Local stability criterion for the Saturnian ring system

The self-gravitating particulate disk of the Saturnian ring system is studied using linear theory to determine its evolution and stability against gravitational Jeans-type perturbations. The analysis is carried out in approximation of the basically homogeneous and two-dimensional system. In addition, the case is considered with rare collisions between particles when the squared epicyclic frequency κ² as well as the squared orbital angular frequency Ω² greatly exceeds the squared frequency of interparticle physical collisions ν_c²; that is, the analysis presented here is valid only in the regime of low optical depth in Saturn's rings, τ ≃ ν_c/Ω < 1. According to observations, such low optical depth regions can be found in the C ring, the inner portions of the B ring at distances r < 100 000 km and the A ring at distances r > 123 000 km from the planetary center. The analogy with magnetized plasma problems is utilized by applying the so-called single particle dynamics method (the Lagrangian description): The motion of an "average" particle is considered. In the framework of this analytical method the local dispersion relation for small-amplitude oscillations is derived. Using the dispersion relation, an analysis is given of the dispersion law both for axisymmetric (radial) and nonaxisymmetric (spiral) Jeans perturbations. The main result, which follows from the dispersion relation, is the local stability criterion. The criterion generalizes the well-known Toomre's one (Astrophys. J. 139, 1217-1238, 1964) for spiral gravity perturbations. The dynamical behavior of the different models of Saturn's ring disk is studied by N-body computer simulations in order to confirm the validity of the generalized stability criterion. The numerical method of local simulations (or N-body simulations in a Hill's approximation) is applied. It is shown that the stability criterion obtained from the computer models is in general agreement with the theoretical prediction. It is proposed that as a direct result of the Jeans instability of nonaxisymmetric perturbations the Saturnian ring disk is subdivided into numerous irregular ringlets, with size and spacing of the order 50-100m. Forthcoming Cassini spacecraft high-resolution images of Saturn's rings may reveal this kind of structure.