Hadronic supercriticalities are radiative instabilities that appear when large amounts of energy are stored in relativistic protons. When the proton energy density exceeds some critical value, a runaway process is initiated resulting in the explosive transfer of the proton energy into electron-positron pairs and radiation. The runaway also leads to an increase of the radiative efficiency, namely the ratio of the photon luminosity to the injected proton luminosity. We perform a comprehensive study of the parameter space by investigating the onset of hadronic supercriticalities for a wide range of source parameters (i.e., magnetic field strengths of 1 G$- 100$ kG and radii of $10^{11}$ cm$-10^{16}$ cm) and maximum proton Lorentz factors ($10^3-10^9$). We show that supercriticalities are possible for the whole range of source parameters related to compact astrophysical sources, like gamma-rays bursts, cores and jets of active galactic nuclei. We also provide an in-depth look at the physical mechanisms of hadronic supercriticalities and show that magnetized relativistic plasmas are excellent examples of non-linear dynamical systems in high-energy astrophysics.
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