Dynamics and Mission Design Near Libration Points, Vol. II: Fundamentals: The Case of Triangular Libration Points

r

orld Scientific Monograph Series Mathematics - Vol. 3

Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular Libration Points

T World Scientific

Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular 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. 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. 6:

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

Forthcoming 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

World Scientific Monograph Series in Mathematics - Vol. 3

Dynamics and Mission Design Near Libration Points Vol. II Fundamentals: The Case of Triangular Libration Points

G. Gomez & C. Simo Departament de Matemdtica Aplicada i Analisi Universitat de Barcelona, Spain

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

© World Scientific III

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

Library of Congress Cataloging-in-Publication Data Dynamics and mission design near libration points / Gerard G6mez ... [et al.]. p. cm. — (World Scientific monograph series in mathematics ; vol. 3) Includes bibliographical references. Contents: - v. 2. Fundamentals : the case of triangular libration points ISBN 9810242743 (v. 2 : alk. paper) 1. Three-body problem. 2. Lagrangian points. I. G6mez, Gerard. II. World Scientific monograph series in mathematics ; v. 3. QB362.T5 D96 2000 52r.3--dc21

00-043633

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

Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. Thisbook, 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.

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

Printed in Singapore by Uto-Print

Preface

It is well-known that the restricted three-body problem has triangular equilibrium points. Those points are linearly stable for values of the mass parameter, /i, below the Routh's critical value (ii. It is also known that in the spatial case it is nonlinearly stable, not for all the initial conditions in a neighborhood of the equilibrium points £4, I/5 but for a set of relatively big measure. The fraction of stable motions tends to 1 when the size of the neighborhood tends to 0, for almost all fi £ (0,^i). This follows from the celebrated Kolmogorov-Arnold-Moser theorem. In fact there are neighborhoods of computable size for which one obtains "practical stability" in the sense that the massless particle remains close to the equilibrium point for a big time interval (some millions of years, say). The question is which part of this stability subsists when the idealized RTBP is substituted by the Earth-Moon system with its real motion and under the very strong influence of the Sun and the milder perturbations due to planets, solar radiation pressure, no spherical shape of the Earth and Moon, etc. According to the literature, what has been done in the problem follows two approaches: a) Numerical simulations of more or less accurate models of the real solar system. Usually the starting point is taken at one of the equilibrium points Li, L5. The results are slightly confusing. Depending on the initial epoch chosen, the orbit escapes in a few months or behaves according to the pattern that we proceed to describe. First the particle spirals from the equilibrium point outwards until it reaches a size of the order of magnitude of the Earth-Moon distance. Then the particle spirals inwards going close again to the equilibrium point. The behavior repeats itself several times or, eventually, escapes after some of these big oscillations, when a closed encounter with one of the primaries is produced. b) Study of periodic or quasi-periodic orbits of some much simpler problem. This can be the bicircular model or a coherent system close to the bicircular one and still periodic or a Hamiltonian system retaining a few leading terms

VI

Preface

of the equations. In this case, the methods of perturbation theory, mainly those based on Lie series, lead to much simpler auxiliary Hamiltonians that can be studied analytically. The other cases can be studied in turn numerically or semianalytically. The results are again confusing: small changes in the approach produce big changes in the size of the periodic orbits or they even disappear. This is a consequence of the lack of convergence of the methods used and of the sensitivity to some resonances. The concrete questions that are studied in this book are: a) Is there some orbit of the real solar system which looks like the periodic orbits of the previous item b) ? That is, are there orbits performing revolutions around L4 covering, eventually a thick strip ? Furthermore we would be pleased if those orbits be quasi-periodic, at least if the motion of the bodies of the solar system is assumed to be quasi-periodic with respect to time. The present knowledge of the motion of the main bodies of the solar system ensures that this assumption can be accepted for moderate time intervals, much larger than the ones for present planned missions. However there is no guarantee that the orbits we look for exist nor they be quasi-periodic. b) If the orbit of a) exists and two particles (spacecrafts) are put close to it, how does the mutual distance and orientation change with time ? As a final conclusion of the work, there is evidence that orbits moving in a somewhat big annulus around L\ and L5 exist, that these orbits have small components out of the plane of the Earth-Moon system, and that they are at most mildly unstable. The mutual distance of two points starting close to these orbits changes by an important factor (at most 1 to 100), and the orientation changes in a regular way, unless some small loops are present in the projection of the relative motion on the (x, y)-plane or this projection comes too close to the origin. In any case we believe that it can be a useful place to locate one or two spacecrafts for scientific purposes because of the nice properties concerning stability. The station keeping necessary to maintain the orbit in its right place can be reduced to an unimportant amount. The contents of this book is the final report of the study contract that was done for the European Space Agency in 1987. This report is reproduced textually with minor modifications: the detected typing or obvious mistakes have been corrected, some tables have been shortened and references, which appeared as preprints in the report, have been updated. The layout of the (scanned) figures 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.

Preface

vii

For the ESA's study we also produced software that is not included here, although all its main modules are described in detail in the text. 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 of works on Dynamics and Mission Design Near Libration Points.

Preface

STUDY ON ORBITS N E A R T H E T R I A N G U L A R LIBRATION P O I N T S IN T H E PERTURBED RESTRICTED THREE-BODY PROBLEM FINAL R E P O R T

ESOC C O N T R A C T NO.: 6139/84/D/JS(SC) ESOC T E C H N I C A L SUPERVISORS: Dr. W. Flury and Dr. J. Rodn'guezCanabal 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, 08075 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, February, 1987

Contents

Preface

v

Chapter 1 Bibliographical Survey 1.1 Equations. The Triangular Equilibrium Points and their Stability . . . . 1.2 Numerical Results for the Motion Around L4 and L5 1.3 Analytical Results for the Motion Around L4 and L5 1.3.1 The Models Used 1.4 Miscellaneous Results 1.4.1 Station Keeping at the Triangular Equilibrium Points 1.4.2 Some Other Results Chapter 2 Periodic Orbits of the Bicircular Problem and Their Stability 2.1 Introduction 2.2 The Equations of the Bicircular Problem 2.3 Periodic Orbits with the Period of the Sun 2.4 The Tools: Numerical Continuation of Periodic Orbits and Analysis of Bifurcations 2.4.1 Numerical Continuation of Periodic Orbits for Nonautonomous and Autonomous Equations 2.4.2 Bifurcations of Periodic Orbits: From the Autonomous to the Nonautonomous Periodic System 2.4.3 Bifurcation for Eigenvalues Equal to One 2.5 The Periodic Orbits Obtained by Triplication

1 1 2 6 6 12 12 12

15 15 16 19 21 21 24 26 28

Chapter 3 Numerical Simulations of the Motion in an Extended Neighborhood of the Triangular Libration Points in the E a r t h M o o n System 33 3.1 Introduction 34 ix

x

3.2 3.3

Contents

Simulations of Motion Starting at the Instantaneous Triangular Points at a Given Epoch 35 Simulations of Motion Starting Near the Planar Periodic Orbit of Kolenkiewicz and Carpenter 35

Chapter 4 The Equations of Motion 4.1 Reference Systems 4.2 The Lagrangian 4.3 The Hamiltonian and the Related Expansions 4.4 Some Useful Expansions 4.5 Fourier Analysis: The Relevant Frequencies and the Related Coefficients 4.6 Concrete Expansions of the Hamiltonian and the Functions 4.7 Simplified Normalized Equations. Tests 4.7.1 Tests of the Simplified Normalized Equations

47 47 48 51 52 54 62 65 66

Chapter 5 Periodic Orbits of Some Intermediate Equations 5.1 Equations of Motion for the Computation of Intermediate Periodic Orbits 5.2 Obtaining the Periodic Orbits Around the Triangular Libration Points for the Intermediate Equations 5.3 Results and Comments

71 71 73 74

Chapter 6 Quasi-periodic Solution of the Global Equations: Semianalytic Approach 87 6.1 The Objective 87 6.2 The Algorithm 88 6.3 The Adequate Set of Relevant Frequencies 90 6.4 Avoiding Secular Terms 94 6.5 The Coefficients Related to the Different Frequencies 94 6.6 Determination of the Coefficients of Quasi-periodic Functions Using FFT 95 6.7 Results and Conclusions 103 Chapter 7 Numerical Determination of Suitable Orbits of the Simplified System 107 7.1 The Objective 107 7.2 Description of Two Families of Algorithms. Reduction of the Linearized Equations 108 7.3 Description of the Methods. Comments 112 7.4 Results and Discussion 116 Chapter 8 Relative Motion of Two Nearby Spacecrafts 8.1 The Selection of Orbits for the Two Spacecrafts 8.2 Variations of the Relative Distance and Orientation. Results 8.3 Comments on the Applicability of the Results

121 121 122 135

Contents

xi

Chapter 9 Summary 9.1 Objectives of the Work 9.2 Contribution to the Solution of the Problem 9.3 Conclusions 9.4 Outlook

137 137 138 140 141

Bibliography

143

Chapter 1

Bibliographical Survey

A bibliographical survey of the most important papers about the motion near L\ and £ 5 is presented. This chapter has four sections. In the first one we give the basic definitions and results on the triangular equilibrium points L4 and L$. The second one is devoted to summarize the known results concerning the numerical simulations of motion around the triangular equilibrium points in the Earth-Moon system for different models of motion for the solar system. The third section deals with the motion at some intermediate distance from the equilibrium point, looking for periodic and quasi-periodic orbits. Finally in the last section some miscellaneous results are summarized. We have not reported in this survey all the papers dealing with analytic computation of periodic orbits around the triangular equilibrium points for the restricted three-body problem, because they are not of practical interest in this case.

1.1

Equations. The Triangular Equilibrium Points and their Stability

As a very rough approximation to the real situation, but which shall be useful later on, we start this section summarizing some properties of the triangular equilibrium points of the restricted, circular and elliptic, three-body problem. In a rotating, barycentric, dimensionless coordinate system with the smaller primary on the positive z-axis, the differential equations of motion for the circular three-dimensional restricted problem are:

x-2y

=

ilx,

y + 2i

=

fly,

z =

ttz,

1

2

Bibliographical

Survey

where 2 r\ r\

= =

n 2

2

(x + fi) +y

r2 2

+ z,

2

(x + fi- l) + y2 + z2.

In the above equations, fi denotes the mass of the smaller primary when the total mass of the primaries has been normalized to unity. It is well-known that the above differential system has five equilibrium points. Three of them, denoted by L\, L-i and L3, are collinear with the primaries and the other two are the triangular equilibrium points of coordinates L4 = (1/2 — fi, \/3/2,0), £5 = (1/2 — fi, — -s/3/2,0). Birkhoff (see [40]) showed that the two triangular points are stable, in the linear sense, for values of the mass ratio in the open interval (0,/ii), where fix = 0.038521.... The global (Lyapunov) stability of these points has been studied by several authors: Leontovich [25], Deprit and Deprit [9], Markeev [26], Rvismann [34]. The final conclusion is that, in the planar case, the two equilibrium points are always stable except for two values of the mass ratio for which they are unstable. These two resonant values are: fi% = 0.024293... and n$ = 0.013516.... For the three-dimensional case the same result holds but now leaving aside not only the values fi2 and fi$, but for a fixed fi a set of initial conditions of Lebesgue measure relatively small (see Markeev [27]). For the elliptic restricted three-body problem, the five equilibrium points are also present, with the same coordinates as in the circular case in an adequate frame of pulsating coordinates. Of course now, due to the nonautonomous character of the equations of motion, the analytic computation of the linear stability is not as easy as it is in the circular case. There are also many papers, both analytic and numerical, devoted to this goal (Danby, Benett, Meire, Tschauner, Deprit et al., Giacaglia, Alfriend and Rand, Nayfeh, Kamel, Vinh, Kinoshita,...). The results can be summarized in Figure 1.1, which in the (fi, e)-plane (where e stands for the eccentricity of the primaries) represents the region of linear stability for these three equilibrium points. Finally, it must be said that in the real world the triangular equilibrium points do not exist at all. They must be redefined since the gravitational forces due to the Sun and the planets will disturb the equilibrium forces acting on a body located at the initial equilibrium point.

1.2

Numerical Results for the Motion Around £4 and Z5

Some rough numerical explorations of stability regions near the triangular equilibrium points of the Earth-Moon system were done by McKenzie and Szebehely [29]. The model that they used was the planar and circular restricted problem of three bodies. They computed the regions about L4 and L5 in which a particle, with zero initial velocity, librates about the equilibrium point. The region in the vicinity of

Numerical

0 . 0 2 h1' H-° O.O6 Fig. 1.1

0.08

Stability chart for the equilateral equilibrium points. T h e stable region is dotted ([40]).

iSIS!!MHIi|{iM^!MMi»!HS Fig. 1.2

3

Results for the Motion Around Li and L5

mtnmmi

Regions of stable initial conditions for motions around L4 (left) and L5 (right) ([29]).

the equilibrium points was 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 were then integrated for 480 time units (1 time unit = 4.7 days). If the orbit did not touch or cross the z-axis, the initial position was considered to be one giving libration. Their results are presented in Figure 1.2. Gyorgyey [18] studied numerically the regions of stability around the triangular equilibrium point L5 in the elliptic planar restricted problem in a similar way to McKenzie and Szebehely. He showed that with increasing eccentricity, the width of these libration regions is decreasing. More realistic models using restricted four-body problems, which shall be detailed, were considered by Tapley and Lewallen [41], and Tapley and Schutz [42], [43] and [44]. The results of [42] were included and improved in [43], so we shall report them

4

Bibliographical

Survey

explicitly in this survey. In the first three papers the authors considered the following model of the solar system. The Earth and the Moon were assumed to move in circular orbits about their mutual mass center. The mass center turns in a circular orbit about the Sun. The Earth-Moon orbit plane was taken with an inclination of 5°9' with respect to the ecliptic. It must be said that when the effects of the Sun are included, the system is nonautonomous and L 4 and L5 are no longer equilibrium points. They are redefined as those points where a particle at rest in a synodical system will have zero acceleration when only the actions of the Earth and the Moon are considered. The first paper [41], studies the motion of a satellite placed at the libration points with zero velocity when the Sun is collinear with the Earth and the Moon, and the Moon is between the Earth and the Sun at the initial epoch. 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 100 000 km 240 000 km

Amplitude of y 30 000 km 60 000 km 125 000 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 of 6 000 km. Similar results were obtained for L5. When solar radiation pressure was included, the amplitude of the motion was bigger. The paper includes estimations about the impulse required for forcing a vehicle to remain precisely at L4. For one year 750 m/s are required. Forcing the vehicle to remain at a point near the equilibrium increases the total amount of fuel. For the same model of motion of the primaries as in [41], Feldt and Schulman [13], extended the interval of integration to a larger one. They showed that the envelope of the motion of the spacecraft with the above initial conditions expands up to 270000 km approximately and contracts to some 9 000 km from L 4 . The period of the pulsating motion was about 1 500 days. This was already noticed by Tapley and Schutz in [42]. In [43] Tapley and Schutz showed that after 3 900 days approximately, the spacecraft left a libration-point-centered motion as the result of a near-lunar encounter. The displacement at 5 000 days for this case was over 94 million kilometers from the L4 point, indicating that the spacecraft had escaped from the Earth-Moon system. Then, the problem is to determine the initial angle between the Sun-Earth and Earth-Moon lines so as to minimize the maximum displacement from L4. The same question has already been studied by Wolaver [51] but for a planar model for the motion of the primaries. This study is sensitive to the constant values adopted for the model. In summary, with the choice of constants used by the authors, a libration-point-centered motion continues for a period of at least 8 000 days, and

Numerical Results for the Motion Around L\ and L5

5

the nature of the motion indicates that it may persist for a much longer period. The fourth paper [44] is similar to the above mentioned ones. 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: Earth, Moon and Sun. 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 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 showed that a particle placed at L5 on Julian Ephemeris Date 2 439 796.735 will follow a libration point centered motion for 2 500 days. The envelope of the particle's motion about L5 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. Using the same initial epoch, but starting at the geometrical L4 point and with initial velocity determined in the elliptic restricted three-body problem approximation, a near lunar encounter occurs after 579 days. 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. In summary, many papers deal with numerical computations about the motion of a particle in a (big) vicinity of L\ or L5 under the influence of the Earth, Moon and Sun (and in some cases, all the solar system [17]). All the simulations seem to confirm the lack of stability of the motion very close to L\ and L5. However, some simulations starting at L4 or L5 have a pulsating character (concerning average distance to the equilibrium point) with a long period. The simulations show a strong sensitivity with respect to the model of the solar system and to the initial epoch as well as to initial conditions. Possible approaches to the Moon after one year are found in several cases. Our own simulations show motions, which shall be explained in detail in Chapter 3, for which the particle is confined to less than 0.8 Earth-Moon distance for 7000 days. For instance, the one starting at the instantaneous L5 point of the EarthMoon system the day 16 000 after 1950.0, when the full solar system is used. The simulations also show that a stable periodic orbit exists for some nearby problem (the periodic orbit found by Schechter, Kolenkiewicz and Carpenter, Kamel and Breakwell, Wiesel, etc.; see the next section) because for some months the motion looks like a quasi-periodic motion around a periodic orbit. However, after a somewhat long period the vicinity of this orbit is left. Starting with the initial conditions given by Wiesel or Kolenkiewicz and Carpenter some evidence of this quasi-periodic motion, for several months, is found. Later, an escape is produced. Results concerning periodic orbits of interest for the problem will be reported in the next section.

