I am trying to figure out a specific step in the derivation of the power law spectrum for cosmic ray particles upon diffusive shock acceleration. I am working with Drury 1983 (pdf link) but I have also dug into some other literature on the subject, without a satisfying explanation for my question which is: in the derivation of Equation 2.47 a weighting factor of 2$\mu$ is introduced for the isotropic particle distribution and the explanation given by Drury is, that the probability for a particle to cross the shock is proportional to $\cos(\theta)$. I understand that the probability is proportional to the relative velocity between particle and shock front which in turn is proportional to $\cos(\theta)$. Why this yields a factor of 2$\mu$ in the integration, however, is not conclusive to me.
Does anyone have a somewhat more elaborated explanation for me?