  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
include the results of these computations as data products alongside the
article. We find that the angular distribution of a future signal would depend
on the velocity-dependence of the annihilation cross section more strongly for
cuspy profiles than for cored profiles. 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.
