## 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, 186Appendix 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 thetroffpackage 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.