  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.
