Celestial mechanics and astrodynamics theory and practice

Astrophysics and Space Science Library 436

Pini Gurfil
P. Kenneth Seidelmann

Celestial Mechanics
and Astrodynamics:
Theory and Practice


Celestial Mechanics and Astrodynamics: Theory
and Practice


Astrophysics and Space Science Library
EDITORIAL BOARD
Chairman
W. B. BURTON, National Radio Astronomy Observatory, Charlottesville,
Virginia, U.S.A. (); University of Leiden, The Netherlands
()
F. BERTOLA, University of Padua, Italy
C. J. CESARSKY, Commission for Atomic Energy, Saclay, France
P. EHRENFREUND, Leiden University, The Netherlands
O. ENGVOLD, University of Oslo, Norway
A. HECK, Strasbourg Astronomical Observatory, France
E. P. J. VAN DEN HEUVEL, University of Amsterdam, The Netherlands
V. M. KASPI, McGill University, Montreal, Canada
J. M. E. KUIJPERS, University of Nijmegen, The Netherlands
H. VAN DER LAAN, University of Utrecht, The Netherlands
P. G. MURDIN, Institute of Astronomy, Cambridge, UK
B. V. SOMOV, Astronomical Institute, Moscow State University, Russia

R. A. SUNYAEV, Space Research Institute, Moscow, Russia

More information about this series at />

Pini Gurfil • P. Kenneth Seidelmann

Celestial Mechanics
and Astrodynamics:
Theory and Practice

123


Pini Gurfil
Faculty of Aerospace Engineering
Technion-Israel Institute of Technology
Haifa, Israel

P. Kenneth Seidelmann
Department of Astronomy
The University of Virginia
Charlottesville, USA

ISSN 0067-0057
ISSN 2214-7985 (electronic)
Astrophysics and Space Science Library
ISBN 978-3-662-50368-3
ISBN 978-3-662-50370-6 (eBook)
DOI 10.1007/978-3-662-50370-6
Library of Congress Control Number: 2016943837

© Springer-Verlag Berlin Heidelberg 2016
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, express or implied, with respect to the material contained herein or for any
errors or omissions that may have been made.
Cover illustration: Technion’s Space Autonomous Mission for Swarming and Geo-locating Nanosatellites (SAMSON). Credit: Asher Space Research Institute, Technion
Printed on acid-free paper
This Springer imprint is published by Springer Nature
The registered company is Springer-Verlag GmbH Berlin Heidelberg


I dedicate this book to my children, Eytam,
Oshri, and Ohav, and to my parents, Arie and
Sara.
Pini Gurfil
This book is dedicated to Bobbie Seidelmann
and our family, Holly, Kent, Jutta, Alan,
Karen, and Sarah.
P. Kenneth Seidelmann
Also we dedicate the book to the scientists
that preceded us and taught, mentored, and
inspired us.

Foreword

The early contributions to artificial satellite orbit theory were mostly made by
the celestial mechanicians, e.g., Brouwer, Garfinkel, Vinti and Kozai. Then, as
aerospace engineering curricula emerged, their astrodynamics graduates began to
make contributions. Most of the recent astrodynamics books have been written
by engineering graduates. This book, co-authored by a celestial mechanician, Ken
Seidelmann, and an astrodynamicist, Pini Gurfil, is a welcome addition to the
aerospace community as it merges the two backgrounds.
Chapter 1 begins with a short history of celestial mechanics and then transitions
to introductions to some of the key topics covered in the book. Topics included that
are not usually seen in astrodynamics books are stability, chaos, Poincaré sections,
KAM (Kolmogorov-Arnold-Moser) theory, and observation systems. Chapter 2
covers the basic math and physics concepts needed for the subjects in the book.
Chapter 3 provides an excellent discussion of coordinate systems and introduces relativity, a subject not usually included in astrodynamics books but certainly present
in celestial mechanics, e.g., the precession of Mercury’s perihelion. Chapters 4 and 5
provide a thorough discussion of the central force and two-body problems. Included
is a section on Einstein’s modification of the orbit equation. The focus of Chap. 6 is
initial orbit determination. Chapter 7 provides a thorough discussion of the N-body
problem and the integrals associated with this problem. Chapter 8 then addresses
the special case of the circular restricted 3-body problem (CR3BP). The coverage
of the CR3BP is more comprehensive than found in most astrodynamics books and
includes a discussion of families of periodic orbits. Chapter 9 is an introduction to
numerical procedures used in astrodynamics and celestial mechanics. This chapter
is not a comprehensive coverage and comparison of numerical integration methods
but an introduction to the methods needed to understand numerical methods and
error computation.

