The three terms represent competing physical effects:

* $\Omega^2$ is the stabilizing <epicyclic motion> of a radially displaced fluid element in a <Keplerian disk>.
* $-2\pi G\Sigma|k|$ is the destabilizing self-gravity of a surface-density perturbation, obtained from the <razor-thin disk Poisson kernel>.
* $c_s^2k^2$ is the stabilizing pressure response, strongest at short wavelength.

Solution

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

 Solution

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