Shepherding of the Uranian Rings. I. Kinematics

We identify several orbital resonances involving the newly discovered satellites, 1986U7 and 1986U8, and the Uranian rings. The most important resonances in eccentric rings are known as eccentric resonances and are generalizations of the more familiar Lindblad resonances. In keeping with the notation established for Lindblad resonances, we distinguish inner and outer eccentric resonances by the symbols IER and OER. We show that by reducing the absolute radius scale of the Uranian ring system by 0.0124% the 24:25 OER of 1986U7 and the 14:13 IER of 1986U8 fall at the inner and outer edges of the E ring. The same scale change also brings the 23:22 IER of 1986U7 into coincidence with the outer edge of the {j ring and the 6:5 IER of 1986U8 close to the center of they ring. Furthermore, adopting the latest Voyager value of GM u and our reduced radius scale, we find that the pattern speed of the m = 2 distortion in the {j ring corresponds to that expected for a normal mode excited either by an internal viscous overstability or parametrically by shepherd satellites. These kinematic results make a compelling case for our proposed reduction in the ring radius scale and also imply that 1986U7 and 1986U8 are the inner and outer shepherds for theE ring, that 1986U7 is the outer shepherd for the {j ring, and that 1986U8 is an outer shepherd for they ring.