The hadronic model of active galactic nuclei and other compact high-energy astrophysical sources assumes that ultra-relativistic protons, electron-positron pairs and photons interact via various hadronic and electromagnetic processes inside a magnetized volume, producing the multiwavelength spectra observed from these sources. A less studied property of such systems is that they can exhibit a variety of temporal behaviours due to the operation of different feedback mechanisms. We investigate the effects of one possible feedback loop, where γ-rays produced by photopion processes are being quenched whenever their compactness increases above a critical level. This causes a spontaneous creation of soft photons in the system that result in further proton cooling and more production of γ-rays, thus making the loop operate. We perform an analytical study of a simplified set of equations describing the system, in order to investigate the connection of its temporal behaviour with key physical parameters. We also perform numerical integration of the full set of kinetic equations verifying not only our analytical results but also those of previous numerical studies. We find that once the system becomes 'supercritical', it can exhibit either a periodic behaviour or a damped oscillatory one leading to a steady state. We briefly point out possible implications of such a supercriticality on the parameter values used in active galactic nuclei spectral modelling, through an indicative fitting of the VHE emission of blazar 3C 279.

In this work we propose an innovative estimation method for the minimum Doppler factor and energy content of the γ-ray emitting region of quasar 3C 279, using a standard proton synchrotron blazar model and the principles of automatic photon quenching. The latter becomes relevant for high enough magnetic fields and results in spontaneous annihilation of γ-rays. The absorbed energy is then redistributed into electron-positron pairs and soft radiation. We show that as quenching sets an upper value for the source rest-frame γ-ray luminosity, one has, by necessity, to resort to Doppler factors that lie above a certain value in order to explain the TeV observations. The existence of this lower limit for the Doppler factor also has implications on the energetics of the emitting region. In this aspect, the proposed method can be regarded as an extension of the widely used method for estimating the equipartition magnetic field using radio observations. In our case, the leptonic synchrotron component is replaced by the proton synchrotron emission and the radio by the very high energy γ-ray observations. We show specifically that one can model the TeV observations by using parameter values that minimize both the energy density and the jet power at the cost of high values of the Doppler factor. On the other hand, the modelling can also be done by using the minimum possible Doppler factor; this, however, leads to a particle-dominated region and high jet power for a wide range of magnetic field values. Despite the fact that we have focused on the case of 3C 279, our analysis can be of relevance to all TeV blazars favouring hadronic modelling that have, moreover, simultaneous X-ray observations.

We present a time-dependent approach to the one-zone hadronic model in the case where the photon spectrum is produced by ultrarelativistic protons interacting with soft photons that are produced from protons and low magnetic fields. Assuming that protons are injected at a certain rate in a homogeneous spherical volume containing a magnetic field, the evolution of the system can be described by five coupled kinetic equations, for protons, electrons, photons, neutrons, and neutrinos. Photopair and photopion interactions are modelled using the results of Monte-Carlo simulations and, in particular, from the SOPHIA code for the latter. The coupling of energy losses and injection introduces a self-consistency in our approach and allows the study of the comparative relevancy of processes at various conditions, the efficiency of the conversion of proton luminosity to radiation, the resulting neutrino spectra, and the effects of time variability on proton injection, among other topics. We present some characteristic examples of the temporal behaviour of the system and show that this can be very different from the one exhibited by leptonic models. Furthermore, we argue that, contrary to the wide-held belief, there are parameter regimes where the hadronic models can become quite efficient. However, to keep the free parameters at a minimum and facilitate an in-depth study of the system, we have only concentrated on the case where protons are injected; i.e., we did not consider the effects of a co-accelerated leptonic component.