**DDA2001, April2001**

*Session 4. Posters*

Monday, 8:00pm
[Previous] |
[Session 4] |
[Next]

## [4.07] Long term integration error of KS regularized orbital motion

*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 s^{2} 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.

[Previous] |
[Session 4] |
[Next]