Kepler's Equation


Kepler's equation gives the relation between the polar coordinates of a celestial body (such as a planet) and the time elapsed from a given initial point. Kepler's equation is of fundamental importance in celestial mechanics, but cannot be directly inverted in terms of simple functions in order to determine where the planet will be at a given time.

Let M be the mean anomaly (a parameterization of time) and E the eccentric anomaly (a parameterization of polar angle) of a body orbiting on an ellipse with eccentricity e, then


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

Writing a E as a power series in e gives


where the coefficients are given by the Lagrange inversion theorem as

 a_n=1/(2^(n-1)n!)sum_(k=0)^(|_n/2_|)(-1)^k(n; k)(n-2k)^(n-1)sin[(n-2k)M]

(Wintner 1941, Moulton 1970, Henrici 1974, Finch 2003). Surprisingly, this series diverges for


(OEIS A033259), a value known as the Laplace limit. In fact, E converges as a geometric series with ratio


(Finch 2003).

There is also a series solution in Bessel functions of the first kind,


This series converges for all e<1 like a geometric series with ratio


The equation can also be solved by letting psi be the angle between the planet's motion and the direction perpendicular to the radius vector. Then


Alternatively, we can define e in terms of an intermediate variable phi




Iterative methods such as the simple


with E_0=0 work well, as does Newton's method,


The point at which Laplace's formula for solving Kepler's equation begins diverging is known as the Laplace limit.

See also

Eccentric Anomaly, Kapteyn Series, Laplace Limit

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Kepler's Equation

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Weisstein, Eric W. "Kepler's Equation." From MathWorld--A Wolfram Web Resource.

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