Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 347 / 3

A radiatively inefficient accretion flow radiates only a small fraction of the energy released before the gas crosses the inner boundary. At low Eddington ratio, an optically thin flow has such low density that radiative cooling, commonly proportional to density squared, is slower than inflow; the gas remains hot and forms an advection-dominated accretion flow. At high Eddington ratio, an optically thick slim accretion disk can instead undergo photon trapping in an accretion flow: diffusion is slower than inward motion, so radiation is advected into the hole.

The local accretion-flow advection balancehas three sign classes. If $q_{\mathrm{adv}}=0$, local heating equals local radiative cooling and the flow is a radiatively efficient thin disk. If $q_{\mathrm{adv}}>0$, heating exceeds cooling and inward advection removes the excess; low-rate ADAFs and high-rate slim disks are the two principal realizations. If $q_{\mathrm{adv}}<0$, radiation exceeds local dissipation and compressive advection supplies heat, producing a luminous hot accretion flow branch.

Denote the four terms byso the radial or poloidal momentum equation is $I=P+G+C$.

i) In a thin Keplerian accretion disk, radial inertia and pressure are higher-order in $H/R$, leaving $G+C\simeq 0$.

ii) In a nearly static stellar atmosphere, $I=C=0$ and hydrostatic pressure balance gives $P+G=0$.

iii) In pressureless gravitational collapse, rotation and pressure are negligible, so $I=G$; this is free fall.

iv) A slim accretion disk retains radial inertia and radial pressure together with gravity and centrifugal support, so all four terms generally survive: $I=P+G+C$.

v) A stationary geometrically thick disk or torus has negligible poloidal inertia but order-one pressure support, giving $P+G+C=0$.

vi) Nonrotating Bondi accretion has $C=0$ and $I=P+G$. In a highly supersonic Bondi--Hoyle limit the pressure term is also negligible, reducing this to ballistic $I=G$.

vii) A sub-Keplerian advection-dominated accretion flow has significant pressure support and radial inflow as well as rotation, so again $I=P+G+C$, with $C$ smaller than the Keplerian value and the remaining inward gravity balanced by $P$ and $I$.

The stated standard thin-disk dissipation flux is summed over the two faces, so the total luminosity isThis equals the Newtonian standard thin-disk luminosity and the orbital binding energy delivered per unit time at the inner edge. It is half the magnitude $GM\dot{m}/R_{\mathrm{ISCO}}$ of the potential-energy decrease because the other half appears as orbital kinetic energy. In the zero-torque model that remaining mechanical energy passes through the inner edge rather than being dissipated at larger radii. The associated Newtonian radiative efficiency of black-hole accretion is $GM/(2R_{\mathrm{ISCO}}{c}^2)$.

Steady mass conservation gives $\dot{m}=-2\pi R\Sigma u_R$. With specific angular momentum $l=R^2\Omega$, multiply the angular-momentum equation by $2\pi$ and define the signed viscous torque in an accretion diskThenIntegration from $R_1$ to $R_2$ gives

The viscous power generated in an annulus is $2\pi R{F}_{\mathrm{diss}}dR=\mathcal{G}d\Omega$. Taking the inner torque to vanish and writing $l_{\mathrm{in}}=l(R_{\mathrm{in}})$ gives $\mathcal{G}=-\dot{m}(l-l_{\mathrm{in}})$, henceIntegration by parts yieldsFor steady circular force balance, $de=\Omega dl$; equivalently, the stated equality of gravitational- and rotational-potential differences makes the integral $e_{\mathrm{out}}-e_{\mathrm{in}}$. Therefore

When $R_{\mathrm{out}}\gg R_{\mathrm{in}}$, the outer energy and boundary term vanish. For an approximately Keplerian inner orbit, $e_{\mathrm{in}}=-R_{\mathrm{in}}^2{\Omega }_{\mathrm{in}}^2/2$, soComparison with part (c) givesA Keplerian inner flow generates the standard thin-disk power. A pressure-supported sub-Keplerian slim disk generates less through shear, and its emergent luminosity can be smaller still because photon trapping in an accretion flow carries part of that generated energy through the inner edge.