The three-body problem

Half-title......Page 2
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
1.1 About the three-body problem......Page 12
1.2 The three-body problem in astrophysics......Page 16
1.3 Short period comets......Page 19
1.4 Binary stars......Page 23
1.5 Groups of galaxies......Page 26
1.6 Binary black holes......Page 28
2.1 Newton’s laws......Page 31
2.2 Inertial coordinate system......Page 32
2.3 Equations of motion for N bodies......Page 33
2.4 Gravitational potential......Page 35
2.5 Constants of motion......Page 36
2.6 The virial theorem......Page 38
2.7 The Lagrange and Jacobi forms of the equations of motion......Page 40
2.8 Constants of motion in the three-body problem......Page 42
2.9 Moment of inertia......Page 43
2.11 Integration of orbits......Page 45
2.12 Dimensions and units of the three-body problem......Page 49
2.13 Chaos in the three-body problem......Page 50
2.14 Rotating coordinate system......Page 54
Problems......Page 56
3.1 Equations of motion......Page 58
3.2 Centre of mass coordinate system......Page 59
3.3 Integrals of the equation of motion......Page 60
3.4 Equation of the orbit and Kepler’s first law......Page 63
3.5 Kepler’s second law......Page 64
3.6 Orbital elements......Page 65
3.7 Orbital velocity......Page 68
3.8 True and eccentric anomalies......Page 69
3.9 Mean anomaly and Kepler’s equation......Page 71
3.10 Solution of Kepler’s equation......Page 72
3.11 Kepler’s third law......Page 74
3.12 Position and speed as functions of eccentric anomaly......Page 75
3.13 Hyperbolic orbit......Page 77
3.14 Dynamical friction......Page 79
3.15.1 Series expansion of the eccentric anomaly......Page 81
3.15.2 Series of sin nE and cosnE......Page 84
3.15.3 Distance as a function of time......Page 86
3.15.4 Legendre polynomials......Page 87
Problems......Page 89
4.1 Generalised coordinates......Page 91
4.2 Hamiltonian principle......Page 92
4.3 Variational calculus......Page 93
4.4 Lagrangian equations of motion......Page 96
4.5 Hamiltonian equations of motion......Page 98
4.6 Properties of the Hamiltonian......Page 100
4.7 Canonical transformations......Page 103
4.9 The Hamilton–Jacobi equation......Page 106
4.10 Two-body problem in Hamiltonian mechanics: two dimensions......Page 108
4.11 Two-body problem in Hamiltonian mechanics: three dimensions......Page 114
4.12 Delaunay’s elements......Page 119
4.13 Hamiltonian formulation of the three-body problem......Page 120
4.14 Elimination of nodes......Page 122
Problems......Page 124
5.1 Coordinate frames......Page 126
5.2 Equations of motion......Page 127
5.3 Jacobian integral......Page 130
5.4 Lagrangian points......Page 134
5.5 Stability of the Lagrangian points......Page 136
5.6 Satellite orbits......Page 141
5.7 The Lagrangian equilateral triangle......Page 144
5.8 One-dimensional three-body problem......Page 147
Problems......Page 150
6.1 Scattering of small fast bodies from a binary......Page 152
6.2 Evolution of the semi-major axis and eccentricity......Page 159
6.3 Capture of small bodies by a circular binary......Page 163
6.4 Orbital changes in encounters with planets......Page 165
6.5 Inclination and perihelion distance......Page 168
6.6 Large angle scattering......Page 173
6.7 Changes in the orbital elements......Page 176
6.8 Changes in the relative orbital energy of the binary......Page 180
Problems......Page 181
7.1 Escapes in a bound three-body system......Page 182
7.2 A planar case......Page 190
7.3 Escape velocity......Page 191
7.4 Escaper mass......Page 194
7.5 Angular momentum......Page 195
7.6 Escape angle......Page 199
Problems......Page 206
8.1 Three-body scattering......Page 208
8.2 Capture......Page 214
8.3 Ejections and lifetime......Page 218
8.4 Exchange and flyby......Page 222
8.5 Rates of change of the binding energy......Page 225
8.6 Collisions......Page 227
Problems......Page 230
9.1 Osculating elements......Page 232
9.2 Lagrangian planetary equations......Page 233
9.3 Three-body perturbing function......Page 236
9.4 Doubly orbit-averaged perturbing function......Page 238
9.5 Motions in the hierarchical three-body problem......Page 242
Problems......Page 250
10.1 Perturbations of the integrals k and e......Page 251
10.2 Binary evolution with a constant perturbing force......Page 254
10.3 Slow encounters......Page 257
10.4 Inclination dependence......Page 271
10.5 Change in eccentricity......Page 275
10.6 Stability of triple systems......Page 279
10.7 Fast encounters......Page 285
10.8 Average energy exchange......Page 292
Problems......Page 296
11.1 Binary black holes in centres of galaxies......Page 299
11.2 The problem of three black holes......Page 307
11.4 Three galaxies......Page 321
11.5 Binary stars in the Galaxy......Page 324
11.6 Evolution of comet orbits......Page 331
Problems......Page 338
References......Page 340
Author index......Page 352
Subject index......Page 354