Gravitational rms escape and collapse speeds ($⟨v_{\mathrm{esc}}^2⟩=4⟨v_{\mathrm{eq}}^2⟩,⟨v_{\mathrm{coll}}^2⟩=2⟨v_{\mathrm{eq}}^2⟩$)

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.