result. On the other hand, some reference frames may include terms in the equations of motion which approach the limiting accuracy with which small terms are evaluated by the computer. Aware of the accuracy issues associated with the proper choice of reference frame for state propagation, the current system propagates the equations of motion in either a reference frame that is fixed (non-rotating) to any barycentered frame that can be defined or to any chosen celestial body. If it is attached to a specified celestial body, the frame translates but does not rotate. The celestial body selected to be the center of this frame depends on the nature of the problem being modelled. Each segment in the mission can have a distinct reference frame for propagation if desired. The only restriction is that it be a non-rotating frame. For example, if interplanetary missions are be-

An Architecture for Spacecraft Thjectory Design and Optimization System 551

.

j+

i

,'

. i

Fig. 4. Modelling of a Complex Mission. Different segment types are used t o construct a round trip mission beginning at the interior libration point of the Earth-Moon system, to the Jupiter-Europa System, with an intermediate flyby of a comet, before returning t o the starting point. Some segments will be low thrust arcs, others will be ballistic arcs with impulsive maneuvers at their nodes. All segments are shown initially disconnected. Some of these could have been defined to sequentially connected to other segments initially. The dashed lines between some of the nodes imply that these nodes are required to be connected at the final solution.

ing solved with respect to the Sun, and the other celestial bodies of interest are treated as non gravitating zero point masses, the reference frame should be centered at Sun or the solar system barycenter. For the same mission, if some of the segments are required to operate in the vicinity of a celestial body, such as in the case of a long duration spiral escape or capture from a given celestial body, then that segment should have as its propagation

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C. Ocompo

frame, a frame fixed to the celestial body of interest. In some problems, state definition and targeting can be specified in coordinates of a rotating frame defined between two celestial bodies. Though it is possible to integrate the equations in coordinates of the rotating frame, experimentation has so far determined that this does not provide any significant advantages. However, the use of rotating frame coordinates for state definition and targeting is a key component of the current architecture. The discussion of the rotating frames used for this purpose are reserved for a later section. In a barycentered frame that is assumed to be inertial the second order equations of motion are

where r is the relative position vector of the spacecraft with respect to barycentered frame, rj is the relative position of celestial body j with respect to the barycentered frame, ncb is the number of celestial bodies in the model, and GMj is gravitational parameter for celestial body j . All additional non-control related terms are contained in apertand control related terms are contained in I?.

If the propagation reference frame is centered at a specified celestial body, denoted by C B f ,the equations of motion in this fixed (non-rotating) but translating frame is

where G M ~ isB the ~ gravitational parameter of the celestial body to which the frame is fixed, rj is the position vector of the other celestial bodies with respect to C B f ,and nCbis the number of celestial bodies treated as third bodies. The vector terms associated with the gravitational acceleration due to third bodies include both the direct acceleration vector term and the indirect acceleration term needed to account for the fact the C B f fixed centered frame is not an inertial frame. Eqs. 9 or 10 are used to propagate all of the segments from ti+ + t)- (i = 1, ..., n ) where n is the number of segments in the mission and noting that each segment can use either set of equations and have its own distinct choice for the center of the reference frame.

An Architecture for Spacecraft lhjectory Design and Optimization System 553

The position of the other celestial bodies is obtained from an explicit time dependent ephemeris and hence the time dependence of the force model. The ephemeris provides the positions, and possibly the velocity, of any of the celestial bodies with respect to any other celestial body and can be a highly accurate ephemeris, such as the Jet Propulsion Laboratory’s set of planetary ephemerides, or any user defined ephemeris. The user defined ephemeris is useful when studying the dynamics in simplified force models such as the circular or elliptic restricted three body problem; or user defined model solar systems with one or more arbitrary celestial bodies where the motion of the celestial bodies relative to the frame center can be simple Keplerian type orbits or based on a precomputed ephemeris.

For example, consider the analysis of cycling trajectories between two planets orbiting the Sun. Such a study should begin by assuming that the planets of interest are in circular orbits about the Sun. This model provides the basic properties associated with the construction of these trajectories. Once these properties have been understood, then the model is modified to account for the real motion of the planets. The assumption being that the circular model solutions carry over to the real planetary model with an expected variation. However, it is important to be aware that the model that uses the real planetary model may yield solutions that do not exist in the simplified circular model. If so, then a better approximation to the real planetary model should be used to identify the basic properties of cycling trajectories between these celestial bodies, such as an elliptic model for the motion of the planets of interest. Current and future missions will require targeting relative to one or more solar system minor bodies such asteroids and comets. Assuming an ephemeris for these bodies is available, then these are treated as celestial bodies with an associated gravitational parameter. If only a state vector and epoch is known for any of these bodies then either an ephemeris is computed separately with the current system by treating it as a ballistic segment, or the body is treated as a ballistic segment in the solution process by assuming that its mass does not influence the motion of the other spacecraft segments.

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2.4. Multi-Body Reference h m e s

There are certain spacecraft trajectory design and optimization problems that are best solved in alternate reference frames. In the case where two or more celestial bodies are included in the force model and any two of these bodies are in bounded orbits relative to each other, the dynamics of a spacecraft is best analyzed and understood in a reference frame that can in general translate, rotate, and pulsate in such that the representation of the positions of the celestial bodies remain stationary. This frame is the well known rotating frame that has been used extensively in the study of trajectories in the Circular and Elliptic Restricted Three Body problems lo. For example, consider the motion of a spacecraft in the Earth-Moon system. Analogous t o the barycentered rotating frame of the restricted three body problem, a convenient frame in which both the Earth and Moon remain stationary is one that rotates with the instantaneous Earth-Moon line and pulsates with the varying distance between the Earth and Moon if the motion of the Moon about the Earth is based on a realistic ephemeris. Another example of the use of a rotating frame is in the analysis of Earth-Mars trajectories where a suitable reference frame is one where both Earth and Mars remain stationary. Here, the fundamental axis is the instantaneous Earth-Mars line. The north pole axis of the Earth or the Sun can be used to construct a normal vector to this line, from which the final right-handed coordinate frame is constructed. The advantage of this frame is clearly noted in the visualization and targeting of trajectories connecting both planets but more importantly, the capability then exists to specify initial state components and constraint functions directly in this frame greatly simplifying the targeting and optimization of trajectories relative to any set of moving bodies. These two example reference frames can be examined under a single formulation. In the first example, two celestial bodies are in orbit about their common barycenter. In the other example two celestial bodies are orbiting a common reference center that is not their barycenter. The state, targeting, and maneuver definitions can be given in coordinates of this frame and appropriate transformations are needed to transform position, velocity, and acceleration between this frame and the segment propagation frame.

An Architecture for Spacecraft lhjectory Design and Optimization System 555

Consider the motion of two celestial bodies CBi and CBj. The time varying position vectors for these bodies are known relative to some other fixed frame that can be an inertial frame or a frame centered at another body; i.e., rCBi and r C B j are known vector functions of time. Define a unit vector along the relative position vector of CBj with respect to CBi ?=

(r C B - r C B i ) / IrCBj j

- rCBi

I

(11)

Several options exist to define the remaining two basis vectors for the frame. If CBj is in a closed and bounded orbit about CBi such that its relative angular momentum vector with respect to CBi remains nearly constant, then the remaining basis unit vectors are defined as

where the i unit vector is along the instantaneous specific angular momentum vector and v is the relative velocity vector of CBj with respect to CBi, v = V C B ~- V C B ~This . basis can be used, for example if the two bodies are a planet and the Sun, or a planet and its moon. Figures 5 and 6 illustrate the CBi-CBj rotating frames in both a CBk fixed frame and a rotating frame centered at one of the celestial bodies. Alternatively, if the relative angular momentum vector between the bodies changes in way that the motion alternates between being retrograde to prograde (such as the motion of two celestial bodies around a central body that are not in orbit about each other) then it is convenient to use a nearly constant vector, such as a reference north pole axis to define the remaining basis unit vector. Let this reference unit vector be 4. The remaining unit vectors then are

This set of basis vectors is ideally suited for a rotating-pulsating frame where the Earth is CBi and Mars is CBj (or vice-versa). Figures 7 and 8 illustrate the CBi-CBj rotating-pulsating frames in both a CBk fixed frame and a rotating-pulsating frame centered at one of the celestial bodies.

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C. Ocampo

Earth

Comet

I I i I I I / / / /

, /

Earth

,,

f

L1

'-. '-.._ Seg7

Fig. 5. The CB;-CBj Rotating n a m e with respect to a CBk Fixed Centered Frame. The two celestial bodies, CB; and C B j , are in closed orbits about their common barycenter. In general, the distance between them varies.

The transformation of a position vector referenced in the rst frame to a fixed frame centered at CBi with basis unit vectors i,j, k (referred to as the i j k frame) and scaled with respect to the instantaneous distance between CBj and CBi is

where r = )rl, k is a positive scaling constant with the same units as r , and R is the transformation direction cosine matrix between the rst frame and the i j k frame,

The fixed frame velocity and acceleration vectors are obtained by successive

A n Architecture for Spacecraft Thjectory Design and Optimization System 557

I

Spacecraft Trajectory,

;

Fig. 6. The CBi-CBj Rotating Frame with respect to a CBi Fixed Centered Frame. CBj pulsates along the line connecting CBi and CBj if their absolute motion is noncircular with respect to each other.

time differentiation of Eq. 16,

[ aijk= 1 [ (iR + 2 i R + TR)rrSt+ (2iR + 2.“) k

vijk = 1 k (+R+rR) rrst + rRvTSt]

(18)

vTSt+ rRarSt](19)

where i = dr/dt and i = d2r/dt2. The inverse transformation that provides expressions for rrst,vTSt,and arst in terms of r i j k ,vijk,and aijk is readily available. Note that the higher order time derivatives for T and R will require up to a first order time derivative in the relative acceleration vector between CBj and CBi. If this information is not available from the ephemeris, then it will need to be estimated with any finite difference approximation provided at least the time dependent position vectors are available from the ephemeris. In either the rotating or the rotating-pulsating frame, it becomes possible to define initial state components for any segment and to target and treat as constraints the state components or functions of them at either node

558

C. Ocampo

_--SpacecrafiTrajectory

Fig. 7. The CBi-CBj Rotating Pulsating Frame with respect to a CBk Fixed Centered Frame. The two celestial bodies, CB; and C B j , do not orbit in closed orbits about their common barycenter.

of any segment. For example, finding an equilibrium point in a rotatingpulsating frame centered at the Earth with the Moon along the i. direction requires a three dimensional search for the position components rTstwith the conditions that vTst= aTst= 0, the definition of an equilibrium point. If there are no external accelerations beyond those from the gravitational acceleration due to the Earth and the Moon, then this equilibrium point will remain fixed in the rotating-pulsating frame. If external accelerations are present, such as the gravitational acceleration due to the Sun, then this equilibrium point can be only defined for a particular epoch.

2.5. State l’mnsfonnations Recall that all segment propagations are made in either a barycentered or celestial body fixed frame. It is required that a segment’s initial state vector

An Architecture for Spacecraft If-ajectory Design and Optimization System 559

Fig. 8. The CBi-CBj Rotating Pulsating Frame with respect to a CBi Fixed Centered Frame. CBi and CBj remain fixed in this frame.

and all other quantities that define it be referenced to the segment's fixed reference frame used in its propagation. It is convenient for certain problems to express these quantities in any other reference frame or coordinate set. If this is the case, a suitable transformation to the segment's propagation frame is necessary. Consider, for example, a segment i in a gravity field composed of the celestial bodies, CBi and CBj. Celestial body CB, can be defined to be C B f , the center of the integration frame, and the motion of CBj is assumed known relative to C B f . The initial state of the spacecraft can be referenced to any of the bodies. If it is referenced to CBj, the required transformation is

If there exists a more convenient vector or parametric representation of the initial state vector, say the classical Keplerian orbital elements of the

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spacecraft with respect to CBj at t;, then the required transformation is

.T

where ( a ,e, i, R,w ,v)hBjis a vector whose components are the semi-major axis, eccentricity, inclination, right ascension of the ascending node, argument of periapsis, and true anomaly of the spacecraft at t; relative to CBj. Generally, it can be assumed that an initial state vector, y ( t ; ) , is comprised of any independent set of quantities that uniquely represent the initial state vector of the spacecraft. Ultimately, the transformation y(t;)CBj + (rb,vb)cB, will be needed for any independent state vector representation with respect to any celestial body CBj. There are many possible parameterizations for y(t;)cBj including those that contain departure, capture, or flyby parameters used in practice such as the hyperbolic excess velocity v, or the square of its value referred to as twice the hyperbolic energy, C3, and the direction of the incoming or outgoing asymptote, %-

,

SV,

*

2.6. Open Loop Solution of a l h j e c t o r y or a Mission Given the number of segments and their defining parameters, the open loop solution of the system can be determined. The open loop solution is the first estimate or iterate which will generally be infeasible since the constraints or boundary conditions are not necessarily satisfied. The segments are all propagated independently unless a particular segment is required to be connected sequentially to any other segment in which case the segments that are to be continued with other segments are propagated after the segments to which they are connected to.

If the segments are initially disconnected, i.e., discontinuous in any of the state variables and possibly the time, and these segments form a trajectory for a single spacecraft, the segments will have to be connected by the solution process. Some of the parameters defining these segments will

An Architecture for Spacecraft Rajectory Design and Optimization System 561

be variables that can be adjusted, and constraint conditions are imposed as continuity conditions for values of all or either the position, velocity, mass, and time. For a multiple spacecraft mission where several of the spacecraft need to rendezvous with other spacecraft, some of the segments will have to be connected either in position or velocity, or both, at the appropriate nodes.

If a segment k is forced to be connected sequentially to any other node of any other segment i beforehand, the connectivity is enforced at any of the discrete node times, tt;, ), ti;,;, ti:,;, ti: f), i.e., the continuity of the position, velocity, and mass of segment i can be made prior to or after the initial or final impulsive maneuvers or any of the mass discontinuities, if they exist. For example, if a purely ballistic segment k is required to be connected sequentially in all three state variables (r,v,m) to any other segment i in the model at the t f node after all, if any, of the velocity and mass discontinuities, the state vector of the to node of segment k is then

Here then, it is necessary that segment i be evaluated entirely before segment k can be evaluated. On the other hand, if the final solution requires that segments i and k be connected as in Eq. 22, these can be initially defined independently and Eq. 22 is used as an equality constraint condition that must be satisfied by the final solution. Continuity is not allowed at any time between to+and t7 because only node points can be constrained. But this is not a restriction, because an additional segment is introduced such that one of its nodes can be used as the continuation to another segment. This generality facilitates the modeling of complex trajectories that may be otherwise difficult to obtain as a single forward time or backward time propagated trajectory and in the modeling of multi-spacecraft problems where the other spacecraft are moving along different trajectories.

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3. Solution Methods For any given problem, that in general can be complex and involve multiple constraints, it is possible that no solution exists. A common reason for this is that the constraints are inconsistent meaning the problem definition itself precludes the possibility of a solution. Therefore, a necessary requirement is to formulate a consistent problem where the variables of the problem are independent and the constraints are at least linearly independent. If it is then assumed that a solution should exist, there are two possible solution methods. These methods provide solutions to the majority of the spacecraft trajectory design and optimization problems considered by the current architecture. This section describes these methods which are used to solve trajectory problems that are based on the set of basic segments. Let n be the number of segments in the trajectory or the mission. Let the segment vector, si,be a vector whose components are all of the independent and dependent quantities that uniquely define segment i. Within an individual segment, some of these quantities will be either dependent or independent. For example, m f ,the vehicle mass at tf, is a dependent quantity that depends on any impulsive maneuver at to and/or tf, any and any non-maneuver mass depleting finite maneuver between t i and mass discontinuity that may be present at either or both nodes. But m i is an independent quantity (within the segment), though it may be constrained outside of the segment during the solution process. Depending on the target conditions required to be satisfied for a given mission, further information can be computed from si that can be used as constraints or as a cost function. For example, the state vector at any of the discrete times of the node point relative to any of the other celestial bodies that are present or any function of these can be computed provided the appropriate transformations are available. A common function is the two-body energy of a segment endpoint with respect to a celestial body.

t7,

Let xp be a p vector containing all of the variable parameters that have been identified for any given problem. The parameter vector xp can contain any independent element of any or all of the segments identified in si(i = 1, ...,n ) . It can also contain any parameter that defines the force model. Let c be a q vector containing all of the equality and inequality constraint functions. The constraint vector c is in general a nonlinear function of any of the elements in xp.Let J be a general nonlinear scalar function of any

A n Architecture for Spaceemft lkajectory Design and Optimization System

563

of the elements in x,. The value of J is to be extremized. A description of the two solution methods and their implementation into the current architecture follow. Case 1: Solution of a System of Nonlinear Equations, c(xp)=O. If the number of variables in xp is equal to the number of constraint equations c (p = q ) and all of the elements in c are equality constraints then the problem to solve is a targeting problem with no explicit optimization., i.e., no explicit scalar function is extremized. However, this does not exclude the possibility of implicitly extremizing a cost function if the equality constraint functions include necessary conditions based on optimal control theory. The targeting is completely general and the solution for xp is a solution to a general nonlinear root finding problem for a system of nonlinear functions or equations with an equal number of unknown parameters. F'unctionally, the problem is to solve the set of equations c(xp)= 0.It is assumed that the functions C(X,) be smooth, i.e., at least twice-continuously differentiable, in the independent variable vector x,. Though in general, these equations are not available explicitly, the functions given in c(xp)are determined from a sequence of operations that depend on the specific model and problem. These operations will involve numerical propagations of the segments, transformations between reference frames, and function evaluations of the elements in the constraint vector, c. The system can use any efficient and robust algorithm to solve the nonlinear root finding problem. The current prototype system uses a NewtonRaphson / Steepest descent correction to compute a search direction and is coupled with Broyden's method for computing and updating the Jacobian matrix of the system, dc/bx,. The analytical and numerical issues associated with this and similar methods can be found in Dennis and Schnabel 1 1 , Gill et. a1 1 2 , and Nocedal and Wright 13. The two-body orbital boundary value problem, which is known as the Lambert Problem if the gravitational force model includes only one celestial body with no other perturbations is a subset of this problem. The orbital boundary value problem, for any general gravitational force field which can include more celestial bodies, or a complex potential gravity field for one or

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more celestial bodies, or other perturbations, is a two point boundary value problem which is also a subset of this case. Other specific problems that can be solved as a nonlinear root finding problem in orbital mechanics include periodic orbit searches, Earth-to-Moon trajectories, including free return trajectories, multi-body gravity assisted trajectories in the Solar System, etc. The functional optimization or optimal control problem for either impulsive maneuvers, finite engine burn maneuvers, or non-mass depleting acceleration controlled maneuvers using optimal control theory is also a subset of this case. Here the time varying Lagrange multipliers associated with each of the physical state variables augment the si vector for each segment, and the first order necessary conditions and the transversality conditions lead to a well defined and in general multi-point boundary value problem. If solved as a nonlinear root finding problem, the problem is an implicit optimization. An extremal solution that satisfies the first order necessary conditions from optimal control can be produced by solving the appropriate nonlinear system of functions. This is the well known indirect method for trajectory optimization and has been well documented in the literature. A comprehensive treatment of the optimal control problem and its solution methods is given by Hull 14. Since the system of equations is in general nonlinear, multiple solutions can be expected. If the functional optimization problem is solved indirectly, the solution provided by the nonlinear search will satisfy first order necessary conditions for an optimal solution. If the general targeting problem is solved without the use of the functional optimization conditions, then the solution is only a feasible solution.

'

It will always be necessary to provide a reasonable estimate the for the starting iterate given by xp. For simple problems, this starting guess can be estimated analytically. For more complex problems, it may be necessary to use results available from previous research in specific problems. For example, circular restricted three body trajectories are well understood though closed form solutions for these do not exist. For force models that are not too different from a simplified circular restricted three body force field model, the results from the simple circular restricted three body model can be used as starting iterates for more complex models.

If some of the elements in the constraint vector, c , for a targeting prob-

A n Architecture for Spacecraft 'majectory Design and Optimization System 565

lem are inequality constraints, these can be converted to equality constraints by adding one more variable to the parameter vector x p known as a slack variable such that the final problem is an equality constrained problem and solved as such 14. The underdetermined or overdetermined cases where there are more or less variables, respectively, than equality constraints is solved as a minimax problem l2 commonly used for nonlinear minimax data-fitting, where the constraint function with the maximum absolute value is minimized. The minimax solution to the system of nonlinear equations C(XP)

=0

is the solution that minimizes the function, F defined by F(Xp)

= max I C i ( X p ) l

where c i is the i-th element of the constraint vector c . This technique is robust and efficient for the class of nonlinear targeting problems considered here where there is not an explicit cost function and satisfaction of the constraint equations results in a feasible solution. In the underdetermined case, a solution to the minimax problem can be used as the starting estimate for the parameter optimization problem since all of the constraints are satisfied initially. Case 2: The Constrained Parameter Optimization Problem This is the general problem of nonlinear functional optimization with nonlinear equality and inequality constraints and whose solution methods have been well documented in the literature (see for example Gill et al. l 2 and Nocedal and Wright l 3 ). The problem is to minimize or maximize the objective function

J = J(xp) subject to both equality and inequality constraints, : qeq x 1

ceq(xp)

=0

ceq

Cineq(Xp)

20

Cineq

:qineq x 1

where C e q is the q e q vector of equality constraints and C i n e q is the q i n e q vector of inequality constraints. The functions ceq( x p )and cineq( x p ) can

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represent lower and upper bounds on the individual elements in x,, linear constraints on two or more of the elements in x,, or more generally, nonlinear functions of the elements of x,. The algorithm used in the current system is a sequential quadratic programming (SQP) variable metric method described in 12913. With the given architecture, the formulation of many parameter optimization problems is straightforward. The objective function to extremize (minimum or maximum) can be taken to be a single scalar variable in si(i = 1,...,n) or any scalar function of these elements. For example, a trajectory that begins in an orbit about CBi and terminates in an orbit about CBj can be such that the initial mass of the spacecraft is a minimum if the final m a s has been prescribed. For this case J = min(mh-) where i is first segment of the trajectory representing the spacecraft. The value of the initial mass required is a function of all the segments following it including all of the possible impulsive or finite burn maneuvers used to reach the final orbit, and any non maneuver mass discontinuities. The constraints associated with the initial orbit and the final orbit would be the nonlinear constraint functions in c, and x p would contain the parameters that can be estimated such as the time of the maneuvers and the parameters that define the maneuver and that are allowed to be adjusted.

4. General System Issues

Several main aspects important to a general trajectory design and optimization system are discussed in this section. This includes be benefits of using a system that is modular in all of its subcomponents, the importance of tuning the algorithms and automating the tuning process, and the use of integrated visualization as a key part of the solution process.

4.1. Modular Architecture

A general trajectory optimization system should be modular in the sense that the algorithms used to solve different parts of the overall trajectory problem are independent and can be readily modified or replaced. The basic components include algorithms for explicit numerical integration, solutions to systems of nonlinear equations, and solutions of the nonlinear constrained

An Architecture for Spacecraft hjectory Design and Optimization System 567

parameter optimization problem. The individual algorithms used for each of these functions should be the ‘best available’ and with the added flexibility of being updated or changed entirely as new and more robust and efficient algorithms are developed. Provisions should also be made to allow completely general force models to be used including the use of either realistic ephemeris models or user defined models. In summary, the assumption is to treat each algorithm to each sub-problem as ‘solved’. The continued development should be directed to the system architecture and all that it entails such as refinement of the definition of the basic segment, the maneuver models, and the coordinate frames for state definition, targeting, and maneuver parameterization.

