Orbital Motion, Fourth Edition

Long 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. New to the Fourth Edition: \* 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 n-body problem Extending its coverage to meet a growing need for this subject in satellite and aerospace engineering, Orbital Motion, Fourth Edition remains a top reference for postgraduate and advanced undergraduate students, professionals such as engineers, and serious amateur astronomers. Frontmatter Orbital Motion, Fourth Edition 2 Contents 5 Chapter 1: The Restless Universe 13 1.1 Introduction 13 1.2 The Solar System 13 1.2.1 Kepler’s laws 16 1.2.2 Bode’s law 16 1.2.3 Commensurabilities in mean motion 17 1.2.4 Comets, the Edgeworth-Kuiper Belt and meteors 19 1.2.5 Conclusions 21 1.3 Stellar Motions 21 1.3.1 Binary systems 23 1.3.2 Triple and higher systems of stars 23 1.3.3 Globular clusters 25 1.3.4 Galactic or open clusters 26 1.4 Clusters of Galaxies 26 1.5 Conclusion 27 Bibliography 27 Chapter 2: Coordinate and Time-Keeping Systems 28 2.1 Introduction 28 2.2 Position on the Earth’s Surface 28 2.3 The Horizontal System 30 2.4 The Equatorial System 32 2.5 The Ecliptic System 33 2.6 Elements of the Orbit in Space 34 2.7 Rectangular Coordinate Systems 36 2.8 Orbital Plane Coordinate Systems 36 2.9 Transformation of Systems 37 2.9.1 The fundamental formulae of spherical trigonometry 37 2.9.2 Examples in the transformation of systems 40 2.10 Galactic Coordinate System 47 2.11 Time Measurement 48 2.11.1 Sidereal time 48 2.11.2 Mean solar time 51 2.11.3 The Julian date 53 2.11.4 Ephemeris Time 53 Problems 54 Bibliography 55 Chapter 3: The Reduction of Observational Data 56 3.1 Introduction 56 3.2 Observational Techniques 56 3.3 Refraction 59 3.4 Precession and Nutation 60 3.5 Aberration 65 3.6 Proper Motion 67 3.7 Stellar Parallax 67 3.8 Geocentric Parallax 68 3.9 Review of Procedures 72 Problems 73 Bibliography 73 Chapter 4: The Two-Body Problem 74 4.1 Introduction 74 4.2 Newton’s Laws of Motion 74 4.3 Newton’s Law of Gravitation 75 4.4 The Solution to the Two-Body Problem 76 4.5 The Elliptic Orbit 79 4.5.1 Measurement of a planet’s mass 81 4.5.2 Velocity in an elliptic orbit 82 4.5.3 The angle between velocity and radius vectors 85 4.5.4 The mean, eccentric and true anomalies 86 4.5.5 The solution of Kepler’s equation 88 4.5.6 The equation of the centre 90 4.5.7 Position of a body in an elliptic orbit 90 4.6 The Parabolic Orbit 92 4.7 The Hyperbolic Orbit 95 4.7.1 Velocity in a hyperbolic orbit 96 4.7.2 Position in the hyperbolic orbit 97 4.8 The Rectilinear Orbit 99 4.9 Barycentric Orbits 101 4.10 Classification of Orbits with Respect to the Energy Constant 102 4.11 The Orbit in Space 103 4.12 The f and g Series 107 4.13 The Use of Recurrence Relations 109 4.14 Universal Variables 110 Problems 111 Bibliography 112 Chapter 5: The Many-Body Problem 113 5.1 Introduction 113 5.2 The Equations of Motion in the Many-Body Problem 114 5.3 The Ten Known Integrals and TheirMeanings 115 5.4 The Force Function 117 5.5 The Virial Theorem 120 5.6 Sundman’s Inequality 120 5.7 The Mirror Theorem 123 5.8 Reassessment of the Many-Body Problem 124 5.9 Lagrange’s Solutions of the Three-Body Problem 124 5.10 General Remarks on the Lagrange Solutions 129 5.11 The Circular Restricted Three-Body Problem 130 5.11.1 Jacobi’s integral 130 5.11.2 Tisserand’s criterion 133 5.11.3 Surfaces of zero velocity 134 