Stability of Toomre-Hayashi Disk

We investigate stability of Toomre-Hayashi model for a self-gravitating, rotating gas disk. The rotation velocity, vφ, and sound speed, cs, are spatially constant in the model. We show that the model is unstable against an axisymmetric perturbation irrespectively of the values of vφ and cs. When the ratio, vφ/cs, is large, the model disk is geometrically thin and unstable against an axisymmetric perturbation having a short radial wavelength, i.e., unstable against fragmentation. When vφ/cs is smaller than 1.20, it is unstable against total collapse. Toomre-Hayashi model of 1.7 . vφ/cs . 3 was thought to be stable against axisymmetric perturbations in earlier studies in which only radial motion was taken into account. Thus, the instability of Toomre-Hayashi model having a medium vφ/cs is due to not change in the surface density but that in the disk height. We also find that the singular isothermal perturbation is stable against non-spherical peturbations.