Stability of Toomre-Hayashi Disk

We investigate stability of Toomre-Hayashi model for a self-gravitating, rotating gas disk. The rotation velocity, $ v_\phi $, and sound speed, $ c _{\rm s} $, are spatially constant in the model. We show that the model is unstable against an axisymmetric perturbation irrespectively of the values of $ v_\phi $ and $ c_{\rm s} $. When the ratio, $ v_\phi / c_{\rm s} $, 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 $ v_\phi / c_{\rm s} $ is smaller than 1.20, it is unstable against total collapse. Toomre-Hayashi model of $ 1.7 \la v_\phi / c_{\rm s} \la 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 $ v_\phi / c_{\rm s} $ 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.