  In recent years, the usefulness of astrophysical objects as Dark Matter (DM)
probes has become more and more evident, especially in view of null results
from direct detection and particle production experiments. The potentially
observable signatures of DM gravitationally trapped inside a star, or another
compact astrophysical object, have been used to forecast stringent constraints
on the nucleon-Dark Matter interaction cross section. Currently, the probes of
interest are: at high red-shifts, Population III stars that form in isolation,
or in small numbers, in very dense DM minihalos at $z\sim 15-40$, and, in our
own Milky Way, neutron stars, white dwarfs, brown dwarfs, exoplanets, etc. Of
those, only neutron stars are single-component objects, and, as such, they are
the only objects for which the common assumption made in the literature of
single-component capture, i.e. capture of DM by multiple scatterings with one
single type of nucleus inside the object, is valid. In this paper, we present
an extension of this formalism to multi-component objects and apply it to Pop
III stars, thereby investigating the role of He on the capture rates of Pop III
stars. As expected, we find that the inclusion of the heavier He nuclei leads
to an enhancement of the overall capture rates, further improving the potential
of Pop III stars as Dark Matter probes.
