  If dark matter interacts too strongly with nuclei, it could be slowed to
undetectable speeds in Earth's crust or atmosphere before ever reaching a
detector. For sub-GeV dark matter, analytic approximations appropriate for
heavier dark matter fail, necessitating the use of computationally expensive
simulations. We present a new method of modeling attenuation of light dark
matter in the Earth, based on the approximation that the scattering is
isotropic in the lab frame. We show that this approach agrees well with Monte
Carlo results, and can be much faster when the number of scatterings becomes
large, as the runtime for Monte Carlo methods increases exponentially with
cross section. We use this method to model attenuation for sub-dominant dark
matter--that is, particles that make up a small fraction of the dark matter
density--and show that previous work on sub-dominant dark matter overestimates
the sensitivity of direct detection experiments.
