4. Acceleration at multiple shocks
The subject of multiple shock acceleration has been extensively investigated both analytically and numerically over the past few years (Spruit 1988, Achterberg 1990, Schneider 1993, Pope & Melrose 1993, Melrose & Crouch 1997). Analytic solutions of the diffusion-advection equation including synchrotron losses can be derived if we assume that the time spent by a fluid element between two consecutive shocks is much longer than the acceleration time at a single shock problem (see Schneider 1993). Another way of formulating this condition is that the first-order Fermi acceleration process should be much faster than all other processes, such as escape, decompression and losses. With this hypothesis, a final power-law index can be calculated which takes account of the different generations of particles accelerated at individual shocks. At sufficiently high momentum, this approximation fails, since the time-scale of the losses must eventually become comparable to the acceleration time-scale. The effect of multiple shocks is to increase the acceleration efficiency, by reducing the effective escape rate. In a purely 1D system, the escape rate is formally zero, and the distribution functions tends to . As shown, using spatially averaged equations, by Kardashev (1962) and by Schlickeiser (1984), in the presence of losses, this spectrum extends up to momentum values where the loss effect becomes dominant, and the spectrum piles up at a momentum where the effective acceleration time equals the loss time. More recently, Protheroe & Stanev (1999) (see, however, Drury et al. 1999) have proposed an alternative method of computing the cut-off and pile-up effects in the high energy particle spectrum with various energy dependent diffusion coefficients. They present both an analytical model and a conventional Monte-Carlo simulation (of the same kind as described in the introduction) and show that the effect of the Klein-Nishina regime of inverse Compton losses may modify the results obtained with continuous (synchrotron and inverse Compton in the Thomson regime) losses. In this paper we include neither non-continuous energy losses nor energy dependent diffusion, but postpone an investigation of these effects to future work.
In general for multiple shock system, the picture is somewhat more complicated than depicted in previous analytical or semi-analytical models, since the shocks may be so close together that the acceleration time at a single shock is not short compared to the flow time between the shocks. Also, at high momenta, the synchrotron loss time-scale must be compared not only to the acceleration time at a single shock, but also to the flow time between the shocks.
To investigate these situations, we consider a simple periodic pattern (of period L) including shock fronts and re-expansion regions (see Fig. 4). The flow speed of the pattern () can then be written as
where is the flow speed on the downstream side of the shock.
There are two free parameters (for a given compression ratio) in this problem: the advection time , which is the (dimensionless) time for the fluid to flow through one wavelength of the pattern and is numerically equal to the dimensionless length L, and the loss strength given by , which effectively defines a momentum scale, since the (dimensionless) synchrotron loss time at momentum is
To these we add a third: an escape time , assumed independent of x and p. This can be understood as a crude way of incorporating 2 or 3-dimensional effects into our 1-dimensional simulation, since only in 1 dimension are the particles unable to leave the train of shock fronts. The inclusion of escape effects in the SDE system can be effected by rescaling the number of particles that have crossed the shock with at time by the factor .
There are then two important characteristic times which are important for a discussion of the solutions:
We present, in the following, results for a shock system described by Eq. (24) with a Péclet number . We adopt as typical values of the parameters: compression ratio , escape time , an inter-shock distance , and, in the case of synchrotron losses, (chosen to give a peak momentum of ). All the simulations are run for a time of and with a time step .
4.1. Multiple shock effect: the case without losses
We first consider the case where losses are inefficient (). The results are shown in Fig. 5a. The stationary solution at each shock front (with n an integer) is close to . This confirms the multiple shock effect as an efficient way of producing higher energy particles and harder spectra than isolated shocks. In fact, the stationary index is not exactly 3 but slightly steeper, owing to the finite escape time. Using the general relation between the power-law index of accelerated particles and the escape and acceleration times: , with (Kirk et al. 1994) we find for the effective acceleration time in this particular multiple shock system . Thus, is () in this case.
For momenta lower than that of injection, , the method of Schneider (1993), and Melrose & Crouch (1997), gives a continuation of the power-law down to . This is clearly an artifact of the assumed separation of the acceleration and expansion processes. It does not appear in our simulations, which show instead a hardening to lower frequencies starting at the point . However, because of the transients associated with our method close to the injection momentum, this effect may also be an artifact.
At still lower momentum values, the analytical solution for an escape probability independent of momentum is given by (see Eq. 4.5 in Schneider 1993)
The stationary index is for , but for lower values of the ratio of escape to multiple shock acceleration time, the spectrum hardens, and typically for a ratio , we get , in good agreement (given the accuracy of the time-scales) with the simulations; for down to .
