= Critically rotating self-gravitating cylinder
{title2=$\Omega=kR,\quad R_0^2=2\pi G\rho_0/k^2$}
A uniform-<mass density> self-gravitating cylinder has $\Phi_R=2\pi G\rho_0R$. Radial balance with the displayed rotation law gives $p_R=\rho_0(k^2R^3-2\pi G\rho_0R)$. At central <pressure> $\pi^2G^2\rho_0^3/k^2$, <pressure> becomes $\rho_0k^2(R^2-R_0^2)^2/4$ and the <free surface> has zero effective gravity. The positive radius uses $|k|$ if either rotation direction is allowed.
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