Chapter 10 begins a group of five chapters that this writer considers very
important for astrodynamics and celestial mechanics but is often not found in
astrodynamics books. I believe that the motion under the influence of conservative
perturbations, those derivable from a potential, is best addressed and understood

vii


viii

Foreword

using Hamiltonian mechanics and perturbation methods such as Lie series. Chapter 10 discusses the basics of Hamiltonian mechanics, canonical transformations,
generating functions, and Jacobi’s theorem and applies these to the two-body
problem. The focus of Chap. 11 is perturbation methods, and it begins with an
excellent discussion of the variation of parameters (VOP), which leads to Lagrange’s
planetary equations. Then, with the perturbations expressed as specific disturbing
accelerations instead of the accelerations obtained from a potential, Gauss’ variational equations are derived for the accelerations in the radial, transverse, and
orbit normal directions and the tangential, normal, and orbit normal directions.
Included is a discussion of Lagrange brackets, which are needed for the VOP. Also
in this chapter is the presentation of the Kustaanheimo-Stiefel variables. Using the
foundations developed in Chap. 10, Chap. 11 addresses the solution for the 3rd body
perturbations, atmospheric drag, and gravitational potential. Then Chap. 12 focuses
on the solution for motion about an oblate planet. There are many such solutions
beginning with Brouwer’s 1959 paper, and presenting even a few solutions would
be prohibitive. The solution presented here is the Cid-Lahulla radial intermediary.
Special perturbation (numerical integration) methods are the most accurate and
the general perturbation analytical methods, e.g, Brouwer’s solution, are the most
efficient. Chapter 13 presents the semianalytical approach, which is more efficient
than numerical integration and more accurate than the analytical solution. The

method is then applied to the four problems, a LEO satellite perturbed by drag,
frozen orbits, sun-synchronous and repeat ground track orbits, and the motion of a
geosynchronous satellite.
Chapters 10–13 address the problem of the motion of a space object under the
influence of forces derivable from a potential except for the section on the effects
of atmospheric drag. Chapters 14 and 15 consider the problem of the control of a
space object using both continuous and impulsive control. Chapter 14 considers the
control of specific types of orbits such as sun-synchronous orbits, frozen orbits, and
geosynchronous orbits, as well as gravity assists. Both impulsive and continuous
thrust control are addressed. Chapter 15 provides a very thorough coverage of the
well-known problem of optimal impulsive orbit transfers.
Chapter 16 addresses the problem of orbit data processing and presents batch
least squares and recursive filtering. Also discussed is the use of polynomials for the
compression/representation of ephemerides. Chapter 17 provides a summary of the
problem of space debris including probability of collision and collision avoidance
maneuvers. The book concludes with another discussion of main contributors to
celestial mechanics and the early pioneers of astrodynamics.
Entire books have been written on the subjects presented in many of the chapters
in this book. Thus, when writing a book on astrodynamics, there has to be a
balance between the amount of material presented and the necessary balance of
mathematical rigor and its application to the problem at hand. I believe this book
has achieved such a balance. There is a breadth of topics and each one is presented
with the necessary depth needed for the reader to understand the topic. The book can


Foreword

ix

be used for a senior/1st-year graduate class in astrodynamics and also for a 2nd-year

graduate class in astrodynamics. It is a pleasure for me to write this Foreword and
recommend this book to the astrodynamics community.
Texas A&M University, College Station, TX, USA

