Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers

231 Pages·2004·2.246 MB·English

by  Edward Belbruno

Capture Dynamics and Chaotic Motions in
Celestial Mechanics
Unauthenticated
Download Date | 12/14/18 2:40 PM
Unauthenticated
Download Date | 12/14/18 2:40 PM
Capture Dynamics and Chaotic
Motions in Celestial Mechanics
With Applications to the Construction
of Low Energy Transfers
Edward Belbruno
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Unauthenticated
Download Date | 12/14/18 2:40 PM
Copyright (cid:1)c 2004 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 3 Market Place,
Woodstock, Oxfordshire OX20 1SY
All Rights Reserved
The publisher would like to acknowledge the author of this volume
for providing the camera-ready copy from which this book was printed.
Library of Congress Cataloging-in-Publication Data
Belbruno, Edward, 1951-
Capture dynamics and chaotic motions in celestial mechanics:
with applications to the construction of low energy transfers / Edward Belbruno
p. cm
Includes bibliographical references and index
ISBN 0-691-09480-2 (cl : alk. paper)
1. Many-body problem. 2. Chaotic behavior in systems.
3. Celestial mechanics. I. Title.
QB362.M3B45 2004
521-dc21 2003056475
British Library Cataloging-in-Publication Data is available
Printed on acid-free paper. ∞
www.pupress.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Unauthenticated
Download Date | 12/14/18 2:40 PM
Contents
List of Figures vii
Foreword xi
Preface xiii
Chapter 1. Introduction to the N-Body Problem 1
1.1 The N-Body Problem 1
1.2 Planar Three-Body Problem 9
1.3 Two-Body Problem 11
1.4 Regularization of Collision 16
1.5 The Restricted Three-Body Problem: Formulations 24
1.6 The Kepler Problem and Equivalent Geodesic Flows 35
Chapter 2. Bounded Motion, Cantor Sets, and Twist Maps 49
2.1 Quasi-Periodicity and the KAM Theorem 50
2.2 The Moser Twist Theorem, Cantor Sets 59
2.3 Area-Preserving Maps, Fixed Points, Hyperbolicity 66
Unauthenticated
Download Date | 12/14/18 2:40 PM
vi CONTENTS
2.4 Periodic Orbits and Elliptic Fixed Points 78
2.5 Aubrey-Mather Sets and the Restricted Three-Body Problem 91
Chapter 3. Capture 103
3.1 Introduction to Capture 105
3.2 The Weak Stability Boundary 120
3.3 Existence of Primary Interchange Capture and an Application 131
3.4 A Low Energy Lunar Transfer Using Ballistic Capture 144
3.5 Parabolic Motion, Hyperbolic Extension of W 156
3.6 Existence of a Hyperbolic Network on W˜ 165
H
Bibliography 193
Index 209
Unauthenticated
Download Date | 12/14/18 2:40 PM
List of Figures
1.1 Jacobi coordinates. 10
1.2 Polar coordinates and momenta. 12
1.3 Sectorial area. 13
1.4 Geometry of the Kepler ellipse. 15
1.5 Evolution of ellipses to a consecutive collision orbit. 20
1.6 Geometry of eccentric anomaly. 23
1.7 Jacobi integral surface. 28
1.8 Rotating and inertial coordinates. 29
1.9 Rotating coordinate system showing Lagrange points. 30
1.10 Stereographic projection from S2. 37
1.11 Projection from two-sheeted hyperboloid. 40
1.12 Inverted Lobachevsky disc and geodesics. 41
1.13 Lobachevsky disc and geodesics. 42
2.1 Precessing elliptical motion. 51
2.2 Flow on torus. 53
2.3 Twist map. 61
Brought to you by | Stockholm University Library
Authenticated
Download Date | 12/14/18 2:43 PM
viii LISTOFFIGURES
2.4 Twist map induced on a section of KAM tori. 64
2.5 Classic example of construction of a Cantor set. 65
2.6 Hyperbolic point. 68
2.7 Homoclinic point r. 70
2.8 Transverse homoclinic point r and hyperbolic network. 71
2.9 Heteroclinic connection. 72
2.10 Transverse heteroclinic points r,w and hyperbolic network. 73
2.11 Section of a hyperbolic periodic orbit. 73
2.12 Intersections of invariant manifolds near a hyperbolic orbit. 74
2.13 Orbit moving from stable to unstable manifolds in ex-
tended phase space. 75
2.14 Complicateddynamicsinaneighborhoodofanellipticpoint. 78
2.15 Surface of section. 88
2.16 Periodic point of type (1,3). 99
2.17 Covering space S for lift map. 100
3.1 Three-dimensional elliptic restricted three-body problem. 106
3.2 Temporary ballistic capture. 114
3.3 A restricted four-body problem. 116
3.4 The capture problem in inertial coordinates centered at P . 116
1
3.5 Stable, unstable motion and primary interchange escape. 122
3.6 Hill’s regions. 128
3.7 Moon W surface: (e ,r ,−C),θ =π. 131
2 2 2
3.8 Local invariant manifolds to a Lyapunov orbit. 134
Brought to you by | Stockholm University Library
Authenticated
Download Date | 12/14/18 2:43 PM
LISTOFFIGURES ix
3.9 Transit orbits in neck region. 135
3.10 Proof of primary interchange capture by asymptotic cap-
ture to a Lyapunov orbit. 137
3.11 ProjectionontoEarth–Sunplaneofballisticcapturetrans-
fer arc. 142
3.12 Projection onto plane normal to Earth–Sun plane in ro-
tating coordinates. 143
3.13 RepresentationofcompletetransferfromQ tolowcircu-
0
lar lunar orbit. 144
3.14 Procedure to obtain transfer using backwards integration
from the stability boundary using connecting segments I, II. 146
3.15 Ballistic capture transfer used by Hiten arriving at Moon
on October 2, 1991. 147
3.16 Nominal exterior WSB-transfer. 150
3.17 Capture process. 154
(cid:1)
3.18 Time sections t to periodic orbit ψ havin(cid:1)g a transverse
homoclinic orbit φ. A map σ is defined on with trans-
t
verse homoclinic point r. The sections are all identified. 170
3.19 Intersection of Wu(0),Ws(0) on Q =0. 173
3
3.20 Region near infinity in McGehee coordinates. 174
3.21 Map ψ near infinity. 175
3.22 Construction of the transversal map, φ˜. 177
3.23 Kepler flow in McGehee coordinates. 181
3.24 Hyperbolic invariant set Λ˜ in W˜ ∩D . A pseudoballistic
H o
capture orbit shown passing through Λ˜ on its way to near
infinity. It is lying on the invariant hyperbolic set Λ˜×{t}
intheextendedphasespace. Thisisanalogoustotheorbit
φ shown in Figure 3.18. 188
Brought to you by | Stockholm University Library
Authenticated
Download Date | 12/14/18 2:43 PM
Brought to you by | Stockholm University Library
Authenticated
Download Date | 12/14/18 2:43 PM