Gravitational radius ($R_g$)

In the Newtonian virial theorem convention, the gravitational radius of a self-gravitating system of total mass $M$ and gravitational potential energy $W<0$ isIt converts global binding energy into a characteristic length without selecting a physical outer boundary. For example, a uniform-density sphere of radius $R$ has $W=-3G{M}^2/(5R)$, so $R_g=5R/3$, larger than its surface radius. For an isolated system in virial equilibrium, $2K+W=0$ gives the mass-weighted mean-square speed $⟨v^2⟩=GM/R_g$ in the centre-of-mass frame.

The same phrase is also used for relativistic lengths proportional to $GM/c^2$, including the Schwarzschild radius. The definition here uses the system's Newtonian binding energy and depends on its mass distribution; specifying the convention avoids confusing these different scales.

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.