Self-gravity in a thin spherical shell expansion ($W_{\mathrm{self}}=-G{M}_s^2/(2R)$)

A shell of fixed mass $M_s$ and radius $R$ has self-energy $-G{M}_s^2/(2R)$. Its inward generalized force opposing an increase of radius is $G{M}_s^2/(2R^2)$. Including this in a swept-gas shell threshold changes an interior-only factor $f(1-f)$ to $f(1-f/2)$ when the original total mass is held fixed.