4.2. Algorithm and System Tuning

Provided an estimate for a convergent solution is available for both the nonlinear equation solvers or the parameter optimization algorithms based on gradient information, there exists a sequence of perturbations step sizes for the parameter vector, xp,that achieves a solution in a minimum number of function evaluations for the constraint vector, c , or both c and the cost function, J. If the method of finite differences are being used to estimate gradients, typically the vector of perturbation step sizes, Bxp, is set as a constant during the iteration process. Regardless of the method used to compute the finite difference based gradients, determining the ‘best’ value for the perturbation vector is a process referred to as tuning and which is problem, algorithm, and processor dependent. A given value of Bx, that provides accurate estimates for the gradients at the beginning of an iteration process may not be the best choice at other parts of the search space, therefore this vector should be allowed to change and recomputed periodically to produce accurate gradients at different points in the search space. Any system should have in place an automatic tuning algorithm that can adjust the perturbation vector over the course of the iteration process to achieve convergence using the minimum amount of function and constraint vector evaluations. Included in the tuning process is the proper scaling of the parameter vector and the constraint functions if their values differ by large orders of magnitude. Also, a change to each of the elements in the parameter vector x p per iteration should be bounded to avoid evaluating subsequent

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iterates that cause the solution to diverge. This is common in trajectory problems where the endpoints of a trajectory operating in a complex force field are highly sensitive to changes in the parameters of earlier parts of the trajectory. But for well behaved solutions, if this upper bound is too small, many iterations will be needed for convergence. The proper choice of Sx, per iteration, the scaling, and the maximum allowable changes to x p per iteration, are three factors that influence the convergence rate of a solution and have to be considered carefully in an automated tuning process.

A procedure that leads to accurate derivative estimates is based on the calculation of the state transition matrix along the ballistic or controlled accelerated arcs for all segments between t i and t7 and for the variables that are numerically integrated. The state transition matrix is the time varying fundamental matrix solution of the linearized variational equations evaluated along the trajectory arcs. If the state vector x ( t )is composed only of the physical state variables r,v,mthe state transition matrix at t7 for a given statement provides the Jacobian matrix of the final state with respect to the initial state. It is a 7 x 7 matrix with the gradients a x ( t , ) / O x ( t $ ) . If the Lagrange multiplier vector is part of the state vector, so that all of the variables r,v,m,X, ,A, ,A, are numerically integrated along a trajectory arc, then the state transition matrix will be a 14 x 14 matrix. The state transition matrix is integrated along with the state vector thereby requiring the integration of n+n2 first order equations where n is the size of the state vector. The information available in the state transition matrix represents some of the terms necessary to compute the required gradients. Given a constraint vector, c , or a scalar objective function J , that depends on an independent parameter vector, x,, there exists analytical expressions for both the gradient vector, d J / d x , , and the Jacobian matrix, & / a x p , that are linearly valid near any nominal solution of a segment, trajectory, or mission. These expressions are based on state transition matrices along all of the segments, and the gradients across impulse points, discontinuous mass points, and any state and maneuver transformations used. Though numerical integration of the state vector and the state transition matrix is required to evaluate the quantities necessary to evaluate both a J / a x p and d c / d x , , these provide accurate values at x p necessary for any gradient based solution method. The disadvantage of this process is that it is problem specific and requires the derivation of these expressions for each type of problem, a time consuming process if many classes of problems

A n Architecture for Spacecraft !hjectory Design and Optimization System 569

are to be considered. Nevertheless, these expressions can be evaluated and used as the actual gradients necessary in the solution methods. Alternatively, these expressions can be evaluated at discrete points in the iteration process, and used to tune the perturbation vector SX, at these points. The perturbation vector is then used to estimate the derivatives with a finite difference approximation.

4.3. Integmted Visualization

A general trajectory design and optimization system should include interactive visualization capability not only for presentation of intermediate and final results but as a key part of the solution process. The system currently provides the capability to visually display a three dimensional graphics representation of the trajectory design and optimization process in real processor time; i.e. immediate visual feedback during the targeting or optimization process is available. The reference frame used to visualize the dynamics and solution process can be independent from the reference frames used in the trajectory problems. The information shown in the graphics representation typically includes the positions of each point in the trajectory. However, other phase space variables can be visualized as well if the visualization of their evolution provides better information. Though visualization has generally not been accepted as a necessary capability in the solution process, it is used here as a critical component in the design and optimization process for several reasons. First, it is immediately possible to determine if there is an error in the input information thereby allowing for the termination of the process and correction of the error. Secondly, it is possible to determine whether the solution is converging, and if so, the rate of convergence. If convergence is slow, the speed of convergence can be increased by re-tuning the algorithms. Thirdly, an intuitive understanding of the dynamics associated with complex trajectories in multi-body gravity fields or rotating frames can be obtained, thereby facilitating future investigation of more difficult problems. All of these benefits can be obtained from a non-graphics based system in a post execution manner by examining the output data generated and plotting any desired information. But clearly, the time saved by not having to perform these post execution tasks is a reason that supports the use of integrated graphic visualization in the solution process.

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5. Conclusions

The key aspects associated with a general trajectory optimization system have been presented. The continuing effort is an attempt to construct a system that can be used to analyze a large range of problems that are currently required in trajectory design and optimization. Future spacecraft missions will use innovative trajectory dynamics that take advantage of the natural and complex dynamics associated with multi-body gravity fields, multiple and hybrid propulsion systems, and multiple spacecraft. A main objective is the development of a system that can facilitate the solution to these problems using a general framework that is applicable to all of the sub-problems required for the overall solution. Such a system should produce results that can be used in research topics involving both trajectory optimization theory and numerical methods for the solution to these problems. The system should also be able to produce results that can be used in actual spacecraft operations by accounting for actual mission operation constraints; detailed propulsion system models; attitude control, constraints, and requirements; and some level of information regarding the observability of the solutions. A deterministically optimal solution based on propellant or mass performance only may not be the best solution given a measurement model to observe the state of the spacecraft. The capability to produce an integrated solution that offers an acceptable compromise between performance and the ability to accurately navigate it is desirable. Caution must be exercised in not having a system so general that it solves many problems only superficially and with many restricting assumptions. There will always be a balance between how general a system is and how detailed and complex the solutions produced by such a system are for any given problem. A measure of the usefulness of such a system is the scope of the problems it can solve while solving them to the level of detail needed to be valid enough. A useful system should produce solutions of practical interest to spacecraft missions that will require only minor adjustments when and if the mission is actually flown.

No claim has been made concerning the superiority of the described architecture over existing architectures and systems. The only claim made is that the given architecture facilitates the modeling and optimization of many classes of trajectories. The system remains evolutionary, meaning that it can be changed and enhanced as needed to address problems whose

An Architecture for Spacecmft hjectory Design and Optimization System 571

requirements can not currently be met with the current system architecture. If this is the case, then it is postulated that the majority of the changes needed will be directed to the modification and enhancement of definition of the basic segment; and not so much in the solution methods. References 1. Gbmez, G., Koon, W.S., Lo M.W., Marsden, J.E., Masdemont , J., Ross, S.D., ”Invariant Manifolds, The Spatial Three-Body Problem and Petit Grand Tour of Jovian Moons”, In these Proceedings. 2. Azimov, D.M., Bishop, R.H. “Extremal Rocket Motion with Maximum Thrust in a Linear Central Field”, Journal of Spacecraft and Rockets, Vol. 38, No. 5, Sep-Oct 2001. 3. Whiffen, G.J., Sims, J.A., “Application of the SCD Optimal Control Algorithm to Low-Thrust Escape and Capture Trajectory Optimization” , Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, San Antonio, Texas, February 2001. 4. Lawden, D.E., Optimal Dajectories for Space Navigation, London Butterworths, 1963. 5. Betts, J.T., “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, MarchApril 1998. 6. Hull, D.G., “Conversion of Optimal Control Problems into Parameter Optimization Problems”, Journal of Guidance, Control, and Dynamics, Vol. 20, No. 1, Jan-Feb 1997. 7. Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Education Series, American Institute of Aeronautics and Astronautics, Reston, Virginia, 1999. 8. Belbruno, E.A., Miller, J.K., “Sun-Perturbed Earth-to-Moon Transfer with Ballistic Capture”, Journal of Guidance, Control, and Dynamics, Vol. 16, No.4, 1993. 9. Byrnes, D.V., Bright, L.E., “Design of High-Accuracy Multiple Flyby Trajectories Using Constrained Optimization”, Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Halifax, Nova Scotia, Canada, 1995. 10. Szebehely, V., Theory of Orbits, Academic Press, 1967. 11. Dennis, J.E., Schnabel, R.B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983. 12. Gill, P.E., Murray, W., Saunders, M.A., Wright M.H., Practical Optimization, Cambridge University Press, 1998. 13. Nocedal, J., Wright, S.J., Numerical Optimization, Springer Series in Operations Research, Springer, 1999. 14. Hull, D.G., Calculus of Diflerentials and Optimization, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, www.ae.utexas.edu/-dghull,2002.

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Libration Point Orbits and Applications G. G 6 m q M. W. Lo and J. J. Masdemont ( 4 s . ) @ 2003 World Scientific Publishing Company

RESTRICTED FOUR AND FIVE BODY PROBLEMS IN THE SOLAR SYSTEM F. GABERN and A. JORBA Departament de Matemdtica Aplicada i Andisi Universitat de Barcelona Gmn Via 585, 08007 Barcelona, Spain.

We focus on the dynamics of a small particle near the Lagrangian points of the Sun-Jupiter system. To try to account for the effect of other planets, such as Saturn or Uranus, we develop specific models based on the numerical computation of periodic and quasi-periodic (with two frequencies) solutions of the N-body problem and write them as perturbations of the Sun-Jupiter restricted Three Body Problem.

1. Introduction

The dynamics around the Lagrangian Lq and L5 points of the Sun-Jupiter in the Restricted system have been studied by several authors Three Body Problem using semi-analytical tools such as normal forms or approximate first integrals. 5,109396,11

On the other hand, it is known that Trojan asteroids move near the triangular points of the Sun-Jupiter system. The dynamics of these jovian Trojan asteroids has been studied by many authors using the Outer Solar System model, where the Trojans are supposed to move under the attraction of the Sun and the four main outer planets (Jupiter, Saturn, Uranus and Neptune). This is a strictly numerical model, so the semianalytical tools mentioned above cannot be used in principle. 8f9312

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A.

Jorba

In this paper, we briefly present three intermediate models for the motion of a Trojan asteroid. These models try to simulate in a more realistic way the relative Sun-Jupiter motion and are written as perturbations of the RTBP, such that the semi-analytical tools can be applied. The first model that we present is a natural improvement of the SunJupiter RTBP that includes the effect of Saturn on the motion of Sun and Jupiter. In this model, Sun, Jupiter and Saturn move in a periodic solution of the (non-restricted) planar three body problem, with the same relative period as the real one. Then, it is possible to write the equations of motion of a fourth massless particle that moves under the attraction of these three. This is a restricted four body problem that we call Bicircular Coherent Problem (BCCP, for short). Its detailed construction and study can be found in Ref 4. In the second model, the periodic solution of the BCCP is used as the starting point of the computation of a 2-D invariant torus for which the osculating eccentricity of Jupiter’s orbit is the actual one. In this sense, the Sun-Jupiter relative motion is better simulated by this quasi-periodic solution of the planar three body problem. Afterwards, the equations of motion of a massless particle that moves under the attraction of these three main bodies (supposing that they move in the quasi-periodic solution) are easily derived. We call this restricted four body problem as the Bianular Problem (BAP, for short). The third model is based on the computation of a quasi-periodic solution (with two basic frequencies) of the planar four body problem Sun, Jupiter, Saturn and Uranus. A restricted five body model can be constructed by writting the equations of a massless particle that moves under the influence of the four bodies. We call it Tricircular Coherent Problem (TCCP, for short).

2. The Bicircular Coherent Problem

It is possible to find, in a rotating reference frame, periodic solutions of the planar three body Sun-Jupiter-Saturn problem by means of a continuation method using the masses of the planets as parameters (see Ref for details). The relative Jupiter-Saturn period can be chosen as the actual one, and its

Restricted Four and Five Body Problems in the Solar System 575

related frequency is wsat = 0.597039074021947. Assuming that these three main bodies move on this periodic orbit, it is possible to write the Hamiltonian for the motion of a fourth massless particle as:

+ + +

+ +

where q i = (z- p)2 y2 z2, q: = (z- p + 1)2 y2 z2 and q:at = (z- a 7 ( e ) ) 2+ (y - as(e))2 z2. The functions ai(t9)are periodic functions in t9 = wSatt and can be explicitly computed with a Fourier analysis of the numerical periodic solution of the three body problem. At that point, we want to mention that a Bicircular Coherent problem was already developed in Ref for the Earth-Moon-Sun case to study the dynamics near the Eulerian points.

3. The Bianular Problem

In this section, we compute a quasi-periodic solution, with two basic frequencies, of the planar Sun-Jupiter-Saturn three body problem. This quasiperiodic solution lies on a 2-D torus. As the problem is Hamiltonian, this torus belongs to a family of tori. We look for a torus, on this family, for which the osculating eccentricity of Jupiter’s orbit is quite well adjusted to the actual one. First, we rewrite the Hamiltonian of the planar three body problem in which the computations are done and we revisit a method for computing 2-D invariant tori. Afterwards, the desired quasi-periodic solution is found and the Hamiltonian of the Bianular Problem is explicitly obtained.

3.1. The reduced Hamiltonian of the Three Body Problem

We take the Hamiltonian of the planar three body problem written in the Jacobi coordinates in a uniformly rotating reference frame and we make a canonical change of variables (using the angular momentum first integral)

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F. Gabern and A. Jorba

in order to reduce this Hamiltonian from 4 to 3 degrees of freedom. We obtain:

angular momentum of the system.

3.2. A method for computing 2 - 0 invariant tori

We are interested in finding a quasi-periodic solution (with two frequencies) of a given vector field. We reduce this problem to the one of finding an invariant curve of a suitable PoincarC map. This invariant curve is seen as a truncated Fourier series and our aim is to compute its rotation number and a representation of it. We follow roughly the method developed by Ref 2 . Numerical computation of invariant curves Let be j . = f(z)(z,fE R") an autonomous vector field of dimension n (for example, the reduced field of the three body problem given at 3.1) and a(., t ) 3 at(.) its associated flow. Let us define the PoincarC map as the time T-flow @ T ( - ) ,where T is a prefixed value (T = Tsat, the period of Saturn in the Sun-Jupiter system, in our case).

Rn) Let w be the rotation number of the invariant curve. Let, also, C(T1, be the space of continuous functions from T1in R", and let us define the linear map T, : C(T1, Rn)+ C(T1, Rn)as the translation by w , (T,cp)(8) = cP(6 + w). Let us define F : C(T1,R")

-+ C(T1,R")

as

F(cp)(O)= aT(cP(e))- (LcP)(q VcP E C(T1, Rn). (3) It is clear that the zeros of F in C(T1,Rn) correspond to invariant curves of rotation number w. The equation satisfied is

aT(p(e))= cp(e + w ) ve E T.

(4)

Restricted Four and Five Body Problems in the Solar System 577

The method we want to summarize in this section boils down to looking numerically for a zero of F.Hence, let us write (p(8) as a real Fourier series, cp(6') = A0 4-c ( A k COS(k8)

+ Bk Sin(k8))

Ak, Bk E Rn k E N.

k>O Then, we will fix in advance a truncation value N f for this series (the selection of the truncation value will be discussed later on), and let us try to determine (an approximation to) the 2Nf 1 unknown coefficients Ao, Ak and Bk, 1 5 k 5 N f . To this end, we will construct a discretized version of the map F , as follows: first, we select the mesh of 2Nf 1points on T1,

+

+

and evaluate the function (3) on it. Let F N ~be this discretization of F :

+w),

0 Ij I 2 N f . (6) So, given a (known) set of Fourier coefficients Ao, Ak and Bk (1 5 k 5 N f ) ,we can compute the points (p(8j), then aT((p(8j)) and next the points @T(cp(Bj))- cp(8j + w ) , 0 5 j 5 N f . jfiom these data, we can immediately obtain the Fourier coefficients of aT((p(8))- (p(8 w ) . * T ( c p ( @ j ) )- P(8j

+

To apply a Newton method to solve the equation F N ~= 0, we also need to compute explicitly the differential of F N ~This . can be done easily by applying the chain rule to the process used to compute F N ~Note . that the number of equations to be solved is (2Nf 1). and that the unknowns are (Ao,A1, B1,. . ., AN^, B N)~ ,w and the time T for which we fix the Poincar6 That is, we deal with ( 2 N f 1). 2 map associated to the flow (@T(-)). unknowns. In each step of the Newton method, we solve a non-square linear system by means of a standard least-squares method. We want to mention that this system is degenerated unless we fix (keep constant during the computation) some of the unknowns.

+

+

+

Note that in the case of the reduced three body problem, an integral of motion is still left: the energy. We can easily solve the problem of the degeneracy, induced by it, fixing the time T of the Poincar6 map. Discretization error Once we have solved equation ( 6 ) with a certain tolerance (error in the Newton method; we take tipically lo-"), we still don't have any information on the error of the approximated invariant curve. The reason, as

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A.

Jorba

explained in Ref 2, is that we have not estimated the discretization error; i.e., the error when passing from equation (4) to equation (6). In order to do it, we compute

E(cp,w) =

IW(P(8, z ) ) - cp(8 + w , .>I

in a mesh of points, say, 100 times finer than the mesh ( 5 ) and consider it as an estimation of the error of the invariant curve. the solution obtained is not considered good enough and another one with the same initial approximation for the Newton method but with a greater discretization order N f is computed. The process is repeated until the sub-infinity norm of the discretization error is smaller than

If llE1lm >

3.3. Finding the desired t o m s

The initial approximation to the unknowns in the Newton method is given by the linearization of the Poincark map around a fixed point (a periodic orbit, for the flow) X O .We use the periodic orbit computed in Section 2 for the BCCP model:

xo = %Lt(XO), where, ~ P T , , (.) ~ is the time T,,t-flow corresponding to Hamiltonian (2).

It is easy to see, by looking at the eigenvalues of D@T~,,(XO), that there are two different non-neutral normal directions to the periodic orbit X O . Thus, two families of tori arise from it. We call them Family1 and Family2. We compute a first torus for each family (they are called Torus1 and Torus2) with the method described in 3.2. Once we have a first torus, we want to continue the family it belongs to. This family can be parameterized by the angular momentum K . It is straightforward from the computations that there is a strong relationship between the angular momentum, K , and the osculating orbital elements of Jupiter’s and Saturn’s orbits. As we want to simulate in a more realistic way the Sun-Jupiter relative motion, we are more interested in adjusting Jupiter’s orbital elements than Saturn’s ones. As we have one degree of freedom (we are allowed to set K ) , we select the

Restricted Four and Five Body Problems in the Solar System 579

osculating eccentricity of Jupiter’s orbit as targeting value. Thus, by means of a continuation method, we try to find another torus inside Familyl and Family2 for which the osculating eccentricity of Jupiter (at a given moment) is exactly 0.0484. In order to continue the families, we add to the invariant curve equations the following one: eccen(Q1, Q2

,Q 3 , 9 , p 2 , p3, K ) = e,

where eccen(.) is a function that gives us Jupiter’s osculating eccentricity at a given moment (we evaluate it when Sun, Jupiter and Saturn are in a particular collinear configuration), and e is a fixed constant that is used as a control parameter. We try to continue each family increasing the parameter e to its actual value. In Familyl, we start from Torus1 increasing little by little the parameter e in order to have a good enough initial point for the Newton method in each step of the continuation process. What is observed is that when e increases, the number of harmonics ( N f ) has also to be increased if we want the discretization error of the invariant curve to be smaller than a certain tolerance (tipically we take lo-’). We stop the continuation whm the number of harmonics is about 180. At this moment, if we look at the orbital elements of Jupiter’s and Saturn’s orbits, we see that they do not evolve in the desired direction, but they are getting farther from the real ones. Thus, increasing Jupiter’s eccentricity inside Familyl forces us to move away from the desired solution and we fail in finding an adequate torus in Familyl.

For Family2, we proceed in the same way as before but starting from Torus2. In this case, we are able to increase e up to its actual value (e = 0.0484),the number of harmonics doesn’t grow up very much (actually, if we ask the invariant curve to have an error smaller than lo-’, N f increases from 6 to 9) and the solution obtained is of the planetary type. In Figure 1, we plot the variation of the angular momentum K of the planar SJS Three Body Problem when the parameter e is increased during the continuation process. We can see the projection of the final torus into the configuration space in Figure 2. This solution of the planar Sun-Jupiter-Saturn TBP is what we call the Bianular solution of the TBP. This torus is parameterized with

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Jorba

0.rnlSYlI

0

O

'

W

'

0.01

'

0015

'

001

'

0.025

'

0.W

'

0.W

'

0.01

'

O W

I

Om

Fig. 1. Plot of the evolution of the angular momentum K when the parameter e is increased from 0.00121 (corresponding t o Torus2) to 0.0484 (the desired final value) in the continuation of Family2.

+

+

the angles (el,&) = ( w l t Oy, w2t @)-,where the frequencies are w1 = w,,t = 0.597039074021947 and w2 = = 0.194113943490717 (ais the rotation number of the invariant curve), and @y,2 are the initial phases.

Fig. 2. Projection into the configuration space of the Bianular solution of the planar three body problem Sun-Jupiter-Saturn in the rotating reference frame (left plot) and in an inertial system (right plot).

Restricted Four and Five Body Problems in the Solar System 581

3.4. The Hamiltonian of the BAP Model

Finally, it is possible to obtain the equations of a massless particle that moves under the attraction of the three primaries. The corresponding Hamiltonian is:

where q t = (z-

+ y2 + z 2 , q:

= (z- p

+ 1)2 + y2 + z2, qZat

=

( ~ - a 7 ( e 1 , e 2 ) ) 2 + ( y - - ~ 8 ( e 1 , e 2 ) ) 2=wlt+e: +~2,e1 m d e 2 =w2t+e:. The auxiliar functions ai(O1,02)ii=1+8) are quasi-periodic functions that can be computed by a Fourier analysis of the solution found in 3.3.

4. The Tricircular Coherent Problem

In this section, we describe the computation of a quasi-periodic solution with two basic frequencies of the planar four body problem Sun, Jupiter, Saturn and Uranus (SJSU). We adapt the method described in Section 3.2 for computing invariant curves of maps to this case. First, we write the reduced Hamitonian of the four body problem in which the computations are done. Second, we give an heuristic approximation of the initial point used in the Newton method for computing an invariant curve when the mass of Uranus is equal to zero. Then, we compute a first torus for this case. Finally, by means of a continuation method (taking the mass of Uranus as the parameter), we compute a quasi-periodic solution of the SJSU planar problem.

4.1. The Hamiltonian of the

SJSU problem

We take the Hamiltonian of the planar four body problem written in the generalized Jacobi coordinates (Figure 3) in a uniformly rotating reference

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frame and we make a canonical change of variables (using the angular momentum first integral) in order to reduce this Hamiltonian from 6 to 5 degrees of freedom. We obtain the following Hamiltonian:

( P i +P:)

1 a +(P: + P i ) - K - 27 r

SUN

Fig. 3.

Jacobi coordinates for the Sun, Jupiter, Saturn and Uranus four body problem.

4.2. Computation of a first torus

As a first approximation, we suppose that Uranus has mass equal to zero and that it is moving in a Kepierian orbit around the Sun. Using the method for computing invariant curves, described in 3.2, we obtain a 2-D invariant torus for this case.

