CHAOTIC ORBITS WITHIN THE 3 2-JOVIAN MEAN MOTION RESONANCE/

Since the minor planet population as a whole shows a mean proper eccen tricity, e(p), near 0.1, this paper starts by asking why the Hilda gro up, some sixty objects whose mean motions lie very close to 3/2 times that of Jupiter, all have e(p)>0.1. The Hildas execute small oscillati ons, usually called librations, about this 3/2 ratio and these oscilla tions have a well defined frequency v, which can, for example, be dete rmined from periodic variations in a minor planet's semimajor axis or eccentricity. Another basic frequency, omega, is given by the apsidal motion. We show that orbits for which the two frequencies differ by sm all integral or half integral multiples [i.e., v=(n/2)omega, n=3,4,5.. ..] are clearly chaotic, with Lyapunov times short enough to suggest i nstability. Such orbits lie within the 3:2 mean motion resonance and a ll have e(p)<0.1. Although the low e(p) region is streaked with chaoti c zones, they are generally narrow, hence generating many chaotic orbi ts, and possibly depopulating the entire e(p)<0.1 region, would seem t o require an added process that would slightly alter the loci of the r esonances over time. At higher multiples of the two frequencies, parti cularly v=6 omega, orbits are remarkably regular, even when the amplit ude of libration of successive conjunctions of Jupiter and a minor pla net about the latter's pericenter is even as large as 90 deg. In fact, numerical results show that all of the real Hildas having large libra tions lie at or very close to such commensurabilities, while hypotheti cal bodies that do not are markedly chaotic. (C) 1995 American Astrono mical Society.