Mathematical analysis of a model for moon-triggered clumping in Saturn's rings

摘要

Spacecraft observations of Saturn's rings show evidence of an active aggregation-disaggregation process triggered by periodic influences from the nearby moons. This leads to clumping and break-up of the ring particles at timescales of the order of a few hours. A mathematical model has been developed to explain these dynamics in Saturn's F-ring and B-ring (Esposito et al., 2012) [4], the implications of which are in close agreement with the empirical results. In this paper, we conduct a rigorous analysis of the proposed forced dynamical system for a class of continuous, periodic and zero-mean forcing functions that model the ring perturbations caused by the moon flybys. In specific, we derive the existence of at least one periodic solution to the dynamic system with the period equal to the forcing period of the moon. Further, conditions for the uniqueness and stability of the solution and bounds for the amplitudes of the periodic solution are derived.