Kyle T. Alfriend



Preface

While astrodynamics is a relatively new science, celestial mechanics, dealing with
the motion of planets, satellites, comets, stars, and galaxies, is over three centuries
old, dating back to Kepler’s laws and Newton’s Principia. Celestial mechanics has
evolved into a myriad of approaches, methods, and results, some of which are the
bases for astrodynamics. Indeed, celestial mechanics and astrodynamics share some
fundamental tools, ranging from analytical dynamics to computer programs, used
for the calculation of spacecraft and planetary orbits.
In recent years, an unprecedented interest in celestial mechanics and astrodynamics has risen due to new space programs. Astrophysicists, astronomers,
space systems engineers, mathematicians, and scientists have been cooperating to
develop and implement groundbreaking space missions. Progress in the theory
of dynamical systems and computational methods has enabled development of
low-energy spacecraft orbits; significant progress in the research and development
of electric propulsion systems promises revolutionary, energy-efficient spacecraft
trajectories; and the idea of flying several spacecraft in formation may break
the boundaries of mass and size by creating virtual spaceborne platforms. The
problems with debris have been recognized and studied. All of these factors have
generated a growing interest in astrodynamics, a science devoted to understanding
and controlling the interaction between a spacecraft and the space environment.
Whereas there are many books dealing separately with celestial mechanics and
astrodynamics, one rarely finds a book dealing with these two topics in a unified

manner. The juxtaposition of celestial mechanics and astrodynamics is a unique
approach that is expected to be a refreshing attempt to discuss both the dynamics of
celestial objects and the mechanics of space flight. The purpose of this book is to
holistically describe methods and applications common to celestial mechanics and
astrodynamics. The book includes classical and emerging topics, manifesting the
state of the art and beyond. The book contains homogenous and fluent discussion
of the key problems, rendering a portrayal of recent advances in the field together
with some basic concepts and essential infrastructure in orbital mechanics. The
text contains introductory material followed by a gradual development of ideas
interweaved to yield a coherent presentation of advanced topics. The book presents
xi


xii

Preface

the main challenges and future prospects for the two fields, in an elaborate,
comprehensive, and mathematically rigorous manner.
This book is designed as an introductory text and reference book for graduate students, researchers, and practitioners in the fields of astronomy, celestial
mechanics, astrodynamics, satellite systems, space sciences, and astrophysics. The
purpose of the book is to emphasize the similarities between celestial mechanics and
astrodynamics and to present recent advances in these two fields, so that the reader
can understand the interrelations and mutual influences.
This book is of value to graduate students and academic researchers, for
its introduction of concepts in the field for future work and its comprehensive
discussion of the scientific and engineering state of the art; to university professors
teaching courses on orbital and/or celestial mechanics; to aerospace engineers, for
its discussion of advanced trajectory analysis and control techniques; to mathematicians, for its discussion of nonlinear dynamics and mechanics; and to astronomers,
for its presentation of perturbation methods and orbit determination schemes. It is

also of value for commercial, economic, and space policymakers, as it presents the
forefront of space technology and science from a broad and innovative perspective.
Some of the developments in the book are based on the classical books by Danby,
McCuskey, Brouwer and Clemence, Kovalevsky, and Hildebrand. We cite these
authors throughout the text.
Haifa, Israel
Charlottesville, VA, USA

Pini Gurfil
P. Kenneth Seidelmann


Acknowledgments

The authors wish to acknowledge all the celestial mechanicians and astrodynamicists that preceded them in written works, in research, and in teaching and
mentoring.
Pini Gurfil wishes to acknowledge his collaborators and students, who contributed to various chapters of this book: Dr. Vladimir Martinusi, Dr. Martin Lara,
Dr. David Mishne, Dr. Dmitry Pisarevsky, Ohad Ben-Yaacov, Alex Galperin, Elad
Denenberg, Ariel Vaknin, Gali Nir, Anton Jigalin, Sofia Belyanin, Changxuan Wen,
and Weichao Zhong. Pini Gurfil’s contribution to this book was supported by grants
from the European Research Council, the German-Israeli Foundation, and the Israeli
Ministry of Science and Technology.
Special thanks go to Dr. Michael Efroimsky, whose far-reaching vision and true
commitment to science paved the way for the creation of this book.

