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Fig. 4.24.2. Low energy portions of the spectra shown in Fig. 4.24.1 where the solid histograms are the Monte Carlo results and the dashed curves are the semi-analytical results. The heavy dotted histograms show the distribution of particles that were injected in the Monte Carlo model. The solid vertical lines are drawn at pjnj, the momentum at which particles are injected in the semi-analytical model. For vtflCes = 0, nMc = 5.7x 10—2 and nSA = 4.6 x 10—2 , while for vMiis = 4m2 , nMC = 4.1 x 10—3 and nSA = 2.5 x 10—3 . The lightweight dotted curve shows the Maxwellian (TDs = 2.2 x 109 K) that would have resulted if no diffusive shock acceleration occurred. According to Ellison et al., 2005.

(v) For the particular parameters used in these examples, the overall compression ratio is relatively insensitive to vthres, but shocks having other parameters may show a greater sensitivity. Note that Alfvén wave heating is assumed in all of the results presented here. If only adiabatic heating was assumed, the compression ratios would be much higher (in agreement with Ellison et al., 2000).

(vi) In all cases, the Monte Carlo model injects more particles than the semi-analytical model but the average energy of the injected particles is less, as indicated by the peak of the curve labeled 'MC inj' vs. the 'SA inj' energy in Fig. 4.24.2.

(vii) In contrast to vMCs = 0, the 'thermal' part of f (p) with vt^^es = 4«2 (Fig. 4.24.2) shows large differences in the two models. While both conserve particle, momentum, and energy fluxes so that the broad-band f (p) matches well for a wide range of parameters, the different treatments of the sub-shock lead to large differences in the critical energy range 2 < E/kT^s < 5. This offers a way to distinguish these models observationally.

Using two approximate acceleration models, Ellison et al. (2005) have shown that the most important features of non-linear diffusive shock acceleration, i.e., at r >> 4 and concave spectra, are robust and do not strongly depend on the injection model as long as injection is efficient. If injection is weak, as might be the case in highly oblique shocks, accelerated spectra will depend more on the details of injection, at least in the transition range between thermal and super-thermal energies. Also, the relative efficiencies for injecting and accelerating electrons vs. protons or protons vs. heavier ions may require a more detailed description of injection, as may be provided by future PIC simulations.

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