An Introduction to Celestial Mechanics PDF

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An Introduction to Celestial Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 2 Newtonian Mechanics 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Newton’s Laws of Motion . . . . . . . . . ....

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An Introduction to Celestial Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction

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2 Newtonian Mechanics 2.1 Introduction . . . . . . . . . . . . . . . 2.2 Newton’s Laws of Motion . . . . . . . . 2.3 Newton’s First Law of Motion . . . . . 2.4 Newton’s Second Law of Motion . . . . 2.5 Newton’s Third Law of Motion . . . . . 2.6 Non-Isolated Systems . . . . . . . . . . 2.7 Motion in a One-Dimensional Potential 2.8 Simple Harmonic Motion . . . . . . . . 2.9 Two-Body Problem . . . . . . . . . . . 2.10 Exercises . . . . . . . . . . . . . . . . .

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7 7 8 8 11 13 16 17 20 22 23

3 Newtonian Gravity 3.1 Introduction . . . . . . . . . . . . . . 3.2 Gravitational Potential . . . . . . . . 3.3 Gravitational Potential Energy . . . . 3.4 Axially Symmetric Mass Distributions 3.5 Potential Due to a Uniform Sphere . . 3.6 Potential Outside a Uniform Spheroid 3.7 Potential Due to a Uniform Ring . . . 3.8 Exercises . . . . . . . . . . . . . . . .

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25 25 25 27 28 31 32 35 35

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37 37 37 38 38 40

4 Keplerian Orbits 4.1 Introduction . . . . . 4.2 Kepler’s Laws . . . . 4.3 Conservation Laws . 4.4 Polar Coordinates . . 4.5 Kepler’s Second Law .

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2

CELESTIAL MECHANICS 4.6 4.7 4.8 4.9 4.10 4.11 4.12

Kepler’s First Law . Kepler’s Third Law . Orbital Energies . . Kepler Problem . . Orbital Elements . . Binary Star Systems Exercises . . . . . .

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5 Orbits in Central Force-Fields 5.1 Introduction . . . . . . . . . . . . . . . 5.2 Motion in a General Central Force-Field 5.3 Motion in a Nearly Circular Orbit . . . 5.4 Perihelion Precession of the Planets . . 5.5 Perihelion Precession of Mercury . . . . 5.6 Exercises . . . . . . . . . . . . . . . . .

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41 42 43 45 48 51 53

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55 55 55 55 58 61 62

6 Rotating Reference Frames 6.1 Introduction . . . . . . . . . . . . 6.2 Rotating Reference Frames . . . . 6.3 Centrifugal Acceleration . . . . . 6.4 Coriolis Force . . . . . . . . . . . 6.5 Rotational Flattening of the Earth 6.6 Tidal Elongation of the Earth . . . 6.7 Tidal Torques . . . . . . . . . . . 6.8 Roche Radius . . . . . . . . . . . 6.9 Exercises . . . . . . . . . . . . . .

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65 65 65 66 69 71 74 78 82 83

7 Lagrangian Mechanics 7.1 Introduction . . . . . . . 7.2 Generalized Coordinates 7.3 Generalized Forces . . . 7.4 Lagrange’s Equation . . . 7.5 Generalized Momenta . . 7.6 Exercises . . . . . . . . .

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85 85 85 86 86 89 90

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93 93 93 94 95 96 97 99

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8 Rigid Body Rotation 8.1 Introduction . . . . . . . . 8.2 Fundamental Equations . . 8.3 Moment of Inertia Tensor . 8.4 Rotational Kinetic Energy . 8.5 Principal Axes of Rotation 8.6 Euler’s Equations . . . . . 8.7 Eulerian Angles . . . . . .

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CONTENTS 8.8 8.9 8.10 8.11

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Free Precession of the Earth . . . . . . . . . McCullough’s Formula . . . . . . . . . . . . Forced Precession and Nutation of the Earth Exercises . . . . . . . . . . . . . . . . . . . .

9 Three-Body Problem 9.1 Introduction . . . . . . . . . . . . . . . 9.2 Circular Restricted Three-Body Problem 9.3 Jacobi Integral . . . . . . . . . . . . . . 9.4 Tisserand Criterion . . . . . . . . . . . 9.5 Co-Rotating Frame . . . . . . . . . . . 9.6 Lagrange Points . . . . . . . . . . . . . 9.7 Zero-Velocity Surfaces . . . . . . . . . . 9.8 Stability of Lagrange Points . . . . . . . 10 Lunar Motion 10.1 Historical Background . . . . 10.2 Preliminary Analysis . . . . . 10.3 Lunar Equations of Motion . 10.4 Unperturbed Lunar Motion . 10.5 Perturbed Lunar Motion . . 10.6 Description of Lunar Motion

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11 Orbital Perturbation Theory 11.1 Introduction . . . . . . . . . . . . . . . 11.2 Disturbing Function . . . . . . . . . . . 11.3 Osculating Orbital Elements . . . . . . 11.4 Lagrange Brackets . . . . . . . . . . . . 11.5 Transformation of Lagrange Brackets . 11.6 Lagrange’s Planetary Equations . . . . 11.7 Transformation of Lagrange’s Equations 11.8 Expansion of Lagrange’s Equations . . . 11.9 Expansion of Disturbing Function . . . 11.10 Secular Evolution of Planetary Orbits . 11.11 Hirayama Families . . . . . . . . . . . . A Vector Algebra and Vector Calculus A.1 Introduction . . . . . . . . . . . . A.2 Scalars and Vectors . . . . . . . . A.3 Vector Algebra . . . . . . . . . . . A.4 Cartesian Components of a Vector A.5 Coordinate Transformations . . . A.6 Scalar Product . . . . . . . . . . .

