Dynamics and Mission Design Near Libration Points, Vol. IV: Advanced Methods for Triangular Points

orld Scientific Monograph Series Mathematics - Vol. 5

Dynamics and Mission Design Near Libration Points Vol. IV Advanced Methods for Triangular Points

I. Gomez V

A. Jorba J. Masdemont . Simo

T World Scientific

Dynamics and Mission Design Near Libration Points Vol. IV Advanced Methods for Triangular 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, Ft. Martinez and C. Simo

Vol. 4:

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

Vol. 5:

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

Vol. 6:

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

World Scientific Monograph Series in Mathematics - Vol. 5

Dynamics and Mission Design Near Libration Points Vol. IV Advanced Methods for Triangular Points

G. Gomez, A. Jorba & C. Simo Universitat de Barcelona, Spain

J. Masdemont Universitat Politecnica de Catalunya, Spain

©World Scientific •

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DYNAMICS AND MISSION DESIGN NEAR LIBRATION POINTS VOL. 4 Advanced Methods for Triangular Points Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

The final objective of this work is to understand, analyze and compute the kinds of motion that appear on a vicinity (may be a large vicinity) of the geometrically defined equilateral points of the Earth-Moon system, as a source of possible nominal orbits for future space missions. In Chapter 1 we describe numerical simulations carried out for the RTBP, for the elliptic restricted and bicircular ones and, finally, for the real motion (using the JPL model). We present the results concerning sets of points which subsist, in an extended neighborhood of the triangular equilibrium points, for a long time span. Furthermore a frequency analysis of several orbits has been carried out. The chapter ends with a discussion of the results. These ones suggest some reasonable explanations. Just looking at the frequency analysis we can detect the key role played by the inner frequencies of the system (that is, what can be seen as a generalization of the vertical, short period and long period modes of the circular RTBP), together with the solar synodic frequency and, to a minor extent, the Moon's mean anomaly frequency. The second chapter is devoted to the analysis of the circular RTBP. The Normal Form around the triangular libration points is obtained. This allows to see, already in this model, the variation of the inner frequencies with respect to the corresponding amplitudes. Chapter 3 describes the reduction of the bicircular model to some normal form. The essential part is to convert the periodically dependent Hamiltonian in an autonomous one. This is done around the small unstable periodic orbit described in 0.2.3. Then, one obtains a clear explanation for the instability found in the numerical simulations of that problem for small values of the z-amplitude. An analytic description is given for the 2-dimensional unstable tori. The source of the instability is the 1-1 resonance between the short period frequency and the solar one. Furthermore "big tori" have been found, around some family of periodic orbits of a modified problem, displaying stable behavior, and Fourier analysis has been used v

Preface

VI

to detect the main resonances. The fourth chapter is devoted to obtain a suitable analytic model for the real motion. This requires, as a basic tool, a refined Fourier analysis. In Chapter 5 some solutions have been obtained satisfying the previous analytic model. This has been done to obtain a dynamical equivalent to the libration point for the present problem. As an approximation, a quasi-periodic solution has been obtained. It has, at most, 5 basic frequencies, the ones appearing in the model. To this end a suitable symbolic manipulator has been implemented and used. The orbit is then refined to a "true" orbit of the JPL model by parallel shooting. The local properties of the orbit have been studied. As a conclusion the orbits are only mildly unstable. Chapter 6 deals with the study of the transfer orbits from the vicinity of the Earth (more concretely, from a given family of GTO orbits) to two classes of nominal paths: One of them is a rather planar orbit (in the synodic system) like the ones found in the previous chapter. The other class contains orbits of moderate and relatively big ^-amplitude, among the ones described in the first chapter, with good stability properties. In any case, the results are fairly good. The transfers from the GTO to the "planar" orbits can be achieved for a total cost under 900 m/s, and for the ones departing in a significant way of the Earth-Moon plane, one can have transfers for less than 800 m/s. The book ends with four Appendices, the last one devoted to the conclusions.

The contents of this book is the final report of the study contract that was done for the European Space Agency in 1993. 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. 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.