xiii



Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1
Definitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2
History .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3
Properties of Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3.1
The Ellipse, 0 < e < 1.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3.2
The Parabola, e D 1 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3.3
The Hyperbola, e > 1. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.4
Astronomical Background .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.5
Stability and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.5.1
Three-Body Problem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.5.2
Solar System . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.5.3
Resonances, Singularities and Regularization .. . . . . . . . .
1.6
Stability Determination .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.6.1
Poincaré Surface of Section . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1.6.2
Hill Stability . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.6.3
Lyapunov .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.6.4
Kolmogorov-Arnold-Moser Theorem . . . . . . . . . . . . . . . . . .
1.6.5
Spacecraft Orbit Stability. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.7
Chaos Determination . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8
Observational Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.1
Transit Circle . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.2
Photographic . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.3
Radar Observations . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.4
Laser Ranging .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.5
VLBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.6
CCDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.7
Optical Interferometry .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.8
Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8.9
GNSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1.8.10 Satellite Observations.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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

21
22
23
23
xv


xvi

Contents

2

Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2
Scalar Product .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3
Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4
Triple Scalar and Vector Products .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.5
Velocity of Vector .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.6
Rotation of Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.7
Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.8
Rotating Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

2.9
Gradient of a Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.10 Momentum and Energy .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.10.1 Simple Harmonic Motion . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.10.2 Linear Motion in an Inverse Square Field . . . . . . . . . . . . . .
2.10.3 Foucoult’s Pendulum . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25
25
27
28
29
30
31
32
33
37
38
42
42
43
44

3

Reference Systems and Relativity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1
Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2

Relativistic Coordinate Systems . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2.1
Newtonian Coordinates .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2.2
Relativistic Coordinates . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2.3
ICRS, BCRS, GCRS . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2.4
Geodesic Precession and Nutation . .. . . . . . . . . . . . . . . . . . . .
3.3
Reference Frames .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.1
Celestial Reference Frames . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.2
CIP and CIO . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.3
Equation of Equinoxes.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.4
Equation of Origins .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.5
Terrestrial Reference Frames . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.6
Terrestrial Intermediate Origin .. . . . .. . . . . . . . . . . . . . . . . . . .
3.3.7
ECEF, ECI, ECR . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.8
Satellite Geodesy . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.9
GNSS Reference Systems . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.4

Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5
Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.1
Origins and Planes . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.2
Horizon Reference Frame . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.3
Geocentric Coordinates .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.4
Geodetic Coordinates .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.5
Geographic Coordinates .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.6
Astronomical Coordinates .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

45
45
46
47
48
49
50
51
51
51
53
54
55
55

56
57
58
59
62
62
63
63
64
64
66


Contents

xvii

3.6

Kinematics of the Earth.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.1
Earth Orientation.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.2
Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.3
Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6.4
Polar Motion . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7
Observation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.7.1
Aberration .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.2
Proper Motion .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.3
Radial Velocities . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.4
Parallax .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.5
Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.6
Relativistic Light Deflection . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.7
Space Motion . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7.8
Tidal Effects .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.8
Earth Satellite Equations of Motion in GCRS . . . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

67
67
67
68
69
69
69
71
71
72

72
73
74
74
75
77

4

Central Force Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.2
Law of Areas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3
Linear and Angular Velocities .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.4
Integrals of Angular Momentum and Energy... . . . . . . . . . . . . . . . . . . .
4.5
Equation of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.6
Inverse Square Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.6.1
Eccentricity Vector.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.6.2
From Orbit to Force Law . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.7
Einstein’s Modification of the Orbit Equation .. . . . . . . . . . . . . . . . . . . .
4.8
Universality of Newton’s Law. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

