Space Trajectories

About the Author xv Foreword xvii Acknowledgments xix Introduction xxi Part I Free Orbital Motion 1 1 Two-Body Problem 3 1.1 Introduction 3 1.2 Keplerian Motion 3 1.2.1 Dynamic Model 4 1.2.2 Prime Integrals 5 1.2.3 Orbit Shape 7 1.3 Motion Time Law 12 1.3.1 Elliptical Orbit 12 1.3.2 Hyperbolic Orbit 14 1.3.3 Parabolic Orbit 15 1.3.4 Lagrange Coefficients 16 1.3.5 Universal Variable 18 1.4 Orbital Parameters 23 1.4.1 Classical Orbital Parameters 23 1.4.2 Relation to Position and Velocity 23 1.4.3 Equinoctial Orbital Parameters 25 1.4.4 Earth Orbits 25 1.5 Conclusion 31 1.5.1 The Key Points 31 1.5.2 To Go Further 31 2 Perturbed Motion 33 2.1 Introduction 33 2.2 Unperturbed Motion 34 2.2.1 Keplerian Model 34 2.2.2 Orbital Parameters 34 2.2.3 Useful Frames 36 2.3 Perturbed Motion 36 2.3.1 Osculating Orbit 37 2.3.2 Derivation Formula 37 2.3.3 Gauss Equations 38 2.3.4 Lagrange Equations 41 2.3.5 Equinoctial Parameters 43 2.3.6 Integration Methods 44 2.4 Gravitational Perturbations 48 2.4.1 Gravitational Potential 49 2.4.2 Spherical Body 49 2.4.3 Nonspherical Body 51 2.4.4 First Zonal Term 54 2.5 Other Perturbations 58 2.5.1 Third-Body Attraction 58 2.5.2 Atmospheric Friction 64 2.5.3 Radiation Pressure 65 2.6 Conclusion 65 2.6.1 The Key Points 65 2.6.2 To Go Further 66 3 Three-Body Problem 67 3.1 Introduction 67 3.2 Circular-Restricted Three-Body Problem 68 3.2.1 Motion Equations 68 3.2.2 Accessible Region 69 3.2.3 Lagrange Points 71 3.3 Periodic Orbits 74 3.3.1 Linearized Solution 74 3.3.2 Periodic Solution 80 3.3.3 Halo Orbits 83 3.4 Transfers 86 3.4.1 Linearization in the Orbit Vicinity 86 3.4.2 Invariant Manifolds 88 3.4.3 Transfer Strategies 89 3.5 Conclusion 93 3.5.1 The Key Points 93 3.5.2 To Go Further 93 4 Orbit Determination 95 4.1 Introduction 95 4.2 Measurements 96 4.2.1 Observation System 96 4.2.2 Measurements 100 4.2.3 Directions of Stars 102 4.3 Preliminary Orbit Estimation 104 4.3.1 Position Measurements 104 4.3.2 Direction Measurements 107 4.4 Continuous Orbit Estimation 110 4.4.1 Least Squares 111 4.4.2 Differential Correction 113 4.4.3 Kalman Filtering 116 4.5 Conclusion 119 4.5.1 The Key Points 119 4.5.2 To Go Further 119 5 Collision Risks 121 5.1 Introduction 121 5.2 Orbit Uncertainties 122 5.2.1 Orbital Motion 122 5.2.2 Ellipsoid of Uncertainty 124 5.2.3 Gaussian Model 126 5.3 Conjunction 129 5.3.1 Numerical Simulation 129 5.3.2 Orbit-to-Orbit Distance 129 5.3.3 Trajectory-to-Orbit Distance 133 5.3.4 Combined Covariance 134 5.4 Risk of Collision 136 5.4.1 Short Conjunction 137 5.4.2 Probability of Collision 138 5.4.3 Analytical Formula 140 5.4.4 Maximum Probability 141 5.4.5 Long Conjunction 143 5.5 Conclusion 145 5.5.1 The Key Points 145 5.5.2 To Go Further 145 Part II Controlled Orbital Motion 147 6 Impulsive Transfer 149 6.1 Introduction 149 6.2 Orbit Target 150 6.2.1 Problem Formulation 150 6.2.2 Transfer in One Impulse 151 6.2.3 Transfer in Two or Three Impulses 155 6.3 Point Target 160 6.3.1 Problem Formulation 160 6.3.2 