At the end of the book, after the bibliography, updates with respect to the work done for the European Space Agency and collected in the four Volumes on "Dynamics and Mission Design Near Libration Points" are gathered. These are the contributions made by the authors and collaborators to these topics, either from a theoretical, numerical or applied point of view. For completeness, references to papers of the same authors cited in the different Volumes, are also included.

Contents

Preface

v

Introduction 0.1 Detailed Objectives 0.2 Known Results About the Stability of L 4 , 5 0.2.1 For the Circular RTBP 0.2.2 For the Elliptic RTBP 0.2.3 The Bicircular Problem 0.2.4 Intermediate Models 0.3 Main Difficulties of the Problem

1 1 1 1 3 3 4 5

Chapter 1 Global Stability Zones Around the Triangular Libration Points 7 1.1 Equations of Motion 7 1.2 Results for the Restricted Circular Problem 9 1.3 Simulations for the Bicircular Problem 17 1.4 Results for the Simulations Using the JPL Model 38 1.5 Discussion and Tentative Explanations 44 Chapter 2 The Normal Form Around L 5 in the Three-dimensional RTBP 47 2.1 Checks of the Normal Form 50 Chapter 3 Normal Form of the Bicircular Model and Related Topics 53 3.1 The Equations of the Bicircular Problem 53 3.2 Expansion of the Hamiltonian 55 3.2.1 Expansions of the Potentials and Recurrences 57 3.3 Cancelling the Terms of Order One 60 3.3.1 Cancelling Order 1 Using the Lie Transformation 61 3.3.2 Cancelling Order 1 by Computing the Periodic Orbit 63 vii

viii

3.4

3.5

3.6 3.7

3.8

Contents

3.3.3 Test of the Results 64 Normal Form to Order Two 65 3.4.1 Diagonalization of H2{x,y,6) Using Floquet Theorem 66 3.4.2 Practical Implementation of Proposition 3.1 73 3.4.2.1 Case of the Hamiltonian Expanded in Complex Variables 74 3.4.2.2 Case of the Hamiltonian Expanded in Real Variables . 75 3.4.3 Applying the Obtained Change of Variables 78 3.4.4 Second Expansion of the Hamiltonian 79 Normal Form of Terms of Order Higher than Two 81 3.5.1 The Method 82 3.5.1.1 Selecting Ck 84 3.5.2 Implementation 84 3.5.2.1 The Algebraic Manipulator 84 3.5.2.2 The Main Algorithm 85 3.5.3 Results 85 3.5.4 The Change of Variables 86 3.5.5 Going Back to Real Coordinates 86 3.5.5.1 The Hamiltonian 87 3.5.5.2 The Change of Variables 88 3.5.6 Checks of the Software 89 A Test Concerning the Normal Form Around the Small Periodic Orbit Near L5 in the Bicircular Problem 89 On the Computation of Unstable Two-dimensional Tori 92 3.7.1 The Equations and the Algorithm 92 3.7.2 Truncated Power Series Results 95 3.7.3 Discussion 99 Big Tori and Stability Zones 100 3.8.1 An Autonomous Intermediate Vector Field and its Vertical Periodic Orbits 100 3.8.2 Simulations Around the Periodic Orbits. Region of Stability . . 101 3.8.3 Frequency Analysis 101

Chapter 4 The Quasi-periodic Model 4.1 The Lagrangian and the Hamiltonian 4.2 Some Useful Expansions 4.3 The Fourier Analysis

111 113 115 118

Chapter 5 Nominal Paths and Stability Properties 5.1 Introduction 5.2 Idea of the Resolution Method 5.3 The Algebraic Manipulator 5.3.1 High Level Routines 5.3.1.1 Input/output Routines

133 133 134 136 136 136

Contents

5.4 5.5

5.6

5.3.1.2 Evaluation of the Function 5.3.1.3 Evaluation of the Jacobian Matrix 5.3.1.4 The Newton Method Results with the Algebraic Manipulator Numerical Refinement 5.5.1 The Program 5.5.2 Nominal Paths The Neighbourhood of the Almost Planar Nominal Paths