79
79
79
81
82
83
85
90
91
92
93
94

5

The Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2
Classical Orbital Elements . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.1
Osculating Orbital Elements . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.2
Nonsingular Orbital Elements .. . . . . .. . . . . . . . . . . . . . . . . . . .
5.3
Motion of the Center of Mass . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.4
Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5.5
The Integral of Areas . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.6
Elements of the Orbit from Position and Velocity.. . . . . . . . . . . . . . . .
5.7
Properties of Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.8
The Constant of Gravitation .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.9
Kepler’s Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.9.1
Series Expansion .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.9.2
Differential Method .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.10 Position in the Elliptic Orbit. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.11 Position in the Parabolic Orbit . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

95
95
95
97
98
98
99
101
102
104
105
106
108

109
110
111


xviii

Contents

5.12
5.13

Position in a Hyperbolic Orbit . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Position on the Celestial Sphere.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.13.1 Heliocentric Coordinates . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.13.2 Geocentric Coordinates .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

113
115
115
118
119

6

Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2

Known Radius Vectors . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3
Laplace’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.4
Gauss’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5
Lambert’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.6
Parabolic Orbits, Olber’s Method . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.7
Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

121
121
121
124
131
136
138
141
142

7

The n-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7.3
Angular Momentum, or Areal Velocity, Integral . . . . . . . . . . . . . . . . . .
7.4
Integral of Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5
Stationary Solutions of the Three-Body Problem . . . . . . . . . . . . . . . . .
7.6
Generalization to n Bodies . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.7
Equations of Relative Motion . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.8
Energy Integral and Force Function.. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

143
143
144
146
149
150
155
156
160
161

8

The Restricted Three-Body Problem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

8.2
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3
The Jacobi Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.4
Zero Velocity Curves . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.5
The Lagrangian Points . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.6
Stability of Motion Near the Lagrangian Points .. . . . . . . . . . . . . . . . . .
8.7
Hill’s Restricted Three-Body Problem .. . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.7.1
Equations of Motion .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.7.2
Hill’s Equations of Motion . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.7.3
Families of Periodic Orbits . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

163
163
163
165
167
169
173
180
181
183

184
194

9

Numerical Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.1
Differences and Sums. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.2
Interpolation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.3
Lagrangian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.4
Differentiation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.5
Integration .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.6
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

197
197
199
200
203
204
205


Contents

9.7
9.8
9.9
9.10
9.11

xix

Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Numerical Integration by Runge-Kutta Methods .. . . . . . . . . . . . . . . . .
Accumulation of Errors in Numerical Integration .. . . . . . . . . . . . . . . .
Numerical Integration of Orbits .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.11.1 Equations for Cowell’s Method . . . . .. . . . . . . . . . . . . . . . . . . .
9.11.2 Equations for Encke’s Method . . . . . .. . . . . . . . . . . . . . . . . . . .
9.11.3 Comparison of Cowell’s and Encke’s Methods . . . . . . . .
9.12 Equations with Origin at the Center of Mass . .. . . . . . . . . . . . . . . . . . . .
9.13 Integration with Augmented Mass of the Sun .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

208
211
215
218
219
220
222
225
225
227

228

10 Canonical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.2 Canonical Form of the Equations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.3 Eliminating the Time Dependency .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.4 Integral of a System of Canonical Equations ... . . . . . . . . . . . . . . . . . . .
10.5 Canonical Transformation of Variables . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.5.1 Necessary Condition .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.5.2 Sufficient Condition . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.6 Examples of Canonical Transformations . . . . . .. . . . . . . . . . . . . . . . . . . .
10.6.1 Change of Variables by Means of a
Generating Function . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.6.2 Conjugate Variables to Qj . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.7 Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.8 Canonical Equations for the Two-Body Problem . . . . . . . . . . . . . . . . .
10.9 Application of Jacobi’s Theorem to the Two-body Problem.. . . . .
10.9.1 Meaning of the Constants a . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.9.2 Variables Conjugate to Qi . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.9.3 Application to the General Problem . . . . . . . . . . . . . . . . . . . .
10.10 The Delaunay Variables . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.11 The Lagrange Equations .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.12 Small Eccentricity and Small Inclination .. . . . .. . . . . . . . . . . . . . . . . . . .
10.12.1 Small Eccentricity . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.12.2 Small Inclination .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.12.3 Universal Variables . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