5.11.4 The stability of the libration points 138 5.11.5 Periodic orbits 142 5.11.6 The search for symmetric periodic orbits 144 5.11.7 Examples of some families of periodic orbits 146 5.11.8 Stability of periodic orbits 148 5.11.9 The surface of section 150 5.11.10 The stability matrix 151 5.12 The General Three-Body Problem 152 5.12.1 The case C < 0 153 5.12.2 The case for C = 0 154 5.12.3 Jacobian coordinates 155 5.13 Jacobian Coordinates for the Many-Body Problem 156 5.13.1 The equations of motion of the simple n-body HDS 157 5.13.2 The equations of motion of the general n-body HDS 159 5.13.3 An unambiguous nomenclature for a general HDS 163 5.14 The Hierarchical Three-Body Stability Criterion 163 Problems 164 Bibliography 164 Chapter 6: The Caledonian Symmetric N-Body Problem 166 6.1 Introduction 166 6.2 The Equations of Motions 166 6.3 Sundman’s Inequality 169 6.4 Boundaries of Real and Imaginary Motion in the Caledonian Symmetrical N-Body Problem 174 6.5 The Caledonian Symmetric Model for n = 1 176 6.6 The Caledonian Symmetric Model for n = 2 180 6.6.1 The Szebehely Ladder and Szebehely’s Constant2 185 6.6.2 Regions of real motion in the ρ1, ρ2, ρ12 space 186 6.6.3 Climbing the rungs of Szebehely’s Ladder 189 6.6.4 The case when E0 < 0 194 6.6.5 Unequal masses μ1 is not equal to μ2 in the n = 2 case 194 6.6.6 Szebehely’s Constant 195 6.6.7 Loks and Sergysels’ study of the general four-body problem 196 6.7 The Caledonian Symmetric Problem for n = 3 197 6.8 The Caledonian Symmetric N-Body Problem for Odd 203 Bibliography 205 Chapter 7: General Perturbations 206 7.1 The Nature of the Problem 206 7.2 The Equations of Relative Motion 207 7.3 The Disturbing Function 209 7.4 The Sphere of Influence 210 7.5 The Potential of a Body of Arbitrary Shape 213 7.6 Potential at a Point Within a Sphere 218 7.7 The Method of the Variation of Parameters 220 7.7.1 Modification of the mean longitude at the epoch 224 7.7.2 The solution of Lagrange’s planetary equations 226 7.7.3 Short–and long-period inequalities 229 7.7.4 The resolution of the disturbing force 232 7.8 Lagrange’s Equations of Motion 235 7.9 Hamilton’s Canonic Equations 238 7.10 Derivation of Lagrange’s Planetary Equations from Hamilton’s Canonic Equations 243 Problems 244 Bibliography 245 Chapter 8: Special Perturbations 246 8.1 Introduction 246 8.2 Factors in Special Perturbation Problems 247 8.2.1 The type of orbit 247 8.2.2 The operational requirements 247 8.2.3 The formulation of the equations of motion 247 8.2.4 The numerical integration procedure 247 8.2.5 The available computing facilities 247 8.3 Cowell’s Method 248 8.4 Encke’s Method 249 8.5 The Use of Perturbational Equations 251 8.5.1 Derivation of the perturbation equations (case h is not equal to 0) 253 8.5.2 The relations between the perturbation variables, the rectangular co-ordinates and velocity components, and the usual conic-section elements. 256 8.5.3 Numerical integration procedure 258 8.5.4 Rectilinear or almost rectilinear orbits 261 8.6 Regularization Methods 263 8.7 Numerical Integration Methods 265 8.7.1 Recurrence relations 267 8.7.2 Runge–Kutta four 267 8.7.3 Multistep methods 268 8.7.4 Numerical methods 268 Problems 273 Bibliography 273 Chapter 9: The Stability and Evolution of the Solar System 275 9.1 Introduction 275 9.2 Chaos and Resonance 276 9.3 Planetary Ephemerides 278 9.4 The Asteroids 278 9.5 Rings, Shepherds, Tadpoles, Horseshoes and Co-Orbitals 281 9.5.1 Ring systems 281 9.5.2 Small satellites of Jupiter and Saturn 282 9.5.3 Spirig and Waldvogel’s analysis 285 9.5.4 Satellite-ring interactions 293 9.6 Near-Commensurable Satellite Orbits 296 9.7 Large-Scale Numerical Integrations 298 9.7.1 The outer planets for 120000 years 298 9.7.2 Element plots for 1000000 years 298 9.7.3 Does Pluto’s perihelion librate or circulate? 