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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.