  Compact astrophysical objects have been considered in the literature as dark
matter (DM) probes, via the observational effects of annihilating captured DM.
In this respect, Population III (Pop III) stars are particularly interesting
candidates, since they form at high redshifts, in very high DM density
environments. It is customary to assume such a star would form roughly at the
center of a DM halo, and, as such, have no rotational velocity. In this paper,
we break from this assumption and explore the effects we can expect to observe
if a Pop III star forms at some distance away from the center of the halo and
thus has a non-zero rotational velocity. The capture rate of DM in such a star
is suppressed by a predictable amount. We develop and validate an analytical
expression for the capture rate suppression factor and re-evaluate the bounds
placed on the DM-nucleon cross section as a result of DM capture. We find that
our previous results, obtained under the assumption of star formed within the
central 10 AU of the DM mini-halo are essentially unchanged, even when
considering the possible rotational velocities for those central stars.
