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