## Chapter 1 The n - Body Problem

1.1 Introduction, 1

1.2 Equations of Motion for the n - Body Problem, 4

1.3 Justification of the Two-Body Model, 8

1.4 The Two-Body Problem, 11

1.5 The Elliptic Orbit, 13

1.6 Parabolic, Hyperbolic, and Rectilinear Orbits, 16

1.7 Energy of the Orbit, 17

References, 20

Problems, 20## Chapter 2 Position in Orbit as a Function of Time

2.1 Introduction, 25

2.2 Position and Time in an Elliptic Orbit, 25

2.3 Solution for the Eccentric Anomaly, 29

2.4 The f and g Functions and Series, 31

2.5 Position versus Time in Hyperbolic and Parabolic Orbits: Universal Variables, 36

References, 41

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 The Three-Body Problem

4.1 Introduction, 60

4.2 Stationary Solutions of the Three-Body Problem, 60

4.3 The Circular Restricted Problem, 65

4.4 Surfaces of Zero Velocity, 66

4.5 Stability of the Equilibrium Points, 66

4.6 Periodic Orbits in the Restricted Case, 70

4.7 Invariant Manifolds, 72

4.8 Special Solutions, 75

References, 76

Problems, 77## Chapter 5 Lambert's Problem

5.1 Introduction, 79

5.2 Transfer Orbits Between Specified Points, 79

5.3 Lambert's Theorem, 84

5.4 Properties of the Solutions to Lambert's Equation, 86

5.5 The Terminal Velocity Vectors, 91

5.6 Applications of Lambert's Equation, 94

5.7 Multiple-Revolution Lambert Solutions, 95

References, 99

Problems, 99## Chapter 6 Rocket Dynamics

6.1 Introduction, 101

6.2 The Rocket Equation, 101

6.3 Solution of the Rocket Equation in Field-Free Space, 103

6.4 Solution of the Rocket Equation with External Forces, 107

6.5 Rocket Payloads and Staging, 108

6.6 Optimal Staging, 113

References, 117

Problems, 117## Chapter 7 Impulsive Orbit Transfer

7.1 Introduction, 119

7.2 The Impulsive Thrust Approximation, 119

7.3 Two-Impulse Transfer between Circular Orbits, 121

7.4 The Hohmann Transfer, 123

7.5 Coplanar Extensions of the Hohmann Transfer, 127

7.6 Noncoplanar Extensions of the Hohmann Transfer, 130

7.7 Conditions for Intercept and Rendezvous, 133

References, 135

Problems, 135## Chapter 8 Continuous-Thrust Orbit Transfer

8.1 Introduction, 138

8.2 Equation of Motion, 139

8.3 Propellant Consumption, 139

8.4 Quasi-Circular Orbit Transfer, 142

8.5 The Effects of Nonconstant Mass, 144

8.6 Optimal Quasi-Circular Orbit Transfer, 145

8.7 Constant Radial Thrust Acceleration, 146

8.8 Shifted Circular Orbits, 151

References, 152

Problems, 153## Chapter 9 Interplanetary Mission Analysis

9.1 Introduction, 154

9.2 Sphere of Influence, 154

9.3 Patched Conic Method, 155

9.4 Velocity Change from Circular to Hyperbolic Orbit, 162

9.5 Planetary Flyby (Gravity- Assist) Trajectories, 163

9.6 Flyby Following a Hohmann Transfer, 166

9.7 Gravity-Assist Applications, 171

References, 175

Problems, 175## Chapter 10 Linear Orbit Theory

10.1 Introduction, 179

10.2 Linearization of the Equations of Motion, 179

10.3 The Hill-Clohessy-Wiltshire (CW) Equations, 182

10.4 The Solution of the CW Equations, 184

10.5 Linear Impulsive Rendezvous, 189

10.6 State Transition Matrix for a General Conic Orbit, 192

References, 196

Problems, 196## Chapter 11 Perturbation

11.1 Introduction, 155

11.2 The Perturbation Equations, 155

11.3 Effect of Atmospheric Drag, 164

11.4 Effect of Earth Oblateness, 164

11.5 Effects of Solar-Lunar Attraction, 209

11.6 Effect on the Orbit of the Moon, 214

References, 215

Problems, 215## Chapter 12 Canonical Systems and Lagrange Variational Equations

12.1 Introduction, 219

12.2 Hamilton's Equations, 220

12.3 Canonical Transformations, 220

12.4 Necessary and Sufficient Conditions for a Canonical Transformation, 222

12.5 Generating Functions, 224

12.6 Jacobi's Theorem, 226

12.7 Canonical Equations for the Two-Body Problem, 228

12.8 The Delaunay Variables, 231

12.9 Average Effects of Earth Oblateness Using Delaunay Variables, 232

12.10 Lagrange Equations, 234

References, 235

Problems, 235## Chapter 13 Perturbations Due to Nonspherical Terms in the Earth's Potential

13.1 Introduction, 238

13.2 Effect of the Zonal HArmonic Terms, 239

13.3 Short-Period Variations, 241

13.4 Long-Period Variations, 242

13.5 Variations at O(J_{2}^{2}) , 243

13.6 The Potential in Terms of Conventional Elements, 245

13.7 Variations Due to the Tesseral Harmonics, 247

13.8 Resonance of a Near-Geostationary Orbit, 249

References, 251

Problems, 251## Chapter 14 Orbit Determination

14.1 Introduction, 253

14.2 Angles-Only Orbit Determination, 253

14.3 Laplacian Initial Orbit Determination, 255

14.4 Gaussian Initial Orbit Determination, 259

14.5 Orbit Determination from Two Position Vectors, 263

14.6 Differential Correction, 263

References, 267

Problems, 267Appendix 1 Astronomical Constants, 270

Appendix 2 Physical Characteristics of the Planets, 271

Appendix 3 Elements of the Planetary Orbits, 272

Index, 273

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 42 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, the three-body equations of motion, and the Gauss and Laplace methods of orbit determination, from first principles.

NEW TO THIS EDITION

A considerable amount of new material has been added to this second edition:

A new chapteronthe three-body problem(Chapter 4). Since the original book was published in 1993, this subject has received much more attention; the Lagrange points are now being considered as destinations, and thhe low-energy and periodic orbits in the system have been used for several space missions.

A new chapteron continuous-thrust transfer(Chapter 8)recognizes that since teh original publication, low-thrust electtricx propulsion has now been used for several space missions, such asDeep Space 1andDawn, so familiarity with the techniques for generating their trajectories is important.

A new chapterdescribing canonical systems and the derivation of the Lagrange variational equations(Chapter 12). These are advanced topics that appear infew texts but are useful when the disturbances on a spacecraft are derivable from a potential function, such as the coomplete earth gravitational potential, which is now described in thenew Chapter 13.

New sectionson multiple-revolution Lambert solutions, gravity-assist applications, and the state transition matrix for a general conic orbit.

Many new examples and problemshave been added. This was a suggestion of the book's reviewers to make it more useful as a text. These are primarily of the practial type, i.e., where numerical results are obtained, for there were already many problems that required derivinig an analytical result. Note that problems preceded by an asterik are considered by the authors as being more difficult to solve than other problems.

J.E.P.

B.A.C.

Urbana, Illinois

February 2012