Dynamics and Mission Design Near Libration Points, Volume I : Fundamentals : The Case of Collinear Libration Points (World Scientific Monograph Series in Mathematics)

World Scientific Monograph Series in Mathematics - Vol. 2

Dynamics and Mission Design Near Libration Points Vol. I Fundamentals: The Case of Collinear Libration Points

. Gomez . Llibre R. Martinez

.

T World Scientific

Dynamics and Mission Design Near Libration Points Vol.1 Fundamentals: The Case of Collinear Libration Points

World Scientific Monograph Series in Mathematics Eds.

Ron Donagi (University of Pennsylvania), Rafael de la Llave (University of Texas) and Mikhail Shubin (Northeastern University)

Published

Vol. 1:

Almgren's Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2 Eds. V. Scheffer and J. E. Taylor

Vol. 2:

Dynamics and Mission Design Near Libration Points Vol. I Fundamentals: The Case of Collinear Libration Points by G. Gomez, J. Llibre, R. Martinez and C. Simo

Vol. 3:

Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular Libration Points by G. Gomez, J. Llibre, R. Martinez and C. Simo

Vol. 4:

Dynamics and Mission Design Near Libration Points Vol. Ill Advanced Methods for Collinear Points by G. Gomez, A. Jorba, J. Masdemont and C. Simo

Vol. 5:

Dynamics and Mission Design Near Libration Points Vol. IV Advanced Methods for Triangular Points by G. Gomez, A. Jorba, J. Masdemont and C. Simo

Vol. 6:

Hamiltonian Systems and Celestial Mechanics Eds. J. Delgado, E. A. Lacomba, E. Perez-Chavela and J. Llibre

World Scientific Monograph Series in Mathematics - Vol. 2

Dynamics and Mission Design Near Libration Points Vol.1 Fundamentals: The Case of Collinear Libration Points

G. Gomez Universitat Politecnica de Catalunya, Spain

J. Llibre & R. Martinez Universitat Autbnoma de Barcelona, Spain

C. Simo Universitat de Barcelona, Spain

©World Scientific •

Singapore »New Jersey'London* Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

World Scientific Monograph Series in Mathematics - Vol. 2 DYNAMICS AND MISSION DESIGN NEAR LIBRATION POINTS Vol. I: Fundamentals: The Case of Collinear Libration Points Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Foreword

Already Leonhard Euler and Joseph-Louis Lagrange knew about the libration points in the restricted three-body problem in the second half of the eighteenth century. The real importance of these special locations became clear very soon in the Space Age. In the days of the Apollo flights to the Moon consideration was given for placing a relay satellite near the translunar libration point. Fundamental investigations were carried out by Giuseppe Colombo, John Breakwell, Robert Farquhar, and others on the practical use of these points and on quasi-periodic orbits. A milestone is the seminal compendium of Victor Szebehely "Theory of Orbits: the Restricted Problem of Three Bodies", Academic Press, 1967. It took, however, more than 20 years after Sputnik to use these privileged positions in the solar system for a satellite project: the launch on August 12, 1978 of NASA's International Sun-Earth Explorer (ISEE-3) into a Halo orbit around L\ in the Earth-Sun system. Later followed the ESA-NASA spacecraft SOHO (solar and heliospheric observatory), launched on Dec. 2, 1995, and in August 1997 NASA's Advanced Composition Explorer (ACE), both for the same destination near L\. Several fundamental astronomy missions will benefit from the unique properties of Halo and Lissajous orbits near L 2 in the Earth-Sun system: in 2007 ESA's Cornerstone mission FIRST (far infrared astronomy) will be launched jointly with Planck (mapping of the microwave background). In 2008 the successor to the Hubble Space Telescope, the New Generation Space Telescope (NGST) will be placed near Li. ESA's successor of the highly successful astrometry satellite HIPPARCOS, the GAIA spacecraft, is planned to be placed in the same location. Pioneering work for the understanding of the dynamical properties and the construction of quasi-periodic orbits in the vicinity of the collinear and triangular libration points was carried out by a group of mathematicians led by Carles Simo of the University of Barcelona. With a series of research contracts *, initiated in 1983 and sponsored by the European Space Agency a thorough analysis of quasiperiodic orbits in the Earth-Moon and the Earth-Sun system, and the construction *ESA CR(P)2272, ESA CR(P)2440, ESA CR(P)2288, ESA CR(P)3742

vi

Foreword

of quasi-periodic orbits using sophisticated and powerful algorithms, was accomplished. Real-world models for the ephemerides of the planets and the Moon were adopted. The insight into the dynamical properties of the quasi-periodic orbits is also of paramount importance for the design of fuel-optimum orbit control strategies and optimum transfer strategies from the Earth. An important outcome of this research is the discovery of transfer trajectories with zero velocity requirement for the injection into the final orbit. ESA's future scientific missions such as FIRST, Planck, Gaia and possibly DARWIN will greatly benefit from the sophisticated methods and tools developed by the Barcelona research group. ESOC, Darmstadt, Fall 2000

Walter Flury Head of Mission Analysis Section

Preface

The present work is the final report of a study contract that was done for the European Space Agency and was finished in 1985. At that moment it was part of the mission analysis studies done by ESA for the SOHO mission. Unfortunately, none of the techniques developed in our work was used for the real mission, and the tools used for the determination of the nominal trajectory, the transfer from the Earth and the station keeping, were the same ones used, at the end of the seventies, for the ISEE-C mission, requiring a larger cost. Now the scenario has changed. Most of the missions to the Lagrange points are studied using the dynamical systems tools and ideas of our work. In general, they can be used when the basic model is a (perturbed) restricted three-body problem or some similar simple model which can be studied extensively. They have been shown to give a global and geometric picture of the problem, a clear understanding of the phase space and, at the same time, they provide a systematic approach that avoids the "trial and error" procedures, so widely used in the study of this kind of missions. In this book the problem of station keeping is studied for orbits near libration points in the solar system. Main attention is devoted to orbits near halo ones in the Sun-Earth+Moon system. Taking as starting point the restricted three-body problem, the motion in the full solar system is considered as a perturbation of this simplified model. All the study is done with enough generality to allow an easy application to other primary-secondary systems as an easy extension of the analytic and numerical algorithms. First, the families of halo orbits and the motion near them are studied. Then, the general equations of motion are stated and integrated to get the final nominal quasiperiodic orbit. Once the nominal orbit is available, an on/off station keeping method is developed, founded on geometrical considerations that use ideas of dynamical systems theory. The feasibility of radiation pressure and low thrust station keeping has been shown. It has been proved that the proposed on/off method of station keeping is very cheap, in terms of fuel consumption, and the numerical simulations can be done in a very fast way, when the nominal orbit and projection factors are available. vii

