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

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1. Introduction

The nonthermal, highly variable X-ray emission of blazars is usually interpreted as the synchrotron emission of relativistic electrons accelerated in a jet which itself moves at relativistic speed towards the observer. In this scenario, the higher energy gamma-rays detected from several such sources arise from the inverse Compton scattering of soft photons by these electrons (for a review, see Sikora  1994). Two different types of model of the emitting region can be found in the literature: the `inhomogeneous' and the `homogeneous' models. In the inhomogeneous model, the emitting part of the jet is assumed to be cylindrically symmetric with a cross-section which varies with distance from the central object. The magnetic field is also taken to vary with distance from the central object. It is usually assumed that relativistic electrons are `injected' at a shock front which moves through the emission region, starting close to the central object. These electrons also move outwards with the jet material, and undergo energy losses due both to synchrotron radiation and to the adiabatic expansion of the jet (Marscher & Gear  1996, Maraschi, Ghisellini & Celotti 1992, Marscher & Travis 1996). The resulting synchrotron spectrum is found by integrating over the emission region, within which both the magnetic field and the particle distribution is inhomogeneous.

In the homogeneous model, on the other hand, both the magnetic field and the particle distribution function are assumed homogeneous throughout the emission region (Inoue & Takahara  1996, Chiaberge & Ghisellini  1997, Mastichiadis & Kirk  1997). The relativistic electrons are injected with a specified distribution and are assumed to escape on a timescale [FORMULA]. After escape, a particle no longer radiates. Although at first sight somewhat arbitrary, there is a good observational reason to believe that escape, or, equivalently, sudden energy loss by adiabatic expansion, is important: the spectral index of the radio emission of blazars is typically hard ([FORMULA]) and cannot be produced by particles which have been allowed to cool completely by synchrotron emission. If we adopt a strictly homogeneous model, electrons must escape from the emission region and thus be prevented from cooling completely. Such a picture departs from the standard explanation, in which the flat spectrum is thought to be the result of a variation of the self-absorption frequency within the source in an inhomogeneous model (the `Cosmic Conspiracy' Marscher  1980). Depending on the magnetic field strength and Doppler boosting factor, self-absorption may also become important in a homogeneous model. Additional components emitting at low frequency would then be required.

The homogeneous model with escape corresponds approximately to the plausible physical situation in which particles are accelerated at a shock front, provided there is a region of relatively high magnetic field just behind the shock. In this case, radiation from this region may be expected to dominate the observed emission. This requires that on leaving the region (on a timescale [FORMULA]), the particles encounter a magnetic field that is so weak, that no significant further contribution to the emitted spectrum arises, despite the fact that particles accumulate there over the entire life of the source. However, a problem arises in the homogeneous model concerning time variability - only if the physical dimensions of the source are such that the light crossing time is short compared with the synchrotron cooling time is it reasonable to assume homogeneity. If this condition is not fulfilled, the observed variability is dominated by the shape and orientation of the source, rather than the intrinsic cooling and acceleration timescales. Until recently, models of the rapid variability of blazars have either fulfilled this condition or have made specific assumptions about the source geometry (Mastichiadis & Kirk  1997, Chiaberge & Ghisellini  1997).

In this paper we present a model in which particles are accelerated at a shock front and cool by synchrotron radiation in the homogeneous magnetic field behind it. The plasma downstream of the shock front moves relativistically towards the observer; the shock front is nonrelativistic when seen from the rest frame of the downstream plasma. The kinetic equations are solved for the time, space and energy dependences of the particle distribution function, and the resulting synchrotron emission is calculated. `Escape' is accounted for by assuming that the magnetic field strength drops suddenly at a finite distance behind the shock, so that the radiation from the adiabatically cooled electrons in the weaker field can be neglected. Thus, this model is homogeneous in the sense that the magnetic field does not vary through the emission region, but contains an inhomogeneous electron distribution. The variability is computed by assuming the observer lies in the direction of the normal to the shock front. This preferred orientation does not affect the observed variability provided the light travel time across the face of the source (i.e., over the surface of the shock) is short compared to the synchrotron cooling time as measured in the rest frame of the plasma. At the highest electron energies considered in the application to X-ray blazars, this condition may be violated, in which case the computed variability will be smoothed out over the longer timescale. Whether or not this is expected to occur could be decided by comparing the predicted emission by inverse compton scattering with gamma-ray observations. In principle, observations of the frequency dependence of the variation timescale could also resolve the question.

We consider only the synchrotron radiation of the accelerated particles, leaving the more involved computation of the inverse Compton emission to future work. This is sufficient for comparison with the observed radio to X-ray emission of blazars, provided the energy losses of the electrons are not dominated by inverse Compton scattering, which is usually the case, at least for BL Lac-type objects (Comastri et al.  1997).

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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