 Gravitational binding-energy lower bound for a spherical star

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.