1. The Origin of the Kirkwood Gaps: A Mapping for Asteroidal Monitor Near the 3/1 Commensurability. 2. The Resonance Overlap Criterion and the Onset of Stochastic Behavior in the Restricted Three-Body Problem

Citation

Wisdom, Jack Leach (1981) 1. The Origin of the Kirkwood Gaps: A Mapping for Asteroidal Monitor Near the 3/1 Commensurability. 2. The Resonance Overlap Criterion and the Onset of Stochastic Behavior in the Restricted Three-Body Problem. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/s1r9-ff38. https://resolver.caltech.edu/CaltechTHESIS:08282017-145742321

Abstract

Paper 1:

A mapping of the phase space onto itself with the same low order resonance structure as the 3/1 commensurability in the planar elliptic three-body prob­lem is derived. This mapping is approximately one thousand (1000) times faster than the usual method of numerically integrating the averaged equations of motion (as used by Schubart, Froeschlé and Scholl in their studies of the asteroid belt). This mapping exhibits some very surprising behavior that might provide the key to the origin of the gaps. A test asteroid placed in the gap may evolve for a million years with low eccentricity (< 0.05) and then suddenly jump to large eccentricity (> 0.3) becoming a Mars crosser. The asteroid can then be removed by a close encounter with Mars. To test this hypothesis a dis­tribution of 300 test asteroids in the neighborhood of the 3/1 commensurability was evolved for two million years. When the Mars crossers are removed the dis­tribution of initial conditions displays a gap at the location of the 3/1 Kirkwood gap. While this is the first real demonstration of the formation of a gap, the gap is too narrow. The planar elliptic mapping is then extended to include the incli­nations and the secular perturbations of Jupiter's orbit. The two million year evolution of the 300 test asteroids is repeated using the full mapping. The resulting gap is somewhat larger yet still too small. Finally the possibility that over longer times more asteroids will become Mars crossers is tested by studying the evolution of one test asteroid near the border of the gap for a much longer time. A jump in its eccentricity occurs after 18 million years indicating that indeed it may simply be a matter of time for the full width of the gap to open.

Paper 2:

The resonance overlap criterion for the onset of stochastic behavior is applied to the planar circular-restricted three-body problem with small mass ratio (µ). Its predictions for µ = 10^-3, µ = 10^-4 and µ = 10^-5 are compared to the transitions in the numerically determined Kolmogorov-Sinai entropy and found to be in remarkably good agreement. In addition, an approximate scaling law for the onset of stochastic behavior is derived.

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