2013A&A...556A..88M

Shocks in jets and hot spots of active galactic nuclei (AGN) are one prominent class of possible sources of very high-energy cosmic-ray particles (above 10¹⁸eV). Extrapolating their spectrum to their plausible injection energy from some shock implies an enormous hidden energy for a spectrum of index ~-2. Some analyses suggest the particles' injection spectrum at source to be as steep as -2.4 to -2.7, which exacerbates the problem, by a factor of 10⁶. Nevertheless, it seems implausible that more than at the very best 1/3 of the jet energy goes into the required flux of energetic particles, thus one would need to allow for the possibility that there is an energy problem, which we would like to address in this work. Sequences of consecutive oblique shock features, or conical shocks, have been theoretically predicted and eventually observed in many AGN jets. Based on that, we use by analogy the Comptonization effect and propose a scenario of a single injection of particles consecutively accelerated by several oblique shocks along the axis of an AGN jet. We developed a test-particle approximation Monte Carlo simulations to calculate particle spectra by acceleration at such a shock pattern while monitoring the efficiency of acceleration by calculating differential spectra. We find that the first shock of a sequence of oblique shocks establishes a low-energy power-law spectrum with ∼E^–2.7. The following consecutive shocks push the spectrum up in energy, rendering flatter distributions with steep cut-offs, and characteristic depletion at low energies, which could explain the puzzling apparent extra source power. Our numerical calculations show a variation of spectral indices, a general spectral flattening, and starved spectra, which connect to the relativistic nature of the shocks, the multiple shock acceleration conditions, and the steepness of the magnetic field to the shock normal. This helps in understanding the jet-magnetic field geometry and the irregular or flat spectra observed in many AGN jets (e.g., CenA, 3C279, PKS 1510-089). Furthermore, the E^–2.4-E^–2.7 ultra-high-energy cosmic-ray injected source spectra claimed by many authors might be explained by the superposition of several, perhaps many, emission sources, all of which end their particle shock-acceleration sequence with flatter, starved spectra produced by two or more consecutive oblique shocks along their jets. It might also imply a mixed component of the accelerated particles above 10¹⁹ eV. Moreover, the present acceleration model can explain the variability of inverted gamma-ray spectra observed in high redshifted flaring extragalactic sources.