6

1.3

Bibliographical

Survey

Analytical Results for the Motion Around L4 and i 5

When we study the motion of a spacecraft near the triangular equilibrium points in the Earth-Moon system, the first question that arises is the adopted model of motion used for the bodies producing the field of forces acting on the infinitesimal one. Some intermediate "mathematical" models, shall be reported here due to its usefulness from a practical point of view. Of course other technical questions, as for example the adopted theory for the motion of the Moon (de Pontecoulant's, HillBrown's,. ..) can be of importance in order to explain quantitative discrepancies between similar results. This is especially true if we take into account the sensitivity of the problem, already displayed in the preceding section, but shall not be explained here in detail. Once the equations have been obtained, the second question deals with the tools to be used to analyze the problem (literal expansions of the equations, perturbation theories,...). Of course these tools are closely related to the kind of searched result, i.e.: the qualitative picture of the phase space near the equilibrium (displaying its periodic orbits), the computation of some particular trajectories, etc. In this section we shall report the results found in the literature concerning these topics.

1.3.1

The Models

Used

It has been conventional in treating this problem to use a very restricted four-body problem, or bicircular problem, in which two masses move in circular orbits about their barycenter, while the barycenter also describes a circular path about the third mass. This model has been widely used in the literature ([46], [38], [40], [51]). Nevertheless its conclusions are not valid for applications, mainly because it does not take into account the indirect influence between the primaries. Mohn and Kevorkian [31] developed the following model: the motion of the three most relevant primaries (Earth, Moon and Sun) is given in terms of asymptotic solutions of the RTBP, for the limiting case where the particle (Moon) remains close to one primary (Earth). They use for this goal both Hill's and de Pontecoulant's lunar theories. As the distances of the Moon and the spacecraft (if it moves near the Lagrangian points) are both 0(y}l3), the variables (including the independent one) are scaled adequately. The final asymptotic equations include the leading terms of the solar perturbation in a dynamically consistent manner. As a final remark we must say that we have not seen any applications of these equations in any other paper. Having in mind the re-computation of the periodic orbits obtained by several other authors, Wiesel [49] produced a model in which the periodic orbits (of the adequate period) known for the genuine RTBP can be analytically continued into periodic orbits of a restricted four-body problem. The major idealization, according

7

Analytical Results for the Motion Around L4 and L$

to the author, is that it neglects the eccentricity of the Sun's orbit. Going from one model to the other can be done roughly in the following steps: (1) Add eccentricity to Moon's orbit (elliptic RTBP). (2) Include the Sun and, optionally, some terms of the lunar motion, to make it coherent. Then, we obtain a restricted four-body problem with the eccentricities of the Moon and the Sun as perturbations. Avoiding the first step, for the moment, in the usual system of units of the RTBP and introducing as a constant the mean motion of the Sun, ns, in an adequate set of coordinates and momenta the Hamiltonian of the problem can be written as: ~{PX + Py +Pz) + VPx - XPy - y 2 rPE

rPM

4/3

—-f ^N(ysm6-xcos6) (1 + ms)2'3

+ rPS

where rpE, rpM and rp$ are the distances from the satellite to the Earth, Moon and Sun respectively, and 9 = (1 — n$)t. The first terms of the above Hamiltonian (outside brackets) correspond to the RTBP. For the motion of the Sun, the author considered a circular one given by:

fl + ms\1/3 xs

=

5— 1 ,

cos l

\

n

~

u

n

s)t,

\ 1/3

ys

=

x 1 +2ms - (— —) '

zs

=

0.

( ~ nsK

sin 1

The motion of the Earth and the Moon can be written, in general, as: rE = -firEM,

rM = (1 - fi)rEM-

Then, for TEM three possibilities are considered (1) Circular motion: TEM = (1,0,0) T . (2) The periodic orbit, r]^'M given by Kolenkiewicz and Carpenter [23], and also reproduced by the author in the paper. (3) The periodic orbit given above plus some Floquet modes: TEM = r^lf + 5 3 V — ~ (LiR(*)

sin

QiM + Lu(t) cos QiM)

i=\

in order to take care of coherence. This orbit is given explicitly in Wiesel [48], and it is good enough so that further analytic refinement becomes unnecessary.

8

Bibliographical

Survey

For the purpose of explicit computation of periodic orbits, some intermediate "mathematical models" are introduced by the author. The general idea is to start at some limit problem, embedded in a two-parameter family of problems, and then continue it, by changing the parameters, till a more realistic one. Two possible choices are displayed in the paper: (1) Writing rEM = (l-e)(-l,0,0)T

+

er%(t),

when e = ms = 0, the RTBP is obtained. If e = 1, ms = actual mass of the Sun, then we get a model equivalent to the second one previously mentioned. In this way resonant orbits of the RTBP can be continued, with some difficulty, across e and ms(2) The second procedure is developed in the following way. Writing the bracket term of the Hamiltonian as: H

T

= ^n2s[(l-3cos2

e)x2+6xysmecose+(l-3sm2

e)y2+z2]+0{ms1/3),

the full Hamiltonian can be written in the form:

K(p, q, e) = hp2x+p2+ p2z) + ypx - xpy - ^ where H'g'jj is equal to H™ deleting the 0(ms

- -JL- + tHfM,

' ) term.

For e = 0 we have again the restricted three-body problem, while for e = 1 we have the asymptotic form (e = l,ms = 0) of the standard system. The above Hamiltonian, K, is known as Hill's limit of the problem. In the paper it is shown that the continuation from the restricted problem to Hill's limit is a far better conditioned process than the direct continuation across the (e, ms)-plane. Once Hill's limit has been reached, the continuation up to the actual mass ms becomes easy and can be done generally in a single step. For this Hill's limit problem Figure 1.3 (first computed by Wiesel [49]) shows the evolution of the yo initial condition for some periodic orbits with the e continuation parameter. The infinitesimal solution at L 4 evolves with e to yield a periodic orbit at point B, which is periodic orbit II of Kolenkiewicz and Carpenter [23]. Constructing their other stable orbit in Hill's limit model yields the orbit at point A. Evolving this orbit with e yields their third (unstable) orbit at point C. The complete evolution of the second family has not been determined. Finally, starting the continuation process across the (e, ms)-plane requires the solution of a bifurcation problem in the infinitesimal e regime near the RTBP. For e small, the Hamiltonian K can be written as K — KRP + ei^pert where KRP is the Hamiltonian of the RTBP and K Pert

- u°o,,n " f l s + W

_,\ 1

Mj

\ (x - n)(l + xp) + yyp _ (x + 1 - /i)(l + xp) + yyp ' [ ((i - /i)2 + y 2 + * 2 ) 3 / 2 ((x + 1 - AO2 + y2 + * 2 ) 3 / 2 .

Analytical

Results for the Motion Around L4 and L5

0.85

9

0.9 Vo

Fig. 1.3 Evolution with the continuation parameter e (y-axis) of the initial j/o coordinate (i-axis) for some periodic orbits.

where xp and yp are the periodic components of r^M (t). For the obtained results dealing with the analytic study of the Hamiltonian in the vicinity of the triangular points in the Earth-Moon system, standard perturbation techniques have been proven to be unsuccessful because of the occurrence of small divisors in many terms of the assumed series solution. One of the pioneering works in this direction was the one of Breakwell and Pringle [4]. They analyzed a two-dimensional approach of the motion which took into account the dominant nonlinear resonances by examining only the slowly varying terms. In [4] 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 the fourth order, and comparable solar effects, are included in its model. The von Zeipel method was only carried up to the second approximation to remove short period terms. This work was extended and improved by Schechter [35]. The main idea of [35] is to analyze the stability of slow variations around the periodic orbits found to exist in the problem (one of them stable and the other unstable). For the relations giving the Earth-Moon distance and its angular velocity, a classical simplified model (the eccentricity of the Moon is taken equal to zero) was used. Due to the near resonance ui\ w 3o>2 of the coplanar frequencies (uii = ±0.95459, w2 = 0.29791), fourth order terms in the Taylor expansion of the Hamiltonian were retained. The Hamiltonian is split into two parts H = H° + H1, where H° contains the second and third order terms which are linear and quadratic in position and

10

Bibliographical

Survey

momenta, and the perturbation H1 contains the cubic terms in position (H3) and the terms of global order four {Hi). The solution of the linear homogeneous differential equations, governed by H°, consists of a part depending on six constants a*, Pi, i = 1,2,3 and a forced response. The constants ctj, /?$ are taken as independent variables when H1 is included. Two new sets of canonical variables are introduced, one in order to get a slowly varying Hamiltonian and the other to get a time independent one. Let K(a,P) be the final Hamiltonian. If for small a.\, a2, 0:3, there is a stable motion then, in order to determine long-term effects, it is enough to retain only linear terms in the development of K. For this linear expression we have: 0.02425ai + 0.02412a2 + 0.07899a3 - 0.02563ai cos(2/31), which is of Mathieu type and leads to parametric resonance in the a.\ motion. Since 0.02563 > 0.02425, the motion falls into the unstable region of the Mathieu plane, and therefore no motion can exist for which a.\ remains very small. In order to look for periodic solutions, it is seen that the 0:3 variable does not affect very much the a i , 0.2 ones, so an equilibrium is searched for 03 = 0. Using normal coordinates defined by

Qz = (2 ai ) 1/2 sin ft 1 P = (2a i ) 1 /2 cos/3 . J '

. ,1 9 *- '^

two equilibria are found of the form: Q1=Q2= Q1=Q2=

0, 0,

P1 = 0.1093, Pi = 0.1106,

P2 = 0, P 2 = -0.003675,

the first one linearly stable and the second one unstable. These equilibria, when seen as periodic orbits, are quite unrelated to the periodic orbits found by Wiesel and Kolenkiewicz and Carpenter which shall be described later. In a later paper, Kamel and Breakwell [21] outline, without giving any details of its computations, which are in [20], a higher order two-dimensional theory. The small parameter introduced in order to apply perturbation theories, is the ratio, m = 0.074801, of the sidereal month to the sidereal year. The Hamiltonian is expanded in powers of x, y up to 6th degree and re-scaled and split as follows —-r = H0 + \

—rHq,

9=1

where the "unperturbed" Hamiltonian H0 includes all the quadratic terms in x, y, px, py with constant coefficients. After introducing action-angle variables, a perturbation theory based on Lie series [8] with the practical modifications introduced by Kamel [20] is applied in order to skip short periodic terms. The new Hamiltonian is partially given in the paper. Again, looking for the critical points of this

Analytical

Results for the Motion Around L4 and L$

11

Hamiltonian, both the stable and unstable periodic orbits, in this planar situation, are found. The periodic orbits found in [21] are in good agreement (3%) with the ones computed numerically by Kolenkiewicz and Carpenter [22] using an algorithm which shall be now summarized. The standard equations of motion for Moon, Sun and particle are written in a planar model. Then, for the position of the Moon (and also the Sun) a solution with period equal to the synodic period of the Sun in the Earth-Moon system is searched. It is tried f = (1 + a)f0 + 0w, where a — ^ o ° a% cosk9 + ask sinkO, /? = X^o° @k cosk9 + /?| s'mk8, 6 = (UM — ns)t. fo is the position vector in a fixed reference ellipse (usually taken as a circle) and w = ^dfo/dt. Let us suppose first that the motion of the Sun is circular around the Earth. Starting with a = /3 = 0, substitution into the equations of motion and integration gives a new set a,j3. By iteration, they obtain a coherent motion for the Moon, with suitable period and the Sun as stated. Then, the roles of Sun and Moon are changed. The new model of the Moon is kept fixed and one iteration is done for the a,/3 related to the Sun. By iteration of the procedure a periodic solution of the three-body problem is found (in synodical coordinates). Then the same thing is done for the particle, taking r 0 as a vector advanced 60° with respect to the Moon. Two different orbits are found both stable and roughly equal but with the phase changed in 180°. They are larger than the periodic orbit found by Schechter. The shape is that of a 1 : 2 ellipse with semimajor axis « 145 000 km. Another periodic orbit near the Li point describing two loops in one synodic month is found. It agrees with the unstable orbit found by Schechter. The discrepancies between the theoretical results of Schechter and the numerical ones of Kolenkiewicz and Carpenter was one of the motivations of the more accurate theory developed by Kamel and Breakwell previously mentioned ([19], [21]). Finally we shall mention the results of Wiesel concerning periodic solutions. (1) Starting at e = 0, solving first a bifurcation problem, and by continuation of L4 they reach a periodic orbit: the periodic orbit II of Kolenkiewicz and Carpenter. Bifurcations of this family are not analyzed (mainly the resonance 3:1). (2) Starting at Hill's limit problem, another stable orbit is found. It can be continued till the actual mass of the Sun, ms- Further continuation gives another periodic orbit, much smaller and unstable. The author suggests the following extensions: (1) To propagate periodic orbits from RTBP for model (2) and to analyze the stability using model (3). (2) The continuation of periodic orbits bifurcating from short and long periodic orbits near L4.

12

Bibliographical

1.4

Survey

Miscellaneous Results

1.4.1

Station

Keeping

at the Triangular

Equilibrium

Points

De Fillipi [14] in 1978 studied the station keeping at the libration point L 4 for the Earth-Moon system. He considered a model given by a restricted four-body problem in the space in which the Earth and the Moon describe an elliptic orbit around their center of masses, the barycenter of the Earth-Moon moves in an elliptic orbit about the Sun, and the center of the Sun is the center of masses of the full system. In order to control the motion he introduced a periodic gain matrix as the solution of the adequate Riccati's equation. The results that he obtained are: (1) Using the averaged gain matrix, which makes sense due to its periodic character, the computational effort decreases and the results for the control are not worse. The following table displays this comparison for the AV cost: Final Time Optimal System Averaged System

15 0.064286 0.066905

25 0.103100 0.105890

45 0.17738 0.17635

In the above table, 1 time unit is equal to 4.384 mean solar days and 1 velocity unit is equal to 1023 m/s. (2) The cost of the station keeping decreases when the particle is allowed not to be exactly at the equilibrium but in a neighborhood of it, as it should be. (3) Some initial configurations of the primaries, which are computed in the work, minimize the station keeping. The average thrust variation due to different initial configurations is about a 9%. 1.4.2

Some Other

Results

The study of the libration points in the Earth-Moon system has been parallel to the search of natural or artificial bodies near these points. The first observational evidence was anonymous but reported by Kordylewski in 1961. Two "faint cloudlike satellites" in the neighborhood of L 5 were reported by this author. Later, the discovery of such a "cloud" near L 4 was also reported. Simpson, in 1967, confirmed the existence of these clouds as having diameters of the order of 1 to 5 degrees. But subsequent ground based observations: Roosen (1966,1968), Roosen and Wolff (1969), Bruman (1969), Munroe et al. (1975), Roach (1975), Freitas and Valdes (1980) failed to confirm these results. Olszewski [32], in 1971, studied the region of "practical stability" around the triangular points in the elliptic planar restricted problem. Practical stability means

Miscellaneous

Results

13

that there exists a sufficiently small neighborhood of the equilibrium point, where the solutions of the system remain for sufficiently long time. For practical purposes his conclusions are useless. Kunitsyn and Perezhogin [24] in 1978 studied numerically the motion near the triangular points of the circular planar restricted problem by including the solar radiation pressure. These libration points are shown to be Lyapunov stable in the region where the necessary stability conditions are satisfied. They showed that for all the planets of the Solar system the libration points L4 and L5 are Lyapunov stable. Mignard [30] studied the motion near the triangular points of the circular planar restricted problem by including the solar radiation pressure. The triangular equilibrium points disappear. However libration orbits are not completely destroyed.

