Regularized formulations in relative motion

Abstract

Variational methods based on orbital elements depend strongly on the eccentricity of the reference orbit. The resulting Jacobian matrix typically becomes singular when the reference orbit is parabolic or hyperbolic. This singularity can be avoided if the problem is formulated using sets of variables that do not depend on the eccentricity of the reference orbit. The solution to the linear equations of relative motion is derived in this paper from the Sperling-Burdet regularization and the Kustaanheimo-Stiefel transformation. A unified description for circular, elliptic, parabolic, and hyperbolic reference orbits is provided by means of the Stumpff functions. The independent variable is the fictitious time introduced by the Sundman transformation. An asynchronous solution is derived and corrected a posteriori. The first order correction recovers the synchronism, while a second order correction introduces nonlinear effects and improves the accuracy of the algorithm.