Minimum-Energy Transfer 161 6.3.3 Minimum-Eccentricity Transfer 162 6.3.4 Noncollinear Transfer 163 6.3.5 Collinear Transfer 168 6.4 Point and Time Target 171 6.4.1 Lambert’s Problem 171 6.4.2 Lambert’s Theorem 171 6.4.3 Transfer Time Equation 175 6.4.4 Universal Variable 181 6.4.5 Solution Methods 183 6.5 Conclusion 184 6.5.1 The Key Points 184 6.5.2 To Go Further 184 7 Orbital Rendezvous 187 7.1 Introduction 187 7.2 Phasing and Transfer 188 7.2.1 Orbital Model 188 7.2.2 Phasing 189 7.2.3 Transfer 190 7.2.4 Visibility 197 7.3 Target in Circular Orbit 198 7.3.1 Hill–Clohessy–Wiltshire Equations 198 7.3.2 Free Motion 200 7.3.3 Maneuvers 204 7.3.4 Approach Scenario 206 7.4 Control Laws 206 7.4.1 Optimum Control 206 7.4.2 Specific Controls 210 7.5 Conclusion 214 7.5.1 The Key Points 214 7.5.2 To Go Further 214 8 Optimal Thrust Level 215 8.1 Introduction 215 8.2 Problem Formulation 216 8.2.1 Optimal Control Problem 216 8.2.2 Conditions for Optimality 217 8.2.3 Property of the Velocity Costate 218 8.3 Analytical Solution 221 8.3.1 Direction of Thrust 221 8.3.2 Costate Vector 225 8.3.3 Injection Point and Direction 226 8.3.4 Reduced Problem 228 8.3.5 Performance Estimate 229 8.4 Application 230 8.4.1 Optimized Thrust Profile 230 8.4.2 Fixed Thrust Level 232 8.5 Conclusion 236 8.5.1 The Key Points 236 8.5.2 To Go Further 237 9 Low-Thrust Transfer 239 9.1 Introduction 239 9.2 Problem Formulation 240 9.2.1 Dynamics 240 9.2.2 Optimal Control Problem 242 9.2.3 Local Control Laws 245 9.2.4 Edelbaum’s Solution 249 9.3 Transfers to the Geostationary Orbit 251 9.3.1 Dynamic Model 251 9.3.2 Optimal Control Problem 253 9.3.3 Solution Method 255 9.3.4 Application Cases 258 9.4 Transfers Between Circular Orbits 262 9.4.1 Dynamic Model 262 9.4.2 Optimal Control Problem 263 9.4.3 Form of Optimal Trajectories 265 9.4.4 Solution Method 268 9.4.5 Application Case 270 9.5 Conclusion 273 9.5.1 The Key Points 273 9.5.2 To Go Further 273 9.5.3 Authorizations 274 10 Space Debris Cleaning 275 10.1 Introduction 275 10.2 Problem Formulation 276 10.2.1 Debris Orbits 276 10.2.2 Cleaning Program 277 10.2.3 Optimization Problem 279 10.3 Transfer Problem 280 10.3.1 Generic Transfer Strategy 280 10.3.2 High-Thrust Propulsion 281 10.3.3 Low-Thrust Propulsion 282 10.3.4 Reduced Formulation 284 10.4 Solution Method 285 10.4.1 Simulated Annealing 285 10.4.2 Cost Function 287 10.4.3 Solution Process 290 10.5 Application Case 290 10.5.1 High-Thrust Case 292 10.5.2 Low-Thrust Case 294 10.5.3 Comparison of High and Low-Thrust Solutions 296 10.6 Conclusion 297 10.6.1 The Key Points 297 10.6.2 To Go Further 297 10.6.3 Authorizations 297 Part III Launch and Reentry 299 11 Launch into Orbit 301 11.1 Introduction 301 11.2 Launcher Dynamics 302 11.2.1 Force Models 302 11.2.2 Motion Equations 304 11.3 Launcher Configuration 307 11.3.1 Propulsive ΔV 307 11.3.2 Staging 308 11.3.3 Preliminary Design 310 11.3.4 Versatility 311 11.4 Trajectory Optimization 312 11.4.1 Constraints 312 11.4.2 Risk Mitigation 315 11.4.3 Optimal