299 9.7.4 The outer planets for 108 years--and longer! 300 9.7.5 The analytical approach against the numerical approach 302 9.7.6 The whole planetary system 303 9.8 Empirical Stability Criteria 303 9.9 Conclusions 307 Bibliography 308 Chapter 10: Lunar Theory 311 10.1 Introduction 311 10.2 The Earth-Moon System 311 10.3 The Saros 313 10.4 Measurement of the Moon’s Distance, Mass and Size 315 10.5 The Moon’s Rotation 316 10.6 Selenographic Coordinates 318 10.7 The Moon’s Figure 318 10.8 The Main Lunar Problem 319 10.9 The Sun’s Orbit in the Main Lunar Problem 321 10.10 The Orbit of the Moon 322 10.11 Lunar Theories 323 10.12 The Secular Acceleration of the Moon 325 Bibliography 326 Chapter 11: Artificial Satellites 327 11.1 Introduction 327 11.2 The Earth as a Planet 327 11.2.1 The Earth’s shape 329 11.2.2 Clairaut’s formula 330 11.2.3 The Earth’s interior 333 11.2.4 The Earth’s magnetic field 333 11.2.5 The Earth’s atmosphere 334 11.2.6 Solar-terrestrial relationships 336 11.3 Forces Acting on an Artificial Earth Satellite 338 11.4 The Orbit of a Satellite About an Oblate Planet 339 11.4.1 The short-period perturbations of the first order 342 11.4.2 The secular perturbations of the first order 345 11.4.3 Long-period perturbations from the third harmonic 345 11.4.4 Secular perturbations of the second-order and long-period perturbations 346 11.5 The Use of Hamilton-Jacobi Theory in the Artificial Satellite Problem 347 11.6 The Effect of Atmospheric Drag on an Artificial Satellite 349 11.7 Tesseral and Sectorial Harmonics in the Earth’s Gravitational Field 354 Problems 355 Bibliography 355 Chapter 12: Rocket Dynamics and Transfer Orbits 357 12.1 Introduction 357 12.2 Motion of a Rocket 357 12.2.1 Motion of a rocket in a gravitational field 358 12.2.2 Motion of a rocket in an atmosphere 359 12.2.3 Step rockets 360 12.2.4 Alternative forms of rocket 362 12.3 Transfer Between Orbits in a Single Central Force Field 362 12.3.1 Transfer between circular, coplanar orbits 363 12.3.2 Parabolic and hyperbolic transfer orbits 366 12.3.3 Changes in the orbital elements due to a small impulse 367 12.3.4 Changes in the orbital elements due to a large impulse 369 12.3.5 Variation of fuel consumption with transfer time 370 12.3.6 Sensitivity of transfer orbits to small errors in position and velocity at cut-off 372 12.3.7 Transfer between particles orbiting in a central force field 376 12.4 Transfer Orbits in Two or More Force Fields 380 12.4.1 The hyperbolic escape from the first body 380 12.4.2 Entry into orbit about the second body 382 12.4.3 The hyperbolic capture 384 12.4.4 Accuracy of previous analysis and the effect of error 385 12.4.5 The fly-past as a velocity amplifier 388 Problems 390 Bibliography 391 Chapter 13: Interplanetary and Lunar Trajectories 392 13.1 Introduction 392 13.2 Trajectories in Earth-Moon Space 392 13.3 Feasibility and Precision Study Methods 393 13.4 The Use of Jacobi’s Integral 394 13.5 The Use of the Lagrangian Solutions 394 13.6 The Use