Ulrich Walter Astronautics The Physics of Space Flight 2nd, Enlarged and Improved Edition
Preface to Second Edition Preface XVII Acknowledgments XIX List of Symbols XXI XV 1 Rocket Fundamentals 1 1.1 Rocket Principles 2 1.1.1 Momentum Thrust 2 1.1.2 Effective Exhaust Velocity 4 1.1.3 Pressure Thrust 5 1.1.4 Momentum versus Pressure Thrust 9 1.2 Rocket Equation of Motion 12 1.3 Relativistic Rocket 13 1.3.1 Space Flight Dynamics 14 1.3.2 Relativistic Rocket Equation 16 1.3.3 Exhaust Considerations 18 1.3.4 External Efficiency 20 1.3.5 Space-Time Transformations 21 2 Rocket Flight 25 2.1 General Considerations 25 2.2 Rocket in Free Space 26 2.3 Rocket in a Gravitational Field 27 2.3.1 Impulsive Maneuvers 28 2.3.2 Brief Thrust' 29 2.3.3 Gravitational Loss 29 2.4 Propulsion and Fuel Demand 30 2.4.1 Propulsion Demand 30 2.4.2 Fuel Demand - Star Trek Plugged 32 2.5 Rocket Performance 33 2.5.1 Rocket Power 34
VIII Contents 2.5.2 Rocket Efficiency 34 2.5.3 Payload. Considerations 37 3 Rocket Staging 41 3.1 Serial Staging 42 3.1.1 Definitions 42 3.1.2 Rocket Equation 45 3.2 Serial-Stage Optimization 45 3.2.1 Road to Stage Optimization 45 3.2.2 General Optimization 46 3.3 Analytical Solutions 50 3.3.1 Uniform Staging, 53 3.3.2 Uniform Exhaust Velocities 53 3.3.3 Uneven Staging 54 3.4 Parallel Staging 55 3.5 Other Types of Staging 57 4 Thermal Propulsion 59-4.1 Engine Thermodynamics 60 4.1.1 Physics of Propellant Gases 60 A.I2 Flow Velocity 64 4.1.3 Flow at the Throat 65 4.1.4 Flow in the Nozzle 66 4.2 Ideally Adapted Nozzle 71 4.2.1 Ideal-Adaptation Criterion 71 4.2.2 Ideal Nozzle Design 72 4.2.3 Ideal Engine Performance 74 4.3 Engine Thrust 75 4.3.1 Characteristic Thrust Coefficients 76 4.3.2 Thrust Performance 77 4.3.3 Nozzle Efficiency 80 4.4 Engine Design 81 4.4.1 Combustion Chamber 82 4.4.2 Nozzle 83 4.4.3 Design Guidelines 86 5 Electric Propulsion 89 5.1 Overview 89 5.2 IonThruster 90 5.2.1 Ion Acceleration and Flow 91 5.2.2 Engine Thrust 94 5.2.3 Thruster Efficiency 95 5.3 Electric Propulsion Optimization 97
6 Ascent Flight 101 6.1 Earth's Atmosphere 101 6.1.1 Density Master Equation 101 6.1.2 Homosphere (Barometric Formula) 104 6.1.3 Heterosphere 105 6.1.4 Piecewise-Exponential Model 106 6.2 Equations of Motion 107 6.3 Ascent Phases 113 6.4 Ascent Optimization 115 6.4.1 Optimization Problem 115 6.4.2 Gravity Turn 118 6.4.3 Pitch Maneuver 120 6.4.4 Constant-Pitch-Rate Maneuver 120 6.4.5 Optimum-Ascent Trajectory 123 7 Orbits 125 7.1 Equation of Motion 325 7.1.1 Gravitational Potential 125 7.1.2 Gravitational Field 128 7.1.3 Conservation Laws 129 7.1.4 Newton's Laws 130 7.1.5 General Two-Body Problem 133 7.2 Motion Principles 136 7.2.1 Angular Momentum Conservation 136 7.2.2 Motion in the Orbital Plane 137 7.2.3 Kepler's Second Law 138 7.2.4 Energy Conservation 139 7.2.5 Effective Potential 140 7.3 Motion in a Gravitational Field 141 7.3.1 