= Gravitational radius
{title2=$R_g$}
In the Newtonian <virial theorem> convention, the <gravitational radius> of a self-gravitating system of total <mass> $M$ and <gravitational potential energy> $W<0$ is
$$
R_g=-\frac{GM^2}{W}.
$$
It converts global <binding energy> into a characteristic length without selecting a physical outer boundary. For example, a uniform-density <sphere> of radius $R$ has $W=-3GM^2/(5R)$, 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> $\langle v^2\rangle=GM/R_g$ 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.
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