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Celestial Mechanics

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Celestial Mechanics

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.

Astronomy CosmologyPhysics
Author
Jeremy Tatum
Publisher
University of Victoria
Source
LibreTexts
License
CC BY-NC
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 181 sections
  1. 01: Numerical Methods
  2. 1.01: Introduction
  3. 1.02: Numerical Integration
  4. 1.03: Quadratic Equations
  5. 1.04: The Solution of f(x) 0
  6. 1.05: The Solution of Polynomial Equations
  7. 1.06: Failure of the Newton-Raphson Method
  8. 1.07: Simultaneous Linear Equations N n
  9. 1.08: Simultaneous Linear Equations N n
  10. 1.09: Nonlinear Simultaneous Equations
  11. 1.10: 1.10-Besselian Interpolation
  12. 1.11: Fitting a Polynomial to a Set of Points - Lagrange Polynomials and Lagrange Interpolation
  13. 1.12: Fitting a Least Squares Straight Line to a Set of Observational Points
  14. 1.13: Fitting a Least Squares Polynomial to a Set of Observational Points
  15. 1.14: Legendre Polynomials
  16. 1.15: Gaussian Quadrature - the Algorithm
  17. 1.16: Gaussian Quadrature - Derivation
  18. 1.17: Frequently-needed Numerical Procedures
  19. 02: Conic Sections
  20. 2.01: The Straight Line
  21. 2.02: The Ellipse
  22. 2.03: The Parabola
  23. 2.04: The Hyperbola
  24. 2.05: Conic Sections
  25. 2.06: The General Conic Section
  26. 2.07: Fitting a Conic Section Through Five Points
  27. 2.08: Fitting a Conic Section Through n Points
  28. 03: Plane and Spherical Trigonometry
  29. 3.01: Introduction
  30. 3.02: Plane Triangles
  31. 3.03: Cylindrical and Spherical Coordinates
  32. 3.04: Velocity and Acceleration Components
  33. 3.05: Spherical Triangles
  34. 3.06: Rotation of Axes Two Dimensions
  35. 3.07: Rotation of Axes Three Dimensions. Eulerian Angles
  36. 3.08: Trigonometrical Formulas
  37. 04: Coordinate Geometry in Three Dimensions
  38. 4.01: Introduction
  39. 4.02: Planes and Straight Lines
  40. 4.03: The Ellipsoid
  41. 4.04: The Paraboloid
  42. 4.05: The Hyperboloid
  43. 4.06: The Cylinder
  44. 4.07: The Cone
  45. 4.08: The General Second Degree Equation in Three Dimensions
  46. 4.09: Matrices
  47. 05: Gravitational Field and Potential
  48. 5.01: Introduction
  49. 5.02: Gravitational Field
  50. 5.03: Newton's Law of Gravitation
  51. 5.04: The Gravitational Fields of Various Bodies
  52. 5.4.01: Field of a Point Mass
  53. 5.4.02: Field on the Axis of a Ring
  54. 5.4.03: Plane discs
  55. 5.4.04: Infinite Plane Laminas
  56. 5.4.05: Hollow Hemisphere
  57. 5.4.06: Rods
  58. 5.4.07: Solid Cylinder
  59. 5.4.08: Hollow Spherical Shell
  60. 5.4.09: Solid Sphere
  61. 5.4.10: Bubble Inside a Uniform Solid Sphere
  62. 5.05: Gauss's Theorem
  63. 5.06: Calculating Surface Integrals
  64. 5.07: Potential
  65. 5.08: The Gravitational Potentials Near Various Bodies
  66. 5.8.01: Potential Near a Point Mass
  67. 5.8.02: Potential on the Axis of a Ring
  68. 5.8.03: Plane Discs
  69. 5.8.04: Infinite Plane Lamina
  70. 5.8.05: Hollow Hemisphere
  71. 5.8.06: Rods
  72. 5.8.07: Solid Cylinder
  73. 5.8.08: Hollow Spherical Shell
  74. 5.8.09: Solid Sphere
  75. 5.09: Work Required to Assemble a Uniform Sphere
  76. 5.10: Nabla Gradient and Divergence
  77. 5.11: Legendre Polynomials
  78. 5.12: Gravitational Potential of any Massive Body
  79. 5.13: Pressure at the Centre of a Uniform Sphere
  80. 06: The Celestial Sphere
  81. 6.01: Introduction to the Celestial Sphere
  82. 6.02: Altazimuth Coordinates
  83. 6.03: Equatorial Coordinates
  84. 6.04: Conversion Between Equatorial and Altazimuth Coordinates
  85. 6.05: Ecliptic Coordinates
  86. 6.06: The Mean Sun
  87. 6.07: Precession
  88. 6.08: Nutation
  89. 6.09: The Length of the Year
  90. 6.10: Problems
  91. 6.11: Solutions
  92. 07: Time