If we suppose that, at t = 0, Sun, Jupiter, Saturn and Uranus are in

Restricted Four and Five Body Problems in the Solar System 583

a particular collinear configuration, it is easy to obtain an approximate initial condition for the Newton method by taking the Keplerian one for Uranus, and the periodic orbit XO (used in the construction of the BCCP in Section 2) for Sun, Jupiter and Saturn. As we have already seen, the period where ws,t = 0.597039074021947 of this (periodic) orbit is T = Tsat = is the relative frequency of Saturn in the Sun-Jupiter rotating system. This will be the first frequency of the 2-D invariant torus.

&,

If we integrate the flow corresponding to Hamiltonian (7) in the time taking as initial condition a point of the periodic orbit interval t E [0,Ts,t] XOfor the Sun-Jupiter-Saturn system and the Keplerian approximation for Uranus, we find that the orbit corresponding to Sun, Jupiter and Saturn obviously closes (it is a periodic orbit of period Tsat) and the one corresponding to Uranus turns one lap-odd (see Figure 4).

4

a

-1

0

1

2

3

Fig. 4. Heuristic first approximation of the Uranus motion (exterior orbit) around the Sun. The inside orbit corresponds to a periodic orbit of Saturn. The lines show the starting and final points of the Uranus orbit for an interval of time of Tsat.

We are interested in measuring the angle that Uranus covers in its trajectory (actually, the angle between the two straight lines in Figure 4). We relate this angle with the rotation number of the invariant curve that we are looking for. It is possible, from the initial an final points of the integration, to compute the value of the angle: wo = 2.749448441. Let us note that this angle is very close to the following number: w- = - 2rwura Wsat

mod 27~= 2.750807556,

(8)

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where wUTa = 0.858425538978989 is the relative frequency of Uranus in the Sun-Jupiter rotating system. Thus, if we impose 6 to be the rotation number of the invariant curve, the second frequency of the 2-D invariant torus that we are computing will be w,,, Uranus’s frequency.

4.3. The l k i r c u l a r solution of the SJSU problem

Once we have computed an invariant torus for the case muTa = 0, we proceed by a continuation method to increase the parameter muTaup to its actual value. During the continuation, the two frequencies, w1 = Wsat and w2 = wUTDare kept constant. Thus, we obtain a quasi-periodic solution (that moves on a 2-D torus parameterized by the two angles 81 = w l t + @ and O2 = w 2 t 8;) of the reduced four body field (7).

+

In Figure 5, we can see the projection into the configuration space of this torus in a rotating frame (left plot) and in an inertial frame (right plot). We call this solution “Tricircular Coherent Solution” of the SJSU four body problem.

a,

,

,

,

,

,

.

,

,

Fig. 5 . Quasi-periodic solution for the four body Sun-Jupiter-Saturn-Uranus Problem. The exterior orbit concerns to Uranus, the one in the middle to Saturn and the interior one (seen as a small point that librates around the point ( - l , O ) , in the left plot) is the relative Sun-Jupiter’s orbit. The left plot is represented in the rotating coordinates and the right one in an inertial reference frame.

Restricted Four and Five Body Problems in the Solar System 585

4.4. The Harniltonian of the TCCP Model

Finally, it is possible to write the equations of motion of a massless particle that moves under the attraction of the four primaries, supposing that they move on the tricircular solution. The corresponding Hamiltonian is:

: [

-a6(e1,e2)

qJ

+ + +

1-

+P +msat + mura qsat

,

qura

+ +

+ +

where q i = (z y2 z2, q: = (z- p 1)2 y2 z2, q:at = (z- ~ 7 ( & , 0 2 )+ ) ~(Y - a ~ ( 0 1 , 8 2 )+ ) ~z 2 , = (z - a9(81,e2))2 (Y alo(el,e2))2 z 2 , el = wsatt ey and e2 = wurst e;.

+

+

The auxiliar functions ai (el,82)(i=l+10) are quasi-periodic functions that can be computed by a Fourier analysis of the tricircular solution found in 4.3.

5. Conclusions

We have seen a particular case of a methodology for constructing semianalytic models of the Solar System and write them as “perturbations” of the Sun-Jupiter RTBP. For instance, if a quasi-periodic solution of the N-Body Problem with m frequencies is known, it is then possible to write the Hamiltonian of the Restricted Problem of ( N 1) bodies as:

+

1

H = -al(e)(p: +Pi + P 3 2

+ az(O)(zPz+ YP, + ZP,)

where the functions ai(f3)are also quasi-periodic with the same m frequencies (0 E Tm) and pi is the distance between the particle and the i-th body written in a “rotating-pulsating” reference system. All these models (as BCCP, BAP and TCCP) are specially written in order that semi-analytical tools (such as Normal Forms or numerical First Integrals techniques) can be applied.

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Acknowledgements

This work has been partially supported by the Spanish CICYT grant BFM2000-0623, the Catalan CIRIT grant 2001SGR-00070 and DURSI.

References 1. M.A. Andreu. The quasi-bicircular problem. PhD thesis, Univ. Barcelona, 1998. 2. E. Castella and A. Jorba. On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem. Celestial Mech., 76( 1):35-54, 2000. 3. A. Celletti and A. Giorgilli. On the stability of the Lagrangian points in the spatial Restricted Three Body Problem. Celestial Mech., 50(1):31-58, 1991. 4. F. Gabern and A.Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete and Continuous Dynamical Systems - series B. Volume 1, Number 2. 143-182, 2001. 5. A. Giorgilli, A. Delshams, E. Fontich, L. Galgani, and C. Sim6. Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem. J. Differential Equations, 77:167-198, 1989. 6. A. Giorgilli and C. Skokos. On the stability of the Trojan asteroids. Astron. Astrophys., 317:254-261, 1997. 7. A. Jorba. A methodology for the numerical computation of normal forms, centre manifolds and f i s t integrals of Hamiltonian systems. Exp. Math., 8( 2):155-195, 1999. 8. H.F. Levison, E.M. Shoemaker, and C.S. Shoemaker. The long-term dynamical stability of Jupiter’s Trojan asteroids. Nature, 385:42-44, 1997. 9. E. Pilat-Lohinger, R. Dvorak, and CH. Burger. Trojans in stable chaotic motion. Celestial Mech., 73:117-126, 1999. 10. C. Sim6. Estabilitat de sistemes Hamiltonians. Mem. Real Acad. Cienc. Artes Barcelona, 48(7):303-348, 1989. 11. C. Skokos and A. Dokoumetzidis. Effective stability of the Trojan asteroids. Astron. Astrophys., 367:729-736, 2000. 12. K. Tsiganis, R. Dvorak, and E. Pilat-Lohinger. Thersites: a ‘jumping’ Trojan? Astron. Astrophys., 354:1091-1100, 2000.

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

INVARIANT MANIFOLDS, THE SPATIAL THREE-BODY PROBLEM AND PETIT GRAND TOUR OF JOVIAN MOONS G. GOMEZ IEEC €4 Departament de Matemcitica Aplicada i Anhlisi Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain W.S. KOON Control and Dynamical Systems, California Institute of Technology, M C 107-81, Pasadena, California 91125, USA M.W. LO Navigation and Mission Design, Jet Propulsion Laboratory, California Institute of Technology, M / S 3Ol-l4OL,Pasadena, California 91 109, USA

J.E. MARSDEN Control and Dynamical Systems, California Institute of Technology, M C 107-81, Pasadena, California 91125, USA J.J. MASDEMONT IEEC €4 Departament de Matemcitica Aplicada I, Universitat Polittcnica de Catalunya, E. T.S.E.I. B., Diagonal 647, 08028 Barcelona, Spain S.D. ROSS Control and Dynarnical Systems, California Institute of Technology, M C 107-81, Pasadena, California 91125, USA

This paper is a summary of a longer paper, “Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design,”’ which received the award for the Best Paper at the AIAA Astrodynamics Specialist Conference, Quebec City, Canada, August 2001. 587

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The invariant manifold structures of the collinear libration points particular, the stable and unstable invariant manifold “tubes” associated to for the spatial restricted three-body problem provide the framework for understanding complex dynamical phenomena from a geometric point of view. In libration point periodic orbits are phase space structures that provide a conduit for orbits between the primary bodies in separate three-body systems. These invariant manifold tubes can be used to construct new spacecraft trajectories, such as a “Petit Grand Tour” of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. The current work extends the results t o the spatial case.

1. Introduction

New space missions are increasingly more complex, requiring new and unusual kinds of orbits to meet their scientific goals, orbits which are not easily found by the traditional conic approach. The delicate heteroclinic dynamics employed by the Genesis Discovery Mission dramatically illustrates the need for a new paradigm: study of the three-body problem using dynamical systems theory as laid out by P 0 i n c a r 6 . ~ ~ ~ ~ ~ It appears that the dynamical structures of the three-body problem (such as stable and unstable manifolds, and bounding surfaces), reveal much about the morphology and transport of particles within the solar system, whether they are asteroids, dust grains, or spacecraft. The crossfertilization between the study of the natural dynamics in the solar system and engineering applications has produced a number of new techniques for constructing spacecraft trajectories with desired behaviors, such as rapid transition between the interior and exterior Hill’s regions, resonance hopping, temporary capture, and c~llision.~ The invariant manifold structures associated to the collinear libration points for the restricted three-body problem, which exist for an interesting range of energies, provide a framework for understanding these dynamical phenomena from a geometric point of view. In particular, the stable and unstable invariant manifold tubes associated to L1 and Lz orbits are phase space structures that conduct particles to and from the smaller primary body (e.g., Jupiter in the Sun-Jupiter-comet three-body system), and between primary bodies for separate three-body systems (e.g., Saturn and Jupiter in the Sun-Saturn-comet and the Sun-Jupiter-comet three-body

Invariant Manifolds, the Spatial Three-Body Problem and Petit Gmnd Tour 589

systems). Furthermore, these invariant manifold tubes can be used to produce new techniques for constructing spacecraft trajectories with interesting characteristics. These may include mission concepts such as a low energy transfer from the Earth to the Moon7 and a “Petit Grand Tour” of the moons of Jupiter.g Using the phase space tubes in each 3-body system, we were able to construct a transfer trajectory from the Earth which executes an unpropelled (i.e., ballistic) capture at the Moon. An Earth-to-Moon trajectory of this type, which utilizes the perturbation by the Sun, requires less fuel than the usual Hohmann t r a n ~ f e r . ~ Moreover, by decoupling the Jovian moon n-body system into several three-body systems, we can design an orbit which follows a prescribed itinerary in its visit to Jupiter’s many moons. In an earlier study of a transfer from Ganymede to Europa,8 we found our transfer AV to be half the Hohmann transfer value. As an example, we generated a tour of the Jovian moons: starting beyond Ganymede’s orbit, the spacecraft is ballistically captured by Ganymede, orbits it once and escapes, and ends in a ballistic capture at Europa. One advantage of this Petit Grand Tour as compared with the Voyager-type flybys is the “leap-frogging” strategy. In this new approach to space mission design, the spacecraft can circle a moon in a loose temporary capture orbit for a desired number of orbits, perform a transfer AV and become ballistically captured by another adjacent moon for some number of orbits, etc. Instead of flybys lasting only seconds, a scientific spacecraft can orbit several different moons for any desired duration. The design of the Petit Grand Tour in the planar case is guided by two main ideas. First, the Jupiter-Ganymede-Europa-spacecraft four-body system is approximated as two coupled planar three-body systems. Then, as shown in Figure 1, the invariant manifold tubes of the two planar threebody systems are used to construct an orbit with the desired behaviors. This initial solution is then refined to obtain a trajectory in a more accurate 4-body model. The coupled 3-body model considers the two adjacent moons competing for control of the same spacecraft as two nested 3-body systems (e.g., Jupiter-Ganymede-spacecraft and Jupiter-Europa-spacecraft). When close to the orbit of one of the moons, the spacecraft’s motion is dominated by the 3-body dynamics of the corresponding planet-moon system.

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Between the two moons, the spacecraft’s motion is mostly planet-centered Keplerian, but is precariously poised between two competing 3-body dynamics. In this region, orbits connecting unstable libration point orbits of the two different 3-body systems may exist, leading to complicated transfer dynamics between the two adjacent moons. We seek intersections between invariant manifold tubes which connect the capture regions around each moon. In the planar case, these tubes separate transit orbits (inside the tube) from non-transit orbits (outside the tube). They are the phase space structures that provide a conduit for orbits between regions within each three-body systems as well as between primary bodies for separate threebody systems.‘

Extending Results from Planar Model to Spatial Model. Previous work based on the planar circular restricted three-body problem (PCRSBP) revealed the basic structures controlling the dynamic^.'^^^'^^ But current missions (such as Genesis2) and future missions will require three-dimensional capabilities, such as control of the latitude and longitude of a spacecraft’s escape from and entry into a planetary or moon orbit. For example, the proposed Europa Orbiter mission desires a capture into a high inclination polar orbit around Europa. Three-dimensional

Fig. 1. The Coupled 3-Body Model. (a) Find an intersection between dynamical channel enclosed by Ganymede’s L1 periodic orbit unstable manifold and dynamical channel enclosed by Europa’s Lz periodic orbit stable manifold (shown in schematic). (b) Integrate forward and backward from patch point (with AV to take into account velocity discontinuity) to generate desired transfer between the moons (schematic).

Invariant Manifolds, the Spatial Three-Body Problem and Petit Grand Tour 591

capability is also required when decomposing an n-body system into threebody systems that are not co-planar, such as the Earth-Sun-spacecraft and Earth-Moon-spacecraft systems. These demands necessitate the extension of earlier results to the spatial model (CR3BP). In our current work on the spatial three-body problem,’ we are able to show that the invariant manifold structures of the collinear libration points still act as the separatrices for two types of motion, those inside the invariant manifold “tubes” are transit orbits and those outside the “tubes” are non-transit orbits. We have also designed an algorithm for constructing orbits with any prescribed itinerary and obtained some initial results on the basic itinerary. Furthermore, we have applied the new techniques to the construction of a three dimensional Petit Grand Tour of the Jovian moon system. By approximating the dynamics of the Jupiter-EuropaGanymede-spacecraft 4-body problem as two 3-body subproblems, we seek intersections between the channels of transit orbits enclosed by the stable and unstable manifold tubes of different moons. In our example, we have designed a low energy transfer trajectory from Ganymede to Europa that ends in a high inclination orbit around Europa. See Figure 2. Focus of this Paper. In this paper, we will mainly focus on the key ideas that lead to the construction of the Petit Grand Tour. For more details of this work, the reader can consult our full paper published in Advances in the Astronautical 5’ciences.l

2. Invariant Manifold as Separatrix

Review of Planar Case. Recall that in the planar Jupiter-Moonspacecraft 3-body system (PCRSBP), for an energy value just above that of L2, the Hill’s region contains a ‘heck” about L1 and L2 and the spacecraft can make transition through these necks. More precisely, in each equilibrium region around L1 and L2, the dynamics of the spacecraft is of the form saddle and center and there exist 4 types of orbits: loill

(1) an unstable periodic orbit (black oval); (2) four cylinders of asymptotic orbits that wind onto or off this periodic orbit; they form pieces of stable and unstable manifolds;

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Ganymede's orbit Europa'sorbit

Close approach nvmede I

.;..

\\

.

.

Injection into high inclination orbit around Europa I

Fig. 2. The three dimensional Petit Grand Tour space mission concept for the Jovian moons. (a) We show a spacecraft trajectory coming into the Jupiter system and transferring from Ganymede t o Europa using a single impulsive maneuver, shown in a Jupiter-centered inertial frame. (b) The spacecraft performs one loop around Ganymede, using no propulsion at all, as shown here in the Jupiter-Ganymede rotating frame. (c) The spacecraft arrives in Europa's vicinity at the end of its journey and performs a final propulsion maneuver to get into a high inclination circular orbit around Europa, as shown here in the Jupiter-Europa rotating frame.

(3) transit orbits which the spacecraft can use to make a transition from one region to the other; for example, passing from the exterior region (outside moon's orbit) into the moon temporary capture region (bubble surrounding moon) via the neck region; (4) nontransit orbit where the spacecraft bounces back to its original region.

Invariant Manifolds, the Spatial Three-Body Problem and Petit Grand Tour 593

h

$

b

M

0

c m c

.C

e

v

h

K

(rotating frame) (4

x (rotating frame) (b)

Fig. 3. (a) Hill’s region (schematic, the region in white), which contains a “neck” about L1 and L z . (b) The flow in the region near L z , showing a periodic orbit around L I Z , an asymptotic orbit winding onto the periodic orbit, two transit orbits and two non-transit orbits. A similar figure holds for the region around L1.

Furthermore, these two-dimensional tubes partition the threedimensional energy manifold and act as separatrices for the flow through the equilibrium region: those inside the tubes are transit orbits and those outside the tubes are non-transit orbits. For example in the Jupiter-moon system, for a spacecraft to transit from outside the moon’s orbit to the moon capture region, it is possible only through the L2 periodic orbit stable manifold tube. Hence, stable and unstable manifold tubes control the transport of material to and from the capture region.

Results of the Spatial Case. This planar result generalizes readily to the spatial case.12 For the dynamics near the equilibrium point, instead of the form saddle and center, we have saddle, center, and center. The last part corresponds to the harmonic motion in the z-direction. Since it is more difficult to draw spatial figures, we will still use the planar case to do the illustration. Again, there are 4 types of orbits, as depcited in Figure 4: (1) a large number of bounded orbits, both periodic and quasi-periodic, which together form a 3-sphere, i.e., instead of a periodic orbit S’ in the planar case, you have a S3 of bounded orbits in the spatial case; it is an example of a normally hyperbolic invariant manzfold (NHIM)13 where the stretching and contraction rates under the linearized dynam-

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ics transverse to the 3-sphere dominate those tangent to the 3-sphere; (2) four cylinders of asymptotic orbits that wind onto and off this 3sphere; the only difference from the planar case is that, instead of two-dimensional invariant manifold tubes, one has four-dimensional invariant manifold tube: S3 x R; (3) transit and nontransit orbits.

x (Jupiter-Moon rotating frame)

(4

in the region Fig. 4. (a) The projection of invariant manifolds Wi;:,o. and WiL;:,o, M of the position space. (b) A close-up of the intersection region between the Poincarb cuts of the invariant manifolds on the U3 section (z = 1 - p , y > 0). (c) Location of Lagrange point orbit invariant manifold tubes in position space. Stable manifolds are lightly shaded, unstable manifolds are darkly. The location of the Poincar6 sections ( U i , U2, U3, and U4)are also shown. (d) A close-up near the moon.

Invariant Manifolds, the Spatial Three-Body Problem and Petit Grand Tour 595

Now, since the invariant manifold tubes are four-dimensional tubes in a fivedimensional energy manifold, they again act as separatrices for the flow through the equilibrium region: those inside the tubes are transit orbits and they transit from one region to another; those outside the tubes are non-transit orbits and they bounce back to their original region. In fact, it can be shown that for a energy value just above that of L1 ( L 2 ) , the nonlinear dynamics in the equilibrium region R1 (R2) is qualitatively the same as the linearized picture that we have shown a b o ~ e This. geometric ~ ~ insight ~ will ~ be ~ used ~ below ~ to~guide ~ our~ numerical explorations in constructing orbits with prescribed itineraries.

3. Constructing Orbits with Desired Itinerary

A key difficulty in the spatial case is to figure out how to link appropriate invariant manifold tubes together to construct orbit that visits the desired regions in a desired order. Review of Planar Case. In the planar case, it is quite straightforward. Let us take constructing an ( X ;M , I ) orbit as an example. This orbit goes from the exterior region ( X ) to the interior region ( I ) passing through the moon region ( M ) . Recall that for the planar case: the invariant manifold tubes separate two types of motion. The orbits inside the tube transit from one region to another; those outside the tubes bounce back to their original region. Since the upper curve in Figure 4(b) is the Poincark cut of the stable manifold of the periodic orbit around L1 in the U , plane, a point inside that curve is an orbit that goes from the moon region to the interior region, so this region can be described by the label (; M , I ) . Similarly, a point inside the lower curve of Figure 4(b) came from the exterior region into the moon region, and so has the label ( X ;M ) . A point inside the intersection AM of both curves is an ( X ;M , I ) orbit, so it makes a transition from the exterior region to the interior region, passing through the moon region. Other more complicated orbits can be constructed by choosing appropriate Poincare sections and linking invariant manifold tubes in right order.

~

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Extension to Spatial Case. Since the key step in the planar case is to find the intersection region inside the two Poincard cuts, a key difficulty is to determine how to extend this technique to the spatial case. Take as an example the construction of a transit orbit with the itinerary ( X ;M , I ) that goes from the exterior region to the interior region of the Jupitermoon system. Recall that in the spatial case, the unstable manifold “tube” of the NHIM around L2 which separates the transit and non-transit orbits is topologically S3 x R. For a transversal cut at 2 = 1 - p (a hyperplane through the moon), the Poincark cut is a topological 3-sphere S3 (in R4).It is not obvious how to find the intersection region inside these two Poincark cuts ( S 3 )since both its projections on the (9,y)-plane and the ( z ,,?)-plane are (2-dimensional) disks D 2 . (One easy way to visualize this is to look at the equation: E2 q2 rj2 = r2 = r i r:. that describes a 3-sphere in R4.Clearly, its projections on the ((,()-plane and the (q,i)-plane are 2-disks as re and r9 vary from 0 to r and from r to 0 respectively.)

+ i2+ +

+

However, in constructing an orbit which transitions from the outside to the inside of a moon’s orbit, suppose that we might also want it to have other characteristics above and beyond this gross behavior. We may want to have an orbit which has a particular z-amplitude when it is near the moon. If we set z = c, i = 0 where c is the desired z-amplitude, the problem of finding the intersection region inside two Poincark cuts suddenly becomes tractable. Now, the projection of the Poincark cut of the above unstable manifold tube on the (y,y)-plane will be a closed curve and any point inside this curve is a ( X ; M )orbit which has transited from the exterior region to the moon region passing through the La equilibrium region. See Figure 5. Similarly, we can apply the same techniques to the Poincark cut of the stable manifold tube to the NHIM around L1 and find all ( M , I ) orbits inside a closed curve in the (y,y)-plane. Hence, by using z and i as the additional parameters, we can apply the similar techniques that we have developed for the planar case in constructing spatial trajectories with desired itineraries. See Figure 5(a).

Invariant Manifolds, the Spatial Three-Body Problem and Petit Grand Tour 597

0.6 0.4

0.2 "

0 -0.2 -0.4

-0.6 0

0.010

0.005

0.015

V

Z

(4

N

0.W 0.006 0.008 0.010 0.012 0.014

Y

(a) Shown in black are the y?j (left) and z i (right) projections of the Fig. 5 . 3-dimensional object C t u 2 , the intersection of W t ( M E ) with the Poincar6 section z = 1 - p. The set of points in the y?j projection which approximate a curve, y z r i r , all have ( z , i ) values within the small box shown in the z i projection (which appears as a thin strip), centered on (%',if). This example is computed in the Jupiter-Europa are shown, the intersections of system for C = 3.0028. (b) The curves Gan$i and Eur-y:i GanW:(M1) and EurW$(M2)with the Poincar6 section 171 in the Jupiter-Europa rotating frame, respectively. Note the small region of intersection, int(Gan&) rl int(Eury:i), where the patch point is labeled. (c) The ( X , M , I ) transit orbit corresponding to the initial condition in (b). The orbit is shown in a 3D view. Europa is shown to scale.

