Spherical binding-energy mass integral
= Spherical binding-energy mass integral
{title2=$W=-\dfrac G2\int_0^\infty M(r)^2r^{-2}\,dr$}
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.