34th Meeting of the AAS Division on Dynamical Astronomy, May 2003
6 Poster Papers
Posters, Monday, May 5, 2003, 8:00pm,

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[6.03] Efficient Orbit Integration by Scaling for Kepler Energy Consistency

T. Fukushima (NAOJ)

Extending the idea of manifold correction (Nacozy 1971) by using the concept of integral invariant relation (Szebehely and Bettis 1970), we propose a new approach to integrate the quasi-Keplerian orbits numerically. The method integrates the time evolution of the Kepler energy and the usual equation of motion simultaneously. Then it adjusts directly the integrated position and velocity by a space scale transformation in order to satisfy the Kepler energy relation rigorously at every integration step. The scale factor is determined by solving an associated cubic equation precisely with help of the Newton method. In treating multiple bodies, the Kepler energies are integrated for each body and the scale factors are adjusted separately. The implementation of the new method is simple, the additional cost of computation is little, and its applicability is wide. Numerical experiments showed that the scaling reduces the integration error drastically. In case of pure Keplerian orbits, the truncation error grows linearly with respect to time and the round-off error does slower than that. When the perturbations exist, a component growing in a quadratic or higher power of time appears in the truncation error but its magnitude is reduced significantly when compared with the case without scaling. The manner of decrease is roughly 5/4 to 5/2 power of the strength of the perturbing acceleration where the power index depends on the type of perturbation. The method seems to suppress the accumulation of round-off errors in the perturbed cases although the details remain to be investigated. In conclusion, the new approach provides a fast and high precision device to simulate the orbital motions of major and minor planets, natural and artificial satellites, comets, and space vehicles at negligible increase of computational cost.

The author(s) of this abstract have provided an email address for comments about the abstract: Toshio.Fukushima@nao.ac.jp

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Bulletin of the American Astronomical Society, 35 #4
© 2003. The American Astronomical Soceity.