ANZIAM J. 46(E) pp.C1086--C1103, 2005.

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

K. R. Grazier

W. I. Newman

James M. Hyman

Philip. W. Sharp
(received 28 October 2004, revised 18 October 2005)

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 t2. If the round-off error is stochastic, the error grows as t1/2 and t3/2 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 2o after 109 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 t3/2, conforming with existing models for stochastic error growth in an otherwise well-behaved system of ordinary differential equations.

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Authors

K. R. Grazier
The Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA. mailto:krg@anlashok.jpl.nasa.gov
W. I. Newman
Departments of Earth and Space Sciences, Physics and Astronomy, and Mathematics, University of California, Los Angeles, CA 90095, USA. mailto:win@ucla.edu
James M. Hyman
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. mailto:hyman@lanl.gov
Philip. W. Sharp
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand. mailto:sharp@math.auckland.ac.nz

Published October 19, 2005. ISSN 1446-8735

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Copyright Austral. Mathematical Soc.; prepared October 19, 2005, from Gra2.tex with anziamje.sty (v2.3i)