Electron injection in collisionless shocks

Electron acceleration in collisionless shocks is considered. It is shown that injection of thermal electrons may be controlled by self-generated whistlers, provided that the Alfvén Mach number ahead of the shock exceeds 43/(β_e)^1/2, where β_e is the ratio of thermal electron pressure to magnetic pressure inside the shock. Acceleration of subrelativistic electrons by whistlers at lower Mach number shocks may still be possible if the thermal electrons are preinjected by some other mechanism. The confinement of the upstreaming electrons to the background fluid is accomplished by off-axis whistlers. Using quasi-linear theory the diffusion coefficient is calculated self-consistently and appears to increase with increasing electron energy and with decreasing efficiency. Wave damping due to induced scattering and whistler mode coupling is considered. Mode coupling is shown to be unimportant when M_A > 13, whereas induced scattering is even less important, except maybe for waves having frequencies near the ion gyrofrequency.