Celestial Mechanics

by Jeremy Tatum

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This book presents a treatment of celestial mechanics, integrating numerical methods, geometry, and Newtonian gravitation to model and predict celestial motion. It progresses from mathematical foundations to orbital theory, ephemeris computation, perturbations, and the analysis of planetary systems and binary stars, tightly linking theory with observational astronomy.

Description from LibreTexts.

Publisher
University of Victoria
Subjects
Physics, Astronomy Cosmology
Licence
CC BY-NC 4.0, as LibreTexts states it. Non-commercial use only.
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. 1: Numerical Methods
  7. 1.1: Introduction to Numerical Methods
  8. 1.2: Numerical Integration
  9. 1.3: Quadratic Equations
  10. 1.4: The Solution of f(x) = 0
  11. 1.5: The Solution of Polynomial Equations
  12. 1.6: Failure of the Newton-Raphson Method
  13. 1.7: Simultaneous Linear Equations, N = n
  14. 1.8: Simultaneous Linear Equations, N > n
  15. 1.9: Nonlinear Simultaneous Equations
  16. 1.10: 1.10- Besselian Interpolation
  17. 1.11: Fitting a Polynomial to a Set of Points - Lagrange Polynomials and Lagrange Interpolation
  18. 1.12: Fitting a Least Squares Straight Line to a Set of Observational Points
  19. 1.13: Fitting a Least Squares Polynomial to a Set of Observational Points
  20. 1.14: Legendre Polynomials
  21. 1.15: Gaussian Quadrature - the Algorithm
  22. 1.16: Gaussian Quadrature - Derivation
  23. 1.17: Frequently-needed Numerical Procedures
  24. 2: Conic Sections
  25. 2.1: The Straight Line
  26. 2.2: The Ellipse
  27. 2.3: The Parabola
  28. 2.4: The Hyperbola
  29. 2.5: Conic Sections
  30. 2.6: The General Conic Section
  31. 2.7: Fitting a Conic Section Through Five Points
  32. 2.8: Fitting a Conic Section Through n Points
  33. 3: Plane and Spherical Trigonometry
  34. 3.1: Introduction
  35. 3.2: Plane Triangles
  36. 3.3: Cylindrical and Spherical Coordinates
  37. 3.4: Velocity and Acceleration Components
  38. 3.5: Spherical Triangles
  39. 3.6: Rotation of Axes, Two Dimensions
  40. 3.7: Rotation of Axes, Three Dimensions. Eulerian Angles
  41. 3.8: Trigonometrical Formulas
  42. 4: Coordinate Geometry in Three Dimensions
  43. 4.1: Introduction
  44. 4.2: Planes and Straight Lines
  45. 4.3: The Ellipsoid
  46. 4.4: The Paraboloid
  47. 4.5: The Hyperboloid
  48. 4.6: The Cylinder
  49. 4.7: The Cone
  50. 4.8: The General Second Degree Equation in Three Dimensions
  51. 4.9: Matrices
  52. 5: Gravitational Field and Potential
  53. 5.1: Introduction
  54. 5.2: Gravitational Field
  55. 5.3: Newton's Law of Gravitation
  56. 5.4: The Gravitational Fields of Various Bodies
  57. 5.4.1: Field of a Point Mass
  58. 5.4.2: Field on the Axis of a Ring
  59. 5.4.3: Plane discs
  60. 5.4.4: Infinite Plane Laminas
  61. 5.4.5: Hollow Hemisphere
  62. 5.4.6: Rods
  63. 5.4.7: Solid Cylinder
  64. 5.4.8: Hollow Spherical Shell
  65. 5.4.9: Solid Sphere
  66. 5.4.10: Bubble Inside a Uniform Solid Sphere
  67. 5.5: Gauss's Theorem
  68. 5.6: Calculating Surface Integrals
  69. 5.7: Potential
  70. 5.8: The Gravitational Potentials Near Various Bodies
  71. 5.8.1: Potential Near a Point Mass
  72. 5.8.2: Potential on the Axis of a Ring
  73. 5.8.3: Plane Discs
  74. 5.8.4: Infinite Plane Lamina
  75. 5.8.5: Hollow Hemisphere
  76. 5.8.6: Rods
  77. 5.8.7: Solid Cylinder
  78. 5.8.8: Hollow Spherical Shell
  79. 5.8.9: Solid Sphere
  80. 5.9: Work Required to Assemble a Uniform Sphere
  81. 5.10: Nabla, Gradient and Divergence
  82. 5.11: Legendre Polynomials
  83. 5.12: Gravitational Potential of any Massive Body
  84. 5.13: Pressure at the Centre of a Uniform Sphere
  85. 6: The Celestial Sphere
  86. 6.1: Introduction to the Celestial Sphere
  87. 6.2: Altazimuth Coordinates
  88. 6.3: Equatorial Coordinates
  89. 6.4: Conversion Between Equatorial and Altazimuth Coordinates
  90. 6.5: Ecliptic Coordinates
  91. 6.6: The Mean Sun
  92. 6.7: Precession