This page is intentionally left blank

Chapter 2

Periodic Orbits of the Bicircular Problem and Their Stability

In this chapter we look for low period periodic orbits of the bicircular problem. The periods found are equal to the period of revolution of the Sun in the Earth-Moon synodical system and to three times this period. The period must be a multiple of the period of revolution. The bicircular problem is far from the real one. It is even not coherent, that is, the three massive bodies do not satisfy Newton's equations. However it gives us insight concerning the periodic orbits of a system much closer to the real system, as we shall show in Chapter 5. These last periodic orbits will be the basic ones to build up the final quasi-periodic solutions.

2.1

Introduction

The bicircular problem is a very simplified model for the four-body problem that we are considering. In this model we suppose that the Earth and the Moon are revolving in circular orbits around their center of masses, and the Earth-Moon barycenter moves in a circular orbit around the center of masses of the Sun-EarthMoon system. The first thing to state is that this model is not coherent, i.e., the positions of Sun, Earth and Moon, as we have described, do not satisfy Newton's equations (see [40] p. 288). Despite this fact we can consider the equations of the bicircular problem as an approximation to the true equations of motion. We consider it as a test model and in this chapter we study some of its periodic orbits. Using synodical coordinates with respect to the Earth-Moon system, the equations are periodic with period, Ts, equal to the synodic period of the Sun in the EarthMoon system. In Chapter 5 we will see that there is a similar model with periodic equations obtained by keeping, in the general equations of motion, only the terms independent of time and the periodic terms with period T$15

16

Periodic Orbits of the Bicircular Problem and Their

Fig. 2.1

2.2

Stability

Geometry of the bicircular problem.

The Equations of the Bicircular Problem

Let /i be the mass of the Moon, 1 — // the mass of the Earth and m j the mass of the Sun. Let (X, Y, Z) be the coordinates of a particle of infinitesimal mass with respect to the barycenter of the Earth-Moon system, B. Let the distance from the Earth to the Moon be taken as unity. Then, the distance from B to the Sun is asFigure 2.1 displays the relevant geometry. The coordinates of Earth, Moon and Sun with respect to B are given by XE YE ZE

= /icosf, = /usini, = 0 ,

XM YM ZM

= = =

(/x —l)cost, (fi —1) sin t, 0,

Xs Ys Zs

= as cos nst, — assinnst, = 0,

where ns is the mean motion of the Sun, and o | n | = 1. The coordinates (X, Y, Z) are not referred to an inertial frame. However, if (Xi,Yi,Zi) denote inertial coordinates (with respect to the barycenter of the SunEarth-Moon system), we have v Xi

=

Yi

=

Zi

=

v rns X — ascosnst, 1 + ms Y-— assinnst, 1 + ms Z.

The Equations

of the Bicircular

17

Problem

Hence ms

ir

X

*

Y

— Xi 2~ cosn s r, a ss . v> rn = Yi 2~ sinn s r, as

Z

= Zi,

(2.1)

because msasns 1 + ms

ms a2s

For the inertial acceleration we have ^

{X-XE)(1-IM)

=

(X-XM)H

r<J

T

PE

fi

=

' PE M)

-^(1 ~

'PS

(Y-YM)»

mO

Z-

/rt<J

r

PM

(Y-YE)(l-y)

=

(X-Xs)ms

tv*0

z

fi-

« J

' PM

' PS

*•

'

Zms

mO

nfO

rrtii

1

' PM

' PS

PE

(Y-Ys)ms

„o

where rpE, TPM and rps denote the distance from the particle to Earth, Moon and Sun, respectively. We introduce synodical coordinates with respect to the Earth-Moon system by x

=

X cos t + Y sin t,

y

= -Xsint

+ Ycost.

(2.3)

From (2.1), using (2.2) and (2.3), the equations of the bicircular problem are obtained: x — 2y — x —

3—{x - \i) r

PE

•• -

S

TTIQ

r

TYIQ

+ ass'm6) + -^-sinfl,

r

PM

^-cosO, a

PS

j — V ~ ~~3—V- -3-(y r

Z

LI

— (a; - as cos6) r

PM

1 — 11

y + 2x-y =

3—(x - fi + 1) r

PE

(2.4)

a

PS

S

•*•O~ ^Z _ o ^Z _ H±s_ q Z. /fO

' PE

irt<J

ir"J

'PM

'PS

where 9 = (1 - ns)t and rps, rPM and rPs are computed by means of rPE = {x - fi)2 +y2 + z2,r2PM = {x-Li + l ) 2 +y2+z2, r2PS = (x - xs)2 + {y- ys)2 + z2. In the last formula we have xs = ascos9, ys = - a s s i n # , and as can be computed as as

~

if ms and ns are the basic quantities.

yS s

n

18

Periodic Orbits of the Bicircular Problem and Their

Stability

Defining momenta px = x — y, py = y + x, pz = i , equations (2.4) correspond to the Hamiltonian equations with Hamiltonian function: H

1 = 9 (Px + Py + Pz) + VPx ~ xpy 2 -„ - . ---

m

s rPS

P rpE

TpM

rns Triysmff — xcos6). ai

Equations (2.4) have the apparent symmetries Si : 52 :

{x,y,z,x,y,z,t) (x,y,z,x,y,z,t)

(x,-y,z,-x,y,-z,-t), (x,y,-z,x,y,-z,t),

which can be useful in order to find periodic orbits. The variational equations associated to (2.4) are

A =

(2.5)

DF-A,

with A: A(t) and A(0) = I§, where

f

1

DF

\ 1

0 9n

912

921

922

923

\ 931

932

933

913

1 0 2 0 -2 0 0 0 0 0 )

and gij are used for shortness to denote the following functions (we suppose that we are interested in planar orbits, but the vertical stability is also checked): 3(1 -/x) (x - M ) 2

1-H 3n

1

q

~r

' PE

' PE r

n)y

3fi(x + 1 - fi)y

3ms{x - xs){y - ys)

' PM

' PS

921 1

931

922

1-

' PM

PS

3(1 - n){x913

nY

r

PS

PE

= 0, r

923

' PM

3/J,(X + 1 -

+

Zms{x - xs)2

ms

912

fj,

c

PE 932 - 0, 1-/X

+

3(1 -

' PE

r

PM

3/i^2 _ m s ' PM

'PS

3ms(y - ys)2 'PS

ms

933 ' PE

M

M)2/2

' PM

' PS

As it is well-known, the planar and vertical modes are uncoupled for planar periodic orbits. Hence, if A = (ay), i,j — 1,...,6 and A4 = (a^), i,j = 1,2,4,5

Periodic Orbits with the Period of the Sun

19

we have that (2.5) reduces to 0 0

1

3n

312

521

322

0 -2

0 0

d_ ( a33 dt \ a 63

a 36 \ _ / 0 a66 J V 333

1 \ 0 y

1 2 0 , f a33 V a63

a 36 \ ^66 / '

with A(0) = h, i.e., A4(Q) = h, a 33 (0) = a 66 (0) = 1, a 36 (0) = a 63 (0) = 0. In order to compare results with the ones obtained by Wiesel [49] we have adopted the following values: H = 0.012150298,

ms = 328900.12,

ns = 0.07480132816,

all the other parameters being computed from these three magnitudes.

2.3

Periodic Orbits with the Period of the Sun

We look for planar periodic orbits of system (2.4) that, in synodical coordinates, have the same period as the Sun, i.e., Ts = 27r/(l — ns). From now on we write (2.4) with ms replaced by ems, where e is a parameter ranging from 0 to 1, but we do not change the value of as- For e = Owe have the restricted three-body problem, and for e = 1 we get the bicircular problem. One possibility is to start with periodic orbits of the restricted three-body problem. Unfortunately the short periodic orbits of the family Cs around the equilateral points 1/4,5 lying on a big vicinity of £4,5 (mean radius about 1/2 of the EarthMoon distance) have a period shorter than Ts- The long periodic orbits of Ci in the same vicinity have a period slightly bigger than 2>Ts (see section 2.5). Hence, the only simple orbits with the right period, when e = 0, are the equilibrium points £4,5. From now on we only consider the orbits around L5. As the Earth is placed to the right and the Moon to the left in synodical coordinates, the L5 point has positive y coordinate. (We note that this point is called L4 by Wiesel [49], but we have used the notation of Szebehely [40].) As we look for periodic orbits with a fixed period, Ts, the equation to solve is F{x, y, x,y)

=

$ T s (x, y,x,y)

- (x,y, x,y) = 0,

(2.6)

where $t denotes the flow under time t. The monodromy matrix M = D$TS c a n be computed by integration of the variational equations (2.5). Newton's method is used to solve (2.6). This requires that the matrix M—I be regular, i.e., 1 ^ spec(M). We recall that for e = 0 the eigenvalue 1 of M has, at least, multiplicity equal to

20

Periodic Orbits of the Bicircular Problem and Their

Stability

Fig. 2.2 Evolution with the continuation parameter e of the periodic orbits of the RTBP to the bicircular problem.

2. This means that small values of e can produce troubles. We shall return to this point in section 2.3. The first family numerically computed is the one starting at L5 for e = 0. If e is increased, it is easy to do numerical continuation with big steps (of 0.1 units) in e till 0.8. Then, the step should be reduced because det (M — I) reaches a minimum at e RS 0.877. The values of x against e are given, for this family, in Figure 2.2. The orbits in the family are linearly stable even if we consider the vertical stability, i.e., the six eigenvalues are grouped in 3 pairs of complex conjugate values with modulus 1 and different from ± 1 . Comparing with the results of Wiesel [49] we see that the branch emerging from L5 connects with the orbit called B of Figure 4 in [49] instead of connecting with the orbit called A. If now we start with orbit A of [49] when e = 1 we found a nearly periodic orbit for system (2.4). This orbit is again linearly stable. Decreasing e, the orbit can be continued in a natural way till a value of e = ec close to 0.905. For e = ec a saddle-node bifurcation is found. Two of the complex conjugate eigenvalues of the monodromy matrix for the previous family become equal to the unity. Increasing again e an unstable orbit is found. This can be continued till e = 1 with unstable character. It has a reasonable agreement with the orbit called C by Wiesel. The last family is also shown in Figure 2.2. The fact that even qualitatively our results do not agree with those of [49] can be explained because the models are slightly different and in the neighborhood of

The Tools: Numerical

Continuation

of Periodic Orbits and Analysis

of Bifurcations

21

e = 0.9 the matrix M — I is rather close to singular. Small differences in the model can, then, produce a shift from a connection BC to a connection AC. Concerning stability we must add that in both families, ending at A and B, there are points where a couple of eigenvalues related to planar stability become cubic roots of unity. These points are located, in family A, at e = ei « 0.9351784438, and in family B at e = e2 ~ 0.9254693472, and they are denoted by D and E, respectively. The study of the vertical stability shows that all the periodic orbits found are stable concerning the vertical coordinate.

2.4

2.4.1

The Tools: Numerical Continuation of Periodic Orbits and Analysis of Bifurcations Numerical Continuation of Periodic and Autonomous Equations

Orbits for

Nonautonomous

Let us put equation (2.4) in the form *

=

f(v,t,e),

veR6,

(2.7)

where ems is used instead of ms- The function / is Tg-periodic and we look for orbits having a period equal to kTg, k small positive integer. Let $4 denote the flow of (2.7) under time t, starting at t = 0. If the initial condition is v = VQ at t = 0, then we are asking for a solution of the equation G{v0,e)

=

$kTs(vo,e)-vo

= 0.

(2.8)

For a fixed e, equation (2.8) can be solved by Newton's method. Starting at some approximate value VQ ' we define recursively v{0n+1) = vin) - (DV0*kTs(vin),e)

- Id)-H*kTsUn\e)

- «< n) ).

In this formula DVo$kTs(vo >e) denotes the differential of the final point with respect to the initial conditions. This can be obtained either by numerical differentiation or using the variational equations (2.5). We have used the last approach. The implicit function theorem ensures that no troubles should appear in Newton's method unless 1 6 spec(DVo $kTs (vo >e)) (or if there is some eigenvalue rather close to 1). When 1 is an eigenvalue we are faced with a bifurcation. It is also interesting to study the case of an eigenvalue being an mth root of unity. Then the bifurcated orbits have period equal to mTs. This topic will be studied later on in this section. When vo is obtained for a given e, the parameter e can be changed by some amount Ae. As a first approximation we have G{v0 + Av0,e + Ae) ~ DVoG(v0,e)Av0

+

DtG(v0,e)Ae.

22

Periodic Orbits of the Bicircular Problem and Their

Stability

The differential DVoG is the same as that in Newton's method. The differential DeG can again be obtained either by numerical differentiation or by integration of the variational equations, which in this case can be written as

w = Dvf-w

+ DJ,

weR6,

with w(0) = 0 and where w =

Dt$kTs(v0,e).

In the next section we refer also to some periodic orbits for e = 0, i.e., the RTBP. As this system is autonomous, at least one of the eigenvalues of Dv$kTs (v) ls equal to one (in fact at least two due to the Hamiltonian character). This is due to the fact that we can select as initial point any point in the orbit. Hence, in order to get periodic orbits we proceed in the following way. As we are looking for periodic orbits around the triangular points, we use a Poincare section through y = ±\fi/2. Starting at one point (x0,yo = ± - \ / 3 / 2 , z 0 , x 0 , y 0 , i 0 ) we compute the next intersections of the forward flow with the Poincare surface looking for a point coinciding with the initial one. One should be careful concerning the right number of intersections due to the loops which appear in some orbits. Let -P(uJo) be the image of some initial point Wo under the Poincare map. We suppose that the required time is tj. We ask for Q(w0) = P(w0) - W0 = 0. As the energy is preserved, DP(WQ) has an eigenvalue equal to 1. Therefore, we cannot Use directly Newton's method. One possibility is to keep one of the variables fixed and ask only for 4 components of Q(WQ) be equal to 0, the fifth one must be zero due to the conservation of the energy. However this approach has the drawback of the lack of symmetry and, what is worse, one does not know beforehand if the variable kept fixed during the iteration procedure reaches some extremum and hence, cannot be used as local parameter for the family of periodic orbits. Let P be 4 components of P (for instance Xf,Zf,if,Zf). We ask for Q(W0) - P(W0) - (W0)4 = 0, where (u>o)4 contains 4 components of Wo, exactly the same as those we use in P. We note that Q sends a five-dimensional space into a four-dimensional one. The generic Q = 0 is a curve in the five-dimensional space. Given an approximate Wo we refine it to some WQ + Awo such that 0 = Q(W0 + Aw0) « Q{w0) +

DQ(W0)AW0,

with the additional condition that the norm of Awo (with respect to some metric) be minimum.

The Tools: Numerical

Continuation

of Periodic Orbits and Analysis

of Bifurcations

23

The prediction of a point in the Poincare section corresponding to another orbit of the family is done by differentiation of Q(wo) = 0. We have D^0Q • Aw0 = 0. From this (AiiTo)i _ (AtiJo), Ai

Aj

where (AWo)i means the ith component of AWQ and Ai is the minor obtained from Dw0Q when the ith column is suppressed. Finally, using as a convenient parameter the arclength s we can write d(w0)i ds

Ai

= 1,...,5.

