Shock crossing probability for isotropic particle flux

[le chat]

Hi there,

I am currently trying to understand the theoretical frame work of diffusive shock acceleration. I am having trouble understanding a step in the derivation given by drury 1983 (http://www.oa.uj.edu.pl/user/mio/Ast-Wys-En/Literatura/drury.pdf). In the derivation of eq. 2.47 it is stated (just below the equation), that for an isotropic flux the shock crossing probability is proportional to cos(theta) and hence a factor of 2 cos(theta) is needed. I tried to tinker this together by using other literature and googling it a bit but i don't seem to find a satisfying answer for where exactly this comes from. I am sure the solution is rather simple but i don't see it. I would appreciate it if someone could explain it to me or perhaps refer me to a piece of literature where it is better described.

Thx

 

[lippyka]

in advance!The factor of 2 cos(theta) comes from the assumption that the diffusive flux is isotropic. This means that the average flux observed over a sphere is independent of the direction in which it is measured. To calculate this average flux, one needs to integrate the flux over the solid angle of a sphere. Since cos(theta) is the projection of the flux on the sphere surface, the integral should be multiplied by this factor. This is because the shock crossing probability depends on the magnitude of the flux and not in which direction it is directed. In other words, the integration over the solid angle takes into account the fact that the flux is equally likely to come from any direction. Therefore, the factor 2 cos(theta) is needed to calculate the average flux.