IX

136 136 136 137 138 139 140 159

Chapter 6 Transfer to Orbits in a Vicinity of the Lagrangian P o i n t s l 6 3 6.1 Computations of the Transfer Orbits 165 6.1.1 Looking for Arrival Conditions at the QPO 165 6.1.1.1 Case of the Target Orbits of Section 5.5.2 166 6.1.1.2 Case of the Target Orbits of Section 1.4 166 6.1.2 Computing the Successive Swingbys 168 6.1.3 Departure from the GTO. Change of Inclination 171 6.1.4 Global Optimization of the Transfer 173 6.2 Summary of the Results 174 6.2.1 On the Magnitudes 174 6.2.2 On the Shapes of the Transfer Orbits 194 6.2.2.1 Looking at the Orbits in the Sidereal Frame 194 6.2.2.2 Looking at the Orbits in the Synodic Frame 194 Appendix A Global Stability Zones Around the Triangular Libration Points in the Elliptic R T B P 213 Appendix B Fourier Analysis B.l Introduction B.2 The Method B.2.1 The DFT of Simple Inputs B.2.2 Several Filterings B.2.3 Determination of the Frequencies: First Approximation B.2.4 An Alternative Method to Compute the Coefficients B.2.5 The Effect of Errors on the Frequencies B.2.6 Determination of the Frequencies: Improvement B.3 An Application to the Analysis of Orbits of the Restricted Problem

227 227 228 228 232 234 234 235 236 . . 237

Appendix C Geometrical Bounds for the Dynamics: Codimension 1 Manifolds 241 C.l The Center, Center-stable and Center-unstable Manifolds 241 C.2 On the Analytic Computation of Invariant Manifolds 242 C.3 The Center-stable and Center-unstable Manifolds for L 3 in the RTBP . 243

x

Contents

Appendix D Conclusions 249 D.l Summary of Achievements 249 D.2 On the Methodology 252 D.3 On the Application of the Results to the Design of Spacecraft Missions . 253 D.4 Outlook 253 Bibliography

255

Updates with Respect to the Work Done for the European Space Agency

257

Introduction

In this introduction we summarize the present knowledge and the detailed objectives of the work. We shall also discuss the main difficulties of the problem and we shall describe shortly the contents of the chapters of this book. The simplest model for the motion near the triangular libration points of the Earth-Moon system is the spatial Restricted Three-Body Problem (RTBP). Concerning the RTBP, the main perturbations are due to the eccentricity of the lunar orbit and to the presence of the Sun. This is the reason to devote some special attention to these two problems: elliptic and bicircular.

0.1

Detailed Objectives • For the RTBP, we have detected the zone of stable motion, the fast escape region and the origin of the transition from one to the other. • The same topics are studied for a model with one external frequency (i.e. the elliptic restricted or the bicircular). Later on, a model containing more external frequencies (five) is developed and studied. This model should be a better approximation of the real system. • Finally, we do the refinement of some selected (families of) orbits for the real system. The local behavior around these nominal orbits, the station keeping strategies (if necessary) and the transfer from the vicinity of the Earth (for instance, from a GTO orbit) to the target orbit is studied.

0.2 0.2.1

Known Results About the Stability of £4,5 For the Circular

RTBP

• They are linearly stable for the mass parameter \i £ [0,/ii], where w

long 1

2

Introduction

The values of a ^ Q ^ , wj o n K are the frequencies at £4,5 in the planar case ( [19]). Their values are i ( l ± (1 - 27M(1 - »))1/2)

•

• The problem has nonlinear stability for fi £ [0,/JI] \ {^2,^3} in the planar case [14]. • The nonlinear stability of the spatial problem is unknown. Probably there is some amount of Arnol'd diffusion. • There is a domain of "practical stability", around £4,5, that is not too small. We summarize how this result is obtained ( [6], [17]). Let H = H2+ H3 + ... be the power expansion of the Hamiltonian around £4,5, where Hk contains the homogeneous terms of degree k in positions and momenta. By a linear transformation, H2 can be reduced to -ujs{ql + p\) - -ui(ql +pl) + -uz(q%

