Orbital Mechanics
by
John E. Prussing
and
Bruce A. Conway
Oxford University Press
1993
Contents
Chapter 1 The n - Body Problem
1.1 Introduction, 3
1.2 Equations of Motion for the n - Body Problem, 6
1.3 Justification of the Two-Body Model, 9
1.4 The Two-Body Problem, 12
1.5 The Elliptic Orbit, 15
1.6 Parabolic, Hyperbolic, and Rectilinear Orbits, 17
1.7 Energy of the Orbit, 19
References, 22
Problems, 22
Chapter 2 Position in Orbit as a Function of Time
2.1 Introduction, 26
2.2 Position and Time in an Elliptic Orbit, 26
2.3 Solution for the Eccentric Anomaly, 30
2.4 The f and g Functions and Series, 32
2.5 Position versus Time in Hyperbolic and Parabolic Orbits: Universal Variables,
36
References, 42
Problems, 42
Chapter 3 The Orbit in Space
3.1 Introduction, 46
3.2 The Orbital Elements, 46
3.3 Determining the Orbital Elements from r and v , 49
3.4 Velocity Hodographs, 54
Reference, 59
Problems, 59
Chapter 4 Lambert's Problem
4.1 Introduction, 62
4.2 Transfer Orbits Between Specified Points, 62
4.3 Lambert's Theorem, 67
4.4 Properties of the Solutions to Lambert's Equation, 70
4.5 The Terminal Velocity Vectors, 75
4.6 Applications of Lambert's Equation, 78
References, 79
Problems, 79
Chapter 5 Rocket Dynamics
5.1 Introduction, 81
5.2 The Rocket Equation, 81
5.3 Solution of the Rocket Equation in Field-Free Space, 83
5.4 Solution of the Rocket Equation with External Forces, 87
5.5 Rocket Payloads and Staging, 88
5.6 Optimal Staging, 92
References, 97
Problems, 97
Chapter 6 Impulsive Orbit Transfer
6.1 Introduction, 99
6.2 The Impulsive Thrust Approximation, 99
6.3 Two-Impulse Transfer between Circular Orbits, 102
6.4 The Hohmann Transfer, 104
6.5 Coplanar Extensions of the Hohmann Transfer, 108
6.6 Noncoplanar Extensions of the Hohmann Transfer, 112
6.7 Conditions for Intercept and Rendezvous, 114
References, 117
Problems, 118
Chapter 7 Interplanetary Mission Analysis
7.1 Introduction, 120
7.2 Sphere of Influence, 121
7.3 Patched Conic Method, 124
7.4 Velocity Change from Circular to Hyperbolic Orbit, 128
7.5 Planetary Flyby (Gravity- Assist) Trajectories, 129
7.6 Flyby Following a Hohmann Transfer, 134
References, 137
Problems, 137
Chapter 8 Linear Orbit Theory
8.1 Introduction, 139
8.2 Linearization of the Equations of Motion, 139
8.3 The Hill-Clohessy-Wiltshire (CW) Equations, 142
8.4 The Solution of the CW Equations, 144
8.5 Linear Impulsive Rendezvous, 150
References, 153
Problems, 153
Chapter 9 Perturbation
9.1 Introduction, 155
9.2 The Perturbation Equations, 155
9.3 Effect of Atmospheric Drag, 164
9.4 Effect of Earth Oblateness, 164
References, 168
Problems, 168
Chapter 10 Orbit Determination
10.1 Introduction, 170
10.2 Angles-Only Orbit Determination, 172
10.3 Laplacian Initial Orbit Determination, 173
10.4 Gaussian Initial Orbit Determination, 176
10.5 Orbit Determination from Two Position Vectors, 180
10.6 Differential Correction, 181
References, 185
Problems, 186
Appendix 1 Astronomical Constants, 188
Appendix 2 Physical Characteristics of the Planets, 189
Appendix 3 Elements of the Planetary Orbits, 190
Index, 191
Preface
This text takes its title from an elective course at the University
of Illinois at Urbana-Champaign that has been taught to senior undergraduates
and first-year graduate students for the past 22 years. Many of these students
chose aerospace engineering because of their keen interest in space exploration.
For them, the senior-year elective courses, such as orbital mechanics, rocket
propulsion, and spacecraft design, are the reason they came to the university.
We attempt to develop the subject of orbital mechanics starting from the
first principles of Newton's Laws of Motion and the Law of Gravitation.
While it is not unusual in an introductory book to derive Kepler's Laws
of Planetary Motion from Newton's laws, we also derive the other important
results: Lambert's equation, the rocket equation, the hyperbolic gravity-
assist relations, the Hill-Clohessy-Wiltshire equations of relative motion,
the Lagrange perturbation equations, and the Gauss and Laplace methods of
orbit determination, from first principles.
There is more material in the text than we can present in one semester.
We customarily cover Chaps. 1 through 7, after which the student is well-versed
in the basic fundamentals and ready to study advanced topics. Orbit transfer
receives special emphasis because it is a favorite research area of the
authors and their graduate students. There is usually time remaining to
cover at least one of the remaining three chapters: Chap. 8, on linear orbit
theory, which is important to problems of spacecraft rendezvous, Chap. 9,
on the effect of perturbations such as atmospheric drag and earth oblateness,
and Chap. 10, on orbit determination from observations. All of these subjects
are important, even for an introduction to orbital mechanics, but each instructor
can choose which topics to emphasize.
It is customary in the last paragraph of a preface for the authors to thank
those who did the typing, the graphics, and the proofreading. With computer
word processing and drawing programs, it is virtually as easy to do such
things yourself as supervise someone else's work, so we have only ourselves
to thank or blame. The typesetting was done using the troff package
under the Unix operating system, and the graphics were done on a Macintosh
computer using the SuperPaint program. We do, however, wish to thank the
many years worth of AAE 306 students who provided the inspiration to write
this book, those who brought errors in previous versions to our attention
over the past six years, Dan Snow and Denise Kaya who gave us valuable information
on how orbit determination is actually done, and several anonymous reviewers
whose suggestions were helpful.
J.E.P.
B.A.C.
Urbana, Ill.
December 1992
Back Cover Description
One of the major challenges of modern space mission design is the
orbital mechanics -- determining how to get a spacecraft to its destination
using a limited amount of propellant.
Recent missions such as Voyager and Galileo required gravity assist
maneuvers at several planets to accomplish their objectives.
Today's students of aerospace engineering face the challenge
of calculating these types of complex spacecraft trajectories.
This classroom-tested textbook takes its title from an elective course
which has been taught to senior undergraduates and first-year graduate
students for the past 22 years.
The subject of orbital mechanics is developed starting from first
principles, using Newton's laws of motion and the law of gravitation
to prove Kepler's empirical laws of planetary motion.
Unlike many texts the authors also use first principles
to derive other important results including Kepler's equation,
Lambert's time-of-flight equation, the rocket equation, the
Clohessy-Wiltshire equations of relative motion, Gauss' equations
for the variations of the elements, and the Gauss and Laplace
methods of orbit determination.
The subject of orbit transfer receives special attention. Optimal
orbit transfers such as the Hohmann transfer, minimum-fuel transfers using
more than two impulses, and non-coplanar orbital transfer are discussed.
Patched-conic interplanetary trajectories including gravity-assist
maneuvers are the subject of an entire chapter and are particularly
relevant to modern space missions.
About the authors
John E. Prussing and Bruce A. Conway are Professors of Aeronautical
and Astronautical Engineering at the University of Illinois at
Urbana-Champaign.