Ruffolo and Channok (2003) present analytic and numerical models of finite duration shock acceleration. For a given injection momentum po, after a very short time there is only a small boost in momentum, at intermediate times the spectrum is a power law with a hump and steep cutoff at a critical momentum, and at longer times the critical momentum increases and the spectrum approaches the steady-state power law. The composition dependence of the critical momentum is different from that obtained for other cutoff mechanisms. The spectral form of Ellison and Ramaty (1985), a power law in momentum with an exponential rollover in energy, has proved very useful in fitting spectra of solar energetic particles. The composition dependence of the rollover energy depends on the physical effect that causes the rollover. For interplanetary shocks traveling well outside the solar corona, observations typically indicate a rollover at ~ 0.1 to 1 MeV/nucleon. Ruffolo and Channok (2003) consider what physical mechanism could explain this. If there is a cutoff for k/u on the order of the shock thickness (Ellison and Ramaty, 1985), where k is the parallel diffusion coefficient and u is the fluid velocity along the field, the observed long mean free paths for pickup ions (Gloeckler et al., 1995) would imply an extremely low cutoff energy. On the other hand, a cutoff owed to shock-drift acceleration across the entire width of a shock (such as that inferred for anomalous CR) is of the order of hundreds of MeV per unit charge. Ruffolo and Channok (2003) propose that the physical origin of such rollovers is the finite time available for shock acceleration. The typical acceleration timescale tac corresponding to observed mean free paths is of the order of several days, so the process of shock acceleration at an interplanetary shock near Earth should usually give only a mild increase in energy to an existing seed particle population. Indeed, the analyses of ACE observations argue for a seed population at substantially higher energies than the solar wind (Desai et al., 2003). On the other hand, finite duration shock acceleration should yield the standard power-law spectrum in the limit of a long duration t relative to the acceleration timescale. As a corollary of this idea, for an unusually strong shock (unusually short acceleration timescale) it is possible to obtain power-law spectra up to high energies (e.g., as observed by Reames et al., 1997). Therefore, Ruffolo and Channok (2003) derives a simple theory of finite duration shock acceleration and explores implications for the composition dependence of the spectrum. They consider a combinatorial model of finite duration shock acceleration assuming a constant acceleration rate ra (i.e., the rate of a complete cycle returning upstream, or 1/At of Drury, 1983) and a constant escape rate re . After a time t the distribution of residence time T is
P(T) = re exp(- reT) + exp(- ret)S(T -1). (4.21.84)
The Poisson distribution of the number of acceleration events n during T is
The overall probability of n acceleration events is t r ( r Y+1 - (t(r + r ))
P(n,t)= JP(n,T)P(T)dT = exp(-t(ra + re)) £ KKa e>>
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