Newtonian gravitational potential energy

Two point masses $m_1,m_2$ separated by distance $r$ have potential energyThe corresponding attractive force is $-\nabla V$.

This is the Newtonian point-mass specialization of gravitational energy, with the reference value chosen at infinite separation.

Assembling a spherical mass distribution in shells gives gravitational potential energy $W=-G\int M(r)dM(r)/r$. Integration by parts gives the displayed identity when $M(r{)}^2/r$ vanishes at both endpoints. It includes each interacting pair once; no extra factor of one half belongs in the original shell-assembly integral.

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.