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<2- In these orbits a spacecraft can be placed to observe also the Sun, and the requirement about the angular distance to the Sun is not required. Partial eclipses will be continuously observed, however. They can be avoided using again halo orbits. A solar observatory placed near L3 would allow to follow the changing details of the Sun surface continuously. It should be coupled with L\, L^ or Earth based observations. As it will be shown, the behavior near that point is quite smooth. Something similar to halo orbits should be used again, for the same reason as for L\. A drawback of this location is that the transference cost is much larger than for L\ and L 2 . For these ones the cost is similar to a lunar mission. A spacecraft placed near points L4 or L 5 of the Sun-Barycenter, is better seen as a planet almost in the orbit of the Earth. Concerning the Earth-Moon system, the more attracting point is £2- A spacecraft placed near a halo orbit (in fact, in a perturbation of a halo orbit) near L2 will allow to establish a permanent communications link between the Earth and the hidden part of the Moon. A radio-observatory placed on this part, will never receive radio-electrical noise from the Earth. The triangular points have a remarkable position because they are relatively near the Earth and they have better stability properties than the halo orbits. The effects of the noncircular motion of the Moon and of the Sun, produce only mild instabilities. Zones of regular motion seem to exist and it looks possible to place in them several satellites with slowly changing mutual distance.

0.1.4

The Real Nominal

Quasi-periodic

Orbit

For a given mission near one of the libration points in the Sun-Barycenter or EarthMoon systems (or in any other such system: Sun-planet or planet-natural satellite), the first thing to do is to define a nominal orbit suitable for the purposes of the mission. This definition can be strict or not, that is, we can force the spacecraft to follow the nominal orbit closely, or we can only ask the spacecraft to be not too far from a given path. For the ISEE-C mission the second procedure was adopted. In this work we propose to follow the orbit closely. This means a substantial reduction

4

Introduction:

Libration Points and Station

Keeping

in the expected fuel consumption for station keeping. In the real world periodic orbits no longer subsist. They are replaced, hopefully, by quasi-periodic ones. A quasi-periodic motion can be seen as a superposition of harmonics with different incommensurable frequencies. A halo orbit should be replaced by a nearby quasi-periodic orbit, and the possible escaping components should be avoided. In this work we show that it is feasible, and in fact we do all the computations, to obtain nominal halo orbits for halo type missions. Those orbits are obtained both analytically and numerically. Both nominal orbits agree quite well. For instance, for the L\ case, Sun-Barycenter problem, they differ at most in 100 km. At a distance of 1.5 • 106km from the Earth, this means a small fraction. We remark at this point that the quasi-periodic orbits replacing halo orbits belong to a two-parameter family. The values of the parameters can be chosen in the more convenient way for the mission purposes. For the Earth-Moon case, the deviations from halo to quasi-periodic are larger. A rough analytic expression has been obtained and the main difficulties to refine this expression are pointed out. The numerical refinement is also possible, and it has not been implemented because it requires a model, for the motion of the Moon, better than the one available to us.

0.2 0.2.1

Station Keeping of Libration Point Orbits Unstable Nominal

Orbits

The halo orbits of small and medium size are unstable. This means that a particle starting at a distance, d, from a halo orbit, leaves this orbit at an exponential rate. After a time interval t, it will be, roughly, at a distance d • exp(mi). Here m > 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)
Numerical

Results for Three-dimensional

Periodic Orbits Around L\, L2 and L3

9

A(t) is a 6 x 6 matrix which is divided into 4 sub-matrices, each 3 x 3

* = U0where 0 is the zero matrix, / is the identity matrix, / L

=

0 2 0 \ -2 0 0

V 0 0 0 / and Qxx is the symmetric matrix of second partial derivatives of fi with respect to xi,X2,xs, evaluated along the orbit. The initial condition, 0(0,0), is equal to the identity matrix. The transition matrix at the end of a full cycle (in some cases a half-cycle or a quarter of it) of a nearly periodic orbit, can be used to adjust the initial conditions so as to obtain periodicity. See [1-1] and [1-2]. Its eigenvalues determine the stability of the orbit. Because of the autonomous character of the differential system and the existence of an integral, two of the eigenvalues are always equal to unity. It can be shown that the other four form a self reciprocal set: (Ai, 1/Ai, A2,1/A2). The periodic orbit is unstable unless both Ai and A2 lie on the unit circle. The stability analysis of symmetric and doubly symmetric periodic orbits, can be done by integrating the equations of variation for half a period or one quarter of the period of the orbit, depending on the symmetries, see [1-1] and [1-2]. The equations of motion have two singularities, corresponding to collisions with the primaries, which can be removed independently by means of the KustaanheimoStiefel (KS) regularization. See [1-5] for the expressions of the regularized equations. From the inspection of the terms occurring in the differential equations, it follows that if for a fixed value of /U, (x(t),y(t),z(t)) is a particular solution, then : (i) (x(—t), —y(—t),±z(—t)) is also a solution for the same value of /x. (ii) (—x(—t),y(—t), ±z(—t)) is also a solution for a value of the mass parameter equal to 1 — /x. 1.1.3

Linear

Theory Around

the Equilibrium,

Points

In order to study the motion near the collinear equilibrium points, the linearized equations are considered for the perturbations:

i = M, where£ = (€,v,C,Lv>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

<

-

/

\

J

0 0)

A,= 100,000 KM ~ f

1 1DO0

,

1

I

2000

3000

TIME IN ORBIT (DAYS)

4000

2000

3000

TIME IN ORBIT (DAYS)

Fig. 1.2 Minimum solar elongations for Lissajous orbits of amplitudes Ay and Av • = 200 000 km (right) ([1-4]).

500 000 km (left)

For the mass ratios explored, the orbits of the families are symmetric both with respect to the xi-axis and the (xi,a;3)-plane. So, they cannot avoid the "exclusion zone". In what follows we shall summarize the results about three-dimensional periodic orbits emanating from vertical critical ones. We shall consider the vertical critical orbits of families (a), (b) and (c) of plane retrograde periodic orbits around L3, L2 and Li, respectively. As it has already been said, Henon ([1-6]) showed that each vertical critical orbit

Numerical Results for Three-dimensional

Periodic Orbits Around L\, L2 and L3

13

corresponds to an intersection of the family of plane periodic orbits with a family of three-dimensional periodic ones. He computed the vertical stability index along the families (a), (b) and (c) for \x = 0.5 and vertical critical orbits were refined for this mass ratio. For this particular value of n, families (a) and (6) are symmetric of each other with respect to the origin. As a consequence, they have the same vertical stability curve.

Fig. 1.3 Vertical stability curve for family (6), showing the vertical stability index, av, as a function of the Jacobi constant C. Parts of the curve between av = —1 and av = + 1 correspond to vertical stable orbits. Intersections with av — ± 1 , are vertical critical orbits ([1-6]).

1 Fig. 1.4

2

3

4C

Vertical stability curve for family (c) ([1-6]).

14

Bibliographical

Survey

Starting from the vertical critical periodic orbits b\v, b2V, c\v, C2„, of the Copenhagen problem, computed in [1-6], by numerical continuation with respect to /x, four one-parameter families of vertical critical periodic orbits, of the planar restricted problem, were computed in [1-9] for values of /j, E [0.00003,0.5]. In [1-10], Michalodimitrakis computed the family of three-dimensional periodic orbits starting at c\v for fi = 0.5. The periodic orbits of this family are symmetric with respect to the (£i,:E3)-plane and, in fact, connect the vertical critical orbits C\v and m,2V already computed by Henon. The orbits of this family, because of the symmetry, are in fact halo orbits. The stability of the orbits is not studied in the paper. In a remarkable paper, [1-11], Zagouras and Kazantzis, computed the families arising from the vertical critical periodic orbits a\v, a^v, biv, b^v, ci„, c-^v for the Sun-Jupiter case (/J = 0.00095). Those coming from a\v, b\v, c\v are of type A, symmetric with respect to the (xi,ar3)-plane (halo orbits) and the other three of type C, symmetric with respect to the ari-axis. Since no regularization was used in the work, the computation of the families was stopped when the orbits passed close to a collision orbit. Although the stability indices were computed for the members of the families, there is no much attention paid to the stable regions. Numerical computations for halo orbits for the Earth-Moon system and its evolution with the mass parameter have been done in [1-2], [1-7] and [1-8]. In [1-2], and for the Earth-Moon system (circular case), the halo families originating in the vicinities of both L\ and L2 are computed. The stability curves for these families are given in Figures 1.8 and 1.9. The stability parameters i/j are defined by vi

=

^ i + K1)-

Orbits are stable if both \vi\ and | ^ | < 1 and unstable if one has some \i>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
Numerical

Results for Three-dimensional

Periodic Orbits Around L\, L2 and £3

15

Fig. 1.5 The family A\v of orbits symmetric with respect to the Oxz-plane in three conical projections. The origin of the coordinate system is at the point (1.07,0,0). The projections are made on a plane normal to the line joining the viewpoint V, with the origin O of the coordinate system, at point P. The distance OP is 0.15. (a) Family A\v. Viewpoint: (0,0,1.07). (b) Family Alv. Viewpoint: (0,1.07,0). (c) Family Aiv. Viewpoint: (0.8,0.8,0.8) ([1-11]).

16

Bibliographical

Survey


Fig. 1.6 The family B\v of type A in three conical projections. The origin of the coordinate system is at the point ( - 1 , 0 , 0 ) . The distance OP is 2.10. (a) Family Blv. Viewpoint: (5,0,0). (b) Family B i „ . Viewpoint: (0,5,0). (c) Family Blv. Viewpoint: (3.75,3.75,3.75) ([1-11]).

Numerical Results for Three-dimensional

Periodic Orbits Around L\, L2 and L3

17

(a)

miir

VII

+0.1

(c)

X

0.1

Fig. 1.7 The family C\v of type A in three conical projections. The origin of the coordinate system is at the point (0.93,0,0). The distance OP is 0.15. (a) Family C i „ . Viewpoint: (0,0,0.93). (b) Family Ci„. Viewpoint: (0,0.93,0). (c) Family Clv. Viewpoint: (0.7,0.7,0.7) ([1-11]).

18

Bibliographical

2

M.„

Survey

(10000 km)

—

PARAMETERS » PERIOD

10

1000

10

11

12

PERIOD IDAYSI

Fig. 1.8 Stability curve for the halo family around L\ for the Earth-Moon system. In the x-axis the stability index is represented, while in the j/-axis the maximum z-amplitude is displayed ([1-2]).

Fig. 1.9 Stability curve for the halo family around L2 for the Earth-Moon system. In the x-axis the maximum x-amplitude is displayed, while in the y-axis the stability index is given. The period of the orbits is also displayed in Figure ([1-2]).

Numerical Results for Three-dimensional

Periodic Orbits Around L\, L2 and L3

19

Fig. 1.10 Exact apocenter points for the L\ and L% halo family in the Earth-Moon case, as well as the almost rectilinear approximation ([1-7]).

12r

10

o o o o EXACT Lj-FAMILY < 4

s EXACT ALMOST RECTILINEAR APPROXIMATION

-10

•5-4 -3 -2

0

2

3 4 5

10

Fig. 1.11 Stability indices v,, for the same families displayed in Figure 1.10. The approximation forms a bridge between the two families ([1-7]).

Bibliographical

20

Survey

EXACT ?Nfl 0ROER APPR0X

x

<

m2

1-3

0.5

1

1.5

MAXIMUM X"

Fig. 1.12

(x,z) profile for the halo family around Lz for \i — 0.96 ([1-7]).

1

^^ ^**<::r^:::^

// / 0

'/,/ '/f

1 1

1

//

• EXACT 2ND ORDER APPROX

f!

1 0.1

1

1 0.2

i 0.3

MAXIMUM X"

Fig. 1.13

Stability indices Vi, for the families displayed in Figure 1.12 ([1-7]).

Numerical Results for Three-dimensional

Periodic Orbits Around L\, L2 and £3

3-10

u--06_ Q N 5 .04

IQ2

\QI

3-10"

MINIMUM X Fig. 1.14

Stability indices for the L\ families ([1-8]).

0.4

-0.2 MINIMUM X Fig. 1.15

Stable regions for the L\ families ([1-8]).

21

Bibliographical

22

Survey

MAXIMUM X Fig. 1.16

Stability indices for the Li families ([1-8]).

rM 1.0

0.5

0.0

MAXIMUM X Fig. 1.17

Stable regions for the L2 families ([1-8]).

Numerical

Results for Three-dimensional

Periodic Orbits Around L\, Li and L3

MAXIMUM X Fig. 1.18

-

Stability indices for the £3 families ([1-8]).

^ ^ _ _

_____

W-.88 (INC0MPLETE)

w-.50

L?

m"2

MAXIMUM X Fig. 1.19

—

/u-,76

Stable regions for the L3 families ([1-8]).

23

24

Bibliographical

Survey

These two families, for the Earth-Moon system were recomputed in [1-7] using the Kustaanheimo-Stiefel regularization in order to eliminate the singularities. The numerical results are compared with a 1-st order theory in the in-plane components of the small angular momentum vector and the small in-plane components of the eccentricity vector. The comparison of the results obtained in [1-7] is shown in Figures 1.10 and 1.11. A similar analysis is done, but up to the second order, for the family of halo orbits near L3. The resulting orbits and their stability are compared with those obtained by the regularized numerical integration for n = 0.96. The results are shown in Figures 1.12 and 1.13. The evolution of the halo families with the mass parameter fj, is done in [1-8]. Special attention is paid to stable regions which are computed carefully. The results are summarized in Figures 1.14 to 1.19.

1.2

Analytic Results for Halo Orbits Associated to L±, L2 and L3

1.2.1

References

The references summarized in this section are the following: [2-1] R.W. Farquhar. "Station-keeping in the vicinity of collinear libration points with an application to a lunar communications problem". In Space Flight Mechanics, Science and Technology Series. American Astronautical Society, New York. [2-2] R.W. Farquhar. "The Moon's influence in the location of the Sun-Earth exterior libration point". Celestial Mechanics, 2 (2), 131-133, 1970. [2-3] R.W. Farquhar. "The control and use of libration point satellites". Technical Report TR R346, NASA, 1970. [2-4] R.W. Farquhar and A.A. Kamel. "Quasi-periodic orbits about the translunar libration point". Celestial Mechanics, 7 (4), 458-473, 1973. [2-5] 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. [2-6] R.W. Farquhar, D.R Muhonen, C.R. Newman and H.S. Heuberger. "Trajectories and orbital maneuvers for the first libration-point satellite". Journal Guidance and Control, 3 (6), 549-554, 1980. [2-7] D.L. Richardson. "Analytical construction of periodic orbits about the collinear points". Celestial Mechanics, 22 (3), 241-253, 1980. [2-8] D.L. Richardson. "Halo orbit formulation for the ISEE-3 mission". J. Guidance and Control, 3, 543-548, 1980. [2-9] D.L. Richardson. "A note on the Lagrangian formulation for motion about the collinear points". Celestial Mechanics, 22 (3), 231-235, 1980.

Analytic Results for Halo Orbits Associated

25

to L\, L2 and L3

[2-10] D.L. Richardson and N.D. Cary. "A uniformly valid solution for motion about the interior libration point of the perturbed elliptic-restricted problem". In AAS/AIAA Astrodynamics Specialist Conference, AAS Paper 75-021., July 1975. 1.2.2

Equations

of

Motion

Consider the motion of an infinitesimal body in the gravitational field of two finite bodies (Pi,P2) revolving around their common barycenter in elliptic orbits. The following quantities are denned to be equal to the unity: (1) The mean distance between Pi and P2, (2) The mean angular rate of the finite bodies about their barycenter (6), (3) The sum of the masses of the two finite bodies. The larger mass at Pi is taken as 1 — fi, while the smaller mass at P2 is \i.

Fig. 1.20

Geometry of the restricted three-body problem.

Because the motion of the two finite bodies can be obtained from the solution of the two-body problem, it is possible to write the distance R —P1P2 and the angular rate 0 as a power series of the eccentricity, which are convergent if e < 0.6627.... e2

-•

R

=

\p1p2\

= l + p=l-eCos{t

+
(/>))+0(e3),

26

Bibliographical

9

=

Survey

l + i / = l + 2ecos(i + <j>) + - e 2 cos 2 (t +
where 4> 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
29

Analytic Results for Halo Orbits Associated to L\, L2 and L3 r

1 t

--ej2

-I Q'7

cos( - 2r)) + -7-me2 cos(2£) + — - m 2 e cos(2f - 0)

+ ^mee'[cos(2^ - 0 - ') - cos(2£ - 0 + 0')] + ~e' 4 a' 00

cos(£ + ')

1 nc

- m 2 e cos 0 + m2e — cos(2£ + 0) _ — cos(2£ - e0) — -me

cos0 -i-m e ^ c o s ( 2 £ - 0 ' ) - ^ c o s ( 2 £ + 0')

21 +—mee'[cos(0 - 0') - cos(0 + 0')], 8 i/z

=

2e cos 0 + - e 2 cos(20) + -^-m 2 cos(2f) + - m e cos(2£ - 0) e3 , 13 , 85 , ._ 15 a / 0 ,s - — cos0 4- — e cos(30) + j^m cos(2£) — V m ~ i c o s £ 5 75 143 - - e 7 2 cos(0 - 277) + —me 2 cos(2£) + —-m 2 ecos(2£ - 0) 4 8 16 35 15 + — mee' cos(2£ — 0 — 0') —— mee' cos(2£ — 0 + 0') oa . , .. Oo + -2— e cos(£ -f 0 ) — - m ecos0 a' 51 lf)5 +m 2 e - c o s ( 2 £ + 0 ) - — c o s ( 2 £ - 3 0 ) —3m2e' cos 0' + m2e' ^ cos(2£ - 0') - ^ cos(2£ + 0') 21 +—mee'[cos(0 - 0') - cos(0 + 0')],

vx

=

3m 2 7C0s£sin(*/' - fi),

with 225

=

i -4""V - ~32~ m

£

=

(l-m)i + e-e', /, 3 , 9 ,

71

=

{1

0'

=

rat + e' — £>',

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(<£-Ti) +0.493213—A v sin(0 + 7i) -0.09588405^cos(2Ti) + 0.128774.42 Cos(2T2) -0.268186A 2 - 0.205537^, y2 = -1.90554—A y cos{4>-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(<£ -T2-

Survey

2£) - 3.831413 sin(<£ + T2 - 2£)]

+A^ [0.017664 sin(3T2)] +AzA2y[-0.86684sin(T2

- 2Ti) - 0.044724sin(T2 + 2Ti)]

+A*[-1.487917 sin(T2 - 2£) + 0.475507 sin(T2 + 2£)]. In a posterior paper [2-10], Richardson and Cary give analytic solutions for the motion of a spacecraft in the vicinity of the L\, L 2 points of the Sun-Earth+Moon system, considering a restricted four-body problem. The system is treated as a circular restricted three-body problem (Sun, Earth-Moon barycenter spacecraft) with corrections due to the eccentricity, e, of the barycenter motion around the Sun and to the Earth-Moon mass ratio. The perturbing accelerations are constructed up to order 3, where e is assumed to be of order 1 and the departures x,y,z from the libration point are taken to be of order 3/2. For orbits as far as 4 x 105 km from L\ they get bounded solutions which include all perturbations having amplitudes > 150 km. An intermediate orbit solution is determined neglecting all eccentricities and the effect of the Earth-Moon mass ratio. They are included after as perturbations of the intermediate orbit. A more convenient formulation of the equations, from the point of view of algebraic manipulation, and in order to obtain halo orbits, is given by Richardson [2-7]. Although his model is a circular restricted one, which does not take into account the perturbations of a "third primary", his results where used, in fact, as a first approximation for the ISEE-C mission. In [2-7], Richardson gives the following Lagrangian formulation for the motion about the collinear points. In a reference system, centered at any of the three collinear equilibria (E = £1,2,3, and with the x-axis, in all cases, oriented with —7*12, as in Figure 1.23). The Lagrangian can be written in the form L

=

fa**? z

r

r\f> + 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
z" + X z

=

0,

=

0,

=

0,

and taking c2 = A2 — A, the z-equation then reads as oo 2

Z" + X Z =

/

\

C Z n 2p

J2 n P ~ "[-)+Az-

It is assumed that the correction constant A is A = 0{A\) (at least for the SunBarycenter system), so it will first appear in the expressions for the third order corrections.

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

35

In order to remove any secular terms which appear, as a result of the successive approximations, a new independent variable r is introduced via the relation r

=

u>s,

where w =

1+ ^ w

, w„ < 1,

n

n>l

where the un are assumed to be of order 0(AX). Each of the un is chosen during the development of the solution to remove the secular terms. Using the perturbation method, already mentioned, the third order solution for periodic motion about any of the collinear point is x

=

0,21 Al + a22A2z - Ax c o s r i + ( 0 2 3 ^ - a2iA2z) 3

+ (a31A

x

COS(2T X )

- 0 3 2 ^ ^ ) cos(3ri), (b2iAl-b22A2z)sm(2T1)

y

=

kAx sinn+

z

=

SnAz cosri + Snd2iAXAZ(COS(2TI

{b3iAl-b32AxA2z)sm(3Tl),

+

- 3)) + 6n(d32AzAl

- d31A3z) cos(3n),

where the amplitude relationship is hAl + l2A2z + A

=

0,

the constants 5n and T\ are defined as Sn = 2 — n, n = 1,3, T\ = \T + <j>, and a^, bij, dij, h and k are adequate constants depending only on the mass parameter. The ISEE-C spacecraft was targeted to a halo orbit corresponding to an amplitude Ax — 206000 km and Az = 110000 km around the collinear point L\ in the Sun-Barycenter system. To ascertain the effects of higher-order corrections on the halo orbits, a differential correction scheme was developed to produce these periodic orbits numerically. Once the appropriate initial conditions, for a particular periodic orbit, have been determined, then, through numerical integration, the orbit itself is known and can be expressed as the Fourier series x

=

y^an

cos(nflt),

n>0

y =

y^ft n sin(nflt), n>0

z

=

yjc„cos(nfii). ra>0

The comparison of these expressions, up to order n = 13, and those obtained up to the third order, already mentioned, is done in [2-9]. For the nominal orbit, for the ISEE-C, the maximum differences in the coordinates are Azmax

=

6 641 km,

36

Bibliographical

Ay r o a x

Survey

=

7468 km,

k-z-max =

3 909 km,

where the coordinates were obtained from numerical integration of the circular restricted equations, after correcting the initial conditions, given by the third order theory, in order to have a periodic solution. As noted in [2-2], it is much more convenient for all the purposes to use the barycenter of the Earth-Moon system when dealing with L\ and L 2 halo orbits of the Sun-Barycenter system than the Earth itself. In references [2-4] and [2-5] nothing is added to what it has been said until now, except some technical parameters about the orbit of the ISEE-C spacecraft.

1.3

Motion Near L\ and L5

1.3.1

References

We have not included those references which deal with the motions near i 4 i 5 in the restricted, planar and circular three-body problem. The references summarized in this section are the following [3-1] K.B. Bhatnagar and P.P. Hallan. "Effect of perturbed potentials on the stability of libration points in the restricted problem". Celestial Mechanics, 20 (1), 95-103, 1979. [3-2] J.V. Breakwell and R. Pringle. "Resonances affecting motion near the Earth-Moon equilateral libration points". Progress Astronautics and Aeronautics, 17, 1966. [3-3] H.M. Dusek. "Optimal station keeping at collinear points". Progress Astronautics and Aeronautics, 17, 1966. [3-4] R.W. Farquhar. "The control and use of libration point satellites". Technical Report TR R346, NASA, 1970. [3-5] T.R. Kane and E.L. Marsh. "Attitude stability of a symmetric satellite at the equilibrium points in the restricted three-body problem". Celestial Mechanics, 4 (4), 468-489, 1971. [3-6] R. McKenzie and V. Szebehely. "Non-linear stability motion around the triangular libration points". Celestial Mechanics, 23 (3), 223-229, 1981. [3-7] W.J. Robinson. "Attitude stability of a rigid body placed at an equilibrium point in the restricted problem of three-bodies". Celestial Mechanics, 10 (1), 17-33, 1974. [3-8] B.D. Tapley and J.M. Lewallen. "Solar influence on satellite motion near the stable Earth-Moon libration points". AIAA Journal, 2 (4), 728-732, 1964. [3-9] B.D. Tapley and B.E. Schultz. "Numerical studies of solar influenced particle motion near triangular Earth-Moon libration points". In G.E.O.

37

Motion Near L4 and L5

Giacaglia, editor, Periodic Orbits, Stability and Resonances, pages 128-142, 1970. [3-10] J.P. De Vries. "The Sun's perturbing effect on motion near the triangular Lagrange point". In Proceedings XHIth I AC, pages 432-450, 1964. [3-11] L.E. Wolaver. "Effect of initial configurations in libration point motion". Progress Astronautics and Aeronautics, 17, 1966. 1.3.2

The Triangular

Equilibrium

Points,

Location

and

Stability

The equilateral-triangle equilibrium points L4 and L5 are known to exist for the restricted three-body problem and for the restricted elliptic three-body problem. The stability, in the linear sense, of these points has been studied by many authors (Danby, Benett, Kinoshita,...) and the results are summarized in Figure 1.24.

0.0 2 ^ ' Mo 0.06 Fig. 1.24 ([3-6]).

0.08

Stability chart for the equilateral equilibrium points. The stable region is crosshatched

Bhatnagar and Hallan [3-1] studied the location and the linear stability of these points in the restricted circular problem when there are perturbations in the potentials of the primaries. They considered the following three cases: (i) The larger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of motion of the primaries. (ii) Both primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of motion of the primaries. (iii) The primaries are spherical in shape and the larger one is a source of radiation. In all three cases, the range of stability in the fi parameter, for the triangular points, decreases.

38

Bibliographical

Survey

If a satellite moving in the mutual gravitational field of two finite bodies is equipped with a continuous thrust device, it is possible to artificially generate new equilibrium positions. This generalized libration point concept was introduced by Dusek [3-3], who developed general conditions for the existence of these points and examined the stability of a variable thrust satellite in their vicinity. The geometry for a general isosceles-triangle point is given in Figure 1.25.

Fig. 1.25

Geometry for a general isosceles-triangle point.

If the equations of motion are augmented with a planar thrust acceleration {Fcx,Fcy), they become x-2(l

+ u)y

=

-pcos9

+ 2(l + u)psm9 + (l + p)vsin9

+ (l + p){l + u)2cos9 ^2„ , .....

+(1 + vyx + vy

!-A»,

-((1 + p) cos 9 + x)

4 ( ( 1 + P){cos6 - 1) + x)„ f „ 2 + Fcx, nr2 (I + P) y + 2(l + v)x

=

-psin6-2(l

+ u)pcos9 - (1 +p)i/cos6

+ (1 + p)(l + v)2 sin6 —vx + (1 + v)2y — z

= —

.3

'1

+

„3

r?l

r?2 J

((l+p)sm9

+

y)+Fcy,

Z,

'2J

where the distance R and the angular rate 6 are denned as R=l

+ p(t),

6 = l + v{t).

As long as p(t) and u(t) are functions of the eccentricity only, but they do not

39

Motion Near L4 and L5

depend on gravitational perturbations from other bodies, it can be verified that the isosceles triangle point is an equilibrium solution if Fc

=

2

2(l + p) (l-cos6»)

1-[2(1-cos6>)]

3/2

0* e4 - sin -1 + cos -j

A coordinate rotation of angle 6/2 uncouples the thrust acceleration terms in the equations of motion. Denoting by (x',y',z') the new variables after the rotation, taking p = v = 0, expanding the equations in power series, and retaining only linear terms, the equations become [3-4] a;"-2y'-/i1(/i,fl)a;'-|[(l-Ai)sin%'

=

0,

y" + 2 i ' - | [ ( l - M ) s i n 0 ] a ; ' - / i 2 ( / i , % '

=

0,

=

0,

z" + h3{ji,e)z' where hi(ii,9)

=

^ ( l + c o s 6 » ) - | { H - 3 c o s 6 l + 2[2(l-cos6i)]- 3 / 2 }, | ( 1 - costf) - | { 1 - 3cos0 - 4[2(1 - cos^)]- 3 / 2 },

h3(n,6)

=

l +

M{[2(l-cos^)]-

3 2

/ -l}

The values of these coefficients hi as functions of 6 are given in Figure 1.26 for the Earth-Moon system (/u = 0.0121507) and for the Sun-Barycenter system (// = 3.0404 -10- 6 ), see [3-4].

\

A, h

1

*3

I V/

v/3

i(ili)*0

2 id-o Sh0

" ^ 40

50

60

e<

Fig. 1.26 Behavior of the functions hi for the Earth-Moon system (left) and the Sun-Barycenter system (right) ([3-4]).

40

Bibliographical

Survey

The thrust accelerations in Earth gravity units (g = 9.81 m/s 2 ) for these two systems are given in Figure 1.27 ([3-4]). For a fixed value of /x the isosceles-triangle points are linearly stable for values of 8 greater than a critical value 6C. The approximate values of 8C for the Earth-Moon and Sun-Barycenter systems are 38°.9 and 2°.45, respectively.

30 0 (OEG)

40

50

60

0

1

2

3 4 B (DEG)

Fig. 1.27 Values of the thrust accelerations for the Earth-Moon system (left) and the SunBarycenter system (right) ([3-4]).

1.3.3

Numerical

Explorations

Some rough numerical explorations for the Earth-Moon system were done by McKenzie and Szebehely [3-6]. The model that they used is the plane and circular restricted problem of three-bodies. They computed the regions about £4 and L5 in which a particle, with zero initial velocity, librates about the equilibrium point. The region in the vicinity of the equilibrium point is divided into grids, with a mesh size of 0.005 nondimensional units (1920 km). The trajectories of particles placed at each node point with zero initial velocity are then integrated for 480 tine units (1 unit = 4.7 days). If the orbit does not touch or cross the x-axis, the initial position is considered to be one which gives libration. Their results are presented in Figure 1.28. More realistic models were considered by Tapley and Lewallen [3-8] and Tapley and Schutz [3-9]. In the first paper, the Earth and the Moon are assumed to move in circular orbits about their mass center, and the Earth-Moon orbit plane is assumed to be inclined at an angle of 5°9' to the ecliptic. The direct gravitational effect of the Sun and the solar radiation pressure are considered.

Motion Near L4 and £5

Fig. 1.28

41

Regions of stable initial conditions for motions around L4 (left) and L5 (right) ([3-6]).

It must be said that when the effects of the Sun are included, L4 and L5 are no longer equilibrium points. They are redefined as those points where a particle at rest with respect to L4 and L5 will have zero acceleration in a synodic system. So, for a fixed value of time, the position of the Sun must be calculated. Initial conditions were chosen so that the Sun, Earth and Moon were collinear, and the satellite was placed at the "equilibrium" point with synodic velocity equal to zero. The results obtained for the size of the envelope of motion around L4 are the following After 8 months 15 months 23 months

Amplitude of x 50 000 km 100000 km 240000 km

Amplitude of y 30000 km 60000 km 125000 km

In the z direction they found a near periodic motion (but with increasing amplitude) of period 27.6 days. The amplitude of the motion after 8 months was 6 000 km. Similar results were obtained for L5. When solar radiation pressure was included, the results where worse. The paper includes estimations about the impulse required for forcing a vehicle to remain precisely at L 4 ; for one year 750 m/s are required. The second paper [3-9] is similar to the above mentioned. In it, the equations of motion of the restricted problem of four-bodies were numerically integrated using the JPL Ephemeris Tapes DE3 to provide the position of the three primaries: Sun, Earth and Moon. Using initial conditions for the particle which satisfied the elliptic restricted problem of three-bodies (this means that, if the effects of the Sun were

42

Bibliographical

Survey

neglected, the Earth and the Moon would move in elliptic orbits and the particle would have the proper velocity to maintain the equilateral triangle configuration), the numerical results of the restricted problem of four-bodies indicate that a particle placed at L5 on Julian Ephemeris Date 2439796.735 will follow a libration point centered motion for 2 500 days. The envelope of the particle's motion about I/5 expands and contracts with a period of approximately 650 days. Additional computations show that the motion will persist for a period in excess of 5 000 days. For the same initial data, a close lunar encounter occurs after 579 days for a particle placed at L4 with initial conditions satisfying also the elliptic restricted problem. This encounter causes a sudden change of motion. Similar results were obtained when the velocity of the particle was set equal to zero. The Julian Date mentioned was chosen in order to have the Earth's center relatively close (less than one Earth radius) to the Sun-Moon line. 1.3.4

Analytic

Results

Several papers are devoted to study the motion of a small particle near the Lagrange equilateral libration points for the Earth-Moon system when it is perturbed by the Sun. In [3-10] the Sun and the Moon are assumed to move in circular coplanar orbits. The equations of motion for a particle in the neighborhood of L4, are derived by the expansion of the force function in the neighborhood of the equilibrium point. The force function is first written in Jacobi coordinates and the Legendre polynomials are used to express the inverse of the several distances. According to the notation of Figure 1.29 the expression obtained for the force function is the following

F& = e +

m0 m3k2

ri cos Si 2

+ m2

rl [ - cos2 Si3

!»"1

+ rl — I o COs3 ^13 - 7; c o s Si3 r3

+ -A;2r2 — (3 cos S23 + 6 cos S13 cos S12 - 15 cos2 S13 cos S23) + • 2 r3 where k2 is the gravitational constant and A2

=

r\ - 2nr2 cos Si2 + r\

is the distance between m\ and 7712. If the equations of motion are written in a rotating coordinate system centered at L4 they become x-2y

=

Ciz + C2y + C 4 -l-C , 6 cos(2<^) + C7sin(2 < ^) + C 8 cos(i) + C 9 sin(£) + Cni/(a;cos(2^i) - ysin(20*)) +Ci2^(3xz, cos(0i) - yL sin(<j)t))x + C\iv(ZyL cos(cf>t) - xL sin(<j)t))y,

43

Motion Near L4 and L$

m, ( s a t e l l i t e ) ">,(Sun)

(Earth) m

t (Moon)

Fig. 1.29

Jacobi's coordinates of the Sun, Earth, Moon and satellite in a bicircular model.

y + 2x

=

C2x + C3y + C5 + C7 cos(2(pt) - C 6 sm(2<j>t) - C 9 cos(£)) -r-Ci2^(3yi cos(<j>t) - xL sm{4>t))x + C\2v{ZxL cos(t) - yL sm((pt))y,

where the constants are denned by 3

1

C2 = 5v3(i + 2/z),

c3

9

c6

1

= 4 r>

Ci =

c5

+

1

=

* ^xL, \VML,

3

C7 =

—^vyL,

XL

=

C8 = -l-(3xl+y2L),

2/L-JV3,

C9 =

1, M

8r3

-—xLyL, 8r3

Cl0 = l-(3yl or3

+ x2L),

-3

r

r -

— n nir>ir>oc;fir>

—

~ 82.450002 ~ ° - 0 1 2 1 2 8 5 6 2 '

v = ^ r

= 0.00566779, 3

m 3 = 3.33432 x 10 5 . 31

r 3 = 388.9237.

The above equations are linear with variable coefficients. As all the terms with time-dependent coefficients contain the parameter u, a solution can be obtained, in principle, in the form

= £*/*%), i>0

y

= Y,»iy{i)(t)i>0

44

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Survey

In [3-10] a second order solution is obtained, whose expression is zW = -0.000043827+ {Ax + 0.018490) cosait

+ (—

- 0.011189) cosa 2 t + (X°~

+(x0-Ai

- 0.030532^) s i n a i t

* + 0.048085 J sina 2 t

-0.0057357 cos 2 ^ + 0.0075861 sin 20t -0.0015216 cos(f>t - 0.00039342 sint, 2/(°) = -0.0010647+(A 2 -0.025796) cosa x ^ + ( — + 0.0011842 X )sina 1 i +(2/o -A2+

0.018950) cos a 2 t + (Vo~

2

- 0.022898 ] sin a 2 t

+0.0075232 cos 2
= [1.7798cos(ait) +0.44239sinait + 0.076915cosa 2 t + 0.067517sina 2 t -1.8767 cos 0t + 0.47526 sin <j>t] x 10" 3 + [0.61849 cosait + 0.21851 sin a i t - 3.8764 cosa 2 t - 0.71859 sina 2 t +3.25791 cos( - ax)t] x 1 0 - 6 ,

I/J/ 1 )

= [-0.45899cos(ait) - 1.2589sinait - 0.023284cosa 2 t - 0.057020sina 2 t -0.48229 cos 0t - 1.3073 sin *] x 1 0 - 3 + [-0.11778cosajt-0.45971 sinait + 1.8953cosa 2 t +1.42885 sina 2 t - 1 . 7 7 7 5 c o s ( 0 - a i ) t +0.82492sin(
with A,

=

a\+c3-

4)x0 - 2y0 - c2y0 - « 2 a*if -a.%

Ao

=

Bi

=

Bo

=

-1.0734347a;o - 2.5174708j/0 - 1.5954815j/0, a\ + ex - 4)j/0 + 2±Q - c2x0 -2.9615272yo + 2.5174708i 0 - 1.5954815a;o, [a\ + c3)x0 - 2C2:EO - c2y0 - 2c2y0 a\ -a\ 3.9615168i 0 - 3.1909630a;o - 1.5954815jr0 - 5.6714422j/0, {a\ + ci)y0 + 2ciar0 - c2i;o + 2c2y0 2.0734144j/o + 3.1909630y0 - 1.5954815y0 + 1.8952373zo.

To test the validity of the analytic solution, the results are compared with the numerical integration of the restricted four-body problem.

Station

Keeping

45

The two solutions are quite close for about the first 90 days. If ephemeris positions for the Moon and the Sun are used, results are worse. The most important reason for the discrepancy seems to be the fact that the Moon has an elliptic orbit. In [3-11], Wolaver uses the first order solution obtained by de Vriesin order to find the initial conditions which will reduce the maximum excursions of both x and y. Three sets of initial conditions are found which correspond to three different values of the initial configuration angle a formed by the Sun, Earth-Moon barycenter line and the Earth-Moon line, and which are the following a = 133°, a = 210°, a = 337°,

x0 = -0.00646, y0 = -0.00385, x0 = -0.01029, y0 = 0.01260, x0 = 0.00517, y0 = 0.00424, x0 = 0.01582, y0 = -0.01016, x0 = -0.01073, y0 = 0.00211, x0 = 0.00123, y0 = 0.01463.

A different approach is used in [3-2]. In the paper the restricted problem of three-bodies is extended to include direct and indirect influence of the Sun on a particle near the L4 point. Perturbations in the displacements from the Lagrange point, up to fourth order, and comparable solar effects are included. The Von Zeipel perturbation method is carried to the second approximation and the slow variations of parameters are studied. Some periodic orbits around the points are computed. Papers [3-5] and [3-7] deal mainly with attitude stability which is not of direct interest for this work. We merely mention them as a subject for future work. 1.4

Station Keeping

1.4.1

References

The references summarized in this section are the following [4-1] J.V. Breakwell, A.A. Kamel and M.J. Ratner. "Station-keeping for a translunar communications station". Celestial Mechanics, 10 (3), 357-373, 1974. [4-2] J.V. Breakwell. "Investigation of halo satellite orbit control". Technical Report, NASA-CR-132858, 1970. [4-3] G. De Fillipi. "Station keeping at the L4 libration point: A threedimensional study". Master's thesis, Dept. Aeronautics and Astronautics, Air Force Institute of Technology Wright-Patterson AFB, Ohio, USA, 1978. [4-4] E.A. Euler and E.Y. Yu. "Optimal station keeping at collinear points". Journal Spacecraft and Rockets, 8, 513-517, 1971. [4-5] R.W. Farquhar. "Station-keeping in the vicinity of collinear libration points with an application to a lunar communications problem". In Space Flight Mechanics, Science and Technology Series, volume 11, pages 519535. American Astronautical Society, New York, 1967. [4-6] R.W. Farquhar. "The control and use of libration point satellites". Technical Report TR R346, NASA, 1970.

46

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Survey

[4-7] R.W. Farquhar. "Limit cycle analysis of a controlled libration point satellite". Journal of the Astronautical Sciences, 17, 267-291, 1970. [4-8] R.W. Farquhar. "Comments on 'Optimal controls for out of plane motion about the translunar libration point' ". Journal Spacecraft and Rockets, 8, 815-816, 1971. [4-9] 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. [4-10] R.W. Farquhar, D.P. Muhonen, C. Newman and H. Heuberger. "The first libration point satellite. Mission overview and flight history". In AAS/AIAA Astrodynamics Specialist Conference, 1979. [4-11] T.A. Heppenheimer. "Optimal controls for out of plane motion about the translunar libration point". Journal of Spacecraft and Rockets, 7, 10881092, 1970. [4-12] H. Heuberger. "Halo orbit station keeping for ISEE-C". In AAS/AIAA Spaceflight Mechanics Conference, 1977. [4-13] S.S. Lukjanov. "Control of a spacecraft motion in the vicinity of a collinear libration point of the circular three-body problem by means of light pressure". Cosmic Research, 19, 518-527, 1980. [4-14] D.P. Muhonen. "Accelerometer enhanced orbit control near the Sun-Earth L\ libration point". In Space Flight Mechanics, Science and Technology Series, AIAA Paper 83-0018, 1983. [4-15] M.J. Ratner. Station keeping for a translunar communication station. PhD thesis, Stanford University, Stanford, USA, 1973. [4-16] D.L. Richardson. "Halo orbit formulation for the ISEE-C mission". J. Guidance and Control, 3, 543-548, 1980. [4-17] W. Wiesel and W. Shelton. "Modal control of an unstable periodic orbit". Journal of the Astronautical Sciences, 31 (1), 63-76, 1983. 1.4.2

The Control

of Libration

Point

Satellites

In a remarkable paper [4-6] Farquhar presents many of the significant results about missions to orbits near libration points. Both the study of the orbit and the control are presented. The information in [4-5] and [4-7] can also be found in [4-6]. First some conclusions of Colombo for a collinear libration point of the EarthMoon system are recalled: (a) To the first order, this point is an exact solution of a restricted four-body problem (Sun, Earth, Moon, spacecraft). (b) It is possible to use a simple linear feedback control for position stabilization. (c) It is possible to obtain the acceleration needed for position and attitude stabilization of a libration point satellite by simply varying the magnitude and direction of a solar sail.

Station

47

Keeping

In [4-5] it is shown that near a collinear libration point a single-axis linear control, using only range and range-rate measurements, will guarantee asymptotic stability. The cost of the station keeping is given as a function of the measurement noise. Returning to [4-6], first the general equations of motion in the vicinity of a libration point are given. Let R = l + p{t),

6 = l + v(t),

be the distance and angular rate of the primaries, where p and v arise from the orbital eccentricity of the finite bodies and perturbations from bodies external to the basic two-body system. Let r = ((l±"/L)(l

p)+x,y,z)T,

+

be the position of the satellite near L2 respect to the larger body arjd R = (l +

p,0,0)T,

be the position of the smaller body. For a satellite of unit mass the kinetic and potential energies are, respectively, T = -r

• r,

U =

2

;—;

u

|r|

=

*\\r-R\

5-

R3

The Lagrange equations can be easily obtained and developed to a suitable order. For equilateral points we put

r = l^(l + p)+x,±-£(l + p)+y,z\

,

instead of the previous value. Therefore in p and v, the indirect perturbations on the satellite are included. 1.4.2.1 Perturbations and Nominal Path Control Besides the terms coming from Lagrange equations, several perturbations act on the satellite, mainly due to gravitational influence of other finite bodies and to solar radiation pressure. The basic strategy for the control about a nominal path can be explained for the motion near the collinear points. The addition of control terms and perturbations to the linearized equations of motion gives x - 2y - (2BL + l)x y + 2y + (BL-l)y z + BLz

=

Px(t) + Fcx,

=

Py(t) + Fcy,

=

Pz(t)+Fcz,

48

Bibliographical

Survey

where P is a periodic perturbing acceleration and Fc is a control acceleration. The nominal path (xn,yn,zn)T, is determined with Fc = 0. Using the change (£> V, 0T = fa, V, z)T - (xn, yn, zn)T the resulting equations are of the same type with the P terms suppressed. This is correct if the size of the nominal path is small. Periodic orbits (first in a linear approximation; after, to high order using the Lindstedt-Poincare method) can be obtained. The linear and nonlinear responses to periodic inputs P(t) are given. In this way it is easy to consider the effect due to several contributions in P: eccentricity correction, gravitational perturbations and solar radiation pressure. For instance, the solar perturbation near the Earth-Moon collinear points is studied using: (a) The de Pontecoulant's expressions for the motion of the Moon. They introduce suitable values for p and v. (b) The direct solar effect, given through a potential energy (for motion relative to the Earth) ( — ! - = , - T~^)6 , \\rs-r\ \rs\ J

UD = -Ms where r = ( ( l ± 7 L ) ( l + p ) +x,y,z)T,

rs = rs(coswst,

-

smwst,0)T.

The effect of Sun in the Earth-Moon libration points and the effect of the Moon in the Sun-Barycenter collinear ones is roughly studied, as well as some Jupiter effects. 1.4.2.2

Linear Feedback Control

At collinear points and for small eccentricities, the linearized equations of motion at a collinear point (or relative to a small nominal path about the point) are, approximately x-2y-(2BL

+ l)x

y + 2y + (BL-l)y z + BLz

=

Fcx,

=

Fcy,

=

Fcz.

The z-axis motion is bounded and can be damped by simply taking FC2 = -k[z, with gain k[ > 0. The control synthesis for the (x,y) motion (unstable) can be accomplished with classical methods (Routh's and root locus criteria). For x-axis control it is enough to take Fcx = -k\i — k2x, Fcy = 0, with suitable hi, k2 values depending on B>L (for instance fci = 4, fc2 = 12 for BL = 4 ) . If the y-axis is used the required control is Fcx = 0, Fcy = —k5x-k6x-k7y + ksy.

Station

49

Keeping

If the periodic coefficients of the linearized equations are not too small then Floquet theory can be used. It is required that all the eigenvalues of the fundamental matrix be less than one in module. Using the x-axis control, if e is large (e is not too small for the Earth-Moon system) stability charts can be obtained giving the regions of the (fci,/^)-space for which stability is obtained. The analysis of the control near a periodic orbit requires also Floquet methods. At the equilateral triangle points a simple damping Fcxi = —kix'+kiy', Fcyi — 0, where prime refers to coordinates relative to the equilibrium point, can be used. Routh's criterion can show how to select k\ and k±. 1.4.2.3

Station Keeping

The cost of station keeping for a collinear point satellite depends on measurement noise and random accelerations. An z-axis control system with noise inputs is studied, looking for the Routh's and root locus criteria. Estimations of the cost of noise at Earth-Moon collinear points are given. The noises in a;, a; are supposed independent, white, Gaussian processes. The costs of a constant displacement and of a sinusoidal control acceleration with noise are also estimated. As an example, a solar sail is proposed for station keeping in the vicinity of a collinear point of the Earth-Moon system. Some variation of the area of the solar sail should be used. Concerning the on/off control, the related equations near the collinear libration points are x - 2y - (2BL + l)x

=

y + 2y + {BL~l)y

=

-F(0, 0,

where £ = x + \x. The function F has a dead zone ^ and a hysteresis loop of width 28. Limit cycles of this system, as well as their stability, are found using harmonic analysis. An example is again a solar sail control, this time at the Sun-Barycenter L\ point. Several applications for lunar and planetary missions are proposed. However the concept of halo orbit is still not present in the proposed missions. 1.4.3

Station

Keeping for a Translunar

Station

In [4-1] Breakwell and coworkers (see also [4-3] and [4-15]) study a model for the control of a translunar station, i.e., a station near the L2 point of the Earth-Moon system in a halo orbit. The orbit is periodic except for the effects of Sun, radiation pressure and eccentricity. The error due to the approximated analytic orbit used as a nominal orbit (with an y-amplitude of 45000 km), is of the order of 10_6<7 (residual acceleration).

50

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Survey

One needs to provide feedback to overcome instability and recovery from initial errors. A quadratic cost criterion is introduced. There are several possible controls, for instance: 3-axes control; towards Earth; towards (polar axis of) Moon. The station keeping can be reduced even taking into account execution and tracking errors. 1.4.3.1

Dynamics

The standing equations are x-2ny

=

Ux+ux,

y + 2ny

-

Uy+uy,

z

=

Uz+uz,

where U

=

a

=

^

d-E

+ ^L+n2((x

<1EM

2

+

l2))

+y2),

,

fe^U

72 =

+ a(l-n

d,M

o.1678,

a n

=

HEM,

and u is the control vector. If u = 0, a nominal orbit r/v(i) = (xjv(t),yiv(t), zjv(t))T, with the origin at L2, is found analytically up to a given order. Order four is selected in the work. Due to the symmetry, one has (2irnt\

E a cos I —^- 1 , . (2imt\ E 6„sin \—f~) ' n

ZN

=

^-^ (2imt\ ± 2 ^ C n C0S 1 ~f~ ) '

where the period T depends on the amplitudes The coefficients an, bn, cn and T are functions of some initial coefficient, for instance, b\. Let 5r(t) be the deviation from the nominal orbit. Then Sf

=

2nk A5f + Urr(rN(t))Sr

+ e(t) + 0(6r2),

where k is the unitary vector along the z-axis and e(t) = VU(rN(t))

- 2nkArN

-

rN,

is the (T-periodic) error due to the fact that the nominal orbit does not satisfy the equations of motion. One intends to avoid an excessive control cost.

Station Keeping

51

The authors claim that there is no need to improve the periodic orbit because there exist other noncommensurable perturbations and the improvement to get a quasi-periodic orbit is hard. The cost to minimize is min lim / (\u\2 + k\6r\2)dt, u(t) t/-x» y0 where A; is a tightness coefficient. As stated before, the noncommensurable perturbations are ignored. It is supposed that they are included in the nominal orbit and not in the feedback control law. 1.4.3.2

Linear Control

The differential equation of the control is x

=

F(t)x + G(t)u + E(t),

T

where x = (Sr, Sr) ,

7

°

F=(

V UTr{rN)

^ n=

2uQ J '

E = (0,e(t))T, F and G are T-periodic. The possible controls are specified by 0

ueR3,

(a) G = ( ^ 3 j ,

(b) G = (0,0,0,u,0,0) T , u£ ¥E, 0Q) )TT, -u ceR,ro ,„w<, ~* -=-y „,„„ _i_ „ . , (c) G = (0,0,0, — —,— where x* 2a + xN, p' P ' The functional to minimize is J

-(xTAx

=

+ uTu),

2

it Jtn

where 71

~

'

0

o3

starting at a given xo for t = to. Let J(x0,t)

— min /

L(x,u),

"(0 Jto /to

with a;(io) = £o> where L

=

-(z T ,4:r + w T u).

„2 _p2„*2 =x*2+y2N.

52

Bibliographical

Survey

Then J satisfies (use dynamic programming or maximum principle to show this) Jt + min(L + Jx • x)

=

0.

u(t)

We need to minimize -uTu+JxGU, and therefore Jt + ^xTAx

i.e. uT + JxG = 0. This implies u —

+ Jx(Fx + E)-^JxGGTJj

=

-GTJj

0.

The solution is of the type J(x,t)

=

-xTS(t)x

+ bT(t)x + n(t),

where S, b and n satisfy suitable differential equations, with the end condition of being zero at £/. Let A = jj. Then (x,\)T satisfies the differential equation,

The corresponding transition matrix <j> associated to the homogeneous part is symplectic. For the application the eigenvalues have modulus different from the unity and are simple. For tf -» oo, the value u = — C(t)x — /3(t) is obtained, where C(t) = GTS(t), /?(£) = GTb(t), S satisfies a Ricatti matricial equation S + SF + FTS + A-SGGTS

=

0,

with S(tf —> oo) = 0 and bT + bT(F-CG)

=

ETS,

b(t) being T-periodic, i.e. bT(t)

6T(O)0-1(t,O)- f

=

ET{T)S{T)4>*-\t,T)dT,

Jo

for 0 < t < T and 6 T (0)

=

Jo

fT(ET(T)S(r)*-1(T,T)drW1(T,0)-Ie)-1.

The transition matrix 4>*(t,r) for x = (F - GC)x is regular. After some manipulation it is possible to get a suitable expression for S(t). Let \vi\ be the eigenvalues of 4>(T,0) inside {\z\ < 1}. Then T • log(min|vj|) is the settling time (time needed to reduce errors to 1/10 of the original values).

Station

1.4.3.3

53

Keeping

Effect of Errors

In fact the control equation is x

=

Fx + G(u +

w)+E,

where w is a dynamic noise. Then in the formula for u the function x should be replaced by x, the Kalman estimate of the state x. A discussion follows in the paper about the attainment of this estimate. Numerical results for bx = 45 000 km , T = 14.4 days are given. For a settling time of 2.5 months a three-axes control requires 10~ 8 g. This means a low amount of roughly 3 m/s per year. The observation errors from the Earth are supposed to be: 3 m in range and 1 mm/s in range-rate. Therefore the continuous feedback control introduced succeeds stabilizing the orbit. 1.4.4

Control of an Unstable Periodic

Orbit

In [4-17], Floquet's variables are introduced near a periodic orbit of unstable type. The control is only applied to suppress secular deviations. The theory obtained is applied to a planar periodic orbit that closes after two revolutions near the L3 point of the Earth-Moon system. The model used does not take into account the lunar and solar eccentricities, nor the lunar inclination. From Hamilton's equations x

=

JDXH,

the variational equations Sx

=

JDxxH(t)Sx

= A(t)5x

are obtained. The matrix A(t) is T-periodic. Using Floquet's theory, the fundamental matrix (£, 0) is written in the form A ( i ) e J i A _ 1 ( 0 ) , where A is a T-periodic matrix and J is a constant matrix. For the orbit under consideration I

0

-M J

= \

where UJX = 0.077 ...i,u2

0 0 0 0

1^11 0 0 0 0 0 0

0 w2 0 0 0

0 0 0 -w2 0 0

0 0 0 0 0

-M

= 0.1789..., w3 = 0.084 ...i.

0 0 0 0 |w3| 0

54

Bibliographical

Survey

The eigenvectors of (j)(T,0) are the columns of A(0) and the eigenvalues A; are related to uii (characteristic exponents through w» =

^log(Ai).

In order to use Floquet's variables (77 = A~1(t)5x) to get a linear system 77 = J77, the matrix A(t) is needed through a whole period. The differential equation A =

A(t)A(t) - A(t)J,

is easily obtained. The authors recommend to use A(i) and A _1 (£) by means of its Fourier series. The effect of the Floquet transform is to reduce the system to an uncoupled linear system with linear coefficients. For the selected orbit around L3, the flow is of center type in the (171,772) and (f?5;r)6)-planes, and of saddle type in the (773,7/4)-plane. The phase portrait obtained by integration of the full flow, computing 8x and going back to 77, is projected in the (773,774)-plane. The result is roughly a deformation of the flow near a saddle. Therefore some control is needed. We wish to control the 773 variable (the one which has positive eigenvalues). We return to the variational equations in physical variables, adding a control A(t)Sx +

5x

B(t)u(t),

where u(t) is a scalar and B(t) = (0,1,0,1,0,0) T (components x, x, y, y, z, i ) , corresponding to two fixed thrusters operating at a 45° angle to the Earth-Moon line. Other choices of control are, of course, possible. Rewriting the last equation in Floquet's variables we get V =

J77 +

A~\t)Bu.

If we intend to control 773, then u = krjz is taken, where k is a suitable parameter. Then (

0

-M 77

=

V sie A = A-x(t)B. In the equation for

0 0 0 0

0 0

M 0 0 0 0 0

W2

+ k\{t) 0 0 0

0 0 0 -U)2

0 0

m m

=

(w2

+kX3(t))ri3,

A3 is split into the constant and the periodic parts A3(t)

=

A3c + A 3p (i).

0 0 0 0 0

-lw3l

0 \ 0 0 0 1^3 0

Station

55

Keeping

Then »?(*)

=

773(0)exp((w2 + k\3c)t)exp

( / kX3p(s)ds)

.

For different values of k numerical examples are offered. The value k = 0.3, is just above what is needed to stabilize the system. For k = 0.5 the pattern is that of a node. For a gain k — 0.8 the nodal character is still present and furthermore the basin of attraction is large. Control accelerations applied in these cases are O(10~5g) during the suppression of the transient. No long-term control cost is mentioned, since once the satellite reaches the reference trajectory no further control is needed, ignoring outside perturbations. The previously ignored perturbations (eccentricities, inclinations, etc.) can be studied under two different approaches: The first approach would be to add damping to the two stable Floquet modes. The second, and more interesting, is to continue to teach the controller to ignore any stable effect. For this, classical perturbation techniques can be extended. The variables r]i (t) are then expressed in the form Vi(t)

-

^2

Ck e x

P0'i w i + 32 w2 + •••)•

k

If r\i is well behaved (free of resonances and positive real) the controller can be taught to suppress only the free component (i.e., the component already present without perturbations). 1.4.5

Station

Keeping

for the ISEE-C.

Algorithm

and

Results

A large number of papers deal with the pre- and post-studies related to the ISEE-C mission [4-9], [4-10], [4-12], [4-14] and [4-16]. Some information is redundant. Then, we summarize here mainly the results of [4-9], [4-12] and [4-14]. 1.4.5.1 Mission Design First, in [4-9], an orbit is selected with a low cost for the transfer. The required station keeping should be studied (moment of on/off control and suitable control). Due to the telemetry requirements a minimum angular distance of 3°.5 to the Sun was required. Vertically (to the ecliptic) the maximum allowable angle was 6° due to antenna requirements. A halo orbit near L\ for the Sun-Barycenter system was selected with amplitudes Az = 120000 km, Ay = 666672 km. For the station keeping an on/off control is scheduled. To return to the nominal orbit some pulses of amount Av (usually split into Avxy and Avz) are used. The value Av depends, mainly, on the precision of tracking. A rough estimation of the divergent part of the solution is x = keat,

y = -0.5345fceQt,

56

Bibliographical

Survey

where a = 2.5327, k = 0.6257z0 - 0.1122y0 + 0.1737i;o + 0.9286y0It is supposed that the errors are mainly due to the application of the control Av

=

(x2 + y2)1/2

~ (0.1737i 0 + 0.9286£ 0 )2.872e aAf ,

where Ai is the time between two controls. If 5v = (±1 + yl)1/2, taking x0 = yo, we have for At (in days) At 1.4.5.2

=

22.95log ( l . 8 5 - r ^ J .

The Algorithm

With the operational constraints in mind, a station keeping algorithm was developed [4-12] to compute a velocity correction at any point on the actual ISEE-C trajectory to drive the spacecraft back to the vicinity of a pre-computed numerical path. A third order analytic theory, or a numerical periodic orbit for the RTBP, were used as nominal paths. The idea of the algorithm is quite simple (but requiring a large numerical effort). Let R(t) be a point in the "real" orbit obtained by high precision numerical integration method of a good model of the real problem. Then, the nearest point R* in the nominal path is obtained, using a Newton-Raphson scheme. For a time t, \R(t) — R* is called the normal distance. Let F be an approximation of the Li norm of R(t) — R* during a time span [to, to + T], for instance / i

F

_

\ 1/2

2

= (JVETO-^I )

.

with ti = to+iAt, At = T/N. Then F is a function of the initial velocity (±o, yo, io) which can be minimized. Actually this was minimized using an iterative conjugate gradient method where the partial derivatives were computed numerically. We note that this is a kind of orbital stabilization and not a Lyapunov one. The station keeping problem is not to keep the spacecraft on the halo orbit, but to keep it inside some torus centered about this path. We note also that it is perhaps not necessary to get a minimum of F. A smaller Av can still maintain an acceptable orbit. Associated with this idea of loose versus tight control, is the choice of a time span. The shorter the time span the better the numerical trajectory agrees with the nominal path. However, due to the unstable character of the orbits, the time span should not be too large. In the full mission several controls should be included, with medium size time span. For the simulations the execution errors were chosen to be 10 percent of each component of Av and tracking errors were specified as x = 3.0 km, x = 15 mm/s,

y - 30.0 km, y — 15 mm/s,

z = 30.0 km, z = 30 mm/s.

Station

Keeping

57

The cost of the control was taken as Avxy + Avz. A simulation was performed to mimic the operational approach. The operational plan was as follows: After one Av correction is applied to the trajectory, execution and tracking errors are factored in, and the station keeping algorithm is used to determine what the next Av would be at different times advanced from the maneuver (every 3 days starting at 30 days after the first maneuver). The smallest Av exceeding some minimum criteria is chosen, and the related time is selected for the next maneuver. The time span was 180 days and Avmi„ = 1.5 m/s in the (x,y)-plane. The process is repeated to develop a station keeping history. We note that Av is not strictly increasing with time. Often it is beneficial to delay a maneuver and take advantage of a smaller Av correction. The average time between maneuvers was 60 days at an average cost of 13.3 m/s per year in the simulations. We remark that the methods of [4-1] and [4-6] are not particularly applicable to the ISEE-C mission due to constraints on the type and frequency of station keeping maneuvers. 1.4.5.3 A Posteriori Information on the Mission In [4-14], Muhonen reports that the ISEE-C was maintained in a halo orbit for nearly four years. Control was used to keep it close to the pre-computed nominal orbit with maneuvers every three months. Execution errors were lessened by processing on-board accelerometer telemetry data in real time (with suitable smoothing) and adjusting the maneuvers. Fifteen maneuvers were executed between November 20,1978 and June 10,1982. The average time between maneuvers was 82 days and the average value was Av = 2 m/s. All thrusting was made in the ecliptic plane. The average relative error in the station keeping maneuvers was 2.1 % and the maximal error attained was so low as 4.4 %. On June 10, 1982 a small maneuver (Av = 4.5 m/s) was executed to go to a new mission. 1.4.6

Additional

Work

Beyond the more important papers referred to in 1.4.2 to 1.4.5.3, many additional work can be found in the literature related to station keeping for libration point orbits. We summarize here some of them, but the information contained is redundant with what has been said or even irrelevant. In [4-11] (a pre-halo history) some jumps are proposed to maintain visibility from the Earth, of a translunar station near (too near in fact) Li- The cost of the proposed control is enormous and, therefore, the approach is useless. It is criticized in [4-8]. In paper [4-4], the control is intended to keep the spacecraft near the libration

58

Bibliographical

Survey

point using continuous (low thrust) control and minimizing the amount of control. For the L2 point of the Earth-Moon system to keep the station at some 300 km of the equilibrium point an amount of 1.5 m/s per year is required. The model taken is the RTBP and solar effects are not included. In [4-13], Lukjanov made a study of control using solar radiation pressure. Several simplifications are made: Earth, Moon and Sun move in the same plane and eccentricities are neglected. Furthermore, only planar motions near collinear points are considered. It is supposed that the spacecraft has an opaque spherical envelope of radius R that can be changed. In modal variables, the equations (for an Li of the Earth-Moon system) are z = Az + ub, where ( A 0 0 -A A = 0 0 V 0 0

0 0 0 u

^ 0 J

( qk2 cos ip + sin ip \ qk2 cos ip — sin tft b= — sinV' \ qki cos tp J

where q, hi, k2 are constants and tp measures the position of the Sun. A control u = UQ + Su, with |<5u| < u* < uo is searched. First a quasi-periodic solution of i = Az + uob is taken as a reference orbit. One wishes to control using Su(t) in order to go from some initial z°, near the reference orbit, to this orbit. It is shown that the system is completely controllable. The function 6u is taken as —usign 6{t) where 8 is a switch function. The controllability region is included between two hyperplanes of the z\ variables. It is enough, in a one-dimensional approach, to study z\ and do the switch every half period (with respect to Sun: 7r/i/, where v = UM — ns). Plots of the controllability region as a function of the transfer time T are given for the cases Earth-Moon and Sun-Barycenter. Another possibility (two-dimensional) asks for a transfer, in time T, from the initial phase vector to a vector in the plane z\ = z2 = 0, giving rise to a conditionally periodic motion. Then the controllability region is cylindrical. For the Sun-Barycenter system it is possible to synthesize the control, giving explicitly the switch line. As an application for the case Sun-Barycenter, to put the spacecraft in orbit near L\, the error in position should be less than 300 km and in velocity less than 15 cm/s. Then for a mass of 1000 kg the maximum and minimum radii of the opaque spherical envelope are 6 and 3 m, respectively.

Chapter 2

Halo Orbits. Analytic and Numerical Study

The aim of this chapter is to describe the halo orbits of the restricted three-body problem (RTBP), circular case. The analytic representation of the halo orbits is given. The complete families of halo orbits have been computed numerically for the Sun-Barycenter and Earth-Moon cases. The termination of these families becomes clear after studying the halo orbits of Hill's problem and of the two-body rotating problem. The elliptic RTBP is a more approximated model. In this problem halo orbits have been computed numerically as bifurcated from some halo orbits of the circular case which have a period commensurable with 2TX, this means that they are also periodic orbits in the sidereal system. The RTBP is studied as a first approximation to the real problem. 2.1 2.1.1

The Restricted Three-body Problem Description

of the

Problem

We consider in the space X,Y,Z, two primaries with masses mi and m^ respectively. We assume that, under the influence of their mutual gravitational attraction, they revolve in the (X,Y)-plane around the common center of masses with angular velocity equal to n (see Figure 2.1). A third body of mass mz much smaller than either mi or m-2, moves in the space under the influence of the two primaries. In fact, we consider that m.3 does not act on mi and 7712To study the motion of m^ the equations can be written in a suitable rotating (synodic) coordinate system x, y, ~z, such that the plane x, y rotates with the two primaries in order to have mi and m,2 fixed on the 3>axis. In the computations we use the equations referred to a synodic system x, y, z, of dimensionless coordinates defined by x

X=

d>

y

y

=d> 59

Z =

~z d'

60

Halo Orbits. Analytic and Numerical

Study

Z=z

N

a

Fig. 2.1

^

Sidereal and synodic reference frames for the restricted three-body problem.

where d — a + b (see Figure 2.1). In this system mi and m 2 are located at Pi = (/i,0,0) and P 2 = Qu — 1,0,0) respectively, where fx = rri2/M and M = m\ + rri2- From now on we consider fi < 1/2. For the applications \i is rather small. Furthermore the unit of time is taken such that n = 1. Then the equations are the following (see [15]) x-2y

=

y + 2x

= Cty,

z

=

flx, ft2,

where 2,

T\ 2

2

T2

r\

=

{x - n) + y + z ,

r\

=

(x-n

+ l)2+y2

I

2

+ z2.

These equations have a first integral, denoted as Jacobi integral, given by x2+y2+z2

=

2n(x,y,z)-C,

where C is the so-called Jacobi constant. It is well-known that the system above has five equilibrium points: the collinear points Li, L/2 and L3 and the triangular ones L4 and L5. All of them are on the (x, y)-plane. L\, L2 and L3 are located on the x-axis and L4 and L$ form an equilateral triangle with the two primaries as it is shown in Figure 2.2. The collinear equilibrium points are unstable and every one has a family of planar hyperbolic periodic orbits (the Lyapunov families associated to it).

The Restricted

Three-body

Problem

61

NT-'U£-1,0,0)

Fig. 2.2

The five equilibrium points of the restricted three-body problem.

A halo orbit (see [3]) is a spatial periodic orbit of the RTBP in synodic coordinates x,y,z such that its projection on the (y, z)-plane is homeomorphic to a circumference, with the origin contained in the bounded component (see Figure 2.2). This kind of orbits can be obtained from the Lyapunov families associated to the equilibrium points, when the frequencies on the (x, y)-plane and the vertical one are identical. In other cases, the projection of the orbit on the (y,z)-pla,ne describes a Lissajous's curve. The halo orbits are symmetric with respect to the (x, z)-plane. Besides one of these orbits, there exists the symmetric one with respect to the z = 0 plane. In fact, for each collinear point, there is one family of halo orbits emanating from a planar periodic orbit belonging to the Lyapunov family, the so-called Lyapunov orbit with critical vertical stability. 2.1.2

Values Adopted

for the

Constants

The values of the mass parameter /x used in this chapter are the following li = 0.012150668 for the Earth-Moon system, fj, = 0.000003040357143 for the Sun-Barycenter system. For the eccentricity e used in the elliptic model, the adopted figures are e = 0.054900489 e = 0.0167

for the Earth-Moon system, for the Sun-Barycenter system.

To obtain coordinates (resp. velocities) in physical units one should multiply by

Halo Orbits. Analytic and Numerical

62

Study

the semimajor axis a (resp. the mean motion n).

2.2

2.2.1

Analytic Determination of the Families of Halo Orbits for the Restricted Three-body Problem Introduction

The goal of this section is to obtain halo orbits, given a value of fi, some amplitude specifying the selected orbit in the family and the case L\, Li or L3 under consideration. We recall that we are looking for periodic solutions of the spatial RTBP in synodic coordinates. It is more convenient to write the equations of motion with respect to a system having the origin at the libration point and such that the distance to the nearest primary is taken as unity (see [11], [12]). The positive a>axis is directed from the larger primary to the smaller one, i.e., with the orientation reversed with respect to the one of the preceding section. The y-axis is contained in the plane of the sidereal motion of the primaries at TT/2 from the x-axis in counterclockwise sense. The z-axis completes a positively oriented coordinate system. Let 7 = d(m2,Li) for i = 1,2, 7 = d{m\,Lz). Then a new independent variable r is introduce through r = 7 3 / 2 £. If only the linear terms near Li are taken into account, there are periodic orbits in the (a;,2/)-plane of period 27r/A, where A is the eigenvalue related to the planar equations in the neighborhood of Li. We look for periodic orbits for the spatial problem. Then the z-equation is modified by subtracting from the linear term an amount A to get the same frequency for the z-oscillations. The same amount is, of course, added to the nonlinear terms. The Lindstedt-Poincare method (see [9]) is used to get a periodic orbit. As the frequency of the orbit can change, depending on the amplitude, it is more convenient to introduce a new independent variable TU, where OJ =

l + ^W

2

i,

and uij are terms of order j in the amplitudes a, /?. The basic amplitudes, a for the x and /? for the z, are taken as the coefficients of cos(wrA) of x and z, respectively. In the y variable the main term is A;asin(u;rA) where A; is a suitable constant such that (-acos(wrA),fcasin(a;rA)) is the linearized Lyapunov orbit. However, there appear terms of higher order in a and /? in the coefficient of sin(wrA) for y. This is crucial to perform the Lindstedt-Poincare method, despite the fact that these terms are not given in [10]. We note that A = A(a,/3). Therefore, to get a halo periodic orbit some minimum amplitude, a m in is needed, where amin is the solution of A = A(a, 0). For this value of a m ; n a halo orbit bifurcates from the planar Lyapunov orbit. The relation

63

Analytic Determination of the Families of Halo Orbits

A = A(a,/3) gives the possibility of having a family of halo periodic orbits. A requirement for the validity of the analytic expansions is that, for all points in the orbit, the distance to the related Lj should be smaller than the distance to both primaries. We shall loosely refer to halo orbits associated to Lj when we refer to halo periodic orbits bifurcated from a Lyapunov p.o. around Lj. 2.2.2

Analytic

Computations

Here we present a method for the computation of the coefficients in the Fourier series expansions of the coordinates x, y, z of the halo p.o. as a function of time. The coefficients are given as power series in the basic amplitudes a and /3. The coefficients of these power series are only functions of /z for a given case Lj. They can be obtained recursively as explained below. Also the related coefficients in A and ui are obtained. The method given here has been implemented as stated in 2.2.3. We note that the implementation is semianalytic in the sense that, given a value of /i, the numerical values of the coefficients of the power series are computed. They have to be computed only once for a given /i and Lj, and are independent of the concrete orbit. The standing equations are w2x" -2uy'-(l

+ 2c2)x

=

5 Z c n + 1 (n + l)p"P„

[-]

n>2

u2y" + 2ux' + (c2 - l)y

=

M,

=

£

(2.1) cn+1y £

n>2

pn~x (1 + 41- 2n)P„_ 2 i _ 1 ( - J

1=0

TAr,

(2.2) -1 2

Ul

2

Z" + X2Z

=

Yl

C

n>2

«+l* E

P^1^1

+ 4l -

2n P

) n-2l-l

+Az = P + Az,

(2.3)

where d(TUl)

,

Ti=wrA,

0J = l + ^Tw2i,

p2 = x2+y2+z2, A =

(-) ^P'

(=0

A = ^A

2 i

A = A2-c2,

,'((2 - c2) + ((2 - c 2 ) 2 + 4(c 2 - 1)(1 + c . ) ) 1 / ^

V2

,

64

Halo Orbits. Analytic and Numerical

Study

and c„ are constants given by c„

=

7-

3

[(±)»/,+ ( - l ) n ( l - A ) 7 » + 1 ( l T 7 ) - ' - i ] I

with upper sign for L\ and lower sign for L2 and

for the L3 case. We set 7 = n for Li, L 2 and 7 = r 2 for L3. Computation of n, r 2 : Case Lj: n is the positive root of r\ - (3 - ji)r? + (3 - 2/u)r3 - nr\ + 2y,rx -fj, = 0, that can be found using Newton's method starting at n = (fi - 3) 1//3 . Case L2: r\ is the positive root of r\ + (3 - fi)r$ + (3 - 2,u)r3 - firf - 2fin - [i = 0, that can be found using Newton's method starting at n = (fj, — 3) 1 / 3 . Case L3: r 2 is the positive root of r\ + (2 + M H + (1 + 2fx)r32 - (1 - /i)r| - 2(1 - fi)r2 - (1 - /1) = 0, that can be found using Newton's method starting at r 2 = 1 — 7fi/12. The expansions of the right-hand side terms require the expressions p

n

n

^

(x\ Jp)

_ _

1 2" on

^ £-*> 2^

(2n-2si-2s2-2s3)! (n-2si -2s2 -2s3)!

0<S1+S2+S3<[f] Sl,S2,«2 > 0 f _ 1 \ S l + S 2 + S3

I

v

I

L

l

„n-2s2-2s3„2s2 -2s3 M

I

1

l

X

»

'

(n - S i - s 2 - S3)!si!s2!s3'

VP/

i=0

Z=0

(n - 21 - 2 - 2si - 2s 2 - 2s 3 )! (n - / - 1 - si - s2 - s 3 )!(si + s 2 + s 3 - l)\

£ 0<Sl+S2+S3<[f] Sl,S2,S2 > 0

(*1 + *2 +

S3

)!(-1)S1+S2+S3-'

(n - i - 2si - 2s 2 - 2s 3 )!si!s 2 !s 3 !

n

-l-2.a-2M..2.a.2„

Analytic Determination

65

of the Families of Halo Orbits

Therefore ?w _

M

n + 1

V

V*

~ 1^^TC"

(2n-2si - 2 s 2 -2s3)!

1^

"^2

(„ _ 2 s i - 2s2 - 2s3)!

0<S1+S2+S3<[f] Sl,S2,S2 > 0 C_l'\Sl+*2 + 83 x

^

/

(n - si - s 2 - s 3 ) ! s i ! s 2 ! s 3 f ,—...

^

l

n>2

1=0

^Jl+4/-2n 2T(n-2l-2-2s1 -2s2 -2s3)\ (n - I - 1 - s\ - s 2 - s 3 )!(si + s2 + s3 - l)\

£*

x 0
~n-2s2-2s3,.2s2r2s3

2

+ as<[f]

Si, S2, S2 > 0

(«1 + *2 + S3)K-l)« + *>+»-1 (n-/-2si-2s2-2s3)!Sl!s2!s3! V-

V^ 1 + 41 - 2n

\>2

1=0

n-j-2,,-2,

2,a + 1

a,,

y

2*

(n - 2/ - 2 - 2si - 2s 2 - 2s 3 )! ( n - / - 1 - s i - s2 -s3y.(si + s2 +s3

E

-l)\

0<Sl+S2+S3<[f] Sl,S2,S2

(Si + *2 +

x

> 0 83

)!(-l)'i+"+'»-'

!.,,.,

8

2 | | + 1

y

(n-i-2si-2s2-2s3)!si!s2!s3!

Taking into account the symmetries of the problem and setting a = Ax, (3 = Az, 7 = exp(jri) and 7' = i\f, we look for a solution of the type

x= J2

a

iJk<*iPjlk, r ^ J

6«*a'^7*, * = E

i,j>0

i,j>0

i,j>0

k ez

kez

kez

M>0 i+j>2

i,j>0 i+j>2

with aook = 0,ajjfc = ciijt-k, a yi = -l^ii^oj, book = 0,6yfc = -bijt-k, 6101 = - | ,

Vfc, i + J > \k\,j = 0(2),i - fc = 0(2), Vfc, » + j > |fc|,3 = 0(2), i - fc = 0(2),

c

^aW,

66

Halo Orbits. Analytic and Numerical

Cook = 0,Cijk

= Cijt-k,

Study

Vfc,

Ciji - \8oihj, rfy = 0 , fij = 0 ,

i+j> \k\,j = 1(2),» - k = 1(2), if i or j is odd, if i or j is odd,

where 6ij means the Kronecker symbol. For n — i + j = 1 the solution is

where k~ = 2A/(A2 + 1 - c 2 ). Let us see a method for obtaining the coefficients a's, b's, c's, d's, / ' s . Let us suppose that we have found all the coefficients up to i + j < n — 1, |fc| < n - 1 , for a given n > 2. By substitution in M, N, P we obtain the right-hand side up to order n (i + j < n, \k\ < n). The terms of (2.1) in a* ft"/, i+ j = n give 2A6yi + di.itj(X2

- Xk)

=

Mij! A2 i i + i 2 = i - l , Jl+J2=j,

-

2A

(ti.ji) #(0,0) (i2,J2)#(0,0)

2^

rf«iii^»2i2,i

*1 + *2 = *, ( * l , J l ) / ( 0 , 0 ) Jl + J 2 = j , (l2,j2) # (1,0)

= The terms of (2.2) in alP^,

Myi.

(2.4)

i + j = n give

(-A 2 + c2 - l ) 6 y i + di_1):,(A2fc - A) = Niji (iiji)#(0,0)

i 2 + i 3 = i!,

(t2lja)#(0,0)

J 2 + J 3 = j l , (*3,J3) # (0,0)

+2A

2_, (
d

hjibi-h,j-h,l=

N

ijl-

(2-5)

( * i . J i ) 7^ ( * ' - l , j )

From (2.4) and (2.5) we get 6

= ijl

Mjji(X2k - A) - J y ( A 2 - Xk) _ 2A(A2fc - A) + (A2 - c2 + 1) (A2 - A*)' (2.6) 2

d

= i-lj

'

~

2XNin + (A - c2 + l)M<ji 2A(A2fc-A) + ( A 2 - c 2 + l)(A 2 -Afc)

Analytic Determination

The terms of (2.3) in a'^j, X2( —^~ \ 2 « i , j - i +

67

of the Families of Halo Orbits

i+ j = n give \

2_^ ",iih<*i2J2J ii + * 2 = i, ( » i , J i ) 7^ ( 0 , 0 ) Jl+J2=j-1, (t2,j2)^(0,0)

=

**ji

+•

fij-l o

'

and therefore fij-i

=

—

£

2 P y i — 2A dj,j_i — A

dj2j2.

ii + 1 2 == ». & .J'i)#(0,0) Ji + h : = J -- i , (f8 , J 2 ) # ( 0 , 0 ) For given fc 7^ 1 t h e t e r m s of (2.1), (2.2) in alfijyk - ( A 2 A;2 + 1 + 2c2)aijk + A K

y ^

+ 2Xkbijk

Mijk

\2urs-f-

/

(r,.)*(0,0)

-2AA;

=

2J

j

give

di1j1di2J2

y

1 + i 2 = = : r !

(tl.j'l) / (0,0)

jl+J2=S,

(l2,j2)/(0,0)

d r s 6j_ r j_ s , f c = M y t ,

I &i—r,j — s,k

(2.7)

(r,«)#(0,0)

- (X2k2 - e 2 + l)6ijfc = iVijjt

-Xkaijk •"

/

\ ^"rs 1

y

(r,.)#(o,o)

-2AA;

2J drSai-r,j-s,k (r,s)*(0,0)

/

"'iiji(^i2J2

y

«i+ta=r,

(ii,ji)^(0,0)

J1+J2=S,

(»2,j2)/(0,0)

= Nijk-

(2.8)

From (2.7) and (2.8) we get My* (A2 A2 - c2 + 1) + JVyt2Afc &tifc

—

2 2

(X k

+ 1 + 2c2)(A2fc2 - c2 + 1) - 4A2fc2'

Nijk(X2k ijk

~

2

+ 1 + 2c2) +

2

2

Cijk —

Mijk2Xk

(A fc + l + 2c 2 )(A fc -c 2 + l ) - 4 A 2 f c 2 '

In a similar way from the terms in a*^jk 1 k \*ijk + y^ffcl __ -<\ ] \ \"iJ +

/ , 2-j

J ®i—r,j — s,k

2

in (2.3) we get

i-r,jJrsri-r,j-s,k

68

Halo Orbits. Analytic and Numerical

+ A K

/ J \ £dTS "+" (r,s)^(0,0) i\

£ /

+ i2 == r

h + h--= s

Study

di1j1di2j2 "iiji"J2J2 I ci—r,j — s,k

J

(n, 3i)* (*2,

(0,0) j'2)/(0,0)

P iijfc

A 2 (fc 2 -1)' 2.2.3

Description alytic Halo

of the Program for the Computation Orbits

of the An-

A program for the analytic construction of halo orbits in the RTBP has been implemented under the name ANACOM. The program computes the coefficients a^k, hjk and Cijk (see 2.2) of the periodic solution up to a given order. We explain in this section a compact form of the program. We give the most important ideas used in the principal routines. 2.2.3.1

Storage of the Elements

The series involved in the analytic construction have the following form F

=

a

ijkaiPJ7k,

£

aijk £ R,

(i,j,k)€l

where I = {(i, j,k) £ Z x Z x Z | i,j > 0, (i,j,k) ^ (0,0,0),i + j > \k\} . We consider four types of series (a), (6), (c) and (d) which have a pair of indices (isi, is2) associated to them in the following way (a) (6) (c) (d)

(isi, is2) (isuis2) (isi, is2) (isi, is2)

= = =

(1,0), (-1,0), (1,1), (-1,1).

We say that F is a type (isi, is2) series if it satisfies the following property P: a^k = 0 if the condition j = is2(mod2),

i — k = is2(mod2),

is not satisfied and Prototypes of the (a), (b) and (c) types are x, y and z respectively. The following table holds for the multiplications concerning type (a) (b) (c) (d)

(a) (a) (b) (c) (d)

(b) (b) (a) (d) (c)

(c) (c) (d) (a) (6)

(d) (d) (c) (6) (a)

Analytic Determination

69

of the Families of Halo Orbits

Let

(i,j,k)eln

where In

=

{(i,j,k)eZxZxZ\i,j>0,(i,j,k)^(0,0,0),0
+

j
be a finite sum up to order n (f.s.n). We say that Fn is of one of the types above if its elements a^k verify the property P for some pair of indices (isi, IS2). Let Fn be an f.s.n of type (a) or (6). We order the elements of Fn, a^k, according to the indices (i,j,k) as follows (1,0,1), (2,0,0), (0,2,0), (2,0,2), (0,2,2), (3,0,1), (1,2,1), (3,0,3), (1,2,3), (4,0,0), (2,2,0), (0,4,0), (4,0,2), (2,2,2), (0,4,2), (4,0,4), (2,2,4), ( 0 , 4 , 4 ) , . . . . The order can be represented as in Figure 2.3.

^ s\ s \ •

\ \ \ , N\ <^

^

1

1

0,2 1,3 1,3,5. 0,2,4 Fig. 2.3

possible values of k on one diagonal

Order for the elements of Fn for a series of type (a) or (b).

We note that using the property P , the element a^,-*;, with k > 0 can be obtained from a^k if the type of the series is known. Therefore, we only need to keep the terms with k > 0. This fact is especially taken into account in the subroutine PROD which multiplies two series. It is easy to see that the element (i,j, k) is in the place Wijk

m 3 + 3m 2 + (2 + 3k)m + 6j + k 12

where m = i + j and

-{

6k if m is even, 3fc if m is odd.

70

Halo Orbits. Analytic

and Numerical

Study

The last element of order n is in the place „2, rt"+6n''+14n 12

if n is even,

n"+6n* + lln12

if n is odd.

Wr,

For a given order n, the number wn is a bound of the memory used. We note that an increment in j of two units increases W^k in one unit, and an increase in k of two units increases Wijk in [(m + 2)/2] units. If Fn is an f.s.n of type (c) or (d) we order its elements in the following way: (0,1,1), (1,1,0), (1,1,2), (2,1,1), (0,3,1), (2,1,3), (0,3,3), (3,1,0), (1,3,0), (3,1,2), (1,3,2), (3,1,4), (1,3,4), (4,1,1), (2,3,1), (0,5,1), (4,1,3), (2,3,3), (0,5,3), (4,1,5), (2,315), (0,5,5),.... In this case we have Figure 2.4.

^

5

\ \

> _ ^ ^-^. -A_i

*v^M<S'vJ> 0,2 1,3

1

1,3,5... 0,2,4

Fig. 2.4

possible values of k on one diagonal

Order for the elements of Fn for a series of type (c) or (d).

The element (i, j , k) is in the place

wijk = I(2m (2m

3 3

+ 3m 2 + m(6k - 2) + 12j + 12)/24 + 3m 2 + m{Qk -8) + 6k + 12j + 3)/24

if m = i + j is even, if m = i + j is odd.

An increment in j of two units increases W^k in one unit and an increase in k of two units increases Wijk in [(m + l)/2] units. The bound of the necessary memory for this kind of series is given by the place wn of the last element of order n u>n =

2n 3 + 9n 2 + 10n + 3" 24

The place where some element a ^ is stored is computed by the function INDE.

Analytic Determination of the Families of Halo Orbits

n 1

x,v

./'— x y

M

,xz

v:

Az24-

ix- , 21

•• / x

!

, 2 ix z

y

•j

2

3 y

iv ( x z Jv --- „. J y z xy

<

V* 1 1

li 2

-'

3 2 x y 3 2 x z

1/1

if i x z !,'x

4 2 x y 4 2 x 2 2 4 * y 2 2 2 x2y z ,) x 24

*, ',

z

(xz

/

.

1

4

<=

\|x y ' 2 3

\! 2 2 «x y z

* y 2 2 x yz

i 2 3 I x z

3 2 z

' 4 !y z i 2 3 •y z

Xy

s xy 2 z 2

2

V

1--Vk

\\

2

(xyz

2

x3z \ xy

I

2 2 2 2 y -Z \ 4-

--...'•

jxy

2 2* x y

)K

3

2

v

u

\—•-. i y z J

ix

a

y

-• -

"•S! Ix 3y 3 x

3

yz

2

,'x5z

3

^ 2

x y z 3 3 x z 4

j xy

2 2 3 x y 7xy

i 3 2 I xy z /

li x y z 4/ /

ixz

/ / 4 2 y 2 2 A

Y z 6^Table 2.1 Monomials appearing in M, N and P and its generation.

72

2.2.3.2

Halo Orbits. Analytic

and Numerical

Study

The Series M, N and P

The series M, N and P (see equations (2.1), (2.2) and (2.3) of 2.2) are computed using a previous reduction to the form M

= (a,b,c)el

N

=

E /aW*C, (a,b,c)el

P

=

E /aW*C(a,b,c)6l

The computation of fj£c, f™bc and f%bc is done in the subroutine FMNP taking into account that /™ f c + l c = fa,b,c+i- & function FAC is used in FMNP for the computation of the factorial. In FMNP some numerical coefficients cn are also used. They depend on /J through the location of the related collinear point. This is computed in the subroutine GAM. From the definition of M, N and P it is obvious that they are of types (a), (b) and (c) respectively. Also from the definition we note that in the computation of f£lc and /" 6 c every factor y2 introduces a negative sign through y/^1. It is easy to see that the element f^c is stored in the place 24

Wabc

=

+ n

if n — a + b + c is even,

< n3+6„2

+ lln-18

+

ifn

n

=

a +

6+

c i S odd,

where (n - a) 2 + 2(n - a) n

=

+

c 2

+ L

The elements /™6c and f^bc are stored in the places number WQ,&_i,c + 1 and Wa,b,c-i + 1 respectively. Table 2.1 gives the monomials which appear in M, N and P for some orders n — a + b + c. Therefore the number of elements (monomials) of order n, involved in the computation of M, N and P is

=

3n 2 + 10n+8

+ n

3n +12^+9

+

j

fn

ig

eye^

< ^

i f n i s o d d -

We realize that all the monomials of order n can be obtained from the monomials of order n - 1 by multiplication by x, y or z. This is also sketched in Table 2.1 where the solid line, the line-dot-line and the dotted line mean, multiplication by x, y and z,respectively. This is done in subroutine MNP where only the terms of

Analytic

Determination

of the Families of Halo Orbits

73

order n are computed, because the terms of order n — 1 are already known. Note that in this case we should store all the terms up to order n - 1 for every one of the previously computed monomials. A different approach consists in computing the terms of lower order every time they are required. This produces an increase of computation time but a decrease in the amount of the storage. It is the second approach that has been implemented. The reason was the extra time added in the access to mass storage due to the limitations in our system. 2.2.3.3

Addition of Series

In some stages of the computations it is necessary to add to a series F l another series like coef x F2, where coef is some given coefficient. Of course the addition is only done for series of the same type. The subroutine SUM performs the addition. In fact it only adds the terms of order n. For a fixed n, first the term with j = is2, k = n, i = n — is2 is computed. Then k is decreased in two units successively and so the index is decreased in [(m+2—is2)/2] units where m = i + j . When k reaches the value 0 or 1 according to whether n is even or odd, we reset k = n, increase j in two units and the index is the initial one plus 1. 2.2.3.4

Multiplication of Series

The subroutine PROD multiplies two series F l by F2 and puts the result in F 3 , taking into account the type of each series.

Let Fl =

J2

"iiiitia^V1,

(«iji*i)€/

be of ( i s l i , i s l 2 ) type,

be of (is2i,is22)

type and

F3 =

J2

"Ws^'V8-

The elements of F 3 are computed in the same order that they are stored, up to a fixed order n = n^ = i% + jz. So the key of this subroutine is the computation of one term Wi3j3k3 of F 3 . We assume that the indices 13, j'3 and £3 are fixed. For every value of ni = i\ + j \ going from 1 to n — 1, the subroutine computes all the possible factors which contribute to F3 with a term of indices (13,^3,^3).

74

Halo Orbits. Analytic and Numerical

Study

(a) or (b) (h,ji) *i (8,0) (s-2,2) (s-4,4) ni

(*2,J2)

fci

(s-1,1) (s-3,3) (s-5,5)

"i

(ni,s) (s,0) (s-2,2) (s-4,4)

(1 -ni,s (8-1,1) (s-3,3) (s-5,5)

(c) or (d)

nx + 2

l+ni)

(ni,s)

(1 - ni,s - l + ni)

(8,0) (8-2,2)

(8-1,1)

(s-4,4)

(a--3,3) (S-5,5)

s

s

(1 - n i , s - 1 + n x )

(ni,s) Table 2.2

ni + 2

For a fixed value of n\ = s, factors which contribute to F 3 .

These factors are represented in Table 2.2 where n\ — 0 if n\ is even and ni = 1 if ni is odd. For a fixed value of n\ the terms Ui1j1k1Vi2j2k2 are computed moving hi from n\ to n\ with a step of two units. In this way the elements u^^^ are considered in the same order that they are stored. The subroutine computes the products corresponding to k\ and —fciat the same time, except for the case k\ = 0, because there are no elements with —ki. We remark that not all the values of fci in the range above are allowed. In the same way for a given k\ not all the values of i\ will produce a term Wi3j3k3 because «2, h > 0. Then the following constraints on fci and i\ are imposed k\f < h < ku, where ku = min(A;3 + n 3 — n i , n i ) and k\f = max(fc3 + n\ — n 3 , - n i ) and hf < h < hi, where

{

i3

if 23 and ni + isl2

have the same parity, i3 - 1 in other cases,

Analytic Determination of the Families of Halo Orbits

75

and

{

max(ni - J3,0)

if max(ni - J3,0) and i%i + i s l 2

have the same parity, max(ni — j'3,0) + 1 in other cases. Therefore the terms with ii > in must be skipped. This implies an increase of ni -is\2 +hi 2 ' units of the place (index LI) where u^j^ is stored. If in < i\f, then any ii is allowed and we increase n\. Following the order of the storage, if we increase n\ in one unit, the index LI increases in ni+2 "ni + 2 — isl.2 2 2 units. In this way the function INDE is never used to compute the index LI. The index L2 for v^j2k2 is computed by INDE for the first allowed element corresponding to a fixed value of k%, that is for (12,72,^2) = (*3 — *ii,^2 — h + in,kz — k\). For the other elements with the same value of k2, L2 decreases or increases in two units depending on the sign of fc2An additional subroutine, PRODT, produces the terms up to order n of a series F 3 = Fl x F2 when F l and F2 are known up to order n - 1. We recall that the lowest order in a series (with respect to the amplitudes a, /?) is one. As it has been mentioned in 2.2.3.2, this is only needed to decrease the amount of required storage. 2.2.3.5

Computation of the Amount of Elements Used in the Program

Table 2.3 gives the number of terms involved in x, y, z, and the total amount depending on the order. We note that in x or z, the only term in 7 is 07 or $7, respectively. This produces a reduction in the number of (a priori) terms different from zero in x, y. Furthermore, there are no terms without 7 in y.

Item

Dimensions for n Even

Maximum number of terms in a series x, y or z Number of monomials in x, y and z in M up to degree n Number of monomials in x, y and z in M or P up to degree n Number of terms in to or A Auxiliary dimension Total number of monomials in M , N and P of degree n and n — 1 Total number of monomials in M, N and P up to degree n Total number of terms accumulated up to order n

jw-i _ n J + 6 n +14n

1V1 1

~~

12

JT/o = " + 9 " +26n

M 3 = "J+«2«-+8"

^ g -

Odd \fi

1V1 L

~~

jifo _ ^ ~

Jva

lln-6

12

n3+9n2+23n-9 24

JM-Q _ n"+6n-' + l l n + 6

M4 = ^ - ± ^ where m = [a±^ M5= [2±1 3n*+8n+4 Af6 = 3 "*+ 8fl + 5

Ml

—

SrS+lOn+lT

\f£

— n a +5n : i +8ra+8

3n'z + 12n+9

Ml-

M8

Dimensions related with the maximum order n: LTER(n), Table 2.3

— n"+6rS +

_ ~

na+5n'+7n+7 4

ITAM(n),

(routin

Summary of dimensions to be used in program ANACO

Order # terms in x # terms in y # terms in z # terms in w of order n — 1 # terms in A of order n—1 # total # total accumulated up to n

2 4 2 2

4 9 6 6

8 13

12 25

5 6 9 6 3

6 16 12 12

3

2

1 5 5

3 2 4 2 2

21 46

Table 2.4

27 73

7 12 16 12 4

8 25 20 20

4 40 113

48 161

9 20 25 20 5

10 36 30 30

5 65 226

75 301

96 397

Number of terms at different orders.

78

Halo Orbits. Analytic and Numerical

order time (s) Table 2.5

2.2.4

Results

5 3

6 7

7 17

8 40

9 86

10 179

Study

11 337

15 2100*

Number of terms at different orders. (*) under VAX 11/785.

of the Analytic

Computations

Table 2.6 gives the values of the coefficients for the Sun-Barycenter L\ up to order 15 obtained by ANACOM. The computing time as a function of the order, under VAX 11/780 (see however the last paragraph in the explanation of the computation of M, N, P) is given in Table 2.5. For small values of/i the coefficients for the cases L\, Li are coef(Hill) +0(/i 1 / 3 ) and all coef(Hill) are different from zero. Concerning L% the coefficients have a limit when /J -> 0. However L3 is meaningless for fi — 0 (in fact the program is unable to compute the L$ case for /x = 0 because of some zero divisions). The values for fi small differ from the limiting value in an amount of order at most u. The limiting values are rational (for instance: 01,0,1 = —1/2, 02,0,0 = 1/2, ^0,2,0 = —1/4, ao,o,2 = 1/8, a3,0,3 = 3/16, 01,2,3 = - 1 / 1 6 , a4,0,o = -23/64, a2,2,o = 23/32, a0,4,o = 1/16, 04,0,2 = 17/48, o2,2,2 = - 7 / 3 2 , o0,4,2 = 1/32, o4,0,4 = -67/384, 02.2.4 = 3/64, 05,0,3 = -53/128, 03,2,3 = 21/64, 01,4,3 = - 7 / 1 2 8 , a5,o,5 = 17/96, 03.2.5 = —1/24, etc.) and some of them are zero (for instance: fij, a\j, for all i,j, i + j > 0, Co,3,3, 00,4,4, 6o,4,4, C3,i,o, C i , 3 , 0 , 01,4,5, 61,4,5, Co,5,3, C2,3,5, Q),5,5, ^0,6,4, 02,4,6, Oo,6,6, 6o,6,4, 62,4,6, 60,6,6, C 1,5,4, c 3,3,6, Cl,5,6, • • • ) •

Looking for the first element of the halo family, i.e. the Lyapunov periodic orbit around L3 with critical vertical stability, we should impose A = A(a,0) = /2,0a 2 + /4,0a 4 + /6,0a 6 + . . . . The values A/u, fijfl/n have a finite limit when H —> 0. Figure 2.5 gives the plot of

'Vi for k = 1,2,3,4 and A/u illustrating the lack of convergence of the series for the values of the independent variable where the possible value of a should be found to obtain A(a,0) = A. For a as large as 0.5 if k = 1,2,3,4 we get a = 0.3360, no solution, 0.4262 and 0.4130, respectively. For /x = 0.5 there is a slow convergence towards the right solution. This worsens when u decreases. For the Earth-Moon case the numerically obtained orbit has a point at x = 0.3096, y = z = 0. The distance at L3 is larger than the one at m i , showing that the theory is no longer applicable. The output of the program ANACOM can be stored on a file (see Table 2.6). Then a program (TESTAN) has been implemented to test the analytic expansions. First the file of results is read. Then the program asks for a value of the amplitude of z. From this value and the relation A = A(a,/3), the value of a is obtained. We note

Analytic

Determination

79

of the Families of Halo Orbits

o_ a

Fig. 2.5

Behavior of the series (l//i) ! C , = i f2j,oa2-', for different values of k.

that this value of a is critical, in the sense that most of the errors in the analytic orbit are introduced at this step, because the series giving A is slowly convergent. After writing the value of a a new a value can be introduced, if desired, for the computation of the Fourier coefficients of x, y and z. They are produced in the form

x = J2 Aj cos(jt/T),

y = ^2Bj sin(ji/T),

z = £ Cj cos jt/T),

where T is the period. Finally the program produces initial conditions x, y, z in both normalized coordinates (the ones used in the analytic theory of the halo orbits) and adimensional coordinates (the usual ones in the RTBP).

80

Halo Orbits. Analytic CASE LI LAMBDA = DELTA =

ORDER = 15 0.2086453455276052D+01 0.2922144542546494D+00

and Numerical

Study

MU = 0.3040357143000000E-05 GAMMA = 0.1001090475489518D-01 TERMS = 1153

( 1 ( 0 ( 0

0 1 0

1) -0.5000000000000000D+00 1) 0.5000000000000000D+00 0) 0.OOOOOOOOOOOOOOOOD+00

B( 1 D( 0

0 0

1)=-0.1614634048091902D+01 0)= 0.1000000000000000D+01

( ( ( (

2 2 2 1

0 0 0 1

0) 0.2092695580695451D+01 2) -0.4529823976983048D+00 2) 0.2462229375506568D+00 0) 0.1040596381457606D+01

A( A( B( C(

0 0 0 1

2 2 2 1

0)= 0.2482976702637035D+00 2)= 0.5223205820371600D-01 2)= 0.3037323358663149D-01 2)=-0.1734327302429343D+00

( ( ( ( ( (

3 3 3 2 2 2

0 0 0 1 0 0

3) 0.3969100975569138D+00 1) 0.1422540573647265D+01 3) -0.4428503881079669D+00 3) 0.1990477125836558D+00 0) -0.8246605234726857D+00 0) 0.1596559878224751D+02

A( B( B( C( D( F(

1 1 1 0 0 0

2 2 2 3 2 2

3)=-0.4134269264659037D-01 1)= 0.2158462883436975D+00 3)= 0.1150991368886276D-01 3)=-0.9521935026325322D-02 0)= 0.1210986086886631D+00 0)=-0.1740900545935829D+01

( ( ( ( ( ( ( ( ( ( (

4 0 2 4 0 2 4 0 1 1 1

0 4 2 0 4 2 0 4 3 3 3

0) 0) 2) 4) 4) 2) 4) 4) 0) 2) 4)

-0.4013683102602195D+01 0.2066050137329525D-03 0.3170484851704876D+00 -0.5950722377459613D+00 -0.2611206967035164D-02 -0.3112867482350429D+00 0.5373563863006729D+00 -0.9570234679330889D-03 -0.3628883717797557D+00 0.3808264149657870D-01 0.1665836365677682D-01

A( 2 A( 4 A( 0

2 0 4

0)= 0.7367284964230373D+00 2)= 0.8702523463634248D+00 2)=-0.1516702481261075D-01 4)= 0.8583525454169250D-01 2)=-0.9602066054440655D+00 2)=-0.1411620147750674D-01 4)=-0.2119382173259504D-01 0)= 0.5579707364350662D+01 2)=-0.5852617844163607D+00 4) =-0.2393891847888100D+00

( ( ( ( ( ( ( ( ( ( ( ( ( (

5 1 3 5 1 3 5 1 2 4 0 2 4 0

0 4 2 0 4 2 0 4 3 1 5 2 0 4

3) 3) 5) 1) 1) 3) 5) 5) 3) 5) 5) 0) 0) 0)

-0.1288359895027210D+01 0.2840005065521092D-02 -0.1392591533750437D+00 -0.1933414620390834D+01 0.5127782166626836D+00 -0.2233628709658197D+00 -0.8169824737276291D+00 -0.4202707340991322D-03 -0.6948842442756333D-01 0.3421806625433890D+00 0.6910570379278959D-03 0.1249884256559071D+00 -0.4193325174140423D+02 -0.9694037731575121D+00

( 6 ( 2 ( 6

0 4 0

0) 0.2710980997194095D+01 0) -0.7924857240578073D+00 2) -0.5129057343505673D+00

A( B( B( B( C( C( C(

2 2 4 0 0 2 3 3 3

A( A( A( B( B( B( B( C( C( C(

3 2 3)=- 0.9558282479578856D-01 5 0 5) = 0.8436465974865815D+00 1 4 5) = 0.5518323044841954D-02 3 1)=- 0.3162370152105307D+01 5 3) = 0.1551690740561202D+01 1 3) = 0.8893678366419844D-02 3 5) = 0.6840622513706198D-01 3) = 0.5015859248901828D+00 4 3) = 0.1053601245507077D-02 0 5)=- 0.3558655401101440D-01 2

D( 4 D( 0 F( 2

0 4 2

0)= 0.6284608999650463D-01 0) = 0.7362838875346794D-01 0) = 0.6073575647603767D+01

A( 4 A( 0 A( 4

2 6 2

0)= 0.1004675896141445D+02 0)=-0.2132645394541060D-01 2)=-0.2669695101042291D+01

Analytic

Determination

of the Families of Halo Orbits

81

A[ 2 AC 6 A( 2 A( 6 A[ 2 B: 6 B[ 2 BC 6 B: 2 B: 6 B: 2 C: 5 C i C: 3 C: 5 C' i C 3

4 0 4 0 4 0 4 0 4 0 4 1 5 3 1 5 3

2) 4) 4) 6) 6) 2) 2) 4) 4) 6) 6) 0) 0) 2) 4) 4) 6)

0.4305325034344030D+00 0.2393231791342900D+01 -0.6876911459890523D-02 -0.1316113331609642D+01 -0.1502895391895493D-01 0.8904712899101077D+00 -0.2441093298597426D+00 -0.2457065108246097D+01 -0.3169362927122516D-01 0.1252252243254906D+01 0.2384605932874955D-02 0.8343096552668855D+01 -0.5989080840442173D+00 -0.1522072224333391D+01 -0.4052889381245967D+00 0.1614904672852111D-02 0.6809153696339198D-01

A( 0 A( 4 A( 0 AC 4 AC 0 B{ 4 B: o BC 4 BI 0 B: 4 B: o C: 3 C' 5 C: i C: 3 C 5 C i

6 2 6 2 6 2 6 2 6 2 6 3 1 5 3 1 5

2) 4) 4) 6) 6) 2) 2) 4) 4) 6) 6) 0) 2) 2) 4) 6) 6)

-0.9895078333592311D-02 -0.1588820032471286D-01 0.5259147259391397D-05 0.2588683014479830D+00 0.2254906872733710D-03 0.2514203853840293D+01 -0.6766143190158090D-02 0.4392923193453057D+00 0.2112921974996023D-03 -0.1417302684528179D+00 0.6692006656181552D-04 0.3271374060292969D+01 0.3799310551999121D+01 0.1430289205956307D+00 0.5727885018777525D-01 -0.5123487868335589D+00 -0.2129902629006471D-02

A' A A A A A B C B B( B B B( B( C( C( C C( C( C( D( D( F( F(

3 7 3 7 3 7 3 7 3 7 3 7 3 6 2 6 2 6 2 6 2 6 2

0 4 0 4 0 4 0 4 0 4 0 4 0 4 1 5 1 5 1 5 0 4 0 4

3) 3) 5) 5) 7) 7) 1) 1) 3) 3) 5) 5) 7) 7) 3) 3) 5) 5) 7) 7) 0) 0) 0) 0)

0.1188612898048618D+01 -0.9417722474955654D+00 -0.4277063682628160D+01 -0.2164096943635063D-01 0.2102124841038076D+01 0.3406353780377439D-01 -0.3335185787190698D+01 0.3077770924814405D+01 -0.1438145007988841D+01 0.4911175407297458D+00 0.4375518433614990D+01 0.6251194562856077D-01 -0.2024272232172943D+01 -0.1076236749118598D-01 -0.4166166205901074D+01 -0.2526291459936012D+00 0.2504881878192462D+00 -0.3581170470918928D-02 0.8097619214566230D+00 0.6263414608900488D-02 0.2777792852769958D+01 0.1653832177807962D+01 -0.1828312241452823D+03 -0.1954553879455016D+02

A' A A A A A B B B( B( B B B( B( C( C( C< C( C( C( D( D( F( F(

1 5 1 5 1 5 1 5 1 5 1 5 1 4 0 4 0 4 0 4 0 4 0

2 6 2 6 2 6 2 6 2 6 2 6 2 6 3 7 3 7 3 7 2 6 2 6

3) 3) 5) 5) 7) 7) 1) 1) 3) 3) 5) 5) 7) 7) 3) 3) 5) 5) 7) 7) 0) 0) 0) 0)

0.4928551189857202D+01 0.3719597148500192D-01 0.3730333436675225D+00 0.1416680142346882D-02 -0.4727951909632446D+00 -0.7749646088636261D-03 -0.8565313754823189D+01 0.3869843504815316D-01 -0.4423875027941504D+01 0.3394955143497154D-02 -0.8176686455662081D+00 -0.8315602597988779D-03 0.2978093077000148D+00 0.5732214316586692D-05 0.2326297409060391D+01 0.2216920897476010D-02 -0.4988330745680017D-01 0.9803005157590981D-04 -0.1330947590693516D+00 -0.6824400831267295D-04 -0.8755884366341286D+01 0.1282930827333395D-01 0.1651995555280459D+03 -0.2162904494665954D+00

A( A( A( A( A( A( A(

8 4 0 6 2 8 4

0 4 8 2 6 0 4

0) 0) 0) 2) 2) 4) 4)

0.1695385661898966D+02 0.1379976568898341D+01 -0.1130650185542458D-01 -0.1074992381762671D+02 -0.2554316630931724D+00 -0.2721796914826760D+01 0.1938104251295075D+01

A( A( A( A( A( A( A(

6 2 8 4 0 6 2

2 6 0 4 8 2 6

0) 0) 2) 2) 2) 4) 4)

0.8580655208286363D+01 -0.5354253432952137D+00 -0.4762508000790642D+01 0.4734700896408471D+01 0.2276213552147612D-03 -0.9154171659410125D+01 -0.1034947085963011D+00

7

5

Halo Orbits. Analytic and Numerical Study

82

8 2

8 2

0 4 8 3

8 8 8 0

0.6532994710860325D-03 -0.1122922992794590D+01 -0.4487156149244651D-02 -0.3472691333021193D-02 -0.7794031767117611D-01 -0.2394215332525664D-04 -0.1152848543909099D+02 -0.5189853244569546D+00 0.3272466281446759D+01 -0.1039667157995541D+01 0.3429938366492446D-03 0.1684540622003119D+01 0.4479145472588786D-02 0.3347357739275778D+01 0.3069912947262455D-01 -0.6285439750926862D-05 0.9985170526390104D+02 -0.3947947383619969D-02 -0.1326575561573411D+02 0.2871247305835570D-01 -0.4263233175125608D+01 -0.1411889920576044D-01 -0.2444065872702838D-01 -0.8299019104386267D-03 0.2570336972287643D+00 0.2996518863935818D-03

A 8 A 4 A 0 A 6 A 2 B 8 B 4 B>0 B: 6 B 2 B: 8 B: 4 B' 0 B 6 B 2 C 7 C 3 C 7 C 3 C 7 C 3 C 7 C 3 C 7 C 3

0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 1 5 1 5 1 5 1 5 1 5

6) 6) 6) 8) 8) 2) 2) 2) 4) 4) 6) 6) 6) 8) 8) 0) 0) 2) 2) 4) 4) 6) 6) 8) 8)

0.7909199009426922D+01 0.8338028853590628D-01 0.5955965002534692D-04 0.8864077243239778D+00 0.2636306589019560D-02 0.4233976284451412D+01 0.4378448974650873D+01 0.2347780492320902D-02 0.7500407641948444D+01 0.8911045937343276D-02 -0.7904132803725337D+01 -0.1548946924193637D+00 0.1121998447304225D-05 -0.5976977537749855D+00 -0.2778038438684634D-03 -0.9998178458584117D+02 -0.1310111929935706D+02 0.9147918686600128D+01 0.1564063824877135D+01 0.6078823074231050D+01 0.5649863210751497D+00 0.1257111528361984D+00 0.1591112952325133D-01 -0.1319245152896688D+01 -0.1582241367785139D-01

0.8068079798934447D+01 -0.3397323067388822D+01 0.1323557747304551D-01 0.1626070980305299D+02 0.2753894450069139D+00 -0.1471654301093932D+02 -0.2697971800367871D+00 -0.4180939842460980D-03 -0.1669566554496361D+01 -0.7489404479478278D-02 0.2288573164819599D+02 -0.5191674462226457D+02 -0.3724702762388579D+00 0.1197480810728607D+02 0.6386711615281698D+00 -0.7320428323018683D+01 0.2278310784051264D+01 -0.8407312556641665D-03 -0.3493438491684357D+01 -0.1415484920731468D-01 -0.5666985871214039D+01 -0.8075438030877770D-01 0.3331793736625268D-05

A 7 A> 3 A: 9 A: 5 A: i A: 7 A: 3 A: 9 A[ 5 A; i B: 7 BI 3 B: 9 B: 5 B: i B[ 7 B: 3 B[ 9 B: 5 B: i B: 7 B: 3 C: 8

2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 0 4 8 2 6 1

3) 3) 5) 5) 5) 7) 7) 9) 9) 9) 1) 1) 3) 3) 3) 5) 5) 7) 7) 7) 9) 9) 3)

0.2163259999340063D+01 0.8741478491928709D-01 0.6697902337306950D+01 -0.3965143358086419D+01 -0.4651618699562770D-02 0.2823974692523106D+01 0.1662465530268899D-01 0.5854976349387342D+01 0.1719889992961506D+00 0.1136913907131755D-03 0.5493934796665167D+02 0.1024123559294611D+02 -0.9472354425315620D+01 -0.4462239619522923D+01 -0.4969870408673344D-02 -0.1349086264892303D+02 -0.5942221712122789D-01 0.1463952228008301D+02 0.3601109912209110D+00 0.8290462144729400D-04 0.1193218273432096D+01 0.1690081721976145D-02 -0.7770826159550040D+01

Analytic Determination of the Families of Halo Orbits

83

6 2 8 4 0 6 2 8 4 0 6 2 8 4 0

3 7 1 5 9 3 7 1 5 9 2 6 0 4 8

3 )= 0.1674239758848576D+02 3 )=-0.3153427008028988D-01 5 )=-0.9275665444690048D+01 5 =-0.1195868897768637D+01 5 =-0.1749785530748982D-03 7 = 0.2308962751985966D+00 7 = 0.2833409875596201D-02 9 = 0.2206307792439324D+01 9]= 0.3826445590486699D-01 9 = 0.7769681446189371D-05 0 =-0.3920499204721087D+02 0 = 0.1251083136365141D+01 0 = 0.6054765090860736D+03 0 = 0.1247661479528764D+03 0 = 0.2636553503259624D+00

C[ 4 C: o C: 6 C 2 C 8 C 4 C 0 C 6 C 2 D 8 D 4 D 0 F 6 F 2

5 9 3 7 1 5 9 3 7 0 4 8 2 6

3) 3) 5) 5) 7) 7) 7) 9) 9) 0) 0) 0) 0) 0)

-0.2292141516142383D+01 0.8787372086235314D-03 0.7745312697205909D+01 0.4612797432356615D-01 -0.9951106538447188D+00 -0.4618254733535148D-01 -0.2740242987241492D-04 -0.4981525864392229D+00 -0.1101487554365056D-02 -0.6426049026624542D+01 0.5149867207866891D+01 -0.2626601728556573D-01 -0.2919909961955180D+03 -0.2296979737062554D+02

A( 10 A( 6 A( 2 A(10 A 6 A 2 A 10 A 6 A 2 A 10 A 6 A( 2 A( 10 A( 6 A( 2 A 10 A 6 A 2 B 10 B 6 B 2 B 10 B 6 B 2 B 10 B< 6 B 2 B :io C: 5 C: i C: 7 C: 3 C: 9 C: 5

0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 5 9 3 7 1 5

0 =-0.7129755617116606D+02 0 = 0.1970050063271232D+03

A 8 2 A 4 6 A( 0 10 A 8 2 A 4 6 A ' 0 10 A 8 2 Ak 4 6 A 0 10 A 8 2 A 4 6 A 0 10 A 8 2 A 4 6 A 0 10 A 8 2 A[ 4 6 A : o 10 B: 8 2 B: 4 6 B : o 10 B: 8 2 B 4 6 B 0 10 B 8 2 B 4 6 B , 0 10 B[ 8 2 C{ 3 7 CC 9 1 C[ 5 5 C[ 1 9 C{ 7 3 C[ 3 7

0) 0) 0) 2) 2) 2) 4) 4) 4) 6) 6) 6) 8) 8) 8) 10) 10) 10) 2) 2) 2) 4) 4) 4) 6) 6) 6) 8) 0) 2) 2) 2) 4) 4)

-0.2468771338813701D+03 -0.2737111772560102D+02 0.1218347572202061D-02 0.5656071272515059D+02 0.9405033703986818D+01 0.4734933348138903D-02 0.8752034216607006D+01 0.2807085703176711D-01 0.3591256683951602D-03 -0.2910734948853540D+02 -0.6860035665086741D+00 -0.5342765999515769D-04 -0.6529337262230226D+01 -0.5002673146036473D-01 -0.1257598559765208D-04 0.3172353650391458D+01 0.2924800738127445D-01 0.2835876839796149D-05 -0.8593346035670890D+02 -0-8101228338857177D+01 0-4601953911952184D-02 -0.2835343836188911D+02 -0.1517065071071456D+01 0.5456944198390840D-04 0.2390968512940711D+02 0.2080937318946674D+00 -0.2854579234928528D-04 0.7313916003371146D+01 -0.1457818407055739D+02 -0.6923366609846084D+02 -0.2976494822110474D+02 -0.9975165397695985D-01 -0.2183139982938703D+02 0.7930318813711787D-01

c c c c c C( C( C( C(

c D( D( F( F( F(

o: =

0.5751747472611738D+00 2] = 0.1856634818139609D+02 2 =-0.4326020751448914D+02 2 =-0.5895241801773740D+00 4 =-0.1658871184518462D+02 4 = 0.2840502312371445D+01 4 =-0.3762669472587895D-01 6 =-0.1568001261841928D+02 6 = 0.8093919959637775D+01 6 = 0.1832237644955751D-01 8 = 0.2769046052557364D+02 8 = 0.7335323051244024D+00 8 = 0.1646705063688345D-02 10 =-0.1005425975632199D+02 10 =-0.3757630001628763D+00 10 =-0.4630143172622403D-03 2 )=-0.2015231269161201D+02 2 )= 0.6423957894279708D+02 2 )=-0.1743091149374989D-02 4 )= 0.1585001173849887D+02 4 = 0.1011589860424446D+02 4 = 0.4219630544564031D-01 6 = 0.1662433507567487D+02 6 =-0.4856001897623320D+01 6 = 0.5578032757812470D-03 8 )=-0.2740518288320943D+02 0 )=-0.2147396245023263D+01 0 )= 0.4480125665299781D+00 2 )= 0.7163732399220788D+02 2 )= 0.4195352875935355D+01 4 )= 0.4918789989867344D+01 4 )= 0.3312904413912443D+01

84

Halo Orbits. Analytic and Numerical

c: c: c: c: c c c c:

i 7 3 9

9 3 7 1 5 5 i 9 7 3 3 7

A ii A' 7 A 3 A '11 A 7 A 3 A 11 A 7 A 3 A 11 A 7 A 3 A( 11 A( 7 A( 3 B( 11 B( 7 B( 3 B( 11 B( 7 B( 3 B( 11 B( 7 B< 3 B( 11 B 7 B 3 B( 11 B 7 B< 3 B( 11 B 7 B 3 C 10 C 6 B 6 B 2 B 10 B 6 B 2 C 9

0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4 8 1 5 4 8 0 4 0 1

4) 6) 6) 8) 8) 8) 10) 10)

-0.7749064973557996D-02 -0.1432547847641771D+02 -0.1376430733289433D+00 0.2927811965783194D+01 0.1306476587214277D+00 0.1999096205228858D-03 0.9651816550577877D+00 0.3397990397483984D-02

c: 9 c' 5 c' 1 c 7 c 3 c 9 c 5 c 1

3) 3) 3) 5) 5) 5) 7) 7) 7) 9) 9) 9) 11) 11) 11) 1) 1) 1) 3) 3) 3) 5) 5) 5) 7) 7) 7) 9) 9) 9) 11) 11) 11) 3) 3) 8) 8) 10) 10) 10) 0)

-0.3783945271179436D+02 0.1111256959501345D+03 0.1689801536629679D+01 0.2926936668309299D+02 0.3390677219605319D+01 0.1372268146049836D+00 0.3589239653308103D+02 -0.1639183370007276D+02 -0.6143446787299062D-01 -0.5244171844832913D+02 -0.1877891608972293D+01 -0.5984186688930447D-02 0.1751216408311896D+02 0.8102811007714873D+00 0.1562076972107717D-02 -0.3358729219198546D+02 -0.5790037326622555D+03 -0.2161127538816210D+01 0.4447296523626907D+02 -0.1090861641399769D+03 -0.2422624136757493D+00 -0.2864087487245985D+02 -0.2116292533144421D+02 -0.1360891431514417D+00 -0.3712847741887627D+02 0.1024759003653316D+02 0.5122573851407475D-02 0.5180435787293881D+02 0.1981852798339388D+01 0.2913130870590113D-02 -0.1703105124876990D+02 -0.4617610044081054D+00 -0.2625690322061074D-03 0.7494111170561709D+02 0.5160215871421180D+02 -0.8510664600967772D+00 -0.6730490343208339D-03 0.9754266634648484D+01 0.1967688889173996D+00 0.3149530356089375D-04 -0.5242587025388775D+03

A A A A A A A A< A( A( A( A( A( A( A( B( B( B( B( B( B( B( B( B( B( B( B( B( B( B( B( B( B( C( C( B( B( B( B( B( C(

9 5 1 9 5 1 9 5 1 9 5 1 9 5 1 9 5 1 9 5 1 9 5 1 9 5 1 9 5 1 9 5 1 8 4 4 0 8 4 0 7

Study

1 5 9 3 7 1 5 9

6) 6) 6) 8) 8) 10) 10) 10)

0.1468547188244113D+02 0.2566381086752193D+01 0.1687949974822940D-02 -0.7796991055789970D+00 -0.9369174513999351D-02 -0.3761636159395907D+01 -0.8870430148950659D-01 -0.4420360090932645D-04

2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 2 6 10 3 7 6 10 2 6 10 3

3) 3) 3) 5) 5) 5) 7) 7) 7) 9) 9) 9) 11) 11) 11) 1) 1) 1) 3) 3) 3) 5) 5) 5) 7) 7) 7) 9) 9) 9) 11) 11) 11) 3) 3) 8) 8) 10) 10) 10) 0)

-0.1276309676855210D+03 -0.2405853397844817D+02 -0.2406303861936567D-01 -0.4047891528745884D+02 -0.9711289124223957D+00 -0.3518069827853312D-02 0.5158963695638334D+02 0.1653147623051860D+01 0.5743763258601394D-03 0.1452102381459467D+02 0.1447770109889411D+00 0.9210023171222330D-04 -0.6057604411668585D+01 -0.5186336747335993D-01 -0.1716254954911322D-04 0.5829015658578981D+03 0.1131589208127538D+03 -0.2339562614180808D+00 0.1354472648329044D+03 0.1549218917970710D+02 -0.6095737734589860D-02 0.6223317645707080D+02 0.3269511377107820D+01 0.1814101038476934D-03 -0.4249749291205375D+02 -0.6106536360411758D+00 0.1426036601249386D-03 -0.1531785311535529D+02 -0.1223269973359190D+00 -0.7573356031488052D-05 0.4658799500159618D+01 0.1926603296435307D-01 -0.9027117942522344D-06 -0.1056788605655875D+03 -0.7932741768493653D+01 0.4361530901601613D-01 -0.1619641521916067D-05 -0.2361011058375921D+01 -0.6127746559259348D-02 0.6868850992211061D-06 0.4978238813027554D+03

Analytic C( 2 C(10 C( 6 C( 2 C(10 C( 6 C( 2 C(10 C( 6 C( 2 C(10 C( 6 C( 2 D(10 D( 6 D( 2 F(10 F( 6 F( 2

Determination

of the Families of Halo Orbits

9 3) = 0.2640682010665975D+00 1 5) = 0.2764300543677512D+01 5 5) =-0.4629754762747476D+01 9 5) = 0.2797995508945524D-01 1 7) =-0.2347822550568181D+02 5 7) =-0.5429611559642320D+01 9 7) =-0.7661950215242871D-02 1 9) =-0.7080944048984284D+01 5 9) =-0.3438092858266359D+00 9 9) =-0.8462957710180211D-03 1 11) = 0.6516523830786322D+01 5 11) = 0.2007556316299077D+00 9 11) = 0.1935218614600998D-03 0 0) =-0.6441210449609173D+01 4 0) =-0.2036681009325649D+03 8 0); =-0.1022127732532939D+01 0 0); = 0.5750413980677357D+04 4 0); = 0.4455787554520020D+04 8 0); = 0.5457402008484339D+01

C( 0 C( 8 C( 4 C( 0 C( 8 C( 4 C( 0 C( 8 C( 4 C( 0 C( 8 C( 4 C( 0 D( 8 D( 4 D( 0 F( 8 F( 4 F( 0

85

11 3)= 0.5471285880212955D-03 3 5): 0.2790034098450088D+02 7 5)' -0.2173681106142922D+00 11 5)= -0.1550334808824581D-03 3 7)- 0.2658413941629577D+02 7 7) = 0.3690990915413692D+00 11 7)= 0.1393851388929327D-04 3 9) = 0.2106315841513603D+01 7 9) = 0.2745780440412474D-01 11 9)= 0.5134056514269108D-05 3 11)= -0.1873375849047157D+01 7 11)= -0.9675683327607945D-02 11 11)= -0.9605921540905053D-06 2 0)= 0.2009793351122411D+03 6 0)= 0.4188953698698820D+02 10 0)= -0.2776973164498690D-01 2 0)= -0.1100238911857800D+05 6 0)= -0.5363553066166753D+03 10 0)= 0.3247578540599317D+00

Table 2.6 Results of ANACOM, computed to order 15, for a halo orbit around L \ , case SunBarycenter. The table has been shortened to order 11.

