On the shepherding of Uranian epsilon ring

To explain the long-term existence of Uranian narrow rings, we study the dy namical evolution of Uranian epsilon ring under the influence of its two sh epherding satellites: Cordelia and Ophelia. The model of planar-elliptic re stricted three-body problem is employed, in which the three bodies are Uran us, a satellite and a test particle. Dissipations due to interparticle coll isions are also taken into consideration. A mapping system has been obtaine d based on the differential equation system. Numerical results show that on ly those particles originally lying in the libration region of the resonanc e can be shepherded. The size of the libration region depends mainly on the orbital eccentricity and mass of the shepherd satellite. For the outer she pherd Ophelia with larger orbital eccentricity, if its mass is a third of t he nominal mass 2.5 x 10(19) g, most of the orbits are regular and the oute r edge of the ring can be shepherded, but more fuzzy than the inner edge. T hus we infer that Ophelia has a smaller mass.