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

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2. The "box" model of diffusive shock acceleration

The main features of the "box" model, as presented in the literature (see references above) and exemplified by Protheroe & Stanev (1998)) can be summarised as follows. The particles being accelerated (and thus "inside the box") have differential energy spectrum [FORMULA] and are gaining energy at rate [FORMULA] but simultaneously escape from the acceleration box at rate [FORMULA]. Conservation of particles then requires

[EQUATION]

where [FORMULA] is a source term combining advection of particles into the box and direct injection inside the box.

In essence this approach tries to reduce the entire acceleration physics to a "black box" characterised simply by just two rates, [FORMULA] and [FORMULA]. These rates have of course to be taken from more detailed theories of shock acceleration (eg Drury 1991). A minor reformulation of the above equation into characteristic form,

[EQUATION]

is useful in revealing the character of the description. This is equivalent to the ordinary differential equation,

[EQUATION]

on the family of characteristic curves described by

[EQUATION]

giving the formal solution,

[EQUATION]

The number of particles at energy E and time t in the "box" is given simply by an exponentially weighted integral over the injection rate at earlier times and lower energies. Of particular interest is the steady solution at energies above those where injection is occuring which is easily seen to be a power-law with exponent

[EQUATION]

At first sight (to one familiar with shock acceleration theory) it appears odd that the exponent depends not just on the ratio of [FORMULA] to [FORMULA] but also on the energy dependence of [FORMULA]. However, as remarked by Protheroe and Stanev, the physically important quantity is not the spectrum of particles inside the fictitious acceleration "box" but the escaping flux of accelerated particles [FORMULA] and this is a power-law of exponent

[EQUATION]

Thus provided the ratio of [FORMULA] to [FORMULA] is fixed, the power-law exponent of the spectrum of accelerated particles escaping from the accelerator is determined only by this ratio whatever the energy dependence of the two rates.

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999
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