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levels. This grid spacing is much larger than the diffusion length for pmin = 10 , ¡dif = 2.5 x 10—9, which is contrary to what was found in previous simulations in

Eulerian grid (Kang et al., 2001). The faster convergence at lower resolution seems to result from the fact that the shock stays in the same grid zone in the comoving frame. It was used 230 uniformly spaced logarithmic momentum zones in the interval log(p/mc) = [log pmm,log Pmax ]=[" 3.0,+6.0]. (4.27.8)

The CR modified shock structure and the CR momentum distribution inside the simulation box,

The density in the precursor, p, and in the post-shock region, P2, immediately before and after the sub-shock, respectively, are shown in the top panel of Fig. 4.27.2. In Fig. 4.27.2 the middle panel shows the Mach number of the sub-shock, while the bottom panel shows the CR pressure and gas pressure in units of

the ram pressure of unmodified Sedov-Taylor similarity solution, pousj x t .

Kang and Jones (2005a) note the following important observations from Fig. 4.27.1 and Fig. 4.27.2:

1. The CR protons are accelerated to the proton knee energy in the spectrum of galactic CR (about 1014 —1015 eV, p/mc = 105 —106) in several thousand years, as expected from the standard estimate (Lagage and Cesarsky, 1983).

2. The ratios of both post-shock Pc and Pg relative the shock ram pressure approach to time-asymptotic values quickly. The post-shock Pc is about 50 % of the shock ram pressure, while the gas pressure takes only 20 %.

3. Both the CR momentum distribution at the shock, g (rs, p ) = f (rs, p )p4, and the integrated distribution, G (p ), exhibit characteristic concave curvature, reflecting the nonlinear velocity structure in the precursor.

Fig. 4.27.1. Evolution a typical supernova remnant expanding into the uniform interstellar

medium. The model parameters are Eo = 10 ergs, Msn = 10MSun, nH = 3 x 10 cm , with Pco = 0.5Pg and = 5. It assumes a preexisting CR population of f (p)x p 4 5, y,±^ ^ co go.

but thermal leakage is not included. The lower left panel shows the integrated particle spectrum according to Eq. 4.27.9. The time t = 1 corresponds to 6100 years. The initial condition at t/to = 1.0 (solid line) is set by the Sedov-Taylor similarity solution. According to Kang and Jones (2005a).

Fig. 4.27.2. Pre-shock density, px, post-shock density, p2, the shock Mach number, Ms, the post-shock CR Pc2 and gas pressure Pg2 in units of the ram pressure of Sedov-Taylor 2 _1 2

solution, pouST x t ■ . According to Kang and Jones (2005a).

Fig. 4.27.2. Pre-shock density, px, post-shock density, p2, the shock Mach number, Ms, the post-shock CR Pc2 and gas pressure Pg2 in units of the ram pressure of Sedov-Taylor 2 _1 2

solution, pouST x t ■ . According to Kang and Jones (2005a).

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