229
229

231
232
233
234
234
235
236
236
237
238
240
242
243
244
246
247
249
251
251
252
252
253

11 General Perturbations Theory . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.2 Variation of Parameters .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.3 Properties of the Lagrange Brackets . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.4 Evaluation of the Lagrange Brackets . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.5 Solution of the Perturbation Equations . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.6 Case I: Radial, Transverse, and Orthogonal Components . . . . . . . .

11.7 Case II: Tangential, Normal, and Orthogonal Components .. . . . . .

255
255
256
264
266
271
273
276


xx

Contents

11.8

Expansion of the Third-Body Potential .. . . . . . .. . . . . . . . . . . . . . . . . . . .
11.8.1 The Factor .r=r0 /2 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.8.2 The Factor P2 .cos / . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.9 The Earth-Moon System. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.10 Expansion of the Gravitational Potential . . . . . .. . . . . . . . . . . . . . . . . . . .
11.11 Atmospheric Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.12 Regularization of Perturbed Motion . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

278
280
282

286
290
293
294
297

12 Motion Around Oblate Planets . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.2 Axially-Symmetric Gravitational Field . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.3 Equatorial Motion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.3.1 The Orbital Angle and Radial Period . . . . . . . . . . . . . . . . . . .
12.3.2 New Orbital Elements . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.3.3 Open Orbits and the Escape Velocity . . . . . . . . . . . . . . . . . . .
12.3.4 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.4 The Cid-Lahulla Approach . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.4.1 Polar-Nodal Coordinates . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
12.4.2 The Cid-Lahulla Radial Intermediary .. . . . . . . . . . . . . . . . . .
12.4.3 Comparison with Brouwer’s Approximation .. . . . . . . . . .
12.5 Solution for Motion in a Cid-Lahulla Potential . . . . . . . . . . . . . . . . . . .
12.5.1 Main Steps Towards a Solution . . . . .. . . . . . . . . . . . . . . . . . . .
12.5.2 New Independent Variable . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

299
299
300
301
303
306
307

308
310
310
311
316
316
319
323
325

13 Semianalytical Orbit Theory . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.3 Semianalytical Models.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.3.1 The Zonal Part of the Geopotential .. . . . . . . . . . . . . . . . . . . .
13.3.2 Second-Order Effects .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.3.3 The Tesseral-Sectorial Part of the Geopotential .. . . . . . .
13.3.4 Atmospheric Drag . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.4 Frozen Orbits .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.5 Sun-synchronous and Repeat Ground-track Orbits . . . . . . . . . . . . . . .
13.6 Geostationary Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.6.1 In-Plane Motion .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.6.2 Out-of-Plane Motion.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.6.3 Averaged Solution . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
13.6.4 The Perturbed Problem . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

327
327
328

331
331
334
336
342
345
349
351
354
355
357
360
365

14 Satellite Orbit Control .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 369
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 369
14.2 Stability and Control of Dynamical Systems . .. . . . . . . . . . . . . . . . . . . . 370


Contents

14.3
14.4

xxi

Impulsive and Continuous Maneuvers .. . . . . . . .. . . . . . . . . . . . . . . . . . . .
Gravity Assist Maneuvers . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.4.1 Multiple Gravity Assists . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.4.2 Concatenation Rules . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

14.5 Optimization of Orbits .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.5.1 Static Optimization . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.5.2 Dynamic Optimization . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.6 Linear Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.7 Low Earth Orbit Control.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.7.1 Altitude Correction . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.7.2 Frozen Orbit Control.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.7.3 Sun-synchronous Orbit Control .. . . .. . . . . . . . . . . . . . . . . . . .
14.7.4 Repeat Ground-track Orbit Control .. . . . . . . . . . . . . . . . . . . .
14.8 Geostationary Orbit Control .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.8.1 North-South Stationkeeping .. . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.8.2 East-West Stationkeeping . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.8.3 Eccentricity Correction . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.9 Nonlinear Feedback Control of Orbits . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
14.10 Fixed-Magnitude Continuous-Thrust Orbit Control . . . . . . . . . . . . . .
14.11 Comparison of Continuous-Thrust Controllers .. . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

