Long-Term Integration Error of Kustaanheimo-Stiefel Regularized Orbital Motion. II. Method of Variation of Parameters

Abstract

We have discovered that application of the method of variation of parameters to the Kustaanheimo-Stiefel (K-S) regularization drastically reduces the orbital integration errors of the perturbed two-body problem for arbitrary types of perturbations. This is because not only the errors of position, whose linear growth was determined previously (Paper I), but those of the physical time grow only linearly with respect to the fictitious time even if using traditional integrators such as the Runge-Kutta, extrapolation, or Adams methods. Further, we introduce the concept of a time element in the framework of Stiefel's approach and develop a complete set of K-S elements for the first time, which leads to a smaller error in the physical time than those of Stiefel and Stiefel & Scheifele. Numerical experiments show no significant increase in the CPU time due to the introduction of variation of parameters in practical problems.