## 3. Physical interpretation of the box modelWe 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 we also choose to work in
terms of particle momentum The fundamental assumption of diffusive shock acceleration theory
is that the charged particles being accelerated are scattered by
magnetic structures advected by the bulk plasma flow and that, at
least to a first approximation, in a frame moving with these
structures the scattering changes the direction of a particle's
motion, but not the magnitude of its velocity, energy or momentum. If
we measure where is the unit shock normal and the integration is over all directions of the velocity vector . Notice that this flux is localised in space at the shock front and is strictly positive for a compressive shock structure. This spatially localised flux in momentum space is the essential mechanism of shock acceleration and in our description replaces the acceleration rate . The other key element of course is the loss of particles from the shock by advection downstream. We note that the particles interacting with the shock are those located within about one diffusion length of the shock. Particles penetrate upstream a distance of order where is the diffusion tensor and the probability of a downstream particle returning to the shock decreases exponentially with a scale length of . Thus in our picture we have an energy dependent acceleration region extending a distance upstream and downstream. The total size of the box is then . Particles are swept out of this region by the downstream flow at a bulk velocity . Conservation of particles then leads to the following approximate description of the acceleration, that is the time rate of change of the number of particles involved
in the acceleration at momentum Substituting for and simplifying we get the equation which is our version of the "Box" equation. Note that this, as is readily seen, gives the well known standard results for the steady-state spectrum and the acceleration time-scale. In fact our description is mathematically equivalent to that of Protheroe and Stanev as is easily seen by noting that However our version has more physical content, in particular the two rates are derived and not inserted by hand. It is also important to note that in our picture the size of the "box" depends on the particle energy. © European Southern Observatory (ESO) 1999 Online publication: June 18, 1999 |