Global Instability of Polytropic Gaseous Disk Galaxies with Toomre's Density Distribution

The origin of spiral and/or bar structures of galaxies is studied from a point of view that they are manifestations of gravitationally unstable eigenmodes of galactic disks. Equilibrium models having the surface-density distribution identical to that of Toomre's (1963) first model and with various amount of internal energy due to gaseous pressure of the polytropic exponent Γ = 4/3 are analyzed. By the use of expansion of variables in terms of associated Legendre functions, the problem of small-amplitude oscillations of disk models is reduced to an eigenvalue problem, which is solved numerically.

Main results obtained are summarized as follows: (1) Gaseous pressure generally serves to stabilize the disks. (2) Not only two-armed spiral modes but also one-armed and multi-armed spiral modes as well as ring modes are expected to grow for relatively cool models. (3) The growing spiral patterns are trailing unless the model is extremely cold. (4) There is no tightly wound smooth spiral pattern as far as the most rapidly growing modes are concerned. (5) The amplitude of disturbance undulates significantly along the spiral arms of some modes. (6) The most rapidly growing modes for the first several azimuthal Fourier components have nearly the same pattern speed. The growth time of these modes is comparable with the period of rotation of the system.