Vlll

Preface

All the above items have been studied numerically and, as far as possible, analytically. For the ESA original study, we produced a large amount of software to perform the analytic and numerical computations and to do simulations of the real behavior. This software is not included in the text, but all its main modules are described with detail. The ESA report is reproduced textually with minor modifications: the detected typing or obvious mistakes have been corrected and some tables have been shortened. The layout of the (scanned) figures and tables has changed slightly, to accommodate to latex requirements. The last page of this preface reproduces the cover page of the report for the European Space Agency showing, in particular, the original title of the study. This is the first of a series of four reports done by persons of our research team for ESA. These four reports are now reprinted by World Scientific Pub. Co., to make available this information. Together they form a collection of works on Dynamics and Mission Design Near Libration Points. All the data concerning performance of programs correspond to the time when the programs were developed, tested and executed. Some of them are now 17 years old. In this period the facilities available to us have improved by a factor ranging between 5 x 102 and 5 x 104. The theoretical understanding, the algorithms and their implementation have also been largely improved, leading to a gain in speed, in some cases, by a factor up to 103. It is not surprising that computations requiring 2 weeks of CPU time in 1983 take now 1 second. Updates on the state of the art, both concerning theoretical and practical studies, can be found at the end of Volume IV of this collection. These updates refer to contributions done, in this domain, by our research team. We are indebted to all the staff of World Scientific Pub. Co. for the cooperation and support in the preparation and publication of these Volumes, and to Prof. Rafael de la Llave for encouraging us in this task and his unvaluable help. Our deep acknowledgement to Dr. Walter Flury from the European Space Operations Center in Darmstadt for his kindness in writing the Foreword. Barcelona, Fall 2000

On behalf of the authors Carles Simo

Preface

ix

STATION K E E P I N G OF LIBRATION P O I N T ORBITS FINAL R E P O R T ESOC C O N T R A C T NO.: 5648/83/D/JS(SC) ESOC T E C H N I C A L SUPERVISORS: Dr. E. A. Roth A U T H O R S : G. Gomez \ J. Llibre 2 , R. Martinez 2 , C. Simo 1

2

3

3

Departament de Matematica Aplicada I, ETSEIB, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. Departament de Matematiques, Facultat de Ciencies, Universitat Autonoma de Barcelona, Bellaterra, 08193 Barcelona, Spain. Departament de Matematica Aplicada i Analisi, Facultat de Matematiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain.

C O M P A N Y : Fundacio Empresa i Ciencia, Pomaret 21, 08017 Barcelona, Spain EUROPEAN SPACE AGENCY. CONTRACT REPORT The work described in this report is done under ESA contract. Responsibility for the contents resides in the authors that prepared it

Barcelona, November, 1985

To the memory of Dr. Ernest A. Roth A Space Science Man

Contents

Foreword

v

Preface

vii

Introduction: Libration Points and Station Keeping 0.1 The Neighborhood of Libration Points as a Useful Place for Spacecrafts 0.1.1 The Libration Points in the Restricted Three-body Problem . . . 0.1.2 The Libration Points in Perturbations of the Restricted Threebody Problem 0.1.3 Possible Missions Around the Libration Points 0.1.4 The Real Nominal Quasi-periodic Orbit 0.2 Station Keeping of Libration Point Orbits 0.2.1 Unstable Nominal Orbits 0.2.2 Requirements for an On/off Control Chapter 1 Bibliographical Survey 1.1 Numerical Results for Three-dimensional Periodic Orbits Around L\, L2 and L3 1.1.1 References 1.1.2 Equations. General Properties 1.1.3 Linear Theory Around the Equilibrium Points 1.1.4 Description of the Results 1.2 Analytic Results for Halo Orbits Associated to L\, L^ and L3 1.2.1 References 1.2.2 Equations of Motion 1.2.3 Construction of Halo Periodic Solutions 1.3 Motion Near L 4 and L 5 1.3.1 References 1.3.2 The Triangular Equilibrium Points, Location and Stability . . . 1.3.3 Numerical Explorations 1.3.4 Analytic Results xi

1 1 1 2 2 3 4 4 5 7 7 7 8 9 11 24 24 25 34 36 36 37 40 42

xii

1.4

Contents

Station Keeping 1.4.1 References 1.4.2 The Control of Libration Point Satellites 1.4.2.1 Perturbations and Nominal Path Control 1.4.2.2 Linear Feedback Control 1.4.2.3 Station Keeping 1.4.3 Station Keeping for a Translunar Station 1.4.3.1 Dynamics 1.4.3.2 Linear Control 1.4.3.3 Effect of Errors 1.4.4 Control of an Unstable Periodic Orbit 1.4.5 Station Keeping for the ISEE-C. Algorithm and Results 1.4.5.1 Mission Design 1.4.5.2 The Algorithm 1.4.5.3 A Posteriori Information on the Mission 1.4.6 Additional Work

Chapter 2 Halo Orbits. Analytic and Numerical Study 2.1 The Restricted Three-body Problem 2.1.1 Description of the Problem 2.1.2 Values Adopted for the Constants 2.2 Analytic Determination of the Families of Halo Orbits for the Restricted Three-body Problem 2.2.1 Introduction 2.2.2 Analytic Computations 2.2.3 Description of the Program for the Computation of the Analytic Halo Orbits 2.2.3.1 Storage of the Elements 2.2.3.2 The Series M, N and P 2.2.3.3 Addition of Series 2.2.3.4 Multiplication of Series 2.2.3.5 Computation of the Amount of Elements Used in the Program 2.2.4 Results of the Analytic Computations 2.3 Numerical Determination of the Families of Halo Orbits for the RTBP. Analysis of Bifurcations and Terminations 2.3.1 Description of the Program for the Numerical Computation of Symmetric Periodic Orbits for the RTBP 2.3.1.1 The Continuation Method 2.3.1.2 Refinement 2.3.1.3 Implementation

45 45 46 47 48 49 49 50 51 53 53 55 55 56 57 57 59 59 59 61 62 62 63 68 68 72 73 73 75 78 93 93 93 94 95

