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 10^18eV). Extrapolating their spectrum to their plausible injection energy from some shock, impli
es an enormous hidden energy for a spectrum of index ~-2. Some analyzes suggest the particles injection spectrum at source to be as steep as -2.4 to -2.7, making the problem much worse, by a factor of order 10^6. 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 theorized and eventually observed in many AGN jets. Based on that, we use by analogy the Comptonisation effect and we propose a scenario of a single injection of particles which are accelerated consecutively by several oblique shocks along the axis of an AGN jet. We use detailed test-particle approximation Monte Carlo simulations in order to calculate particle spectra by acceleration at such a shock pattern while monitoring the efficiency of acceleration, 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 consecutive shocks push the spectrum up in energy, rendering flatter distributions with steep cut-offs and characteristic depletion at low energies, an effect which could explain the puzzling apparent extra source power as well as the flat or inverted spectra from distant flaring sources.
The flux of Ultra High Energy Cosmic Rays (UHECRs) at $E>10^{18.5}$ eV is believed to arise in plasma shock environments in extragalactic sources. In this paper, we present a systematic study of particle acceleration by relativistic shocks, in partic
ular concerning the dependence on bulk Lorentz factor and the angle between the magnetic field and the shockflow. For the first time, simulation results of super- and subluminal shocks with boost factors up to $Gamma=1000$ are investigated and compared systematically. While superluminal shocks are shown to be inefficient at the highest energies ($E>10^{18.5}$ eV), subluminal shocks may provide particles up to $10^{21}$ eV, limited only by the Hillas-criterion. For the subluminal case, we find that mildly relativistic shocks, thought to occur in jets of Active Galactic Nuclei (AGN, $Gammasim 10-30$) yield energy spectra of $dN/dEsim E^{-2}$. Highly relativistic shocks expected in Gamma Ray Bursts (GRBs, $100<Gamma<1000$), on the other hand, have spectra as flat as $E^{-1.5}$. The model results are compared to the measured flux of Cosmic Rays at the highest energies and it is shown that, while AGN spectra are well-suited, GRB spectra are too flat to explain the observed flux. The first evidence of a correlation between the Cosmic Ray flux above $5.7cdot 10^{10}$ GeV and the distribution of AGN by Auger are explained by the model. Neutrino production is expected in GRBs, either in mildly or highly relativistic shocks and although these sources are excluded as the principle origin of UHECRs, superluminal shocks in particular may be observable via neutrino and photon fluxes, rather than as protons.
It is well accepted today that diffusive acceleration in shocks results to the cosmic ray spectrum formation. This is in principle true for non-relativistic shocks, since there is a detailed theory covering a large range of their properties and the r
esulting power-law spectrum, which is nevertheless not as efficient to reach the very high energies observed in the cosmic ray spectrum. On the other hand, the cosmic ray maximum energy and the resulting spectra from relativistic shocks, are still under investigation and debate concerning their contribution to the features of the cosmic ray spectrum and the measured, or implied, cosmic ray radiation from candidate astrophysical sources. Here, we discuss the efficiency of the first order Fermi (diffusive) acceleration mechanism up to relativistic shock speeds, presenting Monte Carlo simulations.
The non-linear back reaction of accelerated cosmic rays at the shock fronts, leads to the formation of a smooth precursor with a length scale corresponding to the diffusive scale of the energetic particles. Past works claimed that shocklets could be
created in the precursor region of a specific shock width, which might energize few thermal particles to sufficient acceleration and furthermore this precursor region may act as confining large angle scatterer for very high energy cosmic rays. On the other hand, it has been shown that the smoothing of the shock front could lower the acceleration efficiency. These controversies motivated us to investigate numerically by Monte Carlo simulations the particle acceleration efficiency in oblique modified shocks. The results show flatter spectra compared to the spectra of the pressumed sharp discontinuity shock fronts. The findings are in accordance with theoretical predictions, since the scattering inside the precursor confines high energy particles to further scattering, resulting in higher energies making the whole acceleration process more efficient.
