$J$-factors for Velocity-dependent Dark Matter
  If dark matter annihilates with a velocity-dependent cross section within a
subhalo, then the magnitude and angular distribution of the resulting photon
signal will change. These effects are encoded in the $J$-factor. In this work
we compute the $J$-factor for a variety of choices for the cross section
velocity-dependence, and for a variety of choices for the dark matter profile,
including generalized Navarro-Frenk-White (NFW), Einasto, Burkert and Moore. We
find that, for cuspy profiles, the angular distribution of a future signal can
potentially be used to determine the velocity-dependence of the annihilation
cross section, and that these results are robust to changes in the form of the
profile. For a cored profile, determining the velocity-dependence of the cross
section is more difficult, but potentially still possible. Interestingly, we
find that for a density profile with an inner slope power law steeper than 4/3,
Sommerfeld-enhanced annihilation in the Coulomb limit leads to a divergence at
the center, requiring a more detailed treatment of departure from the Coulomb
limit.
