Electron Injection by Whistler Waves in Non-relativistic Shocks

Electron acceleration to non-thermal, ultra-relativistic energies (~ 10-100 TeV) is revealed by radio and X-ray observations of shocks in young supernova remnants (SNRs). The diffusive shock acceleration (DSA) mechanism is usually invoked to explain this acceleration, but the way in which electrons are initially energized or 'injected' into this acceleration process starting from thermal energies is an unresolved problem. In this paper we study the initial acceleration of electrons in non-relativistic shocks from first principles, using two- and three-dimensional particle-in-cell (PIC) plasma simulations. We systematically explore the space of shock parameters (the Alfv\'enic Mach number, M_A, the shock velocity, v_{sh}, the angle between the upstream magnetic field and the shock normal, theta_{Bn}, and the ion to electron mass ratio, m_i/m_e). We find that significant non-thermal acceleration occurs due to the growth of oblique whistler waves in the foot of quasi-perpendicular shocks. The obtained electron energy distributions show power law tails with spectral indices up to alpha ~ 3-4. The maximum energies of the accelerated particles are consistent with the electron Larmor radii being comparable to that of the ions, indicating potential injection into the subsequent DSA process. This injection mechanism, however, requires the shock waves to have fairly low Alf\'enic Mach numbers, M_A <~ 20, which is consistent with the theoretical conditions for the growth of whistler waves in the shock foot (M_A <~ (m_i/m_e)^{1/2}). Thus, if the whistler mechanism is the only robust electron injection process at work in SNR shocks, then SNRs that display non-thermal emission must have significantly amplified upstream magnetic fields. Such field amplification is likely achieved by the escaping cosmic rays, so electron and proton acceleration in SNR shocks must be interconnected.