4. Spatial Petit Grand Tour of Jovian Moons

We now apply the techniques we have developed to the construction of a fully three dimensional Petit Grand Tour of the Jovian moons, extending an earlier planar result.8 We here outline how one systematically constructs a spacecraft tour which begins beyond Ganymede in orbit around Jupiter, makes a close flyby of Ganymede, and finally reaches a high inclination

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orbit around Europa, consuming less fuel than is possible from standard two-body met hods. Our approach involves the following three key ideas: (1) treat the Jupiter-Ganymede-Europa-spacecraft 4-body problem as two coupled circular restricted 3-body problems, the Jupiter-Ganymedespacecraft and Jupiter-Europa-spacecraft systems; (2) use the stable and unstable manifolds of the NHIMs about the JupiterGanymede L1 and LZto find an uncontrolled trajectory from a jovicentric orbit beyond Ganymede to a temporary capture around Ganymede, which subsequently leaves Ganymede’s vicinity onto a jovicentric orbit interior to Ganymede’s orbit; (3) use the stable manifold of the NHIM around the Jupiter-Europa LZ to find an uncontrolled trajectory from a jovicentric orbit between Ganymede and Europa to a temporary capture around Europa. Once the spacecraft is temporarily captured around Europa, a propulsion maneuver can be performed when its trajectory is close to Europa (100 km altitude), taking it into a high inclination orbit about the moon. Furthermore, a propulsion maneuver will be needed when transferring from the Jupiter-Ganymede portion of the trajectory to the JupiterEuropa portion, since the respective transport tubes exist at different energies.

Ganymede to Europa Transfer Mechanism. The construction begins with the patch point, where we connect the Jupiter-Ganymede and JupiterEuropa portions, and works forward and backward in time toward each moon’s vicinity. The construction is done mainly in the Jupiter-Europa rotating frame using a Poincar6 section. After selecting appropriate energies in each 3-body system, respectively, the stable and unstable manifolds of each system’s NHIMs are computed. Let GanWt(M1)denote the unstable manifold of Ganymede’s L1 NHIM and EurW$(M2)denote the stable manifold for Europa’s. L2 NHIM. We look at the intersection of GanWT(M1) and EurW;(M2) with a common Poincar6 section, the surface Ul in the Jupiter-Europa rotating frame, defined earlier. See Figure 5(b). Note that we have the freedom to choose where the Poincar6 section is with respect to Ganymede, which determines the relative phases of Europa

Invariant Manifolds, the Spatial Three-Body Problem and Petit Grand Tour 599

and Ganymede at the patch point. For simplicity, we select the U1 surface in the Jupiter-Ganymede rotating frame to coincide with the Ul surface in the Jupiter-Europa rotating frame at the patch point. Figure 5(b) shows the curves Gany:i and Euryzi on the (z,k)-plane in the Jupiter-Europa rotating frame for all orbits in the Poincar6 section with points ( z ,2 ) within (0.0160 f 0.0008, f0.0008). The size of this range is about 1000 km in z position and 20 m/s in z velocity. From Figure 5(b), an intersection region on the zk-projection is seen. We pick a point within this intersection region, but with two differing y velocities; one corresponding to GanWu +(M1),the tube of transit orbits coming from Ganymede, and the other corresponding to EurW$(M2), the orbits heading toward Europa. The discrepancy between these two y velocities is the AV necessary for a propulsive maneuver to transfer between the two tubes of transit orbits, which exist at different energies.

Four-Body System Approximated by Coupled PCR3BP. In order to determine the transfer AV, we compute the transfer trajectory in the full 4-body system, taking into account the gravitational attraction of all three massive bodies on the spacecraft. We use the dynamical channel intersection region in the coupled 3-body model as an initial guess which we adjust finely to obtain a true 4-body bi-circular model trajectory. Figure 5(c) is the final end-to-end trajectory. A AV of 1214 m/s is required at the location marked. We note that a traditional Hohmann (patched 2-body) transfer from Ganymede to Europa requires a AV of 2822 m/s. Our value is only 43% of the Hohmann value, which is a substantial savings of on-board fuel. The transfer flight time is about 25 days, well within conceivable mission constraints. This trajectory begins on a jovicentric orbit beyond Ganymede, performs one loop around Ganymede, achieving a close approach of 100 km above the moon's surface. After the transfer between the two moons, a final additional maneuver of 446 m/s is necessary to enter a high inclination (48.6") circular orbit around Europa at an altitude of 100 km. Thus, the total AV for the trajectory is 1660 m/s, still substantially lower than the Hohmann transfer value.

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5. Conclusion

In our current work on the spatial three-body problem, we have shown that the invariant manifold structures of the collinear libration points still act as the separatrices for two types of motion, those inside the invariant manifold “tubes” are transit orbits and those outside the “tubes” are non-transit orbits. We have also designed a numerical algorithm for constructing orbits with any prescribed finite itinerary in the spatial three-body planet-moonspacecraft problem. As our example, we have shown how to construct a spacecraft orbit with the basic itinerary (X;M, I ) and it is straightforward to extend these techniques to more complicated itineraries. Furthermore, we have applied the techniques developed in this paper toward the construction of a three dimensional Petit Grand Tour of the Jovian moon system. Fortunately, the delicate dynamics of the Jupiter-EuropaGanymede-spacecraft 4-body problem are well approximated by considering it as two 3-body subproblems. One can seek intersections between the channels of transit orbits enclosed by the stable and unstable manifold tubes of the NHIM of different moons using the method of Poincark sections. With maneuvers sizes (AV) much smaller than that necessary for Hohmann transfers, transfers between moons are possible. In addition, the three dimensional details of the encounter of each moon can be controlled. In our example, we designed a trajectory that ends in a high inclination orbit around Europa. In the future, we would like to explore the possibility of injecting into orbits of all inclinations.

References 1. Gbmez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross [2001],Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design, Advances in the Astronautical Sciences, volume 109,part 1, p. 3-22, AAS 01-301. 2. Howell, K., B. Barden and M. Lo, Application of Dynamical Systems Theory to Trajectory Design for a Libration Point Mission, Journal of the Astronautical Sciences, 45,No. 2, April-June 1997, pp. 161-178. 3. Gbmez, G., J. Masdemont and C. Simb, Study of the Transfer from the Earth to a Halo Orbit around the Equilibrium Point L1, Celestial Mechanics and Dynamical Astronomy 56 (1993) 541-562 and 95 (1997), 117-134. 4. Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross, The Genesis Trajectory

Invariant Manifolds, the Spatial Three-Body Problem and Petit Grand Tour 601

5.

6.

7.

8.

9.

10. 11. 12.

13. 14. 15. 16. 17.

18.

19.

and Heteroclinic Connections, AAS/AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, 1999, AAS99-451. Koon, W.S., M.W. Lo, J.E. Marsden, and S.D. Ross, Heteroclinic Connections between Periodic Orbits and Resonance Transitions in Celestial Mechanics, Chaos 10(2), 2000, 427-469. Koon, W. S., M.W. Lo, J. E. Marsden and S.D. Ross [2001], Resonance and Capture of Jupiter Comets, Celestial Mechanics and Dynamical Astronomy 81(1-2), 27-38. Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2001], Low Energy Transfer to the Moon, Celestial Mechanics and Dynamical Astronomy 81(12), 63-73 Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2002], Constructing a Low Energy Transfer between Jovian Moons, Contemporary Mathematics 292, 129-145. Belbruno E.A. and J.K. Miller, Sun-Perturbed Earth-to-Moon Transfer with Ballistic Capture [1993], Journal of Guidance, Control and Dynamics 1 6 , 770-775. Conley, C.,Low Energy Transit Orbits in the Restricted Three-Body Problem. SIAM J . Appl. Math. 16, 1968, 732-746. McGehee, R. P., Some Homoclinic Orbits for the Restricted Three-Body Problem, Ph.D . thesis, 1969, University of Wisconsin. Appleyard, D.F., Invariant Sets near the Collinear Lagrangian Points of the Nonlinear Restricted Three-Body Problem, Ph.D. thesis, 1970, University of Wisconsin. Wiggins, S. [1994] Normally Hyperbolic Invariant Manifolds in Dynamicat Systems, Springer-Verlag, New York. Moser, J., On the Generalization of a Theorem of A. Liapunov, Comm. Pure Appl. Math., XI, 1958, 257-271. Hartman, P.,Ordinary Differential Equations, Wiley, New York, 1964. Wiggins, S., L. Wiesenfeld, C. J a E and T. Uzer [2001] Impenetrable Barriers in Phase Space, Phys. Rev. Lett. 86, 5478. Gdmez, G. and J. Masdemont, Some Zero Cost Transfers between Libration Point Orbits, Advances in the Astronautical Sciences (2000), Volume 105, Part 2, p. 1199-1216, AAS 00-177. Jorba, A. and J. Masdemont, Dynamics in the Center Manifold of the Collinear Points of the Restricted Three Body Problem, Physica D 132, 1999, 189-213. Gbmez, G., A. Jorba, J. Masdemont and C. Simd, Dynamics and Mission Design near Libration Points, Vol 111, Advanced Methods for Collinear Points, World Scientific, 2001.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

PERTURBING ACTION OF THE EARTH’S THIRD-DEGREE HARMONICS ON PERIODIC ORBITS AROUND GEOSTATIONARY EQUILIBRIA J. KACZMAREK, I. WYTRZYSZCZAK Astronomical Observatory of A . Mickiewict University, ul. Stoneczna 36, 60-286 Potnali, Poland

I. GACKA Astronomical Institute of Wroctaw University, ul. M. Kopernika 11, 51-622 Wroctaw, Poland

We investigate the influence of third-degree and order harmonics on periodic orbits around the stable and unstable points of the geostationary orbit. Although higher harmonics displace an orbit primarily along the equator and change its period, they preserve the orbital stability induced by the J2 and 522 terms of the geopotential.

1. Introduction Since the 1960s (Blitzer et al. 2; Musen and Bailie 7; Morando 6 ; Blitzer 3 , it has been widely known that the geostationary orbit has two stable and two unstable equilibria. The motion around the unstable points is a short-period libration; near stable points diurnal and long-period ( w 2.3-year) oscillations exist. Blitzer has also shown that higher-degree harmonics cause a longitude shift of stationary points. Recently Lara and Elipe generated families of planar periodic orbits emanating from both kinds of geostation603

604

J . Kaczmarek, I. Wytrzyszczak and I. Gacka

ary points. They have shown that, in the vicinity of stable points, periodic orbits are stable, while around unstable points families of periodic orbits are generally unstable. However, for an unstable point it is possible to find stable periodic orbits, but bifurcation occurs far from equilibrium. Their calculations have been performed with a truncated geopotential, limited to J 2 and J 2 2 terms. Our goal is to check the influence of the third-degree harmonics on the periodicity of these orbits. We limite calculations to orbits lying not far from a given equilibrium.

2. Method of calculation

We consider the motion of a satellite in a synodic reference frame rotating with the Earth. The origin at the center of mass of the Earth, with the xaxis directed toward the equator at the longitude of a selected geostationary point. Table 1. The GRIM5 coefficients after rotation to appropriate unstable or stable points. The position of each point is indicated by the value of X given in the top row. ~

coef.

GRIM5

-1409288

-1104038

~

~~

7500711

7500594

-2.1026e-10 1.7104e-09 C22 1.5746e-06 S Z Z -9.0388e-07

-6.4380e-10 -5.4430e-10 1.6351e-09 1.5985e-09 1.8018e-06 1.8156e-06 O.eO -2.2284e-07

1.5985e-09 1.5984e-09 6.4380e-10 6.4413e-10 -1.8156e-06 - 1.8156e-06 O.eO -7.4447e-10

2.1930e-06 2.6799e-06 3.0906e-07 -2.1143e-07 1.0057e-07 1.9720e-07

2.0499e-06 2.0967e-06 6.9631e-07 8.2390e-07 3.6685e-07 3.7330e-07 -2.9502e-08 -7.5096e-08 -6.7539e-08 -2.7704e-08 2.1962e-07 2.1081e-07

8.2390e-07 -2.0499e-06 -3.7329e-07 2.9502e-08 -2.1081e-07 -6.7539e-08

Czl S21

c31 s31 c32 532 C33 S33

8.2432e-07 -2.0498e-06 -3.7331e-07 2.9349e-08 -2.1078e-07 -6.7669e-07

The values of the harmonic coefficients have been recomputed with respect to the new axis. They are presented in Table 2. The second column in the table gives the coefficients of the GRIM5 Earth Potential Model (Biancale et 01. l ) . The third (A = -1409288), and fifth (A = 7500711) columns show the same coefficients rotated to the Earth’s principal axes of inertia - i.e., to the unstable and stable points when only J 2 , C 2 2 and 5 2 2 har-

Perturbing Action of Earth’s Third-Degree Harmonics 605

monics are taken into account. The columns labeled as X = -11?4038 and X = 75?0594 contain the GRIM5 harmonics after rotation to the unstable and stable points shifted by the influence of the third-degree geopotential terms. Zonal harmonics remain invariant under rotations along the equator and are not presented here. Rotations have been performed only along the equator even though all tesseral harmonics with odd L m displace equilibria along the z-axis (see Table 2). However, in the case of the Earth this shift is so small that one can use the continuation method to refine initial conditions so that the orbit regains periodicity.

+

Table 2. Displacement of equilibria caused by the third-degree harmonics of the geopotential. X is the longitude of an equilibrium point along the equator, z describes its shift normal to the equator plane, and AT is the radial displacement with respect to the position determined assuming a spherical Earth. Harmonics SC,m m = 0,...,e

CC,,

e=2

e = 2,3

unstable point

stable point

x

a

[cml

Ar [ml

Peg1

[cml

Ar [ml

-0.2 +25

+527.50 +526.77

75?0711 75?0594

$0.4 -137

+516.99 +516.14

x

z

Peg1

-1409288 -1104038

3. Results Using the method of numerical continuation described in detail by Gacka 4 , we generate families of periodic orbits in a neighborhood of the stable and unstable equilibria, including only the J2 and 1 7 2 2 terms of the geopotential. Next we repeat calculations with all terms of the geopotential up to degree and order three. We look for symmetric planar orbits that cross the x-axis with velocities j: = i = 0 after a half of a period. Results are presented in Figures 1-8. Figures 1 and 2 show the behavior of a long-period orbit in the neighborhood of the stable equilibrium under the action of selected harmonics: first c 3 1 , s 3 1 and then C 3 3 , S 3 3 . The tesseral harmonics C 3 1 , S 3 1 move the

606 J . Kaczmarek, I . Wytrzyszczak and I . Gacka

0 15

Fig. 1. The smooth orbit at z = 0 is a long-period libration around the stable point X = 7500711 caused by the J2 and C22 harmonics. The oscillatory motion arises from the influence of C 3 1 and S31 terms. Initial conditions for both orbits are: z = 1.000993, t j = -0.1298. The unit of length is the geostationary radius.

Fig. 2. Oscillatory motion characterizes the orbit with Initial conditions are the same as above.

C33

and S33 coefficients included.

position of the stationary point along the equator by 1P2 degrees west, and about 1 meter south. This is the reason for the displacement of the orbit along the equator as well as of its small vertical oscillations.

It happens (Fig. 2) that the action of the Earth’s (333 and S33 geopotential terms almost cancels the influence of the C 3 1 and S 3 1 coefficients. The stable point together with a long-period orbit are moved in this case by about 1 9 east.

Perturbing Action of Earth’s Third-Degree Harmonics

607

Clm Slm 1=2,3 Clm Slm 1-2 -

1e-08

0 -1e-08 -2e-08

-38-08 -48-08 -5e-08 -68-08 -78-08

0.15

0.9

Fig. 3. The oscillating curve is the long-period orbit calculated with all second- and third-degree terms of the geopotential.

Clm Slm 1-2 3 Clm Slm 1-2

-5e-09 -1e-08 5e-09 0;

-

-

- - l b

48-06 -358-08 -3.58-08

-1.5e-08 -2e-08 -258-08 -2.58-08

0 15 0.15

Fig. 4. The upper curve shows the orbit with J2 and Czz; the lower curve takes into account all harmonics, and its initial conditions were refined by the continuation method.

Geopotential terms proportional to the C32 and S32 coefficients do not shift the orbit along the equator but act in the radial direction instead, changing the orbital period. Figure 3 shows the influence of all third-degree and order terms of the Earth’s potential. The resulting orbit is displaced only slightly along the equator, but there is some change of its period. The same orbit with the initial conditions adjusted by the continuation method is plotted in Figure 4. The initial data for the long-period orbit in Figure 3 were: 2 = 1.000993

608 J . Kaczrnarek, I. Wytrzyszczak and I. Gacba

[geo radii/time unit]. The unit of time [geo radii] and y = -0.1298. was chosen so that the gravitational constant would have a value of unity ( p = 1); the geostationary radius a = 42164 km was the unit of a distance. After improvement the initial position and velocity were 2 = 1.000993 [geo [geo radii] and y = -0.1298. loF3 [geo radii/time radii], z = -0.324. unit]. Figure 5 shows short-period orbits around the unstable equilibrium. The closed curve represents ten revolutions of an unstable orbit in the potential field limited to J2, C22 harmonics. The helix results from the same 10 orbital periods when all third-degree harmonics are included. The geostationary point was located approximately at (1.00001, 0,O). It is easily seen that the third-degree terms of the geopotential force the unstable orbit to lose its periodicity. New initial conditions causing the orbit to be closed again can be found; the orbit, however, as shown in the stability plot (Figure 6), remains unstable.

C'm S rn 1-23 Clm Slm l a

-

1. w e 1.2e-08 19-00

6649 6e-09 4e-09 29-09

0 -29-09 ,0002

Fig. 5. Short-period orbits around the unstable point X = -1409288, initial conditions: x = 1.0001, y = -0.175.

Figure 6 presents a sample of 100 orbits of a short-period family around the unstable point. Orbits calculated with the second- and third-degree harmonics are shown using vertical crosses; the tilted crosses represent orbits under the action of JZ and C22 harmonics. Each orbit is described by a pair of stability coefficients P and Q (eigenvalues of the monodromy matrix). An orbit is stable when its orbital stability parameters lie in the range -2 5 P, Q 5 2.

Perturbing Action of Earth's Third-Degree Harmonics 609

In Figure 7 the x-axis is directed toward the stable point X = 7299288. The curve parallel to the xy-plane shows 10 orbital periods generated by the geopotential restricted to J2 and C22 harmonics. The inclined orbit represents 10 revolutions under the combined action of the second- and thirddegree terms of the geopotential. Periodicity disturbances are too small to be observable. The stability graph of 100 orbits around the stable equilibrum (Figure 8) demonstrates the stability of the perturbed orbit.

Fig. 6. Stability indices of the short-period family around the unstable point X = -1409288.

Clm Slm 1=2,3 Clrnl-2

-

>.0002

o.wo12 5 Y

Fig. 7. Short-period orbits around the stable point X = 7500711, initial conditions: 2 = 1.0001, $ = -0.1755.

610

J . Kaczmarek, I . Wytrzyszczak and I. Gacka

-1 99994 199995

g

-199996 199997 199998

199999 2

lww6

lmw7

1woo8

1-

1c

1

Fig. 8. Stability indices P , Q of the short-period family near the stable point X = 750071 1.

4. Conclusions

The third-degree harmonics of the Earth's gravitational field have small coefficients and cause no significant effect on the stability of either shortor long-period orbits. However, they affect the location of orbits, largely through the displacement of equilibria, and they change the orbital period.

Acknowledgments The work was supported by the grant nr 5 T12D 026 23, and the project nr 4612/PB/IA/01 of the Polish Committee of Scientific Researches.

References 1. Biancale, R., Balmino, G., Lemoine, J.-M., Marty, J.-C., Moynot, B., (CNES/GRGS, Toulouse, France), Barlier, F., Exertier, P., Laurain, 0. (OCA/CERGA, Grasse, France), Gegout, P. (EOST, Strasbourg, France), Schwintzer, P., Reigber, Ch., Bode, A., Gruber, Th., Knig, R., Massmann, F.-H., Raimondo, J.C., Schmidt, R., Zhu, S.Y.: 2000, " A New Global Earth's Gravity Field Model from Satellite Orbit Perturbations: GRIM5-Sl" . Geophys. Res. Let. 27,3611-3614.

Perturbing Action of Earth’s Third-Degree Harmonics

611

2. Blitzer, L., Boughton, E. M., Kang, G. and Page, R. M.: 1962, ”Effect of Ellipticity of the Equator on 24-Hour Nearly Circular Satellite Orbits”, J. Geophys. Res., 67, 329-335. 3. Blitzer, L.: 1965, ”Equilibrium Position and Stability of 24-hour Satellite Orbits”, J. Geophys. Res., 70, 3987-3992. 4. Gacka, I.: 2002, ”Halo Orbits in the Sun-Mars System”, In this Proceedings. 5. Lara, M. and Elipe, A.: 2002, ”Periodic Orbits around Geostationary Positions” , Celest. Mech. Dyn. Astron., 82, 285-299. 6. Morando, B.: 1963, ”Orbites de rksonance des satellites de 24 heures”, Bull. Astron., 24, 47-67. 7. Musen, P. and A. E. Bailie, 1962, ”On the Motion of a 24-hour Satellite”, J. Geophys. Res., 67, 1123-1132.

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Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

ONE KIND OF COLLISION ORBITS RELATED TO LAGRANGIAN LIBRATION POINTS A. E. ROSAEV FGUP NPC “NEDRA” Svobody 8/38, Yaroslavl 150000, Russia

In the context of the problem of asteroid hazard, the determination of potentially dangerous orbits -the orbits of collision with the Earth- has a significant interest. One kind of collision orbits is related to the Lagrangian libration slution of the three body problem. It is known that (temporary) capture into a satellite orbit from an orbit near a libration point is possible. In case that the captured orbit has a large inclination its eccentricity increases in a fast way leading to collision. An example of this kind of collision orbit is presented in this work. It is not excluded that the Earth (temporary) captured a small satellite during its geologic evolution. Due to time reversibility, the constructed orbit of collision can be considered as a launch orbit that could be of interest for low-cost mission design.

1. Introduction

The origin of near Earth objects (NEO) and its possibility of Earth’s collisions has broadly considered and catalogued Some of such objects may become co-orbital. In this case, the significant maximum in the flux of interplanetary mass, related with libration points L1 and L2 may be expected. The neighbourhood of libration points L4 and L5 can collect minor bodies and increase lifetime of NEO. On the other side, the area of stability for such orbits is small, and in a relatively easy way can become a nonresonant and very dangerous objects. During their evolution, these objects 613

614 A . E . Rosaev

will have a number of catastrophic, low velocity encounters with Earth. A number of potential impactors in the group of co-orbital, and close to coorbital, objects may be present. So, dealing with the problem of asteroid hazard, their study is very important since a similar sort of dangerous orbits really exists. Moreover, it is proved that the possibility of switching between retrograde satellite (temporary) orbits and heliocentric orbits is possible and it is related to libration points regime. In certain cases, some of captured orbits will have high inclination. Lidov has shown that a satellite on a polar orbit turns inevitably towards Earth after several revolutions as a consequence of the fast increase of its eccentricity. Even only because of probabilistic facts, we must assume the existence of orbits of capture with large inclinations which evolve to fast collision. The numeric investigation of such orbits is a main target of this brief investigation. Few important notes, which can increase capture possibility, are given. It is also shown that the breakup of the minor body inside of planet’s action sphere makes the capture easier. The calculation of the Jacobi constant for real near Earth asteroids is done, and the area of their possible motion is also considered. As an additional argument, we may note a few long crater chains at the Earth. It is not excluded that the object that caused an ecological catastrophe 40 Myr. ago on the border of the Mesozoic and Cenozoic era moved as an Earth’s satellite orbit before collision.