  93. 08: Planetary Motions
  94. 8.01: Introduction to Planetary Motions
  95. 8.02: Opposition Conjunction and Quadrature
  96. 8.03: Sidereal and Synodic Periods
  97. 8.04: Direct and Retrograde Motion and Stationary Points
  98. 09: The Two Body Problem in Two Dimensions
  99. 9.01: Kepler's Laws
  100. 9.02: Kepler's Second Law from Conservation of Angular Momentum
  101. 9.03: Some Functions of the Masses
  102. 9.04: Kepler's First and Third Laws from Newton's Law of Gravitation
  103. 9.05: Position in an Elliptic Orbit
  104. 9.06: Position in a Parabolic Orbit
  105. 9.07: Position in a Hyperbolic Orbit
  106. 9.08: Orbital Elements and Velocity Vector
  107. 9.09: Osculating Elements
  108. 9.10: Mean Distance in an Elliptic Orbit
  109. 10: Computation of an Ephemeris
  110. 10.01: Introduction to an Ephemeris
  111. 10.02: Elements of an Elliptic Orbit
  112. 10.03: Some Additional Angles
  113. 10.04: Elements of a Circular or Near-circular Orbit
  114. 10.05: Elements of a Parabolic Orbit
  115. 10.06: Elements of a Hyperbolic Orbit
  116. 10.07: Calculating the Position of a Comet or Asteroid
  117. 10.08: Quadrant Problems
  118. 10.09: Computing an Ephemeris
  119. 10.10: Orbital Elements and Velocity Vector
  120. 10.11: Hamiltonian Formulation of the Equations of Motion
  121. 11: Photographic Astrometry
  122. 11.01: Introduction to Photographic Astrometry
  123. 11.02: Standard Coordinates and Plate Constants
  124. 11.03: Refinements and Corrections
  125. 11.3.01: Parallaxes of the Comparison Stars
  126. 11.3.02: Proper Motions of the Comparison Stars
  127. 11.3.03: Refraction
  128. 11.3.04: Aberration of Light
  129. 11.3.05: Optical Distortion
  130. 11.3.06: Errors Mistakes and Blunders
  131. 12: CCD Astrometry
  132. 13: Calculation of Orbital Elements
  133. 13.01: Introduction to Calculating Orbital Elements
  134. 13.02: Triangles
  135. 13.03: Sectors
  136. 13.04: Kepler's Second Law
  137. 13.05: Coordinates
  138. 13.06: Example
  139. 13.07: Geocentric and Heliocentric Distances - First Attempt
  140. 13.08: Improved Triangle Ratios
  141. 13.09: Iterating
  142. 13.10: Higher-order Approximation
  143. 13.11: Light-time Correction
  144. 13.12: Sector-Triangle Ratio
  145. 13.13: Resuming the Numerical Example
  146. 13.14: Summary So Far
  147. 13.15: Calculating the Elementss
  148. 13.16: Topocentric-Geocentric Correction
  149. 13.17: Concluding Remarks
  150. 14: General Perturbation Theory
  151. 14.01: Introduction to General Perturbation Theory
  152. 14.02: Contact Transformations and General Perturbation Theory
  153. 14.03: The Poisson Brackets for the Orbital Elements
  154. 14.04: Lagrange's Planetary Equations
  155. 14.05: Motion Around an Oblate Symmetric Top
  156. 15: Special Perturbations
  157. 15.01: Introduction
  158. 15.02: Orbital elements and the position and velocity vector
  159. 15.03: The equations of motion
  160. 16: Equivalent Potential and the Restricted Three-Body Problem
  161. 16.01: Introduction
  162. 16.02: Motion Under a Central Force
  163. 16.03: Inverse Square Attractive Force
  164. 16.04: Hooke's Law
  165. 16.05: Inverse Fourth Power Force
  166. 16.06: The Collinear Lagrangian Points
  167. 16.07: The Equilateral Lagrangian Points
  168. 17: Visual Binary Stars
  169. 17.01: Introduction to Visual Binary Stars
  170. 17.02: Determination of the Apparent Orbit
  171. 17.03: The Elements of the True Orbit
  172. 17.04: Determination of the Elements of the True Orbit
  173. 17.05: Construction of an Ephemeris
  174. 18: Spectroscopic Binary Stars
  175. 18.01: Introduction to Spectroscopic Binary Stars
  176. 18.02: The Velocity Curve from the Elements
  177. 18.03: Preliminary Elements from the Velocity Curve
  178. 18.04: Masses
  179. 18.05: Refinement of the Orbital Elements
  180. 18.06: Finding the Period
  181. 18.07: Measuring the Radial Velocity