On the statistical convergence of N-body simulations of the Solar System

Most direct N-body integrations of planetary systems use a symplectic integrator with a fixed timestep. A large timestep is desirable in order to speed up the numerical simulations. However, simulations yield unphysical results if the timestep is too large. Surprisingly, no systematic convergence study has been performed on long (Gyr) timescales. In this paper we present numerical experiments to determine the minimum timestep one has to use in long-term integrations of the Solar System in order to recover the system’s fundamental secular frequencies and instability rate. We find that timesteps of up to 32 days, i.e. a third of Mercury’s orbital period, yield physical results in an ensemble of 5 Gyr integrations. We argue that the chaotic diffusion that drives the Solar System’s long-term evolution dominates over numerical diffusion and timestep resonances. Our results bolster confidence that the statistical results of most simulations in the literature are indeed physical and provide guidance on how to run time and energy efficient simulations while making sure results can be trusted.