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Non-linear diffusive shock acceleration: A recipe for injection of electrons

Prescriptions for electron injection into the diffusive shock acceleration process are required in many practical considerations of cosmic-ray astrophysics, particularly in modeling of the synchrotron emission of astrophysical sources. In particle-in-cell simulations of quasi-parallel magnetized collisionless shocks, we analyse the evolution of particle spectra. We find that in the later stages of shock evolution, the initially strong suprathermal part in the ion spectra fades, thus leaving the spectra composed of a Maxwellian and a power law. Once the electron and ion spectra flatten and become parallel, we find that the amounts of cosmic ray ions and electrons become similar. We make the step towards relating the micro and macro-scale physics by applying this injection rule to Blasi's semi-analytical model of non-linear diffusive shock acceleration, in order to obtain the particle spectra and electron-to-proton ratio $K_{\mathrm{ep}}$ at high energies. By using shock jump conditions that include the electron heating, we find $K_{\mathrm{ep}}$ as a function of Mach number. For Mach number $\sim$ 100, our model finely reproduces the typically observed ratio for Galactic cosmic-rays $K_{\mathrm{ep}} \sim$ 1:100 in the test particle regime.