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Astron. Astrophys. 347, 370-374 (1999)

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3. Physical interpretation of the box model

We 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 p and the distribution function [FORMULA] rather than E and [FORMULA].

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 p, the magnitude of the particle's momentum, in this frame, it is not changed by the scattering and the angular distribution is driven to being very close to isotropic. However if a particle crosses a shock front, where the bulk plasma velocity changes abruptly, then the reference frame used to measure p changes and thus p itself changes slightly. If we have an almost isotropic distribution [FORMULA] at the shock front where the frame velocity changes from [FORMULA] to [FORMULA], then it is easy to calculate that there is a flux of particles upwards in momentum associated with the shock crossings of


where [FORMULA] is the unit shock normal and the integration is over all directions of the velocity vector [FORMULA]. 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 [FORMULA]. 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 [FORMULA] where [FORMULA] is the diffusion tensor and the probability of a downstream particle returning to the shock decreases exponentially with a scale length of [FORMULA]. Thus in our picture we have an energy dependent acceleration region extending a distance [FORMULA] upstream and [FORMULA] downstream. The total size of the box is then [FORMULA]. Particles are swept out of this region by the downstream flow at a bulk velocity [FORMULA].

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 p plus the divergence in the accelerated momentum flux equals the source minus the flux carried out of the back of the region by the downstream flow. The main approximation here is the assumption that the same [FORMULA] can be used in all three terms where it occurs. In fact in the acceleration flux it is the local distribution at the shock front, in the total number it is a volume averaged value, and in the loss term it is the downstream distribution which matters. Diffusion theory shows that in the steady state all three are equal, but this need not be the case in more elaborate transport models (Kirk et al. 1996).

Substituting for [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999