7E¥

An Adams-Bashforth method (with increasing order in the first steps) allows to predict the successive points. The differential of the Poincare map can be obtained numerically or using variational equations. In the last approach let M be the 6 x 6 variational matrix obtained when going from wo to P(WQ)- As the time is not fixed and the initial and final values of y are kept fixed we have ( Ax 0 \ 0 Az 0 M Ai0 Ay0

(

x \

( - \ 0

y

z

+

At

X

(2.9)

=

y

V Ai0 J

S

\

\- J

I t=t,

Hence At =

y(tf)

(m 2 i AZQ + m2zAzQ + m2iAx0

+ m2hAy0 + m2&Az0).

The five components left free on the right-hand side of (2.9) should be equal to ( Az 0 \ A^o DP{w0)

A±0 Ayo

V Ai0 J Let DP(w0) = (riij), i,j = l,...,5. n

u

=TOU

h

-m2i-

nu = mi+1,1 - m 2 i - ^ — ,

Then we have, by substitution of At in (2.9)

n\j = m-ij+i

h

- m2,j+i:

riij =mi+1j+i-m2j+i^j—,

2,...,5,

i,j =

2,...,5,

24

Periodic Orbits of the Bicircular Problem and Their

where f1: f2, / 3 , fi, h and f6 denote x(tf), respectively. 2.4.2

y(tf),

z(tf),

Stability

x(tf),

y(tf)

and

Bifurcations of Periodic Orbits: From the Autonomous the Nonautonomous Periodic System

z{tf),

to

Now we turn to the problem of bifurcation. We will consider two different kinds of problems: (1) How a periodic orbit of an autonomous system can be continued when a periodic perturbation is included. (2) How to detect the branches produced when one of the eigenvalues (of a nonautonomous system) becomes equal to the unity. For the first one we consider the system x =

f{x)+eg(x,t),

with g T-periodic with respect to t and x £ E 4 , as a model of the perturbed planar RTBP. Let xo(t) be a periodic solution of x = f(x) with period T. We wish to know if the periodic orbit can be continued for small e and which one is the right initial point (in the autonomous system every point in xo(t) can be taken as the initial point for t — 0). Proceeding to the first order in e we put x = XQ + exi and the equation for X\ is xx{t)

=

D/(x„(t))3;i+ 9 (x 0 ( 1 • I N I T I A L AMP FINAL EPOCHS OF INTECtATIOHt I 7 S S 7 . S 7 4 0 M M 19IBB.SM31131 I N I T I A L CONDITIONS FINAL CONB1T10MS V(l> - - • . S 2 t S S « M « t * t « 0 * * t YFtl> - - * . l t 7 4 M 3 B S t 5 C t D * » Y • - • . l * S 1 3 7 M « t » t « 0 t » l V F I J I • - • . I 1 5 9 1 S 3 M 1 7 4 8 B + A I Y(J) « -t.llltS12f«24f,!0-15 VFI3) • -•.88*4*74B4BlMB-*3 V(4> • - • . 2 9 4 S B * * * « M * * 0 * M V F t l l - - • . 3 3 7 I 0 7 X t t M S I D * « t V(5) * . 1 5 1 7 S * * « 0 « # * B * M V f ( S ) • -t.39«144fi«S*3SSD-01 VIH • B . l f i 7 3 9 S 7 9 7 7 4 2 £ B - l f V F I t l • -•.1SSC35B43B7S2D-01

1

•

•

l

I

i

t

I

'(1.2)

i_47B98*tO

J.*7320jrtjfc^

-*'7Sirtff / :.777SlfHl | |

L-I7SIDUM%

-9«BD»«tw!

ul«l4D+Bt

L.1IIC0*S1

L-ISI9D+BI

...,.._ ..__ ^ L . I » Bp3 D - A 1 _r_ _ ^ _ 1

,

T ^

^

,

'(1,3)

-.MBjSD***

,

-.(73eBt*»

1

1

1

L.1BB3B-BI

O

i

^

y

^

"

s,

]

'

-

a^iotM 1

t.LB2«D*BB 1

• -•O^lDtfi 1

'(2.3)

* F^lT"

•~4t**ty£ii€^H£lKnSSESj?y**^L

^^

"*"\ _

L.147fVl>->M

-

^*^^^^^t^^-^^

utTtio-^a^^.

,

I

L.»7B1Q)>*2\

) 1 ~

uU

U4S3lp-p7S

\

/y£

^42SSV-OsK

J

&) ~

_

-xwib^i^f ,

•^im\ a-+i/ -.919,3D***

,

-.S72^D*M

,

-.23^lD+«*

(

• .1*2|«D+M

,

• . 439,1V**!

)

:. 15*20-01

_

L~2KBD-tl -,ia^«*»i

-.ItB^D'tl ,

'T'1*

-.B7yo«*t

(

-.479,9B***

Fig. 3.5 Motion starting near the planar periodic orbit around the equilibrium point L5 computed by Kolenkiewicz and Carpenter. The first component of the velocity x has been modified by an amount of - 0 . 0 1 . For the top figure the simulation goes from JD50 = 17 997.574 to JD50 = 18 360, and for the bottom one from JD50 = 17 997.574 t o JD50 = 19 100.

42

Numerical Simulations

of the Motion

i

SIHUlAriOlt OF NOTION I N THE S O U * SVtTEN SVSTCfl OF U M T S ! ADIflENSlOMAL C M T t f - M O f l •DOTTED IWDEL OF PWTtOM FOR THE; SOLAR SVSTtfli .{ « M M I I t 1 IS 1 1 M • I « t t J » t AND FINAL EPOCHS OF M T E C M T I O N I 17997.574MOO t l 3 U . t I I 3 N I 4 I N I T I A L CONDITIONS F l M » i CONBITlwtS V U I • -•.S211S*Mt««MO**0 v r i n • -•,43l7IJH*S«ll>«M V(2) • ••.t«S137M***MD*01 * F ( 2 I • - • . t 7 9 * 5 « l T S 7 S l D * M V O I « ••.12*?3*331Ht3D-lE VFOI • • .22SMIBIM2939-K V M 1 • - I . I h M H H H H D H I VFt4> • • .C4522$7iM*Ct&~ll VI5I • t , l 6 t ? S 0 t t M t M D t M VFCSI • - • . 3 2 l S H 2 7 ( 7 7 f l B 6 - i l VtSl • - t . l f » 3 l 5 I I Z I I 7 D - l t V M S ) • - t . 7 l 7 S « 1 7 4 3 B t S O - «

1

I

:.fi7$2B»**

I

1

f

1

'(1,2>

,

i_734i0*»/r/

x.7S4cV»ltl

\

-15410A*

V \

L~813BD*««\

:~8734D*M

/)) "

-l»33D*«l

1

I

1

1

I

I

1

I

'u.3l

UW71»-W

\?\^

y^^

1 /£~)

LZtMKKJ^Oy V ^ S v \ U1M3D-H

-f "\

i«t27si-M /

/^

^IH4l-M l

I ^ - ^ ^ ^

H

I /

/

^ - s ^

-.4«4M**I

-.444^0+**

,

-.574^0+**

L

1

i

1

1

'(2.3>

X J V

-.2?48D*H

~

—

^l«T*frW(_

_

- . i»at»»M

^aCMfii«I

~ ~ -

\

1.40MD-92V

;.f442S-9l -.1M3MM

.

-.97340**9

,

-.KilB^M

,

-.TS^tD***

(

-.tl^S0*M

V^,

SldULATIOtt OF NOTION I N TMC 5 0 L M Jv$TE« SYSTEM OF U H I I S i AlinEHSIOMtL CMTH-IIOON MOFTEII HODCL OF NOTION FOR THE SOLA* SVSTCNI • * I 1 It • I I I N I T I A L AND FINAL EPOCHS OF INTEGRATION! 17097. S T 4 0 M O 1 9 M 2 . 9 7 C 1 I 2 S 5 I N I T I A L CONDITIONS FINAL CON! I T IONS V(l> - - • . S 2 I I S t * M * * * * 0 * * * VFID • • .S92*mi*«22IO**l VIE) - -*.l*S13?«**«»ltD**l VFI2) • *.2SU3I99«eS9»*«J V I } ) > -*.12»73*331«*S3D-1S VFI3) > *.1*9749I2249*70**1 V - - * . U l S H « I H M « l t M VFMft«MMmt4> YFC4I - - • • I 2 3 « « 7 i a 7 M » ^ « t VIS» - « . 1 4 1 ? | M « « M M S * M VFfS> • - • • * • « > * > S 5 H U S ! ! ? Vt*> • t . l » S I 3 C « L 3 4 « ? B - l f t V F t i ) • -«.»7S?SS«*23D9I>-R1

1^90*—

-.«^S»*—

|

A.SJ^ID***

,

-.7tR7D-tl

A.tHjlO»4>l

t

••3T^TB*4W)

Fig. 3.7 Motion starting near the planar periodic orbit around the equilibrium point L5 computed by Kolenkiewicz and Carpenter. The second component of the velocity y has been modified by an amount of —0.01. For the top figure the simulation goes from JD50 = 17 997.574 to JD50 = 18 360, and for the bottom one from JD50 = 17 997.574 to JD50 = 19 100.

44

Numerical

Simulations

of the

Motion

SIMULATION OF NOTION I N THC SOLA* svsrcn SvSTEN Or u t t l T $ i ADINENSIOMAL EARTH-NOON A 6 0 » U B NOBEL Or NOTION FOR THE SOLA* $VSTEN i i i i i i i i t i ti # r ' I N I T I A L AMP FINAL EPOCHS OF INTtCRATIOni 11*12.*••*«•»* 2IIM.t2SS7IS» I N I T I A L CONDITIONS FINAL C0H8ITI0HS YC11 • - • . 5 2 l S 5 t * « M » M I > * 0 V F f l J • - * . * » B ? 3 Z C X I 9 3 a D * M V ( 2 ) • - « . i « « i 3 ? » M « M a O * t l YF « . 1 S 1 ? I M * M * R * D * 4 B VF(5> • - • - 1738S18S3B321B-R1 Vi«) • - * . 1 2 7 V M 7 B 1 I 7 « 0 - U VF > A . * l 7 » 3 7 1 I t 7 4 2 4 0 t | « VIS) • - f . l 2 7 V 7 4 7 « U 7 2 B £ - U VF(I> • t . 1 1 1 2 M l l t 4 7 3 £ B - S l

MOaitt*

f

-.S17j6B»B»

t

-.M^ID*—

|

«.«4|3&-B1

l.3T»B
• •f.lM324B3f9CS3D*tl V(3> • *.1**«3**3?2l*S0-tt VTI)) • • .2M32ECE247UB-tl V(41 - -*.K115*ICStt»B-*C YF • - » . l t * M M S * « 4 M * * * VIS) • 9.|9124ieM4I32D4M VFIII - -•a2S7S977I2323B*»* V - -•.229CI44SI383T9-*! V14) . -*.427*«71313I7»D+M VF«4) • • .22144SSM4293B*** V«S1 - a.UlS93«113M*B»M VFfS) - - * . 1S3BS3IS439I«*M V(Sl • -t.H74S*47M»33ft~*l VF • -*.3ESB»4f1IU84B-II

\i

u4W*BVM/

* . 39-f7»*«l - . I I ^ I » * « I

;

-.

1 H

-.«(«*.. , - .SC^TS***

.D..I

1

I

1

1

1

-.«,...#.

l

'(1.2.

:_4fi22S***^

L.EC7i»tf/ ^7S9«wLI|

^9744B**«V :.1«77D**1 _tl7«C+«I -12IZB**! -.M91D+** , - . I T y f B * * * , 1

1

1

1

1

1

1

1

'(1.3>

1

T

1

1

•.23^30***

(

1

1

* . B^tBtM , 1

*.44*t»«M

'

'iJ^JC

-

UMt3y#4/VW L11MB-4&&

L :m D

- * T(yC

\

L.1MUM j \

-

L>34*SB-»VJC

fc"SJ

-

L>22*3B-*I

, \

^7l7K\/ OOvH

"

UIIMB\4/

XlfvA/V ^^^7

L.37MD-u\. t.J*tSD-*f^ ^U*U-*lV

t

i.n*t^*rVKQa&

^333*B^U^06 (

'^^-i_a«W"' -.«Ty?P*M ( - . 1 3 4 , 3 B * * *

t-3»3*o{tT^

4 f ^

i c-44E3fr-*i • , 44*|C&4«* -.121^84*1

i

L.44HM1 -.9^10+M

(

*.l*3|l»+l* ,

-.lBTjTDttl

,

-.•7|*D*M ,

- . M^ifr^** ,

-.4i«a»**«

Fig. 3.9 Pieces of the trajectory displayed in Figure 3.8. In the top figure time varies between JD50 = 19 200 and JD50 = 20 700 and in the bottom one between JD50 = 20 700 and JD50 = 21800.

This page is intentionally left blank

Chapter 4

The Equations of Motion

The equations of the real motion near L 4 or L5 are written. The full solar system and the radiation pressure are considered. The Hamiltonian of the problem is also given. The terms of the Lagrangian and the Hamiltonian which involve Legendre polynomials are expanded as power series in x, y and z with coefficients periodic functions of time (with incommensurable frequencies). A Fourier analysis of such functions, as well as the other functions needed for the equations and the Hamiltonian, is performed. The big size of the region where the motion takes place leads to convergence problems in the expansions. The number of terms to be retained in the expansions of the Hamiltonian and the equations, even with large tolerances, is given. This suggests to write down the equations of motion in a simplified way. The full coefficients which appear in the equations finally used are given. Checks against the full vector field are done. 4.1

Reference Systems

We consider an inertial reference frame with the origin at the center of masses of the solar system and the axes parallel to the ecliptic ones. The equations of motion of a spacecraft in the solar system can be written as £

=

y

GA(RA - R)

A€{S,E,M,Pi,--,Pk}

I

A

~

'

where G is the gravitational constant, and R and RA are the position vectors of the spacecraft and the body of mass A respectively. The summation is taken for the Sun (5), Earth (E), Moon (M) and the planets (Pk). However, the above reference system is not convenient to study the motion of a spacecraft in the vicinity of the libration points L4 or L5 corresponding to the Earth-Moon system. As usual, the libration points are defined as the ones that form an equilateral triangle with the Earth and the Moon. These points are placed at the instantaneous plane of the motion of the Moon around the Earth (see Figure 4.1). 47

48

The Equations

of Motion

TEM

Fig. 4.1

M

Reference systems at the triangular equilibrium points.

We define a normalized reference system centered at the libration points and given by the unitary vectors ei, e2

,n-2 a"~"

is

x x +TaH( - s) J2 ^+r p «( c o s 5 i) k3' n>2 S r

+ y + 2x-y

K

E

+ (X- XA) ^2 T^+T-Pn(COS A i )

?EA

A€{S,Pu...,Pk,M}

= C(ll)x

an~2^

XEA

MA

i>2

r

A

+ C(12)y + C{13)z + C(U)x + C(15)y + C{16)z + C(17)

+ —(1 -fiM 3

A;

+ HE)(V - VE) ] T a " " 2 P „ ( C O S S I ) n>2

-,n-2

K

x p cos5 +T3t i) k3 s(y - vs) J2 ^+r «( n>2 rS

K 1\ k3

v ^

-W^ + ty-y^Yl ' EA

Ae{S,Pu...,Pk,M}

an~2^ -^+iPn(cosAl)

n>2

r

A

C(21)x + C{22)y + C(23)z + C(25)y + C(26)i + C(27) + ^ ( 1 - HM + VE)z J2 a " ~ 2 p n ( c o s S i ) n>2 +

an~2^ P

tfV§(Z ~ ZS) J2 ^ + T n ( c O s 5 i ) n>2 VS

K_

+

¥

£

fJ-A

A6{S,Pi,...,P fe ,M}

ZEA 3 rf EA

+ (Z - zA) Y, n>2

a " " 2 ^ P COsA ^+T n( ^)

T

A

where C(l) C(3) C(5) C(12) C(16) (7(23) C(ll)

= = = = = = =

2e + e2+A-kk~1, D, 2(e + E), 2e + e2 + £ - M r 1 , 2F, C-JtA;" 1 , -C(2),

C{2) = (7(4) = C{7) = C(13) = C(17) = C(27) = C(14) =

2kk~1E + E, -2kk-\ (kk-1 - A)xE - {2kk~lE + E)yE, 2kk'1F + F, (2kk-1E + E)xE + (kk-1-B)yE, l - £ > i B + {2kk~ F + F)yE, -C(5),

The Hamiltonian

C(15) (7(22) C(26) e 5

= (7(4), = -C(13), = C(4), = (9—1, = B-(l + e)2,

and the Related

Expansions

51

C(21) == C(3), C(25) == -C(16), A

2?