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102 103 104 112

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115 115 115 116 117 119 121 124 127

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133 133 134 135 138 139 144

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149 149 149 150 153 154 158 160 161 164 172 179

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185 185 185 186 187 188 190

4

CELESTIAL MECHANICS A.7 A.8 A.9 A.10 A.11 A.12 A.13 A.14 A.15 A.16 A.17 A.18

Vector Product . . . . . Rotation . . . . . . . . Scalar Triple Product . Vector Triple Product . Vector Calculus . . . . Line Integrals . . . . . Vector Line Integrals . Volume Integrals . . . Gradient . . . . . . . . Grad Operator . . . . . Curvilinear Coordinates Exercises . . . . . . . .

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192 194 196 197 198 198 201 202 203 206 207 208

B Useful Mathematrics 211 B.1 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 B.2 Matrix Eigenvalue Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Introduction

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1 Introduction The aim of this book is to bridge the considerable gap between standard undergraduate treatments of celestial mechanics, which rarely advance much beyond two-body orbit theory, and full-blown graduate treatments, such as that by Murray & Dermott. The material presented here is intended to be intelligible to advanced undergraduate and beginning graduate students. A knowledge of elementary Newtonian mechanics is assumed. However, those non-elementary topics in mechanics that are needed to account for the motions of celestial bodies (e.g., gravitational potential theory, motion in rotating reference frames, Lagrangian mechanics, Eulerian rigid body rotation theory) are derived in the text. This is entirely appropriate, since these many of these topics were originally developed in order to investigate specific problems in celestial mechanics (often, the very problems that they are used to examine in this book.) It is taken for granted that readers are familiar with the fundamentals of integral and differential calculus, ordinary differential equations, and linear algebra. On the other hand, vector analysis plays such a central role in the study of celestial motion that a brief, but fairly comprehensive, review of this subject area is provided in Appendix A. Celestial mechanics is the branch of astronomy that is concerned with the motions of celestial objects—in particular, the objects that make up the Solar System. The main aim of celestial mechanics is to reconcile these motions with the predictions of Newtonian mechanics. Modern analytic celestial mechanics started in 1687 with the publication of the Principia by Isaac Newton (1643–1727), and was subsequently developed into a mature science by celebrated scientists such as Euler (1707–1783), Clairaut (1713–1765), D’Alembert (1717–1783), Lagrange (1736–1813), Laplace (1749–1827), and Gauss (1777– 1855). This book is largely devoted to the study of the “classical” problems of celestial mechanics that were investigated by these scientists. These problems include the figure of the Earth, the tidal interaction between the Earth and the Moon, the free and forced precession and nutation of the Earth, the three-body problem, the orbit of the Moon, and the effect of interplanetary gravitational interactions on planetary orbits. This book does not discuss positional astronomy: i.e., the branch of astronomy that is concerned with finding the positions of celestial objects in the Earth’s sky at a particular instance in time. Nor does this book discuss the relatively modern (but extremely complicated) developments in celestial mechanics engendered by the advent of cheap fast computers, as well as data from unmanned space probes: e.g., the resonant interaction of planetary moons, chaotic motion, the dynamics of planetary rings. Interested readers are directed to the book by Murray & Dermott. The major sources for the material appearing in this book are as follows: An Elementary Treatise on the Lunar Theory, H. Godfray (Macmillan, London UK, 1853). An Introductory Treatise on the Lunar Theory, E.W. Brown (Cambridge University Press, Cambridge UK, 1896).

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CELESTIAL MECHANICS

Lectures on the Lunar Theory, J.C. Adams (Cambridge University Press, Cambridge UK, 1900). An Introduction to Celestial Mechanics, F.R. Moulton, 2nd Revised Edition (Macmillan, New York NY, 1914). Dynamics, H. Lamb, 2nd Edition (Cambridge University Press, Cambridge UK, 1923). Celestial Mechanics, W.M. Smart (Longmans, London UK, 1953). An Introductory Treatise on Dynamical Astronomy, H.C. Plummer (Dover, New York NY, 1960). Methods of Celestial Mechanics, D. Brouwer, and G.M. Clemence (Academic Press, New York NY, 1961). Celestial Mechanics: Volume II, Part 1: Perturbation Theory, Y. Hagihara (MIT Press, Cambridge MA, 1971). Analytical Mechanics, G.R. Fowles (Holt, Rinehart, and Winston, New York NY, 1977). Orbital Motion, A.E. Roy (Wiley, New York NY, 1978). Solar System Dynamics, C.D. Murray, and S.F. Dermott (Cambridge University Press, Cambridge UK, 1999). Classical Mechanics, 3rd Edition, H. Goldstein, C. Poole, and J. Safko (Addison-Wesley, San Fransisco CA, 2002). Classical Dynamics of Particles and Systems, 5th Edition, S.T. Thornton, and J.B. Marion (Brooks/Cole—Thomson Learning, Belmont CA, 2004). Analytical Mechanics, 7th Edition, G.R. Fowles, and G.L. Cassiday (Brooks/Cole—Thomson Learning, Belmont CA, 2005). Astronomical Algorithms, J. Meeus, 2nd Edition (Willmann-Bell, Richmond VA, 2005).

Newtonian Mechanics

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2 Newtonian Mechanics

2.1 Introduction Newtonian mechanics is a mathematical model whose purpose is to account for the motions of the various objects in the Universe. The general principles of this model were first enunciated by Sir Isaac Newton in a work entitled Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). This work, which was published in 1687, is nowadays more commonly referred to as the Principa.1 Up until the beginning of the 20th century,...