4.1.1. Variations of the escape time
If the inter-shock distance L is kept constant, increasing (decreasing) the escape time leads to a harder (softer) stationary spectra. This is clearly seen in Fig. 5b where we have reduced the escape time by a factor of 2 compared to the fiducial case of Fig. 5. The resulting spectrum has an index of 3.2, which again gives . For momentum , from Eq. (27), the relation gives a spectrum with . We obtained down to an index of . If , the particles escape the system before being advected to the next shock. The stationary solution tends to the single shock result of Sect. 2.1 without losses, which is a power-law spectrum with an index of 4 (for ).
4.1.2. Variations of the inter-shock distance
We now keep constant and equal to the fiducial value of 100. For intershock distances , no multiple shock effect is seen (Fig. 5c). As in the case of short escape time (Fig. 5b), the stationary solution tends to that from a single shock. As decreases, the spectrum hardens and the spectral index can take all values between 3 and 4 (for ). This is consistent with our finding that . If the inter-shock distance is reduced to below one diffusion length , the assumptions normally used to derive the basic transport equation (Skilling 1975) are violated. We have not investigated this regime; although it potentially interesting, since if the distribution remains almost isotropic, the diffusion approximation may in fact remain adequate. For low momentum , the spectrum steepens with decreasing L and tends to .
4.2. Multiple shock effect: the case with synchrotron losses
The stationary spectrum computed with the fiducial parameters is given in Fig. 6a. The inclusion of losses creates a pile-up at a momentum where losses equal gains i.e., . From the simulation, we find this occurs at roughly , which implies close to . This provides an independent check on the estimate made from the results presented in Figs. 5b and 5c. The width of this hump on the low momentum side is determined by the momentum at which , where escape intervenes to overwhelm the effects of synchrotron losses. Since , the hump makes its appearance about one order of magnitude before it peaks.
At momenta higher than the peak of the hump, the losses dominate over all other processes, since there , and, in this example, the acceleration at a single shock is approximately equal to the advection time (see Eq. 26). Consequently, the spectrum cuts off exponentially. This conclusion is consistent with the results of Melrose & Crouch (1997) found using an iterative method in which the synchrotron cut-off is estimated in advance.
4.2.1. Variations of the loss rate
A decrease (increase) of with L and kept unchanged, leads to stronger (weaker) losses, and lower (higher) momenta at the peak of the pile-up. This is confirmed in Fig. 6b, which also demonstrates that the peak momentum is directly proportional to the loss time . The width of the hump is unaffected, since its lower bound also moves in proportion to . Radiative losses are unimportant below the injection momentum, except when the loss rate is so strong () that particle are prevented from being accelerated at all.
4.2.2. Effects of the other parameters
We stress here the effects of both escape and adiabatic losses on the shape of the pile-up.
4.3. Synchrotron spectrum
The optically thin synchrotron emission produced by the particle distribution within a shock pattern described by Fig. 4, assuming constant magnetic field (i.e., parallel shocks) is given by
where, for a relativistic particle of charge q, mass m, and pitch-angle ,
The characteristic frequency depends on p as , where the cyclotron frequency is . The integration over the particle position is done by computing the number of particles at a given momentum p at each time step inside the pattern. This gives the quantity contributing at a given frequency by the factor to the synchrotron spectrum produced by the pattern. Fig. 7 shows the resulting synchrotron spectrum.
The results, shown in Fig. 7, display a great variety of synchrotron spectra. For large ratios the spectrum is flat at low frequency, and extends over a range which depends on the relative strengths of loss and acceleration, which can be quite sufficient to explain the flat radio spectra observed in radio-loud quasars. Such an explanation provides an alternative to the inhomogeneous self-absorbed models usually advanced as the origin of flat radio Quasar spectra (Marscher & Gear 1985). Fig. 7 also shows that inverted spectra over 2-3 decades are also quite possible, where the cooling time is still shorter than the escape time. In fact the pile-up effect, as stressed by Melrose (1996), is essential to explain the inverted spectra observed in galactic centre sources such as Sagittarus (Beckert et al. 1996), and also may be associated with the flaring states of some extragalactic sources, such as the recent X-ray bursts of Mrk 501 (Pian et al. 1998). Such flares can arise from a variation of either the inter-shock distance or the escape time from the system. A change in the magnetic field stength does not mimick the multi-wavelength behaviour adequately (Mastichiadis & Kirk 1997).
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999