The mechanism responsible for the natal kicks of neutron stars continues to be a challenging problem. Indeed, many mechanisms have been suggested, and one hydrodynamic mechanism may require large initial asymmetries in the cores of supernova progenitor stars. Goldreich, Lai, & Sahrling (1997) suggested that unstable g-modes trapped in the iron (Fe) core by the convective burning layers and excited by the $epsilon$-mechanism may provide the requisite asymmetries. We perform a modal analysis of the last minutes before collapse of published core structures and derive eigenfrequencies and eigenfunctions, including the nonadiabatic effects of growth by nuclear burning and decay by both neutrino and acoustic losses. In general, we find two types of g-modes: inner-core g-modes, which are stabilized by neutrino losses and outer-core g-modes which are trapped near the burning shells and can be unstable. Without exception, we find at least one unstable g-mode for each progenitor in the entire mass range we consider, 11 M$_{sun}$ to 40 M$_{sun}$. More importantly, we find that the timescales for growth and decay are an order of magnitude or more longer than the time until the commencement of core collapse. We conclude that the $epsilon$-mechanism may not have enough time to significantly amplify core g-modes prior to collapse.
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