= A-(l + e)2, = £ - ( l + e) 2 .

We have also used Pn(a) = - - r - P „ - 2 ( a ) , where P„ denotes the nth Legendre polynomial. 4.3

The Hamiltonian and the Related Expansions

We start with the expression of the Lagrangian (given in [17]) of the problem when normalized coordinates centered at the instantaneous L45 point of the Earth-Moon system are used. The momenta px, py, pz are introduce through r\ -r

px =

—7ox

= k2x + kkx — k2Ey — kkxE + k2EyE,

TTT- — k2y + kky + k2(Ex - Fz) - kkyE -k2ExE, oy dL, pz = ^ = k2z + khz + k2Fy - k2FyE. dz From these relations it is easy to express x, y, z as functions of the positions and momenta, the coefficients being functions of time. The Hamiltonian is obtained as H = Y^Pi' Qi ~ L, pM-

9 =

where in qi, as well as in x1 + y2 + z2, xx + yy + zz, E(xy — yx) + F(yz — zy), xEx + yEy and —EyEx + ExEy + FyEz. The variables x, y, z are substituted by their expressions in terms of positions and momenta. After a somewhat lengthy computation the following Hamiltonian is obtained: H

=

-k~2(p2x+p2y+p2z)-kk~1(xpx+ypy +F(zpy - ypz) + (kk^XE + l^k2(E2 -k2{EF

- EyE)px

- A)\ x2 + ^k2(E2+F2

+ D)xz

+ zpz) + + (kk^ys

E(ypx-xpy) + ExE)py

- B)j y2 + \^k\F'

+ Fyspz - C)

52

The Equations

+k2(A - E2)xEx

of

Motion

+ k2{B - E2 - F2)yEy

+ k2(EF + D)xEz

+term purely depending on time -Kk~l(l

-HM + n&) Yl

"•"PnicosEi)

*>i

_^-i^^(^y Pn(cos5l ) -Kk-1

Y,

acosA 2

^

1 v ^ / « V n /

T

r A

EA

A€{S,M,Pi,---,Pk}

n > l ^

,. \

r A j

The time-dependent functions which appear as coefficients have been partly computed in the development of the Lagrangian. The only additional functions are k~2,E, k2(E2 - A), k2(E2 + F2 - B), k2(F2 - C), k2(EF + D). The computation of the function A;-1 is also useful.

4.4

Some Useful Expansions

In order to get a quasi-periodic solution of the problem, it is convenient to develop the right-hand side of the equations as series of the following types 2_.a,ijkrXly^zkF(vrt

+ 4>r), or it y ^ a r F ( ^ r ^ + 4>r),

where i) = x,,y or i and F stands for one of the trigonometric functions sine or cosine. A Fourier analysis of the functions that appear in the equations will give the coefficients (aijkr, ar), the frequencies (vr) and the phases (4>r) involved in the dominant terms. We deal separately with the terms of the equations which involve the Legendre polynomials. These terms come through derivation with respect to x, y or z, of the corresponding terms of the Lagrangian. So it is more convenient to develop the summations of the Lagrangian and then to derivate. Using the normalized system of coordinates we have xxE +yyE + zzE cos £1

=

;

arE XXM+WM

cos Mi

=

+

1 /= - — (x + sVoy), Za

ZZM

1 ,

= —(x -

:

^a

Q.TM

Therefore, it is not difficult to see that

n>l

n>l

k=0

V

'

/r

x

sV3y).

53

Some Useful Expansions

^—' m\(n — 2k — m)\ v

m=0

'

*-?

ni!n 2 !n 3 !

2ni+n—2k—m2n2+m d, y

2ns z j

ni+n2+ri3=k

where ( 1 ii A A= =E A= M \ 0 if i4 M ''

_ ff 1 if A 4=E £7 ~ \ \ -- 1 if A A= =M M ' l if

rr _

S

_ f 1 for Li \ - l for L5

_

A routine, EXPEA, gives the coefficients of each monomial xly^zk up to a given order n = i + j + k. This routine has been checked comparing the development with the generating function 1

y/1 — la cos A\ + a2 where A = E or M. It has been seen that to get a difference of the order of 1 0 - 5 , terms up to order 20 are required if the magnitude of a is 0.5. We note that the coefficients in the development of J2 anPn(cos Ar), n>l

for A = E or M are constants and so no Fourier analysis is needed for these terms. For the planets and the Sun, similar expansions are obtained where the coefficients of the monomials are functions of time. We have For the planets

Kk-^A^Tf^-)

P n (cosAi).

For the Sun 1

Kk-\ns+»s)—

/

L

n

\

'E(^ ) rs^VsJ

PnicosSt).

In general, we have the following expansion 1/2]

k

k — ni

n—2k n—2k—l\

S£S!F^^)|£ v f-±\

' fy±\

2

(*±\

3

S zlWa!(„_2Jb - Ix - «a)! (1_\

T2n1+h

U J {rA) {rA) U-J *

2n2+l2y2ns+h

V *>

54

The Equations

of Motion

where fa = n — Ik — fa — fa.

The terms of the expansion can be collected in the form /(9i.92,92,93

where

The coefficients (c 9ig293 i 1 i 2 i 3 ) and the exponents (qi, q2, q3, fa, fa, fa) are computed up to a given order n = qi + q^ + q% in the routine EXPEAT. The routine is checked using the generating function 1 \Jf\

1 2

- 2CL?A cos Ai + a

rA

Some computations for the Sun, Venus and Jupiter show that order 3 is needed for the Sun, and order 2 is sufficient for the planets in order to have a difference of order 1CT5. A routine, FUN4, has been done to compute the functions at a given day. This routine will be used by program FURI to do the Fourier analysis. We classify the functions in the following way (1) From the Lagrangian: f{qi,q2,

to to

1 o o o o o o o o o o o

-a to o

I—1

o to o o o o o o o o 00 o o o

O to o

Ol

h-» Cn

H-* l—i ~J OS

1—>

OS

a> as

o

CD

h-1

o to

O

as

h-»

-~4

1—>

o to as 1—> o o to o 1 to to o o 0 0 0 0 to I—

h->

oo

to

CO

h-1

to to to to to to to to to to

1

o

to

I-" O C O O to C O

t—*

to to to

4^ to

W

OS Cn

CO

Fourier Analysis:

i 7

7 7 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 Table 4.3

3

4 13 16 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 0 1 2 4 5 6 7 9 10 11 12 13 14 15 16

Au

The Relevant Frequencies and the Related Coefficients

• 103

Bn • 103

0.5

0.8

-4.1 -0.5

-7.1 -0.8

6.0 2.2 0.4

0.0 -0.3

39.9

-0.8

0.3

1.2 0.0

-1.0 208.4 -0.8 -0.2

0.0 33.7 17.0

1.0 5.3 1.4 10.7

3.6 0.7 71.9

0.5 -1.9 370.5 -1.4 -0.4

1.1 0.8 51.6 30.3

1.6 8.5

1.7

-0.6

0.0 1.0 2.3 8.3 0.0 0.2 0.9 0.0 0.0 -0.4

3.0 0.1 2.2 0.0 -0.6 -0.1

1.7 0.0 3.4 0.0 0.0 0.0 0.0

i

3

9 9 9 10 10 10 10 11 11 11 11 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13

17 18 19 3 8 20 21 3 8 20 21 4 7 11 13 16 18 19 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17

Mi • 103

Bn • 103

2.4 1.0 1.4 0.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.4

-0.7

3.9 1.2 0.0 0.0 0.0 0.0 0.9 0.5 0.6

-1.4

2.0 0.6 -0.5 -0.1

-7.1 -0.8

0.2 4.1 0.4

0.6 0.7

-0.1 -0.2

-6.4 -2.2 -0.4 -39.5 -0.3

0.0 0.2

1.1 -218.3

0.8 0.2 -0.8 -53.5 -20.8 -1.7 -8.8 -1.8

-1.6

0.0 -1.2

0.0 0.4 0.0 -1.0 -3.4

0.0 0.0 0.0 0.0 0.0

Coefficients of the Fourier expansion of SO(i) (continuation of Table 4.2).

61

62

The Equations

i

1 1 1 1 1 1 2 2 2 2 2 2 3 3 4 4 4 Table 4.4

J Aij • 103 0 1.4 4 -0.5 12 0.5 4.1 13 16 0.5 0.1 18 0 2.5 4 0.8 12 0.2 -7.2 13 16 -0.8 18 -0.1 0.6 3 8 -0.6 0 1.4 4 -0.5 4.1 13

Bij

• 10 3

0.0 0.8 0.2 -7.1 -0.8 -0.1

0.0 0.5 -0.5 -4.1 -0.5 -0.1

0.4 -0.4

0.0 0.0 0.0

of Motion

i

4 4 5 5 5 5 5 6 6 7 7 7 7 7 8 8 9

J 16 18 4 12 13 16 18 3 8 0 4 13 16 18 3 8 0

Coefficients of the Fourier expansion of SOS(i)

Aj

• 10 3

0.5 0.1 0.0 0.5 0.0 0.0 0.0 0.0 0.0 1.4 0.5 -4.1 -0.5 -0.1

0.7 -0.7 -2.8

B^

• 10 3

0.0 0.0 0.9 -0.1 -8.3 -0.9 -0.1

0.7 -0.7

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

= ^ • Aij cos(vjt + 4>j) + Bij sm{yjt +

2,

with A either Earth or Moon, is carried out by a suitable routine. However at this point we introduce a piece of information that in fact, will only be available in section 5.3 of the next chapter. This is the real size of the periodic orbits obtained from the previous equations when only the frequencies 6, 13 and 18 retained. These orbits have as period the synodical period of the Sun in the Earth-Moon system (or a multiple of it). They can be considered as perturbations of the orbits of the bicircular problem. But the change in size is big, leading to orbits of the types A, B, F and G (see Chapter 2) far away from Lii5. The terms mentioned at the beginning of this section come purely from the RTBP. We analyze now the number of terms to retain in the expressions of the Hamiltonian and in the equations. The worst case corresponds obviously to the equations, because a term like xk in the Hamiltonian produces a term kxk~1 in the equations and, hence, it'is multiplied in modulus by k/\x\. We have selected 11 points which are close to the periodic orbit of type A of the intermediate equations. As this orbit is planar we have selected a small uniform

Concrete Expansions

of the Hamiltonian

and the

Functions

63

value of z (z = 0.1) in the computations but the contribution of z is not significant. The selected points have normalized coordinates as follows: (-0.8,0.1), (0.4,-0.3), (-0.6,-0.1).

(-0.6,0.3), (0.2,-0.4),

(-0.4,0.4), (0..-0.4),

(0.0,0.4), (-0.2,-0.3),

(0.2,0.2), (-0.4,-0.2),

In the Lagrangian or in the Hamiltonian, there appears the function 1 - V-M (l + x + sVSy + a2)1/2

(1-x

RA£ + sVSy +

a2)1/2

(4.1)

with s = 1 for L 4 , s = - 1 for L 5 and a2 — x2 + y2 + z2. To have some idea about the suitable order to be reached in the expansions we have computed: i) The value of (4.1) directly, ii) The value of (4.1) expanded in powers of the coordinates up to a certain order, iii) The same as in ii) but skipping those terms whose absolute value is less than a given threshold. The objective is to obtain small differences i)—iii) but keeping as few terms as possible. The study has been done for the 11 points above mentioned and looking for the worse case. We show a sample of results in Table 4.5.

Order 25

30

35 40 Table 4.5

Threshold 2.0 xlO" 4 1.2 xlO" 4 1.0 x l 0 ~ 4 0.8 xlO" 4 0.6 xlO" 4 2.0 xlO" 4 1.2 xlO" 4 1.0 x l O - 4 0.8 x l 0 ~ 4 0.6 xlO" 4 1.0 xlO" 4 1.0 xlO" 4

i)-iii) 19.5 xlO" 4 6.4 xlO" 4 5.0 xlO" 4 6.0 xlO" 4 8.0 xlO" 4 18.0 xlO" 4 5.0 x l O - 4 2.7 x l O - 4 5.9 xlO" 4 3.6 x l O - 4 3.0 xlO" 4 2.7 xlO" 4

Number of terms to retain 314 347 363 392 422 372 422 446 484 521 510 556

Results of the different truncations of the expansions of function (4.1).

If we are satisfied with errors less than or equal to 5 • 1 0 - 4 one should use at least 363 terms in the expansion of the Hamiltonian. This is much worse in the equations.

64

The Equations

Order 40

35

30 32 34 36 38 Table 4.6

Threshold 1.0 x 10" 4 1.2 x 10- 4 1.5 x 10" 4 2.0 x 10" 4 3.0 x 10- 4 4.0 x 10~ 4 5.0 x 10- 4 6.0 x 10" 4 8.0 x 10" 4 10.0 x 10- 4 2.0 x 10" 4 3.0 x 10" 4 4.0 x 10~ 4 5.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4 3.0 x 10" 4

of Motion

Maximum error 1.7 x 10~ 3 1.9 x 10" 3 1.7 x 10" 3 1.8 x 10~ 3 2.4 x 10~ 3 2.9 x 10" 3 3.4 x 10~ 3 5.7 x 10" 3 5.0 x 10" 3 6.6 x l O - 3 3.1 x 10~ 3 1.7 x 10~ 3 3.6 x 10~ 3 3.4 x 10" 3 6.2 x 10~ 3 7.5 x 10" 3 3.2 x 10" 3 2.2 x 10" 3 3.0 x 10" 3

Total number of terms to retain 1428 1375 1320 1233 1140 1086 1033 1000 943 887 1037 962 918 879 776 853 925 994 1070

Bounds of the errors of the expanded equations.

We have expanded the equations in powers of x, y, z. Then we have compared the direct evaluation of the derivatives of (4.1) against the evaluation of the expansion keeping only the terms greater than some threshold in absolute value. The maximum of the differences at the 11 preselected points is displayed. We remark that the terms retained in the expansions are not the same for the 11 points. So we count the terms that have been used at least in one of the points and give the total number of terms to be used. We merely note that the expansions up to order 40 (the maximum order allowed in our program) require 6390 terms. Some unavoidable error is due to that truncation at order 40. We present some results in Table 4.6. Hence one of the most appealing parameters are: order 35, threshold 3.0 x 10~ 4 giving errors as small as 1.7 x 10~ 3 and requiring 962 terms. Even taking these "excellent" parameters the number of terms in the equations becomes unmanageable. As a conclusion, the approach which uses the expansions of the Hamiltonian or of the equations is not feasible. Hence we should look for other forms of the equations which are more useful for the applications. Furthermore we wish to comment the results of Kamel in [19]. There, a Lie series method is used to study the motion near 1*4,5 in the Earth-Moon system, taking into account some of the terms of the lunar and solar motion. The perturbation method is used to eliminate short and long

Simplified Normalized Equations.

