Orbital and Celestial Mechanics

Transcription

1 Orbital and Celestial Mechanics John P. Vinti Edited by Gim J. Der TRW Los Angeles, California Nino L. Bonavito NASA Goddard Space Flight Center Greenbelt, Maryland Volume 177 PROGRESS IN ASTRONAUTICS AND AERONAUTICS Paul Zarchan, Editor-in-Chief Charles Stark Draper Laboratory, Inc. Cambridge, Massachusetts Published by the American Institute of Aeronautics and Astronautics, Inc Alexander Bell Drive, Reston, Virginia

2 Table of Contents Preface xv Introduction 1 Chapter 1 Newton's Laws 7 I. Newton's Laws of Motion 7 II. Newton's Law of Gravitation 7 III. The Gravitational Potential 8 IV. Gravitational Flux and Gauss' Theorem 10 V. Gravitational Properties of a True Sphere 11 Chapter 2 The Two-Body Problem 13 I. Reduction to the One-Center Problem 13 II. The One-Center Problem 14 III. The Laplace Vector 15 IV. The Conic Section Solutions 17 V. Elliptic Orbits 19 VI. Spherical Trigonometry 24 VII. Orbit in Space 24 VIII. Orbit Determination from Initial Values 29 Chapter 3 Lagrangian Dynamics ".". 31 I. Variations 31 II. D'Alembert's Principle 32 III. Hamilton's Principle : 32 IV. Lagrange's Equations \ 34 Reference 35 Chapter 4 The Hamiltonian Equations 37 I. An Important Theorem 39 II. Ignorable Variables 39 Chapter 5 Canonical Transformations 41 I. The Condition of Exact Differentials 41 II. Canonical Generating Functions 44 III. Extended Point Transformation 47 IV. Transformation from Plane Rectangular to Plane Polar Coordinates 47 V. The Jacobi Integral 49 References 51 Chapter 6 Hamilton-Jacobi Theory 53 I. The Hamilton-Jacobi Equation 53 ix

3 II. An Important Special Case 54 III. The Hamilton-Jacobi Equation for the Kepler Problem 55 IV. The Integrals for the Kepler Problem 58 V. Relations Connecting ^2 and /S3 with w and Q, 61 VI. Summary 69 Bibliography 70 Chapter 7 Hamilton-Jacobi Perturbation Theory 71 Bibliography 74 Chapter 8 The Vinti Spheroidal Method for Satellite Orbits and Ballistic Trajectories 75 I. Introduction 75 II. The Coordinates and the Hamiltonian 75 III. The Hamilton-Jacobi Equation 77 IV. Laplace's Equation 78 V. Expansion of Potential in Spherical Harmonics 79 VI. Return to the HJ Equation 81 VII. The Kinematic Equations 82 VIII. Orbital Elements 83 IX. Factoring the Quartics 84 X. The p Integrals 85 XI. The r] Integrals 90 XII. The Mean Frequencies 96 XIII. Assembly of the Kinematic Equations 99 XIV. Solution of the Kinematic Equations 99 XV. The Periodic Terms 101 XVI. The Right Ascension (p 102 XVII. Further Developments 103 References 105 Chapter 9 Delaunay Variables.\ 107 % Reference 108 Chapter 10 The Lagrange Planetary Equations 109 I. Semi-Major Axis 110 II. Eccentricity 110 III. Inclination 110 IV. Mean Anomaly :... Ill V. The Argument of Pericenter. 112 VI. The Longitude of the Node 112 VII. Summary 113 Reference 114 Chapter 11 The Planetary Disturbing Function 115 Bibliography. 117 Chapter 12 Gaussian Variational Equations for the Jacobi Elements 119 References 125

4 XI Chapter 13 Gaussian Variational Equations for the Keplerian Elements I. Preliminaries 127 II. Equations for dj and a 130 III. Equations for a.2 and e 132 IV. Equations for a 3 and / 133 V. Equations for $ 3 = & 135 VI. Equations for fi 2 = d> 136 VII. Equations for ft and I 140 VIII. Summary 144 Chapter 14 Potential Theory 145 I. Introduction 145 II. Laplace's Equation 147 III. The Eigenvalue Problem 151 IV. The R(r) Equation 153 V. The Assembled Solution 153 VI. Legendre Polynomials 154 VII. The Results for P n (x) 154 VIII. The Solution for m > References 156 Chapter 15 The Gravitational Potential of a Planet 157 I. The Addition Theorem for Spherical Harmonics 157 II. The Standard Series 161 III. Orthogonality of Spherical Harmonics 166 IV. The Normalized Coefficients and Harmonics 168 V. The Figure of the Earth 169 VI. Geoid as an Oblate Spheroid 172 References Chapter 16 " Elementary Theory of Satellite Orbits with Use of the Mean Anomaly 175 I. A Few Numbers \ 175 II. The Disturbing Function 175 III. Elliptic Expansions 177 IV. Solution of the Lagrange Variational Equations 184 V. Motion of Perigee, First Approximation 184 VI. Motion of the Node, First Approximation 186 VII. The Semi-Major Axis 187 VIII. The Inclination. 187 IX. The Eccentricity 188 X. Variation of the Mean Motion 189 XI. Variation of the Mean Anomaly 189 References 191 Chapter 17 Elementary Theory of Satellite Orbits with Use of the True Anomaly 193 I. Introduction 193 II. Derivatives with Respect to e 195 III. The Semi-Major Axis a 195

