For a steady axisymmetric disk,
$$
\dot M=4\pi RHm_pn|u_R|,
$$
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 $t_{\rm in}\sim R/(\alpha v_K)$ and
$$
n\sim\frac{\dot M}{4\pi R^2m_p\alpha v_K}.
$$
Substitution into the given <electron--proton thermal equilibration time> yields
$$
\frac{t_{e-p}}{t_{\rm in}}
\simeq
\frac{0.24\pi\alpha^2GMm_p^2}{\dot M\sigma_Tcm_e}
\left(\frac{k_BT_e}{m_ec^2}\right)^{3/2}.
$$
Using $\dot M_{\rm Edd}=10L_{\rm Edd}/c^2=40\pi GMm_p/(\sigma_Tc)$ and a mildly relativistic electron temperature $k_BT_e/(m_ec^2)\simeq0.1$ gives
$$
\boxed{\frac{\dot M}{\dot M_{\rm Edd}}\lesssim0.4\alpha^2.}
$$
Below this rate, <Coulomb collision>[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 $\eta\simeq0.1$ of a thin alpha disk and decreases with accretion rate.

Solution

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

{c}

A Coulomb collision is the deflection or energy exchange of charged particles through their electric interaction. In a dilute plasma, the cumulative effect of many small-angle encounters controls transport and interspecies thermal equilibration.


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