Tests

65

periodic terms of the Hamiltonian. At the end, a formally integrable Hamiltonian is found and the author looks for fixed points. Those will give rise, going back in the transformation, to quasi-periodic solutions. It is found that there are fixed points, which appear if the theory is carried out to a given order, but not if the order is increased. This can be due to the big amount of terms required to have convergence. We will also refer to convergence problems in Chapters 6 and 7. 4.7

Simplified Normalized Equations. Tests

Prom what has been said in section 4.6 it is concluded that a simpler form should be used for the equations. In this new form we retain the terms called SO(i), SOS(i) in section 4.5 and the terms coming from the RTBP are left as they are (with respect to the point L4 or L5 taken as origin and multiplied by the factor Kk~3). Let us introduce some auxiliary functions P(i), i = 1, • • •, 20 which account for the terms appearing in SO(i) and SOS(i) according to the following definition: P(l) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(10) P(ll) P(12) P(13) P(14) P(15) P(16) P(17) P(18) P(19) P(20)

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

5 0 5 ( 1 ) + 50(7), 1 + 2 5 0 5 ( 4 ) + 0.2550(1)+ 50(2), 5 0 5 ( 5 ) + 50(3) + a 50(1), 5 0 5 ( 6 ) + 50(4), 50(5), 2 + 50(6), 1 + 50(1), 5 0 5 ( 2 ) + 50(12), 5 0 5 ( 5 ) + 50(8) + a 50(1), 1 + 2505(7)+ 50(9)-1.2550(1), 5 0 5 ( 8 ) + 50(10), -2-50(6), 50(5), 50(11), SOS(3) + yESO(lQ)-xESO(4), 5 0 5 ( 6 ) + 50(4), 505(8)-50(10), 2 5 0 5 ( 9 ) + 5 0 ( 1 3 ) + 50(1), -50(11), S0(5),

where a stands for expressed as

Q^MXEVE-

With these definitions the functions P(i) can be

m

m

P(i) = Aito + ^2 Ai,j cos6j + ^2 Bid sindj, j=i

j=i

66

The Equations

of Motion

where Oj = Ujtn + j, Vj, fa being the frequency and phase as given in section 4.5 and tn the normalized time. The different values of i, j , Aij, Bij, are listed in Tables 4.7, 4.8 and 4.9. With this notation the equations of motion written in a simplified form, to be used from now on are (we shall denote them by S.E.):

=

P(7)

X-XE •—^

(1 -

X + XE 3 MM -

MM)

' PE

.

.

xE{l

-

2/J.M)

' PM

+P(1) + P(2)x + P(3)y + P(4)z + P(5)± + P(6)y, V

=

P(7)

y-VE,, •—^ U ' PE

—

. VM) 1

y-yE 3 MM PM

+P(8) + P(9)x + P(10)y + P(ll)z =

^(7)

--o—(l-MM) 1 PE

-

VE

+ P(12)x + P(13)y + P(14)i,

-fJ-M ' PM

+P(15) + P(16)x + P(17)y + P(18)z + P(19)y + P(20)z, where rps, TPM, denote the distances from the particle to the Earth and Moon, respectively, given by rpE = (x - xE)2 + (y- yEf We recall (xE,yB) 4.7.1

+ z2,

rPM = (x + xEf

= (-1/2,--y/3/2) for L4 and {xE,yE)

Tests of the Simplified

Normalized

+ (y -

VE?

+ z2.

= ( - 1 / 2 , ^ 3 / 2 ) for L5.

Equations

To be sure that the simplified system is meaningful we have written a test program which computes the vector field on the particle in ecliptic coordinates and then performs the transformation to the normalized ones as described in section 4.1. On the other hand we can evaluate directly the simplified vector field (already in normalized coordinates). Then we can select either a given point in the neighborhood of L4 in the phase space (introducing the position and velocity) or some points from the periodic orbits of the intermediate system, to be found in the next chapter. The relevant angle in these orbits is the mean elongation of the Sun which is expressed, as stated before, as 6 = 0.9251959855 • TN + 5.0920835091. We recall that TN = (JD50 - 18 262)/n M , JD50 = Julian Date since 1950.0 and nM mean motion of the Moon longitude. The biggest difference between the two vector fields in normalized coordinates, both in absolute terms and relative to the modulus of the vector field, is of the order 10~ 3 . The vector fields have been computed on points along the periodic orbit called A, following the notation of Chapter 5. Hence it seems reasonable to believe that the orbits found using the simplified equations can be easily modified to satisfy the complete system of equations.

Simplified Normalized Equations.

i 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 5 5 5 5 5 5 5

j 0 4 12 13 16 18 0 1 2 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 18 20 2 4 5 6 7 10 12

Ajj 0.140000e-2 0.000000e+0 0.500000e-3 0.000000e+0 0.000000e+0 0.100000e-3 0.100757e+l 0.142500e-2 0.275000e-3 0.314750e-l 0.250000e-3 -0.800000e-3 0.165000e+0 -0.650000e-3 -0.150000e-3 0.800000e-3 0.300000e-3 0.266500e-l 0.134500e-l 0.200000e-3 0.425000e-2 0.107500e-2 0.100000e-2 0.100000e-2 0.424867e-4 0.451547e-4 -0.802668e-4 -0.280141e-4 0.464887e-4 0.140448e-4 0.137926e-4 -0.394663e-4 0.535112e-4 0.992977e-3 -0.169897e-4 0.147495e-4 -0.140448e-4 -0.237352e-4 0.568839e-5 0.000000e+0 0.000000e+0 0.600000e-3 0.000000e+0 0.600000e-3 0.000000e+0 -0.200000e-3 0.600000e-3 0.190000e-2 Table 4.7

Bij 0.000000e+0 0.160000e-2 0.200000e-3 -0.142000e-l -0.160000e-2 -0.100000e-3 0.000000e+0 -0.500000e-4 0.132500e-2 0.000000e+0 0.950000e-3 0.000000e+0 -0.275000e-3 0.000000e+0 0.800000e-3 0.000000e+0 0.122500e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.535112e-4 0.521771e-4 0.180000e-2 -0.675560e-4 0.000000e+0 0.180266e-3 0.000000e+0 -0.140448e-4 -0.154845e-3 -0.165000e-l 0.000000e+0 -0.200000e-3 -0.180000e-2 0.000000e+0 -0.100000e-3 -0.100000e-2 -0.100000e-3 -0.169000e-l -0.100O00e-3 0.500000e-3 -0.107900e+0 0.100000e-3 -0.400000e-3

i 5 5 5 5 5 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 8 8 8 8 8 8 8 8 9 9 9 9 9

j 13 14 15 16 17 0 2 4 5 6 7 9 10 11 12 13 14 15 16 17 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 0 4 7 11 12 13 16 18 19 0 1 2 4

Tests

Mi 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.200000e+l 0.300000e-3 0.382000e-l 0.300000e-3 -0.110000e-2 0.217600e+0 -0.800000e-3 -0.200000e-3 0.500000e-3 0.600000e-3 0.429000e-l 0.148000e-l 0.100000e-2 0.530000e-2 0.100000e-2 0.100470e+l 0.170000e-2 0.300000e-3 0.315000e-l 0.200000e-3 -0.800000e-3 0.164400e+0 -0.600000e-3 -0.200000e-3 0.400000e-3 0.266000e-l 0.134000e-l 0.800000e-3 0.420000e-2 0.110000e-2 0.250000e-2 0.170000e-2 0.500000e-3 0.600000e-3 0.200000e-3 -0.143000e-l -0.160000e-2 0.500000e-3 0.700000e-3 0.424867e-4 0.451547e-4 0.197331e-4 -0.280141e-4

67

^±L -0.295000e-l -0.880000e-2 -0.700000e-3 -0.380000e-2 -0.700000e-3 0.000000e+0 0.150000e-2 0.000000e+0 0.120000e-2 0.000000e+0 -0.400000e-3 0.000000e+0 0.100000e-2 0.000000e+0 0.280000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.200000e-3 0.130000e-2 0.000000e+0 0.100000e-2 0.000000e+0 -0.300000e-3 0.000000e+0 0.800000e-3 0.170000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.100000e-3 0.200000e-3 -0.500000e-3 0.000000e+0 -0.100000e-3 -0.200000e-3 -0.200000e-3 0.000000e+0 -0.464887e-4 0.521771e-4 0.100000e-3

Cosine and sine coefficients of the functions P(i), i = 1 , . . . , 20.

68

The Equations i

3

Ajj

Jjjj

9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12

5 6 7 9 10 12 13 14 15 16 17 18 0 1 2 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 3 8 20 21 0 2 4 5 6 7 9 10 11 12 13 14 15 16

0.464887e-4 0.140448e-4 0.137926e-4 -0.394663e-4 0.535112e-4 -0.702240e-5 -0.169897e-4 0.147495e-4 -0.140448e-4 -0.237352e-4 0.568839e-5 0.000000e+0 0.100762e+l 0.147500e-2 0.325000e-3 0.335250e-l 0.250000e-3 -0.900000e-3 0.165000e+0 -0.650000e-3 -0.150000e-3 0.110000e-2 0.300000e-3 0.101500e-l 0.135500e-l 0.600000e-3 0.225000e-2 0.102500e-2 0.800000e-3 0.140000e-2 0.130000e-2 -0.140000e-2 0.390000e-2 0.120000e-2 -0.200000e+l -0.300000e-3 -0.382000e-l -0.300000e-3 0.110000e-2 -0.217600e+0 0.800000e-3 0.200000e-3 -0.500000e-3 -0.600000e-3 -0.429000e-l -0.148000e-l -0.100000e-2 -0.530000e-2

-0.675560e-4 0.000000e+0 -0.219733e-3 0.000000e+0 -0.140448e-4 0.451547e-4 0.000000e+0 0.000000e+0 0.200000e-3 0.000000e+0 0.000000e+0 -0.100000e-3 0.000000e+0 -0.150000e-3 0.137500e-2 0.100000e-3 0.950000e-3 0.000000e+0 -0.225000e-3 -0.100000e-3 0.700000e-3 0.000000e+0 0.127500e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.150000e-2 0.000000e+0 -0.120000e-2 0.000000e+0 0.400000e-3 0.000000e+0 -0.100000e-2 0.000000e+0 -0.280000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0

of Motion i

12 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 15 15 15 15 16 17 17 17 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19 19 19 20 20 20 20

Table 4.8 Cosine and sine coefficients of the P(i), Table 4.7.)

j

Ajj

r>ij

17 2 4 5 6 7 10 12 13 14 15 16 17 3 8 20 21 3 8 20 21 20 3 20 21 0 1 2 4 5 6 7 9 10 12 13 14 15 16 17 3 8 20 21 2 4 5 6

-0.100000e-2 0.600000e-3 0.000000e+0 0.600000e-3 0.000000e+0 -0.200000e-3 0.600000e-3 0.190000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.803847e-4 0.621778e-5 -0.337749e-2 -0.103923e-2 0.000000e+0 0.100000e-3 -0.390000e-2 -0.120000e-2 -0.730000e-2 -0.500000e-3 -0.100000e-3 -0.800000e-2 -0.100000e-3 0.300000e-3 -0.539000e-l 0.200000e-3 0.000000e+0 -0.400000e-3 -0.269000e-l -0.740000e-2 -0.900000e-3 -0.460000e-2 -0.700000e-3 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.600000e-3 0.000000e+0 0.600000e-3 0.000000e+0

0.000000e+0 -0.100000e-3 -0.169000e-l -0.100000e-3 0.500000e-3 -0.107900e+0 0.100000e-3 -0.400000e-3 -0.295000e-l -0.880000e-2 -0.700000e-3 -0.380000e-2 -0.700000e-3 0.140000e-2 -0.140000e-2 0.200000e-2 0.600000e-3 0.500000e-4 -0.500000e-4 -0.500000e-3 0.000000e+0 -0.100000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.300000e-3 0.000000e+0 -0.200000e-3 0.000000e+0 0.100000e-3 0.000000e+0 -0.200000e-3 -0.170000e-2 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0 -0.140000e-2 0.140000e-2 -0.200000e-2 -0.600000e-3 -0.100000e-3 -0.169000e-l -0.100000e-3 0.500000e-3

i = 1 , . . . , 2 0 functions.

(Continuation of

Simplified Normalized

l

J

•'^•ij

20 20 20 20

7 10 12 13

-0.200000e-3 0.600000e-3 0.190000e-2 0.000000e+0

Equations.

ij

-0.107900e+0 0.100000e-3 -0.400000e-3 -0.295000e-l

20 20 20 20

Table 4.9 Cosine and sine coefficients of the P(i), Table 4.8.)

Tests

69

J

~^ij

14 15 16 17

0.000000e+0 0.000000e+0 0.000000e+0 0.000000e+0

i = 1 , . . . , 2 0 functions.

ij

-0.880000e- -2 -0.700000e- -3 -0.380000e- -2 -0.700000e- -3 (Continuation of

This page is intentionally left blank

Chapter 5

Periodic Orbits of Some Intermediate Equations

The aim of this chapter is first to describe the model closer to the simplified equations that still has periodic orbits trained by the Sun, i.e., whose period is the one of the Sun in the Earth-Moon system. Second, to determine those periodic orbits and relate them to those of the bicircular problem. A continuation method has been used.

5.1

Equations of Motion for the Computation of Intermediate Periodic Orbits

We look for equations of motion such that they are as close as possible to the simplified ones but still have periodic solutions around L4 or L§ in a suitable region (not too far from the libration point). What we propose is merely to keep only those terms in the simplified equations of Chapter 4, section 4.7, which are either independent of time or have the same period of the solar elongation. This means to retain in the time-dependent functions P(i) the independent term and the frequencies number 6, 13 and 18. Before going into the details of the procedure we consider worthy a short discussion. Keeping those periodic terms, the passage to the simplified equations is done adding an important number of terms. Furthermore, some of those terms are much more important than the retained periodic ones. For instance, looking at the table giving the functions SO{i) (Chapter 4, section 4.5) there are only 4 frequencies with coefficients larger than 1 0 - 2 . They correspond to frequencies number 4, 7, 13 and 14. It is easily checked that frequency number 14 is the double of frequency number 7, and that frequency number 4 is the difference between frequencies numbers 13 and 7. Hence, the really important frequency besides the retained ones (6, 13, 18) is number 7. The related coefficients are of the order of 1 0 - 1 . As it is natural, this frequency is the mean motion of the mean anomaly of the Moon and the coefficients are closely related to the lunar eccentricity. After realizing this fact it looks natural to consider the following approach: 71

72

Periodic Orbits of Some Intermediate

Equations

a) To consider as intermediate equations the simplified ones keeping only the periodic terms whose period is the one of the mean anomaly of the Moon, that is, the terms with frequencies number 7, 14 and 17. b) These equations should be rather close to the planar elliptic RTBP. c) To look for periodic orbits of the planar elliptic RTBP. Their period should be the anomalistic period of the Moon. d) When those orbits are available, we consider the perturbations coming from the previously skipped terms. As they are relatively unimportant, the effect should be small. However, the program just described has some difficulties, which make clear that the terms to retain are those with period equal to the lunar synodical one: (1) To obtain a periodic orbit of the planar elliptic RTBP the natural process is: (a) To obtain, for the selected value of /i, a 27r-periodic orbit. As we look for orbits around L4 it should belong to the short period family. (b) When this orbit is available to perform natural continuation with respect to the eccentricity of the primaries. (2) As it is well-known (see, for instance [10]) for the Earth-Moon system, considered as a planar circular RTBP, the short periodic family contains one orbit with period 2n. But this orbit is placed between the quadruplication orbit (for which the short and long periodic families meet) and the orbit labeled B45, which belongs to the short periodic families associated to L4 and L5 (and which is also an orbit of the Lyapunov family around L3). This orbit is extremely large for our purposes. (3) Furthermore, looking at the simulations done in Chapter 3 using the full solar system or a nice approximation to it, the "natural" period of the orbits, obtained by dividing the time interval between the number of revolutions around L4 or L5 is placed between 29 and 30 days. This is close to the synodical period, roughly 29 d 12 /l 44 m . What precedes makes reasonable our election of the intermediate equations. However we wish also to relate them to the RTBP and to the bicircular problem. Hence we write down the intermediate equations (I.E.) as follows: q

=

acceleration of the RTBP +pi x terms coming from the noncircular motion of the Moon +P2 x terms coming from the Sun + (1 — P2)

x

terms coming from the Sun in the bicircular problem,

The Periodic Orbits Around the Triangular

Points

73

where q stands for one of the coordinates (x, y, z), and pi and p2 are continuation parameters ranging from 0 to 1. Then for p1 — p2 = 0, we obtain the bicircular problem, for pi — P2 = 1, we obtain the intermediate equations. The experience has shown that it is better to use 2 parameters rather than using a single one p. At this point we have already introduced all the systems of equations to be used in this work. The logical sequence for increasing difficulty is:

RTBP —> Bicircular Problem —> Intermediate Equations —>

—> Simplified Equations —> Real Equations.

