Pressure-confined isothermal sphere

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.

At fixed temperature, the particle count of an isothermal sphere is proportional to ${\rho }_c^{-1/2}f(\xi )$, where $f={\xi }^2{\Psi }^{\prime }$ and $f^{\prime }={\xi }^2{e}^{-\Psi }$. A fixed-mass variation therefore satisfies $\delta {\rho }_c/{\rho }_c=2e^{-\Psi }\delta \xi /{\Psi }^{\prime }$. The boundary density changes by $\delta \rho /\rho =\delta {\rho }_c/{\rho }_c-{\Psi }^{\prime }\delta \xi$. Its first stationary point occurs when the title condition holds. This follows a sequence of equilibria, rather than differentiating the radial profile of one equilibrium. The associated boundary radius obeys $\delta r/r=(1/\xi -e^{-\Psi }/{\Psi }^{\prime })\delta \xi$.

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.