86

Halo Orbits. Analytic and Numerical VALUES OF THE AMPLITUDES : AZ = 7.0000000000000000E-02 A(O) A(2) A(4) A(6) A(8) B(0) B(2) B(4) B(6) B(8) C(0) C(2) C(4) C(6) C(8)

= -6.0130604562285994E-02 = 2.6368846369389016E-02 = 9.4892055048656622E-04 = 5.8192640794425940E-05 = 5.2815494790399486E-06 = O.OOOOOOOOOOOOOOOOE+OO = 1.5218477777309812E-02 = 8.7148395920941715E-04 = 5.57090647240313S1E-05 = 5.1416463968222462E-06 = -1.6929794507769324E-02 = 5.2801104585i82952E-03 = 1.9375725375404332E-04 = 1.1630071549371914E-05 = 8.7766948177589847E-07

MU = 1.2150668000000E-02 GAMMA = 0.167833150324792 OMEGA = 1.840429776847764

AX A(l) A(3) A(5) A(7) A(9) B(l) B(3) B(5) B(7) B(9) C(l) C(3) C(5) C(7) C(9)

Study

0.1861549320384853 -0.1861549320384853E+00 4.0197640597225242E-03 2.2867890990096072E-04 1.5463541141930938E-05 1.5177414964427816E-06 0.5292140899791237E+00 4.0214211529683352E-03 2.1806824746352986E-04 1.4940500800322727E-05 1.4837413638083438E-06 7.0000000000000000E-02 9.3830062779383926E-04 4.5686562969892025E-05 3.0554448513217993E-06

LAMBDA = 1.862645423166393 DELTA = 0.279024362315073 PERIOD = 3.413977205879187

INITIAL CONDITIONS (NORMALIZED) X YDOT = 1.061532118053657 Z INITIAL CONDITIONS (ADIMENSIONAL)X YDOT =-0.1781602795438952 Z

=-0.214638871238360 = 5.9543873529674861E-02 =-1.119658964382701 =-9.9934358770263779E-03

Table 2.7 Results of TESTAN. Fourier coefficients, some relevant parameters and initial conditions for a halo orbit around L2 of the Earth-Moon system.

Tables 2.7 and 2.8 and Figures 2.6 and 2.7 show several examples of TESTAN and the numerically integrated orbit obtained from the given initial conditions using a 9th order theory. For instance, the L\ halo orbit shown for the Sun-Barycenter case, with /? = 0.07 has the minimal angular distance to the Sun at the initial point. Its value is of 4 degrees. Even for this small amplitude the difference in the initial conditions when a 10th order theory is used is Ax — 0.45 km, Ay = 66 mm/s, (in the synodic system), Az = 0.26 km. For this orbit the maximal eigenvalue is 1737. This agrees with the fact that after one revolution with this initial condition, the error is small; after one revolution and one half the error is visible and the orbit is absolutely lost before two revolutions if no control is used. Tables 2.9 to 2.12 give the values of the initial conditions (in synodic adimensional coordinates) and the period for the L\ case in the Sun-Barycenter and the Li case Earth-Moon systems using a 11th order theory for the values /3 = 0.07 and P = 0.14. The differences in initial conditions and period are also given with respect to the 11th order theory when theories of order n, n = 1 0 , . . . , 3 are used. The corrections in km, mm/s and seconds appear in parentheses. As a check the

Analytic Determination

of the Families of Halo Orbits

VALUES OF THE AMPLITUDES : AZ = 7.0000000000000000E-02 A(0) A(2) A(4) A(6) A(8) B(0) B(2) B(4) B(6) B(8) C(0) C(2) C(4) C(6) C(8)

= = = = = = = = = = = = = = =

4 .1402433047156001E-02 -1 .6793309652150408E-02 -4 .1889052696504178E-04 -1..7377627937742212E-05 -1..0238568065701968E-06 0 .OOOOOOOOOOOOOOOOE+OO -9..4242522673025244E-03 -3,.8624446082072600E-04 -1..6856504145393948E-05 -1.,0059194592954273E-06 1,.1386889972066265E-02 -3,.6245275834040434E-03 -9..4799171942959011E-05 -3,.8620245592661923E-06 -1..9589653763408635E-07

' 3..040357143000000-06 .001090475489518E-02 GAMMJIL == 1, OMEGJ!L =•• 2 .053525580497667

MU

0.1861549320384853

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87

= = = = = = = = = = = = = = =

-0.1410952182640390E+00 2.0314922357062628E-03 8.1998795442213766E-05 3.8035789073553467E-06 2.4175945239856949E-07 0.4476542988341897E+00 2.3103434691554959E-03 8.0653252617624683E-05 3.7334925491417583E-06 2.3838256529322255E-07 7.OO0OO0OOO0OO0OO0E-02 5.7104944052604618E-04 1.8601548035474034E-05 8.3799549431300649E-07 4.5871203878438020E-08

LAMBDA = 2.086453455276052 DELTi!l = 0.292214454254649 PERIOD = 3.059706373687768

INITIAL CONDITIONS (NORMALIZED) X YDOT = 0.8922860384192243 Z INITIAL CONDITIONS (ADIMENSIONAL)X YDOT = - 0 . 9 3 2 5 9 0 5 4 4 7 3 7 5 9 6 E - 0 3 Z

=-0.1148061505112346 = 7.8254040150882076E-02 =-1.119658964382701 =-7.8339374263622370E-04

Table 2.8 Results of TESTAN. Fourier coefficients, some relevant parameters and initial conditions for a halo orbit around Li of the Sun-Barycenter system.

values obtained numerically are included as well as the differences (in adimensional and physical units) between the 11th and 15th order analytic theories and the numerically determined periodic orbit. We remark that program TESTAN should not be used for L3 halo orbits. In fact for the interesting values of (I(K 0.012, 0.3E-5) the analytic theory is useless for L 3 .

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a IV 00

™ CM

«

00 C O CM CM O i-H

CO i-H

o d

d

i-H

CO 00

i-H

CO CO Oi CM CO CO

i-H

IV

o d Tf d d

r-H

i

i-H

in Oi

•*

o m p p di d i

aa

CO 00 00 CM CM Oi CM

d d

1

d

Oi

a

CO

CO

CO

of the Families of Halo Orbits

CO

a t~ CM OO CM Oi CO

d

o i—i o 00 m i-H

IV CO 00 Oi

Oi IV

IV CO

co o i-H o 00 m I-H !-H

m CO CO

T—1

Oi tv

m CO CO o 00 CO CO in 00

o 00 CO CO in 00 i—i

i-H

o

o CO

CO

00

->#

Ifa

r- CO I V

O

i—(

U0 COCO

i-H

d

1

i

i-H

i-H

-*

i

CO CM CM m tv CM

i—l

CO

i

1

a 3

i

^

i-H

1

3

a

O CO in CO Oi CO r-H r-5 i-H o CO 00 i—5 d

O

a

5 0

rn

O

CM

00 Oi

o d

i-H

^

1 lf5

3

a

p d d

CO CM

CM CO

a

CO in Htf o CO CO GO CM H* Oi Tt< o •*$< in o 00 t^F o i-H o 00 Oi o CO o CM -^ IV CM IV Oi o CO i-H Oi p CM p O oi d di d d <x> d CM d i-H d 1

-<#

00 CO co ° d O0 O

CO

. CO O i-H

1

a

a

CM

in

in

M CO 00 CO CO 00 C O 00 in CM

d

a

i—i

_'

CO o ** •& m i—i l—1 IV m IV C O Tt< CD CO o

1

s/rara

i

a T* 00 i-H H*

m o O CM

a

CM U U C O CO CM "* 00 r H 00 IV IV 00

CO

m

CM CO d d co m O. m co- d CO d « i—i d o rH • i i O

Iv

999

CM

1

CO

S 2!

CO

o

o

i

d

IV CM CM CQ 00 o CM CO Oi t-5 CO *tf

CO

Determination

•* 1 1

a Oi Oi

"* CO C M CM

a **

Analytic

CO ^

CO •

o in i—i

^

O in t•

*

(•-

CO Iv CO CM CO 00

•

i-H

i—i

i-H

CO IV CO CM CO 00

-* in CO 00

o CM CO Oi

00 CO

CO Tf i—l C O i—l CO o CO O "* O *tf o

i—i

CO

1-H

CO

o

1—1 1—1

*

a

CO

1—i

CO

O

CO CO 00 00

CM 00 o CO CM CO o CM in CO CO Oi ft CO n< CO m

a C«

H

en

o

•

^

m

•

O

-tf t-~ m

OO

,E-5 81E-5 mm/s

o o

i—i

CM

00 in O iv i—i O i—i

1

o o

•5 S-i

< O

•s»

• M

< •»a

N

t--. CM i-H

Oi

c_»

i—i co M t_> ^ j t~— •* 72. CS m O Oi Iv Oi IV in O i i—l co 00 O CM O IV IV i-H iv o OCM o O

oi o o

1-H

o

-^ o

CO us 2 M OS CO •x 00 CO OI C O i—l i—1

1

1—5

i-H i—1

62952 726 7E-6

o

< H

O

e

02E-2

481 6338! E-3 2E-7

i0 mm/

70Ekm 724E-

79E-3 km 69E-2 1km 85E-2

I4E-5

719 I15E-2 7362. 19 mm/ 141490 3E-5

47E-:

22915 81E-

88E-3

5342 E-3 I1E-6

63559296 i3E-5

8.4

768 138629356 064E-4 5 mm/; 502E-4 2.1 2 mm/: 925E-3 28 mm 098E-3 82 mm 795 OOE-3 .4.1 06E-3 I8E-4 i km I79E-4

Tor

o

1 .5 °

B 6 o

d

rt

"^

O •*

~;

«3

^

B ^

c

a S

3

a -o o

a;

a j-.

E

13 o "u

- -

3 5 a O w• C5 J 2 .rf I TH

la t: 9° QJ

CO £ 5

^^ W II

' \

O o

91

U 00

g lO

a e

in o

v. •S

•^3

1| C J3

Q 3 ^

aj

•a o

co

£ 3 £ m S -a "! II ^ CM

a o- 3 O 3 A .2 1

js

n 11 10 9 8 7 6 5 4 3 Num An-Num. yli 5 -Num

Xi or Axi -1.117495457292 0.116010E-5 0.446 km 0.140720E-5 0.541 km 0.181878E-4 6.991 km 0.181387E-4 6.973 km -0.101663E-3 -39.080 km -0.122604E-3 -47.129 km -0.128804E-2 -495.127 km -0.116732E-2 -448.722 km -1.117495105092 -0.35220E-6 -0.1354 km 0.00269E-6 0.00103 km

zi or Azi -1.9875790811E-2 0.153855E-6 0.059 km 0.750796E-6 0.289 km 0.252732E-5 0.972 km 0.661255E-5 2.542 km -0.788864E-5 -3.032 km -0.505637E-4 -19.437 km -0.136610E-3 -52.513 km -0.580915E-3 -223.306 km -1.9875717636E-2 -0.73175E-7 -0.0281 km 0.20064E-7 0.0077 km

Vi or Ayi 0.184402738444 -0.597614E-5 -6.115 mm/s -0.591889E-5 -6.056 mm/s -0.849758E-4 -86.945 mm/s -0.790935E-4 -80.926 mm/s 0.531066E-3 543.371 mm/s 0.603546E-3 617.530 mm/s 0.627473E-2 6420.12 mm/s 0.667314E-2 6827.76 mm/s -0.184404416242 0.16778E-5 1.717 mm/s -0.00227E-5 -0.023 mm/s

x 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.18

0.19

Table 2.12 Differences between the analytically computed halo orbit (at different orders) and its n sional and physical units. The halo orbit corresponds to the equilibrium point £2 of the Earth-Moo

Numerical Determination

2.3

93

of the Families of Halo Orbits

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 of Symmetric Periodic Orbits for the RTBP

Computation

The goal of the PO program is to compute families of symmetric periodic orbits with respect to the (a;,z)-plane, in particular halo orbits in the RTBP, as well as their stability parameters. An enlargement of this program under the name of PAPUS will be described in Chapter 3. In this section we describe the main analytic methods which are used in the program. 2.3.1.1

The Continuation Method

We consider the initial point Po = {xo,yo,zo,x0,yo,zo)

=

(x,y,z,x,y,z)0,

=

(x,y,z,x,y,z)f,

with y0 = 0, io = 0 and io = 0. Let Pf = (xf,yf,Zf,Xf,yf,zf)

with yj = 0, the first intersection of the orbit which passes by Po with y = 0. In order to have a periodic orbit (p.o.), it is sufficient, due to the symmetries of the problem, that f1(x,z,y)0 2

f (x,z,y)0

=

xf=0,

=

zf = 0.

(2.9)

We define X = (x, z, y)T and F = (f1, f 2 ) T • Then the solutions of the nonlinear system 2.9 are the zeros of F(X0) = 0.

(2.10)

Let us suppose that the rank of the Jacobian matrix of F, G = DF,

(2.11)

is two. Then, 2.10, describes a curve (characteristic curve) in the space (x,z,y)oIn the next, X = (x,z,y) will represent the coordinates (x0,zo,yo) of some initial point Po. In order to obtain a family of p.o. the program follows the characteristic curve of the family using the continuation method. The idea of that method is to integrate that curve along the arc parameter s, that means to integrate the following system

94

Halo Orbits. Analytic

and Numerical

Study

of differential equations dx_ _ M_ dz_ _ M_ dy _ A3 ds Ao' ds Ao' ds A0' where A0 = {A\ + A\ + A\fl\ and

[

Ax = ( £ / ? - / * / ? ) , A2 = -(fU? is t h e

artial

'

- ft ft),

1( 2

A3 = -(fxfz - flfx) flfz,y) P derivative of f - 1 with respect x(z,y). For the integration of (2.12), the program uses an Adams predictor method of one, two, three or four steps, depending on the number of p.o. belonging to the family which are available at the moment. Therefore, if we suppose that some points Xi, X2,..., Xn, n ^ 1 on the characteristic curve are known, the Adams method gives a new point X%+1 near the curve. The point X°+1 must be refined in order to obtain a new p.o. Xn+i. 2.3.1.2

Refinement -1

Let X * be an approximation to a symmetric p.o. whose initial conditions are given by X. In order to obtain a new approximation we define the following iteration Xk ^Xk~1+AXk-\ where we require that AXk~x

(2.13)

verifies the relation

GtAX*- 1 ) =

-F(Xk~l),

and minimizes the norm defined by AXTQAX, where Q is a given diagonal positive definite weight matrix. With this modified Newton's method, if we have an approximation Xk~1, we minimize the correction AX f c _ 1 . It is known that the solution of this problem of minima is given by A X * - 1 = -Q-1GT(GQ-1GT)~1F{Xk-1).

(2.14)

After some iterations we have the p.o. with a given precision. Let B be the differential matrix of the Poincare map after one revolution at the p.o. It is known that B has one eigenvalue equal to 1 (associated to the first integral) and the others are Ai, 1/Ai, A2 and I/A2. The matrix B is obtained from the integration of the variational equations. Using the coefficients of A4 and A3 of the characteristic polynomial p(X) of B, the program computes the stability parameters Tr\ and Tr2 denned by Tn = \i + —, t = l , 2 . Aj

The periodic orbits for which either Tr\ or Tr2 is equal to 2 are associated to possible bifurcations (see 2.3.5). The program can detect the passage of the stability

Numerical Determination of the Families of Halo Orbits

95

parameters through the value 2. In this case it can be given a good approximation of the corresponding orbit using the secant method. 2.3.1.3

Implementation

The program integrates at the same time the equations of the RTBP 2/1 =

2/4,

2/2 =

2/5,

2/3 =

2/6,

2/4 =

2j/5 + 2/i -

2/5

=

o , -22/4 + 2/2

•

_

2/2(1-M) -3 r

2/2M - r3 - ,

l

2/3(1 - tx)

where j-J = (2/1 - (J-)2 +2/2+2/3 equations

an

(2/1-M + 1 )/*

(2/1 - M ) ( 1 - / * )

2

y3fi

d r 2 = (2/1 _ A* + I) 2 + 2/2+2/3, a n d t n e variational

A =

DX-A,

where A = (a^-) and £>X = (dyi/dyj) are 6 x 6 matrices. So there are 42 equations of first order. The variables j/j for i = 7 , 8 , . . . , 42 are ordered in the following way y6j+i=a,ji,

i = 1,2,... ,6, i = l,2, . . . , 6 .

The integration is performed with a Runge-Kutta-Fehlberg method (subroutine RK78). The components of the vector field are computed in the subroutine DERAE. We note that if fi = 0, the vector field corresponds to Hill's problem, which has also been implemented (see 2.3.6 for the related equations). Given some initial point PQ on 2/2 — 0, the subroutine POINC computes the first intersection Pf, of the orbit which passes by PQ, with a given surface of section, in this case 2/2 = 0, through the subroutine SECCIO. At the same time, we have the matrix A = A{T) after the time r spent to go from PQ to Pf. We note that we must add to A(T) some correction given by

f Ayl ^ A2/2 A2/3 A2/4 A2/5

^ Aye J

( Aj/i

\

( vi

0

=

At/3 0

Mr)

+ AT

A2/5

^

0

\

V2

«3 Vi V5

I

\v6

j

96

Halo Orbits. Analytic and Numerical

Study

where {vi,v2,v3,V4,v5,ve)T is the vector field at the point Pf, in order to have (Ay2)f = 0. This equation gives AT =

(a2iAy1 + a 2 3 Ay 3 + v2 Then the matrix (2.11) is given by

a25Ay5.

_ / a 4 i + /34a2i

o 43 + 04a23

a 45 + /i4a25

\

V "61 + /?6a21

063 + /36G23

a 6 5 + /?6<225 / '

where /?4 = —Vi/v2 and (3& — —ve/v2. The computation of the matrix G is done in the FCTN subroutine. This subroutine is used in the TRACA one to obtain a p.o. and its stability parameters. The subroutine DELTAX computes the vector AX of (2.14) as well as its norm. In this subroutine we decide when the iteration (2.13) is stopped. For each approximated p.o. the computation of the three components of the vector field (2.12) is done in the subroutine CAMP. We remark that when the characteristic curve crosses a bifurcation point, this vector field passes by zero and so it changes its sign. This change is done (if it is necessary) in the subroutine CAMP. The prediction of a new p.o. is given in the main program using the subroutine ADAMS which contains the Adams predictor methods. The computation of the possible bifurcations is done in the subroutine REFINA. The required precision is that \Tri — 2| < l.E-10. We note that for each orbit, the computation of the Jacobi constant is performed in the function CJACOB. The output is taken in care by the subroutine WRI. For most of the orbits the value of the precision required for the modified Newton's method (see 2.3.1.2) has been taken equal to l.E-11. The bound for local errors in the RK78 routine was set to l.E-13. The same value was used for the Poincare section routine. 2.3.2

Tables of Numerical Values of the Families Associated to L\ and L?

of Halo

Orbits

In this section we present the numerical values of some of the computed periodic orbits. The halo families associated to L\ and L2 in the Sun-Barycenter case are given in Tables 2.13 and 2.14 respectively. Tables 2.15 and 2.16 contain the corresponding families for the Earth-Moon system. The initial conditions (I.C.), xo, zo and yo, the x, z and y coordinates after half a period (F.C.), the half of the period (T), and the stability parameters (P and Q) are given for some selected halo orbits. The choice has been made mainly taking into account the maxima and minima of the parameters which characterize the periodic orbit. The behavior of the complete families can be seen in the corresponding graphical representations.

Numerical Determination

of the Families of Halo Orbits

97

For each family two graphics (Figures 2.8 to 2.25) have been displayed. The parameters are referred to the small primary, that means, in the x-axis we represent some variable xmax which is the larger distance in the x coordinate from the orbit to the small primary with a suitable scaling. For the four cases mentioned (and also Hill's problem, see 2.3.6) scalings are made to keep to a fixed value the distance from 77i2 to Li, i = 1,2, in order to make the comparisons easier. The xmin is defined as xmax but for the minimum distance. 1 I.C. F.C. T 2 I.C. F.C. T 3 I.C. F.C. T 4 i.e. F.C. T 5 I.C. F.C. T 6 I.C. F.C. T 7 I.C. F.C. T 8 I.C. F.C. T 9 I.C. F.C. T 10 I.C. F.C. T 11 I.C. F.C. T 12 I.C. F.C. T 13 I.C. F.C. T 14 I.C. F.C.

-0.991576589968E+0 -0.988831379586E+0 0.153007263022E+1 -0.991579380330E+0 -0.988831722416E+0 0.153005833973E+1 -0.991583872482E+0 -0.988832276788E+0 0.153003529868E+1 -0.995429337720E+0 -0.989976674638E+0 0.148534639931E+1 -0.996621528996E+0 -0.990505824504E+0 0.145221787965E+1 -0.998190089705E+0 -0.991391600827E+0 0.137386863437E+1 -0.998285439016E+0 -0.991458636022E+0 0.136693806807E+1 -0.998377654256E+0 -0.991525811053E+0 0.135988381336E+1 -0.998466708152E+0 -0.991593101814E+0 0.135271525987E+1 -0.999822870105E+0 -0.993369965635E+0 0.114860801557E+1 -0.999842323196E+0 -0.993426377506E+0 0.114217710969E+1 -0.100000601369E+1 -0.994095570253E+0 0.106803486587E+1 -0.100001353735E+1 -0.994142285467E+0 0.106302991555E+1 -0.100008051838E+1 -0.994805498490E+0

P

P

P

P

P

P

P

P

P

P

P

P

P

-0.971825310000E-4 0.120455194813E-3 0.176512866359E+4 -0.186561599388E-3 0.231287703436E-3 0.176326829906E+4 -0.274988862298E-3 0.341031904049E-3 0.176027580756E+4 -0.490160816125E-2 0.822302273076E-2 0.287327766978E+3 -0.506202771034E-2 0.955130734186E-2 0.142162592433E+3 -0.473422474195E-2 0.110359073279E-1 0.457422339550E+2 -0.468938185317E-2 0.111168291095E-1 0.420539565967E+2 -0.464250898402E-2 0.111940482296E-1 0.386598702045E+2 -0.459377233684E-2 0.112676379316E-1 0.355358672291E+2 -0.305491560173E-2 0.122529613021E-1 0.223804892371E+1 -0.300596050355E-2 0.122645651658E-1 0.192981236213E+1 -0.244789763825E-2 0.123540443462E-1 0.463122428408E+0 -0.241068407176E-2 0.123575448928E-1 0.466767561136E+0 -0.190971777199E-2 0.123797991308E-1

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

0.962818894763E-2 -0.883616935565E-2 0.199987111819E+1 0.963783189824E-2 -0.884249769900E-2 0.199952407041E+1 0.965334829266E-2 -0.885267436064E-2 0.199896232533E+1 0.211001081385E-1 -0.142083259191E-1 -0.212462018367E-1 0.245309269243E-1 -0.149553177264E-1 -0.105046032904E+1 0.298348296745E-1 -0.153567934168E-1 -0.196796070311E+1 0.302247801735E-1 -0.153528493463E-1 -0.198540034888E+1 0.306147750663E-1 -0.153447355807E-1 -0.199605062803E+1 0.310048536959E-1 -0.153325580937E-1 -0.200015072309E+1 0.419018392984E-1 -0.139094186509E-1 -0.101745062488E+1 0.422969520075E-1 -0.138382833818E-1 -0.100806723974E+1 0.474442218264E-1 -0.129071536864E-1 -0.192178729593E+1 0.478413630609E-1 -0.128365530780E-1 -0.202976496030E+1 0.541972560357E-1 -0.117629713187E-1

Halo Orbits. Analytic and Numerical

T 15 I.C. F.C. T 16 I.C. F.C. T 17 I.C. F.C. T 18 I.C. F.C. T 19 I.C. F.C. T 20 I.C. F.C. T 21 I.C. F.C. T 22 I.C. F.C. T

0.994847458962E+0 -0.100009502279E+1 -0.995381531047E+0 0.940519793028E+0 -0.100007961703E+1 -0.996172491694E+0 0.874485837266E+0 -0.100001669623E+1 -0.997921087009E+0 0.770996025823E+0 -0.100000142229E+1 -0.998751936460E+0 0.748257497055E+0 -0.100000068158E+1 -0.998824337861E+0 0.747467785356E+0 -0.999999451874E+0 -0.998968339747E+0 0.746725416174E+0 -0.999996964424E+0 -0.999782878057E+0 0.957851758883E+0 -0.999996961943E+0 -0.999757104235E+0 0.109193589784E+1 Table 2.13

P

P

P

P

P

P

P

P

P

0.764157525382E+0 -0.151809591340E-2 0.123691421646E-1 0.107486869755E+1 -0.104910871615E-2 0.123306470570E-1 0.142711742231E+1 -0.3087117886832-3 0.122567314763E-1 0.188399665879E+1 -0.108125037904E-3 0.123116951140E-1 0.199377553555E+1 -0.954512231554E-4 0.123250656542E-1 0.200183965198E+1 -0.725622794942E-4 0.123598082384E-1 0.201795555469E+1 -0.105159197096E-5 0.162514842180E-1 0.277793661118E+1 -0.647913511710E-6 0.194011479095E-1 0.191839416527E-1

Study

Q -0.325115223508E+1 0.611590597719E-1 -0.107294515101E-1 Q -0.378956377578E+1 0.741616326551E-1 -0.916887907152E-2 Q -0.400088095300E+1 0.139042986314E+0 -0.520200835215E-2 Q -0.363376998418E+1 0.236338220153E+0 -0.311799186754E-2 Q -0.350888310320E+1 0.251639233257E+0 -0.293204756205E-2 Q -0.351050815422E+1 0.288819779208E+0 -0.256051367037E-2 Q -0.352454789295E+1 0.240462465207E+1 -0.346212179088E-3 Q -0.620097023235E+1 0.306350696490E+1 -0.322639467875E-3 Q 0.247223360252E+1

Periodic halo orbits for the L\ case Sun-Barycenter.

Fig. 2.8 Period, T, and stability indices, P, Q, for the periodic halo orbits of the L\ point, SunBarycenter system.

Numerical Determination

Fig. 2.9

of the Families of Halo Orbits

99

Physical parameters of the periodic halo orbits of the L\ point, Sun-Barycenter system.

Fig. 2.10 Projections of the periodic orbit number 4 of Table 2.13, corresponding to a halo orbit around L\ case for the Sun-Barycenter system.

100

Halo Orbits. Analytic

fttmiMTIWi

and Numerical

i

OF COftTIOl FOR TTC It TIF«

I N I T I A L CONlITtonf FIHnL Ml) • -*).BSS4Cr>*Vt YF<J> YI8> • I . I H M I H H YFI«> V1JI - -•,»»4SB377«3 YF(3) V(4) • * . * M « M * * « « VF<4) Y(St t . « 3 l * M K 3 7 YF«» Vfll • • . • * * * » * • • • * VTffi) FINAL TIflC < 2.7«S43»S«

7 « . m -».tM j » . « 7

v

COHDITIOHS -•.Bft*4fS7Ml ••.MIHItHt * -•.•4MSB37?a4 > ~ " > •

Study

1

1

rr,

t.Mfi

i . n t ^ . n t ? « . » 4 ~#.B.*3 ra.sss ?«.w<

Fig. 2.11 Projections of the periodic orbit number 9 of Table 2.13, corresponding to a halo orbit around L\ case for the Sun-Barycenter system. SIMULATION «F CMTtOL FOR THE RTBPi I N I T I A L CONMTIOKS FINAL 419S374 VF(1> 9.———9*9 YFtai - » . * M 4 t t C * 4 1 VF(3) t . M l l l l l l t l VF14J t . » 4 ? 8 4 1 3 « ] i YF<5I • . M « f t t « * * M VFtC) IDE • S.ia«t5»3

YC2J < Y<3t •

CONDITIONS -l.«Mt!3S373 • • • - « . 1*241 KS4S • • •

Fig. 2.12 Projections of the periodic orbit number 13 of Table 2.13, corresponding to a halo orbit around L\ case for the Sun-Barycenter system.

Numerical Determination of the Families of Halo Orbits

10 11 12 13 14 15 16 17

I.C. F.C. T I.C. F.C. T I.C. F-C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T I.C. F.C. T

-0.100841650141E+1 -0.101127600558E+1 -0.155130400165E+1 -0.100841645317E+1 -0.101127601837E+1 0.155130464533E+1 -0.100841506015E+1 -0.101127582432E+1 0.155129512150E+1 -0.100522538832E+1 -0.101033360395E+1 0.151330206611E+1 -0.100340906546E+1 -0.100951124354E+1 0.146484745260E+1 -0.100187070402E+1 -0.100860215012E+1 0.138615493641E+1 -0.100151252067E+1 -0.100833004882E+1 0.135804693037E+1 -0.100140488602E+1 -0.100824034396E+1 0.134843465170E+1 -0.100018703160E+1 -0.100653488091E+1 0.115245620428E+1 -0.100016292645E+1 -0.100646443609E+1 0.114437485184E+1 -0.100001497688E+1 -0.100588177665E+1 0.107896334280E+1 -0.999994411581E+0 -0.100576294309E+1 0.106599639548E+1 -0.999950688084E+0 -0.100543105500E+1 0.103057346284E+1 -0.999906016810E+0 -0.100440232257E+1 0.929445016371E+0 -0.999913431613E+0 -0.100390589312E+1 0.886048086861E+0 -0.999996742223E+0 -0.100039321477E+1 0.682101205150E+0 -0.999996913628E+0 -0.100022058771E+1 0.658249496180E+0

P P P P P P P P P P P P P P P P P

0.000000000000E+0 0.000000000000E+0 0.169309197193E+4 -0.131712296861E-4 0.165157447761E-4 0.169307431150E+4 -0.113018152676E-3 0.141731582267E-3 0.169220423499E+4 -0.464310896952E-2 0.745553551464E-2 0.406000866425E+3 -0.502493728030E-2 0.958669381005E-2 0.145059363093E+3 -0.471847905190E-2 0.110624814649E-1 0.481797365755E+2 -0.453806252323E-2 0.113643518122E-1 0.347350918908E+2 -0.447280531057E-2 0.114517576939E-1 0.311683305247E+2 -0.302852517614E-2 0.123038027924E-1 0.220125421301E+1 -0.296867921386E-2 0.123164554124E-1 0.181919046583E+1 -0.248955439366E-2 0.123818197139E-1 0.385216611075E+0 -0.239581697756E-2 0.123879012149E-1 0.379789732883E+0 -0.214201405113E-2 0.123945720219E-1 0.499631126892E+0 -0.143631593740E-2 0.123446184006E-1 0.106213298152E+1 -0.114219761199E-2 0.122954359845E-1 0.129204713407E+1 -0.144169691275E-4 0.114924646536E-1 0.191390332764E+1 -0.515619885264E-5 0.111589135151E-1 0.188007509065E+1

Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q

-0.979149314605E-2 0.895014911709E-2 0.199999999999E+1 -0.979171636507E-2 0.895032261412E-2 0.199999761844E+1 -0.979641558917E-2 0.895333636475E-2 0.199982447192E+1 -0.193059074154E-1 0.135712737234E-1 0.056205519726E+1 -0.244921306580E-1 0.148198893936E-1 -0.973641861894E+0 -0.296955121606E-1 0.152055165041E-1 -0.193893149607E+1 -0.312210555816E-1 0.151705666816E-1 -0.199952411156E+1 -0.317242727716E-1 0.151461497488E-1 -0.199825199330E+1 -0.420515821101E-1 0.137546617228E-1 -0.105970188194E+1 -0.425428553645E-1 0.136677101959E-1 -0.104533649792E+1 -0.469651397961E-1 0.128808614831E-1 -0.173260180417E+1 -0.479605410589E-1 0.127065079406E-1 -0.200323634917E+1 -0.509494338767E-1 0.121964602104E-1 -0.269325365309E+1 -0.629279255105E-1 0.104150756731E-1 -0.379492648923E+1 -0.709224693220E-1 0.945444512507E-2 -0.391072201655E+1 -0.649099994713E+0 0.114507926796E-2 -0.251502648566E+1 -0.108574155440E+1 0.687701685731E-3 -0.214992415346E+1

Table 2.14 Periodic halo orbits for the L2 case Sun-Barycenter.

102

Halo Orbits. Analytic

and Numerical

Study

Fig. 2.13 Period, T, and stability indices, P,Q, for the periodic halo orbits of the Li point, Sun-Barycenter system.

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Physical parameters of the periodic halo orbits of the Li point, Sun-Barycenter system.

Numerical Determination

of the Families of Halo Orbits

I

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104

Halo Orbits. Analytic and Numerical

1

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1 I.C. F.C. T 2 I.C. F.C. T 3 I.C. F.C. T 4 I.C. F.C. T 5 I.C. F.C. T 6 I.C. F.C. T 7 I.C. F.C. T 8 I.C. F.C. T 9 I.C.

-0.823390261900E+0 -0.854812338909E+0 0.137150569147E+1 -0.823389443928E+0 -0.854865160541E+0 0.137154160766E+1 -0.823388096847E+0 -0.854957215442E+0 0.137160420673E+1 -0.823379386317E+0 -0.856214890974E+0 0.137245988160E+1 -0.829443928469E+0 -0.904213375092E+0 0.139376722421E+1 -0.830341488531E+0 -0.908965133163E+0 0.139348075865E+1 -0.836282846507E+0 -0.936503063012E+0 0.137587103659E+1 -0.842069723120E+0 -0.955677924640E+0 0.133878883696E+1 -0.842890321377E+0

P

P

P

P

P

P

P

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0.162067799667E-2 -0.141372917727E-2 0.236024059268E+4 0.361713113775E-2 -0.315447911096E-2 0.235660165019E+4 0.559739334482E-2 -0.487937785449E-2 0.235026740063E+4 0.1683780948S0E-1 -0.145932446758E-1 0.226471023110E+4 0.110441386023E+0 -0.771825896776E-1 0.436658437097E+3 0.116209384902E+0 -0.793540905687E-1 0.365993125618E+3 0.145499036502E+0 -0.852043477016E-1 0.125996840141E+3 0.162612218834E+0 -0.823774351416E-1 0.546691023928E+2 0.164379682543E+0

Q

Q

Q

Q

Q

Q

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-0.126370453567E+0 0.133790346930E+0 0.199989653529E+1 -0.126544572792E+0 0.134017984484E+0 0.199948304236E+1 -0.126847445183E+0 0.134414235833E+0 0.199875547442E+1 -0.130913822446E+0 0.139770140504E+0 0.198791951209E+1 -0.225927098203E+0 0.298589462244E+0 0.247977160058E+0 -0.231397144062E+0 0.311835027219E+0 -0.270270303162E-1 -0.254768122149E+0 0.387738465859E+0 -0.150747673920E+1 -0.263138387394E+0 0.446033386964E+0 -0.199530390099E+1 -0.263588135045E+0

Numerical Determination of the Families of Halo Orbits

F.C. -0.957803913863E+0

T 0.133269668405E+1 P 10 I.C. -0.843773901470E+0 F.C. -0.959950473225E+0

T 0.132601258164E+1 11 I.C. -0.845740854727E+0 F.C. -0.964244026939E+0 T 0.131080495024E+1 12 I.C. ^0.868254310680E+0 F.C. -0.986701454497E+0 T 0.114023723299E+1 13 I.C. -0.871454972056E+0 F.C. -0.987827452958E+0 T 0.111839309017E+1 14 I.C. -0.872498182455E+0 F.C. -0.988136296965E+0 T 0.111143230854E+1 15 I.C. -0.880481701119E+0 F.C. -0.989748730752E+0 T 0.106085425339E+1 16 I.C. -0.881439616370E+0 F.C. -0.989868474545E+0 T 0.105511629664E+1 17 I.C. -0.893238618951E+0 F.C. -0.990472419019E+0 T 0.990757886898E+0 18 I.C. -0.912420238006E+0 F.C. -0.989638925948E+0 T 0.916791033889E+0 19 I.C. -0.913151332309E+0 F.C. -0.989590070563E+0 T 0.914969677782E+0 20 I.C. -0.922698785741E+0 F.C. -0.988940295698E+0 T 0.901837421864E+0 21 I.C. -0.933393440199E+0 F.C. -0.988190634961E+0 T 0.961893247486E+0 22 I.C. -0.933517735883E+0 F.C. -0.988150458619E+0 T 0.976217033717E+0 23 I.C. -0.933291465203E+0 F.C. -0.988107996286E+0 T 0.996295105463E+0 24 I.C. -0.930183012085E+0 F.C. -0.988026599254E+0 T 0.106205733475E+1 25 I.C. -0.929831537549E+0 F.C. -0.988022594208E+0 T 0.106690957252E+1

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

-0.816444119816E-1 0.493122748992E+2 0.166142678086E+0 -0.808021777031E-1 0.442824964005E+2 0.169612049616E+0 -0.787776467802E-1 0.352456550443E+2 0.188544799810E+0 -0.533958011815E-1 0.319672145513E+1 0.189960961862E+0 -0.500965603582E-1 0.212621230448E+1 0.190396887864E+0 -0.490453786932E-1 0.187604188271E+1 0.193455796820E+0 -0.413906273122E-1 0.120493081025E+1 0.193801192419E+0 -0.405177290543E-1 0.120601655027E+1 0.198017053317E+0 -0.305296896409E-1 0.147973134509E+1 0.206929784909E+0 -0.170738010005E-1 0.199115740808E+1 0.207399494839E+0 -0.166232211928E-1 0.200870734165E+1 0.215639841606E+0 -0.110962389716E-1 0.224202507139E+1 0.247408780664E+0 -0.467307722641E-2 0.272582884080E+1 0.252844067032E+0 -0.427938238641E-2 0.276990583887E+1 0.260402576647E+0 -0.385016603104E-2 0.279795757788E+1 0.285892818005E+0 -0.298070331535E-2 0.212440746425E+1 0.287861395978E+0 -0.293615374411E-2 0.193626835419E+1

0.453299342744E+0 Q -0.200024359104E+1 -0.263920560950E+0 0.460900638579E+0 Q -0.199302170601E+1 -0.26416850655BE+0 0.477095616204E+0 Q -0.193901365866E+1 -0.242933485319E+0 0.636967529378E+0 Q -0.661584283677E+0 -0.238060510612E+0 0.660540846184E+0 Q -0.726145184038E+0 -0.236410760544E+0 0.668436407107E+0 Q -0.791386371872E+0 -0.222889579789E+0 0.733385541700E+0 Q -0.188730547442E+1 -0.221168215734E+0 0.741825370872E+0 Q -0.20404460804 7E+1 -0.198389761430E+0 0.861804331941E+0 Q -0.352657045589E+1 -0.155322769806E+0 0.116739519947E+1 Q -0.464515960642E+1 -0.153521420136E+0 0.118372714652E+1 Q -0.467247260172E+1 -0.128547961001E+0 0.145908606150E+1 Q -0.503691489319E+1 -0.903891707269E-1 0.227103144184E+1 Q -0.574030584277E+1 -0.878577138595E-1 0.237491183263E+1 Q -0.570406761698E+1 -0.852627291426E-1 0.250585480773E+1 Q -0.553205077297E+1 -0.817550306131E-1 0.285339504197E+1 Q -0.325948697794E+1 -0.817221703249E-1 0.287528625879E+1 Q -0.288000764739E+1

Halo Orbits. Analytic and Numerical

106

26 I.C. F.C. T 27 I.C. F.C. T 28 I.C. F.C. T 29 I.C. F.C. T 30 I.C. F.C. T 31 I.C. F.C. T 32 I.C. F.C. T 33 I.C. F.C. T 34 I.C. F.C. T 35 I.C. F.C. T

-0.929467558855E+0 -0.988018802294E+0 0.107171801812E+1 -0.929091616228E+0 -0.988015208319E+0 0.107648134194E+1 -0.928546151017E+0 -0.988010482969E+0 0.108307215768E+1 -0.928506364014E+0 -0.988010158313E+0 0.108353939977E+1 -0.788270767649E+0 -0.987903977718E+0 0.144341155864E+1 -0.746079486520E+0 -0.987899025385E+0 0.146862332325E+1 0.199990846092E-1 -0.987878376642E+0 0.155639031234E+1 0.881210778703E+0 -0.999383714294E+0 0.146608100773E+1 0.869770335170E+0 -0.100408753420E+1 0.144891511954E+1 0.845786038723E+0 -0.101670220039E+1 0.141468119506E+1 Table 2.15

P

P

P

P

P

P

P

P

P

P

0.289827970160E+0 -0.289382476809E-2 0.168799457640E+1 0.291792225491E+0 -0.285357405885E-2 0.133653300636E+1 0.294537783157E+0 -0.280046529714E-2 0.342315695803E-1 0.294733683481E+0 -0.279680890847E-2 -0.955766759073E-1 0.590624539917E+0 -0.177653825815E-2 0.105563010614E+3 0.644502470861E+0 -0.178059208622E-2 0.146924755718E+3 0.994956323273E+0 -0.259061704589E-2 0.510179117041E+3 0.184358708066E+0 -0.200215615353E-1 -0.740266695867E+1 0.147924200157E+0 -0.191876976723E-1 -0.363314768365E+1 0.266926831825E-4 -0.467189787277E-5 0.199999963694E+1

Study

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Q

Q

-0.817150624388E-1 0.289655762200E+1 -0.242995052811E+1 -0.817328882513E-1 0.291722686647E+1 -0.186616029725E+1 -0.817975635885E-1 0.294518604560E+1 -0.248700853101E+0 -0.818038889883E-1 0.294714058196E+1 0.413271510597E+0 -0.206953539966E+0 0.374438935067E+1 0.463269922875E+2 -0.248360315112E+0 0.375213490563E+1 0.521683762150E+2 -0.101176298211E+1 0.336926137105E+1 0.381074716503E+3 -0.199037526885E+1 0.222519827819E+1 0.267224281388E+1 -0.200220132673E+1 0.221084541694E+1 -0.913106770880E+1 -0.202847382748E+1 0.219370596795E+1 -0.477004389318E+1

Periodic halo orbits for the L\ case Earth-Moon.

Numerical

Determination

107

of the Families of Halo Orbits

2.4

2.0

Fig. 2.18 Period, T, and stability indices, P,Q, for the periodic halo orbits of the L\ Earth-Moon system.

Fig. 2.19

point,

Physical parameters of the periodic halo orbits of the L\ point, Earth-Moon system.

108

Halo Orbits. Analytic

and Numerical

Study

SWUUTION Of CONTROL FOR THE RT3P1 KJ • •.»121S*C4« INITIAL CONDITIONS FINAL CONDITIONS V(i» . -«.*394<43BZK V F t I ) • -«.f£B443.Be*S • . • • • • • • • • • • WIS) • **—*»•—• V(2) • V(3> • • • I l « 4 4 1 3 « 6 « VF<1I • «.lia441316« Vt4* • I . I H M M M * YF<4> • *•••*••••••• Y(S> - * . e e s M 7 t 9 u VF<S> * -».a«Ba7t9M 7(«l • • . •• • • • • • »2 .— • .*4*««MM4> FINAL TIME 7 m >V3 F 4 4I iE) •

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Fig. 2.20 Projections of the periodic orbit number 5 of Table 2.15, corresponding to a halo orbit around L\ case for the Earth-Moon system.

1

SIMULATION OF CONTROL, FOR THE RTB*i W • • .•1215«M« I N I T I A L CONDITIONS FINAL COnDITIOHS Vtl) - « . 1 4 2 « 8 « 3 g I 4 YF(1> -•.t4ZI0B3X14 V(8) • • • • • • • • • • • • » VF(ai • •>.••••••«••• V(3) • «.1C4T7»CS£S YF<3> • «.1C43?9C12S Y(4> • • • • • • • • • • • • • VFI4) • .9——— ¥(S» - * . » 3 S * a i 3 S « VFES) • ~t.aS3fitll3S* YCO • • . • — 9 — VF(«1 • ••••MIM*— FINAL TINE • 2.KE3933?

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Fig. 2.21 Projections of the periodic orbit number 9 of Table 2.15, corresponding to a halo orbit around L\ case for the Earth-Moon system.

Numerical Determination

109

of the Families of Halo Orbits

••"i

SinuLATIOM OF CONTROL FOR TtC RTlPi RU • 4.412144SS4 INITIAL COnDITlOHS FINAL C0M0IT10H3 Y«I* - -4.4914399194 VFU) • -9.I1143K144 V(2l • t . H I I M W M VF(2J • I.IIWIMM Y(]l • 9.1438411924 VFO) • 4.193MUB44 YC4> I . N M M N I I YF(41 • *.99999999— VfSI - -4.UUSW1I7 VF(SI • -4.44119U157 VISI • 9 . i a M M M I » VF(S) * I.HtMMtN FIMAL TIRE I.UMWII

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Fig. 2.22 Projections of the periodic orbit number 13 of Table 2.15, corresponding to a halo orbit around L\ case for the Earth-Moon system.

SlHULflTIOK OF CONTROL FOR TtC *TB*t RU • • .flit MM* INITIAL COHSITIOMS FIKAL C0K8ITI0NS Y<1> • • . 9 4 S 7 9 C 4 3 I 7 VFC1 > > •,t4S79C93R7 VCR) • I . I I I N I I I M VFIB) * 4.W9M9M44 Y(33 • t . M ( H H » 7 VFI3) I.WMHIKT V(4» • 4 . 4 4 * 4 4 * « * 9 4 YF14) <* 11*1 -2.4eB4?3IITS VFO) • -2,aa*4T3at7» V(S) • 4.94999999a* YFlSt I FINAL TIRE • 1,48434234

1.417 y9.744 y 4 . K l T 9.314 j4.44« ,9.147 ,0.349 ,9.913 ,9.944 _j , j j j , j -

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974 7 9.444 79.M4 ,4.444

4.4*4 ,,4.444 ,B.9T» ,4.444

Fig. 2.23 Projections of the periodic orbit number 35 of Table 2.15, corresponding to a halo orbit around L\ case for the Earth-Moon system.

Halo Orbits. Analytic and Numerical

110

1 I.C. F.C. T 2 I.C. F.C. T 3 I.C. F.C. T 4 I.C. F.C. T 5 I.C. F.C. T 6 I.C. F.C. T 7 I.C. F.C. T 8 I.C. F.C. T 9 I.C. F.C. T 10 I.C. F.C. T 11 I.C. F.C. T 12 I.C. F.C. T 13 I.C. F.C. T 14 I.C. F.C. T 15 I.C. F.C. T 16 I.C. F.C. T 17 I.C.

-0.118089511282E+1 -0.112037148096E+1 0.170774971609E+1 -0.118088801584E+1 -0.112034381488E+1 0.170771992114E+1 -0.118087724164E+1 -0.112030186201E+1 0.170767472022E+1 -0.114898661205E+1 -0.105309395487E+1 0.159274289022E+1 -0.114321496984E+1 -0.104467050783E+1 0.156846772162E+1 -0.112658009755E+1 -0.102369480122E+1 0.148668224458E+1 -0.111007449759E+1 -0.100779735821E+1 0.138659185591E+1 -0.110927532844E+1 -0.100715762232E+1 0.138129806077E+1 -0.110847462865E+1 -0.100652881472E+1 0.137595720802E+1 -0.108053455548E+1 -0.991745145541E+0 0.117346696855E+1 -0.107918887905E+1 -0.991344051442E+0 0.116329000920E+1 -0.107852259599E+1 -0.991154242571E+0 0.115824847178E+1 -0.106840929028E+1 -0.988920259749E+0 0.108183346495E+1 -0.106602642316E+1 -0.988550732976E+0 0.106394546754E+1 -0.104466646043E+1 -0.987090470303E+0 0.908450103847E+0 -0.104083229529E+1 -0.987068331126E+0 0.881734480675E+0 -0.991607395535E+0

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

0.198645095312E-2 -0.144012258042E-2 0.121186250606E+4 0.335240233937E-2 -0.243006606374E-2 0.121119311645E+4 0.472241469437E-2 -0.342243801020E-2 0.121017842568E+4 0.148379569759E+0 -0.749733151624E-1 0.203972501748E+3 0.158377381070E+0 -0.755822116039E-1 0.151150883853E+3 0.180490075039E+0 -0.723596622178E-1 0.621874255984E+2 0.194183638215E+0 -0.640024433144E-1 0.237887458866E+2 0.194659883984E+0 -0.635078502219E-1 0.226223730362E+2 -0.195120799300E+0 -0.630059279008E-1 0.215002756742E+2 0.202361120974E+0 -0.43298574S000E-1 0.125345869245E+1 0.202345771148F+0 -0.423195555533E-1 0.882812853789E+0 0.202328070060E+0 -0.418359342556E-1 0.716733243136E+0 0.201301069799E+0 -0.346352301459E-1 -0.168760225111E+0 0.200868810077E+0 -0.329895073342E-1 -0.107965801746E+0 0.194275188917E+0 -0.195088331160E-1 0.726260408438E+0 0.192613336450E+0 -0.173769266795E-1 0.860086865911E+0 0.132905896838E+0

Study

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

0.155879545911E+0 -0.176084402691E+0 0.199988102662E+1 0.155921856657E+0 -0.176164823433E+0 0.199966089397E+1 0.155985982709E+0 -0.176286758355E+0 0.199932632529E+1 0.218859772746E+0 -0.364176383985E+0 -0.385187895865E-1 0.222120858304E+0 -0.389525786375E+0 -0.489472593253E+0 0.225514906799E+0 -0.461408849340E+0 -0.156877529403E+1 0.221464992466E+0 -0.536222711594E+0 -0.199867234518E+1 0.221098276276E+0 -0.540044223004E+0 -0.200005787632E+1 0.220715663227E+0 -0.543897630040E+0 -0.199971002377E+1 0.198454152028E+0 -0.702509720587E+0 -0.129169430329E+1 0.196974642356E+0 -0.711899770666E+0 -0.127024722342E+1 0.196229157487E+0 -0.716633420995E+0 -0.126542109814E+1 0.183883790846E+0 -0.796509330148E+0 -0.204412831256E+1 0.180699145927E+0 -0.817921802125E+0 -0.232378725588E+1 0.147568949659E+0 -0.108055932031E+1 -0.338488794070E+1 0.140747235495E+0 -0.114781842272E+1 -0.335197921203E+1 0.234786807020E-1

Numerical Determination

of the Families of Halo Orbits

111

-0.305994794206E-3 -0.890197540512E+1 F.C. -0.987845245172E+0 T 0.462158993417E+0 P 0.171183090455E+1 Q 0.349974222563E+0 0.970051005838E-1 0.914960289336E-2 18 I.C. -0.988475762462E+0 -0.290498859622E-4 -0.289187203287E+2 F.C. -0.987849236868E+0 T 0.297243002691E+0 P 0.181244026053E+1 Q -0.143579237351E+1 Table 2.16

Periodic halo orbits for the L2 case Earth-Moon.

Fig. 2.24 Period, T, and stability indices, P,Q, for the periodic halo orbits of the Li Earth-Moon system.

point,

112

Fig. 2.25

Halo Orbits. Analytic

and Numerical

Study

Physical parameters of the periodic halo orbits of the L2 point, Earth-Moon system.

Fig. 2.26 Projections of the periodic orbit number 4 of Table 2.16, corresponding to a halo orbit around L2 case for the Earth-Moon system.

Numerical Determination

of the Families of Halo Orbits

1

t

SIMULATION OF CONTROL FOJt THE HT1PI PHI • •.•leiMSSI I N I T I A L CONDITIONS FINAL CONDITIONS Y<1) • -1.1BBS753E14 VF ( 1 1 • -l.l»M7S32tfi V<2) • •.—****•+* VTC8I * •.»•»••••>•• Y<3> I . I » 4 « » I I 4 « VTI3> «.1&4«SV«S4* ¥14) • • . » — » « • • » • « YFI4I • • .**t*t*»«** Y<5> « . 8 a i * 9 I E 7 S 3 YF(E> • .8210BM7C3 YIC) • . • • • • — • — VF((> • ••••—MW1 FINAL TIME • S.TSSSSStZ

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Fig. 2.27 Projections of the periodic orbit number 8 of Table 2.16, corresponding to a halo orbit around L 2 case for the Earth-Moon system.

1

SIMULATION! OF CONTROL FOR THE R T l » i Ml • • .•181S4MSI I N I T I A L CONDITIONS FINAL CONDITIONS V(t) • -l.*C84»929*3 VF(I) * -1.•C«4«9EB*3 Y(2> * • . • • • • • • • • • • YF(2J • •••••••••••• VC3) • . 2 * l 3 « i a S B S YF(3) • *.2«ia*ltC9t YC4) • . • • • • • « * • * « YF(4) •.•*•••••••• Y(S) • • . 1 I 3 M 3 7 S M VF(S) • a.lS3M3TB*t YCO • • . • • • • » • • • » • YF(S> •.•*«*•••••• FINAL TINE • 8.1S3SMS3 OKI.

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Fig. 2.28 Projections of the periodic orbit number 13 of Table 2.16, corresponding to a halo orbit around Li case for the Earth-Moon system.

114

Halo Orbits. Analytic and Numerical

Study

Fig. 2.29 Projections of the periodic orbit number 17 of Table 2.16, corresponding to a halo orbit around Li case for the Earth-Moon system.

We note that xmax decreases along the families except for the families associated to L\. Figures 2.8, 2.13, 2.18 and 2.24 represent the period and the stability parameters P and Q as a function of xmax. For P and Q, which have large variations, we plot a magnitude linearly related to the argument of hyperbolic sine. The behavior of Xmin and the z coordinates are represented in Figures 2.9, 2.14, 2.19 and 2.25. We note that the vertical scale for z is one half of the horizontal one. Table 2.15 needs some comments. In this Table the orbits numbers 29, 30, 31 and 32 have complex stability parameters P ± iQ. In Figures 2.18 and 2.19, only the orbits before the orbit 29 are represented. It can be seen that the maximum value of z increases through the family but for the last orbits it decreases to zero. This family ends in a large plane orbit surrounding the two primaries. The same behavior is observed for the L\ case in the Sun-Barycenter system. However, the amount of computations is enormous if some regularization is not used (see 2.3.6). Figures 2.10-2.12 display halo orbits Li case for the Sun-Barycenter system. Orbits numbers 4, 9 and 13 of Table 2.13 are given. Last one belongs to the range of stable orbits. The last two orbits of Table 2.13 (numbers 21 and 22) collide with the surface of the Earth. However, as it has been stated before, the family can be continued and produce orbits which have no collision after this range of orbits with very close encounters. In Figures 2.15-2.17, orbits numbers 4, 7 and 11 of Table 2.14 are presented. As

Numerical

Determination

of the Families of Halo Orbits

115

before last one is linearly stable. We remark that the last orbit given in the Table collides with the Earth surface. Figures 2.20-2.23 give a representation of orbits numbers 5, 9, 16 and 35 of Table 2.15. Number 16 is a stable periodic orbit. Number 35 is a very large orbit near the end of the family at a counterclockwise orbit around the two primaries. Orbits from numbers 22 to 32 shall produce collisions with the lunar surface. Finally Figures 2.26-2.29 present a plot of orbits numbers 4, 8, 13 and 17 of Table 2.16. The last two are linearly stable. We note that last one intersects the lunar surface. Some of these results were obtained by Breakwell et al. (see [1], [7] and [8]). 2.3.3

The Halo Families

Associated

to L3

First we discuss the Earth-Moon case. The corresponding family has been computed numerically. Table 2.17 shows the results which are presented in the same way as Tables 2.13-2.16. Figures 2.30 and 2.31 represent the global behavior of the parameters. In this case Xfyifxx {xmin) is the larger (smaller) value of x in the orbit. As in the figures above, the vertical scale for the z coordinate is one half of the horizontal one. Linear scaling is used in this case for P and Q. In Figures 2.32 and 2.33 the orbits numbers 4 and 8 from Table 2.17 are displayed. From orbit number 8 onwards, they are linearly stable. However for Table 2.17, from orbit number 6 onwards, they have collisions with the Earth surface. Concerning the L3 halo family for the Sun-Barycenter case, there are severe numerical difficulties. This is due to the fact that these orbits, which are far from Earth and Sun, are essentially orbits of the two-body problem (Sun+small mass) under the effect of the Earth (and all other bodies in the solar system in reality). If the semimajor axis is equal to the Earth one, these orbits will be seen as a p.o. in synodic coordinates. In fact the motion can be well approximated by the rotating two-body problem. For this one all the orbits in a suitable level of energy are periodic. Hence their stability parameters are equal to 2. See also 2.2.4 for the analytic difficulties involved. The halo family certainly exists but the orbits are very large and not useful for the applications.

116

Halo Orbits. Analytic and Numerical

1 I.C. F.C. T 2 I.C. F.C. T 3 I.C. F.C. T 4 I.C. F.C. T 5 I.C. F.C. T 6 I.C. F.C. T 7 I.C. F.C. T 8 I.C. F.C. T 9 I.C. F.C. T 10 I.C. F.C. T 11 I.C. F.C. T 12 I.C. F.C. T 13 I.C. F.C. T

0.169673185272E+1 0.309569463601E+0 0.311957347458E+1 0.169643807553E+1 0.309421428709E+0 0.311957112742E+1 0.169579908109E+1 0.309099593106E+0 0.311956601773E+1 0.137330374763E+1 0.172713387633E+0 0.311603903077E+1 0.484181734142E+0 0.195633624966E-1 0.307855109974E+1 0.244676390793E+0 0.132110773072E-1 0.302544735117E+1 0.181066771661E+0 0.126050976952E-1 0.298865021529E+1 0.177126868837E+0 0.125779034989E-1 0.298552695588E+1 0.149887122760E+0 0.124183566774E-1 0.295957747550E+1 0.377699045030E-1 0.121560709823E-1 0.248456037081E+1 0.208124934483E-1 0.121513174082E-1 0.196165476636E+1 0.160570906994E-1 0.121508172151E-1 0.155837949240E+1 0.132672001836E-1 0.121506829077E-1 0.103335218803E+1 Table 2.17

P

P

P

P

P

P

P

P

P

P

P

P

P

0.810094281818E-2 -0.142739686605E-2 0.276569205646E+1 0.364904154663E-1 -0.642757875548E-2 0.276546500423E+1 0.639099878129E-1 -0.112494547168E-1 0.276497291268E+1 0.113567556433E+1 -0.133661391900E+0 0.254976896047E+1 0.187998762881E+1 -0.292637557905E-1 0.220135452814E+1 0.192384672198E+1 -0.859694037070E-2 0.206928057742E+1 0.191839526936E+1 -0.501902976933E-2 0.200350541460E+1 0.191759215405E+1 -0.482632709583E-2 0.199806123683E+1 -0.109650075951E+0 -0.358901841201E-2 0.195236594785E+1 -0.170658424886E+1 -0.325503005237E-3 0.552523368120E+0 -0.145772745099E+1 0.962786250991E-4 ^0.139454530845E+1 -0.125005231958E+1 0.424163400260E-4 -0.198540607689E+1 -0.950086039711E+0 0.118033932419E-4 -0.907773954392E+0

Study

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

0.127954846944E+1 0.208177763528E+1 0.199999994610E+1 -0.127932327995E+1 0.208231058317E+1 0.199999890617E+1 -0.127893347999E+1 0.208347013194E+1 0.199999664228E+1 -0.103342200395E+1 0.275059374690E+1 0.199845904519E+1 -0.363098827563E+0 0.802115581718E+1 0.198052622974E+1 -0.181223121911E+0 0.150682732792E+2 0.193836508137E+1 -0.133146476510E+0 0.197739877721E+2 0.189652114354E+1 -0.130174637369E+0 0.201677164348E+2 0.189249817405E+1 -0.109650075951E+0 0.234079229153E+2 0.185625847377E+1 -0.255730082109E-1 0.778954211450E+2 0.475465059209E+0 -0.125931750745E-1 0.143243934691E+3 -0.144178343757E+1 -0.855279322590E-2 0.215816719931E+3 -0.201297532233E+1 -0.535529629410E-2 0.409123148275E+3 -0.994596870522E+0

Periodic halo orbits for the L% case Earth-Moon.

Numerical Determination

117

of the Families of Halo Orbits

—

.6 _1_

.8

1.0

-V—

1.2 _l

1.4

Q

1.6 _J

Fig. 2.30 Period, T, and stability indices, P, Q, for the periodic halo orbits of the L\ point, Sun-Barycenter system.

Fig. 2.31

Physical parameters of the periodic halo orbits of the L\ point, Sun-Barycenter system.

118

Halo Orbits. Analytic

and Numerical

Study

Fig. 2.32 Projections of the periodic orbit number 4 of Table 2.17, corresponding to a halo orbit around Lz case for the Earth-Moon system.

SWULATIOM OF COHTTWL FOR TMG KTIPl

I

'

'

'

'

'

'

'

'

'ci,2>

Fig. 2.33 Projections of the periodic orbit number 8 of Table 2.17, corresponding to a halo orbit around L3 case for the Earth-Moon system.

Numerical Determination

2.3.4

Interest

of the Families of Halo Orbits

of These Orbits for Spacecraft

119

Missions

In this subsection we describe the possible use of the orbits obtained so far for spacecraft missions. Instead of studying the L\, L2 and L3 cases separately, we prefer a different approach considering first small size halo orbits for the L\, L 2 cases, after halo orbits in or near the linear stability zone for the same two cases, and finally the L3 case. The special conditions of the Sun-Barycenter system and the Earth-Moon system are taken into account. A large number of previous works are concerned with applications of halo orbits to spacecraft missions (see mainly [2], [3], [4], [5], [12] and [13]).

2.3.4.1

Small Size Halo Orbits Related to L\ and L2

Most of the effort done till the present has been devoted to these orbits. From the scientific and related points of view they are useful for performing experiments concerning the solar structure (L\ case, Sun-Barycenter system), the geomagnetic tail (L2 case, Sun-Barycenter system), for communications with the far side of the Moon (L2 case Earth-Moon system), etc. We are sure that new applications will be discovered for these cases in the near future as well as for the L\ case, Earth-Moon system. One of the technical problems is the need of control because these orbits are all unstable. As has been shown in 2.3.2 the instability decreases when the amplitude increases. Another technical problem is the convenience of constant communications with the Earth stations (or with the far side of the Moon stations). A spacecraft in a halo orbit related to L\ for the Sun-Barycenter system should avoid a narrow cone around the Sun-Barycenter direction because the solar interferences can produce noise in the communications. If we suppose that the spacecraft is stabilized with instruments facing the Sun and a fixed antenna is facing opposite to the Sun, we cannot allow large elongations from the Sun when the spacecraft is seen from the Earth, because this requires an extremely large beam-width and the power of the antenna should be increased beyond the limits available to the power supply. Figure 2.34 shows for these orbits (L\ case, Sun-Barycenter system) the values of the minimal (7 m m ) and maximal (jmax) angular distance from the Sun and the maximal stability parameter (|Ai| m0 x) against the value xmax of the orbit. The minimal angular distance from the Sun is attained at a point with zero solar elongation (contained in the (a;,z)-plane). There are two points in the orbit with zero solar elongation. The one giving minimal angular distance to the Sun is the more distant to the Earth (despite the fact that it has a maximum value of \z\). Therefore the selection of the best orbit should be done trying to minimize l-^ilmoa:, to reduce jmax to allowable values and to increase 7 m j n to avoid passages too close to the Sun. For the other cases (L2, Sun-Barycenter system and L±, L2, Earth-Moon sys-

120

Halo Orbits. Analytic and Numerical

Study

max

QO

i

-•9892

..g 8 91

i

-.9890

i

-.9889

-.9888

Fig. 2.34 Minimal and maximal angular distances from the Sun and the maximal stability parameter against the value xmax of the orbit. {L\ case, Sun-Barycenter system.)

tem) the requirements about the values of jmin, Jmax are not so severe. Concerning the transfer orbit, it should be stated here that for all of these orbits some previous numerical experiments seem to give the following evidence: the stable manifold (two-dimensional) of these unstable orbits comes close to the Earth. Therefore, a good policy can be to inject the orbit into the stable manifold and perform some control along the transfer to be sure that the departures from this manifold are corrected. This study will be done in the next chapters. We can point out, however, that this method of transfer can be too slow. Another important point is that the real problem is much more complicated than the RTBP. However, for those orbits an analytic theory for quasi-periodic motions is developed in future chapters. Because of the small size, the orbits are amenable with analytic methods. The theory produced is an extension of the one presented in 2.2. The perturbations for the halo orbits of the RTBP model in the Sun-Barycenter system are not too strong, but they are in the case of the Earth-Moon system, mainly due to the solar effect. 2.3.4.2

The Stable Zone of Halo Orbits in the L\ and L2 cases

For both cases and in both systems (Sun-Barycenter and Earth-Moon) stable halo orbits appear. They can be considered in some sense as satellites of the smaller primary slightly distorted towards the larger one. In this case |Ai| m a x is small (1

Numerical Determination

121

of the Families of Halo Orbits

in the stable zone and close to 1 for nearby orbits) but the values of 7TOj„ and jmax are very large. In fact they are near 90 degrees showing that the orbits are near a plane parallel to the (y, z) one passing through the center of the smaller primary. Let us consider just one case: halo orbits of the family associated to L\ for the Sun-Barycenter system. Figure 2.35 displays the values of 7 m j n , ^max and |Ai|maa; against xmax. These values are large but we should be aware that the spacecraft is always at a large angular distance from the Sun. Technical solutions for telecommunications problem seem not too hard. 100*

50'

10

, -.998

max f -.996

(

-.994

-.992

Fig. 2.35 Minimal and maximal angular distances from the Sun and the maximal stability parameter against the value xmax of the orbit, for orbits near the stable zone. (L\ case, Sun-Barycenter system.)

Concerning these orbits the following experiment has been performed: Suppose that the spacecraft is rotating with constant angular velocity around an axis parallel to the x-axis (Sun-Barycenter line). Then all the instruments can face the Sun. An antenna is rigidly attached to the spacecraft but the angle between the z-axis opposite to the Sun direction and the antenna is not zero as in 2.3.4.1 but a. When the spacecraft is in the (x, z)-plane the central point of the beam is also contained in this plane. Table 2.18 gives the maximal beam-width required for several values of a and for several orbits. Orbits numbers 3, 4, 11 and 13 of Table 2.13 are used. The first one is a very small size (in z) halo orbit. Orbit number 4 is a medium size one, and the values given show the improvement obtained if a ^ 0 is allowed (compare with Figure 2.34, i.e., the best value with the one for a = 0). Orbits 11 and 13 are extrema of a stability zone. Another problem of these stable orbits is the possibility of close passages near

122

Halo Orbits. Analytic and Numerical

Orbit number 3 Angle Max. beam-width Orbit number 4 Angle Max. beam-width Orbit number 11 Angle Max. beam-width Orbit number 13 Angle Max. beam-width Table 2.18

Study

0 25.4

10 16.0

15 13.8

20 18.2

30 28.2

0 51.4

40 23.2

45 22.2

50 22.1

55 23.1

60 25.1

90 50.6

0 87.1

45 64.9

50 64.3

55 64.0

60 63.9

65 64.0

70 64.4

0 90.4

45 72.5

50 72.3

55 72.3

60 72.5

90 76.7

75 65.0

90 68.1

Maximal beam-width required for several values of a.

the smaller primary. This can produce collisions with the surface of this primary, as noted in [1], for the Earth-Moon case. The orbits are almost vertical with large eccentricities when considered as satellites of the smaller primary. For the case under consideration {L\, Sun-Barycenter) an additional problem is the possibility of close encounters with the Moon. In a rather simple model we can consider that the Moon can reach any point contained in a set A around the barycenter of the Earth-Moon system defined as follows A={(z,y,z)

| r = (x2 + y2 + z2)1/2

€ [rure],

\axcsin(z/r)\

< iM] ,

where (x,y,z) are the coordinates with respect to the barycenter, and the ranges are given by r* = OM(1 - MM)(1 - CM), re = a M ( l - MM)(1 + £M)- Here aM, P-M, &M and IM refer to the values of the semimajor axis, mass ratio, eccentricity and inclination (with respect to the ecliptic) of the Moon taken as a satellite of the Earth, respectively. Figure 2.36 shows the values of pm%n, Pmax and dmin against xmax, where pmin and pmax refer to the minimal and maximal distances to the barycenter and dmin to the minimal distance to A, of points of the orbit. We note that there are interesting halo orbits of the stable type far enough from Earth and Moon (pmin > 350 000 km, dmin > 160000 km). These orbits are exterior to the orbit of the Moon. After, when xmax decreases, there is a zone of mild instability, but the orbit is too close to the Moon. Finally, a new zone, still of mild instability, of orbits far from the Earth and the Moon is found, but this time they are orbits interior to the orbit of the Moon (e.g. pmin = 60000 km, dmin = 160000 km or pmin = 35000 km, dmin = 240000 km). Of course, some problem in the transfer can be foreseen for these orbits because of their almost polar character. When the perturbations are considered, these stable halo orbits present a formidable problem. The analytic theory for the halo orbits of the RTBP is no longer

Numerical

Determination

of the Families of Halo Orbits

123

^012

E-M d i s t a n c e .O02

Fig. 2.36 Minimal and maximal distances to the barycenter and to A against the value for orbits near the stable zone. {L\ case, Sun-Barycenter system.)

xmax,

available because the distance to one primary is much less than the distance to L\. It is then natural to take as starting point a theory centered at the smaller primary. However if the effect of the Moon is considered, as it should be, we have dmax (maximal distance to points of A of points of the orbit) > pmin and dmin < pmax • Therefore, the perturbation is produced by a body that cannot be considered interior nor exterior. Furthermore, the only nice way to obtain quasi-periodic solutions is the use of analytic theories. 2.3.4.3

The L3 Case

Missions in orbits around L3 have also been considered of interest, especially for the Sun-Barycenter system. A spacecraft near the L3 point for this system, at a convenient angular distance from the Sun, allows a continuous observation of the evolution of the solar surface when used simultaneously with Earth based observatories. The halo orbits in the Earth-Moon system associated to L3 have not been considered till now. They have the drawback of being of large amplitude. They are strongly perturbed by the Sun and the minimal distance to the Earth is relatively small. Then we analyze here only the case of the Sun-Barycenter system. As in the Earth-Moon case, the amplitude is quite large. They are poorly

124

Halo Orbits. Analytic

and Numerical

Study

numerically conditioned (all the eigenvalues are close to 1; they are exactly 1 for the two-body problem). Furthermore, the effect of Jupiter and Venus is much more important than the one due to the Earth. We propose a different approach: to consider the motion of the spacecraft as a two-body problem around the Sun perturbed by the influence of the planets (the Earth is merely seen as another perturbing body).

yaxis

Fig. 2.37 Two-body problem periodic orbit, around L3, with e' = .088, i' = 12°. The orbit stays roughly at 6° of angular distance to the Sun.

Let us choose a two-body orbit with the same semimajor axis and the same argument of the perihelion than the Earth. Let e, e' be the eccentricities of the Earth and the spacecraft orbits and i' the inclination, with respect to the ecliptic, of the spacecraft orbit. If the spacecraft is at his aphelion when the Earth is at the perihelion, then the orbit of the spacecraft seen from the Earth, is roughly an ellipse centered at the Sun with horizontal semiaxis e + e' and vertical one «'/2, provided e', i' are not too large. If both, the Earth and the spacecraft, are simultaneously at the perihelion but the arguments of the perihelion differ in n, then e + e' should be replaced by |e' — e|. In the first case taking, for instance, e' = 0.088, i' = 12°, produces an orbit which stays roughly at 6° of angular distance to the Sun, except for the small perturbations due to the planets. This is a starting orbit for a quasiperiodic orbit near this one. Figure 2.37 shows this orbit, seen from the Earth, when two-body motions are considered for the Earth and the spacecraft. The Sun is placed at the origin and each division corresponds to one degree. 2.3.5

Bifurcations

As in 2.2.3, let' B be the differential matrix of the Poincare map after one revolution. In order to clarify the results, we take (a;, z, x, C, z), where C is the Jacobi constant, as coordinates, and we suppose that B is expressed in these coordinates. Therefore,

Numerical

Determination

125

of the Families of Halo Orbits

if B = {bij), using the Jacobi integral, we have 64^ = 0 for i = 1,2,3 and 5 and bu = 1. We suppose that we have a periodic orbit such that one of the stability parameters Tri equals 2. Then, the corresponding eigenvalue Aj equals 1. In this case, it could be a bifurcation of the family. The possible bifurcated directions will be given by the eigenvectors of B with eigenvalue equal to 1. If we look for these eigenvectors, we must solve the following linear system (611 - l ) A z + b21Ax+ b31Ax+ 641AZ+

bX2Az+ (622 - l ) A z + b32Az+ b42Az+

bl3Ax+ b23Ax+ (633 - 1)A±+ 643Ai;+

&14AC+ 6i 5 Ai &24AC+ b25Az 634AC+ b35Az 614AC+ (615 - l ) A i

= = = =

0, 0, 0, 0.

The matrix associated to this system has been computed numerically for the orbits which have one of the stability parameters equal to 2. Table 2.19 gives a sample of results,

M

Case

Initial conditions

Stability and period

a

(Tn,Tr2,T) 0.3040357143E-5 (S-B)

£1

0.3040357143E-5 (S-B)

L2

0.012150668 (E-M)

£1

0.012150668 (E-M)

L2

-0.999837825844 -0.3017484368E-2 0.4220329826E-1 -1.00017447374 -0.2997772215E-2 -0.4230266200E-1 -0.871955999839 0.190171717932 -0.237271805283 -0.912788259977 0.207164259496 -0.154417672012

Table 2.19

2 -0.10094872644 1.14369183704 2 -1.0507775776 1.1483055703 2 -0.75442706734 1.11504012900 2 -4.65895383148 0.91586251842

0.39243499

-0.40243634

2.3921474

1.5564530

Bifurcation periodic halo orbits.

Let a, c, d, e, g, h, m, n, p, q and a be constants depending on the orbit. In each case it has been observed numerically that the matrix has the following form / a e h \ ah

aa ae ah a2h

c d a aa

m n p q

d \ g e ae J

and the rank is equal to 4. The parameter a is given in Table 2.19. Therefore, the

Halo Orbits. Analytic

126

and Numerical

Study

previous system gives only one direction to obtain periodic orbits. This direction corresponds to the "actual" family and no other branches can be found. This fact can be easily explained: For this periodic orbit the eigenvalues are Aj, 1/Ai, 1 (triple). Then, the generic Jordan form will be ( Xi 0 0 B = 0

V o

0 1/Ai 0 0

o

0 0 0 0 1 1 0 1

0 \ 0 0 1

o o iJ

Therefore, there is only one eigenvector of eigenvalue 1 and B — Id has rank equal to 4. 2.3.6

Termination of the Families of Halo Orbits. lem. The Rotating Two-body Problem

2.3.6.1

The Hill

Prob-

Introduction

We recall that Hill's problem is obtained from the RTBP using the change of variables xi = (x + {l-n))fj, where (x,y,z)

1/3

, x2 = W

-1/3

X3 = Z\l

-1/3

are the synodic coordinates and /i the mass ratio. Then we obtain Xi

+ ^{3xl-±(xl+xl))

x\

=

2x2 + 3a;i

x2

=

-2x1-°^+»-1/3(-Zx1x2)

x3

-

+

+ 0(^%

0(^3),

1/3 ( - 3 x 1 x 3 ) + 0(M2/3), -xE3 3 - ^ + M -

where r 2 = x\ + x\ + x\ and the expressions in /i 2 / 3 are uniformly bounded in a neighborhood of the small primary, i.e., a finite neighborhood of the origin in the {x,\, X2, ^-coordinates. In a more compact form we get Xl

'

X2 X3

/

\

( + 11

Xl

j ^ l K Xz X2

-^/ 3 V {-X\ +

^(Xyxl+X^+O^3).

The goal of this subsection is to analyze first the case /i = 0 studying the stability of the so-called vertical periodic orbits (a similar study has been done by Breakwell et al., see [1],[7] and [8]), and after, to detect the bifurcation to halo periodic orbits for Hill's problem. By symmetry reasons the halo families for Hill's

Numerical Determination

127

of the Families of Halo Orbits

problem, associated to L\ and L2 are symmetric. The numerical continuation of these orbits is also presented. Then, the perturbations due to /i are included to recover the initial RTBP. A breaking of symmetry is obtained (as noted in [1] for the Earth-Moon mass ratio) and is fully analyzed here. The passage of the halo families near vertical orbits is described and the analysis done allows us to show how the family related to L2 terminates and how the family related to L\ passes near vertical orbits. Further numerical work allows us to determine the termination of this family. 2.3.6.2

The KS Transform

Extensive use is made in the analysis of the KS transform (see [14]). Let u e E 4 and

L(u) = \

-u2

-1*3

Ui

U2

U\

—U4

-u3

«3

Ui

Ml

U2

u2

-Ui

"1

(

"4

-u3

\

/

We make the transformation x = L(u)u where x is embedded in E 4 taking the fourth component as zero. Then x\ = u\ — u\ — u\ + u\,

x2 = 2(uiu2 — U3U4),

xz — 2{u\uz + u2Ui),

r = \u\2 and the time t is scaled to a new time s through t' = |u| 2 , where ' denotes differentiation with respect to s. There is, of course, some freedom in obtaining u from x: each point x gives rise to a circle in the u variables. The related changes of velocities are x = 2 fis , or u' = \LT{u)x. The governing differential equations are h ""+0^

1 d = i4du t:

4 + u\f

MaK(«5f

+l \

U1P1 + U2p2 ^ -u2pi

+

uip2

-U3P1

-

Uip2

+fi

(

+

-U3Pl -

V t' = |u|

A P)„ . H Adu

V

2-,3

(-(«l-«2-«3+«4)

+ u\u\ + u\u\ + ulul))] ) + 0 ( / i 2 / 3 ) ,

UiP! + U2p2 -U2P1

2{u[,u'2,u3,u'i) :

1/3

U4P1 - U3p2 J

+6(ul - u\ - u\ + u\){u\ul

ti =

+ 2(ui« 3 + u2m)2) J

^

Uip2 Uip2

uip1 - u3p2

j

128

Halo Orbits. Analytic

and Numerical

Study

where h is given by -2

h

=\~

„2

\\*\2 + ^T - y - /*1/3 (- x ? + l^i^l + xixD^j ,

and 4 Pi

=

pj^(

« 2 " i + U i U 2 - U 4 U 3 - U3U4),

P2

=

p j 2 ( - U i u ' i + U2U2 +

4 u

3 « 3 - W4U 4 )-

This implies h' = 0, making things easier. 2.3.6.3

Vertical Periodic Orbits for Hill's Problem. Local Analysis

If we consider fi = 0 and look for solutions such that x\ = X2 = 0, then it is enough to put iti = U3, U2 = U4 = 0. We get the simple equation u'l + -Ul

+ 3uf = 0,

with the first integral u'l +-huj

+ u*

=-.

This gives rise to vertical collision orbits in the physical space. The KS transform regularizes these equations. As it is a one-dimensional problem this regularization can also be achieved using the Levi-Civita transformation, but we are interested in the spatial perturbations. Let X3 = xzmax be the maximum distance to the origin of the collision orbit. Then x — 0, and therefore 1 1

_

•*•

X3max

T2

^"imax

^

Initial conditions are obtained through m = {xzmax/2)ll2, u[ = 0. The equation can be solved by using elliptic functions. Let us analyze the behavior near these vertical periodic orbits. Let WQ = (ui,0,ui,u'1,0,u[,0)T be the positions and velocities in the E 8 space giving the vertical periodic orbits of Hill's problem. We write w = WQ + jU1//3u as the first order solution in (j,1/3 of the general equation. The variational equation associated to (u',u",h')T along the vertical periodic 9 9 orbits \sA' = M-A, A(0) =I,Ae £ ( E , E ) with M

=

Numerical Determination

129

of the Families of Halo Orbits

where _ i » _ , ,U4

( Ai

( =

V

-14uf

2u]u[ — h2 u\

0

2u\ 0 -2u\ 0

W

2

-2u\

-2u\ 0 2u\

—2u\' A \ -2u\

2uiUi

2u u[

-\\u\ 2u\u[

\

A,

l

2

—2u\u'x

-t-«!

—2uiu'1

0

-2«f \

2uf

0

0

2u\

)

/ -«i/2 \

0

, b =

oy

-2u\

1

2MIU'1

l

-«i/2

{

0

/

We denote as M, the matrix M with the last row (related to the nonexistent variation of h) suppressed. A is similarly defined. Inserting w in the general equations, we get for the terms of order fj,1/3 the equation 7 0) n\T v' = Mv + (0,0,0,0, -6u[, 0,6u[,

Let vp be a solution of this last equation with vp(0) = 0. Then the general solution is v

v(s)=A(s)[

2.3.6.4

j®

Hill's Problem Bifurcations.

)+vp(s).

Perturbations

Due to fi

We look for values of v giving rise to periodic orbits for the perturbed problem and to bifurcated periodic orbits of Hill's problem. Starting from wo(0) + /i1/3w(0) and performing the integration for half a period, s/, in the u-space (recall that periods are doubled in the KS-space) we should impose final conditions equal to —wo(0) — ^i 1 / 3 u(0). The final conditions are

- Wo (0)+M 1/3 U(*/)

v(0) Ah

+ vp{sf) ) +As/x 1 / 3 z,

where A s ^ 1 / 3 allows for the variation of period and z is the main part of the vector field at the point wo(sf) = —wo(0), i.e. x = (0,0,0,0,-^Ul

- ul,0,-^Ul

~

ul,0)J=Sf.

Therefore the condition for periodic orbit is (A(sf)+I)[

v(0) "^ )+vp(sf)

+ zAs = 0.

130

Halo Orbits. Analytic

and Numerical

Study

We look for symmetric halo orbits. They are specified by values of x\, x3, ±2 and X2 = x\ = x3 = 0 as initial conditions. Taking v(0) = (ui,0,U3,0,0,u 6 ,0,

-v6)T,

we get, to the first order in /4 1 / 3 xi

=

X2 = x3

n1/32u1(vi x3max

-v3), 1/3

+ n 2m(vi

+ v3),

1/3

=

fi 2v6/ui.

We rearrange the order of the components of v as follows (vi, v3, V2, «4, V5, v-j, vg, v$). Then A(sj) has the form

/

V

a -I-a a -a 9 -9 -9 -9 m P m P -t t t -t 0 0

-b —e -d d b e -b d -d b e —e h -l-h e / k —e -l-h h I —e e k —r r n —n q -1-q r —n n -1-7 —r q —X X u —u y -1-2/ X —x —u u -1-2/ 2/ 0 0 0 0 0 0

o\ 0 0 0 z z 0 0 1/

where a,...,z are suitable numbers which can be obtained by numerical integration. Then the conditions for periodic orbit, taking into account that due to the symmetries we have [vP3, vP3, vP3, vP3)

=

— (vpi, vP2, vP5, vP6),

are / a+1 9 m P \ t

-I-a -g p m -t

2e k-1 2r —2r 2(1 + 2/)

0 \ 0 z z 0/

v3 «P5

+

V

P6

V Ah J

\ vP7 )

( +

As

-fui -3uf

= 0.

- | u i -3uf 0 \

First we remark that the role of Ah and As is similar and nonessential, because

Numerical

Determination

of the Families of Halo Orbits

131

we can subtract the fourth equation from the third; after recover zAh + ( - - u i - 3 u ? ) A s . In fact Ah is obtained through

Ah=-(v1-rv3)(-^

+4u\

Now we suppose no perturbations in /x 1 / 3 . Therefore the term vp disappears and it remains only / a+ 1 2e \ 9 k-l m —p Ar V t 2(1 + y ) /

vi-v3 V6

\ J

= N

\

Vl-v3 v6

If rank N = 2, the only solution is vi = V3, VQ — 0. This means variation only in x3, i.e., the nearby periodic orbits to the given orbit are collision orbits. If rank N = 1 the solution is 2e vi-v3 = 7-rV6. a+1 This happens only for the value X3max = 0.832111052 and then, the half period is 0.7175304941. Then we have vi - v3 = -0.91642485u 6 or, equivalently x2 = —2.6227195x1. Taking VQ as a parameter, the linear terms in As, Ah and v\ + V3 are zero. This is also true for X3 and AT as functions of x\. Only quadratic terms are present. On the opposite side, if we consider the terms in /i 1 / 3 , a solution is obtained if and only if rank N = 2. The values of v\, V3 and ^6 can be obtained for any value of X3max ^ x^max. If we look for initial conditions of the perturbed problem with the same x3max, we impose v\ + v3 = 0. Then the behavior of the initial conditions xi and i 2 as a function of x3max has an asymptote at X3max = x3max. For x3max < x3max, we get xx < 0, x2 > 0, and for x3max > x\max, xx > 0, x2 < 0 is obtained. For large values of x3max we get Xl

x2

= ^0(xlmax) = ^0(xlmax)

+ +

0(^% 0(^3).

In a similar way, for small values of X3max we get Xl

= ^30(xlmax)

x2

=

+

0(^3),

M 1 / 3 0^3max) (xLai)+0(^3).

Figure 2.38 summarizes the results of the initial conditions leading to a periodic orbit.

132

Halo Orbits. Analytic

x

and Numerical

Study

3|-'o(M 1 / 3 ) halo .orbits for Hill's problem

family of halo orbits associated to L for the RTBP

Fig. 2.38

family of halo orbits associated to L. for the RTBP

Periodic orbits of the perturbed Hill's problem.

The integration up to the value of ui = 0 allows us to study the minimal distance (essentially xzmin) to the small primary along the periodic orbit of the RTBP, seen as 2/3 a perturbation of Hill's one. For large values of xzmax we get xzr, = »,*2/3 o(xZax) 2//3 and for small values of xzmax we have xzmin = M 0(a;| m a x ). This shows that the 1/2 families end in a collision orbit with the small primary, while the L\ families approach the small primary only up to a finite distance. 2.3.6.5

Numerical Results

Table 2.20 offers data for the family of halo orbits of Hill's problem related to L\. As it has been stated this family, starting at a Lyapunov periodic orbit, terminates at a vertical collision orbit. The related plots are given in Figures 2.39 and 2.40. Concerning the termination of the halo families for fi ^ 0, the data have already been given in 2.3.1. The analytically predicted behavior is easily checked. Figures 2.41 to 2.44 show the orbits numbers 4, 8, 13 and 16 of Table 2.20.

1 I.C. F.C. T 2 I.C. F.C. T 3 I.C. F.C.

0.100240756040E-1 -0.614033477276E+0 0.774643925678E+0 -0.804023963284E-2 0.670476922815E+0 0.581161387194E+0 0.154071229048E+1 P 0.172817252861E+4 Q 0.199981416849E+1 0.774603431186E+0 0.200100483023E-1 -0.614715429796E+0 -0.160444126621E-1 0.671528400279E+0 0.580853295063E+0 0.154068553152E+1 P 0.172529390216E+4 Q 0.199925749978E+1 0.299539450312E-1 -0.615841755183E+0 0.774535981628E+0 -0.240039837002E-1 0.673266779274E+0 0.580343546825E+0

Numerical Determination

T 4 I.C. F.C. T 5 I.C. F.C. T 6 I.C. F.C. T 7 I.C. F.C. T 8 I.C. F.C. T 9 I.C. F.C. T 10 I.C. F.C. T 11 I.C. F.C. T 12 I.C. F.C. T 13 I.C. F.C. T 14 I.C. F.C. T 15 I.C. F.C. T 16 I.C. F.C. T 17 I.C. F.C. T

0.154064116128E+1 0.690794494953E+0 0.310165907220E+0 0.149050839924E+1 0.656942041229E+0 0.236352680621E+0 0.145956881632E+1 0.597849885650E+0 0.132674672848E+0 0.138565007566E+1 0.585037319529E+0 0.114729824684E+0 0.136668465370E+1 0.578296025660E+0 0.105948124723E+0 0.135638076039E+1 0.571358058289E+0 0.973792155330E-1 0.134557028752E+1 0.452999137162E+0 0.121028797889E-1 0.114828126758E+1 0.446822705466E+0 0.100429629650E-1 0.113806917214E+1 0.405879526334E+0 0.145182538013E-3 0.107206084246E+1 0.400317152490E+0 -0.771388120737E-3 0.106335677735E+1 0.368181009067E+0 -0.452076434263E-2 0.101452007602E+1 0.313740466807E+0 -0.651911556977E-2 0.938216303160E+0 0.263681524618E+0 -0.565642572377E-2 0.876305074160E+0 0.175970872842E-1 -0.307645152160E-5 0.718270500300E+0 Table 2.20

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

of the Families of Halo Orbits

0.172053702793E+4 0.575898597664E+0 -0.338600018351E+0 0.274720202813E+3 0.658349082233E+0 -0.348129482699E+0 0.146373561328E+3 0.757834465344E+0 -0.328657018909E+0 0.503290042441E+2 0.773160108057E+0 -0.320382983515E+0 0.400746969547E+2 0.780447616405E+0 -0.315639193060E+0 0.355479493984E+2 0.787420577609E+0 -0.310517463045E+0 0.314153536523E+2 0.847817722579E+0 -0.208802062288E+0 0.210984939367E+1 0.848960429997E+0 -0.203514668135E+0 0.164792148711E+1 0.853757156983E+0 -0.169719317059E+0 0.420266089909E+0 0.854097132381E+0 -0.165316895899E+0 0.424480434172E+0 0.854945224969E+0 -0.140863013578E+0 0.618364087489E+0 0.853189134423E+0 -0.103489727142E+0 0.105023808895E+1 0.849461830804E+0 -0.738595756455E-1 0.137883036218E+1 0.832215599486E+0 -0.339513206544E-3 0.199813366128E+1

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

0.199832876080E+1 -0.981507276758E+0 0.147279082491E+1 -0.626363687674E-1 -0.102661648745E+1 0.168573054521E+1 -0.987010749017E+0 -0.105476211040E+1 0.203231599900E+1 -0.193156429849E+1 -0.105408352183E+1 0.210551098084E+1 -0.198914983642E+1 -0.105289695319E+1 0.214393355082E+1 -0.199959219168E+1 -0.105111024099E+1 0.218346369778E+1 -0.199709922501E+1 -0.953097615766E+0 0.290691989052E+1 -0.103394590755E+1 -0.945316524849E+0 0.295041333081E+1 -0.102560308296E+1 -0.888776825034E+0 0.326562247209E+1 -0.185560394645E+1 -0.880473575208E+0 0.331272711975E+1 -0.204057385503E+1 -0.829819861771E+0 0.361045690096E+1 -0.295874610955E+1 -0.734271320619E+0 0.424834746044E+1 -0.377626687745E+1 -0.636400838509E+0 0.506590642223E+1 -0.395606611614E+1 -0.461333445654E-1 0.767387536931E+2 -0.311442615359E+1

Periodic halo orbits for the L\ case of Hill's problem.

134

Halo Orbits. Analytic

Fig. 2.39 problem.

and Numerical

Study

Period, T, and stability indices, P, Q, for the periodic halo orbits of the L\ point, Hill's

Fig. 2.40

Physical parameters of the periodic halo orbits of the L\ point, Hill's problem.

Numerical

Determination

SlnULATlOK OF CONTROL FO« THE RTlPi

V(4) • « V ( S ) - - # , 9«1S*TK7C« Vf(5> • V<«) • • • » — • • — • VF • 2.»!•!£•• FINAL TIKE 0*f.

135

of the Families of Halo Orbits

~T

_ «.StlE*727Sl

i

i

r~

rr

t.4«4

i.zn

• -Is*

n. 4

— *1* _

-».U

_ -•.*» _

-».44M V -•.531 .•.31*

_,

,

«.3S>

*.4«5 .».453 . » . » • •

,t.S4t ,•-59* ,•.643

B.ttl

,_

—v.v-v* j » i i " T » . » " » T-w<arf* . v * v n

,w*a*v , v * w « ,

Fig. 2.41 Projections of the periodic orbit number 4 of Table 2.20, corresponding to a halo orbit around L3 for Hill's problem.

S l W U I T I o n OF CONTROL FOII THE R T 1 H I N I T I A L CONDITIONS FINAL CONDITIONS rrmaKvasT w t n •.57129S92SI . • * • « • • « • • • VF<2) • .7M447CIS4 . 7 M 4 4 7 C 1 S 4 VF<31 . — — • » • • • • YF<41 -t.»5a«M»38 .•C2SBSPC32 YF«S1 • — • • » » • • VFCi) FINAL TIKE a.7isTsm OKI. N. •

*.1K

-l

1

1

r

. I . 1 M ,«.1CS ,».224 ,«.2S3 ,9.348

"I

1

1

,*.4*1 . ! . « • .•-K10

T7.

».571

n

«.1SS

*.B24

1

I T«'«M r > - M t

«.at3 .•.342

1

7«.143

1

•.Stt

«.4»1

1

|».E71

r-

, • • • » • , > . I 4 3 , • . « • ,».43« ,1.57;

Fig. 2.42 Projections of the periodic orbit number 8 of Table 2.20, corresponding to a halo orbit around L3 for Hill's problem.

136

Halo Orbits.

Analyti\ic and Numerical

Study

sinuiATian OF CONTJOL FOR THE B T I F I

lHiTi*t"co«>itl»Nf.„ FINAL .4*43171MS YFUl . » • • • • • • • * • VFtai .BS4M7I324 VFO) . • • • • • * • • • • VF(4> .BS*47?S7C* VF4S> . — • « » » • • • YFCBI FINAL T I M • 2.1ZS7135S ¥(1> •

•commons • .4M31TJ52S • ••••»••—•• • «.SE4*971324 •.»»••••»»•» • -•••••473S7M • • .••»«••——

• ••49 • . ! • • I . I U . I . 2 H , • . < « • , I . J M , « . 3 M ,*.4«H

1 7«.319 r B . t i a ? • . ! • ? , » . — •

r l . H l , • . • 4 9 , « . 1 M . • . ! » . • . » • . « . » • . I . 3 M .•.3S» .•.4*4

, • • ! • » ,•.113 ,t.31B ,».<

Fig. 2.43 Projections of the periodic orbit number 13 of Table 2.20, corresponding to a halo orbit around L3 case for Hill's problem.

1

1

SimilATION OF COMTJOL FOB THI K W I INITIAL CONDITIONS FINAL CONDITIONS V<1) « •.SS3M1H4C VF<1) «.M3E»1S*4« VtZl • . • » • • • • • • * • Vffa> • I.MHMItM V(3) • •.S4»4S1«3*I VFO) • *.I494«I>3M V(4) • $.——•——• YT14I •.••••»••••« V(S> - -•.S3S4SM3K VF1S1 - -•.•3I444M31S V<«> 1. H « VFCC) • ••••••»•»••• FINAL TIHC 1.7WK1B1S OBI. H. 1»

-

4.2B9

-

•.130

-

• ••T*

1

1

1

1

1

- •]•••

'u.ai

J-

_ -•••7* _ -•.lii

_ -•.»*»\ _ -0.279

7 * - « ,*.w* t » . M > 1-

-i

r

1

1

1

1

^^

_

t.734

_

t.B19

_

I.M1

_

t.3M

_

•.171

_

•.197

_

•••4«//

^ * ^S^ ^S^ JT

yr

i

'<,.3>

1

1

,...»« , • . 1 2 9 i

1

, • . 1 * 3 j*.IBS I

1

~ - _ •.734 - _ %.%\jf - _ • .jfco - _ •Jm - — ••*?»

S

_

-

\ \ /

-

J -

/ -

— a.»4B

r..CT

-

.

•.isrx.

— -».»74 _ -•-•74 ; « . M S ,«.«2S , • • • * * , » . * « ,*.129 ,1.113 ,».I9S • . 2 3 * ,«.2S'

,..,„,...« • '(t.i>

T ».2*9

T*.I39

!••«. , • . • * > •

, • • • 7 * ,«.139

,..*•• ,..m

Fig. 2.44 Projections of the periodic orbit number 16 of Table 2.20, corresponding to a halo orbit around L% for Hill's problem.

Numerical Determination of the Families of Halo Orbits

2.3.6.6

137

The Rotating Two-body Problem

The restricted three-body problem when fi is small, can be considered as a perturbation of the rotating two-body problem. The perturbation becomes singular when the massless body approaches the position of the smaller primary. However, for the halo orbits associated to L3, there are no such approaches and it is not necessary to introduce changes, as we did before (using Hill's problem as limiting case). This is due to the fact that the perturbation introduced by the smaller primary is uniformly bounded, except in a region of finite size which contains this small body. For fi = 0, the restricted problem has a family of vertical symmetric collision orbits on the a;-axis. Pierre Guillaume, in [6], studied the continuation of this family to symmetric periodic orbits of the restricted problem for small values of /i. In order to prove this result, it is necessary to regularize the restricted problem by using again the KS variables. The regularization will be accomplished in the homogeneous Hamiltonian formalism. The Hamiltonian H is thus put in the homogeneous form H

syn = V

f\\y\? \

—7, I

1

(^12/2 - ^22/1)

T

\- flR + y0

where t = XQ and yo are considered as canonical conjugate variables, as well as Xi, 2/i, i = 1,2,3 and v is a function of the Xi, yi, % = 0,1,2,3. Hsyn is identically zero on each real orbit of the restricted problem. R is the perturbing function and r = ||a;|| = {x\ + x\ + xj)172. First, we rewrite the homogeneous Hamiltonian in sidereal variables: Hsid = Hsyn

=

f\\Y\\2 v ( ^-^2

1 --+nR r

+ Y0

After, we rewrite it in KS variables Xi+iX2

=

2(ui - m 4 )(u 3

+iu2),

X3

— \ui +iui\2

X0

=

Y1+iY2

=

2u0, 2 -((U3 + iu2)(vi - ivi) + (ui - iui)(v3 + iv2)),

= =

2 ~((ui+iu4)(vi-iv4) 2v0,

y3 Y0

- \u3 + iu2\2,

-

(u3-iu2)(v3+iv2)),

getting 1 /1 HKs = ^Hsid = v I - Y,(vt V fc=i

1 1 + vou2k) - - + -ixRr

138

Halo Orbits. Analytic and Numerical

Study

This KS transformation is not exactly the classical one. Now, we modify the KS variables: Uk

=

ukvl'4,

Vk

=

vkVQ1//l,

k =

l,...,4,

A; = 1 , . . . ,4 4

.

,

ukvk,

4vnU ^—' k=\

V0

=

v0

— 1/2

If we put v = rV0

' we get as new Hamiltonian HMKS

— HKS = H0 + fiHi,

where 4 H

V o = jT,( k+U2k)--^ • -*, A^:-k

*=1

1/2

and —

H"x 1=_

^"O

4

rR i ^ '

i"0

Finally, we consider the following transformation Uk

=

Vi

=

Uo = ^o

=

Ak cosE + Bk sinE, -A fc sinE 4-5^ A0 +

COSJB,

A; = 1 , . . . ,4 A: = 1 , . . . ,4 3

^ 3

82?03/2'

B0.

The new Hamiltonian He is numerically different from puted that it is He = HMKS

HMKS-

It can be com-

— H0 = /u#i •

It is therefore no longer identically zero on the real orbits. On the other hand, since it reduces to 0 if M = 0, Ak and Bk (k = 0 , . . . ,4) are all constant in the two-body problem. The symmetry equations x2 = y\ = y% = 0, in the modified KS variables give Ui cos 7 — U\ sin 7

=

0,

U2 cos(7 + c) -U3 sin(7 + c)

=

0,

V\ cos 7 + V4 sin 7

=

0,

V3 cos(7 + c) + V2 sin(7 + c)

=

0,

Uo =

c/2,

where 7 is a parameter and c is a simple time shift. So, a solution which verifies the given symmetry equations for two different values E = 0 and E — 77 of the independent variable E, will be periodic of period 2?? in E.

Numerical

Determination

of the Families of Halo Orbits

139

If the symmetry equations are to be satisfied at E = 0, we get the following expressions for Ui, V* as a function of E Ui

— Ai cos70 cosE + Hi sin70 sin E + O(p),

U2

— A2sin7ocosE - ^2COS7osin£J + 0 ( ^ ) ,

[73

=

A2 cos70 cos E + H2 sin 70 sin E + O(p),

Ui

=

Ai sin70 cos E - pi cos70 sin E + 0(p),

Vk

=

dUkldE, fc = l , . . . , 4 ,

where A», pi, i = 1,2 are suitable parameters. Writing again the symmetry equations for E = 77, after some manipulation the following system of equations is obtained Si

(Ai + m) sin(?7 + S) + 0{JJL) = 0,

S2

(Ai-^i)sin(r/-(5)+O0u) = 0 ,

S3

(A2 + /x2) sin(r? + 5 + T) + O(n) = 0,

SA

(A2 - /i 2 ) sin(r? - S + T) + 0(/x) = 0,

S5

r//(4B03/2(0))-T + O ( M ) = 0 ,

S6

(A 2 + A2 +

^

+

^2) _

1/(25^

2

( 0 ) ) + 0( M ) = 0,

where 77 is the period in the E variable and T the period in the t variable. Furthermore S = 7 — 7o- The last equation is HMKS = 0. For fj, = 0 we obtain the family of vertical periodic orbits taking 2T]

= kw, 2<5 = kir,

\2 = H2=Hi—

0,

Ai ^ 0,

because this implies £i

£2 = 0,

x3 — 2a cos £ ,

with a = Af. In order to verify the applicability of the Implicit Function Theorem to the above system of equations, for the particular solution obtained, we have that det

d(Si,...,S6) = -8p3sini(T), d{T, 77, S, p, a, a)

where Ai = pcosa, A2 = psina, p,i = a sin a, p,i = crcosa. As long as sin(T) 7^ 0, the determinant does not vanish and we can, apply the Implicit Function Theorem to get a two-parameter analytic symmetric periodic orbits by analytic continuation. The two parameters the initial value of the total energy (related with Bo(0)). The function sin(T) vanishes only when a 3//2 = j = l / | n | . For these shall have bifurcations. That is, the bifurcations are characterized by a of the synodic period of the orbit.

therefore, family of are p, and values we value 2JTT

140

Halo Orbits. Analytic and Numerical

Study

The natural termination of halo orbits related to L3 tends to the above bifurcation orbit for a = 1 when fi goes to 0. This follows from our numerical computation. max the natural termination of halo orbits related to L,

2(a-l)

max Fig. 2.45

Natural termination of halo orbits related to L3.

Some information about the termination of families of halo orbits in the L^ and L3 cases using almost vertical p.o. has also been obtained in the last years by Breakwell and coworkers (see [1], [7] and [8]). 2.3.7

The Elliptic

Restricted

Three-body

Problem

A simple model which takes into account the eccentricity e of the primaries is the elliptic RTBP. In that model mi and 7712 describe an elliptic orbit. Introducing a nonuniformly rotating and pulsating dimensionless coordinate system, the differential equations of the motion of the infinitesimal body are the following x - 2y

=

wx,

y + 2x

=

wy,

—

wz,

z+z where

to(x,y,z) 1 + e cos / ' with Z

r\

=

r\

=

'1

'2

^

{x-ii)2+y2+z2, (x-

/i + 1)2 +y2 + z2,

and the dot denotes derivation with respect to / , the true anomaly (for more details see [15]).

References

Case

e

Initial Conditions

141

Stability Parameters

Minimal Period

(TruTr2), 0

Li

.0167

Li

0

L2

.0167

L2

0

L2

.0167

L2

Table 2.21

-0.9942916393 0.0123668069 -0.0126062598 -0.9941967081 0.0125929230 -0.0123943055

-1.3883761492 -6.0949990833 (after time = 27r) -1.8533997644 -3.7775307995

2TT/3

-1.0055882256 0.0123931862 0.0124421639 -1.0056724211 0.0126248144 0.0122153129

-1.2057539436 -6.41473565 (after time = 2ir) -1.626954995 -4.2680692091

2TT/3

-1.0025013548 0.0121136604 0.0064124132 -1.0025243951 0.0122726284 0.0063604518

115.9618185 -1.0052614652 (after time = 2ir) 115.82620684 -1.0183058024

TT/2

2TT

2TT

2TT

Periodic halo orbits of the elliptic problem for the Sun-Barycenter system.

Some halo orbits for the elliptic problem have been obtained by prolongation of the halo orbits of the circular problem, which have a period commensurable with 1-K. We computed these halo orbits for the values of the eccentricities given in 2.1.2. The numerical results are given in Tables 2.21 and 2.22.

2.4

References [1] 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. [2] A. C. Clarke. "Stationary orbits". Journal of the British Astronomical Association, 57, 232-237, 1947. [3] R.W. Farquhar. "The control and use of libration point satellites". Technical Report TR R346, NASA, 1970. [4] R.W. Farquhar. "A halo-orbit lunar station". Astronomy and Aeronautics, 10, 59-63, 1972.

142

Halo Orbits. Analytic

and Numerical

e

Case

Initial Conditions

0

ii

0.054900489

Li

-0.8827822600 0.1942812082 -0.2187219101 -0.8756808042 0.2013026802 -0.2152146991

0

L2

0.054900489

L2

(x,z,y)

Table 2.22

-1.0637860021 0.2004019429 0.1776105379 -1.0637124071 0.2124599143 0.1631186925

Study

Stability Parameters (Trx.Tra), -1.8463890469 -4.6312124914 (after time = 2-K) 4.6227609249 -8.7977322238

Minimal Period

-0.0311902477 -2.5592470348 (after time = 27r) -1.8122048638 ± 0.4317680325 i (complex)

2TT/3

2TT/3

2TT

2TT

Periodic halo orbits of the elliptic problem for the Earth-Moon system.