5.2

Obtaining the Periodic Orbits Around the Triangular Libration Points for the Intermediate Equations

We have set up a program, TRIANG, for the computation of those periodic orbits. The leading idea is to start with the periodic orbits of the bicircular problem of types A, B, C, F and G and to perform continuation till the I.E. First of all we remark that, at least with the tolerance which we are using in the coefficients of SO(i), SOS(i), the change of variables (x,y,z,x,y,z,t)

—>

(x,-y,z,-x,y,-z,~t),

leaves the equations invariant. Hence, we only need to work with periodic orbits around L4, the ones around L5 being obtained by the symmetry. The program starts at some initial conditions. After one period, which is equal to k times the synodical period of the Moon, with k — 1 for orbits A, B and C and k = 3 for orbits F and G, a point is obtained. We require that this point coincides with the initial one in the phase space. The difficulties mentioned in the bicircular problem appear also here. That is, the system is a small periodic perturbation of an autonomous Hamiltonian system. Hence, at least two of the eigenvalues of the monodromy matrix D$T are close to 1. As Newton's method requires the inversion of D$T ~ I, this produces some problems unless the initial point, in Newton's method, is rather close to the solution. This means that even using some linear or quadratic extrapolation, when several orbits are available for different values of the active parameter pi, i = 1 or 2 the parameter step should be small (even less than 10- 2 ).

74

Periodic Orbits of Some Intermediate Equations periodic orbit of the intermediate equations (1.1)

(0.0) periodic orbit of the bicircular problem

(1.0)

Fig. 5.1 Evolution in the parameter space from the bicircular problem to the intermediate equations.

For the evaluation of the vector field, the coefficients SO(i) are multiplied by Pi, the SOS(i) by pi and the terms ms ,

as cos 6)

" -r 3 — \x

ps

m s

1

,

• a\

--3— {y + a s sinfj)

ms cos 8, a s ms al suit

r

PS

ms 3 'PS

^">

expressing the difference between the bicircular problem and the RTBP (see Chapter 2, section 2.2), are multiplied by (1 — P2). The variational equations are integrated simultaneously. This allows to apply Newton's method looking for the initial conditions and also to know the eigenvalues of the monodromy matrix, reflecting the stability properties of the orbit. The program also computes, optionally, the Fourier analysis of the orbit found. It is useful, for requirements of Chapter 7, to know the point of the orbit and the corresponding value of 9, when the projection of the orbit in the (x, y)-plane meets the segment joining L\ with the Earth. The periodic orbits for the I.E. have been found using different paths on the parameter plane (pi, p^). Small steps have to be used, especially with p\, due to the bad condition number of the linearized problem. However, the results obtained, to be presented in section 5.3, are independent of the followed path. The diagram in Figure 5.1 shows the paths in the parameter space.

5.3

Results and Comments

The orbits obtained by continuation of orbits A, B, C, E and F of the bicircular problem are given in the following pages as well as the corresponding plots. In the plots the origin is placed at L4 and the dots in the lower part correspond to the positions of Earth (left) and Moon (right). For comparison purposes we present also in the same plot the bicircular and intermediate periodic orbits of type A (note the

Results and

Comments

75

change in the sign of x: now the Moon is on the left and the Earth on the right). As it is natural we denote the periodic orbits by the same letters used in the bicircular case. At the light of the results, several comments should be done: a) Orbits A, B, F and G have increased the size in a significant way. For orbits F and G the maximum distance to L4 is close to the Earth-Moon distance. This increase has already been mentioned in Chapter 4, section 4.6, and produces problems of convergence in the expansions. In the plots each division corresponds to 0.1 normalized units, except in the magnification of orbit C where each division means 0.0025 units. b) The orbit C is quite small and for the I.E. looks like a periodic path traveled twice. c) The plots of orbits A and B and of F and G are almost indistinguishable except in what concerns the initial point (marked with a dot on the orbit). They are almost the same orbits but with a time delay of roughly 1/2 synodical period. d) The Fourier analysis of orbits A and B shows again the almost coincidence of both orbits with a suitable time shift. It can also be said, with the level of error in the equations, that the first 4 or 5 harmonics are enough to represent the orbits accurately. For these orbits the main harmonic is the first one giving as result that the shape is a distorted ellipse. For the orbit C the two first harmonics are enough and the dominant one is the second in agreement with b). e) The stability of the periodic orbits is quite close to that of the bicircular problem. The five orbits are now unstable but the instability of all of them (especially in what makes reference to orbits A and B) is not significant. The values of the modulus of the dominant eigenvalues of the monodromy matrix, scaled to the synodical period Ts, are 1.0011 for A and B, 1.1666 for C, 1.8012 for F and 1.8140 for G. Even for the worse case, G, to have an amplification of the errors by a factor of 1000 requires near 12 synodical periods (roughly 1 year).

76

Periodic Orbits of Some Intermediate

PERIODIC ORBIT

A

INITIAL CONDITIONS 0.3446348364766992 0.1131305748645592

Equations

AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O. -5.5680350668244804E-02 -0.3702379416714056

CONTINUATION PARAMETERS

1.000000000000

1.000000000000

PERIOD = 6.791193871917804 EIGENVALUES FOR THE STABILITY OF THE P.O. (-0.8042860126098644, 0.5960657272228537) (-0.8042860126098644, ••0.5960657272228537) ( 0.9744318402714565, 0.2198111634920723) ( 0.9744318402714565,-•0.2198111634920723) ( 0.8315322727454194, 0.5554764435894994) ( 0.8315322727454194,-•0.5554764435894994)

Orbit A

Fig. 5.2

Periodic orbit A of the intermediate equations. LP. denotes the initial point for 0 = 0.

Results and

Comments

77

P.O. of the intermediate equations

* M

Fig. 5.3 Periodic orbit A of the bicircular and the intermediate equations. LP. denotes the initial point for 0 = 0.

78

Periodic Orbits of Some Intermediate Equations VARIABLE X COEFFICIENT OF COSINE

0 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-0 .1820373734425437D+00 0 .5473727418033956D+00 -0 .2612533555332382D-01 0 .3596130222497096D-02 0..2344569225597054D-02 -0,.4575103314521871D-03 -0 .1283596091750251D-03 0 .8115733431706914D-04 -0 .6006697781577388D-05 -0 .7998225804570571D-05 0..2902385293734465D-05 0..2671358749239722D-06 -0..4466534961041022D-06 0..8072182940984220D-07 0..3628311894839477D-07 -0..2085549226832394D-07 0..9006514180829629D-09 VARIABLE Y COEFFICIENT OF COSINE

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.3074625593793142D-01 -0.8108614861426111D-01 -0.1353602163256157D-01 0.9323372828908902D-02 -0.6696404904180181D-03 -0.6781769862557938D-03 0.2243659953483073D-03 0.1784110670955425D-04 -0.2865629165719000D-04 0.5521312671056611D-05 0.2034387741816418D-05 -0.1259385321574008D-05 0.6153485844562423D-07 0.1461539099788795D-06 -0.4661939029472928D-07 -0.6300820342498891D-08

16

0.8235503779228720D-08

COEFFICIENT OF SINE

0.1021536482035979D+00 0.1969261999112519D-01 -0.8006804264988614D-02 0.7371063032005168D-03 0.6051812116823743D-03 -0.2131976043251629D-03 -0.1378901338908217D-04 0.2723903977540330D-04 -0.5546731166285052D-05 -0.1915504731193367D-05 0.1225612300955243D-05 -0.6720753096896170D-07 -0.1407865417703950D-06 0.4621568855238718D-07 0.5862414195848801D-08 -0.8079707750053004D-08

COEFFICIENT OF SINE

-0.3279162252541900D+00 -0.3144486918061280D-01 -0.5644561891475041D-02 0.2620597115032109D-02 -0.5087362805546621D-03 -0.1465742159558731D-03 0.8538414503175204D-04 -0.5310073093758154D-05 -0.8427090278480657D-05 0.2955041510378497D-05 0.2963664177241574D-06 -0.4614450552628417D-06 0.8033867877525882D-07 0.3795713981675681D-07 -0.2116564775316567D-07 0.8161916230497732D-09

Table 5.1 Fourier coefficients of th x and y coordinates of orbit A.

Results and

PERIODIC ORBIT

B

INITIAL CONDITIONS

Comments

79

AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O.

-0.7582100971505726 -3.6463329908348591E-02

8.9087947614105868E-02 0.2717079806443636

CONTINUATION PARAMETERS

1.000000000000

PERIOD

1.000000000000

6.791193871917804

EIGENVALUES FOR THE STABILITY OF THE P.O. (-0.8083524564940744, 0.5905803225873873) •0.5905803225873873) (-0.8083524564940744, • ( 0.9744672713141360, 0.2195441596004288) ( 0.9744672713141360,- 0.2195441596004288) ( 0.8314962415092812, 0.5555303775275825) ( 0.8314962415092812,-•0.5555303775275825)

Orbit B

Fig. 5.4

Periodic orbit B of the intermediate equations. LP. denotes the initial point for 8 = 0.

80

Periodic Orbits of Some Intermediate

VARIABLE X COEFFICIENT OF COSINE

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-0,.1829095046378868D+00 -0..5485886673345659D+00 -0,.2578065911946599D-01 -0,.3542603775514775D-02 0,. 2368594561928400D-02 0,.4520705863907640D-03 -0,. 1326625497700241D-03 -0,.8151340436942062D-04 -0..5460716124152157D-05 0,.8209793896144044D-05 0,.2881079098107230D-05 -0..3069937620521880D-06 -0,.4546225570443204D-06 -0..7676823204406415D-07 0..3859534953274974D-07 0., 2093160189132245D-07 0.. 5623238525211474D-09 VARIABLE X COEFFICIENT OF COSINE

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.3083494565531459D-01 0.8113074533596854D-01 -0.1375549583216786D-01 -0.9339186468253033D-02 -0.6443154890681684D-03 0.6885485971883162D-03 0.2234431872573132D-03 -0.1962622749851291D-04 -0.2901338989277210D-04 -0.5364937796184402D-05 0.2135853165996285D-05 0.1265255429078654D-05 0.4712719912378712D-07 -0.1510518557543697D-06 -0.4581665863319156D-07 0.7269941138508557D-08 0.8405390861468149D-08

Table 5.2

Equations

COEFFICIENT OF SINE

-0.1020528523635009D+00 0.1983618130474269D-01 0.8036535305076123D-02 0.7159895565795747D-03 -0.6149682407757302D-03 -0.2125403078480607D-03 0.1544898101060362D-04 0.2760222564002016D-04 0.5402964622344150D-05 -0.2014544666400152D-05 -0.1232061712601718D-05 -0.5335713298378076D-07 0.1456208343812417D-06 0.4547668119814228D-07 -0.6811210312841061D-08 -0.8251535600915437D-08

COEFFICIENT OF SINE

0.3284296996209018D+00 -0.3177683546152209D-01 0.5733817517258685D-02 0.2633667402750067D-02 0.5022993202481919D-03 -0.1512532488421611D-03 -0.8568456295413692D-04 -0.4721802148857870D-05 0.8641196629111997D-05 0.2930565983724559D-05 -0.3375223421981380D-06 -0.4694078630035761D-06 -0.7618065057130223D-07 0.4031175252572723D-07 0.2122615936322471D-07 0.4695181133461478D-09

Fourier coefficients of th x and y coordinates of orbit B.

Results and

PERIODIC ORBIT

C

INITIAL CONDITIONS

Comments

81

AROUND THE EQUILIBRIUM POINT L4 x,y,xdot,ydot FOR THE P.O.

-4.5395552497062299E-03 1.3665787366715615E-02 2.9866888241593641E-02 -3.8408486363138898E-03 CONTINUATION PARAMETERS PERIOD =

1.000000000000

1.000000000000

6.791193871917804

EIGENVALUES FOR THE STABILITY OF THE P.O. (-0.4707137878243862, 0.8819740228870461) (-0.4707137878243862,- 0.8819740228870461) ( 1.166584855221183, 0.0000000000000000) ( 0.8576749916489031, 0.0000000000000000) ( 0.8491731641193959, 0.5281145115781721) ( 0.8491731641193959,- 0.5281145115781721)

Orbil C (magnification) Orbit C

- 1

1

1

Fig. 5.5

1

1

1

1

1

1



i=i

where vi, I = 1,2,..., m is a preselected set of fixed and known frequencies. The known periodic solution, Xo(t), is considered as the first approximation to the final solution in our algorithm. After inserting it in the right-hand side of (6.1), a Fourier analysis is done of the function f(t, xo,xo). It is assumed that the main frequencies appearing in this analysis belong, all of them, to our preselected set {v{\. If this is not the case, then the set of frequencies must be enlarged. This point is not foreseen to be done automatically. Two exceptions must be considered. If the frequency is too large and with a small, or not,

The

Algorithm

89

amplitude associated to it, it will produce rapidly oscillating terms in the solution of small amplitude, so it can be skipped. If a small frequency, UJ, appears in the analysis of the function / , after the integration, the corresponding coefficients will be divided by w2. As these coefficients have been computed with a certain degree of accuracy, it may happen that the final figure is not meaningful, so these frequencies are skipped too. At this point it must be said that if we allow an independent term in the Fourier development of f(t, xo,x0), it will produce a secular term after the integration of the trigonometric sum, which is, of course, the next step in the algorithm. This has been avoided by adding the adequate "integration constants" to x0{t) in order to fulfill the purpose. Once the integration has been done we get our next approximation, xi(t), to the desired solution. This procedure completes one loop of the iterative algorithm, which is schemed in the following diagram

nth approximation to the solution: xn(t)

Fourier analysis of

f(t,xn(t),xn(t)) Integration of the equations when their right-hand sides are replaced by suitable trigonometric approximations. The result, xn+i(t), plays the role of xn (t) in the next step

Fig. 6.1

Diagram of the iterative algorithm.

Of course this loop is stopped when two consecutive approximations are close enough. The role of the parameter e, introduced in the differential equations, is relevant in two senses: - The first one is that, taking xo(t) as the first approximation to the solution, we can have a small rate of convergence to it, or, in the worst of the cases, we cannot have convergence at all. Introducing intermediate models of the differential system, via the continuation parameter e, and varying slowly

90

Quasi-periodic

Solution of the Global Equations:

Semianalytic

Approach

the parameter, these two facts can be avoided. - The second one, in certain sense related to the first, is the following. Assume, as it is the case in our real situation, that we can compute numerically a quasi-periodic solution, during a certain time interval and for a given value of e greater than zero. After performing its Fourier analysis, this solution can be taken as the starting point, xo(t), in the algorithm, which shall be much closer to the desired one that the periodic solution computed for e = 0. The procedure that has been explained has four important critical points which are: (i) The definition of the adequate set of relevant frequencies Vi, i = 1,2,..., m. (ii) The adequate selection of the integration constants of xn(t) in order to avoid secular terms in the solution. (iii) The adequate identification of the frequencies appearing in the Fourier developments of f(t,xn(t),xn(t)) according to the preestablished set. (iv) The good computation of the coefficients appearing in the Fourier developments, when these coefficients are firstly, computed by an FFT routine and after are shifted to those corresponding to the adequate set of frequencies. These questions have been solved in the way that shall be discussed in detail in the following sections. A modular program has been developed which performs all the above mentioned questions. The program tries to minimize memory and CPU time and the only external routine that it uses is the Fast Fourier Transform in Sine and Cosine (FFTSC) of the IMSL library. The flow chart of the program mainly follows the one which has been displayed for the algorithm. The results obtained shall be discussed in the last section of this chapter.

