A new timestep criterion for N-body simulations

We derive a new criterion for estimating characteristic dynamical timescales in N-body simulations. The criterion uses the second, third, and fourth derivatives of particle positions: acceleration, jerk, and snap. It can be used for choosing timesteps in integrators with adaptive step size control. For any two-body problem the criterion is guaranteed to determine the orbital period and pericenter timescale regardless of eccentricity. We discuss why our criterion is the simplest derivative-based expression for choosing adaptive timesteps with the above properties and show its superior performance over existing criteria in numerical tests. Because our criterion uses lower order derivatives, it is less susceptible to rounding errors caused by finite floating point precision. This significantly decreases the volume of phase space where an adaptive integrator fails or gets stuck due to unphysical timestep estimates. For example, our new criterion can accurately estimate timesteps for orbits around a 50m sized Solar System object located at 40AU from the coordinate origin when using double floating point precision. Previous methods where limited to objects larger than 10km. We implement our new criterion in the high order IAS15 integrator which is part of the freely available N-body package REBOUND.