Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 2 / c / Solution

Put $x=kH$, with $H=c_s/\Omega$ and $Q=c_s\Omega /(\pi G\Sigma )$. The dispersion relation becomesThe two neutral roots areReal roots enclosing a range with ${\omega }^2<0$ exist precisely whenwhich is the Toomre stability criterion for axisymmetric instability. For $Q\ll 1$, the Taylor expansion of the square root gives