4. Results for strong shocks
First we consider a strong shock with an initial Mach number of . Unlike an ordinary hydrodynamic simulation, the simulation of the CR shock acceleration requires specification of three physical parameters, , , and the shock Mach number in addition to the diffusion coefficient. We adopted the following nominal physical scales for physical parameters: , , s, cm, . We use for the simulations presented here, and a magnetic field of G. The initial conditions are specified as follows: , , and in the upstream region, while , , downstream. These values reflect the shock jump conditions in the rest-frame of the shock.
We define the diffusion length and time at a given momentum as and . For a proper convergence, the spatial grid size should be smaller than the diffusion length of the injection pool particles ( in upstream for ). On the other hand, the spatial region of the calculation in upstream and downstream should be larger than the diffusion length-scale of the particles with the highest energies reached at the end of our simulation period ( for ). So we used 51200 uniform grid zones for , with the shock initially at and the grid size . We use 128 uniform grid zones in for . We integrate the solutions until which corresponds to for , so the CR particles became only mildly relativistic by the end of our simulations.
4.1. Dynamical evolution
Fig. 4 shows the normalized gas density , gas pressure , plasma velocity and the cosmic-ray pressure over the spatial length x, for different times. This shows clearly the basic features of the shock modification by a diffusive component; that is, the adiabatic precursor compression and the sub-shock. The CR pressure is responsible for the deceleration and compression of the plasma flow in the precursor region upstream of the sub-shock, which still remains strong. As a result, the gas is compressed to higher density downstream of the sub-shock.
The cosmic-ray pressure immediately downstream of the sub-shock has not reached a steady state yet. The reason is that for a the non-thermal particles with a momentum distribution with , the energy density is an increasing function of . This applies even if the injection is shut down completely, like for an -function type injection in time, as shown by Drury (1983). We expect that will continue to increase after our integration time , which leads to a significant modification of the shock structure and to the steepening of the power-law distribution of suprathermal particles. The simulations of such non-linear evolution, however, require much greater spatial region and grid zones and also longer integration time than what we could afford in our simulations.
In real astrophysical shocks, the energy density is limited by radiation losses in the case of electrons or more generally by particle escape due to the finite extent of the acceleration region. For the maximum energy of particles () achieved by in this simulation, neither effect is important, and, therefore, not included.
4.2. Energy distribution
The phase-space distribution immediately (three zones) behind the sub-shock is shown in Fig. 5 for three different times. Initially this distribution is given by a Maxwell distribution, as shown by the dotted line. At the thermal part of the distribution the cooling of the postshock gas due to the energy flux into the CR particles is responsible for the shift of the Maxwellian distribution towards lower energies. We have also plotted the transparency function at the same simulation times. According to Eq. (16) the injection rate into the non-thermal distribution depends on overlap of and that determines the injection pool. One can see that initially the injection rate is high and so the postshock gas cools quickly, resulting in narrowing down of the injection pool. This causes the injection rate to decrease. But then the transparency function also shifts toward lower momenta, because the downstream plasma velocity decreases as the postshock gas cools. The combination of the shift of toward lower momenta and the decrease of the particles in the Maxwellian tail due to the gas cooling leads to the self-regulation of the injection rate at a quite stable value. According to the plot of at and , the Maxwell distribution turns into a power-law at an almost constant "effective injection momentum" which determines the magnitude of the CR distribution function at a stable value (about of the thermal peak). The value of this constant effective injection momentum can be translated into the parameter (where ) defined by Kang & Jones (1995). But this is somewhat larger than what they used ().
The narrow injection pool also leads to a rather sharp transition from the Maxwell distribution to the non-thermal part starting shortly above the effective injection momentum (see Fig. 5). The canonical result in the test particle limit, constant, for a strong shock with is reproduced very well in our simulations. The same energy spectrum is shown in Fig. 6 in the form of the omni-directional flux vs. proton kinetic energy downstream of the shock normalized to . At energies above the injection pool we expect, for the strong shock () simulated here, the result , with , which is reproduced with high accuracy.