Contents

2.3.2

2.4

Tables of Numerical Values of the Families of Halo Orbits Associated to L\ and L 2 2.3.3 The Halo Families Associated to L 3 2.3.4 Interest of These Orbits for Spacecraft Missions 2.3.4.1 Small Size Halo Orbits Related to Lx and L2 2.3.4.2 The Stable Zone of Halo Orbits in the Lx and L 2 cases 2.3.4.3 The L3 Case 2.3.5 Bifurcations 2.3.6 Termination of the Families of Halo Orbits. The Hill Problem. The Rotating Two-body Problem 2.3.6.1 Introduction 2.3.6.2 The KS Transform 2.3.6.3 Vertical Periodic Orbits for Hill's Problem. Local Analysis 2.3.6.4 Hill's Problem Bifurcations. Perturbations Due to /i . 2.3.6.5 Numerical Results 2.3.6.6 The Rotating Two-body Problem 2.3.7 The Elliptic Restricted Three-body Problem References

xiii

96 115 119 119 120 123 124 126 126 127 128 129 132 137 140 141

Chapter 3 The Neighborhood of the Halo Orbits: Numerical Study and Applications 143 3.1 Numerical Study of the Local Invariant Manifolds 143 3.1.1 The First Order Variational Equations 143 3.1.2 Floquet Theory for Halo Orbits. The Floquet Modes 145 3.1.3 Numerical Computation of the Unstable and Stable Manifolds . 148 3.1.4 Numerical Computation of the Remaining Floquet Modes . . . . 149 3.1.5 Results of the Computations. Consequences and Discussion . . . 150 3.2 The Proposed Method for the On/off Control 159 3.2.1 On/off Control Using Invariant Manifolds 159 3.2.2 The Projection Factors 160 3.2.3 A First Approach Towards the Optimal Maneuvers 161 3.2.4 Numerical Computations of Projection Factors and Unitary Controls 163 3.2.5 Discussion of Results: the Different Gains and the No Delay Rule 177 3.2.6 A Second Approach Towards the Optimal Maneuvers 178 3.3 On the Use of Solar Radiation Pressure for Station Keeping 179 3.3.1 Using Radiation Pressure: Applicability and Limitations 179 3.3.2 On the Determination of Optimal Parameters 179 3.3.3 The "Always Towards the Sun" Rule 181 3.4 Globalization of the Invariant Manifolds 184 3.4.1 The Computation of the Global Stable Manifold 184

xiv

3.5

Contents

3.4.2 Results of Interest for the Transfer Orbit References

184 188

Chapter 4 Analytic Solution of the Variational Equations. Analytic Computations for Control Parameters 189 4.1 Analytic Solution of the Variational Equations 189 4.1.1 The Analytic Method 189 4.1.2 Description of the Program to Do the Analytic Computations . . 194 4.1.3 Results and Numerical Tests 196 4.1.4 The Resonance Between Modes and the Related Problems of Convergence 197 4.2 Analytic Computations for Control Parameters 204 4.2.1 The General Method to be Used 204 4.2.2 On the Fast Computation of the Involved Parameters 205 4.2.3 Description of the Program to Obtain the Control Parameters . 206 4.2.4 Comparison with Direct Numerical Computations 207 4.3 References 216 Chapter 5 The Equations of Motion for Halo Orbits Under the Effect of Perturbations and Near Triangular Points 217 5.1 Analysis of the Perturbations 217 5.1.1 Description and Justification of the Method. The Problems of Dynamical Coherence 217 5.1.2 Determination of the Magnitudes to be Associated to the Different Weights 218 5.1.3 The Reference Systems and the Related Changes of Coordinates 220 5.1.4 A Program for the Simulation of the Solar System and the Computation of Residual Accelerations 222 5.1.5 The Adopted Model for the Motion Near Halo Orbits in the Cases Under Consideration 226 5.2 Equations of Motion for Perturbed Halo Orbits 237 5.2.1 Introduction 237 5.2.2 The Lagrangian in Ecliptic Coordinates 237 5.2.3 The Collinear Points for the Sun-Barycenter Problem 239 5.2.4 Expressions of the Lagrangian in Different Coordinate Systems . 240 5.2.5 The Lagrange Equations for the Sun-Barycenter Problem, Collinear Points Case 246 5.2.6 The Collinear Points in the Earth-Moon Problem: Several Expressions for the Lagrangian 250 5.2.7 The Lagrange Equations for the Earth-Moon Problem: Collinear Points 253 5.3 Equations of Motion Near the Triangular Points 254 5.3.1 The Triangular Points for the Sun-Barycenter Problem 254

Contents

5.3.2

5.4

5.5

The Lagrangian in Normalized Coordinates with Origin at L4 or L5 5.3.3 The Lagrange Equations for the Sun-Barycenter Problem, Triangular Case 5.3.4 The Equations in the Earth-Moon Problem Numerical Tests of the Equations of Motion 5.4.1 Description of a Program to Check the Equations of Motion . . . 5.4.2 Numerical Results References

xv

255 256 258 259 259 260 264

Chapter 6 Expansions Required for the Equations of Motion. Collinear Points Case 265 6.1 Analytic Expansion of the Coefficients Due to the Noncircular Motion of the Earth-Moon Barycenter 266 6.1.1 The Coefficients Appearing in the Equations of Motion 266 6.1.2 A Program for the Computation of the Coefficients 267 6.1.3 Numerical Results and Tests 268 6.2 Preliminary Exploration of the Functions to be Kept 276 6.2.1 Expansion of the Perturbing Functions as Sum of Monomials in the Coordinates 276 6.2.2 A Rough Exploration of the Functions to be Kept 277 6.3 Computation of the Approximate Frequencies and Magnitudes of the Perturbations Using FFT 278 6.3.1 The Functions to be Analyzed 278 6.3.2 The Filtering Procedure for the Output of FFT 279 6.3.3 Numerical Results 279 6.3.4 The Identification of the Frequencies 280 6.4 The Final Computation of the Perturbing Terms 290 6.4.1 Determination of the Coefficients 290 6.4.2 Conversion to a Suitable Form for the Equations of the Quasi-periodic Orbit. Numerical Results 290 6.5 References 304 Chapter 7 The Quasi-periodic Orbits: Equations, Method of Solution and Results 305 7.1 The Final Equations of Motion for the Collinear Case 305 7.1.1 Expansion of the Equations of Motion in a Suitable Form for the Integration 305 7.1.2 A Program for Producing the Equations of Motion 307 7.1.3 Results and Numerical Checks 309 7.2 The Method of Solution of the Equations for the Quasi-periodic Orbit . 323 7.2.1 General Expression of the Equations. The Recurrent Procedure . 323 7.2.2 A Lindstedt-Poincare Device for the Quasi-periodic Orbit . . . . 324

xvi

Contents

7.2.3

7.3

7.4

Description of the Program to Obtain the Analytic Quasi-periodic Orbit 326 The Results. Problems Related to Small Divisors 329 7.3.1 Sample of Results in Several Cases 329 7.3.2 The Problem of Small Divisors 330 7.3.3 The Analytic Expression of the Nominal Orbit 331 References 344

