At the equilibrium thickness $H_0$,
$$
\Omega_z^2H_0=CH_0^{-\gamma}.
$$
Set $H=H_0+h$ and retain terms linear in $h$. The <linearization> is
$$
\ddot h+(\gamma+1)\Omega_z^2h=0.
$$
Thus the <vertical breathing mode of an astrophysical disk> has angular frequency
$$
\boxed{\omega_H=\sqrt{\gamma+1}\,\Omega_z}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 3 / f / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-321/3/f/solution