The Energetics of Cusp Destruction

We present a new analytic estimate for the energy required to create a constant density core within a dark matter halo. Our new estimate, based on more realistic assumptions, leads to a required energy that is orders of magnitude lower than is claimed in earlier work. We define a core size based on the logarithmic slope of the dark matter density profile so that it is insensitive to the functional form used to fit observed data. The energy required to form a core depends sensitively on the radial scale over which dark matter within the cusp is redistributed within the halo. Simulations indicate that within a region of comparable size to the active star forming regions of the central galaxy that inhabits the halo, dark matter particles have their orbits radially increased by a factor of 2--3 during core formation. Thus the inner properties of the dark matter halo, such as halo concentration, and final core size, set the energy requirements. As a result, the energy cost increases slowly with halo mass as M$_{rm{h}}^{0.3-0.7}$ for core sizes $lesssim1$ kpc. We use the expected star formation history for a given dark matter halo mass to predict dwarf galaxy core sizes. We find that supernovae alone would create well over 4 kpc cores in $10^{10}$ M$_{odot}$ dwarf galaxies emph{if} 100% of the energy were transferred to dark matter particle orbits. We can directly constrain the efficiency factor by studying galaxies with known stellar content and core size, such as Fornax. We find that the efficiency of coupling between stellar feedback and dark matter orbital energy need only be at the 1% level or less to explain Fornaxs 1 kpc core.