Orbit Equation 141 7.3.2 Orbital Velocity 144 7.3.3 Orbital Energy 246 7.3.4 Orbital Elements (Keplerian Elements) 147 7.3.5 Position on the Orbit 150 7.4 Keplerian Orbits 151 7.4.1 'Circular Orbit 152 7.4.2 Parabolic Orbit 153 7.4.3 Elliptic Orbit 155 ' 7.4.4 Hyperbolic Orbit 161 7.5 Radial Orbits 164 7.5.1 Radial Elliptic Orbit 166 7.5.2 Radial Hyperbolic Orbit 167 7.5.3 Radial Parabolic Orbit 168 7.5.4 Free Fall 169 7.5.5 Bounded Vertical Motion 170
7.6 life in Other Universes? 272 7.6.1 Equation of Motion in n Dimensions 173 7.6.2 4-dim Universe 176 7.6.3 Universes with dim > 5 277 7.6.4 Universes with dim < 2 2 79 Orbital Maneuvering 287 8.1 One-Impulse Maneuvers 188 8.1.1 Basic Principles 288 8.1.2 Maneuvers in Elliptical Orbits 190 8.1.3 Maneuvers in Circular Orbits 195 8.2 Lambert Transfer 196 8.2.1 Orbital Boundary Value Problem 197 8.2.2 Lambert Transfer Orbits 200 8.2.3 Lambert's Problem 204 8.2.4 Minimum Effort Lambert Transfer 206 8.3 Hohmann Transfer 207 8.3.1 The Minimum Principle 207 8.3.2 Transfer between Circular Orbits 220 8.3.3 Transfer between Near-Circular Orbits 224 8.3.4 Sensitivity Analysis 215 8.4 Other Transfers 228 8.4.1 Bi-elliptic Transfer 228 8.4.2 n-impulse Transfers 220 8.4.3 Continuous Thrust Transfer 220 8.5 Relative Orbits 222. 8.5.1 General Equation of Motion 222 8.5.2 Hill's Equations 226 8.5.3 Flyaround Trajectories 229 8.6 Orbital Rendezvous 234 8.6.1 Launch Phase 236 8.6.2 Phasing 240 8.6.3 Homing Phase 241 8.6.4 Closing Phase 247 8.6.5 Final Approach 250 8.6.6 Shuttle-ISS Rendezvous 255 8.6.7 Plume Impingement 257 Interplanetary Flight 263 9.1 Patched Conies 263 9.1.1 Sphere of Influence 264 9.1.2 Patched Conies 266 9.2 Departure Orbits 267 9.3 Transit Orbits 270 * 9.3.1 Hohmann Transfers 270 9.3.2 Non-Hohmann Transfers 273
9.4 Arrival Orbit 279 9.5 Flyby Maneuvers 282 9.5.1 Overview 281 9.5.2 Flyby Framework 282 9.5.3 Planetocentric Flyby Analysis 285 9.5.4 Heliocentric Flyby Analysis 290 9.5.5 Transition of Orbital Elements 292 9.6 Weak Stability Boundary Transfers 296 10 Re-entry 302 10.1 Introduction 302 10.1.1 Thermal Challenges 302 10.1.2 Entry Interface 304 10.1.3 Deorbit Phase 305 10.2 Equations of Motion 309 10.2.1 Normalized Equations of Motion 322 10.2.2 Reduced Equations of Motion 325 10.3 Elementary Results 317 10.3.1 Drag-Free Phase 327 10.3.2 Ballistic Re-entry 329 10.3.3 Heat Flux 322 10.4 Re-entry with lift 324-10.4.1 Lift-Only Case 324 10.4.2 General Results 326 10.4.3 Near-Ballistic Re-entry 329 10.5 Reflection and Skip Re-entry 334 10.5.1 Reflection 334 10.5.2 Skip Re-entry 338 10.5.3 PhygoidMode 341 10.6 Lifting Re-entry 345 10.6.1 Re-entry Trajectory 347 10.6.2 Critical Deceleration 348 10.6.3 Heat Flux 349 10.6.4 Space Shuttle Re-entry 352 11 Three-Body Problem 359 11.1 Overview 359. 11.2 Synchronous Orbits 361 11.2.1 Isomass Configurations 361 11.2.2 Collinear Configuration 361 11.2.3 Equilateral Configuration 367 11.3 Restiicted Three-Body Problem 369 11.3.1 Collinear Libration Points 3 71. 11.3.2 Equilateral Libration Points 374