[5] R.W. Farquhar, D.P. Muhonen, C. Newman and H. Heuberger. "The first libration point satellite. Mission overview and flight history". In AAS/AIAA Astrodynamics Specialist Conference, 1979. [6] P. Guillaume. "New periodic solutions of the three-dimensional restricted problem". Celestial Mechanics, 10 (4), 475-495, 1974. [7] K.C. Howell and J.V. Breakwell. "Almost rectilinear halo orbits". Celestial Mechanics, 32 (1), 29-52, 1984. [8] K.C. Howell. "Three-dimensional periodic halo orbits". Celestial Mechanics, 32 (1), 53-72, 1984. [9] H. Poincare. Les Methodes Nouvelles de la Mecanique Celeste. GauthierVillars, 1892, 1893, 1899. [10] D.L. Richardson. "Analytical construction of periodic orbits about the collinear points". Celestial Mechanics, 22 (3), 241-253, 1980. [11] D.L. Richardson. "A note on the Lagrangian formulation for motion about the collinear points". Celestial Mechanics, 22 (3), 231-235, 1980. [12] D.L. Richardson. "Halo orbit formulation for the ISEE-3 mission". J. Guidance and Control, 3, 543-548, 1980. [13] L. Steg and J.P. De Vries. "Earth-Moon libration points: Theory, existence and applications". Space Science Reviews, 5 (3), 210-233, 1966. [14] E.L. Stiefel and G. Scheifele. Linear and Regular Celestial Mechanics. Springer Verlag, 1971. [15] V. Szebehely. Theory of Orbits. Academic Press, 1967.

Chapter 3

The Neighborhood of the Halo Orbits: Numerical Study and Applications

The main objective of this chapter is to introduce the basic ideas concerning the proposed method for station keeping in the case of perturbed halo orbits. To reach this goal we have to analyze, in the simplest case, i.e., halo orbit, the behavior of the solutions of the first order variational equations. In this chapter this is done numerically, in order to have insight for the analytic methods used in Chapter 4. The main topics used are the Floquet modes, the invariant manifolds and what are called projection factors. The ideas introduced in this chapter are used later in Chapters 4 and 8, and applied to the simulation in Chapter 9. The execution of maneuvers using the solar radiation pressure as acting force has been proposed by several authors in science and in fiction. Letting aside technical difficulties, this possibility is also discussed here. The global stable manifolds of the halo orbits can be used to minimize the transfer step. Their computation ends the chapter.

3.1

Numerical Study of the Local Invariant Manifolds

3.1.1

The First Order Variational

Equations

The understanding of the local behavior near the halo orbit is basic for studying the effect on the orbit of errors due to the model, tracking and maneuvers. As it has been shown in 2.3.2, the halo orbits have two-dimensional stable, Ws, and unstable, Wu, manifolds. We consider the equations of motion for the spatial RTBP as given in 2.3.1.3. In a compact form they can be written as y = f(y),

yeE6.

(3.1)

To study the neighborhood of a given orbit, the first step is done by means of 143

144

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

the first order variational equations A = Df(y(t))A,

A(0) = J, Ae £ ( E 6 , R 6 ) .

(3.2)

Let T{y) denotes the point, in the orbit passing through y at t = 0, at a moment t = r. The solution A(T) of 3.2, is the differential matrix, DT, of
Mv + h) =

My) + D^T(y)-h + 0(\h\2).

Therefore, <j)T (y) + A(r)h is a good approximation to <j>T(y + h), provided h is small enough. Higher order variational equations are easily obtained by Taylor expansion. For instance, the second order ones are B = Df(y(t))B

D2f(y(t))-(A,A),

+

with B(0) = 0, B € £(E 6 x E 6 , E 6 ) . This means that for any h £ E6 we should have B(h, h) = Df(y(t)) • (B(h, h)) + D2f{y(t)){Ah, Ah), or componentwise

r

r,s

However, as it will be seen later, for this application we do not require variational equations of order greater than or equal to two. From the symmetry (a;, y, z, t) -¥ (x, -y, z, -t)

(3.3)

of the equations we get a similar relation for the variational equations. In vectorial form, equation (3.2) can be written as yi=Df{y{t))yu

yi(0)

= h,

6

yieR

.

Let yi (t) = (£{t),T)(t),((t),£(t),ri(t),((t))T be a solution of the variational equations associated to the RTBP. Then, (£(-*), -/?(-*), <(-*)> -£(-*),»)(-*), ~C(-*)) is also a solution. Before going to the analytic construction of the solutions of the variational equations, we have performed several numerical computations to get some hints on the behavior. Let T be the period of the halo orbit. Then, the matrix M = A(T) is called the monodromy matrix and plays a crucial role in what follows. The matrix M has been obtained numerically in the course of the computation of the periodic orbits (see 2.3.1).

Numerical Study of the Local Invariant

3.1.2

Floquet Theory for Halo Orbits.

145

Manifolds

The Floquet

Modes

First we analyze the eigenvalues of the monodromy matrix M. They are Ai > 1,A2 < 1, A3 = A4 — 1,A5 = A6 complex of modulus 1. We remark that in 2.3.1.2 two traces, T r i ( = P) and XV2(= Q), are given. They are related to the eigenvalues through 7>i = Ai + A2, Tr2 = A5 + A6. The three pairs of eigenvalues have the following geometrical meaning a) The first pair (Ai, A2), with Ai • A2 = 1, is associated to the unstable character of the small and medium size halo orbits. They describe a hyperbolic character. The value Ai is the dominant eigenvalue. The eigenvector associated to Ai (we call it ei(0)) gives the most expanding direction. Using Dcf)T, we can obtain the image of ei(0) under the variational flow: ei(r) = D4>T • ei(0). At each point of the halo orbit, the vector ei(r), together with the vector tangent to the orbit, span a plane. This plane is tangent to the local unstable manifold Wi" . By forward continuation of points Wi" under the flow, we can produce the full unstable manifold. Using the symmetry, as explained in 3.1.1, we obtain the stable manifold and, in particular, the vector e2(r) = D(j>T • e2(0). Of course e2(0) has eigenvalue A 2 under M. We recall here the result given in 2.3.2 concerning the fact that as the zamplitude, j3, increases, then, the maximal eigenvalue, Ai, decreases. For instance, for j3 = 0.08 we have Ai = 1727, but for /3 = 0.199 its value is 1540. b) The second pair, (A3,A4) = (1,1), is associated to neutral variables (i.e., nonunstable modes). However, there is only one eigenvector of M with eigenvalue equal to one. This vector is, obviously, the tangent vector to the orbit. We call it e3(0) at time t = 0 and e${t) in general. The other eigenvalue equal to 1 is associated to variations of the energy, or of any equivalent variable. In the range of small size halo orbits, a suitable variable associated to this eigenvalue is a parameter of the family of halo orbits, for instance the z-amplitude, /?. Along the orbit, these two vectors generate a family of planes which are sent one into another by the variational flow. The monodromy matrix restricted to this plane has the form 1 e 0 1 The fact that e is not zero is due to the variation of the period when the orbit changes across the family. c) The third pair is associated to the complex eigenvalues, A5 = A~6, of modulus one. The monodromy matrix, restricted to the plane spanned by the real

146

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

and imaginary parts of the eigenvectors associated to A5, Ag, has the form cosT sinT

-sinT cosT

i.e., it is a rotation. Therefore, in a suitable basis, the monodromy matrix has the form

A1

f

0

A(T) = M =

0 1

K

\ 1 0

e 1

V

(3.4) cosT sinT

-sinT cosT

where all the elements out of the boxes are zero. With this matrix in mind we define the Floquet modes. First of all, we recall that Floquet theory deals with the solution of linear differential equations with periodic coefficients. This is, indeed, the case of (3.2) when y(t) is a halo solution. We look for 6 T-periodic vectors, ej(r), i = 1 , . . . ,6, such that we can recover ej(£) easily from them. To define the first mode we write down ei(r) = ei(r)exp I - r

InAi

(3.5)

Obviously ei(0) = ei(0) = ei(T), as desired. The second mode is obtained, in a similar way, starting at the eigenvector e2(0) with eigenvalue Af1 and multiplying A(r) e2(0) by exp(r-lnAi/T). From the symmetry (3.3), it follows that if ei(r) = (fi(r),7Ji(r),Ci(r),|i(r),Tyi(r),C{(r)) T , then e 2 (r) = ( e i ( - r ) , - ^ i ( - r ) , C i ( - r ) , - ^ ( - r ) , r ? i ( - r ) , - C ; ( - r ) ) T .

(3.6)

The third mode is the tangent vector to the orbit. It is already a periodic function ez{r). To obtain the fourth mode we proceed as follows: We start with the tangent vector to the family at the initial point. We recall from Chapter 2, that the initial point is chosen in the symmetry plane y = 0. Then, the initial conditions are x(0), 2/(0) = 0, z(0), i(0) = 0, 2/(0), i(0) = 0. So, it is enough to specify x,z,y at the initial point, which are all of them functions of the parameter /?. Hence, the tangent vector to the family is written as

Then, we apply the matrix A(T). The vector obtained in this way is decomposed as ei(T) = ei(T)+e(T)e3(T),

(3.7)

Numerical Study of the Local Invariant

Manifolds

147

where e ^ r ) is chosen orthogonal to e 3 (r), such that it is continuous and with e 4 (0) = e4(0). e(r) is an auxiliary function; it equals the value e, appearing in (3.4), when T = T. In this way, ^ ( r ) is a periodic function, and e(r) is the sum of a linear function, e • T/T, and a T-periodic function. The fifth and sixth modes are obtained as follows: Let A5 = r + is. We denote by e 5 (0) + i • e 6 (0) the complex eigenvector associated to A5. From M (es (0) + ie% (0)) = A5(e 5 (0)+ie 6 (0)) we get two real equations by equating the real and imaginary parts. By elimination of e6(0) we have that e 5 (0) belongs to the kernel of M2 -2r-M + I, and it is clear that 2r = Tr2, using the previous remark. Of course, also e 6 (0) belongs to this kernel. Then e 5 (0) is determined looking for a solution of the type e 5 (0) = ( - , - , 1, - , - , 0 ) r . The reason for this choice is that for planar Lyapunov orbits, the z variable becomes uncoupled (in the variational equations) from the x, y variables and, therefore, z satisfies an equation of the type of the Hill's equation. All the solutions are of the type (0,0, —,0,0, — ) T , and because of the freedom in the election of the initial vector (the solutions are rotations), we are allowed to take e5(0) as (0,0,1,0,0,0) T . The small size halo orbits are not far from some planar Lyapunov ones and, therefore, our choice is reasonable. When es(0) is obtained, ee(0) follows from Me5(0)=re5(0)-se6(0). Starting with es(0), ee(0) we obtain e 5 (r), e§{r) by applying the matrix A(T). The fifth and sixth modes are denned as e 5 (r) = cos ( - 1 ^ ) e 5 (r) - sin ( ~ r ^ ) e 6 (r), (3.8) e 6 (r) = sin ( ~ r £ ) e 5 (r) + cos ( - 1 ^ ) e 6 (r). We remark that an analysis of the planar Lyapunov orbits shows that the vectors h{T)i e&iT) rotate in a clockwise sense. This follows from the equation

e

C"+¥>(r)C = 0. Denning £i = C, C2 = C' we have Ci = C2, C2 = - °Therefore the angle T appearing in (3.4) is negative and, due to the fact that for the bifurcation orbit it has an absolute value of 2-K, T is determined as T = arccos(r) — 2ir, where the arccos is found in the first quadrant. In this way e5(T) = e 5 (0) and e 6 (T) = e 6 (0) as desired. Suppose that e ^ r ) , i = 1 , . . . ,6, Ai and T are known, either by means of numerical computations or by analytic expansions. Then we can recover from (3.5) to (3.8) the vectors ej(r), i = 1 , . . . ,6. In this way we know everything about the local behavior of the solutions.

148

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

We should remark that the analytic computation of e^ proceeds in a different way as it will be explained in Chapter 4. It is also true that ei,e2,e3,e5 and e§ are not the same in the numerical and analytic computations. They can differ by a constant factor, but this is unessential. Returning to e^ the only thing we request is that the plane spanned by {e%,e%) coincides with the plane spanned by ( e ^ e j ) , where the superscript a or n refer to analytic or numerical, respectively. 3.1.3

Numerical folds

Computation

of the Unstable

and Stable

Mani-

The local computation of these invariant manifolds is equivalent to the computation of ei and e%The matrix M has been numerically obtained in the course of the computation of the halo periodic orbits (program PO, see 2.3.1). An enlargement of this previous program has been produced under the name PAPUS. It includes all the previous features of program PO, i.e. computation of periodic orbits along a halo family, plus all the variational computations to which we refer in this chapter. The initial vector ei(0) = ei(0) is computed using the power method. A few number of iterations is required because max|Aj/Ai|, for i — 2 , . . . ,6, is less than 10" 3 . The symmetry (3.3) produces immediately the vector e2(0) = e2(0). With the final data for the periodic orbit we start the integration of equation (3.2). As explained in 3.1.2, we proceed to the computation of ei(r)=exp(-AiT/r)i4(T)e x (0)

and e 2 (r) = exp(AiT/T)i4(r)e 2 (0).

A Fourier analysis of the vector ei(r) shows that the symmetry appearing in x, y, z, (i.e. x,z have only cosine terms and y only sine ones) for the halo orbit is no longer preserved. This is easily predictable because ei(0) does not satisfy the symmetry relation: y-component equal to zero. This will be useful for the analytic theory to be presented in Chapter 4. In what follows £,fj,C refer to the first 3 components of e~\ and £, 77, £ to the first three of e\. A rough examination of the Fourier analysis, made for different values of j3, shows several facts that are independent of (3: a) £ and f) have constant components which are close to the values obtained at the point L\ in a linear analysis. The ^-component is rather small. b) The most important periodic term is the first one. The other periodic terms contribute less than 1/4 to the global periodic term. The periodic term is small compared with the constant term. Hence a rough approximation is £ = 00 + ay COS(TW -
(3.9)

Numerical Study of the Local Invariant

149

Manifolds

Therefore the maneuvers should be performed, essentially, in the (x,y,x,y)subspace. We return to this in the next section. In (3.9) u> represents the frequency of the halo orbit. Let a = - In Ai/T. Then we get the components of ei(r) as £

=

exp(er"nj)(a0 + a\ cos(ra; — {) + ...),

7]

=

exp(2) + . . . ) ,

£

=

<JLJ£(—aiuism(Tu> — \) + ...),

fj

=

aujTj(—biwsm(T(jj — fa) + • • •),

where • = jf- We recall from Chapter 2 that r is linearly related to the physical time. More concretely, r is the time of the RTBP, i.e., it increases 2ir units in a revolution of the two main bodies. The angle between the invariant manifolds Wu and Ws is also computed. It is equal to the angle between e\ and &2- Using a) and b) we see that it is almost constant in the (x,y)-plane and that its value is, approximately, 2arctan(&o/ao)For instance, for /? = 0.08 the value of the angle is roughly 1 rad. This is not seriously affected by the value of /? nor by the small periodic terms. These small variations are also found when the angle is computed in the phase space. We remark that due to the freedom in the election of sign for the (normalized) eigenvectors, either an angle U(T) or 7r — V(T) can be found. Of course U(T) is T-periodic and even. 3.1.4

Numerical

Computation

of the Remaining

Floquet

Modes

The third mode is obtained starting with £3(0) equal to the tangent vector to the orbit at the initial point. Their components are (0,y,0,x,0,z)t=oThen ^ ( r ) = A(r)e3(0). A Fourier analysis of the vector ez shows that it has no independent terms and the six components have terms only in sin, cos, sin, cos, sin and cos, respectively, as it should be. They agree with the values computed analytically. As explained in 3.1.2, §4(0) is taken as the vector tangent to the family at the initial point. We recall from 2.3.1 that the family is defined through two equations: / 1 (a;,2;,y)o =: if = 0 and f2(x,z,y)0 =: Zf = 0. Using X for (x,z,y)T and F for 1 2 7 ( Z ; / ) the characteristic curve is F(X) = 0. Let G = DxF. Expressions for G are given in 2.3.1.3. Put G

_ ( 9n 3i2 3i3 "\ V 921 922 923 J '

Then, the tangent vector to the family is defined through (gn,gi2,gi3)T

A

{g2i,922,923)T-

150

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

The matrix G is available because it has been used in the process of continuation of the halo family. Then, e4 is defined by A(r)e 4 (0) = e 4 (r) + e(r)e 3 (r), and we have that e ^ r ) _L e~z(r). The fourth mode has independent terms in the first, third and fifth components. Concerning the purely periodic components they have, alternatively, only terms in cosine and sine. As it was mentioned in 3.1.2 e(r) is the sum of a linear function and a T-periodic function. The values of e(r) are rather large. For instance, for P = 0.08 the slope of the linear part is close to 1.5. The purely periodic part is odd and with large coefficients. The dominant term is a sin(rw) term, with amplitude « -3.7. The vectors £5, e§ are computed as described in 3.1.2. Concerning the parity properties shown by Fourier analysis, we obtain that es has components alternatively even and odd in r. The reverse holds for e%. 3.1.5

Results

of the Computations.

Consequences

and

Discussion

All the previously mentioned computations become part of the program PAPUS. First we compute the dominant eigenvalue and the normalized dominant eigenvectors of the monodromy matrix M and of the differential, DP, of the Poincare map P (see Table 3.1). Then, several checks are allowed, using numerical differentiation of P, to see up to which distance from the periodic orbit the linear behavior gives a reasonable approximation. This is done at the beginning of routine VARCOM. The next step is (optionally) to compute the nonlinear character of the stable/unstable manifold. Due to the symmetry, it is enough to compute the unstable one. The computations proceed as follows. An initial distance from the periodic orbit is required. This distance should be small enough in order that the product of this distance by Ai still gives points in a region where the linear and nonlinear manifolds are quite close. However, it cannot be too small to prevent rounding errors (and the fact that the initial conditions for the periodic orbit are not exactly known). For the purpose of selecting this initial factor, it is useful to make several checks as described before. We wish to produce points in the intersection of Wu with y = 0. The initial points are obtained using w = I.C. + V • factor • (1 + n • step), where I.C. means the initial conditions for the periodic orbit and V is the dominant eigenvector of DP. Factor is the small initial factor given, step is related to the relative distance between points in the nonlinear invariant manifold and n ranges from 0 to a given value N. For each value of n, starting at w the full Poincare map is used to produce points on the invariant manifold. We note that when n increases,

MAXIMAL EIGENVALUE COMPUTED FROM THE DIFFERENTIAL OF THE POINCARE MAP AND 0.1727955175697D+4 0.1727955175718D+4 DOMINANT EIGENVECTOR OF THE DIFFERENTIAL OF THE POINCARE MAP: 0.3335358054443D+0 0.8627300388346D-2 0.8684184942643D+0 -0.357077 DOMINANT EIGENVECTOR OF THE MONODROMY MATRIX: 0.3640187581334D+0 -0.1220085535615D+0 0.9415778222749D-2 0.8353988914446D+0 -0.3897114649572D+0 0.5243344313626D-1 FACTOR AND QUOTIENTS OF THE :[INCREMENTS, COMPONENTWISE 0.1706056457947D+4 0..1708579361595D+4 l.E-13 0..1708985018726D+4 0..1746885143570D+4 0.1749283624996D+4 0.. 1746745738708D+4 -l.E-13 0.1725792813808D+4 0..1726044966332D+4 l.E-12 0..1726043272031D+4 -l.E-12 0..1729905550470D+4 0.1730188577433D+4 0., 1729934476372D+4 0.1727741686229D+4 0., 1727769487212D+4 l.E-11 0..1727770186154D+4 -l.E-11 0..1728149260624D+4 0.1728179584292D+4 0. 1728151742688D+4 0.1727894009433D+4 0..1727921902095D+4 l.E-10 0..1727924180497D+4 0..1727985928561D+4 0.1728016433598D+4 0..1727988542722D+4 -l.E-10 0.1727555447147D+4 0..1727809314222D+4 l.E-09 0..1727832397123D+4 0..1728078036054D+4 0.1728355194077D+4 0.1728101176552D+4 -l.E-09 l.E-08 0.. 1726747072081D+4 0.1723982995171D+4 0.1725516556725D+4 -l.E-08 0..1729165608115D+4 0.1731936444599D+4 0.1729397060169D+4 0.1688638986850D+4 0. 1713715887210D+4 l.E-07 0..1715979883257D+4 0.1768189565042D+4 0.1742521410574D+4 -l.E-07 0..1740164233423D+4

0..1709 0..1745 0.,1726 0..1729 0..1727 0..1728 0..1727 0..1727 0,.1727 0..1728 0..1726 0..1729 0..1713 0,.1743

Table 3.1 Output of program PAPUS. Maximal eigenvector computed from the trace using t Poincare map and the monodromy matrix are used. Sun-Barycenter problem. Case L\. fi = 0

152

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

I + n • step can be larger than Ai. Then, to prevent large initial errors, factor is replaced by factor/Ai and, instead of applying P , we apply P 2 . This device is used as many times as required, each time dividing the current value of factor by Ai and increasing by 1 the number of iterates of P . A sample output of program PAPUS is given in Table 3.2. After working with positive values of factor, the negative branch is computed changing sign to factor. The output are linear and nonlinear approximations to Wu, with optional drawings. This computation is carried out by routine WUNL. The next step, done in part of the routine PRECON, computes the six Floquet modes as described before and performs the Fourier analysis. The integration is done with constant step, to get equally spaced (in time) points for the harmonic analysis. Usually 256 steps per revolution are enough. Also the angle v{r) between Wu and Ws and the function e(r) are given. Other magnitudes computed by PRECON are required for station keeping and shall be discussed later. The number of harmonics has been kept at a maximum of 15. The maximal error between the function and the trigonometric sum is also given. Output of this part of the program is given in Table 3.3 (Floquet modes), Table 3.4 (functions V(T), e(r) and p(r)), Table 3.5 (projection factors), Table 3.6 (unitary controls) and Table 3.7 (unitary signed gain factors). They were computed for a halo orbit of the Sun-Barycenter system around the L\ equilibrium point. The value of /? was taken equal to 0.08, that corresponds to the initial conditions x

=

-0.988838492596434,

z

=

0.895742025400000 • K T 3 ,

y

=

-0.896233211995794-10~ 2 .

The half period of this orbit is T = 1.529782262453289 and its Jacobi constant C = 3.000829263532742. The values of the stability indices are: Trx~ 1727.95575441572, Tr2 = 1.992624897950193. This orbit was obtained refining (by the own program) the analytically computed halo orbit using a 15th order theory. For all the Fourier developments that appear in the Tables, first the number of the harmonic is given; for each harmonic, the cosine and sine coefficients are given, respectively, in the first and second lines. The maximal eigenvalue computed from the trace, from the monodromy matrix or from DP, gives relative differences O ( 1 0 - u ) . We summarize results: a) Values from 1 0 - 1 2 to 10~8 in adimensional units, are suitable for the numerical differentiation of the Poincare map. b) The nonlinear Wu has been computed with factor = 1 0 ~ u , step = 50 and N = 20. The results are shown in Figure 3.1. For this Figure the size of the window is (-0.988941,-0.988744) in x and (0.0011178,0.0011242) in z, i.e., roughly 29400 km in x and 950 in z. The quadratic behavior of the

Numerical Study of the Local Invariant

Manifolds

153

error as function of the distance to the halo orbit is clearly seen from the list of results. For points at some 800 km of the halo orbit in the x direction, the differences between the real manifold and the linear approximation are of the following orders of magnitude: 0.5 km in x, 0.05 km in z, 0.3 mm/s in x, 0.15 mm/s in y and 0.04 mm/s in i. Everything is measured in the plane y = 0. Hence, as a conclusion, up to some 1000 km at least, the nonlinear behavior of the local manifolds is not required. c) The Fourier analysis of the Floquet modes (Table 3.3) and of the functions e and v (Table 3.4) is given. Also the plots of v and e are given in Figures 3.2 and 3.3, respectively. The parity properties of the modes have been explained before. They will be useful in the next chapter.

ii = 0 . 0 8 Linear and nonlinear unstable nanifold

Fig. 3.1 Comparison of the linear and nonlinear unstable manifolds of a halo orbit around Li of the Sun-Barycenter system with 0 = 0.08. The size of the displayed window is (-0.988941,-0.988744) in x and (0.0011178,0.0011242) in z, i.e., roughly 29 400 km in x and 950 in z.

154

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

Fig. 3.2 For the halo orbit of amplitude /3 = 0.08, angle between the stable and the unstable manifolds, as a function of time, along a period.

Fig. 3.3 For the halo orbit of amplitude f) — 0.08, behavior of the function e(r), as a function of time, along a period.

NONLINEAR UNSTABLE MANIFOLD 1D-10 STEP/IN.FAC. = 0.5D+02 NUMBER OF POINTS = 20 INITIAL FACTOR CURRENT ITERATIONS AND FACTOR: POINT NUMBER 1 NONLINEAR -0.988838486833D+00 0.895742174457D-03 0.150042697652D-07 LINEAR -0.988838486833D+00 0.895742174475D-03 0.150058823183D-07 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 2 NONLINEAR -0.988838198676D+00 0.895749627359D-03 0.765265805366D-06 LINEAR -0.988838198665D+00 0.895749628270D-03 0.765299998235D-06 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 3 NONLINEAR -0.988837910539D+00 0.895757078546D-03 0.151546493378D-05 LINEAR -0.988837910498D+00 0.895757082064D-03 0.151559411415D-05 POINT NUMBER 4 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0.988837622423D+00 0.895764528019D-03 0.226560150150D-05 LINEAR -0.988837622330D+00 0.895764535858D-03 0.226588823007D-05 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 5 NONLINEAR -0.988837334326D+00 0.895771975776D-03 0.301567555906D-05 LINEAR -0.988837334163D+00 0.895771989652D-03 0.301618234598D-05 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 6 NONLINEAR -0.988837046250D+00 0.895779421821D-03 0.376568717835D-05 LINEAR -0.988837045995D+00 0.895779443446D-03 0.376647646190D-05 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 7 NONLINEAR -0.988836758194D+00 0.895786866150D-03 0.451563627201D-05 LINEAR -0.988836757828D+00 0.895786897240D-03 0.451677057782D-05 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 8 NONLINEAR -0.988836470158D+00 0.895794308767D-03 0.526552290817D-05 LINEAR -0.988836469660D+00 0.895794351035D-03 0.526706469374D-05 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 9 NONLINEAR -0.988836182142D+00 0.895801749670D-03 0.601534715134D-05 LINEAR -0.988836181493D+00 0.895801804829D-03 0.601735880965D-05 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 10 NONLINEAR -0.988835894146D+00 0.895809188861D-03 0.676510889209D-05 LINEAR -0.988835893325D+00 0.895809258623D-03 0.676765292557D-05 CURRENT ITERATIONS AND FACTOR: POINT NUMBER 11

-0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1

NONLINEAR -0 .988835606170D+00 0.895816626338D-03 0.751480820540D-05 LINEAR -0 .988835605158D+00 0.895816712417D-03 0.751794704149D-05 POINT NUMBER 12 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0 .988835318215D+00 0.895824062103D-03 0.826444511631D-05 LINEAR -0 .988835316991D+00 0.895824166211D-03 0.826824115741D-05 POINT NUMBER 13 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0 .988835030280D+00 0.895831496156D-03 0.901401951407D-05 LINEAR -0 .988835028823D+00 0.895831620006D-03 0.901853527332D-05 POINT NUMBER 14 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0 .988834742364D+00 0.895838928498D-03 0.976353171314D-05 LINEAR -0 .988834740656D+00 0.895839073800D-03 0.976882938924D-05 POINT NUMBER 15 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0 .988834454469D+00 0.895846359128D-03 0.105129814080D-04 LINEAR -0 .988834452488D+00 0.895846527594D-03 0.105191235051D-04 POINT NUMBER 16 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0 .988834166594D+00 0.895853788047D-03 0.112623687688D-04 LINEAR -0 .988834164321D+00 0.895853981388D-03 0.112694176210D-04 POINT NUMBER 17 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0 .988833878739D+00 0.895861215255D-03 0.120116937247D-04 LINEAR -0 .988833876153D+00 0.895861435182D-03 0.120197117369D-04 POINT NUMBER 18 CURRENT ITERATIONS AND FACTOR: NONLINEAR -0 0.895868640752D-03 0.127609563566D-04 .988833590904D+00 LINEAR -0 0.895868888976D-03 0.127700058529D-04 .988833587986D+00 POINT NUMBER CURRENT ITERATIONS 19 AND FACTOR: NONLINEAR -0 0.895876064538D-03 0.135101565305D-04 .988833303090D+00 LINEAR -0 0.895876342771D-03 0.135202999688D-04 .988833299818D+00 POINT NUMBER CURRENT ITERATIONS 20 AND FACTOR: NONLINEAR -0 0.895883486615D-03 0.142592944442D-04 .988833015295D+00 LINEAR -0 0.895883796565D-03 0.142705940847D-04 988833011651D+00

-0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8 1 -0.8 -0.8

CURRENT ITERATIONS AND FACTOR: POINT NUMBER 1 NONLINEAR -0.988838498360D+00 0.895741876304D-03 -0.150075893424D-07 LINEAR -0.988838498359D+00 0.895741876324D-03 -0.150058823183D-07

1 -0.8 -0.8

POINT NUMBER 2 NONLINEAR -0.988838786538D+00 LINEAR -0.988838786527D+00 POINT NUMBER 3 NONLINEAR -0.988839074736D+00 LINEAR -0.988839074694D+00 POINT NUMBER 4 NONLINEAR -0.988839362954D+00 LINEAR -0.988839362862D+00 POINT NUMBER 5 NONLINEAR -0.988839651193D+00 LINEAR -0.988839651029D+00 POINT NUMBER 6 NONLINEAR -0.988839939451D+00 LINEAR -0.988839939197D+00 POINT NUMBER 7 NONLINEAR -0.988840227730D+00 LINEAR -0.988840227364D+00 POINT NUMBER 8 NONLINEAR -0.988849516029D+00 LINEAR -0.988845515532D+00 POINT NUMBER 9 NONLINEAR -0.988840804348D+00 LINEAR -0.988840803699D+00 POINT NUMBER 10 NONLINEAR -0.988841092688D+00 LINEAR -0.988841091866D+00 POINT NUMBER 11 NONLINEAR -0.988841381047D+00 LINEAR -0.988841380034D+00 POINT NUMBER 12 NONLINEAR -0.988841669427D+00 LINEAR -0.988841668201D+00

CURRENT ITERATIONS 0.895734421618D-03 0.895734422529D-03 CURRENT ITERATIONS 0.895726965216D-03 0.895726968735D-03 CURRENT ITERATIONS 0.895719507098D-03 0.895719514941D-03 CURRENT ITERATIONS 0.895712047265D-03 0.895712061147D-03 CURRENT ITERATIONS 0.895704585715D-03 0.895704607353D-03 CURRENT ITERATIONS 0.895697122447D-03 0.895697153559D-03 CURRENT ITERATIONS 0.895689657463D-03 0.895689699764D-03 CURRENT ITERATIONS 0.895682190760D-03 0.895682245970D-03 CURRENT ITERATIONS 0.895674722341D-03 0.895674792176D-03 CURRENT ITERATIONS 0.895667252205D-03 0.895667338382D-03 CURRENT ITERATIONS 0.895659780349D-03 0.895659884588D-03

AND FACTOR: -0.765334238858D-06 -0.765299998235D-06 AND FACTOR: -0.151572341618D-05 -0.151559411415D-05 AND FACTOR: -0.226617513205D-05 -0.226588823007D-05 AND FACTOR: -0.301668933296D-05 -0.301618234598D-05 AND FACTOR: -0.376726618033D-05 -0.376647646190D-05 AND FACTOR: -0.451790572033D-05 -0.451677057782D-05 AND FACTOR: -0.526860770336D-05 -0.526706469374D-05 AND FACTOR: -0.601937238277D-05 -0.601735880965D-95 AND FACTOR: -0.677019957931D-05 -0.676765292557D-05 AND FACTOR: -0.752108936385D-05 -0.751794704149D-05 AND FACTOR: -0.827204189522D-05 -0.826824115741D-05

1 - 0 .896 - 0 .896 1 - 0 , .896 - 0 . .896 1 - 0 .896 - 0 .896 1 - 0 , .896 - 0 , .896 1 - 0 .896 - 0 .896 1 - 0 , .896 - 0 , .896 1 - 0 .896 - 0 .896 1 - 0 , .895 - 0 , .895 1 - 0 .895 - 0 . .895 1 - 0 , .895 - 0 .895 1 - 0 .895 - 0 , .895

POINT NUMBER 13 NONLINEAR -0 .988841957827D+00 LINEAR -0 .988841956369D+00 POINT NUMBER 14 NONLINEAR -0 .988842246247D+00 LINEAR -0 .988842244536D+00 POINT NUMBER 15 NONLINEAR -0 .988842534687D+00 LINEAR -0 .988842532704D+00 POINT NUMBER 16 NONLINEAR -0 .982842823148D+00 LINEAR -0 .988842820871D+00 POINT NUMBER 17 NONLINEAR -0 .988843111629D+00 LINEAR -0 . 988843109039D+00 POINT NUMBER 18 NONLINEAR -0 .988843400130D+00 LINEAR -0 .988843397206D+00 POINT NUMBER 19 NONLINEAR -0 .988843688651D+00 LINEAR -0 .988843685374D+00 POINT NUMBER 20 NONLINEAR -0 .988843977192D+00 LINEAR -0 .988843973541D+00

CURRENT ITERATIONS 0.895652306774D-03 0.895652430793D-03 CURRENT ITERATIONS 0.895644831481D-03 0.895644976999D-03 CURRENT ITERATIONS 0.895637354469D-03 0.895637523205D-03 CURRENT ITERATIONS 0.895629875737D-03 0.895630069411D-03 CURRENT ITERATIONS 0.895622395285D-03 0.895622615617D-03 CURRENT ITERATIONS 0.895614913115D-03 0.895615161823D-03 CURRENT ITERATIONS 0.895607429222D-03 0.895607708028D-03 CURRENT ITERATIONS 0.895599943610D-03 0.895600254234D-03

AND FACTOR: -0.902305713095D-05 -0.901853527332D-05 AND FACTOR: -0.977413492201D-05 -0.976882938924D-05 AND FACTOR: -0.105252753758D-04 -0.105191235051D-04 AND FACTOR: -0.112764785333D-04 -0.112694176210D-04 AND FACTOR: -0.120277444163D-04 -0.120197117369D-04 AND FACTOR: -0.127790728749D-94 -0.127700058529D-94 AND FACTOR: -0.135304641633D-04 -0.135202999688D-04 AND FACTOR: -0.142819181236D-04 -0.142705940847D-04

-0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89 1 -0.89 -0.89

Table 3.2 Output of program PAPUS. Sun-Barycenter problem. Ca

The Proposed Method for the On/off

3.2

Control

159

The Proposed Method for the On/off Control

3.2.1

On/off

Control

Using Invariant

Manifolds

In this section we explain, for the RTBP, the proposed method of station keeping using on/off maneuvers. The small modifications to account for the perturbations are more technical than conceptual. The basic idea proceeds as follows: a) Due to errors in the injection maneuver, in the tracking, in the execution of the station keeping maneuvers and in the used model of the solar system, the spacecraft is not in its nominal orbit. b) Local errors change in time, according to the solutions of the variational equations. Modes 3, 5, 6 are neutral, i.e., they do not increase very much in size during a large time span. Mode 2 is stable and we do not care about it. Mode 4 can produce, on the average, departures from the nominal point at a linear rate. This rate is simply the slope of the linear part of function e. However, this effect can be annihilated taking at every moment as nominal point, not the point in the nominal orbit at this exact time, but the closest point in the nominal orbit to the actual point (obtained by tracking). The search for closest point should be done locally, i.e., for values of time near the actual one. This is not relevant for the RTBP, where the nominal orbit is periodic, but has full sense for the quasi-periodic nominal orbits to be introduced in Chapter 7. As a conclusion, the only dangerous mode is number 1. The component of the error along this mode is increased by a factor Ai in one revolution. c) An on/off maneuver acting on a short time, can be considered as a jump of the point in the phase space only in the velocity components. For instance, a constant acceleration of 1/6 cm/s 2 acting during 1 minute produces a change of 10 cm/s in velocity and 3 m in position. For the Sun-Barycenter problem, in adimensional units, these figures are 3.4 • 10~ 5 and 2 • 1 0 ~ n respectively. Hence, we can consider the maneuver as a jump. d) When it is detected that the real point has a significant departure from the nominal one, it is not necessary to return to the nominal orbit. The only thing to do is to cancel the unstable component. This should be done in the best way, i.e., with the minimum possible jump (implying minimum fuel expenditure). At this moment it is not clear what "significant departure" means. It should not be too small because perhaps the departure is only due to tracking errors. Another reason to prevent from being too small is that in this case the maneuvers should be done at short time intervals. On the other hand, it should not be too large because this increases the value of the jump in an exponential way. Delaying 1 month a maneuver of 10 cm/s will

160

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

require to increase it to some 35 cm/s. Other considerations, as operational constraints, also have an important effect on the election of the maneuver epoch. These ideas are developed in the next subsections. 3.2.2

The Projection

Factors

Let 6 = (8x,Sy,8z,Sx,8y,,5z)T be the error vector, i.e., the difference between the actual coordinates and the nominal ones. At the current moment, r, 6 can be expressed in the local basis {^(r)} obtained from the Floquet modes: 6

6-

^Cigi(r). j=i

We are interested in ci, the component along the unstable direction. Incidentally, the components C3 and C4 have also been computed to see the variations along the orbit and along the family. For c\ we have V

det(ei(r) > ...,e 8 (T))

;

When the determinant appearing in the numerator is developed, we found ci = n\(T)Sx + -Kl2{T)8y + ir\ (T)SZ +

K\(T)8X

+ nl(T)6y +

TTI(T)8Z.

(3.11)

The magnitudes TT}(T) are the projection factors. 7T|(T) is the (signed) minor of the ith element in the first column of the matrix appearing in the numerator of (3.10) divided by the determinant of the Floquet basis. Let 7T1 = (7r£,... ,n\)T. Then, the unstable component is the scalar product of 7i"i and 6. As the vector ei(r) changes in time, it is better to use normalization. To get the normalized unstable component we should multiply the previous value of c\ by the modulus of e~\ (r). In a similar way, we can obtain the vectors ?r 3 (r), 7r4(r) and the normalized components. When the projection factors are available, it is interesting to see the effect on them of the tracking errors. A rough computation proceeds as follows: Let dx, dy, dz, dx, dy, dz be the tracking errors; it is supposed that they are bounded by dx,..., dz, respectively, in absolute value. We define p(r) = max^ 1 • (dx, dy,...,

dz) ,

where \dx\ < dx,... ,\dz\ < dz. Of course, the maximum is reached in a corner of the parallelepiped and is obtained taking as sign of the ith component of the tracking error vector the sign of 7T|(T).

The Proposed Method for the On/off

Control

161

In fact dx,... ,dz are usually not bounds of tracking errors but a (or 2a) deviations. Supposing centered normal behavior for each error and independence, the variable p = -K1 • (dx,..., dz) is centered normal with typical deviation a = ((n{nd-xf 3.2.3

A First Approach

(nl)i(dW)1/2-

+ ... +

Towards the Optimal

Maneuvers

We look for the unitary controls, i.e., the value of the controls to be applied when the unstable component is the unity. We call them Si, S2, S3, i.e., the x, y and z controls. According to d) in 3.2.1, we write

p ^ - + (0,0,0,Si,S2,S3)T Mr) I

=

J2ajej(T). ^

We split this system into two different systems taking the first 3 and the last 3 equations.

| i ± = X> AlM ,

A = -g&l+X>^H,

(3.12)

where A = (Ai, A 2 , A 3 ) r . In (3.12), for any vector e, € E 6 we write e 1

ej = I J3 1 \j

.„;+u ^ = ^ ra>3 with eji,ej2 S

In the first equation of (3.12) it is easily seen that e2i, £31, em are linearly independent. Hence 02, 03, 04 can be obtained as functions of a$ and ae- By substitution in the second equation of (3.12) we obtain A in the form

(

aio + a n Q 5 +ai2a6

\

020 + 0210:5 + ^220:6

J ,

030 + 03105 + 03206

/

(3.13)

where ay, i = 1,2,3, j = 0,1,2, are functions of the time. We try to obtain values of 05 and ae such that the following norm of A be minimized: ||A|| = (Af -I- A2) 1 / 2 + IA3I. This is linearly related to the amount of fuel to be spent if we suppose that the spacecraft has a spinning motion of rotation around the z-axis and maneuvers can be done in the (x, y)-plane (selecting the direction by choosing the right orientation) and in the z direction. A little algebra shows that the optimum is one of the following two possibilities (due to convexity reasons): either Ai = A 2 = 0 (which gives d 5 , d6 and, by substitution, A3), or A 3 = 0 (which gives a relation between 05 and ag, that

162

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

substituted in Ai, A 2 produces ||A|| as a quadratic function in ag, if this one is chosen as independent variable). Other types of controls are possible. The only one-axis control (x, y or z) is obtained from (3.13). For instance, the a:-axis control is obtained putting A2 — A 3 = 0 and obtaining a§ and a%. By substitution in A l 5 the unitary control is obtained. Let ip(r) = ci(r)|ei(r)|, where ci(r) is obtained from (3.11). We also introduce the gain function 9(T)

= TAT;,

where ||A|| denotes the minimal norm, either if optimal 3-axis control or any other type of control (1-axis or 2-axes) are allowed. Suppose that the last previous maneuver was done at r = To and that, due to tracking errors, we got some value rp(ro). The function ip(r) is increasing exponentially IP(T) = exp ( -=^(T

- T0)) ip(T0).

Let (A x , A 2 , As)^ be the real maneuver done at r = r 0 and ( A i , A 2 , A 3 )f the theoretical one. The difference is supposed to be A£° - A[° = WAJ0, where W is some "error in the execution of the maneuver" weight matrix. It is supposed that this matrix is known from performance experiments with the control device. Their elements need not be constants and they can depend on the components of A. The error WA[° should be added to tracking errors. Then, the total unstable component at r is ^tot(r)

=

(V(To)+7r1(ro)WA[°)exp^(r-r0; +7T 1 ( 7 ")- f - a t o t ( T - T o ) 2 , a t o t ( T - T o ) j

,

where a to t accounts for the averaged residual acceleration due to unknown and neglected perturbations. We note that Vtot('r) should also be obtained directly from tracking. The comparison can allow to detect systematic errors in the tracking or a better knowledge of a to tTherefore, the cost of the maneuver at the epoch r is: V'totM/flMThe choice of the time r for the current execution maneuver can be done by trying to minimize the average amount of fuel per unit time, i.e. /(r) =

^t(r)(g(r))_ T -

T0

(314)

The Proposed Method for the On/off

163

Control

Let us try to learn something about (3.14) in a simple case. Bound ^(TO)+7T 1 (TO)WAJ° by some amount V> and the unstable component of the residual acceleration effect by a i ( r - r 0 ) +a2(T - r 0 ) 2 . Furthermore, suppose that g(r) is constant (not very unrealistic as we shall see later). By substitution in (3.14) and derivation, we get the following condition for the minimum: ( N $ ( T - TO) - 4>) e N ( T - r o ) + a2(T - T0)2 = 0

(3.15)

where K = lnAi/T. From (3.15), if we put a 2 = 0, we obtain (r - T 0 ) = 1/H. Then, the suitable time, under these simplifying hypothesis, between two maneuvers is at least 1/K adimensional time units. The effect of a2 is to decrease this time. The main result, of this simple approach, is that the maneuvers should be performed when the unstable component is roughly e times greater than the maximum unstable component due to tracking and execution of maneuver errors. Up to now, we discussed local optimization. For long time optimization, one should determine the number of maneuvers, k, and the moments at which they should be done, between an initial time To and a final time r/, such that

] T V'tot(Ti) (0(Ti)) *=i

be a minimum (TO < T\ < ... < Tk < Tf). However, this gives only an upper estimate of the fuel consumption. The true maneuvers should be done trying to minimize the function / ( T ) defined above, where the unstable component at the epoch r is estimated from tracking.

3.2.4

Numerical Controls

Computations

of Projection

Factors

and

Unitary

This is also done in routine PRECON, part of which has already been described. When the Floquet modes are available, the formulas of 3.2.3 are used to generate the projection factors, unitary controls, gains and some auxiliary functions. Then, the results are Fourier analyzed to obtain a compact representation for all time. Usually the results, up to the 15th harmonic, have relative errors less than 1 0 - 5 . The components of the projection factors, related to the tangent vector to the orbit, are alternatively odd and even, while for the tangent vector to the family, e4, they are alternatively even and odd. The projection factors, concerning the unstable direction, have full Fourier developments. We include a computation of the maximal component of the tracking errors, p{r), as described at the end of section 3.2.2. For the Sun-Barycenter system near L\ the following tracking errors have been used:

164

The Neighborhood of the Halo Orbits: Numerical Study and

3 km in the x coordinate 5 km in the y coordinate 30 km in the z coordinate

Applications

2 mm/s in the x coordinate 2 mm/s in the y coordinate 6 mm/s in the z coordinate.

Fig. 3.4 Behavior of the function p(r) for a halo orbit around L\ of the Sun-Barycenter system with /3 = 0.08.

This function is nonsmooth (because it is the maximal projection of a product of segments on a varying line). The maximum is of the order of 2-10 - 7 , in adimensional units. This value will be important for control in the light of 3.2.3. As it will be explained in the next point, of all the possible controls ((x, y, z), (x, y), (x, z), (y, z), (x), (y), (z)) it is enough to study the {x,y), (x) , (y), (z) ones. The values of Ai and A2 for the (x,y) control are given in Figure 3.5, as well as the related gain factor 9{T)

=

(A?(r) + A|(r))V2

in Figure 3.6. For the one-axis controls, the relation between unitary control and gain is easy. For instance, for the (x), case they are Ai(r) and g(r) = 1/Ai(r). Hence, only the gains (with its sign) are given for one-axis controls. Also quotients of one-axis gains divided by full control gain are given (see Figures 3.7 and 3.8).

The Proposed Method for the On/off

Control

165

Fig. 3.5 x- and y-components of the unitary controls. Halo orbit around L\ of the Sun-Barycenter system with 0 = 0.08.

166

The Neighborhood of the Halo Orbits: Numerical Study and

Fig. 3.6

Applications

For /3 = 0.08, behavior of the gain factor.

Fig. 3.7 For fi = 0.06, behavior of the gain factor using x control divided by the gain factor using full control.

The Proposed Method for the On/off

Control

167

Fig. 3.8 For j3 = 0.06, behavior of the quotients of the gain factor using y control (top) and using z control (bottom) by the gain factor using full control.

168

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o tH tH O o o O tH tH tH tH o a a a a a •* CO 00 CO 00 o CO LO o tH co o oo o CN LO CO CN r~ o r~ n CN CO to o 00 O) c CN in CN c n o o tH CN o > cn LO 00 LO to cn CN r~ CN tH T H tH LO to fcn cn O o o o o o o 1 1 1

tH tH o o a> tH O tH tH CN tH CN CN CO CN CO on T H O tH O tH O tH o o •tH o o O tH o o o + 1 + i a a a a a Q Q o a a a a a a a a a LO •* oo LO tH CN CN cn to CN to IH cn to to m CN N- O CN o r- CD tH • * cn o tH O in en co CN tH O ) 00 o oo to cn O ) 00 CN s *j< s- cn oo cn CN CN tH CN L O CN C O oo on LO T H CO to co C O r cn o to 00 CN r> L O CN LO cn CO f o cn o> cn LO oo to co • * 0 0 LO LO LO CO O •sf cn LO LO 00 LO M" o 00 o cn OO CN tH CN H •* to tH tH CO LO tO CN oo o o o o o o o o o o o o o o o o o I I I I I I I I

Orbits:

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The Neighborhood

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The Proposed Method for the On/off Control

v 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15

0 0 0 0 0 -0 0 0 0 -0 0 0 0 -0 0 0 0 -0 0 0 0 -0 0 0 0 -0 0 0 0 -0 0 ERROR 0

1971987776D+01 1779525025D-01 6333977335D-16 2797530920D-01 6622447199D-17 1871343839D-02 4237202419D-17 6831993316D-03 1524185447D-16 1443080860D-03 1689526278D-16 3825157524D-04 1049846997D-16 1009585276D-04 1179951257D-16 2627849420D-05 1342581583D-16 6893557690D-06 1331739561D-16 1785084096D-06 2177417256D-16 4621328310D-07 2621940147D-16 1191247283D-07 1147425192D-16 3067128850D-08 2535203973D-16 7884770775D-09 1461843822D-16 2026644941D-09 2480993864D-16 6996409615D-10 ERROR UP TO THE SEVENTH

e -0.3484368439D-08 0.1117282145D-08 -0.3652755474D+01 0.6515830516D-09 0.3384946203D+00 0.4950955891D-09 0.1110959665D+01 0.2499317279D-09 -0.5176442197D+00 0.2074125817D-09 0.4828468981D+00 0.1121794958D-09 -0.2707230420D+00 0.1018316033D-09 0.1992232403D+00 0.6182691884D-10 -0.1244356770D+00 0.5817516173D-10 0.8536356607D-01 0.3962661419D-10 -0.5538068093D-01 0.3720781838D-10 0.3711245472D-01 0.2781958153D-10 -0.2439952256D-01 0.2580787614D-10 0.1622152956D-01 0.2065219618D-10 -0.1071402213D-01 0.1896912545D-10 0.7103651184D-02 0.1329840634D-01 HARMONIC:

171

p 0 -0 0 0 -0 -0 0 -0 -0 0 -0 -0 -0 0 0 -0 -0 -0 0 -0 -0 0 -0 -0 0 0 0 -0 -0 -0 0 0

1320132098D-06 3009465464D-07 1622848706D-07 1187191824D-07 9380017493D-98 9537275933D-09 4033566099D-08 2806285825D-09 1006401493D-08 2353063842D-09 4085392264D-09 6917642048D-09 3724701728D-09 8643876277D-09 8149385183D-09 5322033053D-10 2066799725D-09 1647925141D-09 1231947452D-09 1324337373D-09 1504911767D-09 1616443569D-09 6414353821D-11 1806801192D-09 3995393036D-10 1259512493D-09 2076111850D-09 6525730508D-10 2808544106D-09 6878739215D-11 3806203203D-10 1426022443D-08

0.3557967589D-05 0.3507208554D+00 0.2462591384D-08 SLOPE OF THE LINEAR PART OF EPSILON = 1.467321176828839 Table 3.4 Output of routine PRECON. Fourier coefficients of the functions v(j), e(r) and p(r)). Sun-Barycenter problem. Case L\. /J = 0.08.

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CN t H CN t H C N 1 H C O 1 H C O 1 H C O 1 H C O 1 H C O 1 H C O O tH O TH O tH O i H O i H O i H O - r H O t H O t H O I I I I I I I I I I I I I I I I I I I Q Q Q Q Q Q Q C I Q Q Q a a c n T - I CN • * rf o O i l O O l L O l D L O O l C O O O M H S C O • * c n CO LO t H CD r o ( - ( O L O < f m o i i £ i c i i H C T S o O t H c n C N CN CO ( O O I O C O O O O C N ^ ^ L O C N T H C N C D > * CO L o n c n i O L O ^ o i N L O t o r o o s O O o t O C I > * LO CO O < * n o ) c O L O i o o L o i o ( D C i i c i L O cn • * t H 1-- t - T H C N C N C N C O t H L O - * i H L O 0 0 C N r ~ - C N CN * CN 0 0 CN CD s c O L O H c o o i o c o s i o O L O L O i - l CO CO CN CO CN H C S O J C S L O r l C O H d H T H H ^

a aa a a a aa a a a

O H tH O tH O iH H O I I I I I I I Q Q ^f oo e n o o r - CN CO ^ LO CN • > * t— r - CN t H LO > * i H LO y-> CN LO CO C - LO CO CN t LO O
The Neighborhood of the Halo Orbits: Numerical Study and

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ENTH HARMONIC:

The Neighborhood of the Halo Orbits: Numerical Study and

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D+0 D+0

THE 3265

The Proposed Method for the On/off

Unit, control (x) 0 -0.1383245047D+01 1 -0.3585519946D+00 1 -0.6575653419D-01 2 0.4100338138D-01 2 0.4397466767D-01 3 -0.9088664529D-02 3 -0.1722858077D-02 4 0.1701837508D-02 4 0.2203446502D-03 5 -0.2300745774D-03 5 0.8714367325D-04 6 0.3770136026D-04 6 -0.3509531636D-04 7 -0.3811023273D-05 7 0.8754694998D-05 8 0.7704749281D-06 8 -0.3291676003D-05 9 0.2599492095D-06 9 -0.6707037120D-07 10 0.2290481438D-06 10 -0.7757296750D-06 11 0.2158108737D-06 11 -0.5343750312D-06 12 0.1906472601D-06 12 -0.5360609576D-06 13 0.1762473368D-06 13 -0.4861746063D-06 14 0.1631740879D-06 14 -0.4554375401D-06 15 0.1529849448D-06 15 -0.4252679649D-06 ERROR 0.1015621200D-04 ERROR UP TO THE SEVENTH 0.1231414878D-04

Unit, control (y) -0.7284190957D+00 0.1365036996D+00 0.2644186317D+00 0.2163106855D-01 0.8835943967D-03 -0.3676236561D-02 -0.4125025907D-02 0.1676017351D-02 0.1026454686D-02 -0.3773873506D-03 -0.2738047617D-03 0.8956126243D-04 0.4959809585D-04 -0.1897535371D-04 -0.1185702015D-04 0.4593458659D-05 0.9441414813D-06 -0.5345173234D-06 -0.1426673571D-05 0.5120766559D-06 -0.8286706920D-06 0.2435024688D-06 -0.8559725107D-06 0.2695591304D-06 -0.7680596085D-06 0.2378763155D-06 -0.7158600823D-06 0.2243250188D-06 -0.6654740394D-06 0.2105897953D-06 -0.6223265630D-06 0.1465094412D-04 HARMONIC: 0.2040781484D-04

Control

Gain factor 0.6386305119D+00 -0.1032834300D+00 0.2577307996D-01 0.2600527140D-01 0.1894336456D-02 -0.8111583384D-02 0.3767837278D-03 0.2248154825D-02 -0.8562292551D-04 -0.6159793478D-03 0.2662902900D-04 0.1710385185D-03 -0.7685429183D-05 -0.4713817820D-04 0.1510482890D-05 0.1327972708D-04 -0.8063464473D-06 -0.3540875482D-05 -0.1795060482D-06 0.1118427275D-05 -0.2997814919D-06 -0.1937250898D-06 -0.2413385138D-06 0.1604527051D-06 -0.2306008953D-06 0.5285735660D-07 -0.2118606258D-06 0.7574803007D-07 -0.1977715586D-06 0.6362069222D-07 -0.1848720872D-06 0.4402224538D-05 0.1828484058D-04

Table 3.6 Output of routine PRECON. Fourier coefficients of the unitary controls (x- and ^-components; the z one is zero) and gain factor. Sun-Barycenter problem. Case h\. 0 = 0.08.

175

176

The Neighborhood of the Halo Orbits: Numerical Study and

x-control ~0 -0.5485385294D+00 1 0.2850928612D-01 1 -0.7362748995D-01 2 -0.8868070703D-02 2 0.1655640282D-01 3 0.3347214665D-02 3 -0.5370685104D-02 4 -0.8611518383D-03 4 0.1535465635D-02 5 0.2228048502D-03 5 -0.4463920484D-03 6 -0.6008078988D-04 6 0.1290849449D-03 7 0.1583539634D-04 7 -0.3638049938D-04 8 -0.4505320738D-05 8 0.1091391473D-04 9 0.1050094060D-05 9 -0.2645454153D-05 10 -0.4423642087D-06 10 0.1176117805D-05 11 -0.1627780039D-07 11 0.5240148859D-07 12 -0.1202279253D-06 12 0.3411433989D-06 13 -0.8126887085D-07 13 0.2327284777D-06 14 -0.8378249146D-07 14 0.2405157311D-06 15 -0.7641296194D-07 15 0.2181811031D-06 ERROR 0.5220474750D-05 ERROR UP TO THE SEVENTH 0.1600944008D-04

y-control -0.3059947839D+00 0.1556603154D+00 0.8042545705D-01 -0.3351665833D-01 -0.1302978559D-01 0.9532284887D-02 0.2423733532D-02 -0.2636270034D-02 -0.6310421619D-03 0.7406886174D-03 0.166744725ED-03 -0.2071812826D-03 -0.4393804211D-04 0.5832103880D-04 0.1168503880D-04 -0.1630289971D-04 -0.3341039466D-05 0.4649879087D-05 0.7679830325D-06 -0.1249853937D-05 -0.3495097081D-06 0.4023891154D-06 -0.2928272571D-07 -0.6720338973D-07 -0.1085499753D-06 0.6083508165D-07 -0.7309530275D-07 0.2166268103D-07 -0.7923661283D-07 0.3007673061D-07 -0.7243788669D-07 0.1780136919D-05 HARMONIC: 0.2316279196D-04

Applications

z-control 0.3622247966D-02 -0.1829991513D-01 0.3056056222D-01 0.1367079606D-02 -0.7043851514D-02 -0.3710877619D-03 0.1788979331D-02 0.8899533314D-04 -0.5053576245D-03 -0.2380661893D-04 0.1425262383D-03 0.6068598995D-05 -0.4016329582D-04 -0.1630916590D-05 0.1135286424D-04 0.4307289028D-06 -0.3191127981D-05 -0.1243267113D-06 0.9117926295D-06 0.2858318588D-07 -0.2471105832D-06 -0.1291229159D-07 0.7916759243D-07 -0.8404392390D-09 -0.1355158175D-07 -0.3772795079D-08 0.1200636253D-07 -0.2605011983D-08 0.4274989414D-08 0.2648564908D-08 0.5983854359D-08 0.1441348600D-06 0.4459207106D-05

Table 3.7 Output of routine PRECON. Fourier coefficients of the unitary signed gain factors when control is applied only in the x, y or z directions. Sun-Barycenter problem. Case L\. /3 = 0.08.

The Proposed Method for the On/off

3.2.5

Discussion Rule

of Results:

the Different

177

Control

Gains and the No

Delay

First of all the results obtained are not very sensitive to the values of the z-amplitude P (below P — 0.2 for instance) nor to changes in /J, (for instance from Earth-Moon to Sun-Barycenter systems). We summarize the results about different axes controls and gains: a) The 3-axes control always has zero component in the ^-direction. This is the best possible control. b) The 2-axes controls using the (x,z) or the (y,z) variables always have zero component in the z-direction. c) Concerning the one-axis controls, the a;-axis one is always better than the ?/-axis. The z-axis control should not be used. At some points of the orbit the controllability is lost in the z-axis control. This is the only control for which this happens. For all points the gain is very poor in the z-axis control. d) Comparing the useful controls: (x, y)-plane, z-axis and y-axis, it is found that the x-axis control has an average of efficiency of 0.84 with respect to the (x, y) one, ranging from 0.72 to 0.96. For the y-axis control the figures are average efficiency 0.48 and range from 0.26 to 0.70. Therefore the z-only control is almost as good as full control and the y-only control has a cost that is roughly the double. For the full control (i.e. (x,y) control), the gain factor is rather uniform: the quotient of the maximal to the minimal values is roughly 1.43. We should add that the gain factors refer to the efficiency of a given type of maneuvers to cancel the unitary unstable component. This component is obtained using the projection factors and the local errors. As the projection factors change along the orbit (but in a quite smooth way) the same error in position and velocity has different unstable components. What is dynamically important is the unstable component and not the local errors. This explains the choice of the gain factors. Now we go to the no delay rule. A natural question is to ask whether it is recommendable to wait for a given maneuver trying to reach another epoch with a better gain. Of course the gain can increase but also the unstable component increases. We should look for the function exp((lnAi/T)r)/ f l (r) in every case. It turns out that this function is always increasing as Figure 3.9 shows. Therefore, it is never good to wait for a maneuver except for operational reasons. Short delays are not very important. In the Sun-Barycenter problem (halo orbits near L\ or L2) the unstable component, and therefore the cost of the maneuver roughly doubles every two weeks. This is dramatically changed for the case of small size 1/2 halo orbits in the Earth-Moon problem where this time is reduced to 1.4 days.

178

The Neighborhood of the Halo Orbits: Numerical Study and Applications

.1 = 0.06 log[exponential instability / g a i n factor using full control) It shows that it is never reccorenaable to valt for naneuvers Window in y (,-.1211,8.7965]

Fig. 3.9 Logarithm of the exponential instability divided by the gain factor, using full control. Halo orbit around L\ of the Sun-Barycenter system with /3 = 0.06.

3.2.6

A Second Approach

Towards the Optimal

Maneuvers

Prom what has been said, it is clear that we can restrict ourselves to (x,y), (x) and (y) maneuvers. We suppose that the Floquet modes e;(r), i = 1,...,6 are known. As it has been obtained in 3.2.2, we have TT}(T) = mj(r) • (det("r)) -1 , where rrii is the (signed) minor of the ith element of the first column of the matrix A(T) whose columns are e ^ r ) , . . . ,e 6 (r), and det(r) stands for det(j4(-r)). To get the normalized component we multiply TT}(T) by |ei(r)|. To get the unitary controls in the (x,y) case we put ei(r) + (0,0,0, A1; A 2 ,0) T = J2ajCj(T)> |ei(r)| j=2 and we try to minimize Af + A 2 . The first relation is equivalent to det(r)/|ei(r)| + + A2TB5(T) = 0, and it follows

AI?714(T)

Ai

=

-det(T)m 4 /((mij+Tnjj)|ei(T)|),

A2

=

-det(T)mB/((ml+m§)|ei(T)|).

(3.16)

In the (a;) case, we have Ai = - det(T)/(m 4 • |ei(r)|), A 2 = 0. For the (y) case the formula is similar. The results obtained using (3.16) are the same as that using the formulas of 3.2.3. However, the approach taken here is more compact. It will be used in this form for the analytic computation of projection factors and unitary controls in Chapter 4.

On the Use of Solar Radiation Pressure for Station Keeping

3.3

179

On the Use of Solar Radiation Pressure for Station Keeping

3.3.1

Using Radiation

Pressure:

Applicability

and

Limitations

In this section we discuss the theoretical possibility of using solar radiation pressure for station keeping. We suppose that a device is available such that the cross section of the spacecraft can be changed at will by a considerable amount. We can imagine a couple of (almost) specular rectangles that can rotate around a ^-oriented axis. Of course, the spacecraft should not spin and it is supposed to be attitude stabilized with respect to the three axes. If the rectangles are pointing to the Sun, their cross section is negligible. If the normal to the rectangles, points towards the Sun, the cross section is maximum. The rectangles are supposed to have the same direction, i.e., their normals are parallel. This is done to prevent undesirable torques. They should be symmetric with respect to the center of masses of the spacecraft. The operating procedure is as follows: a) Usually the rectangles (if you wish they can be called the wings) are oriented towards the Sun. Then, the effect can be neglected or can be included in the fixed part of the spacecraft cross section. b) When the unstable component is significant, the wings are rotated in some way (for instance with normals parallel to the rc-axis or oriented in some optimal direction) and an interval of time is spent in this status. During this time the solar radiation pressure exerted on the wings is the force used to perform the maneuver. At the end of the maneuver, when the unstable component is suitable, we return to phase a). See later for the discussion about the final suitable unstable component. Several questions appear: (1) (2) (3) (4)

Is it always possible to perform a radiation pressure maneuver ? What is the optimal orientation of the wings ? What is the required surface of the wings ? Which is the effect of the time interval spent by the maneuver ?

The answer to these questions concerning applicability and limitations is given in the next subsections. 3.3.2

On the Determination

of Optimal

Parameters

Let Pr be the constant of the solar radiation pressure. We have adopted the value Pr = 4.56 • l ( r 6 Newton/m 2 at 1 AU. We suppose that the unstable component is the unity in the normalized ei(-r) vector, and we ask for the required value of s/m (surface of the wings divided by mass of the spacecraft) and the optimal orientation of the normal to the wings, to

180

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

get a minimum value of s/m if the maneuver has to be performed in a time interval At. We assume At is small and that the effect of the motion of the spacecraft during this interval can be neglected. We shall return later to this point. In any case, it is supposed that the normal to the wings is contained in the (x,y)-plane. Let At be the time interval, 7Ti, 7r2, 7T4, TT5 the 1st, 2nd, 4th and 5th projection factors and (x,y,z) the coordinates of the spacecraft. Let 8 be the angle between the x-axis and the normal to the wings and d the distance from the Sun to the spacecraft. We define / i = (x - n)/d, f2 = y/d, e = a r c t a n ( / 2 / / i ) . The value e measures the angle between the (x,y) projection of the spacecraft, seen from the Sun, and the negative x-axis. Neglecting z, because it is small, the angle between the Sun direction and the normal to the wings is 8 — e. Letpi = 7r 4 Ai-|-i7ri(Ai) 2 ,p2 = iT5At+^ir2(At)2. Then, the maneuver condition is 1 + -Prfl^9d2

COS2 (8 - c)(pi COS/? +p2 sin 8) = 0,

m dl where one of the factors, cos(/3 — e), comes from the specular condition and the other from the cross section when the wings are seen from the Sun. The last factor can be written as mcos(/3 — a), where m 2 = p\ + p\ and a = arctan(p2/pi)To get s/m minimal, we should maximize
On the Use of Solar Radiation Pressure for Station

Keeping

181

efficiency of the radiation pressure maneuvers will be poor if the unstable component is directed outwards the Sun. As the results show, m/s even changes sign along the orbit. This means that for some intervals the controllability is lost. Even in the best case it is s/m ss 10 2 . Therefore this case should be avoided. 3.3.3

The "Always

Towards the Sun"

Rule

As pointed out in 3.3.2, the radiation pressure maneuvers require that the unstable component be always directed towards the Sun. To reach this end, we should require several things: a) When the spacecraft is injected into the nominal orbit, one should add some component towards the Sun (for instance, a small change in the x variable). This change should not be too small, in order not to be canceled by tracking and model errors. In adimensional units, values around 3 — 4 • 1 0 - 7 are enough (equivalent to 40 km in x). b) As the unstable component increases with time, a maneuver is foreseen after a roughly computable time. It is recommended to perform the maneuver for unstable components around 2 - 4 - 1 0 - 6 adimensional units. Smaller values imply too frequent controls. Larger ones imply either a too large s/m ratio or a too large time interval for the maneuver. In fact, the unstable component during the maneuver follows a differential equation of the type u — Ku — a\ — a^t, u(0) = UQ. Therefore, in the maneuver condition (see 3.3.2) we should replace 1 by eN*. If UQ increases, t should be increased to reach a final value u = Uf, Uf < «o- Even if u 0 is large enough, in such a way that Nu > oi, the control is not possible and an escape is produced. As a last comment on the subject, we wish to point out that the "always towards the Sun" rule can be skipped in the following way. When the nominal orbit is designed, we can include in the section of the spacecraft, as seen from the Sun, one half of the value of the maximal section of the wings. Then, pointing the wings towards the Sun (and therefore reducing the effective section), will produce a net force towards the Sun, which is able to cancel outwards escapes.

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•rH CN CN CN CO CO CO CO * CD r~- ^"^ CN cn CN CN CN cn rH CO o LO oo tn LO 00 CO CD LO o o ^f CO <* cn r H rH r H 00 CO • * c n f CN 1 ^ 00 I CO in 00 00 00 l^ r- CO oo r cn no o rH i-H r H CN r H T H cn 00 CO cn o CO CN i-- T-I cn m cn CO CO o h- < * t~- LO tn > CO CO cn L O CI l>- f- T-i CO to to « "3 t^ 0 0 CN c n c n o 'S< oo >x in tn • * c n o o •* o f00 i~CN i-- CN ^f 00 LO CN r- rH CD CN cn r- o 00 cn co LO r H cn c n c n c n r H oo •* 00 OO P) 00 i ~ 00 CO CO cn co cn cn c n o c n >x r- tn 00 r H « * * 00 T-iI-- •LO* en LO CO •* CN I 00 M LO t- cn < LO ^f I CO CO o cn co LO o a > **CD •rH r H cn f-- CO CD 00 CN •* to CD CI cn LO •=f cn m CD CD tn CN no CO 00 •* i-icn •< *r•H*rCNH CO LO i-H r H LO LO r H rH LO
W

O r H r H C N C N 0 0 C 0 ' * < ' L 0 L 0 C 0 t 0 t ~ - t ~ - 0 0 0 0 c n C n O O r H r H C N C N 0 0 C 0 ' l ' ' * L 0 L 0 p 5 T H r H r H r H r H r H r H r H r H r H r H r H O

I

Q LO fCO LO LO CN 1-- CO CD LO CO CN CO r H I-- •t-i 00 i-H •=)• rH CN CN LO r H

LO CN !~rH tO •* LO

o o o o o o o o o o o o o o o o o o o o o o o o o o o o

cn r~-

o o o I I I a a Q oo CD CN r H rt o co cn CN -rH CD cn r - CD oo CO i-H co •*? r H co •* « * CN cn

o o I I

T-i T-i O (N CN CN 00 CN <* oo LO *m (N CO • * a) co cn cn CN cn cn i-H o co CN cn CD LO t~ <#•rH o cn CN ** CN I-- o CO c 00 co cn LO LO r- CN cn cn Io-o- CO n"in 00 r- CO i- CN to CO CO CN LO oo oo t»- CD 1-- 00 •rH cn *m o r H to t-i o CO i-H cT-to h- t- CN f~ co cn CO i-i 00 LO CO cn • * rH CO • * LO 00 CD « t f LO CD OO CN CN 00 in o c CN CN LO in CN hsn m o c n o c n > * in o o in CN o •rH f~ cn CN CO co CN CO 1-- CO LO 00 LO MP CD en CN oo a> 00 LO cn o CO * i-i > LO ICN o to oo ^p oo co i-H i-H i-H CO a> •rH CN i~- r H LO cn ^ CO LO LO cn oo cn cn i H •rH cn **CO t-i •si* •* LO CN i H LO CN LO cn cn o C O • * tn o cn CO cn CN CD n o c n cn o CN ^ LO oo LO •rH LO CO • H* CO CN OO LO I-- ^ CO CO o • * 0 0 CN o i-- O CN cn O) CO cHo CN m 00 oo O r- o 00 •rH r rH CO 1-1 r H CN 00 CO r- CN >- i-H CO < # i-H 00 to 1- i-H LO i-H LO LO r H ^ •*CO

D+0

184

3.4 3.4.1

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

Globalization of the Invariant Manifolds The Computation

of the Global Stable

Manifold

In 3.1.5 it is explained how to compute the intersection of the stable and unstable manifolds of the halo orbit with the y = 0 hyperplane. Using again the symmetry, it is enough to compute the unstable manifold. To compute the full manifold (outside of a neighborhood of the halo orbit) it is enough to pick up an interval of the unstable manifold on y = 0 such that the image, under the Poincare map, of the extremum nearer to the periodic orbit, be the other extremum. Taking this set of points (in fact a lattice in it) as initial condition, the forward flow of the restricted three-body problem produces the full manifold. More precisely, the initial conditions on Wu can be taken in the form Q + ee\, where Q is the fixed point of the Poincare map (i.e., the intersection of the halo orbit with y — 0) and e\ is the normalized dominant eigenvector of the differential of the Poincare map at Q. The parameter e should be taken in a range [eo, Aieo). eo is chosen small enough, in such a way that Q + coAie^ be very close to Wu. Let z be a parameter in [0,1). We take as initial conditions Iz — Q + eo exp(z • lnAi)e^. As — e\ is also a normalized dominant eigenvector, one should be careful with the right choice of the sign of e. One of the signs will produce orbits which approach the Earth and the other will produce escapes. For every value of z, we can study the behavior of the orbit starting at Iz by forward integration in time. The orbit approaches the Earth and after that goes far away. Next encounters in future time are possible. The nearest point to the Earth is located as a function of z. For every z in [0,1) this is carried out for a given halo orbit. In this way we obtain the minimal distance from Ws to the Earth in the first close approach.

3.4.2

Results

of Interest

for the Transfer

Orbit

First we perform the preceding computation for a periodic halo orbit near the nominal one used for the ISEE-C mission. A value /? (^-amplitude) equal to 0.08 has been used. For this halo orbit, e0 = —3 • 1 0 - 1 1 the minimal distance is reached for z = 0.629. The distance has a value of 11327 km and the velocity with respect to the Earth is 8 396.8 m/s. The inclination of the osculating orbit at this point is 6°.3. The effect of the size of the halo orbit is the following: when the parameter /3 is decreased the minimal distance from Ws to the Earth increases. Further computations show that for /3 = 0.2 the minimal distance is 6514 km and the related velocity is 11099 m/s. By slightly changing j3 we can find a minimal distance to the center of the Earth which coincides with the distance to which the ISEE-C spacecraft was injected into the transfer orbit. For the ISEE-C, the distance was 6 564.1 km and the injection velocity was 10990 m/s. Taking /? = 0.199, we get 6 562.2 km for the distance

Globalization of the Invariant

Manifolds

185

and 11057 m/s for the velocity (the related value of z is 0.672). However, in our program the mass of the Moon is included in the mass of the Earth. Performing the corresponding reduction (this produces an error because the mass of the Moon has been considered in the computation of Ws, but is not considered now) we get the same value 10 990 m/s. The inclination of the osculating orbit at the computed point is 15°.7, in good agreement with the real value adopted for the ISEE-C. As a conclusion, we have that changing the target orbit (the z-amplitude is « 300000 km, but the ^-amplitude is roughly the same as that used for the ISEE-C mission) the transfer can be done using the stable manifold. No corrections along the transfer are required, except small maneuvers to take care of errors of injection. The traveling time will be the same as that used for the ISEE-C mission. When the spacecraft approaches the target orbit, the station keeping can start. This produces an important saving in the fuel consumption of the spacecraft. A sample of results is given in Table 3.10. However, as the Moon is not taken into account, the results are of little practical interest. But we remark that the result will be true, with slight modifications, for the transfer to the real quasiperiodic nominal orbit. The "stable manifold" (to be suitably defined in this case in Chapter 8) will give an easy way for the transfer to the target orbit.

PARAMETERS FOR THE TRANSFER ROUTINE: INITIAL FACTOR = -l.D-11 NUMBER OF DIVI INITIAL AND FINAL POINTS = 7,9 RMAX = 0.015 RMIN = 0.005 RCOL = 4.D-5 POINT NUMBER 7: TRAVELING TIME = 0.6641289026D+1 MINIMAL DISTANCE = 0.12431 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.11034911484D-3 0.55783047699D-4 -0.12907410606D-4 -0.10096032449D+0 -0 INCLINATION OF THE OSCULATING PLANE = 0.2092816307 POINT NUMBER 8: TRAVELING TIME = 0.6356371431D+1 MINIMAL DISTANCE = 0.80333 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: 0.78979405977D-4 0.11915747345D-4 -0.85911427879D-5 -0.39040885533D-1 -0 INCLINATION OF THE OSCULATING PLANE = 0.1222328160 POINT NUMBER 9: TRAVELING TIME = 0.6082963533D+1 MINIMAL DISTANCE = 0.23542 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.23286855011D-3 -0.28399167254D-4 -0.19795385982D-4 0.21373455225D-1 -0 INCLINATION OF THE OSCULATING PLANE = 0.2164371110 PARAMETERS FOR THE TRANSFER ROUTINE: INITIAL FACTOR = -l.D-11 NUMBER OF DIVI INITIAL AND FINAL POINTS = 77,79 RMAX = 0.015 RMIN = 0.005 RCOL = 4.D-5 POINT NUMBER 77: TRAVELING TIME = 0.6441150658D+1 MINIMAL DISTANCE = 0.7599 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73304424297D-4 0.18279072013D-4 -0.82653272814D-5 -0.68945870849D-1 -0 INCLINATION OF THE OSCULATING PLANE = 0.1130816660 POINT NUMBER 78: TRAVELING TIME = 0.6412795539D+1 MINIMAL DISTANCE = 0.7584 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73698173176D-4 0.15902199248D-4 -0.82231771890D-5 -0.59412473593D-1 -0 INCLINATION OF THE OSCULATING PLANE = 0.10863954310 POINT NUMBER 79: TRAVELING TIME = 0.6384531866D+1 MINIMAL DISTANCE = 0.7727 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.75565492639D-4 0.13824089025D-4 -0.83314370560D-5 -0.49302639148D-1 -0 INCLINATION OF THE OSCULATING PLANE = 0.1120997344 PARAMETERS FOR THE TRANSFER ROUTINE: INITIAL FACTOR = -l.D-11 NUMBER OF DIVI INITIAL AND FINAL POINTS = 775,778 RMAX = 0.015 RMIN = 0.005 RCOL = 4.D-5 POINT NUMBER 775: TRAVELING TIME = 0.6426962395D+1 MINIMAL DISTANCE = 0.757

POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73320649688D-4 0.17045087345D-4 -0.82254702143D-5 -0.64276708517D-1 -0. INCLINATION OF THE OSCULATING PLANE = 0.10989459527 POINT NUMBER 776: TRAVELING TIME = 0.6424127276D+1 MINIMAL DISTANCE = 0.7571 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73367058676D-4 0.16809755174D-4 -0.82220069805D-5 -0.63317918649D-1 -0. INCLINATION OF THE OSCULATING PLANE = 0.10948377441 POINT NUMBER 777: TRAVELING TIME = 0.6421293006D+1 MINIMAL DISTANCE = 0.7572 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73427963514D-4 0.16577931149D-4 -0.82200459502D-5 -0.62351695015D-1 -0. INCLINATION OF THE OSCULATING PLANE = 0.10915199497 POINT NUMBER 778: TRAVELING TIME = 0.6418459627D+1 MINIMAL DISTANCE = 0.7574 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73503414601D-4 0.16349485504D-4 -0.82195871853D-5 -0.61378445451D-1 -0. INCLINATION OF THE OSCULATING PLANE = 0.10890026213 PARAMETERS FOR THE TRANSFER ROUTINE: INITIAL FACTOR = -l.D-11 NUMBER OF DIVIS INITIAL AND FINAL POINTS = 7761,7762 RMAX = 0.015 RMIN = 0.005 RCOL = 4.D-5 POINT NUMBER 7761: TRAVELING TIME = 0.6423843803D+1 MINIMAL DISTANCE = 0.757 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73372496169D-4 0.16786416460D-4 -0.82217432713D-5 -0.63221621697D-1 -0. INCLINATION OF THE OSCULATING PLANE= 0.109447019949 POINT NUMBER 7762: TRAVELING TIME = 0.6423560344D+1 MINIMAL DISTANCE = 0.757 POSITION AND VELOCITY WITH RESPECT TO THE EARTH: -0.73378078578D-4 0.16763113109D-4 -0.82214945890D-5 -0.63125252461D-1 -0. INCLINATION OF THE OSCULATING PLANE = 0.10941105791 Table 3.10

Output of program PAPUS concerning transfer to the Earth using the stable manifol

188

3.5

The Neighborhood of the Halo Orbits: Numerical Study and

Applications

References [1] R. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin, 1978. [2] E. Fehlberg. "Classical fifth, sixth, seventh and eighth order Runge-Kutta formulas with stepsize control". Technical Report TR R287, NASA. [3] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer Verlag, 1983. [4] P. Hartman. Ordinary Differential Equations. John Wiley & sons, 1964. [5] S.S. Lukjanov. "Control of a spacecraft motion in the vicinity of a collinear libration point of the circular three-body problem by means of light pressure" Cosm. Res., 19, 518-527, 1980. [6] S. Smale. "Differentiable dynamical systems". Bull. American Mathematical Society, 73, 747-817, 1967. [7] V. Szebehely. Theory of Orbits. Academic Press, 1967.

Chapter 4

Analytic Solution of the Variational Equations. Analytic Computations for Control Parameters

Trying to go as far as possible in the analytic approach to the problem, this chapter is devoted to redo the computations numerically performed in Chapter 3 in a fully analytic way. In the same spirit as the analytic theory for halo orbits in Section 2.2, all the relevant magnitudes are expressed as series in the x- and z-amplitudes, a and p\ and in the angle XLOT along the orbit. The coefficients of the series are numbers which depend only on the mass parameter, n, and on the equilibrium point, L\ or L2. The main task will be done using routines for the formal manipulation of series. The algorithms are designed to perform this task in an efficient way. The numerical results of the previous chapter give a hint for the types to be expected for the series representing the Floquet modes. Checks against the numerical results of Chapter 3 are done for the Floquet modes as well as for the projection factors. Some difficulties are present due to resonances between modes when the zamplitude goes to zero. This produces problems of convergence. They are discussed and their importance is analyzed.

4.1 4.1.1

Analytic Solution of the Variational Equations The Analytic

Method

It is useful for several reasons to know the local behavior near the periodic orbit. Here we give the method to obtain analytic expansions (depending on the amplitudes a and (i of the orbit) of the unstable and stable eigenvectors, the eigenvector related to the complex eigenvalue, and the vectors tangent to the orbit and to the family at any point of the orbit. From the equations of motion for the RTBP as used in Section 2.2 we get immediately the variational equations u2£," - W

- (1 + 2c2)i = Mxi + Myr] + MZQ = XR+

w V + 2w<£' + (c2 - 1)7? = (Nx£ + Nyrj + NZQV=1 189

y/^lXI,

= YR + V=1YI,

(4.1)

190

Analytic Solution of Variational Equations.

w2C" + c2( = Px£ + PVT) + PZQ = ZR +

Control

Parameters

V^IZI,

where' = ^ Mx = dM/dx, ...,PZ = OP/dz. The functions Mx,My,... ,PZ are known periodic functions of n = Awr of period T (the period of the halo orbit). The function u> is known as a function of a and /3. We shall also use w — COT. Let e = (£,77, (,£,fj, C) T . Then (4.1) can be written e' = A{r\)e, with A a T-periodic matrix. As we know from Section 2.3, for moderate amplitudes of the halo orbit only one of the eigenvalues of the monodromy matrix is greater than one in modulus. This is true till orbits numbers 8, 9, 8 and 7 of Tables 2.13-2.16, respectively. Even before these orbits are reached the analytic theory for the halo orbit is no longer valid. From our knowledge on the form of the Floquet modes acquired in Section 3.1, for the unstable eigenvector e\ = (£, 77, ()T we look for an expression of the type:

£ = evw(G + V=lH), r, = euw{J+y/^lK), C =

vu,

e (L +

(4.2)

y/=lQ),

where v = J2i ,->o vijalft with Vij = 0 if i or j is odd, and G, H, J, K, L and Q are power series in a,/3 and 7 — exp(«Ao;r) of the types (a), (b), (a), (b), (c) and (d), respectively (see subsection 2.2.3). Besides the results of Section 2.2, the reasons for this choice are (as it is easy to show by induction) that G, H, J and K are even functions in /3, while L and Q are odd functions in /3 and 1/ is even in both a and The freedom in the normalization of e\ allows us to take the time-independent term of G equal to one, i.e. J2i j Gijoofft — 1- This implies Gij0 = Si0SjoBy substitution of (4.2) in (4.1) and from the terms without a,/3 nor 7 of the first two equations we get "00 = \ ((<* - 2) + [(ca - 2) 2 + 4(c2 - 1)(1 + 2c 2 )] 1 / 2 ) , and T

-*ooo —

^o2o ~ (1 + 2c2) 7, • 2^oo

When the terms of order n — 1 are available we obtain the terms of order n = i+j with I A; I < n, solving the following systems: / 2is0o — 2 Jooo — 2i>oo A ( vij \ V 2i/00Jooo + 2 ul0 + c2-l J \ Jij0 J

_

( XRijQ \ \ YRij0 J '

. ^,

191

Analytic Solution of the Variational Equations

/

\

Vi 2Xi/ook

-2Xv00k i>i

-2i/ 0 0

2i/0o

—2Xk

v2

2Xk

2VQQ

2Xv00k

2\k

—2Xk

\ I Gijk \

— 2VQQ

Hijk

-2Xu0ok

i>2 J \ Kijk J

X2k2 + c 2 2Xi/00k

-2Xv00k ulQ

\

(4.4)

I -K'ijk

Jijk

for k ^ 0, where i>\ = VQ0 - X2k2 - (1 + 2c 2 ) a n d i>2 = vl0 ^oo

/ XRijk Xlijk

\ YIijk J X2k2 + c 2 - 1 a n d

Lijk

2 2

X k + c2

ZRijk Zlijk

(4.5)

T h e determinant Dx of (4.3) is equal t o i / 0 0 1 ( 3 ^ 0 + ^ 0 ( 2 - c 2 ) + (c 2 - l ) ( l + 2c 2 )) which is always different from zero if c 2 > 1. T h e determinant of (4.4) has t h e following expression D2 = b8k8 + b6k6 + b4ki + b2k2, where b2

d(36cl - 3 2 c 2 ) ,

bi

d(9c22 - 8c2) + A 2 (7c| - Uc\ - 8c2 + 16),

be

d(6c22 + Ac2 - 12) + X2{-2c\

h

2

d(3cl - 5c2 + 3) + X (-5c

3 2

- 2c\ + 22c2 - 20), + 16c?, - 14c2 + 4),

and d = (c2 — 1)(1 -f 2c 2 ). For all c2 between 4 and 8 we have that b2,64, be and b8 are positive numbers, so the determinant is different from zero. In the same way, the determinant D$ of (4.5) can be written in the following form £> 3 =fc 4 (A 2 (2-c 2 ) + (C2-1)(1 + 2c2)) + fc22AVoY • c 2 ) + c | + ( c 2 - l ) ( l + 2c 2 ). It is easy to see that D3 is different from zero for values of c2 > 1. The second Floquet mode is obtained from the first one using the symmetry (x,y,z,t) ^ (x,-y,z,-t). Explicitly if e i ( r ) = ( £ ( T ) , T K T ) , C ( T U ( T ) , 7 K T ) , C(T))T , then e 2 (r) = (£(—r), - r ? ( - r ) , C ( - r ) , - ^ ( - r ) , 7 ) ( - r ) , -((-T))T. It is, of course, of the type e~vwg2(T), g2 being T-periodic. The third vector e 3 to get a local basis is the tangent vector to the orbit. Let 631,632,633 be their first three components. The remaining ones are obtained by time derivation 63^+3 = 63^, k = 1,2,3. This vector is obtained by derivation with respect to r of the three components x, y and z, of the orbit. We get k lXu)y^/kaijkal(3J-f i~,

e3i

e 32

=

-Xoj^kbi:jkQiPi^k,

(4.6)

e33

where a,ijk,bijk,Cijk are coefficients of the development of x,y and z, respectively (see Sections 2.2 and 3.1). As the ^-amplitude, /?, is a parameter for the halo family,

192

Analytic

Solution of Variational Equations.

Control

Parameters

let e\ be the vector obtained from the halo orbit by derivation with respect to /?. Putting eli,el2isl3 for the first three components we get

du 1 = e4i + r — - e 3 i , dp u

dw 1 = e 42 + T— - e 3 2 , dp a; = ] > > * ( i a ' - 1 ^ ' ^ + i a ^ - 1 ) 7* + r ~ ( v ^ A w £ =

. (4.7) fcW*

dw 1 43 + T 3 ^ - e s s dp w

e

In the above expressions % is obtained by derivation of the relation given by A = Ydijtfpi. The vector e4 is defined having as first three components e 4 i,e 4 2,e 4 3 as defined by (4.7). The remaining part of e\ is a multiple of e^. We should remark that e4)fc+3 = ei}k + %ue3k ^ or ^ = 1?2,3. Notice that the vector e 4 (r) defined in this way is not collinear with the vector e 4 (r) defined in subsection 3.1.4. However both couples (e3,e 4 ) and (eJ3,e4) define the same two-dimensional plane. We turn to the last couple of Floquet modes. Let 115 = e& + ieg be the complex eigenvector associated to the eigenvalue A5 — r + is. We consider iw

u5 = e ^

Yrijk cos(k\w) + V-~lJ2uijk sin(fcAw) \ Y Sijk sin(fcAw) 4- \[-\ Y Vijk cos{k\w) , Y Ujk cos(k\w) + y/^1 Y wijk sin(fcAu;) /

where ip is a parameter. In order to get the coefficients r^k, Uijk, s^k, v^k, tijk, Wijk and V which depend on a and j3, we write these series as power series in a, f3 and 7 and we look for a solution of the variational equations of the following form u5 = &iw^

where tp — YijXj^ij^ft Satisfy ±iijtk

+ V=i an(

— tli,j, — ki &i,j,k

^

=

tne >

(4.8)

coefficients of the series R,S,T,U,V

~^ i1jt — ki -Li,j,k

=

J-i,j, — kj Ui,j,k

=

and W

Ui,j, — ki *i,j,k

=

Vij:-k and Wijtk — —Witjt-k- It has been checked by induction that the series R,S,T,U,V and W are series of types (c), (d), (d), (c), (a) and (b) respectively, and the coefficients tpij of tp are equal to zero if i or j is odd. As a normalization the time-independent term of T is taken equal to 1, so Tijo = SioSjQ.

Analytic Solution of the Variational

Equations

193

By substitution of U5 in the variational equations we get LJ2C

-

wV +

2WT)'

-

(1 + 2c2)£

2w£'

+ +

( c 2 - 1)77

2

u C"

c2C

= = =

XC + XD, (YC + YD)V=1, ZA + ZB,

(4.9)

where XC, YC and ZA are the part in cosine of the first, second and third equations respectively, and XD,YD and ZB contain the corresponding parts in sine. From the terms with i = j = k = 0 of the third equation we get ^00 = —\/c2Let us suppose that the terms of order n — 1 = i + j with \k\ < n — 1 are known. Equating the terms with % + j — n, \k\ < n, in the first and second equations we get the following system /

-f/>i -2Xip00k 2ipoo \ 2Xk

2\ip00k $1 -2Xk -2^oo

2^oo 2Afc -2Xipook

2Xk \ ( Rijk 2ipoo Uijk V -2Xip00k ijk J \ Sijk i>2

where ipi = tpQ0 + X2k2 + (1 + 2c2) and V>2 = —ip{ If fc 7^ 0 we obtain from the third equation - ^ 0 - X2k2 + c2 -2Xt/j0ok

2Xipo0k ^00 + A2fc2 - c2

\

)

( XCijk XDijk YCijk

^

V YDijk

J

,(4.10)

X2k2 + ca - 1.

-* ijk

wijk

ZAijk ZBijk

(4.11)

and equating the time-independent terms of order n in the last equation of (4.9) we get the coefficients of ip in the following way

ipij =

ZA

ijO

2
The determinant Dt of system (4.11) equals A2fc2(4c2 - X2k2), so it is different from zero in the cases of interest. The determinant of (4.10) has the expression D8

=

A 8 fc 8 -(2c 2 +4)A 6 fc 6 + (c2 + 2c 2 +6)A 4 fc +(-36cl + 34c?. + 2c2 - 4)A2fc2 + c\ - 2c2 + 1.

For c2 between 4 and 8 and any value of fc, D5 is different from zero. For fc = 1 it is negative, and for fc = 2 it is relatively small. This is due to the almost resonance with the halo orbit. This fact produces a slow convergence of the series. We shall return later to this point.

194

Analytic Solution of Variational Equations.

4.1.2

Description

Control

Parameters

of the Program to Do the Analytic

Computations

In the analytic computation of the projection factors and the unitary controls, the first three components of ei, e$, e^, e$ and e$ are written in the following form (G vw

e1=e

+ ^r=\H\ J + yf=\K

/ ^TlA ,

e3 =

\ L + ^TlQ )

\ B

I ,

e4 =

\ V=1C J

D\ y/^lE

\

,

FJ

iwip

e5 = e

V sf=iw J

where G, H, J, K, L, Q, A, B, C, D, E, F, R, S, T, U, V and W are power series in a and p of types (a), (b), (a), (b), (c), (d), (b), (a), (d), (c), (d), (a), (c), (d), (a), (d), (c) and (b) respectively. The functions v and ip are power series in a and /?. We note that all the series involved in the analytic computation of ei,e3,e4,es and ee are power series in a, j3 and 7 of types (a), (b), (c) or (d), or power series in a and j3 of the same type of to = a>oo + ]C» j>o i+j>2 uijaiP^ with Wy = 0 if i or j is odd. The series above are computed in the program ANAVAR up to a given order n. In this program one has to read, first, the results of program ANACOM (through subroutine LECT12). The series A, B and C are computed in the subroutine CE3 by derivation of x, y and z with respect to r, that is, A

=

£fcojiifca\0V,

B

=

-^khjka'F^,

C

=

^fcc^aW-

We note that we do not need to multiply these series by Aw (which is a constant) in order to get the direction of e$. The subroutine CE4 computes the series D,E and F of e\ in the following form

E = /V/*)£iW+1/?'~V-/W)£iW_10i+V, F

=

/2(a,/?)^jcij,ai+1^-17fc-/1(a!/3)E^^ai-1^+17fc,

where

4 | | = /~+ E «,j>0,i+j>2

f2(a,P)

a da

i,j>0,i+j>2

Wf'

Analytic Solution of the Variational

Equations

195

with //• = f?j = 0 if i or j is odd. Some auxiliary subroutines, DEAAC and DECAA, are used to get series

Y.iaijka'-1^1^

= £)(* + l ) a i + i , i - i , * a W ,

from a given series A = Yl ^ijkQ-lPj'yk- We note that if the initial series A is of type (a) ((b), (c) or (d)), then the final series will be of type (c) ((d), (a) or (b)). In order to solve the variational equations it is necessary to compute the partial derivatives of the right-hand side of the restricted three-body problem (M, TV and P) with respect x, y and z, along the halo orbits. The series M, N and P are power series in x,y and z (see 2.2.3.2) whose coefficients are computed by the subroutine FMNP. In the partial derivatives of M, N and P (Mx,My, Mz,Nx,Ny, Nz,Px,Py and Pz) some extra monomials appear for order n > 2. We get the extra monomials multiplying the monomials of P by y. The series Mx,My,Mz,Nx,Ny,Nz,Px,Py and Pz for the family of halo orbits are computed in the subroutine DMNP according to Table 4.1. The numbers (i,j,k) in the table represent the monomials xly^zk. The monomials of M, N and P are listed in column 1. Columns x, y and z contain the partial derivatives of the series in the right part with respect to x, y and z respectively. The table shows how the monomials of degree n of the partial derivatives can be obtained from the monomials of degree n — 1 of M, N, P and the extra ones (noted by (*)). The exponents of x and y of these monomials up to a given degree are generated by the subroutine PONENT following the same order used for M, N and P and adding for each degree the extra terms at the end. The computation of ei,es and e$ is done in a recurrent way. The terms of order n of the series involved can be obtained from terms of order less than n. The subroutines PE1E2 and DE1E2 compute the terms of order n of the right-hand side of the equations when order n — 1 is available. In this step some auxiliary subroutines (PASPRO, MIXX, MIXT, ABAC) are used to compute the products of the different types of series (series in a and /?, or series in a, /3 and 7) which appear in the computations up to a given order. Also the subroutines PROD and SUM developed for program ANACOM are required. For each term of order n of G, H, J, K, L, Q, R, S, T, U, V and W, systems (4.3), (4.4), (4.5), (4.10) and (4.11) are solved. The solution of (4.4)and (4.10) is computed by the subroutine LINEAR. In that routine the solution of a linear system is computed using Gauss method. The series G, H, J, K, L, Q and v are written to some output channel, A, B, C, D, E and F to a second output channel, and R, S, T, U, V, W and tp to a third output channel.

196

Analytic Solution of Variational Equations.

order of the derivatives

[(1,0,0) {(0,1,0) 1(0,0,1) "(2,0,0) (0,2,0) (0,0,2) {(1.1.0) 1(1.0,1) '(3,0,0) (1,2,0) (1,0,2) '(2,1,0) (0,3,0) .(0,1,2) (2,0,1) (0,2,1) (0,0,3) "(4,0,0) (2,2,0) (2,0,2) (0,4,0) (0,2,2) .(0,0,4) (3,1,0) (1,3,0) .(1,1,2) (3,0,1) (1,2,1) (1,0,3)

1

2

3

Table 4.1

4.1.3

Results

1

X

Control

y

Parameters

z M N P

[(1,0,0) M

\[(0,1,0) {(0,1,0) 1(0,0,1) "(2,0,0) (0,2,0) (0,0,2) [(1,1,0)

[(1,0,0)

[(1,0,0) ^[(1,1,0) (2,0,0) (0,2,0) (0,0,2)

1(1,0,1) 0(0,1,1) "(3,0,0) (1,2,0) .(1,0,2)

N P M

11(1,0,1) N 0(0,1,1) (2,0,0) (0,2,0) .(0,0,2)

P

'(2,1,0)

(2AD (0,3,0) .(0,1,2)

(2,1,0) i (0,3,0) .(0,1,2) (2,0,1) (0,2,1) (0,0,3)

11(0,0,1)

(3,0,0) (1,2,0) (1,0,2) (* (1,1,1)

M

(0,2,1) (0,0,3) N (* 1(1,1,1) (3,0,0) (1,2,0) (1,0,2)

P

Scheme for computing the derivatives of M, N and P.

and Numerical

Tests

The results of program ANAVAR are an intermediate step for the analytic computation of projection factors and unitary controls. A sample for the L\ case in the SunBarycenter problem is given at the end of the section. We restrict the sample to order 5. Besides the series G, K, H, J, L, Q, A, B, C, D, E, F, R, S, T, U, V and W, the series corresponding to v (written as TV in the output) and if: (written as P in the output) are given. Furthermore the values of Jooo(JO), voo(NU0) and tpoo(PSO) are written. We remark that it follows from (4.1) that if the analytic theory is available up to order n, then Mx,..., Pz are available up to order n — 1 and this is the higher order for the solutions of the variational equations. For the applications we have run the program ANAVAR up to order 13. The CPU time for this run was 470 s under VAX

Analytic Solution of the Variational

197

Equations

11/785. In order to perform some checks a complementary program PROVAR has been developed. This program reads the results of the subroutine ANAVAR, the amplitudes a and /? of a given orbit, and computes the Fourier coefficients of the first three components of e±, e%, e,j, es and e&. These results should be compared with the output of the routine PRECON as given in Section 3.2. Taking care of the suitable normalizations the comparison shows the following facts:

a) The analytic first mode agrees perfectly with the numerical results. The relative error is O(10~ 7 ). b) The agreement is also 0(1O - 7 ) for the tangent vector to the orbit. This is not true for ei for which the relative error is of the order of 1 0 - 2 . However, as it has been stated before the analytic and numerical fourth modes are computed in a different way and it is not required that they coincide. The only thing to be required is that they span the same plane together with e3-

c) Concerning modes 5 and 6 the situation is much worse. The relative errors between numerical and analytic results for ft = 0.08 are roughly 1/4. This is improved when /? increases. The reason for this will be explained in 4.1.4. However the induced error in the projection factors is smaller as we shall discuss in 4.2.4. d) From the values of the traces of the periodic orbit Ai and T can be computed (see Section 3.1). Furthermore the period of the orbit and the value of the function u> are available (using results of program TESTAN, Section 2.2). From this information it is possible to obtain numerical values of u and ip. The agreement with the analytically obtained v is excellent (relative error less than 10~ 8 ). For ip the relative error is worse (in the range 1 0 - 3 to 10 - 4 ) as it can be foreseen from c).

4.1.4

The Resonance Convergence

Between

Modes and the Related Problems

of

The difficulties associated to es and ee are easily explained. For /? = 0 the halo orbit is reduced to a planar Lyapunov orbit. At this point a bifurcation is produced. Concerning the eigenvalues of the monodromy matrix, M, we obtain Ai 3> 1 (unstable mode), A2
198

Analytic Solution of Variational Equations.

Control

Parameters

The vector associated to the Lyapunov family, — {x(a, P, r),y(a, /?, r ) , z(a, P, r),x{a, P, r),y(a, P, r),z{a, P, r ) ) r at /? = 0, a = a* such that A(a*, 0) = A2 — C?, and r = 0, is not an eigenvector of M (it is, however, an eigenvector with eigenvalue 1 of the differential of the Poincare map when we restrict it to y = 0). To see this it is enough to check that the period changes in the Lyapunov family at a = a*. This follows from fj(a*,0) ^ 0, easily obtained from the expression of w. Therefore, in the Jordan decomposition of M the block associated to the eigenvalue 1 has nonzero out of the diagonal elements. On the other hand when j3 —• 0, e\ goes to a solution of the variational equations, because e\ - e 4 contains the factor %, and % = ^% + %, where % = - f f / § £ • But both | £ and ^ | go to zero when P does. A similar thing happens with e 5 and ee- When /? -> 0 the rotation exp(iipcoT) has a period going to the period T of the bifurcating orbit. Therefore es and e§ tend to be solutions of the variational equations. If they were independent together with e% we would obtain 4 independent periodic solutions e3(r),e4(r),es(r) and e§{r) of the variational equations. This would imply 4 independent eigenvectors of M with eigenvalue 1, which is an absurdity. Another way to explain the difficulties is that when ft goes to zero the six Floquet modes go to the following forms: e\ — ( - , —,0, - , - , 0 ) T , e-i = (—, —,0, —, - , 0 ) r , e 3 = ( - , - , 0, - , - , 0) T , e 4 = (0,0, - , 0,0, - ) T , e 5 = (0,0, - , 0,0, - ) T , e 6 = (0,0, - , 0,0, —) T , according to the type of their elements. Hence e^,e^ and e§ cannot be independent. We remark also that at the branching orbit e ^ r ) can be chosen as •^ (x(a,0,r),... , i ( a , 0 , r ) ) | a = a » , which is of the form ( - , —,0, - , — ,0) T , showing that the continuity is lost. As an additional illustration we mention that for the L\ case in the SunBarycenter problem using a 15th order theory it is obtained a* = 0.1391564 for P = 0, with a relative error of 10~ 6 at most. The rotation exp(itpu>T) should be the identity, i.e. \tp\uT = 2TT, but XOJT = 2?r and this implies |V>| = A = 2.086453455. By substitution of the given x and ^-amplitudes for P = 0 in the expression of V> given by program ANAVAR we obtain \tp\ =

2.015210551 + 0.036494495 + 0.008557552 + 0.004297167 +0.002718250 + 0.001908684 + 0.001433824 + . . .

up to order 12 (the last available term). The given partial sum amounts to 2.07062 . . . still far from the correct sum. This is due to the fact that we are in the boundary of the convergence region. The formulation given for the analytic Floquet modes is not valid for P going to zero. The convergence is lost. The discussion above shows also why the results improve when /? is increased within the domain of convergence of the analytic halo orbits.

Analytic Solution of the Variational

CASE HU = NN = JO =

uo (1 (1 (1 (1 (0 (0 (2 (0 (2 (0 (2 (0 (2 (0 (2 (0 (1 (1 (1 (2 (0 (3 (1 (3 (1 (3 (1 (3 (1 (3 (1 (3 (1 (3 (1 (3 (1 (2 (0 (2 (0 (2 (0 (2 (0

LI 0.3040357143000000D-05 5 -0.5345736537149110D+00 = 0 . 2532658995564170D+01 0.1808611697444885D+00 0 1) 0.4954206606046848D+00 0 1) 0.5425752192716954D+00 0 1) 0.8125982118531568D+00 0 1) -0.1859120783228149D+00 1 1) = 0.3209366083928439D+00 1 1) -0.4086103589925799D+00 0 2) = 0.2721328431881468D-01 2 2) -0.1153760845415829D-02 0 2) = -0.2768883474794640D-02 2 2) = -0.1285266335620483D+01 0 2) = 0.116 527573121993D+00 2 2) -0.3319434138362190D+00 0 0) = 2 0) = -0.3914010804067391D-01 -0.7942773196008624D+00 0 2) = 2 2) = -0.5261469961823192D-01 1 0) 0.5185860051066978D+00 1 2) = -0.1488253921689322D-02 1 2) = -0.4253708300612538D+00 0 0) = -0.2666495570309598D+01 2 0) = -0.8130751988271776D+00 0 1) = -0.6618019259405915D+00 2 1) 0.1614746070884018D+00 0 3) 0.6174773316028679D+00 2 3) = -0.5658168201740319D-01 0 1) = -0.5820048510573394D+00 2 1) 0.1546860035983995D+00 0 3) = -0.5827049941730015D+00 2 3) 0.3206072489608153D-01 0 1) = -0.8427549935317539D+00 2 1) = -0.2917903581005164D+00 0 3) 0.1491333699495434D+01 2 3) = -0.1392762380010141D+00 0 1) = -0.2124739872197645D+01 2 1) 0.4222133120050080D+00 0 3) 0.1644494147705355D+01 2 3) = -0.3987904255676670D-01 1 1) = -0.1064334312196404D+01 3 1) 0.9058192731774575D-01 1 3) 0.2317018812855133D+00 3 3) = -0.1459057497588900D-01 1 1) 0.1622000980453661D+01 3 1) = -0.4263374444908390D-02 1 3) 0.6738814464777059D+00 3 3) = -0.2805815701422004D-01

G (4 G (2 G (0 G (4 G (2 G (0 K (4 K (2 K (0 K (4 K (2 K (0 H (4 H (2 H (0 H (4 H (2 H (0 J (4 J (2 J (0 J (4 J (2 J (0 J (4 J (2 J (0 L (3 L (1 L (3 L (1 L (3 L (1 Q (3 Q (1 Q (3 Q (1 N (4 N (2 N (0 G (5 G (3 G(1 G (5 G (3 G (1 G (5 G (3 G (1 K (5

0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 1 3 1 3 1 3 1 3 1 3 0 2 4 0 2 4 0 2 4 0 2 4 0

2) 2) 2) 4) 4) 4) 2) 2) 2) 4) 4) 4) 2) 2) 2) 4) 4) 4) 0) 0) 0) 2) 2) 2) 4) 4) 4) 0) 0) 2) 2) 4) 4) 2) 2) 4) 4) 0) 0) 0) 1) 1) 1) 3) 3) 3) 5) 5) 5) 1)

Equations

199

0.1106160271738590D+01 -0.9874814112100292D-01 0.2524723144624028D-01 -0.1384821346687992D+01 0.2013713197634864D+00 -0.6614618582184655D-02 -0.1860984947130167D+01 -0.7008416740860990D-01 0.2508330093138730D-01 0.1126246755465649D+01 -0.5529635117522212D-01 -0.2069439938165449D-02 0.3196632678030173D+01 0.4516856158246034D+00 -0.1586460315180400D-01 -0.2525494671579871D+01 -0.3257946306542989D+00 -0.8399921469441334D-02 -0.1947235892604816D+01 -0.5942958087732516D+00 -0.1713810039912929D-01 0.2654866341211914D+01 -0.8987387206716629D+00 0.9448759178972720D-02 -0.2437335461152137D+01 0.1012950290895219D+00 0.2767392063167708D-02 -0.1877939180544048D+01 0.5942485479930130D+0 0.4584653142886637D+00 -0.4810641361738689D-01 -0.4783142393577762D+00 0.3618425013715348D-01 -0.1948949152483246D+01 0.9102471081467338D-02 -0.9890326895369906D+00 0.6273388172443814D-01 0.4414210812583315D+01 -0.1707571462182482D+01 -0.2382403359865240D+00 0.9473018680448164D+00 0.2124743478531757D+01 -0.1590522083296336D+00 -0.3962175242726377D+01 0.2054716940517980D+00 -0.1405041844135964D-01 0.2497016085109489D+01 -0.4143726090839239D+00 0.1677515525928288D-01 -0.1080773346389202D+01

200

K(3 K(l K(5 K(3 K(l K(5 K(3 K(l H(5 H(3 H(l H(5 H(3 H(l H(3 H(l J(5 J(3 J(l J(5 J(3 J(l

Analytic Solution of Variational Equations.

2 4 0 2 4

0 2 4

0 2 4 0 2 4 2 4 0 2 4 0 2 4

= = = = 3) = 5) = 5) = 5) = 1) = 1) = 1) = 3) = 3) = 3) = 5) = 5) = 1) = 1) = 1) = 3) = 3) = 3) = 1) 1) 3) 3)

0 -0 0 -0 0 -0 0 -0 0 0 -0 -0 0 -0 -0 0 -0 0 0 -0 0 -0

2301929754722930D+01 1318069927535936D+00 3965741803619705D+01 3433369099237444D+00 1144026973024185D-01 2365512354924140D+01 2193156513250643D+00 2361543782279740D-02 1961184786006026D+01 3350878672363064D+01 6970255954520934D+00 4660929776898495D+01 7755389806700120D-01 2065290953211215D-01 6301170592670368D+00 2261881293230590D-01 1289172710224523D+01 2598428402696061D+01 6621283666156592D-01 5737518111597422D+01 1284997910180245D+01 3113740608597905D-01

Table 4.2

J(5 J(3 J(l L(4 L(2 L(0 L(4 1,(2 L(0 L(4 L(2 L(0 Q(4 Q(2 Q(0 Q(4 Q(2 Q(0 Q(4 Q(2 Q(0

0 2 4 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5

5) 5) 5) 1) 1) 1) 3) 3) 3) 5) 5) 5) 1) 1) 1) 3) 3)

3) 5) 5) 5)

Control

= = = = = = = = = = = = = = = = = = = = =

0 -0 0 0 -0 0 -0 -0 -0 0 -0 0 -0 0 -0 0 -0 0 0 -0 0

Parameters

4122849529758202D+01 3233770865005978D+00 1891895583537684D-02 1829591690211440D+01 6911803355110367D+00 8241147375488192D-01 1153060210102002D+00 1088370871514493D-01 3223078000668421D-02 9563393525623444D+00 1052980120423679D+00 2258228174168213D-02 2885012772216766D+01 1581734599200697D+01 5359117841731795D-01 2263275322521081D+01 9881675839618288D-01 2089890835512201D-02 1608823538605654D+01 1519579468554130D+00 2504263005975283D-02

Output of program ANAVAR: Series G, K, H, J, L, Q and N.

CASE LI MU = 0.3040357143000000D-05 NN = 5 A(l 0 1) = -0.5000000000000000D+00 B(l 0 1 ) = 0.1614634048091902D+01 C(0 1 1) = 0.5000000000000000D+00 D(0 1 1) = -0.1740900545935829D+01 ECO 1 1 ) = -0.5621834591619539D+01 F(l 0 1 ) = 0.1596559878224751D+02 A(2 0 2) = -0.9059647953966096D+00 A(0 2 2) = 0.1044641164074320D+00 B(2 0 2) = -0.4924458751013136D+00 B(0 2 2) = -0.6074646717326298D-01 C(l 1 2) = -0.3468654604858686D+00 D(l 1 0) = 0.3042958344362914D+02 D(l 1 2 ) = 0.1812751255969184D+00 E(l 1 2) = 0.3654306030270019D+01 F(2 0 0 ) = 0.3322748864122144D+02 F(0 2 0 ) = 0.3623149617156789D+01 F(2 0 2) = -0.5537914773536906D+01 F(0 2 2) = -0.6038582695261314D+00 A(3 0 3) = 0.1190730292670741D+01 A(l 2 3 ) = -0.1240280779397711D+00 B(3 0 1) = -0.1422540573647265D+01

B(l B(3 B(l C(2 C(0 D(2 D(0 D(2 D(0 E(2 ECO E(2 E(0 F(3 F(l F(3 F(l A(4 A(2 A(0 A(4 A(2 A(0 B(4

2 0 2 1 3 1 3 1 3 1 3 1 3 0 2 0 2 0 2 4 0 2 4 0

= = = = = = = = = = = = = = = = = = = = = = 4) = 2) = 1) 3) 3) 3) 3) 1) 1) 3) 3) 1) 1) 3) 3) 1) 1) 3) 3) 2) 2) 2) 4) 4)

-0 0 -0 0 -0 0 -0 0 -0 0 -0 -0 0 -0 0 0 0 0 0 -0 -0 0 -0 0

2158462883436975D+00 1328551164323901D+01 3452974106658828D-01 5971431377509674D+00 2856580507897597D-01 6073575647603767D+01 1938807546315024D+01 1505642659652701D+01 1439470323978127D+00 4825667498956720D+02 5509395511523980D+01 3890700238651595D+01 4007523004923090D-01 8386650348280846D+02 6073575647603767D+01 6355831835269535D+01 4739487204510206D+00 1740504692726970D+01 6340969703409752D+00 3033404962522150D-01 2380288950983845D+01 3433410181667700D+00 1044482786814066D-01 1920413210888131D+01

Analytic Solution of the Variational Equations

(2 2 (0 4 (4 0 (2 2 (0 4 (3 1 (1 3 (3 1 (1 3 (3 1 (1 3 (3 1 (1 3 (3 1 (1 3 (3 1 (1 3 (3 1 (1 3 (4 0 (2 2 (0 4 (4 0 (2 2 (0 4 (4 0 (2 2 (0 4 (5 0 (3 2 (1 4 (5 0 (3 2 (1 4 (5 0 (3 2 (1 4 (5 0

2) 2) 4) 4) 4) 2) 2) 4) 4) 0) 0) 2) 2) 4) 4) 2) 2) 4) 4) 0) 0) 0) 2) 2) 2) 4) 4) 4) 3) 3) 3) 5) 5) 5) 1) 1) 1) 3)

-0.6225734964700858D+00 0.2823240295501348D-01 -0.2149425545202692D+01 0.8477528693038016D-01 0.3828093871732036D-02 -0.1170523568832721D+01 0.7616528299315740D-01 -0.9575567391552400D+00 0.66633454627i0728D-01 -0.1429861478737300D+03 0.2741822741841397D+02 0.2585046898688398D+02 -0.1973446876442709D+01 -0.2806047726966592D+01 0.2642067037839506D+00 -0.9664478465891418D+01 0.3012100291825212D+01 0.6130383992403205D+01 -0.2698209650126737D+00 0.3624378103724396D+01 0.2351991270108950D+02 0.2771527104985947D+01 0.1040228368475635D+02 -0.2465222311819647D+01 -0.5399091896018933D+00 -0.7643983354294898D+01 -0.9047520718626013D+00 0.5800110876896068D-01 -0.3865079685081630D+01 -0.2867484743873657D+00 0.8520015196563276D-02 0.4218232987432908D+01 -0.6962957668752185D+00 0.2759161522420977D-01 0.1933414620390834D+01 0.3162370152105307D+01 -0.5127782166626836D+00 -0.4655072221683606D+01 Table 4.3

B (3 2 B (1 4 B (5 0 B (3 2 B (1 4 C (4 1 C (2 3 C CO 5 C [4 1 C [2 3 C [0 5 D [4 1 D [2 3 D 10 5 D (4 1 D (2 3 D (0 5 D (4 1 D (2 3 D [0 5 E [4 1 E [2 3 E [0 5 EA 1 E 12 3 E :o 5 E [4 1 E [2 3 E 10 5 F [5 0 F :3 2 F :i 4 F [5 0 F :3 2 F :i 4 F :5 0 F :3 2 F :i 4

3) 3) 5) 5) 5) 3) 3) 3) 5) 5) 5) 1) 1) 1) 3) 3) 3) 5) 5) 5) 1) 1) 1) 3) 3) 3) 5) 5) 5) 1) 1) 1) 3) 3) 3) 5) 5) 5)

201

0.6700886128974591D+00 -0.2668103509925653D-01 0.4084912368638145D+01 -0.3420311256853099D+00 0.2101353670495661D-02 0.1504757774670548D+01 -0.2084652732826900D+00 0.3160803736521231D-02 0.1710903312716945D+01 -0.1779327700550720D+00 0.3455285189639480D-02 0.1651995555280459D+03 -0.3909107758910032D+02 -0.6488713483997862D+00 -0.2912812528032309D+02 0.3479335711693285D+01 -0.1504227162383318D+00 0.5793625143247275D+01 -0.7497913633448329D+00 0.1921370320283125D-01 0.1736097556034969D+03 -0.7460319916155322D+02 0.5269810361186367D+00 0.2502583178147209D+02 -0.6208969443204541D+01 0.7559703408175591D-01 -0.9854266966338632D+01 0.6608516167547436D+00 -0.1463299100868062D-02 -0.5484936724358469D+03 0.3303991110560918D+03 -0.1954553879455016D+02 -0.1737063209804607D+02 0.2702709559846036D+01 0.8809916763057747D+00 0.1092623833842276D+02 0.1356656157547936D+01 -0.1374792111909475D+00

Output of program ANAVAR: Series A, B, C, D, E and F.

