Put $x=kH$, with $H=c_s/\Omega$ and $Q=c_s\Omega/(\pi G\Sigma)$. The dispersion relation becomes
$$
\frac{\omega^2}{\Omega^2}=1-\frac{2x}{Q}+x^2.
$$
The two neutral roots are
$$
\boxed{k_{1,2}=\frac1{QH}
\left(1\mp\sqrt{1-Q^2}\right)}.
$$
Real roots enclosing a range with $\omega^2<0$ exist precisely when
$$
\boxed{Q<1},
$$
which is the <Toomre stability criterion> for axisymmetric instability. For $Q\ll1$, the <Taylor expansion> of the square root gives
$$
\boxed{k_1\simeq\frac{Q}{2H},
\qquad
k_2\simeq\frac{2}{QH}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-321/2/c/solution