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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.

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.