The Structure and Stability of Self-Gravitating Disks

Abstract

A method is presented for relating the law of rotation and density distribution of a plane disk of self-gravitating material. The method involves the use of oblate spheroidal coordinates, and the superposition of elementary solutions of Laplace's equation due to distributions of matter on the disk. General expressions for the law of rotation and the density-distribution are found as related series of Legendre functions. Simple particular disk-models involving only a finite number of terms in these series can be found, the most simple being the well-known one of a uniformly rotating disk with surface density σ = constant × |$(1-r^2/R_0^2)^{1/2}$| , where r is a radial coordinate and R0 is the radius of the disk. A small-perturbation stability-analysis is carried out for this particular model by solving the zero-pressure hydrodynamic equations of compressible flow. The disk is found to be unstable to an infinite number of axisymmetric and non-axisymmetric modes of oscillation.