  The observed central densities of Milky Way dwarf spheroidal galaxies (dSphs)
are significantly lower than the densities of the largest (Vmax about 35 km/s)
subhalos found in dissipationless simulations of Galaxy-size dark matter hosts.
One possible explanation is that gas removal from feedback can lower core
densities enough to match observations. We model the dynamical effects of
supernova feedback through the use of a time-varying central potential in high
resolution, idealized numerical simulations and explore the resulting impact on
the mass distributions of dwarf dark matter halos. We find that in order to
match the observed central masses of M_star about 10^6 M_sun dSphs, the energy
equivalent of more than 40,000 supernovae must be delivered with 100%
efficiency directly to the dark matter. This energy requirement exceeds the
number of supernovae that have ever exploded in most dSphs for typical initial
mass functions. We also find that, per unit energy delivered and per cumulative
mass removed from the galaxy, single blow-out events are more effective than
repeated small bursts in reducing central dark matter densities. We conclude
that it is unlikely that supernova feedback alone can solve the "Too Big to
Fail" problem for Milky Way subhalos.
