Kang and Jones (2003) studied the CR injection and acceleration efficiencies at cosmic shocks by performing numerical simulations of CR modified, quasi-parallel, shocks in ID plane-parallel geometry for a wide range of shock Mach numbers and pre-shock conditions. According to the diffusive shock acceleration theory populations of CR particles can be injected and accelerated to very high energy by astrophysical shocks in tenuous plasmas (Malkov and Drury, 2001), and a significant fraction of the kinetic energy of the bulk flow associated with a strong shock can be converted into CR protons (Kang and Jons, 2002; Kang, 2003). Kang and Jones (2003) developed a numerical scheme that self-consistently incorporates a 'thermal leakage' injection model based on the analytic, nonlinear calculations of Malkov (1998a,b). This injection scheme was then implemented into the combined gas dynamics and the CR diffusion-convection code with subzone shock-tracking and multi-level adaptive mesh refinement techniques (Gieseler et al., 2000; Kang et al., 2002). Kang and Jones (2002), Kang (2003) applied this code to studying the CR acceleration at shocks by numerical simulations of CR modified, quasi-parallel shocks in 1D plane-parallel geometry with the physical parameters relevant for the e cosmic shocks emerging in the large scale structure formation of the Universe. In Kang and Jones (2003) are presented new simulations with a wide range of physical parameters. They calculated the CR acceleration at 1D quasi-parallel shocks that were driven by accretion flows in a plane-parallel geometry. Two sets of models are presented: 1) To = 104K and uo = (15km/s)Mo and 2) To = 106K and uo = (150km/s)Mo, where To, uo, and Mo = 5-50 are the temperature, accretion speed, and Mach number of the accretion flow, respectively. The Bohm diffusion model was adopted for the CR diffusion coefficient

where the particle momentum is expressed in units of mpC . The choice of Ko is arbitrary, since Kang and Jones (2003) present the results in terms of the diffusion time and length scales defined by to = Kofu^2 and xo = Ko/uo . The gas density normalization constant, po, is arbitrary as well, but the pressure normalization constant depends on M o as

They adopted an injection scheme based on a 'thermal leakage' model that transfers a small proportion of the thermal proton flux through the shock into low energy CR (Malkov, 1998a,b; Gieseler et al., 2000). This model has a free parameter, e = Bo/B, defined to measure the ratio of the amplitude of the post-shock MHD wave turbulence B^ to the general magnetic field aligned with the shock normal, Bo (Malkov, 1998a,b). In Kang and Jones (2003) are presented models with e = 0.2 only.

The injection efficiency in Kang and Jones (2003) are defined as the fraction of particles that have entered the shock from far upstream and then injected into the CR distribution:

£(t ) = J dx J 4nfcR (p, x, t )p 2dp/( noUsdt), (4.21.96)

where fcR (p, x, t) is the CR distribution function, no is the particle number density far upstream, and us is the instantaneous shock speed. As a measure of acceleration efficiency they define the 'CR energy ratio', namely the ratio of the total CR energy within the simulation box to the kinetic energy in the initial shock frame that has entered the simulation box from far upstream,

where uso is the initial shock speed before any significant nonlinear CR feedback occurs. Although the sub-shock weakens as the CR pressure increases, the injection rate decreases accordingly and the sub-shock does not disappear. Kang and Jones (2003) found that the post-shock CR pressure reaches an approximately time-asymptotic value and the evolution of the CR shock becomes 'self-similar' owing to a balance between fresh injection/acceleration and advection/diffusion of the CR particles away from the shock. So the CR energy ratio O also changes asymptotically to a constant value, as shown in Fig. 4.21.4 (except in the model with To = 106 K and Mo = 50 which has not reached the time-asymptotic state up to t/to = 40). The time-asymptotic value of O increases with Mo, but it converges to ® « 0.5-0.6 for Mo > 20.

To = 104 K, while right panels for models with To = 106 K. Accretion speed of each model is given by uo = (15km/s) Mo for models with To = 104 K and by uo = (150km/s) Mo for models with To = 106 K. Time is given in terms of the diffusion time to =Koj u02 MO

According to Kang and Jones (2003).

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