Chapter 8 Numerical Refinement of the Quasi-periodic Orbit: The Final Numerical Determination of the Orbit and of the Projection Factors 345 8.1 A Parallel Shooting Method for the Numerical Refinement of the Quasiperiodic Orbit 345 8.1.1 The Input Parameters to Get the Nominal Orbit 345 8.1.2 A First Shooting Procedure. Description of the Program Used to this End 348 8.1.3 Numerical Results and Discussion of the First Procedure . . . . 356 8.1.4 A Second Shooting Procedure 358 8.1.5 Numerical Results and Discussion of the Second Procedure . . . 359 8.1.6 Comments on Different Shooting Approaches 360 8.2 The Final Nominal Orbit and Projection Factors 370 8.2.1 The Problem of the Floquet Modes for a Quasi-periodic Orbit . 370 8.2.2 The Adopted Procedure to Obtain the Projection Factors . . . . 371 8.2.3 A Program for the Final Nominal Computations. Results and Discussion 372 8.3 References 379 Chapter 9 The On/off Control Strategy: Simulations and Discussion 381 9.1 A Simulation Program for the Motion of the Controlled Spacecraft . . . 381 9.1.1 The Equations of Motion. Introduction of Random and Systematic Noise in the Solar Radiation Pressure Effect 381 9.1.2 The Effect of Tracking Errors and Errors in the Execution of Maneuvers 382 9.1.3 How to Decide When a Control Should be Applied 383 9.1.4 Description of the Program 384 9.2 Numerical Results of the Simulations 385 9.2.1 A Summary of Results 385 9.2.2 Discussion on the Effect of the Different Errors and Magnitudes 390 9.3 The Feasibility of the Control Using Solar Radiation Pressure 402 9.3.1 The Determination of the Maneuver Interval. Discussion of Feasibility 402

Contents

9.3.2

Simulations of the Radiation Pressure ControLResults and Discussion

xvii

403

Chapter 10 Other Cases and Further Simulations 409 10.1 The L 3 Case for the Sun-Barycenter Problem 409 10.1.1 Computation of Initial Conditions 409 10.1.2 Results of Simulations 410 10.2 The L3 Case for the Earth-Moon Problem 410 10.2.1 Results of a Simulation 410 10.3 The Triangular Cases for the Sun-Barycenter Problem 413 10.3.1 The Adopted Model for the Motion Near L 4 and L5 413 10.3.2 Development of the Simplified Equations of Motion 415 10.3.3 Analytic Solution. On the Existence of Almost Tori 419 10.3.4 Simulations Starting at the Instantaneous Equilibrium Point or with Suitable Initial Conditions 420 10.4 Simulations for the Triangular Cases in the Earth-Moon Problem . . . . 428 10.4.1 Results of the Simulations 428 10.5 References 430 Chapter 11 Summary and Outlook 433 11.1 Summary of the Achieved Results 433 11.1.1 Concerning the Halo Orbits 433 11.1.2 Concerning the Quasi-periodic Nominal Orbit 435 11.1.3 Concerning Station Keeping 436 11.2 Possible Extensions of the Work 439 11.2.1 The Use of Better Models for the Solar System 439 11.2.2 The Translunar Problem Using the Real Solar System 439 11.2.3 Stable Manifolds and the Transfer Problem 440 11.3 Theoretical Problems 440 11.3.1 The Small Divisors Problem for Quasi-periodic Orbits 440 11.3.2 The Large Size Orbits in the Halo Family: Analytic Study and Perturbation to a Quasi-periodic Orbit 440 11.3.3 The Floquet Modes for a Quasi-periodic Orbit. Searching for an Analytic Approach 441 Acknowledgments

443

Introduction: Libration Points and Station Keeping

0.1

0.1.1

The Neighborhood of Libration Points as a Useful Place for Spacecrafts The Libration

Points

in the Restricted

Three-body

Problem

As it is well-known since Euler and Lagrange epoch, the three-body problem and, in particular, the restricted three-body problem, has five libration points. The restricted three-body problem describes the motion of a particle, of very small mass, under the gravitational attraction of two massive bodies, that are in circular motion around the common center of masses. For space missions, the two massive bodies can be considered the Sun and the Earth-Moon barycenter or the Earth and the Moon, and the particle is a spacecraft. Synodic coordinates can be introduced to keep fixed both main bodies. This is accomplished by introducing axes which rotate with the same angular velocity as that of the main bodies. The libration points have the nice property that a particle initially in them, remains in them forever. In synodic coordinates, the libration points have zero velocity. In the physical space, the attraction of the massive bodies is exactly canceled by the centrifugal force and they also describe circles. In particular, the position of the libration points with respect to the two main bodies is always the same. As it has been said, there are five libration points. Three of them are in the line joining both primaries. One of them, that we denote by L\, between the primaries, and the other two on each one of the half lines, in which the line joining the primaries is split by the own primaries. Of these last two points, the one nearest to the smaller primary is called Li and the third one, £3. The two remaining libration points are in the plane of motion of the primaries and they form, with the two primaries, an equilateral triangle. The two triangles are symmetric with respect to the line joining the primaries.

1

2

Introduction:

0.1.2

Libration Points and Station

The Libration Points in Perturbations body Problem

Keeping

of the Restricted

Three-

The previous situation is quite idealistic. However, real world can be considered as a not too large perturbation of this ideal scenario in many cases. We will focus on the Sun-Barycenter and Earth-Moon systems, mainly. Barycenter means, of course, the barycenter of the Earth+Moon system. In the Sun-Barycenter system the approach consists in taking the motion of the Barycenter around the Sun as circular (it is already almost planar), with a mean motion equal to the mean motion of the Barycenter longitude at a given epoch. The distance from the Barycenter to the Sun is taken as the semiaxis and the mass of the Sun is slightly changed in order to satisfy Kepler's third law. Furthermore, the planets, radiation pressure, and any other type of forces, have been neglected. Going back to the real world, we should add the following perturbations: a) Those due to the noncircular motion of the Barycenter, b) Those due to the fact that there is the Earth+Moon system instead of the Barycenter, c) The effect of the planets, mainly Venus and Jupiter, d) The solar radiation pressure, e) Other forces. In this complex field of forces, the dynamical libration points do not subsist. However, we can introduce geometrical libration points, with respect to Barycenter and Sun, given by the same relations that we have obtained in the restricted problem. The distance between primaries is changing, the rate of the angular motion is not constant, etc. However, if the perturbations are not too large (and they are not), the libration points in the perturbed system, inherit a good property of the restricted problem: a particle placed on them moves slowly, provided it is not too far from them. With some modifications, this holds for the Earth-Moon system. In this case the perturbations from the reference restricted problem are: a) The effect of the noncircular motion of the Moon around the Earth. b) The effect of the Sun. c), d) and e) are the same as before. Effects a) and b) are stronger in this case but we can still define the geometrical libration points and study the motion near them. 0.1.3

Possible

Missions

Around

the Libration

Points

For the exposed nice properties of the libration points, they are suitable for space missions. Let us consider the L\ point of the Sun-Barycenter system. This is an ideal site for a solar observatory. The Sun surface is always available, the Earth is far enough

The Neighborhood of Libration Points as a Useful Place for Spacecrafts

3

to have low noise coming from it, and near enough to allow for good communications. However, the exact geometrical L\ point is not suitable because, at this point, the radio-electric signals from the spacecraft almost disappear in the solar noise. Some minimum angular deviation from the Sun is required for the communications link. Fortunately, there are periodic orbits in the restricted problem which do exactly what we want: they are not too far from L\ and the angular distance to the Sun is large enough. These orbits are known as halo orbits. In this work we deal with this type of orbits and we study how they are modified when the perturbations a) to d) are considered. Perturbations in e) should be specified or they can be considered as noise. Similar orbits exist near I 0, means the maximal characteristic multiplier. For the L\ case in the Sun-Barycenter problem, if t is measured in days, a typical value of m is 0.042. In one year an initial error will be multiplied by more than 3 • 10 6 . The result is similar for the case L2. For the Earth-Moon problem, the situation is much worse, because a typical value for m is 0.46. This means an amplification of the errors by a factor 106 in one month. It is clear that these orbits require station keeping. It is certainly true, as it has been found in the work, that there are stable halo orbits in both cases. Probably, this stability is modified when the perturbations are included, but the total instability is mild. However, these orbits are of very large size and have large oscillations. For the Lx case, in the Sun-Barycenter problem, for example, the first halo orbit which is stable reaches an ecliptic latitude near - 8 6 ° and +64°. The situation is worsened when we progress along the family on the stable range. Similar behavior is observed in the Earth-Moon case. Then we are faced with unstable nominal orbits. An important thing is that, in

Station Keeping of Libration Point

Orbits

5

the range of interest, only one unstable direction appears. There is an associated stable direction. The other directions are neutral, i.e., they are linearly stable for both positive and negative time. Hence, we only expect deviations from the nominal orbit in a given direction (that changes slightly with time with a periodic behavior). This result is essentially true for the real quasi-periodic orbits. The basic idea to cope with the escaping direction is to perform on/off maneuvers to annihilate the unstable component of the motion. This should be done in the most effective way. As we do not care about neutral or stable components, it is not necessary to act a control that returns the orbit to the nominal path, but one that cancels the unstable component leaving the other components free. This method has shown to be quite effective. 0.2.2

Requirements

for an On/off

Control

On/off maneuvers are done by thrusting for a short time in a given direction. In practice, if the thrusters act on a fixed plane and the spacecraft rotates around the axis normal to the plane, several short pulses can be required to complete the maneuver. For the SOHO project three axes stabilization is scheduled. Due to the short time in the execution of the maneuvers they can be considered as variations of velocity without variation in position. That is, the maneuvers are of impulsional type. The on/off control should satisfy the following requirements: a) The maneuvers should not be too small. If the component along the unstable direction is very small, its value can be due, exclusively, to tracking errors. b) The maneuvers should not be too frequent. There is a lower bound in time, after a maneuver, that allows the tracking procedure to reach suitable expected errors. c) The control strategy should allow for delays or advances in the execution of the maneuvers in order not to interact with running or planed observations. These changes will lead to an increase in cost, but it should be reduced at the minimum. d) If the unstable component is larger than a given value, a maneuver should be done (except for operational reasons). If not, the additional cost of the delay is high. e) In the intermediate zone between too small and too large controls, a maneuver can be done if the time since last maneuver is large enough and there are not operational constraints. However it must be checked that the dominant behavior of the unstable component is exponential, with exponent near the one of the unstable escape. Otherwise, the unstable component can be due to random forces plus large tracking errors, and the correction would not be really necessary.

6

Introduction:

Libration Points and Station

Keeping

The on/off method should be readily adaptable to radiation pressure maneuvers and/or low thrust acting for several days. Letting aside technical questions, we can imagine that the spacecraft has some relatively large surfaces, whose orientation with respect to the Sun can be changed in an easy, cheap and reliable way. Hence, we can substitute the short pulses producing changes in velocity of, say, 5 cm/s, by solar radiation pressure maneuvers lasting for 4 or 5 days. Instead of the radiation pressure maneuvers low thrusters (i.e., ionic thrusters) can be used. They can easily produce these low impulses. A further requirement for the control is that it should be continued for long mission durations. This is especially critical for the Moon case, where the time scale is short. A typical mission, using the L2 halo orbits of the Earth-Moon system, 6 years long, means near 150 revolutions. It seems feasible that the total cost of the station keeping for this mission can be reduced to less than 25 m/s. As a final remark, the proposed method for the nominal orbits and station keeping, can be easily applied to any binary perturbed system (Sun-planet or planetsatellite) in the solar system, provided that the time scales of the motion are not too small.

Chapter 1

Bibliographical Survey

A bibliographical survey of the most important papers about halo orbits (numerical and analytic treatment) and station keeping is done. Papers related to motion near Li and L5 are also included. In each subsection the corresponding references are given at the beginning. The comments are concentrated on the most relevant papers.

1.1

Numerical Results for Three-dimensional Periodic Orbits Around Li, L2 and L3

1.1.1

References

The references summarized in this section are the following: [1-1] T.A. Bray and C.L. Goudas. "Three-dimensional periodic oscillations about L\, 1/2; L3". Advances in Astronomy and Astrophysics, 5, 71-130, 1967. [1-2] J.V. Breakwell and J. Brown. "The halo family of three-dimensional periodic orbits in the Earth-Moon restricted three-body problem". Celestial Mechanics, 20 (4), 389-404, 1979. [1-3] R.W. Farquhar and A.A. Kamel. "Quasi-periodic orbits about the translunar libration point". Celestial Mechanics, 7 (4), 458-473, 1973. [1-4] R.W. Farquhar, D.P. Muhonen and D.L. Richardson. "Mission design for a halo orbiter of the Earth". Journal Spacecraft and Rockets, 14 (3), 170-177, 1977. [1-5] C.L. Goudas. "Three-dimensional periodic orbits and their stability". Icarus, 2, 1-18, 1963. [1-6] M. Henon. "Vertical stability of periodic orbits in the restricted problem I. equal masses". Astronomy & Astrophysics, 28, 415-426, 1973. [1-7] K.C. Howell and J.V. Breakwell. "Almost rectilinear halo orbits". Celestial Mechanics, 32 (1), 29-52, 1984. [1-8] K.C. Howell. "Three-dimensional periodic halo orbits". Celestial Mechan7

8

Bibliographical

Survey

ics, 32 (1), 53-72, 1984. [1-9] S. Icthiaroglou, K. Katopodis and M. Michalodimitrakis. "Restricted problem: Families of vertical critical periodic orbits". Astronomy & Astrophysics, 90, 324-326, 1980. [1-10] M. Michalodimitrakis. "A new type of connection between the families of periodic orbits of the restricted problem". Astronomy & Astrophysics, 64 (1), 83-86, 1978. [1-11] C.G. Zagouras and P.G. Kazantzis. "Three-dimensional periodic oscillations generating from plane periodic ones around the coUinear Lagrangian points". Astrophysics Space Science, 61 (4), 389-409, 1979. 1.1.2

Equations.

