The recent detection of high energy gamma -rays coming from supernova remnants and active galactic nuclei has revived interest in the diffusive shock acceleration of electrons. In the present paper we examine the basis of the so-called ``box'' model for particle acceleration and present a more physical version of it. Using this we determine simple criteria for the conditions under which ``pile-ups'' can occur in shock accelerated electron spectra subject to synchrotron or inverse Compton losses (the latter in the Thompson limit). An extension to include nonlinear effects is proposed.

Recent observations of blazars have established that their γ-ray emission is associated, as a rule, with very fast variability (as short as ~15 minutes for the TeV photons of Mrk 421); as such, these observations push the theoretical models for the production of the required relativistic electrons to their limits. Herein we investigate the possibility that ``blobs'' loaded with relativistic protons could produce such an activity. We show that, if the proton number density in a blob exceeds a certain critical value, then reflection of its own synchrotron produced photons on some external ``mirror,'' such as a line-emitting cloud, can initiate a feedback process in which the protons can lose most of their energy content in a blob crossing time, resulting in a flare of the same duration. By performing a dimensional analysis, we find the necessary conditions for such an instability to occur, and we show that the conditions required are consistent with those usually assumed to prevail within the relativistic jets of this class of active galactic nuclei.

The recent High-Energy Gamma-Ray Array (HEGRA) observations of the blazar Mrk 501 show strong curvature in the very high energy γ-ray spectrum. Applying the γ-ray opacity derived from an empirically based model of the intergalactic infrared background radiation field to these observations, we find that the intrinsic spectrum of this source is consistent with a power law: dNγ/dE~E-α, with α=2.00+/-0.03 over the range 500 GeV-20 TeV. Within current synchrotron self-Compton scenarios, the fact that the TeV spectral energy distribution of Mrk 501 does not vary with luminosity, combined with the correlated, spectrally variable emission in X-rays as observed by the BeppoSAX and Rossi X-Ray Timing Explorer instruments, also independently implies that the intrinsic spectrum must be close to α=2. Thus, the observed curvature in the spectrum is most easily understood as resulting from intergalactic absorption.

The acceleration of electrons at a shock front can produce characteristic patterns in the variation of the spectral index of the synchrotron emission as a function of flux. Using a simple model of the acceleration process, we present a discussion of these patterns and show how they compare with the variations in the emission of the same electrons via inverse Compton scattering of isotropically distributed target photons from an external source. The "soft lag" behaviour is observed in synchrotron emission, and should also be present in the inverse Compton flux. Shock models can also show "hard lag" behaviour of the synchrotron emission, but this is more difficult to achieve in the inverse Compton emission, because of Klein-Nishina effects. In some cases, the time scales of rise and fall of both the synchrotron and inverse Compton fluxes can depend on the acceleration mechanism.

There is increasing observational evidence that relativistic particles of energies ∼1 TeV provide a significant pressure component of the plasma which powers at least some of the relativistic jets associated with AGN. Furthermore, observations of flares with duration ∼15 min at TeV energies indicate that the associated electrons are accelerated to the required energies on these or shorter time scales, which are comparable to the synchrotron loss time for the values of the magnetic fields thought present in these jets. As such, they push the potential acceleration mechanisms to their limits and prompt us to examine the conditions under which it may be possible for hadronic processes to provide the electrons of the requisite energies. Relativistic hadrons could presumably exist within the flow, having been accelerated efficiently near the compact object and then transported along with it, releasing their energy by an instability due to p- γ reactions once a well-defined threshold is reached.