## 4. Acceleration at multiple shocksThe 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 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 To these we add a third: an There are then two important characteristic times which are important for a discussion of the solutions: -
the single shock acceleration time , which is the characteristic time of momentum increase at an isolated shock front: (Lagage & Cesarsky 1981) with -
the effective multiple shock acceleration time which is controlled by both the advective time and the single shock acceleration time and is inferred from our numerical results.
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 lossesWe first consider the case where losses are inefficient
(). The results are shown in
Fig. 5a. The stationary solution at each shock front
(with
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 timeIf the inter-shock distance ## 4.1.2. Variations of the inter-shock distanceWe 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 ## 4.2. Multiple shock effect: the case with synchrotron lossesThe 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 rateA decrease (increase) of with
## 4.2.2. Effects of the other parametersWe stress here the effects of both escape and adiabatic losses on the shape of the pile-up. -
Keeping the inter-shock distance constant and allowing to vary causes a change in both the spectral slope at low momenta, where losses are unimportant, and a change in the width of the pile-up Fig. 6c. As increases, the low momentum spectrum hardens, and the width of the hump increases, as it shortens, the spectrum softens, and the hump becomes less prominent, until it disappears completely once . At this point, the power-law index of the low momentum spectrum is approximately 4, in agreement with the findings of Kardashev (1962). -
Keeping the escape time constant, but varying the advection time (and inter-shock distance) enables one to distinguish the regimes of single and multiple shock acceleration. For larger values of , than our fiducial case, four regions in momentum space can be found. Starting at low momentum, these are (Fig. 6d) -
For , acceleration proceeds by the multiple shock process without losses. -
For the spectrum is affected by losses, and starts to form a pile-up. Acceleration at multiple shocks takes place. -
For particles are prevented from reaching the next shock before cooling. An interesting phenomenon appears in this case: the spectrum shows a power-law index appropriate to acceleration at a single shock front (see the dotted line) -
For losses dominate over all other processes and the spectrum cuts off.
In Fig. 6d all four regions can be distinguished in the case . At higher *L*, escape prevents multiple shock acceleration, and at lower values the shocks are too close together to allow the emergence of the single-shock power-law spectrum. These effects do not appear in previous works, which had to assume a separation in the time-scales of the different. The high momentum power-law tail due to the single shock process may be important and contribute a substantial fraction of the total pattern luminosity. -
## 4.3. Synchrotron spectrumThe 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 The characteristic frequency depends on
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 |