= Gravitational rms escape and collapse speeds
{title2=$\langle v_{\rm esc}^2\rangle=4\langle v_{\rm eq}^2\rangle,\quad\langle v_{\rm coll}^2\rangle=2\langle v_{\rm eq}^2\rangle$}
For identical point masses of total mass $M$, let $W=-GM^2/R_g$ exclude self-interactions. The <virial theorem> gives $\langle v_{\rm eq}^2\rangle=GM/R_g$. 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>.
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