Equilibrium and stability of a self-gravitating torus in the field of a large central mass

A model is developed for a homogeneous, self-gravitating fluid torus subject to the potential field of a central mass. This model generalizes the classical Laplace treatment to a differentially rotating configuration in equilibrium, where the relative mass of the central body may be large. The stability of the model is tested, with emphasis on radial perturbations. Two types of disturbances can exist: either symmetric or antisymmetric about the outer contour of the torus. Only the symmetric perturbations, however, can be unstable. The instability may be dynamical, or it may be secular because of dissipation in the system. The secular-type instability is universal, in that it can develop even in a dynamically stable system. Stability limits are established for all the radial modes. Nonradial perturbations are investigated as well, for the case where the azimuthal wavelength much exceeds the scale of the torus cross section. If the section is flattened enough, the nonradial disturbances will not engender any new instabilities of their own. The stability analysis is pertinent to the fine radial divisions observed in Saturn's rings.