  Neutron stars harbour matter under extreme conditions, providing a unique
testing ground for fundamental interactions. We recently developed an improved
treatment of dark matter (DM) capture in neutron stars that properly
incorporates many of the important physical effects, and outlined useful
analytic approximations that are valid when the scattering amplitude is
independent of the centre of mass energy. We now extend that analysis to all
interaction types. We also discuss the effect of going beyond the
zero-temperature approximation, which provides a boost to the capture rate of
low mass dark matter, and give approximations for the dark matter up-scattering
rate and evaporation mass. We apply these results to scattering of dark matter
from leptonic targets, for which a correct relativistic description is
essential. We find that the potential neutron star sensitivity to DM-lepton
scattering cross sections greatly exceeds electron-recoil experiments,
particularly in the sub-GeV regime, with a sensitivity to sub-MeV DM well
beyond the reach of future terrestrial experiments.
