Binding energy

Binding energy is the energy required to separate a bound state into specified free constituents. For an electron bound in a chemical atom, neglecting this cost is an assumption of the free-electron Compton scattering model.

In Newtonian gravity, assembling a spherical mass distribution from shells gives the gravitational potential energy in the title formula, with zero at infinite separation. Its magnitude measures the gravitational contribution to binding. The full energy accounting of a hot self-gravitating body also includes its internal energy; the virial theorem relates the two in equilibrium. A uniform-density sphere has $\Omega =-3G{M}^2/(5R)$.

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.