Long-Term Integration Error of Kustaanheimo-Stiefel Regularized Orbital Motion

Abstract

We confirm that the positional error of a perturbed two-body problem expressed in the Kustaanheimo-Stiefel (K-S) variable is proportional to the fictitious time s, which is the independent variable in the K-S transformation. This property does not depend on the type of perturbation, on the integrator used, or on the initial conditions, including the nominal eccentricity. The error growth of the physical time evolution and the Kepler energy is proportional to s if the perturbed harmonic oscillator part of the equation of motion is integrated by a time-symmetric integration formula, such as the leapfrog or the symmetric multistep method, and is proportional to s2 when using traditional integrators, such as the Runge-Kutta, Adams, Stormer, and extrapolation methods. Also, we discovered that the K-S regularization avoids the step size resonance/instability of the symmetric multistep method that appears in the unregularized cases. Therefore, the K-S regularized equation of motion is useful to investigate the long-term behavior of perturbed two-body problems, namely, those used for studying the dynamics of comets, minor planets, the Moon, and other natural and artificial satellites.