2. Area of Motion

The regions of possible motions may be estimated by Hill’s zero-velocity curves based on the Jacobi integral in the restricted three body problem (see Fig.1). In the 3-dimensional case, the equation of Hill’s surfaces (in dimension-less variables) is 4:

R ( z , y , z ) = -1 z 2 2

1-CL + -21 y 2 +-+-+ r1

P 7-2

1 p ( 1- P I ,

One Kind of Collision Orbits Related to Lagrangian Libmtion Points

615

t’

Fig. 1. Hill’s zero velocity curves XY-projection.

The projections of these surfaces are shown in Fig 1. In most cases the inclination is enough small to have this planar consideration in mind. In the figure we see that for an object with Jacoby constant in range 3 < C < C2 motion both around Earth as well as around Sun is possible. For Earth-Sun System C2 = 3.0009 4 . Jacoby constants and orbital elements of the near Earth minor planets close to libration points have been computed and it is not excepted temporary capture some of these objects onto Earth’s satellite orbit.

3. Notes on Capture Possibility

The discovery of Earth-co-orbital object (3753)Cruithne lead to the possibility of the existence a wide group of objects, related to Lq and Lg. During their evolution under the planetary perturbations, these orbits may be transformed into orbits related to L I , and then, into orbits captured as Earth’s satellite orbits. But in most cases only temporary satellite capture is possible. As a result of this work, it is shown that the possibilities of a capture, and the lifetime of the captured object, are significantly increased when taking into account the phenomena of splitting, up to point that the temporary capture can become permanent.

616 A.E. Rosaev

Richardson et al. assume that some asteroids may consist of a number of weakly consolidated fragments. Evidently, such structures are easy destroyed. Preliminary estimates of such motion circumstances are possible on the base of the solution of the two body problem. Let us assume that a small body B moves deeply inwardly inside the gravitational sphere of influence of the planet. It can be expected that the change of velocity, AV, of the body B when splitting is rather small. The condition of capture has the form,

H

P r

=--

+ -12 v , <~0,

v , =~(vT- A .12 + (v,- A v,)’ ,

where VT,Vf are the radial and tangential components of velocity before breakup. Doing the necessary substitutions, we get the condition,

+

e2 - 1 A V” - 2 A V’ (sin( 4 )

AV‘=AV

+ esin( + + 9 )) < 0,

(1)

d:-,

where: p is the parameter of orbit B, m is the gravity parameter of the planet, 4 is the true anomaly of the body in orbit, e is the eccentricity of this orbit, AV is the value of breakup velocity and 9 is the angle between AV and the planet’s radius-vector OA. Condition (1) on AV’ defines a parabola whose branches are directed upwards, consequently, capture is possible under the conditions,

b-

d G
(2)

where,

-b = sin(+) + e s i n ( + +

a),

c = e 2 - 1.

In case that the “breakup velocity” is normal to the planetocentric vector the condition of capture can be approximately rewritten as, 1-2-

fi

fi

l+e l+e As can be seen from this condition of capture, the required value for changing the velocity of B can be small. The splitting of the small body becomes easier in case it has a fast axis of rotation.

One Kind of Collision Orbits Related to Lagmngian Libmtion Points

617

4. Numeric Research

Because the time of interaction at the encounter is relatively small, a simple integration method like Runge -Kutta may be used. The integration is performed in the restricted three body problem using the constants provided in table 4 and with step-size of a 1/100 day or smaller. Table 1. Parameters for the numeric integration. Parameter Sun mass Earth mass Sun-Earth distance Time steD

Value 1.9891030 kg 5 loz5 kg 1.49 lo1' m 0.001 dav

There are some points of interest in these numeric experiments. The estimation of the area of Earth satellite existence, the dependence of lifetime of high inclined orbits of collisions with respect to the inclination and other orbital elements, and the conditions of capture. Some of the results of the numeric investigations are: The maximal geocentric distance amaz at which (temporary) satellite capture is possible is determined. For retrograde orbits amas=3000000 km, for direct orbits a,=1700000 km and for polar orbits amaz = 2000000 km. The high inclined collision orbit (Lidov's orbits) is confirmed by nu-. meric integration. There is also a number of collision orbits close to an inclination of 90" (in the range 80" - 90"). The orbits with smaller inclination are save from collision. The lifetime of a collision orbit is inversely proportional to the magnitude of the semi-major axis (Fig. 2 Table 4). For large values of a , however, collision does not take place, but the orbit becomes retrograde. The behaviour of the orbit with respect to the initial inclination and orientation (node) is considered too. The lifetime is independent from the initial inclination, but it strongly depends on the orbit orientation (node longitude). There are three types of final (asymptotic) orbits depending on the initial inclination. For small inclinations the orbits are stable and a relatively slow evolution takes place. At intermediate ranges, the orbital evolution is very fast, and orbits may become retro-

618

A.E. Rosaev

grade. Finally, a small region of inclination, near polar orbits, leads to collision with Earth. The lifetime region decreases when increasing the initial semi-major axis.

Table 2.

Range of inclinations leading to collision and lifetimes

a,th. km (e=O, i=90, w=O) Lifetime (years) Limit I (degrees)

1284

1084

884

684

2.0

3.04 87

4.95 84

8.4 82

No collision

a, h.m

Fig. 2.

Time before collision vs semi-major axis.

Orbits with high geocentric inclination and arbitrary eccentricity and orientation have been integrated and the resulting heliocentric orbital elements computed. Due to time reversibility, these elements give an orbit of temporary capture from heliocentric to a (temporary) Earth’s satellite orbit. The ranges for the elements of these orbits are given in Table 3. It seems, that all them are close to co-orbital. Taking into account the changes of the orbits when splitting (as described above), the family of collision orbits can be constructed. An example of these orbits is given in Fig.3.

One Kind of Collision Orbits Related to Lagmngian Libmtion Points

619

Table 3. Data of highly inclined orbits Geocentric orbits characteristic Symmetric Z = f90, e = 0.7 q = 386000 km, w = 0 Asymmetric Z = -91(+89), e = 0.7 4 = 386000 km, w = 0

a 1.128

Initial/final heliocentric orbit 1 a e e 0.085 0.584 0.908 0.068

0.713

1.088

0.057

0.907

1.197

0.926

0.053

1

Fig. 3. Projections and 3 D representation of an example of highly inclined collision orbit.

5. Geological Evidence

It is not excluded that an object which previously had been an Earth’s satellite caused an ecological catastrophe 40 Myr. ago, at the MesozoicCenozoic boundary (K-T boundary). In it is supposed that the craters of the Mesozoic-Cenozoic boundary were formed by the impact of an object

620

A.E. Rosaev

(comet) with an inclination about 75 degrees with respect to the Earth's equator. However, assuming a common origin of these craters, a satellite parent orbit seams more feasible due to the long distances between the craters. Unfortunately, the ages of most craters, close to the K-T boundary, are known with insufficient accuracy. The craters having an age about 39 Myr. (Popigai, Wanapitei, Mistastin and Beyechime-Salaatin) present a concentrated distribution too. This concentration is slightly more remarkable for the Cretaceous-Paleogen case and, moreover, ages of craters close to 40 Myr. are known with higher accuracy. According to geological data, the impact structure Shunak may be added to these group of craters. Collecting all similar objects with an age near 40 Myr. it is seen that they are situated near meridian 90" E -90" W and close to an orbit with inclination about 75 degrees with respect to Earth's equator (as in the case of the K-T boundary). Note, that after taking into account the inclination of Earth's equator with respect to the ecliptic plane, the orbit of these impactors is almost orthogonal with respect to the ecliptic plane.

6. Conclusions

The problem of the transition from heliocentric to satellite orbits around a planet, appears in a natural way in the study of the close approaches of comets and asteroids to large planets, and also when considering possible scenarios for the origin of some natural satellites (external satellites of Jupiter, Phobos and Deimos and others.).

A numeric investigation for high inclined satellite orbits has been done in this work. It is shown that the splitting at close encounter can make temporary capture permanent. By taking into account this fact, the construction of special kind of a collision orbit is possible. Initially, the capture from quasi-co-orbital orbit, related to the libration point L1, to an Earth's satellite orbit with high inclination takes place. After that, a fast increase of the geocentric eccentricity takes place and a collision with the Earth surface is encountered. Due to time reversibility, the constructed collision orbits can be considered as launch orbits. This means that it is relatively easy to reach some Near Earth Objects by using such trajectories.

One Kind of Collision Orbits Related to Lagmngian Libmtion Points 621

In our opinion, some long craters chains may due to an impact of some objects moving on the satellite orbit. As an example we point the attention on two known craters chains near 40 Myr. ago and at the KT-boundary which may have a satellite nature '. The extension of these investigations is required to describe all the features of the motion in the problem.

Acknowledgements This work was supported by the Swedish Natural Science Research Council, the Goran Gustafsson Foundation, and by the Human Capital and Mobility programme of EU. I would like to thank those who provided preprints or unpublished material that was used in this article, and A. Suzor-Weiner, G. H. Dunn, and W. Shi for valuable comments on the preliminary drafts.

References 1. A. E. Rosaev, Catalogue of semicrossing orbits of the Earth Approach Aster-

oids. Vologda, p. 124 (1997). 2. F. Namouni, Secular interactions of Coorbiting Objects. Icarus, 137, 293-314 (1999). 3. M. L. Lidov, On the approximate analysis of evolutions of orbits artificial satellites. Problems of moving artificial celestial bodies, Nauka, 119-134 (1963). 4. V. Szebehely, Theory of Orbits. The Restricted Problem of Three Bodies, Academic Press (1967). 5. D. C. Richardson, W. F. Bottke and S. G. Love, Tidal Distortion and Disruption of Earth-Crossing asteroids, Icarus 134,47-76 (1998). 6. V. L. Masaitis, M. A. Nazarov, D. D. Badukov and B. A. Ivanov, Impact Event on the Mesozoic and Cenozoic boundary: data interpretation. Impact craters on the border Mesozoic and Cenozoic. Nauka, 146-185 (1990). 7. A. E. Rosaev, Paleogen disaster: how many satellites did the Earth have in the Late Mesozoic 30-Int. Geolog. Congr. Vol. 3 Beijing, China, p. 508 (1996).

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Libration Point Orbits and Applications G. G h e z , M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

THE INVESTIGATION OF STATIONARY POINTS IN CENTRAL CONFIGURATION DYNAMICS A.E. ROSAEV FGUP NPC “NEDRA” Svobody 8/38, Yaroslavl 150000, Russia

A system of N points, each having mass m, and a central mass M forming a planar central configuration, is considered. The equations of motion of a test particle are given and compared using different coordinates. For large values of N , even or odd, the equations are given too. The method used for the computation of some relevant sums can be important for practical applications. It is shown that the stationary solutions (libration points) can be determined from an algebraic equation of 5-th degree. If the mass of the particles is large, the outer libration points disappear. For the inner libration points the limit minimal distance has been detected. If m/M is small the solution obtained for the libration has as limit a solution similar to the for 3-body collinear points. We have also constructed a central configuration with 2N + 1 bodies, in which N new particles, each of mass ml # m, are placed at the libration points of our first system. The main analytic results of this work have been verified with the computer algebra system Maple and by numeric calculations.

Main Equations

A system of N + 1 particles is considered, one body is placed a t the center of the system and the remaining N bodies are set a t the vertexes of a regular polygon. The stationary solution of the N-body problem, with the masses placed a t the vertexes of a regular polygon is well known, see Wintner ’. The system of K similar rings was studied by Seidov ti and the equilibrium 623

624

A.E. Rosaev

condition was given. The ability of capture of such a system is shown in Pustylnikov 3 , where the final motions are considered. So, this model is not new. Maxwell was the first to apply this model to the study of the planetary rings dynamics. There are some evidences of presence large bodies in Saturn’s ring obtained recently, Showalter ’. On the other side, as it known at present, the planetary rings are a very complex collisional system, which are more suitably described by kinetic equations, see Burns l .

A system of N points, each of mass m j , and a central mass M , forming a central configuration, is considered. The test particle is a particle of the ring whose motion is perturbed by the remaining N - 1 ring particles. It is easy to consider the case in which the test particle is placed at equal distance between two ring particles (in non-coilinear libration points). The equations of motion can be given in different coordinates, all of them can be derived by differentiation of the potential energy. According to Ollongren 2 , the perturbation function in this system is:

u=-G M R

+Ncl j=O

J(X

- Xj)2

G mj

+ (Y - y 3 ) 2 + (2 - z j y

(1)

where G is the gravity constant, ( X , Y , Z ) , ( X j , Y j , Z j ) are the rectangular coordinates of a testing particle m and the particles of the ring mj respectively, R is the distance between M and m. The equations of motion of the ring particle can be obtained from the partial derivatives of U . The equations for n rings easily follow from the one ring case. Ollongren and some other authors use different expressions for the perturbing function if the number of particles in the ring is odd or even. Perov, in Rosaev et al. 4 , suggests very complicated expressions that use the oblateness of the central body in additions to mutual ring particle perturbations. In our opinion, the equations of motion in polar coordinates are more suitable to study central configurations. They can be derived by differentiation of (1). We shall consider the planar motion of a particle with mass m in the gravitational force field of the other N - 1 particles, placed at the vertexes of a regular polygon, and a central body of mass M . Evidently, for the

The Investigation of Stationary Points in Central Configuration Dynamics

625

radial and tangential forces of interaction between the particles we have:

where rj is the distance between particles, Cpj is the angle between Om and Omj and 0 is the center of the ring. For a circular ring: ~ j ~ = x ~ + 4 R ~ s i n ’ ( (l+:), ?) COS(q5j)

= - 2 R x + 2 x 2 + 4 R 2 s i n 2 ( 5 ) (1+:) 2

2+4R2sin2(?)

(3)

(l+:),

where R is the ring radius, cq is the angle between particles, and x is the distance between the test particle and the ring. The equation of motion can be easily obtained using (2) and (3). One can see that these equations are valid both for a number of particles either odd or even:

fi-R($+fi)2=---

GM

Gmj (2& sin2 aj

N-l

+ x)

(x2 + 4R; sin2 a j ) (4)

+

where R = R, x is the distance from the testing particle to the origin, R, is the distance from the j-particle to the origin, IR is the angular velocity and 4 is the angular declination from stationary position. Now, we derive the equations of motion in the usual way. Using cylindrical coordinates:

d dt

- (R2i)

= Ul,

(5)

626

A.E. Rosaev

The perturbation function in cylindrical coordinates is:

c d ~ ( +i ) ~ s=-c + N-1

u=

Gmmi R2 - 2r(i)Rcosa

i=l

N-1

i=l

N-1

u.=-c

i=l

+ (z-~ ( i ) ) ~ ’ Gmmi (R + x) R sin a

(6) ( ( R + x ) ~ R2 - 2 ( R + x ) R c o ~ a + ( z - z ( i ) ) ~ ) ~ / ~ ’

+ x - Rcosa) ((R + x ) + ~ R2 - 2(R + x)Rcosa + ( z - ~ ( i ) ) ~ ) ~ / ~ Gmmi(R

’

These equations were derived with the use of Maple software for symbolic computations. After insignificant manual simplifications, the motion equations can be rewritten as in (4).Evidently, for large values of N , these equations are valid both for an odd or an even number of particles. Let us assume the masses of all particles to be equa1,mj = m. In this case the second equation in (4)reduces to the conservation of the angular momentum:

L=

(6+ R)R2.

To study this problem we have used Maple symbolic computation software. It allows to get a solution in symbolic form and to consider different asymptotic cases. Different solutions for this equation are compared. As before, we assume x to be the deviation of the particle from the ring. The symbolic computations allow us to represent the solution as an x power series (in case of small x). This enables t o study the dependence of the libration point position on initial conditions. In case of insignificant perturbations, x reduces t o the form:

..

2Gm 3L2 x-x+-x=R4 R3 Denoting by Di = d 2 R 2

N-1 3=1

<<

R, the first equation (4)

Gm (2R sin(aj/2))3x’

+ ( z - ~ ( i )-)2R2 ~

COSQ,

(7)

the expansion of (7) in

The Investigation of Stationary Points in Central Configuration Dynamics 627

powers of x up to order O(z2), obtained with Maple, is: N-1

U,=-Gmx i=l

miR( 1 - cos a ) 0;

N-1

3miR2(1 - cos a)2 20:

i=l

+Gmy x +Gmy (i=l

N-1

Ur = - G m

i=l

i= 1

+

Gm

( 9 m i R ( l - c 0 s a ) - 15miR3(1-cosa)3 20: 207

mi R2 sin a 0; miR sin a 0:

(3miR2

-

3miR3( 1 - cos a )sin a

0:

( 3 - 2 cos a ) sin a - 15miR4( 1 - cos a)2sin a 20; 207

i=l

)

x2.

It seems that, close to equilibrium, the radial force depends on x as U, = a0 (a1 a2)x + a3x2 .... In the coefficient of x, the term a2 is much smaller than al. Close to the equilibrium position the radial perturbation is much larger than the tangential, as can be seen in Figure 1

+

+

+

Fig. 1.

Radial/tangential force ratio (N=10).

As a result, after substituting ( 9 ) in ( 6 ) we get (7). So, in the linear

628

A.E. Rosaeu

approximation, we get oscillations with frequency: 2Gm

3L2

N-l j=1

Gm (2Rsin(aj/2))3'

The computation of Ul according (9),shows that the tangential force is much smaller then the radial (Fig.1) and the value of the ratio between both increases with N . Table 1. Coefficients of the linear term in the expansion of U,..

N

a1 39.88 38789 3.877 lo' 3.102 lo8 24.804 10' 83.790 10' 198.80 10' 386.17 10'

10 100 1000 2000 4000 6000 8000 10000

a2 7.26 129.3 1844.4 4020 8701 13635 18733 23919

al/a2 5.5 300 2.1019 lo4 7.7164 lo4 28.5068 lo4 61.4520 lo4 106.1340 lo4 161.4500 lo4

The second term of the expansion of U, gives the conditions of appearance of oscillations: N-1 j=1

6Mx mj >(2Rsin(aj/2))3 R4 '

The perturbations caused by a moonlet with mass m, moves the particles of the ring: first those close to m and after the most distant ones. As the ring is not absolutely rigid, a phase angle 6 will appear, and in the general case:

x = -w2x - 2kx 2kw tan6 =

+f(t),

w; - w2'

+

where k is a resistance factor and wo = J-2Gm/R3 3L2/R4 the fundamental frequency. The difference between w and wo enables to describe stationary oscillations within the ring. Their existence is conditioned by:

C (Gmj/(2R~in(aj/2))~) wo

1

2Gm

3L2

+ 1= integer.

The Investigation of Stationary Points in Central Configuration Dynamics

629

The character of the motion is conditioned by the value of the parameter k. For large values of k the oscillations damp quickly, forming a non-vibrating component.

Fig. 2.

Stationary and non-stationary radial oscillations.

The time of existence of this kind of motion is determined by collisions with non-vibrating component and may be long. One possibie reason of these oscillations is a collision with the nearby moonlet. From & = R = const follows that the time dependent part of the angular velocity = 0 and the first equation of (4)reduces to: $J

fi-RR2=--G - M

+

Gmj (2 & sin2aj z)

N-l (x2

+ 4 h 2sin’aj)

In case of a regular rotation of all the system

(fi = 0) one can determine

fl (for non-interacted particles evidently Sli = G M / R i ) :

By using the approximate formula:

one can estimate the increment in the angular velocity of the ring with the number of particles. By expressing the mass of ring in central mass units, the dependence R(N) can be established. So, the the particles in addition to the movement of attraction, tend to move away from the planet due to

A.E. Rosaeu

630

the resonances. This movement increases with the mass of the ring and the number of particles. For N = 10l2 and m = 0.01. M the displacements reach 2 %. If the mass of the ring is small this effect is negligible.

1. The Libration Point Coordinates

Before considering any question related with the stability in this problem, the libration points should be determined. In this section we will study only planar central configurations. The libration points coordinates can be determined from (4) and there here are two kinds of solutions:

Ncl

+

Gmj (2& sin2(aj/2) z) GM -RR2 = -R2 j=1 [(z2+4Risin2(aj/2)) (1 +z/&)I3/”

and

c

+ +

GM - Gm - N-’ Gmj (2% sin2(aj/2) z) -RR2 = -R2 x2 j=1 [(z2 4Ri sin2(aj/2)) (1 z / R , , ) ] ~ / ~ ’ 27r a . - --j ++o, (9) ’ - N In the first case 40 = p / N , z o = 0 and we have non-collinear libration point. In the second case 40 = 0,lzol > 0 and the z-coordinate of the collinear points must be determined. So, the equations (9) for the collinear libration points are the most interesting. They can be reduced to a fifth degree polynomial. For small zo

+

<< &, we have that approximately:

Gmj (2& sin2(aj/2) + z)

+

[(z2 4Ri sin2(aj/2)) (1

x

Gmj (2& sin2(aj/2) + z)

+z/&)]~/~

After the substitution R = &+zo ~ 0 we ) get ~

[2& sin(aj /2)13/2

and multiplying all the terms by xi(&+

- ( R ~ + z o ) ~ z @=~- G M x Z - G ~ ( R ~ +- ~ ( A~z)+~B ) ( & + z ~ ) ~ (10) zi, where N-1

A=

C

Gm

N-1

Gm B=C 4Ri sin(aj/2)

j=1 (2fi ~ i n ( a j / 2 >’ > ~

j=1

‘

The Investigation of Stationary Points in Centml Configuration Dynamics 631

Setting xo = x , & = R and after some simplifications we get

(R2 + A)x5 + (3R2R + 2AR + B)x4 +(3R2R2 R2A 2BR)x3+ ( B R + Gm)x2+ GmR2 = 0.

+

+

(11)

Some simplifications can still be done. The coefficient B is very small, due to the symmetry of the sinus function, so the coefficient of the quadratic term in (11) can be neglected. If R is large and x small ( x << 1 << R ) then

3GM

+ AR2 + 2BR

In general case, taking into accounting that R2 = G M / R i , an approximate solution (for m / M << 7.84048/N3and x / R < p / N ) is

m

xi

= MR3(3 - k) ’

k = 0.3005125000-

m N3 MT3‘

At the limit M >> m , the expression (10) reduces to the well-known solution of the 3-body problem (see Szebehely *). Increasing the masses of the particles of the ring, the libration points move away from axis of the ring (see Table 2 ) . A similar behaviour is observed when the number of particles decreases. For k close to 3 and for large values of k the equations (10) are unsuitable. Numerical solution for large m/M show that the inner libration point coordinate has a limit, s, that depends on N . It cannot get close to the center of system, even if M = 0. The outer libration point goes away from the ring’s axis when increasing the m / M ratio. When the mass of the ring is large ( m * N/M > 1) the outer libration points disappear (see Tables 2-3) In these tables the mass of the ring (in central mass units) is given in the first column and the remaining ones give, for different values of the number of particles N , the ratios xL/R (for N 1 bodies) and xo/R (for the 3-body problem).

+

As is seen in Table 2 , for the inner points the ratio xL/R becomes constant in the case of N 1 bodies when me N / M > 100, in contrast with the 3-body situation in which xo/R increases with m . N / M . For outer libration points, we have an asymptotic increasing of x h / R for 0.1 < m . N/M < 1 (see Table 3).

+

The result (12) was obtained analytically and, using Maple, it was seen

632

A.E. Rosaev Table 2.

n .N / M

Dependence of the inner libration point with the mass ratio.

