Extension of the solution of Kepler's equation to high eccentricities

Abstract

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e−e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJn(ne). The expansion is convergent for values of the eccentricity such that |e−e*|<ρ(e*), where the radius of convergence ρ(e*) is a positive real number, which is calculated numerically.