  A promising probe to unmask particle dark matter is to observe its effect on
neutron stars, the prospects of which depend critically on whether captured
dark matter thermalizes in a timely manner with the stellar core via repeated
scattering with the Fermi-degenerate medium. In this work we estimate the
timescales for thermalization for multiple scenarios. These include: (a) spin-0
and spin-$\frac{1}{2}$ dark matter, (b) scattering on non-relativistic neutron
and relativistic electron targets accounting for the respective kinematics, (c)
interactions via a range of Lorentz-invariant structures, (d) mediators both
heavy and light in comparison to the typical transfer momenta in the problem.
We discuss the analytic behavior of the thermalization time as a function of
the dark matter and mediator masses, and the stellar temperature. Finally, we
identify parametric ranges where both stellar capture is efficient and
thermalization occurs within the age of the universe. For dark matter that can
annihilate in the core, these regions indicate parametric ranges that can be
probed by upcoming infrared telescopes observing cold neutron stars.
