= Solution
Using the supplied first critical value $\xi_c\simeq6.5$, the <critical radius of a pressure-confined isothermal sphere> is
$$
\boxed{r_c=6.5\sqrt{\frac{K}{4\pi G\rho_c}}
\simeq1.83\frac{c_s}{\sqrt{G\rho_c}},\qquad c_s^2=K.}
$$
No radius in metres can be specified without $K$ and a central <mass density>. Here $c_s$ is the <isothermal sound speed>, consistent with the fixed-temperature variations in part (d). At a given central <mass density>, equilibria truncated at $\xi<\xi_c$ lie on the first stable branch, while increasing the dimensionless outer radius beyond this first threshold gives an unstable configuration. Numerical integration gives $\xi_c\simeq6.451$, consistent with the paper's rounded $6.5$.
For a transparent comparison with the given <Jeans length>, set the uniform comparison density $\rho_0=\rho_c$ and use the same <speed of sound>. Then
$$
\boxed{\frac{r_c}{\lambda_J}=\frac{6.5}{2\pi}\simeq1.03.}
$$
Both scales express the competition between <pressure> support and <self-gravity>, with characteristic length $c_s/\sqrt{G\rho}$. Their numerical closeness under this density choice does not mean that the models impose the same threshold: $r_c$ is a boundary radius, whereas $\lambda_J$ is the wavelength of a perturbation. Comparing $r_c$ with $\lambda_J/2$ would instead give a factor about $2.07$. Using the sphere's mean or boundary density instead of its central density changes the coefficient again, so the reference density must always be stated.
The <pressure-confined isothermal sphere> gives an actual finite, centrally concentrated <hydrostatic equilibrium> and an explicit confining <pressure>, making its radial stability threshold more self-consistent for a roughly spherical isothermal cloud. The uniform <Jeans instability> model gives a simpler local criterion, includes perturbations at different <wavelengths>, and does not require constructing a spherical equilibrium. Its ideal uniform static infinite background requires the <Jeans swindle> or an explicitly specified treatment of background gravity. Neither model includes rotation, <magnetic fields>, turbulent support or a realistic temperature evolution, so neither is a universal collapse criterion.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2021-q5-isothermal-sphere.png]
{title=Numerically integrated isothermal-sphere density profile and the first pressure turning point along a fixed-mass sequence}
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