AN INTRODUCTION TO PLANE ASTRONOMY
1 Some basic concepts 1 11 Newton’s laws of motion 1 12 Inertial reference frames 3 13 Deterministic and chaotic motion 6 14 Newton’s law of gravitation 8 15 Gravity field of a thin spherical shell and a sphere 13 16 External gravity field of a body with arbitrary mass distribution 15 17 Maneuvers with rocket thrust 20 18 Astronomy and the solar system 24 2 Many-body problem 25 21 Integrals of motion 26 22 Motion relative to the barycenter 31 23 Polar moment of inertia, angular momentum and energy 33 24 Evolution of n-body systems 36 25 Total collision 41 26 Pseudo-inertial reference frames 42 27 Angular momentum in the two-body problem 43 3 Three-body problem 45 31 Equations of motion 45 32 Central configuration solutions 49 33 Circular restricted three-body problem 55 34 Jacobi’s integral 60 35 Copenhagen problem 61 36 Surfaces of Hill 62 37 Lagrange libration points 64 38 Motion after leaving a surface of Hill 68 39 Stability in the libration points 70 310 Motion about the libration points 74 311 Application of Jacobi’s integral for lunar trajectories 88 312 Ballistic capture, weak stability boundary and invariant manifold 91 313 Phenomena at and use of libration points 97 4 Relative motion in the many-body problem 103 41 Equations of motion 103 42 Relative perturbing acceleration of the Earth and of an Earth satellite 107 43 Sphere of influence 112 5 Two-body problem 117 51 Conservation laws 118 52 Shape of the orbit 122 53 Conic sections 124 54 Kepler’s laws 126 55 From geocentrism to heliocentrism 128 56 Velocity components 135 57 Eccentricity vector 137 58 Stability of Keplerian orbits 139 59 Roche limit 143 510 Relativistic effects 145 511 Solar radiation pressure and the Poynting-Robertson effect 149 6 Elliptical and circular orbits 157 61 Geometry, energy and angular momentum 157 62 Circular orbit 160 63 Velocity and orbital period 162 64 Kepler’s third law 165 65 Kepler’s equation 167 66 Graphical and analytical solution of Kepler’s equation 171 67 Lambert’s theorem 174 7 Parabolic orbits 181 71 Escape velocity 181 72 Flight path angle, total energy and velocity 183 73 Barker’s equation 184 74 Euler’s equation 187 8 Hyperbolic orbits 191 81 Geometry, energy and angular momentum 191 82 Velocity 193 83 Relation between position and time 194 84 Numerical and graphical solution of the transcendental equation 198 85 Comparison of the expressions for elliptical and hyperbolic orbits 201 9 Relative motion of two satellites 203 91 Clohessy-Wiltshire equations 203 92 Analytical solution of the Clohessy-Wiltshire equations 206 93 Characteristics of unperturbed relative motion 209 94 Relative motion after an impulsive shot 214 10 Regularization 219 101 Singularity and numerical instability 219 102 Method of Burdet for a Keplerian orbit 223 103 Generalized time transformation 226 104 Method of Burdet for a perturbed Keplerian orbit 227 105 Method of Stiefel for perturbed two-dimensional motion 230 106 Method of Stiefel for perturbed three-dimensional motion 234 11 Reference frames, coordinates, time and orbital elements 243 111 Position on the Earth’s surface 243 112 Astronomical concepts 245 113 Topocentric, geocentric and heliocentric reference frames 250 114 Definition and measurement of time 252 115 Orbital elements 260 116 Relation between geocentric and geodetic latitude 263 117 Transformation from rectangular coordinates to geodetic coordinates 267 118 Transformation from orbital elements to rectangular coordinates 268 119 Transformation from spherical coordinates to orbital elements 274 1110 Transformation from rectangular coordinates to orbital elements 276 1111 f- and g-series 278 12 Transfer between two coplanar orbits 285 121 Optimum transfer between two circular orbits 285 122 Faster transfer between two circular orbits 291 123 Multiple Hohmann transfer between two circular orbits 294 124 Hohmann transfer from a circular orbit to an elliptical orbit 299 125 Hohmann transfer from an elliptical orbit to a