 Fixed-mass turning point of an isothermal sphere

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 $\delta r/r=(1/\xi -e^{-\Psi }/{\Psi }^{\prime })\delta \xi$.