of Two-Body Solutions 395 13.7 Artificial Lunar Satellites 399 13.7.1 Relative sizes of lunar satellite perturbations due to different causes 400 13.7.2 Jacobi’s integral for a close lunar satellite 404 13.8 Interplanetary Trajectories 405 13.9 The Solar System as a Central Force Field 406 13.10 Minimum-Energy Interplanetary Transfer Orbits 407 13.11 The Use of Parking Orbits in Interplanetary Missions 413 13.12 The Effect of Errors in Interplanetary Orbits 418 Problems 419 Chapter 14: Orbit Determination and Interplanetary Navigation 420 14.1 Introduction 420 14.2 The Theory of Orbit Determination 421 14.3 Laplace’s Method 423 14.4 Gauss’s Method 425 14.5 Olbers’s Method for Parabolic Orbits 427 14.6 Orbit Determination with Additional Observational Data 429 14.7 The Improvement of Orbits 433 14.8 Interplanetary Navigation 436 14.8.1 Stabilized platforms and accelerometers 436 14.8.2 Navigation by on-board optical equipment 438 14.8.3 Observational methods and probable accuracies 440 Bibliography 441 Chapter 15: Binary and Other Few-Body Systems 442 15.1 Introduction 442 15.2 Visual Binaries 444 15.3 The Mass-Luminosity Relation 447 15.4 Dynamical Parallaxes 448 15.5 Eclipsing Binaries 449 15.6 Spectroscopic Binaries 454 15.7 Combination of Deduced Data 457 15.8 Binary Orbital Elements 457 15.9 The Period of a Binary 459 15.10 Apsidal Motion 459 15.11 Forces Acting on a Binary System 460 15.12 Triple Systems 460 15.13 The Inadequacy of Newton’s Law of Gravitation 463 15.14 The Figures of Stars in Binary Systems 464 15.15 The Roche Limits 465 15.16 Circumstellar Matter 466 15.17 The Origin of Binary Systems 468 Problems 469 Bibliography 469 Chapter 16: Many-Body Stellar Systems 470 16.1 Introduction 470 16.2 The Sphere of Influence 470 16.3 The Binary Encounter 471 16.4 The Cumulative Effect of Small Encounters 474 16.5 Some Fundamental Concepts 476 16.6 The Fundamental Theorems of Stellar Dynamics 477 16.6.1 Jeans’s theorem 479 16.7 Some Special Cases for a Stellar System in a Steady State 480 16.8 Galactic Rotation 481 16.8.1 Oort’s constants 482 16.8.2 The period of rotation and angular velocity of the Galaxy 484 16.8.3 The mass of the Galaxy 485 16.8.4 The mode of rotation of the Galaxy 487 16.8.5 The gravitational potential of the Galaxy 491 16.8.6 Galactic stellar orbits 492 16.8.7 The high-velocity stars 496 16.9 Spherical Stellar Systems 497 16.9.1 Application of the virial theorem to a spherical system 498 16.9.2 Stellar orbits in a spherical system 499 16.9.3 The distribution of orbits within a spherical system 501 16.10 Modern Galactic Studies 501 Problems 504 Bibliography 505 Orbital Motion, Fourth Edition......Page 2 Contents......Page 5 1.2 The Solar System......Page 13 1.2.2 Bode’s law......Page 16 1.2.3 Commensurabilities in mean motion......Page 17 1.2.4 Comets, the Edgeworth-Kuiper Belt and meteors......Page 19 1.3 Stellar Motions......Page 21 1.3.2 Triple and higher systems of stars......Page 23 1.3.3 Globular clusters......Page 25 1.4 Clusters of Galaxies......Page 26 Bibliography......Page 27 2.2 Position on the Earth’s Surface......Page 28 2.3 The Horizontal System......Page 30 2.4 The Equatorial System......Page 32 2.5 The Ecliptic System......Page 33 2.6 Elements of the Orbit in Space......Page 34 2.8 Orbital Plane Coordinate Systems......Page 36 2.9.1 The fundamental formulae of spherical trigonometry......Page 37 2.9.2 Examples