3. The test case of acceleration at a single shock
We consider an infinite 1D plasma in which particles propagate with diffusion coefficient . The flow velocity of the plasma is constant for (the upstream region) and equal to in the shock restframe. Similarly, in the downstream region, , the velocity is constant, , where r is the compression ratio of the shock. Position, time and momentum are normalised to the diffusion length and time scales upstream: (, and respectively) and the injection momentum of a particle . Thus we define the following dimensionless variables and coefficients:
where controls the synchrotron losses, and is the Péclet number, since the shock transition is confined to the region .
Using the scheme (15,16) we have simulated trajectories in shocks of different . Each simulation runs with a given number of particles N, injected at momentum , for a given computation time . Particles can escape at by crossing a boundary at . The particle distribution is measured at the shock front. Each crossing with an initial momentum increases by unity the differential logarithmic distribution in the bin of momentum bracketing .
It is easily seen from Eq. (16) that for large values of occur if the advective and diffusive steps almost cancel: . In this case the result is particularly sensitive to our assumption that the trajectory between the points and can be interpolated linearly. A partial solution to this problem is to reduce the time step which automatically decreases the probability of choosing a diffusive step which cancels the advective step. However, this procedure increases the computation time. The problem can be avoided completely by replacing the Wiener process by a different random process possessing the same mean and variance, but which effectively prevents the rare steps with . For a large number of trials, the random step produced by any such distribution tends to the Gaussian form of W. The simplest choice is to take with equal probability for each sign.
We first test this prescription in the most difficult case of infinitely thin shocks where , in which case the profile shows a discontinuous change in velocity between the up and downstream regions.
Fig. 1 gives the stationary spectrum obtained with the modified scheme described above. We found over at least two decades of energy a stationary power-law distribution function of index in the case of strong shocks. Larger computation times are needed to reach the same accuracy at larger momentum. A test against the analytic solution gives a value .
We consider now the case of a shock with Péclet number . We compare our results with the semi-analytical derivation of the spectral index by Schneider & Kirk (1987). The simulations here use a linear velocity profile
For larger Péclet number, the particles experience a smaller velocity jump at each step. The Fermi process is thus less efficient, leading to softer stationary distributions. For a Péclet number of we get an index of (see Fig. 2) in good agreements with the results of Schneider & Kirk (1987). The discrepancies seen at small momenta are not statistical errors, but arise because our method assumes zero particle flux in space at the injection point, whereas the scale-independent power-law distribution implies a finite diffusive flux. This `transient' effect disappears at momenta slightly above that of injection.
3.1. Synchrotron losses
Synchrotron losses in the diffusive shock acceleration process have been investigated analytically by Webb et al. (1984). Their main effect is to soften the spectrum at momenta greater than where the loss rate equals the acceleration rate. The inclusion of loss terms in the scheme modifies the way the momentum gain is calculated. Returning to Eq. (7), and using the linear interpolation of the trajectory (14) we arrive at the ordinary differential equation
where the first-order Fermi term () and the loss term are
This solution is exact for the linearly interpolated trajectory (14). Having determined from Eq. (15), the new value of the momentum is given by Eq. (21) with .
In Fig. 3, we show the results of this implicit scheme. Following Webb et al. (1984) and KA94, we define upstream and downstream synchrotron coefficients (), given by . This gives for the characteristic momentum , with . For a compression ratio , we have and , and the distribution cuts-off for (i.e., ). The solution showed in Fig. 3 is in good agreement with the analytical result of Webb et al. (1984).
The implicit scheme is, of course, much faster than the explicit scheme for flows with small Péclet numbers, since the latter are accurate only when , whereas for the former suffices (see Table 1). However, it is also interesting to compare the implicit and explicit schemes in flows with large Péclet numbers, where both should be accurate. In this case, the explicit and implicit schemes give similar results for single shocks for . For larger time steps the particles tend to be advected prematurely from the acceleration zone and both schemes show an unphysical softening of the spectrum. For a given , the explicit scheme is faster by a factor , since all quantities entering into the computation of the position and momentum increments are calculated only once per time step.
Table 1. Summary of the results of the explicit and implicit schemes. The power-law index of the distribution is compared with the value given by an analytic calculation (Schneider & Kirk 1987).The typical error on the index estimates is of order of .
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999