XII Contents 11.4 Circular Restricted Three-Body Problem 374 11.4.1 Equation of Motion 376 11.4.2 Jacobi's Integral 377 11.4.3 Stability at Libration Points 380 11.4.4 Invariant Manifolds 382 11.5 Dynamics about Libration Points 386 11.5.1 Equation of Motion 386 11.5.2 Collinear Libration Points 388 11.5.3 Equilateral Libration Points 400 12 Orbit Perturbations 409 12.1 General Problem 409 12.1.1 Problem Setting 409 12.1.2 Gaussian Variational Equations 412 12.2 Gravitational Perturbations 412 12.2.1 Geoid 412 12.2.2 Gravitational Potential 414 12.2.3 Lagrange's Planetary Equations 422 12.2.4 Numerical Perturbation Methods 422 12.3 Perturbation Effects 425 12.3.1 Oblateness Perturbations 427 12.3.2 Higher-Order Perturbations 431 12.3.3 Perturbation Orbit Design 436 12.A Resonant Orbits 438 12.4.1 Resonance Conditions 440 12.4.2 Resonance Dynamics 443 12.4.3 GPS Orbits 446 12.4.4 Geostationary Orbit 450 12.5 Solar Radiation Pressure 455 12.5.1 Effects of Solar Radiation 456 12.5.2 Orbital Evolution 459 12.5.3 Correction Maneuvers 462 12.6 Drag 464 12.6.1 Drag Coefficient and Perturbations 465 12.6.2 Orbit Circularization 468 12.6.3 Circular Orbits 472 12.6.4 Orbit Lifetime 475 12.7 Celestial Perturbations 478 12.7.1 Lunisolar Perturbations 478 12.7.2 Relativistic Perturbations, 482 13 Reference Frames 489 13.1 Space Frames 489 13.2 Time Frames 495
14 Orbit Determination 499 14.1 Orbit Measurements 499 14.1.1 Radar Tracking 499 14.1.2 Other Tracking Systems 500 14.2 Methods of Orbit Determination 502 14.3 Orbit Estimation 503 14.3.1 Simple Orbit Estimation 503 14.3.2 Lambert's Method 504 14.4 Conversion of Orbital Elements 506 14.4.1 State Vector to Keplerian Elements 506 14.4.2 Keplerian Elements to State Vector 508 14.5 State Vector Propagation 508 14.5.1 Propagating State Elements 509 14.5.2 Vector Propagation 522 14.5.3 Universal Propagator 512 15 Rigid Body Dynamics 513 15.1 Fundamentals of Rotation 513 15.1.1 Elementary Physics 513 15.1.2 Equations of Rotational Motion 519 15.1.3 Coordinate Systems 522 15.1.4 Rotation-to-Translation Equivalence 523 15.2 Torque^Free Motion 524 15.2.1 Stability 525 15.2.2 Nutation 527 15.2.3 Nutation under Energy Dissipation 530 15.2.4 General Torque-Free Motion 533 15.3 Gyro under External Torque 534 ' 15.4 Gravity-Gradient Stabilization 536 15.A.I Gravity-Gradient Torque 536 15.4.2 Gravity-Gradient Oscillations 538 Appendix A Planetary Parameters 543 A.I Mean Orbit Radius 543 A. 1.1 Titius-Bode Law 543 A. 1.2 Average over True Anomaly 544 A. 1.3 Time Average 544 A.2 Mean Orbital Velocity 545 A.2.1 Average over True Anomaly 545 A.2.2 Time Average 546 Appendix B Approximate Analytical Solution for Uneven Staging 547 References 551 Index 555