in the transformation of systems......Page 40 2.10 Galactic Coordinate System......Page 47 2.11.1 Sidereal time......Page 48 2.11.2 Mean solar time......Page 51 2.11.4 Ephemeris Time......Page 53 Problems......Page 54 Bibliography......Page 55 3.2 Observational Techniques......Page 56 3.3 Refraction......Page 59 3.4 Precession and Nutation......Page 60 3.5 Aberration......Page 65 3.7 Stellar Parallax......Page 67 3.8 Geocentric Parallax......Page 68 3.9 Review of Procedures......Page 72 Bibliography......Page 73 4.2 Newton’s Laws of Motion......Page 74 4.3 Newton’s Law of Gravitation......Page 75 4.4 The Solution to the Two-Body Problem......Page 76 4.5 The Elliptic Orbit......Page 79 4.5.1 Measurement of a planet’s mass......Page 81 4.5.2 Velocity in an elliptic orbit......Page 82 4.5.3 The angle between velocity and radius vectors......Page 85 4.5.4 The mean, eccentric and true anomalies......Page 86 4.5.5 The solution of Kepler’s equation......Page 88 4.5.7 Position of a body in an elliptic orbit......Page 90 4.6 The Parabolic Orbit......Page 92 4.7 The Hyperbolic Orbit......Page 95 4.7.1 Velocity in a hyperbolic orbit......Page 96 4.7.2 Position in the hyperbolic orbit......Page 97 4.8 The Rectilinear Orbit......Page 99 4.9 Barycentric Orbits......Page 101 4.10 Classification of Orbits with Respect to the Energy Constant......Page 102 4.11 The Orbit in Space......Page 103 4.12 The f and g Series......Page 107 4.13 The Use of Recurrence Relations......Page 109 4.14 Universal Variables......Page 110 Problems......Page 111 Bibliography......Page 112 5.1 Introduction......Page 113 5.2 The Equations of Motion in the Many-Body Problem......Page 114 5.3 The Ten Known Integrals and TheirMeanings......Page 115 5.4 The Force Function......Page 117 5.6 Sundman’s Inequality......Page 120 5.7 The Mirror Theorem......Page 123 5.9 Lagrange’s Solutions of the Three-Body Problem......Page 124 5.10 General Remarks on the Lagrange Solutions......Page 129 5.11.1 Jacobi’s integral......Page 130 5.11.2 Tisserand’s criterion......Page 133 5.11.3 Surfaces of zero velocity......Page 134 5.11.4 The stability of the libration points......Page 138 5.11.5 Periodic orbits......Page 142 5.11.6 The search for symmetric periodic orbits......Page 144 5.11.7 Examples of some families of periodic orbits......Page 146 5.11.8 Stability of periodic orbits......Page 148 5.11.9 The surface of section......Page 150 5.11.10 The stability matrix......Page 151 5.12 The General Three-Body Problem......Page 152 5.12.1 The case C < 0......Page 153 5.12.2 The case for C = 0......Page 154 5.12.3 Jacobian coordinates......Page 155 5.13 Jacobian Coordinates for the Many-Body Problem......Page 156 5.13.1 The equations of motion of the simple n-body HDS......Page 157 5.13.2 The equations of motion of the general n-body HDS......Page 159 5.14 The Hierarchical Three-Body Stability Criterion......Page 163 Bibliography......Page 164 6.2 The Equations of Motions......Page 166 6.3 Sundman’s Inequality......Page 169 6.4 Boundaries of Real and Imaginary Motion in the Caledonian Symmetrical N-Body Problem......Page 174 6.5 The Caledonian Symmetric Model for n = 1......Page 176 6.6 The Caledonian Symmetric Model for n = 2......Page 180 6.6.1 The Szebehely Ladder and Szebehely’s Constant2......Page 185 6.6.2 Regions of real motion in the ρ1, ρ2, ρ12 space......Page 186 6.6.3 Climbing the rungs of Szebehely’s Ladder......Page 