  Neutron stars provide a cosmic laboratory to study the nature of dark matter
particles and their interactions. Dark matter can be captured by neutron stars
via scattering, where kinetic energy is transferred to the star. This can have
a number of observational consequences, such as the heating of old neutron
stars to infra-red temperatures. Previous treatments of the capture process
have employed various approximation or simplifications. We present here an
improved treatment of dark matter capture, valid for a wide dark matter mass
range, that correctly incorporates all relevant physical effects. These include
gravitational focusing, a fully relativistic scattering treatment, Pauli
blocking, neutron star opacity and multi-scattering effects. We provide general
expressions that enable the exact capture rate to be calculated numerically,
and derive simplified expressions that are valid for particular interaction
types or mass regimes and that greatly increase the computational efficiency.
Our formalism is applicable to the scattering of dark matter from any neutron
star constituents, or to the capture of dark matter in other compact objects.
