Math 2.0

Mechanics of particles and systems Particle systems and celestial mechanics Concept

Kepler's equation

Builds on Kepler's laws of planetary motionEllipseEccentricity

Summary

Kepler's equation is M=E−esin⁡EM = E - e \sin E, which links the time along an elliptic Kepler orbit to the position on it. Here ee is the eccentricity of the ellipse, the mean anomaly MM grows linearly with time, and the eccentric anomaly EE fixes the position. Solving it for EE is the step that turns Kepler's laws into a prediction of where a planet is at a given time. It has no solution in elementary functions, and it has been a classical test case for numerical methods and series expansions.

Reading path

To understand this page from the ground up, read these 67 pages in order. The foundations come first.

  1. Conic section concept
  2. Cross product concept
  3. Derivative concept
  4. Differential equation concept
  5. Eccentricity concept
  6. Ellipse concept
  7. Euclidean space object
  8. Euclidean vector concept
  9. First integral concept
  10. Focus of a conic concept
  11. Function concept
  12. Hamiltonian system concept
  13. Interval concept
  14. Ordinary differential equation concept
  15. Velocity concept
  16. Kepler problem example
  17. Fréchet derivative definition
  18. Total derivative concept
  19. Ordinary differential equation definition
  20. Cross product definition
  21. Function of several variables concept
  22. Vector-valued function concept
  23. Classical mechanics concept
  24. Lipschitz condition definition
  25. n-body problem problem
  26. Momentum concept
  27. Differentiable function of several variables concept
  28. Double integral concept
  29. Limit of a function of several variables concept
  30. Partial derivative concept
  31. Vector field concept
  32. Inertial frame of reference concept
  33. Solution of an ordinary differential equation definition
  34. Angular momentum concept
  35. Gradient concept
  36. Triple integral concept
  37. Continuity in R^n definition
  38. Differentiable function of several variables definition
  39. Newton's laws of motion statement
  40. Partial derivative definition
  41. Initial value problem definition
  42. Scalar potential concept
  43. Centre of mass concept
  44. Differentiable on an open set definition
  45. Gradient definition
  46. Higher partial derivatives definition
  47. Picard–Lindelöf hypotheses definition
  48. Reduced mass concept
  49. Conservative vector field concept
  50. Iterated derivative definition
  51. Picard operator definition
  52. Central force concept
  53. C^k function characterised by continuous derivatives statement
  54. An iterate of the Picard operator is a contraction statement
  55. Integral equation form of an initial value problem statement
  56. Vector field definition
  57. Picard–Lindelöf theorem statement
  58. Conservative vector field definition
  59. Classical mechanics definition
  60. Momentum definition
  61. Potential energy definition
  62. Angular momentum definition
  63. Central force definition
  64. Newton's law of universal gravitation statement
  65. Two-body problem problem
  66. Kepler's laws of planetary motion statement
  67. Kepler's equation concept

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Kepler%27s_equation
About this page
TagMSB4ZTG422
URIm2://MSB4ZTG422
Kindconcept
WritingW0 stub
MSC70F05, 70F15, 65H05
UsesKepler's laws of planetary motion, Ellipse, Eccentricity
Same as
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Definitions

Used by