189 6.6.5 Unequal masses μ1 is not equal to μ2 in the n = 2 case......Page 194 6.6.6 Szebehely’s Constant......Page 195 6.6.7 Loks and Sergysels’ study of the general four-body problem......Page 196 6.7 The Caledonian Symmetric Problem for n = 3......Page 197 6.8 The Caledonian Symmetric N-Body Problem for Odd......Page 203 Bibliography......Page 205 7.1 The Nature of the Problem......Page 206 7.2 The Equations of Relative Motion......Page 207 7.3 The Disturbing Function......Page 209 7.4 The Sphere of Influence......Page 210 7.5 The Potential of a Body of Arbitrary Shape......Page 213 7.6 Potential at a Point Within a Sphere......Page 218 7.7 The Method of the Variation of Parameters......Page 220 7.7.1 Modification of the mean longitude at the epoch......Page 224 7.7.2 The solution of Lagrange’s planetary equations......Page 226 7.7.3 Short–and long-period inequalities......Page 229 7.7.4 The resolution of the disturbing force......Page 232 7.8 Lagrange’s Equations of Motion......Page 235 7.9 Hamilton’s Canonic Equations......Page 238 7.10 Derivation of Lagrange’s Planetary Equations from Hamilton’s Canonic Equations......Page 243 Problems......Page 244 Bibliography......Page 245 8.1 Introduction......Page 246 8.2.5 The available computing facilities......Page 247 8.3 Cowell’s Method......Page 248 8.4 Encke’s Method......Page 249 8.5 The Use of Perturbational Equations......Page 251 8.5.1 Derivation of the perturbation equations (case h is not equal to 0)......Page 253 8.5.2 The relations between the perturbation variables, the rectangular co-ordinates and velocity components, and the usual conic-section elements.......Page 256 8.5.3 Numerical integration procedure......Page 258 8.5.4 Rectilinear or almost rectilinear orbits......Page 261 8.6 Regularization Methods......Page 263 8.7 Numerical Integration Methods......Page 265 8.7.2 Runge–Kutta four......Page 267 8.7.4 Numerical methods......Page 268 Bibliography......Page 273 9.1 Introduction......Page 275 9.2 Chaos and Resonance......Page 276 9.4 The Asteroids......Page 278 9.5.1 Ring systems......Page 281 9.5.2 Small satellites of Jupiter and Saturn......Page 282 9.5.3 Spirig and Waldvogel’s analysis......Page 285 9.5.4 Satellite-ring interactions......Page 293 9.6 Near-Commensurable Satellite Orbits......Page 296 9.7.2 Element plots for 1000000 years......Page 298 9.7.3 Does Pluto’s perihelion librate or circulate?......Page 299 9.7.4 The outer planets for 108 years--and longer!......Page 300 9.7.5 The analytical approach against the numerical approach......Page 302 9.8 Empirical Stability Criteria......Page 303 9.9 Conclusions......Page 307 Bibliography......Page 308 10.2 The Earth-Moon System......Page 311 10.3 The Saros......Page 313 10.4 Measurement of the Moon’s Distance, Mass and Size......Page 315 10.5 The Moon’s Rotation......Page 316 10.7 The Moon’s Figure......Page 318 10.8 The Main Lunar Problem......Page 319 10.9 The Sun’s Orbit in the Main Lunar Problem......Page 321 10.10 The Orbit of the Moon......Page 322 10.11 Lunar Theories......Page 323 10.12 The Secular Acceleration of the Moon......Page 325 Bibliography......Page 326 11.2 The Earth as a Planet......Page 327 11.2.1 The Earth’s shape......Page 329 11.2.2 Clairaut’s formula......Page 330 11.2.4 The Earth’s magnetic field......Page 333 11.2.5 The Earth’s atmosphere......Page 334 11.2.6 Solar-terrestrial relationships......Page 