372
374
375
376
379
379
381
382
390
390
392
393

395
396
396
397
399
399
403
407
409

15 Optimal Impulsive Orbit Transfers . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
15.2 Modified Hohmann Transfer . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
15.3 Modified Bi-Elliptic and Bi-Parabolic Transfers . . . . . . . . . . . . . . . . . .
15.3.1 Definitions.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
15.3.2 Modified Bi-Elliptic Transfer . . . . . . .. . . . . . . . . . . . . . . . . . . .
15.3.3 Modified Bi-Parabolic Transfer . . . . .. . . . . . . . . . . . . . . . . . . .
15.4 Comparison Between the Modified Bi-Parabolic
and the Modified Hohmann Transfers . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
15.4.1 Bi-Elliptic Transfer . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
15.4.2 Bi-Parabolic Transfer .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

411
411
412
423
423
424
431

16 Orbit Data Processing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.2 Principle of Least Squares .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.3 Least Squares Approximation .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.4 Orthogonal Polynomials .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.5 Chebyshev Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.5.1 Chebyshev Approximation . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.5.2 Other Polynomial Approximations .. . . . . . . . . . . . . . . . . . . .
16.6 Fourier Approximation: Continuous Range . . .. . . . . . . . . . . . . . . . . . . .
16.7 Fourier Approximation: Discrete Range . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.8 Optimum Polynomial Interpolation .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

441
441
442
444
446
448
449
451
452
456
460

432
433
436
439

xxii

Contents

16.9
16.10
16.11
16.12

Chebyshev Interpolation .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Economization of Power Series . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Recursive Filtering .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Mean Elements Estimator . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.12.1 Initial Conditions and Parameter Values .. . . . . . . . . . . . . . .
16.12.2 Uncontrolled Orbits, Single Run .. . .. . . . . . . . . . . . . . . . . . . .
16.12.3 Orbits with No Control Inputs, Monte-Carlo Runs .. . . .
16.12.4 Impulsive Maneuvers . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
16.12.5 Continuous Thrust . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

462
465
468
470
472
473
477
477
480

487

17 Space Debris. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
17.2 SGP4 Propagator and TLE . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
17.3 Sizing the Debris .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
17.4 Time of Closest Approach .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
17.5 Probability of Collision . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
17.6 Calculating the Required v . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

489
489
491
493
493
495
498
499

18 People, Progress, Prospects .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.1 People and Progress.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2 Future Prospects: Exoplanets .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2.1 History.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2.3 Types of Exoplanets . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2.4 Orbit Determinations . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2.5 Planetary Systems . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2.6 Habitable Zone .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
18.2.7 Observing Program . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

501
501
505
505
506
508
509
510
510
511
511

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 513
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 515


List of Figures

Fig. 1.1
Fig. 1.2
Fig. 1.3
Fig. 1.4
Fig. 1.5
Fig. 1.6
Fig. 2.1
Fig. 2.2
Fig. 2.3
Fig. 2.4

Fig. 2.5
Fig. 2.6
Fig. 2.7
Fig. 2.8
Fig. 2.9
Fig. 2.10
Fig. 2.11
Fig. 2.12
Fig. 2.13
Fig. 2.14
Fig. 2.15
Fig. 2.16
Fig. 2.17
Fig. 3.1
Fig. 3.2

A conic .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
An ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A circle and an ellipse .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A hyperbola. The case a D b yields a rectangular hyperbola . . . . .
The positions with respect to the Earth in orbits of
planets as seen from the north pole of the ecliptic .. . . . . . . . . . . . . . . .

