# Laplace limit

In mathematics, the Laplace limit is the maximum value of the eccentricity for which the series solution to Kepler's equation converges. It is approximately

0.66274 34193 49181 58097 47420 97109 25290.

Kepler's equation M = E − ε sin E relates the mean anomaly M with the eccentric anomaly E for a body moving in an ellipse with eccentricity ε. This equation cannot be solved for E in terms of elementary functions, but the Lagrange reversion theorem yields the solution as a power series in ε:

$E = M + \sin(M) \, \varepsilon + \tfrac12 \sin(2M) \, \varepsilon^2 + \left( \tfrac38 \sin(3M) - \tfrac18 \sin(M) \right) \, \varepsilon^3 + \cdots$

Laplace realized that this series converges for small values of the eccentricity, but diverges when the eccentricity exceeds a certain value. The Laplace limit is this value. It is the radius of convergence of the power series.