336 11.3 Forces Acting on an Artificial Earth Satellite......Page 338 11.4 The Orbit of a Satellite About an Oblate Planet......Page 339 11.4.1 The short-period perturbations of the first order......Page 342 11.4.3 Long-period perturbations from the third harmonic......Page 345 11.4.4 Secular perturbations of the second-order and long-period perturbations......Page 346 11.5 The Use of Hamilton-Jacobi Theory in the Artificial Satellite Problem......Page 347 11.6 The Effect of Atmospheric Drag on an Artificial Satellite......Page 349 11.7 Tesseral and Sectorial Harmonics in the Earth’s Gravitational Field......Page 354 Bibliography......Page 355 12.2 Motion of a Rocket......Page 357 12.2.1 Motion of a rocket in a gravitational field......Page 358 12.2.2 Motion of a rocket in an atmosphere......Page 359 12.2.3 Step rockets......Page 360 12.3 Transfer Between Orbits in a Single Central Force Field......Page 362 12.3.1 Transfer between circular, coplanar orbits......Page 363 12.3.2 Parabolic and hyperbolic transfer orbits......Page 366 12.3.3 Changes in the orbital elements due to a small impulse......Page 367 12.3.4 Changes in the orbital elements due to a large impulse......Page 369 12.3.5 Variation of fuel consumption with transfer time......Page 370 12.3.6 Sensitivity of transfer orbits to small errors in position and velocity at cut-off......Page 372 12.3.7 Transfer between particles orbiting in a central force field......Page 376 12.4.1 The hyperbolic escape from the first body......Page 380 12.4.2 Entry into orbit about the second body......Page 382 12.4.3 The hyperbolic capture......Page 384 12.4.4 Accuracy of previous analysis and the effect of error......Page 385 12.4.5 The fly-past as a velocity amplifier......Page 388 Problems......Page 390 Bibliography......Page 391 13.2 Trajectories in Earth-Moon Space......Page 392 13.3 Feasibility and Precision Study Methods......Page 393 13.5 The Use of the Lagrangian Solutions......Page 394 13.6 The Use of Two-Body Solutions......Page 395 13.7 Artificial Lunar Satellites......Page 399 13.7.1 Relative sizes of lunar satellite perturbations due to different causes......Page 400 13.7.2 Jacobi’s integral for a close lunar satellite......Page 404 13.8 Interplanetary Trajectories......Page 405 13.9 The Solar System as a Central Force Field......Page 406 13.10 Minimum-Energy Interplanetary Transfer Orbits......Page 407 13.11 The Use of Parking Orbits in Interplanetary Missions......Page 413 13.12 The Effect of Errors in Interplanetary Orbits......Page 418 Problems......Page 419 14.1 Introduction......Page 420 14.2 The Theory of Orbit Determination......Page 421 14.3 Laplace’s Method......Page 423 14.4 Gauss’s Method......Page 425 14.5 Olbers’s Method for Parabolic Orbits......Page 427 14.6 Orbit Determination with Additional Observational Data......Page 429 14.7 The Improvement of Orbits......Page 433 14.8.1 Stabilized platforms and accelerometers......Page 436 14.8.2 Navigation by on-board optical equipment......Page 438 14.8.3 Observational methods and probable accuracies......Page 440 Bibliography......Page 441 15.1 Introduction......Page 442 15.2 Visual Binaries......Page 444 15.3 The Mass-Luminosity Relation......Page 447 15.4 Dynamical Parallaxes......Page 448 15.5 Eclipsing Binaries......Page 449 15.6 Spectroscopic Binaries......Page 454 15.8 Binary Orbital Elements......Page 457 15.10 Apsidal Motion......Page 