Electron accelerations at high Mach number collision-less shocks are investigated by means of two-dimensional electromagnetic Particle-in-Cell simulations with various Alfvén Mach numbers , ion-to-electron mass ratios , and the upstream electron \beta _ { e } ( the ratio of the thermal pressure to the magnetic pressure ) .
We found electrons are effectively accelerated at a super-high Mach number shock ( M _ { A } \sim 30 ) with a mass ratio of M / m = 100 and \beta _ { e } = 0.5 .
The electron shock surfing acceleration is an effective mechanism for accelerating the particles toward the relativistic regime even in two dimensions with the large mass ratio .
Buneman instability excited at the leading edge of the foot in the super-high Mach number shock results in a coherent electrostatic potential structure .
While multi-dimensionality allows the electrons to escape from the trapping region , they can interact with the strong electrostatic field several times .
Simulation runs in various parameter regimes indicate that the electron shock surfing acceleration is an effective mechanism for producing relativistic particles in extremely-high Mach number shocks in supernova remnants , provided that the upstream electron temperature is reasonably low .