Efficient Orbit Integration by Linear Transformation for Kustaanheimo-Stiefel Regularization

Abstract

We extended the quadruple scaling method of the Kustaanheimo-Stiefel(K-S) regularization by adding three independent components of thefour-dimensional fictitious angular momentum tensor to the quasi-conservedquantities to be monitored during the numerical integration. By usinga linear transformation in the four-dimensional fictitious spaceto make the newly introduced components and the full harmonic energiesapproximately consistent, we adjust the four amplitudes and the threephase differences for the four-dimensional harmonic oscillator associatedwith the K-S regularization at every integration step. The determinationof transformation parameters is unique, simple, and universal. Thenew method is superior to the quadruple scaling method in the sensethat the errors in all unperturbed orbital elements except the meanlongitude at epoch are reduced to the machine epsilon level independentlyof the precision of the numerical integration. For perturbed orbits,the errors increase more slowly than the quadruple scaling method.Although the number of variables to be integrated is increased to16 per celestial body, the new method provides the best performanceamong the manifold correction methods we developed for K-S-regularizedorbital motions.