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For the regular isothermal Lane-Emden equation, the first fixed-mass pressure turning point occurs at ${\xi }_{\mathrm{crit}}\simeq 6.451$. Thus the boundary radius there is $R_{\mathrm{crit}}\simeq 1.820c_s/\sqrt{G{\rho }_c}$, where $c_s^2=K$ is the squared isothermal sound speed. The approximate choice ${\xi }_{\mathrm{crit}}=6.5$ gives coefficient $1.83$. Comparing with the Jeans length requires specifying the reference density and remembering that the latter is a wavelength rather than a boundary radius.

Set $\rho ={\rho }_c{e}^{-\Psi (\xi )}$ and $r=a\xi$. Then $d\ln \rho /dr=-a^{-1}{\Psi }^{\prime }$ and the equation from part (a) becomesWith $a^2=K/(4\pi G{\rho }_c)$, the coefficients cancel, giving the isothermal Lane-Emden equationThe regular central boundary conditions areThe first makes the central mass density equal to ${\rho }_c$. The second follows from smooth spherical symmetry and the vanishing central gravitational acceleration; a preferred radial slope at the origin would be singular. Substitution of a regular power series giveswhich supplies nonsingular starting data for numerical integration. The central conditions select the regular solution, not the singular isothermal sphere.

The particle count is the enclosed mass divided by the particle mass $m$:Integrating the isothermal Lane-Emden equation giveswhere regularity makes the lower endpoint vanish. ThusThe prefactor is a mass divided by a mass, and the remaining factor is dimensionless. In particular, at fixed $K$ the dependence on central mass density is $N\propto {\rho }_c^{-1/2}f(\xi )$, with $f(\xi )={\xi }^2{\Psi }^{\prime }(\xi )$. Both changes must be retained when comparing equilibria at fixed $N$.

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.