A Fluid Dynamical Treatment of Common Action of Self-Gravitation, Collisions, and Rotation in Saturn'sB-Ring

Abstract

We study the generation of structures in Saturn's B-ring under the common action of self-gravitation, inelastic collisions, and rotation. Folowing a hydrodynamical approach the ring corresponds to a viscous fluid. Then the state of the rings is governed by the surface density dependence of the dynamic viscosity, d(vo)/do, with v being the kinematic viscosity and o the surface density. Several authors have calculated v and e: &#61 d ln(vo)/d ln o. In all of their models e takes a positive quantity. Then two different instabilities may arise. For all e below a small positive number the ring is securlarlyunstable. The wavelengths of the fastest growing modes are of the order of one Jeans length, i.e. 80-120 m in our case. For a sufficient large positive e another unstable branch appears: The ring is overstable. Here individual particle oscillate and the hydrodynamic picture is a standing wave with rapidly growing amplitudes. Again, the wavelength of the fastest growing mode is in the range of about 80 m. Since the ring deviates considerably from a classical gas, it is rather difficult to determine the quamtity e. Although this problem has already been an outstanding matter of investigation for a long time, a generally accepted solution is stillmissing. For that reason we restrict our stability analysis to the parameterized representation of v with v x o_ and &#61 e -1 &#61 const and discuss both the secular instability, which sets already in for small positive e if self-gravitation is included, and the overstable modes that are produced by sufficiently large e>0, even without self-gravitation. In this paper we derive the quasi-linear perturbation equations and solve toe corresponding initial-boundary value problem. The ring's gravitational potential is also treated in a self-consistent way. The aim of the paper is to show that the disk is probably unstable or overstable and small-scale perturbations exhibit a remarkable intensification within a few orbital periods.