In using the standard cosmic-ray transport equation, we have, of course, made use of the diffusion approximation, which may introduce an error especially for . Using an eigenfunction method, Kirk & Schneider (1989) have explicitly calculated the angular distribution of accelerated particles and accounted for effects of a strong anisotropy especially at low particle velocities. They were able to calculate the injection efficiency without recourse to the diffusion approximation, and found always lower efficiencies compared to those in the diffusion approximation. Using the initial thermal distribution, we have estimated an effective injection momentum from the peak of the distribution function, . For the shock parameters considered here and for we get an effective initial injection velocity of about (in the shock frame). For this injection velocity, and , they estimate a reduction effect of , leaving the diffusion approximation as quite reasonable even in this regime.
4.3. Injection and acceleration efficiencies
To describe the injection efficiency often a parameter is used for the fraction of the in-flowing plasma particles that are instantaneously accelerated to a fixed injection momentum (e.g. Falle & Giddings 1987; Dorfi 1990; Jones & Kang 1990; Zank et al. 1993; Berezhko et al. 1994). The injection energy flux I transferred to CRs is then given by
where is the upstream plasma velocity in the shock frame, and is the upstream density. From the fact that the injected energy flux I must be equal to the spatial integral of the injection energy loss term , that is, and by assuming momentarily a step function for the transparency , we get:
where is the upstream number density. This is equivalent to the injection parameter used by Kang & Jones (1995). The so-defined injection parameter is, however, not an exact measure of the number of particles contributing to the population of cosmic rays, because the acceleration process cannot be described by shifting particles instantaneous from thermal energies to an injection momentum . Furthermore, depends strongly on the chosen injection momentum , which is not a fixed single parameter in our numerical simulation.
A method to measure the injection efficiency without specifying the injection momentum, is to compare the number of particles in the CR part to the number of particles swept through the shock. According to our definition of the CR population we can write for the CR number density
The fraction of particles that has been swept through the shock after the time t, and then injected into the cosmic-ray distribution is then given by
The time development of this injection efficiency is plotted in Fig. 7 for three values of the inverse wave amplitude . Recall that Malkov (1998) found . In the very beginning of the simulation the injection does depend strongly on the wave-amplitude, because of the very steep dependence of the Maxwell distribution at the injection energies. However, as described above, a strong initial injection leads to a temperature decrease of the plasma, and to a shift of the Maxwell distribution, which balances this effect. Therefore at later times the fraction of injected particles, , does not depend strongly on the initial wave-amplitude. At time (or s) we get a fraction of injected particles of for the interval .
To measure the efficiency of the particle acceleration at a shock front, we compare the energy flux in cosmic rays to the total energy which is available from the downstream plasma flow. This energy consists of the sum of kinetic energy and the gas enthalpy. The fraction of this initial energy flux, which is transferred to CRs is given by
where is the initial downstream plasma velocity in the upstream rest frame. The definition of the efficiency is similar to the definition from Völk et al. (1984). However, Eq. (22) compares the energy flux in CRs not only to the kinetic energy flux of the gas, but also includes the gas enthalpy flux. We measure the CR pressure immediately downstream of the sub-shock, where it will first reach the constant downstream value, in case a steady state does exist (see below). The time dependent values are averaged over the interval in the shock frame to avoid influence of small scale modifications of the cosmic-ray pressure and plasma velocity on the injection efficiency. When the quantities , and have reached steady-state distributions downstream of the sub-shock, is also no longer time dependent.
The evolution of the energy efficiency, , is plotted in Fig. 7 for three different magnetic-field wave amplitudes. See Fig. 4 and the description in Sect. 4 for the corresponding parameters. The case corresponds to the highest injection efficiency and therefore leads to the highest cosmic-ray pressure. To assure a vanishing value of the cosmic-ray pressure at the spatial grid boundaries at all times, the calculation for was done on a somewhat larger grid with 60416 uniform zones for . For the value , which was calculated by Malkov (1998), we see that about 20% of the available energy in this shock is transferred into the cosmic-ray population. The acceleration efficiency has, however, not reached a real steady state value, but is increasing with with . The acceleration efficiency achieved by this time is given by for . Thus a substantial amount of the initial energy flux at a shock front can be transferred to a high energy part of the distribution, during the relatively short time we have simulated here.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001