202

Analytic Solution of Variational Equations.

CASE LI NU = 0.3040357143000000D-05 NN = 5 PSO .2015210551475635D+01 R (0 1 1) = 0.3555491785032116D+00 S (0 1 1) = 0.5255626768139885D-01 T CI 0 1) = 0.3809615156516177D+00 U CO 1 1) = -0.1404090194763038D+00 V CO 1 1) = -0.7568030245445969D-01 W (1 0 1) = -0.7359068222738914D+00 R Cl 1 0) = 0.6441981072035151D+01 R CI 1 2) = -0.3062419937854517D+00 S Cl 1 2) = -0.5219158068158726D+00 T C2 0 2) = 0.6285867060601335D+01 T CO 2 2) = -0.6084152806466143D+00 U Cl 1 2) = 0.1395961323514941D+00 V [1 1 0) = -0.2117024583273942D+02 V Cl 1 2) = -0.5658888325104418D+00 C2 0 2) = -0.5879787771943789D+01 2 2) = 0.5511133843024713D+00 P C2 0 0) = -0.1884607948507593D+01 p CO 2 0) = 0.5784555513222096D+00 R C2 1 1) = -0.2520868634389511D+01 R CO 3 1) = -0.3009517595629857D+00 R [2 1 3) = 0.1176239384641776D+01 R [0 3 3) = -0.1321691221902800D+00 S [2 1 1) = -0.1774713159145644D+01 S 3 1) = -0.8778949980093376D-01 S '2 1 3) = 0.9467348674210815D+00 S 0 3 3) = -0.7252745407458637D-01 T 3 0 1) = 0.7596710720570268D+01 T 1 2 1) = -0.2753522637026878D+01 T C3 0 3) = -0.2629753886402562D+01 T : i 2 3) = 0.1915882425249975D+00 U [2 1 1) = 0.1484314328242942D+02 U 0 3 1) = 0.2815014518882014D+00 U 2 1 3) = -0.8321961824412727D+00 U 0 3 3) = 0.1111956535336291D+00 V 2 1 1) = 0.3546362444227999D+01 V 0 3 1) = -0.7844145060429415D-01 V 2 1 3) = 0.1027382163945612D+01 V 0 3 3) = -0.6471472028380583D-01 3 0 1) = -0.7749764260123540D+01 '1 2 1) = 0.4961099472286703D+01 3 0 3) = 0.2142242059317589D+01 1 2 3) = -0.9176029847319209D-01 R 3 1 0) = 0.3407320374101070D+03 R 1 3 0) = -0.2880451202609797D+02 R 3 1 2) = -0.1246336480202328D+03 R 1 3 2) = 0.1492055902854416D+02

w w:o

:o

w w

w

w

R (3 1 R (1 3 S (3 1 S (1 3 S (3 1 S (1 3 T (4 0 T (2 2 T (0 4 T (4 0 T (2 2 T (0 4 U (3 1 U (1 3 U (3 1 U (1 3 V (3 1 V (1 3 V (3 1 V (1 3 V (3 1 V (1 3 W (4 0 W (2 2 W (0 4 W (4 0 W (2 2 W (0 4 P (4 0 P (2 2 P (0 4 R (4 1 R (2 3 R (0 5 R (4 1 R (2 3 R (0 5 R (4 1 R (2 3 R (0 5 S (4 1 S (2 3 S (0 5 S (4 1 S (2 3 S (0 5 S (4 1 S (2 3 S (0 5 T (5 0

4) 4) 2) 2) 4) 4) 2) 2) 2) 4) 4) 4) 2) 2) 4) 4) 0) 0) 2) 2) 4) 4) 2) 2) 2) 4) 4) 4) 0) 0) 0) 1) 1) 1) 3) 3) 3) 5) 5) 5) 1) 1) 1) 3) 3) 3) 5) 5) 5) 1)

Control

Parameters

-0.8607979272709691D+00 0.1006934154338758D+00 -0.3773381277887974D+03 0.4756770192837946D+02 -0.1766479262595035D+00 -0.2283219966606174D-01 0.1662716353540178D+03 -0.6641099044983831D+02 0.4500285091427183D+01 0.3106669147459936D+01 -0.5423695146083907D+00 0.3625147633799059D-01 0.1093650099999283D+03 -0.1440517625497601D+02 0.3059639853883218D+00 -0.5670851991686953D-01 -0.1067418792384887D+04 0.9415184466574465D+02 -0.3934835584384589D+03 0.4768001949048987D+02 -0.4435142700523801D+00 -0.1975654230817772D-01 -0.1663922527299533D+03 0.6118475272623466D+02 -0.4496328528469434D+01 -0.2411124653263710D+01 0.3294513828473766D+00 -0.2937009781187618D-01 -0.2282111404838287D+02 0.2921448970812386D+02 0.3695779392844654D-01 0.2150438086724619D+03 -0.6590496494955985D+02 0.2144188095182672D+01 -0.1634258486416097D+03 0.1163615729347325D+02 0.9733005473734387D+00 0.1695781593531514D+01 -0.3148490901413368D+00 0.1319209990100450D-01 -0.1550847853342221D+03 0.1556266473971390D+02 -0.4377869017458173D-01 0.1364372235741325D+03 -0.2413457558005557D+02 0.5809887125398270D+00 0.1866478057223303D+00 -0.1156596518968369D-01 0.4650915051162774D-02 0.1939972742018780D+03

Analytic Solution of the Variational

T(3 T(l T(5 T(3 T(l T(5 T(3 T(l U(4 U(2 U(0 U(4 U(2 U(0 U(4 U(2 U(0 V(4

2 4 0 2 4 0 2 4 1 3 5 1 3 5 1 3 5 1

1) = -0 .7486472009796719D+02 1) = -0 .1518360174505468D+01 3) = -0 .5972778963315799D+02 3) = -0 .1305861216545816D+02 3) = 0,,3433561905330682D+01 5) = -0,.3857414624873869D+01 5) = 0,.8161216523564926D+00 5) = -0,.6011513530854667D-01 1) = 0,.3691358793918864D+03 1) = 0..1199206195889770D+02 1) = -0..2193661921934581D+01 3) = 0..1933598445433581D+03 3) = -0..1383982907802805D+02 3) = -0..9745127192809568D+00 5) = -0..6674366804966338D+00 5) = 0..1953597789783172D+00 5) = -0..1049664507793955D-01 1) = 0..1435713470020519D+03 Table 4.4

V(2 V(0 V(4 V(2 V(0 V(4 V(2 V(0 W(5 W(3 W(l W(5 W(3 W(l W(5 W(3 W(l

3 5 1 3 5 1 3 5 0 2 4 0 2 4 0 2 4

1) 1) 3) 3) 3) 5) 5) 5) 1) 1) 1) 3) 3) 3) 5) 5) 5)

Equations

= = = = = = = = = = = = = = = = =

203

-0 .1149683923639975D+02 0,.6362304069589682D-02 0 .1623055282719583D+03 -0 .2449836391865280D+02 0-.5807053480503643D+00 0,.7402539997217207D+00 -0..2975398610385267D-01 0,.3840862333939001D-02 -0..1990943048800168D+03 0..2051878845660479D+03 -0..8412254952370170D+01 0.,6028832948482691D+02 0..2224735767139009D+02 -0..3624685475206499D+01 0.2817258365076908D+01 -0. 4094066183775335D+00 0.3897495165605431D-01

Output of program ANAVAR: Series R, S, T, U, V, W and P.

204

Analytic Solution of Variational Equations.

4.2

Analytic Computations for Control Parameters

4.2.1

Control

Parameters

The General Method to be Used

The analytic solutions of the variational equations being now available, we wish to obtain the analytic expressions of the projections factors and the unitary controls. Using the approach given in Section 3.2, all the computations depend only on the Floquet modes. Analytic expressions for them were partially given in the last section. The first step is to obtain the full components of the six Floquet modes. Using the notation of 4.1.1 the first three components of the unstable solution of the variational equations is exp{vw)(G+ \/^\H,J+ ^f^\K,L +^/^1Q)T with w = LOT. From this the full solution follows exp(vw) [G + V^H,

J+V^1K,L

u{vJ - XKl) + y/^lw{pK

+ T/^IQ,UJ(VG

+ XJl),u{vL

- XH1) +

lw(i/# + AG1),

- XQl) + V^lco(isQ + XL1)]

T

In this formula (and through this section) the number 1 after the name of a series means the derivative with respect to ri = XOJT. The first mode ei is obtained from the previous expression deleting the factor exp(vw). In a similar way we obtain the remaining modes. The expressions are + \T:iK,L-V:-iQ,u(-vG

e 2 = [G-V^lH,-J +V^lu(isH u{-vL

+ XGI),OJ{VJ

+ XHl)

- XKl) +
+ XQl) + y/^loj(vQ

+

- XJ1),

XLl)f,

(4.12) r

e3 =

[v^^,5,v^C,-AwAl,y^AwJBl,-AwC*l] ,

e 4 = [D,^f^E,F,^J^Xu}Dl where

v

e5

=

e6

=

+ yB,<J:-iXuFl

+ y^f^A,-XujEl

+ ip

icf,

, <9A du = A— - , [R,V^lS,T,Vzlto(XRl-^U),to(-XSl-^V)yV:ilio{XTl-ijW)]T, [V^IU, V, y/^lW,u^R

- XUl),V:zluj(ipS

matrix containing the vectors e\, / b+a a +b d+c M = a+b b+a \ c+d

e-fi

+ \Vl),u(\l>T

- Aiyi)] T .

is

a+b b c c d\ b+a a d d c b c+d d a a c b+ a d d a +b b c c d c b b a) d+c

(4.13)

where, for shortness, only the type of the series appears and not the full expression. Furthermore the underlined letters refer to series with independent term coming

Analytic

Computations

for Control

Parameters

205

from the normalizations already described. We note that in (4.13) the first two columns have mixed terms, i.e., terms of different types. Each one of these columns has been split into two parts. This splitting is crucial for what follows and is unique except by permutations (i.e., we can exchange the full left-hand side of the first column with the right-hand side one, and the same for the second column). Each one of the elements of (4.13) is a series of one of the four known types. What we need for the control parameters is the determinant of (4.13) and the (signed) minors of M, rrii,i — 1 , . . . , 6. The elements m*, i = 1 , . . . , 6 are of types d + c, c + d, b + a,c + d, d + c and a + b, respectively, and the determinant is of type c + d. Here, as in M, the different elements are broken as a suitable sum to make easier the computations.

4.2.2

On the Fast Computation

of the Involved

Parameters

To compute the determinant of a matrix such that their elements are power series, some of them without independent term, only products and sums are allowed. For the determinant of an n x n matrix, (n — l)n! products are required in principle. Each one of the products is time consuming if the series are truncated at high order. To obtain a fast computation of m,, i — 1 , . . . , 6 and det(M) we proceed as follows: a) First we compute the I

J = 15 2 x 2 minors obtained from the columns

5 and 6 of M and the 15 2 x 2 minors corresponding to columns 3 and 4. Each minor requires 2 products. The total number of products at this step is 60. b) The next step is the computation of the I

I = 15 4 x 4 minors obtained

from columns 3 to 6 by using Laplace rule and the previously computed 2 x 2 minors. Each 4 x 4 minor requires I

J products. The total number

of products is 90. c) The minors m* are again computed using Laplace rule. For instance, for mi we write: d a a b a d d c mi = (b + a) b c c d c b b a

•Hb+a)

~{c + d)

a d d c d a a b -{a+b) b c c d c b b a

a d a c

a d d c a d d c b c c d c b b a

d d a a d d b b

c a d d b d a a + (d+c) c a d d a b c c

206

Analytic Solution of Variational Equations.

Control

Parameters

where, as usual, we only put the type of the elements. The five 4 x 4 minors in the above expression are of types c, b, c, d and a, respectively. We have broken e2 in such a way that now the five products obtained from the left part of e2 are of the same type, and the same is true for the right part. In the example, and putting only the types, we have b-c — c-b + b-c — a-d+d-a

= d

and

a-c — d-b + a-c — b-d + c-a = c.

In this way it is easy to obtain separately the two components of each m;. The types of rrii were given in 2.1. This step requires 5 x 2 x 6 = 60 products. d) To obtain the determinant we write (for the types) det(M) =

(b + a)-(d + c)-{a

+ b)-(c + d) + (d + c)-(b + a)

- ( a + b) • (c + d) + (b + a) • {d + c) - (c + d) • (a + b). If we take either the first term of each factor or the second term of each factor we obtain series of the same type: c. The remaining products are of type d. The number of products in this last step is 24. Therefore the total number of products is 234, a quite moderate amount of work. We remark that if the Gauss method is used to compute each of the minors (for instance with numerical entries) we require 44 operations (product or division). The total amount is then 6 x 44 + 10 = 274. The minors rrii have lower degrees 2,2,4,2,2 and 4, respectively and the lower degree of det(M) is also 2. 4.2.3

Description ters

of the Program

to Obtain the Control

Parame-

A program under the name of CONA has been developed to perform the previously mentioned computations. The starting point are the first three components of Floquet modes numbers 1,3,4, 5 and 6, and the functions v(a, /?) and ip(a, j3). All these elements have been computed by program ANAVAR. It is also required to know the constant A and the functions A and u) obtained in the program ANACOM. There are two steps in the program: 1) The first step is the computation of the full components of the 6 Floquet modes. The components of modes 1 and 2 consist of two series each. The remaining components consist of just one series. To obtain the full components we apply the formulae (4.12). We remark that some of the series (the underlined ones in (4.13)) have independent term. Therefore we are dealing with series of the form ^2 C(i,j, k)aiP^k, with 7 = exp(irwA) where the couple (i,j) = (0, 0) is allowed. Elementary modifications

Analytic

Computations

for Control

207

Parameters

of the routines PRODT and SUMT of program ANACOM produce the routines PR0DT0 and SUMTO to be used now. It has been quite useful to produce a routine, C0P0 which copies one series to (from) another series or element of an array of series. Each series is represented by a list of coefficients and its type ((a), (b), (c) or (d)) as usual. The computation of derivatives with respect to T\ = TOJ\ is done by routine DERO. Several products of "long" series of the form shown before by "short" series, of the form Yl ^(w)" 2 */? 2 ' 7 are required. This is done by the routine MIXTO where first a "short" series is expanded to a "long" one. The computation of function

u(a,p)

= £

wya^'

it follows that Ai = „ § ^ and U\ = 4 | £ are short series. Then tp = Aa/?u;iAi. The derivatives Ai and uii are computed by routine DERS. Then ui\ is expanded to a long series and multiplied, by Ai using MIXTO. The product wiAi is a short series in expanded form. This is multiplied by a, (3 and A in routine PRABL. As wi Aj is a type (a) series, a/JwiAi is of type (c). The conversion is easily done because an element in a2*/?2-7 in u>iA.x is stored in the place (m 3 + 3m 2 + 2m + 12j)/12, where m = 2i + 2j. The element in a2l+1 j32^+l in a/?wiAi is stored in the place (2m 3 + 3m 2 - 2m + 12(2.7 + 2))/24, where m = 2i + 2j + 2. One should be careful because elements in wiAi of degree n will produce elements of degree n + 2 in a/3u>iAi and must be skipped if the degree is too large. 2) The second step is the computation of minors (of size 2 x 2 , 4 x 4 , 5 x 5 and 6 x 6). This is done using the algorithm given in 2.2. The only routines we require at this step are C0P0, PR0DTO and SUMTO. This step is carried out in routine D56. The final result are the 5 x 5 minors of the first column and the determinant of the Floquet modes. In the output the full expression of the first Floquet mode is also included because, as explained in Chapter 3 (3.2.6), it is required for the projection factors and unitary controls. In fact it is only required for the normalized projection factors given by wj = mi • ( d e t ) - 1 • |ei|, because then Ai = —n\/ ((7f|)2 + {K\)2), A 2 = -7f5V((7f])2 + ( ^ ) 2 ) .

4.2.4

Comparison with Direct Numerical

Computations

At the end of this section a short sample of results is given. As mentioned earlier, the first mode contains components of types b,a,a,b,d,c,... (see (4.13)). These are the successive elements in Table 4.5, each one as coefficients of power of a,f) and 7. Similar conventions are used for the 6 minors (see the end of 4.2.1) and the

208

Analytic Solution of Variational Equations.

Control

Parameters

determinant. As the system is conservative, the determinant should be constant for every halo orbit, i.e., only 0th order harmonics should appear. This is not true in the results, but we must take into account that the relation A ( Q , / 3 ) = A has not been used in the final expression of the determinant. When a couple (a, /3), for which the mentioned relation holds, is substituted in the expression of the determinant, this one is almost constant. It has harmonics of positive order (cosine terms) but with relatively small amplitude (a few thousandths). We remark also that from (4.13) it follows that the dominant terms in the determinant have the factor a/3. Hence, for small amplitude halo orbits, some numerical difficulties are foreseen. However they do not appear in the usual range of /?. To present in a short form the comparison with the results of program PAPUS concerning projection factors and gain (in the (x, y) maneuvers) we have drawn the corresponding functions for two revolutions. In the figures given at the end of this section the results are shown for /3 = 0.08 and 0.16 in the Sun-Barycenter problem. Together with the numerical results of PAPUS and the analytic ones of C0NA we also display the refined values for /? = 0.08, to be explained in Chapter 8. From the pictures it is easily seen that for /3 = 0.08 the relative errors in the ff1 vector and in the gain are a few hundredths, while for 0 = 0.16 they are a few thousandths. These errors are, at most, of the order of magnitude of the random errors we have been using in the simulations of the control maneuvers, to be described in Chapter 9. In this chapter we also present the results of the simulations for a fully analytic station keeping. In 1.3 we have mentioned the bad agreement between the analytic and numerical results for the Floquet modes numbers 5 and 6. To make clear what is happening we have done the following check: For a given point in the halo orbit we have two Floquet bases at our disposal (the numerical and the analytic one). We can express each one of the vectors of the analytic basis in the numerical one. The results are: a) For ei,e2,e3 the agreement is excellent: the ith analytic vector only has component in the ith numerical one (except rounding errors O(10 - 7 ) at most), for i — 1,2,3. b) The vector e 4 has an important component in the third numerical vector, moderate components in the fifth and sixth and small in the first and the second. c) Vectors e 5 and e6 have important components in the third numerical vector and moderate components in the other vectors. Of course, in a) and b) the vectors e4,e 5 ,e 6 have important components in the respective numerical vectors. As in ff1 the important things are the first mode and the subspace spanned by the remaining vectors, the influence of the bad determination of e4,e5 and eg is lowered.

Analytic

Computations

for Control

Parameters

209

ANALYTIC EXPANSIONS OF THE PARAMETERS REQUIRED FOR CONTROL, HALO CASE ORDER = 5 UNSTABLE VECTOR -1 0 0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+00 -0.842754993531759D+00 -0.139276238001011D+00 0.OOOOOOOOOOOOOOOD+00 -0.158646031518040D-01 -0.839992146944134D-02 -0.697025595452094D+00 -0.206529095321125D-01 0.226188129323050D-01 1 0 0.100000000000000D+01 0. OOOOOOOOOOOOOOOD+OO -0.661801925940591D+00 -0.565816820174031D-01 0.OOOOOOOOOOOOOOOD+OO 0.252472314462402D-01 -0.661461858218465D-02 -0.159052208329633D+00 -0.140504184413596D-01 0.167751552592828D-01 1 0 -0.534573653714911D+00 -0.391401080406739D-01 -0.212473987219764D+01 -0.398790425567667D-01 -0.171381003991292D-01 0.944875917897272D-02 0.276739206316770D-02 0.662128366615659D-01 -0.311374060859790D-01 0.189189558353768D-02 -1 0 0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+00 -0.582004851057339D+00 0.320607248960815D-01 0.OOOOOOOOOOOOOOOD+OO 0.250833009313873D-01 -0.206943993816544D-02 -0.131806992753593D+00 0.114402697302418D-01 -0.236154378227974D-02 -1 1 0. OOOOOOOOOOOOOOOD+OO -0.425370830061253D+00 0.673881446477705D+00 0.OOOOOOOOOOOOOOOD+OO -0.989032689536990D+00 0.158173459920069D+01 -0.988167583961828D-01

0.542575219271695D+00 -0.128526633562048D+01 -0.291790358100516D+00 0.OOOOOOOOOOOOOOOD+OO 0.319663267803017D+01 -0.252549467157987D+01 0.i96118478600602D+01 -0.466092977689849D+01 0.415259155139893D+01

0.OOOOOOOOOOOOOOOD+OO 0.116052757312199D+00 0.149133369949543D+01 0.OOOOOOOOOOOOOOOD+OO 0.451685615824603D+00 0.325794630654298D+00 0.335087867236306D+01 0.775538980670012D-01 -0.630117059267036D+00

0.180861169744488D+00 -0.408610358992579D+00 0.161474607088401D+00 0. OOOOOOOOOOOOOOOD+OO 0.110616027173859D+01 -0.138482134668799D+01 0.947301868044816D+00 -0.396217524272637D+01 0.249701608510948D+01

0.OOOOOOOOOOOOOOOD+OO 0.272132843188146D-01 0.617477331602867D+00 0.OOOOOOOOOOOOOOOD+OO -0.987481411210029D-01 0.201371319763486D+00 0.212474347853175D+01 0.205471694051798D+00 -0.414372609083923D+00

0.812598211853156D+00 -0.794277319600862D+00 0.422213312005008D+00 -0.194723589260481D+01 0.265486634121191D+01 -0.243733546115213D+01 -0.128917271022452D+01 -0.573751811159742D+01 0.412284952975820D+01

-0.331943413836219D+00 -0.526146996182319D-01 0.164449414770535D+01 "0.594295808773251D+00 -0.898738720671662D+00 0.101295029089521D+00 0.259842840269606D+01 0.128499791018024D+01 -0.323377086500597D+00

0.495420660604684D+00 -0.115376084541582D-02 0.154686003598399D+00 0.OOOOOOOOOOOOOOOD+OO -0.186098494713016D+01 0.112624675546564D+01 -0.108077334638920D+01 0.396574180361970D+01 -0.236551235492414D+01

0.OOOOOOOOOOOOOOOD+OO -0.276888347479464D-02 -0.582704994173001D+00 0.OOOOOOOOOOOOOOOD+OO -0.700841674086099D-01 -0.552963511752221D-01 0.230192975472293D+01 -0.343336909923744D+00 0.219315651325064D+00

0.320936608392843D+00 0.162200098045366D+01 -0.280581570142200D-01 -0.194894915248324D+01 0.627338817244381D-01 -0.535911784173179D-01 0.208989083551220D-02

0.OOOOOOOOOOOOOOOD+OO -0.426337444490839D-02 0.OOOOOOOOOOOOOOOD+OO 0.910247108146733D-02 -0.288501277221676D+01 0.226327532252108D+01 0.160882353860565D+01

210

Analytic

Solution of Variational Equations.

-0.151957946855413D+00 1 1 0.000000000000000D+00 -0.148825392168932D-02 0.231701881285513D+00 0.594248547993013D+00 -0.478314239357776D+00 -0.691180335511036D+00 -0.108837087151449D-01 -0.105298012042367D-00 1 0 0.253265899556417D+01 -0.506373718181529D+00 0.155804727440972D+00 0.728478058099669D+00 -0.150227010218387D+00 0.577187343313840D-01 0.533516074414575D-01 0.932923804472644D+00 0.227911926359752D+00 -0.193479754114853D+00 -1 0 0.000000000000000D+00 0.000000000000000D+00 -0.640641079902468D+01 -0.706904354893229D+00 0.000000000000000D+00 0.201604270857720D-01 -0.764785118561426D-01 -0.196235412499067D+01 -0.112616488375175D+00 0.232288643309602D+00 -1 0 0.000000000000000D+00 0.000000000000000D+00 -0.966112459962440D+01 0.168418415094630D+00 0.000000000000000D+00 0.777703744038109D-01 0.178549532540246D-01 -0.116914319016461D+00 -0.212388959520230D+00 0.137557752828591D-01 1 0 -0.135389277287267D+01 0.171565501956938D+00 -0.717845597162893D+01 -0.301679646580134D+00 0.567219596349565D-01 -0.526978849827665D-01 0.242800208409208D-01 -0.835983575352494D-02 -0.154577527560710D+00 0.294277822899310D-01 1 1 0.000000000000000D+00 0.177126363662744D+01 -0.363124496342260D+01

Control

Parameters

0.250426300597528D-02 -0.185912078322814D+00 -0.106433431219640D+01 -0.145905749758890D-01 0.458465314288663D+00 0.361842501371534D-01 0.824114737548819D-01 -0.322307800066842D-02 0.225822817416821D-02

0.518586005106697D+00 0.905819273177457D-01 -0.187793918054404D+01 -0.481064136173868D-01 0.182959169021144D+01 -0.115306021010200D+00 0.956339352562344D+00

-0.673998272494953D+00 0.432842607242743D+01 0.789093132464872D+00 0.677233216059789D+01 -0.130176482283584D+02 0.175700280942576D+02 0.115784151625806D+01 0.239014835374731D+02 -0.369968547038112D+02

-0.475507946336915D+01 -0.415355383647811D+00 -0.777093553240463D+01 -0.104341628615520D+01 -0.100857698549691D+01 -0.220901644690856D+01 -0.266223786823187D+01 -0.185802533912045D+01 0.552408506176867D+01

0.175151642239731D+01 -0.496023433736832D+01 -0.631144475803597D+00 0.000000000000000D+00 0.202295507854446D+02 -0.179536779331749D+02 0.157878024655170D+02 -0.468839624075968D+02 0.365666375408371D+02

0.000000000000000D+00 0.407480561959568D+00 0.764205284564742D+01 0.000000000000000D+00 0.530752245126898D+00 0.250573424568921D+01 0.137700293081030D+02 0.214975006555647D+01 -0.591871744838563D+01

0.295017993954095D+01 -0.331736739864111D+01 0.122714375613312D+01 0.000000000000000D+00 0.910404826857448D+01 -0.174891490018958D+02 0.445801668556014D+01 -0.315871410958222D+02 0.370196320899194D+02

0.000000000000000D+00 -0.226568881273650D+00 0.881768844473083D+01 0.000000000000000D+00 -0.413441716970738D+01 0.705342652558683D+00 0.924233445809208D+01 0.876870069466651D+01 -0.281810444023493D+01

0.102436202189546D+01 -0.200681906185476D+01 0.209922558981633D+00 -0.697357753701957D+01 0.182624694797808D+02 -0.155723853186334D+02 0.135300713621047D+02 -0.501817185303616D+02 0.351194290810479D+02

0.170124072936740D+01 -0.121700799300537D+00 0.781230344183521D+01 -0.593164025051652D+00 -0.134030109746995D+01 0.718038818524312D+00 0.282451469783100D+00 0.536761321954160D+01 -0.310696337960156D+01

-0.114047119305413D+01 -0.464359011581622D+01 0.138673164989853D+00

0.131340151080716D+01 0.251359490716884D+00 -0.722209722160079D+01

Analytic 0.124243060707266D+01 0.704287582884577D+01 -0.524938514405512D+01 -0.112617369161528D+00 0.131858195895152D+01 -1 1 0.000000000000000D+00 -0.108352960427970D+01 0.315701747977191D+01 0.000000000000000D+00 -0.649680412772595D+01 0.954681233929422D+00 -0.275266986061919D+00 -0.148335466634831D+01 MINORS -1 1 0.OOOOOOOOOOOOOOOD+OO -0.151455459366213D-14 -0.137026993680817D+01 0. OOOOOOOOOOOOOOOD+OO 0.397137150300560D+01 0.582359759881045D+03 -0.109859148532478D+03 0.139391156045965D+01 1 1 0.OOOOOOOOOOOOOOOD+OO 0.143522323027123D-14 -0.168706093929012D+00 -0.338333884629264D+04 -0.650786476968026D+00 0.216944144630744D+04 -0.707200171761561D+02 -0.387570337158193D+00 1 1 0.OOOOOOOOOOOOOOOD+OO -0.268479978445897D-14 -0.132023718264934D+01 0.638386273071464D+03 0.330378893167362D+01 0.141725252898068D+04 -0.625457756895754D+02 0.110464591032274D+01 -1 1 0. OOOOOOOOOOOOOOOD+OO 0.283320096891614D-14 -0.811672700015922D+00 0.OOOOOOOOOOOOOOOD+OO 0.185976406325590D+01 -0.382154813055468D+04 0.106880079160730D+02 0.722759897732445D+00 -1 0 0. OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO 0.108321786805821D-15 0. OOOOOOOOOOOOOOOD+OO 0.464038114689835D+00

Computations

for Control

Parameters

211

0.783719691379984D+01 -0.431922930539726D+00 0.254370590642196D+00 0.741202463573805D-02 -0.204058191093420D-01

0.558867186983263D-01 0.172038701603975D+02 0.120819743497864D+02 -0.143615956911870D+02

0.424926090057802D+00 0.681095554214881D+00 -0.162389410479740D+00 -0.995096999929633D+00 0.460870544777004D+00 -0.155081573761328D-01 -0.117331659508570D-01 0.299008841133168D-01

0.OOOOOOOOOOOOOOOD+OO -0.312903367800989D-01 0.OOOOOOOOOOOOOOOD+OO 0.369544141165548D-01 -0.722171725405909D+01 0.609993087741313D+00 0.140513891401758D+02

0. OOOOOOOOOOOOOOOD+OO 0.914234547673684D+02 0.149415234192236D+00 -0.630959793747229D+03 -0.433041247747244D+00 -0.465674777865388D+01 0.129279741462377D+01 -0.330356222297392D-01

0.OOOOOOOOOOOOOOOD+OO -0.717387325089227D+00 0.OOOOOOOOOOOOOOOD+OO 0.517485869898476D+02 0.389160200710921D+04 0.119376167845610D+04 -0.100049372990490D+02

0.OOOOOOOOOOOOOOOD+OO 0.275454107285173D+03 0.183958356356978D-01 -0.661072335553477D+03 0.709622325158910D-01 -0.146404070443865D+02 0.262979655321866D+00 0.427184889628613D-02

-0.439379469626167D+03 -0.219262579827908D+01 -0.208206851328277D+05 0.137148469621858D+02 0.140183461258137D+05 0.125224552725064D+04 0.319507859157373D+01

0.OOOOOOOOOOOOOOOD+OO 0.171065300622894D+03 0.143959626155371D+00 -0.374515741463518D+03 -0.360247556840933D+00 -0.461845652597413D+00 0.935439096195555D+00 -0.971726819105999D-02

0.788177450133445D+02 -0.455868586891861D-01 0.410569095022475D+04 -0.207361596264845D+01 0.689626889064416D+04 0.119025180763064D+04 -0.931330853053706D+01

0.OOOOOOOOOOOOOOOD+OO -0.488183115224217D+03 0.885053837222893D-01 0.657383701358575D+03 -0.202790031065674D+00 0.687025214310611D+00 0.668774361621462D+00 -0.182508526891491D-01

0.OOOOOOOOOOOOOOOD+OO 0.885053837222893D-01 0.OOOOOOOOOOOOOOOD+OO -0.274581586877134D+01 -0.227489003750351D+05 -0.117721182702471D+04 -0.509335755357864D+01

0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO 0.518361578535528D+02 0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO

0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOD+OO -0.131961766259766D+03 0.551989115486356D+00

212

Analytic Solution of Variational Equations.

-0.601892960988937D-01 0.378497346460280D+03 -0.139443294887495D+02 0.250355954637660D+00 1 0 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 -0.384083248177705D-15 0.138736751412112D+01 0.688445305181666D+00 -0.523845187889477D-02 0.106659012678644D+04 -0.121698532514425D+02 -0.304479083437008D-01 1 1 0.000000000000000D+00 0.605821837464852D-15 -0.568212834832062D+00 -0.953708786672230D+03 0.772979855714525D-01 -0.192741133924676D+03 0.165373274440876D+01 -0.449906998293943D-01 -1 1 0.000000000000000D+00 0.598009679279682D-15 -0.171321461605415D+00 0.000000000000000D+00 0.157733380149887D+00 0.492677091781588D+03 -0.171596768023156D+02 0.522915139789056D-01 -1 1 0.000000000000000D+00 -0.111866657685790D-14 -0.101209635228405D+01 0.000000000000000D+00 0.766296227861854D+00 -0.503353010159490D+03 -0.198594487726374D+02 0.130785468329751D+00 1 1 0.000000000000000D+00 -0.113328038756646D-14 0.417033941252438D+00 -0.513142129796986D+03 -0.754164029203628D+00 0.802487307568476D+03 0.126412172820186D+02 -0.234416859890363D+00 1 0 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 -0.278475957007121D-15 0.547790885607190D+00 -0.809109431562565D-02

Control

Parameters

0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00

0.256831318093476D+04 0.358883782301608D+03 -0.229598568041279D+01

0.000000000000000D+00 0.000000000000000D+00 0.139124535239862D+03 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00

0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 -0.158961622848531D+03 -0.213934610882802D+03 0.480412399971907D-01 0.727281933886191D+04 0.438587228961811D+03 0.279234267292885D+00

0.000000000000000D+00 -0.225139933249004D+02 0.619583422994942D-01 0.447139579094357D+02 -0.842862877342883D-02 -0.507347637466270D+01 0.391988830575143D+00 0.436771620981848D-03

-0.121988644769968D+03 -0.727167360825521D+00 -0.593249919112045D+04 -0.110012627300423D+02 -0.701173416912637D+03 -0.252388579614980D+02 0.375869801185505D+00

0.000000000000000D+00 0.670494787691698D+02 0.186810172363236D-01 -0.166548942601069D+03 -0.171993629152561D-01 0.218633088572070D+00 0.214274793807920D+00 -0.435829476414996D-02

0.000000000000000D+00 0.186810172363236D-01 0.000000000000000D+00 0.144168853662054D+02 0.305898554703444D+04 0.191081382373625D+03 -0.113004504868181D+00

0.000000000000000D+00 -0.614477913304919D+02 0.110359725072768D+00 -0.197119867627952D+02 -0.835575000742569D-01 -0.367316624882559D+01 0.723294840768583D+00 -0.268136434335982D-02

0.000000000000000D+00 -0.529870119305743D+00 0.000000000000000D+00 -0.137513231922346D+00 -0.283324984660872D+04 0.125195419898568D+03 -0.973901992238996D+00

0.000000000000000D+00 0.970437182454245D+02 -0.454736853845667D-01 -0.602704435706455D+02 0.822345962761911D-01 -0.192632384502494D+01 -0.395461514915795D+00 0.913593349587341D-02

-0.652119155464124D+02 -0.277221639508547D+00 -0.314731475877946D+04 -0.101076451692409D+02 0.465930302219873D+04 0.109322743765263D+03 0.138143117743076D+01

0.000000000000000D+00 0.000000000000000D+00 0.226791624809684D+02 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00

0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 -0.627647161054614D+02 0.371031143278801D+00 -0.589628627898442D-01

Analytic 0.642935360087567D-02 0.172953679245870D+03 0.123833294418754D+01 -0.233316669292475D-01 -1 0 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 -0.183725590031283D-15 0.000000000000000D+00 0.169890511142578D+00 -0.257875535845191D-02 0.302019443820539D+03 -0.244531827974051D+01 0.274563975666184D-02 DETERMINANT 1 1 0.000000000000000D+00 0.593912671327162D-14 -0.146665066606660D+01 -0.543890619857743D+04 0.293828776049668D+01 -0.897823788773588D+01 -0.535707752391015D+01 0.654422547727651D+00 -0.726937369764287D+00 -0.212866155128093D-01 -1 1 0.000000000000000D+00 0.000000000000000D+00 -0.277555756156289D-16 0.000000000000000D+00 0.555111512312578D-16 -0.115463194561016D-12 0.133226762955018D-14 0.954097911787243D-17 -0.726937369764287D+00 -0.212866155128093D-01 Table 4.5 Output of explanations.

Computations

for Control

Parameters

213

0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00

0.122016441420395D+04 -0.146666064423079D+02 0.213972035325640D+00

0.000000000000000D+00 0.000000000000000D+00 0.391506219149131D+02 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00

0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 -0.514927149949368D+02 0.236494689525658D-01 0.205808934594104D+04 0.861730976626764D+02 -0.251799465844185D-01

0.000000000000000D+00 -0.439995199820012D+01 0.159924659267497D+00 0.853995946780794D+01 -0.320393042323791D+00 -0.247976440510047D+02 0.111397957023344D+01 -0.147196977453509D-01 0.000000000000000D+00

-0.702179056925392D+03 -0.363459484812910D+01 -0.332417627345543D+05 -0.396918885457232D+02 0.171586701684228D+04 0.267869920652504D+03 -0.476363182685990D+01 -0.127611229098999D+02 0.195217103873552D+00

0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 -0.124344978758017D-13 -0.383995017167159D-17 0.000000000000000D+00 0.286229373536173D-16 -0.108420217248550D-18 0.000000000000000D+00

0.000000000000000D+00 0.000000000000000D+00 0.000000000000000D+00 0.133226762955018D-14 -0.113686837721616D-12 0.159872115546022D-13 0.111022302462515D-15 -0.127611229098999D+02 0.195217103873552D+00

CONA, Sun-Barycenter

, case L\. See the text for

214

Analytic Solution of Variational Equations.

Control

Parameters

Fig. 4.1 First projection factor (top) and gain of the (x,y) maneuvers (bottom). Two revolutions are shown. The three lines correspond to numerical results (dots), analytic results (points) and refined values (see Chapter 8). Sun-Barycenter problem, case L\, z-amplitude /3 = 0.08.

Analytic

Fig. 4.2

Computations

for Control

Parameters

Plots similar to previous figure for /3 = 0.16.

215

216

4.3

Analytic Solution of Variational Equations.

Control

Parameters

References [1] R. Abraham and J. E. Marsden. Foundations of Mechanics. Benjamin, 1978. [2] E. W. Brown and C. A. Shook. Planetary Theory. Dover, 1964. [3] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer Verlag, 1983. [4] P. Hartman. Ordinary Differential Equations. John Wiley & Sons, 1964. [5] S. Smale. "DifFerentiable dynamical systems" Bull. Am. Math. Soc, 73, 747-817, 1967 [6] V. Szebehely. Theory of Orbits. Academic Press, 1967.

Chapter 5

The Equations of Motion for Halo Orbits Under the Effect of Perturbations and Near Triangular Points

In this chapter we concentrate on the real problem for the motion of a spacecraft taking the solar system as a perturbation of the restricted three-body problem. First, we expose the general method to be followed, how to decide about the terms to be retained and the reference systems to be used in the sequel. As a result we have a final model to be used in the computation of the quasi-periodic orbits. This semianalytic task (analytic theory with numerical coefficients) will be carried out in Chapters 6 and 7. Then we go to the more technical part: to obtain the final equations of motion. They are (in form) slightly different in the case Sun-Barycenter, where the barycenter plays a special role, and in the Earth-Moon case. As a first step the Lagrangian is expressed in ecliptic coordinates. Then we perform a change to normalized coordinates, i.e., suitable coordinates which are well adapted to the halo orbits. After some manipulations with Lagrange's equations we reach the final equations of motion. The same steps are applied later to the triangular points. Finally we have done checks of the obtained equations to be certain that no mistakes are introduced in the computations.

5.1 5.1.1

Analysis of the Perturbations Description and Justification of Dynamical Coherence

of the Method.

The

Problems

We consider two models of the solar system acting on the spacecraft. The first one is the real model. The second one will be the adopted model. This one has to be given in analytic form, because afterwards we shall look for quasi-periodic solutions (to a certain degree of approximation) in an analytic way. We request that the residual acceleration along the orbit be small. The residual acceleration is defined as the difference between the true acceleration acting on the body and the acceleration acting when only the terms contained in the adopted 217

218

Perturbed Equations

of Motion: Halo and Triangular

Cases

model are considered. If the analytic solutions of the adopted model were exact then this difference would be equal to the difference between the true acceleration and the one obtained by differentiation of the solution twice. An extra residue is introduced due to the fact that only an approximation of the solution is obtained. As the solution is not known we compute the residual accelerations in the first way along the halo orbit which is a rough approximation of the quasi-periodic solution. The differences due to this approximation are not relevant. Let ar be the residual acceleration. Several norms can be computed to estimate the importance (the weight, as denned in 5.1.2) of the different terms. The norm of interest for space missions is the L\ norm. It gives the average acceleration required for a continuous control designed to balance exactly the errors of the model. Therefore it measures the fuel consumption or Av required per unit time. As a complementary information the L2 and L^ norms have been obtained. Some checks are also made about the average of the residual acceleration (not in norm). In several cases an increase of one unit in the weight of the terms has been obtained due to cancellations along the time. However this has not been taken into account. As a last remark we note that some terms with a very small residual acceleration can produce large deviations in the quasi-periodic solution. This is the case when small amplitude and very large period periodic forcing terms are present. The adopted procedure has been to skip these terms from the solution and to cancel their effect through rather small additions to the maneuvers. An important point to be reminded when we deal with analytic theories for the motion of the planets and Moon is the lack of dynamical coherence. Let fi(t) be the function giving the position of the ith body of the solar system and rrij the mass of the jth. body. Then the function

should be identically zero. However this is not true and some residual acceleration is present in the analytic theory. Fortunately in the analytic model we use, the values of a\{t) are small and can be neglected. However this effect has been considered in the checking program. Their importance is much greater for the Earth-Moon problem because we have used as model a truncation of Brown's theory and not a coherent approximation to the real motion. 5.1.2

Determination of the Magnitudes ferent Weights

to be Associated

to the Dif-

The analytic solutions up to a given order in a, /3 are available from Chapter 2 (item 2.2.4). Therefore the acceleration up to a given order can be obtained analytically. The parameter describing the orbit is an angle ranging from 0 to 27r. Then, for

219

Analysis of the Perturbations

the same parameter we can compute two different accelerations which give the magnitude of the terms included in an approximation and not in the other. For a given order we can also compute the residual acceleration as the difference between the one obtained analytically and the acceleration at the same point when the restricted three-body model is used. Only odd order terms need to be computed because the even order terms following the ones of odd order do not improve the solution (see Chapter 2). The magnitude of the terms of a given order is the weight associated to this order. We note that for the orbits of interest the amplitudes of x and z are, roughly, 0.15 and 0.08, respectively, for L\ and Li halo orbits in the Sun-Barycenter case. This means that for individual terms and disregarding the coefficients going from one order to the other the importance of successive terms decreases by a factor around 10. However when the full number of terms and their coefficients are considered, the reduction going from one order to the next one is smaller. A program has been produced to compute the residues analytically (TESTRANA). Tables 5.1 to 5.4 give the magnitudes of some weights for different ^-amplitudes, /?, in the L\ and L2 cases for the Sun-Barycenter and Earth-Moon problems. All the results are given in adimensional units (i.e., the units of the RTBP). We recall that for the Sun-Barycenter system 1 adimensional unit of acceleration 5.93 mm/s 2 and for the Earth-Moon system it is 2.72 mm/s 2 . Weight

p

0.04 0.08 0.12 0.16

17

15

13

11

9

7

5

3

0.61E-9 0.79E-9 0.11E-8 0.17E-8

0.56E-8 0.69E-8 0.96E-8 0.14E-7

0.51E-7 0.60E-7 0.80E-7 0.12E-6

0.47E-6 0.52E-6 0.64E-6 0.90E-6

0.43E-5 0.47E-5 0.54E-5 0.69E-5

0.40E-4 0.43E-4 0.47E-4 0.54E-4

0.36E-3 0.38E-3 0.41E-3 0.46E-3

0.24E-2 0.24E-2 0.25E-2 0.27E-2

Table 5.1 Weights for the h\ case of the Sun-Barycenter problem. Weight 15

13

11

9

7

5

3

0.74E-8 0.89E-8 0.12E-7 0.18E-7

0.65E-7 0.75E-7 0.98E-7 0.14E-6

0.58E-6 0.63E-6 0.72E-6 0.10E-5

0.51E-5 0.52E-5 0.56E-5 0.65E-5

0.45E-4 0.45E-4 0.46E-4 0.50E-4

0.39E-3 0.41E-3 0.44E-3 0.48E-3

0.25E-2 0.25E-2 0.26E-2 0.28E-2

13 0.04 0.08 0.12 0.16

Table 5.2 Weights for the L2 case of the Sun-Barycenter problem. Weight

p

0.04 0.08 0.12 0.16

15

13

11

9

7

5

0.12E-7 0.18E-7 0.33E-7 0.71E-7

0.14E-6 0.20E-6 0.34E-6 0.63E-6

0.17E-5 0.22E-5 0.34E-5 0.58E-5

0.21E-4 0.26E-4 0.35E-4 0.54E-4

0.26E-3 0.29E-3 0.35E-3 0.46E-3

0.31E-2 0.33E-2 0.37E-2 0.44E-2

Table 5.3 Weights for the L\ case of the Earth-Moon problem.

220

Perturbed Equations

of Motion: Halo and Triangular

Cases

Weight

7

p

17

15

13

11

9

0.04 0.08 0.12 0.16

0.15E-6 0.17E-6 0.21E-6 0.27E-6

0.10E-5 0.11E-5 0.13E-5 0.17E-5

0.66E-5 0.69E-5 0.78E-5 0.99E-5

0.43E-4 0.44E-4 0.47E-4 0.57E-4

0.28E-3 0.29E-3 0.31E-3 0.34E-3

Table 5.4

5

0.18E-2 0.11E-1 0.19E-2 0.11E-1 0.20E-2 0.12E-1 0.21E-2 0.13E-1

Weights for the L2 case of the Earth-Moon problem.

For the Sun-Barycenter problem the importance of the different weights is similar for L\ and £2- This is quite different for the Earth-Moon problem. This is due to the fact that the value of 7, the distance from the equilibrium point to the secondary, changes much more in the Earth-Moon problem when we go from the L\ case to the L2 one. Furthermore, for a given value of /? the z-amplitude a is larger in Earth-Moon L2 case than in Earth-Moon L\ case. This means that the domain of convergence of the halo orbit theory is smaller in the Earth-Moon, L2 than in the Earth-Moon, L\. Therefore, the coefficients increase faster in the L2 case. When the weight is increased by two units the reductions in the magnitudes of the residual accelerations are: 8 to 9 for the Sun-Barycenter problem (both L\ and L2); 11 to 12 for the Earth-Moon problem Lx case, and 6 to 7 for the L2 case in the same problem. In Chapter 7 an assignation of weights to coefficients will be used. It is such that the addition of one unit to the weight means a reduction of e = exp(l) in the magnitude of the coefficient in the Sun-Barycenter problem, and a reduction of exp(0.8) in the Earth-Moon problem, L2 case. The Earth-Moon problem L\ case, is not analyzed. Comparing the L2, Earth-Moon problem with the L\,L2, Sun-Barycenter problems, we conclude that to reach the same precision the theory should be developed, in the first case, to a weight 4 units larger. 5.1.3

The Reference nates

Systems

and the Related

Changes

of

Coordi-

Both for the purpose of comparison of numerical results and to obtain the equations of motion in a suitable way we have introduced two systems of reference. The first one is the natural extension of the coordinate system used in the RTBP. We call it the adimensional system. We denote by e and a the coordinates of a point in the ecliptic and adimensional systems of reference, respectively. We go from one system to the other via a transformation of the form e = kCa + b,

(5.1)

where A; (change of scale factor), C (orthogonal matrix) and b (translation) must be computed.

Analysis of the

221

Perturbations

We note that we start with some given origin of coordinates. As the positions of the planets are given, according to Newcomb's theory, with respect to the Sun, this one is taken at the origin. When we have selected a given model, b means the position of the center of masses (c.o.m.) of the two bodies with respect to the c.o.m. of the solar system. This last c.o.m. depends on the selected model. In this way the indirect effects of the perturbing bodies, due to their effect on the Sun, are taken into account. Of course, k, C and b as they appear in (5.1) are functions of time. From now on, when we refer to ecliptic coordinates, we understand that they are centered at the c.o.m. of the solar system. This is taken as an (approximate) inertial frame. We denote by Rp and Rs the positions of the primary and the secondary in the ecliptic system and by Rps the position of the barycenter. Vs will denote the velocity of the secondary with respect to the primary. Since the barycenter is fixed at the origin in the synodic system, it follows that b = RpsBecause in the adimensional system the primaries are fixed at the points (fi, 0,0) and (/i — 1,0,0), where fi is the mass parameter, it follows that k = \\RP — Rs\\. Let c\, C2, cz denote the three column vectors of the matrix C. We have taken Cl =

Rp — Rs —k—>

_ Vs A (Rs - Rp) >=\\VsA(Rs-RpW

C

_ C2 = C 3 A C l

_ -

, '

.

(5 2)

The determination of c\ is trivial if we ask for the primary being at (fi, 0,0) and the secondary at (fi — 1,0,0). We have selected c
222

Perturbed Equations

of Motion: Halo and Triangular

Cases

systems the unit of time is the same one. It is defined in such a way that the mean time that the secondary pends to perform a revolution around the primary equals 2n in the new units. For the cases Sun-Barycenter and Earth-Moon this is equivalent to ask for units such that the frequency of the mean longitude of the Sun and the Moon, respectively, becomes equal to the unity. At this point a problem of coherence appears. We wish to express the equations of motion as a perturbation of the RTBP. If we consider the primaries in both problems, Earth and Moon or Sun and Earth-Moon barycenter, the Kepler relation n2a3 = G(mp + ms) should hold. In this relation n is the mean motion of the related mean longitude and a is the mean distance between the primaries. It turns out that this relation is not satisfied. Then we have modified the mass of the more massive primary, m p , to recover Kepler's third law. The remaining part of the mass (and, when required, the radiation pressure effect) is considered as a perturbation. 5.1.4

A Program for the Simulation of the Solar System Computation of Residual Accelerations

and the

The program SSI simulates the motion of a spacecraft in the solar system by integration of Newton's equations. The model of the solar system is selected by the user. It produces graphical output that can be represented in ecliptic, adimensional or normalized coordinates referred to any equilibrium point in the Sun-Barycenter problem or the Earth-Moon problem. Other main goal of the program is to compute the residual accelerations, along a halo orbit of the restricted circular three-body problem, when the influence of the selected model of the solar system is taken into account. By comparison of several models of the solar system, it is possible to estimate the order of magnitude of the effect of the planets, on the motion along a halo orbit, and for each planet, the order of magnitude of the effect of its different elements. For the simulation of the motion of a body in the solar system, several models are taken into account for each of them. For all the planets, except for the Earth-Moon barycenter, the models considered are the following four: a) b) c) d)

Newcomb's theory, Circular motion on the ecliptic plane, Elliptic motion on the ecliptic plane, Osculating model.

In the last three models the elements are taken at an osculating epoch TAU. The value of TAU is a parameter and can be easily changed. In the runs we have used the epoch January 1st year 2000 (Julian Day = 18 262 with the modification of the Julian Day used in the programs). The periods are the tropic ones at the osculating date.

Analysis of the

223

Perturbations

For the motion of the center of masses of the Earth-Moon barycenter, to the above models we have added the following: e) f) g) h)

Osculating with linear variation of the eccentricity, Osculating with linear variation of the argument of the perihelion, Osculating with the above two variations, Newcomb's theory with the periodic perturbations in longitude, radius vector and latitude due to the effect of the planets, also given by Newcomb, i) Same as h) with several periodic perturbations which can be selected and varied during the execution of the program.

The osculating epoch is also TAU. The rates of the linear variations in e), f), g) are computed at the same epoch. The computation of the elements of a planet, as well as its first and second time derivatives, is performed by routine UPLNET. For the Earth-Moon barycenter the third derivatives are also computed. The periodic terms for the Earth-Moon barycenter, models h) and i) are computed by routine NEWPER. We remark that the routine NEWPER requires the mean anomalies of the Earth-Moon barycenter and the perturbing planet. Hence, models b), c), d) are not allowed for planets because in b) the mean anomaly is replaced by the mean longitude with only linear variations, and in c), d) the mean motion includes the time derivatives of to and fl. We remark that when the Sun-Barycenter system is studied, the mass of the Sun has been modified in order that dynamical coherence be obtained. The remaining part of the mass of the Sun (as well as the radiation pressure "mass") is considered as another perturbing body located at the place of the Sun. The mass of the Earth is modified in a similar way when the Earth-Moon system is studied. The computation of a generic Keplerian element, a, at a given epoch T, is carried out for Newcomb's model using the relationship: aT = a0 + axT + a2T2 + <j3T3, where the coefficients, CTJ, that have been used are those given by Escobal. For all the other models, first of all the elements and its first derivatives are evaluated at the epoch r = 1 — 1 — 2000 and the ones that are taken are the following: b) Circular plane motion: a, where

e = 0,

i = 0,

fl = 0,

Mi = MT + ClT + CJT,

w = 0,

M = M0 + Mit,

M0 = MT + fir + UJT - MXT.

c) Elliptic plane motion: a, where

e = eT,

i = 0,

O = 0,

Mi = M r + fir + uiT,

u> =

LOT

+ Or,

M0 =: MT - MXT.

M = M0 + Mit,

224

Perturbed Equations

of Motion: Halo and Triangular

Cases

d) Osculating: a, where

e = er,

i = iT, ft = ftT, u—u>T,

Mi = MT + ftT + w r ,

M = M0 + Mit,

M 0 = MT - M^T.

e) Osculating with linear variation of the eccentricity: a,

e = e 0 + e i i , ft =

ftr,

u> = u>T,

M = M0 + Mit,

where e\ = eT, eo = eT — reT. f) Osculating with linear variation of the argument of the perihelion: a, e = eT, i = iT, ft = ftr, w = UJQ + u)\t, M = M0 + Mit, where wi = 6JT, UJ0 = uT — TLJT, MO = MT — (MT + ftr)r. g) Osculating with linear variations of the eccentricity and argument of the perihelion: The same as f) and the eccentricity as in e). We note that for the Earth, models c) and d) coincide. To perform tests of the results given in Chapter 2, 2.3.7 the model c) has been used with the same value of the eccentricity used in Chapter 2, i.e., e = 0.0167. We recall that there the independent variable was the true anomaly. To get the velocity with the time as independent variable, and taking into account that the origin of the independent variable is at the passage through the pericenter, we should multiply the physical velocity by (1 + e) 1 / ' 2 (l — e ) - 3 / 2 . For the motion of the Moon the following models have been used: a) The theory of Brown as given by Escobal. b) The same theory deleting the linear variation of the argument of the perihelion of the Earth and using osculating linear expressions for the remaining four arguments. c) Same as b) suppressing all the periodic terms. d) Same as b) including several periodic terms which can be selected and varied during the execution of the program. Once the Keplerian elements of a planet have been computed, its ecliptic coordinates (position and velocity, acceleration and overacceleration if they are needed) can be computed by means of the routine EPHPLA. Given the position of the planet by: =

cos ft cos w — sin ft cos i sin LJ a(cosE — e) [ sin ft cos w + cos ft cos i sin LJ sin i sin to

+

- cos ft sin CJ — sin ft cos i cos u> a(l — e 2 ) 1 / 2 sinE [ - sin ft sin OJ + cos ft cos i cos u sin i sin UJ

Analysis of the

Perturbations

225

the routine computes the velocity, acceleration and overacceleration of the planet by differentiation of the above formula, and taking into account that the derivatives of the elements have already been computed by subroutine UPLNET. The residual accelerations are computed in the following way: For points in an analytic halo orbit the analytic acceleration is computed. For the selected points (equally spaced in the angle of the halo orbit) the acceleration exerted by a given model of the solar system is computed. To do this we transfer from the adimensional coordinates to the ecliptic ones. Then the acceleration is computed according to Newton's law and it is transformed back to adimensional coordinates. The transformations are done, using the formulas given in 1.3, by routine TRANS. We remark that the transformation of accelerations requires the overacceleration of the secondary with respect to the primary. Then the acceleration obtained numerically is compared with the one obtained analytically. This gives the residual acceleration. In fact, two different models are considered simultaneously. Then, not only the acceleration due to each one of them is compared with the analytic acceleration, but a comparison between both models is done. In this way we can easily detect the influence of changes in any element (for instance, the effect of considering e — 0 or not, i = 0 or not, etc.). The computation of the residual accelerations is done by routines RESIDU and RESIDUS. The routine RESIDUS computes the L\,L2 and Loo norms of the three residual accelerations along a halo orbit started on a given day. Then, the routine RESIDU looks for the maximum value of these three norms when the starting epoch of the halo orbit is changed. The results are given for a prefixed set of z-amplitudes of the halo orbit. To carry out the simulation of the motion of the spacecraft in a given model of the solar system, several possibilities are offered: To compute and draw an orbit, only compute and store, draw a previously computed orbit, etc. For each simulation the input and output systems of units can be chosen at will. The available systems are: ecliptic coordinates, adimensional coordinates for the Sun-Barycenter system or Earth-Moon system, and normalized coordinates for one of both systems and for any of the equilibrium points. Outputs of these simulations are scattered along the forthcoming chapters. Other possibilities offered by this program are the computation of ephemerides and several checks of the analytic computations of the ephemerides (for instance evaluation of velocity, acceleration and overacceleration by numerical differentiation, etc.). At the end of this section we give samples of the output concerning residual accelerations.

226

Perturbed Equations of Motion: Halo and Triangular

5.1.5

The Adopted Model for the Motion Cases Under Consideration

Cases

Near Halo Orbits in the

The runs done using the program described in 5.1.4 allow us to detect the weights of the different terms for the residual accelerations. As a general rule terms producing residual accelerations of weight greater than or equal to 12 are skipped. The adopted standard amplitude of z has been 0.08. No large differences appear in the range of interesting z-amplitudes. However see the comments at the end of the description of the adopted model concerning some higher weight terms. First we discuss the different effects of a perturbing body (for instance, a planet) on the spacecraft. Besides the direct terms there are several indirect terms. The first of them is the effect of the Sun. This is taken into account through the displacement of the Sun by means of the change of coordinates. The indirect effects due to global perturbations (direct and indirect) on the Earth-Moon barycenter are of two types: secular and periodic. The first ones enter in the mean motions of the barycenter, mean anomaly and argument of the perihelion. The second ones enter through the periodic terms of Newcomb's theory (or any other planetary theory). In order to keep the longitude mean motion of the different models of the Earth constant (this gives angularly coherent models) in the change of coordinates, the scalings in the velocity and acceleration to go to adimensional units, have been determined taking as period of the Earth, in any model, the true tropic period. Therefore, the mass of the Sun has been changed as explained in 1.4 to get a coherent model. Similar changes are done in what respects the Earth-Moon system. First we explain the results for the Sun-Barycenter system L\ case. As a result of several runs of the program SSI the following weights have been obtained: a) Mercury can be considered in an elliptic planar motion (global weight 10; eccentricity 11). The same is true for Mars. b) Venus can be considered in an elliptic motion (global weight 8; eccentricity 11) with inclination (weight 10). c) Jupiter can be considered in an elliptic motion (global weight 8; eccentricity 10) with inclination (weight 11). d) Saturn can be considered in a circular planar motion (global weight 11). e) No other planets have been considered with weight less than 12. See g) however. f) The Moon has global weight 6. In fact this weight is the difference between the effect of the Barycenter and the effect of Earth+Moon. In Brown's theory the angles should be considered with linear variation in time, except the argument of the perihelion of the Earth, whose variation can be neglected. The periodic terms to be retained are the first six in longitude (of weights 8,10,10,11,11,12), the first four in latitude (of weights 9,11,11,12) and the first four in parallax (of weights 9,11,11,12). As a conclusion the adopted

Analysis of the

227

Perturbations

model is: Longitude = £ + 22639.580 s i n / - 4586.438 sin(l-2D)

+ 2369.899 sin(2D)

+769.021 sin(2Z) - 668.944sin(Z') - 411.614sin(2F), Latitude = 18461.480 sinF + 1010.180 sin(Z + F) -999.695 sin(F -I)-

623.658 sin(F - 2D),

Parallax = nt + 186.5398cos/ + 34.3117cos(Z - 2D) +28.2333 cos(2£>) + 10.1657 cos(2Z). g) The global weight of the Earth is 1 (according to the scale of weights of 5.1.4, the weight of the Sun would be —2). The weight of the effects of the eccentricity is 5. In the mean anomaly linear terms in time are enough. The linear variation of the argument of the perihelion has weight 10. The effect of the periodic terms in the motion of the Earth (indirect periodic effect of the planets) appears in longitude, latitude and radius vector. The effects on the latitude are small but they have to be considered if terms up to weight 11 or slightly higher are kept. There are terms with relatively large values of the amplitude and for which jrip + iriE is small. These terms are large because of the resonant effects. To these ones we apply the last remark in 5.1.1. The list of terms and weights up to order 10 is the following: Terms in the longitude: mean motion ny — TIE 2ny

— 2TIE

nE 2nE -

nj nj

3riv — 3TIE

2>ny — 4 n £ 2UE — 2UM nE - 2nj 2nE - 3 n j 2 n y — TIE 2nv— 3 n s 4ny

— 4TIE

5nv — 5riE 2>UE — 4TIM

2nE nj SUE — 3 n j nE ns

weight 7 7 7 7 8 8 8 8 8 9 9 9 9 9 9 9 9

mean motion ny 3 n y — 5TIE

4 n y — 5UE 5 n y — 7TIE 6 n y — 6TIE 7nv — 7TIE 8 n y — 8TIE

ns ~

nM

3riE — 3TIM 2TIE — 3njvf 4TIE — 5UM 4TIE — 6n.M 5n_E — 7TIM

-

nj

2>nE -

2nj

ns - 3nj 3nE 2>nE 2nE -

^nj ^nj 2ns

weight 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

228

Perturbed Equations

of Motion:

Halo and Triangular

Cases

Terms in the radius vector: mean motion

weight

mean motion

weight

2ny

7 8 8 9 9 9 9 9 9 9

2 n y — TIE

10 10 10 10 10 10 10 10 10

— 2TIE

nE - nj 2nE - 2nj ny — nE 3riv — iriE 2>ny — 4TIE Any

— ATIE

2UE — 2TIM

nE - 2nj 2nE - 3 n j

2ny

— 3TIE

5nv

—5UE

6nv —6nE 3UE — AriM 2nB - nj 3UE —2nj 2>TIE —

nE -

ns

3nj

There are no terms of weight less than or equal to 10 in the latitude. There are only three terms of weight 11: the ones of mean motion ny - 2UE, 3ny — AUE and nE -

2nj.

In the previous list nv,nE,riM,nj,ns refer to the mean motions of Venus, Earth, Mars, Jupiter and Saturn, respectively. Of course, several other terms have to be retained, at least the ones giving weight 11 and 12. An exhaustive run of all the terms given in the routine NEWPER shows that 100 terms have to be retained if we only skip those giving a residual acceleration (Li norm) less than 0.275 • 10~ 6 . The index of these terms in the routine NEWPER are 1, 5, 6, 10, 11, 12, 14, 15, 16, 17, 20, 21, 22, 24, 25, 26, 28, 29, 32, 35, 36, 38, 39, 41, 43, 44, 45, 48, 49,51, 52, 53, 56, 57, 59, 60, 63, 67, 84, 85, 86, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98, 104, 107, 108, 110, that is, a total of 55 terms in longitude, 121, 125, 126, 130, 131, 132, 135, 136, 137, 140, 141, 144, 145, 146, 148, 152, 156, 161, 165, 167, 168, 172, 173, 176, 177, 180, 185, 208, 209, 210, 212, 213, 214, 217, 218, 219, 220, 221, 222, 229, 233, 234, that is, a total of 42 terms in radius vector, and 243, 251, 270, that is, a total of 3 terms in latitude. Concerning the errors in the adopted model we have made several tests. If only the effects coming from the planets and the Moon but not the periodic ones due to the planets are considered, the total residual acceleration (difference between the two models) has weight 11.5. There is no strong difference if we pick up the 55 more important periodic terms (the ones up to order 10) and the 100 more important. In any case the residual acceleration is close to 0.5-10~5 in adimensional units. Going back to physical quantities this means 3-10~5 mm/s 2 . The yearly cumulative effect is less than 100 cm/s. However, these residual accelerations have changing directions, so there are cancellations which decrease the bound. This is seen in the simulations of the control. No systematic search has been done for the L2 case in the Sun-Barycenter

Analysis of the

Perturbations

229

system. Taking as a model for the solar system the one previously obtained for the L\ case and evaluating along a Li halo orbit the residual acceleration, a value of 0.3315-10 -6 (in adimensional units) was obtained (for the ^-amplitude 0.08). In this run the periodic perturbations of the planets on the Earth-Moon barycenter were skipped. For that model the total error is of order roughly 11.5, as it was the case for L\. For the Earth-Moon system the weight of the different planets has been computed as well as the weight of the first and last periodic term in the expressions of Moon's longitude, latitude and parallax of Brown's theory as given by Escobal. The results are the following: Sun, weight 5; Mercury, Mars and Saturn, weight 14; Jupiter, weight 11; Venus, weight 10, eccentricity of the Earth, weight 8; first and last terms of Moon's longitude, weights 2 and 8; second and last terms of Moon's latitude, weights 2 and 11, first and last terms of Moon's parallax, weights 3 and 10. (Note: The first term in latitude as well as the fourth one in longitude must always be included in order to consider the Moon in a circular orbit around the Earth.) A sample of results of program SSI for the computation of the above weights is given below. Assuming an expense of fuel of 1 m/s/year, in order to correct the errors in the modeling of the solar system, a residual acceleration of the order 10~ 5 (in adimensional units) can be corrected, which requires keeping in the model terms up to weight 12. This should require to retain hundreds of periodic terms in Moon's theory. The proposed approach is to keep only terms up to weight 8, computing after their contribution to the equations by Fourier analysis. The extension to a higher order theory is easy using the JPL ephemerides and not so expensive in CPU time.

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

FIRST MODEL = 1 0 10 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = 2 0 10 0 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = ANALYTIC - M0DEL1 ANALYTIC - M0DEL2 N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.12023D-5 0.12671D-5 0.21353D-5 0 . 11873D-5 0..12033D-•5 0 .16254D-5 FIRST MODEL = 3 0 10 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = 1 0 10 0 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.12045D-5 0.12695D-5 0.21355D-5 0.12023D-5 0 12671D-5 0.21353D-5 FIRST MODEL = 0 1 10 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 2 10 0 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 NORMINF N0RM2 0.16399D-4 0.20100D-4 0.38750D-4 0.16830D-4 0 20005D-4 0.39374D-4 FIRST MODEL = 0 1 10 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 4 10 0 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.16399D-4 0.20100D-4 0.38750D-4 0.16399D-4 0 20100D-4 0.38750D-4 FIRST MODEL = 0 2 10 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 3 10 0 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.16830D-4 0.20005D-4 0.39374D-4 0.16665D-4 0 20532D- 0.40017D-4 FIRST MODEL = 0 0 10 1 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 10 2 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 18262.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.11178D-5 0.12393D-5 0.19894D-5 0.59195D-6 0 65189D-6 0.10917D-5 FIRST MODEL = 0 0 10 1 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 10 3 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 18262.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.11178D-5 0.12393D-5 0.19894D-5 0.11242D-5 0 12477D-5 0.20080D-5 FIRST MODEL = 0 0 10 3 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 10 1 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 18082.00, INCREMENT = 180.0, NUMBER OF EPOCHS =

N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF Z AMPL 0.46140D-6 0.51851D-6 0.89522D-6 0.46120D-6 0. 51833D-6 0.89514D-6 0.08 FIRST MODEL = 0 0 10 0 1 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 10 0 3 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 4 N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF Z AMPL 0.15325D-4 0.16070D-4 0.21842D-4 0.15328D-4 0, 16072D-4 0.21847D-4 0.08 FIRST MODEL = 0 0 10 0 4 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 10 0 1 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 4 N0RM1 N0RM2 NORMINF N0RM1 Z AMPL N0RM2 NORMINF 0.15324D-4 0.16068D-4 0.21840D-4 0.15325D-4 0. 16070D-4 0.21842D-4 0.08 FIRST MODEL = 0 0 10 0 2 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 10 0 3 0 0 0 0 1 0 0 0 INITIAL EPOCH = 17902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 4 N0RM1 N0RM2 NORMINF N0RM1 Z AMPL N0RM2 NORMINF 0.13092D-4 0.13709D-4 0.18565D-4 0.15328D-4 0. 16072D-4 0.21847D-4 0.08 FIRST MODEL = 0 0 10 0 0 1 0 0 0 1 0 0 0 SECOND MODEL = 0 0 10 0 0 2 0 0 0 1 0 0 0 INITIAL EPOCH = 17902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 4 N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF Z AMPL 0.61892D-6 0.64715D-6 0.94119D-6 0.54474D-6 0. 56797D-6 0.84239D-6 0.08 FIRST MODEL = 0 0 10 0 0 0 1 0 0 1 0 0 0 SECOND MODEL = 0 0 10 0 0 0 2 0 0 1 0 0 0 INITIAL EPOCH = 17902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 4 N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF Z AMPL 0.60731D-7 0.73743D-7 0.12700D-6 0.61007D-7 0. 73873D-7 0.12801D-6 0.08 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 2 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 1 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 8, N0RM1 N0RM2 NORMINF N0RM1 Z AMPL N0RM2 NORMINF 0.54005D-4 0.67598D-4 0.25573D-3 0.54005D-4 0.04 67598D-4 0.25573D-3 0.08 0.53983D-4 0.68033D-4 0 25947D-3 0.53983D-4 68033D-4 0.25947D-3 0.12 0.54087D-4 0.68525D-4 0.26345D-3 0.54087D-4 68525D-4 0.26345D-3 0.16 0.54185D-4 0.69074D-4 0.26759D-3 0.54185D-4 69974D-4 0.26759D-3 0 0 10 0 0 0 0 0 0 1 3 0 0 FIRST MODEL = SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 8, INITIAL EPOCH

AMPL z0 .08

AMPL z0 .08

AMPL z0 .08

AMPL

z0 .08

AMPL z0 .08

AMPL z0 .08

AMPL

z0 .08

NORMl

N0RM2

NORMINF

NORMl

N0RM2

NORMINF

0 .52413D-4 0.64038D-4 0.23210D-3 0.52834D-4 0 .64859D-•4 0.23455D-3 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 0 MOON PERIODICS : 2 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF NORMl N0RM2 NORMINF NORMl N0RM2 0 .52565D-4 0.64253D-4 0.23144D-3 0.52413D-4 0 . 64038D-•4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 0 MOON PERIODICS : 3 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF NORMl N0RM2 NORMINF NORMl N0RM2 0 .52413D-4 0.64038D-4 0.23210D-3 0.52173D-4 0 .63721D--4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 0 MOON PERIODICS : 4 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF NORMl N0RM2 NORMINF NORMl N0RM2 0 .52413D-4 0.64044D-4 0.23187D-3 0.52413D-4 0 .64038D-•4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 5 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF NORMl N0RM2 NORMINF NORMl N0RM2 0 .52413D-4 0.64038D-4 0.23210D-3 0.52412D-4 0 . 64038D-•4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 0 MOON PERIODICS : 6 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF NORMl N0RM2 NORMINF NORMl N0RM2 0 .52417D-4 0.64021D-4 0.23212D-3 0.52413D-4 0 .64038D--4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 7 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF NORMl N0RM2 NORMINF NORMl N0RM2 0 .52413D-4 0.64038D-4 0.23210D-3 0.52413D-4 0 .64038D--4

EPOCHS = 8 NORMINF 0.23210D-3

EPOCHS = 8 NORMINF 0.23198D-3

EPOCHS = 8 NORMINF 0.23210D-3

EPOCHS = 8 NORMINF 0.23229D-3

EPOCHS = 8 NORMINF 0.23210D-3

EPOCHS = 8 NORMINF 0.23216D-3

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

MOON PERIODICS : 22 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 0.52489D-4 0.64324D-4 0.23508D-3 0.52413D-4 0.64038D-4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 23 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 0.52413D-4 0.64038D-4 0.23219D-3 0.52406D-4 0.64018D-4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 24 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 0.52413D-4 0.64038D-4 0.23210D-3 0.52416D-4 0.64053D-4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 24 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 0.52405D-4 0.64016D-4 0.23189D-3 0.52413D-4 0.64038D-4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 26 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 0.52413D-4 0.64038D-4 0.23210D-3 0.52412D-4 0.64036D-4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 33 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 0.53465D-4 0.66839D-4 0.25285D-3 0.52413D-4 0.64038D-4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 34

EPOCHS = 8 NORMINF 0.23210D-3

EPOCHS = 8 NORMINF 0.23187D-3

EPOCHS = 8 NORMINF 0.23227D-3

EPOCHS = 8 NORMINF 0.23210D-3

EPOCHS = 8 NORMINF 0.23205D-3

EPOCHS = 8 NORMINF 0.23210D-3

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.04 0.08 0.12 0.16

Z AMPL 0.08

Z AMPL 0.08

INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF .52413D-4 0.64038D-4 0.23210D-3 64531D-4 0.23645D-3 52579D-4 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 35 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 1 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM2 NORMINF N0RM1 N0RM2 NORMINF N0RM1 0.52207D-4 0.63504D-4 0.22727D-3 0.52413D-4 0 64038D-4 0.23210D-3 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS : 36 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 N0RM2 NORMINF NORMINF N0RM1 .52413D-4 0.64038D-4 0.23210D-3 0.52416D-4 0 64052D-4 0.23263D-3 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 1 MOON PERIODICS 37 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 3 0 0 INITIAL EPOCH 1 7 5 4 2 . 0 0 , INCREMENT = 1 8 0 . 0 , NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF .52415D-4 0.64051D-4 0.23235D-3 0.52413D-4 0.64038D-4 0.23210D-3 FIRST MODEL = 0 0 2 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 4 0 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 117902.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 N0RM1 NORMINF N0RM2 NORMINF 0.77425D-4 0.77428D-4 0.78042D-4 0.46905D-3 51595D-3 0.83489D-3 0.77429D-4 0.77431D-4 0.78052D-4 0.48568D-3 53167D-3 0.84761D-3 0.77433D-4 0.77436D-4 0.78084D-4 0.51164D-3 55713D-3 0.86984D-3 0.77437D-4 0.77440D-4 0.78135D-4 0.54535D-3 59146D-3 0.90261D-3 0 0 1 0 0 0 0 0 0 1 0 0 0 FIRST MODEL = 0 0 7 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = INITIAL EPOCH = 16102.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.44188D-3 0.47782D-3 0.81070D-3 0.44188D-3 0 47782D-3 0.81071D-3 FIRST MODEL = 0 0 5 0 0 0 0 0 0 1 0 0 0 SECOND MODEL = 0 0 1 0 0 0 0 0 0 1 0 0 0 INITIAL EPOCH = 16102.00, INCREMENT = 180.0, NUMBER OF EPOCHS = N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.44173D-3 0.47753D-3 0.80992D-3 0.44188D-3 0 47782D-3 0.81070D-3

FIRST MODEL = SECOND MODEL =

0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0

INITIAL EPOCH = 16102.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 2 Z AMPL N0RM2 NORMINF N0RM1 N0RM2 NORMINF N0RM1 0.44188D-3 0.47782D-3 0.81070D-3 0.44198D-3 0.47793D-3 0.81094D-3 0.08

FIRST MODEL = SECOND MODEL = Z AMPL 0.08

INITIAL EPOCH = 16102.00, INCREMENT = 180.0, NUMBER OF EPOCHS = 1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF N0RM1 .43159D-3 0.46794D-3 0.78966D-3 0.43155D-3 0.46786D-3 0.78952D-3

FIRST MODEL = SECOND MODEL = Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

Z AMPL 0.08

1 1 1 1 1 1 1 1 0 1 0 0 0 3 4 6 3 4 2 0 0 0 1 0 0 0

3 4 6 3 4 2 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0

INITIAL EPOCH = 18262.00, INCREMENT = 180.0, NUMBER OF N0RM2 N0RM1 N0RM2 NORMINF N0RM1 .44857D-3 0.48281D-3 0.81263D-3 0.44844D-3 0.48266D-3 FIRST MODEL = 0 0 10 0 0 0 0 0 0 1 1 0 0 SECOND MODEL = 0 0 10 0 0 0 0 0 0 1 4 0 14 MOON PERIODICS : 1 2 3 4 5 6 22 23 24 25 33 34 35 36 INITIAL EPOCH = 17542.00, INCREMENT = 180.0, NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 .53983D-4 0.68033D-4 0.25947D-3 0.53953D-4 0.67951D-4 FIRST MODEL = 1 1 1 1 1 1 1 1 0 1 1 0 0 SECOND MODEL = 3 4 6 3 4 2 0 0 0 1 4 0 14 MOON PERIODICS : 1 2 3 4 5 6 22 23 24 25 33 34 35 36 INITIAL EPOCH = 16102.00, INCREMENT = 180.0 NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 .44294D-3 0.47765D-3 0.82241D-3 0.44295D-3 0.47762D-3 FIRST MODEL = 1 1 1 1 1 1 1 1 0 1 1 0 0 SECOND MODEL = 3 4 6 3 4 2 0 0 0 1 4 0 14 MOON PERIODICS : 1 2 3 4 5 6 22 23 24 25 33 34 35 36 INITIAL EPOCH = 17542.00, INCREMENT = 180.0 NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 .43267D-3 0.46581D-3 0.76360D-3 0.43255D-3 0.46582D-3 FIRST MODEL = 1 1 1 1 1 1 1 1 0 1 1 0 0 SECOND MODEL = 3 4 6 3 4 2 0 0 0 1 4 0 14 MOON PERIODICS : 1 2 3 4 5 6 22 23 24 25 33 34 35 36 INITIAL EPOCH = 18982.00, INCREMENT = 180.0 NUMBER OF N0RM1 N0RM2 NORMINF N0RM1 N0RM2 .46135D-3 0.49429D-3 0.82598D-3 0.46153D-3 0.49449D-3

Z AMPL 0.08 FIRST MODEL =

1 1 18

1 1 1 1 1 0 1 0 0 0

EPOCHS = 1 NORMINF 0.81249D-3

EPOCHS = 8 NORMINF 0.25887D-3

EPOCHS = 8 NORMINF 0.82235D-3

EPOCHS = 8 NORMINF 0.76358D-3

EPOCHS = 8 NORMINF 0.82613D-3

SECOND MODEL = 3 4 19 3 4 2 0 0 0 1 0 55 0 EARTH PERIODICA : 5 6 10 11 12 15 16 17 20 21 24 26 28 32 35 41 44 48 49 89 90 93 94 95 97 98 104 107 126 130 131 132 135 136 140 144 148 168 176 INITIAL EPOCH = 1 8 1 6 2 . 0 0 , INCREMENT = 1 0 0 . 0 , NUMBER OF EPOCHS = 3 , NUMBE Z AMPL N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.08 0 . 4 6 1 6 2 D - 3 0.51894D-3 0 . 8 3 6 6 9 D - 3 0 . 4 6 1 7 7 D - 3 0.51925D-3 0.83726D-3 FIRST MODEL = 1 1 11 1 1 1 1 1 0 1 1 0 0 SECOND MODEL = 3 4 16 3 4 2 0 0 0 1 4 0 14 MOON PERIODICS : 1 2 3 4 5 6 22 23 24 25 33 34 35 36 INITIAL EPOCH = 1 8 1 6 2 . 0 0 , INCREMENT = 1 0 0 . 0 , NUMBER OF EPOCHS = Z AMPL N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.08 0 . 4 6 7 9 2 D - 3 0 . 5 2 8 0 7 D - 3 0 . 8 2 8 4 5 D - 3 0 . 4 6 7 8 7 D - 3 0.52796D-3 0.82789D-3 FIRST MODEL = 1 1 18 1 1 1 1 1 0 1 1 0 0 SECOND MODEL = 3 4 19 3 4 2 0 0 0 1 4 55 14 EARTH PERIODICS : 5 6 10 11 12 15 16 17 20 21 24 26 28 32 35 41 44 48 49 88 89 90 93 94 95 97 98 104 107 126 130 131 132 135 136 140 144 148 168 1 INITIAL EPOCH = 1 8 1 6 2 . 0 0 , INCREMENT = 1 0 0 . 0 , NUMBER OF EPOCHS = 3 , NUMBER Z AMPL N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.08 0.46515D-3 0.52679D-3 0 . 8 4 2 1 4 D - 3 0.46535D-3 0.52706D-3 0.84282D-3 FIRST MODEL = 3 4 19 3 4 2 0 0 0 1 4 1 1 0 100 14 EARTH PERIODICS : 1 5 6 10 11 12 14 15 16 17 20 21 22 24 25 26 28 45 48 49 51 52 53 56 57 59 60 63 67 84 85 86 87 88 89 90 93 94 95 121 125 126 130 131 132 135 136 137 140 141 144 145 146 148 152 15 176 177 180 185 208 209 210 212 213 214 217 218 219 220 221 222 22 MOON PERIODICS : 1 2 3 4 5 6 22 23 24 25 33 34 35 36 SECOND MODEL = 1 1 18 1 1 1 1 1 1 1 1 1 1 0 0 0 INITIAL EPOCH = 1 7 2 6 2 . 0 0 , INCREMENT = 1 0 0 . 0 , NUMBER OF EPOCHS = 20 Z AMPL N0RM1 N0RM2 NORMINF N0RM1 N0RM2 NORMINF 0.08 0.50390D-3 0 . 5 5 0 8 1 D - 3 0 . 9 1 9 6 0 D - 3 0 . 5 0 3 4 9 D - 3 0.54985D-3 0.91983D-3 Table 5.5 Output of program SSI, for global residual accelerations in the Sun-Barycenter s N0RM1 N0RM2 NORMINF refer to the differences, in the norms | | i , | |2 and | |oo as follows: AN M0DEL1 - M0DEL2.

Equations

5.2

of Motion for Perturbed Halo Orbits

237

Equations of Motion for Perturbed Halo Orbits

5.2.1

Introduction

First, we shall use ideas of Richardson and Farquhar to obtain the equations of motion in the vicinity of the three collinear points, for the Sun-Barycenter problem and for the Earth-Moon problem, in a general form so that: 1) It is readily adaptable to the perturbation analysis. 2) All orders of the nonlinear developments of the motion are shown to be easily obtainable using recursive relationships. So, the form of these equations is good for being used in the computer. 3) Furthermore, the equations shall be of the form obtained by Richardson, for the restricted circular three-body problem, plus perturbations. Point 1) was solved by Farquhar without take into account point 2). Richardson solved point 2) for the circular restricted problem omitting point 1). 5.2.2

The Lagrangian

in Ecliptic

Coordinates

We study the motion of a spacecraft in a solar system formed by the Sun, the Earth, the Moon and several planets P\,... ,PkFirst, we consider an inertial frame of reference with origin at the center of masses of the solar system, whose axes are parallel to the ecliptic ones, the time t is given in Modified Julian Days (MJD) and the distances are given in km. We have not considered the effect of the variation of the equinoxes in the equations, but can be easily taken into account by introducing a change of coordinates. an Let R, RS,RE'SLM'BLB d Rpt (resp. O, S, E, M, B, Pi) be the position vector with respect to the inertial frame (resp. the masses) of the spacecraft, Sun, Earth, Moon, Earth-Moon barycenter and the ith planet, respectively. From Newcomb's theory we know RS,RB and RP.. Furthermore, using Brown's theory we obtain RE and RM. Therefore, the motion of a spacecraft is governed by a restricted (3 + k) + 1-body problem whose equations are:

Ae{s,E,M,Pi,-,Pk}

\~A

_

—'3

where G is the gravitational constant, dots denote derivatives with respect to t and | | denotes the Euclidean norm in R 3 . Let a be the radiation pressure of the Sun in N/m 2 at one astronomical unit, i.e. a = 4.56 • 1(T 6 . We denote by m (in kg) and a (in m 2 ) the mass and the effective cross section of the spacecraft to the radiation pressure. Then, the contribution of the radiation

238

Perturbed Equations

of Motion: Halo and Triangular

Cases

pressure to the equations of motion is

Rs-E

~

—a

\RS-R\3'

where 149 597 871.41055842 • 86 400 : W km 3 a = 1000 m (Modified Julian Days) 2 ' We denote by as and UB the semimajor axis and the mean motion of the EarthMoon barycenter around the Sun. The mean motion takes into account the secular terms in the ecliptic argument of perihelion and in the mean anomaly, due to the effect of the planets. Then, using Kepler's third law we have that nBaB is close to G(S + E + M). Following the idea given in 5.1.3 let S be the modification of the Sun mass to have nBa3B = G(S + E + M). In short, taking into account the radiation pressure we can write the equations of motion in the following form GS(Rs - R) —

]

GS(RS -R) +

I\R 7 ?-R\ „ - 3/ ? | 3

|\R R „-R\ _ 73? | 3

S

—b

^ +

,

S

'—b

—'

—'

GA(RA

^ 2^

'~A

Ae{E,M,P!,...,Pk}

- R)

\RA ~R\3

,

'

—'

where S = S — S — a/G. The contribution of S — S to S in the above equation is called the coherence term (or Kepler coherence terms to distinguish from the problem of coherence mentioned in Section 5.1). The position of the Sun Rs satisfies

Rs=

GA{RA - Rg)

£

,

\RA-RS\3

Ae{E,M,Pi,--,Pk]

So, if r = R — Rs then the equations of motion of the spacecraft with respect to the ecliptic frame of reference with origin at the Sun are r=

GSr j=

GSr ^=+ r

v^ > AeiE,^,..,^

„ . ( r
r%A

where r_SA denotes the vector from S to A. Also, as it is usual we write r and TSA instead of \r\ and \rSA\ respectively. We note that the above equations of motion correspond to the Lagrange equations given by the Lagrangian T

1.

.

GS

L= -r 2 •r+ r

GS

+ r

+

v>1 t— A€{E,M,Pu--,Pk}

„.(

1

, GA V k s 4 X |

-

M

—r

\

_l

LSA • L T %A bA

Equations

239

of Motion for Perturbed Halo Orbits

or equivalently 1 .

.

T

L=-r-r

. J 1 -

JL4B

+ K(

/X5

v^

?- + ^+

J2

Ae{E,M,Pu-,Ph}

(

r

1

^A '

SA

r

"

~S'SA

-SA

where K = G(S + E + M) and HA = A(S + E + M)~1 for Ae{S, S,E, M, Pu .. Note that /J,S = 1 - \iB where \iB = ( # + M ) ( 5 + £ + M ) _ 1 . 5.2.3

27ie Collinear

Points for the Sun-Barycenter

.,Pk}.

Problem

The collinear point spacecraft geometry for the Sun and the Earth-Moon barycenter is illustrated in Figure 5.1. r

?

SB

s

?

=

B

M

-?L

2

L/y s.

x

/

Spacecraft ?

/ B /

s

S

1=

B

\P 1^

' /

y

\/

x L2

E * r

?

SB ?

B="?L

s

Spacecraft 2

VI=BI //

4

X

^^ r

^\. ?

S

B

s ?

B

=

"?L

E «

Fig. 5.1 Geometry for the spacecraft under the action of Sun, Earth and Moon, near a collinear point of the Sun-Barycenter system.

240

Perturbed Equations of Motion: Halo and Triangular Cases

Let Li (for i = 1,2,3) be located with respect t o t h e E a r t h - M o o n barycenter by t h e vector rL. T h e expression determining r_L is (J-BLL r

=1 + ~SB

=-(!-/**)

r

SB

r

S

SB

This equation is equivalent t o t h e quintic of Euler for t h e determination of t h e collinear points of the circular restricted three-body problem. Let 71 = r^r^g, 72 = r L2rsB a n d 73 = T 'S' 7 "SB- So, t h e locations of t h e collinear points Li are obtained from t h e solutions of t h e quintics 7i - (3 - fxBht

+ (3 -

2

M B ) 7 I - M B 7 I + 2 M B 7 I - MB = 0,

l l + (3 - M B ) 7 2 + (3 - 2 M B ) 7 2 3 - MB7 2 2 ~ 7f

5.2.4

2

M B 7 2 - MB = 0,

3

+ (2 + VBht

+ (1 + 2// B )7 3 - (1 - ^ B ) 7 3 2 - 2(1 - ^ 7 3 - (1 - / i B ) = 0.

Expressions tems

of the Lagrangian

in Different

Coordinate

Sys-

First of all we consider a translation of our reference system t o t h e point Li of t h e Sun-Barycenter system, i.e. let p — Z~LSL be t h e position vector of t h e spacecraft in t h e ecliptic frame of reference with origin a t L j . Positions of t h e Sun, E a r t h , Moon, a planet Pi a n d t h e E a r t h - M o o n barycenter with respect t o Li are denoted by rs,rE,rM,jip. a n d rB, respectively. In this new frame of reference t h e Lagrangian goes over t o L

=

J<^

1

1

1. . . . oP • 9- P-Ls + K (1 -

9B)

r.sB-9

+ 9B

LS-P\

'SB •LSA-9

+K

\Ls - P\

Ae{E,M,Pi

\LA

Pk,B}

~ P\

' SA

where we omit t h e terms in t h e Lagrangian which only depend on time a n d 1 -1

i{A) =

if if

Ae{E,M,Pu...,Pk}, A = B.

T h e Lagrangian can b e written in t h e form

1. . . . +K

1

„

(1-MB)

S

+•

\ts ~ P\

„ sLs'9 , (LB-9 (1 - M B ) — 3 - + MB — 3 r

K9s

+

Ml

LS- 9

r

V B

K

Ml

LSB-9 -3

r

SB LSA-9

i(A)[iA Ae{E,M,Pi,--,Pk,B}

LB

+ 9-B

\LA

- P\

'SA

-9

241

Equations of Motion for Perturbed Halo Orbits

Using the definition of Lj we have - (1 -

(1 - / i f i W + HB-I- =

PB)LSB

'SB

So, I 1 - VB)—— + HB —3 r S \ rB LB ~ (1 ~ VB)LSB PBLSB\ 'SB

'SB

-3 SB

r

_

Ls-p

J

'SB

Hence, the Lagrangian becomes +K n

L = -p-p-p-rs Kr

+ -S-P +,

^ 'SB

KHs

\I • r,

+K T^i IL5-£l

L

\LS-P\

s-P\ ,

f

r

l J

\ -

1

LB-P

\\r.B-p\

'B

^ (^L

E ' (i(A A^ Ae{S,M^r,...,P ,B} fc

*±£ MLA-fll

'54

The next step is to express the Lagrangian in normalized coordinates. First, we consider a frame of reference with origin at the equilibrium point Li, the z-axis points along the line formed by the two primaries (the Sun and the Earth-Moon barycenter) away from the larger primary, the (x, y)-plane coincides with the instantaneous plane of motion of the primaries, and the z-axis completes the right-handed system (see the figures in 2.3). Let 9 and 6 be the longitude and latitude of the Earth-Moon barycenter in ecliptic coordinates centered at the Sun. Then, the unitary vector on the z-axis is e, = = £ a , the unitary vector on the 2-axis is e, = |T gg ~ ?B , and the unitary vector -1

J

rSB'

-3

\rSBArSB\

•>

on the y-axis is e 2 = e 3 A e x . Let C be the matrix whose columns are the three vectors ex, e2 and e 3 , i.e.

(

cos S cos 9 cos S sin 8 sin S

— (sin 0 + sin S cos 6 R)R\ (cos 9 — sinSsin9R)Ri cos SRRi

(—cos9sinS + sin9R)Ri — (sin 9 sin 6 + cos 9 R)R\ cos SRi

where R = 5(6 cos 6)-1 and R\ - (1 4- R2)~1/2. Note that the matrix C is orthogonal. Now, we consider the change of variables from the ecliptic frame of reference with origin at Li,p, to the frame of reference with origin at Li, a = (x,y,z), given by p = kC a, { where k is a scaling factor defined by k = <

rL

= =

I rs =

ursB 7« r 5B

for for

i = 1,2, « = 3.

242

Perturbed Equations

of Motion: Halo and Triangular

Cases

In the system (a;, y, z) units of mass, time and distance are chosen in the following way: - The unit of distance is taken as 7 times the semimajor axis as • - The new unit of time s is the time t in Modified Julian Days scaled by 7 s / 2 , so s = 7 3 / 2 i. In what follows, primes will denote derivatives with respect to s: ' ~ Is- ^ n i s t ' m e s w a s c a n e d r m Chapter 2 and will be denoted usually t in the subsequent chapters. - The unit of mass is chosen in such a way that K = G(S + E + M)= n2Ba3B = 1, where nB is the mean motion. System (x,y,z) with these new units is the normalized system centered at Li, as defined in 5.1.3. We remark that now the passage from ecliptic to normalized coordinates is done performing first the translation to Li. This is more convenient for the analytic developments. The vectors p, ZA>?-SA> m the ecliptic coordinate system with origin at L\ or L2, are denoted by a = (x,y,z),rA = {xA,yA,zA) and rSA = (xSA,ysA,zsA) in the normalized one. As it is usual a,fA and ?SA denote \a\, \rA\ and |£ S y l |, respectively. The angle from the vector r_A (resp. LSA) t o the vector a will be denoted by Ai (resp. A2). In what follows, we usually denote a (resp. a) by a (resp. o). The Lagrangian in the normalized system becomes

1

I

LB

'£

•p\

A€{E,M,Pi

|

3

K-)

rs-P

rB

r

Pk,B}

X

|

SB '~A

K~f-3fi§ ks - P\

-'

bA

7

where p = kCa and r_A = kC^AWe shall need the following relations to compute explicitly the Lagrangian in the normalized system: i) CTC = Id, so Ca-Cb = a-b, ii) (C')TC + CTC = 0, so Ca • C'b = 0 if a = 6. / 0 -E 0 \ / A 0 D hi) CTC = \ E 0 -F and (C')TC = I 0 B 0

V 0

F

0 / + 62),

VD

0 C

where A

=

7-

3

(62cos28

B

=

7-

3

{[(cos 2 (5 + sin 2 <5ii 2 ^ 2 + (2 + i?2)(52 + (-R + s i n ^ ) 2 ] ( l + i ? 2 ) - 1

-R2R2(1

+

R2)-2},

Equations

{[R262-S2

C

=

^

D

=

7-

E

=

j-^idcosS

F

=

7-

3

243

of Motion for Perturbed Halo Orbits

+ (R +sin 56)2](l + R2)-1-R2R2(l

R2y2},

+

[-sin(5cos^2(l + / ? 2 ) 1 / 2 - c o s ( 5 ^ ( l + i?2)-1/2], R2)-1/2,

+ SR^l +

3 2

/ [ ^ s i n ^ + i?(l + J R 2 )- 1 ],

where R = 5(9 cos J ) - 1 . iv) If p = kCa and 77 = kCb then fc7_3a

p' . 77' =

• 6 + fcfc7-3/2(a • b + a' • b)

+k2aT(C')TC'b

+ k2{aT{C')Td

- ~a'(C')TCb) + k2c± • &'.

v) From iv) it follows that p'.p'

=

k2>y-3(x2+y2+z2) 2

2

2

+k (Ax 2

+ 2kkf-3/2(xx'+yy'+ 2

zz')

2

+ By + Cz + 2Dxz) + 2k [E(xy' - yx') + F{yz' - zy')]

2

l2

+k (x' +y +z'2). vi) We have that r s = (xs,0,0) where xs = 1 — 7 _ 1 , or xs = —1 — 7 - 1 , or x$ = 1 if we consider the normalized system centered at 1/1,1/2 or L3, respectively. So r 5 = 0. From iv) it follows that 7~3/V-rs

k27~3xsx

=

+ kk'y~z/2xsx'+

k2(Axsx

+Dxsz)

+

k2Exsy'.

vii) The following relation holds £|2

\LA

= =

(IU - £) ' (IA -a) r\

1

=fA-

2(xAx + yAy + zAz) + 1

_ 2 1 XAX + VAV + zAz\ J|_ , / _£_ r^a

r\

1-2COSJ4I

TA

TA

+

7\4

V r >t

Hence -1/2

\lA-a\

x

=

f/

1 - 2 cos Ax — + TA

\TA

= ^ £ ( £ ) p»(cos^), n>0

where Pn(cosAi)

is the Legendre polynomial in the variable cos A\ of degree n and

we assume that — < 1 for every A G {E, M, Pi,..., TA

Pk, B}. For more details see

244

Perturbed Equations of Motion: Halo and Triangular

Cases

"Methods in Celestial Mechanics" of D. Brouwer and G. Clemence. Therefore, \LA-P\-1

k-1\fA-a\-1

=

= (kfA)-iy2(^)

PnCcosAx).

n>0

\VA-

Note that LA-P =

= k~

(kfA)-1

^3

(•?-) cosA, = (kfA)-1

(•?-)

rA

Pi (cos 4 0

rA

viii) Concerning the Barycenter f_B — (XB,0,0) where XB = 1, or XB = —1 or _1 XB = 1 + 7 according to the normalized system centered at l a , £ 2 or L3, that we consider. From vii) it follows that (1 - PB)

Ls ' P

ks ~ P\

= (1 - nB)(krs)~l

LB

- 3 " I +P-B

\LB - P\

UBikrB)'1

+

+ (l-fiB)(kfS)-1J2(jL)

PnicOsS^+VBikfB)-1

+PB\XB\~1

1

J2

(j-j

PnicOsBt)

n>2

n>2

= constant + k

• P

(l-/x

B

)|

a ; s

|-1^(^) n>2 \I XS I

P n (cos5i)

Pn[cosBi)

^2

\XB

n>2

= constant + AT1 V ^2 = constant + k~xj3

1 - n+1 i g . + ( T 1 ) « - J £ n+1 L - anPn (cos 5i) [\xS\ \xB\ 2_. cn^nPn ( —) , n>2

where we have used - 3 [/

c„,

=

Pn(-*)

_ =

4_Cx1"l n

^

for

Li and L 2 ,

for

L3,

<

7" cos5i

-i\n 1-MB 1-MB

LS -a _ xsx fsa \xs\a n (-l) Pn(z).

,

MB

for for

L\ and L2, L3

In these expressions the upper sign is used for motion about L\ while motion about L 2 requires the lower sign. We have also used Figure 5.1 to compute cos P i

Equations

245

of Motion for Perturbed Halo Orbits

as a function of cos Si. Note that -3 3

77 "

I

[(-!)"(1(1^)n+r + (±1)"^]

for

v"+ l

U

and L 2 ,

for

which agrees with Chapter 2. ix) We have that X Ls ' P_ _ T_s ' & SX 3 kf , kf\ 'SB ""SB "• ' SB

7 xsx "

x) Now, we develop the last two expressions of the Lagrangian. First, from vii) it follows kqf-3fi§ k-y-3n§ i 1 = u-

^

/ 5 \" 2_> ~ ^n(cos5i).

We have that £s/l " P _ f 5,4 • a _ a cos ,4 2 'SA

kf

kf2

SA

SA

Then, again from vii) we obtain 1 kA - P\

'P _ 1 k r SA

LSA

a cos A 2

3

S A

rAfri\r J ' "• n>0A

Finally, the Lagrangian in normalized coordinates becomes L = 1 {k2(x'2

+ y'2 + z'2) + 2kk~f-3l2(xx'

+ yy' + zz') + 2k2[E(xy' - yx')

+F(yz' - zy')] + k2j-3(x2

+y2 + z2) + k2(Ax2 + By2 + Cz2 + 2Dxz)}

-k2-Y~3xsx

- kk^~3/2xsx'

- k2(Axsx

+Kk~1 £

cnanPn

(J) + Kk~lxsx

+ Dxsz) +

—y--3(1§

krs

n>2

+Kk-11~3

-

k2Exsy'

^£

U\rs

n>l

J2

Ae{E,M,Pi,--,Pk,B}

*(^W

-(

)

x

a cos A2 + 1 -=-H(-?-) r SA

p cos51

FAVOSA!)

246

Perturbed Equations

5.2.5

The Lagrange Equations Collinear Points Case

i)

| ^ ox1 —

for the Sun-Barycenter

=

k2x' +

=

k2y' + kkj-3/2y

+ k2 (Ex - Fz) -

k2z + kk-y-3/2z

+ k2Fy,

^ = dz> 4 | ^

of Motion: Halo and Triangular

=

=

k2x"+3kk1-3/2x'

k2y"+ 3kkj-3/2y'+ +k2-f-3/2(Ex-Fz)

A g

=

k2z"+3kk1-3/2z' + (2kF +

Problem,

kk1-3/2x-k2Ey-kkj~3/2xs,

-{2kE + kE)k^-3l2y — —7

Cases

+ (k2 +

k2Exs,

kk)1-3x-k2Eyl

- (k2 +

kk)j-3xs,

(k2 + kk)j-3y + k2Ex' -k2Fz' + (k2 + kk)1-3z

2kkj-3/2(Ex-Fz)

+

-(2kE+kE)k-y-3/2xs, + k2Fy'

kF)kj~3/2y.

ii) The following relations about Legendre polynomials will be used (see, for more details, Handbook of mathematical functions, M. Abramowitz and LA. Stegun). (a) P0(z) = 1 , ^ ( 2 ) = z,Pn(z) = 2-^zPn.x(z) - ^ P n _ 2 ( . ) forn = 2 , 3 , . . . (b) (z2 - l)P'n(z) = n[zPn{z) - P n _i(z)] for n = 1,2,... From (a) and (b) we obtain (c) nPn(z)-zPn{z) = -P^1{z). If we define Pn(z)=n\

f zPn-!(z) 5

Pn(z)\

for n = 1 , 2 , 3 . . .

then we have (d) Pn(z) =-P^iz) forn = 1,2,3,... It is easy to prove (e) Pn(z) - Pn-2{z) = - ( 2 n - 3)P„_ 2 (z). From (e) it follows easily (f) Pn{z) = sum\^\z + Ak- 2n)Pn-2k-2(z) brackets denote the integer part function. (f) is equivalent to

for n = 2 , 3 , . . . where the [ ]

Pn(z) = (3 - 2n)P n _ 2 (^) + (7 - 2n)P n _ 4 (*) + . . . + (-5)P 2 (*) + (-1)*U*), or Pnz) = (3 - 2n)P n _ 2 (z) + (7 - 2n)Pn-A(z) according to whether n is even or odd.

+ ... + (-7)P 3 (z) + (-3)Pi(z),

Equations of Motion for Perturbed Halo Orbits

247

We shall also use the following formulas:

«sS"" p -(;)=E«^' , -(;)' n>2

n>2

" n>2

n>2

and a similar expression for the derivative with respect to z. Q an (J) J^Yl^+l 0 1 „>l rA

an—2 ^ n ( c O s A i ) = {X - XA) E „ > 2 T^+T r A

^(cOsAj),

and similar expressions for the derivatives with respect to y and z. a cos A2

<

1 v-^ ^ a \ ™ r-, / ^i + — V — Pn(cosAi) r A *—*

' S4

n>l

\x r A I 2

= ~W-

+ (X~

X

A) Y

'SA

^ + 1 ^ ( 0 0 8 Ai),

>2

^

and, again, similar expressions for the derivatives with respect to y and z. Formula (g) follows from (b). From (c) and (d) the relations (h) and (j) follow. Finally (k) follows from (j). Using (g), (h), (j) and (k) it follows iii)

dx

= kk-y-3/2x' - k2Axs

+ k2Ey' + k2j~3x

+ Kk-1

+ k2(Ax + Dz) -

ncnan-lPn^

Y

(-) +

k2j-3xs

Kk~lxs

n>2

,n-2

+ jr*-s-Vs(*-*5)£j^pr P„(cos5i) n>2

+ Kk-1^-3

Y,

WVA

'SA

Ae{E,M,Pi,...,Pk,B}

dy

n-2

~f3- + (x-XA) Y

- k2Ex' + k2Fz' + k27-3y

= kh-^y'

+

n>2

JnTTP^COsA^

T

A

+ k2By

Kk-'yY^a^Pn^) n>2 n-2

+ IttrWsJ/E

ThnTT^icosSx)

n>2

+ Kk-1^3

XS\

Y. Ae{E,M,Plt...,Pk,B}

dL — = kk~f-3/2z' oz

^VA

n-2

v

^+{y-VA) Y ^+r^n(cosA!) SA

- k2Fy' + k2^~3z + k2Cz + k2Dx -

„ > , rA

k2Dxs

248

Perturbed Equations

of Motion: Halo and Triangular

Cases

n>2 an-2

+ Kk-^itgz^

m^PnicosSx) |X5

n>2

I

A€{E,M,Pi

J ^ + (z-ZA) r SA

Pk,B}

^ ^ 2

rn-flPnicOsAi) r

A

In short, from i) and iii) the Lagrange's equations for the motion in normalized coordinates with origin at Li for i = 1,2,3 are x" = ~2kk-1j-3/2x' - kk-li~3x + 2Ey' +(2kE + ktyk^-y^^y + kk'^^xs 3

*

V

+ Ax + Dz - Axs

/X\

n x ,

( - ) + Kk~3xs

+Kk~ + J2 ncna - Pn^ n>2

+Kk-31'3H{x-xs)

53 i „•>•>

+Kk-37-3

Y

\

. n+1 A,(cos5i) x s

<

*^W

Ae{B,M,Pi,-,Pk,B}

y" = ~2kk-1j-3/2yl 3/2

SA

L

n>2

- 2kk~1j~3/2(Ex

- kk^^y

rA

- Fz)

- Fz) - 2Ex' + 2Fz' + (2kE + A£0*- _11 7- 3 / 2 'XS

-~/- (Ex

+By + Kk~3y £ cnan-2Pn

Q

t>2

+Kk-31-3»§y £

\~\n+rp^cosS^

n>2

+Kk~31-3

*( A w

Yl

A£{E,M,Pi,-,Pk,B] x 3 2

z" = -2kk- ^ l z'

l

3

- kk- i~ z

r L

SA

„-^o TA

- 2Fy' - (2kF +

+Cz + Dx- Dxs + Kk~3z ] T cnan'2Pn

kF)k~l^-3l2y

(-)

n>2

+Kk-3r3^T,iz^n+ip^cosS^ n>2 I 1 5 1

+Kk~37-3

Y A£{E,M,Pi,---,Pk,B}

i{A)nA

- ^ + SA

(z-zA)Y^JrPn(cosA1) n>2

T

A

If the motion of the Earth-Moon barycenter around the Sun were circular then

Equations

249

of Motion for Perturbed Halo Orbits

in normalized coordinates the semimajor axis as would be 7 - 1 , the mean motion us = #and #7~ 3 / 2 = 1. As this motion is not far from circular, we have that e = ^ 7 ~ 3 / 2 - 1 is small. To generalize Richardson's formulation of the motion near Li (i = 1,2,3) for the circular restricted three-body problem, as given in Chapter 2, to the motion near Li for the Sun-Barycenter in the solar system we introduce the functions E

=

Ej3/2

-0 = [E-{l 2

+ e)]-f^, + e)2]13,

A

=

A^-9

= {A-(l

B

=

B73-62

= [ B - ( l + e) 2 ] 7 3 .

So, the equations of motion around Li for i = 1,2,3 are: x" - 2y' - (1 + 2c2)a; = £ ( n + l)cn+lanPn

Q

n>2 1

+C(0)^ncna"- P„_1(^) n>2

+C(l)a: + C{2)y + C(3)z + C(4)x' + C(5)j/' + C{7) +Kk-3j-3ii§(x-xs)

Y

n>2

+^fc-37-3

( c o s 5 l )

^TT^

^A^A

Y

Ae{E,M,Pi,---,Pk,B}

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

,n-2 r

SA

n

c ^ " " 2 ^ ( ^ ) + C(0)y Y

n>3

>2

r

A

cna"-2P„ ( | )

n>2

+ C ( l l ) z + C(12)y + C(13)z + C(14)z' + C(15)t/' + C(16)z' + C(17) -,n-2

+Kk-31-3Ȥy

Y n>2

+Kk-37~3

j^^Pnicos xs\

S1)

^A^A

Y,

A€{E,M,Pu--,Pk,B]

" + c2z = zY

-,71-2 r

SA

Cnan-2Pn ( ^ ) + C(0)z Y

n>3

n

cna"-2P„

>2

^

g )

n>2

+C(21)z + C(22)i/ + C(23)z + C(25)y' + C(26)z' + C(27) +M-37-3^^-^-TIP„(cos51) n>2

+Kk-37-3

Y

*(A)^

Ae{B,Af,Pi,...,P*,B}

^SA

,n-2

^ 3 - + (z-zA) 2 ^ ' SA

fn+1 >2 M

P„(cosAi)

250

Perturbed Equations of Motion: Halo and Triangular Cases

where C(0) C(l) C(2) (7(3) (7(4) C(5)

= = = = = =

Kk~3 - 1 , 2e + e2 4- A7-3 -- k k ' 1 ^ - 3 (2kk~ 1E + E)j--3/2 D, -2kk- - i 7 - 3 / 2 ) 2(e + S 7 - 3 / 2 ) ,

(7(7) (7(12) (7(13) (7(16) (7(23)

xsikk-1^-3 -A + Kk-3), 2 2e + e + J B 7 - 3 - f c A ; - 1 7 - 3 , ( 2 k - 1 F + F)7-3/2, 2F, C-Jbfc"^-3,

= = = = =

and (7(11) = -(7(2), (7(14) = -(7(5), (7(15) = (7(4), (7(17) = xsC(2), (7(21) = (7(3), (7(22) = -(7(13), (7(25) = -(7(16), (7(26) = (7(4), (7(27) = -xsC{3). 5.2.6

T/ie Collinear Points in the Earth-Moon Expressions for the Lagrangian

Problem:

Several

Using the inertial frame of reference as in 5.2.2, the equations of motion of the spacecraft are: B

=

-

GS(RS-R)

^

\RS-R\3

,

GA(RA-R)

^

,

\RA -R?

'

where 5 = —a/G, i.e. S takes into account the radiation pressure of the Sun. The position vector of the Earth RE satisfies

-

RE

GA(RA-RE)

J2 yie{s,M,Pi,...,pfc}

\RA-RE\3

'

Then if r = R — RE we have that the equations of motion of the spacecraft with respect to the ecliptic frame of reference with origin at the Earth are r

=

-gJE+Gffc**-E>+

\LES-r\

~ A e { s

^_

GA(LEAJZL_LEA

\\rEA~r\3

P k }

r3EA

where rEA = RA - RE. We denote by ajv/ and UM the semimajor axis and the mean motion of the Moon around the Earth. The mean motion takes into account the secular terms in the ecliptic argument of ascending node, of perihelion and in the mean anomaly, due to the effects of the Sun and planets. Then, using Kepler's third law we have that n a M M IS close to G(E + M). Let E be the modification of the Earth mass to have n2MaM = G{E + M). Therefore, we write the equations of motion in the form GEr r=-

GEr

GS(r -r)

ES + ™**s-L) +

[LES-'Q

^

J2

GA

. w e , , P . P.X Ae{S,M,Pu-,Pk}

/ rEA-r \\tEA

rEA\

-r-13

L\~

<EA.

251

Equations of Motion for Perturbed Halo Orbits

where E = E - E. These equations of motion are the Lagrange equations given by the Lagrangian 1. . r L=-r-r+ 2

GE GE GS + +1 r r r \rES - r\

£

+

LEA-L

GA '-BA

Ae{S,M,Pu-,Pk}

EA

which can be written as

2

"

"

'

r

r

£

fJ-A Ae{S,M,Pi,.,P,}

' ^ - = 1

lEA-TXL.EA-

' EA

where K = G(E + M),/iE = E(E+M)-\ns = SiE+M)-1 and / ^ = AiE + M)'1 for A e{S,M,Pu...,Pk}. Let if (i = 1,2,3) be located with respect to the Moon by the vector r_L. As in 5.2.3 the equation which defines Li is T-L Is 'EM

VMT-L _ /, ,, N ( LE , T-EM Is— - - U - VM) I T- 3 - + - 3 — 7 L \ E 'EM

and 7 = rL rEM for Lx and L 2 or 7 = r B r ^ for L3. Let p = r_ — r_Eh be the position vector of the spacecraft with respect to the collinear point Lf, i.e. the position of the spacecraft in the ecliptic frame of reference with origin at Lj. In a similar way to 5.2.4 we denote the positions of the Sun, Earth, Moon, a planet P, and the Earth-Moon bary center with respect to L; by Ls>LE>LM'LPi and r B respectively. If we replace r by p — r_E the Lagrangian becomes

L

=

2 ^ ' ^ ~ £'^E

+

(1 - HM)-,

K

+K

\LE

r T

-S

-D\ P\

P\

+ /*M

£

^

A€{S,Pi,..,ft}

?-EM ' P

\ZM - P\

'EM "LEA

KlLA

P\

•P

' EA

where we omit in the Lagrangian the terms which only depend on the time t. The Lagrangian can be written in the form L = -p-p-p-rE

LE'P

+ K (1-PM)

1

, ,

\HM-P\

\LE~P\ ,1

+K

,LE-P

,

(LM'P

'M

+

Kp,E '-E

P\

+ fracKp,§\rs-

p\+K

LM-P

'M

LEM-P

'EM

^ A€{S,Pu...,Pk}

LEA'P

p,A ( \LA

- P\

1

EA

252

Perturbed Equations of Motion: Halo and Triangular

Cases

Using the definition of Li we have /,

^E-P

,

[l-HM)——

(LM-P

LEM-P\

+P-M I —3

'E

\

LE-P

-g

'M

'EM

I = -3 /

'EM

For more details see 5.2.4. Hence, the Lagrangian becomes 1

L =

L-E KL +

E'E

K +

< EM

P-E

r

P\

E

K +

P-s

(

/

1

LM-P

(

' " ^{5^...,Pfc} ^

r

P\

WL-M

y^

+ K

\LE - P\ ' \Ls ~P\

,

£E-£\

-p-p-p-rE+K

M

!

^A'£

V \LA ~ P\

r%A

As in the Sun-Barycenter problem we consider now a frame of reference with origin at the equilibrium points Li (i = 1,2,3) defined by

TEM

\LEM^LEM\

The change of variables form the ecliptic system with origin at Li, p, to the system with origin at Li, a, defined by e 1 ,e 2 ,e 3 , is given by p = kC a, where k is a scaling factor defined by k = <

.

{ rE

-

JTEM

for

' ' 1 = 3.

