Electron Heating in Hot Accretion Flows

Local (shearing box) simulations of the nonlinear evolution of the magnetorotational instability in a collisionless plasma show that angular momentum transport by pressure anisotropy (p_⊥ ≠ p_∥, where the directions are defined with respect to the local magnetic field) is comparable to that due to the Maxwell and Reynolds stresses. Pressure anisotropy, which is effectively a large-scale viscosity, arises because of adiabatic invariants related to p_⊥ and p_∥ in a fluctuating magnetic field. In a collisionless plasma, the magnitude of the pressure anisotropy, and thus the viscosity, is determined by kinetic instabilities at the cyclotron frequency. Our simulations show that ~50% of the gravitational potential energy is directly converted into heat at large scales by the viscous stress (the remaining energy is lost to grid-scale numerical dissipation of kinetic and magnetic energy). We show that electrons receive a significant fraction [~(T_e/T_i)^1/2] of this dissipated energy. Employing this heating by an anisotropic viscous stress in one-dimensional models of radiatively inefficient accretion flows, we find that the radiative efficiency of the flow is greater than 0.5% for ≳ 10^-4_Edd. Thus, a low accretion rate, rather than just a low radiative efficiency, is necessary to explain the low luminosity of many accreting black holes. For Sgr A* in the Galactic center, our predicted radiative efficiencies imply an accretion rate of ≈3 × 10^-8 M_☉ yr^-1 and an electron temperature of ≈3 × 10¹⁰ K at ≈10 Schwarzschild radii; the latter is consistent with the brightness temperature inferred from VLBI observations.