Efficient orbit integration by linear transformation for Kustaanheimo-Stiefel regularization

We extended the quadruple scaling method of the Kustaanheimo-Stiefel (K-S) regularization by adding three independent components of the four-dimensional fictitious angular momentum tensor to the quasi-conserved quantities to be monitored during the numerical integration. By using a linear transformation in the four-dimensional fictitious space to make the newly introduced components and the full harmonic energies approximately consistent, we adjust the four amplitudes and the three phase differences for the four-dimensional harmonic oscillator associated with the K-S regularization at every integration step. The determination of transformation parameters is unique, simple, and universal. The new method is superior to the quadruple scaling method in the sense that the errors in all unperturbed orbital elements except the mean longitude at epoch are reduced to the machine epsilon level independently of 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 to 16 per celestial body, the new method provides the best performance among the manifold correction methods we developed for K-S-regularized orbital motions.