N = 50 XLIR xo/R

0.00001 0.00010 0.00100 0.01000 0.10000 1.0 10.0 100.0 1000.0 10000.0

-0.0045 -0.0085 -0.0185 -0.0375 -0.0705 -0.1055 -0.11 75 -0.1195 -0.1195 -0.1195

Table 3.

-0.0041 -0.0087 -0.0188 -0.0405 -0.0874 -0.1882 -0.4055 -0.8736 -1.8821 -4.0548

-0.0035 -0.0065 -0.0145 -0.0305 -0.0605 -0.0915 -0.0995 -0.1005 -0.1015 -0.1015

-0.0032 -0.0069 -0.0149 -0.0322 -0.0693 -0.1494 -0.3218 -0.6934 -1.4938 -3.2183

N = 1000 XL/R XO/R -0.0015 -0.0015 -0.0035 -0.0032 -0.0095 -0.0069 -0.0 149 -0.0255 -0.0322 -0.0505 -0.0693 -0.0655 -0.1494 -0.0675 -0.3218 -0.0685 -0.6934 -0.0685 -1.4938 -0.0685

Dependence of the outer libration point with the mass ratio.

n .N / M

N = 50

XLIR 0.00001 0.00010 0.00100 0.01000 0.10000 1.0 10.0 100.0

N = 100 XL/R zo/R

0.0045 0.0085 0.0175 0.0475 0.1925

xo/R 0.0041 0.0087 0.0188 0.0405 0.0874

N = 100 XLIR xo/R 0.0035 0.0075 0.0155 0.0425 0.2125

0.0032 0.0069 0.0149 0.0322 0.0693

N = 1000 20

0.0015 0.0035 0.0125 0.0425 0.2975

0.0015 0.0032 0.0069 0.0149 0.0322

The libration points are absent

that was a solution of (11)at the limit B = 0 and for x small. In fact

x=

(R ( A+13Ro) )

113 m113

+0 ( ~ 2 / 3 ) .

Some another asymptotic expansions were obtained with Maple.

2. Generalisation to the Case with 2N

+ 1 Particles

Let us consider the system with 2N masses ml, I = l...n, which form two regular polygons, the vertex of each lying on the radius the circle described. There are two main cases, as is shown in Fig 3. The mass M is in the center of the system. Each pair of masses, m l and lie either on the same ray for the collinear 2N 1 body configuration, or in a middle position for the non-collinear 2N 1 body configuration (Fig.3). The motion equations for the particles of the rings can be derived m j ,

+ +

The Investigation of Stationary Points in Central Configuration Dynamics 633

Fig. 3.

Non-collinear and collinear 2N

+ 1-body configurations.

from Rosaev in the case of only one ring. In polar coordinates we have

where k = o, i and s = i, 0.There are four possibilities for these equations, but to study stability only two of them are of interest. We also need to choose the minus sign in the right part of the equations to describe the motion of the rings. In the above equations F k and f k are given by

In these expressions R k is the ring radius, N the number of particles, m k the mass of a ring’s particle, G the gravitational constant, Z k and $k the deviations of the test particle with respect to the equilibrium point in radial and tangential directions respectively, R the angular velocity of the circular rotation. Evidently, the equation (13) is correct for any value of N , either odd or even. We will assume that N is rather large. We will make use of the expansions:

Here, the coefficients should be obtained by differentiation, and can be

634

A.E. Rosaev

represented as

N-1

ak =

3Gmk sin a.j j=1 16 IRk sin(aj/2)I3 ’

‘

N-1

bk = -

j=1

GM

Gmk sin aj 8 IRk sin(aj/2)I3 ’ N-l

It seems, that the coefficients Ak are the largest ones and we can perform the following numerical approximation:

where a(N) M 2.40410. Its validity, as well as the validity of the linearization are shown in Tables 3 in which the expansion coefficients a’ for the l/j3 sums are given. Table 4. Expansion coefficients for different values of N .

N

4 4

a‘(N)

10 20 50 100 200 1250 5000 10000

2.6580862 2.4808764 2.4196434 2.4086496 2.4054148 2.4041442 2.4041037 2.4041018

2.37132406 2.39506411 2.40257668 2.40372180 2.40401482 2.40410137 2.40410137 2.40410137

As it follows from a Table 4, for large values of N N a=

1

1 sin3(i7r/N)

N

M

C

N3 31 87r3 i=l

-

M

N3

0.3005125000-

7r3

The Investigation of Stationary Points in Centml Configuration Dynamics

635

This means, that only the nearest neighbour particles give main contribution in perturbation. This is a remarkable result, which makes possible to effectively compute the perturbation effect in a close neighbourhood of the ring. This method was successfully applied to determine the coordinates of the libration points. It is easy to see the limit of linear approximation: the error increases as

x / R grows. The range of the linear approximation applicability decreases as the number of particles increase. 3. Results and Conclusions

+

Our final target is to construct the stable kN 1-points central configurations. To solve this problem, first of all, the stationary (libration) points in the N 1 case must be determined.

+

At the beginning, we considered a system of N points, each having mass m, and a central mass M , forming a central configuration. The motion equations for a general case (odd and even number of particles) are obtained by differentiation of the potential function. The proposed method to calculate the mutual perturbation force, given in this work, can be important for practical applications. Coordinates for the libration points of the system are calculated. It is shown that the libration points can be determined from a 5-th degree algebraic equation. Main attention is given to the case of large N . It is obtained that for large values of the mass of the particles, the outside libration point disappear. For inner libration points, the limit distance 1 exist. We used analytical approach, as well as computer algebra method, to study central configuration dynamics. As a result we can conclude that to describe libration points behaviour for different number (N) of particles, the analytical method is better; in case of dependence on a particle mass we obtain the same results using the both methods. The result coincides with the classical solution of the 3-body problem. Nevertheless, some results can be obtained more easily with computer algebra system. In particular, asymptotic solution within the limit of ring’s particle rn + 0 of a

636 A.E. Rosaev

neglectable mass may be obtained only by this way. The computer algebra methods have great advantage to expand the perturbation function in the described problem. By using Maple tools, we can determine the frequency of insignificant oscillations in the system. In all these cases, computer algebra methods can help to solve complex problems, making the solutions easier, as well as to verify the results. We believe, that Maple computer algebra package is a suitable tool for this purpose. But it does not mean, that special computer packages to analyse dynamic systems are not required.

References 1. Burns J.A. The formation of Jupiter’s faint rings. Science, 284 1146-1150 (1999). 2. Ollongren A. On a restricted (2n+3) body problem. Celestial Mechanics, 45 163-168 (1981). 3. Pustylnikov, L.D. On some final motions in N-body problem. Applied Math and Mechanics, 54-2 329-331. (1991). 4. Rosaev, A.E. Perov, N.1, Sadovnikov, D.V. On a central configuration problem. In Astronomical Research, Yaroslavl, 131-142 (1996). 5. Rosaev, A.E. A Simple Example of Stable Central Configuration. In Proceedings of Conference “Problems of Celestial Mechanics”, June 3-6, StPetersburg. p. 145 (1997). 6. Seidov, Z.F. About one particular example of central configurations In Analytic celestial mechanics, Kazan, 96-99 (1990). 7. Showalter, M.R. Detection of Centimeter-Sized Meteoroid Impact Events in Saturn’s F Ring. Science 282 1099-1102 (1998). 8. Szebehely, V. Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press (1967). 9. Wintner, A. The Analytical Foundation of Celestial Mechanics, Princeton University Press (1941).

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

STATISTICAL THEORY OF INTERIOR-EXTERIOR TRANSITION AND COLLISION PROBABILITIES S. ROSS Control and Dynamical Systems California Institute of Technology M C 107-81, Pasadena, C A 91125, USA

The dynamics of comets and other solar system objects which have a threebody energy close to that of the collinear libration points are known to exhibit a complicated array of behaviors such as transition between the interior and exterior Hill’s regions, temporary capture, and collision. The invariant manifold structures of the collinear libration points for the restricted threebody problem, which exist for a range of energies, provide the framework for understanding these complex dynamical phenomena from a geometric point of view. In particular, the stable and unstable invariant manifold tubes associated t o libration point orbits are the phase space structures that provide a conduit for particles travelling to and from the smaller primary body (e.g., Jupiter). Using the structures around libration points, a statistical theory for the probability of interior-exterior transition and the probability of collision with the smaller primary body can be developed. Comparisons with observations of Jupiter family comets are made.

1. Introduction

Several Jupiter-family comets such as P/Oterma, P/Gehrels 3 , and P/Helin-Roman-Crockett make a transition from heliocentric orbits inside the orbit of Jupiter to heliocentric orbits outside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around The Tisserand parameters of these objects, termed the quasi-Hziildas (hereafter QHs) by 3112

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Krestik 13, are slightly in excess of 3. The possible pre-capture orbital history of D/Shoemaker-Levy 9 (henceforth referred to as SL9) also places it within this group 2 . An important feature of the motion of these comets is that during the phase right before and after their encounter with Jupiter, their orbits pass close to the libration points L 1 and L 2 of the sun-Jupiter system. This has been pointed out by many authors, including Tancredi, Lindgren, and Rickman 2 3 , Valsecchi 2 4 , and Belbruno and Marsden Hence objects with low velocity relative to these points (i.e., orbits with apoapse near LZ or periapse near L 1 ) are most likely to be captured lo. During the short time just before an encounter with Jupiter, the most important orbital perturbations are due to Jupiter alone, as suggested by the passages of comets by L 1 and Lz.N-body effects of Saturn and the other large planets surely play a role over significantly longer times, but we concentrate here on the time right before a comet’s encounter with Jupiter. To simplify the analysis, we use the most rudimentary dynamical model, namely, the circular, planar restricted three-body model (PCRSBP), to determine the basic phase space structure which causes the dynamical behavior of the QH comets. Furthermore, since the PCRSBP is an adequate starting model for many other systems, results can be applied to other phenomena in the solar system, such as the near-Earth asteroid (NEA) problem, wherein one considers the motion of an asteroid on an energy surface in the sun-Earth system where libration point dynamics are important. Lo and Ross suggested that studying the L 1 and L 2 invariant manifold structures would be a good starting point for understanding the capture and transition of these comets. Koon, Lo, Marsden, and Ross studied the stable and unstable invariant manifolds associated to L 1 and L 2 periodic orbits. They took the view that these manifolds, which are topologically tubes within an energy surface, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter’s orbit. In the present paper, we wish to extend the results of Koon, Lo, Marsden, and Ross l 1 to obtain statistical results. In particular, we wish to address two basic questions about QHs and NEAs: How likely is a QH collision with Jupiter or a NEA collison with Earth? How likely is a P/Oterma-like interior-exterior resonance transition? With this work, we put SL9, NEA

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impacts, and interior-exterior transitions into the broader context of generic motion in the restricted three-body problem. The paper is broken up into two sections. In section 2, we discuss some phenomena of the QH comets, namely interior-exterior and collisions with Jupiter. In section 3, we frame the above questions as a transport problem, viewing the PCR3BP as the underlying dynamical system. We also summarize the results and suggest future directions.

2. The Quasi-Hilda Group of Comets

The QH group of comets is a small group of strongly Jupiter-interacting comets having a Tisserand parameter slightly above 3, characterized by repeated and long-lasting temporary captures ’. As authors have noted, the capture process frequently moves bodies from orbits outside Jupiter’s orbit to inside Jupiter’s orbit, passing by L1 and L2 in the process of approaching or departing from Jupiter’s vicinity l o . We will refer to this type of transition as an interior-exterior transition.

Interior-Exterior Transition. In Figure l(a), we show the interiorexterior transition of QH P/Oterma in a sun-centered inertial frame. The interior orbit is in an exact 3:2 mean motion resonance with Jupitera while the exterior orbit is near the 2:3 resonance with Jupiter. In Figure l(b), we show a homoclinic-heteroclinic chain of orbits in the PCR3BP as seen in the rotating frame. This is a set of orbits on the intersection of L1 and L2 periodic orbit and unstable manifolds with energies equal to that of P/Oterma. The homoclinic-heteroclinic chain is believed to form the backbone for temporary capture and interior-exterior transition of QHs, as can be seen when the orbit of P/Oterma in the rotating frame is overlayed as in Figure l(c) l l . Collision with Jupiter. At the time of its discovery, SL9 was only 0.3 AU from Jupiter and broken up into several fragments due to tidal disruption on an earlier approach within the planet’s Roche limit

17.

Integrations

‘By ezact, we mean that P/Oterma orbits the sun three times while Jupiter orbits the sun twice, as seen in an inertial frame.

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Fig. 1. (a) Orbit of quasi-Hilda comet P/Oterma in sun-centered inertial frame during time interval AD 1910-1980 (ecliptic projection). (b) A homoclinic-heteroclinic chain for the energy of P/Oterma in the circular, planar restricted three-body problem, as seen in the rotating frame with the sun and Jupiter fixed. (c) The orbit of P/Oterma, transformed into the rotating frame, overlaying the chain.

indicated that it would collide with the planet in July 1994.

4,

which it subsequently did

Likely Pre-Collision Heliocentric Orbit of SL9. Pre-collision integrations of individual SL9 fragmemts suggest that the SL9 progenitor approached Jupiter by passing by L1 or L2 from a short-period heliocentric orbit between either Jupiter and Mars or between Jupiter and Saturn (Figure 2(a)). The distribution of heliocentric a and e determined from these fragment integrations are shown in Figure 2(b). The pre-collision fragments have Tisserand parameters of about T = 3.02 f 0.01. From this value and the similarity of the pre-collision orbits to the known QHs, Benner and McKinnon suggest a QH origin for SL9. Twice as many fragments came from the outer asteroid belt as compared to the inner transjovian region. However, Benner and McKinnon do not conclude that SL9 originated from the outer asteroid belt. Instead, they say that “the chaos in SL9’s orbit is so strong ...that what is being seen is a statistical scrambling of all possible trajectories for an object as loosely bound as SL9.” The bias toward an asteroid origin is a measure of the relative ease of capture (or escape) toward L1 versus L2, a known result ’. The statistical likelihood of a pre-collision interior orbit depends on the relative source comet populations interior or exterior to Jupiter and their energies. If the interior and exterior populations are roughly equal, then a pre-collision interior origin is favored.

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Fig. 2. (a) A typical SL9 trajectory showing the passage past a libration point and subsequent capture. The sun is to the right. (Reproduced from Benner and McKinnon. According to their terminology, their L2 is our L1, and vice versa.) (b) Heliocentric a and e of possible SL9 progenitor orbits, based on fragment integrations. The positions of selected comets and two major outer belt asteroid groups, the Trojans and the Hilda, are shown. The dashed curves are for Tisserand parameter T = 3 (for zero inclination); orbits above the upper curve and below the lower curve have T > 3 and are generally not Jupiter-crossing, while those between the two curves (T < 3) are Jupiter-crossing. (Reproduced from Benner and McKinnon.)

3. Transport in the Planar Circular Restricted Three-Body

Problem When the dynamics are chaotic, statistical methods may be appropriate 25. By following ensembles of phase space trajectories, we can determine transition probabilities concerning how likely particles are to move from one region to another. Following Wiggins 25, suppose we study the motion on a manifold M . Further, suppose M is partitioned into disjoint regions

such that NR

M=URi. i=l

At t = 0, region Ri is uniformly covered with species Si. Thus, species type of a point indicates the region in which it was located initially. The statement of the transport problem is then as follows:

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Describe the distribution of species Si, i = 1,. . . ,N R , throughout the regions R j , j = 1,.. .,NR, for any time t > 0.

Fig. 3. The manifold M is partitioned into the regions Ri,Z = 1,.. . ,N R . If points are distributed uniformly over M at t = 0, we want to compute the movement of points between these regions for all times t > 0.

Some quantities we would like to compute are: Ti,j(t),the amount of species Si contained in region Rj,and Fi,j(t) = q ( t ) ,the flux of species Si into region Rj (see Figure 3). For some problems, the probability of transport between two regions or the probability of an event occurring (e.g., collision), may be more relevant.

Planar Circular Restricted Three-Body Problem. Here we only review the material concerning the PCR3BP which has relevance toward our discussion of transport. See details in Szebehely 22 and Koon, Lo, Marsden, and Ross 1 2 . Consider motion in the standard rotating coordinate system as shown in Figure 4 with the origin at the center of mass, and the sun and Jupiter fixed on the z-axis at the points ( - p , 0) and (1- p , 0) respectively. Let (z,y) be the position of the comet in the plane, then the equations of motion in this rotating frame are:

x - 2y = -Ip 7 y

+ 2a: = -Yp ,

where

is the effective potential, subscripts denote its partial derivatives and r 1 , r 2 are the distances from the comet to the sun and the Jupiter respectively.

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These equations are autonomous and can be put into Hamiltonian form. They have an energy integral: 1 2

E = -(i2 + 5 2 ) + v q z , y > . which is related to the Jacobi integral C by C = -2E. The Jacobi integral can be expressed approximately in terms of the comet’s semimajor axis, a, and eccentricity, e , in a form known as the Tisserand parameter, T , i.e., C = T O ( p ) ,where

+

The energy manifolds, M ( P , E ) = {(.,Y,j.,5>

I E(z,Y,%Y) = E l

where E is a constant are 3-dimensional surfaces foliating the 4-dimensional phase space. The Hill’s regions are the projection of the energy manifold onto the position space is the region in the zy-plane where the comet is energetically permitted to move, M ( P , E ) = ((2,Y) I

ueff(2,Y> L €1.

The forbidden region is the region which is not accessible for the given energy. See Figure 4(b) .

Fig. 4. (a) The rotating frame showing the libration points, in particular L1 and L z , of the planar, circular restricted three-body problem. (b) Energetically forbidden region is gray “C”. The Hill’s region, M ( p , E ) (region in white), contains a bottleneck about L1 and L z . (c) The flow in the region near La, showing a periodic orbit around La (labeled PO), a typical asymptotic orbit winding onto the periodic orbit (A), two transit orbits (T) and two non-transit orbits (NT). A similar figure holds for the region around L1.

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Eigenvalues of the linearized equations at L1 and L2 have one real and one imaginary pair, having a saddle x center structure. Our main concern is the behavior of orbits whose energy is just above that of L2, for which the Hill’s region is a connected region with an interior region (inside Jupiter’s orbit), exterior region (outside Jupiter’s orbit), and a Jupiter region (bubble surrounding Jupiter). We will use the terminology interior, exterior, and Jupiter regions to mean regions in the Hill’s region and the corresponding regions of the energy surface, M ( p ,E ) . Thus, we have a useful partition for our problem for which we can compute transport properties. These regions are connected by bottlenecks about L1 and L2 and the comet can pass between the regions only through these bottlenecks. Inside each bottleneck, adjacent regions of the (e.g., the interior and Jupiter regions) share a common boundary in the energy surface. This common boundary is known as the transition state and has been used previously in astrodynamical transFor our analysis of transport, we must focus on the port calculations bottlenecks.

’.

In each bottleneck (one around L1 and one around Lz), there exist 4 types of orbits, as given in Conley and illustrated in Figure 4(c): (1) an unstable periodic Lyapunov orbit; (2) four cylinders of asymptotic orbits that wind onto or off this period orbit, which form pieces of stable and unstable manifolds; (3) transit orbits which the comet must use to make a transition from one region to the other; and (4) nontransit orbits where the comet bounces back to its original region.

Fig. 5. (a) An example of an interior-exterior transit orbit. This on goes from outside to inside Jupiter’s orbit, passing by Jupiter. The tubes containing transit orbits-bounded by the cylindrical stable (lightly shaded) and unstable (darkly shaded) manifoldsintersect such that a transition is possible. (b) An orbit beginning inside the stable manifold tube in the exterior region is temporarily captured by Jupiter. When the tubes intersect the surface of Jupiter, a collision is possible.

Statistical Theory of Interior-Ezterior l h n s i t i o n and Collision Probabilities 645

McGehee l8 was the first to observe that the asymptotic orbits are pieces of the 2-dimensional stable and unstable invariant manifold tubes associated to the Lyapunov orbit and that they form the boundary between transit and nontransit orbits. The transit orbits, passing from one region to another, are those inside the cylindrical manifold tube. The nontransit orbits, which bounce back to their region of origin, are those outside the tube. Most importantly, to transit from outside Jupiter’s orbit to inside (or vice versa), or get temporarily captured, a comet must be inside a tube of transit orbits, as in Figures 5(a) and 5(b). The invariant manifold tubes are global objects-they extend far beyond the vicinity of the bottleneck, partitioning the energy manifold.

Numerical Computation of Invariant Manifolds. Key to our analysis is the computation of the invariant manifolds of Lyapunov orbits, thus we include some notes on computation methods. Periodic Lyapunov orbits can be computed using a high order analytic expansion l5 or by using continuation methods 6. Their stable and unstable manifolds can be approximated as given in Parker and Chua 20. The basic idea is to linearize the equations of motion about the periodic orbit and then use the monodromy matrix provided by Floquet theory to generate a linear approximation of the stable manifold associated with the periodic orbit. The linear approximation, in the form of a state vector, is numerically integrated in the nonlinear equations of motion to produce the approximation of the stable manifold. All numerical integrations were performed with a standard seventh-eighth order Runge-Kutta method. Interior-Exterior Transition Mechanism. The heart of the transition mechanism from outside to inside Jupiter’s orbit (or vice versa) is the intersection of tubes containing transit orbits. We can see the intersection clearly on a 2-dimensional Poincar6 surface-of-section in the 3-dimensional energy manifold. We take our surface to be E(p,c) = ((9, jl)lz = 1 - p , x < 0}, along a vertical line passing through Jupiter’s center as in Figure 6(a). In Figure 6(b), we plot jl versus y along this line, we see that the tube cross-sections are distorted circles. Upon magnification in Figure 6(c), it is clear that the tubes indeed intersect. Any point within the region bounded by the curve corresponding to the stable tube cut is on an orbit that will go from the Jupiter region into the interior region. Similarly, a point within the unstable tube cut is on an

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Fig. 6. (a) We take a Poincark surface-of-section E ( p , E )= {(y,y)lz = 1 - p , x < 0 } , along a vertical line through the center of Jupiter ( J ) . Both the L1 and L2 periodic transversally. (b) On E ( p , E )we , see the orbit invariant manifold tubes intersect E(@,€) first unstable tube cut for L2 and first stable tube cut for L1. (c) A small portion of the containing ) the interior of the tubes intersect-this set in the energy manifold M ( ~ , E comet orbits which pass from the exterior t o the interior region.

orbit that came from the exterior region into the Jupiter region. A point inside the region bounded by the intersection of both curves (lightly shaded in Figure 6(c)) is on an orbit that makes the transition from the exterior region to the interior region, via the Jupiter region.

Interior-Exterior Transition Probability. Note that since p , = y + 2 and x is constant, the (y, y) plane is a linear displacement of the canonical plane ( y , ~ , ) . F’urthermore, the action integral around any closed loop I’on

%>+ is simply the area enclosed by I’ on the surface-of-section E(p,E)19. The agreement between a Monte-Carlo simulation and a Markov approximation in an earlier paper suggests that for energies slightly above L1 and LI,there are components of the energy surface for which the motion is “well mixed” 19. Thus, the Markov approximation is a good one. Let R1 be the interior region and Rz be the exterior region. In the Markov approximation, the probability of a particle going from region Ri to Rj is

where Aj is the area of the first unstable tube cut on X ( p , E l , containing transit orbits from Rj and Fij = Fj~jiis the area of overlap of the first

Statistical Theory of Interior-Exterior h n s i t i o n and Collision Probabilities 647

unstable tube cut from Rj and the first stable tube cut from Ri on E(,,,c). This transition probability is exact for one iterate of the Poincark map; however, it is typically only qualitatively correct for longer times.

