  Recent X-ray observations of merger shocks in galaxy clusters have shown that
the post-shock plasma is two-temperature, with the protons hotter than the
electrons. By means of two-dimensional particle-in-cell simulations, we study
the physics of electron irreversible heating in perpendicular low Mach number
shocks, for a representative case with sonic Mach number of 3 and plasma beta
of 16. We find that two basic ingredients are needed for electron entropy
production: (i) an electron temperature anisotropy, induced by field
amplification coupled to adiabatic invariance; and (ii) a mechanism to break
the electron adiabatic invariance itself. In shocks, field amplification occurs
at two major sites: at the shock ramp, where density compression leads to an
increase of the frozen-in field; and farther downstream, where the shock-driven
proton temperature anisotropy generates strong proton cyclotron and mirror
modes. The electron temperature anisotropy induced by field amplification
exceeds the threshold of the electron whistler instability. The growth of
whistler waves breaks the electron adiabatic invariance, and allows for
efficient entropy production. We find that the electron heating efficiency
displays only a weak dependence on mass ratio (less than 30 percent drop, as we
increase the mass ratio from 49 up to 1600). We develop an analytical model of
electron irreversible heating and show that it is in excellent agreement with
our simulation results.
