The average injection rate varies in the interval % ~ 10 +10 ' , depending on Mo, uo and e. For two models with the same Mach number but different speeds (or different To ) the injection rate is higher for models with higher speeds, but the CR energy increases more slowly in terms of the normalized time t/to . Fig. 4.21.5 shows the total CR distribution within the simulation box,

Fig. 4.21.5. CR distribution function integrated over the simulation box, G(^), which is determined by Eq. 4.21.98, and its power law slope, q determined by Eq. 4.21.99, at t/to = 20. The curves are labeled with the accretion Mach number Mo . According to Kang and Jones (2003).

For all models shown in Fig. 4.21.5, G(p) has an exponential cut-off at a similar momentum (pmax = 4) regardless of values of uo , since the results are shown at the same values of t/to = 20. The integrated distributions show the characteristic 'concave upwards' curves owed to nonlinear modification to the shock structures, and this 'flattening' trend is stronger for higher Mo models. Kang and Jones (2003) came to the following conclusions:

1. The CR pressure seems to approach a steady-state value and the evolution of CR modified shocks becomes approximately 'self-similar'.

2. Supra-thermal particles can be injected very efficiently into the CR population via the thermal leakage process, so that typically a fraction of 10 -4 -10 -3 of the particles passed through the shock becomes CR.

3. For a given injection model, the acceleration efficiency increases with the shock Mach number, Ms, but it moves asymptotically to a limiting value of the CR energy ratio, ® ~ 0.5-0.6, for Ms > 30.

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