Annu. Rev. Astron. Astrophys. 1997. 35: 445-502
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8.3. High-Frequency Synchrotron Emission: Energy Stratification?

At high frequencies (optical-UV for LBL, X-ray for HBL), the synchrotron spectrum steepens considerably and the largest spectral variability is observed. It is natural to associate the steepening with increasing radiative energy losses. Particle acceleration must also occur or the high-energy emission would vanish in a short time. Therefore the average spectrum probably represents an equilibrium among energy gains, losses, and escape. If the "injected" particle spectrum is a power law, its equilibrium spectrum (in a homogeneous region) is steeper by Deltabeta = 1 above the energy gammab at which the radiative lifetime equals the escape time (Kardashev 1962). The peak in the synchrotron power would occur at nub propto gammab2, and the associated change in spectral slope would be Deltaalpha = 0.5.

Spectral variability above nub (flattening with increasing intensity) could be due to fluctuations in the acceleration process on time scales shorter than those over which equilibrium can be established; the flatter injected spectrum would be visible briefly before energy losses set in. Another possibility is that increased power dissipation causes the injection of particles of higher energy. The second hypothesis is favored in model fits (assuming the simple homogeneous case) to the spectral variability of Mrk 421 (Mastichiadis & Kirk 1997). In both cases, the variations at the highest synchrotron and inverse Compton frequencies are quasisimultaneous, whereas at lower frequencies the time scales are longer and the peaks are delayed.

Acceleration and loss processes can depend on position in the jet, causing different electron energy distributions as a function of location (inhomogeneous model). The steeper spectrum above the peak can be reproduced (even Deltaalpha > 0.5) if higher energy electrons occupy progressively smaller volumes (Ghisellini et al 1985). This occurs naturally behind a shock where the acceleration process is localized at the shock front (Marscher 1996). Thus, a disturbance propagating along an inhomogeneous jet or across a shock front would cause spectral variability, usually with shorter time scales and larger amplitudes at higher energies (Celotti et al 1991, Marscher & Travis 1991).

Indeed, the soft photons do lag the hard photons in both Mrk 421 and PKS 2155-304 (the two brightest HBL). In X rays, the frequency dependence of the measured lags is consistent with a nu-1/2 law, as expected from a radiative lifetime; the EUV and UV could also be consistent within the large uncertainties. At each frequency the decay times are longer than the lags, which is not easy to understand in a pure radiative case. A realistic model likely involves both radiative and travel time effects.

In any case, the data require a fast acceleration process injecting high-energy particles. This is not easy to reconcile with diffusive shock acceleration, whereby high energies build up through a cumulative process (see spectral evolution computed for time-dependent shock acceleration, Fritz & Webb 1990). A viable alternative could be particle acceleration through a large scale electric field (Kirk & Bednarek 1996). Note that the achromatic variability found in the first epoch of multiwavelength monitoring of PKS 2155-304 is not expected from a synchrotron flare in a jet and may be caused by a different physical process such as rotation or microlensing.

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