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Fig. 4.24.1. Phase space distributions f (p) multiplied on p4 for a Monte Carlo model

Fig. 4.24.1. Phase space distributions f (p) multiplied on p4 for a Monte Carlo model

(solid curves) and a semi-analytical model (dashed curves). In the left panel v^^es = 0, and in the right panel vfjh^ = 4u2 in the Monte Carlo model. Here, u2 = uo/r is the downstream plasma speed in the shock frame, is the shock speed, and the Mach numbers, compression ratios, and shocked temperatures are indicated. In all cases, the shock is parallel and Alfven heating is assumed in the shock precursor (Ellison et al., 2000). From Ellison et al., 2005).

(ii) The shocked temperature depends on vMCes with weaker injection (i.e., larger vthres) giving a larger TDS .

(iii) The distribution f (p) is harder near pmax in the Monte Carlo results than with the semi-analytical calculations. A common prediction of semi-analytical

models is that the shape of the particle spectra at pmax has the form ^ p ' if the shock is strongly modified and the diffusion coefficient grows fast enough in momentum. This can be demonstrated by solving the equations in the extreme case of a maximally modified shock and approximating the spectrum with a power law at high momentum. The Monte Carlo model makes no such power-law assumption.

(iv) The minimum in the p4 f (p) plot occurs at p > mpC in both models. The transition between f (p) softer than p~4, and f (p) harder than p~4 varies with shock parameters and increases as pmax increases. This is an important difference from the algebraic model of Berezhko and Ellison (1999) where the minimum is fixed at mpc.

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