Electron and proton heating in trans-relativistic magnetic reconnection

Hot collisionless accretion flows, such as the one in Sgr A$^{*}$ at our Galactic center, provide a unique setting for the investigation of magnetic reconnection. Here, protons are non-relativistic while electrons can be ultra-relativistic. By means of two-dimensional particle-in-cell simulations, we investigate electron and proton heating in the outflows of trans-relativistic reconnection (i.e., $\sigma_w\sim 0.1-1$, where the magnetization $\sigma_w$ is the ratio of magnetic energy density to enthalpy density). For both electrons and protons, we find that heating at high $\beta_{\rm i}$ (here, $\beta_{\rm i}$ is the ratio of proton thermal pressure to magnetic pressure) is dominated by adiabatic compression ('adiabatic heating'), while at low $\beta_{\rm i}$ it is accompanied by a genuine increase in entropy ('irreversible heating'). For our fiducial $\sigma_w=0.1$, the irreversible heating efficiency at $\beta_{\rm i}\lesssim 1$ is nearly independent of the electron-to-proton temperature ratio $T_{\rm e}/T_{\rm i}$ (which we vary from $0.1$ up to $1$), and it asymptotes to $\sim 2\%$ of the inflowing magnetic energy in the low-$\beta_{\rm i}$ limit. Protons are heated more efficiently than electrons at low and moderate $\beta_{\rm i}$ (by a factor of $\sim7$), whereas the electron and proton heating efficiencies become comparable at $\beta_{\rm i}\sim 2$ if $T_{\rm e}/T_{\rm i}=1$, when both species start already relativistically hot. We find comparable heating efficiencies between the two species also in the limit of relativistic reconnection ($\sigma_w\gtrsim 1$). Our results have important implications for the two-temperature nature of collisionless accretion flows, and may provide the sub-grid physics needed in general relativistic MHD simulations.