## Kepler's Equation

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

 (1)

For not a multiple of , Kepler's equation has a unique solution, but is a Transcendental Equation and so cannot be inverted and solved directly for given an arbitrary . However, many algorithms have been derived for solving the equation as a result of its importance in celestial mechanics.

Writing a as a Power Series in gives

 (2)

where the coefficients are given by the Lagrange Inversion Theorem as
 (3)

(Wintner 1941, Moulton 1970, Henrici 1974, Finch). Surprisingly, this series diverges for
 (4)

a value known as the Laplace Limit. In fact, converges as a Geometric Series with ratio
 (5)

(Finch).

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

 (6)

This series converges for all like a Geometric Series with ratio
 (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)

Alternatively, we can define in terms of an intermediate variable
 (9)

then
 (10)

 (11)

Iterative methods such as the simple

 (12)

with work well, as does Newton's Method,
 (13)

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

 (14)

which is 1.1996678640257734... (Goursat 1959, Le Lionnais 1983).

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.