## Orbital Motion, Fourth EditionLong established as one of the premier references in the fields of astronomy, planetary science, and physics, the fourth edition of Orbital Motion continues to offer comprehensive coverage of the analytical methods of classical celestial mechanics while introducing the recent numerical experiments on the orbital evolution of gravitating masses and the astrodynamics of artificial satellites and interplanetary probes. Following detailed reviews of earlier editions by distinguished lecturers in the USA and Europe, the author has carefully revised and updated this edition. Each chapter provides a thorough introduction to prepare you for more complex concepts, reflecting a consistent perspective and cohesive organization that is used throughout the book. A noted expert in the field, the author not only discusses fundamental concepts, but also offers analyses of more complex topics, such as modern galactic studies and dynamical parallaxes.
• Numerous updates and reorganization of all chapters to encompass new methods • New results from recent work in areas such as satellite dynamics • New chapter on the Caledonian symmetrical Extending its coverage to meet a growing need for this subject in satellite and aerospace engineering, |

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### Contents

1 | |

4 | |

5 | |

7 | |

9 | |

11 | |

133 Globular clusters | 13 |

134 Galactic or open clusters | 14 |

93 Planetary Ephemerides | 266 |

95 Rings Shepherds Tadpoles Horseshoes and CoOrbitals | 269 |

952 Small satellites of Jupiter and Saturn | 270 |

953 Spirig and Waldvogels analysis | 273 |

954 Satellitering interactions | 281 |

96 NearCommensurable Satellite Orbits | 284 |

97 LargeScale Numerical Integrations | 286 |

973 Does Plutos perihelion librate or circulate? | 287 |

15 Conclusion | 15 |

Coordinate and TimeKeeping Systems | 16 |

23 The Horizontal System | 18 |

24 The Equatorial System | 20 |

25 The Ecliptic System | 21 |

26 Elements of the Orbit in Space | 22 |

27 Rectangular Coordinate Systems | 24 |

29 Transformation of Systems | 25 |

292 Examples in the transformation of systems | 28 |

210 Galactic Coordinate System | 35 |

211 Time Measurement | 36 |

2112 Mean solar time | 39 |

2113 The Julian date | 41 |

Problems | 42 |

Bibliography | 43 |

The Reduction of Observational Data | 44 |

33 Refraction | 47 |

34 Precession and Nutation | 48 |

35 Aberration | 53 |

36 Proper Motion | 55 |

38 Geocentric Parallax | 56 |

39 Review of Procedures | 60 |

Problems | 61 |

The TwoBody Problem | 62 |

43 Newtons Law of Gravitation | 63 |

44 The Solution to the TwoBody Problem | 64 |

45 The Elliptic Orbit | 67 |

451 Measurement of a planets mass | 69 |

452 Velocity in an elliptic orbit | 70 |

453 The angle between velocity and radius vectors | 73 |

454 The mean eccentric and true anomalies | 74 |

455 The solution of Keplers equation | 76 |

456 The equation of the centre | 78 |

46 The Parabolic Orbit | 80 |

47 The Hyperbolic Orbit | 83 |

471 Velocity in a hyperbolic orbit | 84 |

472 Position in the hyperbolic orbit | 85 |

48 The Rectilinear Orbit | 87 |

49 Barycentric Orbits | 89 |

410 Classification of Orbits with Respect to the Energy Constant | 90 |