5 XII IV. The Eccentricity e 196 V. The Inclination / 197 VI. The Motion of the Node 198 VII. The Motion of Perigee 199 VIII. Variation of the Mean Anomaly 204 Reference 206 Chapter 18 The Effects of Drag on Satellite Orbits 207 I. Introduction 207 II. Components of the Drag in Terms of the Anomalies E and/ 209 III. Equations for a and e in Terms of the True Anomaly 210 IV. Secular Behavior of a, e, ft), and 211 V. Equations for a and e in Terms of the Eccentric Anomaly 212 VI. An Equation for 212 VII. Equations for the Integration 213 References 218 Chapter 19 The Brouwer-von Zeipel Method I 219 I. Introduction 219 II. Splitting F\ into Two Parts 220 HI. Elimination of I 220 IV. Short Periodic Terms of Order J V. Second-Order Terms, General 230 VI. A Second Canonical Transformation 232 VII. Results to This Point 235 VIII. Secular Terms 236 IX. Algorithm 239 References 240 Chapter 20 The Brouwer-von Zeipel Method II 241 I. Introduction 241 II. The Effects of / III. The Effects of 7 4 > 246 IV. The Average A 4 F 247 Reference 251 Chapter 21 Lagrange and Poisson Brackets 253 I. Introduction 253 II. Lagrange Brackets 254 III. The Jacobi Relations 255 IV. Poisson Brackets 257 V. Invariance of a Poisson Bracket to a Contact Transformation 258 VI. Other Relations for Poisson Brackets 259 References 262 Chapter 22 Lie Series 263 I. Introduction 263 II. Hori's Section III. Theorems 263 References 273

6 XIII Chapter 23 Perturbations by Lie Series 275 I. Introduction 275 II. Lie Transformations 275 III. Application to Satellite Orbits 277 IV. Elimination of the Mean Anomaly 278 V. Comparison with Brouwer's Theory 280 VI. A Second Lie Transformation 285 References 289 Chapter 24 The General Three-Body Problem 291 I. Introduction 291 II. Formulation of the General Three-Body Problem 291 III. Momentum Integrals 291 IV. Angular Momentum 292 V. Energy 293 VI. Stationary Solutions 294 VII. The Triangular Stationary Solution 295 VIII. The Collinear Stationary Solution. 296 Reference 298 Chapter 25 The Restricted Three-Body Problem 299 I. Introduction 299 II. Zero-Velocity Curves 304 III. Equilibrium Points 305 IV. Motion near the Equilibrium Points 312 V. Motion in the Plane of the Primaries 313 VI. Further Considerations About L 4 and L$ 320 VII. Further Considerations About the Collinear Points 323 References 327 Chapter 26 Staeckel Systems 329 I. Staeckel's Theorem 329 II. Staeckel Systems 332 III. The Staeckel Integrals \ 333 IV. An Example: The Kepler Problem 334 V. General Remarks About Separable Systems 335 VI. Motion According to*2 = F(x) 335 VII. Conditionally Periodic Staeckel Systems 337 VIII. Action and Angle Variables 341 IX. Keplerian Action Variables X. Conditionally Periodic Staeckel Systems, Continued 347 References 352 Appendix A Coordinate Systems and Coordinate Transformations 353 I. Coordinate Systems 353 II. Coordinate Transformations 364 References 365 Appendix B Vinti Spheroidal Method Computational Procedure and Trajectory Propagators 367 I. The Kepler Problem 368 II. Given Constants 368

7 XIV III. The vinti3 Computation Procedure 369 IV. The vinti6 Computation Procedure 371 V. Summary of the Vinti Trajectory Propagators 374 References Appendix C Examples 377 I. Low-Earth Orbit 378 II. High-Earth Orbit 379 III. Molniya Orbit 379 IV. Geosynchronous Orbit 380 V. Parabolic Orbit of 0 Inclination 381 VI. "Parabolic Orbit" of 0 Inclination in the Oblate Spheroidal System VII. Hyperbolic Orbit of 0 Inclination 382 VIII. Hyperbolic Orbit of 90 Inclination 383 IX. Long-Range Ballistic Missile Trajectory 384 X. Exo-Atmospheric Interceptor Trajectory 384 Appendix D How to Use the Vinti Routines 387 I. The Source Folder 387 II. The Examples Folder 387 III. The Users 388 IV. Some Editing Problems 389 Appendix E Bibliography 391 I. Papers Published by the Author 391 II. Papers Acknowledging Vinti's Work 394 III. Books Acknowledging Vinti's Work 396 Index 397