Models of momentum dependent diffusion

Diffusion is treated very differently in the two models. As just mentioned, the Monte Carlo simulation models pitch-angle diffusion by assuming a A(p) (in according with Ellison and Double, 2004). The semi-analytic model does not explicitly describe diffusion but assumes only that the diffusion is a strongly increasing function of particle momentum p so that particles of different p interact with different spatial regions of the upstream precursor. Eichler (1984) used a similar procedure. With this assumption, particles of momentum p can be assumed to feel some average precursor fluid speed u p and an average compression ratio rp ~ upju2 (see Blasi et al., 2005). The different way diffusion is treated influences not only injection, but also the shape of the distribution function f (p/ where f (p) is the momentum phase space density, i.e., particles/(cm3d3p )]. Both models give the characteristic concave f (p) which hardens with increasing p and, since the overall shock compression ratio r can be greater than 4, this spectrum will be harder than p-4 at ultra-relativistic energies. In the results we show here, the acceleration is limited with a cutoff momentum so f (p) cuts off abruptly at pmax . More realistic models will show the effects of escape from some spatial boundary (e.g., finite shock size) or from a finite acceleration time. In either case, the spectrum will show a quasi-exponential turnover, e.g., f (p)« p~aexp(-a~1(p/pmax)-a), (4.24.3)

where a is included to emphasize that the detailed shape of the turnover depends on the momentum dependence of the diffusion coefficient near pmax (Ellison et al., 2000).

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