SHEPHERDING SATELLITES AND DYNAMIC STRUCTURE OF THE RINGS OF URANUS .2.

The dynamical structures of Uranian alpha-, beta-, and epsilon-rings can be successfully explained by assuming that the rings are surrounded by two shepherding satellites, inner and outer, which keep the ring widths very narrow. In fact, the rings move along non-circular orbits with increasing eccentricities as the semi-major axes and their apsidal motions are common in each ring. The precessional motions of the nodes of the ring particles are also thought to be synchronized, since their orbital planes are very thin in spite of the observable small inclinations to the equator of Uranus. As the outer shepherding satellite produces forced eccentricities of ring particles and the inner satellite, their apsidal motions are proved in this paper to be identical to that of the outer satellite. The outer satellite also produces forced inclinations, so that the ascending nodes of the ring particles and the inner satellite move with the same angular velocities and phase angles as those of the outer satellite. In this study the estimated masses of the order of 10(18) g and the distances from the rings of the order of 10 km as well as the eccentricities and the inclinations taking nearly the same values as the ring particles are estimated for the shepherding satellites of each ring.