Some expansions of the elliptic motion to high eccentricities

Abstract

Some classic expansions of the elliptic motion - cos mE and sin mE - in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e-e*), where e* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence ρ(e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions - (a/r), (r/a), (r/a) sin v, ..., - derived straightforward from the expansions of E, cos E and sin E are also presented. © 1995 Kluwer Academic Publishers.

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