= Critical radius of a pressure-confined isothermal sphere
{title2=$R_{\rm crit}\simeq6.451\sqrt{K/(4\pi G\rho_c)}$}
For the regular <isothermal Lane-Emden equation>, the first fixed-mass pressure turning point occurs at $\xi_{\rm crit}\simeq6.451$. Thus the boundary radius there is $R_{\rm crit}\simeq1.820\,c_s/\sqrt{G\rho_c}$, where $c_s^2=K$ is the squared <isothermal sound speed>. The approximate choice $\xi_{\rm crit}=6.5$ 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.
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