General

Properties

In a rotating, barycentric, dimensionless coordinate system with the smaller primary on the positive z-axis, the differential equations of motion for the circular threedimensional restricted problem take the form x = f(x), where x

=

(x1,x2,X3,xi,x5,x6)T

/

=

{fl,f2,f3,U,h,j6)

(x,y,z,x,y,z)T,

=

T

= \Xi,X5,X6,2x5

+ -—,-2Xi

+ -—,-—

ax\

ox2

\

)

0x3 J

and /-> ri r2

^2^ 1 — A*

1/2 =

(xi +A*)2 +xl

=

2

{xi +n-

A*

+x\,

l ) +xl

+xj.

These equations admit the Jacobi integral

C

=

2n-(x\+x\+xl).

The five equilibrium points are the real roots of the simultaneous equations fi{xi,...,x6)

=

0,

i = l,...,6.

Setting X2 = y = 0, we can evaluate the position of the coUinear Lagrangian points L\, L2 and L3. Here we denote as L\ the point lying between the primaries, while L2 is located to the right of the smaller primary and L3 to the left of the other one. The differential equation for the transition matrix 0(i, 0), usually called first variational equation, is

jt4>(t,o)

= A(t)QT = ( 6 , 6 , 6 , ^ 4 , 6 , ^ 6 ) r , and A is the 6 x 6 matrix already mentioned, with the partial derivatives of ftxx computed at the equilibrium points.

10

Bibliographical

Survey

In fact, at the equilibrium points, the above system reduces to

(ii

\

/

6 6 U

0 0 0 1-

•"'X3Z3

0 0

&

Ue y

0 0 0 0

V

0 0 0 0 0

1 ~T Z i '2:30:3

0

0 \ 0 1 0 0

f 6M

0 0 oj

U° /

1 0 0 0 2

^X3

0 1 0 2 0

6 & &

where fi* ' 1

'2

It is obvious that the third and sixth rows of the matrix A influence only £3 and £6 • This means that in the linear case the motion in the z direction is not influenced by the motion in the (x, 2/)-plane. The solution of this equation for £1, £2, £3 can be written as C

J. , £ ° 2

£

•

j.

?i

=

?oi coswji H

sinojji,

£2

=

a £02 cos Wit - a£ 0 i sin Wji,

£3 = ci cos ojzt + C2 sin uzt, where Wi,i = 1,2,3 is the angular velocity around any of the three collinear libration points, Li, UJZ is the angular velocity in the z direction and 1 20 a = • 'X3a;3 2/3

I+

6,s

with 1

+ nx

1/2^ 1/2

+

-O2

4. Ofi

1 "X3X3 ^

^"x3a:3

The above solutions can be periodic in three cases: (a) £1 = £2 = 0 and £3 as in the 3-rd equation. (b) £ 3 = 0 and £1, £2 as in the 1-st and 2-nd equations. ( c ) £11 £2, £3 as above and the two angular velocities uiz and Wj are commensurable. The infinitesimal periodic oscillations around the collinear points, in the plane of motion of the two primaries, are continued along the families (b), (a) and (c) of plane retrograde periodic orbits around L2, £3 and L\ respectively (following Stromgren's terminology). The vertical stability character of each plane periodic orbit indicates whether or not this orbit is at the same time a member of a family of three-dimensional periodic orbits (see [1-6]). Only what Henon calls vertical critical orbits (\av\ = 1)

Numerical Results for Three-dimensional Periodic Orbits Around L\, L2 and L3

11

can be embedded in three-dimensional families of periodic orbits. Considering small out-of-plane perturbations of the periodic orbit, after one revolution, by using Azi \

_

f av

Aii /

~~ \ cv

bv \ ( Az 0 A

dv J V Ai 0 J '

three different kinds of bifurcate solutions can be considered: Type A: Symmetric periodic orbits with respect to the (a;i,2;3)-plane and generated from planar periodic orbits for which av = 1, c„ = 0. Type B: Symmetric with respect to the Zi-axis generating from planar ones for which av = 1, bv = 0. Type C: Symmetric with respect to both the (a;i,a;3)-plane and the zi-axis, originating from planar periodic orbits with av — — 1, bv or/and cv = 0. Orbits of type A bifurcated from the families (a), (b) and (c) are usually called halo orbits. 1.1.4

Description

of the

Results

As has been said there are three types of infinitesimal periodic oscillations around the collinear points. The literature is mainly concentrated on finite periodic solutions generated from the first two types. Orbits of the third kind (Lissajous orbits) are considered in [1-3] and [1-4], where the orbit selection for the ISEE-C is discussed. In general, these kinds of trajectories spend considerable time within the "exclusion zone" due to solar interference (if their amplitudes are not too large) or, if the "exclusion zone" is avoided, then the amplitude Az becomes too large because of the narrow beamwidth of the spacecraft's antenna. Orbits generated from infinitesimal ones of the first kind are studied in the references [1-1], [1-5], [1-11]. In an early paper, [1-5], Goudas computed some orbits of the three families emanating from L\, L2 and L3 for the following mass ratios n = 0.1,0.2,0.3,0.4,0.5. His investigations are too incomplete to lead to firm conclusions. In a later paper, [1-1], Bray and Goudas computed again these families for \i = 0.4 and in [1-10], they are computed for the Sun-Jupiter case (/J, — 0.00095). For these two mass ratios families coming from L2 and JD3 have as terminal members two planar orbits of family (m) of plane retrograde periodic orbits around both primaries. The orbits of the family coming from L\ increase in size along the family, in both cases, and their end is not known to these authors. For the Sun-Jupiter case there is no stable periodic solution in any of the three families. For /J, = 0.4 there is a narrow band of stable orbits for the families coming from L 2 and L3.

12

Bibliographical

Survey

,z

s^

\

y> ^

§v

^S

t= 0 t y

^

ZONE

Fig. 1.1 Lissajous trajectory as seen from Earth. Ay = 200 000 km, Az = 100 000 km, <j>x = 4>z = 0 (grid size « 60 000 km) ([1-4]).

