2021MNRAS.508.6142M

The process that allows cosmic rays (CRs) to escape from their sources and be released into the Galaxy is still largely unknown. The comparison between CR electron and proton spectra measured at Earth suggests that electrons are released with a spectrum steeper than protons by Δs_ep ∼ 0.3 for energies above ∼10 GeV and by Δs_ep ∼ 1.2 above ∼1 TeV. Assuming that both species are accelerated at supernova remnant shocks, we here explore two possible scenarios that can in principle justify steeper electron spectra: (i) energy losses due to synchrotron radiation in an amplified magnetic field and (ii) time-dependent acceleration efficiency. We account for magnetic field amplification produced by either CR-induced instabilities or by magnetohydrodynamics instabilities my means of a parametric description. We show that both mechanisms are required to explain the electron spectrum. In particular, synchrotron losses can only produce a significant electron steepening above ∼1 TeV, while a time-dependent acceleration can explain the spectrum at lower energies if the electron injection into diffusive shock acceleration is inversely proportional to the shock speed. We discuss observational and theoretical evidences supporting such a behaviour. Furthermore, we predict two additional spectral features: a spectral break below ∼few GeV (as required by existing observations) due to the acceleration efficiency drop during the adiabatic phase, and a spectral hardening above ∼20 TeV (where no data are available yet) resulting from electrons escaping from the shock precursor.