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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.