= Self-gravity in a thin spherical shell expansion
{title2=$W_{\rm self}=-GM_s^2/(2R)$}
A thin uniform <spherical shell> of fixed <mass> $M_s$ and radius $R$ has <Newtonian gravitational potential energy>
$$
W_{\rm self}=-\frac{GM_s^2}{2R}.
$$
This follows from the <spherical binding-energy mass integral>: the enclosed mass is zero inside $R$ and $M_s$ outside. The radial generalized <force>, positive outward, is consequently $-\partial_RW_{\rm self}=-GM_s^2/(2R^2)$, taking the derivative at fixed $M_s$. It is an inward resistance to uniform expansion; individual surface forces cancel as vectors by <spherical symmetry>.
If an interior spherical component has <mass> $M_c$, the <spherical shell theorem> adds inward load $GM_cM_s/R^2$. The total inward load is
$$
\mathcal F_g=\frac{GM_s}{R^2}\left(M_c+\frac{M_s}{2}\right).
$$
For a swept gas fraction $f$ of an original enclosed <mass> $M$, with $M_s=fM$ and $M_c=(1-f)M$, this becomes $GM^2f(1-f/2)/R^2$. Omitting shell <self-gravity> would give $f(1-f)$ instead. In an accreting shell, evaluate the instantaneous load at its current mass; differentiating an assigned $M_s(R)$ in the fixed-mass energy formula would also include the energy of newly acquired material.
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