  Dark matter can capture in neutron stars from scattering off
ultra-relativistic electrons. We present a method to calculate the capture rate
on degenerate targets with ultra-relativistic momenta in a compact astronomical
object. Our treatment accounts for the target momentum and the Fermi degeneracy
of the system. We derive scaling relations for scattering with relativistic
targets and confirm consistency with the non-relativistic limit and Lorentz
invariance. The potential observation of kinetic heating of neutron stars has a
larger discovery reach for dark matter-lepton interactions than conventional
terrestrial direct detection experiments. We map this reach onto a set of
bosonic and fermionic effective contact interactions between dark matter and
leptons as well as nucleons. We show the results for the contact operators up
to dimension 6 for spin-0 and spin-1/2 dark matter interactions with
relativistic as well as non-relativistic Standard Model fermions. Highlights of
this program in the case of vector mediated interactions are presented in a
companion letter [1]. Our method is generalizable to dark matter scattering in
any degenerate medium where the Pauli exclusion principle leads to relativistic
targets with a constrained phase space for scattering.
