  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.
