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Black-hole accretion is the inflow of matter onto a black hole, converting some of its gravitational binding energy into heat, electromagnetic radiation and outflows. Matter with substantial angular momentum generally forms an accretion disk; internal stresses transfer angular momentum outward so that matter can move inward. The released energy can be radiated before the material crosses the event horizon. In a thin-disk model the innermost stable circular orbit marks the transition toward rapid plunging motion, although the detailed inner flow depends on the accretion state.^[ref]

The relation $L=\eta \dot{M}{c}^2$ defines the radiative efficiency of black-hole accretion, where $L$ is emitted luminosity, $\dot{M}$ the mass accretion rate and $c$ the speed of light. A radiatively inefficient accretion flow carries much of its available energy inward or into outflows instead of emitting it, so luminosity alone need not measure the mass supply.

If all mass of a spherical body lies within radius $R$, then $r(m)\le R$ in the shell-assembly integral for its gravitational binding energy. Therefore $|\Omega |={\int }_0^{M}Gm/r(m)dm\ge (G/R){\int }_0^{M}mdm=G{M}^2/(2R)$. The inequality is strict for an extended interior mass distribution; a thin shell gives equality. Dividing by a stipulated luminosity gives a lower bound on that reservoir-to-luminosity ratio. A Kelvin-Helmholtz cooling time also needs a structure model and the accessible fraction of gravitational release.

Build the star by adding spherical shells. With $m(r)$ the enclosed mass, its gravitational binding energy isFor a nondegenerate extended star, $r(m)<R_{\odot }$ on the interior, soThis is a gravitational binding-energy lower bound for a spherical star. The Kelvin-Helmholtz cooling time is of orderThe displayed energy bound gives $|\Omega |/L_{\odot }>1.6\times 10^7$ years; it supplies a lower bound on that reservoir-to-luminosity ratio. A stellar structure model sets its coefficient. For an ideal monatomic, gas-supported star the virial theorem puts about half the released gravitational energy into heat, leaving half available for radiation; a uniform-density model would give about $9\times 10^6$ years. These contraction estimates are all of order $10^7$ years, much shorter than the nuclear lifetime.