Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 44 / 1 / a

Build the spherical system by adding shells from the centre outwards. The shell theorem says that a shell of mass $dM$ at radius $r$ interacts with the already assembled interior mass through potential $-GM(r)/r$. Thus its contribution is $dW=-GM(r)dM(r)/r$. Each pair is counted when its outer member is added, so no further factor of one half is needed:Integration by parts givesFor an isolated finite-mass system without a singular central point mass, the endpoint term vanishes provided the central cusp has finite binding energy. ConsequentlyThis is the spherical binding-energy mass integral. If a central singularity makes the endpoint or integral divergent, the finite-energy assumption must be checked rather than silently discarding that term.