6.3

The Adequate Set of Relevant Frequencies

For the selection of the relevant frequencies which shall be considered in the solution, an examination of the equations of motion is needed. As it has been explained in Chapter 5, the equations in a normalized reference system, centered at one of the triangular equilibrium points, can be written in a compact form as: x

=

x—

P(7) L

XE

,-. (1 - n)x

r

x — g

r

'PE PE

XE

fj, - xE(l - 2/i)

PM ' PM

J

+P(1) + P{2)x + P{3)y + P{4)z + P(5)x + P(6)y, P(7)

y-VE,-. " Ir 3 PE

t1

_

, y-VE W ~ ^ 3r /* - VE PM

The Adequate Set of Relevant

+P(8) + P(9)x + P(10)y + P(ll)z =

Frequencies

91

+ P(12)± + P(13)y + P(14)i,

P(7) +P(15) + P(16)z + P(17)j/ + P(18)^ + P(19)2/ + P(20)i,

where {xE,yE) = ( - 1 / 2 , - V 3 / 2 ) for L 4 and ( - 1 / 2 , 7 3 / 2 ) for L 5 , and where the time-dependent functions P(i), i = 1,2,..., 20, have already been given and studied in Chapter 4. This set of equations has been taken as the definitive one, corresponding to e = l. In the Fourier analysis of the functions P(i) it was detected that only 21 frequencies appear, after keeping only those terms with absolute value of the coefficient greater than 5 • 1 0 - 4 . These 21 frequencies are linear combinations of the four fundamental ones, which correspond to the following angles: 61 = 02 = #3 = 04 =

mean mean mean mean

longitude of the Moon, longitude of the lunar perigee, longitude of the ascending node of the Moon, elongation of the Sun.

In fact, of the above mentioned 21 frequencies, only three of them shall be considered as more relevant, in the sense that the coefficients associated to them, or their multiples, are greater than 5 • 10~ 3 . These three frequencies are related to the angles: 04

=

-81+e2

+ 284,

fa = 64, fa

=

01-02-

Note that the second one corresponds to the frequency of the periodic orbit computed for the intermediate equations. Related to these three frequencies, the greatest coefficients that appear in the Fourier analysis are: 0.072, 0.6, 0.370 respectively. These quantities shall be taken as the weights associated to the three angles fa, fa, fa. We have looked for all the combinations of these angles giving weights greater than or equal to the established bound of 5 • 10~ 4 . The frequencies, and the corresponding phases, are given in Tables 6.1 and 6.2. On the right-hand side of Tables appear four integer numbers which are the coefficients of the linear combination of the fundamental frequencies giving the obtained ones (i.e.: ni0i + 7i202 + "303 + rn9\). The above given set of frequencies is the one which has been adopted as {i>i, i = 1,2,..., 61}. Related to these frequencies, the question that arises is its identification with those that appear in the FFT of f(t,xn(t),xn(t)).

92

Quasi-periodic

Solution of the Global Equations:

Semianalytic

Approach

In the process of identification some care must be taken with respect to the effect termed as leakage. This effect is inherent in the discrete Fourier transform because of the required time domain truncation (see [5]). Recall that even for a periodic function, its truncation at other than a multiple of the period results in a sharp discontinuity in the time domain, or equivalently results in side lobes, such as those of sin(:r)/a;, in the frequency domain. These side lobes produce spurious frequency components which are known as leakage. To reduce leakage it is necessary to employ a time domain truncation function with side lobes characteristics much smaller in magnitude than those of sin(:r)/a;.

Frequency 8.326763869672732 7.467917120105036 7.401567884153539 6.675419606488837 6.609070370537340 6.542721134585843 6.476371898634347 5.882922092872638 5.816572856921141 5.750223620969645 5.683874385018148 5.617525149066651 5.551175913115154 5.484826677163658 4.957726107353446 4.891376871401949 4.825027635450452 4.758678399498956 4.692329163547459 4.625979927595962 4.559630691644465 4.032530121834254 3.978654502824115 3.966180885882757 3.899831649931260 3.845956030921122 3.833482413979763 3.767133178028266 3.700783942076770

Phase 1.846454431531298 0.187006518277412 3.037556229623082 1.960194200858354 4.810743912204023 1.378108316370106 4.228658027714866 3.733381883439295 0.300746287605377 3.151295998951047 6.001845710295807 2.569210114461890 5.419759825807560 1.987124229972733 4.924483681531079 1.491848085697162 4.342397797042831 0.909762201208914 3.760311912553674 0.327676316719757 3.178226028065427 6.115585479622863 1.953886167510574 2.682949883788946 5.533499595134615 1.371800283021871 2.100863999300243 4.951413710645913 1.518778114811541

Table 6.1

n3

"i

n2

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

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

The relevant frequencies.

rii

The Adequate Set of Relevant

Frequencies

Frequency Phase 3.634434706125273 4.369327826157211 3.040984900363564 3.874051681881184 2.974635664412068 0.441416086046812 2.908286428460571 3.291965797392482 2.841937192509074 6.142515508737697 2.775587956557577 2.709879912903780 2.709238720606080 5.560429624248995 2.642889484654584 2.127794028415078 2.115788914844372 5.065153479972968 2.049439678892875 1.632517884139051 1.983090442941378 4.483067595484226 1.916741206989882 1.050431999650122 1.850391971038365 3.900981710995564 1.784042735086888 0.468346115161646 1.775587956557577 5.297714922420715 1.717693499135391 3.318895826506862 1.124243693373683 2.823619682230835 1.066349235951497 0.844800586317209 1.057894457422186 5.674169393576277 1.016492455353406 0.783406365185843 1.004018838412048 1.512470081463988 0.991545221470689 2.241533797742133 0.925195985519192 5.092083509087575 0.916741206989881 3.638267009167057 0.858846749567695 1.659447913253431 0.846373132626337 2.388511629531575 0.792497513616198 4.509997624599100 0.784042735086888 3.056181124678582 0.726148277664702 1.077362028764729 0.199047707854490 4.014721480322619 0.132698471902993 0.582085884488702 0.066349235951496 3.432635595834144 Table 6.2

ni

-1 -4 3 2 1 0 -1 2 -4 -3 2 -1 0 -1 -1 -2 -3 2 -2 1 1 1 0 0 -1 -1 -2 -2 -3 -3 2 -1

93

Tl2

n3

ri4

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

The relevant frequencies. (Cont. of Table 6.1.)

The truncation function that has been used is Hanning's one: 1 2

"• w W = -^ —

1 2

T: COS

2nt N

——,

where N is the number of points used in the FFT analysis. This function, as is well-known, exhibits the desired characteristics and has given us good results. The only disappointing question is that the memory requirements are enlarged

94

Quasi-periodic

Solution of the Global Equations:

Semianalytic

Approach

due to the fact that the FFT must be used twice, first for the functions properly and after for the functions multiplied by H(t). When the frequencies given by the FFT are known, then they are identified, by a minimum distance criteria, with those that appear in the relevant set. As a final question we shall mention that the Fourier analysis has been done using 2 13 = 8192 equally spaced distributed points during 450 revolutions of the Moon. The epochs at which the points are taken are centered at JD50 = 18262.

6.4

Avoiding Secular Terms

Avoiding secular terms in the "solutions" xn(t) is possible due to the freedom that we have in choosing the integration constants of xn-.i(t). The problem is in fact reduced to the computation of the zeros of a system of three nonlinear equations depending on three variables. The variables correspond to the integration constants, one for each component of the solution, and the equations correspond to the independent terms of the Fourier series of each of the three functions that appear as right-hand side in the equations of motion. As the independent terms are the mean values of the functions which define the differential system, we can skip for their computation all the periodic contributions that appear in them. These periodic parts are the most CPU time consuming ones, due to the large number of trigonometric functions evaluations that must be done for their computation. As algorithm for the computation of the zeros we have selected Newton's method. The computation of the partial derivatives has been done by numerical differentiation using a centered difference formula. The results obtained are good in the sense that they are fast and that no more than 4-5 iterations are needed to cancel the constant terms with an adopted tolerance established at 10~ 8 .

6.5

The Coefficients Related to the Different Frequencies

The question to which this section is devoted is the following: Assume that we have a real quasi-periodic function from which we know beforehand the frequencies that appear in it. Its discrete Fourier transform shall give us the coefficients of the cosine and sine terms at equally spaced frequencies that shall not be, in general, equal to those ones that we know that appear. After having filtered those spurious frequencies due to leakage, we want to recover the right coefficients of the cosine and sine terms, when the function is represented in terms of the known set of frequencies instead of those given by an FFT algorithm. There are several ways to solve this question: The conceptually most elemental is a direct computation of the coefficients using some minimization technique. This is not possible with our computational media due to the large size of the linear

Determination

of the Coefficients

of Quasi-periodic

Functions

Using FFT

95

system that, in general, must be solved. Another way is its computation argument by argument, using a trapezoidal rule for the integration and starting at an epoch such that the related argument is zero. The integration interval is then taken equal to an integer multiple of the period related to the analyzed frequency. The large number of frequencies, 61, that appear in our situation does not allow, also by CPU time reasons, this way of computation. The third one follows from a careful study of the coefficients computed by the FFT method. This has been the option adopted and it is explained in the following section.

6.6

Determination of the Coefficients of Quasi-periodic Functions Using F F T

We are faced with the following problem: Given a function f(t) which is supposed to be of the form m

f(t)

=

m

A0 + ^2Aicos(pit)

+

J2Bism{uit),

i=i

i=i

where the frequencies, vi, I = 1,2,... ,m are known, we try to determine the coefficients AQ, AI, Bi, I = 1 , . . . , m. We shall expose in the sequel the FFT approach. We recall the formulae giving the discrete Fourier transform. Let ax,...,ajv be a set of real values. They shall be identified later as the values of the function f(t) at equally spaced points t0,..., ijv-i. Then, we obtain coefficients c\,..., cN/2+i, s2,. . . , SJV/2) N being supposed even, through:

Ck+1 =

S v^

AT 1^

aj+1 cos

2irjk

~W

=

,

°''''' I '

3=0

2 ^v1 Sk+l

=

a

2-Kik lsin

Jsrz2 i+ -N->

k=

l,...,N/2-l,

j=0

where 5 = 1 for k = 0, N/2 and 2 otherwise. The initial values a i , . . . , a j v are recovered from the coefficients c 1 ; . . . , cN/2+i, s 2 , . . . , sN/2 by means of 2irjk

E cjt+icos—— + fc=i

^-^ 2^

2-KJk Sfc+isin—_.

*=i

The given function, / , contains terms with frequencies which are, in general, unrelated to the frequencies k/N which appear in the Fourier transform. Therefore, the first question is to find the values of c^+i, s^+i in the cases aj+\ = 1, a,j+i = COS(WJ) and a,j+i = sin(wj).

96

Quasi-periodic

Solution of the Global Equations:

Semianalytic

Approach

To carry out the computations a preliminary result is useful. Let M

1 eaiNa

=

_ ^

1

Then cos(iVa) — 1 + i sin(Na) cos a — 1 + i sin a cos((N — l ) a ) - cos(Na) — cos a + 1 4 sin2 (a/2) sin((./V — l ) a ) — sin(./Va) + sin a + 4 sin2 (a/2) '

M

and hence Re(M) Im(Af)

1+

sin(^i)a^

=

-

=

1 / c o s f - c ov s (22 ^ ) a ' - ' —-^ "~ 2 1 sh

We return to the computations, presenting separately the three cases. C a s e I: Oj+i = 1.

c\

=

c fc+1

=

1, - ^ c o s - ^

AT

r

= — £(e*»

j=o

+

e - * )

j'=o 027rifc

27V

Sk+i

e ^ - l

+

e~~i$~ — 1

2 ^ . 2njk , 1 ..TV 1 n -jj } _ ; sin - ^ - = 0, fc = l , . . . , — - 1 . 2 AT JV

=

i=o

Case II: Oj+i = cos(wj). We denote 2-Kk/N by ZA JV-l Ck+l

=

JV

^

cos(wj) cos(j/j)

J=0 JV-l

= I7v£( e ^ + e "^)(e ^ j + e - ^ j ) JV-l

=

— V

(ei{u+l/)j

4/V ^—' V j=o

+ conj. +ei^-u)j

+ conj. ) /

Determination

of the Coefficients of Quasi-periodic -ei(u+v)N ei(u+v)

AN

_ i

ei(w-v)N

+ conj. +

_ I

Functions

Using FFT _ i

ei(ui-v)

con].

_ I

2+^(y^y))+^(^

_5_ AN

sin (I ( - + ¥ ) )

97

("-¥))

sin(I(^-^))

* = o,...,f. If k = 0 or AT/2, the second and third terms in the square brackets are equal. N-l

Sk+1

=

AT J2 C0S (^') s i n ( I / i) 3=0 N-l

efc,i + e iuij ei j e ivi = —-YI ( ~ )( " ~) 2Ni 3=0

ei(u+v)N

1 2Ni

_ ^

ei{u+v)

,i(-u+u)N

— conj. + J '

_ I

ei(-u+v)

cos(i(^+^))-cos(^i(g;

1 2N

conj.

_ \ +

^))

Bin ( I (a,+ 2 ^ ) )

cos ( I ( - „ + 2 S * ) ) - C O B (3^=1 ( - a , + 3 ^ ) )

+

,fc=l,...,y-l.

Bin(|(-W+^))

Case III: a,+i = s i n ^ j ) . N-l

Ck+l

=

~M Yl sin ( w i) c o s (^) 3=0 •

AAfi 6 ANi AN

+

N-l

1^^

. e-™i)(eW

+

e-W)

3=0

ei(w+v)N ei(ui+v)

_ i _ J

ei{w-u)N

con

^ ^ J" j -+i

_ j

..v,...,/. gifa — v) _ 7j

con

-

J-

COB ( 1 ( ^ + 3 ^ ) ) - C O B (2^=1 ( a i + 2 ^ ) ) Bin ( ! ( „ + * * ) )

cos (I (w - ^ ) ) - c

0

s(2^1(c-^))'

sin(I(--^))

,

*=0,...,y.

As in case II, the two fractions inside the square brackets are equal if k = 0

98

Quasi-periodic

or k =

Solution of the Global Equations:

Semianalytic

Approach

N/2. N-l

Sk+i

=

-T7 Yl

sin

(wi)

sm

(^i)

i=0 N-l j

)(eivj

2TV f-r v

e~ivj)

-

j=0

1 27V 1 27V

; i(w-i/)iv

_

j

gi^+^JV _

+ conj. —

e i(w-j/) _ ^

s i n ( ^ i ( g , - ^ ) ) s i n ( I ( w _ ¥ ) )

ei(u+v)

x

_ J

-

conj.

sin ( 3 ^ ( 0 , + ^ ) ) s m ( I ( w + 2 ^ ) ) * - l

-

1.

It is readily checked t h a t in case II, if w = 2irm/N, m = 0 , . . . , TV/2 we have Cfc+i = 1 for A; = m and c^+i = 0 otherwise, and Sk+i = 0 for any k. In a similar way, in case III all the Ck+i are zero and Sk+i — 1 for k = m, Sk+i = 0 otherwise. To get t h e values for k = m some limiting procedure is required t o avoid indeterminacy. To discuss the dependency of Ck+i, sjt+i with respect t o u> let us introduce e through to — v = (27r/7V)e, where v = 2irk/N as before. Hence e = 1 when the difference between the frequencies equals the step in the frequency space under the discrete Fourier transform. If we skip the terms related to sums of frequencies in the expressions of c^+i, Sfc+i and approximate sin(7re/TV) by ire/N (both things are allowable if TV is big enough) and suppose e not too big (at most some units) we obtain: Case II. sin 2ixe Cfc+l

2we '

1 — cos 2ire Sk+l —

2TTC

'

Case III. 1 — cos 2we Cjfc+l

2?re

'

sin 2ire Sk+l

-

2ne

It is enough to discuss two of the functions

si 00 =

sin 2ne 2ne

sin 7re cos 7re 7re

,

. . 1 — cos 2-7re 52(e) = 2ire

T h e modulus 93(e) = ((91(e))2+

(g2(e))2)1/2

=

sin7re

is also useful. Figure 6.2 shows the behavior of