circular orbit 301 126 Transfer from a circular orbit to an intersecting elliptical orbit 304 127 Gravity losses 306 13 Transfer between two orbits in different orbital planes 313 131 Geometry of orbital plane changes 313 132 Changing S without changing i 315 133 Changing i without changing S 317 134 Transfer between two circular orbits with different inclinations 318 135 Inclination change at high altitude 322 136 Orbital plane changes in practice 326 14 Phasing orbits 331 141 Geometric aspects of direct transfer orbits 332 142 Geometric aspects of low phasing orbits 336 143 Selection of a low phasing orbit 339 144 Geometric aspects of high phasing orbits 344 145 Selection of a high phasing orbit 347 15 Rendez-vous flights 353 151 General rendez-vous aspects 353 152 Flight from a large distance to the last hold point 357 153 Flight from the last hold point 362 154 Last phase of the rendez-vous flight 366 155 Typical Space Shuttle rendez-vous flight 369 156 Space Shuttle rendez-vous with ISS 374 157 ATV rendez-vous with ISS 376 16 Launching of a satellite 379 161 Launch vehicle ascent trajectories 379 162 Orbit injection and injection constraints 381 163 Launching of the Space Shuttle 385 164 Performance of launch vehicles 387 165 Effects of variations in the in-plane injection parameters 393 166 Effects of injection errors 396 17 Lunar flights 405 171 Earth-Moon system 410 172 Two-body lunar trajectories 413 173 Patched-conic two-dimensional lunar trajectories 417 174 Three-dimensional lunar trajectories 423 175 Non-Hohmann low-energy transfer trajectories 430 18 Interplanetary flights 437 181 General aspects of an interplanetary trajectory 439 182 Launching of interplanetary spacecraft 442 183 Assumptions and approximations 444 184 Optimum two-dimensional direct-transfer trajectories 447 185 Insertion into an orbit about the target planet 451 186 Faster two-dimensional direct-transfer trajectories 456 187 Launch opportunities 459 188 Three-dimensional direct-transfer trajectories 463 189 Gravity losses 469 1810 Approach and flyby of a planet 471 1811 Swingby flights 476 1812 Non-Hohmann low-energy transfer trajectories 492 19 Low-thrust trajectories 497 191 Electric propulsion systems and missions 498 192 Equations of motion 508 193 Trajectories with radial thrust 511 194 Trajectories with tangential thrust 514 195 Initial part of a trajectory with tangential thrust 518 196 Approximate solution of an escape trajectory with tangential thrust 520 20 Perturbing forces and perturbed satellite orbits 527 201 Earth’s gravitational force 527 202 Atmospheric drag 534 203 Gravitational attraction by other celestial bodies 540 204 Radiation force 541 205 Electromagnetic force 544 206 Special and general perturbations methods 549 207 Historical development of the theory of orbit perturbations 553 21 Elementary analysis of orbit perturbations 555 211 Basic equations 555 212 Perturbations due to the J2-term of the gravity field 558 213 Perturbations due to lunar and solar attraction 561 214 Perturbations due to atmospheric drag 567 215 Perturbations due to the solar radiation force 571 216 Perturbations due to the electromagnetic force 575 217 Perturbations due to the J2,2-term of the gravity field 578 22 Method of variation of orbital elements 585 221 Lagrange’s planetary equations 586 222 Canonic form of the planetary equations 594 223 Use of the mean anomaly 595 224 Singularities of the planetary equations 597 225 Gauss’ form of the planetary equations 598 226 Approximate analytical solution of the planetary equations 603 227 Application of Gauss’ equations to orbit maneuvers 604 23 Orbit perturbations due to the Earth’s gravity field 613 231 Application of conservation laws 613 232 Characteristics of the variation of the orbital elements 615 233 First-order secular variation of the orbital elements 619 234 Periodic variations of the orbital elements 628 235 Full spectrum of orbit perturbations 634 236 Application of Kaula’s theory 645 237 Specialized orbits 650