Principles of box models of shock acceleration

Many authors (Volk et al., 1981a,b; Bogdan and Volk, 1983; Moraal and Axford, 1983; Lagage and Cesarsky, 1983; Schlickeiser, 1984; Volk and Biermann, 1988; Ball and Kirk, 1992; Protheroe and Stanev, 1998) have used, under various guises, a simplified but physically intuitive treatment of shock acceleration, sometimes referred to as a 'box' model. Drury et al. (1999) present an alternative more physical interpretation of the 'box' model which can be significantly different when additional loss processes, such as synchrotron or inverse Compton losses, are included. The main features of the 'box' model, as presented in the literature (see references above) and exemplified by Protheroe and Stanev (1998), can be summarized as follows. The particles being accelerated (and thus 'inside the box') have differential energy spectrum N(E) and are gaining energy at rate racE but simultaneously escape from the acceleration box at rate resc . Conservation of particles then requires

where Q(E) 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' characterized simply by just two rates, rac and resc . These rates have, of course, to be taken from more detailed theories of shock acceleration (e.g., above, and Drury, 1991).

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