GENERAL TRANSFORMATION OF KEPLERS EQUATION FOR P-ITERATIVE SOLUTION OF THE LAMBERT GAUSS PROBLEM/

A general transformation of Kepler's time-of-flight equation (with det ailed derivation) is presented in terms of variables suitable for para meter-iterative solution of the Lambert/Gauss orbit determination prob lem. Flight time between two points along an elliptic orbit is obtaine d as a function of the direction of motion (either short way or long w ay), the eccentricity and parameter (semilatus rectum) of the orbit, a nd the orbital radii and connecting chord at the given points, Functio nal dependence of flight time on eccentricity is eliminated through us e of an auxiliary equation relating eccentricity to the parameter, orb ital radii, and connecting chord. The transformed equation for ellipti c orbits is general in the sense that it converts readily to a form ap plicable to hyperbolic orbits. It also provides an expression for flig ht time between two points along a parabolic orbit in the limit as the value of eccentricity goes to unity. In this special case flight time is obtained as a function of the direction of motion, the orbital rad ii, and connecting chord only. Considerable simplification of the tran sformation is achieved by expressing the results in dimensionless form and in terms of the parameter and eccentricity of the symmetric ellip se or minimum-eccentricity orbit associated with the given position ve ctors. The final formulation for elliptic orbits converts readily to t hat for hyperbolic orbits and resolves the problem of quadrant ambigui ty prevalent in solutions of Kepler's equation. The performance of a p arameter-iterative solution algorithm for the Lambert/Gauss problem us ing the present transformation has been reported in an earlier paper. The present note is intended only to document the derivation of the tr ansformation for archival purposes.