Establishment and evolution of satellite-satellite resonances

Abstract

Recent papers by Greenberg and Sinclair have lent support to Goldreich's tidal hypothesis which provides an explanation for the existence of several orbit-orbit resonances among the satellites of Saturn and Jupiter. In this paper I show that near resonance the two-body interaction appropriate to the three resonances observed in Saturn's system can be adequately described by a one dimensional Hamiltonian. A slow variation in the semimajor axes is then introduced to provide the evolution. First I determine the specific relationship between the initial inclination/eccentricity and the masses, etc., which separate direct capture (a la Greenberg) from a probabilistic capture (a la Sinclair). Once a resonance is established, I show that the action integral can be used to analytically follow the evolution of the libration amplitude as a function of the change in the semimajor axes, as was first done by Allan for the Mimas-Tethys case. The secular changes in the semimajor axes suggested by the presently observed libration amplitude are compared with their maximum tidal change using Goldreich's value forQ. I find that the tidal mechanism is insufficient only in the Titan-Hyperion case. I discuss the possibitility that satellite accretion may have produced the requisite evolution in Titan's orbit. Finally I show that both the stability and capture mechanisms for the three-body resonance among Jupiter's inner satellites depend on the observed near commensurability of the inner and outer pairs of satellites.