Stability analysis of the finite, uniformly rotating, gaseous disk

The calculations of this paper use two methods. First, we perform a complete and exact normal modes analysis of linear adiabatic perturbations in a completely-flattened, uniformly-rotating, self-gravitating, gaseous disk of finite radial extent. A ''two-dimensional pressure'' aids the centrifugal force due to rotation in supporting the equilibrium state against self-gravity. In particular, we establish a global stability criterion for each important class of disturbance: compressional, distortional, and convective. Second, we consider various asymptotic approximations to the exact results. In a number of interesting cases, we find that we can improve upon the standard asymptotic analysis used, e.g., in density-wave theory, by including in the dispersion relation certain terms which depend on the azimuthal wavenumber. (Lau and Mark reached similar conculsions in related contexts.) Not surprisingly, however, we find that no asymptotic approximation accurately describes the behavior of barlike distortional modes. Our results have two principal physical implications. First, they reaffirm the importance of barlike distortions as an important mode of instability in finite disks which do not have a strong central concentration of mass or a massive halo of surrounding material. For the barlike disturbance, we find that the relevant criterion for the onset of instability is that proposed by Ostriker and Peebles. Second, we find that Schwarzschild's condition defines the test for convective instability in the uniformly rotating disk