Theory of Electron Injection at Oblique Shock of Finite Thickness

A theory of electron injection into diffusive shock acceleration (DSA) for the generation cosmic-ray electrons at collisionless shocks is presented. We consider a recently proposed particle mechanism called stochastic drift (SSDA). find that SSDA may be understood as an oblique finite thickness. More specifically, it described by solution to diffusion-convection equation particles with characteristic diffusion length comparable On other hand, same yields standard DSA if much longer than Although predicts, in general, spectral index steeper DSA, more efficient low-energy and favorable injection. The threshold energy corresponds transition between two different regimes. It order $0.1\text{-}1$ MeV typical interstellar interplanetary conditions dissipation scale turbulence around determined ion inertial length. high $M_\textrm{A} / \cos \theta_{Bn}$ where $M_\textrm{A}$ $\theta_{Bn}$ are Alfv\'en Mach number obliquity. suggests ultra-relativistic energies will easily realized high-Mach-number young supernova remnant shocks, but not weak or moderate heliosphere unless upstream magnetic field nearly perpendicular normal.