where W is the width of the shock and x are axis points in the direction of the flow in the shock frame; U2. A modified profile:

obtained by fitting the results of a self-consistent Monte Carlo simulation of shock structure (Vainio, 2003). Here H(x) is the step function. To make the two models comparable, the shock width W was adjusted so that the transition from upstream (uj) to downstream (u2) values takes place over the same distance in both models. For this, it was used the width of the region where u(x) is in the range u1-Su < u < u2 + Su with Su = 0.01 c. (4.29.10)

This gives

Virtanen and Vainio (2003a) used the value of 10 for the upstream bulk

Lorentz factor, and U1U2 = c2 /3. Electrons are injected into the acceleration process in the downstream region.

Two models are considered for the injection energy, E1 and E2. E1. A 'kinematics' injection energy:

i.e., the energy of cold upstream electrons as seen from the downstream gas. E2. A 'thermalized' injection energy:

i.e., a fraction of proton thermal energy in the downstream region (a = 1 corresponding to equal-partition). Virtanen and Vainio (2003a) used a = 0.2.

The results of the simulated electron spectrum for all eight models (Q1E1, Q1E2, Q2E1, Q2E2 for two velocity profiles, U1 and U2) are plotted in Fig.

4.29.4, which shows that the two velocity profiles produce different results in the case Q2: for the injection E1, the speed profile U1 produces a significantly harder spectrum than the speed profile U2, and the results are slightly different even for the injection model E2. The reason for the differences is probably that U1 has a larger maximum value of the speed gradient than U2. The fact that the results are so similar in the case Q1, however, indicates that the adjustment of the shock width for U1 is reasonable.

Fig. 4.29.4. The energy spectra of accelerated electrons in parallel relativistic shocks with finite thickness. Solid and dashed curves correspond to speed profile U1 and U2, respectively. See text for a description of the different models. According to Virtanen and Vainio (2003a).

From Fig. 4.29.4 can be seen that the difference between the two turbulence models Q1 and Q2 is significant. The spectrum in the case Q1 is a power law with a spectral index of ~ 3.2 independent of the injection energy, as expected. The spectral shape in the case Q2 is not a power law but hardens as a function of energy, because for a mean free path increasing with energy, the shock seems thinner for electrons at higher energies. In the case E1 the Kolmogorov scattering law Q2 produces accelerated particles much less efficiently than in the case of an energy-independent mean free path. The thermalized injection E2 yields accelerated particle populations in both turbulence models. At the highest energies, the spectral index in the Q2E2 model approaches the value of 2.2 obtained for a step-like shock at TjĀ» 1 (Kirk and Duffy, 1999). Virtanen and Vainio (2003a) came to the conclusion that electron acceleration in parallel relativistic shock waves with nontrivial internal structure is heavily dependent on the rigidity dependence of the particle's mean free path. For a shock thickness determined by ion dynamics and a mean free path increasing with energy the standard power-law electron spectra can be obtained only at very high energies, e.g., at Y > 105 for A ^ y13 .

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