+p\),

where us, wj stand for Wg^Q^, w j o n g , and coz is the frequency in the vertical direction (LJZ = 1). A canonical transformation (q,p) —> (Q,P), obtained by means of a generating function G = G3 + G4 + • • • + Gn, is applied to put H in normal form up to order n. We get the new Hamiltonian U = N2 + N3 + • • • + Nn + 1Zn+u where Nk are terms of degree k in the normal form with N2 = H2- If we only keep N2 + N3 + •• • + Nn, the system is integrable. Let

be the new momenta. Then, the diffusion of the momenta is due to Ij = {Ij,

Rn+i},

where {•, •} stands for the Poisson bracket. As the remainder, 7£„ + i, is of the form Rn+i + Rn+2 + •••, we introduce some norm for || Rk || (for instance YL I coefficients |, but it is better to assign some weight to get better results). Then, one obtains successively bounds for || Hk ||, k > 2, 11 Gk 11, 2 < k < n (only a finite number of small divisors appear) and for || Rk ||, k > n. The bounds of || Rk || are given by means of a recurrence that depends only on 11 Hk 11, the norm of the homogeneous parts of the initial Hamiltonian, and on the current small divisors that appear up to order n.

Known Results About the Stability of Z/4,5

3

In a ball of radius p in the (Q, P) variables, one has

\TZn+1 |< Y,PkWRk\l k>n

where | | denotes the sup norm. In a similar way we can bound the speed of diffusion \Ij\. Given T and S, there exists an initial radius, po, such that if (Q, P)t=o € BPo then (Q,P)t € Bpo(1+S) for all \t\ < T, where Bp denotes the ball of radius p centered at the origin. We remark that one obtains better results if Hk, Nk, Gk are computed explicitly up to some order (we did it for n = 16) for the desired value of p., by means of a symbolic manipulator. As a result one obtains, in general, Nekhorosev type estimates (i.e., for 8 fixed, one has T « exp(c/p d ), for some positive constants c, d). Furthermore, using T = 5 x 109 years, 5 = 0.1, one has po = 10~ 3 adimensional units (1 unit = Earth-Moon distance). • Numerical simulations show a "stable domain" even larger than the one described above for the planar case [15]. • Two main families of periodic orbits are known for the planar case [8]: The short period family: In the limit the period is T = 6.58268 (for the Earth-Moon case). The period decreases locally. The family exists, locally, for values of the Jacobi constant less than 3. The long period family: In the limit the period is T = 21.07007. The period increases locally and the family exists, locally, for values of the Jacobi constant larger than 3. One of the orbits of the long period family is a quadruplication orbit of the short period family. The short period family has a triplication bifurcation. The triplication family has (locally) an increasing period which reaches a maximum of 20.35848. 0.2.2

For the Elliptic

RTBP

• There is a zone in p, e of linear stability of the planar problem [19]. • There are numerical simulations detecting a "stable domain" around £4,5. For p = MMoon t n e d ° m a m shrinks if e increases [20], [4]. 0.2.3

The Bicircular

Problem

• It can be obtained from the circular RTBP by continuation with respect to the mass of the Sun or by means of some intermediate Hill's problem [8]. (See Chapters 1 or 3 for the equations of motion). • There are three simple periodic orbits (two of them linearly stable and another small and slightly unstable) with period equal to the synodic period of the Sun in the Earth-Moon system: Ts = 6.79117. One can think about

4

Introduction

the use of the two stable orbits. However they lie far away from the triangular libration points and when the full set of perturbations is included, they seem to become slightly unstable [8]. They will be not considered in this work. • There are two triple periodic orbits, large, linearly unstable, with period 20.37351. • By continuation, decreasing the mass of the Sun (ms), the small periodic orbit does not connect with the triangular point of the RTBP but it has a turning point and it connects with one of the stable orbits, if we increase the mass of the Sun after reaching the minimum along the continuation curve. In a similar way, the triangular libration point, when Sun's mass is increased, connects with the other stable orbit. The triple periodic orbits can be also continuated for ms decreasing. They are triplication orbits of the ones obtained by continuation of the stable simple periodic orbits, for some intermediate values of ms- The value of 7715 can be decreased again but the family has a turning point for very small values of ms not reaching the zero value. For the minimum, along these families, of 7715, the triple periodic orbits are close to the triple periodic orbits of the RTBP, but with slightly different periods (20.37351 and 20.35848 as we said). • No fixed points, nor autonomous first integral exists for the bicircular problem. The existence of a "stable domain" is, up to here, an open question. But see Chapter 1, where this stable domain will be found.

