Critically rotating self-gravitating cylinder ($\Omega =kR,R_0^2=2\pi G{\rho }_0/k^2$)

A uniform-mass density self-gravitating cylinder has ${\Phi }_R=2\pi G{\rho }_{0}R$. Radial balance with the displayed rotation law gives $p_R={\rho }_0(k^2{R}^3-2\pi G{\rho }_{0}R)$. At central pressure ${\pi }^2{G}^2{\rho }_0^3/k^2$, pressure becomes ${\rho }_0{k}^2(R^2-R_0^2{)}^2/4$ and the free surface has zero effective gravity. The positive radius uses $|k|$ if either rotation direction is allowed.

The regular $m=1$ velocity family satisfies the displayed formulas. The surface pressure condition and gravitational matching give $W_s=-{\Omega }_s^2{R}_0\eta$, while the kinematic condition is $i(\omega +{\Omega }_s)\eta =u_R(R_0)$. If the material radial displacement $u_R/[i(\omega +kR)]$ is continued regularly to the boundary, then $\eta =-ia$ and ${\omega }^2=0$: translation of an isolated cylinder has no restoring force. Using only the Eulerian velocity conditions instead leaves ${\omega }^2(\omega +{\Omega }_s)=0$. The additional surface-corotation case has $\eta =0$, ${\Phi }^{\prime }=0$, $u_R=a(kR-{\Omega }_s)$ and $W=iaR[(kR{)}^2-{\Omega }_s^2]$; it satisfies the velocity equations and zero surface pressure. Excluding that degenerate branch is an extra admissibility assumption, not a valid division by a nonzero frequency at every surface.

For a uniform-mass density cylinder with surface displacement $\eta e^{im\varphi }$, the interior and exterior potential perturbations solve Laplace's equation and are proportional to $R^m$ and $R^{-m}$. Continuity of potential and the Poisson derivative jump ${\Phi }_{R,\mathrm{out}}^{\prime }-{\Phi }_{R,\mathrm{in}}^{\prime }=4\pi G{\rho }_0\eta$ give the displayed surface amplitude. The surface mass density term must be retained even though bulk incompressibility makes the interior mass density perturbation zero.