Global Compton heating and cooling in hot accretion flows
The hot accretion flow is usually optically thin in the radial direction, therefore the photons produced at one radius can travel for a long distance without being absorbed. These photons thus can heat or cool electrons at other radii via Compton scattering. This effect has been ignored in most previous works on hot accretion flows and is the focus of this paper. If the mass accretion rate is described by $\dot{M}=\dot{M}_0(r/r_{\rm out})^{0.3}$ and $r_{\rm out}=10^4r_{\rm s}$, we find that the Compton scattering will play a cooling and heating role at $r\la 5\times 10^3 r_{\rm s}$ and $r\ga 5\times 10^3 r_{\rm s}$, respectively. Specifically, when $\dot{M}_0>0.1L_{\rm Edd}/c^2$, the Compton cooling rate is larger than the local viscous heating rate at certain radius; therefore the cooling effect is important. When $\dot{M}_0>2L_{\rm Edd}/c^2$, the heating effect at $r_{\rm out}$ is important. We can obtain the self-consistent steady solution with the global Compton effect included only if $\dot{M}_0\la L_{\rm Edd}/c^2$ for $r_{\rm out}=50r_{\rm s}$, which corresponds to $L\la 0.02L_{\rm Edd}$. Above this rate the Compton cooling is so strong at the inner region that hot solutions can not exist. On the other hand, for $r_{\rm out}= 10^5r_{\rm s}$, we can only get the self-consistent solution when $\dot{M}_0\la L_{\rm Edd}/c^2$ and $L<0.01L_{\rm Edd}$. Above this accretion rate the equilibrium temperature of electrons at $r_{\rm out}$ is higher than the virial temperature as a result of strong Compton heating, so the accretion is suppressed. In this case the activity of the black hole will likely "oscillate" between an active and an inactive phases, with the oscillation timescale being the radiative timescale of the gas at $r_{\rm out}$.