Chaotic Orbits Within the 3-2 Jovian Mean Motion Resonance

Abstract

view Abstract Citations (2) References (17) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Chaotic Orbits Within the 3-2 Jovian Mean Motion Resonance Franklin, Fred A. Abstract Since the minor planet population as a whole shows a mean proper eccentricity, e (p), near 0.1, this paper starts by asking why the Hilda group, 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 oscillations, usually called librations, about this 3/2 ratio and these oscillations have a well defined frequency, ν, which can, for example, be determined from periodic variations in a minor planet's semimajor axis or eccentricity. Another basic frequency, ωcaron, is given by the apsidal motion. We show that orbits for which the two frequencies differ by small integral or half integral multiples [i.e., ν=(n/2)ωcaron, n=3,4,5... .] are clearly chaotic, with Lyapunov times short enough to suggest instability. Such orbits lie within the 3:2 mean motion resonance and all have e(p) <0.1. Although the low e(p) region is streaked with chaotic zones, they are generally narrow, hence generating many chaotic orbits, and possibly depopulating the entire e(p)<0.1 region, would seem to require an added process that would slightly alter the loci of the resonances over time. At higher multiples of the two frequencies, particularly ν=6ωcaron, orbits are remarkably regular, even when the amplitude of libration of successive conjunctions of Jupiter and a minor planet 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 librations lie at or very close to such commensurabilities, while hypothetical bodies that do not are markedly chaotic. Publication: The Astronomical Journal Pub Date: October 1995 DOI: 10.1086/117659 Bibcode: 1995AJ....110.1879F Keywords: CHAOTIC PHENOMENA; CELESTIAL MECHANICS; INSTABILITIES full text sources ADS |