The three-body problem

How do three celestial bodies move under their mutual gravitational attraction? This problem has been studied by Isaac Newton and leading mathematicians over the last two centuries. Poincaré's conclusion, that the problem represents an example of chaos in nature, opens the new possibility of using a statistical approach. For the first time this book presents these methods in a systematic way, surveying statistical as well as more traditional methods. The book begins by providing an introduction to celestial mechanics, including Lagrangian and Hamiltonian methods, and both the two and restricted three body problems. It then surveys statistical and perturbation methods for the solution of the general three body problem, providing solutions based on combining orbit calculations with semi-analytic methods for the first time. This book should be essential reading for students in this rapidly expanding field and is suitable for students of celestial mechanics at advanced undergraduate and graduate level. 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 Half-title 2 Title 4 Copyright 5 Contents 6 Preface 10 1 Astrophysics and the three-body problem 12 1.1 About the three-body problem 12 1.2 The three-body problem in astrophysics 16 1.3 Short period comets 19 1.4 Binary stars 23 1.5 Groups of galaxies 26 1.6 Binary black holes 28 2 Newtonian mechanics 31 2.1 Newton’s laws 31 2.2 Inertial coordinate system 32 2.3 Equations of motion for N bodies 33 2.4 Gravitational potential 35 2.5 Constants of motion 36 2.6 The virial theorem 38 2.7 The Lagrange and Jacobi forms of the equations of motion 40 2.8 Constants of motion in the three-body problem 42 2.9 Moment of inertia 43 2.10 Scaling of the three-body problem 45 2.11 Integration of orbits 45 2.12 Dimensions and units of the three-body problem 49 2.13 Chaos in the three-body problem 50 2.14 Rotating coordinate system 54 Problems 56 3 The two-body problem 58 3.1 Equations of motion 58 3.2 Centre of mass coordinate system 59 3.3 Integrals of the equation of motion 60 3.4 Equation of the orbit and Kepler’s first law 63 3.5 Kepler’s second law 64 3.6 Orbital elements 65 3.7 Orbital velocity 68 3.8 True and eccentric anomalies 69 3.9 Mean anomaly and Kepler’s equation 71 3.10 Solution of Kepler’s equation 72 3.11 Kepler’s third law 74 3.12 Position and speed as functions of eccentric anomaly 75 3.13 Hyperbolic orbit 77 3.14 Dynamical friction 79 3.15 Series expansions 81 3.15.1 Series expansion of the eccentric anomaly 81 3.15.2 Series of sin nE and cosnE 84 3.15.3 Distance as a function of time 86 3.15.4 Legendre polynomials 87 Problems 89 4 Hamiltonian mechanics 91 4.1 Generalised coordinates 91 4.2 Hamiltonian principle 92 4.3 Variational calculus 93 4.4 Lagrangian equations of motion 96 4.5 Hamiltonian equations of motion 98 4.6 Properties of the Hamiltonian 100 4.7 Canonical transformations 103 4.8 Examples of canonical transformations 106 4.9 The Hamilton–Jacobi equation 106 4.10 Two-body problem in Hamiltonian mechanics: two dimensions 108 4.11 Two-body problem in Hamiltonian mechanics: three dimensions 114 4.12 Delaunay’s elements 119 4.13 Hamiltonian formulation of the three-body problem 120 4.14 Elimination of nodes 122 4.15 Elimination of mean anomalies 124 Problems 124 5 The planar restricted circular three-body problem and other special cases 126 5.1 Coordinate frames 126 5.2 Equations of motion 127 5.3 Jacobian integral 130 5.4 Lagrangian points 134 5.5 Stability of the Lagrangian points 136 5.6 Satellite orbits 141 5.7 The Lagrangian equilateral triangle 144 5.8 One-dimensional three-body problem 147 Problems 150 6 Three-body scattering 152 6.1 Scattering of small fast bodies from a binary 152 6.2 Evolution of the semi-major axis and eccentricity 159 6.3 Capture of small bodies by a circular binary 163 6.4 Orbital changes in encounters with planets 165 6.5 Inclination and perihelion distance 168 6.6 Large angle scattering 173 6.7 Changes in the orbital elements 176 6.8 Changes in the relative orbital energy of the binary 180 Problems 181 7 Escape in the general three-body problem 182 7.1 Escapes in a bound three-body system 182 7.2 A planar case 190 7.3 Escape velocity 191 7.4 Escaper mass 194 7.5 Angular momentum 195 7.6 Escape angle 199 Problems 206 8 Scattering and capture in the general problem 208 8.1 Three-body scattering 208 8.2 Capture 214 8.3 Ejections and lifetime 218 8.4 Exchange and flyby 222 8.5 Rates of change of the binding energy 225 8.6 Collisions 227 Problems 230 9 Perturbations in hierarchical systems 232 9.1 Osculating elements 232 9.2 Lagrangian planetary equations 233 9.3 Three-body perturbing function 236 9.4 Doubly orbit-averaged perturbing function 238 9.5 Motions in the hierarchical three-body problem 242 Problems 250 10 Perturbations in strong three-body encounters 251 10.1 Perturbations of the integrals k and e 251 10.2 Binary evolution with a constant perturbing force 254 10.3 Slow encounters 257 10.4 Inclination dependence 271 10.5 Change in eccentricity 275 10.6 Stability of triple systems 279 10.7 Fast encounters 285 10.8 Average energy exchange 292 Problems 296 11 Some astrophysical problems 299 11.1 Binary black holes in centres of galaxies 299 11.2 The problem of three black holes 307 11.3 Satellite black hole systems 321 11.4 Three galaxies 321 11.5 Binary stars in the Galaxy 324 11.6 Evolution of comet orbits 331 Problems 338 References 340 Author index 352 Subject index 354 How do three celestial bodies move under their mutual gravitational attraction? This problem has been studied by Isaac Newton and leading mathematicians over the last two centuries. PoincarE's conclusion, that the problem represents an example of chaos in nature, opens the new possibility of using a statistical approach. For the first time this book presents these methods in a systematic way, surveying statistical as well as more traditional methods. The book begins by providing an introduction to celestial mechanics, including Lagrangian and Hamiltonian methods, and both the two and restricted three body problems. It then surveys statistical and perturbation methods for the solution of the general three body problem, providing solutions based on combining orbit calculations with semi-analytic methods for the first time. This book should be essential reading for students in this rapidly expanding field and is suitable for students of celestial mechanics at advanced undergraduate and graduate level.

How do three celestial bodies move under their mutual gravitational attraction? This problem has been studied by Isaac Newton and leading mathematicians over the last two centuries. Poincaré's conclusion, that the problem represents an example of chaos in nature, opens the new possibility of using a statistical approach. For the first time this book presents these methods in a systematic way, surveying statistical as well as more traditional methods. This book should be essential reading for students in a rapidly expanding field and is suitable for students of celestial mechanics at advanced undergraduate and graduate level.

"How do three celestial bodies move under their mutual gravitational attraction? It is a problem that has been studied by Isaac Newton and leading mathematicians over the last two centuries. Poincare's conclusion that the problem represents an example of chaos in nature, opens new possibilities of dealing with it: a statistical approach. For the first time such methods are presented in a systematic way. The book surveys statistical methods as well as more traditional methods, suitable for students of celestial mechanics at advanced undergraduate level."--Jacket The book surveys statistical and perturbation methods for the solution of the general three body problem, providing solutions based on combining orbit calculations with semi-analytic methods for the first time. This book is essential reading for students in this rapidly expanding field.