Instability considerations at self-gravitating circumplanetary ringlet structures

Abstract

The aim of this paper has been to study here specific forms of instabilities in circumstellar and circumplanetary dust ringlets in Keplerian rotation around a central gravitating mass without taking shear flow effects into consideration. Due to the presence of a central mass in the disk, an additional force term appears in the linearized equation of motion. Here we investigate the importance of such a term with respect to the onset of gravitational instabilities in both tangential and radial direction of ring-like substructures in the disk. In addition, we compare the instability tendencies of self-gravitating disks with those of fluid layers where perturbation effects are simply controlled by surface tension. In both cases, the material of the layer is treated as an incompressible inviscid fluid. This assumption, however, as shown from our study of the polytropy of dust gases, was proven to be correct for perturbation wavelengths comparable or larger than the thickness of the layer. From our general dispersion relations for symmetric and anti-symmetric perturbation modes, we can retain for the radial wave propagation the results of Lin and Shu, and Goldreich and Ward in the asymptotic case of an infinitely thin layer without shear flow. However, for the tangential waves we find a different stability criterion showing that the onset of the instability depends on the propagation direction. In the ‘finite layer’ case, we derive much more general relations showing different instability ranges for ‘bending’ wave modes and self-excited ‘density’ wave modes pointing to local and global instability forms in ringlets.