In Figure 7, we give the results of the calculations of P 1 2 and P 2 1 for and a variety of enegies in the range of mass paramter p = 9.537 x QH Jupiter-family comets. This is the probability of a comet to move from the interior to the exterior and vice versa during its first pass through the surface-of-section E(,,,E).

Fig. 7. Interior-exterior transition probabilities for quasi-Hilda Jupiterfamily comets. The probability of a comet to move from the interior to the exterior and vice versa during its first pass through the surface-of-section E ( p , c )is plotted as a function of energy in the planar, circular restricted three-body problem. The energy value of P/Oterma is shown for comparison. Note that interior to exterior transitions are slightly more probable than the reverse transition.

A few comments regarding this result are due. (1) Notice that there is a lower limit in energy, Et M -1.517. For E 5 E t , the tube cuts do not overlap and no direct transition is possible. After more loops around Jupiter, transition may be possible l l . (2) The probability increases as a function of energy. (3) Quasi-Hilda P/Oterma is located in the region of M 25% probability. (4)Finally, notice that P 1 2 > Pzl, which is a result of A 1 > A 2 , the slight asymmetry we should expect for a mass parameter of this value or larger 21.

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Collision Probabilities. Collision probabilities can be computed for objects coming through the L1 and LI bottlenecks from the interior and exterior regions, respectively. We augment the procedure for computing interior-exterior transition probalities in the following way. Instead of computing 3 i j , we now compute the overlap of the first unstable manifold cut with the diameter of the secondary (e.g., Jupiter). Since the surface E(p,E) passes through the center of the secondary, any particle located on E(p,E) with IyI 5 R will have collided with the secondary, where R is the radius of secondary in units of the primary-secondary distance. This is illustrated in Figure 8. PoincamSection:Tube lnkrsedng a Planet

Fig. 8. The surface-of-section, E ( p , e ) is , shown, with y vs. y. The area inside the first unstable manifold tube cut with IyI 5 R is shown in in black. These are orbits that collide with the surface of the secondary. The two vertical lines are at y = fR.

There is a singularity at the center of the secondary, y = 0 on E(p,E), so the calculation is actually performed along a nearby parallel surface-ofsection, where 2 = 1 - p 5 c, with c a small number on the order of the integration tolerance (the '+' sign is for orbits coming from the exterior, and the '-' for orbits coming from the interior). Collision probabilities for the sun-Jupiter case ( p = 9.537 x are given in Figure 9. We notice the following. (1) The probability is not monotonically increasing as in Figure 7. (2) The energy range of possible pre-collision Shoemaker-Levy 9 orbits lies in

R = 8.982 x

Statistical Theory of Interior-Exterior Thnsition and Collision Probabilities

649

Collision Probability for QuasiHilda Comets

Energy

Fig. 9. Collision probabilities for quasi-Hilda comets. The probability of collision for orbits making their first pass through the surface-of-section E ( p , E )is plotted as a function of energy. The energy range of possible pre-collision D/Shoemaker-Levy 9 orbits is shown for comparison.

the range of highest collision probability, suggesting the utility of this approach, (3) There is an asymmetry in orbits coming from the interior or the exterior, and now there are two lower energy cutoffs, EE x -1.5173 and E," x -1.5165, below which no collision can occur on the first pass by Jupiter. The asymmetry may be too slight to differentiate an interior origin from an exterior origin for SL9.

As a final computation, we address the NEA collision problem. For a mass parameter corresponding to the sun-Earth-asteroid problem ( p = we compute the collision probability. The 3.059 x R = 4.258 x result is shown in Figure 10. It is interesting that the collision probabilities are nearly twice those for the quasi-Hilda case, even though Jupiter has a much larger mass and radius than the Earth. The asymmetry in interior/exterior originating orbits is not as pronounced as in Figure 9, owing to the smaller value of p, and EE M E," x -1.5 - 4.03 x loM4.

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Energy (E+ 13)

&to4

Fig. 10. Collision probabilities for near-Earth asteroids. Note that the collision probabilities are nearly twice those for the quasi-Hilda case in Figure 9, even though Jupiter has a much larger mass and radius than the Earth.

4. Conclusions

We address some questions regarding nonlinear comet and asteroid behavior by applying statistical methods to the planar, circular restricted three-body problem. In particular, we make a Markov assumption regarding the phase space and compute probabilities of interior-exterior transition and collision with the secondary. Theory and observation are seen to agree for the comets P/Oterma and D/Shoemaker-Levy 9.

References 1. Belbruno, E. and B. Marsden [1997], Resonance hopping in comets. The Astronomical Journal 113(4), 1433-1444. 2. Benner, L.A.M., and W.B. McKinnon [1995], On the orbital evolution and origin of comet Shoemaker-Levy 9. Icarvs 118, 155-168. 3. Carusi, A., L. Kresiik, E. Pozzi, and G.B. Valsecchi [1985], Long term evolution of short period comets. Adam Hilger, Bristol, UK. 4. Chodas, P.W. and D.K. Yeomans [1993], The upcoming collision of comet Shoemaker-Levy 9 with Jupiter. Bull. A m . Astron. SOC.25.

Statistical Theory of Interior-Exterior finsition and Collision Probabilities 651 5. Conley, C. [1968], Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16, 732-746. 6. Doedel, E.J., R.C. PafFenroth, H.B. Keller, D.J. Dichmann, J. Galan, and A. Vanderbauwhede [2002], Continuation of periodic solutions in conservative systems with application to the 3-body problem. Int. J. Bifurcation and Chaos., to appear. 7. Heppenheimer, T.A., and C. Porco [1977], New contributions to the problem of capture. Icarus 30, 385-401. 8. Howell, K.C., B.G. Marchand, and M.W. Lo [2000], Temporary satellite capture of short period Jupiter family comets from the perspective of dynamical systems. AAS/AIAA Space Flight Mechanics Meeting, Clearwater, Florida, USA. AAS Paper 00-155. 9. JafF6, C., S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly, and T. Uzer [2002], Statistical theory of asteroid escape rates, Physical Review Letters 89, 011101. 10. Kary, D.M., and L. Dones [1996], Capture statistics of short-period comets: implications for comet D/Shoemaker-Levy 9. Icarus 121, 207-224. 11. Koon, W.S., M.W. Lo, J.E. Marsden, and S.D. Ross [2000], Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10(2), 427-469. 12. Koon, W.S., M.W. Lo, J.E. Marsden, and S.D. Ross [2001], Resonance and capture of Jupiter comets. Celestial Mechanics and Dynamical Astronomy 81(1-2), 27-38. 13. Kresik, L. [1979], Dynamical interrelations among comets and asteroids. In Asteroids (T. Gehrels, Ed.), pp. 289-309. Univ. of Arizona Press, Tucson. 14. G6mez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont, and S.D. Ross [2001], Invariant manifolds, the spatial three-body problem and space mission design. AAS/AIAA Astrodynamics Specialist Conference, Quebec City, Canada, Aug. 2001. 15. Llibre, J., R. Martinez, and C. Sim6 [1985] Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem. Journal of Differential Equations 58 104-156. 16. Lo, M.W. and S.D. Ross [1997], Surfing the interplanetary tides. Bull. Am. Astron. SOC.29(3), 1021. 17. Marsden, B.G. 119931, IAU Circ. 5726. 18. McGehee, R. [1969], Some homoclinic orbits for the restricted three body problem. Ph.D . Thesis, University of Wisconsin, Madison, Wisconsin, USA. 19. Meiss, J.D. [1992], Symplective maps, variational principles, and transport. Reviews of Modern Physics 64(3), 795-848. 20. Parker, T.S. and L.O. Chua [1989], Practical numerical algorithms for chaotic systems, Springer-Verlag, New York. 21. Sim6, C., and T.J. Stuchi [2000], Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Physica D 140(1-2), 1-32. 22. Szebehely, V. [1967], Theory of orbits, Academic Press, New York. 23. Tancredi, G., M. Lindgren, and H. Rickman [1990], Temporary satellite cap-

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ture and orbital evolution of comet P/Helin-Roman-Crockett. Astronomy and Astrophysics 239,375-380. 24. Valsecchi, G.B. [1992], Close encounters, planetary masses, and the evolution of cometary orbits. In Periodic Comets (J.A. Fernhdez and H. Rickman, Eds.), pp. 143-157. Univ. de la Republica, Montevideo, Uruguay. 25. Wiggins, S. [1992] Chaotic transport in dynamical systems. Interdisciplinary Appl. Math. 2, Springer-Verlag, New York.

Libration Point Orbits and Applications G . Gbmez, M. W. Lo and J. J. Masdemont ( 4 s . ) @ 2003 World Scientific Publishing Company

SMALLER ALIGNMENT INDEX (SALI): DETERMINING THE ORDERED OR CHAOTIC NATURE OF ORBITS IN CONSERVATIVE DYNAMICAL SYSTEMS CH. SKOKOS*, CH. ANTONOPOULOS and T.C. BOUNTIS Department of Mathematics, Division of Applied Analysis and Center f o r Research and Applications of Nonlinear Systems (CRANS), University of Patras, GR-26500 Patras, Greece *also at Research Center for Astronomy, Academy of Athens, 14 Anagnostopoulou str., GR-10673 Athens, Greece

M. N. VRAHATIS Department of Mathematics and University of Patras Artificial Intelligence Research Center (UPAIRC), University of Patms, GR-26110 Patras, Greece

We apply the smaller alignment index (SALI) method for distinguishing between ordered and chaotic motion in some simple conservative dynamical systems. In particular we compute the SALI for ordered and chaotic orbits in a 2D and a 4D symplectic map, as well as a t w d e g r e e of freedom Hamiltonian system due to HBnon & Heiles. In all cases, the SALI determines correctly the nature of the tested orbit, faster than the method of the computation of the maximal Lyapunov characteristic number. The computation of the SALI for a sample of initial conditions allows us to clearly distinguish between regions in phase space where ordered or chaotic motion occurs.

1. Introduction

One of the most important approaches for understanding the behavior of a dynamical system is based on the knowledge of the chaotic vs. ordered nature of its orbits. For Hamiltonian systems with two degrees of freedom (or equivalently for 2D symplectic maps), the inspection of the consequents 653

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of an orbit on a Poincark surface of section (PSS), can give us reliable information for the dynamics of individual orbits. On the other hand, the distinction between ordered and chaotic motion becomes particularly difficult in systems with many degrees of freedom, where phase space visualization is no longer easily accessible. A quantitative method for distinguishing between order and chaos that has been extensively used (also for multidimensional systems), is the computation of the maximal Lyapunov characteristic number (LCN) The LCN is the limit of the finite time Lyapunov characteristic number:

(where 6 and & are the distances between two points of two nearby orbits at times t = 0 and t ) ,when t tends to infinity. In other words, LCN measures the average exponential deviation of two nearby orbits, so if LCN=O the tested orbit is ordered and if LCN > 0 it is chaotic. In maps, t is a discrete variable i.e. the number N of iterations of the map, so the finite time Lyapunov characteristic number can be denoted as L N . An advantage of this method is that it can be applied to systems of any number of degrees of freedom. The main disadvantage of LCN is that the time needed for Lt to converge to its limit can be extremely high and in some cases even unrealistic for the systems under study. A fast, efficient and easy to compute criterion to check if orbits of multidimensional maps are chaotic or not has been introduced in 5 : It concerns the computation of the smaller alignment index (SALI). Recently this method has been successfully applied to a two-degree of freedom Hamiltonian flow In the present communication we first recall the definition of the SALI and show its effectiveness by applying it to a 2D and a 4D symplectic map, comparing it also to the computation of LCN. We then use the SALI to study the dynamics of the Hknon-Heiles Hamiltonian system; and illustrate its ability to distinguish between regions of the phase space where ordered and chaotic motion occurs clearly and faster than LCN.

',

Smaller Alignment Indez (SALI) 655

2. Definition of the smaller alignment index (SALI)

Let us consider the 2n-dimensional phase space of a conservative dynamical system, described by a symplectic map T or a Hamiltonian system defined by the n degrees of freedom Hamiltonian function H . The time evolution of an orbit with initial condition P ( 0 ) = ( q ( O ) , z2(0), . . . , ~ ~ ( 0is )defined ) by the repeated applications of the map T or by the solution of Hamilton's equations of motion. In order to find the LCN one has to compute the limit of the finite time Lyapunov characteristic number Lt (1) for an initially infinitesimal ) ) the , time t or the deviation vector ((0) = ( d q ( O ) , d23(0),. . . , d ~ ~ ( 0 as number of iterations N tend to infinity, for Hamiltonian flows and maps respectively. The time evolution of the deviation vector is given by the equations of the tangent map

for maps, and by the variational equations

J.$=DH.J',

(3) for flows, where (') denotes the transpose matrix and matrices J and DH are defined by

d2H

with i , j = 1,2, ...,2n,

(4)

In being the n x n identity matrix and 0, the n x n matrix with all its elements equal to zero. In order to define the SALI for the orbit with initial conditions P(0) we follow the time evolution of two initial deviation vectors & ( O ) and ( ~ ( 0 ) . At every time step we normalize each vector to 1 and define the parallel alignment index

d - ( t ) = IIG(t) - G(t)ll

(5)

and the antiparallel alignment index

=

d + ( t ) llEl(t> + G(t)ll, (6) following (11 . 11 denotes the Euclidean norm of a vector). The smaller alignment index SALI is given by SALI = min(d-(t),d+(t)).

(7)

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From the above definitions we see that when the two vectors &(t),&(t) tend to coincide we get d,(t)

+ 0,

d+(t)

+ 2,

SALI --t 0,

while, when they tend to become opposite we get

d,(t)

+ 2,

d + ( t ) + 0, SALI * 0.

So, it is evident that SALI is a quantity that clearly informs us if the two deviation vectors tend to have the same direction by coinciding or becoming opposite. The reason why this information is useful for understanding if an orbit is chaotic or not is that, for systems of 2n-dimensional phase space with n 2 2, the two vectors tend to coincide or become opposite for chaotic orbits ', i.e. the SALI tends to zero. This is due to the fact that the direction of the two deviation vectors tends to coincide with the direction of the most unstable nearby manifold. On the other hand, if the tested orbit is ordered it lies on a torus and the two deviation vectors eventually become tangent to the torus, but in general converge to different directions. In that case the SALI does not tend to zero, but its values fluctuate around a positive value.

Although the SALI method is perfectly suited for multidimensional systems, it can also be applied to 2D maps. For 2D maps, whose phase space is 2-dimensional, the SALI tends to zero both for ordered and chaotic orbits, but with completely different time rates which allows us to clearly distinguish between the two cases. This behavior is due to the fact that for ordered orbits the two vectors, as has already mentioned, become tangent to the torus, which now is simply an invariant curve. So, the only possibilities for the two vectors are to become identical or opposite which means that the SALI tends to zero.

3. Applications of the SALI 3.1. Symplectic maps

Following we compute the SALI in some simple cases of ordered and chaotic orbits in symplectic maps with two and four dimensions. In partic-

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ular we use the 2D map:

+

x: = X I 2 2 x i = x2 - vsin(x1

+ x2)

(mod 2n),

and the 4D map:

+

x: = 21 2 2 XI, = x2 - vsin(xl+ x2) - p[1- cos(zl 4 = x3 +x4 x i = 2 4 - ~ s i n ( x 3 24) - p [ l - cos(xl

+

+ x2 + x3 + x4)1 + x2 + 2 3 + x4)]

(mod 2n),

(9) which is composed of two 2D maps of the form (8), with parameters v and K , coupled with a term of order p. All variables are given (mod 2n), so xi E [-n,n), for i = 1,2,3,4. The map (9) is a variant of the 4D map studied by Fkoeschl6 2. Some dynamical structures on the phase space of this map were examined in detail in for small values of the coupling parameter p.

1

.3

.2

.1

0

x;

1

2

3

, \

,

2

3

4

5

6

logN

Fig. 1. (a) Phase plot of the 2D map (8) for v = 0.5. The initial conditions of the ordered orbit A (21 = 2, 22 = 0) and the chaotic orbit B ( 2 1 = 3, 2 2 = 0) are marked by black and light-gray filled circles respectively. (b) The evolution of the SALI, with respect to the number N of iterations of the 2D map (8) for orbit A (black line) and for orbit B (gray line).

In the case of the 2D map (8) we consider the ordered orbit A with initial condition x1 = 2, x2 = 0 and the chaotic orbit B with initial condition x1 = 3, 2 2 = 0 for v = 0.5. The phase plot of map (8) can be seen in figure l(a), where the initial conditions of orbits A and B are marked by

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black and light-gray circles, respectively. The ordered behavior of orbit A and the chaotic nature of orbit B are evident from the distribution of their consequents on the 2D phase space. In particular the successive consequents of orbit A lie on a smooth invariant curve, while the successive consequents of orbit B are scattered in the small chaotic region that surrounds the main stability island around 21 = 22 = 0. The different nature of the two orbits is revealed also by the behavior of the SALI. The initial deviation vectors used for the computation of the SALI are & ( O ) = (1,O) and & ( O ) = ( 0 , l ) for both orbits. These vectors eventually coincide in both cases, but at completely different time rates. This is evident in figure l(b), where the SALI is plotted as a function of the number N of iterations for the ordered orbit A (black line) and the chaotic orbit B (gray line) in log-log scale. For the ordered orbit A the SALI decreases as N increases, following a power after lo7 iterations. On the other hand, law and becomes SALI M the SALI of the chaotic orbit B decreases abruptly, reaching the limit of accuracy of the computer (10-l6) after only about 200 iterations. After that time, the two vectors become identical to computer accuracy. So, it becomes evident that the SALI can distinguish between ordered and chaotic motion even in a 2D map, since it tends to zero following completely different time rates.

0

16 2

3

4

5 logN

6

7

2

3

4

5

6

7

logN

Fig. 2. The evolution in log-log scale of (a) the SALI and (b) the finite time Lyapunov characteristic number L N , with respect to the number N of iterations of the 4D map (9) with u = 0.5, K = 0.1, p = for the ordered orbit C with initial condition 11 = 0.5, 2 2 = 0,2 3 = 0.5,2 4 = 0 (black line) and for the chaotic orbit D with initial condition 21 = 3, 2 2 = 0,2 3 = 0.5,2 4 = 0 (gray line).

Smaller Alignment Indez (SALI)

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we In the case of the 4D map (9) for v = 0.5, n = 0.1 and p = consider the ordered orbit C with initial condition 2 1 = 0.5, x 2 = 0,5 3 = 0.5,2 4 = 0 and the chaotic orbit D with initial condition x1 = 3, 5 2 = 0, 2 3 = 0.5,2 4 = 0. The initial deviation vectors used for the computation of the SALI are (l,l,l,l)and (1,0,0,0), for both orbits. As we see in figure 2(a) the SALI of the ordered orbit C remains almost constant (black line), fluctuating around SALI M 0.28. On the other hand, the SALI of the chaotic orbit D decreases abruptly, reaching the limit of accuracy of the computer (10-l6) after about 4.7 x lo3 iterations (gray line). After that time, the coordinates of the two vectors are represented by opposite numbers in the computer (because the SALI actually coincides with d+ in this case), and any further computation of their evolution is not necessary.

So, in 4D maps the SALI tends to zero for chaotic orbits, while it tends to a positive value for ordered orbits. Thus, the different behavior of SALI clearly distinguishes between ordered and chaotic motion. Another advantage of using the SALI is that we can be sure about the nature of the tested orbit faster than using LCN. This becomes evident by looking at the evolution of the finite time Lyapunov characteristic number LN (1) for orbits C and D in figure 2(b). As expected, for the ordered orbit C, L N decreases as the number of iterations N increases, following a power law, reaching the after lo7 iterations. On the other hand, LN of the value LN M 1.6 x chaotic orbit D, after some fluctuations, seems to stabilize near a constant after lo7 iterations. By comparing panels non-zero value LN M 5 x (a) and (b) of figure 2 we see that, after about 4.7 x lo3 iterations we can be sure that orbit D is chaotic using the SALI, since it has become equal to and the two deviation vectors practically coincide, while we cannot stop the computation of LN at that time, since it is not yet evident whether the LCN for orbit D will ultimately be zero or not.

3.2. The He'non-Heiles Harniltonian system

In order to illustrate the effectiveness of the SALI in determining chaotic vs. ordered orbits in Hamiltonian flows, we consider the two degrees of freedom Hhon-Heiles Hamiltonian

660

Ch. Skokos et al. 0.5 -

P

0.0

-

.n -..,F, -

-0.5

0.0

0.5 Y

Fig. 3. The Poincar6 surface of section for z = 0 of the two degrees of freedom HBnonHeiles Hamiltonian (10) for H=1/8. The projection on the PSS of the initial conditions of the ordered orbit E (z = 0, y = 0.55, p , N 0.2417,py = 0) and the chaotic orbit F (z = 0, y = -0.016, p , N 0.49974,py = 0) are marked by black and light-gray filled circles respectively. The axis py = 0 is also plotted.

where x, y are the generalized coordinates and p,, p , the conjugate momenta. In particular, we consider the case of fixed energy H = 1/8, for which the system exhibits a rich dynamical structure. As it can be seen on the Poincar6 surface of section (PSS) for x = 0 in figure 3 there exist islands of stability, where ordered motion occurs, as well as extensive regions where chaotic motion takes place. In order to apply the SALI method, we consider the ordered orbit E with initial condition x = 0, y = 0.55, p , 11 0.2417, p , = 0 and the chaotic orbit F with initial condition x = 0 , y = -0.016, p , 11 0.49974, p , = 0. The projection of the initial conditions of orbits E and F on the PSS are marked by black and light-gray filled circles respectively in figure 3. The initial deviation vectors (dx(O),dy(O), d p I ( 0 ) ,dp,(O)) used for the computation of SALI are (1,0,0,0) and (O,O, 1,O). As we see in figure 4(a) the SALI of the ordered orbit E remains almost constant, fluctuating around SALI M 1 (black line), while the SALI of the chaotic orbit F decreases abruptly reaching the limit of accuracy of the computer after about 1,700 time units (gray line). The behavior of the SALI is similar to the one encountered in the 4D map (9) (figure

Smaller Alignment Index (SALI)

661

0

.1

-

-I

M

.2

-

-3

-4

.

1

0

1

2 bat)

3

4

5

Fig. 4. The evolution in log-log scale of (a) the SALI and (b) the finite time Lyapunov characteristic number Lt as a function of time t , for the Hamiltonian (10) with H=1/8, for the ordered orbit E with initial condition x = 0, y = 0.55, p, 1~ 0.2417, p, = 0 (black line) and for the chaotic orbit F with initial condition x = 0, y = -0.016, p, N 0.49974, p, = 0 (gray line).

2(a)), since the phase space of the Hamiltonian system is 4-dimensional as in the case of the 4D map. In figure 4(b) we see the time evolution of finite time Lyapunov characteristic number Lt (1) for orbits E (black line) and F (gray line). Lt of the ordered orbit E, after an initial transient time interval where it exhibits large fluctuations, starts to decrease following a for t = lo5. Lt of the chaotic orbit power law, reaching the value Lt M F has larger fluctuations without tending to zero, although even for t = lo5 it does not seem to stabilize around a non-zero value. We underline the fact that in this case, using the SALI we were absolutely sure that orbit F is chaotic at t = 1,700, at which time SALI became practically zero, although at that time the use of Lt could not give us the same information. In figure 3 we see that in a large portion of phase space the motion of system (10) is chaotic. The chaotic region corresponds to the area filled with scattered points on the PSS, while ordered motion corresponds to the islands formed by the invariant smooth curves. Since the SALI tends to completely different values for ordered and chaotic orbits, its computation for a sample of initial conditions can be used to distinguish between regions where ordered or chaotic motion occurs.

