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

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- Solution a

Solution

The three terms represent competing physical effects:

$Ω^{2}$ is the stabilizing epicyclic motion of a radially displaced fluid element in a Keplerian disk.
$- 2 π G Σ∣ k ∣$ is the destabilizing self-gravity of a surface-density perturbation, obtained from the razor-thin disk Poisson kernel.
$c_{s}^{2} k^{2}$ is the stabilizing pressure response, strongest at short wavelength.

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