Control Problem 316 11.4.4 Numerical Methods 318 11.4.5 Trajectory Segmentation 320 11.5 Guidance 322 11.5.1 Implicit Guidance 323 11.5.2 Explicit Guidance 324 11.5.3 Flight Control 326 11.6 Conclusion 327 11.6.1 The Key Points 327 11.6.2 To Go Further 327 12 Launcher Staging 329 12.1 Introduction 329 12.2 Staging Problem 330 12.2.1 Launcher Modeling 330 12.2.2 Trajectory Modeling 332 12.2.3 Global Problem 334 12.3 Impulsive Method 335 12.3.1 Propulsive Velocity Increment 335 12.3.2 Staging Problem 336 12.3.3 Staging Loops 337 12.4 Coupled Method 339 12.4.1 Control Law 339 12.4.2 Iterations and Margins 340 12.4.3 Versatile Configuration 341 12.4.4 Application Case 1: Number of Boosters 342 12.4.5 Application Case 2: Upper Stage Loading 345 12.5 Propellant Reserve 348 12.5.1 Performance and Reserve 348 12.5.2 Iterative Method 349 12.5.3 Embedded Method 350 12.5.4 Application Case 351 12.6 Conclusion 354 12.6.1 The Key Points 354 12.6.2 To Go Further 354 12.6.3 Authorizations 354 13 Flat Earth Solutions 355 13.1 Introduction 355 13.2 Ballistic Launch at Constant Acceleration 355 13.2.1 Modeling 356 13.2.2 Optimal Solution 357 13.2.3 Application Case 358 13.3 Injection into Orbit at Constant Acceleration 359 13.3.1 Modeling 359 13.3.2 Optimal Solution 359 13.3.3 Application Case 361 13.4 Vertical Launch at Variable Thrust 363 13.4.1 Modeling 363 13.4.2 Optimal Solution 364 13.4.3 Application Case 366 13.5 Injection into Orbit at Variable Thrust 367 13.5.1 Modeling 367 13.5.2 Optimal Solution 368 13.5.3 Switching Function 370 13.5.4 Numerical Solution 372 13.6 Transition to the Round Earth 374 13.6.1 Change of Coordinates 374 13.6.2 Continuation Method 376 13.6.3 Application Case 378 13.7 Conclusion 381 13.7.1 The Key Points 381 13.7.2 To Go Further 381 14 Interplanetary Trajectory 383 14.1 Introduction 383 14.2 Trajectory Modeling 384 14.2.1 Patched Conics 384 14.2.2 Sphere of Influence 385 14.2.3 Heliocentric Phase 387 14.2.4 Planetocentric Phase 390 14.3 Escape Conditions 392 14.3.1 Keplerian Motion 392 14.3.2 Hyperbolic Orbit 393 14.3.3 Escape Velocity 395 14.3.4 Fly-by Maneuver 397 14.4 Mission Scenario 399 14.4.1 Launch Strategy 399 14.4.2 Tisserand Criterion 401 14.4.3 Scenario Optimization 405 14.4.4 Correction Maneuvers 406 14.5 Conclusion 409 14.5.1 The Key Points 409 14.5.2 To Go Further 410 15 Atmospheric Reentry 411 15.1 Introduction 411 15.2 Motion Equations 411 15.2.1 Reentry Missions and Vehicles 411 15.2.2 Rotating Geocentric Frame 413 15.2.3 Spherical Coordinates 414 15.2.4 Forces Applied 416 15.2.5 Differential Equations of Motion 419 15.2.6 Trajectory Optimization 420 15.3 Orbital Phase 421 15.3.1 Deorbitation 422 15.3.2 Ballistic Range 424 15.3.3 Return Maneuver 425 15.4 Atmospheric Phase 427 15.4.1 Simplified Equations 427 15.4.2 Reentry Corridor 428 15.4.3 Flight Corridor 430 15.4.4 Accessible Area 432 15.4.5 Steep Reentry 435 15.5 Conclusion 436 15.5.1 The Key Points 436 15.5.2 To Go Further 436 Short Bibliography 437 Web links 438 Index 439