  93. 6.8: Nutation
  94. 6.9: The Length of the Year
  95. 6.10: Problems
  96. 6.11: Solutions
  97. 7: Time
  98. 8: Planetary Motions
  99. 8.1: Introduction to Planetary Motions
  100. 8.2: Opposition, Conjunction and Quadrature
  101. 8.3: Sidereal and Synodic Periods
  102. 8.4: Direct and Retrograde Motion, and Stationary Points
  103. 9: The Two Body Problem in Two Dimensions
  104. 9.1: Kepler's Laws
  105. 9.2: Kepler's Second Law from Conservation of Angular Momentum
  106. 9.3: Some Functions of the Masses
  107. 9.4: Kepler's First and Third Laws from Newton's Law of Gravitation
  108. 9.5: Position in an Elliptic Orbit
  109. 9.6: Position in a Parabolic Orbit
  110. 9.7: Position in a Hyperbolic Orbit
  111. 9.8: Orbital Elements and Velocity Vector
  112. 9.9: Osculating Elements
  113. 9.10: Mean Distance in an Elliptic Orbit
  114. 10: Computation of an Ephemeris
  115. 10.1: Introduction to an Ephemeris
  116. 10.2: Elements of an Elliptic Orbit
  117. 10.3: Some Additional Angles
  118. 10.4: Elements of a Circular or Near-circular Orbit
  119. 10.5: Elements of a Parabolic Orbit
  120. 10.6: Elements of a Hyperbolic Orbit
  121. 10.7: Calculating the Position of a Comet or Asteroid
  122. 10.8: Quadrant Problems
  123. 10.9: Computing an Ephemeris
  124. 10.10: Orbital Elements and Velocity Vector
  125. 10.11: Hamiltonian Formulation of the Equations of Motion
  126. 11: Photographic Astrometry
  127. 11.1: Introduction to Photographic Astrometry
  128. 11.2: Standard Coordinates and Plate Constants
  129. 11.3: Refinements and Corrections
  130. 11.3.1: Parallaxes of the Comparison Stars
  131. 11.3.2: Proper Motions of the Comparison Stars
  132. 11.3.3: Refraction
  133. 11.3.4: Aberration of Light
  134. 11.3.5: Optical Distortion
  135. 11.3.6: Errors, Mistakes and Blunders
  136. 12: CCD Astrometry
  137. 13: Calculation of Orbital Elements
  138. 13.1: Introduction to Calculating Orbital Elements
  139. 13.2: Triangles
  140. 13.3: Sectors
  141. 13.4: Kepler's Second Law
  142. 13.5: Coordinates
  143. 13.6: Example
  144. 13.7: Geocentric and Heliocentric Distances - First Attempt
  145. 13.8: Improved Triangle Ratios
  146. 13.9: Iterating
  147. 13.10: Higher-order Approximation
  148. 13.11: Light-time Correction
  149. 13.12: Sector-Triangle Ratio
  150. 13.13: Resuming the Numerical Example
  151. 13.14: Summary So Far
  152. 13.15: Calculating the Elementss
  153. 13.16: Topocentric-Geocentric Correction
  154. 13.17: Concluding Remarks
  155. 14: General Perturbation Theory
  156. 14.1: Introduction to General Perturbation Theory
  157. 14.2: Contact Transformations and General Perturbation Theory
  158. 14.3: The Poisson Brackets for the Orbital Elements
  159. 14.4: Lagrange's Planetary Equations
  160. 14.5: Motion Around an Oblate Symmetric Top
  161. 15: Special Perturbations
  162. 15.1: Introduction
  163. 15.2: Orbital elements and the position and velocity vector
  164. 15.3: The equations of motion
  165. 16: Equivalent Potential and the Restricted Three-Body Problem
  166. 16.1: Introduction
  167. 16.2: Motion Under a Central Force
  168. 16.3: Inverse Square Attractive Force
  169. 16.4: Hooke's Law
  170. 16.5: Inverse Fourth Power Force
  171. 16.6: The Collinear Lagrangian Points
  172. 16.7: The Equilateral Lagrangian Points
  173. 17: Visual Binary Stars
  174. 17.1: Introduction to Visual Binary Stars
  175. 17.2: Determination of the Apparent Orbit
  176. 17.3: The Elements of the True Orbit
  177. 17.4: Determination of the Elements of the True Orbit
  178. 17.5: Construction of an Ephemeris
  179. 18: Spectroscopic Binary Stars
  180. 18.1: Introduction to Spectroscopic Binary Stars
  181. 18.2: The Velocity Curve from the Elements
  182. 18.3: Preliminary Elements from the Velocity Curve
  183. 18.4: Masses
  184. 18.5: Refinement of the Orbital Elements
  185. 18.6: Finding the Period
  186. 18.7: Measuring the Radial Velocity
  187. Back Matter
  188. Index
  189. Detailed Licensing