1

'

'

1

it 0 w . 0 6

1

«)10

'

_ A = 200,000 KM

Z 0

( \

BEAMWIDTH CONSTRAINT

h z 0 J u


i\ > 1. For the family near L\ there is a narrow band of stable orbits from 73 000 km < Zmax < 74500 km and a second one for 112 000 km < zmax < 113000 km. The first one is formed by orbits which pass very close to the Moon and the second one by orbits which in fact cut the lunar surface. At zmax = 113 000 km the two stability indices coincide and for larger zmax, a pair of complex conjugate eigenvalues outside the unit circle, with their reciprocals lying inside the unit circle, appear. For the family coming from L2 there are also two narrow bands of stable orbits with a qualitative behavior, for the orbits inside them, similar to the one already described for L\. Both families tend to almost rectilinear orbits (x,y ) + - e 2 cos 2 (t + is the mean anomaly at t = 0. The synodic equations for the infinitesimal body are the following

t-20r, = &, + ^ - - ! ^ i i ( e - f c ) - 4 ( * - & ) . n + 2i( = - 9 £ + ^ , - ( l ^ + | ) , ,

(1.1)

For e = 0 the above equations become the standard ones for the restricted circular three-body problem. It is well-known that there are five equilibrium solutions of equations (1.1). The paper of Farquhar [2-3], is devoted to the collinear ones Li, L2 which are located at distances JLR from the finite bodies.

L

Fig. 1.21

l

-7 T R' L 2

Locations of the collinear equilibrium points.

The constants 7^ are the real roots of Euler's quintic equations. In a coordinate system centered at a libration point (Figure 1.21) equations (1.1) become x

=

2(1 + v)y + (1 ±

7L)P

2

(i + v) ((i±yL)(i

- vy

+ P) + x) M

r

y

r

l

2

+

2(l + i / ) ( ( l ± 7 i ) p + ±) + £ ' ( ( l ± 7 i ) ( l + p ) + a : )

=

(l + ^)22/1 - M

l-/i

, /x

+ A* ,

{i+pr

l L

Analytic Results for Halo Orbits Associated to L\, Li and La

27

A restricted three-body model, for the motion of a libration point satellite, is an idealization of the true physical situation. A more realistic model must account, at least, for the gravitational perturbations of other bodies. Papers [2-1] and [2-1] are devoted to study the motion of a spacecraft on a vicinity of Lz in the Earth-Moon system, taking into account the effect of the Sun. As paper [2-1] is mainly devoted to the two-dimensional case we shall concentrate in the other one. Considering both direct and indirect solar perturbations, the equations of motion become, in a coordinate system centered at L-i (as in Figure 1.22) x

-

{l+1l1)p-vzy

2{\ + vz)y +

= (1 + uzf ((1 + 7 7 1 ) ( 1 + p)+x)-(l-

vz)vxz

1-M ILrE

M

(1L{^

ll

+ P)+X

+ (1+P)2

'M

,2„3

o | /"(1 - /x + 7/,)(l + p) + 71,(3; - xs)

1L

+

2(1 + vz) ((1 + 7 £ 1 ) p + x)+uz -2vzz + uxz 1-Ai

=

+ 2vxy + vxy 1

-ri

V

m a

y-ys

ys

v\z - i/B (1 + vz) ((1 + 7 £- 1 )(l + p) + x)

M , M

+ 'M

((1 + 7Z 1 )(1 + P) + x)

(ul + (1 + vx)2)y 2„3

'M

z

M 1 + P) + 1LXS\ + R%

73" IL

•,2„3 ma

'S

zs 3

+

zs D3

EARTH

Fig. 1.22

Reference system, centered at the equilibrium point L2, of the Earth-Moon system.

28

Bibliographical

Survey

where -

fi is mass parameter for the Earth-Moon system (fi = 1/82.30), 7 i is the distance ratio for the L2 point (-JL — d2/RM = 0.1678331476), m is the ratio of mean motions of the Moon and the Sun (m = 0.0748013263), rs, fE, TM are the distances between the small body and Sun, Earth and Moon respectively, - Rs is the distance between the Earth and the Sun, - as is the mean distance between the Earth-Moon barycenter and the Sun, TBS is the distance between the Earth-Moon barycenter and the Sun (the components are denoted by XBS, VBS, ZBS)Expanding the above equations up to the fourth order, they can be written as: -2(l

+ vz)y

=

({l + vz)2 + 2{l + p)-3BL)x

+

vzy-vxvzz

-Z-CL{l+p)-i{2x2-{y2+z2)) +2DL{l + p)-*{2x2-3{y2 --EL{1

=

{(l+vz)2

rC2

BS 1

1-3

=

2

„XBSZBS z

2

BS

+ ul)y-uzz

+ uxz + 2uxz

- | / ? i ( l + p)-5 (4ar2 - (y2 + z2)) y

p)-6{4x2-3(y2+z2))xy 0XBSVBS 6 r2 r

2 ^J5_ r3 ~BS

V~ 3

+ z2) + 6y2z2)

' BS

+ p)-3BL

-(l

+ ^EL(l +

z

0XBSVBS

x - 3—J

' BS

+3CL(l + p)-4xy

+m

z2))x

+ p)-6 (8a;4 + 3(y4 + z 4 ) - 24s V

2 ^S_ —m .r 3 BS

y + 2{\ + vz)x

+

,VBS 2 r BSJ

X

BS

3

[y x - (1 + p) BL) z-i/x(l

Z/ + 3

VBSZBS 1

BS

+ uz)x - vxy - 2vxy

+3C L (1 + p)-Axz - ^DL(1 + p)-5 (4x2 - (y2 + z2)) z + ^EL(l

+m

2 ^S_ 3 rr BS

p)-6{4x2-3(y2+z2))xz

+ Z

0XBS

BS

.

0VBSZBS

3—5—x + ^—o—v ~ ' BS

' BS

Z

1-3

BS

' BS

where BL =3.190423657, CL =2.659334398, DL =2.583009811, EL = 2.572040953 and (1 + p ) _ 1 , vx, vz are given, according to the de Pontecoulant's lunar theory, by (1+p)-1

=

1 15 1 + e cos <j> + e2 cos(20) + - m 2 + m 2 cos(3f) + — me cos(2£ - ) 6 o 1J i , 9 5, .„ .. 19 3 , . ,. 15 a - - e c o s < ^ + -C cos(3) + —m* cos(2',

j/'

=

5 mt + e' + 2e'sin0' + -e' 2 sin(20'),

+

~r

(Z

-32™

2

9

} t + e — u>,

273

A

-128m)t

,

273

+

e Qo

- >

A

and the astronomical constants used in the above equations are e

=

Moon's orbital eccentricity (e = 0.054900489),

30

Bibliographical

Survey

e'

=

Earth's orbital eccentricity (e'=0.0167217),

a/a'

=

modified ratio of semimajor axes for the orbits of Earth and Moon (a/a' = 0.0025093523),

7

=

tangent of the mean inclination of the Moon's orbit (7 = 0.0900463066),

e, e'

=

mean longitudes at the epoch of the mean motions of the Moon and the Sun,

6J,UJ'

=

mean longitudes of the lunar and solar perigees,

fio

=

longitude of the mean ascending node of the Moon's orbit.

