An Exhaustive Search for Stable Orbits between the Outer Planets
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**Session 13 -- Solar System**
*Oral presentation, Monday, June 12, 1995, 10:00am - 11:30am*

## [13.03] An Exhaustive Search for Stable Orbits between the Outer Planets

*K.R. Grazier, W.I. Newman, W.M. Kaula, F. Varadi (UCLA), J.M. Hyman (LANL)*
Using high-order multistep integration
methods optimized to minimize roundoff error propagation, we
performed fully three-dimensional integrations of planetesimal
trajectories for 100 million years to examine possible niches in the
Jupiter/Saturn, Saturn/Uranus, and Uranus/Neptune zones.
We computed the trajectories of 100,000 massless particles in the
Jupiter/Saturn zone, and 10,000 particles each in the Saturn/Uranus
and Uranus/Neptune zones.
A planetesimal situated between two large perturbers
resides near a separatrix that determines the qualitative dynamical
character of its orbit.
Given sufficient time and according to the accuracy of the scheme,
the dynamical character will *artificially*
change in the vicinity
of a separatrix.
The modified Cowell-St\"ormer integrator employed in these surveys
produces a longitude error for Jupiter bounded below $2^\circ$
in 1 billion years, orders of magnitude smaller
than earlier surveys.
Other surveys, particularly those
based on mapping techniques, sacrificed numerical
accuracy in order to permit greater integration times.
We believe that this may have resulted in an incorrect
assessment of the relative significance of the different
physical ejection mechanisms present.

The highly accurate integration of such large numbers of
particles (i.e. employing hundreds to thousands of times
more particles than in previous surveys) allows us to
make statistically significant inferences about the
dynamics and dominant physical mechanisms in these
regions.
Sample sizes should be at least 10,000 so that the
``statistics of small numbers'' is not a factor.
In assessing the reliability of such ``counting experiments,''
the relative error (i.e. $3\sigma$ level)
is on the order of $3/\sqrt{N}$ for a sample of size
$N$---for a
survey containing 100,000 particles, this is 1\% of the sample; for
100, particles, counts with fewer than 30\% the sample are
not significant.
The very large size of our samples,
coupled to the increased numerical accuracy in our
surveys permits us for the first time to explore the
relative significance of the various ejection mechanisms
present.

**Monday
program listing**