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      Time-Symmetrized Kustaanheimo-Stiefel Regularization

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          Abstract

          In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although analytical solutions exist in the unperturbed and weakly perturbed cases, numerical integration is necessary in situations where the perturbation is relatively strong. Kustaanheimo--Stiefel (KS) regularization is widely used to remove the singularity in the equations of motion, making it possible to integrate orbits having very high eccentricity. However, even with KS regularization, long-term integration is difficult, simply because the required accuracy is usually very high. We present a new time-integration algorithm which has no secular error in either the binding energy or the eccentricity, while allowing variable stepsize. The basic approach is to take a time-symmetric algorithm, then apply an implicit criterion for the stepsize to ensure strict time reversibility. We describe the algorithm in detail and present the results of numerical tests involving long-term integration of binaries and hierarchical triples. In all cases studied, we found no systematic error in either the energy or the angular momentum. We also found that its calculation cost does not become higher than those of existing algorithms. By contrast, the stabilization technique, which has been widely used in the field of collisional stellar dynamics, conserves energy very well but does not conserve angular momentum.

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          Author and article information

          Journal
          04 April 1996
          Article
          10.1086/118136
          astro-ph/9604025
          853951ee-606c-40d0-ba2a-d5b8a74e12fa
          History
          Custom metadata
          figures are available at http://grape.c.u-tokyo.ac.jp/~funato/; To appear in Astronomical Journal (July, 1996)
          astro-ph

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