459 15.12 Triple Systems......Page 460 15.13 The Inadequacy of Newton’s Law of Gravitation......Page 463 15.14 The Figures of Stars in Binary Systems......Page 464 15.15 The Roche Limits......Page 465 15.16 Circumstellar Matter......Page 466 15.17 The Origin of Binary Systems......Page 468 Bibliography......Page 469 16.2 The Sphere of Influence......Page 470 16.3 The Binary Encounter......Page 471 16.4 The Cumulative Effect of Small Encounters......Page 474 16.5 Some Fundamental Concepts......Page 476 16.6 The Fundamental Theorems of Stellar Dynamics......Page 477 16.6.1 Jeans’s theorem......Page 479 16.7 Some Special Cases for a Stellar System in a Steady State......Page 480 16.8 Galactic Rotation......Page 481 16.8.1 Oort’s constants......Page 482 16.8.2 The period of rotation and angular velocity of the Galaxy......Page 484 16.8.3 The mass of the Galaxy......Page 485 16.8.4 The mode of rotation of the Galaxy......Page 487 16.8.5 The gravitational potential of the Galaxy......Page 491 16.8.6 Galactic stellar orbits......Page 492 16.8.7 The high-velocity stars......Page 496 16.9 Spherical Stellar Systems......Page 497 16.9.1 Application of the virial theorem to a spherical system......Page 498 16.9.2 Stellar orbits in a spherical system......Page 499 16.10 Modern Galactic Studies......Page 501 Problems......Page 504 Bibliography......Page 505 Annotation Long 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. New to the Fourth Edition:Numerous updates and reorganization of all chapters to encompass new methodsNew results from recent work in areas such as satellite dynamicsNew chapter on the Caledonian symmetrical n-body problemExtending its coverage to meet a growing need for this subject in satellite and aerospace engineering, Orbital Motion, Fourth Edition remains a top reference for postgraduate and advanced undergraduate students, professionals such as engineers, and serious amateur astronomers Long 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. New to the Fourth 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 n -body problem Extending its coverage to meet a growing need for this subject in satellite and aerospace engineering, Orbital Motion, Fourth Edition remains a top reference for postgraduate and advanced undergraduate students, professionals such as engineers, and serious amateur astronomers. A Long Established Core Text For Advanced Undergraduates And Graduate Students In A Range Of Disciplines From Astronomy And Planetary Science To Aerospace And Satellite Engineering, This Fourth Edition Includes Numerous Updates And Reorganisation To All Chapters, New Results From Recent Work And A New Chapter On The Caledonian Symmetrical N-body Problem.--book Jacket. The Restless Universe -- Coordinate And Time-keeping Systems -- The Reduction Of Observational Data -- The Two-body Problem -- The Many-body Problem -- The Caledonian Symmetric N-body Problem -- General Perturbations -- Special Perturbations -- The Stability And Evolution Of The Solar System -- Lunar Theory -- Artificial Satellites -- Rocket Dynamics And Transfer Orbits -- Interplanetary And Lunar Trajectories -- Orbit Determination And Interplanetary Navigation -- Binary And Other Few-body Problems -- Many-body Stellar Systems -- Astronomical And Related Constants -- The Earth's Gravitational Field. A.e. Roy. Previous Ed : 1988. Includes Bibliographical References And Index.