411 The Orbit in Space | 91 |

412 The f and g Series | 95 |

413 The Use of Recurrence Relations | 97 |

414 Universal Variables | 98 |

Problems | 99 |

Bibliography | 100 |

The ManyBody Problem | 101 |

52 The Equations of Motion in the ManyBody Problem | 102 |

53 The Ten Known Integrals and Their Meanings | 103 |

54 The Force Function | 105 |

55 The Virial Theorem | 108 |

57 The Mirror Theorem | 111 |

58 Reassessment of the ManyBody Problem | 112 |

510 General Remarks on the Lagrange Solutions | 117 |

511 The Circular Restricted ThreeBody Problem | 118 |

5112 Tisserands criterion | 121 |

5113 Surfaces of zero velocity | 122 |

5114 The stability of the libration points | 126 |

5115 Periodic orbits | 130 |

5116 The search for symmetric periodic orbits | 132 |

5117 Examples of some families of periodic orbits | 134 |

5118 Stability of periodic orbits | 136 |

5119 The surface of section | 138 |

51110 The stability matrix | 139 |

512 The General ThreeBody Problem | 140 |

5121 The case C 0 | 141 |

5122 The case for C 0 | 142 |

5123 Jacobian coordinates | 143 |

513 Jacobian Coordinates for the ManyBody Problem | 144 |

5131 The equations of motion of the simple 𝙣body HDS | 145 |

5132 The equations of motion of the general 𝙣body HDS | 147 |

5133 An unambiguous nomenclature for a general mis | 151 |

Problems | 152 |

The Caledonian Symmetric NBody Problem | 154 |

63 Sundmans Inequality | 157 |

64 Boundaries of Real and Imaginary Motion in the Caledonian Symmetrical TVBody Problem | 162 |

65 The Caledonian Symmetric Model for 𝙣 1 | 164 |

66 The Caledonian Symmetric Model for 𝙣 2 | 168 |

661 The Szebehely Ladder and Szebehelys Constant2 | 173 |

662 Regions of real motion in the pi P2 pu space | 174 |

663 Climbing the rungs of Szebehelys Ladder | 177 |

664 The case when E₀ 0 | 182 |

666 Szebehelys Constant | 183 |

667 Loks and Sergysels study of the general fourbody problem | 184 |

67 The Caledonian Symmetric Problem for 𝙣 3 | 185 |

68 The Caledonian Symmetric NBody Problem for Odd N | 191 |

Bibliography | 193 |

General Perturbations | 194 |

72 The Equations of Relative Motion | 195 |

73 The Disturbing Function | 197 |

74 The Sphere of Influence | 198 |

75 The Potential of a Body of Arbitrary Shape | 201 |

76 Potential at a Point Within a Sphere | 206 |

77 The Method of the Variation of Parameters | 208 |

771 Modification of the mean longitude at the epoch | 212 |

772 The solution of Lagranges planetary equations | 214 |

773 Short and longperiod inequalities | 217 |

11 A The resolution of the disturbing force | 220 |

78 Lagranges Equations of Motion | 223 |

79 Hamiltons Canonic Equations | 226 |

710 Derivation of Lagranges Planetary Equations from Hamiltons Canonic Equations | 231 |

Problems | 232 |

Bibliography | 233 |

Special Perturbations | 234 |

82 Factors in Special Perturbation Problems | 235 |

83 Cowells Method | 236 |

84 Enckes Method | 237 |

85 The Use of Perturbational Equations | 239 |

851 Derivation of the perturbation equations case h 0 | 241 |

852 The relations between the perturbation variables the rectangular coordinates and velocity components and the usual conicsection elements | 244 |

