  We describe a simple model which explains the qualitative and (approximate)
quantitative features of the distribution of orbital velocities of merging
pairs of dark matter halos. Our model considers a primary dark matter halo as a
perturber in a background of secondary halos with velocities described by
linear theory. By evaluating the ensemble of secondary halos on orbits within
the perturbing halo's "loss cone" we derive the distribution of orbital
parameters of these captured halos. This model is able provide qualitative
explanations for the features of this distribution as measured from N-body
simulations, and is in approximate quantitative agreement with those
measurements. As the velocity dispersion of the background halos is larger on
smaller scales our model predicts an overall increase in the characteristic
velocities of merging halos, relative to the virial velocities of those halos,
in lower mass systems. Our model also provides a simple explanation for the
measured independence of the orbital velocity distribution function on redshift
when considered at fixed peak height. By connecting the orbital parameter
distribution to the underlying power spectrum our model also allows for
estimates to be made of the effect of modifying that power spectrum, for
example by including a truncation at large wavenumber. For plausible warm dark
matter models we find that this truncation has only a small effect on the
predicted distributions.
