SOME EXPANSIONS OF THE ELLIPTIC MOTION TO HIGH ECCENTRICITIES

Some classic expansions of the elliptic motion - cos m E and sin m E - 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 respec t to the eccentricity. The expansions have the same radius of converge nce rho(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.