Perturbation Analysis of a General Polytropic Homologously Collapsing Stellar Core

For dyanmic background models of Goldreich & Weber and Lou & Cao, we examine three-dimensional perturbation properties of oscillations and instabilities in a general polytropic homologously collapsing stellar core of a relativistically hot medium with a polytropic index γ = 4/3. Perturbation behaviours, especially internal gravity g−modes, depend on the variation of specific entropy in the collapsing core. Among possible perturbations, we identify acoustic p−modes and surface f−modes as well as internal gravity g−modes and g−−modes. As in stellar oscillations of a static star, we define g− and g−−modes by the sign of the Brunt-Väisälä buoyancy frequency squared N 2 for a collapsing stellar core. A new criterion for the onset of convective instabilities is established for a homologous stellar core collapse. We demonstrate that the global energy criterion of Chandrasekhar 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 sufficiently high-order g−modes, both corresponding to convective core instabilities. Such instabilities occur before the stellar core bounce, in contrast to instabilities in other models of supernova (SN) 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. Other high-order and high-degree unstable g−modes may shred the nascent neutron core into pieces without an eventual compact remnant (e.g. SN1987A). Formation of binary pulsars and planets around neutron stars might originate from unstable l = 2 g−modes and high-order high-degree g−modes, respectively.