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Perturbation Analysis of a General Polytropic Homologously Collapsing Stellar Core

For dynamic background models of Goldreich & Weber and Lou & Cao, we examine 3-dimensional perturbation properties of oscillations and instabilities in a general polytropic homologously collapsing stellar core of a relativistic hot medium with a polytropic index of 4/3. We identify acoustic p-modes and surface f-modes as well as internal gravity g$^{+}-$ and g$^{-}-$modes. We demonstrate that the global energy criterion of Chandrasehkar is insufficient to warrant the stability of general polytropic equilibria. We confirm the acoustic p-mode stability of Goldreich & Weber, even though their p-mode eigenvalues appear in systematic errors. Unstable modes include g$^{-}-$modes and high-order g$^{+}-$modes. Such instabilities occur before the stellar core bounce, in contrast to instabilities in other models of supernova explosions. The breakdown of spherical symmetry happens earlier than expected in numerical simulations so far. The formation and motion of the central compact object are speculated to be much affected by such g-mode instabilities. By estimates of typical parameters, unstable low-order l=1 g-modes may produce initial kicks of the central compact object.