Astron. Astrophys. 333, 452-458 (1998)

2. The electron distribution function

Consider, then a shock front propagating along a cylindrical jet of constant cross-section. Electrons are accelerated at the shock front, and subsequently drift away from it in the downstream flow. Following Ball & Kirk (1992), we treat two spatial zones: one around the shock front, in which particles are continuously accelerated, and one downstream of it, in which particles emit most of the radiation. In each zone, it is assumed that pitch-angle scattering maintains an almost isotropic particle distribution. The number of particles with Lorentz factor between and in the acceleration zone around the shock is governed by the equation

(Kirk et al.  1994), where

with the Thomson cross section. The first term in brackets in Eq. (1) describes acceleration at the rate , the second describes the rate of energy loss due to synchrotron radiation averaged over pitch-angle (because of the isotropy of the distribution) in a magnetic field B (in Tesla). Particles are assumed to escape from this region at an energy independent rate , and to be picked up (injected) into the acceleration process with Lorentz factor at a rate Q particles per second. Note that the concept of this `acceleration zone' differs from the emission region in the usual homogeneous models in two important respects: a) particles are injected at low energy and continuously accelerated and b) very little radiation is emitted by a particle whilst in the acceleration zone (see the discussion below).

Assuming a constant injection rate after switch-on at time , and setting for , Eq. (1) has the solution

for , and , otherwise. Here is the Heaviside step function,

and the upper bound is determined by

with . The quantities and have been assumed independent of energy in this solution, the more general solution allowing for energy dependence is given in the Appendix.

In the model developed by Ball & Kirk (1992), accelerated particles escape into the downstream plasma, where they radiate. We can formulate the kinetic equation obeyed by the density of particles in the radiation zone most compactly using a coordinate system at rest in the radiating plasma. The shock front then provides a moving source of electrons, which subsequently suffer energy losses, but are assumed not to be transported in space. The kinetic equation governing the differential density of particles in the range , is then

where is the position of the shock front at time t. For a shock which starts to accelerate (and therefore `inject') particles at time and position and moves at constant speed , the solution of Eq. (6) for is

where is given by Eq. (5). To obtain the synchrotron emissivity as a function of position, time and frequency we convolve the density n with the synchrotron Green's function . A convenient approximation to this function is given by Melrose (1980):

where is a constant and , with the electron gyro frequency and the angle between the magnetic field direction and the line of sight. At a point () on the symmetry axis of the source at time t the specific intensity of radiation in the direction depends on the retarded time and is given by

and the integrated particle density can readily be evaluated:

The limits of the spatial integration are given by the retarded position of the shock front

and the retarded position of the point furthest from the shock front at which particles have Lorentz factor at time . This is given either by the solution of the transcendental equation

or by the assumed maximum spatial extent of the emission region (i.e., the point at which the magnetic field declines substantially). Denoting this distance by L, we have

This is most conveniently expressed in terms of the (retarded) time for the plasma to traverse the emitting region, as measured in the plasma rest frame:

Eq. (10) gives the integrated particle density for times greater than the `switch-on' time:

before which it vanishes. The resulting electron spectrum, integrated over the source is depicted in Fig. 1. A characteristic break in the spectral slope appears at a particular Lorentz factor which, at any given time, separates those electrons which cool within the source () from those which do not cool within the source (). At large times, all electrons with leave the source before cooling, and the integrated electron density becomes time-independent.

Fig. 1. The electron distribution integrated over the source, as given by Eq. (10). The three curves correspond to the times (solid line), (dotted line) and , (dashed line). At larger times the distribution does not evolve appreciably. For this plot, (i.e., ) and . At , no particles have had time to cool, since the cooling time at the maximum Lorentz factor of is approximately . At times all particles with cool, whereas those of lower leave the source without significant loss of energy.

A question which remains open in this approach is the synchrotron radiation emitted by a particle whilst in the acceleration region. If the magnetic field ahead of the shock were the same as that behind the shock, the total emission could easily be computed using Eq. (3). For , we would have an extra contribution to the flux:

and the total emission would be given by

However, for oblique shocks, the magnetic field strength is expected to increase upon compression at the shock. Particles undergoing acceleration spend part of the time in the upstream and part in the downstream plasma, so that it is not clear how to evaluate their synchrotron emission, although Eq. (16) certainly gives an upper limit. At oblique shocks, reflection at the front itself is thought to be more important than diffusion in the downstream zone, (Kirk & Heavens  1989), so that accelerating particles spend all their time upstream. In this case, it seems reasonable to neglect the emission from the acceleration zone completely, which is the approach adopted here. An improved treatment of this point demands a full time-dependent solution of the diffusion advection equation, which requires considerable numerical effort (Fritz & Webb  1990).

All quantities calculated so far in this section refer to the frame in which the radiating plasma is at rest (the jet frame). For application to blazars, they must be transformed into the observer's frame. Assuming the observer to lie in the direction of motion of the plasma, and denoting the plasma bulk velocity by , the relevant transformations are

where () is the Lorentz factor of the approaching jet plasma and D the position of the detector with respect to the position of the shock at , as measured in the observer's reference frame.

Several simple qualitative results follow from these expressions. Close to the maximum emitted frequency, the timescale on which the intensity varies in the frame of the plasma is roughly the switch-on time . According to Eqs. (18) and (19), Thus, the observed timescale is shorter than the intrinsic by a factor . Doppler boosting of the flux, is independent of the shock speed, and is given simply by Eq. (21).

© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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