Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 347 / 3 / d / Solution

For a steady axisymmetric disk,where factors of order unity depend on the vertical density profile. In a geometrically thick advection-dominated accretion flow, $H\sim R$, $c_s\sim v_K$, and the alpha disk estimate gives $|u_R|\sim \alpha v_K$. Thus the inflow time is $t_{\mathrm{in}}\sim R/(\alpha v_K)$ andSubstitution into the given electron--proton thermal equilibration time yieldsUsing ${\dot{M}}_{\mathrm{Edd}}=10L_{\mathrm{Edd}}/c^2=40\pi GM{m}_p/({\sigma }_{T}c)$ and a mildly relativistic electron temperature $k_B{T}_e/(m_e{c}^2)\simeq 0.1$ givesBelow this rate, Coulomb collisions cannot transfer the ions' viscously generated heat to radiating electrons before inflow. The resulting two-temperature accretion flow advects most of that energy through the horizon, so its efficiency is well below the canonical $\eta \simeq 0.1$ of a thin alpha disk and decreases with accretion rate.