**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.