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0.5

I

P y 0.0

L

09

0.'

Y

4.5

.0.5

0.0

0.5

Y

Fig. 5 . (a) The value of SALI for t = 4,000 for orbits with initial conditions on the p , = 0 axis of the PSS shown in figure 3, as a function of the y variable of the initial condition. The data are plotted as black points and are connected by gray lines. (b) Regions of different values of the SALI on the PSS (y,py)after 1,000 time steps. Initial conditions that give SALI < lo-'' are marked by black points, initial conditions that 5 SALI < give are marked by deep gray points, initial conditions that are marked by gray points, while initial conditions that give give lo-* 5 SALI < 5 SALI are marked by light gray points.

As a first example, we consider orbits that lie on the p , = 0 line on the PSS shown in figure 3, having initial conditions x = 0, p , = 0, while p , is defined by the Hamiltonian (10). The values of the SALI for all these orbits, after 4,000 time units, are plotted with black points in figure 5(a), as function of the y variable of the initial condition. These points are line connected in order to be easily visible the changes of the SALI as the initial condition moves on the p , = 0 line. We can clearly see regions of ordered motion where the SALI has values larger than corresponding to the islands of stability that are crossed by the line py = 0 in figure 3. There also exist regions of chaotic motion where the SALI has become smaller than or has even reached the limit of accuracy of the computer (10-l6), in agreement to the regions crossed by the p , = 0 line where scattered points exist on the PSS. Although most of the initial conditions give large values for the SALI, there also exist initial (2 or very small (< conditions that give, after 4,000 time units, intermediate values for the SALI 5 SALI < These correspond to sticky orbits existing near the borders of ordered motion, and more time is needed for the SALI to reach very small values and reveal their chaotic nature.

Smaller Alignment Index (SALI)

663

By carrying out the above analysis not only on a line on the PSS but for the whole plane, plotting with different shades of gray initial points that give values for the SALI in different ranges, we can get an image of phase space regions where ordered and chaotic motion are clearly distinguished (figure 5(b)). In figure 5(b) we see that our PSS is practically divided into regions where ordered motion occurs, colored in light gray, which corresponds to 5 SALI, and those colored in black, where chaotic behavior occurs, On the borders between these two regions corresponding to SALI < we see points that give intermediate values for the SALI colored in deep 5 SALI < and in gray (lo-* 5 SALI < loc4), which gray correspond to sticky orbits. The resemblance between figure 5(b) and figure 3 is obvious. We should also mention that in figure 5(b), we can see some very thin regions of ordered motion corresponding to very small islands of stability that cannot be seen easily in figure 3. So, it is evident that starting with any initial condition, the computed value of the SALI rapidly gives a clear view of chaotic vs. ordered motion even for systems described by ordinary differential equations, where surface of section plots are already time consuming for 2 degrees of freedom and practically useless for systems of higher dimensionality.

4. Conclusions

In this paper, we have given some examples of symplectic maps and Hamiltonian systems, where the computation of the smaller alignment index SALI allows us to distinguish in a cost effective way between ordered and chaotic orbits. The computation of the SALI is a fast, efficient and easy to compute numerical method, perfectly suited for multidimensional systems, but it can also be applied to 2D maps. In 2D maps, the SALI tends to zero both for ordered and chaotic orbits, but following completely different time rates which allows us to distinguish between the two cases. In maps of higher dimensionality (and Hamiltonian systems) the SALI tends to zero for chaotic orbits, while in general, it tends to a positive value for ordered orbits. So, we can easily distinguish between regular and chaotic orbits. Our approach, in fact, begins to be truly valuable for Hamiltonian systems of 2 degrees of freedom (where detailed surface of section plots are computationally too costly) and promises to become extremely useful for higher than 2 degree

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of freedom Hamiltonians and higher dimensional symplectic maps. An advantage of using the SALI in Hamiltonian systems or in multidimensional maps is that usually the chaotic nature of an orbit can be established beyond any doubt. This happens because when the orbit under consideration is chaotic, the SALI becomes zero, in the sense that it reaches the limit of the accuracy of the computer. After that time the two deviation vectors, needed for the computation of the SALI, are identical (equal or opposite), because their coordinates are represented by the same or opposite numbers. Thus, they have exactly the same evolution in time and cannot be separated. This practically means that we do not need to continue the computation of the evolution of the two vectors further on. We should also mention that in all cases studied, the use of the SALI helped us decide if an orbit is ordered or chaotic much faster than the computation of the finite time Lyapunov characteristic number Lt.

Acknowledgments Ch. Skokos thanks the LOC of the conference for its financial support. Ch. Skokos was also supported by ‘Karatheodory’ post-doctoral fellowship No 2794 of the University of Patras and by the Research Committee of the Academy of Athens (program No 200/532). Ch. Antonopoulos was supported by ‘Karatheodory’ graduate student fellowship No 2464 of the University of Patras.

References 1. Benettin G., Galgani L. and Strelcyn J. M., 1976, Phys. Rev. A, 14, 2338 2. F’roeschld C., 1972, A&A, 16, 172 3. Froeschld C., 1984, Celest. Mech., 34, 95 4. Hdnon M., Heiles C., 1964, Astron. J., 69, 73 5. Skokos Ch., 2001, J. Phys. A, 34, 10029 6. Skokos Ch., Contopoulos G., Polymilis C., 1997, Celest. Mech. Dyn. Astron., 65, 223 7. Skokos Ch., Antonopoulos Ch., Bountis T. C. and Vrahatis M. N., 2002, in “CD-Rom Proceedings of GRACM 2002, 4th GRACM Congress on Computational Mechanics”, Univ. of Patras, Patras, Greece 8. Voglis N., Contopoulos G., Efthymiopoulos C., 1999, Celest. Mech. Dyn. Astron., 73, 211

Libration Point Orbits and Applications G. Gbmez, M. W. Lo and J. J. Masdemont (eds.) @ 2003 World Scientific Publishing Company

LOCATING PERIODIC ORBITS B Y TOPOLOGICAL DEGREE THEORY C. POLYMILIS Department of Physics, University of Athens, Panepistimiopolis, GR-15784 Zografos, Athens, Greece G. SERVIZI, G. TURCHETTI

Department of Physics, Bologna University, Via Irnerio 46, I-40126 Bologna, Italy and I.N.F.N. Sedone d i Bologna, Via Irnerio 46, I-40126 Bologna, Italy

CH. SKOKOS Department of Mathematics, Division of Applied Analysis and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, GR-26500 Patms, Greece Research Center f o r Astronomy, Academy of Athens, 14 Anagnostopoulou str., GR-10673 Athens, Greece

M. N. VRAHATIS Department of Mathematics and University of Patms Artificial Intelligence Research Center (UPAIRC), University of Patras, GR-26110 Patras, Greece

We consider methods based on the topological degree theory to compute p e riodic orbits of area preserving maps. Numerical approximations of the Kronecker integral and the application of Stenger's method allows us to compute the value of the topological degree in a bounded region of the phase space. If the topological degree of an appropriate set of equations has a non-zero value, we know that there exists at least one periodic orbit of a given period in the given region. We discuss in detail the problems that these methods face, due to the existence of periodic orbits near the domain's boundary and due to the

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discontinuity curves that appear in maps defined on the torus. We use the characteristic bisection method for actually locating periodic orbits. We apply this method successfully, both to the standard map, which is a map defined on the torus, and t o the beam-beam map which is a continuous map on the plane. Specifically we find a large number of periodic orbits of periods up to 40,which give us a clear picture of the dynamics of both maps.

1. The topological degree (TD) and its computation

We consider the problem of finding the solutions of a system of nonlinear equations of the form

K ( 2 )= e n , (1) . . ,f,) : D, c R" -+ R" is a function from a domain

where Fn = (fl, f 2 , . D, into R", 0, = (O,O, . . . ,0) and can be written as:

2 =

( q ,. .~. ,z,). ,

The above system

The topological degree (TD) theory gives us information on the existence of solutions of the above system, their number and their nature Kronecker introduced the concept of the TD in 1869 5 , while Picard in 1892 g succeeded in providing a theorem for computing the exact number of solutions of system (1). Numerical methods based on the TD theory have been applied successfully to numerous dynamical systems (e.g. 5*91101336.

16,17,18,19,4,8

1.

In order to define the concept of the topological degree we consider the function F, of system (1) to be continuous on the closure of D,, satisfying also F,(x) # 0, for 2 on the boundary b(D,) of D,. We also consider the solutions of (1) to be simple i.e. the determinant of the corresponding Jacobian matrix ( J F , ) at the solution, to be different from zero. Then the topological degree (TO)of F, at 0, relative to 0, is defined as: deg[F',

D,,en]=

sgn(detJF,,(z)) = N+ - N - , Z€Fll(Qn)

(3)

Locating Periodic Orbits by Topological Degree Theory 667

where detJF, is the determinant of the Jacobian matrix of F,, sgn is the well-known sign function, N+ the number of roots with detJF, > 0 and N - the number of roots with detJF, < 0. It is evident that if a nonzero value of deg[F,, D,,On] is obtained then there exist at least one solution of system F,(x) = 0, within D, 5 .

A practical way to find the TD is the computation of Kronecker integral 5 . In particular, under the assumptions of the abovementioned definition of the TD the deg[F,, D,, On] can be computed as:

where

and

r(x) is the gamma function.

In order to find the number N of solutions of system (1) we consider the function

where fn+l

= Y detJF,

,

(7)

:~ 1 ~ x 2 . .,,x,, . y and Dn+l is the product of D, with a real interval on the y-axis containing y = 0. Then the exact number N of the solutions of equation F,(x) = 0, is proven to be ':

By applying this result and using the computation of TD by Kronecker integral (4) in the case of a set of two equations:

668 C. Polymilis et al.

we find that the number N of roots in the domain given by:

Dz

= [u,b]x

[c,d] is

where E is an arbitrary positive value, and

with J denoting the determinant of the Jacobian matrix of F2= (fi, f2). Another method for computing the TD of F, at a domain D, is the application of Stenger's theorem Following this approach for finding the TD, we only need to know the signs of functions f l , fi, . . ., fn in a 'sufficient' set of points on the boundary b(D,) of 0,. l2l7.

2. The characteristic bisection method The characteristic bisection method is based on the characteristic polyhedron concept for the computation of roots of the equation (1).The construction of a suitable n-polyhedron, called the characteristic polyhedron, can be done as follows. Let M n be the 2" x n matrix whose rows are formed by all possible combinations of -1 and 1. Consider now an oriented npolyhedron II", with vertices Vk, k = 1 , . . . ,2,. If the 2" x n matrix of signs associated with F and III", whose entries are the vectors %nFn(Vk) = (sgnfl(Vk),sgnfZ(Vk)r.. . 1 sgnfn(Vk)),

(12)

is identical to M,, possibly after some permutations of these rows, then II" is called the characteristic polyhedron relative to F,. If F, is continuous, then, after some suitable assumptions on the boundary of IIn we have: deg[F,,II",Q,]

=fl

# 0.

(13)

This means that there is at least one solution of system Fn(x) = 0, within

II" . To clarify the characteristic polyhedron concept we consider a function Each function fa, i = 1'2, separates the space into a number

F 2 = (fl, f2).

Locating Periodic Orbits by Topological Degree Theory 669

of different regions, according to its sign, for some regions fi < 0 and for the rest fi > 0, i = 1,2. Thus, in figure l(a) we distinguish between the regions where fi < 0 and f2 < 0, f1 < 0 and f2 > 0, fi > 0 and f2 > 0, f1 > 0 and f2 < 0. Clearly, the following combinations of signs are possible: (-, -), (-, +), (+, +) and (+, -). Picking a point, close to the solution, from each region we construct a characteristic polyhedron. In this figure we can perceive a characteristic and a noncharacteristic polyhedron I12. For a polyhedron 112 to be characteristic all the above combinations of signs must appear at its vertices. Based on this criterion, polyhedron ABDC is not a characteristic polyhedron, whereas AEDC is. A characteristic polyhedron can be considered as a translation of PoincarkMiranda hypercube 15.

Fig. 1. (a) The polyhedron ABDC is noncharacteristic while the polyhedron AEDC is characteristic. (b) Application of the characteristic bisection method to the characteristic polyhedron AEDC, giving rise to the polyhedra GEDC and HEDC, which are also characteristic.

Next, we describe the characteristic bisection method. This method simply amounts to constructing another refined characteristic polyhedron, by bisecting a known one, say II", in order to determine the solution with the desired accuracy. We compute the midpoint M of an one-dimensional edge of IIn, e.g. (K, %). The endpoints of this one-dimensional line segment are vertices of I I n , for which the corresponding coordinates of the vectors, sgnF,(K) and sgnF,(Vj) differ from each other only in one entry. To obtain another characteristic polyhedron we compare the sign of F,(M) with that of F,(K) and Fn(%) and substitute M for that vertex for which the signs

670 C. Polymilis

et al.

are identical. Subsequently, we reapply the aforementioned technique to a different edge (for details we refer the reader to 1. 13t1471698

To fully comprehend the characteristic bisection method we illustrate in figure l ( b ) its repetitive operation on a characteristic polyhedron I12. Starting from the edge AE we find its midpoint G and then calculate its vector of signs, which is (-1, -1). Thus, vertex G replaces A and the new refined polyhedron GEDC, is also characteristic. Applying the same procedure, we further refine the polyhedron by considering the midpoint H of GC and checking the vector of signs at this point. In this case, its vector of signs is (-1, -l), so that vertex G can be replaced by vertex H. Consequently, the new refined polyhedron HEDC is also characteristic. This procedure continues up to the point that the midpoint of the longest diagonal of the refined polyhedron approximates the root within a predetermined accuracy.

3. Applications

We apply methods based on the topological degree theory to compute periodic orbits of two area preserving maps, the Standard map (SM) 2 , which is a map defined on the torus:

+

&

z’ = z y sin(2nz) y’ = y sin(2m)

&

and the beam-beam map (BM) z1 = zcos(2nw)

‘,11,

, mod(l),

z,y E [-0.5,0.5),

which is a map defined on R2:

+ (y + 1- e-1’) sin(2nw), + (y + 1- e-z2 cos(2nw).

y l = -z sin(2nw)

(14)

(15)

Given a dynamical map M : {z’ = gl(z,y), y’ = gZ(z, y)}, the periodic points of period p are fixed points of the piteration MP of the map and the zeroes of the function:

where I is the identity. One can use a color map to inspect the geometry of function F (16) and to locate its zeroes. The color map is created by choosing a lattice of m x m points and by associating to each point a color chosen according to the

Locating Periodic Orbits by Topological Degree Theory 671

Fig. 2. sign.

Sketch of the domains where functions

fl

and fz of system (16) have a definite

signs of the functions f i , fi as shown in figure 2. A simple algorithm allows to detect the cells, formed by the lattice of m x m points, whose vertices have different colors. A cell is a candidate to have a zero in its interior if the corresponding topological degree is found to be different from zero. In figures 3 and 4 we construct the color map and apply the above mentioned algorithm for locating periodic orbits of period 3 for the SM and of period 5 for the BM, respectively. In both figures the gray circles at the right panels denote the positions of the found periodic orbits. We see that for both maps some periodic orbits were not found because some of the four color domains close to the fixed point were very thin. On the other hand, due to the discontinuity of function F (16), some zeros that do not correspond to real periodic orbits were found for the SM (right panel of figure 3).

For maps defined on the torus like the SM (14), the computation of the TD using Stenger's method or the Kronecker integral (10) faces problems due to the presence of discontinuity curves. Indeed Kronecker integral is defined on a domain where the function F (16) is continuous. For the SM the discontinuity curves are the lines x = -0.5 and y = -0.5. By applying the SM map M once these lines are mapped on the curves seen in the right panel of figure 5. On the initial phase space there exist also the discontinuity curves that will be mapped after one iteration to the lines x = -0.5 and y = -0.5. These curves are also plotted in the left panel of figure 5. These curves can be produced by applying the inverse SM to

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C.Polymilis

et al.

05

35

*

%

4.5

35

4.5

-05 -05

X

05 X

Fig. 3. Standard map (14) for k = 0.9: color map for p = 3 iterations of the map computed on a square of m x m points for m = 512 (left panel); phase plot of the map (right panel). The gray circles denote the positions of the zeros of the corresponding function (16).

-2

2 X

-2

2 X

Fig. 4. Beam-beam map (15) for w = 0.21: color map for p = 5 iterations of the map computed on a square of m x m points for m = 512 (left panel); phase plot of the map (right panel). The gray circles denote the positions of the zeros of the corresponding function (16).

the discontinuity lines x = -0.5 and y = -0.5. So the discontinuity curves divide the initial phase space in five continuous regions marked as I, 11, 111, IV and V in figure 5. In each region the computation of the TD can be performed accurately by Stenger’s method or by evaluating Kronecker integral. If, however, the boundary of the domain where these procedures are applied, cross a discontinuity curve the results we get are not correct (figure 6). In order to study the dependence of the procedure for finding the TD in a region D, with respect to the distance of a root from the boundary of

Locating Periodic Orbits by Topological Degree Theory 673 0.5

m Y 0.0

4.6

4.5

OR

0.5

i

S

Fig. 5. The discontinuity curves of the standard map M (14) divide the phase space in five continuous regions (I, 11, 111, IV, V). In each region the computation of the TD can be performed accurately.

X

Fig. 6. (a) Number of period 1 fixed points Af1 evaluated for the SM (14) with k = 0.9 using the Kronecker integral ( l o ) , in a rectangular domain whose upper-side moves, as a function of the y coordinate of this side. The rectangle and the discontinuity lines are shown in (b). For the various rectangles, Af1 should be equal to 1 since they contain only 1 fixed point of period 1, namely point (0,O). The two points marked by arrows in (a) where N1 deviates from the correct value Af1 = 1, correspond to y M 0.358 and y M 0.466 respectively, where the upper-side of the rectangular crosses the two discontinuity curves in (b).

I>, we consider the simple map

The lines f1 = 0, fi = 0 are plotted in figure 7(a). The system F* = (0,O) has three roots. The determinant of the corresponding Jacobian 0) matrix ( d e t J p ) is positive for root (0,O) and negative for roots

(-a

674 C. Polymilis et al.

I

$+

Fig. 7. (a) Plot of the curves fl = y I = 0, f2 = y = 0 (b) Dependence of the number of grid points n g p ,needed for computing the correct value of the T D in a domain, on the distance E of a root from the boundary of the domain, for the set of equations of (a) (dashed line) and the SM (continuous line).

(a, a,

and 0). We also consider a rectangular of the form [-a, 21 x [-2,2] with a > shown in figure 7(a). Since this domain contains the three roots of the system the value of the TD is -1. We set a = 8 E with E > 0 so that the boundary approaches the root as E + 0, as shown by the arrow in figure 7(a). We compute the TD for different values of 6 applying Stenger’s method, by using the same number of points m on every side of the rectangle. We denote by ngp= 4 m the smallest number of grid points needed to compute the TD with certainty. In figure 7(b) we plot in log-log scale, ngpwith respect to E (dashed line). The slope of the curve is almost -1 so that m 0: E - ’ . The same result holds for any map when the boundary approaches a root (the solid line in figure 7(b) is obtained for a similar example for the SM (14)).

+

Despite the probIems caused by the discontinuity curves or by roots located very close to the domain’s boundary, the use of the characteristic bisection method can locate a big fraction of the real periodic orbits. Actually by applying the characteristic bisection method on the cells of a lattice formed by 2000 x 2000 grid points we were able to compute a sufficient number of the periodic orbits with period up to 40, for the SM (figure 8, left panel) and the BM (figure 8, right panel). The computed periodic orbits give us a clear picture of the dynamics of these maps.

Locating Periodic Orbits by Topological Degree Theory 675

0.5

-X

0.5

-5

X

5

Fig. 8. Periodic orbits up to period p = 40 for the SM (14) for k = 0.9 (left panel) and for the BM (15) for w = 0.14 (right panel). Different gray-scales correspond to periodic orbits with different kind of stability.

4. Synopsis

We have studied the applicability of various numerical methods, based on the topological degree theory, for locating high period periodic orbits of 2D area preserving maps. In particular we have used the Kronecker integral and applied the Stenger’s method for finding the TD in a bounded region of the phase space. If the TD has a non-zero value we know that there exist at least one periodic orbit in the corresponding region. The computation of the TD for an appropriate set of equations allows us to find the exact number of periodic orbits. We also applied the characteristic bisection method on a mesh in the phase space for locating the various fixed points. The main advantage of all these methods is that they are not affected by accuracy problems in computing the exact values of the various functions used, as, the only computable information needed is the algebraic signs of these values. We have applied the abovementioned methods to 2D symplectic maps defined on Rz and on the torus. The methods for computing the TD are applied to continuous regions of the phase space, so their use for maps on the torus is limited to regions where no discontinuity curves exist. On the

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other hand the characteristic bisection method proved to be very efficient for all different types of maps, as, it allowed us to compute a big amount of the real fixed points of period up to 40 in reasonable computational times.

Acknowledgments Ch. Skokos thanks the LOC of the conference for its financial support. Ch. Skokos was also supported by ‘Karatheodory’ post-doctoral fellowship No 2794 of the University of Patras and by the Research Committee of the Academy of Athens (program No 200/532).

References 1. Bazzani A., Servizi G, Todesco E. and Turchetti G., 1994, CERN Yellow Report 9402. 2. Chiricov B. V., 1979, Phys. Rep., 52, 263. 3. Cronin J., 1964, Fixed points and topological degree in nonlinear analysis, Mathematical Surveys No. 11, Amer. Math. SOC.,Providence, Rhode Island. 4. Kalantonis V. S., Perdios E. A., Perdiou A. E. and Vrahatis M. N., 2001, Cel. Mech. Dyn. Astr., 80, 81. 5. Kronecker L., 1895, Werke, Vol. 1, Leipzig, Teubner. 6. Lloyd N. G., 1978, Degree Theory, Cambridge University Press, Cambridge. 7. Mourrain B., Vrahatis M.N., and Yakoubsohn J.C., 2002, J. Compl., 18,612. 8. Perdios E. A,, Kalantonis V. S. and Vrahatis M. N., 2002, Cel. Mech. Dyn. Astr., 84, 231. 9. Picard E., 1892, J. Math. Pure Appl. (4e s6rie), 8, 5. 10. Picard E., 1922, Truit.4 d’ analyse, 3rd ed., Chap. 4.7., Gauthier-Villars, Paris. 11. Polymilis C., Skokos Ch., Kollias G., Servizi G. and Turchetti G., 2000, J. Phys. A, 33, 1055. 12. Stenger F., 1975, Numer. Math., 25, 23. 13. Vrahatis M. N., 1988, ACM Tkans. Math. Software,14, 312. 14. Vrahatis M. N., 1988, ACM Trans. Math. Software, 14, 330. 15. Vrahatis M. N., 1989, Proc. Amer. Math. SOC.,107, 701. 16. Vrahatis M. N., 1995, J. Comp. Phys., 119, 105. 17. Vrahatis M. N., Bountis T. C. and Kollmann M., 1996, Inter. J. Bifurc. Chaos, 6, 1425. 18. Vrahatis M. N., Isliker H. and Bountis T. C., 1997, Inter. J. Bifurc. Chaos, 7, 2707. 19. Vrahatis M. N., Perdiou A. E., Kalantonis V. S., Perdios E. A., Papadakis K., Prosmiti R. and Farantos S. C., 2001, Comp. Phys. Com., 138, 53.

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