## Physical interpretation of the box model

Drury et al. (1999) prefer a very similar, but more physical, picture of shock acceleration which has the advantage of being more closely linked to the conventional theory. For this reason they also choose to work in terms of particle momentum p and the distribution function f (p) rather than E and N(E). For an almost isotropic distribution f (p) at the shock front where the frame velocity changes from ui to U2, then it is easy to calculate that there is a flux of particles upwards in momentum associated with the shock crossings of

*(p, t) = IP vu1yu2)2 f (p, t)v • ndQ = n f (p, t)n • (U1 - u2), (4.22.2)

v2 3

where n is the unit shock normal and the integration is over all directions of the velocity vector v. This flux is localized in space at the shock front and is strictly positive for a compressive shock structure; in this description it replaces the acceleration rate rac . The other key element is the loss of particles from the shock by advection downstream. Drury et al. (1999) note that the particles interacting with the shock are those located within one diffusion length of the shock. Particles penetrate upstream a distance of order

where K1 (p ) is the diffusion tensor and the probability of a downstream particle returning to the shock decreases exponentially with a scale length of

Thus in this picture there are an energy dependent acceleration region extending a distance Li(p) upstream and L2 (p ) downstream. The total size of the box is then

Particles are swept out of this region by the downstream flow at a bulk velocity n • u2. Conservation of particles then leads to the following approximate description of the acceleration,

A(4np2f (p,t)L(p))+ = Q(p,t)- 4n(n • u2)p2f (p,t). (4.22.6)