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The mechanism responsible for the natal kicks of neutron stars continues tobe a challenging problem. Indeed, many mechanisms have been suggested, and onehydrodynamic mechanism may require large initial asymmetries in the cores ofsupernova progenitor stars. Goldreich, Lai, & Sahrling (1997) suggested thatunstable g-modes trapped in the iron (Fe) core by the convective burning layersand excited by the $\epsilon$-mechanism may provide the requisite asymmetries.We perform a modal analysis of the last minutes before collapse of publishedcore structures and derive eigenfrequencies and eigenfunctions, including thenonadiabatic effects of growth by nuclear burning and decay by both neutrinoand acoustic losses. In general, we find two types of g-modes: inner-coreg-modes, which are stabilized by neutrino losses and outer-core g-modes whichare trapped near the burning shells and can be unstable. Without exception, wefind at least one unstable g-mode for each progenitor in the entire mass rangewe consider, 11 M$_{\sun}$ to 40 M$_{\sun}$. More importantly, we find that thetimescales for growth and decay are an order of magnitude or more longer thanthe time until the commencement of core collapse. We conclude that the$\epsilon$-mechanism may not have enough time to significantly amplify coreg-modes prior to collapse.Comment: 32 pages including 12 color figures and 2 tables, submitted to Ap

Pulsational Analysis of the Cores of Massive Stars and Its Relevance to Pulsar Kicks