Electron Heating in Low Mach Number Perpendicular shocks. II. Dependence on the Pre-Shock Conditions

Recent X-ray observations of merger shocks in galaxy clusters have shown that the post-shock plasma is two-temperature, with the protons being hotter than the electrons. In this work, the second of a series, we investigate by means of two-dimensional particle-in-cell simulations the efficiency of electron irreversible heating in perpendicular low Mach number shocks. We consider values of plasma beta (ratio of thermal and magnetic pressures) in the range $4\lesssim \beta_{p0}\lesssim 32$ and sonic Mach number (ratio of shock speed to pre-shock sound speed) in the range $2\lesssim M_{s}\lesssim 5$, as appropriate for galaxy cluster shocks. As shown in Paper I, magnetic field amplification - induced by shock compression of the pre-shock field, or by strong proton cyclotron and mirror modes accompanying the relaxation of proton temperature anisotropy - can drive the electron temperature anisotropy beyond 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 post-shock electron temperature $T_{e2}$ exceeds the adiabatic expectation $T_{e2,\rm ad}$ by an amount $(T_{e2}-T_{e2,\rm ad})/T_{e0}\simeq 0.044 \,M_s (M_s-1)$ (here, $T_{e0}$ is the pre-shock temperature), which depends only weakly on the plasma beta, over the range $4\lesssim \beta_{p0}\lesssim 32$ which we have explored, and on the proton-to-electron mass ratio (the coefficient of $\simeq 0.044$ is measured for our fiducial $m_i/m_e=49$, and we estimate that it will decrease to $\simeq 0.03$ for the realistic mass ratio). Our results have important implications for current and future observations of galaxy cluster shocks in the radio band (synchrotron emission and Sunyaev-Zel'dovich effect) and at X-ray frequencies.