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Contents

Series Preface xi Preface xiii About the Companion Website xvii

1 Historical Overview 1 1.1 Introduction 1 1.2 Early Modern Period 1 1.3 Early Twentieth Century 3 1.4 Space Age 4

2 Two-Body Orbital Mechanics 7 2.1 Introduction 7 2.2 Two-Body Problem 7 2.3 Constants of Motion 11 2.3.1 Conservation of Angular Momentum 11 2.3.2 Conservation of Energy 13 2.4 Conic Sections 15 2.4.1 Trajectory Equation 15 2.4.2 Eccentricity Vector 20 2.4.3 Energy and Semimajor Axis 21 2.5 Elliptical Orbit 23 2.5.1 Ellipse Geometry 24 2.5.2 Flight-Path Angle and Velocity Components 24 2.5.3 Period of an Elliptical Orbit 31 2.5.4 CircularCOPYRIGHTED Orbit 32 MATERIAL 2.5.5 Geocentric Orbits 33 2.6 Parabolic Trajectory 38 2.7 Hyperbolic Trajectory 42 2.8 Summary 46 Further Reading 46 Problems 47

3 Orbit Determination 55 3.1 Introduction 55 3.2 Coordinate Systems 55 3.3 Classical Orbital Elements 57

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3.4 Transforming Cartesian Coordinates to Orbital Elements 60 3.5 Transforming Orbital Elements to Cartesian Coordinates 66 3.5.1 Coordinate Transformations 68 3.6 Ground Tracks 75 3.7 Orbit Determination from One Ground-Based Observation 79 3.7.1 Topocentric-Horizon Coordinate System 79 3.7.2 Inertial Position Vector 81 3.7.3 Inertial Velocity Vector 82 3.7.4 Ellipsoidal Earth Model 85 3.8 Orbit Determination from Three Position Vectors 88 3.9 Survey of Orbit-Determination Methods 95 3.9.1 Orbit Determination Using Angles-Only Measurements 95 3.9.2 Orbit Determination Using Three Position Vectors 97 3.9.3 Orbit Determination from Two Position Vectors and Time 97 3.9.4 Statistical Orbit Determination 98 3.10 Summary 99 References 100 Problems 100

4 Time of Flight 107 4.1 Introduction 107 4.2 Kepler’s Equation 107 4.2.1 Time of Flight Using Geometric Methods 107 4.2.2 Time of Flight Using Analytical Methods 108 4.2.3 Relating Eccentric and True Anomalies 112 4.3 Parabolic and Hyperbolic Time of Flight 117 4.3.1 Parabolic Trajectory Flight Time 117 4.3.2 Hyperbolic Trajectory Flight Time 119 4.4 Kepler’s Problem 123 4.5 Orbit Propagation Using Lagrangian Coefficients 127 4.6 Lambert’s Problem 135 4.7 Summary 145 References 145 Problems 146

5 Non-Keplerian Motion 151 5.1 Introduction 151 5.2 Special Perturbation Methods 152 5.2.1 Non-Spherical Central Body 153 5.3 General Perturbation Methods 159 5.3.1 Lagrange’s Variation of Parameters 160 5.3.2 Secular Perturbations due to Oblateness (J2) 164 5.4 Gauss’ Variation of Parameters 174 5.5 Perturbation Accelerations for Earth Satellites 180 5.5.1 Non-Spherical Earth 180 5.5.2 Third-Body Gravity 182

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5.5.3 Atmospheric Drag 185 5.5.4 Solar Radiation Pressure 189 5.6 Circular Restricted Three-Body Problem 192 5.6.1 Jacobi’s Integral 194 5.6.2 Lagrangian Points 195 5.7 Summary 203 References 203 Problems 204

6 Rocket Performance 213 6.1 Introduction 213 6.2 Rocket Propulsion Fundamentals 213 6.3 The Rocket Equation 214 6.4 Launch Trajectories 219 6.5 Staging 227 6.6 Launch Vehicle Performance 231 6.7 Impulsive Maneuvers 233 6.8 Summary 234 References 235 Problems 235

7 Impulsive Orbital Maneuvers 241 7.1 Introduction 241 7.2 Orbit Shaping 242 7.3 Hohmann Transfer 245 7.3.1 Coplanar Transfer with Tangential Impulses 248 7.4 General Coplanar Transfer 252 7.5 Inclination-Change Maneuver 256 7.6 Three-Dimensional Orbit Transfer 259 7.7 Summary 264 References 264 Problems 264

8 Relative Motion and Orbital Rendezvous 275 8.1 Introduction 275 8.2 Linear Clohessy–Wiltshire Equations 275 8.3 Homogeneous Solution of the Clohessy–Wiltshire Equations 280 8.4 Orbital Rendezvous Using the Clohessy–Wiltshire Equations 288 8.5 Summary 298 References 298 Problems 298

9 Low-Thrust Transfers 303 9.1 Introduction 303 9.2 Electric Propulsion Fundamentals 304 9.3 Coplanar Circle-to-Circle Transfer 306

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9.3.1 Comparing Impulsive and Low-Thrust Transfers 313 9.4 Coplanar Transfer with Earth-Shadow Effects 315 9.5 Inclination-Change Maneuver 318 9.6 Transfer Between Inclined Circular Orbits 320 9.7 Combined Chemical-Electric Propulsion Transfer 322 9.8 Low-Thrust Transfer Issues 328 9.9 Summary 329 References 329 Problems 330

10 Interplanetary Trajectories 335 10.1 Introduction 335 10.2 Patched-Conic Method 338 10.2.1 Sphere of Influence 339 10.2.2 Coplanar Heliocentric Transfers between Circular Orbits 341 10.3 Phase Angle at Departure 351 10.4 Planetary Arrival 355 10.5 Heliocentric Transfers Using an Accurate Ephemeris 359 10.5.1 Pork-Chop Plots 367 10.5.2 Julian Date 368 10.6 Gravity Assists 370 10.7 Summary 378 References 379 Problems 379

11 Atmospheric Entry 385 11.1 Introduction 385 11.2 Entry Flight Mechanics 386 11.3 Ballistic Entry 390 11.4 Gliding Entry 396 11.5 Skip Entry 404 11.6 Entry Heating 412 11.7 Space Shuttle Entry 418 11.8 Summary 422 References 423 Problems 423

12 Attitude Dynamics 429 12.1 Introduction 429 12.2 Rigid Body Dynamics 430 12.2.1 Angular Momentum of a Rigid Body 432 12.2.2 Principal Axes 438 12.2.3 Rotational Kinetic Energy 439 12.2.4 Euler’s Moment Equations 441 12.3 Torque-Free Motion 442

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12.3.1 Euler Angle Rates 447 12.4 Stability and Flexible Bodies 457 12.4.1 Spin Stability about the Principal Axes 457 12.4.2 Stability of Flexible Bodies 459 12.5 Spin Stabilization 464 12.5.1 Dual-Spin Stabilization 466 12.6 Disturbance Torques 467 12.6.1 Gravity-Gradient Torque 467 12.6.2 Aerodynamic Torque 468 12.6.3 Solar Radiation Pressure Torque 469 12.6.4 Magnetic Torque 470 12.7 Gravity-Gradient Stabilization 470 12.8 Summary 476 References 477 Problems 477

13 Attitude Control 485 13.1 Introduction 485 13.2 Feedback Control Systems 485 13.2.1 Transfer Functions 486 13.2.2 Closed-Loop Control Systems 489 13.2.3 Second-Order System Response 490 13.3 Mechanisms for Attitude Control 497 13.3.1 Reaction Jets 497 13.3.2 Momentum-Exchange Devices 497 13.3.3 Magnetic Torquers 501 13.4 Attitude Maneuvers Using Reaction Wheels 501 13.5 Attitude Maneuvers Using Reaction Jets 513 13.5.1 Phase-Plane Analysis of Satellite Attitude Dynamics 513 13.5.2 Reaction Jet Control Law 518 13.6 Nutation Control Using Reaction Jets 527 13.7 Summary 534 References 535 Further Reading 535 Problems 535

Appendix A: Physical Constants 541

Appendix B: Review of Vectors 543 B.1 Introduction 543 B.2 Vectors 543 B.3 Vector Operations 544 B.3.1 Vector Addition 544 B.3.2 Cross Product 545 B.3.3 Dot Product 546

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B.3.4 Scalar Triple Product 547 B.3.5 Vector Triple Product 547

Appendix C: Review of Particle Kinematics 549 C.1 Introduction 549 C.2 Cartesian Coordinates 549 C.3 Polar Coordinates 551 C.4 Normal-Tangential Coordinates 552

Index 555

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