Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 347 / 2

A singular isothermal sphere has total density and circular speedIf gas traces the gravitating matter, ${\rho }_g=f_{\mathrm{gas}}{\rho }_{\mathrm{tot}}$. A moving black hole gravitationally focuses that gas, so the relevant estimate is the Bondi--Hoyle--Lyttleton accretion rateHydrostatic gas in the isothermal potential has $c_s$ of order $\sigma$, and thereforeup to an order-one coefficient. Taking $c_s=\sigma$ and $v_c=\sqrt{2}\sigma$ gives $2/(3\sqrt{3})$ multiplying the displayed expression. The same $G^2{M}^2\rho /v^3$ scaling that produces Chandrasekhar dynamical friction therefore sets the likely capture rate as the hole spirals inward. Feedback, angular momentum, a gas core, or a multiphase medium can reduce the actual rate.

Well inside the Bondi accretion radius, pressure is negligible and the inflow approaches the free-fall speedSteady spherical mass conservation givesEstimating the outward optical depth across a radial scale $r$ with electron-scattering opacity $\kappa ={\sigma }_T/m_p$,The integral optical depth to infinity differs only by a factor of two in this free-fall approximation.

At the Schwarzschild radius, $\tau =1$ whenIf the conventional Eddington accretion rate is ${\dot{M}}_{\mathrm{Edd}}=L_{\mathrm{Edd}}/({\eta }_0{c}^2)$, then ${\dot{M}}_{\mathrm{crit}}=2{\eta }_0{\dot{M}}_{\mathrm{Edd}}$. This comparison depends on the reference efficiency ${\eta }_0$, while ${\dot{M}}_{\mathrm{crit}}$ itself follows directly from optical depth.

Writing $\dot{m}=\dot{M}/{\dot{M}}_{\mathrm{crit}}$ givesFor $\dot{m}>1$, the inner flow is optically thick and its photosphere lies at

The diffusion speed is $v_{\mathrm{diff}}\sim c/\tau$. Henceand equality defines the photon-trapping radiusInside $r_t$, 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 isThe radiative efficiency of black-hole accretion therefore decreases inversely with supply rate:Order-one coefficients depend on the optical-depth and inner-boundary conventions, but the luminosity saturation and $eta\propto {\dot{M}}^{-1}$ are robust.