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Along a family of equilibria at fixed temperature, $K$ and the dimensionless function $\Psi$ are fixed. The boundary mass density changes through both ${\rho }_c$ and the boundary coordinate $\xi$:From part (c), $N=C{\rho }_c^{-1/2}f(\xi )$, where $C$ is constant at fixed temperature and $f={\xi }^2{\Psi }^{\prime }$. Set its logarithmic variation to zero:The isothermal Lane-Emden equation gives $f^{\prime }={\xi }^2{e}^{-\Psi }$, so the fixed-$N$ relation isSubstitute back into the boundary-density variation:For a nontrivial variation at a finite positive radius, its vanishing gives the fixed-mass turning point of an isothermal sphere:This derivative follows a fixed-mass sequence of equilibria; it is not merely the spatial derivative of density inside one sphere. The corresponding radial variation isAt small $\xi$, ${\Psi }^{\prime }\simeq \xi /3$: the boundary density increases while the radius decreases along increasing $\xi$, giving the stated negative $(\partial p/\partial V{)}_{N,T}$ of the stable branch. The first density turning point marks its first loss of radial stability; it need not be confused with a turning point of radius itself.

A pressure-confined isothermal sphere is a regular isothermal sphere truncated at a boundary where its internal pressure equals an imposed external pressure. The finite central density and the boundary density generally differ. Its radial stability depends on the dimensionless truncation radius as well as the fixed mass and temperature. The first fixed-mass turning point of an isothermal sphere occurs at dimensionless radius about $6.451$; the modestly concentrated branch is stable to the radial isothermal perturbations in this model.