= Axisymmetric gravitational instability of an incompressible cylinder
{title2=$\omega^2=4\pi G\rho\kappa I_1(\kappa)[1/2-I_0(\kappa)K_0(\kappa)]/I_0(\kappa)$}
An inviscid uniform-density self-gravitating cylinder in vacuum, with no rotation or <surface tension>, has the displayed axisymmetric <dispersion relation>, where $\kappa=|k_z|R$. To derive it, let $W=\psi+\pi/\rho=\epsilon\Pi_0I_0(kr)e^{i(kz-\omega t)}$. The linear <kinematic boundary condition> gives $\Pi_0=\omega^2/[kI'_0(\kappa)]$. The <Lagrangian pressure perturbation> vanishes on the moving <free surface>, giving $\pi(R)=2\pi G\rho^2R\eta$. Substitute the <axisymmetric gravitational potential of a displaced cylindrical interface> and use $I'_0=I_1$ to obtain the relation. Its nonzero neutral root is $\kappa_c\simeq1.0667$: wavelengths longer than $2\pi R/\kappa_c\simeq5.89R$ have $\omega^2<0$ and grow. The fastest-growing mode has $\kappa\simeq0.58$, so its <wavelength> is about $11R$. Negative $\omega^2$ must be interpreted through an imaginary <frequency>, not through a real oscillatory ansatz.
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