0.2.4

Intermediate

Models

• For some intermediate model keeping only the Sun frequency, the previous periodic orbits are preserved. The size increases but not for the small orbit. This model is closer to the real solar system. • For some analytic approximation of the real equations, (weakly unstable) quasi-periodic solutions of large size have been obtained by numerical methods in an approximate form. The dominant eigenvalue of the variational equations equals A = 12.5 after a time interval of 163 days [8]. • For some approximate model of the preceding one, which keeps only terms of the Hamiltonian around £5 containing frequencies that are multiples of the ones of the Moon and the Sun, a quasi-periodic solution has been found analytically (in an approximate form) by using symbolic manipulation. This orbit "replaces" the libration point. It plays a role similar to the small periodic orbit of the bicircular problem but it is several times larger. This orbit has been numerically refined to a solution of the real (JPL) model for several years. It is weakly unstable, the dominant eigenvalue being A = 3.8 per year, on average [3], [10], [9]. • Theoretical results [11] assure the existence of a quasi-periodic solution with

Main Difficulties

of the Problem

5

basic set of frequencies equal to the one of the perturbation for the equation x = (A + eQ(t, e))x + eg(t, e) + f(x, t, e), around an elliptic point, for almost all e, provided it is sufficiently small. Here A is a matrix with purely imaginary eigenvalues, Q is a matrix depending on t, g is also a function which depends on t and / contains terms in x of degree higher than linear and it depends also on t. The dependence with respect to t is quasi-periodic and the related set of frequencies should satisfy, together with the frequencies associated to A, some Diophantine condition. Also the equation should satisfy some condition preventing locking at resonance. In our present problem e is related to the mass of the Sun and to the Earth-Moon distance. In practice, it is not sufficiently small and, in fact, it is so large that it produces a bifurcation and the stability changes. This is related to the fact that the libration point is linearly stable in the RTBP but the orbit mentioned in the previous item is weakly unstable. But for the problem under consideration the algorithm of the proof works in practice, despite e is too large. At least it works sufficiently well as to allow to obtain the orbit described in the previous item.

0.3

Main Difficulties of the Problem

We can summarize the difficulties to be found in our problem after the previous results are known: (1) Several external frequencies appear together with the (amplitude varying) inner frequencies. (2) Many resonances appear. (3) The effect of the Sun is uniformly large in the vicinity of £4,5. The effect of the Moon is small near the libration point but can be large sufficiently far from it. (4) At the geometrical £4,5 the frequencies of: the proper short period, the vertical oscillations, the perturbations due to the Sun and the perturbations due to the eccentricity of the Moon, are rather close. In the circular RTBP we are faced with a three degrees of freedom autonomous Hamiltonian. In the elliptic and bicircular cases the system is periodic instead of autonomous. Finally, in some analytic intermediate models and in the final model to be used, it is even quasi-periodic, with a basic set of frequencies whose cardinality ranges from 2 to 5 (this is enough to have a very accurate description of our problem). Even in the 3-dimensional autonomous case, to describe to some extent the dynamical behavior in big regions of the phase space can be a formidable task.