4
5
5
6
8

Ends of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A mutually-perpendicular right-hand triad
with positive directions.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Components of a vector .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
An angle between two vectors .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Illustration of the right-hand convention . . . . . . .. . . . . . . . . . . . . . . . . . . .
Time-varying path of a particle . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A rotation about the x-axis by an angle  . . . . . .. . . . . . . . . . . . . . . . . . . .
A rotation about the y-axis by an angle ! . . . . . .. . . . . . . . . . . . . . . . . . . .
A rotation about the z-axis by an angle i . . . . . . .. . . . . . . . . . . . . . . . . . . .
A rotation by a small angle ıÂ . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Finding the components of velocity in polar coordinates . . . . . . . . .
Illustration of a conical pendulum .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Finding the acceleration of a point above the surface
of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Mass, position and velocity . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Velocity and position of a mass m along a path C . . . . . . . . . . . . . . . . .
Rotating motion with respect to the z axis . . . . .. . . . . . . . . . . . . . . . . . . .
Foucoult’s pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

26

Equatorial and ecliptic reference planes (reproduced
from Urban and Seidelmann (2012), with permission) .. . . . . . . . . . .
Ecliptic and equatorial planes (reproduced from
Urban and Seidelmann (2012), with permission) .. . . . . . . . . . . . . . . . .

9

26

26
27
28
30
31
32
32
33
34
36
36
38
39
41
43
46
52
xxiii


xxiv

Fig. 3.3
Fig. 3.4

Fig. 3.5

Fig. 3.6
Fig. 3.7
Fig. 3.8

Fig. 3.9
Fig. 3.10
Fig. 3.11

Fig. 3.12
Fig. 3.13
Fig. 3.14
Fig. 3.15

List of Figures

Relationships of origins (reproduced from Urban and
Seidelmann (2012), with permission) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Schematic representation of the relationships between
the different fiducial meridians encountered in
the celestial and terrestrial coordinate systems
(reproduced from Urban and Seidelmann (2012), with
permission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Variation in the equation of time through the year
(reproduced from Urban and Seidelmann (2012), with
permission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Horizon system altitude and azimuth (reproduced
from Urban and Seidelmann (2012), with permission) .. . . . . . . . . . .
Geocentric and geodetic coordinates (reproduced
from Urban and Seidelmann (2012), with permission) .. . . . . . . . . . .
Relation between geographic latitude and the latitude
of the celestial pole (reproduced from Urban and
Seidelmann (2012), with permission) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Geocentric ( 0 ) and geodetic ( ) latitude (reproduced

from Urban and Seidelmann (2012), with permission) .. . . . . . . . . . .
Astronomical latitude and longitude (reproduced from
Urban and Seidelmann (2012), with permission) .. . . . . . . . . . . . . . . . .
The general precession connects the mean equinox
of epoch, 0 , to the mean equinox of date, M
(reproduced from Urban and Seidelmann (2012), with
permission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Light-time displacement (reproduced from Urban and
Seidelmann (2012), with permission) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Stellar aberration (reproduced from Urban and
Seidelmann (2012), with permission) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Parallax of an object (reproduced from Urban and
Seidelmann (2012), with permission) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Gravitational light deflection (reproduced from Urban
and Seidelmann (2012), with permission).. . . . .. . . . . . . . . . . . . . . . . . . .

56

57

60
63
64

65
65
66

68
70

70
72
73

Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4

The areal velocity due to a central force . . . . . . .. . . . . . . . . . . . . . . . . . . .
Understanding linear and angular velocities . . .. . . . . . . . . . . . . . . . . . . .
Geometry of an ellipse . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Mass particle on a circular orbit . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Fig. 5.1

Definitions of the right ascension of the ascending
node, , the argument of periapsis, !, and the
inclination, i. Also shown is the true anomaly, f .. . . . . . . . . . . . . . . . . 96
Motion of the center of mass. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99
Motion of two masses relative to the center of mass . . . . . . . . . . . . . . 100
The construction of the eccentric anomaly, E . .. . . . . . . . . . . . . . . . . . . . 107
Motion in a parabolic orbit.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 112

Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig. 5.5

80

81
86
91