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H. Arakida (Department of Astronomical Science, Graduate University for Advanced Studies), T. Fukushima (National Astronomical Observatory of Japan)
Long term integrations of highly eccentric orbits are necessary to study the orbital evolution of comets and some minor planets. We confirmed that the positional error of a perturbed two body problem expressed in the KS variable is proportional to the fictitious time s, which is the independent variable in the KS transformation. This property does not depend on the type of perturbations, on the integrator used, nor on the initial conditions including the nominal eccentricity. This phenomenon is based on the fact that the equations of motion in the KS variables are those of perturbed harmonic oscillators. 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 are integrated by the time symmetric integrators such as the leapfrog or the symmetric multistep method, and to s2 when using the traditional integrators such as the Runge-Kutta, Adams, Störmer, or extrapolation methods. Further KS regularization reduces the stepsize resonance/instability of symmetric multistep methods observed in integrating Kepler problem, and the harmonic oscillator potential is the only case that the step size instability does not appear. We applied the method of variation of parameter to KS regularization and found this approach leads to linear error growth of both the position and physical time even if using traditional integrators. Further we introduced the concept of time element in the framework of Stiefel's approach and developed a complete set of KS elements for the first time, Therefore the KS regularization is useful to investigate the long term behavior of perturbed two body problem for studying comets, minor planets, moon, and artificial satellites.