6

Introduction

Starting at some integrable problem and adding a perturbation with increasing parameter we find, successively: a) For the integrable problem the phase space is completely foliated by invariant manifolds, mainly 3-dimensional tori. b) For small perturbations (or, equivalently, when we look very close to the totally elliptic equilibrium point) most of the 3-dimensional tori subsist, as assured by the celebrated KAM theorem [1], and very small zones of chaotic motion appear. They are hardly seen in practice if the perturbation is small, because of the exponentially small character of the splitting of separatrices [13], [5]. Furthermore, the 3-dimensional tori do not separate the levels of energy (which are 5-dimensional). Arnol'd diffusion can appear as a wandering motion between the tori. This prevents, in general, the existence of true barriers for the motion of the momenta. However, this motion is, at most, extremely slow, as assured by the results of Nekhorosev [16]. c) When the perturbation is increased (or, equivalently, when we look at a relatively large distance from the equilibrium point), the 3-dimensional tori are destroyed by some not yet fully understood mechanism. However, cantorian families of normally hyperbolic tori still subsist. They constitute a kind of skeleton of the motion. In some sense, one can consider the motion as a sequence of passages near lower dimensional tori, where they stay for some time interval before reaching the vicinity of next tori, following closely a heteroclinic orbit. The observed behavior in the solar system simulations for very long time intervals seems to be of this type [12]. In the present case we have, as an additional difficulty, that the system is not autonomous but quasi-periodic. In principle, we can assume that the system is reduced to autonomous form by means of time-dependent canonical changes. But this is purely formal and, furthermore, we can be faced with resonances in this process. If no resonances appear and we are satisfied with a study for moderate time intervals, we can describe the motion as the one of the autonomous system obtained by a transformation, which is shaken by the quasi-periodic change of variables.

Chapter 1

Global Stability Zones Around the Triangular Libration Points

1.1

Equations of Motion

First we give the equations and values of the parameters that we have used for the simulations in the RTBP circular and elliptic and in the bicircular problem. The values of the parameters used are: Mass parameter for the Earth-Moon problem: pA*

=

* n„ « 0.012150582. nn 82.300587

Lunar eccentricity: e = 0.054900489. Sun mass (1 unit = Earth+Moon mass): _ 0.29591220828559 x 10" 3 ~ 0.89970116585573 x 1 0 " 9 '

ms

Mean angular velocity of the Sun in synodic coordinates: WS

~

_ 129602770.31 1732564371.15'

Semimajor axis of the Sun (1 unit = Earth-Moon distance): i

as

,

1 + ms

\ 1/3 x

'

Perturbation parameter of the Sun: 6S =

ms 72"'

To keep a Hamiltonian form we use synodic coordinates x, y, z, but instead of the velocities we use momenta px, py, pz defined by px = x — y, py = y + x, pz — z. In this way the equations of the elliptic restricted problem are: 7

Global Stability Zones Around the Triangular Libration

X

=

y

=

Z

=

Pz,

Px

=

Py-X

Py

=

Points

Px+V, py

-X,

-Px-y

+ tj) (x V

r

l

+ ^ (y-

3^(z-£l) - ^(x-H r 2

+ 1) ) ,

( - ^ p + ;|)2/),

*> = ~z+'p(z-(iwi

+

%)z

where

• -

l + ecos(/ + /o) :

r\ = {x- n)2 +y2 + z2,

l ) 2 + y2 + z2.

rl = (x-v+

Here / is the independent variable. It is the true anomaly of the Moon around the Earth. Some initial phase, fo, has been used because at t = 0 the true anomaly can have any value in [0,2n]. The dot denotes derivation with respect to / . If e = 0 we have ip = 1, / = t, one can use / 0 = 0 and we recover the circular restricted problem For the bicircular problem the independent variable is t, the dot means the derivative with respect to t and the equations of motion are: x

=

y

=

Z

=

Px

=

Px+y, Py-x, Pz,

Py

3—(x-/it) r1 ,'1 - (i n\

g(a:-/u+l)

T{x

'2

-xs)

-escos0,

'3

ms .




-r~:$mr+i—>

Fig. 1.31 Representation of the stable region for z = 0.8. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.180,-0.100).

Global Stability Zones Around the Triangular Libration Points

'V

-si

* »

*.-»

v

*

Fig. 1.32 Representation of the stable region for z = 0.9. The horizontal variable is the angle a and ranges in (0.4305, 0.2005). The vertical variable is p and ranges in (-0.220,-0.140).

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