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 \sim 30 \% drop , as we increase the mass ratio from m _ { i } / m _ { e } = 49 up to m _ { i } / m _ { e } = 1600 ) .
We develop an analytical model of electron irreversible heating and show that it is in excellent agreement with our simulation results .