Analytic Approximations for the Velocity Suppression of Dark Matter Capture

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.