The matrix C is computed as in 5.2.4 but the values of the longitude and latitude of the Moon should be used. In this new system, units of mass, time and distance are chosen in the following way: - The unit of distance is taken as 7 times the semimajor axis OM- The new unit of time s is the time t in Modified Julian Days scaled by 7 s ' 2 , i.e. s — 7 3 / 2 i. - The unit of mass is chosen in such a way that K = G(E + M) = n2MaM = 1, where UM is the mean motion. Again, system a = (x,y,z) will be called the normalized system centered at Li for i = 1,2,3. The Lagrangian in the normalized system becomes

2 LM • P

+MM M

+Ky

-Pi y\

. ^

A€{S,PU-,Pk}

'

M

K~f-3rE

•p

'EM

.MET^—ZT)

Kj-3nE \LE

- p\

Kj-3fi§

ks - P\

253

Equations of Motion for Perturbed Halo Orbits

where p = kCa and r_A = kCf_A. Note that rE = (XE,0,0),ZB = (xj5,0,0), where XE = 1 - 7 _ 1 , or XE = _1 — 1 - 7 , or XE = 1 and XB = 1, or XB = —1, or XB = 1 + 7 - 1 , according to the normalized system centered at Li,I<2 or L3 that we consider. Using the notation of 5.2.4, the Lagrangian in normalized coordinates goes over to L

=

l{k2(x'2+yl2+z'2)

+ 2kk1-^2(xx'+yy'

F(yz' - zy')] + k2j-3(x2 -k2j~3xEx

cnanPn

- yx1)

+ y2 + z2) + k2(Ax2 + By2 + Cz2 + 2Dxz)}

- kk-y~3/2xEx'

+Kk~1 £

+ zz')+2k2[E(xyl

- k2(AxEx

( ^ ) + Kk-'xEX

+ DxEz)

-

k2ExEy'

+ ^ Q ^

£

{j~)

p

n(^sE1)

n>2

* £ * £ &'•<«•*> +Kk-^-3

Yl

a cos Ao

VA

' EA

Ae{s,Pi,...,pfc}

5.2.7

The Lagrange Equations linear Points

1 v^ + TA — " } \TA

for the Earth-Moon

Pn(cosAi)

Problem:

Col-

In a similar way to 5.2.5 we obtain x" - 2y' - (1 + 2c2)x = J2(n + l)cn+1anPn

Q

+ (7(0) £

n>2

nCna^Pn-!

n>2

+ C ( l ) x + (7(2)2/ + C(S)z + C(4)x' + C(5)y' + C(7) an~2 +Kk-3j-3fiE(x - xE) ^ I \Z+iP*(<x>*Ei) n>2 n-1

+Kk

3

3 7

fl§(x

-

XS)Y,^nTlPn{cOsS1)

>2rS

+Kk~31~3

Y,

M

Ae{s,Pi,...,pfc}

cnan-2Pn

y" + 2x' - (1 - c2)y = y £

XEA 1

n-2

+ (X- XA) Y

Q

+ C(0)y £

n>3

+C(ll)x

T^+iPnicOsAi)

n>2 TA

EA

cnan-2Pn

(^)

n>2

+ C(12)y + C{lZ)z + C(U)x' + C(15)y' + C(l6)z' + C(17)

+Kk~3j-3nEy

,n-2

Y

I a |n+i^n(cos£;i)

n>2

|:rB|

Q

254

Perturbed Equations of Motion: Halo and Triangular Cases n-2

+Kk-3j-3n§(y

- ys) ^

^ + r ^ n ( c o s Si) r

n>2

+Kk-3-/-3

Yl

S

AM

Cnan-2Pn

(y - VA) Y, T^+rPnicosAi)

?EA

Ae{s,pu...,pk}

" + c2z = zJ2

n-2

VEA

(J) + C(0)z J2 cnan-2Pn

n>3

T

n>2

A

(l)

n>2

+C(21)x + C{22)y + C{23)z + C{25)y' + C(26)z' + C(27) ,n-2

+Kk-37'3^Ez

Y

r—r^PnicosEr) XE\

n>2

n-2

+Kk-31~3fi§(z

- zs) ^

T^r^n(cos5i)

>2rS

+Kk~31-3

]T

VA

Ae{s,Pi,...,p<:}

n-2

ZEA

' EA

+ (z- zA) J2 T^+rP«(C0S^i) ^>2

V

A

where C ( 0 ) , . . . , C(27) are denned as in 5.2.5.

5.3

Equations of Motion Near the Triangular Points

5.3.1

The Triangular

Points for the Sun-Bary

center

Problem

The triangular point spacecraft geometry for the Sun and the Earth-Moon barycenter is illustrated in Figures 5.2.

Spacecraft

S (Sun)

S (Sun)

P

-SB

-SB

B (Barycenter)

B (Barycenter) Spacecraft

Fig. 5.2 The geometry for the Sun-Barycenter triangular points.

Equations

of Motion Near the Triangular

255

Points

The points L4 and L5 are such that Li (for i — 4 and 5), the Sun and the Earth-Moon barycenter form an equilateral triangle. Let p — r - r_sLi ^>e t n e position vector of the spacecraft in the ecliptic frame of reference with origin at Lt for i = 4,5. In a similar way to 5.2.4 the Lagrangian becomes L

=

-p-p-p-rs I- - -

+K

+ K(l-pB+iis)-

^

\Ls ~ £1

/HkT^f""^!"

A€{E,M,Pi,-,Pk}

-

where we use the notation of 5.2.2. 5.3.2

The Lagrangian L4 or L5

in Normalized

Coordinates

with

Origin

at

As in the collinear points we consider a frame of reference with origin at L4 or L5 defined by the unitary vectors: _e,1

= =^-, rSB'

e, = f£2—^^^ \rSBArSB\

and

e2 = e,3 A e,. ~ ~ -1

The change of variables from the ecliptic system with origin at Li, p, to the system with origin at Li, a, defined by e ^ g j , ^ , is given by p = kC a, where k = TSB is a scaling factor. Here, we use the notation of 5.2.4. In this new system units of distance, time and mass are chosen as in 5.2.4. The new Lagrangian has the same expression as that in 5.2.4 with p = kCa, and r_A = kCf_A. Again the system a = (x,y,z) will be called the normalized system centered at Li for i = 4,5. For L4 we have rs = (xs,ys,zs)

= ( - ^ - ^ . O J ,LB = ( | > - ^ r > ° ) a n d r S B =

(1,0,0); and for L 5 these vectors have the same expression but changing — - ^ by 2

•

We shall need the following relations to develop the Lagrangian in the normalized system: i) The expression for p • p is the same as that given in v) of 5.2.4. ii) Using iv) of 5.2.4 we have P'is

=

k2a-rs+

=

k2(xsx 2

kM-rs+

k2aTCTCts

+ ysy) + kk(xsx

+k (-Eysx

+ Exsy

+

-

k2&CTCts

+ ysy) + k2(Axsx Fysz).

+ Bysy + xsz)

256

Perturbed Equations

of Motion: Halo and Triangular

Cases

In a similar way to 5.2.4 the Lagrangian in the normalized system becomes L = - ik2(x2

+ y2 + z2) + 2kk(xx + yy + zz) + 2k2[E{xy - yx)

+F(yz - zy)) + k2(x2 + y2 + z2) + k2(Ax2 + By2 + Cz2 + 2Dxz)} -k2(xsx

+ ysy) - k2(Axsx

+ ysy) - kk(xsx

2

+ Bysy +

l

-k (-Eysx

+ Exsy + Fysz)+Kk- {\-^B+^§)

Y

Dxsz)

a"F„(cos5i)

n>l

+Kk~l

Y,

f*

a cos A2

5.3.3

+

'SA

AE{E,M,Pu--,Pk}

The Lagrange Equations Triangular Case

i-T(~yPn(cOsA1) rA^\rA)

for the Sun-Barycenter

Problem,

We need some preliminary computations i)

—ox r\ T—ay — oz £8L dt dx

=

=

k2x + kkx - k2Ey - kkxs +

=

k2y + kky +

=

k2z + kkz + k2Fy -

k2(Ex-Fz)-k2Exs-kkys,

=

k2Fys,

k2x + Zkkx + (k2 + kk)x - k2Ey - (2kE + kE)ky -{k2 + kk)xs + (2kE +

dt dy

kE)kys,

k2y + 3kky + (k2 + kk)y + 2kk(Ex - Fz) + k2 (Ex - Fz) +k2E± - k2Fz - (2kE + kE)kxs

ddL —^ dt oz

=

k2Eys,

- (k2 + k'k)ys,

k2z + Zkkz + (k2 + kk)z + k2Fy + (2kF + kF)ky -(2kF +

kF)kys.

Using (j) and (k) of 5.2.4 it follows that r\ y

ii) — ox

= kkx + k2Ey + k2x + k2(Ax + Dz) - k2xs +Kk~1{\

-HB+

Vs)(x ~ xs) Y

-k2Axs

a" _2 -Pn(cos5i)

n>2

+Kk~1

Y

A€{E,M,Pi,---,Pk}

dy

»A

n-1

- ^

<

+ (X- XA) Y

'SA

= kky - k2Ex + k2Fz + k2y + k2By - klys

>2

-

^+TPn(COsAl)

r

A

kzBys

Equations of Motion Near the Triangular

+Kk~l(l

-fiB+

Points

257

o n - 2 P„(cos5i)

n§)(y - ys) Y i>2

+Kk~l

Yl

USA

PA

n-2

?SA

Ae{E,M,Pu---,Pk}

V

n>2

dL ^ = kkz - k2Fy + kz + k2Cz + k2Dx dz +Kk~l(l

-^+Tpn(cosA1)

+ (y~ VA) Y A

k2Dxs

Vs)z ] C o n_2 -Pn(cos5i)

-HB+

n>2

+Kk~X

J2

M

,n-2

ZSA

+ (Z- ZA) Y

' SA

Ae{E,M,Pi,-,Ph}

-^+iPn(cOsA1) T

n>2

A

In short, from i) and ii) the Lagrange equations for the motion in normalized coordinates with origin at Li for i = 4,5 are: x = -2kk~1x

- kk~lx + 2Ey + (2kE + kfyk^y 1

-Axs-(2kE+kE)k- ys

+ kk^xs

+ Ax + Dz

3

Kk- (l-fiB+Vs)(x-xs)J2an~2pn(cosS^

+

n>2

+Kk~3

J^

^

Ae{E,M,Pu...,Pk}

y = -2kk-xy

- kk~ly - 2kk~l(Ex

+ (2kE + k^k^xs 3

+Kk~ (l

XSA 3

' rSA r

n-2

+ (X- XA) Y

-^+iPn(cOsA1)

x>2TA

- Fz) - (Ex + Fz) - 2Ex + 2Fz

+ By + kk^ys

-

Bys

n 2

- / i B + p§)(y - ys) Y, a - P„(cOs5!) n>2

+Kk~3

Y

^

Ae{s,M,Pi,...,p f c }

z = -2kk~lz

VSA 3 "frSA

n-2

+ (y -

n>2

- kk~lz - 2Fy - (2kF + kF)k~ly

+(2kF + kF)k-1ys

+ Kk~3(l

Y

VA)

T

_An + i Pn(cosAi) + Cz + DxDxs a

- MB + Hs> £

"~ 2 -P«(cosSi)

n>2

+Kk~3

Y A€{E,M,Pu-,Pk}

PA

ZSA 3 rr SA

-,n-2

-^+TPn(cOsA1)

+ {Z ~ ZA) Y i>2

T

A

In a similar way to the collinear points, we define e = 6-l,E = E-(l + e),A = A— (1 + e)2 and B = B - (1 + e)2. So, the equations of motion around L 4 or L5 are: x-2y-x

= C(l)x + C(2)y + C(3)z + C(4)x + C(5)y + C(7)

258

Perturbed Equations

+Kk~3{l

of Motion: Halo and Triangular

V§)(x ~ xs) Y

-HB+

Cases

o n - 2 -Pn(cosSi)

n>2

+Kk~3

Y

XSA

, ( - ^ 3 — + {X' SA

^

A£{E,M,Pi,-,Pk}

y + 2x-y

n-2

\ S ^ a i> r A ^ XA) 2 ^ ^T+ T - P n ( c O S Ax) >2 A

= C(U)x + C{l2)y + C(13)z + C(U)x + C(15)y + C(16)i tffc-3(l

+C(17) +

- ^ B + / i s ) ( y ~ Vs) Y ,

«n"2^n(cOs5!)

n>2

+Kk~3

J2

M

Ae{E,M,Pu-,Pk}

n-2 - f r 1 + (2/ - I/A) Y 'SA n>2

^nTTPnicOsAi) T

A

z = C(21)x + C(22)y + C(23)z + C(25)y + C(26)i + C(27) +Kk~3(l

Ȥ)(z ~ zs) Y

-HB+

«"~ 2 -Pn(cos5i)

n>2 n-2

+^fc"3

£

" 3 d + (* - **) H ^+r^«(C0S ^i)

HA

'S4

Ae{E,M,Pi,--;Pk}

n>2

r

^

where C ( l ) , . . . , C(27) are defined as in 5.2.5 except C(7)

=

( k - 1 - A ) a ; 5 - ( 2 k - 1 J E + i;)i/ s ,

C(17)

=

( 2 k - 1 E + £)a; s + (fcfc- 1 - B)ys,

C(27) 5.3.4

=

The Equations

and

l

- £ > z s + {2kk~ F + F)ys. in the Earth-Moon

Problem

We proceed as in 5.3.1 to 5.3.3. The only differences are that the coherence term refers to the Earth and not to the Sun, but the radiation pressure term is still present. The units are chosen in a similar way and in the matrix C the longitude and latitude of the Moon should be used. We obtain x-2y-x

= C(\)x + C(2)y + C(3)z + C(4)x + C(5)y + C(7) +Kk~3(l

-HM+

VE)(X ~ xE) Y

an_2P„(cosEi)

n>2 n-2 3

Pn cosS H£,(x-xs)Y^nTT ( ^ V

+Kk

>2 S

+Kk~3

Y A€{S1,P1

y + 2x-y

= C(ll)x

VA Pk,M)

XEA r

EA

n-2

+ {X- XA) Y n>2

-^+iPn(cOsA1) r

A

+ C{12)y + C(13)z + C{U)x + C(15)y + C(16)i

Numerical

259

Tests of the Equations of Motion

+C(17) + Kk~3(l -HM + HE)(y - VE) Y

an~2Pn(™*Ei)

n>2 an-2

+Kk-3n§(y - vs) Y ^+r^"( cos5 i) r

n>2

+Kk~3

S

Y Ae{S,P!

VEA

VA

"EA

Pk,M}

2 a"~ "n~2 P C0S A + (V- VA) n~>2 Y ^nTT n( l) r A

z = C(2l)x + C(22)y + C(23)z + C(25)y + C(26)i + C(27) +Kk~3{l

-HM + HE)Z J2

an~2Pn(casEi)

n>2

+Kk

nn-2

3

P cosS Hs(z-zs)Y,-^+T "( i) r

n>2

+Kk~3

S

Y, Ae{Si,Pi,...,P*,M}

»A

ZEA

,n-2

+ (Z~ ZA) Y

EA

n>2

^nTTPn(COsAl)

r

A

where C ( l ) , . . . , C(27) are defined as in 5.3.3.

5.4 5.4.1

Numerical Tests of the Equations of Motion Description

of a Program

to Check the Equations

of

Motion

Since several changes of coordinates, rearrangements and recombinations of terms have been done to obtain the final equations from the Newton ones, we have made a program, ET, in order to check the final equations. The test is done comparing the acceleration of the spacecraft, computed directly in normalized coordinates and time from the final resulting equations, with the acceleration computed using Newton's equations and the suitable changes of coordinates and time to go from the ecliptic to the normalized system. The tests are performed, either at a point given by the user or, for the L\ and L2 cases, at a prefixed number of points in a halo orbit of given ^-amplitude. All of them are done at a given number of equally spaced epochs. It is convenient to introduce a correction due to the lack of coherence of the analytic model of the solar system that we are using. When this is done, the errors found between the two accelerations are very small and due to rounding errors. We remark that for the Li,2 cases in the Sun-Barycenter system the normalized unit of acceleration is, roughly, 0.06 mm/s 2 ; for the ^3,4,5 cases is close to 6 mm/s 2 . For the Earth-Moon system the figures are 0.5 mm/s 2 and 3 mm/s 2 , respectively. Given a point in the phase space, in normalized coordinates, it is changed to ecliptic ones, Newton's equations are used to compute the acceleration, and we go back to normalized coordinates. From the other side, the ecliptic coordinates

260

Perturbed Equations of Motion: Halo and Triangular

Cases

of the bodies of the solar system with respect to the center of masses of the own system, are changed to normalized coordinates by routine EFEM. To compute the coefficients C ( 0 ) , . . . , C(27) used in the final equations, several auxiliary parameters are needed. They are the elements and derivatives of the orthogonal matrix used in the transformations, the scaling factor k and its first and second derivatives, the longitude and latitude of the primary with respect to the secondary as well as its first and second derivatives, etc. This is done in routine PARAM. Furthermore the coefficients Cn are required. The computation of them is done in routine CEN. The routines PSUM, PBSUM and PBPSUM are in charge of the computation of the diverse series involved in the equations. These series have also been checked against the direct value before expansion.

5.4.2

Numerical

Results

At some preliminary step partial checks were done, taking only into account some of the perturbations. The runs that we present here are done with the full solar system, including radiation pressure. Runs have been done for cases L\ to L$ in the Sun-Barycenter and Earth-Moon problems. In all the cases, at all the epochs and points the residues obtained are less than 1 0 - 1 0 . They are due to rounding errors. It is important to state that if the terms accounting for the lack of coherence of our model of the solar system are not included, the errors are much larger. They are of the order 1 0 - 6 in the Sun-Barycenter problems and 10~ 3 in the Earth-Moon problem. We refer here to L\ and L?- A very small sample of results is given below in Tables 5.6 and 5.7. In these tables the different codes P, V, N, A and D denote POSITION, VELOCITY, NUMERICAL ACCELERATION, ANALYTIC ACCELERATION and DIFFERENCES, respectively.

Numerical

Tests of the Equations

of Motion

261

CURRENT VALUE OF SPACECRAFT SECTION IN m**2 DIVIDED BY MASS IN kg: O.Ol THE MODEL OF THE SOLAR SYSTEM IS: 1 1 8 1 1 1 1 1 1 1 11 1 0 0 0 0 EARTH-MOON SYSTEM THE EQUILIBRIUM POINT IS L2: Z-AMPLITUDE OF THE HALO ORBIT = 0.08 CORRECTION DUE TO LACK OF COHERENCE OF THE MOTION OF THE PRIMARY INCLUDED INITIAL JD = 17762 N. OF EPOCHS = 6 STEP IN DAYS = 200 N. OF POINTS = 1

EPOCH

17762.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES 6.80150562329442E-02 -4. 89668060246066E-15 -1.29942557663951E-16 1.065243759575357 -0.40529129162314 -7. 90464251723612E-02 -0.40529129167518 -7. 90464251772684E-02 5.20398932279952E-11 4.90725862722918E-12 EPOCH 17962.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES 6.80150562329440E-02 P -0.2159341817654990 7. 42655970388849E-15 V -8.76033457265997E-15 1.065243759575337 2.05400362559781E-16 N -3 30754429192645E-02 0.1473989936776778 -0.3477346086349386 -3 30754429205688E-02 A 0.1473989936149495 -0.3477346086617204 D 6.27282947807117E-11 1 30436113299214E-12 2.67817434895789E-11 EPOCH = 18162.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES -0.2159341817655137 -3 82424542044651E-15 P 6.80150562329441E-02 V -4.47907113617948E-15 1.065243759575353 3.19971679733317E-17 7.60786012686369E-02 N 0.1310570269415729 -0.4130982600113180 7.60786012429490E-02 0.1310570269433896 A -0.4130982600218986 D 2.56879136018017E-11 -1 81676895749660E-12 1.05806127748131E-11 EPOCH = 18362.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES -0.2159341817655041 2 51563683797158E-14 P 6.80150562329428E-02 .74630755865822E-14 V 2 1.065243759575342 1.27684315692931E-17 N -1.13775160292638E-02 -1 63318094311868E-02 -0.4691788244725444 A -1.13775159992819E-02 -1 63318094309396E-02 -0.4691788244605368 D -2.99819485644753E-11 -2 47167301984996E-13 -1.20076032450455E-11 EPOCH = 18562.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES -0.2159341817655028 -1 15578086947763E-14 P 0.80150562329440E-02 V -1.03475140800010E-14 1.065243759575344 -8.73253290913403E-17 N 7.51593005432500E-02 -8 27608060370502E-02 -0.4111263652511284 A 7.51593006293423E-02 -8 27608060342255E-02 -0.4111263652193898 D -7.70922700676912E-11 -2 82474911517116E-12 -3.17385909220924E-11 EPOCH = 18762.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES -0.2159341817655023 -2 81266433003390E-15 P 6.80150562329447E-02 V -2.07029382735476E-15 1.065243759575341 2.34847461726517E-17 0.1528921771910376 9 98142426550300E-04 N -0.3395129837640657 0.1528921772942718 9 98142426183967E-04 A -0.3395129837204182 3 66333157385267E-13 D -1.03234122639239E-10 -4.36474745235670E-11

P V N A D

-0 21593418176551E+00 -6 85507705049992E-15 7 61041991265082E-02 7 61041989988855E-02 1 27622657233872E-10

Table 5.6

Output of program ET. Case Li Earth-Moon.

262

Perturbed Equations

of Motion: Halo and Triangular

Cases

CURRENT VALUE OF SPACECRAFT SECTION IN m**2 DIVIDED BY MASS IN kg: O.Ol THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 1 1 1 1 1 1 1 1 1 0 0 0 0 EARTH+MOON-SUN SYSTEM THE EQUILIBRIUM POINT IS LI: Z-AMPLITUDE OF THE HALO ORBIT = 0.08 CORRECTION DUE TO LACK OF COHERENCE OF THE MOTION OF THE PRIMARY INCLUDED INITIAL JD = 18262 N. OF EPOCHS = 1 STEP IN DAYS = 0 N. OF POINTS = 5 EPOCH = 18262.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P 1.01963550684844E-02 0.4207314348755143 4.06163748780382E-02 V 0.3214430737628846 0.3052075793433056 -0.1446380209754839 N 2 .24145798465631E-02 -1.749426993364965 -0.1485542825383776 A 2 .24145798465702E-02 -1.749426993364973 -0.1485542825383783 D -7 .03170160987198E-15 7. 63278329429795E-15 6.90419943438769E-16 POINT NUMBER 2 ADIMENSIONAL LI COORDINATES P 0.1526361820666371 0.2757152978251848 -5.26175334267068E-02 V 9 .17755513673964E-02 0.7507366206869046 -0.1171431936735233 -0.4389373314119561 -1.425579946758043 N 0.2949045155888308 -0.4389373314119655 -1.425579946758043 A 0.2949045155888305 D 9 .48546796664118E-15 -1.66533453693773E-16 2.77555756156289E-16 POINT NUMBER 3 ADIMENSIONAL LI COORDINATES P 0.2757152978251854 0.1526361820666371 -5.26175334267068E-02 V -9 .17755513673951E-02 0.7507366206869057 0.1171431936735233 N -0.4492821262472033 1.430221653225935 0.2932566205529722 -0.4492821262472038 A 1.430221653225933 0.2932566205529722 D 4 .92661467177413E-16 85962356624713E-15 2.08166817117216E-17 POINT NUMBER 4 ADIMENSIONAL LI COORDINATES P 1.01963550684844E-02 0.4207314348755148 4.06163748780382E-02 -0.3214430737628843 0.3052075793433049 V 0.1446380209754839 N 1.48800869148124E-02 1.753602270330053 -0.1482554992488390 A 1.48800869148223E-02 1.753602270330057 -0.1482554992488395 D -9 .90939101608301E-15 -4. 10782519111307E-15 5.20417042793042E-16 POINT NUMBER 5 ADIMENSIONAL LI COORDINATES -0.1146312230589805 -8. 53459468722368E-17 8.94768437229016E-02 P 0.8952570170338835 -1.18858739276885E-17 V 2 .25395069657320E-16 -0.3026475201019374 N 0.8419834962407535 4. 43706976138599E-03 -0.3026475201019377 A 0.8419834962407530 4. 43706976138503E-03 2.70616862252381E-16 9. 63205210036122E-16 D 5 .27355936696949E-16 THE EQUILIBRIUM POINT IS LI: Z-AMPLITUDE OF THE HALO ORBIT = 0 . 0 8 CORRECTION DUE TO LACK OF COHERENCE OF THE MOTION OF THE PRIMARY INCLUDED INITIAL JD = 17262 N. OF EPOCHS = 11 STEP IN DAYS = 200 N. OF POINTS = 1 17262.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES EPOCH 8.94768437229016E-02 2.74953756902971E-16 P -0.1146312230589860 2.30803833616518E-18 0.8952570170338844 5.63025049309300E-17 -2.50719695170269E-02 -0.2862592459622015 0.8308703050994398 -2.50719695077002E-02 -0.2862592456987819 0.8308703051573733 -9.32665273137289E-12 -2.63419529245023E-10 5.79335052153240E-11 EPOCH = 17462.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589833 4.79592285373015E-16 8.94768437229016E-02 V 2.59528085298067E-16 0.8952570170338813 2.30048358410650E-18

Numerical

Tests of the Equations of Motion

263

N 0.8331313228719657 2.98100741054282E-02 -0.2912781553774492 A 0.8331313229243336 2.98100740940855E-02 -0.2912781551630238 D -5.23678600483634E-11 1.13427544266853E-11 -2.14425345534952E-10 EPOCH = 17662.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589833 -5.09735580226378E-16 8.94768437229016E-02 V -9.23336967466776E-16 0.8952570170338840 2.28640150395261E-18 N 0.8329992924983034 -2.05462122161357E-02 -0.2781748804360803 A 0.8329992925229601 -2.05462122113091E-02 -0.2781748802824957 D -2.46567072315073E-11 -4.82661957276753E-12 -1.53584596429556E-10 EPOCH = 17862.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589847 -5.90588671457213E-16 8.94768437229016E-02 V 7.75592950515096E-16 0.8952570170338846 2.28243545575292E-18 N 0.8304210393573760 1.87948340884625E-02 -0.2990810865536457 A 0.8304210393900451 1.87948340850099E-02 -0.2990810864435596 D -3.26691312890758E-11 3.45262490472619E-12 -1.10086044446955E-10 EPOCH = 18062.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589860 2.94093210596822E-16 8.94768437229016E-02 V 7.11540099395429E-16 0.8952570170338793 2.27142097150020E-18 N 0.8268437578119842 -6.16931531266036E-03 -0.2727670831562782 A 0.8268437578194670 -6.16931531243319E-03 -0.2727670831060759 D -7.48279216367109E-12 -2.27164424423942E-13 -5.02022451387418E-11 EPOCH = 18262.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589805 -8.53459468722368E-17 8.94768437229016E-02 V -6.18150899553124E-17 0.8952570170338835 2.26843787731140E-18 N 0.8419834962407535 4.43706976138651E-03 -0.3026475201019374 A 0.8419834962407530 4.43706976138562E-03 -0.3026475201019377 D 5.27355936696949E-16 8.90997345348587E-16 2.70616862252381E-16 EPOCH = 18462.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589847 7.48273188697798E-16 8.94768437229016E-02 V 2.37574376417046E-15 0.8952570170338838 2.26916030248766E-18 N 0.8209302079794383 7.25815380880589E-03 -0.2721766984006765 A 0.8209302079709893 7.25815380950265E-03 -0.2721766984507671 D 8.44903313979017E-12 -6.96757213557164E-13 5.00906122136157E-11 EPOCH = 18662.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589888 3.05148590961130E-16 8.94768437229016E-S2 V -1.18722328985158E-15 0.8952570170338839 2.24187393331493E-18 N 0.8382088467796591 -1.14672907855469E-02 -0.3001709455383223 A 0.8382088467491298 -1.14672907867862E-02 -0.3001709456488073 D 3.05293151869889E-11 1.23934521499557E-12 1.10484982274172E-10 EPOCH = 18862.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589860 -2.34175732267714E-16 8.94768437229016E-02 V 1.27601414216911E-15 0.8952570170338839 2.25034958121785E-18 N 0.8282800052808199 2.30946004210394E-02 -0.2757371961949853 A 0.8282800052548035 2.30946004278268E-02 -0.2757371963472253 D 2.60163834919779E-11 -6.78737101245108E-12 1.52239949813282E-10 EPOCH = 19062.00 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589819 -2.20981602724861E-16 8.94768437229016E-02 V 1.50517948179768E-16 0.8952570170338781 2.21094874860764E-18 N 0.8354124160884162 -2.62289319166143E-02 -0.2938122741162987 A 0.8354124160355899 -2.62289319256993E-02 -0.2938122743325946

264

Perturbed Equations

of Motion: Halo and Triangular

Cases

D 5.28263266463824E-11 9.08497929663631E-12 2.16295883981310E-10 EPOCH = 1 9 2 6 2 . 0 0 POINT NUMBER 1 ADIMENSIONAL LI COORDINATES P -0.1146312230589819 1.11845433578585E-16 8.94768437229016E-02 V 4.17581764478418E-17 0.8952570170338843 2.24009665508986E-18 N 0.8294114270383331 3.24679577073166E-02 -0.2845171364220191 A 0.8294114269818352 3.24679577236557E-02 -0.2845171366838316 D 5.64979174555446E-11 -1.63391166915771E-11 2.61812502233560E-10 Table 5.7

5.5

Output of program ET. Case L\ Earth+Moon-Sun.

References [1] R. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin, 1978. [2] American Ephemeris and Nautical Almanac. U.S. Government Printing Office. Washington D.C., 1984. [3] V. Arnold. Les Methodes Mathematiques de la Mecdnique Classique. Ed. Mir, 1967. [4] A.E. Brigham. The Fast Fourier Transform. Prentice Hall, 1974. [5] E.W. Brown. " Motion of the Moon". Memoirs of the Royal Astronomical Society, 57, 129-145, 1905. [6] Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. Her Majesty's Stationary Office, London, 1961. [7] R.W. Farquhar. "The control and use of libration-point satellites". NASA TR R-346(1976). [8] E. Isaacson and H.B. Keller. Analysis of Numerical Methods. John Wiley & Sons, 1966. [9] D.L. Richardson. "A note on a Lagrangian formulation for motion about the collinear points". Cel. Mech., 22, 231-236, 1980. [10] V. Szebehely. Theory of Orbits. Academic Press, 1967.

Chapter 6

Expansions Required for the Equations of Motion. Collinear Points Case

The final equations of motion, as they are given along Chapter 5, require some preparation to be used for the computation of the analytic quasi-periodic solutions of the motion. The essential tasks to do are expansions and evaluation of coefficients and frequencies. First of all there is a not difficult task: to compute the coefficients C(i), i — 0 , . . . , 27, which appear in the differential equations. For the Sun-Barycenter problem this is done using the expansions of the elliptic motion in terms of the mean anomaly and adding the periodic terms coming from the planetary perturbations. For the Earth-Moon problem the task is more involved. Any attempt to expand the perturbing forces as functions of the anomalies which appear in Brown's theory of the Moon, generates an enormous number of terms. We use a different approach. This approach has also been used for the planetary terms, both in the Sun-Barycenter and in the Earth-Moon problems, for the effects of the Earth and Moon instead of the Barycenter, in the Sun-Barycenter, L\ and L2, near halo orbits, and for effects of Sun (and radiation pressure) in both problems. The perturbing functions depend on time, through the position of the celestial bodies, and on the coordinates, x,y,z, of the spacecraft in the normalized system. The procedure that we have used is to expand the perturbing functions as power series in x,y,z where the coefficients are well-known functions of the coordinates of the perturbing bodies. Instead of trying to develop each of these functions in terms of the elements (or parameters) of the orbit, we have done a Fourier analysis of them. This analysis has been split into several parts: An exploration to look for the important functions, a Fast Fourier Analysis to detect frequencies and order of magnitude of the coefficients, and a refined Fourier Analysis by classical methods of the different harmonics. As the frequencies should be linear combinations of the frequencies of the anomalies appearing in the position of the bodies, it is easy to recover the exact frequency when an approximation is available. The effect of slowly varying anomalies can be neglected. Finally, the terms are converted to a suitable form to be read by program QPO to be explained in the next chapter. 265

266

Expansions

6.1

Required for the Equations

of Motion.

Collinear Points

Case

Analytic Expansion of the Coefficients Due to the Noncircular Motion of the Earth—Moon Barycenter

6.1.1

The Coefficients

Appearing

in the Equations

of

Motion

Several functions are involved in the right-hand side of the equations of motion in normalized coordinates centered at Li for i — 1,2,3, see Chapter 5, 5.2.5, which need to be developed in a suitable way, in order to obtain their analytic solution up to the convenient order. We denote by e, M, CJ and n the eccentricity, the mean anomaly, the derivative of the argument of the perihelion and the mean motion of the orbit of the EarthMoon barycenter around the Sun. E r , E#, and T,s denote the periodic perturbations due to the effect of the planets on the radius vector, longitude and latitude of the Earth-Moon barycenter according to Newcomb's theory. Then, if we develop the functions C'$ which appear in the equations of motion we obtain C(0)

=

C(l)

=

3 / 9 \ 9 53 - e 2 + 3e 1 + - e 2 cos M + - e 2 cos 2M + — e 3 cos 3M 2 \ o J 2 8 - 3 ( l + 3ecosM)E r , - e 2 + e (4n - n 2 + — e2 ) cos M + - e 2 cos 2M + — e 3 cos 3M 2 \ 8 y 2 8 +2(1 + 2ecosM)S e - 2e sin Mtr

-

tr,

C(2)

=

2en(l - n) sin M + 2(1 + 2e cos M)tr

C(3)

=

- ( 1 + 4ecosM)Y,S - 2 e s i n M t s -

C(4)

=

- e | 2n - - e 2 ) sin M - 3e2 sin 2M

(7(5)

=

2e ( 2 - e— j ncosM + 5e 2 cos2M + — e 3 cos3M + 2±0,

C(7)

-

xs

+ 2e sin Mt$ ts, 17

+ t8,

e3sin3M-2Sr

e ( n - 3 ) ( n - l ) c o s M - 3(1 + 3 e c o s M ) S r + 2 e s i n M S r

- 2 ( l + 2ecosM)£fl + II r C(12)

=

C(l),

C(13)

=

(l + 4 e c o s M ) S i + 6esinMS ( 5 + ( l - 2 e c o s M ) S 5 ,

C(16) C(23)

(1 + 2ecosM)Sj + 2esinMSa + (1 - 2 e c o s M ) s J , 9

\

5

39

n + ^2 ' cos M - -e2 cos 2M - —e3 cos 3M - 2 e s i n M S r - Er

It must be noted that in the above relations only terms up to weight 9 have been retained, so powers of e greater than or equal to 4, products of efcEQ for k > 2, a € {r,8,6}, and products of E^ between themselves have been skipped.

Expansions

Due to the Noncircular

Barycenter

Motion

267

We also remark that according to the used units (w + n)7 3 / 2 = 1. 6.1.2

A Program

for the Computation

of the

Coefficients

A program (CES) for the computation of the terms involved in fc~3-l, C(l), xsC(2), xsC(B), C(4), C(5), C(7), C(12), C(13), C(16) and (7(23), where z s = 1 - 7 - 1 for the Li case, and xs = — 1 - 7""1 for the i 2 case, has been implemented. We note that in the Sun-Barycenter case and according to the relationships between the coefficients of the equations (see Chapter 5, 5.2.5) only these need to be computed. Each term can be written as coef F(vt + ip),

(6.1)

where t is the normalized time, v is a frequency,

. If v = 0 and tp = 0, then we put 11=0. We consider the order k~3 - l,C(l),xsC(2),xsC(3),C(4),C(5),C(7),C(U),C(16),C(23),

(6.2)

and we number them from 1 to 10. We note that in fact C(12) = C(l) and so C(12) does not need to be computed. In any case, it will be given in the output of the program in order to be coherent with the Earth-Moon case, where C(12) is different from C(l). The terms of (6.2) are stored in two arrays, CF(N1,N2), NF(N1,N2,N3), where Nl=10, N2=500, N3=2, such that CF(L,J)=COEF, NF(L,J,1)=I, NF(L,J,2)=I1. This means the term number J of the coefficient number L according to the order given in (6.2). The longitude, latitude and radius vector terms represented in (6.1) by £#, £,5 and £ r respectively, are computed by the subroutine SIGMES. The terms of £,5 and £ r are of the following type ^

Cij cos [(-ine - jnp)t + (dij - iM0 -

jMg)},

where ne,np, dij, Mo, and MQ are obtained from the subroutine NEWPER. That gives some frequencies (—ine — jnp) which are stored in GMA and some phases (d^ — IMQ—JMQ) stored in PHA. The longitude terms are computed as sine terms after the suitable modification of the phase. That change is done in order to have only cosine

268

Expansions

Required for the Equations

of Motion.

Collinear Points

Case

terms in the first and third equations and sine terms in the second equation. The (3*1

derivatives £#, £9, £,5, S<5, £^ , £ r and £ r are also computed in the same subroutine. As usual (3) stands for the third derivative. When the longitude, latitude and radius vector terms are available, the terms of (6.2) are computed and stored in the corresponding array. The products coef • F(i/jt + ipj) • £, where £ stands for £#, £5 or £ r or some of its derivatives, is performed by the subroutine SPCS. We assign a weight to each term (6.1) according to the formula [ - l n | c o e / | + 0.5], where [ ] means the integer part function. The computation of the weight of one term is done by the subroutine PROPES. In the subroutine WEINF, the terms which have a weight greater than a maximum value NN are deleted. The subroutines GUARD, STEN, REOR and BUGA, are used for the storage and manipulation of the terms. After the computation of the terms of (6.2) it is possible to perform a test. The test is done by the subroutine TESCE. For a given Julian day, the difference between the values of the coefficients (6.2) using the terms computed by the program CES and the numerical values, is obtained. The subroutines PARAM, EFEM, VEC, TRANSLI, TRANS, COSS and EPHPLA are used for the computation of the numerical values. 6.1.3

Numerical

Results

and

Tests

The program CES has been run for the Sun-Barycenter problem in the following cases (1) L\ case, (2) L2 case, (3) L\ case, simplified model. For the simplified model, only the eccentricity of the Earth, the dynamical coherence of the Earth's motion and the effect of the radiation pressure has been taken into account. This simplified model has been useful to perform simulations of control (see Chapter 9). In order to make a reduction of the number of terms that appear in the computations, the terms such that, the absolute value of its coefficient is less than a given bound e are neglected. In the three cases above the value of e has been taken equal to l.D-10. In Tables 6.1 and 6.2 we give the results for the case (1) up to order 12. The corresponding check (Table 6.3) shows a good agreement between analytic and numerical values. We note that in the model of the solar system used for these checks only the Sun and the Earth's periodic terms have been taken into account.

Expansions Due to the Noncircular Barycenter CASE LI , ORDER = 1 2 GAMMA = 0.1001090475489518E-01 NUMBER OF FREQUENCIES = 162 1 0..9999522207284640D+00 2 0..1999904441456928D+01 3 0..2999856662185392D+01 4 0..3151973342338853D+01 5 0..1625459877970818D+01 6 0..6255076572423544D+00 7 0.2250967535213173D+01 8 0.1251015314484709D+01 9 0.2510630937562449D+00 10 0.2876475192455527D+01 11 0.1876522971727063D+01 12 0..8765707509985993D+00 13 0..1233814697298647D+00 14 0.2502030628969418D+01 15 0.1502078408240954D+01 16 0.5021261875124897D+00 17 0.3127538286211772D+01 18 0..2127586065483308D+01 19 0.1127633844754844D+01 20 0.3753045943454126D+01 21 0.2753093722725662D+01 22 0..4378553600696481D+01 23 0..5004061257938835D+01 24 0..1004252375024979D+01 25 0..5629568915181190D+01 26 0..6255076572423544D+01 27 0..4683008389644087D+00 28 0.1936553898657281D+01 29 0..9366016779288175D+00 30 0..6335054279964647D-01 31 0..1404902516893226D+01 32 0.4049502961647623D+00 33 0.1873203355857635D+01 34 0..8732511351291710D+00 35 0..1267010855992929D+00 36 0..1341551974093580D+01 37 0..3415997533651158D+00 38 0.1809852813057989D+01 39 0..8099005923295245D+00 40 0.2658256908820276D+01 41 0.7465500495298781D+00 42 0.1915609282611617D+01 43 0..9156570618831535D+00 44 0.8429515884531046D-01 45 0..1084247379573774D+01 46 0..2831266344494771D+01 47 0..1831314123766307D+01 48 0..8313619030378430D+00 49 0..2746971185649461D+01 50 0..1747018964920997D+01 51 0..7470667441925326D+00 52 0..3662628247532614D+01 53 0..2662676026804150D+01

Motion

0 6231395117299221E+00 0 6179604927418856D+01 0 6127814737538490D+01 0 2595581545179894D+01 0 1330068567545066D+01 0 1425520961190873D+01 0 5953003745923967D+01 0 5994554612681657D+01 0 2688854920500531D+01 0 1060403842429783D+01 0 1142039162519251D+01 0 4118012100655008D+01 0 2586425363398768D+01 0 2556884642807375D+01 0 2420179273472353D+01 0 5062038073312304D+01 0 ,3987612619288647D+01 0 3798547372393796D+01 0 .3387825310495664D+01 0 . 5402632632501971D+01 0 ,5260691275410965D+01 0 ,5589019480636293D+00 0 1965195315016981D+01 0 , 1076289304286003D+01 0 .3388941974490275D+01 0 .4777782048923683D+01 0 4984774559577462D+01 0 5034529440171451D+00 0 5183875396819217D+00 0 1934137094660285D+01 0 ,5548744339734944D+01 0 ,9716233222678938D-01 0 1267565900474510D+01 0 ,1915909628686537D+01 0 .3207758966480621D+00 0 .6475185442187157D+00 0 ,1505650848520461D+01 0 .2555277632530175D+01 0 .3329634132736192D+01 0 3718394402892522D-01 0 . 4730792266171488D+01 0 .4584756406481247D+01 0 .4303741280471877D+01 0 .4670750522958893D+00 0 .1869375022576490D+00 0 .2128284922737904D+01 0 .2290816253657310D+01 0 . 2732658384359900D+01 0 .3345161826763161D+01 0 .1863680268766507D+01 0 .2192105145087974D+01 0 .4489304589884120D+01 0 .2956335818953634D+01

269

Expansions 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109

Required for the Equations

0.1662723806075686D+01 0.9660209891320313D+00 0.1932041978264063D+01 0.9320897575355987D+00 0.1898110746667630D+01 0.3744445634861095D+00 0.8765707509985993D+00 0.8313619030378430D+00 0.3151973342338853D+01 0.1625459877970818D+01 0.6255076572423544D+00 0.2250967535213173D+01 0.1251015314484709D+01 0.2510630937562449D+00 0.1876522971727063D+01 0.8765707509985993D+00 0.1233814697298647D+00 0.2502030628969418D+01 0.1502078408240954D+01 0.3127538286211772D+01 0.2127586065483308D+01 0.1127633844754844D+01 0.3753045943454126D+01 0.4378553600696481D+01 0.5004061257938835D+01 0.5629568915181190D+01 0.4683008389644087D+00 0.1936553898657281D+01 0.9366016779288175D+00 0.1404902516893226D+01 0.4049502961647623D+00 0.8732511351291710D+00 0.1267010855992929D+00 0.1341551974093580D+01 0.8099005923295245D+00 0.1915609282611617D+01 0.9156570618831535D+00 0.8429515884531046D-01 0.2831266344494771D+01 0.1831314123766307D+01 0.8313619030378430D+00 0.2746971185649461D+01 0.1747018964920997D+01 0.7470667441925326D+00 0.3662628247532614D+01 0.2662676026804150D+01 0.1662723806075686D+01 0.9660209891320313D+00 0.1932041978264063D+01 0.9320897575355987D+00 0.1625459877970818D+01 0.3744445634861095D+00 0.2250967535213173D+01 0.2510630937562449D+00 0.1251015314484709D+01 0.7488891269722191D+00

of Motion.

Collinear Points

0. 1428603035779131D+01 0. 4024941930115337D+01 0. 1756924709239920D+01 0. 1861074776680115D+01 0. 2479483072752919D+01 0. 3678420431785595D+01 0. 5454003523592576D+01 0.3884692104137265D+01 0. 2630488130219780D+01 0. 4443735953102950D+01 0.4566735576464684D+01 0. 2802684446074202D+01 0. 2853049225554463D+01 0. 5843653956941389D+01 0.4281188355156252D+01 0.9763613276011232D+00 0. 2562863418496844D+01 0. 5699873559798764D+01 0.5558281268558158D+01 0. 8425293071948659D+00 0. 6639360358119795D+00 0.2479779861578648D+00 0. 2264530637416166D+01 0. 3691767955393451D+01 0. 5141694553646660D+01 0. 2473493209004818D+00 0. 1843181905987669D+01 0. 3634573622094972D+01 0.3658932995720519D+01 0.2394934381381191D+01 0. 3240849380918976D+01 0. 5057327749351131D+01 0.2886618384113662D+00 0.3789111197808509D+01 0. 1862961498944049D+00 0. 1502504947459261D+01 0. 1162381453804300D+01 0. 7221838949404437D+00 0. 5264641588571715D+01 0. 5431768895010397D+01 0. 5873785558638186D+01 0. 2157864779373282D+00 0. 5003702126029506D+01 0. 5335443127929762D+01 0. 1365165228814270D+01 0.6127599069827331D+01 0. 4570195689368925D+01 0. 8836983423759428D+00 0.4898517362829713D+01 0. 4993940784009937D+01 0. 1079813069376390D+02 0. 1664659540834537D+01 0. 9084444342853685D+01 -0. 3378345891744758D+01 0. 1207504907424061D+02 0. 3877411603578316D+00

Case

Expansions

110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 169 161 162

Due to the Noncircular Barycenter

0 .2876475192455527D+01 0 .8765707509985993D+00 0 .1876522971727063D+01 0 .1233814697298647D+00 0 .1502078408240954D+01 0,.1936553898657281D+01 0..6335054279964647D-01 0..1267010855992929D+00 0..1915609282611617D+01 0..8429515884531046D-01 0..2831266344494771D+01 0,.8313619030378430D+00 0,.1831314123766307D+01 0..1685903176906209D+00 0..2746971185649461D+01 0..7470667441925326D+00 0..3393123159643262D-01 0,.1625459877970818D+01 0,.3744445634861095D+00 0..3250919755941637D+01 0,.1251015314484709D+01 0,.2250967535213173D+01 0,. 2510630937562449D+00 0..1251015314484709D+01 0,.7488891269722191D+00 0..2876475192455527D+01 0..8765707509985993D+00 0..1876522971727063D+01 0..1233814697298647D+00 0..3501982849697882D+01 0..1502078408240954D+01 0..4127490506940236D+01 0,.2127586065483308D+01 0..1936553898657281D+01 0..6335054279964647D-01 0,.1873203355857635D+01 0..1267010855992929D+00 0..2915561503340081D+01 0..9156570618831535D+00 0..1915609282611617D+01 0..8429515884531046D-01 0..2831266344494771D+01 0,.8313619030378430D+00 0,.1831314123766307D+01 0,.1685903176906209D+00 0..3746923406377924D+01 0..1747018964920997D+01 0..2746971185649461D+01 0..7470667441925326D+00 0..1965973209860495D+01 0,.3393123159643262D-01 0,.1876522971727063D+01 0,.1831314123766307D+01 Table 6.1

Motion

0 .1051258347245547D+02 -0,.1950206762142969D+01 0,.7207756444900344D+01 0.,5255033789698098D+01 -0,.5315215575004570D+00 0..9890328113019740D+01 0..2572462121578702D+01 0..1174067367948090D+01 0..7393776571103521D+01 0..5069013663494921D+01 0..1166316401230962D+02 -0..7996262222888242D+00 0..1210518067593741D+02 0,.3576095586610349D+00 0..1123509724332873D+02 -0..1227692991269715D+01 0..5347696774923278D+01 0,.7656916078490094D+01 0..4805874156108349D+01 0..1218439886322319D+02 -0..2783913713752542D+00 0,.1222594972998088D+02 -0..2368405046175639D+00 0..8920250037799152D+01 0..3542540196798690D+01 0..7373434279818472D+01 -0..5089355954779970D+01 0..1034940721795423D+02 0..2113383016644213D+01 0., 8788279760106596D+01 -0..3674510474491846D+01 0..1021900773658787D+02 -0..2243782498010574D+01 0..6749782656981143D+01 0..5713007577617299D+01 0.. 8147304745985758D+01 0..4315485488612684D+01 0..1081615152378047D+02 -0..1646638710817974D+01 0..1053513639777110D+02 0..1927653836827344D+01 0..8522211370956531D+01 -0..3940578863641911D+01 0..8964053501659121D+01 0.,3498736732939321D+01 0.,9576556944062382D+01 -0..2886233290536060D+01 0..8095075386065729D+01 -0..4367714848532714D+01 0..1025633704741456D+02 0..2206453187183884D+01 0..1168539864089180D+02 0..1011608722143649D+02

Output of program CES, first part.

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CO CO CN ^ I i i i a a m cn TO t~- CO CO T H lO T-I C N Cn U5 ^ o CN co r- ai T-I ^ cn a> to CN CN <* in w O co co i- 1-o <# co L O a> in T-I CN in C M C N t- co in C N o co to CN T H in T-I C N [»- T-I

a o a a t~ o y# co •* (O CN CO O) rco r— o in

•*

CO 1 *

in

o in

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a CI

o ho CIO o m to • * CN cn CO o

tH

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1-4

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^

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o

o NCO CN co CN >t CN o i-4 o 00 co T-i to CO CO o fr- o to to

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cn

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o

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o T-I CN CO ^ LO to r-- oo cn o T-I CN CO 1 * LO to r- I-- t- r- t- f- t~- t- r- r- co oo co CO 00 00 00

in CN to CN CN o h- to CN cn 00 ^f CO C) f- CIO to f- in rCN to in LO o o o o CN CO to on CN to to oo

o CO cn cn

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CN CO T f in tr> t- on cn to to to to to to to to

T-i

m o t - i T-i CN LO to to to to

i

CO >* cn o to CO co oo H s; oo o 00 to t> BJ LO Cl f- o TH y w on to co tn CN J3 h-> T H c> LO on CO to u, L b o imH cn r-- CO CO h- to CO to CO o o trs r~- CN CO I-m cn <# T H ito-- CO OJ r ^1" m CN - in bJ in cn CN o i H CO m t-- T H co o o o o o o

m

o o o o o o o o o oo o o o o o o o o O o O o o o o o o o o o o o o ro «j-

• #

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m in to

a a

«n> cn c

^ *• CO < C)
o

• > *

i H CO «tf in m <• >* S Q e> a a Q a a a 00 h*f o cn to LO CN C3 C) CO c n CN cn CN 1-4 ^ CO r- oo to c CN to cn LO i H> o rn co U^ M< CO CO CN in o CO to T-l C) LO ^ o CO cn cn r* co oo i-i T H co co 1-1 i-4

i H i - 4 i H i H C O L O i H i H i H i H i H ^ C O i H T - l i H i H i H i H i - l i H i n t * - T - l i - l T H T - l i H i - l i - l I I I I I I I I I I I i—< o ii o n o I

• *

in LO Q a to to ^ CO co to to CN CN fr- C) co LO LO 00 CO

o

CO o O CO o ^f o to *to 1<

^

o

I

I

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1

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1

I

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1

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I

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TH

•>* in LO m a a a Q1 "S on o yf in to 00 oo 00 sf •=)* c-ni-i cn cn I cn cn C7) o o T-4 o o "5" to CO to CN CN

1

1

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h- 00 co I-O t-- r >- 00 i-l l>- to o o o

• *

i

I

I

• *

I

1

I

I

1

I

I

1

I

o O O o o o O O o O O O O

<* to

LO <-" ^f in in m in in in ^ Tj* i * •
T H C N C o ^ m t o r — co oi o co •*o i O r i M n ^ s c o < f i o o s o tDoocnoiH r o c o c o c o c o c o c o n c o * ^ ^ 1 T j * L o m L O L o m L O L O O o o O T - * Ii H- iT cHNI - cHoI m H I H T H T - I C N C N I

i

* itf a Q a a a Q a a a a a a a a a a a *? - h- CN CN 00 to to CN CN Oi n to to CN CN cn LO c- r-- LO m < * CO C) O o u> c CO m r- h- c- 1- 'j' o o to to LO in «s* r- t-- C-- r-- cn cn LO LO LO in CN CN I * i H ^ 00 CO LO r^ c^ r-- r »< -o on LO TO o TO CO m 00 o o 00 OO 00 CO m in to itoH 00 o o • * 00 in in r to r- h-- f- CO CO 00 on • iH T H •f-i * fh00 00 CN CN in in CN o o 00 00 CN CN i" CN a> cn IN CN 00 r«- f- to to f> * CJ TO CN CN o o cn cn C3 CO TO o in on o Ci o O LO LO O O Ci CN CN LO CN CN o 1-1 <.> in in LO LO ioH o •tf m in o o O o o o o o o o o O o o o o o

274

Cn

to

to CO

>* •Sf ^ Q

co

o a a "S* to 00 to 00 tO

•*

'i'

•iH

•iH

•*

•iH

•*

iH

TH tH

cs

• *

tH

• *

• < *

• *

• *

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to to to to to

cn o

^

• *

o co cn cn CO tH

Required for the Equations

• *

O) *f CO N- 00 CO If

• *

cs

Expansions

<• • *

•*

• *

o tH

cs

>* «tf *f

to to LO to

•*

• *

•*

•*

• *

•*

CO

to •*

^- 00 • > *

•sf

cn o

•* LO

to tH

II

CN

o co

of Motion.

II

o

•>* •*

<* to

O tH

Case

CS CO to o CO to cn en

Collinear Points

II

LO

1

• > *

cn

*1* *f

• *

•*

• *

1

I

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1

LO CJ

o

1—1

tH

tH

o

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cn i -oo t H

tH

tH

a

CN

CN

a a.

~

to to •*

cn

•*

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co OT to LO U CS CJ

CO „

tH

cs LO N-

^ tH o

to

LO to CO CO

i H CN CO CO

ddd

• *

tH

a Cl CO CN LO en 00 i -to to t H t- cn TH s*

II CO

cs m

a

fcL,

c

to to 00 00 CO CO

cn cn oo 00 o o LO LO 00 oo

^ ^ to to

o o

^

r- CO cn o cs cs CN CO

*f to *f to to ^l*

O t H CS CO LO to CS CN cs cs CS cs

• *

tH

a Q to to

*!•LO

cs CO • * LO to r- 00 cn O LO to LO to LO to LO to to to

tH

• *

LO to N- 00

•*

tH

CO

tH

•*

tH

tH

tH

tH

cs CO iH

<* CO ^

•*

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• *

cn o co

tH

"tf

• *

co

•*

•*

•* co cs CN

• *

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< •

CO

t H cs CO to to LO

CO CO CO

cs CO <# LO to t- CO cn o >*
>* co CO •* co CN

LO to I-- 00 CO co co co

•*

• *

1

• *

• *

• > *

• ^

r" 3

a Q a a Q a Q Cl a a taH a Q a Q Q Q o cn r- to O LO to CO cn cs o o H* to * to oo cn cs cs r- t- ttoH tLOH 3 CO 1-- o f- t- o o h- tcs cs tH <* ^f en 00 o LO 00 LO tO to to to to to c n co cs c n i H 1^ o to t H <• LO to CO 00 CO co co t~- r- H ^ > * tH 00 CS 1-- t H >* co oo > r* -•00 00 TCOH tcHo toH toH •>* CN ^o" cs CN 00 tH CS 00 CN c co rt H t H *f <J" to tH co t H CO co << *f ^ «* i H t H o o o o o o o o o o o o o o o o O

•>* to •* to ^ <# to

tH tH

a a Q a Q Q Cl Cl O Cl Cl Cl Q a to to CO CO oo 00 cn cn o o tt HH tt HH • * t H t H CO to LO 00 to to CO cs cs CO co CO o O cs cs t~- t--
LO to LO to CO CO
•*

»* to

•St"

•*

CO

•*

to to to

• *

cs co co

r- 00 cn o t H cs CO *f cs cs cs co co CO CO co

•* <• co

tH

CO t H co •5f LO H* ^> tf o t H it-H cs t~- o to tH ^ LO CS t H i H to LO O r ~ r ~ oo c n o o o o >* ^ > * * ! • CO to t H o o 00 to tCO cn to CN EL) b LO tLl cn cn LO co cn to CS cn o cn CO CO o cn cn -3* s a r- CO a K o CS to cs cs cn to cn <• & to fhto CO PL, r- to cn i H CO o o cn cn to o b pr cn o » * to ^f oo CN *f to r~- t H cs CN t-- t H CN o to co to cn ics H co •si* CO to to to cn cn t- oo co o LO o cn o to O cs cn to co cn st< a> to t H cs r~- CS to cn t~- t-- 1~- t-- CS to 00 CO CO cn CO co o 00 cs cn o co to cn cs o cn CO tH t H to t H to t H cs to >* CO tH cn 5 5 5 t H t H to CN t H LO t H >* o o o o o o o o o o o o o o o o O o o o

o r- Cn 00 co t~CS •sf o o to o to CO LO co o to 00 oo tTHH !--CS o tH to o o

*f co CN tH i H CS «tf cs t H cs en
• *

co • * LO to t~- oo cn CO CO CO co co CO CO

• *

• *

•*

•iH

•*

•*

*1< *J<

to to t-- 00 cn o o o o o o o tH

•*

Q a Q a a a a Q Q a a Q to to CO CO oo 00 * -- cn cn 00 • * • * LO LO to LO cn cn ai o o cn t H t^ h- t H t H t- i-CO CO o CO o 00 00 CO co to to t-- I- to to 00 to to t H t H LO LO t- r- to to cs cs CO CO to LO CO CO t H t H t- t~* cs cs i H o o o CS CS * cs cs co ^* co co o o o o o o o o o o o o

<*

TH

• *

LO to f- oo to LO LO to LO

CO CO CO CO Q

CO •*

• *

•*

Q a a a Q a Q a Q Q Cl Q a a a tH •* T•iHH to ct Hn O to t H cn cn O i H Cl o t- 00 to t to CO 00 00 o c n cs c n CO ^ to o to r- to cs 00 cs to to cn 00 to to cn to LO tcs t~- CO o CN r- CO co CN LO CO to co oo t~- t H ht H co to to f- t H ^ t~- i H c o to o r- to CO to cn to hLO CO o tH tH cs cn t H 00 rH ^ o o a> cn CO cs o t~- co t H t H CS o t H cs cn LO o cs t H cn t- o o cs CO o to CN CO 00 to cn to LO t H CO to TH cn tH CO i H i H CO to to to
sf

CO *}• LO to CN cs CN cs

i H CO cn 00 LO LO 1CO hcs LO 00 CO oo t^- to en o •tf t H 1TH in t H o o t- 00 o o 00 o o cs t H CO cs t H o o o o

T-i

cs cs cs

•*

4BER

a a a a a Cl Cl Cl a Q Cl Cl a Q a Q Q Q a Q a Q Cl a Cl a Cl a a Cl Cl Cl O

•tf

1BER

CO CO CO to *? ^ cn ttoH t H to to cn cn o t^ ^J< cn CO cs to 1^ o oo 00 CO t H to co LO o CS 00 to to CO Cn cs to CO to o o CO CN 1 ~ LO to CO to CO CO o cs cn t H to cn t H cn CO cs t H h- t~- o to cn oo o * t H f- hcs o Ol cn CO CO cn to cn cn c* n•00 cs o cn to tt HH CO >* CO LO CS o cs CN toH o cs LO t H o CO c 00 G> • o ^ tcs o cn CO CO t CO to cn to hH >* o cn to i^ f~ 00 o 't tcn cs o cs cs f- cn CO t H to 00 o cs co co t H to tt HH t H i H to to •* co t H t H t~- t H t H ^ ^ N- t H to co t H t H co cs to CN t~- t H <* i H ** cn to o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

tH o cs I-co cn LO to Cn cn tO CO co oo

econd program Outpu

rERM!

MBER

^BER

Expansions

Due to the Noncircular

Barycenter

Motion

275

CURRENT VALUE OF SPACECRAFT SECTION IN m**2 DIVIDED BY MASS IN kg: O.Ol THE MODEL OF THE SOLAR SYSTEM IS : 0 1 19 0 1 0 0 0 0 1 0 0 0 0 100 0 EARTH PERIODICS: 1 5 6 10 11 12 14 15 16 17 20 21 22 24 25 26 28 29 32 35 36 38 39 41 43 44 45 48 49 51 52 53 56 57 59 60 63 67 84 85 86 87 88 89 90 93 94 95 96 97 98 104 107 108 110 121 125 126 130 131 132 135 136 137 140 141 144 145 146 148 152 156 161 165 167 168 172 173 176 177 180 185 208 209 210 212 213 214 217 218 219 220 221 222 229 233 234 243 251 270 EARTH+MOON-SUN SYSTEM DAY = 17000.00 DIF= 0.586228D-05 W=12 -0.47323223135D-01 0.47317360846D-01 0 -0.47352037297D-01 DIF= 0.328028D-05 W=13 0.47348757010D-01 1 -0.22581500073D-02 DIF= 0.136246D-04 W=ll 0.22445253265D-02 2 W=14 -0.14980665534D-05 0.22703348479D-05 DIF= -0.772268D-06 3 0.72907158388D-02 0.72897958735D-02 DIF= 0.919965D-06 W=14 4 0.63890205566D-01 DIF= 0.289212D-05 W=13 -0.63893097692D-01 5 W=13 0.28511960973D-02 DIF= -0.173268D-05 -0.28494634105D-02 7 W=16 O.OOOOOOOOOOOD+OO DIF= -0.137264D-06 0.13726456587D-06 13 O.OOOOOOOOOOOD+00 DIF= 0.312971D-07 W=17 -0.31297142302D-07 16 DIF= 0.314703D-05 W=13 0.15520478411D-01 23 0.15523625451D-01 DAY= 17500 00 0.42447763370D-01 W=13 DIF= -0.162021D-05 0.42449383589D-01 C 0 0.42434403720D-01 DIF= 0.133870D-06 W=16 0.42434269850D-01 C 1 -0.21177307606D-03 W=12 DIF= -0.418189D-05 C 2 -0.20759118551D-03 0.59078831638D-04 W=13 0.62593230051D-04 DIF= -0.351439D-05 C 3 19473685954D-01 W=12 DIF= -0.830770D-05 0.19481993655D-01 C 4 55887031794D-01 W=12 DIF= -0.722165D-05 0.55894253447D-01 C 5 15056483962D-02 W=ll DIF= -0.110345D-04 -0.14946138207D-02 C 7 OOOOOOOOOOOD+OO W=15 DIF= -0.316697D-06 0.31669713704D-06 C 13 OOOOOOOOOOOD+00 W=14 0.12314860616D-05 DIF= -0.123148D-05 C 16 14234783481D-01 C 23 DIF= 0.624200D-05 W=12 -0.14241025489D-01 DAY= 18000.00 C 0 DIF= -0.206983D-05 -0.83864447118D-02 W=13 -0.83843748798D-02 C 1 -0.82786665903D-02 DIF= 0.149099D-05 W=13 -0.82801575880D-02 C 2 W=14 0.83113327376D-03 0.83195078685D-03 DIF= -0.817513D-06 C 3 0.45574038015D-05 W=13 0.60184118229D-05 DIF= -0.146100D-05 C 4 DIF= -0.371839D-05 -0.32759721147D-01 W=13 -0.32756002748D-01 C 5 -0.11356879515D-01 W=15 DIF= -0.240346D-06 -0.11356639168D-01 C 7 0.10333198911D-01 DIF= 0.270392D-04 W=ll 0.10306159627D-01 C 13 O.OOOOOOOOOOOD+OO DIF= 0.407048D-06 W=15 -0.40704878079D-06 C 16 O.OOOOOOOOOOOD+00 W=16 DIF= -0.122413D-06 0.12241310311D-06 C 23 W=14 0.30452368428D-02 0.30442382676D-02 DIF= 0.998575D-06 DAY= 18500 00 0 C DIF= 0.391254D-05 W=12 30997751585D-01 -0.31001664126D-01 1 C DIF= -0.643908D-05 31048686591D-01 W=12 -0.31042247505D-01 2 C 10932696906D-02 W=14 DIF= 0.515631D-06 -0.10937853225D-02 C 3 15122404059D-04 0.12798829821D-04 DIF= 0.232357D-05 W=13 C 4 25391597752D-01 DIF= 0.464190D-05 W=12 0.25386955846D-01 C 5 W=14 0.41877761045D-01 DIF= -0.130604D-05 -0.41876455002D-01 C 7 W=12 DIF= -0.999022D-05 0.40233239967D-02 -0.40133337759D-02 C 13 0.OOOOOOOOOOOD+OO W=16 DIF= 0.112045D-06 -0.11204567608D-06 C 16 O.OOOOOOOOOOOD+00 W=15 DIF= -0.264382D-06 0.26438273133D-06 0.10391676826D-01 C 23 W=12 DIF= -0.412130D-05 0.10395798126D-01 DAY= 19000.00 C 0 0.51653661865D-01 W=12 DIF= -0.466952D-05 0.51658331389D-01 C 1 0.51611459297D-01 DIF= 0.607769D-05 W=12 0.51605381601D-01 W=ll C 2 0.97131741099D-03 0.98523320557D-03 DIF= -0.139157D-04

276

Expansions

Required for the Equations

of Motion.

Collinear Points

Case

c 34 -0.47199664011D-04 -0.27032153244D-02 c c 5 0.67966155048D-01 c 7 -0.52364740560D-02 c 13 0.00000000000D+00 c 16 0.00000000000D+00 c 23 -0.17510847886D-01 DA'S = 19500.00

-0.50791228875D-04 -0.27036261294D-02 0.67966973302D-01 -0.52362612281D-02 -0.15761879237D-06 -0.99345461584D-06 -0.17516469065D-01

DIF= 0.359156D-05 DIF= 0.410804D-06 DIF= -0.818253D-06 DIF= -0.212827D-06 DIF= 0.157618D 06 DIF= 0.993454D-06 DIF= 0.562117D-05

W=13 W=15 W=14 W=15 W=16 W=14 W=12

C C C C C C C

-0.36266089948D-01 -0.36284451061D-01 0.18729839462D-03 0.13948608114D-04 -0.21881942540D-01 -0.48949166953D-01 -0.18157501517D-02 0.17110937262D-06 0.28917796563D-06 0.12065710655D-01

DIF= DIF= DIF= DIF= DIF= DIF= DIF= DIF= DIF= DIF=

W=15 W=13 W=ll W=13 W=13 W=14 W=13 W=16 W=15 W=13

c c c

0 1 2 3 4 5 7 13 16 23

-0.36266577631D-01 -0.36286160842D-01 0.20022264422D-03 0.15382478862D-04 -0.21880162231D-01 -0.48948321893D-01 -0.18134857972D-02 0.00000000000D+00 0.00000000000D+00 0.12063454072D-01 Table 6.3

6.2 6.2.1

-0.487683D-06 -0.170978D-05 0.129242D-04 0.143387D-05 0.178030D-05 0.845059D-06 0.226435D-05 -0.171109D-06 -0.289177D-06 -0.225658D-05

Checks of the results of program CES.

Preliminary Exploration of the Functions to be Kept Expansion of the Perturbing als in the Coordinates •

Functions

as Sum of

Monomi-

As it has been shown in Chapter 5, 5.2.5, the last terms appearing in the equations are of the form:

J2

i(A)»A

VSA ' SA

Ae{E,M,Pu---,Pk,B}

^2

i(A)fiAGA(v),

. ,

\ S ^ n>2

n-2

a

f> I

A \

T

A

v£{x,y,z}.

Ae{E,M,Pi,-,Pk,B]

These terms must be developed as a sum of monomials in the coordinates (x,y,z), the coefficients of these monomials being some time-dependent functions. The first term in the square brackets is independent of x,y,z, the components of a. The only term to expand is the summation for n > 2. The polynomial Pn{w) can be given explicitly using the formulas Pn(w) = dn -P^iw), Pn{w) = 2 _ n ( n ! ) - 1 - — ( w 2 - l ) n . The argument cosAi is written conveniently as cos Ai = (XXA + WA + zzA)/afA and, as deg Pn = n — 2, the variables x,y,z appear only in polynomial form (i.e., there are no terms in am,m < 0 or m odd, where a = (x2 + y2 + z2)1/2; only terms a2m,m > 0 appear).

Preliminary

Exploration

277

of the Functions to be Kept

Summarizing, we reach the formula

^£pn(cos4i) = Y,A^i^'i^^xhyi2zi3xiAyiA^xifA-.

n>2 TA

m - 2j — 1 v-i"~ 2j — 1 — kit h e i n d i c e s

where £ stands for E ~ = i E}=o J E^1=o ££3i ££"=0 i\,..., iy having the values h = 2ji + /ci,

i 2 = 2j 2 + fo, h=n-

«4 = fci, h = ^2,

i& =n-2j

E^o"

'

2(ii + j 2 ) -fci-fe2- 1,

- 1-fci -fc 2 ,

i7 = 2 ( n - 2 j ) ,

and

,,. . ^^'"",l6>

M_(zlli

(2n-2j)!

2" (n - j)!ji!j 2 !(j - Ji - j2)!fci!fc2!(n - 2j - 1 - h - k2)V

an_2 The coefficients of the first equation (x - XA) E«>2 -n+i Pn(cosAi) lected in the form £ as

f{qi,q2,q3)xqiyQ2zq3.

The function f{qi,q2,q^)

have been colis expressed

^J v^J I?!) (fij The formulas for the second and third equations are obtained using a suitable permutation of indices. The computation of the coefficients cqp and the related exponents is done in routine EXP AN. It has been shown very efficient to compute all the coefficients of the type A and then collect all the coefficients which are related to the same exponents ii,... ,i6. An ordering of the computed coefficients according to a given order of the indices has been used as an intermediate step. This is done by routine ORDRE which calls the comparison routine COMP. 6.2.2

A Rough Exploration

of the Functions

to be Kept

A program has been developed (program TF) in order to compute the maximum degrees up to which the functions Fv must be developed in order to achieve the precision required in the expanded equations of motion. The program computes, for each body of the solar system, the value of its global direct contribution to each of the equations, and the values of the contributions of the direct terms under consideration, when they are expanded up to different degrees, between two previously specified, to the program. As test points, at which the above functions are evaluated, we have taken points on an analytic halo orbit of the RTBP. It must be said that this program has also been used to check the goodness of the automatically computed developments of the functions Fv, performed by routine FUN2 and its related ones.

278

Expansions

Required for the Equations

of Motion.

Collinear Points

Case

The results of this program are summarized in the following table, which gives the maximum degree for the expansions:

Mercury, Mars, Saturn Venus, Jupiter Earth-Moon barycenter Table 6.4

1st eq. 0 1 11

2nd eq. 0 1 11

3rd eq. 0 1 9

Sun-Barycenter problem, Li}2 cases.

Of course not all the terms, up to the computed degrees, shall be kept in the final equations. A second rough exploration of the terms to be retained is performed by program FURI1, routine EXPLOR. The execution of this option of program FURI1 performs the following task: For a given set of epochs, within a certain time interval, all the "a priori" time-dependent functions appearing in the equations are evaluated (3 x 20 functions associated to each planet and 3 x 455 associated to the Earth-Moon barycenter). To the values obtained a correction factor is applied. This factor takes into account the powers of x, y, z related to the function. As significant functions, only those whose corrected value is greater than a certain tolerance (5.E-5 as standard value for the Sun-Barycenter problem in normalized coordinates), have been retained.

6.3

6.3.1

Computation of the Approximate Frequencies and Magnitudes of the Perturbations Using FFT The Functions

to be

Analyzed

When the significant functions, related to the developments, have been selected, the next step is to perform a Fourier analysis of them. This analysis has been mainly divided into two steps: - First a Fast Fourier Transform of all the selected functions has been done, in order to detect the main frequencies involved in its Fourier developments. - Second, when the frequencies have been identified, as linear combination of the mean motions of the planets or of the relevant arguments for the motion of the Moon, the right coefficients associated to these frequencies are computed directly by integration. The results of the FFT give only an approximation of these coefficients. The goodness of the obtained Fourier developments has been tested in the last step. An analysis of the functions C(i),i = 1 , . . . , 27 that appear in the equations for the Earth-Moon problem, L2 case, has also been done following the same steps above mentioned.

Computation

6.3.2

The Filtering

of Frequencies and

Amplitudes

279

Procedure for the Output of FFT

The Fast Fourier analysis of the time-dependent functions previously mentioned is performed by program FURIl, routine FURRAP, which uses the subroutine FFTSC (Fast Fourier Transform Sine Cosine) of the IMSL library. For this analysis the functions are evaluated at 2 M points (the standard value of M used has been 16) equidistributed within a certain time interval centered at the osculating epoch TAU (1/1/2000). This time interval is measured in revolutions of the secondary around the primary and the values that have been used are 2048 (1024 in some cases) for the Sun-Barycenter problem and 128 for the Earth-Moon problem. The output coefficients of the FFT analysis have been filtered using the following criteria: a) Only the modulus of a harmonic, which appears in the function multiplying the monomial xly^zk, is considered. If this coefficient times the factor

\k\ai+j/3k ({i + k)i+kf{i+j

+

k)~(i+j+kA

is greater than a certain tolerance (T0L1), then the term is retained. The meaning of this factor is explained in Chapter 7, 7.1.1. b) This harmonic multiplied by the factor given above can have a large frequency and therefore it can be relatively important but it can produce small amplitude oscillations. This amplitude is estimated and the harmonic is skipped if it is less than a certain tolerance (T0L2). The values which have been used for the above two tolerances, in order to keep terms up to weight 9 have been:

Sun-Barycenter problem Earth-Moon problem Table 6.5

6.3.3

Numerical

T0L1 5.D-5 3.D-4

T0L2 5.D-6 3.D-5

Tolerances of terms to be kept.

Results

The results of the FFT analysis show that there are no contributions of Mercury, Mars and Saturn up to order nine, for the Sun-Barycenter problem both in the L\ and 1/2 cases. For Venus and Jupiter the contribution reduces to the constant term and the major contribution is due to the Earth-Moon barycenter. For the Earth-Moon problem the analysis has been done for the direct terms coming from the Sun and for all the indirect terms involved in the functions C's. The results for the Sun-Barycenter problem, L\ case, are listed below.

280

6.3.4

Expansions

Required for the Equations of Motion.

The Identification

of the

Collinear Points

Case

Frequencies

The numbers in front of the coefficients (coefficients of the cosine and sine terms) of the above mentioned tables are related to the approximated frequencies in the following way: frequency = (number - 1) / (number of revolutions of the secondary), when the frequency of the secondary is taken equal to one. Looking at the obtained frequencies we identified them as follows: For Venus: 1 Independent t e r m 642 Tly — TIE 1282 2riv — 2UE 1923 3riv — 3TIE 2563 4nv — 4-UE 3204 5ny — 5UE 3844 6 n y — QUE 4485 Iny — 7TIE 5125 8nv — 8TIE 6406 10nv — IOTIE

For the Moon: 1 Independent t e r m ni - n3 - n5 115 129 ni n5 136 ni - n3 n5 2n\ — 2ri3 — 2ns 228 - 2 n i + n3 + 3n 5 1343 —2ni + 2ri3 + 3ns 1357 1449 - n\ + n 4 + 2n 5 1471 - n i + n3+ 2n 5 1584 n5 1698 ni n3 3054 - rii + n3 + 3 n 5 2n 5 3167

For Jupit er: Independent t e r m 1 939 nE - nj 1790 2nE - 3 n j 1876 2nE - 3nj 2814 3TIE — 3 n j

where ny,nj and TIE are the derivatives of the mean longitude of Venus, Jupiter and the Earth, and 711,713,714,7x5 are the mean motions of the following arguments: mean longitude of the Moon, mean longitude of the lunar perigee, longitude of the mean ascending node of the Moon and mean elongation of the Sun. These frequencies can be affected by minor modifications due to the fact that some slowly changing anomalies have been neglected. This shall produce not only a very small change in the frequency but a modification of the phase related to it. This last point shall be taken into account in 6.4.2.

Computation

of Frequencies and

281

Amplitudes

CURRENT VALUE OF SPACECRAFT SECTION/MASS IN m**2/kg: 0.00 THE MODEL OF THE SOLAR SYSTEM I S : 0 1 11 0 0 0 0 0 0 1 0 0 0 0 0 0 EARTH+MOON-SUN SYSTEM THE EQUILIBRIUM POINT IS LI BODY NUMBER 2 NUMBER OF FUNCTIONS = 29 NUMBER OF POINTS = 16384 INITIAL DAY = -168742.00 FINAL DAY = 205266.00 NUMBER OF REVOLUTIONS OF THE SECONDARY = 1024 FUNCTION NR. 1: NR. OF TERMS UP TO WEIGHT 9: 1 -4.4939923848751437E-04 .7331954554868247E-05 642 .0642040844543807E-04 1282 .5100932273713361E-05 1923 .9372833164279313E-04 2563 .6658379900449168E-05 3204 .5901120913209079E-04 3844 . 3230066543297976E-05 4485 .0338716303075196E-05 5125 .7591775549788391E-05 6406 FUNCTION NR. 2 NR. OF TERMS UP TO WEIGHT FUNCTION NR. 3 NR. OF TERMS UP TO WEIGHT FUNCTION NR. 4 NR. OF TERMS UP TO WEIGHT FUNCTION NR. 21 NR. OF TERMS UP TO WEIGHT 642 1282 1923 2563 3204 3844 4485 5125 6406 FUNCTION FUNCTION FUNCTION FUNCTION FUNCTION FUNCTION FUNCTION

.4827144987234327E-04 .9483165701272172E-05 .6202991416406071E-04 . 6707410569284612E-05 .1119080347526384E-04 .3911703230731183E-05 .5627015217915499E-05 .8248851646060530E-05 .0866923551690530E-05 NR. NR. OF TERMS NR. NR. OF TERMS NR. NR. OF TERMS NR. NR. OF TERMS NR. NR. OF TERMS NR. NR. OF TERMS NR. NR. OF TERMS

Table 6.6

UP UP UP UP UP UP UP

TO TO TO TO TO TO TO

WEIGHT WEIGHT WEIGHT WEIGHT WEIGHT WEIGHT WEIGHT

10, UP TO WEIGHT 11: 32 0 OOOOOOOOOOOOOOOOE+00 -7, 2615206098866838E-04 9. 3699650886702544E-05 -2. 6423065530096558E-04 1 1288803253358732E-04 -1. 5431044072816770E-04 9. 7523179049771403E-05 -8. 4016640685136371E-05 7 2295340767460178E-05 4, 8792886630977165E-05 0, UP TO 11 0, UP TO 11 o, UP TO 11 9, UP TO WEIGHT 11: 28 -1. 3734852665137244E-05 2. 6366460471883169E-04 4. 5954043355252426E-05 1,9867432207864296E-04 5. 5125262267942052E-05 1. 2000299031923278E-04 4. 9254923424610345E-05 6. 4368298277646131E-05 3. 1185771016455967E-05 0, 0, 0, 0, 0, 0, 0,

UP UP UP UP UP UP UP

TO TO TO TO TO TO TO

11 11 11 11 11 11 11

0 0 0 17 0 0 0

Output of FURIl, routine FURRAP. Terms coming from Venus.

282

Expansions

Required for the Equations

of Motion.

Collinear Points

Case

CURRENT VALUE OF SPACECRAFT SECTION/MASS IN m**2/kg: 0.00 THE MODEL OF THE SOLAR SYSTEM I S : 0 1 11 0 1 0 0 0 0 1 0 0 0 0 0 0 EARTH+MOON-SUN SYSTEM THE EQUILIBRIUM POINT IS LI BODY NUMBER 5 NUMBER OF FUNCTIONS = 29 NUMBER OF POINTS = 16384 INITIAL DAY = -168742.00 FINAL DAY = 205266.00 NUMBER OF REVOLUTIONS OF THE SECONDARY = 1024 FUNCTION NR. 1: NR. OF TERMS 1 .5071768279936979E-04 939 .4973991073471902E-06 1790 .4882859085612199E-04 1876 .0159737325237434E-04 2814 .1592356083398305E-05 FUNCTION NR. 2: NR. OF TERMS FUNCTION NR. 3: NR. OF TERMS FUNCTION NR. 4: NR. OF TERMS FUNCTION NR. 21: NR. OF TERMS 1790 9 .7218336138635408E-05 1876 1 .1610233432350377E-04 2814 -2 .3042035028281357E-04 FUNCTION NR. 22 NR. OF TERMS FUNCTION NR. 23 NR. OF TERMS FUNCTION NR. 24 NR. OF TERMS FUNCTION NR. 41 NR. OF TERMS FUNCTION NR. 42 NR. OF TERMS FUNCTION NR. 43 NR. OF TERMS FUNCTION NR. 44 NR. OF TERMS Table 6.7

UP TO WEIGHT 9:

UP UP UP UP

TO TO TO TO

WEIGHT WEIGHT WEIGHT WEIGHT

9 9 9 9

UP UP UP UP UP UP UP

TO TO TO TO TO TO TO

WEIGHT WEIGHT WEIGHT WEIGHT WEIGHT WEIGHT WEIGHT

9 9 9 9 9 9 9

5, UP TO WEIGHT 11: 18 0 . OOOOOOOOOOOOOOOOE+00 -1 .1890032783201678E-04 9 .8963822524347728E-05 1 . 1787462064385039E-04 -2 .3309029148002041E-04 0 0, UP TO 11 0 0, UP TO 11 o, UP TO 11 0 15 3, UP TO 11 -1 4578965339272261E-04 -8 8760793853527658E-04 -5 1012558764118559E-05 0, UP TO 11 0 0, UP TO 11 0 0, UP TO 11 0 0, UP TO 11 4 o, UP TO 11 0 0 0, UP TO 11 0 UP TO 11 0,

Output of FURI1, routine FURRAP. Terms coming from Jupiter.

CURRENT VALUE OF SPACECRAFT SECTION/MASS IN m**2/kg: 0.00 THE MODEL OF THE SOLAR SYSTEM I S : 0 1 11 0 0 0 0 0 0 1 1 0 0 0 0 0 EARTH+MOON-SUN SYSTEM THE EQUILIBRIUM POINT IS LI BODY NUMBER 11: NUMBER OF FUNCTIONS = 455 NUMBER OF POINTS = 32768 INITIAL DAY = -5113.50 FINAL DAY = 41637.50 NUMBER OF REVOLUTIONS OF THE SEC FUNCTION NR. 1: NR. OF TERMS UP TO WEIGHT 9: 7, UP TO 11: 18 1 1.813270139388E-03 2.67 O.OOOOOOOOOOOOE+00 1471 115 9.578381168986E-05 -3.130628584831E-05 1584 -5.54 129 1.520611156189E-04 3167 -9.84 2.999908084963E-06 228 -4.092671935929E-05 2.903693707059E-05 FUNCTION NR. 2: NR. OF TERMS UP TO WEIGHT 9: 6, UP TO 11: 18 1 1.122219822996E-03 O.OOOOOOOOOOOOE+00 1471 1.65 115 7.435734898833E-05 -2.427451848207E-05 1584 -4.25 129 9.579551786285E-05 3167 -6.04 1.720782158983E-06 FUNCTION NR. 3: NR. OF TERMS UP TO WEIGHT 9: 4, UP TO 11: 15 228 -3.064243930758E-05 -4.324952572406E-05 1584 -3.98 1471 8.570330210166E-04 -3.324906125339E-04 3167 6.17 FUNCTION NR. 4: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 4 1.058534422265E-04 136 -4.626148684080E-05 FUNCTION NR. 5: NR. OF TERMS UP TO WEIGHT 9: 2, UP TO 11: 3 1 4.369088783942E-04 1584 i.97 O.OOOOOOOOOOOOE+00 FUNCTION NR. 6: NR. OF TERMS UP TO WEIGHT 9: 3, UP TO 11: 16 1471 3.279078719711E-04 -1.272193955219E-04 3167 '.35 1584 -1.829349739158E-04 1.497388218227E-04 FUNCTION NR. 7: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 8 136 -3.732133246262E-05 8.538863716984E-05 FUNCTION NR. 8: NR. OF TERMS UP TO WEIGHT 9: 3, UP TO 11: 21 1 -1.042206134195E-03 O.OOOOOOOOOOOOE+00 1357 - 3 96 115 -8.100570165074E-05 2.650719699166E-05 1471 - 3 63 -1.331924530836E-06 129 -8.436483073363E-05 1584 4 80 -3.893187123790E-05 228 5.473293674999E-05 3167 1 37 -6.731097691473E-05 1343 -2.798943302774E-05 FUNCTION NR. 10: NR. OF TERMS UP TO WEIGHT 9: 2, UP TO 11: 4 1 -9.621005515571E-05 O.OOOOOOOOOOOOE+00 1584 4.31 FUNCTION NR. 11: NR. OF TERMS UP TO WEIGHT 9: 2, UP TO 11: 8 1 1.369771303377E-04 O.OOOOOOOOOOOOE+00 1584 -7.13 FUNCTION NR. 12: NR. OF TERMS UP TO WEIGHT 9: 2, UP TO 11: 1471 1.163378737378E-04 -4.513819197066E-05 1584 -7.56 FUNCTION NR. 14: NR. OF TERMS UP TO WEIGHT 9: 4, UP TO 11 18

1 -3.721216569677E -04 0.000000000000E-00 1471 -1. 284288049431E -04 -3.309690220232E-04 FUNCTION NR. 16: NR. OF TERMS UP TO WEIGHT 9: 2, 1 -9.085155274050E -05 0.OOOOOOOOOOOOE-00 FUNCTION NR. 17: NR. OF TERMS UP TO WEIGHT 9: 5, 1343 -5.614062785306E -05 2.342007892272E-05 1357 -7.889757350547E -05 4.317056145094E-06 1471 -7.703729670146E-04 2.989345930936E-04 FUNCTION NR. 19 NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 21 NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 22 NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 24 NR. OF TERMS UP TO WEIGHT 9: 3, 1 -1.313953901498E-04 O.OOOOOOOOOOOOE-00 1471 -4.487231463983E-05 -1.156244439141E-04 FUNCTION NR. 26: NR. OF TERMS UP TO WEIGHT 9: 1, O.OOOOOOOOOOOOE-00 1 -5.040395297243E-05 FUNCTION NR. 27: NR. OF TERMS UP TO WEIGHT 9: 4, 1471 -2.672296690224E -04 1.037022583507E-04 1584 1.276349812562E -04 -1.045472109857E-04 FUNCTION NR. 29: NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 31: NR. OF TERMS UP TO WEIGHT 9: 6, 1 4.188376422942E -04 O.OOOOOOOOOOOOE-00 1357 3.821300675695E -06 7.059178362433E-05 1471 2.374523186035E -04 6.118207816777E-04 FUNCTION NR. 33 NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 37 NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 39 NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 42 NR. OF TERMS UP TO WEIGHT 9: 2, 1471 -9.360777000366E -05 3.632829083831E-05 FUNCTION NR. 46: NR. OF TERMS UP TO WEIGHT 9: 5, 0.OOOOOOOOOOOOE-00 1 1.483266784173E -04 1471 8. 299877925500E -05 2.138273179248E-04 1584 -1.020094581981E -04 -1.247587418520E-04 FUNCTION NR. 48: NR. OF TERMS UP TO WEIGHT 9: 0, FUNCTION NR. 51: NR. OF TERMS UP TO WEIGHT 9: 5, 1357 5.715343993788E-05 -3.219542964396E-06 1471 4.30506366962IE-04 -1.670877885259E-04 1584 -1.660817753460E -04 1.362233432597E-04 FUNCTION NR. 60: NR. OF TERMS UP TO WEIGHT 9: 0,

1584 3167 UP TO 11 1584 UP TO 11

4.7

21 1584 3.2 3167 -5.5

UP UP UP UP

TO TO TO TO

11 11 11 11

2 4 4 10 1584

UP TO 11

4

UP TO 11

15

7.9

1698 3.4 3167 -1.9 UP TO 11 UP TO 11

2 27 1584 -2.5 1698 -1.5 3167 -9.0

UP UP UP UP

TO TO TO TO

11 11 11 11

3 2 5 9 1584

UP TO 11

5.0

17 1698 -5.4 3167 -3.1

UP TO 11 UP TO 11

3 21 1698 -5.5 3167 3.1

UP TO 11

FUNCTION NR. 63: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 67: NR. OF TERMS UP TO WEIGHT 9: 3, UP TO 11: 1 5.363117354167E-05 O.OOOOOOOOOOOOE- 00 1471 2.958292299563E-05 7.620427424884E- 05 FUNCTION NR. 72: NR. OF TERMS UP TO WEIGHT 9: 3, UP TO 11: 1471 1.527416424828E -04 -5.928642651313E- 05 1584 -6.579550158459E -05 5.391670207523E- 05 FUNCTION NR. 78: NR. OF TERMS UP TO WEIGHT 9: 5, UP TO 11: O.OOOOOOOOOOOOE- 00 1 -1.441326636746E -04 -2.984825711335E- 04 1471 -1. 158772743658E -04 1.363949534536E- 04 1584 1. 116486874293E -04 FUNCTION NR. 91: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 95: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 100: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 1471 5.548620020192E- 05 -2.153851626327E-05 FUNCTION NR. 106: NR. OF TERMS UP TO WEIGHT 9: 4, UP TO 11: 1 -5.31078587038IE-•05 O.OOOOOOOOOOOOE- 00 1471 -4.202487604730E-•05 -1.082367388570E- 04 FUNCTION NR. 113: NR. OF TERMS UP TO WEIGHT 9: 3, UP TO 11: 1471 -1.931003458188E-•04 7.496110858058E 05 1584 6.992400143885E- 05 -5.736450029173E 05 FUNCTION NR. 136 NR. OF TERMS UP TO WEIGHT 9 0, UP TO 11: FUNCTION NR. 142 NR. OF TERMS UP TO WEIGHT 9 0, UP TO 11: FUNCTION NR. 149 NR. OF TERMS UP TO WEIGHT 9 2, UP TO 11: 1471 -7. 159550561682E- 05 2.779545288200E- 05 FUNCTION NR. 157: NR. OF TERMS UP TO WEIGHT 9: 3, UP TO 11: 1471 4.816690155803E- 05 1.240378781585E- 04 1584 -4.241178412487E- 05 -5.179900989184E- 05 FUNCTION NR. 187: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 194: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 3167 -1.956351850391E-•04 -4.047196070085E- 05 FUNCTION NR. 202: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 3167 -6.987657799203E-•05 3.377432637896E- 04 FUNCTION NR. 211: NR. OF TERMS UP TO WEIGHT 9: 2, UP TO 11: 3054 -1.292016634430E-04 -7.365209139243E 05 FUNCTION NR. 249 NR. OF TERMS UP TO WEIGHT 9 0, UP TO 11: FUNCTION NR. 257 NR. OF TERMS UP TO WEIGHT 9 0, UP TO 11: FUNCTION NR. 266 NR. OF TERMS UP TO WEIGHT 9 0, UP TO 11: FUNCTION NR. 276 NR. OF TERMS UP TO WEIGHT 9 0, UP TO 11:

4 11 1584 -4.056 13 3167

1.097

1698 3167

7.060 4.422

22

3 5 14 1584 4.480 3167 1.598 19 3167 -1.393 4 12 12 3167 -5.154 19 3167 -1.843 5 7 11 13 3167 4 7 9 12

5.502

FUNCTION NR. 332: NR. OF FUNCTION NR. 342; NR. OF FUNCTION NR. 353 NR. OF FUNCTION NR. 420: NR. OF FUNCTION NR. 431 NR. OF FUNCTION NR. 443: NR. OF FUNCTION NR. 456 NR. OF 1584 -1.608247913607E- 04 FUNCTION NR. 457: NR. OF 1584 -1.226384482797E- 04 FUNCTION NR. 458: NR. OF 1 -8.334247989342E- 04 115 -5.387165301771E- 05 129 -6.630521681319E- 05 228 4.363939691333E- 05 FUNCTION NR. 460: NR. OF 1584 -5.628763275499E- 05 FUNCTION NR. 461: NR. OF 1 -3.206787414494E- 04 1471 -1. 117217803333E- 04 FUNCTION NR. 463: NR. OF 1343 -5.658148038017E- 05 1357 -6.834954253860E- 05 1471 -7.786182567550E- 04 FUNCTION NR. 465 NR. OF FUNCTION NR. 466 NR. OF FUNCTION NR. 467 NR. OF 1 -1.144989501125E- 04 1471 -3.951657523901E- 05 FUNCTION NR. 469: NR. OF 1471 -2.737078978121E- 04 1584 1.140748212161E- 04 FUNCTION NR. 471: NR. OF FUNCTION NR. 472: NR. OF 1 4.778630491958E- 04 228 -4.090754660330E- 05 1343 2. 116263864775E- 05 1357 3.784684324790E- 06 FUNCTION NR. 476: NR. OF

TERMS TERMS TERMS TERMS TERMS TERMS TERMS

UP TO WEIGHT 9: 0, UP TO 11 4 UP TO WEIGHT 9: 0, UP TO 11 8 UP TO WEIGHT 9: 0, UP TO 11 9 UP TO WEIGHT 9: 0, UP TO 11 2 UP TO WEIGHT 9: 0, UP TO 11 3 UP TO WEIGHT 9: 0, UP TO 11 5 UP TO WEIGHT 9: 2, UP TO 11 10 1.320438742647E-04 3167 TERMS UP TO WEIGHT 9: 1, UP TO 11: 1.005072691248E-04 TERMS UP TO WEIGHT 9: 7, UP TO 11: 17 0.000000000000E-00 1471 1.766123906628E-05 1584 -1.165429538026E-06 3167 -3.103812331358E-05 TERMS UP TO WEIGHT 9: 1, UP TO 11: 7 4.607355108958E-05 TERMS UP TO WEIGHT 9: 3, UP TO 11: 16 0.000000000000E-00 1 1584 -2.879513234877E-04 TERMS UP TO WEIGHT 9: 5, UP TO 11: 16 2.357350757641E-05 1584 3.690445660407E-06 3167 3.021149236426E-04 TERMS UP TO WEIGHT 9: 0, UP TO 11 1 TERMS UP TO WEIGHT 9: 0, UP TO 11 3 TERMS UP TO WEIGHT 9: 3, UP TO 11 11 0.000000000000E-00 1584 -1.018366403378E-04 TERMS UP TO WEIGHT 9: 3, UP TO 11: 16 1.062092081275E-04 1584 1.140748212161E-04 -9.353833460232E-05 TERMS UP TO WEIGHT 9: 0, UP TO 11: 1 TERMS UP TO WEIGHT 9: 8, UP TO 11: 21 1471 0.000000000000E-00 1584 2.914315420785E-05 1698 5.071957774306E-05 3167 7.131688385965E-05 TERMS UP TO WEIGHT 9: 0, UP TO 11:

3.12

-2.92 3.22 1.11

1.47 2.77 -5.62

6.10 -9.35

2.74 -2.60 -1.76 -1.04

FUNCTION NR. 477: NR. OF FUNCTION NR. 479: NR. OF 1471 -9.494475429832E--05 FUNCTION NR. 482: NR. OF 1 1.674110732692E--04 1471 9. 491072668524E--05 1584 -1. 032869470778E--04 FUNCTION NR. 486: NR. OF 1357 6. 426705412943E--05 1471 5. 445233716318E--04 1584 -1. 858474227506E--04 FUNCTION NR. 492 NR. OF FUNCTION NR. 494 NR. OF FUNCTION NR. 497 NR. OF 1 5.928676934538E--05 1471 3. 317497704515E--05 FUNCTION NR. 501: NR. OF 1471 1. 895664766738E--04 1584 -7. 313120994269E--05 FUNCTION NR. 506: NR. OF 1 -1.915717655145E--04 1357 -2. 939702916510E- 06 1471 -1. 562745618693E-•04 FUNCTION NR. 515 NR. OF FUNCTION NR. 518 NR. OF FUNCTION NR. 522 NR. OF 1471 6. 725730662768E •05 FUNCTION NR. 527: NR. OF 1 -6.886548111597E •05 1471 -5. 536531175494E- •05 FUNCTION NR. 533: NR. OF 1471 -2.737949427307E •04 1584 8.954643680255E- •05 FUNCTION NR. 546 NR. OF FUNCTION NR. 550 NR. OF FUNCTION NR. 555 NR. OF 1471 -2. 007916254365E •05 FUNCTION NR. 561: NR. OF 1471 -9. 899387908533E 05

TERMS UP TO WEIGHT 9: 0, UP TERMS UP TO WEIGHT 9: 2, UP 3.684462411223E -05 TERMS UP TO WEIGHT 9: 5, UP O.OOOOOOOOOOOOE -00 2.445472582754E -04 1.262291683887E 04 TERMS UP TO WEIGHT 9: 5, UP 3.592602065654E 06 •2.113242321764E •04 1.526122966942E -04 TERMS UP TO WEIGHT 9 0, UP TERMS UP TO WEIGHT 9 0, UP TERMS UP TO WEIGHT 9 3, UP O.OOOOOOOOOOOOE 00 8.546764788241E 05 TERMS UP TO WEIGHT 9: 3, UP 7.357431612583E 05 5.998383105903E 05 TERMS UP TO WEIGHT 9: 6, UP O.OOOOOOOOOOOOE 00 5.296106567376E 05 4.025910377105E- 04 TERMS UP TO WEIGHT 9 0, UP TERMS UP TO WEIGHT 9 0, UP TERMS UP TO WEIGHT 9 1, UP 2.610579095118E 05 TERMS UP TO WEIGHT 9: 4, UP O.OOOOOOOOOOOOE 00 1.426126938159E 04 TERMS UP TO WEIGHT 9: 4, UP 1.062781182764E 04 7.353642083667E- 05 TERMS UP TO WEIGHT 9 0, UP TERMS UP TO WEIGHT 9 0, UP TERMS UP TO WEIGHT 9 1, UP 5.171468065847E- 05 TERMS UP TO WEIGHT 9: 2, UP 3.842919536132E- 05

TO 11 : 5 TO 11 8 1584

TO 11

4 534

16 1698 -6 158 3167 -3 609

TO 11

21 1698 -6 979 3167 3 932

TO 11 TO 11 TO 11

3 4 9 1584 -4 077

TO 11

13 3167

TO 11

20

TO 11 TO 11

3 5

TO 11

9

TO 11

13

TO 11

1 366

1584 1698 3167

1 347 9 462 5 980

1584 3167

5 334 2 112

19 1698 3 510 3167 -1 978

TO 11 TO 11 TO 11

4 4 9

TO 11

13 3167

-7.141

FUNCTION NR. 568: NR. OF 1 6.576301379165E-05 1471 7.093462430717E-05 FUNCTION NR. 591: NR. OF FUNCTION NR. 597: NR. OF FUNCTION NR. 604: NR. OF 1471 2.626869686166E-05 FUNCTION NR. 612: NR. OF 1471 1.157360642614E--04 FUNCTION NR. 642: NR. OF FUNCTION NR. 649: NR. OF 3167 -3.810846419656E--05 FUNCTION NR. 657: NR. OF 3167 3.170862156736E--04 FUNCTION NR. 666: NR. OF 3054 -7.004102344594E--05 FUNCTION NR. 704: NR. OF FUNCTION NR. 712: NR. OF FUNCTION NR. 721: NR. OF FUNCTION NR. 731: NR. OF FUNCTION NR. 787: NR. OF FUNCTION NR. 797: NR. OF FUNCTION NR. 808: NR. OF FUNCTION NR. 875: NR. OF FUNCTION NR. 886: NR. OF FUNCTION NR. 898: NR. OF FUNCTION NR. 911: NR. OF 136 -1.360151430893E--04 FUNCTION NR. 912: NR. OF 136 -8.674026178879E--05 FUNCTION NR. 913: NR. OF 136 -1.647054892450E--04 FUNCTION NR. 914: NR. OF 1 -4.617499985763E--04 FUNCTION NR. 915: NR. OF 136 -3.498873896253E--05 FUNCTION NR. 916: NR. OF 136 -6.480374738577E--05 FUNCTION NR. 917: NR. OF

TERMS UP TO WEIGHT 9: 4, 0.OOOOOOOOOOOOE-00 1.826897123882E-04 TERMS UP TO WEIGHT 9: 0, TERMS UP TO WEIGHT 9: 0, TERMS UP TO WEIGHT 9: 2, 6.764631547104E-05 TERMS UP TO WEIGHT 9 2, -4.493372660120E -05 TERMS UP TO WEIGHT 9 0, TERMS UP TO WEIGHT 9 1, 1.841944388503E -04 TERMS UP TO WEIGHT 9 1, 6.559681868411E-05 TERMS UP TO WEIGHT 9 2, 1.228948208882E -04 TERMS UP TO WEIGHT 9 0, TERMS UP TO WEIGHT 9 0, TERMS UP TO WEIGHT 9 0, TERMS UP TO WEIGHT 9 TERMS UP TO WEIGHT 9 TERMS UP TO WEIGHT 9 TERMS UP TO WEIGHT 9 0, TERMS UP TO WEIGHT 9 TERMS UP TO WEIGHT 9 0, TERMS UP TO WEIGHT 9 0, TERMS UP TO WEIGHT 9 1, 3.112553280943E -04 TERMS UP TO WEIGHT 9 1, 1.984751736206E -04 TERMS UP TO WEIGHT 9 1, -7.21O504680131E -05 TERMS UP TO WEIGHT 9 1, O.OOOOOOOOOOOOE -00 TERMS UP TO WEIGHT 9 1, 8.005183483592E -05 TERMS UP TO WEIGHT 9 1, -2.837178719522E -05 TERMS UP TO WEIGHT 9 2,

UP TO 11:

20 1584 3167

-5.68 -2.72

3167

-1.00

3167

8.37

143167

1.08

UP TO 11: UP TO 11: UP TO 11:

5 9 13

UP TO 11

16

UP TO 11 UP TO 11-

3 7

UP TO 11: 10

UP TO 11:

UP UP UP o, UP o, UP o, UP UP o, UP UP UP UP

TO TO TO TO TO TO TO TO TO TO TO

11 11 11. 11: 11: 11: 11: 11: 11: 11: 11:

UP TO 11: UP TO 11: UP TO 11: UP TO 11: UP TO 11: UP TO 11:

4 6 10 10 4 7 9 2 3 5

•»-»

1 -3.607876656328E--04 O.OOOOOOOOOOOOE- •00 FUNCTION NR. 918: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 136 8.468174599143E--05 -1.937623771502E--04 FUNCTION NR. 919: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 923: NR. OF TERMS UP TO WEIGHT 9: 2, UP TO 11: 1 -1.703466426467E--04 0.OOOOOOOOOOOOE-•00 FUNCTION NR. 924: NR. OF TERMS UP TO WEIGHT 9: 1. UP TO 11: 136 3.104374513392E--05 -7.102657035995E- 05 FUNCTION NR. 925: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 927: NR. OF TERMS UP TO WEIGHT 9: 2, UP TO 11: 136 1.024149225774E--04 4.484099036012E- 05 FUNCTION NR. 928: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 1 1.079149077388E--04 0.OOOOOOOOOOOOE-•00 FUNCTION NR. 933: NR. OF TERMS UP TO WEIGHT 9: UP TO 11: 1584 3.677403434367E--05 4.514010689583E- •05 FUNCTION NR. 935: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 938: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 1584 -4.541909896878E--05 -5.571885350616E-•05 FUNCTION NR. 941: NR. OF TERMS UP TO WEIGHT 9: 1, UP TO 11: 1449 -3.050333781775E-•05 5.936983634480E- •05 FUNCTION NR. 942: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 953: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 957: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 962: NR. OF TERMS UP TO WEIGHT 9: 0, UP TO 11: UP TO 11: FUNCTION NR. 978: NR. OF TERMS UP TO WEIGHT 9: UP TO 11: FUNCTION NR. 983: NR. OF TERMS UP TO WEIGHT 9: FUNCTION NR. 989: NR. OF TERMS UP TO WEIGHT 9: UP TO 11: FUNCTION NR. 1017: NR . OF TERMS UP TO WEIGHT 9: 0, UP TO 11: FUNCTION NR. 1024: NR . OF TERMS UP TO WEIGHT 9: 0, UP TO 11:

o, o, o,

1471

-3.20

1584

8.81

1449

-7.46

7 4 8 6 3 7 8 4 2 6 7 4 3 3 4 2 4 3 3 2

-Table 6.8 Output of FURI1., routine FURRAP. term 1

290

6.4 6.4.1

Expansions

Required for the Equations

of Motion.

Collinear Points

Case

T h e Final C o m p u t a t i o n of t h e P e r t u r b i n g T e r m s Determination

of the

Coefficients

As it has been said the direct effects of the Moon and the planets have been developed as ^2fijk(t)xlyjzk, where fijk(t) are time-dependent functions and x,y,z the normalized coordinates of the spacecraft. Each function fijk has been analyzed using an FFT method as described before. In this way we have detected the main harmonics as already explained. Next step is the computation of the right coefficients. To do this we use not the rough frequency determined by FFT but the right one which is a combination of the mean motions of several arguments related to the motion of the bodies of the solar system. Program FURI1, routine FURLEN, performs these computations. The coefficients are computed argument by argument using the trapezoidal rule for the integration and starting at an epoch such that the related argument is zero (modulus 2-K) . The routine analyzes, for a given frequency, all functions involving the frequency. Tables 6.9 to 6.11 give the values found for the Sun-Barycenter problem, case L\. For the interval of integration we have taken an integer multiple of the period related to the analyzed frequency. On each revolution a fixed number of equally spaced points has been taken. Several different sets of number of revolutions and points per revolution have been used until a good agreement is found. This is done to prevent from aliasing and to filter the influence of too near frequencies. 6.4.2

Conversion to a Suitable Form for the Equations Quasi-periodic Orbit. Numerical Results

of

the

The data obtained from routine FURLEN of FURI1, are used to prepare the files to be used by program EQUAP. This is also done by program FURI1, routine PREPAR. Routine PREPAR not only prepares these files of data but also puts them in a suitable form. First of all the data are filtered in the following way: for those functions which, for a given frequency, have coefficients of the cosine, sine or independent terms, which are less than a given quantity (5.D-5 for the Sun-Barycenter problem and 3.D-4 for the Earth-Moon problem as standard values), these terms are skipped. After this filtering some modifications are done to the phases in order to have only terms of the cosine or sine type for each function. If the modification of the phase is less than a given value (0.05 as standard value) then it is not modified in order to have a reasonable number of different frequencies phases. This modification shall take into account the slow varying frequencies not considered during the identification as mentioned in 6.3.4. A sample of results of this routine is given in Tables 6.12-6.14. This output has also been used to check the goodness of the Fourier developments. Routine FUCHEC of program FURI1 compares, for each function the values obtained by directly and the ones obtained using its Fourier development.

