Expansions of (r/a)m cos jv and (r/a)m sin jv to high eccentricities

Abstract

Expansions of the functions (r/a)cos jv and (r/a)m sin jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e−e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e−e*|<ρ(e*), where the radius of convergence ρ(e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.