Fred Franklin, Paul Soper
Our concern here is the nature of secondary resonances--commensurabilities between apsidal and libration periods lying within first-order mean motion resonances [mmr] in the solar system. At the 4/3 and 3/2 mmr in the asteroid belt, we find in general that it is possible to identify the positions of their considerably large number and to determine the degree of chaos that they develop. The severity of the latter corresponds to the absence of observed asteroids at low eccentricity, e, in both of these mmr and even more so at 2/1. Chaos at higher e is present but weaker so that real bodies can remain over the age of the solar system. This reduced chaos, we suggest, arises because the denser though weaker secondaries have widths that allow them to overlap and so provide a continuous quite constant level of chaos over a broad eccentricity range. We also consider the similar first-order 1/2 mmr with Neptune in the Kuiper belt at 47.8 AU and find considerable chaos, probably once again induced by secondary resonances, among the many orbits with e < 0.10. The instability so generated seems very likely to correspond in realistic models to the slow escape from 1/2 of a fair number of its once captured bodies over the solar system's age.
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http://arxiv.org/abs/1207.3049
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