Chaos and secondary resonances in the mimas–tethys system

Abstract

We have investigated the role of the 200 yr period discovered by Vienne and Duriez (1992) on the tidal evolution of the Mimas–Tethys system through the 2:4 ii′ present resonance. Three terms are found to generate this period. We present a perturbed‐pendulum model in which these terms bring about a perturbation to the ideal ii′ resonance pendulum, which is in a direct ratio to the eccentricity e′ of Tethys. Although e′ is now very small, it is shown that this quantity could have been much greater in the past. We also show, thanks to this model, that these terms may have brought about a stochastic layer of noticeable width at the time of capture in the ii′ resonance, with the consequence that the possible values of the inclination i of Mimas before capture range from 0.4° to 0.6° (these uncertainties arise from the present uncertainties on e′). The role of each one of the three terms is examined in the appearance of chaos. A capture into the 1/1 secondary resonance (between the libration period of the primary ii′ resonance and the period of about 200 yr) is found possible. It means that the system could have experienced several captures in the primary resonance, instead of a single one, and that i could have been, with this assumption, much lower than 0.4°. A probability of capture into this secondary resonance as a function of the eccentricity of Tethys on encounter is derived, using Malhotra's method (Malhotra, 1990). Allan's values of i = 0.42° and e′ ≈ 0 (Allan, 1969) are therefore called into question, and taking e′ ≠ 0 is shown to be absolutely necessary if we want to understand the phenomena at work in the Mimas–Tethys system.