Stability of Toomre-Hayashi Disk

We investigate stability of Toomre-Hayashi model for a self-gravitating, rotating gas disk. The rotation velocity, $v_{\varphi }$, 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 $v_{\varphi }$ and $c_{\mathrm{s}}$. When the ratio, $v_{\varphi }\mathrm{/}{c}_{\mathrm{s}}$, 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 $v_{\varphi }\mathrm{/}{c}_{\mathrm{s}}$ 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 $v_{\varphi }\mathrm{/}{c}_{\mathrm{s}}$ 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.