In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges.
It is approximately 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 gives the solution as a power series in ε: or in general[1][2] Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of M other than a multiple of π if the eccentricity exceeds a certain value that does not depend on M. The Laplace limit is this value.
It is the unique real solution of the transcendental equation[3] A closed-form expression in terms of r-Lambert special function and an infinite series representation were given by István Mező.
The Italian astronomer Francesco Carlini found the limit 0.66 five years before Laplace.