## 2. The electron distribution functionConsider, 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 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 where is given by Eq. (5). To obtain the
synchrotron emissivity as a function of position, time and frequency
we convolve the density 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 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 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.
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 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 |