Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section B / 5 / d / Solution

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.