Discrete spiral modes, spiral waves, and the local dispersion relationship

Numerical calculations of unstable discrete spiral modes for gas disks are presented. These calculations are based on some recent developments of the asymptotic theory of density waves. In addition to Toomre's stability index Q, another local parameter, which depends primarily on the surface density and on the amount of differential rotation, is found to be essential for the growth of spiral modes and waves. A local dispersion relation for these spiral waves is derived, and examined in the light of previous studies.