Growth of a Vortex Mode during Gravitational Collapse Resulting in Type II Supernova

We investigate stability of a gravitationally collapsing iron core against non-spherical perturbation. The gravitationally collapsing iron core is approximated by a similarity solution for dynamically collapsing polytropic gas sphere. We find that the similarity solution is unstable against non-spherical perturbations. The perturbation grows in proportion to $(t - t_0) ^{-\sigma}$ while the the central density increases in proportion to $(t - t_0) ^{-2}$. The growth rate is $\sigma = 1/3 + \ell (\gamma - 4/3)$, where $\gamma$ and $\ell$ denote the polytropic index and the parameter $\ell$ of the spherical harmonics, $Y_{\ell} ^m (\theta, \phi)$, respectively. The growing perturbation is dominated by vortex motion. Thus it excites global convection during the collapse and may contribute to material mixing in a type II supernova.