Long-term planetary integration with individual time steps

We describe an algorithm for long-term planetary orbit integrations, including the dominant post-Newtonian effects, that employs individual timesteps for each planet. The algorithm is symplectic and exhibits short-term errors that are O(εΩ²τ²) where τ is the timestep, Ω is a typical orbital frequency, and ε1 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(ε²Ω³τ²T). A sample 0.8 Myr integration of the nine planets illustrates that Pluto can have a timestep more than 100 times Mercury's, without dominating the positional error. Our algorithm is applicable to other N-body systems.