3. Application to the synchrotron spectra of blazars
The simple model described above provides a remarkably good fit to the radio to X-ray spectra of several blazars. As an example, we show in Fig. 2 observations of the object Mkn 501. The gamma-ray emission of this object, which has been the subject of much recent interest (e.g., Bradbury et al 1997), is not included in this figure, since it is not thought to arise as synchrotron radiation. For the X-ray emission we display the archival data selected by Catanese et al. (1997) and not the data taken during the TeV flare in April 1997.
The stationary emission () found from Eq. (9) by a single numerical quadrature over the synchrotron Green's function is also shown in Fig. 2. The form of the spectrum is very close to that given by Meisenheimer & Heavens (1987), who used an analytic solution to the stationary diffusion/advection equation, including synchrotron losses. Four free parameters are used to produce this fit:
Since we restrict our model to the synchrotron emission of the accelerated particles, it is not possible independently to constrain quantities such as the Doppler boosting factor, or the magnetic field. These can, however, be found using a model for the gamma-ray emission, for example the synchrotron self-compton model (Mastichiadis & Kirk 1997). Similarly, the frequency below which synchrotron self-absorption modifies the optically thin spectrum is not constrained within the current picture. Nevertheless, our model of the synchrotron emission makes predictions concerning the spectral variability in each of the three characteristic frequency ranges which can be identified in Fig. 2. These ranges are generic features of any synchrotron model, so that the predicted variability can easily be applied to the synchrotron emission of other blazars. They are a) the low frequency region, where the particles have not had time to cool before leaving the source (this is the region with in Fig. 2, below the break at Hz) b) the region between the break and the maximum flux, where the particles have had time to cool, but where the cooling rate is always much slower than the acceleration rate and the spectrum is close to , and c) the region around and above the flux maximum at roughly Hz, where the acceleration rate is comparable to the cooling rate.
Variability or flaring behaviour can arise for a number of reasons. When the shock front overruns a region in the jet in which the local plasma density is enhanced, the number of particles picked up and injected into the acceleration process might be expected to increase. In addition, if the density change is associated with a change in the magnetic field strength, the acceleration timescale might also change, and, hence, the maximum frequency of the emitted synchrotron radiation. Considering the case in which the acceleration timescale remains constant, it is a simple matter to compute the emission, since Eq. (6) is linear. An increase of the injection rate by a factor for a time is found by setting
Using , and , we show the resulting emission at a frequency in Fig. 3. In the case of Mkn 501, this corresponds to a frequency of about Hz, which lies towards the high frequency part of region b), between the infra-red and X-ray regions, where the spectral index is close to . Also shown in this figure is the temporal behaviour of the spectral index, as determined from the ratio of the fluxes at and , through the flare. When plotted against the flux at the lower frequency, the spectral index exhibits a characteristic loop-like pattern, which is tracked in the clockwise sense by the system. This type of pattern is well-known and has been observed at different wavelengths in several sources e.g., OJ 287 (Gear et al. 1986), PKS 2155-304 (Sembay et al. 1993) and Mkn 421 (Takahashi et al. 1996). It arises whenever the spectral slope is controlled by synchrotron cooling, (or, in fact, any cooling process which is faster at higher energy) so that information about changes in the injection propagates from high to low energies (Tashiro et al. 1995).
If the system is observed closer to the maximum frequency, where the cooling and acceleration times are equal, the picture changes. Here information about the occurrence of a flare propagates from lower to higher energy, as particles are gradually accelerated into the radiating window. Such behaviour is depicted in Fig. 4, where the same flare is shown at frequencies which are an order of magnitude higher than in Fig. 3. In the case of Mkn 501, the frequency range is close to Hz. This time the loop is traced anticlockwise. Such behaviour, although not as common, has also occasionally been observed, for example in the case of PKS 2155-304 (Sembay et al. 1993).
Finally, in the region of the spectrum below the break, where the spectral index in the case of Mkn 501 is close to , the emission is determined by the finite size of the source region, as expressed by . Here, the flare shown in Figs. 3 and 4 has only a very small effect on the observed flux, since the duration has been chosen to be much smaller than the time taken to fill up the emitting region with radiating particles. However, even in the case of a larger flare, or one of longer duration, no variation of the spectral index is to be expected through the flare at frequencies below the break, because the time taken to fill the effective emitting region is independent of frequency.
This effect of smaller changes in the spectral slope at lower frequencies is also evident from Fig. 5, where the spectrum is shown at times (i.e., the stationary emission as shown in Fig. 2), , , and . Here it can be seen that the rise in emission is rapid at all frequencies, and the subsequent fall sets in as a wave which propagates downwards in frequency.
The results presented in Figs. 3 and 4 are computed for , so that the effects of the finite light travel time between the front and back of the source are negligible. In fact, in the limit , Eqs. (7) to (24) reduce to the equations for a homogeneous source, provided the limit is taken keeping finite, i.e., allowing the maximum spatial extent of the source to vanish.
© European Southern Observatory (ESO) 1998
Online publication: April 20, 1998