= Solution
Define the dimensionless right-hand side of part (d) by $F(\kappa)$. Then $\omega^2=-4\pi G\rho F(\kappa)$. Because $I'_0(\kappa)=I_1(\kappa)>0$ and $I_0(\kappa)>0$ for $\kappa>0$, its sign is the sign of $I_0(\kappa)K_0(\kappa)-1/2$. The original plot is positive for sufficiently long <wavelengths> and negative at short <wavelengths>; the nonzero crossing is approximately $\kappa_c\simeq1.07$. Numerically solving the displayed equation gives $\kappa_c\simeq1.0667$.
For $0<\kappa<\kappa_c$, $\omega^2<0$. Write $\omega=i\sigma$ with $\sigma=\sqrt{4\pi G\rho F}>0$. The time factor $e^{-i\omega t}$ then contains an exponentially growing solution. For $\kappa>\kappa_c$, $\omega^2>0$ and the ideal inviscid modes oscillate stably, without damping. At the crossing the <frequency> is zero and the mode is marginal. In terms of the axial <wavelength> $\lambda_z=2\pi/k$,
$$
\boxed{\begin{aligned}
\lambda_z>\frac{2\pi R}{\kappa_c}\simeq5.89R&:\quad\text{gravitationally unstable},\\
\lambda_z<5.89R&:\quad\text{oscillatory and stable}.
\end{aligned}}
$$
The <growth rate> tends to zero as $kR\to0^+$, so the very longest <wavelengths> are not the fastest growing. The source plot has its maximum near $kR\simeq0.58$, corresponding to $\lambda_z\simeq11R$ and $\sigma\simeq0.25\sqrt{4\pi G\rho}$. Thus a characteristic fragmentation spacing is several cylinder diameters; a quantitative nonlinear fragment mass is beyond this <linear stability> calculation.
For unstable modes the physical displacement must be interpreted as $\operatorname{Re}[\epsilon e^{i(kz-\omega t)}]$ with complex amplitudes handled consistently throughout. The paper's real-part notation is harmless for the squared <dispersion relation>, but it should not be read as imposing real $\omega$ in a <gravitational instability> calculation. The exactly uniform mode $k=0$ is excluded from the finite-$k$ derivation; the limit of the curve alone does not establish a separate volume-preserving uniform-radius mode.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2022-q6-cylinder-rates.png]
{title=Rates computed from the cylinder dispersion relation as a function of wavelength, showing stable oscillation and unstable growth}
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