Setting $\nu=\nu_b=0$, the <dispersion relation> reduces to
$$
\lambda\left(\lambda^2+\Omega^2-2\pi G\Sigma_0k+v_s^2k^2\right)=0.
$$
An axisymmetric density mode grows when
$$
2\pi G\Sigma_0k-v_s^2k^2>\Omega^2.
$$
The left-hand side is a concave <quadratic function> of $k$, with maximum $(\pi G\Sigma_0)^2/v_s^2$ at $k=\pi G\Sigma_0/v_s^2$. Growth is therefore possible precisely when the <Toomre stability criterion> is violated:
$$
\boxed{Q=\frac{v_s\Omega}{\pi G\Sigma_0}<1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 1 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-321/1/d/solution