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?


Drury explains that why the $\cos(\theta)$ factor appears in the text following Equation 2.47:

(assuming isotropy, the probability of crossing the shock at an angle $\theta$ is proportional to $\cos(\theta)$, hence the weighting factor of 2$\mu$).

The factor of 2 comes from the symmetry, i.e., you change the limits of integration under the assumption that things are the same on each side of the boundary. We also know from Equation 2.1 that: $$ \mu = \frac{\mathbf{p} \cdot \mathbf{B}}{p \ B} = \cos(\theta) $$ where $\mathbf{p}$ is the particle momentum and $\mathbf{B}$ is the magnetic field. So we can see from where the 2$\mu$ arises.


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