Long-term planetary integration with individual time steps

Abstract

We describe an algorithm for long-term planetary orbit integrations, including the dominant post-Newtonian effects, that employs individual time steps for each planet. The algorithm is symplectic and exhibits short-term errors that are O(epsilon(Omega2)(tau 2) where tau is the time step, Omega is a typical orbital frequency, and epsilon much less than 1 is a typical planetary mass in solar units. By a special starting procedure long-term errors over an integration interval T can be reduced to O(epsilon2(Omega3)(tau2)T. A sample 0.8 Myr integration of the nine planets illustrates that Pluto can have a time step more than 100 times Mercury's, without dominating the positional error. Our algorithm is applicable to other N-body systems.