853 Numerical integration procedure | 246 |

854 Rectilinear or almost rectilinear orbits | 249 |

86 Regularization Methods | 251 |

87 Numerical Integration Methods | 253 |

871 Recurrence relations | 255 |

873 Multistep methods | 256 |

Problems | 261 |

The Stability and Evolution of the Solar System | 263 |

92 Chaos and Resonance | 264 |

974 The outer planets for 10 yearsand longer | 288 |

975 The analytical approach against the numerical approach | 290 |

976 The whole planetary system | 291 |

99 Conclusions | 295 |

Bibliography | 296 |

Lunar Theory | 299 |

103 The Saros | 301 |

104 Measurement of the Moons Distance Mass and Size | 303 |

105 The Moons Rotation | 304 |

106 Selenographic Coordinates | 306 |

108 The Main Lunar Problem | 307 |

109 The Suns Orbit in the Main Lunar Problem | 309 |

1010 The Orbit of the Moon | 310 |

1011 Lunar Theories | 311 |

1012 The Secular Acceleration of the Moon | 313 |

Bibliography | 314 |

Artificial Satellites | 315 |

1121 The Earths shape | 317 |

1122 Clairauts formula | 318 |

1123 The Earths interior | 321 |

1125 The Earths atmosphere | 322 |

1126 Solarterrestrial relationships | 324 |

113 Forces Acting on an Artificial Earth Satellite | 326 |

114 The Orbit of a Satellite About an Oblate Planet | 327 |

1141 The shortperiod perturbations of the first order | 330 |

1142 The secular perturbations of the first order | 333 |

1144 Secular perturbations of the secondorder and longperiod perturbations | 334 |

115 The Use of HamiltonJacobi Theory in the Artificial Satellite Problem | 335 |

116 The Effect of Atmospheric Drag on an Artificial Satellite | 337 |

117 Tesseral and Sectorial Harmonics in the Earths Gravitational Field | 342 |

Problems | 343 |

Rocket Dynamics and Transfer Orbits | 345 |

1221 Motion of a rocket in a gravitational field | 346 |

1222 Motion of a rocket in an atmosphere | 347 |

1223 Step rockets | 348 |

1224 Alternative forms of rocket | 350 |

1231 Transfer between circular coplanar orbits | 351 |

1232 Parabolic and hyperbolic transfer orbits | 354 |

1233 Changes in the orbital elements due to a small impulse | 355 |

1234 Changes in the orbital elements due to a large impulse | 357 |

1235 Variation of fuel consumption with transfer time | 358 |

1236 Sensitivity of transfer orbits to small errors in position and velocity at cutoff | 360 |

1237 Transfer between particles orbiting in a central force field | 364 |

124 Transfer Orbits in Two or More Force Fields | 368 |

1242 Entry into orbit about the second body | 370 |

1243 The hyperbolic capture | 372 |

1244 Accuracy of previous analysis and the effect of error | 373 |

1245 The flypast as a velocity amplifier | 376 |

Problems | 378 |

Bibliography | 379 |

Interplanetary and Lunar Trajectories | 380 |

133 Feasibility and Precision Study Methods | 381 |

134 The Use of Jacobis Integral | 382 |

136 The Use of TwoBody Solutions | 383 |

137 Artificial Lunar Satellites | 386 |

1371 Relative sizes of lunar satellite perturbations due to different causes | 387 |

1372 Jacobis integral for a close lunar satellite | 390 |

138 Interplanetary Trajectories | 391 |

139 The Solar System as a Central Force Field | 393 |

1310 MinimumEnergy Interplanetary Transfer Orbits | 394 |

1311 The Use of Parking Orbits in Interplanetary Missions | 399 |

1312 The Effect of Errors in Interplanetary Orbits | 405 |

Problems | 406 |

Bibliography | 407 |

Orbit Determination and Interplanetary Navigation | 408 |

142 The Theory of Orbit Determination | 409 |

143 Laplaces Method | 411 |

144 Gausss Method | 413 |

145 Olberss Method for Parabolic Orbits | 415 |

146 Orbit Determination with Additional Observational Data | 417 |

147 The Improvement of Orbits | 421 |

148 Interplanetary Navigation | 424 |

1482 Navigation by onboard optical equipment | 426 |

1483 Observational methods and probable accuracies | 428 |

429 | |

Binary and Other FewBody Systems | 430 |

152 Visual Binaries | 432 |

153 The MassLuminosity Relation | 435 |

154 Dynamical Parallaxes | 436 |

155 Eclipsing Binaries | 437 |

156 Spectroscopic Binaries | 442 |

157 Combination of Deduced Data | 445 |

159 The Period of a Binary | 447 |

1511 Forces Acting on a Binary System | 448 |

1513 The Inadequacy of Newtons Law of Gravitation | 451 |

1514 The Figures of Stars in Binary Systems | 452 |

1515 The Roche Limits | 453 |

1516 Circumstellar Matter | 454 |

1517 The Origin of Binary Systems | 456 |

457 | |

ManyBody Stellar Systems | 458 |

163 The Binary Encounter | 459 |

164 The Cumulative Effect of Small Encounters | 462 |

165 Some Fundamental Concepts | 464 |

166 The Fundamental Theorems of Stellar Dynamics | 465 |

1661 Jeanss theorem | 467 |

167 Some Special Cases for a Stellar System in a Steady State | 468 |

168 Galactic Rotation | 469 |

1681 Oorts constants | 470 |

1682 The period of rotation and angular velocity of the galaxy | 472 |

1683 The mass of the Galaxy | 473 |

1684 The mode of rotation of the Galaxy | 475 |

1685 The gravitational potential of the Galaxy | 479 |

1686 Galactic stellar orbits | 480 |

1687 The highvelocity stars | 484 |

169 Spherical Stellar Systems | 485 |

1691 Application of the virial theorem to a spherical system | 486 |

1692 Stellar orbits in a spherical system | 487 |

1693 The distribution of orbits within a spherical system | 489 |

Problems | 492 |

493 | |

Answers to Problems | 494 |

Astronomical and Related Constants | 502 |

The Earths Gravitational Field | 506 |

Approximate Elements of the Ten Largest Asteroids | 509 |

Ring Systems | 511 |

Satellite Elements and Dimensions | 512 |

514 | |