Radiative energy of virialized gravitational collapse ($E_{\mathrm{rad}}\simeq 3M{\sigma }^2/2$)

For an isolated final stellar population with isotropic one-dimensional velocity dispersion $\sigma$, kinetic energy is $K=3M{\sigma }^2/2$. The virial theorem gives $W=-2K$ and total energy $E=-K$. Collapse from a diffuse nearly unbound gas reservoir therefore requires removal of approximately $K$ of energy. The radiated fraction can differ if there are outflows, appreciable initial binding energy, or boundary stresses.