Let be the mean anomaly and the Eccentric Anomaly of a body orbiting on an
Ellipse with Eccentricity , then

(1) |

Writing a as a Power Series in gives

(2) |

(3) |

(4) |

(5) |

There is also a series solution in Bessel Functions of the First Kind,

(6) |

(7) |

The equation can also be solved by letting be the Angle between the planet's motion and the
direction Perpendicular to the Radius Vector. Then

(8) |

(9) |

(10) |

(11) |

Iterative methods such as the simple

(12) |

(13) |

In solving Kepler's equation, Stieltjes required the solution to

(14) |

**References**

Danby, J. M. *Fundamentals of Celestial Mechanics, 2nd ed., rev. ed.* Richmond, VA: Willmann-Bell, 1988.

Dörrie, H. ``The Kepler Equation.'' §81 in
*100 Great Problems of Elementary Mathematics: Their History and Solutions.*
New York: Dover, pp. 330-334, 1965.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lpc/lpc.html

Goldstein, H. *Classical Mechanics, 2nd ed.* Reading, MA: Addison-Wesley, pp. 101-102
and 123-124, 1980.

Goursat, E. *A Course in Mathematical Analysis, Vol. 2.* New York: Dover, p. 120, 1959.

Henrici, P. *Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros.*
New York: Wiley, 1974.

Ioakimids, N. I. and Papadakis, K. E. ``A New Simple Method for the Analytical Solution of Kepler's Equation.''
*Celest. Mech.* **35**, 305-316, 1985.

Ioakimids, N. I. and Papadakis, K. E. ``A New Class of Quite Elementary Closed-Form Integrals Formulae for Roots
of Nonlinear Systems.'' *Appl. Math. Comput.* **29**, 185-196, 1989.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 36, 1983.

Marion, J. B. and Thornton, S. T. ``Kepler's Equations.'' §7.8 in
*Classical Dynamics of Particles & Systems, 3rd ed.*
San Diego, CA: Harcourt Brace Jovanovich, pp. 261-266, 1988.

Moulton, F. R. *An Introduction to Celestial Mechanics, 2nd rev. ed.* New York: Dover, pp. 159-169, 1970.

Siewert, C. E. and Burniston, E. E. ``An Exact Analytical Solution of Kepler's Equation.'' *Celest. Mech.* **6**, 294-304, 1972.

Wintner, A. *The Analytic Foundations of Celestial Mechanics.* Princeton, NJ: Princeton University Press, 1941.

© 1996-9

1999-05-26