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

摘要

Recent X-ray observations of merger shocks in galaxy clusters have shown thatthe post-shock plasma is two-temperature, with the protons being hotter thanthe electrons. In this work, the second of a series, we investigate by means oftwo-dimensional particle-in-cell simulations the efficiency of electronirreversible heating in perpendicular low Mach number shocks. We considervalues 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 speedto pre-shock sound speed) in the range $2\lesssim M_{s}\lesssim 5$, asappropriate for galaxy cluster shocks. As shown in Paper I, magnetic fieldamplification - induced by shock compression of the pre-shock field, or bystrong proton cyclotron and mirror modes accompanying the relaxation of protontemperature anisotropy - can drive the electron temperature anisotropy beyondthe threshold of the electron whistler instability. The growth of whistlerwaves breaks the electron adiabatic invariance, and allows for efficiententropy 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, $T_{e0}$ isthe pre-shock temperature), which depends only weakly on the plasma beta, overthe range $4\lesssim \beta_{p0}\lesssim 32$ which we have explored, and on theproton-to-electron mass ratio (the coefficient of $\simeq 0.044$ is measuredfor our fiducial $m_i/m_e=49$, and we estimate that it will decrease to $\simeq0.03$ for the realistic mass ratio). Our results have important implicationsfor current and future observations of galaxy cluster shocks in the radio band(synchrotron emission and Sunyaev-Zel'dovich effect) and at X-ray frequencies.