CURRENT VALUE OF SPACECRAFT SECTION/MASS IN m**2/g : 0.00 THE MODEL OF THE SOLAR SYSTEM IS: 0 1 11 0 0 0 0 0 0 1 0 0 EARTH+MOON-SUN SYSTEM THE EQUILIBRIUM POINT IS LI N

FREQUENCY

PHASE

N

FREQUENCY

2 3 4 5

0.6254989350070546D+0 0.1250997870014109D+1 0.1876496805021164D+1 0.2501995740028218D+1 0.3127494675035273D+1

0.1417296496991782D+1 0.2834592993983564D+1 0.4251889490975374D+1 0.5669185987967127D+1 0.8032971777792659D+0

7 8 9

0.37529936100423 0.43784925450493 0.50039914800564 0.62549893500705

IFRE = 1 1 2 21 IFRE = 1 1 2 21 IFRE = 1 1 2 21 IFRE = 1 1 2 21 IFRE = 1 2 21 IFRE = 1 1 2 21 IFRE = 1 1 2 21 IFRE =

1 ENE = 0.6254989350071D+00 N = 512 M = 43 AO = 0.1417296496992D -0.1120886924154275D-02 0.4568607260176857D- 07 -0.449362158 -0.1196479538355846D-06 - 0.2287077047278306D- 03 -0.897863065 2 ENE = 0.1250997870014D+01 N = 512 M = 43 AO = 0.2834592993984D -0.5151970860167298D-03 - 0.1523348026455242D- 06 -0.449298462 0.2684587364771737D- 03 0.1011417690906884D-06 0.192943582 3 ENE = 0.1876496805021D+01 N = 512 M = 43 AO = 0.4251889490975D -0.4043917496626441D-03 - 0.1476774308438172D- 05 -0.448519842 0.2481969087447426D- 03 0.306696901 0.4205589354758795D-06 4 ENE = 0.2501995740028D+01 N = 512 M = 43 AO = 0.5669185987967D -0.3159582413954282D-03 - 0.1114494853507741D- 07 -0.449651794 0.2138022975919758D- 03 -0.2173418973564046D-07 -0.655495224 5 ENE = 0.3127494675035D+01 N = 512 M = 43 AO = 0.8032971777793D -0.2443894462194629D-03 - 0.2463549806484620D- 05 -0.450159077 0.1766164457081246D- 03 -0.124146665 -0.1135706981452381D-06 6 ENE = 0.3752993610042D+01 N = 512 M = 43 AO = 0.2220593674771D 0.5047182570874378D- 06 -0.1869395435249823D-03 -0.447147472 0.1412067412636475D- 03 0.104955293 -0.1413061995308833D-06 7 ENE = 0.4378492545049D+01 N = 512 M = 43 AO = 0.3637890171763D 0.6674104161383844D-0.1433505043383679D-03 07 -0.449084595 0.1124353587183572D03 0.405122784 -0.3262371387560192D-06 N = 512 M = 43 AO = 8 ENE = 0.5003991480056D+01 0.5055186668755D

1 1 -0.1093787811970052D-03 -0.8181604075729483D-07 -0.44965821 2 21 -0.8292081957574832D-07 0.8801665124480810D-04 -0.99142778 IFRE = 9 ENE = 0.6254989350071D+01 N = 512 M = 43 AO = 0.1606594355559 1 1 -0.6236072683115891D-04 0.4765894769308156D-06 -0.44714669 2 21 0.3148931796656982D-07 0.5215829136335666D-04 0.13994959 Table 6.9

Output of FURI1, routine FURLEN. Terms coming f

CURRENT VALUE OF SPACECRAFT SECTION/MASS IN m**2/g : 0.00 THE MODEL OF THE SOLAR SYSTEM IS: 0 0 11 0 1 0 0 0 0 1 0 0 EARTH+MOON-SUN SYSTEM THE EQUILIBRIUM POINT IS LI N

FREQUENCY

PHASE

N

FREQUENCY

1 2

0.9156600625595784D+00 0.1746980187678735D+01

0.1146036806276883D+01 0.1693250981649915D+01

3 4

0.1831320125119 0.2746980187678

IFRE = 1 1 2 21 IFRE = 1 1 2 21 IFRE = 1 1 2 21 IFRE = 1 1 2 21

1 ENE = 0.9156600625596D+00 N = 512 M = 43 AO = 0.1146036806277 0.1497094581445498D-03 - 0.9262426601099436D- 06 0.35038342 -0.8673677030599162D-06 - 0.4916776609049902D- 04 0.67743889 2 ENE = 0.1746980187679D+01 N = 512 M = 43 AO = 0.1693250981650 0.1834791052856497D-03 -0.3465805252685086D- 04 0.35151261 -0.3423626455988194D-04 - 0.1795325882027494D- 03 -0.20413491 3 ENE = 0.1831320125119D+01 N = 512 M = 43 AO = 0.2292073612554 0.1028123963021951D-02 -0.1487783297976936D- 05 0.34998646 -0.1487648962138785D-05 - 0.1012094375141046D- 02 0.12353840 4 ENE = 0.2746980187679D+01 N = 512 M = 43 AO = 0.3438110418831 0.6292336594479343D- 05 0.24007429292973580-03 0.35018770 0.5449584159753695D-05 - 0.2370064468439855D- 03 -0.83907730

Table 6.10 Output of FURI1, routine FURLEN. Terms coming

N

CURRENT VALUE OF SPACECRAFT SECTION/MASS IN m**2/g : 0.00 THE MODEL OF THE SOLAR SYSTEM IS: 0 0 11 0 0 0 0 0 0 1 1 0 0 EARTH+MOON-SUN SYSTEM THE EQUILIBRIUM POINT IS LI FREQUENCY PHASE N FREQUENCY

1 2 3 4 5 6 7 8

0.8869742675313748D+00 0.9999522207438958D+00 0.1053724902883324D+01 0.1773948535062750D+01 0.1048134000056199D+02 0.1059431795377451D+02 0.1131454158595393D+02 0.1148129222130588D+02

0.3432635595834144D+01 0.6231395122225130D+01 0.2703571879556000D+01 0.5820858844887022D+00 0.1711238098207887D+01 0.4509997624599100D+01 0.2388511629531575D+01 0.1659447913253431D+01

IFRE: = i ENE = 0.8869742675314D+00 1 1 -0.1465603760382508D-03 2 2 -0.7580672476865840D-03 3 8 0.5038787102203607D-03 4 458 0.1634594679323987D-03 5 26 0.5820156681546279D-01 6 914 0.5946077797541867D-03 7 928 -0.2087532455016062D-02 IFRE: = 2 ENE = 0.9999522207439D+00 1 1 0.1499880931309437D-03 2 2 0.6273420837422142D-03 3 8 -0.3472220028128979D-03 4 458 -0.1337186066258772D-03 IFRE = 3 ENE = 0.1053724902883D+01 1 4 -0.1879532441927055D-06 2 7 -0.9870870867078197D-06 3 911 -0.4496280151211255D-07 4 912 -0.1879532441927055D-06 5 913 -0.3784652953238619D-03 6 915 -0.4935435433539098D-06 7 916 -0.1985621943854786D-02 8 918 0.1135609467175608D-06 9 924 0.7028507258934257D-06 10 927 0.9904034385888296D-03 IFRE = 4 ENE = 0.1773948535063D+01

9 10 11 12 13 14 15

0.123682664888372 0.132552407563686 0.237365807569306 0.238495587101431 0.247365329776745 0.371047994665117 0.494730659553490

N = 800 M = 43 AO = 0 .3432635595834D+ -0.1359608944934010D- 05 0.18147320 -0.7005563519696952D- 05 0.74879308 0.1485246725045678D-•04 -0.43940706 0.4876246443698255D-•05 -0.17128544 0.2077660135634228D-•03 -0.35026288 0.2129317076000503D--05 -0.57750764 -0.1995586454242680D--04 0.14761114 N = 800 M = 43 AO = 0 .6231395122225D+ -0.1923057975848962D- 06 0.18136856 -0.1028674263119758D--05 0.74833630 0.1019287036218621D- 05 -0.43868749 0.2978236746426478D- 06 -0.17101747 N = 800 M = 43 AO = 0 .2703571879556D+ 0.1480686572052652D- 02 0.31919082 0.7963337706221116D-•02 0.19901140 0.3482857653243684D-•03 0.63204001 0.1480686572052652D-•02 0.31919082 -0.5944864529783911D-•07 0.45732487 0.3981668853110558D- 02 0.99505703 -0.3171761679250696D-•06 0.25444270 -0.9123632340780074D- 03 -0.17789207 -0.5792984540910950D-•02 -0.12838019 0.1620711187674213D-•06 -0.12658823 N = 800 M = 43 AO = 0 .5820858844887D+

to o i a o <* cs i•* oo co

294

cs o i Q o •rf en oo LO cs o oo CO tH

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Expansions

Required

I I I Q Q Q O

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for the Equations

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• * •iH •*<•CN CO t H • ^ o oCO oCN oCO oi H o oCO ot H o•>#oCO oCM oLO oCN o<•oCM o ot H o• * ot H O*F oCM oCO ot H * *•CO CN ^ ^f • * o o o o o o O o o o O o o o o Q Q a a Q Q Q a a a Q a a a a a a a a Q a a a a 1=1 1=1 1=1 1=1 Q a a a a Cl Cl Cl Cl O CN CD CO K t~ t H CI i^ CO LO t H i H o CM CM CN CN LO 00 00 t H r~- i^CM CO CO CN oo LO oo cn CO •iH i H cn ^f t H **oLO tCOH ccnn cn t H 00 **•on CO 1^ CO Ol LO LO cn CN t H CO cn LO CO LO t H Ol 00 cn r-- Ol 1^ * • CO teHn r- * t H t H > * ^" LO LO LO oi i^ CO LO CO C3 CO CO cn i^ CO CD LO Ol co ^ CO co 1^ o co tcn cn t H CO icn cn LO t H LO co co rH cn CN H 00 CO 1^ LO CO CO h- r~ Ol CD oo foi c n c n c o oo en cn *e c n co cn r ^ ^ t H H ) LO LO OO co trH^ C LO O Tcn LO LO tH CM c CO CO CO o CM cn h- cn o t H LO o 1^ LO o tt HH <3< n Ol o o co LO ^ o LO cn co tH CN tH LO CN O O cn CO co t^ LO co LO m Ol CO co o CO o I-- o t H o cn t H 1^ CO o CO t H Oi CM co 00 LO CO •^ct Hn CN T-t CO CN t- LO LO C ) t^ CO CO • * 00 CO CN r^ o cn CO CO cn o LO o o 00 1^ Ol CO CO o ^ " t H t H CN CO LO Ol LO LO CN CN CO LO CM 00 O LO LO h- ^ ^f oo CO CN ^f co •tH co **CN cn 1^ o LO oo OO c> 1^ o hcn co r- CN co ico H 00 1^ t H CO o 00 00 CO O CO LO CO CN t H cn 1^ CO CO CN o ^ OO KC ) CN cn 00 o CN *H*oCN OO t•H*oi^ ^ ^ CN ^ t 00 CO CN 1^ 1^ l»- cn CN t H o co i»- h- CN o co oo 00 o CX) •*co ^ cn co i H CN CN o Ol Ol LO LO ( IN LO < CN CN Ol Ol co LO CO 1^ co OO LO cn CO \T) CO m o> o c n oo cn co < * > * * • ^ C ) cn tH CD CO CN CO LO co CN co CO co LO LO i^ CN CO CO LO oo LO CO Ol CO *f cn c n c o r ^ en r ~ cn o co o o o oo ^ < ^ * t H* C ) LO Ol cn o oo O t H CN CO 1^ LO Ol CO LO CN 1^ CO 1^ en 1^ CO CN t H LO a> cn O tf r~ CO r^ 00 OO o cn > **CO >*cOln ttoHH tO1^H ^ iH tH CO LO C7> CO oi oi CO 'j* CD CO tf CO Ol Ol LO CN CO LO cn l«cn LO t H i^- CO t H CO iH t H t H CO t H CN C ) co CN o CN co t H CO i H cn CN •iH co r- CO oo co co ^ r~ • * CN co CN CN CO *f t H *f LO o o o o o o o o O o o o o o o o o o o o o o o o o o o O o O o O O O o o o o

• *

-iH LO t H •iH to •tH t H LO o *f i H o tH I-- CD CN CO CN CO t H LO CN t H LO o 'J' CN t H LO t H t H o a CN • > * " * • * o o o o o O o o o o o o O1 o o o o o O1 o oi o1 o1 o o o oo o o O o O o o o Q a Q a Q a a Q a a a a a a a a Q a 1=1 a a a1 a a1 a 1=1 a a a CN Q Cl 1=1 a 1 1=1 a1 a LO t H Ol IN 00 + t+H H CO CN O C) CO CN CO LO t+ CO 1^ CN i^ CO •<* co t~ t*t~ cn r~ o oo c n oo cn ^ r- T H o f CO oi ^f *#o LO CN CI CO co *r LO ^< co o CN 00 CM CN CM 00 LO cn *c o ^ cn co CO tH tH t H t H < * CM CO LO CN LO ** co CO CN CN ^f hCN CO LO LO CM CD hCO o c o co c n o co o CO CO i^ ( T-i i H t H *J" OO H Ml 1^ CO CO cn CO 00 LO o cn 1^ cn K r~ CN co Ol •*CM tC3H Ol co o ^~ ^ CO r r~ ^
o en r^ *f LO •* co CN Ol T H

D+0

o o oI oI oI oI oI o oI o o o o o o o o o o o o o o o o o

D+0

D+0

D+0

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• *

•<*

• *

• *

•>*•

• *

• *

1

• *

• *

• *

• *

• *

of the Perturbing

Terms

• *

LO

• *

o

• *

LO

• *

• *

o

• *

• *

LO

1

1

o

• ^

• *

The Final Computation

• *

1

to to o

• *

CN O

• *

• *

• *

• *

• *

• *

• *

• *

800

CN O

• *

• *

• *

• *

• *

• *

• *

• *

• *

• *

• *

• *

800

CN O

• *

• *

• *

• *

297

Q Q CI a + + + + LO to CN en tH CN tID CO tH CO a > CO CO 00 o CO CN cn cn iH O o cn 00 CO CO CD io tH LO CN co tH CO o to CO tH tH co CN CN CN CN CN tH LO h- LO CN tH LO tH CN tH LO CN LO tH CN to CN CN to iH CN LO "** to CN CO > *^ LO iH CN tH to LO LO LO LO 00 o CO co o o o en t o co CO oo r- LO o CO e n cn c o o r cn cn * * LO CN LO 00 oo LO 00 1^ LO t~- o r- O cn LO tH CO CO H tH H LO o H o cn en e n e n oo co oo o o r ~ r H cn n LO co CO LO c i^ LO c l~- cn cn to 00 o tH to en cn O 00 o to CN co co CN 00 cn 00 CO tH CN o CN o CN oo CO tH to LO CO LO CO O tH tH cn r- r- r~ r~ «*to LO o r- LO tH LO CO >*cn CO O to en to oo o CN CO tH tH CO tH CN CN CN t •# LO 00 t^ CO O o •5t< cn to c n o co en tH cn o c n en r ~ o o t o o to tH to t- o CN co 00 o 00 o r~ LO cn CN + + t- oo tH • o o en 00 r- + o o CO LO + CO tH o*CN to LO co •tf » * CN tH CN OO LO to 1-CO tLO f~ Ct CN tH 00 CO CN to to Nco CO o tH LO cn LO oo o CN co en to ^ cn Q LO tH CN tH a to 00 LO LO CN to to O oo co co cn CN to co o co cn e n t o cn ro o tH to to LO r~ LO LO to o CN tH oo r- co o CN tH LO LO CO t CN cn CN tH 00 to 00 t-tH fLO cn c n to r cn oo en r en r o t o to c n r~ > * tH tH CN CO cn 00 to CO to T H cn t*-. CN r- LO T H oo 00 CO 00 LO t-- cn N- LO o t-- r- t-- cn cn CN tH CO cn CO CO o to LO h- to o o CN t- CO tH LO LO LO O O LO LO r- tH CN cn o tH r~ O o o cn rto CO 00 ^ tH iH tH i-CO OS tH CN CN 00 f- CN to CO to to en tH O t-- o LO LO iH oo N- OO o h- tH o CN co tH ^ oo 00 • *00 CO to [-- en CO 00 CO f- co t- iH tH 00 o to co CO O r- tH tH CO t*- CO » h- LO r- tH o tH CN cn tH tf tH cn r~ o ^ tH CN <*CN tH tH T H i^iH tH tH to tH to tH tH CO tH to CO vH rH 00 r t o r co cn **^ tH ^ ^< CO 00 to LO o o d O di d d d di di CN d di d1 di d CO d d di d di o1 o1 oi o d di di o1 d di di d d di cn tH i o LO CO o ^ CN to cn cn CN T H LO tH •"3* o iH CN co CN CO LO LO iH CN CN CN co CN o CN CN CO LO CO LO CN • * < ^ • * " ^ o o o O O o o o o o o o o o*O ^ o ^ o * o o o O o 1 + o o o 1 o o o 1 o 1 1 o o o 1 o i 1 o 1 i o 1 1 o 1 o i i i 1 i i 1 a o CI II a a1 a1 a1 CI Q ai CI ai CI II Q Q CI Qi Q a Q a1 a1 II a a Q Q a a a a1 o a II Q £5 + LO tH CO CO LO LO LO CN to CO CO CO LO LO to to fO to cn CO CN tH en LO oo tH 00 tH co to tH cn *f OO tH LO iH LO O tH tH o oo tH tH t-- o LO co to to 00 to o en CO to r- LO cn to to tH o CN o o CN cn LO o tH to t^ * tH CN CO o LO LO cn to CN 00 CN LO LO to CO t~to l~CO cn 1" cn cn 00 to to o oo to tH o LO r- CN cn CN cn ^ o tH •*to tH fCO CN LO CN fCO o tH oo to o 00 o LO II tH •<*LO oo CO o II >*•to r- LO cn to cn CO -a- CO II II CN CO tH 00 00 r ~ oo c o o r- co r- t^ to to cn t o o to cn r~ *H*CN o LO T tH LO * to tH cn tH CO to CO fCO LO CN to t- 1* LO e n n r r ~ o oo c o £ tH * • * < * iH cn > LO f- £ tH to CN CO CN to £ OO f- to o to to £ •*oo LO CN tH LO c CO oo to e to to t o o oo o cn o o c n o o c n cn n CN iH ^ tH IO tH iH CN o CO oo f- r-- iH LO to o LO co cn oo co CN to co o cn CN to cn r- iH cn < • ^ < < * tH tH tH CN CT1 LO I-00 CO to to 00 hCN to 'l' oo LO LO cn CO f- LO t-- I-ro o f- t^ ^ LO o to tH 00 00 CO CN cn to en CN cn iH tH O LO en 00 >* Hi-- o * T H en l~oo tH CN r~ tH CN cn LO co ii a >* CN iH LO II co cn <*iH tH II T-i tH r- tH h- tH f CN tH tH a o1 o O o o o o o o o1 a oi oi o o o a o o o O o1 o o oi o1 o a o o oi O o1 o1 oi o1 O

• *

• *

r-

800

•=)•

• *

• *

• #

• *

oo

• *

w

• *

• *

• *

• *

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tH CN tH tH to tH CN CN CO tH tH CO 00 00 CO tH tH CO to 00 h- tH 00 CO CN to tH tH CN CO LO to tH tH LO to f- to CO LO to CO 00 II CN CO vf LO fII II CN • * M* to II •a1 T <•
r-

• *

• *

CN tH CN CN tH LO CN CN tH LO tH CO *LO >*tH tH CN CN tH 00 t- o t^ LO LO O to co r- oo ro c n t o r ~ r ~ o e n cn < * ^ > * T H i»- CN to CN CN CO CN CO LO tH CN LO LO cn iH CT> n co CO CO LO o tH cn o O tH r~ ^ 00 r- LO LO c o r- tH c r co 00 tH r~ O tf CN to CO 00 o CN - to to o CO «*•oo O oo tH **"o co co CN tH r- CN CO CN iH CO o tH CN CN LO *J* CN CN to CN CN CO LO LO CN CO LO CO o t o so CN CO O LO CO tH tH LO LO CN o LO t o r ~ o cn tH tH iH CO LO iH CO LO LO LO LO CO ^f LO tH to to cn 1^ to tH o to en o CN CN cn co t^ to r~ tH • oo r^ ^ CN >*OLO N- cn CN •^CO LO 00 CO to LO CN LO tH CN 00 to CO 1^cn « s * co r to to > * < * > * *»o•CN CO CO tH CN t-- cn o CN O «*NCN co to l>- tH 00 CN en CN « LO LO O t to to r- tH co oo en to r-1 fcn co o ^ >* tHa<00 CO T H tH CN LO CO LO CN CO CN tH •a LO CO 1" to 00 CN cn to 00 cn LO oo I-- to r-- co •*CN **•CN t- cn LO CO CN tH o cn tH CN CN to CN LO CO t~- CN to to LO o cn CN LO 00 00 O c o o r ~ e n cn o o o o cn > * ( to O O O 00 CO r~ r- to o co o tH en O LO r- 00 00 o to tH o c n to to CO co to CN iH LO < # tH LO ftH LO f00 CO tH CN 00 00 CO to 00 00 O co CO iH to to oo o Ncn tH o LO en to cn CN o lO o to r~ *» * w ^ o o o o o o o o o o ao o O o o a o O O o O o o o o o a o O o o o O o o o 1 1 1 1 1 1 1 1 1 1 1 a 1 i 1 w u 1 1 1 u w o co CN 1-co iH • <*tH tH

CN O

800

rj< CN CO i-l

298

H

y-i

i

i-i

1

• *

o * to o o oo oo oo o o

Required for the Equations of Motion.

T-t

TH

• *

i 1 1 1 1

r l r H O S 1 O 1CN CN CN i H

to co oo o s co co CN ^ cn <* oo oo to CN «tf O ) •* O l O r t s- oo oo C N O l S O l •* cn co en co to s- LO O cn in to i-i cn cn o to s- s o n r- oo o o to t~- co oo i< co to cn cn o cn I H oo co to to co iH CO s- t- cn to CN P- iH to T H S O CN LO C N o co oo co to o *J< T H CO CN »* L O to 0 0 M J I iH iH co siH *f iH tO LO IO LO co r- cn ^p 00 iH O i* CN CN iH o o to co L O CN rCN CN O LO ** •* iH o to L O "tf oo cs cn S- CO **• LO cs cn oo r~- o iH CO O S 0 1 O CO to oo L O to cn co o tO CO O B S O CO H cn L O to co T-t T-l co t» co o oo CN LO "4< L O r- o •* C O S CN CN i-l ^f CN c n i-i CN CO to co to o o o o o o o o o o o o o o

I a

• *

TH

i

Collinear Points

i 1

1

+

Case

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I

LO

o + CI to

oo rcs rcs *f CO

o CN CO O

a a a o •* en i* to s- co cn CN co to c n •* s - ii ^p «* T H o i o o en oo s- r- iH CN co cs to cs to to to CN co en to o iH CO 00 TH to to o to r-- C N to cn oo to to to o oo co to co s cn CN co s co to cn i-i to E-H o t~- co r- cn <j< I f l L O O I H cn co cs «* cn CN to o cn • C I D I O O cs to cn oo co ,H •* i< cn L O to LO tO 00 CO CN S- to o cn o iH CO M r l S s- cn CN CO CO CN CO to to oo s- iH LO oo to N O I D O CO to O CN i^ LO cs to + s- o N I O S LO i-l O O iH CO co t o t o Q i H • * r - t o i n itf tO tO iH CO CO CN to S 5B CN tO O h - **• to cn o o co o tO ' f • * o in en **• CM cn T H cn cn CN to cn >* es t o t o t o CN S- 1* O iH CS o o s cs < ( * U ) 0 O r t n o o o o o tO o io-l io-l oi-t o o oCo So to o co cn

a

O r t H o o o o o o o o o Ool o o • I

O •* CO iH CN tO CN L O C N tO i-l *f CN rt LO ^ CN i-i « r c o i-i O O O O O I 1 I I I I I I I I I I + I I + i i I I i i i a CI CI ci a ci Ci C I a a Q Q Q a ci Q Q Q Q O Q Q Q

1

o o

Expansions

i

o o oo oo oo o I I I I I I I i + a o Q a Q a Q a a cn to o T-l CO CO CN CN iH to <*• to CN 00 CN LO oo «* o in oo 00 CO iH 00 O CN to ** oo 00 t- s- ^ LO co to to C N s- oo L O en i-l H co en co iH 00 i-l O •* iH CO iH to oo o o i< CN tO S Cn O i-l CN co to s o n O < • *f CN ocn co co O 00 CN co to O CO L O < * O l O S LO 00 o o H t O r H en en co •* r- to o co to o t- iH LO CO o O ) T H to co iH LO 0 0 C n «tf T H iH o oo s- o O iH oics s i-l CO T-H en o oo o o o o o o 1

iH

• *

• *

• *

O i I

00 00

o O I

• *

• *

• *

T~i

• *

y-i

T-l

• *

o

T-i

y-i

T-i

i

o o I I

TH

TH

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o

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1 *

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o

o

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TH

IH TH

o

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TH

IH

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f-i

TH

o

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TH

o I

• *

TH

• *

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TH

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TH

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i-i

i

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TH TH

TH

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T-*

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TH

TH

1 cn p IH 1 t— co

f-i

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o

TH

i *

TH TH

o

T-t

TH

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TH

s U

s

O i H c N c o ^ i o t o s c o c n o T - i c N c o ^ i o t o s ( » c n o i H c s c o ' 4 * L o t o s o o c n o i - i c N c O ' t t , t o t o U , THi-lTHT-liHiHT-lTHTHi-lCSCSCSCNCSCSCSCSCSCNC0COC0C0COCOC0C0COCO'i'^,iS,'d,«a*'3,1,M

^ s - T H C N t o w c N c o t o c o c n H c o i - - c n cs N C N t O T H t o s c o i - i o o » i , c s c, n s t o C S C N C O ^ i ^ t O S - r - O T - l i d t O C n O T H L O L ,O t O t O, t O, t O t O t, D l H - O OOOOOCNCOtOtOOTHi4 tOtO THTHi-lTHTHCNCN«4 iJ'»J '4 <'"*'3 «a 1,"tC'i't010LOLOtOLOtOtOtDtOtD

o I

o co i H co o co o i-i CO cs o T-l i H co to to T-1 nf i H •* o ^f o ^ i CO o i H «5f CO o i H CO cs CN o Q o o o o o o1 oi o o1 o oi o1 o1 o1 o1 o1 o1 o o1 o1 o o1 o1 o o o1 o1 o O o1 o O1 o T-l + io o1 oi o + 1 1 + CI + CI + + + + + Cl + Cl Cl + Q Q+ LO Q+ o CI CI a CI CI Cl Q+ Cl Cl a Cl Cl Cl Cl a a Q a o CI Q a to a Ci Ci 00 a tO i H S CN «5» a a CO to CO to «* cn r~ c n i-l c n cs c s o c o s t o co to c s c n t o o o rt o • iH < •& i-l O LO to i H CN CO CO to «*f- CO CN 00 to CN to CO CN to CO T-l to to o to c n cs cs oo " * T-l cn < * i-i cn CO cs CO to CO rto o cs to o co to o r- s r~ cs s- e n•t H i H cs to CN •*to cs ro cs o co oo cn «* «* CN n to cn cn oo s- T-l oo co CN s- i H to 00 CN c ^ oo t~ to oo o cn o CO 00 to cn •*o i H CO to ^ 1< iH i H CO CN CN « J 1 CO to 00 CN O r- c cn n c n c n t o c s rr « c o t o o rrt o t o o c o o o c n c n o o s o c o CO to 00 t H t- o s- cs o CO I-- cn s- o CO cn to f- o to to to so co LO CO ^ oo ii HH cn tto SCN 00 o CO CN to CN to CN to CO to CO to to s- 00 c o c n s c n o c n t o c n cs r c n cs o o o o s<* « * CN 00 i-to to 00 to 00 O CO CO •a1 00 o CO o t o c o e n c n t o r ~ o o c n e n e i H co o "5" i-i ^ < to « 4n to 00 CO 00 co cn o s- to o cn o to to CN CS 00 en CN cn OS- SS- CO CO CS to CO oo to CS CO o CS CN to i-i co r~ o cn <•CS CO en cn to CN t-- cs o co co o T-l cn to cs to to CO cn cn to CO s- co co <* i H to r- cs o to CO o cn o soo CD T-l r- cn co o to cs i H 00 es to oo s t«* ! • CN CS co to LO O CN en cs 00 co to s to CN cs CN co LO o co CO o CO LO o * co LO ^<00 co co LO •*ctoo ctos <*"to oCO OO i-l • H LO CO CN CO to if to 00 *f cs c n c s c n s c s t o o c o o re n s c n c n c s t o o iH O 00 c CO 00 to to CO c n o cs co "*i-l to cs o cn to o OO es •tf o cn o to o CO r- CO CN T-l co cn co to oo T-l n II s to to i H «# co co cs 1-1 •tf i H <*to s co CO cn cn cs CN cs CO u l«-

T-l > CO O *f 1-1 CO o CO T-l o co cs o co cs CN o cs to tf o co o i-i co cs o T-H co o1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o + 1 + 1 + + 1 i + + i + 1 + 1 1 1 1 1 1 1 + 1 i 1 + 1 1 + + i i + 1 + 1 CI Q Q CI CI Q Cl Cl Q •* a LO o Ci i-i n a a Q Q CO a CI 00 a Cl a Cl Cl Cl a O a Cl a a 00 a Q a Cl Cl II a LO o CN S- to to to LO T H CO co CO cn to oo co c to 00 CO c to to CO to to o o t H So cs CO cn iH CN O CO LO ioHo LO to CO t14* to CO CO to i-n - s- S icHo to CN to o o t o c n c n sscn CN o ^ < iH 00 cs CO to CO s CO c CO cs oo t I-- to o s- cn CO cn to o co S CO o CN n co LO 00 co CN o o S t H cs to cn ^ i to s to LO to
, 270987! 800 +02

0.1813599 -0.4296001500950472D-05 0.7482296 -0.1746457701489708D-04 0.5722690 -0.5647773376616640D-02 -0.!5347609550109189D-05 0.3051915 -0.3435390242743685D-01 -0.:2386014530002394D-04 -0.4394483 0.2143166922440479D-04 -0. 1475003516322373D-01 -0.2717347 -0. 1045664699694068D+00 0.1258395133862314D-03 -0.2222402 0.1389370246286312D-04 0.3051050966554530D-01 -0.1662568 0.2469317521874746D+00 0.7710288846286750D-04 0.7445776 -0.5124613455732084D-04 0.5594097789939282D-01 0.6140892 0.!5090696973344658D+00 -0.3938993733018322D-03 0.5584313 -0. 1569375055990127D-04 -0.9314486676242559D-01 -0.1080505 0.8968831044210816D-04 -0. 1457787959735107D+00 0.1352331 -0. 1472045056631368D-05 -0.1116157605227783D-02 -0.1713378 -0.'4862475698548866D-02 0.8779442961347989D-05 -0.8788967 -0. 2950007032644746D-01 0.4286333844880958D-04 8272482274555480D-05 -0.1047704 0.1294199838470504D-01 0.4142064 0.2769546717299560D-01 -0.3046381520661902D-04 IFRE: = 15 ENE = 0.4947306595535D+02 N = 800 M = 43 A0 = 0.1518778114812 0.4621899 1 17 0.!5495126019831779D-06 0.1480773874148775D-01 0.7471148 2 0.:3058685060092381D-01 0.4131688182853522D-05 31 -0.8224339 0.1459330695493099D-04 -0.5685811618357794D-01 3 51 9801401709497803D-01 4 78 -0.!1351607816500874D-01 -0.3623948250338392D-04 -0.1084274 0.4153952 5 472 0. 0.2767869916985371D-06

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 6 8 14 17 27 31 46 51 78 456 458 461 463 472

0.1507534997960027D-02 0.'7673080032467071D-02

o.;

Table 6.11 Output of FURI1, routine FURLEN. Terms coming fro

300

Expansions Required for the Equations of Motion.

Collinear Points

Case

CASE L I : ORDER = 10 GAMMA = 0.1001090475489518D-01 TERMS COMING FROM BODY 2: NUMBER OF FREQUENCIES = 9 1 0.62549893500705D+00 0.14172964969918D+01 2 0.12509978700141D+01 0.28345929939836D+01 3 0.18764968050212D+01 0.42518894909754D+01 4 0.25019957400282D+01 0.56691859879671D+01 5 0.31274946750353D+01 0.80329717777927D+00 6 0.37529936100423D+01 0.22205936747712D+01 7 0.43784925450494D+01 0.36378901717629D+01 8 0.50039914800564D+01 0.50551866687547D+01 9 0.62549893500705D+01 0.16065943555585D+01 EQUATION NUMBER 1: NUMBER OF TERMS = 10 -0.4493621586696200D-03 0 0 100 0 -0.1120886925085356D-02 0 0 100 1 -0.5151971085381031D-03 0 0 100 2 -0.4043944461260186D-03 0 0 100 3 -0.3159582415919906D-03 0 0 100 4 -0.2444018627201997D-03 0 0 100 5 -0.1869402248683973D-03 0 0 100 6 -0.1433505198749962D-03 0 0 100 7 -0.1093788117964714D-03 0 0 100 8 -0.6236254796301988D-04 0 0 100 9 EQUATION NUMBER 2: NUMBER OF TERMS 9 -0.2287077360246153D-03 0 0 -100 1

0 0.2684587555297388D-03 0 0 0.2481972650539546D-03 0 0 0.2138022986966805D-03 0 0 0.1766164822231288D-03 0 0.1412068119664962D-03 0 0 0.1124358320145184D-03 0 0 0 0.8801669030480799D-04 0 0.5215830086881653D-04 0 0 EQUATION NUMBER 3: NUMBER OF TERMS = 0 Table 6.12

-100 -100 -100 -100 -100 -100 -100 -100

2 3 4 5 6 7 8 9

Output of program FURIl, routine PREPAR. Terms coming from Venus.

The Final Computation

of the Perturbing

Terms

CASE L I : ORDER = 10 GAMMA = 0.1001090475489518D-01 TERMS COMING FROM BODY 5 : NUMBER OF FREQUENCIES = 4 1 0.91566006255958D+00 0.11460368062769D+01 2 0.17469801876787D+01 0.16932509816499D+01 3 0.18313201251192D+01 0.22920736125538D+01 4 0.27469801876787D+01 0.34381104188306D+01 EQUATION NUMBER 1: NUMBER OF TERMS = 5 0.3503834212090000D-03 0 0 100 0 0.1497123234186157D-03 0 0 100 1 0.1867237603557100D-03 0 0 100 2 0.1028125039496269D-02 0 0 100 3 0.2401567397045788D-03 0 0 100 4 EQUATION NUMBER 2 : NUMBER OF TERMS = 3 -0.1827678090851681D-03 0 0 -100 2 -0.1012095468467025D-02 0 0 -100 3 -0.2370690908008197D-03 0 0 -100 4 EQUATION NUMBER 3 : NUMBER OF TERMS = 0 Table 6.13

Output of program FURIl, routine PREPAR. Terms coming from Jupiter.

301

302

Expansions

Required for the Equations

of Motion.

Collinear Points

Case

CASE LI: ORDER = 10 GAMMA = 0.1001090475489518D-01 TERMS COMING FROM BODY 11: 1 0.,88697426753137D+00 2 0..99995222074390D+00 3 0 .10537249028833D+01 4 0,. 17739485350627D+01 5 0..10481340000562D+02 6 0., 10594317953775D+02 7 0..11314541585954D+02 8 0..11481292221306D+02 9 0.,12368266488837D+02 10 0..13255240756369D+02 11 0..23736580756931D+02 12 0..23849558710143D+02 13 0.24736532977675D+02 14 0..37104799466512D+02 15 0..49473065955349D+02

EQUATION NUMBER 1: NUMBER 0.18147320740023D-2 0 0 0.74879308939471D-2 1 0 0.43940706312415D-2 0 2 -0.35026288078635D+0 2 0 0.62734208374221D-3 1 0 -0.14806865720527D-2 0 0 0.50212727377061D-4 0 0 -0.28120925209644D-3 0 2 -0.74673399932669D-3 0 3 0.71855196740396D-3 0 3 0.13229188284290D-2 0 4 -0.96547150776979D-3 0 0 0.24645465446583D-2 0 1 0.55230773658313D-2 0 2 -0.27087744114162D-1 1 2 -0.93225458461149D-2 0 3 0.12119987232546D+0 2 2 -0.15198570965991D-1 0 4 0.61067075646442D-1 1 4 0.21922449825337D-1 0 5 -0.56754314634276D+0 2 4 -0.10714311250374D-1 0 6 0.10189689968394D+1 2 5 0.31684114902349D+0 1 6 -0.46156602894062D+0 1 7 0.90075921044331D-3 0 0 -0.10835871360657D-2 0 1 0.14273442445416D-1 2 0 -0.32758275038948D-2 0 2 -0.10997614941521D-1 0 0 0.34457447245241D-1 3 0 -0.23472492710197D-1 1 2 -0.79899849025527D-1 1 0 -0.96401390565497D-1 2 2 0.74128351207918D-2 0 4 0.68251606795533D-1 1 4 0.34996309267308D+0 2 4 -0.13473823151034D-1 0 6 0.14153577988937D-1 0 7 -0.13591114135226D-1 1 3

OF TERMS 100 0 100 0 100 0 2 0 100 2 1 3 100 4 100 4 -100 5 -100 6 100 6 100 8 -100 8 100 8 100 0 -100 8 100 8 100 8 100 0 -100 8 100 8 100 0 -100 8 100 8 -100 8 100 9 -100 9 100 9 100 9 2 9 100 9 100 9 2 9 -100 9 100 9 100 9 100 9 100 9 -100 9 -100 10

NUMBER OF FREQUENCIES = 15 0.34326355958341D+01 0.62313951222251D+01 0.42743682063508D+01 0.58208588448870D+00 0.17112380982079D+01 0.45099976245991D+01 0.39593079563264D+01 0.16594479132534D+01 0.50920835090876D+01 0.22415337977421D+01 0.39527718959502D+01 0.46834611516165D+00 0.39009817109956D+01 0.27098799129038D+01 0.15187781148115D+01

= 132 -0.14656037603825D-3 -0.75806724768658D-3 0.50387871022036D-3 0.14998809313094D-3 -0.34722200281290D-3 -0.79633377062211D-2 0.10980182181717D-3 0.43823814136631D-3 -0.35628940765918D-3 0.74223339565492D-2 -0.21696061747327D-2 -0.39774804994313D-2 0.12568796404075D-1 0.38606255527053D-1 0.33816294118149D-1 -0.98172632914111D-1 -0.66370281648429D-1 -0.27070274734079D+0 -0.12368479774093D+0 0.28439939687048D+0 0.20032523574174D+0 0.31203029843370D-1 -0.11058264768508D+0 -0.41377217466672D-1 -0.54562713207951D-1 0.46191053380955D-2 0.19420671967225D-1 -0.66361075000889D-2 -0.15032630639257D-1 0.40587678407588D-1 -0.23776401255852D-1 -0.94627935363846D-1 0.36716088504397D-2 0.29980928625528D-1 0.13802872560195D+0 -0.79899229699075D-2 -0.81583402952670D-1 -0.15162889961594D+0 0.21541344039046D-1 -0.38600243864157D-2

0 1 0 0 0 1 0 0 0 0 0 1 1 2 1 2 1 2 1 2 1 0 1 0 0 1 2 1 0 3 2 1 0 1 2 0 1 1 0 0

0 0 2 0 2 0 1 2 2 4 5 0 1 1 2 2 3 3 4 4 5 6 6 7 8 0 0 1 0 0 1 0 3 3 3 5 5 6 8 4

100 100 100 100 100 1 -100

100 100 100 -100

100 -100 -100

100 100 -100 -100

100 100 -100

100 100 -100

100 100 100 -100

2 100 -100

2 -100 -100 -100 -100 -100

100 100 100

1 1 1 2 2 3 4 5 6 0 6 8 8 8 8 0 8 8 8 0 8 8 0 8 8 9 0 9 0 0 9 0 9 9 9 9 9 9 9 10

The Final Computation -0.31675217360437D-1 1 4 0.74458223433347D-2 0 6 -0.98770719498409D-3 0 1 -0.32419469252641D-3 0 0 0.79088569177183D-2 1 1 -0.71057176571844D-2 0 3 0.55333172153350D-2 0 0 -0.14748274211034D-1 0 1 -0.74929625133010D-1 1 1 -0.22907933520032D+0 2 1 0.55779167615818D-1 0 3 0.39595304235498D+0 1 3 0.16093130316042D+1 2 3 -0.13124928422845D+0 0 5 -0.18542019679684D+0 0 6 0.24807576862353D+0 0 7 0.32540280501338D+0 0 8 0.39520567690162D+1 1 8 0.15075349979600D-2 0 0 -0.56477733766166D-2 0 1 -0.14750035163224D-1 0 2 0.30510509665545D-1 0 3 0.55940977899393D-1 0 4 -0.93144866762426D-1 0 5 0.14807738741488D-1 0 3 -0.56858116183578D-1 0 5 EQUATION NUMBER 2: NUMBER -0.17128544069669D-2 0 1 -0.13371860662588D-3 0 1 0.41268276041702D-2 0 3 -0.67949552599563D-3 0 3 0.65516874934926D-3 0 3 -0.69663122610585D-2 0 5 0.60627797124699D-3 0 0 0.21721278068218D-2 0 1 0.11046154730003D-1 1 -0.27087744114162D-1 2 -0.27967637538345D-1 1 -0.99555422472643D-1 2 -0.60794283863966D-1 1 0.12213415129288D+0 2 0.10961224912669D+0 1 0.50081308935435D+0 2 0.18721817906022D+0 1 0.95052344707046D+0 2 0.10022395225018D-1 0 -0.43650170566361D+0 -0.21290503397099D-3 0 0 -0.10698744601540D-2 0 1 -0.65516550077897D-2 1 1 -0.23472492710197D-1 2 1 0.36315044102104D-2 0 3 0.29651340483167D-1 1 3 0.13650321359107D+0 2 3 -0.79179093037801D-2 0 5 0.88305271728579D-2 0 6 -0.21578956730023D-2 0 3 0.27770077863421D-2 0 4

of the Perturbing

100 10 100 10 -100 11 100 12 -100 12 -100 12 100 13 -100 13 -100 13 -100 13 -100 13 -100 13 -100 13 -100 13 100 13 -100 13 100 13 100 13 100 14 -100 14 100 14 -100 14 100 14 -100 14 -100 15 -100 15 OF TERMS 100 0 100 100 100 100 100 -100 100 100 100 -100 -100 100 100 -100 -100 100 100 8 100 0 8 100 9 -100 9 100 9 100 9 100 9 100 9 100 9 100 9 100 9 -100 100 10 -100 10

303

Terms

0.44984357156352D-2 0.37556539975293D-3 -0.21413770540353D-2 0.12711886865180D-2 0.33783704601474D-2 0.10241690239930D+0 0.22637504945201D-1 0.58166013092557D-1 -0.32446946449852D-1 -0.19781075380165D+0 -0.70527252327855D+0 0.89933584675744D-1 0.72951600934986D+0 -0.11965711625367D+1 -0.18780103073705D+1 0.27622438669669D+1 0.16797241502480D+2 -0.41243778809968D+0 0.76730800324671D-2 -0.34353902427437D-1 -0.10456646996941D+0 0.24693175218747D+0 0.50906969733447D+0 -0.14577879597351D+0 0.30586850600924D-1 -0.98014017094978D-1 = 97 0.16345946793240D-3 -0.10961320305915D-3 -0.36543310226163D-3 0.30532359848449D-3 -0.11935396690907D-2 -0.20242469910149D-2 0.24645459393335D-2 -0.87600103680732D-2 -0.45943436095890D-2 0.33816294118149D-1 -0.85571562804382D-2 0.29611684747661D-1 0.13519738664322D-1 -0.24736959548186D+0 0.20522208761048D-1 -0.64285867502246D-1 -0.28604397099973D-1 -0.28964052226671D+0 -0.39185015150359D-1 0.50878329114721D-1 -0.10835871360657D-2 -0.33180537500444D-2 0.15450785455382D-2 0.11014826551319D-1 0.44971392938292D-1 -0.43555520736070D-2 -0.39949614849538D-1 -0.80842938906201D-1 0.14056485789453D-1 -0.15440097545663D-1 0.48671898731582D-2

0 0 0 0 0 0 1 2 0 1 2 0 1 1 1 1 2 0 1 1 1 1 1 0 0 0

5 0 2 1 2 9 0 0 2 2 2 4 4 5 6 7 7 9 0 1 2 3 4 6 4 6

0 0 0 0 0 0 1 1 0 2 0 1 0 2 0 1 0 1 0 0 1 2 0 1 2 0 1 1 0 1 0

1 1 2 2 4 5 0 1 2 1 3 3 4 3 5 5 6 6 7 8 0 0 2 2 2 4 4 5 7 3 5

10 100 11 100 11 -100 12 100 12 -100 12 100 13 100 13 100 13 100 13 100 13 100 13 100 13 -100 13 100 13 -100 13 -100 13 -100 13 100 14 -100 14 100 14 -100 14 100 14 100 14 100 15 100 15 -100

100 100 -100 -100 -100

100 -100

100 -100

100 100 100 -100

100 100 100 -100 -100

100 -100 -100 -100 -100 -100 -100 -100 -100

100 100 100 100

1 4 5 6 6 6 8 0 8 8 8 0 8 8 8 0 8 8 8 8 9 9 9 9 9 9 9 9 9 10 10

304

Expansions

Required for the Equations

-0.58773606507448D-2! 0 6 0.18166631093185D-2! 0 2 -0.29686073666850D-2! 0 2 0.97380417476031D-1. 0 9 -0.14748272955688D-1. 1 0 -0.37464809333102D-1 2 0 0.27554733284100D-1 0 2 0.16733750284745D+0i 1 2 0.35973433870297D+0i 1 3 -0.65624642114223D+0i 1 4 -0.11125211807810D+1 1 5 0.17365303803647D+1 1 6 0.26032224401070D+1 1 7 0.15808227076065D+2! 2 7 -0.39257641135829D+0i 0 9 -0.48624756985489D-2! 0 1 0.12941998384705D-1 0 2 0.13516078165009D-1 0 3 EQUATION NUMBER 3: NUMBER -0.57750764869803D-2: 0 0 -0.34828576532437D-3i 0 0 -0.37846529532386D-3: 0 1 -0.19856219438548D-2: 1 1 0.57929845409110D-2 1 2 0.81436275067354D-3: 0 3 -0.30047941875164D-1 1 0 -0.94627935363846D-1 2 0 -0.23330952182028D+0i 3 0 0.10759129489155D+0i 1 2

-100 10 -100 11 -100 12 100 12 -100 13 -100 13 -100 13 -100 13 100 13 -100 13 100 13 -100 13 100 13 100 13 100 13 100 14 -100 14 100 15 OF TERMS 1 0 100 3 -100 3 -100 3 100 3 -100 7 1 0 1 0 1 0 1 0

of Motion.

Collinear Points

-0.85326654505731D-3 0.10884532922865D-2 -0.64280226490409D-2 -0.36388176012653D-2 -0.12804929043611D-1 -0.64893890940634D-1 -0.19781074702511D+0 0.50769939627863D-1 -0.81097929480023D-1 -0.12220957425971D+0 0.17175638558071D+0 0.23408684256622D+0 -0.30605935435997D+0 -0.37119400928971D+1 -0.11161576052278D-2 -0.29500070326447D-1 0.27695467172996D-1

Case

0 0 0 0 0 1 2 0 0 0 0 0 0 1 0 1 0

1 1 3 0 1 1 1 3 4 5 6 7 8 8 0 1 3

0 1 2 0 0 0 1 2 3 1

2 0 0 2 3 4 0 0 0 2

100 100 100 -100 100 100 100 100 -100 100 -100 100 -100 -100 -100 100 100

11 12 12 13 13 13 13 13 13 13 13 13 13 13 14 14 14

= 20 0.14761114930456D-1 -0.14806865720527D-2 -0.39816688531106D-2 0.91236323407801D-3 0.99040343858883D-3 0.13275826358880D-2 0.95749394410103D-2 -0.79899849025527D-1 -0.22223333984529D+0 0.10384875968149D+0

100 100 100 -100 100

1 0 3 3 3 3 7 1 8 1 9 1 9 1 9

Table 6.14 Output of program FURI1, routine PREPAR. Terms coming from the Moon.

6.5

References [1] A.E. Brigham. The Fast Fourier Transform. Prentice Hall, 1974. [2] E.W. Brown and C.A. Shook. Planetary Theory. Dover, 1964. [3] R. Bulirsh and J. Stoer. Foundations of Numerical Analysis. Springer Verlag, 1981. [4] E. Isaacson and H.B. Keller. Analysis of Numerical Methods. John Wiley & Sons, 1966. [5] H. Poincare. Les Methodes Nouvelles de la Mecanique Celeste. GauthierVillars, 1892, 1893, 1899. [6] C.L. Siegel and J.K. Moser. Lectures on Celestial Mechanics. Springer Verlag, 1971. [7] V. Szebehely. Theory of Orbits. Academic Press, 1967.

Chapter 7

The Quasi-periodic Orbits: Equations, Method of Solution and Results

The objective of this chapter is to obtain the analytic expression of the nominal quasi-periodic orbit. It is in this chapter where the expressions found in Chapter 6 are used. This chapter is also a necessary step towards the numerical refinement of the orbit to be described in Chapter 8. First, the equations of motion should be put in a suitable form to be handled by the program carrying out the computations giving the analytic solution. Then the method of solution is presented. The successive terms are determined by recurrence as in Chapter 2. The Lindstedt-Poincare method used for the halo orbits has been extended to allow to obtain the quasi-periodic orbit. This method is suitable when we deal with the exactly resonant terms. The difficulties are associated to the nearly resonant terms, giving rise to small divisors. In a first approach they have been skipped because its size is very small (especially for the L\ case in the Sun-Barycenter problem).

7.1 7.1.1

The Final Equations of Motion for the Collinear Case Expansion of the Equations for the Integration

of Motion

in a Suitable

Form

The general equations have been developed in a suitable form in order to design a recurrent method for solving these equations up to a given weight. That method will be explained in section 7.2. We recall that the halo terms of the equations, that is, M, N and P in 7.2.2 have the following form: qijkxlyizk, where q^ is a coefficient and i,j, k € NU{0}. In the perturbing terms (we call them "on halo" terms), some frequencies (v) and phases {if) are involved. So, if we consider the terms of the equations up to a given weight, then, there is a finite list of pairs {u, if) related. The first frequency is taken as the basic one for the perturbing terms, that is, the mean motion of the secondary. 305

306

The Quasi-periodic

Orbits: Equations,

Method of Solution and Results

The nonhalo terms of the equations will be written in one of the following ways c1xiyjzkF(vt

+
or

c2v'F(vt +
(7.1) (7.2)

where i,j, k > 0, cx and c2 are two real coefficients, v and ip are the frequency and the phase respectively associated to the corresponding term, F means one of the trigonometric functions sine or cosine, and v' stands for one of the derivatives x',y' or z'. It has been useful to take only cosine terms in the first and the third equations, and sine terms in the second equation. This goal was taken into account in the computations of the coefficients C(1),C(2),... of the equations, as well as in the determination of the perturbing terms due to planets and Moon (see Chapter 6). Once all these terms have been collected from previous computations, a weight is assigned to each term. Let us consider a term (7.1) or (7.2). We define its weight according to the formula:

•I°{MI*I'(0*'(0*{(/+^

(7.3)

where ce is the corresponding coefficient, k is the amplitude of the sine term of order one of the halo y-component, tz — 0.08, for the Sun-Barycenter problem, for the Earth-Moon problem, case L\, for the Earth-Moon problem, case L2, A /

=

for a term (7.1), for a term (7.2), for the Sun-Barycenter problem, for the Earth-Moon problem

and [ ] means the integer part function. The reason for this choice is the following: Individual terms in the halo orbit as given in (7.1) and (7.2) can be approximated by c(a cos ipy(ka sin ip)i{fi cos ip)k x CS, where CS means cosine or sine of vt + p, tp is the angle in the halo orbit and x, y, z have been approximated by the first term. The constant c includes A in the case (7.2). The values of tx and tz are typical values for a and j3\ k depends on /i and is determined at the beginning of program ANACOM. We have to bound (cos ip)l+k (sin V0J, because CS is easily bounded by 1 and no improvement is possible. The bound for the trigonometric expression is (i+k)l+k j^(i+k+j)~^l+k+^. Finally one takes In suitably scaled by the factor / . This factor is determined from the behavior of the terms in Chapter 5, 5.1.2. The result is rounded to the nearest integer. The term \.\l+i+k is taken for safety reasons.

The Final Equations

7.1.2

A Program

of Motion for the Collinear

for Producing

the Equations

of

Case

307

Motion

The program EQUAP generates a file with the nonhalo terms of the equations in the suitable form to be read by the program QPO. First, the routine READCE reads the terms of k~3 - 1, C(l), xsC(2), xsC(3), C(4), C(5), C(7), C(12), C(13), C(16) and C(23) where

(

1 — 7_1

for the Sun-Barycenter problem, case L\,

-1 - 7_1 for the Sun-Barycenter problem, case L2, 1 for the Earth-Moon problem. We recall that for the Sun-Barycenter problem, these terms are computed by the program CES. The terms due to the lack of dynamical coherence of the motion of the secondary around the primary plus the terms due to radiation pressure, are computed and added to the previous ones. The perturbing terms due to the planets and the Moon are read from different files in the routine READPL. After some manipulations, all these terms are given, up to a fixed weight, in the appropriate form to be implemented in the analytic computation of the quasiperiodic orbit, that is, each term that appears in the equations is given as in (7.1) or (7.2). In the program EQUAP, the values of the involved frequencies are stored in the vector GMAR(400) and the corresponding phases in PHAR(400). The terms (7.1) are represented by C(I,J,L-K1,I1), where C= c\, I and J are the exponents i and j respectively, L is a sign equal to 1 for a cosine term and equal to —1 for a sine term,

r k

if

k^o,

\

if

jfe = 0,

100

and II is an index such that GMAR(Il)=i/ and PHAR(Il)=y. If v = 0 and ip = 0, we take 11=0. The terms (7.2) are represented in a similar way as C(0,J,L-K1,I1), where C= c 2 , J = 1,2 or 3 for a term in x',y' or z', Kl = 100, and L and II have the same meaning as before. For the computation of the terms that come from the part with Legendre polynomials, the coefficients of the halo terms of the equations, that is, M,N and P are required. The routine FMNP computes these coefficients in the exponential form of M, N and P (see Chapter 2). If the developments in cosine and sine terms are required, some coefficients given by FMNP change their sign according to the exponent of y. These changes are performed in the routine SIGMNP. In this step, the routine

308

The Quasi-periodic

Orbits: Equations,

Method of Solution and Results

PONENT is required to obtain the exponents of x, y and z in the monomials of M, N and P. The computation of the weights associated to each term is done in the routine RETO via the routines PROPES (for the terms with i = j = k = 0) and PR0PC3. We note that in some cases the formula (7.3) gives a weight less than or equal to the degree of the term. This happens for some of the terms due to the Moon or terms which come from the nonhalo terms of the equations where the Legendre polynomials are involved. In that cases, we assign to these terms a weight equal to its degree plus 1. In order to distinguish these terms in the output, their weight is taken as a negative integer number. In any case the real weight is also written. Therefore, if we want to compute the terms of the equations up to a given weight NN, then some terms with a negative weight NP such that |NP| >NN can appear. So, in order to get the solution up to a given real weight NN it will be necessary to compute the solution up to the maximum of the absolute values of the weights that appear in the terms of the equations. In order to reduce the number of frequencies and phases, and the number of terms of the equations, it has been useful to combine some of the terms. This compactification which takes place in the routine REDUIR is done as follows. Let us consider two terms

*i =

cixlyjzkF(uit

+ fi),

*2 =

c2xiyjzkF{p1t

+ ip2),

and

with the same frequency v\. Then, we can write tl+t2

= c3xiy:>zkF{vlt

+
where 1 /2 c

3 = {c? + c\ + 2cic 2 cos(v?i - ip2)}

and tan

The Final Equations

7.1.3

Results

and Numerical

of Motion for the Collinear Case

309

Checks

The terms of the equations up to weight 9 have been computed in the following cases: (1) Sun-Barycenter, case L\, given in Table 7.1, (2) Sun-Barycenter, case Li, (3) Earth-Moon, case L2. A simplified model of the case (1) has been useful to perform different simulations of control with a short computing time. In this simplified model, only the eccentricity of the Earth, the dynamical coherence of the Earth's motion, and the effect of the radiation pressure have been taken into account. Of course, in this case, there is only one frequency which is the mean motion of the Earth. Some checks of the terms obtained by the program EQUAP have been performed at different points and at different epochs. The points were taken on a halo orbit with z-amplitude /3 = 0.08 (see Table 7.2). The results show a good agreement between analytic and numerical accelerations in the Sun-Barycenter problem. Indeed, in the worst case the difference has a weight equal to 6. We remark that this error can be accepted because in the equations, perturbing terms due to the planets and the Moon, of that order of magnitude were neglected. We recall that terms with relatively large amplitude (i.e., weights from 6 to 9) were skipped in the filtering procedure of Chapter 6, 6.3.2. Those terms have large frequencies, and despite the large value of the accelerations due to them, they produce only short amplitude terms in the position. For the Earth-Moon problem, the approximation is poorer. In that case, the weight of the maximum difference is equal to 5. For the simplified model of (1) the agreement is very good. The differences have weights greater than or equal to 9. We remark that in this case the numerical accelerations have been computed using the same simplified model.

The Quasi-periodic

Orbits: Equations,

EARTH+MOON-SUN SYSTEM CASE LI WEIGHT = 9 NUMBER OF FREQUENCIES = 82 1 0.9999522207284640D+00 2 0.1251015314484709D+01 3 0.9156570618831535D+00 4 0.1831314123766307D+01 5 0.2473653297767500D+02 6 0.3710479946651200D+02 7 0.1999904441456928D+0i 8 0.6255076572423544D+00 9 0.9366016779288175D+00 10 0.1148129222130600D+02 11 0.4947306595534900D+02 12 0.1876522971727063D+01 13 0.8765707509985993D+00 14 0.2510630937562449D+00 15 0.8313619030378430D+00 16 0.1747018964920997D+01 17 0.6254989350070500D+00 18 0.1831320125119200D+01 19 0.1236826648883700D+02 20 0.2373658075693100D+02 21 0.2384955871014300D+02 22 0.2502030628969418D+01 23 0.2746971185649461D+01 24 0.2250967535213173D+01 25 0.1502078408240954D+01 26 0.8732511351291710D+00 27 0.1915609282611617D+01 28 0.8429515884531046D-01 29 0.9660209891320313D+00 30 0.1250997870014100D+01 31 0.1876496805021200D+01 32 0.2501995740028200D+01 33 0.3127494675035300D+01 34 0.2746980187678700D+01 35 0.1325524075636900D+02 36 0.1404902516893226D+01 37 0.2831266344494771D+01 38 0.1932041978264063D+01 39 0.1625459877970818D+01 40 0.6335054279964647D-01 41 0.4049502961647623D+06 42 0.2658256908820276D+01 43 0.1084247379573774D+01 44 0.7470667441925326D+00 45 0.3752993610042300D+01 46 0.4378492545049400D+01 47 0.5003991480056400D+01 48 0.9156600625595800D+00 49 0.1746980187678700D+01 50 0.8869742675313700D+00 51 0.1053724902883300D+01

Method of Solution and Results

0.6231395117299221D+01 -0.2884338825126906D+00 -0.1979167429597718D+01 0.2293632183012776D+01 0.3900981710995600D+01 0.2709879912903800D+01 0.6179604927418856D+01 0.1425662661422312D+01 0.5191770785557459D+00 0.1659447913253400D+01 0.1518778114811500D+01 0.1150705458671403D+01 -0.2167779041033939D+01 0.2643633021351981D+01 0.2714619818594807D+01 0.1881342399767400D+01 0.1417296496991800D+01 0.2292073612553800D+01 0.5092083509087600D+01 0.3952771895950200D+01 0.4683461151616500D+00 0.2553870424499513D+01 -0.3074291866679034D+01 -0.3283894176739517D+00 0.2430331539214389D+01 0.1916038532792506D+01 -0.1827920526552135D+01 -0.3095417103290732D+01 -0.2258506631948617D+01 0.2834592993983600D+01 0.4251889490975400D+01 0.5669185987967100D+01 0.8032971777792700D+00 0.3438110418830600D+01 0.2241533797742100D+01 -0.7233664097316860D+00 0.2182875486824612D+01 0.1756924709239920D+01 0.1359984598049204D+01 -0.1145897920449518D+01 0.9572104907461318D-01 0.3718394402892522D-01 0.1869375022576490D+00 0.2170478728628369D+01 0.2220593674771200D+01 0.3637890171762900D+01 0.5055186668754700D+01 0.1146036806276900D+01 0.1693250981649900D+01 0.3432635595834100D+01 0.4274368206350897D+01

The Final Equations of Motion for the Collinear Case 0.5820858844887000D+00 52 0.1773948535062700D+01 0.1711238098207900D+01 53 0.1048134000056200D+02 0.4509997624599100D+01 54 0.1059431795377500D+02 0.2612227390392102D+01 55 0.3151973342338853D+01 -0.2891284191112373D+00 56 0.12510i5314484709D+01 0.2289217823911790D+01 57 0.1831314123766307D+01 -0.1979628239660955D+01 58 0.9156570618831535D+00 0.1425700153600997D+01 59 0.6255076572423544D+00 0.1138912762564505D+01 60 0.1876522971727063D+01 -0.2167620986141730D+01 61 0.8765707509985993D+00 0.2556299801139196D+01 62 0.2502030628969418D+01 0.5192220030676837D+00 63 0.9366016779288175D+00 -0.2914075454189641D+01 64 0.2746971185649461D+01 0.1864493202009641D+01 65 0.1747018964920997D+01 -0.3185521467599869D+00 66 0.2250967535213173D+01 -0.2294371900809253D+01 67 0.3127538286211772D+01 -0.8816036368470227D+00 68 0.3753045943454126D+01 -0.1557393366956654D+01 69 0.1915609282611617D+01 0.2080900989865821D+01 70 0.2831266344494771D+01 0.2713696048062953D+01 71 0.8313619030378430D+00 0.2429935316830200D+01 72 0.1502078408240954D+01 0.5610543688337673D+00 73 0.4378553600696481D+01 0.1957542532778765D+01 74 0.5004061257938835D+01 -0.2894243332689311D+01 75 0.5629568915181190D+01 -0.7268767491484566D+00 76 0.1404902516893226D+01 0.1916053366219000D+01 77 0.8732511351291710D+00 -0.1798357254562880D+01 78 0.3662628247532614D+01 2945883921131993D+01 79 0.2662676026804150D+01 2258514577062989D+01 80 0.9660209891320313D+00 2603256991840835D+01 81 0.3151973342338853D+01 3959307956326497D+01 82 0.1131454158595400D+02 FIRST EQUATION: NUMBER OF TERMS IN X,Y, AND Z 212 NUMBER OF TERMS IN DERIVATIVES = 4 SECOND EQUATION: NUMBER OF TERMS IN X,Y, AND Z 173 NUMBER OF TERMS IN DERIVATIVES = 4 THIRD EQUATION: NUMBER OF TERMS IN X,Y, AND Z 38 NUMBER OF TERMS IN DERIVATIVES = 2 EQUATION = 1 0.4580383756090650D+00 (1 0 100 1) WEIGHT= 3 3 -0.2274962028893635D+00 (0 2 100 1) WEIGHT= 4 0.4542979617731012D+00 (2 0 100 1) WEIGHT= 4 -0.9117631731075615D+00 (1 2 100 1) WEIGHT= 5 0.1047033318986893D-01 (0 0 100 0) WEIGHT= 5 -0.2279327863620046D+01 (2 2 100 1) WEIGHT= -5 0.2849159829525058D+00 (0 4 100 1) WEIGHT= 5 0.5533317215335000D-02 (0 0 100 5) WEIGHT= 5 -0.1474827421103400D-01 (0 1 -100 5) WEIGHT= 5 -0.3244694644985200D-01 (0 2 100 5) WEIGHT= 5 0.5577916761581800D-01 (0 3 -100 5) WEIGHT= 5 0.8993358467574400D-01 (0 4 100 5) WEIGHT= 5 -0.1475003516322400D-01 (0 2 100 6) WEIGHT= 5 0.3051050966554500D-01 (0 3 -100 6) WEIGHT= 5 0.5594097789939300D-01 (0 4 100 6) WEIGHT= 6 0.3540572732474960D-02 (0 0 100 2) WEIGHT= 6 0.3651006063447805D-02 (0 0 100 3) WEIGHT= 0.2277769325483898D-02 (0 0 100 4) WEIGHT= 6

311

312

The Quasi-periodic

Orbits: Equations,

0.1130821099126589D-01 0.1146084029195635D-01 0.2110626269654990D-02 -0.6291203927079598D-02 0.5691399887514294D-02 -0.2271489808865506D+00 0.6078421154050419D+00 0.1709496504967650D+01 0.5523077365001300D-02 0.2263750494520100D-01 0.5816601309255700D-01 -0.7492962513301000D-01 0.1978107538016500D+00 0.3959530423549800D+00 0.7295160093498600D+00 -0.1312492842284500D+00 0.1507534997960008D-02 -0.5647773376616600D-02 -0.3435390242743700D-01 -0.1045664699694100D+00 0.2469317521874700D+00 0.5090696973344700D+00 -0.9314486676242600D-01 0.1480773874148800D-01 0.3058685060092400D-01 -0.5685811618357800D-01 0.1046085509068239D-02 0.7437105243275871D-03 0.7861733999018258D-03 0.2321493855890120D-01 -0.3470272995680272D-01 -0.9117631731075615D+00 0.7597759545400155D+00 0.9801933440843620D-02 0.7138798452883258D-02 -0.4558657346580400D+01 0.5983237745894601D+01 -0.3324020969941445D+00 -0.1120886925085356D-02 0.1028125039496269D-02 -0.9654715077697900D-03 -0.3977480499431300D-02 0.2464546544658300D-02 0.3381629411814900D-01 -0.9322545846114900D-02 -0.1519857096599100D-01 0.2192244982533700D-01 0.3120302984337000D-01 0.9007592104433100D-03 0.4619105338095500D-02 -0.1083587136065700D-02 -0.3275827503894800D-02 0.3671608850439700D-02 0.7412835120791800D-02 -0.2141377054035300D-02

(1 (1 (0 (0 (0 (0 (3 (1 (0 (1 (2 (1 (1 (1 (1 (0 (0 (0 (1 (1 (1 (1 (0 (0 (0 (0 (0 (0 (0 (2 (1 (1 (4 (0 (0 (3 (2 (0 (0 (0 (0 (1 (0 (1 (0 (0 (0 (0 (0 (1 (0 (0 (0 (0 (0

0 0 0 2 2 0 0 4 2 0 0 1 2 3 4 5 0 1 1 2 3 4 5 3 4 5 0 0 0 0 2 0 0 4 4 2 4 6 0 0 0 0 1 2 3 4 5 6 0 0 1 2 3 4 2

Method of Solution and Results

100 100 100 100 100 2 100 100 100 100 100 -100 100 -100 100 -100 100 -100 -100 100 -100 100 -100 -100 100 -100 100 100 100 100 100 2 100 100 100 100 100 100 100 100 100 100 -100 100 -100 100 -100 100 100 100 -100 100 -100 100 100

0) 7) 8) 0) 7) 1) 1) 1) 10) 5) 5) 5) 5) 5) 5) 5) 6) 6) 6) 6) 6) 6) 6) 11) 11) 11) 9) 13) 15) 0) 0) 1) 1) 0) 7) 1) 1) 1) 17) 18) 10) 10) 10) 10) 10) 10) 10) 10) 19) 19) 19) 19) 19) 19) 2e)

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

6 6 6 6 6 6 6 -6 6 6 6 6 6 6 6 -6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 -7 -7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

The Final Equations of Motion for the Collinear 0.1271188686518000D-02 0.3378370466147400D-02 -0.7105717657184400D-02 -0.2290793352003200D+00 -0.7052725232785500D+00 0.1609313031604200D+01 -0.1196571162536700D+01 -0.1854201967968400D+00 0.7673080032467100D-02 -0.1457787959735100D+00 -0.9801401709497800D-01 0.5118382036713971D-03 0.2935381352951189D-03 0.4511452629764966D-03 0.4308829882251162D-03 0.2408129836217583D-03 0.2247188447278992D-03 0.2212539052766846D-03 0.1138279977502859D-01 -0.2284495752792216D-01 0.2279327863620046D+01 0.5698319659050116D+00 0.3418993009935300D+01 -0.7977650327859469D+01 0.1595530065629849D+02 -0.2659216776049748D+01 -0.5151971085381031D-03 -0.4043944461260186D-03 -0.3159582415919906D-03 -0.2444018627201997D-03 0.2401567397045788D-03 0.1256879640407500D-01 0.3860625552705300D-01 0.1211998723254600D+00 -0.6637028164842900D-01 -0.2707027473407900D+00 0.6106707564644200D-01 -0.1236847977409300D+00 0.2003252357417400D+00 -0.1071431125037400D-01 0.3168411490234900D+00 -0.4137721746667200D-01 0.1427344244541600D-01 -0.6636107500088900D-02 -0.2347249271019700D-01 0.2998092862552800D-01 0.6825160679553300D-01 -0.7989922969907500D-02 -0.1347382315103400D-01 -0.3860024386415700D-02 0.3755653997529300D-03 -0.9877071949840900D-03 -0.3241946925264100D-03 0.7908856917718300D-02 -0.1878010307370500D+01

(0 (0 (0 (2 (2 (2 (1 (0 (1 (0 (0 (0 (0 (0 (0 (0 (0 (0 (2 (1 (2 (0 (1 (4 (3 (1 (0 (0 (0 (0 (0 (1 (2 (2 (1 (2 (1 (1 (1 (0 (1 (0 (2 (1 (1 (1 (1 (0 (0 (0 (0 (0 (0 (1 (1

1 2 3 1 2 3 5 6 0 6 6 0 0 0 0 0 0 0 0 2 0 2 2 2 4 6 0 0 0 0 0 1 1 2 3 3 4 4 5 6 6 7 0 1 2 3 4 5 6 4 0 1 0 1 6

-100

100 -100 -100

100 -100 -100

100 100 100 100 100 100 100 100 100 100 100 100 100 2 2 2 100 100 100 100 100 100 100 100 -100 -100

100 -100 -100

100 100 -100

100 100 -100

100 -100

100 -100

100 -100

100 100 100 -100

100 -100

100

21 ) 21 ) 21 ) 5) 5) 5) 5) 5 6 6 11 12 14 16 1 26 27 29, 7' 7 1,

r1 1, i; l, 30] 31: 32: 33] 34] 10:

io: io: io: 10, 0, 10, 10 0 10 10 19 1919 19 19 19 19 35 20 20 21 21 5)

313

Case WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

7 7 7 7 7 7 -7 -7

6 5

7 -7 -7

6 6

8 8 8 8 8 8 8 8 8 8 8 8 8 8 -8

6

8 a 8 8 a 8 8 a 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 -8

6

The Quasi-periodic

Orbits: Equations,

0.2480757686235300D+00 0.1594984461962005D-03 0.1408353428629889D-03 0.1967666355979878D-03 0.1095274825095838D-03 0.1482182801231775D-03 0.9115856929868940D-04 0.1373162469330937D-03 0.9221772998549920D-04 0.9786339670281754D-04 0.1175416426466345D-03 0.9775501762545559D-04 0.1248978536630959D-03 -0.7589500348085246D-04 0.9623107521221693D-04 -0.5711038762306606D-01 0.9117314693160802D+00 -0.4558657346580400D+01 0.4283280593250083D-01 0.1196647549172920D+02 -0.8328601123624959D-02 -0.9972062909824335D+00 -0.7977658328149244D+01 0.3589942647665841D+02 -0.1196647549221947D+02 0.3739523591318585D+00 -0.1869402248683973D-03 -0.1433505198749962D-03 -0.1093788117964714D-03 0.1497123234186157D-03 0.1867237603557100D-03 -0.1465603760382500D-03 -0.7580672476865800D-03 0.5038787102203600D-03 -0.1480686572052700D-02 0.7963337706221100D-02 -0.2812092520964400D-03 0.4382381413663100D-03 -0.7467339993266900D-03 0.3562894076591800D-03 0.7185519674039600D-03 0.1322918828142900D-02 -0.2169606174732700D-02 -0.9817263291411100D-01 0.2843993968704800D+00 -0.5675431463427600D+00 0.1018968996839400D+01 -0.1195826476850800D+00 -0.4615660289406200D+00 -0.5456271320795100D-01 -0.1503263063925700D-01 -0.1099761494152100D-01 0.4058767840758800D-01 0.3445744724524100D-01 -0.2377640125585200D-01

(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (2 (5 (3 (1 (2 (0 (0 (1 (4 (2 (0 (0 (0 (0 (0 (0 (0 (1 (0 (0 (1 (0 (0 (0 (0 (0 (0 (0 (2 (2 (2 (2 (1 (1 (0 (0 (0 (3 (3 (2

7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 4 2 6 4 4 4 6 8 0 0 0 0 0 0 0 2 0 0 2 2 3 2 3 4 5 2 4 4 5 6 7 8 0 0 0 0 1

Method of Solution and Results

-100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 2 100 2 100 2 2 100 100 100 100 100 100 100 100 100 100 100 1 1 100 100 -100 100 -100 100 -100 100 100 100 -100 100 -100 100 2 2 100 100 -100

5) 22) 23) 24) 25) 28) 36) 37) 38) 39) 40) 41) 42) 43) 44) 7) 1) 1) 7) 1) 7) 1) 1) 1) 1) 1) 45) 46) 47) 48) 49) 50) 50) 50) 51) 51) 52) 53) 53) 54) 54) 54) 54) 0) 0) 10) 10) 0) 10) 10) 0) 19) 0) 19) 19)

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

-8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 -9 -9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 -9 -9 9 9 9 9 9

The Final Equations

of Motion for the Collinear

2 0 -0.9462793536384600D-01 (1 2 0 -0.7989984902552700D-01 (1 2 100 (2 -0.9640139S56549700D-01 (2 3 -100 .1380287256019500D+00 4 100 . 3499630926730800D+00 (2 5 -100 . 8158340295267000D-01 (1 100 6 .1516288996159400D+00 (1 (0 7 -100 .1415357798893700D-01 (0 100 8 .2154134403904600D-01 3 -100 -0.1359111413522600D-01 (1 4 100 -0.3167521736043700D-01 (1 (0 5 -100 0.4498435715635200D-02 100 6 (0 0.7445822343334700D-02 7 -100 0.2762243866966900D+01 (1 100 (0 8 0.3254028050133800D+00 100 (0 0 0.7509530129439980D-04 101 (3 6 -0.3988825164073172D+02 100 8 .3739523591318598D+01 (1 9 -100 (0 .1024169023993000D+00 (2 7 -100 .1679724150248900D+02 100 8 .3952056769016200D+01 (1 9 -100 (0 -0.4124377880996800D+00 100 8 (2 0.2056737975225229D+02 100 (0 10 -0.4113475950450458D+00 100 8 (3 0.8226951980900916D+02 100 -0.4936171140540555D+01 (1 10 100 (2 10 -0.3208511241351357D+02 100 (0 12 0.4456265612987996D+00 100 0.6238771858183195D+01 (1 12 TERMS WITH DERIVATIVES, EQUATION == 1 2 100 (0 0.6683093247027213D-01 1 -100 (0 -0.3341313363288495D-01 100 (0 2 0.1395972549512748D-02 1 -100 (0 -0.8375835297076488D-03 EQUATION = 2 100 (0 1 -0.1536269978504090D+00 1 100 -0.4542979617731012D+00 (1 100 (0 3 0.2279407932768904D+00 1 100 (2 -0.9117631731075615D+00 3 100 0.1139663931810023D+01 (1 1 100 (0 -0.1280492904361100D-01 2 -100 (0 0.2755473328410000D-01 100 (0 3 0.5076993962786300D-01 4 -100 (0 -0.8109792948002300D-01 100 (0 3 0.2769546717299600D-01 0 -100 (0 0.2458370377799402D-02 0 -100 (0 .2919317234866309D-02 0 -100 (0 .1617588490630532D-02 1 100 . 2994806808345556D-02 (0 100 1 (0 .3845857204135967D-02 100 (2 3 0.3418993009935300D+01 (0 5 100 -0.2849160841612750D+00 0 -100 (0 -0.3638817601265300D-02 0 -100 -0.1474827295568800D-01 (1 1 100 -0.6489389094063400D-01 (1

0) 19) 19) 19) 19) 19) 19) 19) 19) 35 ) 35 ) 35 ) 35 ) 5) 5) 55 ) 1 1 21 5 5 5)

5] 5] 5] 5] 6) 56) 57) 58) 0) 7) 1) 1) 5) 5) 5)

Case

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

315

9 9 9 9 9 9 9 9 9 9 9 9 9 -9 -9 9 -10 -10 -10 -10 -10 -10 -11 -11 -12 -12 -13 -13 -14

WEIGHT= WEIGHT= WEIGHT= WEIGHT=

3 4 6 8

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

3 4 -4 5 5 5 5 5 5 5 6 6 6 6 6 6 -6 9 6 6

7 6 8 7 8 8 7 7 8 7 9 8 9 8 9

3

4

316

The Quasi-periodic

Orbits: Equations,

0.1673375028474500D+00 0.3597343387029700D+00 -0.6562464211422300D+00 -0.1222095742597100D+00 -0.4862475698548900D-02 0.1294199838470500D-01 0.1351607816500900D-01 0.6156924805948844D-03 0.7514671727477477D-03 0.6030574064830381D-02 0.5711239381980541D-02 -0.1519551909080031D+01 0.7977650327859468D+01 -0.1994412581964867D+01 -0.1012095468467925D-02 0.6062779712469900D-03 0.2172127806821900D-02 -0.4594343609589000D-02 -0.8557156280438200D-02 0.1351973866432200D-01 0.2052229876104800D-01 -0.2860439709997300D-01 -0.1069874460154000D-02 0.1088453292286500D-02 -0.2968607366685000D-02 -0.6428022649040900D-02 -0.3746480933310200D-01 -0.1978107470251100D+00 -0.1112521180781000D+01 0.1717563855807100D+00 -0.1116157605227800D-02 -0.2950007032644700D-01 0.3113386612948233D-03 0.4442183297899274D-03 0.4781868229738445D-03 0.4403244496417846D-03 0.5361238216393576D-03 0.2931014014351335D-03 0.3081852125997494D-03 -0.1255427695974940D-01 -0.1138279977592859D-01 0.2279407932768904D+00 0.3913008268483868D-01 0.2855519381153303D-01 0.1139663931810023D+01 -0.2279328673290200D+01 -0.9345912590641880D-02 -0.7138800988750139D-02 -0.7977650328149244D+01 0.3324020970062185D+00 -0.2287077360246153D-03 0.2684587555297388D-03 0.2481972650539546D-03 0.2138022986966805D-03 -0.2370690908008197D-03

(1 (1 (1 (0 (0 (0 (0 (0 (0 (0 (0 (3 (3 (1 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (2 (2 (1 (0 (0 (1 (0 (0 (0 (0 (0 (0 (0 (1 (1 (0 (1 (1 (1 (4 (0 (0 (2 (0 (0 (0 (0 (0 (0

2 3 4 5 1 2 3 0 0 3 3 1 3 5 0 0 1 2 3 4 5 6 1 1 2 3 0 1 5 6 0 1 0 0 0 0 0 0 0 1 1 1 3 3 1 1 5 5 5 7 0 0 0 0 0

Method of Solution and Results -100

100 -100

100 100 -100

100 -100 -100

100 100 100 100 100 -100 -100

100 -100

100 -100

100 -100

100 100 -100

100 -100

100 100 -100 -100

100 -100 -100 -100 -100 -100 -100 -100

100 100 2 100 100 2 100 100 100 100 100 -100 -100 -100 -100 -100

5) 5) 5) 5) 6) 6) 11) 59) 60) 0) 7) 1) 1) 1) 18) 10) 19) 10) 10) 10) 10) 10) 19) 21) 21) 21) 5) 5) 5) 5) 6) 6) 61) 62) 63) 64) 65) 67) 71) 0) 7) 1) 0) 7) 1) 1) 5) 7) 1) 1) 17) 30) 31) 32) 34)

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

6 6 6 -6 9 6 6 7 7 7 7 7 7 -7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 -7 -7 7 7 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 -8 -8 8 8 8 8 8

The Final Equations of Motion for the Collinear 0.2464545939333500D-02 0.1104615473000300D-01 -0.2796763753834500D-01 -0.6079428386396600D-01 0.1096122491266900D+00 0.1872181790682200D+00 -0.2896405222667100D+00 0.3918501515035900D-01 -0.2129050339709900D-03 0.6551655007789700D-02 0.1545078545538200D-02 0.3631504410210400D-02 0.2965134048316700D-01 -0.4355552073607000D-02 -0.7917909303780100D-02 -0.2157895673002300D-02 0.2777007786342100D-02 0.4867189873158200D-02 -0.8532665450573100D-03 0.1816663109318500D-02 0.1736530380364700D+01 0.2340868425662200D+00 0.1269075842284760D-03 0.1972333825124775D-03 0.1393010792414695D-03 0.1436276529930975D-03 -0.1578916951208013D-03 0.1016797989571835D-03 0.1474626583711266D-03 0.1083240736399351D-03 0.7669155481266617D-04 0.7539581511020320D-04 0.1002446755001841D-03 0.1660052911983742D-03 0.8190161772268982D-04 0.1106145972656179D-03 0.1054990644629008D-03 0.1313768559709016D-03 -0.2284495752792216D-01 0.3418993009935300D+01 -0.5698321683225501D+00 -0.4997160674174975D-01 -0.3988825163929734D+01 0.1595530065629849D+02 0.8328601123927482D-02 -0.2393295098443894D+02 0.2991618873054868D+01 0.1766164822231288D-03 0.1412062119664962D-03 0.1124358320145184D-03 0.8801669030480799D-04 -0.1827678090851681D-03 0.1634594679324000D-03 -0.3654331022616300D-03 -0.6794955259956300D-03

(1 (1 (1 (1 (1 (1 (1 (0 (0 (1 (0 (0 (1 (0 (0 (0 (0 (0 (0 (0 (1 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (2 (2 (0 (1 (1 (4 (0 (3 (1 (0 (0 (0 (0 (0 (0 (0 (0

0 1 2 3 4 5 6 7 0 1 2 3 3 4 5 3 4 5 1 2 6 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 5 3 3 7 5 7 0 0 0 0 0 1 2 3

-100

100 -100

100 -100

100 -100

100 -100

100 -100

100 100 -100

100 100 -100

100 100 -100 -100

100 -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 -100

100 2 2 100 2 100 100 100 100 -100 -100 -100 -100 -100

100 -100

100

10) 10) 10) 10) 10) 10) 10) 10) 19) 19) 19) 19) 19) 19) 19) 35) 35) 35) 20) 20) 5) 5) 66) 68) 69) 70) 1) 72) 73) 74) 75) 76) 77) 42) 78) 79) 80) 38) 7) 1) 1) 7) 1) 1) 7) 1) 1) 33) 45) 46) 47) 49) 50) 53) 53)

317

Case WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 -8 -8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 -9 -9 9 9 9 9 9 9 9 9

The Quasi-periodic

Orbits: Equations,

(0 0.3053235984844900D-03 2 (0 0.6551687493492600D-03 3 (0 -0.1193539669090700D-02 4 (0 -0.2024246991014900D-02 5 (2 -0.2708774411416200D-01 1 (2 0.3381629411814900D-01 1 (2 -0.9955542247264300D-01 2 (2 0.1221341512928800D+00 3 (2 -0.2473695954818600D+00 3 (2 0.5008130893543500D+00 4 -0.6428586750224600D-01 5 (1 (2 0.9505234470704600D+00 5 (0 0.1002239522501800D-01 7 -0.4365017056636100D+00 7 (1 (0 0.5087832911472100D-01 8 -0.1083587136065700D-02 0 (1 (2 -0.3318053750044400D-02 0 (2 -0.2347249271919700D-01 1 0.1101482655131900D-01 2 (1 (2 0.4497139293829200D-01 2 0.1365032135910700D+00 (2 3 -0.3994961484953800D-01 4 (1 -0.8084293890620100D-01 5 (1 0.8830527172857900D-02 (0 6 0.1405648578945300D-01 (0 7 -0.1544009754566300D-01 3 (1 -0.5877360659744800D-02 (0 6 0.2603222440107000D+01 7 (1 -0.3060593543599700D+00 (0 8 0.7936836143444494D-04 (0 0 -0.5983237746109758D+02 (4 5 0.1495809436527439D+02 (2 7 -0.3739523591318598D+00 (0 9 0.9738041747603100D-01 (0 9 0.1580822707606500D+02 (2 7 -0.3711940092897100D+01 8 (1 -0.3925764113582900D+00 (0 9 0.5484634600600611D+02 7 (3 -0.4113475950450458D+01 9 (1 -0.2468085570270275D+02 (2 9 0.4113475950450458D+00 (0 11 0.5347518735585596D+01 (1 11 -0.4456265612987997D+00 (0 13 TERMS WITH DERIVATIVES, EQUATION = -0.3341313363288495D-01 (0 2 -0.6683093247027213D-01 (0 1 -0.8375835297076488D-03 (0 2 -0.1395972549512748D-02 (0 1 EQUATION = 3 -0.2263475370678822D+00 (0 0 -0.4542979617731012D+00 0 (1 -0.9117631731075615D+00 (2 0 0.2279407932768904D+00 (0 2 -0.7615417908164055D-02 (0 0 -0.1519551909080031D+01 (3 0

Method of Solution and Results -100

100 -100

100 100 100 -100

100 100 -100

100 100 100 100 -100 -100 -100

100 -100 -100

100 -100

100 -100

100 100 -100

100 -100 -100

100 100 100 100 100 -100

100 100 100 100 100 100 100 2 -100

100 -100

100

54) 54) 54) 54) 0) 10) 10) 0) 10) 10) 0) 10) 0) 10) 10) 19) 19) 19) 19) 19) 19) 19) 19) 19) 19) 35) 35) 5) 5) 81) 1) 1) 1) 21) 5) 5) 5) 1) 1) 1) 1) 1) 1)

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

9 9 9 9 9 9 9 9 9 9 9 9 9 9 -9 9 9 9 9 9 9 9 9 9 9 9 9 -9 -9 9 -10 -10 -10 -10 -10 -10 -10 -11 -11 -12 -12 -13 -14

1) 1) 7) 7)

WEIGHT= WEIGHT= WEIGHT= WEIGHT=

3 4 7 8

1) 1) 1) 1) 0) 1)

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

4 5 6 6 7 7

8

7 6 9 8 7 8 8 7 7 9 8 9 8 9 9

The Final Equations

319

of Motion for the Collinear Case

2 0.1139663931910023D+01 (1 4 (0 -0.2849160841612750D+00 0 (0 -0.5800218773453814D-02 2 (2 0.3418993009935300D+01 4 -0.1994412581964867D+01 (1 0 (0 -0.3482857653243700D-03 0 -0.1480686572052700D-02 (1 2 (0 0.9123632340780100D-03 0 -0.3004794187516400D-01 (1 (2 0 -0.9462793536384600D-01 0 -0.1138279977502859D-01 (1 (0 0 0.2279407932768904D+00 0 0.1139663931819923D+01 (1 (4 0 -0.2279328673290200D+01 2 (3 0.7977650327859468D+01 (2 4 -0.7977650328149244D+01 (0 6 0.3324020970062185D+00 6 0.2991618873054868D+01 (1 (0 2 0.1476111493045600D-01 (0 1 -0.3784652953238600D-03 (2 0 -0.3981668853110600D-02 1 -0.1985621943854800D-02 (1 2 0.5792984540911000D-02 (1 3 (0 0.9904034385888300D-03 3 (0 0.8143627506735400D-03 4 0.1327582635888000D-02 (0 0 0.9574939441010300D-02 (1 (2 0 -0.7989984902552700D-01 (3 0 -0.2333095218202800D+00 0 (3 -0.2222333398452900D+00 2 0.1075912048915500D+00 (1 2 0.1038487596814900D+00 (1 TERMS WITH DERIVATIVES, EQUATION = (0 3 0.3341313363288495D-01 (0 3 -0.8375835297076488D-03

1 1 1 1 1 100 100 100 1 1 1 3 3 1 1 1 1 1 1 -100

100 -100

100 -100 -100

1 1 7 1 1 51 51 51 0 0 7 1 1 1 1 1 i; i:

19:

WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT= WEIGHT=

7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

i: 7)

WEIGHT= WEIGHT=

5 9

o;

51] 51;

si: 5i: 51: 82; 82] 10] i9:

o:

19]

o:

3 -100 -100

Table 7.1 Output of program EQUAP: Nonhalo terms in the equations for the analytic qpo. See 7.1.2 for the meaning of the different indices.

CURRENT VALUE OF SPACECRAFT SECTION IN m**2 DIVIDED BY MASS IN kg. : 1.00 THE MODEL OF THE SOLAR SYSTEM IS: 1 1 19 1 1 1 1 1 1 1 1 0 0 0 100 0 EARTH PERIODICS: 1 5 6 10 11 12 14 15 16 17 20 21 22 24 25 26 28 29 32 35 3 56 57 59 60 63 67 84 85 86 87 88 89 90 93 94 95 96 97 98 104 107 108 110 12 137 140 141 144 145 146 148 152 156 161 165 167 168 172 173 176 177 180 185 219 220 221 222 229 233 234 243 251 270 EARTH+MOON-SUN SYSTEM Z-AMPLITUDE = 8.00E-02 CORRECTION DUE TO LACK OF COHERENCE OF THE MOTION OF THE PRIMARY INCLUDED DAY = 0.1750D+05 T = -0.1310852716498577D+02 POINT NUMBER 1:(X,Y,Z)= 0.1019635506848024D-01 0.4207314348755144D (XD,YD,ZD)= 0.3214430737628839D+00 0.3052075793433040D ANALYTIC ACCELERATIONS EQ1=0.2787791196449789D-01 EQ2=-0.17380478540454 NUMERICAL ACCELERATIONS 0.2831617623458477D-01 -0.17397499774892 DIFFERENCES -0.4382642700868841D-03 8 0.1702123443829584D-02 6 POINT NUMBER 2:(X,Y,Z)= 0.1526361820666357D+00 0.2757152978251851 (XD,YD,ZD)= 0.9177555136739562D-01 -0.7507366206869103 ANALYTIC ACCELERATIONS EQ1= -0.4318683308097236D+00 EQ2=-0.1444924831707 NUMERICAL ACCELERATIONS -0.4316207753353331D+00 -0.1446113075735 DIFFERENCES -0.2475554743905373D-03 8 0.1188244028094726D-02 7 POINT NUMBER 3:(X,Y,Z)= 0.1526361820666357D+00 -0.2757152978251851 (XD,YD,ZD)= -0.9177555136739563D-01 -0.7507366206869102 ANALYTIC ACCELERATIONS EQ1= -0.4366294640467704D+00 EQ2= 0.1414894200729 NUMERICAL ACCELERATIONS -0.4363296620688768D+00 0.1416893347240 DIFFERENCES -0.2998019778936226D-03 8 -0.1999146511140248D-02 6 POINT NUMBER 4:(X,Y,Z)= 0.1019635506848024D-01 -0.420731434875514 (XD,YD,ZD)= -0.3214430737628839D+00 0.305207579343304 ANALYTIC ACCELERATIONS EQ1= 0.1571091346738041D-01 EQ2= 0.17490606556383 NUMERICAL ACCELERATIONS 0.1524813977426549D-01 0.17513982571868 DIFFERENCES 0.4627736931150086D-03 8 -0.2337601548421053D-02 6 POINT NUMBER 5:(X,Y,Z)= -0.1146312230589833D+00 -0.944650880822392 (XD,YD,ZD)= -0.1208028119014905D-16 0.895257017033881 ANALYTIC ACCELERATIONS EQ1= 0.8461889947950317D+00 EQ2= 0.17129275419439

NUMERICAL ACCELERATIONS 0.8468285564866052D+00 0.174874015957949 DIFFERENCES -0.6395616915734897D-03 7 -0.3581261763556044D-03 8 DAY = 0.1850D+05 T = 0.4094264390113667D+01 POINT NUMBER 1:(X,Y,Z)= 0.1019635506848024D-01 0.4207314348755144D+ (XD,YD,ZD)= 0.3214430737628839D+00 0.3052075793433040D+ ANALYTIC ACCELERATIONS EQ1= 0.4168312552713130D-01 EQ2=-0.162728495900233 NUMERICAL ACCELERATIONS 0.4239755839318815D-01 -0.162495370007617 DIFFERENCES -0.7144328660568531D-03 7 -0.2331258926155599D-02 6 POINT NUMBER 2:(X,Y,Z)= 0.1526361820666357D+00 0.2757152978251851D (XD,YD,ZD)= 0.9177555136739562D-01 -0.7507366206869103D ANALYTIC ACCELERATIONS EQl=-0.4409516146104189D+00 EQ2=-0.134093114340757 NUMERICAL ACCELERATIONS -0.4397904713516222D+00 -0.134072549114796 DIFFERENCES -0.1161143258790659D-02 7 -0.2056522596116106D-03 8 POINT NUMBER 3:(X,Y,Z)= 0.1526361820666357D+00 -0.2757152978251851D (XD,YD,ZD)= -0.9177555136739563D-01 -0.7507366206869102D ANALYTIC ACCELERATIONS EQl=-0.4571794416683768D+00 EQ2= 0.130864443015044 NUMERICAL ACCELERATIONS -0.4578616407634685D+00 0.130679003751472 DIFFERENCES 0.6821990950917206D-03 7 0.1854392635724317D-02 6 POINT NUMBER 4:(X,Y,Z)= 0.1019635596848024D-01 -0.4207314348755144 (XD,YD,ZD)= -0.3214430737628839D+00 0.3052075793433041 ANALYTIC ACCELERATIONS EQ1= 0.1809046070186403D-01 EQ2= 0.164182870609200 NUMERICAL ACCELERATIONS 0.1897933354781448D-01 0.163993424237570 DIFFERENCES -0.8888728459584509D-03 7 0.1893863716291744D-02 6 POINT NUMBER 5:(X,Y,Z)= -0.1146312230589833D+00 -0.9446508808223920 (XD,YD,ZD)= -0.1208028119014905D-16 0.8952570179338817 ANALYTIC ACCELERATIONS EQ1= 0.8220081330937653D+00 EQ2= 0.248567059422373 NUMERICAL ACCELERATIONS 0.8214130610115142D+00 0.242763241334202 DIFFERENCES 0.5950720822510414D-03 7 0.5803818088171279D-03 7 DAY= 0.1950D+05 T= 0.2129705594521311D+02 POINT NUMBER 1:(X,Y,Z)= (XD,YD,ZD)=

0.1019635506848024D-01 0.3214430737628839D+00

0.4207314348755144D+ 0.3052075793433040D+

ANALYTIC ACCELERATIONS EQ1= 0.2434152595495066D-01 EQ2=-0.1626485518502 NUMERICAL ACCELERATIONS 0.2417051764236660D-01 -0.1629117121623 DIFFERENCES 0.1710083125840657D-03 9 0.2631603121184489D-02 6 POINT NUMBER 2:(X,Y,Z)= 0.1526361820666357D+00 0.2757152978251851 (XD,YD,ZD)= 0.9177555136739562D-01 -0.7507366206869103 ANALYTIC ACCELERATIONS EQl=-0.4546581708825895D+00 EQ2=-0.1296631136615 NUMERICAL ACCELERATIONS -0.4552989346734423D+00 -0.1297761905444 DIFFERENCES 0.6407637908527261D-03 7 0.1130768829031931D-02 7 POINT NUMBER 3:(X,Y,Z)= 0.1526361820666357D+00 -0.275715297825185 (XD,YD,ZD)= -0.9177555136739563D-01 -0.750736620686910 ANALYTIC ACCELERATIONS EQl=-0.4467273914706543D+00 EQ2= 0.1329909903477 NUMERICAL ACCELERATIONS -0.4481043757875935D+00 0.1330683058488 DIFFERENCES 0.1376984316939230D-02 7 -0.7731550107331342D-03 7 POINT NUMBER 4:(X,Y,Z)= 0.1619635506848024D-01 -0.420731434875514 (XD,YD,ZD)= -0.3214430737628839D+00 0.305207579343304 ANALYTIC ACCELERATIONS EQ1= 0.4141001635079894D-01 Eq2= 0.1617215500774 NUMERICAL ACCELERATIONS 0.3970674229298756D-01 0.1616624863459 DIFFERENCES 0.1703274057811320D-02 6 0.5906373141474630D-03 7 POINT NUMBER 5:(X,Y,Z)= -0.1146312230589833D+00 -9.944650880822392 (XD,YD,ZD)= -0.1208028119014905D-16 0.895257017033231 ANALYTIC ACCELERATIONS EQ1= 0.8201433189314190D+00 EQ2=-0.2032483053048 NUMERICAL ACCELERATIONS 0.8200655554961056D+00 -0.1978060725010 DIFFERENCES 0.7776343531337715D-04 9 -0.5442232803807936D-03 8 Table 7.2 Checks of the analytic model, Sun-Barycenter problem, case L\. Point

323

The Method of Solution of the Equations for the Quasi-periodic Orbit

7.2

7.2.1

The Method of Solution of t h e Equations for t h e Quasi-periodic Orbit General dure

Expression

of the Equations.

The Recurrent

Proce-

In this section, a recurrent way to obtain the coordinates x, y and z of the quasiperiodic solutions near a family of halo periodic orbits around the equilibrium point L\ ox Li is given. We recall that for a given value of the mass parameter /J, and a given equilibrium point Lj, j = 1, 2, the program ANACOM computes x,y and z for the halo orbits as Fourier series in the variable LOT such that each coefficient is a power series in the basic amplitudes a and /?. Each halo family has associated two quantities UJ = 1 + ui and A = A (see Chapter 2, 2.2.2) which can be obtained as power series in a and p. In the next, we will assume that p, and Lj are fixed and so, the coefficients of u> and A are known up to a certain order. We consider the equations in the following way u2x" - 2wj/' - (1 + 2c2)a; 2

u y" +

2LJX'

- (1 - c2)y 2

u?z" + X z where ' = — — r , M = ^

P =

YJ

'

=

M + IEQ1 + wIQXl + wIQYl,

=

N + IEQ2 + wIQY2 + wIQZ2,

=

P + Az + IEQ3 + wIQY3 + wIQZ3,

mijkxiyjzk,

^

N-

n=i+j+k>2

pijk xlyizk

^

nijkxiyjzk

(7.4)

and

n=i+j+k>2

are the halo right terms of the equations, IEQi for

n=i+j+k>2

i = 1,2,3 admit developments of the following type ^2 n>h

Yl

cijkrnxiyjzkF(urt

+ <pr)

(7.5)

i,j,k,r>0

and IQXi, IQYi and IQZi for i = 1,2,3 are developed in the form

v'^2

^crnF{vrt

+ <pr),

(7.6)

n>h r>0

where v' = x' for IQXi, v' = y' for IQYi and v' = z' for IQZi. In (7.5) and (7.6) t is the normalized time (taken in radians per tropic year for practical purposes), F stands for one of the trigonometric functions cosine or sine, v\,i>2,---,Vm, a n d <£i,
324

The Quasi-periodic

Orbits: Equations,

Method of Solution and Results

than or equal to their degree, the absolute value of the negative weight computed by the program EQUAP (that is, the degree plus 1) is taken as their weight. The solution of (7.4) can be written as the halo solution plus other terms due to the perturbations in the equations. We write the coordinates of the quasi-periodic solution in the following way

x = xH+xN = y^ xHn + y~^ xNn = y^ xn, n>l

V

=

n>h

n>l

VH + VN = ^ 2 / t f n + ^2 VNn = n>l

n>h

^Vn, n>l

z = zH+zN = y^ zHn + y ] zNn = y zn, n>l

n>h

n>l

where xHn {yHn,ZHn) contains all the terms of weight n corresponding to a; (y, z) of the halo orbit, and in x^n (yNn, ZNn) are included all the nonhalo terms of weight n of the x (y,z) coordinates. In the next subsection we shall explain the recurrent method that gives the terms of weight n > 2, for xn, yn and zn , when all the terms of weight less than n have been computed. 7.2.2

A Lindstedt-Poincare

Device for the Quasi-periodic

Orbit

Let us assume that all the terms of x, y and z are known up to weight equal to n — 1. By substitution of the coordinates x,y and z in the equations we get in the right-hand side some terms which have a weight equal to n. These are cosine or sine terms of the following type m

caiPiF{k\uJT

+ J2 seiyTJ + VrJ)

= c a1 j3j F {kXojT + ar + v),

(7.7)

e=l

where c is a coefficient, i, j , k > 0, F is one of the trigonometric functions cosine or sine, i/ n > • • • i vrm and
The Method of Solution of the Equations for the Quasi-periodic

Orbit

325

The first step of the recurrence uses the halo terms of weight one XHI , ym and ZHI (given by the program ANACOM). It is assumed h > 2 in (7.5) and (7.6). We note that if a = 0 and \k\ = 1 we have a resonant nonhalo term in (7.7). A modified Lindstedt-Poincare method is introduced to deal with the resonant nonhalo terms. We add a nonhalo part u t o u and a nonhalo part A to A such that

U =

l + U} + LO = l+

Y^

d ai J

ij P

i,j>0,i+j>2

A = A + A=

Y,

n>h

ho'? + £

i,j>0,i+j>2

+ Y1 ^2 Uijntf/3j, i,j>0

£ Aijna\0',

n>h i,j>0

where n is the weight associated to each term of Q and A. The coefficients cjijn and Aijn should be computed in a recurrent way according to the resonant terms that appear in the equations. Let us consider a term (7.7) such that a ^ 0 or \k\ ^ 1. Resonant terms will be considered later. We note that we can deal with the first and second equations independently of the third one. So, if (7.7) is a term of the first or second equation, it gives a contribution in the coordinates x and y of the following form x = cxa'P3' Fx(k\ujT + (JT + (p),

y = cyal(33Fy{kXuJT +
The z coordinate is obtained from the integration of the third equation. Therefore from a term (7.7) we have z = czalpjFy(k\ojT

+ (JT + (p).

Table 7.3 gives the coefficients and the character, cosine (s = +) or sine (s = —) of the terms of x,y and z which are obtained from the integration of a term (7.7) depending on the equation which contains that term and its character. In fact, due to the kind of terms that appear in the equations (cosine in the first and third equations, sine in the second one) we have that x and z are series in cosine terms and y is a series in sine terms. In Table 7.3 we put D = AA + A2{c2 - 2) + (1 + 2c 2 )(l - c 2 ), where A = kX + a. If k = 1 and a = 0, then D = 0, and the coefficients in the table are not defined. Let us suppose that we get the following terms in the right-hand side of the first and second equations respectively Cia8/3J coswAr,

C2a'/3J'sinwAr,

with i > 1. In order to avoid terms in cos WAT in a;, we should define a nonhalo term of u of weight n — 1, cDn_ia*-1/?J', and a sine term of weight n in y, ca1^ sinwAr. Let Xtfi = aacosXUJT, yu\ = basinXUIT, and zu\ = c(3cosXLOT, be the halo terms of the solution of weight 1. By substitution of these terms in the equations and equating the terms of weight n in a1 ft, we get the following system for wn-i

326

The Quasi-periodic Orbits: Equations, Method of Solution and Results

X

s

1

+

- c

z

y s

Cy

s

+

2cA

-

i—i

Equations

-

-c(^±i)

-

-2cA

+

2

+

-2cA

-

-c{£*$**)

+

2

-

2cA

+

-c

-

3

+

rC

3

-

C

cx ( * = # * ) •

(£*%**)

-

Table 7.3

Cz

S

1 AS-X2

+

1 A2-A2

The form of the terms obtained by integration.

and c —2A(aA + fo)w„_i — 2Ac = ci, -2A(6A + a) 1 in the third equation. Then we define a term of weight n — 1, A „ _ i a J ^ _ 1 as A„_i = —cz/c. In that way, the resonant nonhalo terms of the third equation are included in A. 7.2.3

Description of the Program periodic Orbit

to Obtain

the Analytic

Quasi-

The program QPO computes up to a given weight, the analytic solution of the general equations of motion near a halo orbit. The three components x, y and z of the solution are given as Fourier series in the basic frequency of the halo orbit plus other frequencies that appear in the equations.

The Method of Solution of the Equations for the Quasi-periodic

Orbit

327

The coefficients of that Fourier developments are power series in a and (3. The series involved in the computations are stored in different arrays according to their type. There are two basic types of series, the general Fourier series (as the ones for x, y or z) and the series in a and /? (like UJ or A). The dimensions of these arrays are related with the number of different terms of these series. This number is unknown a priori. So, the dimensions are taken as parameters and the values used in the program are given in the routine DIMEN. Two arrays, GMA(MQ6) and PHA(MQ6) are used to store the frequencies and the phases respectively. Therefore, each pair of frequency vr and phase y r is represented by an index i\ such that GMA(ii) = vT and PHA(ii) = tpr. The terms (7.7) involved in the computation of the quasi-periodic orbit are represented by a coefficient and 6 indices c(i,j,s

•k,ii,i2,h),

(7.8)

where s is a sign denoting a cosine (+1) or a sine (—1) term, i\,i2 and i$ are the indices of the corresponding frequencies, and i,j and k = k have the same meaning as in (7.7). If k = 0, then we take k = 100. For the storage of a series of that kind of terms (a general series), the program uses three arrays, named generically N(MQ1,6), CF(MQl) and NPN(NPMAX). In that way, each term has associated one index IN such that N(IN,1)= i, N(IN,2)= j , N(IN,3)= s • k, N(IN,4)= ix, N(IN,5)=?2, N(IN,6)= «3, and CF(IN)= c. Therefore, the character sine or cosine of the term is contained in N(IN,3). We note that the indices i\,i2 and i% are ordered such that |ii| < |i 2 | < \iz\ , if i\,i2 and i% are different from zero. The indices equal to zero are stored at the end of the row IN. NPN(I) is the number of terms of N which have a weight less than or equal to I. The terms of the series in a and /3 are represented by the coefficient and 3 indices

c(i,j,n), where i is the exponent of a, j is the exponent of fi, and n is the weight of the term. In the program, they are stored in arrays NW(MQ10,3) and CW(MQIO) such that NW(IN,1)= i, NW(IN,2)= j , NW(IN,3)= n and CW(IN)= c. The coefficients of the halo terms of the equations are stored in FM(MAX4), FN(MAX5) and FP(MAX5) and they are computed in the routine SIGMNP via the routine FMNP. In the program QPO, the terms of IEQ1, IEQ2 and IEQ3 in the equations are represented by c(i,j,s-k,i\), where ii is the index of the corresponding frequency. Each term of one equation L where L=l,2,3, has associated one index IN = 1,2,.. .,MQ7 such that IEQ(L,IN,1)= i, IEQ(L,IN,2)= j , IEQ(L,IN,3)= s • k, IEQ(L,IN,4)= h and CFEQ(L,IN)= c, where s and k have the same meaning as before. The number of terms in IEQ(L, , ) which have a weight less than or equal to I is stored in NPEQ(L,I). We remark that all the monomials that appear in the equations are of the same

328

The Quasi-periodic

Orbits: Equations,

Method of Solution and Results

type of the ones in the halo part. Therefore, for each degree, the monomials are some of the ones given in Table 1 of Chapter 2. The terms in (7.4) that multiply v' where v = x,y or z, are stored in a similar way in IQV, CQV, NQV, where V = X, Y or Z. Therefore, for one term of the equation L we have IQV(L,IN,1)= s • 100, IQV(L,IN,2)= h and CQV(L,IN)= c, where s is a sign and ii is the index of the related frequency. As before, NQV(L,I) gives the position of the last term of IQV(L, , ) which have weight equal to I. The nonhalo terms of the equations as well as the involved frequencies and phases are read in the routine LLEEQ. Let us assume that all the terms of x,y and z are known up to weight NP-1. For the computation of the terms of weight NP, first we fill some arrays NF, CFF, with the terms of weight NP obtained from the terms of the solution which have a weight less than NP. The arrays NF(L,MQ2,6), CFF(L,MQ2) contain the indices and the coefficients respectively of the terms of the equation L. We remark that the main problem (time and memory) is the computation of the monomials of a given degree. It requires to perform some products according to Table 1 of Chapter 2, in order to generate the monomials of degree NG+1 from the ones of degree NG. These computations are done in the routine GRAUT. The series of the monomials of degree NG are stored in NUX(6,MQ1,J), CFUX(MQ1,J) and NPUX(NPMAX,J), where J = l , 2 , . . .,MQ4 if NG is odd, and J = M Q 4 + 1 , . . .,MQ44, if J is even. We note that if we fix some values of MQ4 and MQ44 (for example 30 and 55 respectively), the maximum degree such that the monomials can be obtained in that way is fixed. We name NGDIR that maximum degree (in the example NGDIR=7). The monomials of degree greater than NGDIR should be computed by direct products in the routine RESPRO. The routine QPR0D2 performs the product of two series. For a given weight it has been convenient to compute first all the possible products of two terms that produce a term of that weight. Therefore, a large list of terms is obtained. The list is ordered by the routine ORDRE in such a way that the terms with the same indices become consecutive. Then, the list is reduced comparing and adding if it is possible, each term with the next one. The order used for the terms of the same weight NP is defined as follows. Let us consider the indices i, j , s-k of & term (7.8). First, the degree NG= i + j increases from 0 to NP. For a fixed degree, the index i goes from NG to 0. Once NG and i are fixed, the order for the third index is taken in the following way: 100, - 1 0 0 , 1 , —1,2, —2,... ,NP,-NP. The three indices of the frequencies are ordered in the usual way according to their absolute value, that is, (0,0,0), (1,0,0), (1,1,0), (1,1,1), (1,1,2), (1,1,3),...,(1,1,IGMA), (1,2,0), (1,2,2), (1,2,3),... ,(1,2,IGMA), ( 1 , 3 , 0 ) , . . . , (2,0,0), (2,2,0),(2,2,3),...,(2,2,IGMA),(2,3,0),...

The Results. Problems Related to Small

Divisors

329

In the sequence above, IGMA is the total number of frequencies. If two terms (i,j,s

-k,ii,12,13),

(7-9)

and (i,j,s-k,j1,j2,J3),

(7-10)

satisfy |ii| - \ji\, \i2\ = IJ2I,1*31 = IJ3I, we set (7.9) before (7.10) if \h + i2 + *31 > lii + h + h I • The opposite holds in other case. We remark that due to the large amount of terms that appear in the intermediate computations, some reduction has been made. Let e be a given upper bound. The terms (7.8) such that \c\tlxt{ < e, are neglected. The parameters tx and tz are the same ones used in the program EQUAP (see 7.1.2). As it was explained in the program EQUAP, some terms in the equations stand with a negative (fictitious) weight, that means, its real weight is less than or equal to its degree. In the program QPO we deal with a maximum weight NN that means the maximum fictitious weight to be considered, and a real weight NNR. We remark that the halo monomials of degree NNR need not be computed because the nonhalo terms of the solution have a weight greater than 1.

7.3

The Results. Problems Related to Small Divisors

7.3.1

Sample

of Results

in Several

Cases

The program QPO has been used to obtain an approximation of the quasi-periodic orbit in the following cases: 1) 2) 3) 4)

Sun-Barycenter, case L\, Sun-Barycenter, case L2, Sun-Barycenter, simplified model, case L\, Earth-Moon, simplified model, case L2.

In any of the first 3 cases the solution was computed up to a real weight equal to 9. During the computation of the terms of a weight NP, the terms (7.8) such that \c\txt{ < l.D-10-fNF-\ w h e r e / = 2.5, were neglected. Due to the memory available in our computer, the values of the dimensions MQ4 and MQ44 in program QPO were taken equal to 30 and 55, respectively. Then, the value of NGDIR was taken equal to 7. This means, in particular, that the contribution of the halo terms of the equations which have a weight greater than 7 was not taken into account in the computed solution. In the fourth case, simplified model means that, due to small divisors problems to be explained in 7.3.2, the equations for the QPO have been modified. As a test, only the frequency related to the mean anomaly of the Moon, i.e., the mean longitude of the Moon minus the longitude of the mean perigee of the Moon, is retained. We skip

330

The Quasi-periodic

Orbits: Equations,

Method of Solution and Results

also all the terms in the equations of weight greater than 5. Then a very reduced set of terms appears in the nonhalo part of the equations. With this modified set no problem appeared even running the QPO program up to real weight 9. Table 7.4 displays the results up to weight 5 to prevent from too large lists. We recall that in the displayed results only the nonhalo terms are given. The functions Q and A are also given. As an example, case 1) gives a total of 90 frequencies and 6502 terms (in x, y and z). The functions Q and A have 38 and 35 terms, respectively. The running time under VAX 11/785 is 32 minutes. It reduces to 10 seconds if the weight is reduced to 5. Once the QPO solution is available we can obtain a concrete quasi-periodic orbit if numerical values are assigned to the parameters /3 and t* (see 7.3.3). In this way the solution is given as a Fourier expansion ^2ciF(vit +
7.3.2

The Problem

of Small

Divisors

In the course of the integration of the equations of motion in form (7.4) and (7.5), the Table 7.3 given in 7.2.2 is used to obtain the terms of the solution. The denominator D is zero when we are at exact resonance, i.e., the frequency

The Results. Problems Related to Small

Divisors

331

value (that we have taken as 10). In this case a message appears in an auxiliary file telling us that a small divisor is found and identifying it. 2) In the L\ case for the Sun-Barycenter problem only one of those terms has appeared and it is retained in the solution. It has a frequency equal to the sum of frequencies 1 and 51 in the output given. Frequency 51 is easily identified as coming from the perturbations in longitude introduced in the motion of the Earth by the effect of Jupiter. More concretely, it is associated to a term with frequency (ME+MJ)' where ME and Mj refer to the mean anomalies of Earth and Jupiter. As frequency 1 is exactly ME the total frequency in resonance with A is (2ME + MJ)' = 2.0842006, close to the value A = 2.0864535. In the simplified problem the only perturbing frequencies are multiples of ME, not giving rise to small divisors at low orders. 3) For the L^ case, Sun-Barycenter problem, the results are similar. In this case the unique term in resonance with A up to (real) weight 9 is associated to the sum of frequencies 1 and 59. Frequency 59 comes from the perturbations due to the Moon. It amounts 1.0537249 = (mi —7714 — 771,5)', where m\,m± and 7715 are mean longitude of the Moon, the longitude of the mean ascending node and the mean elongation of the Sun, respectively. Hence, the resonant argument is ME + IE — ^ M , where IE is the mean longitude of the Earth (= mi — 777.5 = ME + wg) and Q M = "^4. The related frequency is 2.0536771 close to the value of A in this case equal to 2.0570143. 4) Finally in the L 2 case, Earth-Moon problem, the influence of small divisors is much larger. They already appear when the solution to fifth order is searched. As the related coefficients (several terms appear with the same resonance) are large and they appear so early, the resulting coefficients in the solution become meaningless. The main resonance is found between the double of the mean elongation of the Sun and the halo orbit angle. We have 2m 5 = 1.850392 and A = 1.862645. These perturbations come from the noncircular motion of the Moon around the Earth and also from the Sun effect. As we have said in 7.3.1 we merely skip those terms and we restrict ourselves to a simplified model in that case.

7.3.3

The Analytic

Expression

of the Nominal

Orbit

Values of a and /? should be substituted in the analytic expressions to get a defined quasi-periodic orbit. The amplitudes a and /3 are related by A(a,/3) = A, where in the A function both the halo and nonhalo terms are included. Given a value of the z-amplitude /3, the x-amplitude a can be determined. Then the basic frequency co{a, (3) is obtained, where in ui the halo and nonhalo terms are also retained. By substitution of a and /3 in the analytic expressions the coefficients of the different terms are obtained. Coefficients multiplying the same cosine or sine functions are collected. From the list of frequencies and the value of UJ already available all the frequencies appearing in the solution are computed. Concerning the phases we recall that the generic argument in the trigonometric functions is of the form

332

The Quasi-periodic

Orbits: Equations,

Method of Solution and Results

k\uj(t — t*) + i>t + ip, where t* is an additional parameter of the quasi-periodic orbit. In the halo part of the solution we have written the arguments as kvur, where VH = Xui and r = t — t*. In the nonhalo part the argument is finally expressed as at + b, with a = kXoj + v, b =

0.6231395117299221D+01 0.6179604927418856D+01 0.6127814737538490D+01 0.6076024547658125D+01 0.6024234357777759D+01 0.5972444167897394D+01 0.5920653978917028D+01 0.5868863788136663D+01 0.5817073598256298D+01 0.5765283408375932D+01 -0.2884338825126906D+00 -0.1979167429597718D+01 0.2293632183012776D+01 0.3900981710995600D+01 0.2709879912903800D+01 0.1425662661422312D+01 0.5191770785557459D+00 0.1659447913253400D+01 0.1518778114811500D+01 0.1150705458671403D+01 -0.2167779041033939D+01 0.2643633021351981D+01 0.2714619818594807D+01 0.1881342399767400D+01 0.1417296496991800D+01 0.2292073612553800D+01 0.5092083509087600D+01 0.3952771895950200D+01 0.4683461151616500D+00 9.2553870424499513D+01 -0.3074291866679034D+01 -0.3283894176739517D+00 0.2430331539214389D+01

The Results.

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Problems Related to Small

0 .8732511351291710D+00 0 .1915609282611617D+01 0 .8429515884531046D-01 0 .9660209891320313D+00 0 .1250997870014100D+01 0,.1876496805021200D+01 0 .2501995740028200D+01 0,.3127494675035300D+01 0,.2746980187678700D+01 0,.1325524075636900D+02 0..1404902516893226D+01 0,.2831266344494771D+01 0..1932041978264063D+01 0..1625459877970818D+01 0..6335054279964647D-01 0.. 4049592961647623D+00 0,.2658256908820276D+01 0..1084247379573774D+01 0..7470667441925326D+00 0,.3752993610042300D+01 0..4378492545049400D+01 0,.5003991480056400D+01 0..9156600625595800D+00 0..1746980187678700D+01 0,.8869742675313700D+00 0..1053724902883300D+01 0..1773948535062700D+01 0..1048134000056200D+02 0,.1059431795377500D+02 0.3151973342338953D+01 0..1251015314484709D+01 0.,1831314123766307D+01 0..9156570618831535D+00 0.6255076572423544D+00 0..1876522971727063D+01 0.8765707509985993D+00 0..2502030628969418D+01 0.9366016779288175D+00 0.2746971185649461D+01 0..1747018964920997D+01 0..2250967535213173D+01 0..3127538286211772D+01 0..3753045943454126D+01 0..1915609282611617D+01 0.. 2831266344494771D+01 0.,8313619030378430D+00 0..1502078408240954D+01 0..4378553660696481D+01 0..5004061257938835D+01 0.5629568915181190D+01 0..1404902516893226D+01 0..8732511351291710D+00 0.,3662628247532614D+01 0..2662676026804150D+01 0.9660209891320313D+00 0..3151973342338853D+01

Divisors

0 .1916038532792506D+01 -0 .1827920526552135D+01 -0 .3095417103290732D+01 -0 .2258506631948617D+01 0 .2834592993983600D+01 0 .4251889490975400D+01 0 .5669185987967100D+01 0 .8032971777792700D+00 0,.3438110418830600D+01 0..2241533797742100D+01 -0..7233664097316860D+00 0..2182875486824612D+01 0,,1756924709239920D+01 0..1359984598049204D+01 -0,.1145897920449518D+01 0,.9572104907461318D-01 0..3718394402992522D-01 0..1869375022576490D+00 0,.2170478728628369D+01 0..2220593674771200D+01 0..3637890171762900D+01 0..5055186668754700D+01 0,.1146036806276900D+01 0,.1693250981649900D+01 0..3432635595834100D+01 0..4274368206350897D+01 0,.5820858844887000D+00 0..1711238098207900D+01 0,.4509997624599100D+01 0.,2612227390392102D+01 -0. 2891284191112373D+00 0..2289217823911790D+01 -0. 1979628239660955D+01 0.,1425700153600997D+01 0. 1138912762564505D+01 -0. 2167620986141730D+01 0.2556299801139196D+01 0.,5192220030676837D+00 -0.,2914075454189641D+01 0..1864493202009641D+01 -0.,3185521467599869D+00 -0. 2294371900809253D+01 -0.,8816036368470227D+00 -0.,1557393366956654D+01 0.,2080900989865821D+01 0..2713996048062953D+01 0.,2429935316830200D+01 0..5610543682337673D+00 0..1957542532778765D+01 -0..2894243332689311D+01 -0.,7268767491484566D+00 0..1916053366219000D+01 -0..1798357254562880D+01 0..2945883921131993D+01 -0. 2258514577062989D+01 0.,2603256991840835D+01

334

The Quasi-periodic

X 1 X' 1 X 2 X 2 X 2 Y 1 Y 1 Y 2 Y 2 Y 2 X 1 X 1 X 2 X 2 X 0 X 0 X 2 X 0 X 3 X 3 X 3 X 3 X 1 X 1 X 1 Y 1 Y 1 Y( 2 Y 2 Y 0 Y 0 Y 2 Y 0 Y 3 Y 3 Y 3 Y 3 Y 1 Y 1 Z 0 Z 0 Z 1 Z 1 Z 1 Z 2 Z 2 Z 2 Z 2 X 3 X 3 X 4 X 4 X 1 X: i X' 2

Orbits: Equations,

Method of Solution and Results

90 = 0.1131454158595400D+02 0.39593079563264970+01 TOTAL NUMBER OF TERMS= 158 0 1 1L 0 0) = -0 2669356593935684D-01 NP 0 1 -1L 0 0) = 0 1252033081780248D-01 NP 0 100 1L 0 0) = 0 ,9833646461186939D-01 NP 0 2 ]L 0 0) = -0 .1891001862953086D-01 NP 0 2 -1L 0 0) = -0 4412153475217745D-01 NP 0 -1 1L 0 0) = 0 8126898113476926D-01 NP 0 -1 -]L 0 0) = -0 5757800552195134D-01 NP 0 -100 1L 0 0) = 0 9541358355130655D-01 NP 0 -2 3L 0 0) = 0 8255662704913419D-02 NP 0 -2 -1L 0 0) = 0 3996125711596286D-01 NP 0 1 1L 0 0) = -0 7952609015785745D-02 NP 1 -3I 0 0) = 0 0 9007524900828336D-02 NP 0 2 1L 0 0) = -0 2376288753568175D-01 NP 2 -3L 0 0) = 0 0 7822524746302241D-01 NP 2 2 1 0 0) = -0 3688256718919798D-03 NP 2 2 -1L 0 0) = -0 1120591959602091D-02 NP 0 100 JL 0 0) = -0 8466828527037358D-01 NP 2 100 1L 0 0) = -0 9428372672550820D-02 NP 0 1 JL 0 0) = -0 2866400430142627D-01 NP 0 0) = 0 1 -J 0 3701839626643600D-01 NP 0 3 1L 0 0) = 0 154324T716056545D-01 NP 0 0 1916496538595158D-01 NP 3 -1L 0 0) = 2 1 1 0 0) = -0 1749527642252451D-02 NP 2 1 -1L 0 0) = -0 1741409742622264D-02 NN 2 3 -1L 0 0) = 0 6911269745932756D-03 NP 0 -1 1 0 0) = 0 1837787471657781D-01 NP NP 0 -1 -1L 0 0) = -0 2666528991327288D-01 0 -2 1 0 0) = -0 4016119726803939D-01 NP 0 -2 -1 0 0) = -0 6032680215587482D-01 NP 2 -2 1L 0 0) = -0 2145939821559522D-03 NP 2 -2 -1 0 9533859147633840D-03 NP 0 0) = NP 0 -100 3L 0 0) = -0 5529834427508559D-01 2 -100 1L 0 0) = -0 9148130627798677D-02 NP 0 -1 3L 0 0) = -0 2252227160286346D+00 NP 0 -1 -1L 0 0) = 0 3316892405619730D+00 NP 0 -3 1L 0 0) = 0 8489999798965142D-02 NP 0 -3 -1L 0 0) = 0 ,1061462300579955D-02 NP 2 -1 1L 0 0) = 0 1670501784932256D-02 NP NP 2 -1 -3L 0 0) = -0 2012182569754218D-02 1 1 0 0) = 0 2129944582128835D-01 NP -1L 0 0) = -0 3472442678152180D-01 NP 100 JL 0 0) = 0 1914632699216435D-01 NP NP 2 1L 0 0) = -0 5397066891656604D-02 0 9925409151016849D-02 NP 2 -1L 0 0) = 1 3L 0 0) = 0 ,6955925824526413D-01 NN NP 1 -1L 0 0) = -0 7740615473686131D-01 NP 3 1L 0 0) = -0 1771928172609215D-02 3 -3L 0 0) = -0 8575189217254399D-02 NP 1 1L 0 0) = 0 1493758465962166D-01 NP 0 0 1 -3L 0 0) = -0 1105196494242014D+00 NP NP 0 2 1L 0 0) = -0 ,1528399737976488D-01 NP 2 -3L 0 0) = -0 7248098572157560D+00 0 0 2409534664114047D-01 NP 2 1 1 0 0) = 1 -1L 0 0) = 0 .4884232796953952D-02 NP 2 0 .1531112884313373D-01 NP 2 2 1L 0 0) =

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5

The Results.

( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

2 2 2 0 0 1 1 3 3 1 1 2 2 2 2 2 0 0 1 1 2 4 4 4 2 2 0 3 3 3 3 2 2 2 2 3 3 3 3 4 4 4 4 3 3 4 4 1 1 2 2 2 2 0 0 1

2 0 0 2 2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 0 0 2 2 0

2 2 2 2 2 1 1 3 3 3 3 100 100 2 2 2 100 100 1 1 100 100 4 4 100 4 100 1 1 3 3 100 2 2 100 1 1 3 1 100 2 2 4 -1 -1 -2 -2 -1 -1 -2 -2 -2 -2 -2 -2 -1

-1 1 -1 1 -1 2 -2 1 -1 1 -1 2 0 2 0 -2 0 14 14 -14

-1 -1 2 -2 0 -2 14 14 -14 15 14 -14 -14 -15 14 14 -14 -14 -1 -1 -1 -1 -1 -1 2

Problems Related to Small

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0)

335

Divisors

-0.2760835214642224D-01 -0.1186298033953153D-01 0.1425961038675736D-01 0.3511323737189227D-02 -0.1126240299134378D-01 -0.4072479028357518D-03 -0.1059625294892354D-03 0.4390937319252455D-01 -0.3049745885122305D-01 -0.2333632281776978D-03 0.2228812101228466D-02 0.1402399775468182D-01 -0.1864345213243881D-02 -0.6395837192119324D-03 -0.1338154780994827D-03 0.9923059331405869D-02 -0.1147792623444093D-02 -0.8968126833019414D-05 -0.3500143992903151D-04 0.4950482330025654D-04 -0.5714246073628670D-02 0.6745130924525374D+00 -0.1590111811422762D-01 0.6501804157556853D-02 0.7124432261788821D-01 -0.5496313757792188D-02 -0.5041683997839682D-02 -0.1530168418226505D-02 0.5440842371230224D-02 -0.4028682187663889D-03 -0.6537444824029897D-03 0.2931221671976148D-03 -0.1065317334916862D-03 -0.2141069188077573D-03 0.5565460801675805D-04 0.1938402376148063D-02 -0.1469427491639834D-02 0.7450097163943381D-03 -0.3291544327809975D-03 -0.6379649990960484D-02 0.3089787051357074D-02 0.6217923871091577D-02 -0.2481559333503040D-02 0.4321680584935495D-01 -0.1965852481170879D+00 0.2661212511772143D+00 0.2094975641268155D+00 -0.4323400510629940D-01 -0.1539017628550321D-01 -0.8566693563945978D-02 0.2998592417363558D-01 0.2097594950836090D-02 0.4116126266101141D-01 -0.2147645230774970D-02 0.8121487136113621D-02 0.1009101621492969D-02

NP = NP = NP = NP = NP = NP = NP = NP = NN = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP = NP =

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The Results. Problems Related to Small

Divisors

337

EXAMPLE OF QUASI-PERIODIC ORBIT FOR THE LI CASE, EARTH+MOON—SUN PROBLEM ONLY COEFFICIENTS GREATER THAN l.D-5 ARE RETAINED FOR THE NONHALO PART ORDER OF THE THEORY FOR NONHALO PART = 11 ORDER OF THE THEORY FOR HALO PART = 15 z-AMPLITUDE = 0.80D-01 x-AMPLITUDE = 0.1406721580148259D+00 OMEGA = 0.9834573393464491D+00 TSTAR =-0.4597131387436053D+01 TAU = T-TSTAR FOURIER DEVELOPMENT OF THE HALO PART x-COMPONENT 0.4157049854675195D-01 * COS (O.OOOOOOOOOOOOOOOOD+00 * TAU) -0.1496721580148259D+00 * COS (0.2051937963795992D+01 * TAU) -0.1652142764859605D-01 * COS (0.4103875927591983D+01 * TAU) 0.1996174144123191D-02 * COS (0.6155813891387975D+01 * TAU) -0.4092680855206494D-03 * COS (0.8207751855183966D+01 * TAU) 0.7948892532718254D-04 * COS (0.1025968981897996D+02 * TAU) -0.1695502931662977D-04 * COS (0.1231162778277595D+02 * TAU) 0.3703708834651033D-05 * COS (0.1436356574657194D+02 * TAU) -0.8365191685526472D-06 * COS (0.1641550371036793D+02 * TAU) 0.1927998899599959D-06 * COS (0.1846744167416392D+02 * TAU) -0.4528808235964133D-07 * COS (0.2051937963795992D+02 * TAU) 0.1678599432413511D-07 * COS (0.2257131760175591D+02 * TAU) -0.2507989213559546D-08 * COS (0.2462325556555190D+02 * TAU) 0.6065508369912068D-09 * COS (0.2667519352934789D+02 * TAU) -0.2033435877789632D-09 * COS (0.2872713149314388D+02 * TAU) 0.5124894637061295D-10 * COS (0.3077906945693987D+02 * TAU) y-COMPONENT 0.4462903402478923D+00 * SIN (0.2051937963795992D+01 * TAU) -0.9485343224882868D-02 * SIN (0.4103875927591983D+01 * TAU) 0.2288953429537220D-02 * SIN (0.6155813891387975D+01 * TAU) -0.3812160243869096D-03 * SIN (0.8207751855183966D+01 * TAU) 0.7909732719394648D-04 * SIN (0.1025968981897996D+02 * TAU) -0.1658351072469369D-04 * SIN (0.1231162778277595D+02 * TAU) 0.3663238302392403D-05 * SIN (0.1436356574657194D+02 * TAU) -0.8280913315098266D-06 * SIN (0.1641550371036793D+02 * TAU) 0.1915949097161194D-06 * SIN (0.1846744167416392D+02 * TAU) -0.4510616993916847D-07 * SIN (0.2051937963795992D+02 * TAU) 0.1076722338164173D-07 * SIN (0.2257131760175591D+02 * TAU) -0.2519075024308943D-08 * SIN (0.2462325556555190D+02 * TAU) 0.6081547916302046D-09 * SIN (0.2667519352934789D+02 * TAU) -0.2034269048673321D-09 * SIN (0.2872713149314388D+02 * TAU) 0.5133044291763707D-10 * SIN (0.3077906945693987D+02 * TAU) z-COMPONENT 0.1296161644970725D-01 * COS (O.OOOOOOOOOOOOOOOOD+OO * TAU) 0.8000000000000000D-01 * COS (0.2051937963795992D+01 * TAU) -0.4128747889169709D-02 * COS (0.410387S927591983D+01 * TAU) 0.6461902427744734D-03 * COS (0.6155813891387975D+01 * TAU) -0.1068102001067884D-03 * COS (0.8207751855183966D+01 * TAU) 0.2084956739123790D-04 * COS (0.1025968981897996D+02 * TAU) -0.4269292404692368D-05 * COS (0.1231162778277595D+02 * TAU) 0.9218892368009590D-06 * COS (0.1436356574657194D+02 * TAU) -0.2052442069937884D-06 * COS (0.1641550371936793D+02 * TAU) 0.4689944336880688D-07 * COS (0.1846744167416392D+02 * TAU) -0.1091057725216870D-07 * COS (0.2051937963795992D+02 * TAU) 0.2582057585115161D-08 * COS (0.2257131760175591D+02 * TAU) -0.6195504530638732D-09 * COS (0.2462325556555190D+02 * TAU)

338

The Quasi-periodic

Orbits: Equations,

Method of Solution and Results

0.1497315418680290D-09 * COS (0.2667519352934789D+02 * TAU) -0.4335384176323811D-10 * COS (0.2872713149314388D+02 * TAU) 0.1089612418841304D-10 * COS (0.3077906945693987D+02 * TAU) FOURIER DEVELOPMENT OF THE NONHALO PART x-COMPONENT -0 481214350648D-2 * COS [0 305189018452D+1 * T + 0 291337850463D+1 0 293599327360D-2 * COS [0 105198574306D+1 * T + 0 301695888439D+1 0 168708819847D-3 * COS Co 999952220728D+0 * T + 0 623139511729D+1 -0 981559283479D-3 * COS [0 510382814832D+1 * T + 0 587854719915D+1 0 861927176494D-3 * COS (0 310392370686D+1 * T + 0 598212757891D+1 0 178746980088D-3 * COS Co 715576611211D+1 * T + 0 256053058649D+1 -0 138299060486D-3 * COS 515586167065D+1 * T + 0 266411096625D+1 0 189236417326D-3 * COS Co 465184240525D+1 * T + 0 286158831475D+1 -0 167194316248D-4 * COS CO 520335223390D-1 * T + 0 306874907427D+1 0 111664213006D-3 * COS Co 199990444145D+1 * T + 0 617960492741D+1 -0 123735441890D-2 * COS CO OOOOOOOOOOOOD+O * T + 0 OOOOOOOOOOOOD+O -0 507278987096D-4 * COS CO 610378036904D+1 * T + 0 582675700927D+1 0 640872851908D-4 * COS CO 410387592759D+1 * T + 0 593033738903D+1 -0 288680392114D-4 * COS 210397148613D+1 * T + 0 603391776879D+1 -0 447077101022D-4 * COS CO 920770407591D+1 * T + 0 552569928100D+1 0 364632311620D-4 * COS 720779963445D+1 * T + 0 562927966076D+1 -0 140790926086D-4 * COS CO 615581389138D+1 * T + 0 261232077637D+1 415590944993D+1 * T + 0 271590115613D+1 0 360232999284D-4 * COS -0 539188850104D-3 * COS Co 125101531448D+1 * T + 0 599475142466D+1 -0 445897653833D-3 * COS Co 915657061883D+0 * T + 0 430491787758D+1 -0 137975502439D-2 * COS CO 183131412376D+1 * T + 0 229363218301D+1 -0 241610286418D-3 * COS 625507657242D+0 * T + 0 142566266142D+1 112596420397D+2 * T + 0 220768266834D+1 0 118562275890D-4 * COS 0 120964903164D-4 * COS Co 110401926540D+1 * T + 0 608570795867D+1 -0 133433217987D-3 * COS 0 936601677928D+0 * T + 0 519177078555D+0 -0 965355655144D-4 * COS 0 876570750998D+0 * T + 0 411540626614D+1 -0 739703135552D-4 * COS 0 831361903037D+0 * T + 0 271461981859D+1 0 124700410407D-3 * COS 0 625498935007D+0 * T + 0 141729649699D+1 0 583223102967D-3 * COS 0 183132012511D+1 * T + 0 229207361255D+1 -0 623627798527D-4 * COS 0 330295327828D+1 * T + 0 267673481200D+1 0 541092990880D-4 * COS 0 800922649311D+0 * T + 0 325360257792D+1 -0 623477223435D-4 * COS CO 113628090191D+1 * T + 0 494433612411D+1 -0 313343876031D-3 * COS Co 398325208756D+1 * T + 0 525880087752D+1 0 570292992740D-3 * COS CO 220623840029D+0 * T + 0 671536511594D+0 0 173663868198D-4 * COS Co 267744562103D+1 * T + 0 439083135593D+1 -0 697938688393D-4 * COS Co 142643030655D+1 * T + 0 153950693309D+1 327354036612D+1 * T + 0 236514374954D+1 0 106837095383D-4 * COS -0 101592156847D-4 * COS CO 416051463365D+1 * T + 0 579777934537D+1 197652297172D+1 * T + 0 113891276256D+1 -0 450365864549D-3 * COS 0 344436299366D-4 * COS Co 251063093756D+0 * T + 0 604654161454D+1 191560928261D+1 * T + 0 425222768770D+1 0 696206043253D-4 * COS 0 259279113211D-4 * COS CO 842951588453D-1 * T + 0 192737723971D+1 -0 420688717192D-4 * COS Co 283126634449D+1 * T + 0 224184199313D+1 174701896492D+1 * T + 0 188134239976D+1 0 179759992768D-3 * COS -0 316384282070D-4 * COS CO 873251135129D+0 * T + 0 191603853279D+1 -0 291330580172D-4 * COS Co 966020989132D+0 * T + 0 402467867523D+1 -0 200278683727D-3 * COS CO 187649689502D+1 * T + 0 425188949097D+1 0 783860315855D-4 * COS Co 250199574002D+1 * T + 0 566918598796D+1 0 281814981694D-4 * COS CO 312749467503D+1 * T + 0 803297177779D+0 274698018787D+1 * T + 0 343811041883D+1 -0 519081124316D-4 * COS -0 131246568684D-4 * COS CO 111533628586D+1 * T + 0 244599161596D+1

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co

The Results.

•0 137298687036D-4 * 0 134158398791D-4 * •0 131800408864D-4 * •0 211718694238D-4 * 0 441726794399D-4 * 0 178181578831D-3 * -0 359249266925D-3 * 0 979584742473D-4 * 0 185113950812D-3 * 0 238163600150D-4 * •0 208270288111D-4 * 0 224065621480D-4 * 0 191465137923D-4 * 0 578971594759D-4 * 0 105123950319D-3 * 0 116551547490D-4 * 0 148109574637D-4 * 0 340739199124D-4 * 0 217615668065D-4 * 0 450520654544D-4 * 0 189732321419D-4 * 0 319829755103D-4 * 0 177771134489D-4 * 0 114838958359D-4 * 0 619469874981D-4 * 0 456392046870D-4 * 0 201311141948D-4 * 0 163629317964D-4 * 0 267159288762D-4 * 0 299641081664D-4 * 0 156924701404D-4 * 0 819593663506D-4 * 0 103963229762D-4 * 0 251785192048D-4 * 0 167924126303D-4 * 0 105487009270D-3 * 0 202034340461D-4 * 0 102764315428D-4 * 0 138822574607D-4 * 0 120089997545D-4 * 0 110421998965D-4 * 0 220716240536D-4 * 0 473529863812D-4 * 0 646981126451D-4 * 0 121652145784D-3 * 0 1173743S9811D-4 * 0 273326272428D-4 * 0 160547521201D-4 * 0 175962330028D-4 * 0 335558755839D-4 * 0 241176825314D-4 * 0 443311393858D-4 * 0 676489600600D-4 * 0 234059194800D-4 * 0 142615595112D-4 *

COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS

Problems Related to Small ((0.117536721279D+1 ((0.288329986683D+1 ((0.122057606075D+1 ((0.267743689880D+1 ((0.142643992878D+1 ((0.388325808891D+1 ((0.220617838676D+0 ((0.392846093552D+1 ((0.175414992068D+0 ((0.535489124207D+1 ((0.285286061310D+1 ((0.501953298947D+1 ((0.318821896570D+1 ((0.593519005135D+1 ((0.227256180382D+1 ((0.472938358483D+1 ((0.347836827034D+1 ((0.210857666985D+1 ((0.230300105755D+1 ((0.180087487003D+1 ((0.396754724640D+1 ((0.136328681184D+0 ((0.488329430829D+1 ((0.566387060610D-1 ((0.259203062896D+1 ((0.274697118564D+1 ((0.312753828621D+1 ((0.193655389865D+1 ((0.283127234584D+1 ((0.288329986683D+1 ((0.227358814539D+1 ((0.222155462305D+1 ((0.287647519245D+1 ((0.150207840824D+1 ((0.140490251689D+1 ((0.193204197826D+1 ((0.633505427996D-1 ((0.404950296164D+0 ((0.375299361004D+1 ((0.915660062559D+0 ((0.886974267531D+0 ((0.330293583381D+1 ((0.800940093781D+0 ((0.392843476881D+1 ((0.175441158774D+0 ((0.455393370382D+1 ((0.450057776232D+0 ((0.695042223982D+0 ((0.455396859276D+1 ((0.450092665173D+0 ((0.695033221853D+0 ((0.379895692871D+1 ((0.304918998874D+0 ((0.842951588453D-1 ((0.185290839237D+1

Divisors T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

0.513294773555D+1 0.567978851311D+1 0.250548875922D+0 0.438246519150D+1 0.154787219752D+1 0.525724230707D+1 0.673095081963D+0 0.410408145708D+1 0.182625593195D+1 0.564190350652D+1 0.621877127154D+1 0.395116995943D+1 0.162631951145D+1 0.194078426486D+1 0.363670520602D+1 0.107281474327D+1 0.450467472761D+1 0.283261065086D+1 0.272852500188D+1 0.320181238714D+1 0.934211075038D+0 0.499612631399D+1 0.520701068764D+1 0.615062726352D+1 0.255629989113D+1 0.336910985298D+1 0.398881340637D+1 0.467386888675D+0 0.224028342267D+1 0.530617670830D+1 0.241693393942D+1 0.563137017232D+1 0.108712257268D+1 0.243933153921D+1 0.555981889744D+1 0.175692470923D+1 0.513728738673D+1 0.957210490746D-1 0.222059367477D+1 0.114603680627D+1 0.343263559583D+1 0.579976168850D+1 0.130575700533D+0 0.933872878312D+0 0.499646451072D+1 0.235116937530D+1 0.270401729345D+1 0.472941724313D+0 0.552146849565D+1 0.587431641380D+1 0.403941158472D+0 0.482966189652D+1 0.110067549250D+1 0.192783804978D+1 0.627956146142D+1

339

340

The Quasi-periodic -0.296554164818D-4 -0.921173067590D-4 0.184940026123D-4 0.467718735163D-4 -0.160797881493D-4 0.309012111697D-4 -0.114559117862D-3 0.110968473139D-4 0.701499665941D-4 0.277426434033D-4 -0.317517569277D-4 -0.173813103845D-4 0.376235986088D-4 0.298103346902D-4 -0.329927063719D-4 0.201624111801D-4 -0.451698784554D-4 -0.213103144892D-4 0.471375099072D-4 0.235344683776D-4 -0.172298228860D-4 0.112483159580D-4 0.155734392073D-4 0.172364570538D-4 0.103875162229D-4 0.186262630004D-4 y-COMPONENT 0.135315277880D-1 -0.113460136872D-1 0.253168139320D-3 -0.561288141247D-3 0.421222325415D-3 0.198737060146D-3 -0.158464971510D-3 0.328728251851D-3 -0.195890791133D-3 0.394189002948D-4 -0.317937876577D-3 -0.304395259206D-4 0.368924459643D-4 0.478008078582D-4 -0.408788087651D-4 0.319417178670D-4 -0.185705499936D-4 0.349549595441D-4 0.836308302559D-3 0.383199747580D-3 0.205052365904D-3 0.125390065223D-3 0.122804708677D-4 -0.193612495844D-4 0.373015394167D-2 -0.180001193226D-4 -0.247440015080D-4 -0.274484217627D-4

Orbits: Equations,

Method of Solution and Results

* * * * * * * * * * * * * * * * * * * * * * * * * *

COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS

(0.593519605271D+1 (0.227255589247D+1 (0.598039889931D+1 (0.222735295586D+1 (0.798712801515D+1 (0.432449976762D+1 (0.191568928261D+1 (0.266267602680D+1 (0.174698018767D+1 (0.213623312264D+1 (0.196764289495D+1 (0.398397994206D+1 (0.119895985531D+9 (0.211528850659D+1 (0.198858742099D+1 (0.379891815147D+1 (0.394957776117D+0 (0.396754724640D+1 (0.136328681184D+0 (0.222737912257D+1 (0.432449376626D+1 (0.427929091966D+1 (0.188232130453D+1 (0.208419960930D+1 (0.217183394932D+1 (0.225096753521D+1

* * * * * * * * * * * * * * * * * * * * * * * * * *

T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + +

0.193922569440D+1) 0.363826377648D+1) 0.786064844418D+0) 0.479142462646D+1) 0.490595295938D+1) 0.318688593358D+0) 0.472579194022D+1) 0.294588392113D+1) 0.169325098164D+1) 0.615293689840D+1) 0.606958579780D+1) 0.472209340375D+1) 0.120824398527D+1) 0.181927077406D+1) 0.411106661496D+1) 0.465841967616D+1) 0.127191771286D+1) 0.140777532756D+1) 0.452256206147D+1) 0.167844789805D+1) 0.320247163917D+0) 0.147340801380D+1) 0.298967216706D+0) 0.135147312377D+0) 0.417341267979D+1) 0.594226669818D+1)

* * * * * * * * * * * * * * * * * * * * * * * * * * * *

SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN

(0.305189018452D+1 (0.105198574306D+1 (0.999952220728D+0 (0.510382814832D+1 (0.318392370686D+1 (0.715576611211D+1 (0.515586167065D+1 (0.405184240525D+1 (0.205193796379D+1 (0.520335223390D-1 (0.199990444145D+1 (0.610378036904D+1 (0.410387592759D+1 (0.210397148613D+1 (0.920776407591D+1 (0.729779963445D+1 (0.615581389138D+1 (0.415590944993D+1 (0.125191531448D+1 (0.915657061883D+0 (0.183131412376D+1 (0.625507657242D+0 (0.112596420397D+2 (0.925973759825D+1 (0.183131412376D+1 (0.947918698389D+0 (0.299985666218D+1 (0.110401926540D+1

* * * * * * * * * * * * * * * * * * * * * * * * * * * *

T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + + +

0.291337850463D+1) 0.301695888439D+1) 0.623139511729D+1) 0.587854719915D+1) 0.598212757891D+1) 0.256053058649D+1) 0.266419096625D+1) 0.286158831475D+1) 0.296510869451D+1) 0.306874907427D+1) 0.617960492741D+1) 0.582675700927D+1) 0.593033738903D+1) 0.603391776879D+1) 0.552569928100D+1) 0.562927966076D+1) 0.261232077637D+1) 0.271590115613D+1) 0.599475142466D+1) 0.430401787758D+1) 0.229363218301D+1) 0.142566266142D+1) 0.220768266834D+1) 0.231126304810D+1) 0.228921782391D+1) 0.316264664392D+1) 0.612781473753D+1) 0.608570795867D+1)

The Results.

Problems Related to Small

0 .123509112151D-3 * SIN ((0.936601677928D+0 0 .110713969657D-3 * SIN ((0.876570750998D+0 0 587319523726D-4 * SIN ((0.831361903037D+0 (0.625498935007D+0 •0 347200263924D-4 * SIN ( -0 217278845601D-2 * SIN ((0.183132012511D+ -0 146583792539D-3 * SIN ((0.330295327828D+ 0 274259321472D-3 * SIN ((0.800922649311D+0 (0.296759502567D+ •0 207267016022D-3 * SIN ( 0 359849667740D-3 * SIN ((0.113628090191D+ -0 203989028933D-3 * SIN ((0.388325209756D+ 0 203127142574D-3 * SIN ((0.229623840029D+0 •0 153452973925D-3 * SIN ((0.267744562103D+ 0 249993037446D-3 * SIN ((0.142643030655D+ 0 115715126022D-4 * SIN ((0.532547332991D+ •0 132105517469D-4 * SIN ((0.327354036612D+ •0 122610439341D-4 * SIN ((0.122169249232D+ 0 131457349536D-2 * SIN ((0.187652297172D+ 0 501030812893D-4 * SIN ((0.225909753521D+ 0 190390972946D-4 * SIN ((0.251063093756D+0 0 757319071646D-4 * SIN ((0.191560928261D+ 0 995918946469D-4 * SIN ((0.283126634449D+ 0 262271712485D-4 * SIN ((0.162545987797D+ 0 959447528174D-4 * SIN ((0.831361903037D+0 0 487411605055D-3 * SIN ((0.174701896492D+ 0 309965852610D-4 * SIN ((0.873251135129D+0 0 180413712723D-3 * SIN ((0.191560928261D+ 0 293307976877D-4 * SIN ((0.966020989132D+0 0 207954137564D-3 * SIN ((0.125099787001D+ 0 759639373180D-3 * SIN ((0.187649680502D+ 0 169041756246D-3 * SIN ((0.250199574002D+ 0 451137264498D-4 * SIN ((0.312749467503D+ 0 107057791354D-3 * SIN ((0.274698018767D+ ( 0 432218126330D-4 * SIN ((0.298853964172D+ ( 0 878637807525D-4 * SIN ((0.111533628586D+ ( 0 357756694095D-4 * SIN ((0.292850871479D+ 0 626952005744D-4 * SIN ((0.117536721279D+ 0 826196677707D-4 * SIN ((0.122057606075D+ 0 885392643234D-4 * SIN ((0.267743689880D+ 0 114344498662D-3 * SIN ((0.142643902878D+ 0 793183339785D-4 * SIN ((0.388325898891D+ 0 251560565866D-4 * SIN ((0.220617838676D+0 ( 0 610903428532D-4 * SIN ((0.392846993552D+ ( 0 454553317292D-4 * SIN ((0.175414992068D+0 ( 0 216220067547D-4 * SIN (0.535489124207D+ « 0 318301810861D-4 * SIN ((0.285286961310D+ ( 0 136007686694D-4 * SIN ((0.591953298947D+ ( 0 194801721643D-4 * SIN ((0.318821886570D+ ( 0 619383023540D-4 * SIN (0.593519005135D+ « 0 463242940782D-3 * SIN ((0.227256189382D+ ( 0 112043464895D-4 * SIN ((0.472938358483D+ ( 0 118665821551D-3 * SIN (0.210857666985D+ « 0 278292898547D-4 * SIN (0.230300105755D+ « 0 624028371830D-4 * SIN (0.180087487003D+ « 0 123227887472D-4 * SIN (0.199029571417D+ « 0 135462823463D-4 * SIN (0.396754724640D+ «

Divisors T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

0.519177078555D+0 0.411540626614D+1 0.271461981859D+1 0.141729649699D+1 0.229297361255D+1 0.267673481200D+1 0.325369257702D+1 0.986001264919D+0 0.494433612411D+1 0.525880087752D+1 0.671536511504D+0 0.439083135593D+1 0.153950603309D+1 0.533031244406D+1 0.236514374954D+1 0.568316036220D+1 0.113891276256D+1 0.594296123478D+1 0.604654161454D+1 0.425222768779D+1 0.224184199313D+1 0.137387247154D+1 0.234100801379D+1 0.188134239976D+1 0.191603853279D+1 0.445526478062D+1 0.402467867523D+1 0.283459299398D+1 0.425188949097D+1 0.566918598796D+1 0.803297177779D+0 0.343811041883D+1 0.348434577307D+1 0.244599161596D+1 0.797389653483D+0 0.513294773555D+1 0.250548875922D+0 0.438246519150D+1 0.154787219752D+1 0.525724230707D+1 0.673095081963D+0 0.410408145709D+1 0.182625593195D+1 0.564190359652D+1 0.621877127154D+1 0.395116995943D+1 0.162631951145D+1 0.194078426486D+1 0.363670529602D+1 0.107281474327D+1 0.283261065086D+1 0.272852500188D+1 0.320181238714D+1 0.297779254026D+1 0.934211075038D+0

341

342

The Quasi-periodic

-0 -0 0 0 0 -0 0 -0 0 -0 0 -0 -0 0 0 0 0 0 -0 0 -0 -0 0 -0 0 -0 0 -0 0 -0 -0 -0 -0 -0 0 0 -0 0 0 -0 -0 0 0 -0 0 0 0 0 -0 -0 0 0 -0 -0 -0

173129897832D-4 118469293706D-4 106340207027D-4 107723477156D-4 122390217042D-4 212745906850D-3 146869559146D-3 557288405408D-4 561053062374D-4 713042141539D-4 554256792474D-4 285389407664D-4 455965918348D-4 266772299595D-3 356533826032D-4 579383252029D-4 353901533135D-4 311314134445D-3 414475525751D-4 207027325723D-4 117019720930D-4 282529537971D-3 104982139995D-4 578767947182D-4 693465636304D-4 137850764960D-4 31i668295107D-4 123112646684D-4 265716289442D-4 395043589299D-4 229937378972D-4 172718704846D-4 178753242097D-4 208534432062D-4 387691407915D-4 245874637708D-4 414623139811D-4 684029773665D-4 152442361615D-4 175298760999D-4 306412371417D-4 305276608801D-3 197532951095D-4 165595632318D-3 132697624628D-4 278876317266D-4 242549381324D-4 198653994954D-4 118551537260D-4 108179003649D-4 191978853611D-4 361977178311D-3 119044407741D-4 337821770801D-4 141381014490D-4

* * * * *

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

* * * * * *

Orbits: Equations,

SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN

SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN

* * * * * * * * * * * * * SIN * SIN * SIN

(0 CO (0 CO CO CO CO CO CO CO CO CO CO

Method of Solution and Results

196764280495D+1 213623312264D+1 136328681184D+0 242638252728D+1 842951588453D-1 250203062896D+1 274697118564D+1 312753828621D+1 193655389868D+1 283127234584D+1 831367904399D+0 288329986683D+1 227358814539D+1 222155462305D+1 CO 287647519245D+1 Co 150207840824D+1 10 140490251689D+1 Co 193204197826D+1 [0 265825690882D+1 0 375299361004D+1 0 437849254504D+1 CO 174698918767D+1 CO 886974267531D+0 '0 230300105755D+1 0 180087487003D+1 0 292518909892D+1 0 117868682866D+1 0 301795895292D+1 0 108591697466D+1 0 800940093781D+0 0 392843476881D+1 0 450092665173D+0 0 695033221853D+0 0 379895692871D+1 0 304918998874D+0 0 831361903037D+0 ( 0 185290839237D+1 0 218826664498D+1 0 127260958309D+1 0 327251402455D+1 0 593519605271D+1 0 227255580247D+1 0 598039889931D+1 0 222735295586D+1 0 798712801515D+1 0 432449976762D+1 0 198858742099D+1 0 177394853506D+1 0 238656609859D+1 0 199529925773D+1 0 375304594345D+1 0 191560928261D+1 ( 0 437855360969D+1 0 266267602680D+1 0 999952229728D+0

co

* * * * * * * * * * * * * * * * * * * *

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 + 0

103779145479D+1 489254593423D+1 499612631399D+1 148771584321D+1 192783804978D+1 255629980113D+1 3369109SS298D+1 398881340637D+1 467386888675D+0 224028342267D+1 234386380243D+1 530617670830D+1 241693393942D+1 563137017232D+1 148712257268D+1 243033153921D+1 555981889744D+1 175692470923D+1 371839440289D-1 222059367477D+1 363789017176D+1 169325998164D+1 343263559583D+1 569880171586D+1 321535673164D+0 488120722730D+1 104913916172D+1 706662062568D+0 522367532646D+1 130575700533D+0 933872878312D+0 587431641380D+1 403941158472D+0 482966189652D+1 110067549250D+1 234542237289D+1 627056146142D+1 167810970133D+1 368849539590D+1 358491501614D+1 193922569440D+1 363826377648D+1 786064844418D+0 479142462646D+1 490595295938D+1 318688593358D+0 353613596295D+1 582085884488D+0 521569346089D+1 309772673817D+1 540158167033D+1 472579194022D+1 561054368833D+0 294588392113D+1 623139511729D+1

The Results. Problems Related to Small

0 250653261701D-4 0 153950264584D-4 -0 203115333986D-4 0 295251676799D-4 -0 146434735767D-4 -0 931639378996D-4 0 989568294694D-4 -0 107616347096D-4 -0 991050794382D-4 0 103741121889D-3 -0 123246308165D-4 0 1338679i4640D-4 0 113262074221D-4 -0 120979335585D-4 0 861408979027D-4 -0 168232621337D-4 -0 292275095901D-4 0 338060064906D-4 -0 106095814964D-4 0 131457755680D-4 -0 199364320687D-4 -0 499048333071D-4 0 560084400405D-4 -0 401695687283D-4 0 263112917938D-4 -0 548442043565D-4 0 542913095797D-4 x— -COMPONENT 0 233997959160D-2 -0 207589577517D-2 0 221925231587D-3 -0 244236848225D-3 0 181886557343D-3 0 575334923941D-4 -0 448144769090D-4 -0 127052637961D-4 0 271183290182D-4 0 278910751506D-4 0 161166153547D-3 0 561566962525D-4 -0 173792163020D-4 0 110952064631D-4 -0 306471352429D-4 0 521713660822D-4 -0 359711645179D-4 0 589335225002D-4 -0 379171135090D-4 0 876969647036D-4 -0 306630295654D-4 0 453426932143D-4 -0 118195318183D-3 0 199763234824D-4 -0 129914193183D-4 0 145742705608D-4 0 164323426118D-4

* * * * * * * * * * * * * * *

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN SIN

COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS COS * COS * COS * COS * COS * COS * COS * COS * COS * COS * COS

Divisors

(0.225095009074D+ (0.287644992574D+ (0.212754245430D+ (0.274697118564D+ (0.212758606548D+ (0.213623312264D+ (0.196764280495D+ (0.398397994206D+ (0.211528850659D+ (0.198858742099D+ (0.245688825996D+ (0.164698766763D+ (0.113627790123D+ (0.127260358174D+ (0.222737912257D+ (0.160188018756D+ (0.160184529862D+ (0.235685696267D+ (0.432449376626D+ (0.279900470798D+ (0.216732202893D+ (0.188232130453D+ (0.208419960039D+ (0.217183394932D+ (0.235689573991D+ (0.218326664498D+ (0.225096753521D+

* * * * * * * * * * * * * * * * * * * * * * * * * * *

T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

0.278280280410D+1 0.420009930109D+1 0.855087367659D+0 0.181270301212D+1 0.404066359625D+1 0.615293689840D+1 0.606058579780D+1 0.472209340375D+1 0.181927077496D+1 0.411106661496D+1 0.306028974359D+1 0.286944764544D+1 0.181913188824D+1 0.369005396636D+1 0.167844789805D+1 0.261151401066D+0 0.337403758789D+1 0.406584418702D+1 0.320247163817D+0 0.489830128416D+1 0.546295054935D+1 0.298967216706D+0 0.135147312377D+0 0.417341267979D+1 0.423708640738D+1 0.129454544881D+1 0.594226669818D+1

(0.305189018452D+ (0.105198574306D+ (0.999952220728D+0 (0.510382814832D+ (0.310392370686D+ (0.715576611211D+ (0.515586167065D+ (0.920770407591D+ (0.OOOOOOOOOOOOD+0 (0.199990444145D+ (0.210397148613D+ (0.405184240525D+ (0.520335223390D(0.415590944993D+ (0.330295327828D+ (0.800922649311D+0 (0.296759582567D+ (0.113628090191D+ (0.388325208756D+ (0.220623840029D+0 (0.267744582103D+ (0.142643030655D+ (0.105372490288D+ (0.310566286667D+ (0.998213060912D+0 (0.111533622586D+ (0.117536721279D+

* * * * * * * * * * * * * * * * * * * * * * * * * * *

T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

0.291337850463D+1 0.301695988439D+1 0.623139511729D+1 0.587854719915D+1 0.598212757891D+1 0.256053658649D+1 0.266411096625D+1 0.552569928100D+1 0.OOOOOOOOOOOOD+O 0.617960492741D+1 0.603391776879D+1 0.286158831475D+1 0.306874997427D+1 0.271590115613D+1 0.267673481200D+1 0.325360257702D+1 0.986001264919D+0 0.494433612411D+1 0.525880087752D+1 0.671536511504D+0 0.439083135593D+1 0.153950603399D+1 0.427436829635D+1 0.956351593683D+0 0.497398579534D+1 0.244599161596D+1 0.513294773555D+1

343

344

The Quasi-periodic 0.118286227469D-4 0.111645004644D-4 -0.198593833103D-4 0.150693276128D-4 -0.362106447788D-4 0.262979978443D-4 -0.146810937057D-4 0.239904042320D-3 0.109051649762D-4 -0.372711147926D-4 0.203169294315D-4 0.121289013228D-4 0.119960762733D-4 0.166127563603D-4 0.203767855962D-4 -0.130341684022D-4 -0.119025506128D-4 0.148782272737D-3 -0.120264677729D-3 0.100495933488D-4 -0.313787737458D-4 0.672412224683D-4 -0.163789788599D-4 0.170543516801D-4 -0.160100204578D-4 0.210639768098D-4 -0.165567843407D-4 -0.372840312448D-4 0.219738389585D-4 -0.124901259471D-4

Orbits: Equations,

* COS (0 122057606075D+1 * COS (0 267743689880D+1 * COS (0 142643902878D+1 * COS (0 388325808891D+1 * COS (0 220617838676D+0 * COS (0 175414992068D+0 * COS (0 125101531448D+1 * COS (0 183131412376D+1 * COS (0 593519005135D+1 * COS (0 227256180382D+1 * COS (0 180087487003D+1 * COS (0 230300105755D+1 * COS (0 213623312264D+1 * COS (0 175441158774D+0 * COS (0 394918998874D+0 * COS (0 225096753521D+1 * COS (0 191560928261D+1 * COS (0 183132012511D+1 * COS (0 187652297172D+1 * COS (0 227255589247D+1 * COS (0 222735295586D+1 * COS (0 205367712361D+1 * COS (0 213623312264D+1 * COS (0 196764280495D+1 * COS (0 211528850659D+1 * COS (0 198858742099D+1 * COS (0 174701896492D+1 * COS (0 187649680502D+1 * COS (0 193204197826D+1 * COS (0 183028778219D+1 Table 7.5

7.4

Method of Solution and Results

* * * * * * * * * *

* * * * * * * * * * * * * * * * * * * *

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

0.250548875922D+0 0.438246519150D+1 0.154787219752D+1 0.525724230707D+1 0.673095081963D+0 0.182625593195D+1 0.599475142468D+1 0.229363218301D+1 0.194078426486D+1 0.363670520602D+1 0.320181230714D+1 0.272852500188D+1 0.489254593423D+1 0.499646451072D+1 0.110067549250D+1 0.594226669818D+1 0.425176687763D+1 0.229207361255D+1 0.113891276256D+1 0.363826377648D+1 0.479142462646D+1 0.422257801647D+1 0.615293689840D+1 0.606058579780D+1 0.181927077406D+1 0.411106661496D+1 0.188134239976D+1 0.425188949097D+1 0.175692470923D+1 0.351340344960D+1

Output of program ANAQPO.

References [1] R. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin, 1978. [2] V. Arnold. Les Methodes Mathematiques de la Mecanique Classique. Mir, 1967. [3] E.W. Brown and C.A. Shook. Planetary Theory. Dover, 1964. [4] H. Poincare. Les Methodes Nouvelles de la Mecanique Celeste. GauthierVillars, 1892, 1893, 1899. [5] C.L. Siegel and J.K. Moser. Lectures on Celestial Mechanics. SpringerVerlag, 1971. [6] S. Sternberg. Celestial Mechanics. Benjamin, 1969.

Chapter 8

Numerical Refinement of the Quasi-periodic Orbit: The Final Numerical Determination of the Orbit and of the Projection Factors

For the station keeping, it is important to have a good nominal orbit (i.e., with a very small residual acceleration) to minimize the fuel consumption due to the maneuvers. The purpose of this chapter is to obtain an orbit, very close to one of the quasi-periodic orbits of the preceding chapter, such that it is a solution of the differential equations of motion in the solar system. In principle, this can be considered as a two-boundary problem. However, for large time intervals, the instability implies an extremely large sensitivity to the initial conditions. For instance, for the L\ case and the orbits near halo ones of the Sun-Barycenter problem, the amplification of errors after 4 years reaches the value 10 26 . Therefore, we have selected the parallel shooting method to solve the problem. Several approaches are possible, concerning the two end equations. Two of them are explained and implemented and several others are discussed. Finally, we describe a program that, starting with the results of the parallel shooting algorithm, produces the nominal orbit. The orbit is given at equally spaced epochs, and it is possible to recover the required position and velocity at any intermediate epoch using Lagrange interpolation. Simultaneously, a kind of numerical Floquet modes for the nominal orbit is obtained. From them we get the instantaneous projection factors. They are also given at equally spaced epochs.

8.1

8.1.1

A Parallel Shooting Method for the Numerical Refinement of the Quasi-periodic Orbit The Input Parameters

to Get the Nominal

Orbit

Till now we have obtained an analytic quasi-periodic orbit (qpo). To improve it numerically, we shall use a parallel shooting algorithm. Two slightly different approaches will be described. In both the initial data are: 345

346

Numerical Refinement

of the Quasi-periodic

Orbit

1) An initial epoch to, 2) A value of z, in normalized coordinates, when y (normalized) equals zero. Therefore, if (xo,yo,zo,xo,yo,zo) are the normalized initial coordinates for t = to, we fix ?/o = 0 and zo and the other variables are left free. We remark that those data determine one quasi-periodic orbit. The only thing to be added, to avoid ambiguities, is if we desire the sign of the z-amplitude, positive or negative. For the halo orbit it is clear that, belonging to a one-parameter family, it is enough to fix the value of z, when y = 0, at a given time. From the characteristic curve of the family, analytically expressed by A(a,8) = 0, the z-amplitude, a, can be obtained from the z-amplitude, 8. It is true that 8 does not coincide with zo, but the expression of z when the halo angle is zero: z = 8 + 0 2 (a,/3), where Oi means terms of second order in a, 8, shows that z and 8 are close. An iterative procedure can be used to determine 8. This is exactly what is done for the quasiperiodic orbit case. The terms added to the halo part of the orbit by program QPO (see Chapter 7) depend on a, 8 and t. All the remaining coefficients depend on the analytic model of the solar system and are kept fixed. Hence, for a given to the values of xo,yo,zo,&o,yo,zo depend on a and B, and a, 8 should satisfy the A condition. They also depend on t*. This is the epoch at which the halo part of the qpo crosses the y = 0 hyperplane, with z having the sign assigned to 8. If the quasi-periodic perturbations were zero, the time t* will coincide with to (modulus the period of the halo orbit) if the sign of B and z0 agree. Otherwise to and t* differ in a half period (also modulus the period). When the quasi-periodic part of the orbit is included, to and t* are close but they are not forcedly equal (or they differ, roughly, in a half period of the halo part). First of all, we have used an iterative procedure to determine 8 and t*, from z and to- This is implemented in the routine ESTQPO. We proceed as follows: a) We start with an initial value of the parameters B and t*. If nothing better is available, we use 8 — ±zo, t* = to + ST/2, where the sign of 8 is chosen in order to agree with the previously assigned sign. In the expression of t*, the parameter S is set equal to zero if 8 and z0 have the same sign, and 5 = 1 otherwise. T refers to the approximate period of the halo orbit. As a and 8 are unknown, this period is not available. However, being concerned with small and medium size halo orbits, the period of them changes slightly along the family (provided we avoid large amplitudes). In any family {L\, L 2 cases, Sun-Barycenter and Earth-Moon problems) we have adopted as approximate value of T, the one which corresponds, roughly, to B — 0.08. When a qpo has been determined, for a given pair (z0,t0), i.e., we have computed 8 and t*, to obtain the parameters 8 and t* corresponding to slight variations of (zo,h), we start with the previously computed values.

Parallel Shooting for the Numerical Refinement

of the Quasi-periodic

Orbit

347

b) When an approximation of (/?,£*) is available, we compute (using the routine ANAQPO) the corresponding values of y and z (in fact, we compute all the coordinates). The standing equations are y(/3,t*,t0) = 0,z(/3,t*,t0) = z0c) The previous equations are solved by Newton's method. For this we need the partial derivatives: dy

dy

dz

dz

dfj' &F' W di*' They are computed using numerical differentiation, using a step, e, given by the user. After some checks, we have adopted the value e = 10~ 6 . d) When the partial derivatives are available, Newton's formula gives the corrections to be applied to (/3,i*). The procedure is iterated till convergence. The value 1 0 - 1 1 , in the corrections of /? and t*, have been used as stopping criteria. e) As we intend to use a parallel algorithm approach, we require initial conditions at the ends of some selected time intervals. Let us suppose that, given to and ZQ for yo = 0, we wish to produce the nominal orbit for a large (4 years or 6 years, for instance) time interval. As mentioned at the beginning of the chapter, the strong instability prevents the use of a direct shooting method. The full time interval is split into N intervals. These intervals have been selected such that each corresponds to a "half revolution" or a "complete revolution". Here, half revolution means the time between two successive passages through y = 0. A complete revolution means the time between two successive passages through y = 0 with the same sign of y. Having already determined /? and t*, we know the full initial conditions (XQ,yo,zo,Ao,Vo,zo) at t = to. Now it is required to determine, for the qpo, the times, t\,... ,tjv, of the successive passages through y = 0. The separation ti+\ — tj is a "half revolution" or a "complete revolution". When ti is known, we determine an approximate value of U+i, by adding to £; either T / 2 or T. The routine PREDYO determines the corrections to be applied to ti+x to get y = 0. This is done using again Newton's method. In this case, the required derivative, y, is already available from routine ANAQPO. By iteration (up to an error less than 10 - 1 0 ) of A.

_

y(U+i)

A*i+1 - - T T 7

r,

y{u+i) we obtain the value tt+iThen, we determine the initial conditions (xi,yi,Zi,Xi,yi,ii) for t = U, i = 1,2,..., N.

(with j/j = 0)

348

Numerical Refinement

of the Quasi-periodic

Orbit

Therefore, the initial conditions, to be entered at the iterative procedure, leading to the solution of the parallel shooting problem, are: Q —

(xo,x0,y0,zo,ti,xi,zi,xi,yi,zi,...,tN,xN,zN,xN,yN,zN)

the total number of variables being 6N + 4. 8.1.2

A First Shooting Used to this End

Procedure.

Description

of the

Program

We define a Poincare-like map, P, (associated to a half revolution or to a complete revolution) through y = 0 (in normalized coordinates), by the forward flow in the solar system. The image of a point (t, x, z, x, y, z) is (t,x,z,x-,y-,i)

=

P(t,x,z,x,y,z).

As the system of differential equations is nonautonomous, the time has to be kept in the representation of an initial and final point. Let Qi = (ti,Xi, Zi,±i,yi, £») be the part of Q which refers to the ith point of the partition of the full time interval. Then, we should impose the matching conditions: Fi(Q) = P(Qi) - Qi+i = 0,

i=

0,...,N-l,

where F denotes the vector of 6 TV + 4 equations to be satisfied by variable Q. F, gives the (i + l)th block of 6 scalar equations. In this way we obtain 6N equations. These matching equations should be imposed in any parallel shooting approach to the problem of obtaining the nominal orbit. The different approaches differ in the election of the remaining four equations. In the first approach that we expose, the last 4 equations are taken as follows. Any quasi-periodic orbit, near a halo orbit, is determined through /3 and t*. For an ideal qpo, we would obtain a solution of the equations of motion. In the real world, the qpo is slightly changing and, hence, the parameters j3 and t* change accordingly. From the values of t and z at the end of the time interval, tw and ZJV, it is possible to determine values fix and t*N. In this way, "theoretical" values of the remaining coordinates at the last point can be determined: X

N

=

x(PN,t*N,tjv),

X% =

x(PN,t*N,tN),

VN

=

y(PN,t*N,tN),

*N

=

z(0N,tN,tN)-

Asking to the final variables to satisfy XN

= xN,

±N

— xN, yjv =

VN> *N

— zN,

Parallel Shooting for the Numerical Refinement of the Quasi-periodic Orbit

349

the final point would be on an (approximated) analytic quasi-periodic orbit. In short, we had left the initial x0 coordinate and the initial velocity (io,yo,z0) free and we impose that the final point be in an analytic qpo. The intermediate matching conditions are a technical matter to help in the solution of the problem, but they are, conceptually, irrelevant. This approach has been slightly modified due to the following reason. We suppose that near the (approximated) analytic qpo, there is a real quasi-periodic solution of the equations of motion of the full solar system. This real solution has stable and unstable directions. As we force the final point to be in an analytic qpo, the real corresponding (with the same tjy and ZN) qpo has final conditions slightly different. If we denote with the superscript r the "real" solution, we have, for t = tjy, differences XN—XN,

XN — XN,

2/jV — 2//\r,

z

N~zNi

such that it can give some unstable and stable components with respect to the real qpo. In this case, the unstable component is not dangerous, but the stable one is. If we go back in time towards the initial epoch, to, the stable component increases. This means that very large corrections to the initial values xo,xo,yo,zo should be applied to reach the desired ending point. If this method is implemented, the linear system involved in the iterations of Newton's method is ill-conditioned. We have left as last 3 equations the ones concerning velocities: FQN+2

=

XN — ±M = 0,

F6N+3

=

VN ~ VN = 0,

FeN+i

=

zN ~ ZN = 0.

The first one of the last 4 equations, has been modified as follows: F6N+I

=x0-x0-

(xN - x*N).

In this equation xQ is the (theoretical) value of x for the estimated analytic qpo such that y(/3o,tQ,to) = 0, z(/?o,*o,£o) = ZQ. AS /?O and t* are determined at the beginning, using the method explained in 8.1.1, x0 is kept fixed through the iterations. The value a;^ is the value of x for the estimated analytic qpo such that y(/3o,tQ,t0) = 0, z(Po,to,to) = ZQ. AS the final point, QN, changes through the iterations, also xlN changes. The equation 6./V -I- 1 means that we allow for some displacement of the x coordinates, at the extrema epochs, with respect to the corresponding analytic quasi-periodic orbits estimated at both ends. Both displacements should be equal. This has an additional advantage. Suppose that the analytic qpo has been obtained with a given value of the s/m parameter (cross section of the spacecraft to the radiation pressure/mass of the spacecraft). If systematic errors are discovered in this s/m value, the parallel shooting program can be run again with the corrected

350

Numerical Refinement

of the Quasi-periodic

Orbit

value of the parameter. An increase (decrease) of its value implies that, globally, the orbit is slightly shifted towards the Earth (towards the Sun), if the L\ case for the Sun-Barycenter problem is considered. Hence, the Xi variables should be displaced by, roughly, the same value. This is what F6N+I requires. From the numerical point of view, the suggested approach is also clearly advantageous. The inclusion of x0 in -F6JV+I produces a feedback effect and the condition number of the matrix DF(Q) is dramatically decreased. The problem is quite well conditioned. As a summary of what we have said and denoting, as before, Q -

(xo,xo,yo,zo,ti,xi,zi,xi,y1,zi,...,tN,xN,zN,iN,yN,ZN),

we have the system of equations: F\{Q)

=

F2(Q)

=

P2(t0,xo,zo,x0,yo,z0)

- xi = 0,

Fe{Q)

=

P6(to,xo,zo,xo,yo,z0)

- zx = 0,

=

Pl(tN-l,XN-l,ZN-l,XN-l,yN-l,ZN-l)

~tN

F6N(Q)

=

Pe(tN-l,XN-l,ZN-l,XN-l,yN-l,ZN-l)

— ZN = 0,

F6N+i(Q)

=

x0 -XoitotZo)

F6N+2(Q)

=

±5V(*JV,ZJV) ~ ^N

F6N+3(Q)

=

yN^NtZN)

-VN = 0,

FeN+4{Q)

=

ZN(tN,ZN)

- ZN = 0,

F6N-5(Q)

Pi{to,xo,zo,xo,yo,zo)-t1=0,

= 0,

- (xN -a;^(tjv,zjv)) = 0, =

°'

where Pi(ti,Xi,Zi,Xi,yi,Zi),...,P6(ti,Xi,Zi,Xi,yi,Zi), mean the components of the image of the point (*»,£», Zi,iri,2/»,ij) under the Poincare map. In the first 6 equations and also in the equation number 6iV + 1, the parameters *o and ZQ appear. They are the initial fixed data. We shall denote the equations in the short form F(Q) = 0. Let Q(°} be the initial value of all the variables as obtained in 8.1.1. Then, the corrected values are obtained by Newton's iteration: DF(QW)(QU+V

-Qij))

= -F(QV>).

Parallel Shooting for the Numerical Refinement of the Quasi-periodic Orbit

351

The differential matrix DF(Q) has the following structure /

^o

-I A!

DF(Q) =

-I A2

-I

iJV-l

V BN

-I AN

j

where A*, i = 1,2,..., N -1 denote the 6 x 6 matrix DP(Qi), the differential of the Poincare map at the point Qi and A0 is the 6 x 4 matrix obtained from DP(Q0) skipping the columns 1 and 3 (related to to and ZQ that do not change). The matrix BN is a 4 x 4 matrix with all entries equal to zero, except the element in the first row and column which is set equal to 1. The matrix AN is a 4 x 6 matrix and it has the structure / ax

\

-1 0 a-2 0 a-z 0 O-A

a5 a6 a7 as

\ -1 - 1 /

where dx N ax = dt N a5

a2 =

dx*N dt JV

a3

= ^k dt N

di

dx N

dx*N dZN '

a6 =

_ dyN a 7 ~ dzN ' dzN'

=

dzN dt N dzN

as = "° dzN

As usual, in the expressions of DF(Q) and AN the boxes or elements left blank mean that they are filled with zeroes. The computation of /?, t* has been explained in 8.1.1. The derivatives en,..., ag could be obtained using relations as ai =

d: '"NdPN

dpN dtN

dxN dt*N dt*N dtN''

but we have chosen to compute them directly using numerical differentiation. We turn to the effective computation of the Poincare map and its differential. Let f, f i , . . . , fk, be the ecliptic coordinates (centered at the center of masses of the solar system) of the spacecraft and of bodies number 1 to A; of the solar system. The lack of coherence and radiation pressure masses, are included in the masses of the corresponding bodies (Sun and Earth, and Sun, respectively). Then, the equations of motion are r = — 2~J Grrii j=i

d?

352

Numerical Refinement

of the Quasi-periodic

Orbit

where rrii is the (total) mass of the ith body, G is the gravitational constant and di =

\\r-ri\\2.

The Poincare map is expressed in normalized coordinates. Hence, given initial conditions t, x, z,x,y, z (y = 0), they are converted to days and ecliptic position and velocity. Then, the numerical integration of the Newton equations is started. First, we perform the integration for a time interval which is close to, but a little bit less than, half the period or the full period of the halo part of the analytic qpo, depending on the type of intervals that we have chosen. Then, after each step in the numerical integration, the point is transformed back to normalized time and coordinates and the condition y = 0 is searched. When at some step a change of sign of y is detected, a Newton method is used, through At = —y/y, to obtain the final time, Pi(t, x, z, x, y, i ) , under the Poincare map. The remaining variables, (x, z, x, y, z) give the other 5 components of the image of the Poincare map. To obtain the differential, DP, of the Poincare map, the equations of motion should be integrated simultaneously with the variational equations. As the system is nonautonomous, we have also variations with respect to the initial time. First, we work in ecliptic coordinates. Let x\, x2, x$, X4, X5, x§ denote ecliptic position and velocity. The equations of the autonomized system are ±0

=

X\

=

1, X4,

X2

=

X5,

£3

=

x6, k

&i =

-xi:1)dl3,

~y^Gmi{x1 i=l k

x5

=

-y^Gmi(x2

x6

=

- ^

-Xit2)d~3,

k

Gmi(x3 -

xit3)d~3.

i-\

The variational equations can be written in the form V = D • V, where V and D are 7 x 7 matrices and V is initialized to the identity matrix. The matrix D has the block structure D =

/ 0 0 \ G

0 0 H

0 \ I3 , 0 /

where I3 denotes the 3 x 3 identity matrix, H is a 3 x 3 matrix and G is a 3 x 1 matrix. The vector G denotes G =

o7(*4, £5,2:6),

Parallel Shooting for the Numerical Refinement

of the Quasi-periodic

Orbit

353

and the matrix H is the Jacobian matrix of X4,xs,x6 with respect to x\,X2,x3. We note that G appears because the coordinates Xij, i = 1,2,... ,k, j = 1,2,3, of the bodies of the solar system depend on t. Let Vij, i,j — 0 , 1 , . . . , 6 be the components of the matrix V. Prom the variational equations, it is immediate that i>oi = 0, i = 0 , . . . , 6. Hence «oo = 1 and v0j = 0 for j = 1 , . . . , 6, and only 42 components of the variational matrix should be computed. Using block matrices, we have «10 «20

(.

«30

Vu

vu

l>13

«21

VIZ

«23

V31

V32

^33

V14

V\h

«16

V24

«25

«26

«34

^35

«36 «40

(.

W 60

U41

ViZ

«43

^51

«52

«53

l>61

^62

«63

Vi4

W45

«46

f54

«55

«56

vu

«65

^66

The components of the vector G are given by

G =

~ I

_

—5

E i = i Gmi[-xia + 3qi{x2 - Zj,2)K : " E i = i Gmi[-±j i 3 + 3%(a;3 - 0^,3)]^

5

where qi denote auxiliary variables defined by
+ (x3 -

and the components of the matrix H by

H

=

h\i

hi2

hi3

h-21

fl22

h23

h3i

h32

h33

Xit3)xit3,

354

Numerical Refinement

of the Quasi-periodic

Orbit

where k

8=1

k

h12

=

-^2Gmi[-3(xi

-

xiA)(x2-Xii2)}d~5,

t=i k

h13

=

/«21

=

h22

=

- ^ 6 , m i [ - 3 ( a ; i - xiA){x3

-x^3)]d^5,

i=l hi2, k

-Y^Gmi[l-Z{x2-xi>2)2]d-\ i=l k

h2z

=

-"^2 Gmi[-3(x2

^31

=

hi3,

h32

=

h23,

h33

=

- xia){xz

-

xi>3)]d~b,

i=l

k

-YJGmi[l-i{x3-x^3f]d-\ i=l

When the integration of the variational matrix, V, is finished we should do two things: a) To obtain the differential matrix of the Poincare map P, in normalized coordinates. b) To obtain, also in normalized coordinates, the matrix of the partial derivatives of x,y,z,x,y and i at the time ij+i, with respect to the values of x,... ,z at t = ti. In the case of a halo orbit this matrix would be the monodromy matrix. The point b) is not used by the parallel shooting algorithm, but will be used in the course of the determination of the numerical projection factors. First of all, we require to express the 7 x 7 matrix V in normalized coordinates, Vn. We have Vn =

DT2-V-DT1,

where DT\ is the differential of the transformation, T\, from normalized variables to ecliptic ones: Ti(t,x,y,z,x,y,z)

=

(td,x1,x2,x3,X4,x5,x6),

Parallel Shooting for the Numerical

Refinement

of the Quasi-periodic

Orbit

355

evaluated at the initial point, and DT2 is the differential of the inverse transformation, T2, at the final point: T2(id,x1,X2,x3,Xi,x5,xe)

-

(t,x,y,~z,x,y~,~z).

The over lined variables mean the image variables, and td refers to the time expressed in Julian days, instead of adimensional (normalized) time units. Using the routines TRANSLI, to go from normalized to adimensional variables, and TRANS, to go from adimensional to ecliptic variables, the transformation T\ is obtained. T2 is given by the inverse. The differentials DT\, DT2 have been evaluated by numerical differentiation. The step used in the variables is a magnitude given by the user. We have used the value 1 0 - 6 as more convenient. We shall come back to this point in 8.1.6. When Vn is available, the matrix referred to in b) is obtained easily by deleting the first row and column. The matrix of a) requires a little bit more work. We put

(At} Vn

Ax 0 Az Ax Ay

{ Az )

(

( X* ^ y z

+ St

c0\ Cl

C2

=

C3

X

C4

y

c5

\<% J

\ 2 Jf

The amount St is left free, in order to have c2 = 0. The components, (x, y, z, x, y, 'z)f, of the vector field at the last point, in normalized coordinates, are obtained again from routines TRANS and TRANSLI. If the element (i, j) of matrix Vn is denoted by V„ij, i, j = 0 , . . . , 6, 6t is determined through Vn20At + K21 Ax + Vn23Az + Vn2iAx + Vn25Ay + Vn26Az + Sty = 0. If this value of St is substituted in the expressions of c 0 , c 3 ) ... ,c 6 , we obtain Co =

VnooAt-rVn01Ax ~ {-)

c6

+ Vn03Az-rVn04Ax

+

Vno5Ay-rVn06Az

(vn2oAt + Vn21Ax + Vn23Az + Vn2iAx + Vn25Ay + Vn2eAz)

=

(vnoo-^f)

=

Vn6oAt-rVneiAx

At + ...(vn06-^f)

Az,

+ Vn63Az + Vn64Ax + Vn65Ay + Vne6Az

356

Numerical Refinement

- ( 4J T/

VnGO

of the Quasi-periodic

Orbit

(Vn20At + Vn21Ax + Vn23Az + Vn24Ax + Vn25Ay + Vn26Az) zVn20 \ :—

At + . . .

y Jf

(

yn66

zVn26\ :

.

—

Az.

y Jf

\

The components of c 0 , c 1; c 3 , . . . , c6 (seen as linear forms) give the matrix DP: (DP)U

=

[Vn00.:....

Vn20 y

n

(DPU = (V„ 66 -^p) . The computation of the image under the Poincare map and its differential, is done in the routine POINCP. The routine DERIVP contains the vector field in dimension 48 obtained adding to the equations of motion, the variational equations (we remark that the equation ±Q = 1 and the related variational equations, Voi = 0, i = 0 , . . . , 6 can be skipped). With the initial assignation of values to Q and the iterative procedure, the parallel shooting program (first approach), PS1, is almost ready. We iterate till convergence and the final value of Q is obtained. In the program PS1 we have added several other computations that are the starting point of program NUNOPF to be explained in Section 8.2. We defer to this section the explanation of the remaining part of PS1. 8.1.3

Numerical

Results

and Discussion

of the First

Procedure

Several cases are studied. First, the program PS1 has been run. Then, program NUNOPF (see 8.2) allows to obtain the nominal orbit and the projection factors at equally spaced epochs. These results are used in program CONSIM, which performs the simulations of the maneuvers, and that will be explained in Chapter 9. The computations have been done in the following cases: a) Sun-Barycenter system, case L\, simplified model. b) Sun-Barycenter system, case L\, c) Sun-Barycenter system, case L2. The L2 case for the Earth-Moon system is very interesting for the applications. However, we have not done runs for this case because, as remarked in 5.1.5, the theory that we have used, i.e., the part of Brown's theory given in the book of Escobal, is a poor approximation to the real motion of the Moon. The neglected terms are not important for L\ or L2 near halo orbits of small and medium size of the Sun-Barycenter system. But they are important for halo orbits of the Earth-Moon system. Going back to the results of Chapter 5, we have that several neglected terms produce residual accelerations of the order of 10~ 3 mm/s. However, preliminary

Parallel Shooting for the Numerical

Refinement

of the Quasi-periodic

Orbit

357

explorations indicate that the program PS1 can produce the parameters for a good nominal orbit if a better model of the motion is used (for instance, JPL tapes). See further comments later. The cases b) and c) refer to the full solar system including radiation pressure. The value s/m = 0.01 has been used. It is a pessimistic value because for the real spacecraft roughly 1/3 of that value is foreseen. The case a) requires an explanation. The computation of the position, velocity and acceleration (and when required, over-acceleration) of the bodies of the solar system is done using analytic formulas. The full model includes 263 periodic terms coming from the perturbations of the planets on the Earth, and 45 periodic terms in the motion of the Moon around the Earth. This is very time consuming and, under VAX 11/785, the simulation represents, on the average, 2 s of CPU time for 1 physical day in the L\ and L% cases of the Sun-Barycenter system. For that reason, we have studied and done the bulk of simulations for a simplified model. In this model we only consider the major perturbations due to the noncircular motion of the Earth, where the periodic planetary perturbations are neglected. The radiation pressure has also been included. The simulation of this model is at least 12 times faster than for the full model. A sample of results is shown at the end of this section in Tables 8.2-8.4. The following input data has been used (see Table 8.1) in cases a), b) and c): -

Initial epoch, t0 = MJD 18000. Initial value of the normalized z, ZQ = 0.08. Positive /? is requested. Number of intervals: 16. Each interval is one half revolution. Lower bound for the terms retained in the analytic qpo: 3 — 5 • 10~ 6 . The procedure has been stopped when final error of the functions ||.F(<3)||oo is less than 10~ 6 , or the final correction ||AQW||oo is less than 1 0 - 7 .

Usually 3 iterations are enough to stop the procedure. The corrections to the foreseen final time are of the order of a few (< 5) minutes. The determinant of the linear system which appears in the Newton iterations is large (see table 8.2). It is close to the value of the dominant eigenvalue, Ai, in the halo orbit approximation, raised to a power equal to the number of revolutions. For 8 revolutions we get the value w 10 26 . Concerning the changes in /? from /30 to fiN, different behaviors have been observed. In the case a) the change is less than 1 %, while in cases b) and c) it is near 10 %. In fact, the equations do not bound these variations in /?. How to modify PS1 to bound variations of will be explained in 8.1.4. However, the observed variations of the z-amplitude using PS1 are admissible. Runs done also for z$ = 0.16 show the same behavior. At this point we wish to comment what happens if the analytic (or numerical) halo orbit is taken, instead of the analytic quasi-periodic orbit. Of course this will imply bad conditions at the final epoch. However, the procedure is still convergent

358

Numerical Refinement

of the Quasi-periodic

Orbit

for the L\ and L2 cases of the Sun-Barycenter problem. The number of iterations is increased to 6-8. A reduction in CPU time is obtained if we run 2 or 3 iterations first with a simplified model. As the remaining perturbations are small, a few iterations first with a simplified model produce a good starting Q for the full system, at low cost. The total number of iterations remains unchanged. For the £2 case in the Earth-Moon system, if we try to start with the analytic halo orbit, even if only a few large terms of the noncircular motion of the Moon are included, the procedure breaks down. For other approaches and recommendations we refer to 8.1.6. 8.1.4

A Second Shooting

Procedure

To prevent from one of the possible failures of the first approach, implemented in program PSl, that is, the possible important changes in the ^-amplitude, we have also analyzed a second procedure that has additional advantages. The equations F{Q) = 0, coincide in this case with the equations of 8.1.2 in the first 6 x N components, i.e., the matching part. For the remaining 4 equations we have done the following modifications: 1) The first of them states that the allowed displacement of the initial and final values of x, with respect to the predictions done by the analytic qpo orbit are equal. Furthermore, in this case the analytic qpo used for both estimations is the same. It has the parameters /?o, t$ estimated from the initial data to and ZQ. 2) The second of them states that the final value of z coincides with the value predicted by the qpo initially estimated. The final time, tjv, however, is left free. 3) Let x0, 2/Q 1 ZQ be the initial values of the components of the velocity at the first point, as predicted by the qpo. Changes of these values are, of course, allowed. Let ±Q, yo, ZQ be the searched values. Then, the third equation is written as 7r4(io - ±0) + TT5(y0 - yl0) + n6(z0 - i£) = 0, where ir4, its and TTQ are the last projection factors. These projection factors are obtained from the analytic theory using as values of a and /? the ones corresponding to the halo part of the estimated qpo. We recall, however, that in the A relation for a and /3, the terms coming from the resonant part of the quasi-periodic equations, have been included (see Chapter 7). The estimation of n^, 775 and TTQ is done at the initial epoch using the routine MAGCON to be described in Chapter 9. 4) The fourth equation is similar to the third one, but the last point is used instead of the first one. The projection factors are the same, because we are using only the part coming from the halo orbit, and this part is periodic.

Parallel Shooting for the Numerical Refinement

of the Quasi-periodic

Orbit

359

Summarizing the full equations, we have that to and ZQ are fixed and then ZN is fixed. This prevents from systematic variations of f3. A link is established between x0 and XN, quite similar to the one used in the first approach. The initial velocities io, 2/o, zo are left free but, with respect to the analytic qpo the changes in the initial velocity should not contribute to the initial value of the unstable component. The same is asked for the final point. Another advantage of this approach is that the end conditions are more favorable to a continuation of the mission. The value of ZN is fixed and the value of tjv, that can vary, changes in fact very few (see 8.1.3 and 8.1.5). Then, if we wish to start a new parallel shooting for N intervals, the conditions z 0 = ZM, to ~ ^N hold, where the over-line denotes the values corresponding to the new parallel shooting process. The variable xjy changes slightly from the initial Q(°) to the final Q^ (see 8.1.5, where changes less than 1 0 - 6 are found in adimensional units). As the variations of the velocities at the end epoch t^ and initial new epoch t have zero unstable component according to 3) and 4), the total variation in the unstable component at the matching epoch is, at most, of the order of 10~ 6 . Hence it can be necessary, at most, a maneuver similar to the usual ones (see 8.3). An improvement in this approach consists in considering nominal orbits starting at epochs to and t0 with some overlap, i.e., to < fjv- Then, the shift from one nominal arc to the other can be done at a point such that the unstable components, with respect to both nominal orbits and related projection factors, are equal. We can state 1) to 4) more formally: F6N+1

= X0 - XQ - (xN

- XlN)

= 0,

where x0 = xa ((30, t^, to) and xN = xa(/30,to,tN) with tN chosen such that yN = ya(Po,to,tN) = 0. Here the superscript a denotes analytic quasi-periodic orbit. FQN+2 = ZN - zN

where zN — za{Po,to^N)

— 0,

with the same t^ used before.

^6JV+3 = 7i"4(a;o - ±o) + ^5(2/0 - 2/0) + ne{zo - z%) = 0, F6N+4 = T^i{XN ~ XN) + 7T5(2/JV where, as usual, x0 = xa(/30,to,t0), similar for yN, zN. 8.1.5

Numerical

Results

VN)

+ KQ(ZN - ZN) = 0,

similar for y0, z\ and xN = xa(P0,to,tN)

and Discussion

of the Second

and

Procedure

The results are quite similar to the ones given in 8.1.3, except that now the initial and final values of /3(/30,/3?v) are very close. For instance, for the L\ case in the Sun-Barycenter problem, using the same data given in 8.1.3, the successive values

360

Numerical Refinement

of the Quasi-periodic

Orbit

of z when the orbit crosses the surface y = 0 (normalized coordinates) with z > 0 are (approximately): 0.08, 0.0813, 0.0819, 0.0827, 0.0827, 0.0827, 0.0822, 0.0813, 0.0803. Hence we have only small fluctuations which, translated to km, are of the order of 4000 for 4 years. Seen from the Earth this means a change in the angular distance to the Sun, when the spacecraft is in conjunction with the Sun, which is less than 10'. For the L\ case in the Sun-Barycenter system, the variation between the starting and the adopted values (after j iterations) of the x at the initial and final epochs: JJ) _ jo) XQ

d CU1U

XQ

JJ) _ _(o) JJ jy

Xjy >

6

are 0.75 • 10~ in adimensional units. The variation of the final epoch t(N' — vlj is near 1/2 hour. In 4 years this represents a relative change less than 1.5 • 1 0 - 5 . These results are slightly better for the simplified L\ case and slightly worse for Z/2- We always refer to the Sun-Barycenter system. The convergence of the second approach is roughly equal to that of the first one. Usually 3-4 iterations are enough to have final errors in the equations less than 10~ 6 (in normalized units, i.e., 1.5 km in position and 0.3 mm/s in velocity). The matrix involved in the computations is well conditioned. 8.1.6

Comments

on Different

Shooting

Approaches

We have presented two slightly different approaches to get the initial conditions for the nominal orbit. Here we discuss some additional alternatives that we have tried and we do some recommendations for future work that will improve the results. As it has already been explained in 8.1.2, the approach that imposes conditions on the final point, i.e., the final values of x,i,y,z are chosen equal to the values predicted by the initially estimated analytic qpo, is ill-conditioned. In 8.1.2 we have given geometrical reasons for this. In general, any procedure that leaves xo, io, yo, zo free and specifies the values of xjy, XJV, i/N, ZN to be equal to given quantities is illconditioned. To get well conditioned methods some feedback is required. A simple approach consists in fixing to, ZQ and to leave XQ, XQ, yo, io, as well as £;v, free. Then we impose XN = XO, ZN — zo, XN = io, VN — VO, ZN = zo- This would produce periodic orbits, if this is possible. But the real quasi-periodic motion is relatively far from this one and the approach seems not too suitable. A handicap of this approach is also that the ending conditions are not appropriate for possible extensions of the mission. Some other tested approaches try to minimize the sum of the squares of the variations in x,x,y,z, with respect to the initially estimated qpo, in the initial and final epochs, that is (x-o - 4 ) 2 + ( i 0 - iof

+ (l/o - 2/o)2 + («d ~ 4 ) 2

Parallel Shooting for the Numerical Refinement

of the Quasi-periodic

Orbit

361

+ (xN - x%)2 + (xN - x%)2 + (yN - ylN)2 + (zN - i ^ ) 2 . This minimum condition should be supplemented with 3 additional equations. But again ill-conditioning has been observed. Now we give some recommendations that can improve the practical application of the algorithms described in 8.1.2 and 8.1.4. As the work has been kept in a general form, ready to be adapted to any perturbed three-body problem, the normalized coordinates have been selected. However, if we study the Sun-Bary center problem, cases L\ or Li-, the plane of the ecliptic can be used as a Poincare plane, instead of the extremely moving y — 0 plane. Then, all the computations can be done in ecliptic coordinates avoiding changes that introduce lots of rounding errors. The algorithms as they are stated, can also deal with the Moon. The use of only ecliptic coordinates for the Moon requires a Poincare section (or some other device, for instance, fixed intermediate time values, t\,..., £;v-i) to carry out the parallel shooting. Another improvement should be done in routine DTRAN and, in general in all the routines which use numerical differentiation. Some tests done with routine DTRAN show that the best election of the step used for the numerical differentiation, e, is near 10~ 6 . This is the step in normalized units (of position, velocity and time), and the equivalent quantities have been used in ecliptic coordinates (i.e. 1.5 km, 0.3 mm/s, 5 s, approximately). Two calls of DTRAN to compute the differentials of the transformation from adimensional to ecliptic and its inverse transformation, give matrices that, by multiplication give a matrix which differs from the identity by less than 10~ 6 . It seems that a better numerical scheme would be required to improve this result to 1 0 - 8 at least.

CURRENT VALUE OF SPACECRAFT SECTION IN m**2 DIVIDED BY MASS IN kg = 0 . 0 1 THE MODEL OF THE SOLAR SYSTEM I S : 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0

EARTH+M00N-SUN SYSTEM: POINT LI IS USED NUMBER OF INTERVALS AND HALF TURNS OF EACH INTERVAL= 16, 1 INITIAL TIME (IN MJD SINCE 1950) = 18000.00 INITIAL Z-AMPLITUDE = 0.08, POSITIVE Z AMPLITUDE USED STEP FOR NUMERICAL DIFFERENTIATION 1.00E-06 BOUNDS FOR THE ERRORS IN EQUATIONS AND CORRECTIONS = l.E-06 l.E-07 BOUND FOR THE COEFFICIENTS TO BE RETAINED IN ANAQPO = 3.00E-06 Table 8.1 Input of program PS1.

362

Numerical Refinement ITERATION NUMBER 0 .

10

11

12

13

14

15

-4.507131387436E+00 -0.115559058913E+00 1.586662621646E-03 -2.944961694502E+00 0.160396465955E+00 3.893129567529E-04 -1.390069945741E+00 -0.116065730501E+00 -1.294266460115E-03 0.112212590003E+00 0.164234910912E+00 -8.385592440359E-05 1.617669477550E+00 -0.115842424716E+00 1.438505383673E-03 3.176024613866E+00 0.160377548387E+00 -4.731114250831E-05 4.735261565475E+00 -0.115781032636E+00 -1.474363536853E-03 6.241662475811E+00 0.164229628817E+00 1.299358200709E-04 7.743149461809E+00 -0.116126700390E+00 1.251755185586E-03 9.297025134860E+00 0.160406829765E+00 -4.816695385722E-04 10.859920052638E+00 -0.115498817096E+00 1.611703307126E-03 12.371063610591E+00 0.164175618640E+00 3.440060280824E-04 13.869326566390E+00 -0.116404589241E+00 1.046162545422E-03 15.418170341974E+00 0.160483500655E+00 -8.931069239843E-04 16.983942434419E+00 -0.115226279337E+00 -1.688641448835E-03 18.500335332837E+00 0.164074322267E+00 5.587165318448E-04

of the Quasi-periodic INITIAL

Orbit

CONDITIONS

-6.634894615582E-13 0.886262554630E+00

8.000000000238E-02 -3.359451465726E-04

-8.315084516352E-12 -0.944520578098E+00

-6.441438439598E-02 1.262505285705E-04

1.713459788021E-11 0.894306514131E+00

8.011265667490E-02 1.646811961546E-04

2.084762028124E-12 -0.100778533910E+01

-6.465065788279E-02 3.226584954547E-05

-1.546079619399E-11 0.890747614487E+00

8.007352369407E-02 -2.413673198202E-64

5.432939212509E-13 -0.944216928013E+00

-6.442740081182E-02 -1.547852926205E-05

3.876921786695E-11 0.889773002479E+00

8.005986675923E-02 2.623513471166E-04

-1.194032310525E-12 -0.100769819133E+01

-6.464701846985E-02 -4.980317810673E-05

-3.540822771433E-11 0.895282019401E+00

8.012047583758E-02 -1.439826177579E-04

1.422349216381E-12 -0.944686977924E+00

-6.440732009544E-02 -1.554428698314E-04

7.129775161465E-11 0.885313393210E+00

7.998090560363E-02 3.549183998220E-04

-6.471857345435E-13 -0.100680667396E+01

-6.461023564241E-02 -1.266748595605E-04

-6.231062182666E-11 0.899748818271E+00

8.014060073148E-02 -5.377988403884E-05

5.071997783184E-13 -0.945919057002E+00

-6.435652150040E-02 -2.778362301887E-04

9.483296070106E-11 0.881037190447E+00

7.987921242033E-02 4.316839062689E-04

-9.560332931332E-14 -0.100513266175E+01

-6.454338139981E-02 -1.903525237791E-04

Parallel Shooting for the Numerical Refinement

16

19.996199454819E+00 -0.116668969914E+00 8.401378212713E-04

-8.764237700257E-11 0.904029289257E+00

ITERATION NUMBER 0. Z( Z( Z( Z( Z(

1) 3) 5) 7) 9)

= = = = = Z 11) = Z 13) = Z 15) = Z 17) = Z( 19) = Z 21) = z 23) = z 25) = z 27) = z 29) = z 31) = z 33) = z 35) = z 37) = z 39) = z 41) = z 43) = z 45) = z 47) = z 49) = z 51) = z 53) = z 55) = z 57) = z 59) = z 61) = z. 63) = z, 65) = z> 67) = z: 69) = z: 7D = z: 73) = z, 75) = z: 77) = z: 79) = z: si) = z; 83) = z: 85) = z[ 87) =

of the Quasi-periodic

-1 0603443386570377E-03 1 7464018840434269E-04 -2 0730854264346676E-03 1 4153459373772437E-04 1 4978835393756294E-04 -4 9399275783292007E-04 -4 4184537736297394E-05 1 2606120916730176E-04 4 8050403322563717E-05 -6 3736676533809877E-05 1 5971436205586796E-04 3 9981384214993199E-04 1 5402817724069484E-04 1 3353602516375310E-04 2 5849541084049799E-04 -2 7258015413300640E-04 1 7861031912340744E-04 7 1424961057554859E-04 1 0962375591306861E-03 7 2704797798670701E-05 2 6140357110502610E-03 -3 3449817715802688E-04 1 8060272276864374E-04 1 1836574281922224E-03 1 3129383670174022E-03 7 9624967664559954E-05 2 5004514218872620E-03 -6 7104350297175763E-04 1 8739868573115076E-04 1 8205708367341755E-03 2 1457300651135380E-03 1 4914551769327697E-05 4 9548048390121169E-03 -5 9653176521257123E-04 1 8909484887316697E-04 1 9509477469835529E-03 2 2867867770570705E-03 -1 9013963828478850E-05 4 4052317353530240E-03 -1 0115089479887374E-03 1 7577534643216895E-04 2 7132688917243519E-03 2 9886065235009696E-03 -4 .1422427459985103E-05

Orbit

8.013616288594E-02 2.120805064254E-05

MATCHING FUNCTIONS

= = = = Z( 10) = Z 12) = Z 14) = Z 16) = Z 18) = Z( 20) = Z( 22) = Z( 24) = Z( 26) = Z( 28) = Z( 30) = Z< 32) = Z( 34) = Z( 36) = Z( 38) = Z 40) = Z 42) = Z 44) = Z 46) = Z 48) = Z( 50) = Z( 52) = Z( 54) = z 56) = z 58) = z 60) = z 62) = z 64) = z 66) = z 68) = z 70) = z' 72) = z: 74) = z 76) = z 78) = z[ 80) = z: 82) = z: 84) = zI 86) = zI 88) = Z( Z( Z( Z(

2) 4) 6) 8)

1 3 -3 4 8 4 5 2 5 -1 -5 5 -1 -5 -1 -6 -1 3 -1 -4 1 -9 -2 1 -1 -4 -1 -1 -4 5 -2 -8 2 -1 -4 3 -2 -7 1 -2 -6 7 -3 -1

1766145477597918E-03 5238906940230457E-03 2146163732526214E-04 1460657619859544E-04 9734025841540387E-04 7731645428449474E-05 5606654756689550E-05 3068848193741626E-04 7542505939712785E-05 8962722335983881E-04 9312291862311345E-04 3927467862016728E-06 6770777732458622E-04 7178466821177676E-04 6775661296775241E-04 8824072804298149E-04 7899428735633036E-03 1620372535266168E-04 2977049432958930E-03 1030988179649815E-03 6738183193372882E-04 3999023560083043E-04 3754961778442477E-03 5299100450927476E-04 4596620205202457E-03 4999405999980309E-03 0961446619640961E-05 7179066475936453E-03 2654477559714776E-03 5834202086318001E-04 5413195759869493E-03 0808085865100804E-03 5669604231053421E-04 6718723100767920E-03 1297676932006070E-03 0785746556620603E-04 5552588833336574E-03 8319178648624242E-03 2993546887955123E-04 5653960019222637E-03 2711884396090345E-03 4777181786813919E-04 5381753014289696E-03 1254912252027549E-02

363

364

Numerical Refinement

of the Quasi-periodic

Z( 89) = 6.8100020657311661E-03 Z( 90) Z( 91) = -8.2329224873900486E-04 Z( 92) Z( 93) = 1.8377095617027814E-04 Z( 94) Z( 95) = 2.6190206354932272E-03 Z( 96) Z( 97) = -8.6080448324921122E-14 Z( 98) Z( 99) = -6.1314842092485833E-13 Z(100) ERROR EQUATIONS = 1.1254912252027549E-02 ITERATION NUMBER 0.

Orbit

3.1607109625912515E-04 -2.3088335672208828E-03 -5.6733447332411573E-03 4.5354694341825527E-04 8.2245270218838512E-11 -2.5544449498696360E-11

MATRIX DZ

L-INF NORM OF DZ = 133.4031159666282 L-l NORM OF DZ = 186.9591118207345 DETERMINANT OF DZ = -4.7245633079049304E+25 ITERATION NUMBER 1.

INITIAL CONDITIONS

ITERATION NUMBER 1.

MATCHING FUNCTIONS

ERROR EQUATIONS = 6.1092839099010927E-05 ITERATION NUMBER 1.

MATRIX DZ

L-INF NORM OF DZ = 133.2354397664145 L-l NORM OF DZ = 186.7804933113461 DETERMINANT OF DZ = -4.8380262451765111E+25 ITERATION NUMBER 2. -4.507131387436E+00 -0.115686035693E+00 1.662145503771E-03 -2.944986885522E+00 0.160462989414E+00 2.410996205515E-04 -1.390101121246E+00 -0.116051504741E+00 -1.392989283615E-03 0.112221634409E+00 0.164246649988E+00 -1.601830841756E-04 1.617685126852E+00 -0.115818192636E+00 1.361232807429E-03 3.176027697283E+00 0.160443483379E+00 -5.388682273332E-05 4.735286970192E+00 -0.115756262473E+00 -1.383197817337E-03

INITIAL CONDITIONS

-6.634894615582E-13 0.886619986829E+00

8.000000000238E-02 -1.248383245332E-03

-8.315084516352E-12 -0.944828894383E+00

-6.432276175955E-02 1.006673130742E-03

1.713459788021E-11 0.894433763998E+00

8.021023385595E-02 -4.450113179182E-04

2.084762028124E-12 -1.007769437558E+00

-6.463682862906E-02 8.899990915711E-04

-1.546079619399E-11 0.890862544613E+00

8.025532591716E-02 -9.499532671739E-04

5.432939212509E-13 -0.944468964591E+00

-6.446792906590E-02 6.600343519173E-04

3.876921786695E-11 0.889877036738E+00

8.031358466230E-02 -1.524633753834E-04

Parallel Shooting for the Numerical Refinement 7

8

9

10

11

12

13

14

15

16

6.241680612742E+00 0.164238018241E+00 1.735901781248E-04 7.743208542115E+00 -0.116088567875E+00 1.363717494576E-03 9.297025862681E+00 0.160470780432E+00 -3.389433917171E-04 10.859966217055E+00 -0.115471188638E+00 -1.339025813437E-03 12.371059677020E+00 0.164184225262E+00 5.025598934296E-04 13.869396506301E+00 -0.116360702705E+00 1.331788891989E-03 15.412134662789E+00 0.160546943009E+00 -6.163694956370E-04 16.983964031346E+00 -0.115203349549E+00 -1.262612584261E-03 18.500275985466E+00 0.164082989988E+00 8.058326846019E-04 19.996177882840E+00 -0.116790032336E+00 8.392487449267E-04

= = = = = = = = = = = = = = = = =

Orbit

-1.194032310525E-12 -1.007626646699E+00

-6.475292532840E-02 5.217763180122E-04

-3.540822771433E-11 0.895310614979E+00

8.043882307934E-02 -6.543388871216E-04

1.422349216381E-12 -0.944902661390E+00

-6.455460479629E-02 2.977289079794E-04

7.129775161465E-11 0.885413348387E+00

8.034039511493E-02 1.402946790738E-04

-6.471857345435E-13 -1.006706360562E+00

-6.480320434719E-02 1.414393276787E-04

-6.231062182666E-11 0.899715421773E+00

8.054298298118E-02 -3.401635232377E-04

5.071997783184E-13 -0.946124973913E+00

-6.458067265322E-02 -7.123822495844E-05

9.483296070106E-11 0.881152594680E+00

8.029037593750E-02 4.255544497979E-04

-9.560332931332E-14 -1.005026401806E+00

-6.478674596167E-02 -2.397015175106E-04

-8.764237700257E-11 0.904142555551E+00

8.056908842050E-02 2.151118683039E-05

ITERATION NUMBER 2. Z( 1) Z( 3) Z( 5) Z( 7) Z( 9) Z( 11) Z( 13) Z( 15) Z( 17) Z( 19) Z( 21) Z( 23) Z( 25) Z( 27) Z( 29) Z( 31) Z( 33)

of the Quasi-periodic

1 2955636563560802E-10 1 6243471845367452E-11 2 2417728784418500E-10 - 6 9661443280466528E-11 1 5343264853084904E-11 2 4831739442454648E-10 88.6184146624868063E-11 - 6 2423243657461214E-12 2 1267548833137084E-10 - 5 9042742917014834E-11 7 9257676810495781E-12 2 2832385981885750E-10 9 3719365601430127E-11 - 88.2706081566685796E-12 2 1932704263871727E-10 - 66.1675997642396396E-11 - 1 4048345819972496E-12

MATCHING FUNCTIONS Z( 2) Z( 4) Z( 6) Z( 8) Z( 10) Z( 12) Z( 14) Z( 16) Z( 18) Z( 20) Z( 22) Z( 24) Z( 26) Z( 28) Z( 30) Z( 32) Z( 34)

= = = = = = = = = = = = = = = = =

-1.2128505144648472E-10 -1 - 3 6700905543332889E-10 - 2 1110258398120141E-11 - 2 0230545190458304E-10 - 5 4565544620123647E-10 5 0857929955079595E-11 - 9 5418180301454214E-11 - 3 4150180042020182E-10 - 2 1087698617470660E-11 - 1 8321299685708503E-10 - 4 9971036962876860E-10 4 5057360132435414E-11 - 1 0807278683078181E-10 - 3 5463608673639413E-10 - 2 8640350169055993E-11 - 11.9557018331173293E-10 - 5 3477865004695167E-10

365

366

Numerical Refinement

of the Quasi-periodic

35) = 2.2573451441409986E-10 Z( 36) 37) = -5.1773696441159700E-10 Z( 38) 39) = 3.3415556779936573E-11 Z( 40) 41) = -1.7317429990004740E-09 Z( 42) 43) = -8.5042084485564828E-11 Z( 44) 45) = -1.8774825444323184E-12 Z( 46) 47) = 3.1208449713382436E-10 Z( 48) 49) = 2.3176194297036545E-10 Z( 50) 51) = -9.5623283596912856E-12 Z( 52) 53) = 4.6364306838242442E-10 Z( 54) 55) = -2.4409112599005126E-09 Z( 56) 57) = 2.6637189329270505E-10 Z( 58) 59) = 1.5408458647891976E-08 Z( 60) 61) = 4.8525161666646000E-10 Z( 62) 63) = -2.2185897216586703E-11 Z( 64) 65) = 1.0794049265516747E-09 Z( 66) 67) = -2.4316637592391999E-10 Z( 68) 69) = 5.0330468959192132E-12 Z( 70) 71) = 7.0046506095433614E-10 Z( 72) 73) = 7.1250916278131626E-10 Z( 74) 75) = -5.0458512368400932E-11 Z( 76) 77) = 1.4901212486151394E-09 Z( 78) 79) = -2.9180391436511854E-10 Z( 80) 81) = -4.6401984685195963E-11 Z( 82) 83) = -1.6497515853419742E-10 Z( 84) 85) = 1.3705037105182782E-09 Z( 86) 87) = -1.3303291974964271E-10 Z( 88) 89) = 3.3974356106192261E-09 Z( 90) 91) = -6.0766192078176573E-10 Z( 92) 93) = 5.1680493218242418E-11 Z( 94) 95) = 1.7392653567371852E-09 Z( 96) 97) = 2.0719412296976714E-11 Z( 98) 99) = 3.3033395463455406E-10 Z(100) ERROR EQUATIONS = 3.1539561302890802E-08

Orbit

3.9367213461804444E-11 7.5081368205998089E-10 2.4348056762534388E-09 -9.2777328070006541E-11 -2.6759293739186951E-10 -7.4355561445926983E-10 5.2035251690732247E-11 -2.4850148327981714E-10 -8.0019358082342824E-10 -5.9246304422774093E-12 -1.2302051405344017E-08 -3.1539561302890802E-08 1.9802161673647747E-09 -5.5791259806481364E-10 -1.6326811463397446E-09 1.0966664978472734E-10 -6.1284008423534431E-10 -1.6831666239003774E-09 9.8060302275947402E-11 -8.1139825067189086E-10 -2.3446966727121848E-09 1.6827905237314382E-10 5.8456772408788105E-11 1.0165694214149187E-10 -1.3370175061218955E-10 -1.7302219552928211E-09 -4.8853910899072494E-09 4.9811485195756918E-10 -1.5268724548690510E-09 -4.1653095477975355E-09 1.7115996247348250E-10 1.8636550080399460E-11 -3.9729996081538552E-12

Table 8.2 Output of program PSl. Results of the iteration towards the final solution. First, the initial epochs and initial conditions, for each interval of the parallel shooting, are given. After, the values of the matching equations, Z(i), are displayed. Finally, information related to the differential of the matching equations, DZ, is given. The table has been shortened. Note that the second iteration already provides the final orbit.

Parallel Shooting for the Numerical Refinement 1 3 5 7 9 11 13 15

0.1809080761671210D+05 0.1826852345487359D+05 0.1844662280886862D+05 0.1862482952058976D+05 0.1880243704667951D+05 0.1898113094089391D+05 0.1915825771569728D+05 0.1933742290018521D+05

2 4 6 8 10 12 14 16

of the Quasi-periodic

Orbit

0.1818119328227665D+05 0.1835603619881526D+05 0.1853726270692905D+05 0.1871211348985509D+05 0.1889329092637503D+05 0.1906822941119053D+05 0.1924927953407722D+05 0.1942437982764563D+05

Table 8.3 Output of program PSl. Final time (in MJD), for each interval of the parallel shooting procedure.

REVOLUTION NUMBER 1: EIGENVALUES 0.1786494138267907D+04 0.0000000000000000D+00 0.OOOOOOOOOOOOOOOOD+OO 0.5637089970989911D-03 0.9169311746184862D-01 0.1001733915293097D+01 0.1001733915293097D+01 -0.9169311746184862D-01 0.9148021367869823D-02 0.1004673821860350D+01 -0.9148021367869823D-02 0.1004673821860350D+01 REVOLUTION NUMBER 1: EIGENVECTORS 0.3626035858133893D+00 0.0000000000000000D+00 -0.1218989788206588D+00 0.0000000000000000D+00 -0.8489172315157018D-02 0.0000000000000000D+00 0.0000000000000000D+00 0.8356180591299494D+00 0.0000000000000000D+00 -0.3913144459056467D+00 0.0000000000000000D+00 -0.4693267681061035D-01 REVOLUTION NUMBER 2: EIGENVALUES 0.1681325825143609D+04 0.0000000000000000D+00 0.5929472683337834D-03 0.0000000000000000D+00 0.5504941972837015D-01 0.9952177776192929D+00 0.9952177776192929D+00 -0.5504941972837015D-01 0.9982010815697566D+00 0.2775102408761454D-01 0.9982010815697566D+00 -0.2775102408761454D-01 REVOLUTION NUMBER 2: EIGENVECTORS 0.3649109427021232D+00 0.0000000000000000D+00 0.0000000000000000D+00 -0.1233126423292467D+00 0.0000000000000000D+00 -0.8517356467805011D-02 0.0000000000000000D+00 0.8350016441270509D+00 0.0000000000000000D+00 -0.3900111034594093D+00 0.0000000000000000D+00 -0.4717037481643692D-01 REVOLUTION NUMBER 3: EIGENVALUES 0.1780248633976915D+04 0. OOOOOOOOOOOOOOOOD+OO 0.5612444044661386D-03 0.OOOOOOOOOOOOOOOOD+OO 0.9951299503032858D+00 0.9103821971873141D-01 0.9951299503932858D+00 -0.9103821971873141D-01 0.9993917766143136D+00 0.8860161915787859D-02 0.9993917766143136D+00 -0.8966161915787859D-02

367

368

Numerical Refinement

1 2 3 4 5 6

REVOLUTION NUMBER 3: 0.3637205874292893D+00 -0.1220160063553767D+00 -0.8594310464509669D-02 0.8355231596326938D+00 -0.3904404373449051D+00 -0.4695124897188449D-01

EIGENVECTORS 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00

1 1 1 1 1 1

REVOLUTION NUMBER 4 0.1687981912964943D+04 0.5952419864938448D-03 0.1003466287800010D+01 0.1003466287800010D+01 0.1001852509031028D+01 0.1001852509031028D+01 REVOLUTION NUMBER 4: 0.3637840413340302D+00 -0.1231908433829183D+00 -0.8472752665024491D-02 0.8350937725591245D+00 -0.3908944483936553D+00 -0.4725802669973034D-01

EIGENVALUES 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.5598872431697979D-01 -0.5598872431697970D-01 0.2753250298303385D-01 -0.2753250298303385D-01 EIGENVECTORS 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO

1 1 1 1 1 1

REVOLUTION NUMBER 5 0.1772532546718421D+04 0.5593012850952188D-03 0.9885971303508935D+00 0 9885971303508935D+00 0.9941590522489543D+00 0.9941590522489543D+00 REVOLUTION NUMBER 5: 0.3648497864551821D+00 -0.1221467306152026D+00 -0.8510023941993586D-02 0.8354218026667695D+00 -0.3895647496889848D+00 -0.4692660700387003D-01

EIGENVALUES 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.9021774004289062D-01 -0.9021774004289062D-01 0.8428068827937931D-02 -0.8428068827937931D-02 EIGENVECTORS 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO

1 1 1 1

REVOLUTION NUMBER 6 0.1695792671058510D+04 0.5979778774538078D-03 0.1011462750422482D+01 0.1011462750422482D+01 0.1005653771719521D+01 0.1005653771719521D+01 REVOLUTION NUMBER 6: 0.3626674958191477D+00 -0.1230532653295172D+00 -0.8419730297104159D-02 0.8351902906402237D+00

EIGENVALUES 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.5799670725351659D-01 -0.5709670725351659D-01 0.2696170655557521D-01 -0.2696170655557521D-01 EIGENVECTORS 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00

1 2 3 4 5 6 1 2 3 4 5 6

1 2 3 4 5 6 1 2 3 4

Orbit

1 1 1 1 1 1

1 2 3 4 5 6 1 2 3 4 5 6

of the Quasi-periodic

Parallel Shooting for the Numerical Refinement 5 6

1 1

-0.3917648157286513D+00 -0.4729478921752832D-01

of the Quasi-periodic

Orbit

0.OOOOOOOOOOOOOOOOD+00 0.OOOOOOOOOOOOOOOOD+00

REVOLUTION NUMBER 7: EIGENVALUES 0.OOOOOOOOOOOOOOOOD+OO 0.1763868446013948D+04 0.OOOOOOOOOOOOOOOOD+00 0.5577963843030978D-03 0.8879965796349758D-01 0.9823194002129516D+00 -0.8879965796349758D-01 0.9823194002129516D+00 0.7303087930321156D-02 0.9891260526050620D+00 -0.7303087930321156D-02 0.9891260526050620D+00 REVOLUTION NUMBER 7: EIGENVECTORS 0. OOOOOOOOOOOOOOOOD+OO 0.3659626246125810D+00 0. OOOOOOOOOOOOOOOOD+00 -0.1222877895376673D+00 0.OOOOOOOOOOOOOOOOD+OO -0.9546147515853732D-02 0.OOOOOOOOOOOOOOOOD+00 0.8353166776324223D+00 0.OOOOOOOOOOOOOOOOD+OO -0.3887092952062111D+00 0.OOOOOOOOOOOOOOOOD+00 -0.4685969761451993D-01 REVOLUTION NUMBER 8: EIGENVALUES 0.1704089203814202D+04 0.OOOOOOOOOOOOOOOOD+OO 0.OOOOOOOOOOOOOOOOD+00 0.5985766460713765D-03 0.5993782065521413D-01 0.1018347644053552D+01 -0.5993782065521413D-01 0.1018347644053552D+01 0.2445714882029143D-01 0.1010102213441894D+01 -0.2445714882029143D-01 0.1010102213441894D+01 REVOLUTION NUMBER 8: EIGENVECTORS 0.OOOOOOOOOOOOOOOOD+OO 0.3616152374044021D+00 0.OOOOOOOOOOOOOOOOD+00 -0.1229266612557326D+00 0.OOOOOOOOOOOOOOOOD+OO -0.8356389257807582D-02 0.OOOOOOOOOOOOOOOOD+00 0.8352534244549467D+00 0. OOOOOOOOOOOOOOOOD+OO -0.3926461689864306D+00 0.OOOOOOOOOOOOOOOOD+00 -0.4726869687872015D-01

Table 8.4 Output of program PSl. Eigenvalues and eigenvectors associated to each revolution of the final nominal orbit. Both for the eigenvalues and eigenvectors the left and right columns display the real and imaginary parts. The table has been shortened.

369

370

8.2 8.2.1

Numerical Refinement

of the Quasi-periodic

Orbit

The Final Nominal Orbit and Projection Factors The Problem

of the Floquet Modes for a Quasi-periodic

Orbit

In the case of the periodic halo orbits, the Floquet modes are associated to the eigenvalues of the monodromy matrix. Now, for a quasi-periodic orbit, as the initial and final points are different, the monodromy matrix properly does not exist, nor the associated eigenvalues (or their logarithms: the characteristic exponents). The characteristic exponents for a periodic orbit are replaced by the Lyapunov exponents for a general orbit. The maximal Lyapunov exponent starting at a point P under a flow (f>t is defined by A ^ l i m l m a x l o g p t : ™ 1 ) & t^oo ti€TPM \ ||£|| ) where £ is a nonzero tangent vector at the base manifold M on the point P. This limit exists under mild hypothesis. The value Am measures the maximal averaged rate of escape from a given orbit. It plays the role of the dominant characteristic exponent, logAi in the case of periodic orbits, where T is the period. The vector £, on which the maximum is attained, depends on t, in general, but it converges to some vector £ m that replaces the maximal eigenvector of the periodic orbits. It corresponds to the fastest escaping direction. If we replace £ by a couple of vectors, and a two-dimensional measure is used instead of a norm, the corresponding limit, A, plays the role of the sum of the two more important characteristic exponents. As the dominant one is already available, we can compute the second Lyapunov exponent. Replacing couple by A;-ple and two-dimensional measure by A;-dimensional measure, we get the sum of the k more important characteristic exponents. From this, the value of the fcth one follows. In this way all the Lyapunov exponents are computed by iteration of the process. The corresponding "eigenvectors" (eventually complex) can be determined as follows. Let (£, T]) define a limit two-plane for which the maximum of the logarithm of the quotient between image area and initial area is obtained. The vector £ m should forcedly be in this plane. If it is not so, taking a suitable vector in the (£, rj) plane and £ m , a faster increase of the area would be obtained. Then, £ m is a linear combination of £ and rj. Unless £ m = 77, £ m and 77 are a basis of this plane. All the directions but one in this plane, have a nonnull £ m -component and, therefore, they escape as fast as £ m in the limit. The remaining direction gives the second "eigenvector" at the point P. Again recurrently, all the "eigenvectors" can be obtained. Let £1, £2, • • •, £n be vectors associated to the Lyapunov exponents Ai, A 2 , . . . , A„.

The Final Nominal

Orbit and Projection

371

Factors

In a parallel way to the periodic orbit case, we can define £(*) = &(t) exp(-Ait),.. .,!„(*) = Ut)

exp(-A„i).

The vectors £ • (i) are quasi-periodic, functions of time and they replace the classical Floquet modes of the periodic orbits. In our problem, a process quite similar to the one used for the analytic qpo orbit in Chapter 7, can be carried out for the solution of the variational equations, as they are given in 3.5. Formal expressions for the quasi-periodic Floquet modes are not difficult to obtain (but it is an enormous task!). Formal means that the diverse terms are obtained by recurrence without taking care of convergence problems. Fortunately, as the quasi-periodic corrections to the analytic Floquet modes (and, therefore, projections factors) are small, the analytic halo approach is efficient enough as we will show in Chapter 9. Concerning numerical values of the projection factors for the numerically refined nominal orbit, we present the adopted procedure in 8.2.2. 8.2.2

The Adopted

Procedure

to Obtain the Projection

Factors

Letting aside theoretical considerations and as going to infinity is a too expensive computer task, we have used an expeditive method. First of all, as the tangent spaces at two different points of M6 can be identified, we have used the variational matrix as if it was a monodromy matrix. In fact, the qpo is not too far from the halo orbit. An important difficulty is that, after A; full revolutions, the dominant eigenvalue is close to A*, where Ai is the dominant one in the halo case. On the other side, the most contractive direction has eigenvalue near Aj~*. For k = 4 and Ai = 1700, the relative values of the maximal and minimal eigenvalues are, roughly, 1026 to 1. Hence, we shall lost all the significant digits in the numerical computation of the minimal eigenvalue, and an important part in the intermediate ones. This has been observed, in fact, in our tests. Then we have adopted the criterion of computing the eigenvalues and vectors just every one revolution. Let Ai, A2,..., A/, be the variational matrices associated to the 1st, 2nd,i... , M i revolution. We have k = N or k = N/2. Then, we have computed (real) eigenbases Bx, JB 2 , . . . , Bk at the beginning of each revolution. It is certainly true that, if we select the first vector of B\, for instance, bu, then we have A1611 ^ &21, where 621 is the first vector of B2 • They will coincide if we were dealing with the periodic orbit. This lack of continuity produces a corresponding lack of continuity on the projection factors, but the (small) discontinuities are innocuous. In the program PS1 (or PS2), when the final results for the unknown variable Q are obtained, we have the matrices A\, A2,... ,A& at our disposal, because they have been computed (except in what refers to some changes of coordinates), by routine POINCP. Then, the eigenvalues and eigenvectors of A\, A2, •. •, Ak are computed. A

372

Numerical Refinement

of the Quasi-periodic

Orbit

list of results can be seen in the output of programs PSl and PS2. We have observed that the variations, with respect to the halo results, are small, and for the L\ or L2 cases of the Sun-Barycenter problem clearly show the effect of the eccentricity in every two revolutions. Let Ai, A2, A3, A4, A5, A6 be the eigenvalues of one of the matrices A{. First of all, the largest and the smallest ones are detected. They correspond to the unstable and stable directions, respectively. The related eigenvectors, suitably normalized, are taken as initial vectors ei and e2- Then we examine the remaining eigenvectors. If there are conjugate complex eigenvectors, a real basis is obtained using the real and imaginary parts. The four vectors obtained are called 1*3,1x4, « 5 and u§. Then, to avoid bad conditioning in the numerical computations, and taking into account, as stated in Chapter 4, that what is relevant is the set of minors of the first column of the Floquet basis, a partial orthogonalization is done. The Gramm-Schmidt procedure has been used to replace 1*3,1x4, u§,u§ by a set e3,e4,e5,e 6 such that, together with e 2 , give an orthonormal basis of the subspace generated by e 2 , U3, U4, 1x5, u§. These vectors are the initial conditions for the variational equations at the beginning of each revolution. If the parallel shooting method uses half revolutions, each one of the matrices Ai can be written as Ai = A\ ' A\ , where A\ ' means the variational matrix corresponding to the first half revolution and A\ ' to the second one. The vectors Uj = A\ 'ej can be defined. Then, as in the previous case, we take the basis ei = u i / | | u i | | , e2 = U2/HU2II and S3, e^, e$, e~e, such that e 2 , . . • ,§6 are an orthogonal basis of the subspace generated by H2,... ,Ue- These new vectors, ej, j = 1 , . . . , 6, are taken as initial conditions, for the variational equations, at the intermediate half revolutions. All those vectors, e\,..., e§, e~i,..., e~e, for each revolution, are computed in programs PSl and PS2 (see Table 8.4) and used in the program NUNOPF to be describe in what follows. 8.2.3

A Program for the Final Nominal and Discussion

Computations.

Results

The available output of PS is organized as follows: a) For each initial epoch of a full revolution or a half revolution, the time ti and the initial conditions Xi, yi = 0, Z{, if, i/i, ii are known in normalized coordinates. b) For the same epochs the (partially orthonormalized) Floquet basis is known. Then we proceed to integrate simultaneously the equations of motion and the variational equations. Using a fixed time step in the integration (small enough), nominal points at equally spaced epochs are computed. The integration of the

The Final Nominal Orbit and Projection

variational equations gives the current Floquet basis, ei(t),... projection factors are computed using the formula m

•Ki =

i|lei|| • •, — , 2=

373

Factors

,e6(t).

Then, the

a l,...,0,

det where m* is the ith (signed) minor of the current Floquet basis and det is its determinant. Hence, also the projection factors are available at equally spaced epochs. When the nominal orbit and the projection factors are available at equally spaced epochs, T0,...,Tq they can be computed for intermediate orbits using Lagrange interpolation:

j= — r

where / is the desired function (a coordinate of the orbit or a projection factor), Tj is the nearest available epoch to the current epoch t, and r is one half of the degree of the interpolation polynomial. Usually r — 5 or 6 has shown to be enough. The Lagrange polynomials fj(t) are given by r <)

yS (*)=

r T

n

(*- <+*)

k = —r k^j

/

n

(Ti+j-Tt+k).

k — —r kjtj

If the nearest point Tj, to the epoch t, in the available lattice has index less than r or greater than q — r then the value of i is changed to r and q — r, respectively. As it will be explained in Chapter 9, the derivatives of some of these functions, with respect to time, are required. They can also be obtained from the Lagrange interpolation formula by

/'(*)« £ /(r!+j)^f(t). j=-r

An explicit formula for the time derivative of the Lagrange polynomials is

jtv?v= £ ( n it-™) / n fo+i-Ti+*) •

/ = —r

k = —r

k = —r

These Lagrange formulas are implemented in routine LAG and will be used in the next chapter. A program called NUNOPF has been implemented to compute, at equally spaced epochs, the nominal orbit and the projection factors. A shortened sample of results is given at the end of this section in Tables 8.5 and 8.6. Usually a time step of 0.6

374

Numerical Refinement

of the Quasi-periodic

Orbit

days has been used for the numerical integration. The points obtained are stored at epochs spaced 1.8 days. This is enough for the interpolation process. Finally we should add a word of caution concerning the unavoidable discontinuities produced at the matching epochs. These discontinuities are not dangerous for the projection factors. However, despite that they are smaller, they can be dangerous for the nominal orbit. An error in the matching influences, with changing sign, epochs ranging from —rh to rh with respect to the matching epoch. Here h refers to the step in the lattice: h = TJ+I — Tj. In the worst case, and only due to these numerical problems, repeated maneuvers with opposite sign would be required by the control algorithm. An easy way to avoid these problems is to perform a smoothing of the data for the nominal orbit. However this has not been implemented.

The Final Nominal Orbit and Projection

0.180000D+5 0.180018D+5 0.180036D+5 0.180054D+5 0.180072D+5 0.180090D+5 0.180108D+5 0.180126D+5 0.120144D+5 0.180162D+5 0.180180D+5 0.180198D+5 0.180216D+5 0.180234D+5 0.180252D+5 0.180270D+5 0.180288D+5 0.180306D+5 0.180324D+5 0.180342D+5 0.180360D+5 0.180378D+5 0.180396D+5 0.180412D+5 0.180432D+5

-0.113101994973D+00 0.950943329824D-03 -0.112700886451D+00 0.249412631348D-01 -0.111596144304D+00 0.487116726452D-01 -0.109689158312D-00 0.720037320238D-01 -0.107106852367D+00 0.946752195529D-01 -0.103832985458D+00 0.116691415248D+00 -0.998897971256D-01 0.137891505760D+00 -0.953022282269D-01 0.158261206749D+00 -0.900989082867D-01 0.177638882583D+00 -0.843127598499D-01 0.195881573093D+00 -0.779811078842D-01 0.212850350534D+00 -0.711453363498D-01 0.228424099302D+00 -0.638501505579D-01 0.242511167767D+00 -0.561425156126D-01 0.255058499448D+00 -0.480703765380D-01 0.266055818934D+00 -0.396814322997D-01 0.275525377401D+00 -0.310227160486D-01 0.283476498784D+00 -0.221421412487D-01 0.289833586382D+00 -0.130912230363D-01 0.294458657320D+00 -0.392474640822D-02 0.297301539285D+00 0.530303597442D-02 0.298436083738D+00 0.145408673673D-01 0.297969293921D+00 0.237403469708D-01 0.295968932290D+00 0.328549604881D-01 0.292490909920D+00 0.418393622700D-01 0.287593349612D+00

Factors

0.139007960301D-11 0.852160016609D+00 0.263574191015D-01 0.849685798829D+00 0.525867647957D-01 0.843891511415D+00 0.785861448874D-01 0.834861379089D+00 0.104257792941D+00 0.822731116261D+00 0.129507013260D+00 0.807595068051D+00 0.154241746224D+00 0.789508729440D+00 0.178371379552D+00 0.768521714691D+00 0.201807089448D+00 0.744702272409D+00 0.224462593325D+00 0.718144383681D+00 0.246254783776D+00 0.688969213674D+00 0.267104924169D+00 0.657321485056D+00 0.286938875984D+00 0.623362328546D+00 0.305687651235D+00 0.587258558662D+00 0.323287321094D+00 0.549166993642D+00 0.339678384933D+00 0.509213001666D+00 0.354804512738D+00 0.467476771133D+00 0.368611545275D+00 0.424031370325D+00 0.381048239921D+00 0.379043818240D+00 0.392072363073D+00 0.332757352003D*00 0.401645213923D+00 0.285372855811D+00 0.409739909192D+00 0.237017845290D+00 0.416314729638D+00 0.187790024449D+00 0.421357390341D+00 0.137791185507D+00 0.424841404466D+00 0.871399974889D-01

375

0.799999999945D-01 -0.997598068523D-02 0.795689837546D-01 -0.178555369929D-01 0.788949401017D-01 -0.256652019383D-01 0.779807026391D-01 -0.333631191971D-01 0.768301651102D-01 -0.409294796397D-01 0.754476696224D-01 -0.483440201468D-01 0.738377838887D-01 -0.556109088943D-01 0.720054090571D-01 -0.627116204168D-01 0.699559504450D-01 -0.696284284419D-01 0.676954395228D-01 -0.763399756641D-01 0.652305897735D-01 -0.828227858817D-01 0.625687937347D-01 -0.890567575074D-01 0.597180677816D-01 -0.950227427095D-01 0.566899517094D-01 -0.100705883963D+00 0.534843713891D-01 -0.106096220715D+00 0.501194853384D-01 -0.111188388240D+00 0.466016025932D-01 -0.115976533194D+00 0.429403691876D-01 -0.120442773918D+00 0.391462294048D-01 -0.124555306989D+00 0.352305739978D-01 -0.128288586199D+00 0.312053290838D-01 -0.131632987903D+00 0.270825982139D-01 -0.134583718194D+00 0.228747003524D-01 -0.137132191574D+00 0.185942880908D-01 -0.139264766290D+00 0.142544913835D-01 -0.140964751879D+00

376 0.180450D+5 0.180468D+5 0.180486D+5 0.180504D+5 0.180522D+5 0.180540D+5 0.180558D+5 0.180576D+5 0.180594D+5 0.180612D+5 0.180630D+5 0.180648D+5 0.180666D+5 0.180684D+5 0.180702D+5 0.180720D+5 0.180738D+5 0.180756D+5 0.180774D+5 0.180792D+5 0.180810D+5

Numerical Refinement

0.506491137621D-01 0.281215913878D+00 0.592408314283D-01 0.273488959597D+00 0.675726865179D-01 0.264444272249D+00 0.756052259690D-01 0.254175151076D+00 0.833024961903D-01 0.242815761373D+00 0.906334206284D-01 0.230547329826D+00 0.975732047544D-01 0.217590831246D+00 0.104103933229D+00 0.204155583530D+00 0.110212655899D+00 0.190335549089D+00 0.115887102229D+00 0.176100370381D+00 0.121114740069D+00 0.161502171450D+00 0.125887952808D+00 0.146770978733D+00 0.130205834823D+00 0.132189230724D+00 0.134078053362D+00 0.117994300421D+00 0.137519035371D+00 0.104360214774D+00 0.140548121380D+00 0.914061649417D-01 0.143187613788D+00 0.792055412390D-01 0.145461465674D+00 0.677925584469D-01 0.147394161908D+00 0.571684966724D-01 0.149009819701D+00 0.473099657849D-01 0.150331596236D+00 0.381806509559D-01

of the Quasi-periodic

0.426748659515D+00 0.359744516653D-01 0.427065596862D+00 -0.155497896093D-01 0.425783704699D+00 -0.672632186769D-01 0.422899891111D+00 -0.118988414300D+00 0.418416449892D+00 -0.170549038000D+00 0.412341123082D+00 -0.221784174529D+00 0.404685997656D+00 -0 272570648582D+00 0.395466137595D+00 -0.322841924612D+00 0.384698212138D+00 -0.372543513388D+00 0.372402190590D+00 -0.421493936712D+00 0.358606193292D+00 -0.469368779920D+00 0.343348373615D+00 -0.515875001595D+00 0.326674197629D+00 -0.560815775331D+00 0.308634766104D+00 -0.604032366455D+00 0.289285974851D+00 -0.645355691155D+00 0.268689430106D+00 -0.684587904999D+00 0.246913352115D+00 -0.721499757157D+00 0.224033529537D+00 -0.755832900049D+00 0.200134137688D+00 -0.787303299376D+00 0.175308435639D+00 -0.815611209410D+00 0.149659125087D+00 -0.840447950797D+00

Orbit

0.986898976117D-02 -0.142213660045D+00 0.545204301167D-02 -0.142992640641D+00 0.101848791333D-02 -0.143283425194D+00 -0.341629231050D-02 -0.143069090141D+00 -0.783641723591D-02 -0.142334485198D+00 -0.122255655822D-01 -0.141066240543D+00 -0.165670202119D-01 -0.139251869710D+00 -0.208436546789D-01 -0.136876676026D+00 -0.250377952271D-01 -0.133919629322D+00 -0.291310527640D-01 -0.130357351806D+00 -0.331044363891D-01 -0.126175708701D+00 -0.369386679639D-01 -0.121398871333D+00 -0.406143098597D-01 -0.115931304291D+00 -0.441117232055D-01 -0.109856697991D+00 -0.474111208603D-01 -0.103141236382D+00 -0.504926684848D-01 -0.957869540257D-01 -0.533397218000D-01 -0.878042529309D-01 -0.559241127892D-01 -0.792137639709D-01 -0.582364892651D-01 -0.700477292157D-01 -0.602566863082D-01 -0.603506009107D-01 -0.619690982594D-01 -0.501784148222D-01

Table 8.5 Beginning of the output of program NUNOPF. Epoch, position and velocity of the nominal orbit. Sun-Barycenter problem, L\ equilibrium point, f) = 0.08.

