= Fixed-mass turning point of an isothermal sphere
{title2=$[\Psi'(\xi)]^2=2e^{-\Psi(\xi)}$}
At fixed temperature, the particle count of an <isothermal sphere> is proportional to $\rho_c^{-1/2}f(\xi)$, where $f=\xi^2\Psi'$ and $f'=\xi^2e^{-\Psi}$. A fixed-mass variation therefore satisfies $\delta\rho_c/\rho_c=2e^{-\Psi}\delta\xi/\Psi'$. The boundary density changes by $\delta\rho/\rho=\delta\rho_c/\rho_c-\Psi'\delta\xi$. 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 $\delta r/r=(1/\xi-e^{-\Psi}/\Psi')\delta\xi$.
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