Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2022 / Section B / 6 / e

Define the dimensionless right-hand side of part (d) by $F(\kappa )$. Then ${\omega }^2=-4\pi G\rho F(\kappa )$. Because $I_0^{\prime }(\kappa )=I_1(\kappa )>0$ 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 ${\kappa }_c\simeq 1.07$. Numerically solving the displayed equation gives ${\kappa }_c\simeq 1.0667$.

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 ${\lambda }_z=2\pi /k$,The growth rate tends to zero as $kR\to 0^{+}$, so the very longest wavelengths are not the fastest growing. The source plot has its maximum near $kR\simeq 0.58$, corresponding to ${\lambda }_z\simeq 11R$ and $\sigma \simeq 0.25\sqrt{4\pi G\rho }$. 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 $\mathrm{Re}[ϵe^{i(kz-\omega t)}]$ 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.