The Final Nominal Orbit and Projection

0.180000D+5 0.180018D+5 0.180036D+5 0.180054D+5 0.180072D+5 0.180090D+5 0.180108D+5 0.180126D+5 0.180144D+5 0.180162D+5 0.180180D+5 0.180198D+5 0.180216D+5 0.180234D+5 0.180252D+5 0.180270D+5 0.180288D+5 0.180306D+5 0.180324D+5 0.180342D+5 0.180360D+5 0.180378D+5 0.180396D+5 0.180414D+5 0.180432D+5

0.157475089845D+01 0.529266452909D+00 0.157331019393D+01 0.532199676197D+00 0.157250729337D+01 0.535260695367D+00 0.157230291005D+01 0.538348203330D+00 0.157256234902D+01 0.541414370158D+00 0.157326158574D+01 0.544483037985D+00 0.157447411327D+01 0.547590779965D+00 0.157630515236D+01 0.550757564814D+00 0.157885416295D+01 0.553982061086D+00 0.158220602429D+01 0.557246361252D+00 0.158642657100D+01 0.560524287632D+00 0.159156148360D+01 0.563782921606D+00 0.159763704647D+01 0.566994712395D+00 0.160466445355D+01 0.570143263952D+00 0.161265562599D+01 0.573234208564D+00 0.162197125487D+01 0.576306581750D+00 0.163190936284D+01 0.579423981221D+00 0.164368470288D+01 0.582607202528D+00 0.165710909007D+01 0.585766477651D+00 0.167201089724D+01 0.588805144474D+00 0.168831318478D+01 0.591727764939D+00 0.170614238943D+01 0.594591335796D+00 0.172569961335D+01 0.597443419234D+00 0.174717565854D+01 0.600305588140D+00 0.177072452542D+01 0.603175924967D+00

Factors

-0.481386384187D+00 0.180731083008D+00 -0.505355647694D+00 0.177049270484D+00 -0.528801625372D+00 0.173710964472D+00 -0.551737533052D+00 0.170683061950D+00 -0.574242420566D+00 0.167990214464D+00 -0.596318897833D+00 0.165560540424D+00 -0.617910523528D+00 0.163503432245D+00 -0.638946451354D+00 0.161800058898D+00 -0.659366102222D+00 0.160453680231D+00 -0.679126180264D+00 0.159463046984D+00 -0.698203665136D+00 0.158825540770D+00 -0.716595091716D+00 0.158540033461D+00 -0.734312767819D+00 0.158609333550D+00 -0.751379974502D+00 0.159042277097D+00 -0.767800708033D+00 0.159859506178D+00 -0.783561363592D+00 0.161074499313D+00 -0.798571176451D+00 0.162717963552D+00 -0.812721728684D+00 0.164829254853D+00 -0.826011270257D+00 0.167363402928D+00 -0.838506936101D+00 0.170321865046D+00 -0.850202249798D+00 0.173729908673D+00 -0.861023647722D+00 0.177622253740D+00 -0.870876329712D+00 0.182034344205D+00 -0.879663740906D+00 0.186988940755D+00 -0.887289229016D+00 0.192501474288D+00

377

-0.671090390847D-01 -0.110203637406D-01 -0.655280939664D-01 -0.976274748340D-02 -0.637804991104D-01 -0.846027478986D-02 -0.619077723875D-01 -0.711187155287D-02 -0.599056969046D-01 -0.571730664503D-02 -0.577497410132D-01 -0.427800569063D-02 -0.554231200387D-01 -0.279614213713D-02 -0.529204210829D-01 -0.127411091808D-02 -0.502431703737D-01 0.285581996588D-03 -0.473998002264D-01 0.188028829598D-02 -0.443959601938D-01 0.350717947179D-02 -0.412389971590D-01 0.516319974789D-02 -0.379318434627D-01 0.684507619827D-02 -0.344708049387D-01 0.854928364765D-02 -0.308422934245D-01 0.102720880663D-01 -0.270218473260D-01 0.120096279816D-01 -0.229860913124D-01 0.137578791383D-01 -0.187437010502D-01 0.155117730271D-01 -0.143343453184D-01 0.172640936810D-01 -0.977128512857D-02 0.190074766765D-01 -0.503164336170D-02 0.207366456724D-01 -0.892585767243D-04 0.224471645861D-01 0.507207288761D-02 0.241338193460D-01 0.104599853416D-01 0.257901317024D-01 0.160766014452D-01 0.274083819023D-01

378

Numerical Refinement

0.180450D+5 0.180468D+5 0.180486D+5 0.180504D+5 0.180522D+5 0.180540D+5 0.180558D+5 0.180576D+5 0.180594D+5 0.180612D+5 0.180630D+5 0.180648D+5 0.180666D+5 0.180684D+5 0.180702D+5 0.180720D+5 0.180738D+5 0.180756D+5 0.180774D+5 0.180792D+5 0.180810D+5

0.179645672409D+01 0.606034629130D+00 0.182443641261D+01 0.608849496995D+00 0.185467894023D+01 0.611581577549D+00 0.188714944530D+01 0.614191897969D+00 0.192176628992D+01 0.616650588719D+00 0.195842368282D+01 0.618950740448D+00 0.199707514529D+01 0.621128609303D+00 0.203791832641D+01 0.623271283918D+00 0.208141513862D+01 0.625432639380D+00 0.212755490428D+01 0.627469113947D+00 0.217545373170D+01 0.629136025160D+00 0.222427152703D+01 0.630337662079D+00 0.227361331512D+01 0.631103267316D+00 0.232321751573D+01 0.631471941745D+00 0.237273001799D+01 0.631448813711D+00 0.242163994310D+01 0.631006605336D+00 0.246925806084D+01 0.630089851625D+00 0.251475952857D+01 0.628624007364D+00 0.255715962105D+01 0.626522982057D+00 0.259537013718D+01 0.623698439881D+00 0.262823265405D+01 0.620071631691D+00

of the Quasi-periodic

-0.893654376759D+00 0.198584455939D+00 -0.898656402494D+00 0.205246481365D+00 -0.902184440264D+00 0.212495456676D+00 -0.904113786355D+00 0.220339995747D+00 -0.904296161115D+00 0.228791474301D+00 -0.902541633463D+00 0.237867212717D+00 -0.898587813376D+00 0.247595502010D+00 -0.892080222948D+00 0.258016384411D+00 -0.882669620088D+00 0.269149484146D+00 -0.870190773835D+00 0.280931706306D+00 -0.854524895460D+00 0.293256027212D+00 -0.835385418877D+00 0.306070293645D+00 -0.812407505403D+00 0.319362657451D+00 -0.785254197200D+00 0.333109716429D+00 -0.753646224157D+00 0.347255701690D+00 -0.717377789387D+00 0.361710046590D+00 -0.676339850310D+00 0.376348883816D+00 -0.630549463746D+00 0.391017423845D+00 -0.580180675129D+00 0.405533553666D+00 -0.525591243035D+00 0.419693735832D+00 -0.467339121205D+00 0.433282096768D+00

Orbit

0.219204076701D-01 0.289797246902D-01 0.279873741470D-01 0.304943182762D-01 0.342719704488D-01 0.319414829489D-01 0.407683367127D-01 0.333099256087D-01 0.474717251119D-01 0.345880962465D-01 0.543804808025D-01 0.357648194349D-01 0.614972837993D-01 0.368303702437D-01 0.688220957528D-01 0.377771714665D-01 0.763266398826D-01 0.385957142921D-01 0.839389496817D-01 0.392654607467D-01 0.915793224179D-01 0.397599555731D-01 0.991859475590D-01 0.400613357528D-01 0.106695801922D+00 0.401592575226D-01 0.114032517499D+00 0.400444442708D-01 0.121098589112D+00 0.397066312844D-01 0.127778569426D+00 0.391350380419D-01 0.133941562894D+00 0.383194306526D-01 0.139445327304D+00 0.372512557933D-01 0.144141959824D+00 0.359247861811D-01 0.147884902336D+00 0.343382769360D-01 0.150536114735D+00 0.324951098599D-01

Table 8.6 Beginning of the output of program NUNOPF. Epoch and coordinates of the projection factors. Sun-Barycenter problem. Li equilibrium point. /3 = 0.08.

References

8.3

379

References [1] R. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin, 1978. [2] R. Bulirsh and J. Stoer. Foundations of Numerical Analysis. Springer Verlag, 1981. [3] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer Verlag, 1983. [4] E. Isaacson and H.B. Keller. Analysis of Numerical Methods. John Wiley & Sons, 1966. [5] J.B. Pesin. "Characteristic Lyapunov exponents and smooth ergodic theory" Russ. Math. Surv., 32, 55-114, 1977. [6] V. Szebehely. Theory of Orbits. Academic Press, 1967.

Chapter 9

The On/off Control Strategy: Simulations and Discussion

This is the closing chapter concerning the quasi-periodic motion of a controlled spacecraft in the vicinity of a halo orbit of the Sun-Barycenter system. Final results for the main goal of the work, i.e., station keeping, are given here. The first step is the election of a control strategy: When and how a maneuver should be done. The second step is the elaboration of a simulation program for the behavior of the controlled spacecraft. Several runs, with different parameters, allow us to detect the influence of the distinct factors in the total fuel consumption. As a last step we discuss the feasibility of the radiation pressure control. 9.1

9.1.1

A Simulation Program for the Motion of the Controlled Spacecraft The Equations of Motion. Introduction of Random and Systematic Noise in the Solar Radiation Pressure Effect

The standing equations of motion for the noncontroUed motion are the classical Newton's equations r = -G >

rnAlF——^

where A runs over all the bodies of the solar system and the coordinates are ecliptic centered at the barycenter of the solar system. The on/off controls are supposed to act in a very short time (except in the case of radiation pressure maneuvers). Then, when a control is applied we merely shift the velocities by the suitable amount. The simulation program is suited to this approach because we have used a onestep method (a Runge-Kutta-Fehlberg algorithm of order 7/8 and automatic step size control). The mass of the Sun has been altered, as explained in Chapter 5, to account for 381

382

The On/off

Control Strategy: Simulations

and

Discussion

the radiation pressure effects. As the effects of the radiation pressure have some uncertainties, we have added a random contribution to the acceleration given above. In order not to produce conflict with the numerical integration scheme, the random forces have been kept fixed at each step of the integration. The "standard" random errors in the radiation pressure consist in a relative error, in the modulus of the radiation pressure acceleration, which is supposed distributed according to a centered normal law with typical deviation 5%. Normally distributed errors, with zero mean and typical deviation equal to 0.05 rad, have been used for both ecliptic latitude and longitude of the radiation pressure acceleration. These standard values can be modified using an amplification (or reduction if less than 1) factor. A systematic noise can be introduced in the following way. The nominal orbit is computed with a given value of s/m (spacecraft section facing to the Sun divided by mass). If the simulation is done with a value of s/m greater (less) than the nominal one, a net acceleration outwards (towards) the Sun will result. In all computations (except for the introduction of systematic errors), the adopted value of s/m has been 0.01. This is pessimistic because, probably, the real figure will be 1/3 of that one.

9.1.2

The Effect of Tracking Errors of Maneuvers

and Errors

in the

Execution

The current point will be known, in the applications, through tracking. As specified by the Agency, for the L\ and L2 cases of the Sun-Barycenter problem, the tracking errors have been supposed to be distributed according to centered normal laws with 2(7 deviations equal to 3, 5 and 30 km, 2, 2 and 6 mm/s in x, y, z, x, y, z, respectively. Here the orientation of the coordinate axes is the same as the adimensional system. When the simulation is started, at a given epoch, the nominal point is computed. Then, random variations in the 6 coordinates are introduced due to tracking errors according to the given laws. The integration of the motion of the spacecraft in the solar system field is done with the random errors of 9.1.1. At every step, the unstable component is computed, but to this computation (not to the current point) again tracking errors are applied. Hence, for a given epoch we have 3 different points: The nominal one, the real one and the estimated point. When a too large unstable component forces to the execution of a maneuver, the x- and y-components of the jump in velocity are computed. Then, again some execution errors are introduced. They can be due to wrong direction for the maneuver or wrong modulus of the same. The "standard" assumptions on those errors are the following. We have supposed that, if an {x,y) maneuver should be done, the x- and ^-components are modified

A Simulation

Program for the Motion of the Controlled Spacecraft

383

with a centered normal deviation with 2a value equal to 5% of the modulus of the scheduled maneuver. For the ^-component (theoretically zero) we have used a 2% of the same amount. If x only or y only maneuvers are done it is supposed, also, that the errors are of 5% in the active component and 2% in the nonactive ones. Both the standard values of the tracking and the execution of the maneuver errors, can be modified using amplifying factors.

9.1.3

How to Decide

When a Control Should be

Applied

It is clear that to do a maneuver when the unstable component is too small is nonsense. The small instability can be due to tracking errors. In Chapter 3 it has been found that, taking a box, centered at the origin, with 2a half sides in each of the six variables, the maximal unstable component, changing from one point in the orbit to another one, is close to 2 • 1 0 - 7 in adimensional units. Furthermore, the random errors in the acceleration should cancel on the average, but locally can have some importance. With the used values, their amount can be estimated in 1 • 1 0 - 7 , approximately, in the unstable component (u.c. for short). Hence, maneuvers with an u.c. less than 2, 3 or 4 times 10~ 7 should not be done. A value is given, by the user of the simulation program, for this lower bound of the u.c. in modulus. Below this value no maneuver should be done. When the u.c. is greater than the given value, a maneuver can be useful. However, too frequent maneuvers should be avoided because some time interval is necessary to settle down the tracking errors. Typically a time interval of one month, at least, has been required, between two consecutive maneuvers. If the u.c. is greater than the lower bound and the time since last control is large enough, we must be sure that the u.c. is increasing in an exponential way. If this is not so, the deviations can be due to the presence of small oscillations in the motion around the nominal orbit, because of a not accurate enough nominal orbit, as happens when an analytic one is taken. If the u.c. increases by a factor Ai (the dominant eigenvalue) in one period, T, in a step of integration, Ai, should increase, in theory by a factor exp [At • ij^). A value, m, slightly smaller than J ^ has been used. If for 3 consecutive steps of integration, the quotient of successive u.c. is greater than exp(mAt), then a clear exponential behavior is present. In any case an upper bound of the u.c. must be given such that, if the u.c. is greater than this upper bound, a maneuver is done unconditionally. From the unstable character of the behavior it follows that the best choice (from the optimal point of view, i.e. the one such that the foreseen fuel used divided by the time interval between two consecutive maneuvers reaches a minimum) is to perform maneuvers at a value of the u.c. roughly equal to e = exp(l) times the sum of the total effect of the different errors. Using a lower bound 0.4 • 10~ 6 , a value of the upper bound of 10~ 6 is suitable.

384

The On/off

9.1.4

Description

of the

Control Strategy: Simulations

and

Discussion

Program

A program (CONSIM) has been produced for the simulation of the control according to what has been said in 9.1.1 to 9.1.3. For the nominal orbit and for the projection factors several possibilities are offered: (1) Numerical halo orbit (in this case, the projection on the Floquet modes 3 and 4 is also available optionally). (2) Analytic quasi-periodic orbit (generated by routine ANAQPO from the output of ANACOM and QPO) and analytic parameters of control (it is enough to use the ones of the RTBP) generated by program CONA from the output of ANAVAR. (3) Refined numerical quasi-periodic orbit and related normalized projection factors. They are generated by programs PS (PS1 or PS2) and NUNOPF. To perform checks, a nominal orbit of any kind can be used with projection factors of any kind. To allow for this possibility several additional parameters can be requested. The determination of a point in the orbit can be done as follows: I) The point in the nominal orbit is computed at the current time. II) It is computed at a time near the current one such that the distance from the actually computed (by integration) point to the selected point in the nominal path be a local minimum. This is done writing d2 = (x - xnf

+ (y- ynf

+ (z - zn)2 ,

where xn, yn, zn are the nominal values and x, y, z the real ones. Then the nominal ones are considered functions of time and the others are constant. By time differentiation - 2 ^ ) ' = (x~

x

n)x + (y- yn)y + {z- zn)z = 0.

This equation is solved for t using Newton's method. The nominal orbit is computed in routine NOMORB. The type of the maneuvers can also be selected: a) b) c) d)

Full optimal control ((z,y)-plane maneuvers). x-axis maneuvers. y-axis maneuvers. Radiation pressure maneuvers.

The random generator used to obtain normally distributed points (for the noise in acceleration and also for tracking and maneuvers errors) is called a given (by the

Numerical Results of the

Simulations

385

user) number of times before starting the proper computations. This allows to do statistics with all the other data fixed. Each time a maneuver is done, written output is produced related to it. At the end of the run a statistics of the maneuvers is done. The available information is: • number of maneuvers, • total amount of Av required (in adimensional and physical units), • minimum and maximum observed time between two consecutive maneuvers (in Julian days), • maximal and minimal maneuver (in adimensional and physical units). The normalized projection factors are computed by routine MAGCON. From them the unitary controls can be computed in any case. The numerical RTBP values are read from the output of program PAPUS. They are given as Fourier series and the only thing to do is the evaluation of the series at a given angle. The analytic values are obtained from the output of CONA. From this output and with the x- and ^-amplitudes determined when the nominal orbit is computed (if the analytic orbit is used) or directly requested otherwise, a Fourier series with numerical coefficients is obtained for the projection factors, determinant and first Floquet mode. From a given angle (note that the period should be given if the orbit chosen is the numerical one) the three magnitudes can be determined and from them the normalized projection factors. Finally, if numerically refined projection factors are used, the only thing to do is to obtain them at a given epoch using the Lagrange interpolation routine LAG.

9.2 9.2.1

Numerical Results of the Simulations A Summary

of

Results

First of all we present tables of a summary of the results in the L\ case SunBarycenter problem for the initial epoch t0 = 18 000 MJD50 (since 1950.0) and z0 = 0.08. Table 9.1 refers to the simplified model using numerically refined nominal orbit and projection factors. The bulk of the simulations is done in this case because of CPU time reasons. Table 9.2 refers to the full model also with numerical values. Some examples of fully analytic control (i.e., analytic qpo and analytic projection factors) are given. The results are quite similar in the full and simplified models. This is due to the fact that, if the nominal orbit is equally good, what matters for the instability are the tracking, perturbing noise and maneuver execution errors. The results are not sensitive to the value of /? (within moderate values) and they are essentially unchanged if Li is used instead of L\ (with a small improvement due

386

The On/off

Control Strategy: Simulations

and

Discussion

to a minor rate of instability). As a summary of the Tables we can say that, for the standard values of the different errors at most 20 cm/s per year are required. The time interval between maneuvers ranges between 2 and 6 months. This is changed if the standard errors are altered as it will be explained in 9.2.2. In the most pessimistic case (errors 2,2 and 4 times the standard ones and a systematic error of 10% in the radiation pressure) at most 50 cm/s per year are required. The analytic case is (of course) worse. Roughly, it requires 1 cm/s per day on the average. Reductions of this result seen not too difficult to obtain if more terms are kept in the analytic expansions, but then the computing time of the solution will be substantially increased. All the computations have been done with the option: minimal distance from the nominal to the current point. In the interactive runs of CONSIM additional information appears which allows to check the values of the changes of the nominal point in time. They rarely exceed 15 m in runs covering 4 years controlled by an on/off algorithm. In the Table Wu max,mm

At ampi, i = 1,2,3

control pr calls man Av -t

min,max

y

min,max

refer

to the upper and lower bounds for the u.c, to the desired minimum time interval between maneuvers, to the amplifying factors of errors in tracking, execution of maneuvers and radiation pressure, to the type: 1 - (x, y), 2 - (x), 3-(y), to the systematic error in the radiation pressure, to the number of previous calls of the random number generator, to the number of maneuvers done, to the total amount of increment of velocity required (cm/s), to the minimum and maximum experimental values of the time interval between maneuvers (jd), to the minimum and maximum values of the maneuvers (cm/s).

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The On/off

Control Strategy: Simulations

and

Discussion

At the end of this section a sample is given of results of CONSIM. 9.2.2

Discussion tudes

on the Effect of the Different

Errors

and

Magni-

The standard values of W ^ a x and W^m in the runs have been 1 0 - 6 and 0.4 • 10~ 6 . An increase in these values (except in the fully analytic models) is not suitable. A decrease is possible without producing too closer maneuvers. This produces a reduction on Av of roughly 25% if W^in m a x are halved. The comparison of results for different values of ampi shows that the tracking errors are responsible (with the standard values taken) of one half of the total Av required. Hence, it is important to reduce as much as possible the tracking errors. The errors in the execution of the maneuvers are responsible for some 10% of the total Av. Therefore, not too strong technical conditions are necessary to be asked to the control device. The random errors in the radiation pressure mean some 15% of the total Av (even in our case whose a large s/m is used). This means that the remaining 25% is due to inaccuracies of the model and to the numerical simulation errors. This can be seen as the action of unknown or neglected terms in the acceleration. Furthermore, as already stated in the results of PAPUS (see Chapter 3), the (a;)only control increases Av in 15%, approximately, and the (y)-only control more than doubles it. In some cases it is even 4 times worse than the full control, again in agreement with Chapter 3. Finally, the systematic errors in the radiation pressure have a strong influence. A 10% error doubles the value of Av.

THE MODEL OF THE SOLAR SYSTEM IS: 0 0 11 O O O O O O I O O O O O O , SPACECRAFT EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 19424 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADIM. SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 60. AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 1.0 TYPE OF MANOEUVRES : (X,Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GENE USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED AT SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+00 INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF TH X = -0.13591794407808D+09 Y = -0.59435605455003D+08 Z = 0.120168487902 XD = x0.98740705136901D+06 YD = -0.23624156175959D+07 ZD = 0.134270560980 APP.CONTROLS(X,Y,Z) INCL EXEC ERRORS TIME W U AMP 18089.10721 0.10344D-05 -0.10413D-05 -0.82399D-06 -0.13446D-07 -0.310 18177.36989 0.81852D-06 -0.13828D-05 -0.56575D-06 -0.49433D-08 -0.411 18250.56504 0.92106D-06 -0.11003D-05 -0.61727D-06 -0.18208D-07 -0.327 18357.75792 0.53203D-06 -0.88446D-06 -0.29374D-06 -0.65669D-08 -0.263 18426.18114 0.45583D-06 -0.54239D-06 -0.31701D-06 -0.34139D-08 -0.161 18489.16308 0.10392D-05 -0.13721D-05 -0.10594D-05 -0.90790D-08 -0.408 18552.82845 0.62040D-06 -0.10327D-05 -0.24911D-06 -0.69310D-08 -0.307 18615.64729 0.10574D-05 -0.10872D-05 -0.77369D-06 -0.20739D-09 -0.323 18724.63105 0.80538D-06 -0.13539D-05 -0.41457D-06 -0.39690D-08 -0.403 18788.78775 0.53632D-06 -0.59559D-06 -0.38209D-06 -0.27080D-08 -0.177 18894.86110 0.10746D-05 0.18339D-05 0.65841D-06 0.15804D-07 0.546 18955.00892 0.10872D-05 -0.13462D-05 -0.66851D-06 -0.11559D-07 -0.400 19065.09675 0.10417D-05 -0.18211D-05 -0.79303D-06 -0.10273D-07 -0.542 19186.86664 0.10101D-05 -0.11626D-05 -0.97816D-06 -0.17867D-08 -0.346 19301.48566 -0.10213D-05 0.14280D-05 0.61501D-06 0.13025D-07 0.425 19355.69985 0.10498D-05 0.10548D-05 0.11009D-05 -0.15147D-08 0.314 19409.02120 0.10498D-05 -0.16800D-05 -0.83260D-06 -0.10598D-08 -0.500 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES = 17 AMOUNT OF DELTA V (ADIM) = 0.23810D-04 = 0.70919D+02 CM/S MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 53.3214, 121.7699 MIN AND MAX MANOEUVRES 0.62825D-06 (ADIM) = 0.18713D+01 CM/S 0.19863D-05 (ADIM

Table 9.3

Output of program CONSIH. Sun-Barycenter problem, simplified model. L

THE MODEL OF THE SOLAR SYSTEM IS: 0 0 11 O O O O O O I O O O O O O , SPACECRA EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 194 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADI SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 6 AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 2 TYPE OF MANOEUVRES : (X.Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GE USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED A SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+00 INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF X = -0.13591794868700D+09 Y = -0.59435604007100D+08 Z = 0.12014524774 XD = 0.98740684976919D+06 YD = -0.23624156458289D+07 ZD = 0.16977645144 TIME W U AMP APP.CONTROLS(X.Y.Z) INCL EXEC ERRORS 18079.92184 -0.20064D-05 0.21142D-05 0.16689D-05 0.61128D-07 0.62 18141.23172 -0.13623D-05 0.18616D-05 0.12787D-05 - 0 . 2 6 5 5 5 D - 0 7 0.55 18244.59704 -0.20779D-05 0.24348D-05 0.13173D-05 0.72 0.78330D-07 18288.84013 -0.20831D-05 0.21986D-05 0.65 0.21686D-05 -0.29777D-07 18362.82336 -0.21243D-05 0.33208D-05 0.98 -0.10287D05 -0.47561D-07 18405.21541 -0.20004D-05 0.28660D-05 0.85 0.10139D-05 - 0 . 4 6 6 5 2 D - 0 8 18460.74088 -0.21767D-05 0.20922D-05 0.62 0.21685D-05 -0.65331D-07 18542.46865 -0.19473D-05 0.30784D-05 0.91 0.13067D-05 0.D60615-08 18604.56545 -0.20318D-05 0.26365D-05 0.78 0.15574D-05 - 0 . 3 3 8 2 7 D - 0 8 18714.21011 -0.21047D-05 0.31651D-05 0.94 0.75685D-06 -0.92166D-07 18752.92893 -0.21973D-05 0.34513D-05 0.10 0.13302D-05 0.44590D-09 18909.41920 -0.15286D-05 0.27307D-05 0.81 0.75246D-06 -0.28675D-07 18976.70766 0.11989D-05 •0.12004D-05 0.35 -0.89576D-06 -0.49584D-07 19053.39523 -0.20790D-05 0.37372D-05 0.11 0.17372D-05 0.75831D-07 19151.13512 -0.97447D-06 0.95642D-06 0.28 0.80717D-06 0.32425D-07 19197.05238 -0.20029D-05 0.25753D-05 0.76 0.20760D-05 - 0 . 2 1 4 1 0 D - 0 7 19262.32596 -0.20932D-05 0.34492D-05 0.10 0.70299D-06 -0.92395D-07 19309.87220 -0.22591D-05 0.29846D-05 0.88 0.14349D-05 -0.35284D-08 19363.10094 -0.20534D-05 0.69 0.21458D-05 0.12972D-07 0.23289D-05 FINAL STATISTICS FOR THE RUN AMOUNT OF DELTA V (ADIM) = 0 : NUMBER OF MANOEUVRES = 19 56715D-04 = 0.16893D+02 CM/S MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 5 3 . 3 2 1 4 , 1 2 1 . 7 6 9 9 MIN AND MAX MANOEUVRES 0.12519D-05 (ADIM) = 0.37290D+01 CM/S 0.41219D-05 (AD Table 9.4

Output of program CONSIM. Sun-Barycenter problem, simplified model.

THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 l l l l l l l l l O O O O , SPACECRAFT EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 19424 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADIM. SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 100 AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 1.0 TYPE OF MANOEUVRES : (X,Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GENE USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED AT SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+00 INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF TH -0.13715466804472D+09 Y = -0.59539470779197D+08 Z = 0.1604883421592 YD= -0.23638836160428D+07 ZD= 0.1214589667611 XD= 0.98781159117598D+06 APP.CONTROLS(X,Y,Z) INCL EXEC ERRORS TIME W U AMP 0.10319D-04 -0.15219D-04 18045.34670 .47952D-05 0.13170D-06 -0 4533 .10162D-04 0.16626D-04 18193.94425 0 4952 .59954D-05 -0.16721D-06 . 10033D-04 -0.12088D-04 18247.00742 -0 3600 .65876D-05 -0.54176D-07 . 10697D-04 -0.16962D-04 -0 5052 18331.65427 .79218D-05 0.37546D-06 .10877D-04 -0.17256D-04 -0 5139 18374.86321 -0.44487D-05 -0.26866D-06 .10043D-04 -0.12029D-04 18424.02042 -0 3582 -0.62405D-05 -0.10449D-06 .10370D-04 0.16124D-04 0 4820 18515.91125 0.78726D-05 -0.28793D-06 -0.10150D-04 0.14530D-04 0 4327 18586.88898 0.47526D-05 0.98306D-07 0.10264D-04 -0.10940D-04 18615.32036 -0.79489D-05 -0.60409D-08 0.3258 .10271D-04 0.11516D-04 18651.00884 0.10371D-04 0.17906D-06 0.3430 .10283D-04 -0.17674D-04 18724.26484 -0.55724D-05 0.66575D-08 0.5264 .10204D-04 0.13215D-04 18846.13370 0.10726D-04 0.17078D-07 0.3936 .10030D-04 -0..80187D-05 -0.16470D-04 18877.83490 0.71693D-07 0.4905 .10996D-04 -0..57933D-05 -0.14831D-04 18947.58869 -0.17241D-06 0.4417 .10895D-04 -0..79999D-05 -0.12173D-04 18968.38169 0.12487D-06 0.3625 .10329D-04 0..98708D-05 0.15784D-04 19035.25517 -0.13673D-06 0.470 .11329D-04 -0..81117D-05 -0.18499D-04 19054.39404 -0.28295D-07 0.5510 .10207D-04 -0..40229D-05 -0.16116D-04 19103.19117 0.15733D-06 0.4800 0.11203D-04 -0..72537D-05 .13500D-04 19136.34014 -0.15512D-06 0.402 0.10903D-04 -0..88470D-05 .10945D-04 19158.22168 -0.37212D-07 0.3260 0.11007D-04 -0..10151D-04 .17669D-04 19218.58767 -0.17679D-06 0.5262 0.10649D-04 -0..58238D-05 17719D-04 19250.38971 0.37417D-06 0.5277 -0.10269D-04 16221D-04 19288.77084 0.53016D-05 0.31778D-06 0.483 0.11191D-04 -0.12259D-04 19323.00640 -0.79567D-05 0.57829D-07 0.365 0.10073D-04 -0.10269D-04 19354.06220 -0.94468D-05 0.85058D-07 0.3058

19403.50755 0.10136D-04 -0.16868D-04 -0.83134D-05 0.17633D-06 -0.50 19422.51196 0.10973D-04 -0.18948D-04 -0.83019D-05 -0.19848D-07 -0.56 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES = 27 AMOUNT OF DELTA V (ADIM) = 0 . 4 5 3 1 0 D - 0 3 = 0.13496D+04 CM/S MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 1 9 . 0 0 4 4 , 1 4 8 . 5 9 7 5 MIN AND MAX MANOEUVRES 0 . 1 3 5 2 3 D - 0 4 (ADIM) = 0.40280D+02 CM/SJ 0.20687D-04 (A Table 9.5

Output of program CONSIM. Sun-Barycenter problem, simplified model.

THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 l l l l l l l l l O O O O , SPACECRAF EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 1942 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADIM SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 60 AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 1. TYPE OF MANOEUVRES : (X,Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GEN USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED AT SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+OO INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF T Y= -0.59541256018711D+08 Z = 0.160408655928 X= -0.13716046038108D+09 XD=0.98750290163574D+06 YD = -0.23631678023408D+07 ZD = -0.229961410683 W U AMP APP.CONTROLS(X,Y,Z) INCL EXEC ERRORS TIME 18118.60671 -0.53969D-06 0.60722D-06 0.53752D-06 0.55303D-08 0.188 18204.69377 0.10040D-05 -0.16480D-05 -0.43639D-06 -0.71202D-08 -0.490 18266.11602 0.81179D-06 -0.83813D-06 -0.63146D-06 -0.17467D-08 -0.249 18335.68565 0.10584D-05 -0.16472D-05 -0.79862D-06 0.91319D-08 -0.490 18385.68293 0.10533D-05 -0.16594D-05 -0.38746D-06 0.12258D-07 -0.494 18449.35155 0.11135D-05 -0.11635D-05 -0.96788D-06 -0.54088D-08 -0.346 18513.87225 -0.74847D-06 0.12001D-05 0.57957D-06 -0.18927D-07 0.357 18612.11158 0.68332D-06 -0.74099D-06 -0.54423D-06 -0.20089D-07 -0.220 18719.17343 0.71723D-06 -0.11952D-05 -0.41158D-06 -0.10288D-07 -0.355 18816.80107 0.50647D-06 -0.48240D-06 -0.46832D-06 -0.92279D-08 -0.143 18890.66459 -0.59666D-06 0.10164D-05 0.34733D-06 -0.74222D-08 -0.302 18946.54703 0.11153D-05 -0.15144D-05 -0.61912D-06 -0.80250D-09 -0.45 19012.58407 0.52557D-06 -0.62742D-06 -0.53585D-06 -0.65867D-08 -0.186 19080.41092 0.10710D-05 -0.18099D-05 -0.59672D-06 -0.11798D-07 -0.539 19193.86704 -0.10932D-05 0.13628D-05 0.10682D-05 0.24395D-08 -0.405 19348.95524 -0.40323D-06 0.40628D-06 0.40073D-06 -0.71948D-08 0.12 19419.48534 0.72569D-06 -0.11967D-05 -0.39809D-06 -0.37669D-08 -0.356 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES = 17 AMOUNT OF DELTA V (ADIM) = 0.21807D-04 = 0.64953D+02 CM/S MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 49.9973, 155.0882 MIN AND MAX MANOEUVRES 0.57070D-06 (ADIM) = 0.16999D+01 CM/S 0.19058D-05 (ADI Table 9.6

Output of program CONSIM. Sun-Barycenter problem, simplified model.

THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 l l l l l l l l l O O O O , SPACECRA EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 194 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADI SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 6 AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 1 TYPE OF MANOEUVRES : (X,Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GE USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED A SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+OO INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF X = -0.13716046264417D+09 Y = -0.59541259054386D+08 Z = 0.16040011240 XD= 0.98750317843358D+06 YD= -0.23631678262200D+07 ZD= -0.22960248861 APP.CONTROLS(X,Y,Z) INCL EXEC ERRORS TIME W U AMP 18098.13458 -0.10905D-05 0.10754D-05 0.94448D-06 -0.25458D-08 0.32 18165.77963 0.84882D-06 -0.14076D-05 -0.64330D-06 0.47942D-08 -0.41 18316.87091 0.58839D-06 -0.83395D-06 -0.55104D-06 -0.87184D-08 -0.24 18406.17686 0.10125D-05 -0.14165D-05 -0.41198D-06 0.11328D-07 -0.42 18466.28574 0.58435D-06 -0.57760D-06 -0.57510D-06 0.35460D-08 -0.17 -0.33 18528.02033 0.68707D-06 -0.11317D-05 -0.45525D-06 0.15493D-07 -0.43 18587.81926 0.10275D-05 -0.14749D-05 -0.52026D-06 -0.47637D-08 18673.15121 -0.61050D-06 0.88606D-06 0.58227D-06 0.15839D-07 0.26 -0.52 18744.85855 0.10539D-05 -0.17661D-05 -0.44069D-06 0.94373D-08 18807.39981 -0.96009D-06 0.94417D-06 0.84290D-06 -0.40000D-08 0.28 -0.40 18953.69594 0.10764D-05 -0.13444D-05 -0.64026D-06 -0.10493D-07 -0.30 18998.91007 0.10039D-05 -0.10330D-05 -0.95614D-06 0.28838D-07 -0.50 19064.27039 0.10149D-05 -0.16859D-05 -0.57471D-06 0.12911D-07 -0.37 19131.32723 0.10244D-05 -0.12634D-05 -0.64064D-06 -0.15290D-08 -0.32 19178.89802 0.10820D-05 -0.10813D-05 -0.99760D-06 0.39466D-08 -0.49 19242.23161 0.94355D-06 -0.16605D-05 -0.60481D-06 -0.84678D-08 -0.28 19310.75131 0.75332D-06 -0.95001D-06 -0.49321D-06 -0.66266D-08 19354.44277 0.10450D-05 -0.10068D-05 -0.10456D-05 0.93763D-08 -0.29 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES = 18 AMOUNT OF DELTA V (ADIM) = 0.25054D-04 -0.74626D+02 MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 43.6915, 151.0913 MIN AND MAX MANOEUVRES 0.81509D-06 (ADIM) = 0.24278D+01 CM/S 0.18203D-05 (AD Table 9.7

Output of program CONSIM. Sun-Barycenter problem, simplified model.

THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 l l l l l l l l l O O O O , SPACECRAF EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L I . INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 1942 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADIM SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 60 AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 1. TYPE OF MANOEUVRES : (X,Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GEN USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED AT SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+00 INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF T Y= -0.59541255324964D+08 Z = 0.16038813842 X = -0.13716046134290D+09 0.98750303964530D+06 YD = -0.23631678291512D+07 ZD = -0.22963174640 XD APP.CONTROLS(X,Y,Z) INCL EXEC ERRORS TIME W U AMP 18074.42179 0.51747D-06 -0.62288D-06 -0.34005D-06 0.18631D-08 -0.18 18133.26163 -0.11248D-05 0.14015D-05 0.10947D-05 0.78141D-08 0.41 18275.27955 -0.10068D-05 0.10459D-05 0.90388D-06 0.86129D-08 0.31 18335.62283 0.79732D-06 -0.13017D-05 -0.63231D-06 0.69312D-08 -0.38 18420.84034 -0.53268D-06 0.66022D-06 0.32175D-06 0.32386D-08 0.19 18528.08706 0.56795D-06 -0.95768D-06 -0.39201D-06 0.12407D-07 -0.28 18602.82803 0.80142D-06 -0.98071D-06 -0.49242D-06 -0.19817D-07 -0.29 18683.93745 0.10716D-05 -0.17030D-05 -0.86417D-06 -0.96606D-08 -0.50 18752.93622 0.85294D-06 -0.12874D-05 -0.38127D-06 0.94128D-08 -0.38 18836.62508 0.71426D-06 -0.92988D-06 -0.75158D-06 0.83032D-08 -0.27 18910.77574 .52764D-06 -0.87065D-06 -0.29212D-06 -0.26351D-08 -0.25 18975.89672 .81386D-06 -0.86698D-06 -0.67508D-06 0.82387D-08 -0.25 19007.01541 .10039D-05 -0.11385D-05 -0.99251D-06 0.16362D-07 -0.33 19077.20568 .52913D-06 -0.87346D-06 -0.26503D-06 -0.30174D-08 -0.26 19148.39770 .80909D-06 -0.84250D-06 -0.59621D-06 -0.35806D-08 -0.25 19196.56075 .10042D-05 -0.12858D-05 -0.10257D-05 0.49406D-08 -0.38 19263.46559 .10351D-05 -0.16762D-05 -0.41730D-06 -0.16629D-07 -0.49 19324.67045 .10708D-05 -0.12149D-05 -0.84502D-06 -0.13733D-07 -0.36 19378.33480 .10653D-05 -0.14142D-05 -0.10255D-05 -0.55396D-08 -0.42 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES = 19 AMOUNT OF DELTA V (ADIM) = 0.24704D-04 = 0.73584D+02 CM/S MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 31.1187, 142.0179 MIN AND MAX MANOEUVRES 0.70966D-06 (ADIM) = 0.21138D+01 CM/S 0.19097D-05 (ADI Table 9.8

Output of program CONSIM. Sun-Barycenter problem, simplified model.

THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 l l l l l l l l l O O O O , SPACECRA EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 187 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADI SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 6 AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 1 TYPE OF MANOEUVRES : (X,Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GE USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED A SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+00 INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF Y = -0.59541246687567D+08 Z = 0.1603872955 X = 0.13716046585275D+09 YD= -0.23631679171694D+07 ZD= -0.2295870161 XD= 0.98750303124843D+06 APP.CONTROLS(X,Y,Z) INCL EXEC ERRORS TIME W U AMP 18091.03148 0.10828D-05 0.11644D-05 0.81463D-06 -0.21678D-07 0.3 18172.33467 0.10463D-05 -0.17151D-05 -0.74013D-06 0.37560D-08 -0.5 18282.92216 -0.10467D-05 -0.94119D-06 -0.25449D-07 0.96849D-06 -0.3 18356.84140 0.14244D-05 0.51489D-06 0.68448D-08 -0.87480D-06 0.4 18422.72615 -0.14043D-05 -0.52671D-06 0.61931D-07 0.11332D-05 -0.4 18487.27727 0.10087D-05 0.81198D-06 -0.22931D-07 -0.81783D-06 0.3 18538.25068 0.20146D-05 0.58477D-06 0.58277D-08 -0.10693D-05 0.6 18595.83755 -0.15347D-05 -0.79755D-06 -0.12153D-07 0.11022D-05 -0.4 18659.63429 0.11438D-05 0.88857D-06 0.17977D-07 -0.95532D-06 0.3 18708.69774 0.16978D-05 0.67749D-06 0.56927D-07 -0.10123D-05 0.5 NUMBER OF MANOEUVRES = 10 FINAL STATISTICS FOR THE RUN AMOUNT OF DELTA V (ADIM) = 0.16113D-04 = 0.47993D+02 CM/S MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 49.0634, 110.5875 MIN AND MAX MANOEUVRES 0.12951D-05 (ADIM) = 0.38577D+01 CM/S 0.20978D-05 (AD Table 9.9

Output of program CONSIH. Sun-Barycenter problem, simplified model

Numerical Results of the 1

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400

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THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 1 1 1 1 1 1 1 1 1 0 0 0 0 , SPACECRAFT EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 18730 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN ADIM. SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD): 50. AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERRORS: 1.0 TYPE OF MANOEUVRES : (X,Y), NUMBER OF PREVIOUS CALLS TO THE RANDOM NUMBER GENE USED NOMINAL ORBIT = 2 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUTED AT PARAMETERS FOR THE QPO: TSTAR = -0.44800D+01 AZ = 0.72000D-01 BOUND FOR THE COEFFICIENTS TO BE RETAINED IN ANAQPO = 1.00E-06 SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: O.OOE+00 INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER OF TH -0.13715467129266D+09 Y = -0.59539453738101D+08 Z = 0.160534272844 XD= 0.98781063082802D+06 YD= -0.23638859703783D+07 ZD= 0.121301907120 W U AMP TIME APP.CONTROLS(X,Y,Z) INCL EXEC ERRORS 0.66117D-05 18051.25062 -0.98570D-05 -0.35974D-05 -0.12905D-07 -0.293 18086.11207 0.11271D-04 -0.11075D-04 -0.88701D-05 0.75151D-07 -0.329 18148.86464 0.68099D-05 -0.10918D-04 -0.69190D-05 -0.15270D-06 -0.325 18199.01975 -0.98302D-05 0.15615D-04 0.43479D-05 -0.70894D-07 0.465 18248.42507 0.10691D-04 -0.12757D-04 -0.74399D-05 -0.19165D-06 -0.379 18303.25557 -0.87770D-05 0.10736D-04 0.91761D-05 0.15112D-06 0.319 18350.32926 0.10585D-04 -0.16838D-04 -0.71077D-05 0.15658D-06 -0.501 18409.24612 0.88456D-05 -0.12803D-04 -0.42967D-05 0.21969D-06 -0.381 18461.12042 0.10633D-04 -0.10611D-04 -0.10797D-04 0.14711D-06 -0.316 18552.05090 -0.62180D-05 0.10339D-04 0.26431D-05 -0.10476D-06 0.307 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES = 10 AMOUNT OF DELTA V (ADIM) = 0 . 1 4 0 3 1 D - 0 3 = 0.41792D+03 CM/S MIN AND MAX TIME INTERVAL BETWEEN MANOEUVRES (IN JD) 3 4 . 8 6 1 4 , 9 0 . 9 3 0 5 MIN AND MAX MANOEUVRES 0.10493D-04 (ADIM) = 0.31254D+02 CM/S 0.18278D-04 (ADIM Table 9.10 Output of program CONSIM. Sun-Barycenter problem, simplified model. L\ equilibrium analytic nominal orbit and projection factors.

402

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9.3 9.3.1

The Feasibility of the Control Using Solar Radiation Pressure The Determination Feasibility

of the Maneuver

Interval.

Discussion

of

In Chapter 3, 3.3.2, a first approach to the control using solar radiation pressure in the RTBP model has been introduced. However, there, it was supposed that the radiation pressure was acting on the wings for a given time interval At and, from this, the value of s/m and the optimal orientation of the wings was determined. With a given spacecraft whose wings have a defined section, the s/m ratio is known, and the unknown variable is At. Furthermore if At is small the exponential instability of the unstable component becomes important. Using the same notation of 3.3.2 we have the condition uf

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The Feasibility of the Control Using Solar Radiation

403

Pressure

From this equation, At should be determined and, simultaneously, the best value of the orientation angle /?. The solution is obtained by iteration. We take a value of At and, using essentially the same method of 3.3.2, we determine /? and an estimated value, (s/m)e, for s/m. If s/m coincides with (s/m)e we are done. If not, the quotient can be used to estimate a new value, (At)m, of At as follows: (Ai)„ =

(S/M)e(m/S)At.

By iteration we solve the equation. We must remark that if u0 is too large for all the other parameters fixed, the equation determining t has no solution. This means that the force towards the Sun is too large to be canceled by the radiation pressure one. It is possible to give lower bounds of the critical value of uo for which a double root appears. In Chapter 3 we have also mentioned the "always towards the Sun" rule. To have a feasible radiation pressure control, the values of the unstable component u should never be negative. In this case, the orbit is escaping outwards the Sun and the radiation pressure cannot cancel this deviation unless, at some good points, very high s/m is used. Hence, u/ should be not reduced to zero, because the tracking errors can produce negative values of u and, later on, an escape. For the same reason, the initial condition, when the spacecraft is injected into the orbit, should be slightly shifted towards the Sun along the x-axis. 9.3.2

Simulations Discussion

of the Radiation

Pressure

Control.

Results

and

Program CONSIM allows also for radiation pressure maneuvers. In this case, the program requires the maximal value of s/m (i.e., with the wings unfolded) compared to the standard one (i.e. with the wings folded). It is required a bound for the time length of the radiation pressure maneuver, if it is desired to use an optimal /? angle or it should be taken as zero, and finally, the x shift used in the initial conditions. The simulation program only uses the a priori time interval, At, computed in 9.3.1 as a hint. The integration is done, with the value of j3 fixed for the full maneuver, with s/m corresponding to the unfolded wings till the unstable component reaches the value Uf. The differences between the true At and the a priori one rarely exceed 10%. For the radiation pressure maneuvers, it has been observed that the real orbit is systematically delayed with respect to the nominal one. This is due to the fact that the escaping part of the motion (which is always present) tends to slow down the motion in other directions. However, the rate of delay is small, of the order of 1 hour per year, at most. Table 9.11 gives a sample of results for Sun-Barycenter problem L\ case is

404

The On/off

Control Strategy: Simulations

and

Discussion

given, where 5 5 stands for the quotient ( s / m ) m a x i m a l / ( s / m ) s p a c e c r a f t and amPl = amp2 = 1. At desired (time between 2 maneuvers) has been set to 60 days in all cases. Now, Av, vmm, u m a x mean sum of active intervals, minimum and maximum active intervals, respectively (all of them in days). In /? the value 1 means that the angle /? has been determined in an optimal way, and 0 means that it has been taken null. The results show that, even with s/m = 2.5 x 0.01 = 0.025, the radiation pressure control is feasible. The value of uf has been taken equal to W£m, values less than 0.3D-6 are not recommended due to the effect of tracking errors. A sample of results is given in Tables 9.12 and 9.13. They are presented also with a simultaneous drawing (done by program DIBUIXET) of the nominal orbit and the real orbit (projection y,z). In the representation of the controlled orbit the initial and final points of a maneuver have been joined by a straight line to visualize them. It is displayed in Figure 9.4. Finally we can say that the approach taken for the radiation pressure maneuvers is essentially the on/off running, say, once every | months for one week. This can also be achieved with a low thrust device (for instance ionic thrusters). A change of 5 cm/s in 5 days is completely feasible. This could be applied with advantage to the translunar problem, where the requirements are higher and long time missions can be foreseen.

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THE MODEL OF THE SOLAR SYSTEM IS: 0 0 11 O O O O O O I O O O O O O , SPAC EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERROR TYPE OF MANOEUVRES : RADIATION PRESSURE, NUMBER OF PREVIOUS CALLS TO THE USED VALUE FOR THE QUOTIENT (CROSS SECTION WINGS)/(CROSS SECTION SPACECR MAXIMAL INTERVAL IN JD ALLOWED TO THIS CONTROL 20.00, RADIATION PRESSURE USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUT SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: 2.00E— INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER X = -0.13591791727460D+09 Y = -0.59435592857363D+08 XD = 0.98740691088587D+06 YD = -0.23624151332852D+07 UNSTABLE COMPONENT RADIATION PRESSURE MANOEUVRE INITIAL FINAL STARTED AT LASTING FOR (DAYS) 4.678488165525323 2.075613E-06 2.003620E-07 18041.39029025868 5.843251204751596 2.115241E-06 18090.62700409627 2.004108E-07 6.349214189700433 2.094798E-06 2.007604E-07 18155.89875583118 5.085643572155277 2.205028E-06 1.996810E-07 18207.48131100370 5.929884514119294 2.002060E-06 1.999997E-07 18260.57530688136 .745898974083048 1.986235E-06 2.000321E-07 18331.01160351493 .275834538201252 2.153925E-06 2.003017E-07 18397.42138380715 .466573821945985 2.015513E-06 1.990140E-07 18450.01303945794 .6894462422010Z9 2.142908E-06 1.987976E-07 18514.36686164792 .957209223081009 2.023522E-06 1.988800E-07 18575.72683107969 .160160199833626 2.024017E-06 2.008441E-07 18622.35345472694 6.924565653112495 2.027361E-06 1.993740E-07 18684.67663366586 6.440517378083314 2.040396E-06 1.995065E-07 18747.71361331227 .288817873336484 2.028806E-06 1.993086E-07 18805.27599690431 .153342961958970 1.938000E-06 1.988359E-07 18872.46867338025 .020324051713033 2.174298E-06 1.999349E-07 18934.98285743509 .219349932423029 2.001494E-06 2.002006E-07 18979.49439871031 . 120838401453966 2.144260E-06 1.999994E-07 19036.01144051052 .271037065484961 2.064357E-06 2.002985E-07 19084.84225980426 .181097507369486 2.162014E-06 1.992722E-07 19151.23800069861 8.326838450753257 2.143713E-06 1.983202E-07 19210.47406980241 5.226769167029488 2.130764E-06 1.998945E-07 19270.08170573782

19321.67358085754 5.533224595569664 2.048041E-06 2.000966E-07 19367.28418219810 4.804782231353329 2.013107E-06 2.003856E-07 19421.17258848335 6.294592806259061 2.109849E-06 1.999339E-07 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES = 2 5 , ACTING CONTROL T MIN AND MAX TIME INTERVAL BETWEEN END OF A MANOEUVRES AND START OF NEXT ON MIN AND MAX OF MANOEUVRE TIME (IN JD)= 0.41602D+01, 0.83268D+01 Table 9.12 Output of program CONSIM. Sun-Barycenter problem, simplified model. L\ equi pressure simulation.

THE MODEL OF THE SOLAR SYSTEM IS: 1 1 18 l l l l l l l l l O O O O , SPAC EARTH+MOON—SUN SYSTEM. EQUILIBRIUM POINT L 1. INITIAL AND FINAL TIME FOR THE SIMULATION (IN MJD SINCE 1950.0) 18000.00 MAX (MIN) AMP OF UNST. COMP. BEFORE UNCONDITIONAL (POSSIBLE) CONTROL (IN SUITABLE MINIMUM TIME INTERVAL BETWEEN TWO CONSECUTIVE MANOEUVRES (IN JD AMPLIFYING FACTORS FOR TRACKING, MANOEUVRES AND RADIATION PRESSURE ERROR TYPE OF MANOEUVRES : RADIATION PRESSURE, NUMBER OF PREVIOUS CALLS TO THE USED VALUE FOR THE QUOTIENT (CROSS SECTION WINGS)/(CROSS SECTION SPACECR MAXIMAL INTERVAL IN JD ALLOWED TO THIS CONTROL 20.00, RADIATION PRESSURE USED NOMINAL ORBIT = 3 (1)=HAL0 ; (2)=QP0 ; (3)=PS, NOMINAL POINT COMPUT SHIFT IN THE ADIMENSIONAL x VARIABLE FOR THE INITIAL CONDITIONS: 3.00E— INITIAL POS. AND VEL. IN ECLIPTIC COORDINATES CENTERED AT THE BARYCENTER X = -0.13716042461726D+09 Y = -0.59541228656201D+08 XD = 0.98750273443797D+06 YD = -0.23631672067518D+07 RADIATION PRESSURE MANOEUVRE UNSTABLE COMPONENT STARTED AT INITIAL LASTING FOR (DAYS) FINAL 18046.73990166762 9.344978755967986 3.181417E-06 2.970087E-07 11.06937566816669 18106.34503324694 3.006147E-06 3.002796E-07 10.80905573389100 18175.45055856023 3.060596E-06 3.001386E-07 7.647450102451330 3.004884E-07 18236.69652677594 3.079406E-06 10.91535710546532 3.026776E-07 18305.60058262063 3.194381E-06 8.641289833321935 18370.65350009813 3.016822E-06 3.011523E-07 8.956379619911786 3.085259E-06 3.000209E-07 18427.34880369345 6.090055467874663 2.202608E-06 2.983608E-07 18497.87872061086 9.865032138072365 3.028778E-06 3.028893E-07 18558.60553130183 8.194840455342273 3.103343E-06 2.995241E-07 18610.65710650033 10.40211550215736 3.017759E-07 18672.20842144599 3.286868E-06 FINAL STATISTICS FOR THE RUN : NUMBER OF MANOEUVRES 11, ACTING CONTROL MIN AND MAX TIME INTERVAL BETWEEN END OF A MANOEUVRES AND START OF NEXT MIN AND MAX OF MANOEUVRE TIME (IN JD)= 0.60901D+01, 0.11069D+02 Table 9.13

Output of program CONSIM. Sun-Barycenter problem. L\ equilibrium point. /3

Chapter 10

Other Cases and Further Simulations

In this chapter we present, for completeness, a few results concerning cases that have been left till the present. We refer to the motion near L3, L4 and L5 points in the Sun-Barycenter and Earth-Moon systems. For the L3 point, in the Sun-Barycenter problem, the perturbations due to the planets (including the Earth among them) in a short time interval (several years) are so small that we can consider it as a two-body problem: Sun and spacecraft. The triangular points for the Sun-Barycenter system have been studied with some detail. The analytic equations have been integrated in a first approximation and, by simulation, the results are compared with the numerical ones. The general result for the £3,4,5 points, in the Sun-Barycenter system, is that the motion behaves in a very stable way. For the L3 case in the Earth-Moon system, the results are worse. This is due to the strong influence of the Sun. A spacecraft left at the equilibrium point, without control, goes away from this point, relatively quickly. However, for the L4 and i 5 cases, it seems, according to the simulations, that there is a chance to find stable orbits. Here stability means that there are bounded orbits at a distance from L4 or L5 less than some 300000 km.

10.1 10.1.1

The L3 Case for the Sun-Barycenter Problem Computation

of Initial

Conditions

As it has been discussed in 2.3.3, the halo orbits in the L3 case cannot be obtained using analytic expansions of the same type of the ones used for L\, L^. This is due to the fact that, before the planar Lyapunov orbit bifurcates to halo orbits, the expansions are no longer convergent. However, a simple approach is enough to get orbits in the region opposite to the Earth with respect to the Sun, such that, when seen from the Earth, look like halo orbits around the Sun. The program L3IN produces initial conditions for a spacecraft such that, seen 409

410

Other Cases and Further

Simulations

from the Earth, stays roughly at a fixed angular distance from the Sun. This distance is an input data. The osculating motion of the Earth at a given epoch is considered. This is a planar elliptic motion. Then, there are several possibilities. Either the Earth and the spacecraft are simultaneously at the perihelion of the respective orbits, or when one of them is at perihelion, the other is at the aphelion. In any case, the semimajor axis of the spacecraft is taken equal to the semimajor axis of the Earth. Two more possibilities are left: the inclination of the spacecraft orbit with respect to the ecliptic can be chosen positive or negative. For a given case the program computes the initial conditions at a given epoch. 10.1.2

Results

of

Simulations

We ran two examples. Both were selected to have positive inclination and angular distance to the Sun equal to « 6°. The initial epoch was MJD50 = 18000 and the time spent was 10 years. One of the examples was taken of the type perihelionperihelion and the other perihelion-aphelion. The results, in Figure 10.1, display, in both cases, a slightly changing halo orbit. The orbits, in the adimensional coordinates, are seen as circles slowly going to the left, both in the perihelion-perihelion case and in the perihelion-aphelion one. This can be due to the slight lack of coherence of the model used for the solar system. A refined study can improve the initial conditions to cancel this slow motion, but this is not necessary. As a conclusion, in L3 halo orbits there is no need of station keeping for the Sun-Barycenter system. 10.2 10.2.1

The L3 Case for the Earth-Moon Problem Results

of a

Simulation

For this case the situation, compared to the one described in 10.1.1 is quite different. Due to the strong influence of the Sun, a motion starting at the equilibrium point goes away and falls towards the Moon after a while. Figures 10.2 and 10.3, show that starting at the L3 point of the Earth-Moon system on MJD50 18000, a deviation (mainly in the adimensional y coordinate) of 35 000 km is produced after one month. This is increased to some 75 000 km after two months. In each month an increasing loop is done. In the third month the spacecraft falls towards the Moon (roughly performing a circle centered at the Earth, that is, near the zero velocity curve of the RTBP). Then, the spacecraft becomes a Moon orbiter. At the end of the fourth month a close encounter with the Moon is detected.

The L3 Case for the Earth-Moon

411

Problem

SimiLATIOn OF NOTION IH ?HC SOIJW JVSTCT INITIAL COrOIITron FIHJL COfttlTIONS i K « . , K » ^ ; | ) Y t t ) - 0.ie?3399fi3Z339D+08 YFU> • t.l87»IW|ia37»+09 vie) . f . « M s s n s 3 T C » o t M V F I | > - ••*Ji5225S75fSSStS5 Y(3> - -0.9833BOC18M31D*07 VF 3> • - • • T f S f K J f l S g f S U ? V(4) - - 0 . 11701130447400*07 VF(4> * -0.11SS54eSBS07«0*f7 V S - 0 817S»IT9»SBO*07 VF<S> • 0.21BS870S1714*0*07 V ( i ! - •0.421SUfiIc»360»OC YFIO) - -0.488 7SU8 838108*06 FINAL T I K - • • 810S304 1450188*05

NEW TftNK 7 , IF MO TIWX-O. ELSE MITE TfMX

SIMULATION OF norion in n c SOLAR SVSTEA IHITlAt CONDITION FINAL CONDITIONS (KB.,Ka'ri»u> Y d ) - 0.14S183978eil50*OB VF(1. - •.1464E1S541S1SD+0B Vlii • 0.30900877838840+00 VF(3) * 0.887S80171443SD+OS V(3) > O.C8487f403*3780+OS VF<3> - 0.87S008073895IO+O? V<4) • -0.734E4S314491CB+0* VF<4) - -0.7O883C6B19US8+0C V<S» > 0.84a473S203802a*07 VFCSJ • l.2413»SMtt?5D+«? YfE> • -«.S3«8S?*08C4STB*0« VFCC) • -O.S3S4?48838378D*«C FINAL TINE - 0.8100849040*400*0*

HEW TFWX ? . IF HO THAX-O. ELSE URITE TfMX

0.877^8*00

L.1IIHHI o.8T^4D»«o

t

o.gayp+QB

|

o.97yD+oo

{

o.ioy.o*ii

|

}

0.08808*00

(

0.97yo*00 , 0.103,18*01

}

0.10y8*01

_ 0.108,80*01

Fig. 10.1 Projections of halo type orbits, around L3, in the Sun-Barycenter problem. In the top three figures, the Earth and the spacecraft are simultaneously at the perihelion of their respective orbits. In the bottom ones, the Earth is at the perihelion and the spacecraft at the aphelion.

412

Other Cases and Further

Simulations

SIMULATION OF NOTION IN THE SOLAR SYSTCT INITIAL CONDITION PINAL CONDITIONS < - • . i e 3 S « 4 2 3 4 t 3 6 S P + « 9 •.ilG791834«23eD+»5 VO> • .S83GG73EG»9e$I>+«5 YFC3) ' •.16461*3E195S7D*«7 V(4> • *.S8'«GC3E£1*43«D**S VFC4) ' V t S I - - » . 3 4 C l » 5 4 4 e » 4 S D + * 7 YFtSJ • - » . 1 8 7 1 3 1 4 i 6 5 1 8 7 D + « 7 •.45sa2Ifl»4aS3D**4 ViSl ' 0 . « 7 7 3 4 4 ? 4 K 6 4 D * » 4 YF(6> « FINAL TINE * •.lS*e»4«ll433«0»*S

NEU TIMX ? .

IF HO THAX-a, ELSE URITE TT1AX

•.nyp+oi

t

t . u y c M i i g,i»^5pt>t , o.iayp+ti , «.IM?C+«I

SIMULATION Of NOTION IN THE SOLAR SVSTEN INITIAL COHOITIOH FINAL CONDITIONS V U > - - t . l 3 B I S a s 6 S 3 I £ l D * 0 » VFtl> « -•.MSSTflSSjSMTO+y* V l » ) - - * . G « 1 S 9 1 8 C 3 4 7 4 1 D + G B VF • - • - 1 5 * » M " J S * | 6 D « M

¥13) - t.»36673EG«9*5D+t5 VF<3» - • •««« 33" » 7*5*5 *•*

Y14> • • . B * 4 6 C 3 S G i a 4 3 * D * « G W ( 4 J V(BI - - * . Z 4 G 1 0 » S 4 4 2 9 4 8 D + t 7 VF(S) • ¥<«) - •>GS7?344?42SS4D*«4 VF(S) FINAL TIME •.ll*G4>tt8a««S40*#S

•-35B388«3S31*1D*«7 -•.4aM*t*M74T7g*M -•.3S!8*6S4S6M1D*»4

NEU TTMX ? . I F NO TNAX-9. ELSE URITE THAX

«.P9yp+M , t - . i t y a m , t.ntj6P+»i

_

• .it«3D*ti

t.fya+ti

t

«.iayp*i

{

|

e.i»yp*»t

t

••ittta+oi

L.45S3S-G3

».t»]|io»ti - . a t y i n i , - . n y p - o i

Fig. 10.2 Coordinate projections of the simulations of motion starting at the equilibrium point L3 of the Earth-Moon system. In the top figures the time interval is of 20 days, and in the bottom ones of 2 months.

The Triangular Cases for the Sun-Barycenter

SIMULATION OF POTION IN THE SOLAR SVSTEN INITIAL COHOITION FINAL CONPITION9 < K « . . K ; ' J f * > „ „

v(i> * •.3«T49i6f?i«4iD««a VFU> • _

..

T VF(l> J YF<3) V t 4 » • -0.a«3«SSS3SSSTB»+«7 Y F ( 4 ) Y I S I - 0.S317OSS31S1930+OG V F ( G I YISI - «.fi8420»C7tlBlD+*4 YFISI FINAL TIflE • 0.1Sa4MltSlS31IH4S

Problem

413

~!

„

TTJl

».3»»?2i«KH?52*!S

- • . I43t373S9t94S0*09 ».S34i4B4t05S73D*OS - -•.241313Sa9376204t7 • •.Sa00S«433tB37D4« • -0.BS71S933731T704O4

NEU TIMX 7 , I F NO TnAX-O, ELSE URITE TfMX

-.13^0*01

(

-.734^Pt»S

(

-.1B1SB+—

t

B.43^SB»SS ,

,19-411

|

B.L0L6P»S1

* . 379,46*00

Fig. 10.3 Coordinate projections of the simulations of motion starting at the equilibrium point L 3 of the Earth-Moon system, during 240 days. After the third month the spacecraft becomes a Moon orbiter.

We note that in the case of the L3 point, for the Earth-Moon system, there is no need of halo orbits due to observational reasons, as it happens in the Sun case. In a similar way to what will be presented in 10.3 for the L 4 and L 5 points of the Sun-Barycenter system, but carried out to a higher order, it seems feasible to obtain a (unstable) quasi-periodic solution. The main perturbations come from the Sun and from the noncircular motion of the Moon around the Earth. We do not pursuit this approach, which is left open for further study.

10.3

The Triangular Cases for the Sun-Barycenter Problem

10.3.1

The Adopted

Model for the Motion

Near LA and L5

We have studied the residual accelerations in the neighborhood of the triangular points. As a result of several runs of program SSI, and being interested in a precision of 6 • 10~ 6 in the normalized acceleration, the following model has been obtained. a) Venus and Jupiter can be considered in circular planar motion. b) No other planets must be considered. c) The barycenter of the Earth-Moon can be considered instead of the Earth

414

Other Cases and Further Simulations and the Moon. d) The eccentricity of the Earth must be taken into account. The effect of the periodic terms in the motion of the Earth appears in longitude, radius vector and latitude. The list of terms that we must consider is given in Table 10.1.

In the Table, as is usual, nv, ns, nM, nj and ns refer to the mean motion of Venus, Earth, Mars, Jupiter and Saturn, respectively. Concerning the errors in the adopted model for _L4 and L5, we have computed the difference between the acceleration of the real solar model and the adopted one. The results have an error of 6.8 • 10~ 6 and 6.5 • 10~ 6 , in the normalized acceleration, for L4 and L$ respectively.

Mean motion nv — ns 2nv — nE 2nv — 2ns 2ny — 3ns 3nv —3nE 3nv — 4ns Mean motion nv nv — ns 2nv — nE 2nv — 2ns 2nv — 3ns 3nv — 3ns 3nv — 4ns 3nv —5n^ 2ns — 2nM 2ns - nM Mean motion 2nv — 3ns

Radius Mean motion 4nv —4nE 5ny — 4ns 5ny — 5ns Qnv — 6ns 2nE — 2nM

vector Mean motion 3ns — 4nM 2nv — nj nE - nj 3ns — 2nj 2nE - 2nj

Longitude Mean motion Mean motion 4nv — 4ns 3nE — 3nM 4nv — 5ns 2nE — 3nM 5nv — 5ns 3ns — 4nM 5ny — 7nE 4nE — hnM 6ny — 6nE 4ns — 6nM 7ny — 7ns 5ns — 7nM 8nv —&nE 2nj5 - nj nE - nj nE - nM 2nE - 2nj - nj ns - 2nj 3ns —2nj Latitude Mean motion Mean motion 5ny — 6n# 3nv — 4njg

Mean motion nE - 2nj 3nE —3nj 2nE - 3nj nE ns 2nE - 2nj

Mean motion 3ns —3nj 2nE - 3nj ns - 3nj 4ns - 4nj 3ns - 4nj 2nE - 4nj nE - ns 2nE - 2ns 2nE - ns

Mean motion 12ny — 8nj3

Table 10.1 Frequencies in the radius vector, longitude and latitude appearing in the simplified model around the triangular equilibrium points.

The Triangular Cases for the Sun-Barycenter

10.3.2

Development

of the Simplified

415

Problem

Equations

of

Motion

The equations of motion around the two triangular points, for the Sun-Barycenter problem, have been obtained in 5.3.3. Using the functions C(l), C ( 2 ) , . . . , C(27), we can write the equations of motion as x-2y-x

=

C(l)x + C(2)y + C{3)z + C{4)x + C(5)y + C(7) + K{1 -

HB

+ K J2

+ V§)(x ~

VA

ZZA)

'SA

+ ..., + C{15)y + C(16)i + C(17)

VSA

+ 02(x,y,z)

rs 3yA {xxA + yyA + zzA)

_ y-VA

'SA

A<E{B,V,J}

=

+ yVA +

C(ll)a; + C(12)t/ + C(13)z + C(U)i

z

rs 3XA (XXA

' SA

+ 02(x,y,z) =

s) - 1

XSA _ X-XA

Ae{B,V,J}

y-2x-y

3(xxs + yys)

x

'A

'SA

+ ...,

C(21)x + C(22)y + C{23)z + C(25)y + C(26)z + C{27) -K{\-

nB+n§)z ZSA 3 1 SA

A€{B,V,J}

+ 02(x,y,z)

_ z - zA ~3

3zA (xxA + yyA + zzA) +

;4 'SA

+

After some rearrangements and recombination of terms, in the above equations of motion, they become 3

3V3,,

n

.

n.

1

/

3 ,\

= - - ^ ^ l + -e2J

-^xT—(l-2fiB)y-2y

+ -

e(4n - nl - 3) - 3/j,§e I 1 + - e '

± \flen{\ - n) cos (M - - ) 9

53

3 - - u ss e 2 cos 2M - -—use cos 3M 4 16 b 1 F\

+

& A€{V,J]

' SA

cosM

416

y=F

Other Cases and Further

3^

(1 - 2nB)x - ?y + 2x

= , V3

Simulations

^ - l ^ h + ^ j g e(4n - n 2 - 3) - 3 ^ e ( 1 + - e

cosM

— en(l — n)cos (M — — J 9^3 „ „ , 53V3 , 2 =F ——[ige* cos 2M ^ yuge3 cos 3M ± —— E r — £ r =p _ 5 _ ^ r ^ V 3£s — -E#

.4G{y,j}

i; + z

=

'S.4

I (E« + E,) T ^

(E,

+ E,)

2

- ( n + 3)ecosMz - 2en sin Mz, where we have canceled the terms of the type 0(/u§E), 0(/zye, fj,je) and 0 ( e £ ) , as well as linear terms in a;, y, z on the right-hand side, which have small coefficients. Here e, n and M denote the eccentricity, mean motion and mean anomaly of the Earth-Moon barycenter, E means E r , £#, E,s and its derivatives. The upper (resp. lower) sign corresponds to L4 (resp. L5). Note that the above simplified equations of motion form a nonhomogeneous linear system of 6 differential equations of first order. The nonhomogeneous part of the linear equations of motion around L4 and L5, are periodic functions of time which have been developed in cosine Fourier expansions. Then, the equations of motion are (remember that the origin, in the present variables, is located at the libration point)

n+l

3 3\/3\ x - -x ± - T - ( l - 2fiB)y ~ 2jf ••

3>/3„

y =F — ( 1

v

9

- 2HB)X

„

--y

-0.1,1 - 2 J a*.l cos(aiM

+ ^,3),

i=2 m+1

-&i,i - ^

+ 2x

Ki cos(bit2t + bit3),

(10.1)

t=2

J+i

z+z

=

- c i , i - y ^ ajti cos(cji2t + Cjfi). i=2

Now, we explain how we have computed the coefficients a^i, a ^ , a;^, biti, bit2, bi,3, Cj,i, c i]2 and 01,3. First, we study the development, in cosine Fourier expansions, of the functions

417

The Triangular Cases for the Sun-Barycenter Problem

/ HA

%SA XA\ - ^r o - + ^r r 5A AJ

, / VSA . VA \ t and fiA \ - rr r - + ^r j , for \ SA A-

„ ,, r T / n A£{K,J}

According to our model of solar system, we can assume, in order to compute the expansions of the above functions, that the motion of Earth, Venus and Jupiter is circular and planar. Then we have , cos# sin# 0 \ / av cos($7y + wy + My) fsv = — I — sin# cos# 0 I ay sin(Hy + wy + My) as 0 0 1 / V 0 that is, C0S(# — fly — LOv — My

\

- sin(# — fly — toy — My

o

J

I

COs(ayt +

ay)

I = — I — sin(ay£ + ay) aB

V

o

where 9 is the longitude of the Earth-Moon barycenter, fly, coy and My are argument of the ascending node, argument of the perihelion and the mean anomaly of Venus; as and ay the semimajor axis of the Earth-Moon barycenter and of Venus, respectively. Since fy = fsv + fs and fs = (— | , T 2 ) >

fy-\

rv

yy

= I f t - fy

nave

^- cos(ayi + ay) — cos j - ^ sin(ayi + av) q= sin §

| = |

1

we

= ^T,{^y

Pr, (COS (ayt +ay ±D).

n>0

By derivation of fv

we obtain n-1

n>l

=

x

'

J^SiCOS ^ a y t + Qy ± - j j . i>0

Similar expressions can be obtained for Jupiter instead of Venus. The routine LEXPLOR computes the coefficients Sj for Venus and Jupiter. A sample of results is given in Table 10.2.

418

Other Cases and Further Simulations

s(2 s(3 s(4 s(5 s(6 s(7; s(8 s(9 s(10 s(ll s(12 s(13 s(14 s(15 s(16 s(17 s(18 s(19 s(20

Venus =0.7377689491340696D-05 =0.4195682483286185D-05 =0.1003371157944280D-05 =0.2244752031252652D-06 =0.4847166319658651D-07 =0.10238868923694 78D-07 =0.2130603136992104D-08 =0.4385585613667933D-09 =0.8952973596931421D-10 =0.1815906550483472D-10 =0.3663977376037354D-11 =0.7361168794680663D-12 =0.1473590001842287D-12 =0.2940909949146466D-13 =0.5853233392468205D-14 =0.1162432254155214D-14 =0.2298041848638178D-15 =0.4546663083126261D-16 =0.8498007296131527D-17 =0.1676413034476548D-17

s(2 s(3 s(4 s(5 s(6 8(7! s(8 s(9 s(10 s(ll s(12 s(13 s(14 s(15 s(16 s(17' s(18 s(19 s(20 s(21 s(22 s(23 s(24 s(25 s(26 s(27 s(28 s(29 s(30

Jupiter =0.1222234283950319D-04 =0.2170463780584770D-04 =0.1806676405456756D-04 =0.1456371258825714D-04 = 0.1150201839987549D-04 =0.8965896451668670D-05 =0.6914136921992453D-05 =0.5296286397454948D-05 =0.4027114837762758D-05 =0.3051735389344124D-05 =0.2297750761958548D-05 =0.1728519627275615D-05 =0.1291105471392541D-05 =0.9661185965714785D-06 =0.7160692874071754D-06 =0.5336821799517928D-06 =0.3919784779731748D-06 =0.2912267883387684D-06 =0.2112189940943199D-06 =0.1565371229477451D-06 =0.1113075333459836D-06 =0.8232435198163356D-07 =0.5659259948480184D-07 =0.4178688080523051D-07 =0.2699293148104656D-07 =0.1990356881180330D-07 =0.1131851174958686D-07 =0.8336241653031183D-08 =0.3411968807173104D-08 =0.2510535742061504D-08

Table 10.2 Output of program L45 (routine LEXPLOR). Fourier cosine coefficients, 3 s(i) of the functions fv and f - 3 Now, it is not difficult to compute the coefficients a's, b's and c's. These computations are done by the routine LPREPAR. For shortness we give in Table 10.3 the results for the z equation in the L\ case. Coefficient 2.4686114735776081E-07 1.2299118996729597E-06 1.6382655021745854E-07 9.9370832543645329E-07 3.1402824274623946E-07 2.5852168794679081E-06 1.4302938322835206E-06 1.9190022036514440E-05

Frequency 2.876475192455527 2.876475192455527 3.501982849697882 3.501982949697882 4.752998164182590 4.752998164182590 7.746215594629126 7.746215594629126

Phase 2.126749972413535 0.555953645618638 0.121819261344875 4.834208241729565 2.290423391267524 0.719627064472627 4.266947481183109 2.696151154388212

Table 10.3 Output of program L45 (routine LPREPAR). Fourier coefficients, frequencies and phases of the z equation of (10.1).

The Triangular Cases for the Sun-Barycenter

419

Problem

The routine LET checks the coefficients a's, b's and c's for L4 and L 5 . Given the initial conditions, x(t0), y(to), z(t0), x(t0), y(t0) and z(t0), LET computes the analytic acceleration through the routine LPROVEQ, and the real acceleration through the routine DERIV. After, it gives the differences, which are less than 6 • 10~ 6 in normalized coordinates. 10.3.3

Analytic

Solution.

On the Existence

of Almost

Tori

The analytic solution of system (10.1) is x(t)

= C\ cos(sii) + 5i sin(sii) + C 2 cos(s2£) + S2 sin(s 2 i)

+

12ai,i=F 4>/3(l - 2/i B )6i,i 9(l-(l-l/i n+l

a»,i

+ ^2 jr-Acos{aifit

B

)

2

)

+ a i)3 )

t 1 i=2 ' m+l ,

+ V] -pp- [-C cos(bU2t + bi,3) + D sm(biat + 6i>3)] i=2

»W

=

Di 2

'

Y, 1 2 1 9\

(2s i 5 i T^(l-2/x B )C i )cos(Sit) 4

t ? (»? + !) LV

/

2 Si Ci ± ^ ( 1 - 2iiB)Si j sin(sif)

+

4V5(1 -2/x B )ai,i 9(l-(l-l/xB)2)

4&I,I=F n+l

+ 5 ^ TT~ {Ecos(ait2t + a i]3 ) - Fsin(a i ) 2 i + a i]3 )l m+l ,

- 5 2 7r-Gcos(6i, 2 t + i)i,3), i=2

*'z H-l

z(£)

= C s c o s t + 5 3 s i n i - Ci - ^ - — ^ - j - C O S ( C J ) 2 * + Cj,3), i=2

where

^ = f a?l2 + 4) C

=

faj.,+3a?.,+

27\ 16;

27 (1 - 2/xs) 2 - 4a? 2 16

3V5(l-2/xB) ± g ( l - 2 M B ) 2 ± 6 2 , 2 T i ^ 2 + 3 6 2 , 2 +

£> = -26 i , 2 (64 2 + 3622 + ^ + y ( l - 2 M B ) 2 6 i , 2 + 86f,2,

|

420

Other Cases and Further Simulations

9 1 / 27 3 ^ ( 1 - 2fiB) ± - ( v1 - 2 ) 2 ± a\ j < + 3o? + MB 2 T 2 2 "64 16, 27 27 (1 - 2»By + 2 ( a*it2 + 3 < 2 + Qj,2 8a?

E ~F

,2

G

3

^ + 36?,2 + g)-^, 2 -g(l-2 M B ) 2

3 \ _ , 3^3

Di,i

= (<2 + - M ±

Di,2

=

E\2

=

±

3\/3

(1 - 2m)C

(1 - 2A»B)-B - 2 a i i 2 F ,

+ 2bii2D +[bt2

+

-\G

-1±[1-27MB(1-MB)]1/2

SI «2

The coefficients Cj, 5j z = 1,2,3 are free and they depend on the initial conditions. The routine LCOMPUTE, computes the analytic solution, (x(t), y(t), z(t), x(t), y(t), z{t)). The routine LS0LUTI0N writes explicitly the analytic solution. Thus, if we only write the coefficients which are greater than l.E-5 the analytic solution near L\ is given by the list of terms given in Table 10.4. Similarly we can obtain the analytic solution near L$. We see that the solution is the superposition of periodic terms with frequencies si, s2 and 1, which correspond to the free modes in the linear approach to the motion near the L4 and L5 points for the RTBP, and quasi-periodic terms. Those terms come from the perturbations due to the effect of the planets. The 3 free modes correspond to motion on a three-dimensional torus. We call almost tori the perturbations of these tori by the quasi-periodic terms. According to the KAM theory, in the spatial RTBP there are three-dimensional tori near 1/4,5. When the perturbations are included, the tori can be destroyed and only tori of lower dimension can subsist. We point out this fact here but we do not pursue the analysis in this work. 10.3.4

Simulations Starting at the Instantaneous or with Suitable Initial Conditions

Equilibrium

Point

The program giving the analytic solution described in 10.3.3 is able to compute initial conditions when values of C*, Si, i = 1,2,3 are assigned, and to compute the values of Ci and Si, when the initial conditions are given. Two sets of examples have been considered: Starting at the exact, L 4 (or L5), geometrical instantaneous equilibrium point, and starting at the initial conditions obtained when d = Si = 0, i = 1,2,3. We summarize the results.

The Triangular Cases for the Sun-Barycenter Problem

X(T)

=

ctantl +ctant2 +ctant3 +ctant4 -0.257890929838D- -4 -0.162713912265D- -3 +0.144588380152D--3 +0.156648647727D--3 -0.119074507409D- -3 -0.112376224887D- -3 -0.269408817410D- -3 -0.218523107982D- -3 +0.308131587355D--3 -0.729614680134D- -3 -0.578571510200D- -3 +0.340925570472D--3 -0.100232806832D- -3 +0.115808032605D--3 -0.255563976769D- -3 +0.309264127310D--3 -0.120358400279D- -3 +0.875228039603D--3 -0.804304334622D- -3 +0.138609640881D--3 -0.133486336651D- -3 +0.164243285280D--3 +0.115828892172D--3 -0.113052106058D- -3 +0.193835300769D--3 -0.279510771990D- -3 -0.307581085922D- -3 +0.531540034311D--3 -0.749341137132D- -3 -0.140500608863D- -3 +0.198071413660D--3 -0.339850224141D- -3 -0.105371611536D- -3 +0.120150457239D--3 -0.225305588439D- -3 +0.317625644251D--3 +0.102529054387D--3 +0.368130871609D--3 -0.518975979490D- -3 -0.338299554340D- -3 +0.476920997707D--3

*cos( 0.453020634230D-2 * sin(0.453020634230D-2 *cos( 0.999989738562D+0 * sin(0.999989738562D+0

* * * *

T) T) T) T)

*cos( 0.625507657242D+0 *cos( 0.125101531448D+1 *cos( 0.125101531448D+1 *cos( 0.251063093756D+0 *cos( 0.876570750998D+0 *cos( 0.936601677928D+0 *cos( 0.936601677928D+0 *cos( 0.633505427996D-1 *cos( 0.915657061883D+0 *cos( 0.915657061883D+0 *cos( 0.842951588453D-1 *cos( 0.966020989132D+0 *cos( 0.936601677928D+0 *cos( 0.915657061883D+0 *cos( 0.915657061883D+0 *cos( 0.625498935007D+0 *cos( 0.915660062559D+0 *cos( 0.915660062559D+0 *cos( 0.625507657242D+0 * sin(0.625507657242D+0 * sin(0.125I01531448D+1 *cos( 0.251063093756D+0 * sin(0.876570750998D+0 *cos( 0.936601677928D+0 * sin(0.936601677928D+0 *cos( 0.633505427996D-1 *cos( 0.915657061883D+0 * sin(0.915657061883D+0 0.915657061883D+0 *cos( 0.915657061883D+0 * sin( 0.842951588453D-1 *cos( 0.966020989132D+0 * sin( 0.936691677928D+0 * sin( 0.915657061883D+0 *cos( 0.915657061883D+0 * sin( 0.625498935007D+0 *cos( 0.915660062559D+0 *cos( 0.915660062559D+0 * sin( 0.915660062559D+0 *cos( 0.915660062559D+0 * sin(

*T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T *T

+ 0.110352965919D+1 + 0.220897935510D+1 + 0.377977568190D+1 + 0.516127357393D+1 + 0.312684679851D+1 + 0.579068566582D+1 + 0.107829668544D+1 + 0.297021567185D+1 + 0.388647501603D+1 + 0.545727134282D+1 + 0.909532712902D+0 + 0.337877020199D+1 + 0.578963846827D+1 + 0.231591151615D+1 + 0.388670784295D+1 - 0.308800462559D+1 + 0.386847780033D+1 + 0.386847780033D+1 + 0.110352965919D+1 + 0.110352965919D+1 + 0.220897935510D+1 + 0.516127357393D+1 + 0.312684679951D+1 + 0.579068566582D+1 + 0.579068566582D+1 + 0.297021567185D+1 + 0.388647501603D+1 + 0.388647501603D+1 + 0.545727134282D+1 + 0.545727134282D+1 0.909532712902D+0 + 0.337877020199D+1 + 0.578963846827D+1 + 0.388670784295D+1 + 0.388670784295D+1 + 0.308800462559D+1 - 0.229768147353D+1 + 0.229768147353D+1 + 0.229768147353D+1 + 0.229768147353D+1 +

421

Other Cases and Further Simulations

422

Y(T) = (-0.577341492422D+0 + (-0.402681335255D-2 + (-0.399704126040D+0 + (-0.615382186606D+0 -0.148893397763D-4 -0.102735432593D-3 +0.111910863077D-3 +0.161375619450D-3 +0.130895129022D-3 -0.177582022755D-3 +0.306884781894D-3 +0.432632307238D-3 +0.243354193045D-3 +0.343069751987D-3 -0.196212618391D-3 +0.107493307640D-3 +0.151539210938D-3 -0.130080242135D-3 -0.183381251209D-3 -0.368130871609D-3 -0.518975979490D-3 +0.338299554340D-3 +0.476920997707D-3 -0.121775162951D-3 +0.151968428598D-3 -0.242806672602D-3 +0.178533359541D-3 -0.649956399972D-3 +0.171801301945D-3 +0.198072786632D-3 +0.104372838750D-3 +0.275499115217D-3 -0.450144539150D-3 +0.413667281739D-3

" sin *cos * sin * sin *cos *cos * sin *cos * sin *cos *cos * sin *cos * sin *cos * sin *cos

"cos 'cos •"cos •"cos

ctantl + 0.402681335255D-2 * ctant2) * * cos(0.453020634230D-2 * T ctantl - 0.577341492422D+0 * ctant2) * * sin(0.453020634230D-2 * T ctant3 + 0.615382186606D+0 * ctant4) * * cos(0.999989738562D+0 * T ctant3 - 0.399704126040D+0 * ctant4) * * sin(0.999989738562D+0 * T

0.125101531448D+1 0.936601677928D+0 0.936601677928D+0 0.936601677928D+0 0.633505427996D-1 0.915657061883D+0 0.915657061883D+0 0.915657061823D+0 0.915657061883D+0 0.842951588453D-1 0.915657061883D+0 0.915657061883D+0 0.915657061883D+0 0.915657061883D+0 0.915660062559D+0 0.915660062559D+0 0.915660062559D+0 0.915660062559D+0 0.625507657242D+0 0.125101531448D+1 0.936601677928D+0 0.633505427996D-1 0.915657061883D+0 0.915657061883D+0 0.842951588453D-1 0.936601677928D+0 0.915657061883D+0 0.915660062559D+0 0.915660062SS9D+0

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

0.377977568190D+1 0.579068566582D+1 0.579068566582D+1 0.107829668544D+1 0.297021567185D+1 0.388647501603D+1 0.388647501603D+1 0.545727134282D+1 0.545727134282D+1 0.909532712902D+0 0.231591151615D+1 0.231591151615D+1 0.388670784295D+1 0.388670784295D+1 0.386847780033D+1 0.386847780033D+1 0.386847780033D+1 0.386847780033D+1 0.110352965919D+1 0.220897935510D+1 0 579068566582D+1 0.297021567185D+1 0.388647501603D+1 0.545727134282D+1 0.909532712902D+0 0.578963846827D+1 0.388670784295D+1 0.229768147353D+1 0.229768147353D+1

Z(T) = ctant5 * cos(T) + ctant6 * sin(T)

Table 10.4 Output of program L45 (routine LSOLUTION). Analytic solution around L4. Only terms with a |coefficient| > 1 0 - 5 were written in the Report (a total of 304 lines). Only periodic terms with a |coefncient| > 1 0 - 5 are reproduced here.

The Triangular Cases for the Sun-Barycenter

423

Problem

For the LA case, when the starting point has been taken at the equilibrium point, the analytic and numerical solutions agree quite well during the full time interval used in the simulation (4 years). In the adimensional system, the spacecraft is moving in the clockwise sense, with small loops, at a rate near 400000 km/year. The results are similar for L5, changing clockwise by counterclockwise. If the initial point is chosen such that the constants C,, Si be zero, the results are much better. For L4, the analytic and numerical results agree, quantitatively, for some half year, but they agree qualitatively for the full time interval (4 years). The separation from the equilibrium point, during the 4 years, has been less than 25000 km. No systematic departure from the equilibrium point has been found. The orbit consists of irregular loops around L4. For L 5 the agreement between numerical and analytic results is better and the separation from the equilibrium point has been less than 15000 km (also for 4 years). In Figures 10.4 to 10.8 we present a sample of results. As a final conclusion, a spacecraft placed near the triangular equilibrium points L4 and L5 of the Sun-Barycenter system, does not require station keeping if suitable initial conditions are used. The angular variation of the position, with respect to the Sun, has an amplitude of the order of 10".

MMN- l . t M M W M WWt- «.79SSUB-«2 Vtmt>-«.S2f44BD-K U

I.U

• .«•

•••<

».«•

•.»

t.tl

».«1

l.«l

Vnuc- t.tSMC2D-»2

Fig. 10.4 (x,y) projection of the numerical and analytic orbits. All the components of the initial condition where taken equal to zero. The time interval of the simulation is 4 years.

424

Other Cases and Further

1

i—i—i—r

i

i

odvi

Simulations

— I "

1

1

i

1

(>t,Y)

-

_e.ee

-

.e.ee _e.ee

-

_e.M~~^

-

.e.ee~~^

-

_«.«•

-

.e.ee

-

.«.««

-

_e.ee

-

_e.ee

-

_e.ee

.e.M

} -

_j.ee

_

_-8.81 _-».»l

1?•",

*•" ,

?•« ,

t-« i

V'1

_e.ee

\

_-e.ei --e.ei e.ee,

.

? J

MOT-RICA- ORJIT 1 .e.ee

1

1

1

1

-

?. e e .

i

f. . i ,

?•"

MMlVTICAl OMIT

1

1

1

tv'.Z)

l

i

1

i

1

i

i •

T

(V.Z)

.e.M

-

_e.ee

-

_e.ee

-

_e.ee

-

_i.ee

-

_e.ee

-

- an

.e.ee

e.ee

-a.ee

_

.e.ee

_

_e.ee

-

.e.ee

-

-e.ee

-

_e.ee

-

_e.ee e.ee ,

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i _e.ee

i

1

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i

i

r

i

.

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,

i

e.ee

i

rx'.Z)

.e.ee e.ee, i

!• ee,

?•M , i

i

1

i

•"• r

T-ee,

e.ee p

i

(X',Z)

-

_e.ee

-

.e.ee

-

_e.ee

-

_e.ee

-

. e.ee

-

.e.ee

-

.e.ee

-

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_

.e.ee

-

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e.M

-

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_e.ee

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f.N.

™

e.w ,

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NUH-RIC«l OR9IT

e.ei ,

?•»'

_e.*e T'Hi

t*±JU

• ee,

?

81

l

e.ei

ANALYTICAL 0R1IT

Fig. 10.5 Coordinate projections of the numerical and analytic orbits. All the components of the initial condition are taken equal to zero. The time interval of the simulation is 4 years.

The Triangular Cases for the Sun-Bary center

l—i-

i—i

i

1

r-j^

i

Problem

-

1

1

425

1

1

1

1

IX'.V)

.....

-

.....

\

..... /

^ 5 v

..... ( ...M

\\ \

...M

\

\ \ \

/

.....

N.

\

"

^ /

| -

^

...M

/

-

...M

t.M

•

H*.

t««.

!•«•

!•»,

t

f.M,

.M,

1

f.M,

f.M,

f.M

MMlVTICAl OMIT

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426

Other Cases and Further

Simulations

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The Triangular Cases for the Sun-Bary center

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428

10.4

10.4.1

Other Cases and Further

Simulations

Simulations for the Triangular Cases in the Earth—Moon Problem Results

of the

Simulations

As a sample of what happens in the neighborhood of the L4 and L5 points of the Earth-Moon system, we have done simulations starting at the instantaneous equilibrium point. The initial epoch was chosen MJD50 = 18 000, both for L4 and for L$. Adimensional coordinates, centered at the equilibrium point, are used. Starting at L5, we left this point and a kind of clockwise spiral is started, when we consider the (x, y) projection. The revolutions have a period close to one month. The spiral is more regular for the simulation started at L4, but the general pattern is the same. The z-component is small. At the epoch 18500, some 17 turns roughly around the equilibrium point are done in both cases. Till this epoch, the maximal departures from the equilibrium point are, roughly, —350000 km and +170000 km in the x-axis and ±150000 km in the y-axis. For the more regular motion, in the case L4, the size of the different "revolutions" increases smoothly with time. In both cases, the z variable is less than 40 000 km, in absolute value, for all the time interval. For the I/5 case, soon after the epoch 18 500, large departures are produced. First, some large loops enclosing the Earth and the Moon are performed. These loops are roughly symmetric with respect to the axis passing through the Earth and the Moon and they are near the (x, ?/)-plane. The shape is roughly circular with a diameter of 1.5 • 106 km. Then, the motion approaches a periodic orbit with four outer loops, but after a little bit more than a revolution of this periodic orbit, a close encounter with the Earth is produced. This encounter introduces a large z-component in the velocity. The subsequent oscillations have z-component which reaches the values ±240000 km. The (x, y) projection of these oscillations looks like a periodic orbit roughly opposite to the Earth with respect to the Moon. Then the simulation is stopped at the epoch 19000. For the L 4 case the behavior is quite different. Soon after 500 days of simulation the x distance to the equilibrium point reaches a maximum value, being equal to 1 adimensional unit. Then, the size of the loops decreases. Near epoch 18700 a new type of motion appears. It looks like quasi-periodic because the (x, y) projection performs almost a periodic orbit, while (x, z) and (y, z) projections display a clear Lissajous behavior. As a conclusion, we can say that an instability is produced, but it seems that it is not so strong that prevents from the existence of regular orbits (quasi-periodic orbits lying on tori). Of course, the sources of the instability are the effect of the Sun and the noncircular motion of the Moon around the Earth.

Simulations

for the Triangular Cases in the Earth-Moon

Problem

429

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430

Other Cases and Further

Simulations

Fig. 10.10 Numerical simulations of motion around the triangular equilibrium point L5 in the Earth-Moon system. The initial conditions are taken at the equilibrium point. The time interval of the simulation is of 1000 days. Comparing this Figure with Figure 10.9, we can see that larger loops, enclosing the Earth and the Moon, have appeared.

10.5

References [1] R. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin, 1978. [2] American Ephemeris and Nautical Almanac. U.S. Government Printing Office. Washington D.C., 1984. [3] V.I. Arnold. Les Methodes Mathematiques de la Mecdnique Classique. Mir, 1967. [4] E.W. Brown. "Motion of the Moon". Memoirs of the Royal Astronomical Society, 57, 129-145, 1905. [5] Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. Her Majesty's Stationary Office, London, 1961. [6] R.W. Farquhar. "The control and use of libration point satellites". Technical Report TR R346, NASA, 1970. [7] D.L. Richardson. "A note on the Lagrangian formulation for motion about the collinear points". Celestial Mechanics, 22 (3), 231-235, 1980.

References

431

[8] B.D. Tapley and J.M. Lewallen. "Solar influence on satellite motion near the stable Earth-Moon libration points". AIAA Journal, 2 (4), 728-732, 1964. [9] B.D. Tapley and B.E. Schultz. "Numerical studies of solar influenced particle motion near triangular Earth-Moon libration points". In G.E.O. Giacaglia, editor, Periodic Orbits, Stability and Resonances, pages 128-142, 1970.

Chapter 11

Summary and Outlook

11.1

Summary of the Achieved Results

11.1.1

Concerning

the Halo

Orbits

In this work the halo orbits have been studied analytically and numerically. We have obtained the following results: a) The equations of motion, in the vicinity of a collinear point, are expanded in a form which is suitable to obtain the solutions recursively. b) An algorithm has been given to obtain the series which give the periodic solutions, called halo orbits, up to any order. The solution is convergent provided the ratio of distances, from points of the orbit to the equilibrium point (Lj or L-z) and to the secondary, is less than 1. c) The solutions are written as trigonometric polynomials in an angular variable, with coefficients which are, in turn, polynomials in two variables a and /?. Those variables are related to the x- and ^-amplitudes. They should satisfy a condition, which follows from the Lindstedt-Poincare method, that is also computed. The coefficients of the polynomials in a, ft are functions which only depend on the mass parameter n and the index (1 or 2) of the equilibrium point. d) A program has been produced which computes, for a given value of /x, all the coefficients which appear in the solution, the relation between a and /? and also the frequency of the angular variable. e) The full family of halo orbits in the L\ and L2 cases for the Sun-Barycenter and the Earth-Moon problems has been computed numerically. As a complementary check, the computations have also been carried out for Hill's problem. Some of the orbits have been continued to the elliptic RTBP. f) The halo family for L\ is born from a bifurcation of a Lyapunov planar orbit. Then the size increases. For moderate sizes, the agreement with the analytic results is excellent. The family has orbits which come very close to the secondary. For practical purposes, these orbits are unsuitable, 433

434

Summary

g)

h)

i)

j)

k)

1)

m)

and Outlook

because collisions with the Earth or Moon surface, respectively, can occur. At some point in the family, the orbits are almost contained in a vertical plane through the secondary. Later on they increase very much the distance to the secondary and they die in a planar periodic orbit encircling both primaries. A symmetric family exists. The halo family for Li is also born from a bifurcation of a planar Lyapunov orbit, but the final expanding size phase of the family is replaced by a decrease in size such that the orbit shrinks to the secondary. A symmetric family also exists. For Hill's case, the families of halo orbits associated to L\ and Li are symmetric. They end in a common orbit, which belongs to a family of vertical collision orbits. A numerical study of the variational equations associated to a halo orbit has been done. Floquet theory has been used to identify the behavior of the solutions of these variational equations. Then, the Floquet modes have been computed by means of their Fourier series. For halo orbits of moderate size, there is an unstable direction. Hence there exists a two-dimensional invariant unstable manifold. This manifold has been computed, starting with a linear local approximation, to see whether or not the second order variational equations are required. The answer is negative. We also point out the possibility of using the stable manifold for transfer. The parameters to perform the station keeping, and that will be described in 11.1.3, are also numerically computed for the halo orbits. From this result we have shown that the (x, y) control is always the optimal one, followed, in efficiency, by the (x) control and the (y) control. The maneuvers using only the z variable are too expensive and controllability is lost at several points. It has also been shown that, provided operational constraints are nonactive, it is never recommended to wait for a future time to execute the maneuvers. The computations of the solution of the variational equations, Floquet modes and parameters related to the control have also been carried out, analytically, for the halo orbits. In this way, the computations are done once for all the halo orbits of moderate size. A good agreement is found for most of the computed magnitudes. For some of the Floquet modes, the agreement is worse because of resonance problems. However, it has been shown that this fact does not affect the parameters used in the control. To perform the symbolic manipulation of power series and trigonometric series, which appear in the solution of the equations and in the variational equations, a package of routines has been developed. They are especially suited for the problem at hand and they prove to be quite efficient.

Summary

11.1.2

Concerning

of the Achieved

the Quasi-periodic

435

Results

Nominal

Orbit

The RTBP is only an approximation to the real problem. In this last one, the searched solutions are quasi-periodic instead of periodic. The main results obtained for those quasi-periodic solutions are: a) The real problem is written in a form that can be seen as a perturbation of the previous equations of motion (the equations of the RTBP). To do this, we have introduced an osculating plane of motion of the two primaries and time-dependent changes of variables, that make the reduction easy. The method can be applied to any primary-secondary couple in the solar system. b) Having in mind that we wish analytic solutions for the quasi-periodic motion and that these solutions will be obtained in a recurrent way as in the halo case, we have given a way to assign weights to the different terms which appear in the equations of motion. c) The residual accelerations obtained when a perturbing body is introduced, or some of its elements are changed, has allowed to detect the influence of each of the terms in the equations, and to select a simplified model to perform the analytic computations. d) A program for the simulation of the solar system has been implemented. Several checks are possible. Graphical output is also possible. e) The equations of motion, in a system of reference adapted to the equilibrium point, called normalized system, are given for all the cases Li, i = 1 , . . . , 5, in the Earth-Moon and Sun-Barycenter problems. These equations are generalizations of the restricted problem equations, such that they reduce to the former ones when the perturbations are skipped. These equations have been tested against direct numerical computation of the acceleration using Newton's law. f) The right form of the equations, for the analytic computation of solutions, requires that the perturbing terms be written as power series in the (normalized) coordinates of the spacecraft. The coefficients of these series are functions of time which depend on the coordinates of the perturbing bodies. Those previous expansions are done and the mentioned functions of time are given explicitly. g) For the functions of time just described, two possibilities have been used: Expansion in terms of elements of the orbits and the related anomalies, and Fourier analysis. For most of the perturbations, the functions have been analyzed using an FFT algorithm. Then, the frequencies are approximately located. They are identified as linear combinations of the motions of several anomalies and the Fourier coefficients with the right frequencies have been obtained. A filtering to skip too small coefficients or terms with too large frequency

436

Summary

and Outlook

(giving small amplitude rapidly oscillating terms in the solution) has been used to reduce the number of terms to something feasible. h) The equations of motion, as they are available at this point, have been transformed to a suitable form by introducing the expansions of functions of the spacecraft coordinates x,y,z in terms of Legendre polynomials. A program (EQUAP) produces the final equations to be solved, where the right-hand side terms are products of powers of x, y and z (eventually x',y' or z' can appear) by trigonometric functions af(bt + c), where b is a frequency, c the related phase and a the amplitude. The function / stands for sine or cosine. i) A Lindstedt-Poincare device has been used to obtain the quasi-periodic solution in a recurrent way. Explicit formulas are given for the components of the solution, the relationship between the x- and ^-amplitudes and the frequency. A program QPO, has been implemented to do this task by manipulation of formal series. It has been successfully run in the Sun-Barycenter problem L\ and Li cases. The nominal orbit, produced in this way, is not far (less than 100 km) from a real quasi-periodic orbit. In fact, a full family of quasi-periodic orbits, depending on two parameters is obtained. Those parameters are determined from the initial conditions (just two conditions determine the orbit if we wish a qpo one). The procedure is easily extended to higher precision if large computer time and memory are available. j) For this work we have taken the decision, to obtain the best possible nominal orbit. This is an opposite approach to the one followed in the ISEE-C mission, where the nominal orbit was only a hint to the path to be followed. With this in mind, we have improved numerically the qpo orbit obtained analytically. Due to the large instabilities for long time intervals, a parallel shooting algorithm is used. Several shooting approaches are introduced and a couple of them are implemented to perform this refinement. The final nominal orbit consists of several arcs with matching errors less than 1.5 km in position, 0.3 mm/s in velocity and 5 s in time. Each of the arcs is a numerical solution of the real solar system. 11.1.3

Concerning

Station

Keeping

For station keeping near halo orbits in the Sun-Barycenter problem, L\ and Li cases, in the real solar system, a method has been proposed based on very simple geometrical ideas. The main achievements are: a) The projection factors and the unitary controls are introduced. Given a nominal path and a point estimated by tracking, the unstable component is computed using the projection factors. If it is too large (see e)) then a maneuver is executed to change the x- and ^-components of the velocity in

Summary

b)

c)

d)

e)

of the Achieved

Results

437

a specified amount. This amount is computed in order that the maneuver be done using the less possible amount of fuel. First, a numerical exploration of the projection factors and unitary controls for the halo orbits has been done. It is learned that the optimal maneuvers should be executed on the (x,y)-p\ane, and that, except for operational constrains, it is never recommended to delay a maneuver. Using (x,y), (x) or (y) maneuvers, the controllability is guaranteed at any point in the orbit. The gains are very similar in the (x,y) and (x) cases (i.e., the (x) case is roughly 16% worse than the (x,y) one). The (y) control is worse, its gain being, on the average, less than 1/2 of the optimal one. The variations along the orbit of the ratio of gains (y)/(x, y) ranges from 1/4 to 3/4. Using the analytic solution of the variational equations, the normalized projection factors are obtained in an analytic way. In fact, they are expressed as quotients of functions given by Fourier series, whose coefficients are power series, times the square root of one of such functions. From them, the unitary controls follow. With this analytic expression in hand, and together with the analytic qpo, it is possible to undertake station keeping in a purely analytic way. The results of simulations show that, on the average, less than 1 cm/s per day is enough. Of course, the maneuvers should not be done every day but at intervals of 2.5-3 weeks at least. The precise dates are detected by the simulation program (see f)) and depend on the random and systematic errors introduced in the model, the random tracking errors and the errors in the execution of the maneuvers. This figure is not quite good compared with the forthcoming results, but it compares favorably with the ISEE-C mission, where more than double was used (in the simulations and in the real world). A refined set of projection factors along the numerical nominal orbit, have also been obtained numerically. This step is done when the parallel shooting results are available, and several starting bases to integrate the variational equations are computed. The computation of these numerical normalized projection factors is done simultaneously with the nominal orbit. The following strategy has been adopted for the simulations of the station keeping: (1) For a given time, a point is obtained by tracking. In the simulation this point has been taken as the point obtained by integration of the equations of motion plus random tracking errors. (2) For that time, the nominal point is computed. Optionally, this point can be taken as the point in the nominal path at minimum (local) distance from the current point. This modified nominal point corresponds to a time different from the current time, but in the simulations with numerical nominal orbit and projection factors, using on/off control, the time difference is less than 15 minutes in time intervals of 4 years.

Summary

and Outlook

This is increased to a few hours in the station keeping using radiation pressure. (In this last case there is a systematic bias between both times.) (3) The residue vector, observed-nominal, is computed. Its unstable component is obtained by doing the inner product with the projection factors. (4) If the unstable component is less than some value (lower bound), the maneuvers have no sense. The unstable component can be due, exclusively, to tracking errors. Some upper bound of the unstable component has been given, such that an unconditional maneuver is done if the current value is greater than the bound. Theoretically, it is shown that the upper bound should be, roughly e = exp(l) times greater than the lower bound. Some freedom is available. Doubling the upper bound can increase the fuel consumption by 20% and to separate the maneuvers some additional two weeks. Using the above strategy, lots of runs of simulation have been done. Under the standard assumptions on the size of the errors, as specified by the ESA, some 20 cm/s per year are enough. Under very pessimistic hypothesis, it is kept below 50 cm/s per year. The solar radiation pressure station keeping has also been studied. It has been shown that, letting aside technical difficulties for the implementation, it is feasible. Values of 0.05 m 2 /kg, the quotient of the section of the spacecraft with suitable moving wings, as it is seen from the Sun, by the mass of the spacecraft, are enough. Hence for a spacecraft of 1000 kg a surface of 50 m 2 would be sufficient. There are some words of caution concerning station keeping: The unstable component should be always directed towards the Sun. Should the unstable component be directed against it, the controllability is lost. Then the spacecraft must be injected in a slightly unstable orbit and the maneuvers must not cancel the full unstable component. Maneuvers lasting for 1 week every two months are typical results. We remark here that for the radiation pressure we have used successfully the same approach as that for the on/off maneuvers. This can be extended to low thrust acting for moderate time intervals. For instance 5 cm/s in maneuvers lasting for 5 days is more than available for ionic thrusters. As complementary results, simulations are done for the L3, L4, L5, SunBarycenter problem, showing that a right election of the initial conditions allows to skip the station keeping. The L 3 , L 4 , L5 cases for the Earth-Moon problem, show mild instability and it should be easy to keep the spacecraft under control.

Possible Extensions

11.2

439

of the Work

Possible Extensions of t h e W o r k

During the realization of this work several problems have presented in a natural way that can be useful to investigate. 11.2.1

The Use of Better

Models for the Solar

System

The model available to us, has been an analytic model based on part of Newcomb's and Brown's developments. The tests against the published ephemeris show an acceptable agreement for the first study. However, the results would be much closer to the real life ones, if the model of the solar system is improved. We suggest to use the JPL ephemeris to modify the computation of the real nominal orbit and projection factors. 11.2.2

The Translunar

Problem

Using the Real Solar

System

A point where the difficulties are increased and that has scientific interest is the problem of a translunar station. It should be placed near a halo orbit around the L2 point of the Earth-Moon system. The orbit can be seen permanently from the hidden part of the Moon and also from the Earth, if the size of the orbit is large enough. Therefore, it can be used to establish a permanent link between the Earth and a lunar basis placed in the hidden part. This is an ideal site to perform radio-electrical observations of the Universe, due to the lack of noise coming from the Earth. The following difficulties are detected concerning this problem: a) The halo orbits of moderate size are highly unstable. The largest eigenvalue of the monodromy matrix is of the order of 103 and the period less than two weeks. This means that a very good nominal orbit is required and that, to prevent from too frequent maneuvers, the total amount of fuel is relatively large. b) To get the nominal orbit analytically a large amount of terms of Brown's theory is required (it is not enough to take the terms given by P. R. Escobal). However, the worse thing is that in the computation of the quasi-periodic solutions there appear strong resonances giving rise to small divisors. c) The numerical refinement of the quasi-periodic orbit and the design of the control strategy require a good enough starting initial approximation to the qpo to get convergence. Despite what has been mentioned, it seems that a nice solution is feasible. For the Sun-Barycenter system, even under pessimistic hypothesis, the rate of fuel consumption for station keeping is less than 50 cm/s per year. Even taking into account that the time scales are roughly 1:13, that is, things are 13 times faster in

440

Summary

and Outlook

the Earth-Moon system than in the Sun-Barycenter, the cost of station keeping would be of 6 m/s per year, which is clearly feasible. 11.2.3

Stable Manifolds

and the Transfer

Problem

For the halo orbits of moderate size there are two-dimensional unstable and stable manifolds. It seems appealing to study whether the stable manifolds could be used for the transfer problem. In the RTBP, it has been shown that increasing the zamplitude, there are orbits coming close to the Earth and belonging to the stable manifold. If a spacecraft is injected in such an orbit, no more maneuvers will be necessary, except to correct small errors in the execution of previous maneuvers.

11.3

Theoretical Problems

Besides the possible extensions mentioned in 11.2, several other problems of a more theoretical character present themselves. We list some of them. 11.3.1

The Small Divisors

Problem for Quasi-periodic

Orbits

The exact resonant terms in the analytic computation of the qpo have been studied using the Lindstedt-Poincare method. However, as it is well-known, the dangerous terms are the near resonant ones. Those terms are associated to small divisors problems, typical in Celestial Mechanics. It seems interesting to work out a procedure that allows to solve analytically the equations of motion in an approximate way and that, for a given time interval, not too large, has provable bounds for the errors. 11.3.2

The Large Size Orbits in the Halo Family: Analytic and Perturbation to a Quasi-periodic Orbit

Study

It has been shown that halo orbits appear such that they are stable. At the beginning of the halo family (starting at the Lyapunov orbit) the instability decreases when the size of the orbit increases. Hence, it seems interesting to explore the use of more stable orbits, especially in the case of the Li point of the Earth-Moon system as we proposed in 11.2.2. However, the bad thing is that the available analytic theory for the halo orbit breaks down, before stable orbits are reached, because of convergence problems. Anyway, it has been found that, using suitable scalings, the halo orbits of large size have projections quite similar to the convergent ones. This opens the possibility of existence of reference systems more adapted to the problem, in which the large orbits could be obtained analytically. Then, it would be possible to extend the orbits to the quasi-periodic case and

Theoretical

Problems

441

see how the stability properties are affected. 11.3.3

The Floquet Modes for a Quasi-periodic for an Analytic Approach

Orbit.

Searching

The Floquet modes (and, then, the projection factors and unitary controls) have been obtained analytically only for the RTBP. This has been proved to be enough for our requirements of precision in the controls. That is, the difference between the values computed analytically for the RTBP and numerically for the qpo is of the order of the expected errors in the execution of the maneuvers. However, it is an interesting problem to give methods to obtain the solution of the variational equations in the quasi-periodic case. A suitable generalization of the Floquet modes should be introduced. A first problem to this end is the improvement of the expansions related to the last two Floquet modes for the RTBP. It seems that the near resonance can be overcome with a good redefinition of the last Floquet modes.

Acknowledgments

For the team that has worked in this study it is a pleasure to express their gratitude to the technical staff of ESOC. Special mention should be done of the late Dr. E. A. Roth and of Dr. J. Rodriguez Canabal. We acknowledge the good disposition to use all the available facilities of the Computing Center of the Universitat Autononoma de Barcelona, as well as the computer facilities of the Institut de Cibernetica of the Universitat Politecnica de Catalunya.

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