Whistler-regulated MHD: Transport equations for electron thermal conduction in the high $β$ intracluster medium of galaxy clusters

Transport equations for electron thermal energy in the high $\beta_e$ intracluster medium (ICM) are developed that include scattering from both classical collisions and self-generated whistler waves. The calculation employs an expansion of the kinetic electron equation along the ambient magnetic field in the limit of strong scattering and assumes whistler waves with low phase speeds $V_w\sim{v}_{te}/\beta_e\ll{v}_{te}$ dominate the turbulent spectrum, with $v_{te}$ the electron thermal speed and $\beta_e\gg1$ the ratio of electron thermal to magnetic pressure. We find: (1) temperature-gradient-driven whistlers dominate classical scattering when $L_c>L/\beta_e$, with $L_c$ the classical electron mean-free-path and $L$ the electron temperature scale length, and (2) in the whistler dominated regime the electron thermal flux is controlled by both advection at $V_w$ and a comparable diffusive term. The findings suggest whistlers limit electron heat flux over large regions of the ICM, including locations unstable to isobaric condensation. Consequences include: (1) the Field length decreases, extending the domain of thermal instability to smaller length-scales, (2) the heat flux temperature dependence changes from $T_e^{7/2}/L$ to $V_wnT_e\sim{T}_e^{1/2}$, (3) the magneto-thermal and heat-flux driven buoyancy instabilities are impaired or completely inhibited, and (4) sound waves in the ICM propagate greater distances, as inferred from observations. This description of thermal transport can be used in macroscale ICM models.