In [2-4] Farquhar and Kamel obtain two different kinds of periodic solutions of the above truncated system by using algebraic manipulators. The first kind of solutions are of the form: x

=

mx\ + m2X2 + m3xz + . . . ,

y

=

myx + m2y2 + m3y3 + ...,

z

=

mz\ + m2Z2 + m3Z3 + . . . ,

where m is the ratio of mean motions of the Moon and the Sun. Using the LindstedtPoincare method, (xi,yi,Zi), i = 1,2,3, can be computed and the solutions correspond to Lissajous trajectories. The amplitudes of these orbits are related to the in-plane (cjxy) and out-of-plane (CJZ) frequencies by the following relation: -LJZ

Uxy

=

0.073672 - 0.000785A£ + 0.000818^.

In order to obtain a halo orbit (ujXy = wz) at least a value of Ay = 46 793 km is needed. For this amplitude the expansion taken for the orbit is no longer valid. In order to eliminate the secular terms, in the perturbation procedure, the expansion is taken of the form: =

m1'2xi

y

=

1 2

z

— m^^zi

x

+ mx^ + m3'2X3 + m2Xi + . . . ,

m / ?/! + my2 + rr?l2yz + m 2 ^ 4 + . . . , + 77102 + m3'2zz + m2Zi + ... .

The condition which appears to second order, on the amplitudes Ay, Az, when eliminating those secular terms is: A\

1.176726A2 + 3.361330.

=

See, however, the related comments in the description of the analytic computations of the halo orbits. The expressions for the (xi,yi,Zi), i = 1,2,3,4, obtained in [2-4] are the following: x1

=

0.341763Aj,sinTi,

Analytic Results for Halo Orbits Associated to L\, L2 and L3 2/1

=

Ay COS T i ,

z\

=

AzsinT2,

31

where Ti=wxy{Av,Az)t

+ 01,

T2 = toz(Ay,Az)t

+ e2,

x2 = 0.554904—i4 y sin(-Tl) + 1.210699—Ay cos(4> + Tx) - 0 . 0 5 5 2 9 6 ^ sin(2Ti) - 0.08659705A2sin(2T2), z2 = 1.052082—Az sin(0 + T2) + 1.856918—AZ sin(0 - T2) m m +0.4241194AyJ42 cos(T2 - Tx) + 0.1339910AyJ4z cos(T2 + Ti), e2 x3 = — Ay[-0.122841

sin(20 - 7\) + 0.643204 sin(20 + Ti)]

+ —A\[0.198388 cos<j> - 0.387184 cos(0 - 2T2) + 0.335398 cos(0 + 2Ti)] + —Al[0.173731 cos + 0.325999 cos(c/> - 2T\) - 0.270446 cos(«/> + 2Ti)l m * +—Ay [-1.10033 sin(0 - Ti - 2 0 - 1.189247 s i n ^ + Ti - 20] +AyA2z [-0.430448 sin(2T2 -Ti)-

0.031302 sin(2T2 + Ti)]

3

+A y [0.027808 sin(3Ti)] + CyAy sinTi +i4 s [-0.38856sin(Ti - 2 0 + 0.455452sin(Ti + 2 0 ] , e2 t/3 = —Ay[0.608685 cos(20 -Tx) + 1.407026 cos(20 + Ti)] +— J 4 2 [-O.116822sin0 - 0.214742 cos( - 2T2) - 0.232503 sin( + 2T2)] +—A\[-0.109499sin<j> - 0.144553 sin(0 - 2Ti) - 0.155751 sin(0 + 2Tx)] +—Ay [2.733367 cos(0 - Ti - 2 0 - 3.848485 cos(0 + TX - 20] +A y yl 2 [-1.191421cos(2T 2 - T i ) -0.000165cos(2T 2 + T i ) ] +i4j [-0.027574 cos(3Ti)]

z3

+Ay[-1.743411 cos(Ti - 2 0 + 0.741825sin(Ti + 2 0 ] , e2 = — A z [ - O . 5 3 6 6 5 2 s i n ( 2 0 - T 2 ) + 1.103381 sin(20 + T2)] + — Ay Az [-0.353754 cos(0 - T2 - Ti) + 0.367360 cos(0 + T2 + Ti) +0.063629 cos( - T2 + Ti) - 0.034729 cos(0 + T2 - Ti)]

32

Bibliographical

+ —AZ [-2.353465 sin( + p

12

2^P)--^yxp)

+ it.

>i-p|

1 \n~P\

np r1

+ (l - p)

r_2P_ 3

rr |r 2 - p\ 2 J J 1 np rip + 0.-11) 3 rr rf J ~ ' ' L|r2 - p\ 2 J

The terms involving the factors p and (1 - p) have the form of a third-body perturbing potential. Using power series developments in terms of Legendre polynomials, the Lagrangian can be rewritten as: =

\(P,P)

P„(cos5i) +

dt

1

n=2

r-i

Pn(cosS2). n=2

Taking the unit of distance as r\

=

1, for the motion about L\ or L2,

Analytic

Results for Halo Orbits Associated

to L\, Li and L$

33

M.

Fig. 1.23

Richardson's reference system centered at the equilibrium point

r2

=

L\.

1, for the motion about L3,

the Lagrangian can be written in the form L

= 5(PV) + EC„P"P„(J;),

where primes denote derivatives with respect to the variable s = 7^ /2 t, and where 1L

_

f nAi2 r2/r12

for for

Li,L2, L3.

The constants c„ are given by the expressions: 1

„(I-A072+1

with the upper sign for L\ and the lower one for L2, and

for L3. The equations of motion have the following compact expressions 00

x" - 2y' - (1 + 2c2)x

.

OO

y" + 2x' + (c2-l)y

=

.

J2(n+1)cn+iPnPn[^),

=

^

,

W

»-

2

V

P„m,

34

Bibliographical

Survey

oo

>.

,

n=3

where l(»-2)/2]

Pn =

1.2.3

Construction

J2

/

s

(3 + 4* - 2n)Pn_2fc_2 ( - ) .

of Halo Periodic

Solutions

The linearized equations are x" - 2y' - (1 + 2c 2 )z

=

0,

y" + 2x' + (c2 - l)y

=

0,

z" + c2z

=

0.

Halo type periodic orbits are obtained if the amplitudes of the in-plane (x, y) and out-of-plane (z) motions are of sufficient magnitude so that the nonlinear contributions to the system produce eigenfrequencies that are equal (i.e. coxy = LJZ). The linearized solution can then be expressed in the form x

=

—Ax cos(\t + ),

y

=

kAxsin(\t

z

— Azsin(Xt + %p).

+ ),

In these expressions, the amplitudes Ax and Az are constrained by certain nonlinear relationship as a result of the application of the perturbation method. The phases