Minimum-separation binding-energy bounds ($B_0\le rU\le A_0$)

For positive masses, let $r$ be the actual minimum pairwise separation and use positive binding energy $U={\sum }_{j<k}G{m}_j{m}_k/r_{jk}$. If $m_{(1)},m_{(2)}$ are the smallest masses, then $G{m}_{(1)}{m}_{(2)}\le rU\le G{\sum }_{j<k}{m}_j{m}_k$. Thus $U^{-1}$ is comparable to $r$ with constants depending only on the masses.

With $E=T-U<0$ and $T\ge 0$, the positive binding energy satisfies $U\ge |E|$. The minimum-separation binding-energy bounds imply $r\le A_0/|E|$. At least one pair stays close; this does not bound every particle's distance or exclude a negative-energy gravitational collision.