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by Dr. J. B. Tatum
jtatum@uvic.ca

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Celestial Mechanics (last updated: 2019 March 10)

Part I. Mathematical Preambles

Chapter 1.    Numerical Methods

 1.1 Introduction 1.2 Numerical Integration 1.3 Quadratic Equations 1.4 The Solution of f(x) = 0 1.5 The Solution of Polynomial Equations 1.6 Failure of the Newton-Raphson Method 1.7 Simultaneous Linear Equations, N = n 1.8 Simultaneous Linear Equations, N < n 1.9 Nonlinear Simultaneous Equations 1.1 Besselian Interpolation 1.11 Fitting a Polynomial to a Set of Points. Lagrange Polynomials. Lagrange Interpolation. 1.12 Fitting a Least Squares Straight Line to a Set of Observational Points 1.13 Fitting a Least Squares Polynomial to a Set of Observational Points 1.14 Legendre Polynomials 1.15 Gaussian Quadrature - the Algorithm 1.16 Gaussian Quadrature - Derivation 1.17 Frequently-needed Numerical Procedures

Chapter 2.    Conic Sections

 2.1 Introduction 2.2 The Straight Line 2.3 The Ellipse 2.4 The Parabola 2.5 The Hyperbola 2.6 Conic Sections 2.7 The General Conic Section 2.8 Fitting a Conic Section Through Five Points 2.9 Fitting a Conic Section Through n Points

Chapter 3.    Plane and Spherical Trigonomtry

 3.1 Introduction 3.2 Plane Triangles 3.3 Cylindrical and Spherical Coordinates 3.4 Velocity and Acceleration Components 3.5 Spherical Triangles 3.6 Rotation of Axes, Two Dimensions 3.7 Rotation of Axes, Three Dimensions. Eulerian Angles 3.8 Trigonometrical Formulas

Chapter 4.    Coordinate Geometry in Three Dimensions

 4.1 Introduction 4.2 Planes and Straight Lines 4.3 The Ellipsoid 4.4 The Paraboloid 4.5 The Hyperboloid 4.6 The Cylinder 4.7 The Cone 4.8 The General Second Degree Equation in Three Dimensions 4.9 Matrices

Chapter 5.    Gravitational Field and Potential

5.1Introduction
5.2Gravitational Field
5.3Newton's Law of Gravitation
5.4The Gravitational Fields of Various Bodies
 5.4.1 Field of a Point Mass 5.4.2 Field on the Axis of a Ring 5.4.3 Plane discs 5.4.4 Infinite Plane Laminas 5.4.5 Hollow Hemisphere 5.4.6 Rods 5.4.7 Solid Cylinder 5.4.8 Hollow Spherical Shell 5.4.9 Solid Sphere 5.4.10 Bubble Inside a Uniform Solid Sphere
5.5Gauss's Theorem
5.6Calculating Surface Integrals
5.7Potential
5.8The Gravitational Potentials Near Various Bodies
 5.8.1 Potential Near a Point Mass 5.8.2 Potential on the Axis of a Ring 5.8.3 Plane Discs 5.8.4 Infinite Plane Lamina 5.8.5 Hollow Hemisphere 5.8.6 Rods 5.8.7 Solid Cylinder 5.4.8 Hollow Spherical Shell 5.8.9 Solid Sphere
5.9Work Required to Assemble a Uniform Sphere
5.10Nabla, Gradient and Divergence
5.11Legendre Polynomials
5.12Gravitational Potential of any Massive Body
5.13Pressure at the Centre of a Uniform Sphere

Part II. Celestial Mechanics

Chapter 6.    The Celestial Sphere

 6.1 Introduction 6.2 Altazimuth Coordinates 6.3 Equatorial Coordinates 6.4 Conversion Between Equatorial and Altazimuth Coordinates 6.5 Ecliptic Coordinates 6.6 The Mean Sun 6.7 Precession 6.8 Nutation 6.9 The Length of the Year

Chapter 7.    Time

Chapter 8.    Planetary Motions

 8.1 Introduction 8.2 Opposition, Conjunction and Quadrature 8.3 Sidereal and Synodic Periods 8.4 Direct and Retrograde Motion, and Stationary Points

Chapter 9.    The Two Body Problem in Two Dimensions

 9.1 Introduction 9.2 Kepler's Laws 9.3 Kepler's Second Law from Conservation of Angular Momentum 9.4 Some Functions of the Masses 9.5 Kepler's First and Third Laws from Newton's Law of Gravitation 9.6 Position in an Elliptic Orbit 9.7 Position in a Parabolic Orbit 9.8 Position in a Hyperbolic Orbit 9.9 Orbital Elements and Velocity Vector 9.1 Osculating Elements 9.11 Mean Distance in an Elliptic Orbit

Chapter 10.    Computation of an Ephemeris

 10.1 Introduction 10.2 Elements of an Elliptic Orbit 10.3 Some Additional Angles 10.4 Elements of a Circular or Near-circular Orbit 10.5 Elements of a Parabolic Orbit 10.6 Elements of a Hyperbolic Orbit 10.7 Calculating the Position of a Comet or Asteroid 10.8 Quadrant Problems 10.9 Computing an Ephemeris 10.1 Orbital Elements and Velocity Vector 10.11 Hamiltonian Formulation of the Equations of Motion

Chapter 11.    Photographic Astrometry

11.1Introduction
11.2Standard Coordinates and Plate Constants
11.3Refinements and Corrections
 11.3.1 Parallaxes of the Comparison Stars 11.3.2 Proper Motions of the Comparison Stars 11.3.3 Refraction 11.3.4 Aberration of light 11.3.5 Optical Distortion 11.3.6 Errors, Mistakes and Blunders

Chapter 12.    CCD Astrometry

Chapter 13.    Calculation of Orbital Elements

 13.1 Introduction 13.2 Triangles 13.3 Sectors 13.4 Kepler's Second Law 13.5 Coordinates 13.6 Example 13.7 Geocentric and Heliocentric Distances - First Attempt 13.8 Improved Triangle Ratios 13.9 Iterating 13.1 Higher-order Approximation 13.11 Light-time Correction 13.12 Sector-Triangle Ratio 13.13 Resuming the Numerical Example

Chapter 14.    General Perturbation Theory

 14.1 Introduction 14.2 Contact Transformations and General Perturbation Theory 14.3 The Poisson Brackets for the Orbital Elements 14.4 Lagrange's Planetary Equations 14.5 Motion Around an Oblate Symmetric Top

Chapter 15.    Special Perturbations

 15.1 Introduction 15.2 Orbital Elements and the Position and Velocity Vector 15.3 The Equations of Motion

Chapter 16.    Equivalent Potential and the Restricted Three-Body Problem

 16.1 Introduction 16.2 Motion Under a Central Force 16.3 Inverse Square Attractive Force 16.4 Hooke's Law 16.5 Inverse Fourth Power Force 16.6 The Collinear Lagrangian Points 16.7 The Equilateral Lagrangian Points

Chapter 17.    Visual Binary Stars

 17.1 Introduction 17.2 Determination of the Apparent Orbit 17.3 The Elements of the True Orbit 17.4 Determination of the Elements of the True Orbit 17.5 Construction of an Ephemeris

Chapter 18.    Spectroscopic Binary Stars

 18.1 Introduction 18.2 The Velocity Curve from the Elements 18.3 Preliminary Elements from the Velocity Curve 18.4 Masses 18.5 Refinement of the Orbital Elements 18.6 Finding the Period 18.7 Measuring the Radial Velocity

Appendix A.    Miscellaneous Problems

Appendix B.    Solutions to Miscellaneous Problems

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