Long simulations of the Solar System: Brouwer's Law and chaos

Abstract

The accuracy of long simulations of the Solar System is limited by the accumulation of round-off error. If the round-off error is systematic, the error in conserved quantities grows as t where t is time, and that in dynamical variables as t 2 . If the round-off error is stochastic, the error grows as t 1/2 and t 3/2 respectively. In a previous study, we showed that it was possible to implement the order thirteen Stormer method so the errors grew stochastically for the two-dimensional Kepler problem. Here we show the implementation gives stochastic error growth on three-dimensional simulations of the Solar System. Our integrations are such that the positions of the major planets are known with an estimated error of no more than 2 o after 10 9 years, a precision unmatched by earlier investigations. Further, our numerical results suggest the outer Solar System is not chaotic as has previously been reported, but rather computational errors in positions grow no faster than t 3/2 , conforming with existing models for stochastic error growth in an otherwise well-behaved system of ordinary differential equations.