## 5. Results for weak shocksWhen 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
and . 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
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; . 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 is
amplified downstream by the factor 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 in the range . We have included the value 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 shown here should be considered as lower limits for the weak shock with () as described above. The physical scales are specified as follows: s, cm, , , . We use for the simulations presented here, and a magnetic field of G. The initial values for the case are , , and in the upstream region, while , , and in the downstream. We have used 44032 uniform grid zones for , with the shock initially at , and 128 uniform grid zones in for . The corresponding Mach number is . Fig. 8 shows the normalized gas density
, gas pressure
, plasma velocity
and the cosmic-ray pressure
over the spatial length
The downstream momentum distribution in Fig. 9 shows clearly the steeper spectrum of the non-thermal part, which asymptotes to the standard result with for . 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.
The energy efficiency , 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, , give the injection efficiency, at time 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.
© European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |