= Solution
Well inside the <Bondi accretion> radius, pressure is negligible and the inflow approaches the <free-fall speed>
$$
v(r)=\sqrt{\frac{2GM_{\rm BH}}r}
=c\sqrt{\frac{r_S}r},
\qquad r_S=\frac{2GM_{\rm BH}}{c^2}.
$$
Steady spherical <mass conservation> gives
$$
\rho=\frac{\dot M}{4\pi r^2v}.
$$
Estimating the outward <optical depth> across a radial scale $r$ with electron-scattering opacity $\kappa=\sigma_T/m_p$,
$$
\boxed{\tau(r)\simeq\kappa\rho r
=\frac{\kappa\dot M}{4\pi\sqrt{2GM_{\rm BH}r}}}.
$$
The integral optical depth to infinity differs only by a factor of two in this free-fall approximation.
At the <Schwarzschild radius>, $\tau=1$ when
$$
\boxed{\dot M_{\rm crit}
=\frac{8\pi GM_{\rm BH}}{\kappa c}
=\frac{2L_{\rm Edd}}{c^2}}.
$$
If the conventional <Eddington accretion rate> is $\dot M_{\rm Edd}=L_{\rm Edd}/(\eta_0c^2)$, then $\dot M_{\rm crit}=2\eta_0\dot M_{\rm Edd}$. This comparison depends on the reference efficiency $\eta_0$, while $\dot M_{\rm crit}$ itself follows directly from optical depth.
Writing $\dot m=\dot M/\dot M_{\rm crit}$ gives
$$
\tau(r)=\dot m\sqrt{\frac{r_S}r}.
$$
For $\dot m>1$, the inner flow is optically thick and its <photosphere> lies at
$$
\boxed{r_{\rm ph}=\dot m^2r_S}.
$$
The diffusion speed is $v_{\rm diff}\sim c/\tau$. Hence
$$
\frac{v}{v_{\rm diff}}
=\frac{v\tau}{c}=\dot m\frac{r_S}r,
$$
and equality defines the <photon-trapping radius>
$$
\boxed{r_t=\dot m\,r_S}.
$$
Inside $r_t$, inward advection outruns outward diffusion, so <photon trapping in an accretion flow> carries most released radiation into the black hole.
Only the binding energy released outside $r_t$ escapes efficiently. Its order of magnitude is
$$
L\simeq\frac{GM_{\rm BH}\dot M}{r_t}
=\frac12\dot M_{\rm crit}c^2
=\boxed{L_{\rm Edd}}.
$$
The <radiative efficiency of black-hole accretion> therefore decreases inversely with supply rate:
$$
\boxed{\eta=\frac{L}{\dot Mc^2}
\simeq\frac{\dot M_{\rm crit}}{2\dot M}
=\eta_0\frac{\dot M_{\rm Edd}}{\dot M}}.
$$
Order-one coefficients depend on the optical-depth and inner-boundary conventions, but the luminosity saturation and $eta\propto\dot M^{-1}$ are robust.
Back to article page
