Unstable Disk Galaxies. I. Modal Properties

I utilize the Petrov-Galerkin formulation and develop a new method for solving the unsteady collisionless Boltzmann equation in both the linear and nonlinear regimes. In the first order approximation, the method reduces to a linear eigenvalue problem which is solved using standard numerical methods. I apply the method to the dynamics of a model stellar disk which is embedded in the field of a soft-centered logarithmic potential. The outcome is the full spectrum of eigenfrequencies and their conjugate normal modes for prescribed azimuthal wavenumbers. The results show that the fundamental bar mode is isolated in the frequency space while spiral modes belong to discrete families that bifurcate from the continuous family of van Kampen modes. The population of spiral modes in the bifurcating family increases by cooling the disk and declines by increasing the fraction of dark to luminous matter. It is shown that the variety of unstable modes is controlled by the shape of the dark matter density profile.