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

K.R. Grazier, W.I. Newman, J.M. Hyman, P.W. Sharp


The numerical integration of Newton's equations of motion for
self-gravitating systems, particularly in the context of our Solar System's
evolution, remains a paradigm for complex dynamics.
We implement Stormer's multistep method
in backward difference, summed form and
perform arithmetic according to a technique we
call ``significance ordered computation.''
We achieve results where the
truncation error of our 13th order
integrator resides well below machine (double)
precision and roundoff error accumulation
is random, not systematic.
In a previous study, we achieved this ``Brouwer's
Law'' in integrations of the 2-D Kepler
Here we show that such error growth can
be attained in 3-D Solar System integrations.
Our integrations are such that the
positions of the major planets are known
with an estimated error of no more than 2 degrees after
one billion years, a precision unmatched by
earlier investigations.
Further, we show 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
ordinary differential equation system.

solar system, long simulations, Stormer, optimal, chaos

Math Review Classification
Primary 65L06 ; Secondary 70F10

Last Updated
October 28, 2004


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