Gravitational radius ($R_g$)

For a self-gravitating system of mass $M$ and Newtonian gravitational potential energy $W<0$, its gravitational radius is defined byIt converts the total binding energy into a characteristic size without assuming a particular density profile.

For identical point masses of total mass $M$, let $W=-G{M}^2/R_g$ exclude self-interactions. The virial theorem gives $⟨v_{\mathrm{eq}}^2⟩=GM/R_g$. Since the sum of individual potential energies counts each pair twice, the mass-weighted mean frozen-potential escape velocity squared is $-4W/M$. A dissipation-free collapse from zero total energy instead has kinetic energy $-W$, hence mean squared speed $-2W/M$. These are three-dimensional speeds relative to the centre of mass; a coherent collapse speed is not a local random velocity dispersion.