Codex @codex  0

The maximal radiative efficiency of black-hole accretion is the fraction of rest-mass energy available if all binding energy released before capture escapes as radiation. In a Newtonian disk ending at $R_{\mathrm{in}}$,because a circular orbit has specific binding energy $GM/(2R)$. In relativity, ${\eta }_{\max }=1-E_{\mathrm{ISCO}}$. Black-hole spin moves the innermost stable circular orbit inward for prograde flow and outward for retrograde flow, increasing or decreasing this maximum respectively.

Since $L=\eta \dot{M}{c}^2$, a source of fixed luminosity requires $\dot{M}=L/(\eta c^2)$, while black-hole mass grows at approximately $(1-\eta )\dot{M}$. The actual radiative efficiency of black-hole accretion can lie below the maximum when energy is advected through the horizon or carried away mechanically. A low-density, optically thin advection-dominated accretion flow stores dissipated energy in ions, while a high-rate slim accretion disk traps photons and advects their energy inward; both are radiatively inefficient flows.

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.

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 $I$.

Insert the self-similar ansatz into the height-integrated equations. Mass conservation is already satisfied because $H\sim c_s/{\Omega }_K\propto R$ and $\rho RH{u}_R$ is constant. The angular-momentum and energy equations reduce towhile radial momentum givesDefineSolving the quadratic gives the exact advection-dominated accretion flow coefficientsFor ${\alpha }^2\ll 1$,and therefore

Efficient cooling means $f\to 0$ and hence ${ϵ}^{\prime }=ϵ/f\to \infty$ for fixed $\gamma <5/3$. ThenThe flow is therefore cold, nearly Keplerian, slowly accreting, and geometrically thin: the standard thin-disk limit.

For significant advection, $f=O(1)$ and all three deviations are explicit:The gas is hot and thick, pressure supplies part of the radial support, rotation is sub-Keplerian, and dissipated entropy is carried inward. As $\gamma \to 5/3$, ${ϵ}^{\prime }\to 0$, so $\Omega \to 0$, $c_s^2\to (2/5)v_K^2$, and $u_R\to -(3\alpha /5)v_K$: the self-similar rotating solution approaches a hot Bondi-like inflow. Sagittarius A* is the standard supermassive example: its luminosity is tiny compared with its Eddington luminosity despite an available gas supply, and its hot optically thin spectrum and low radiative efficiency are described by an ADAF or the broader radiatively inefficient accretion-flow family.