Non-linear Diffusive Shock Acceleration of Cosmic Rays -- Quasi-thermal and Non-thermal Particle Distributions

Diffusive shock acceleration (DSA) of particles at collisionless shocks is the major accepted paradigm about the origin of cosmic rays (CRs). As a theory it was developed during the late 1970s in the so-called test-particle case. If one considers the influence of CR particles at shock structure, then we are talking about non-linear DSA. We use semi-analytical Blasi's model of non-linear DSA to obtain non-thermal spectra of both protons and electrons, starting from their quasi-thermal spectra for which we assumed the $\kappa$-distribution, a commonly observed distribution in out-of-equilibrium space plasmas. We treated more carefully than in previous work the jump conditions at the subshock and included electron heating, resonant and, additionally, non-resonant magnetic field instabilities produced by CRs in the precursor. Also, corrections for escaping flux of protons and synchrotron losses of electrons have been made.