Efficient integration of highly eccentric orbits by scaling methods applied to Kustaanheimo-Stiefel regularization

We apply our single scaling method to the numerical integration of perturbed two-body problems regularized by the Kustaanheimo-Stiefel (K-S) transformation. The scaling is done by multiplying a single scaling factor with the four-dimensional position and velocity vectors of an associated harmonic oscillator in order to maintain the Kepler energy relation in terms of the K-S variables. As with the so-called energy rectification of Aarseth, the extra cost for the scaling is negligible, since the integration of the Kepler energy itself is already incorporated in the original K-S formulation. On the other hand, the single scaling method can be applied at every integration step without facing numerical instabilities. For unperturbed cases, the single scaling applied at every step gives a better result than either the original K-S formulation, the energy rectification applied at every apocenter, or the single scaling method applied at every apocenter. For the perturbed cases, however, the single scaling method applied at every apocenter provides the best performance for all perturbation types, whether the main source of error is truncation or round-off.