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Astron. Astrophys. 364, 911-922 (2000)

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5. Results for weak shocks

When the initial compression ratio decreases for a weak shock, the injection process is influenced in several ways by the change in the plasma and magnetic field properties. To investigate the effects of a lower compression ratio and lower Mach number on the injection process we will consider an example with [FORMULA] and [FORMULA]. At such a shock, the phase space for which the downstream particles can re-cross the shock to upstream is decreased compared to the strong shock case, because the shock velocity in the downstream rest frame [FORMULA] is inversely proportional to the compression ratio. At the same time the plasma is heated less, because the transformation of kinetic energy to thermal energy depends also on the compression ratio; [FORMULA]. This shifts the downstream Maxwell distribution to lower energies, as compared to higher compression, and, therefore, influences strongly the number of particles in the momentum range making the potential injection pool. On the other hand, at quasi-parallel shocks, the amplitude of the magnetic field wave spectrum [FORMULA] is amplified downstream by the factor r. For a decreasing compression ratio, the downstream plasma becomes more transparent. This balances the effects of the phase space and temperature changes described above. The initial downstream (inverse) wave-amplitude [FORMULA] was calculated to be in the interval [FORMULA] in the limit of strong shocks (Malkov 1998; Malkov & Völk 1998). An extrapolation to weak shocks with [FORMULA] of this interval by multiplying [FORMULA] with the factor of (4/2.5) gives [FORMULA]. However, the calculation of the transparency function was based on the assumption of an high amplitude wave spectrum downstream ([FORMULA]). With decreasing wave amplitude the velocity dependence of the transparency function changes towards its asymptotic function, defined by particle kinematics without a wave field: [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]. On the other hand, this limit may be reached in reality only if the resulting beam from downstream to upstream is too weak to produce a magnetic field instability.

As an initial exploration of this behavior, we will present here results for the spatial and momentum distributions and the energy and particle injection efficiency for an inverse magnetic fields amplitude parameter [FORMULA] in the range [FORMULA]. We have included the value [FORMULA] to compare the results directly to the strong shock case. This can demonstrate the principal effect of weaker shocks on the injection process. The resulting injection efficiencies and shock modifications for all values of [FORMULA] shown here should be considered as lower limits for the weak shock with [FORMULA] ([FORMULA]) as described above.

The physical scales are specified as follows: [FORMULA] s, [FORMULA] cm, [FORMULA], [FORMULA], [FORMULA]. We use [FORMULA] for the simulations presented here, and a magnetic field of [FORMULA]G. The initial values for the [FORMULA] case are [FORMULA], [FORMULA], and [FORMULA] in the upstream region, while [FORMULA], [FORMULA], and [FORMULA] in the downstream. We have used 44032 uniform grid zones for [FORMULA], with the shock initially at [FORMULA], and 128 uniform grid zones in [FORMULA] for [FORMULA]. The corresponding Mach number is [FORMULA].

Fig. 8 shows the normalized gas density [FORMULA], gas pressure [FORMULA], plasma velocity [FORMULA] and the cosmic-ray pressure [FORMULA] over the spatial length x, for different times. Because the resulting non-thermal spectrum produced as a result of the injection and particle acceleration is steeper than in the strong shock case, the pressure [FORMULA] in this distribution remains small compared to the gas pressure at all times. As a result, the shock is modified only slightly. Also the temperature of the downstream plasma remains almost constant. Furthermore, because the energy density in non-thermal particles is not an increasing function in time, the shock modification can reach a steady state earlier, as compared to the strong shock case. In fact, at time [FORMULA], shown in Fig. 8, the pressure [FORMULA], [FORMULA], the velocity u and the density [FORMULA] immediately downstream has reached almost a steady state.

[FIGURE] Fig. 8. Gas density [FORMULA], pressure [FORMULA], velocity [FORMULA], and cosmic-ray pressure [FORMULA], at times [FORMULA] (dotted), [FORMULA] (dashed) and [FORMULA] (solid line). The shock Mach number is [FORMULA], [FORMULA], and [FORMULA]. The initial upstream gas pressure is [FORMULA].

The downstream momentum distribution in Fig. 9 shows clearly the steeper spectrum of the non-thermal part, which asymptotes to the standard result [FORMULA] with [FORMULA] for [FORMULA]. It can be seen also, that the thermal part of the distribution is not as much modified as in the strong shock case (compare Fig. 5). Because the modification of the transparency function over time depends only on changes in the downstream plasma velocity, it remains essentially unchanged.

[FIGURE] Fig. 9. Phase-space density [FORMULA] vs. proton momentum immediately downstream of the sub-shock. Also shown is the transparency function [FORMULA]. Both functions are presented for [FORMULA] (dotted), [FORMULA] (dot-dashed), [FORMULA] (dashed), and [FORMULA] (solid line). For the parameters used see Fig. 8 and corresponding text.

The energy efficiency [FORMULA], as defined in Eq. (22), is lower roughly by a factor of two compared to the strong shock case, because of the steeper non-thermal spectrum and the resulting energy density (compare Fig. 7 and Fig. 10). Our results for the wave amplitude, [FORMULA], give the injection efficiency, [FORMULA] at time [FORMULA]s, where the time evolution can be considered as almost a steady state. The number of particles, which are in the non-thermal part is comparable to the strong shock considered above at this time. In addition, we point out that the application of the above described injection model to weak shocks is an extrapolation, and we believe would yield lower limits on the injection efficiency.

[FIGURE] Fig. 10. Energy efficiency [FORMULA] and the fraction of cosmic-ray particles [FORMULA] for four values of the inverse wave-amplitude [FORMULA] at a weak shock. For the parameters used see Fig. 8 and corresponding text.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001