Fig.4.29.5. The value c2 vs. time for a 3D random walk with isotropic injection at the origin at t = 0. Step-lengths l were sampled from an exponential distribution with mean l followed by small-angle scattering with scattering angle 8 sampled from an exponential distribution with mean 9 (the numbers attached to the curves). The dashed line is c2 = 2tvl/3d2 .

Curves for 5°-20° result from 104 simulations; 4° curve results from 8 x 104 simulations (width shows statistical error). According to Protheroe et al. (2003).

For this case it can be seen from Fig. 4.29.5 that a2 ^ 2tvlj362 , and so it can be obtain the connection between small-angle scattering and diffusion theory, namely,

As viewed in the frame of reference of the upstream plasma, ultra-relativistic particles are only able to cross the shock from downstream to upstream if the angle 6 between their direction and the shock normal pointing upstream is

For highly relativistic shocks these particles cross the shock from downstream to upstream traveling almost parallel to the shock normal. Similarly, having crossed the shock, only a very slight angular deflection, by ~ 1/ysh is sufficient to return them downstream of the shock. This change in particle direction gives rise to a change in particle energy E' and momentump', measured in the downstream plasma frame (primed coordinates), of

in an acceleration cycle (downstream ^ upstream ^ downstream), where /3n is the speed of the upstream plasma as viewed from the downstream frame. In 'parallel shocks' the magnetic field is parallel to the shock normal, and so the pitch angle y is the angle to the shock normal and vcosy gives the component of velocity parallel to the shock. Thus the small-angle scattering method described above is used here to simulate particle motion upstream of a parallel relativistic shock, including the effects of pitch-angle scattering, for a given diffusion coefficient. Ultra-relativistic particles were injected at the shock with downstream-frame energy E'o travelling upstream parallel to the shock's normal, i.e., 60 = 0. Let us follow a particle's trajectory until the shock catches up with it, and it crosses from upstream to downstream with an upstream-frame angle 61 to the shock's normal and a downstream-frame energy E '1. The simulation was performed for ysh= 10 and five different mean scattering angles 6 to determine the maximum 6 -value that can safely be used for accurate simulation. The resulting distributions of cos 6 and log(('1/E0) are shown in Fig. 4.29.6.

Fig. 4.29.6. Small-angle scattering simulation of excursion upstream in diffusive shock acceleration at a parallel relativistic shock with ysh = 10. Results are shown for 105

injected particles and 6 = 10-2/ysh (top histogram), 3 x 10-2/ysh , 10-1/ysh , 0.3/ysh , 1/Ysh and 3/ysh (bottom histogram). Note that the top three histograms are almost indistinguishable. According to Protheroe et al. (2003).

Fig. 4.29.6. Small-angle scattering simulation of excursion upstream in diffusive shock acceleration at a parallel relativistic shock with ysh = 10. Results are shown for 105

injected particles and 6 = 10-2/ysh (top histogram), 3 x 10-2/ysh , 10-1/ysh , 0.3/ysh , 1/Ysh and 3/ysh (bottom histogram). Note that the top three histograms are almost indistinguishable. According to Protheroe et al. (2003).

From Fig. 4.29.6 it can be seen that in this application one requires 6 < 0.1/Ysh . The results described are quite consistent with those of Achterberg et al. (2001), who used a diffusive angular step A6st < 0.1/ysh . Protheroe et al. (2003) came to the conclusion that the standard Monte Carlo random walk approach to the simulation of energetic charged particle propagation for a given spatial diffusion coefficient D can be extended to apply accurately to times much less than ff v = 3^ v2 by using a small-angle scattering procedure with steps sampled from an exponential distribution with mean free path / = 6 2f followed at each step by scattering with angular steps sampled from an exponential distribution with mean scattering angle 6 < 0.09 rad (5°). The spatial and angular diffusion coefficients are then D =Jv/(36) and D6^02 v/(2/), and are related by D = v2/(6D6). In the simulation of upstream propagation in relativistic shock acceleration one must use 6 < 0.1/Ysh to obtain accurate results.

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