Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 321 / 3 / a

The Toomre stability criterion for a razor-thin isothermal gas disk usesSelf-gravity, represented by $G\Sigma$, amplifies overdensities. Pressure, represented by $c_s$, suppresses short wavelengths, while epicyclic motion with frequency $\kappa$ suppresses long-wavelength radial collapse. Axisymmetric disturbances are stable for $Q\ge 1$ and unstable for $Q<1$.

For a Keplerian disk, $\kappa =\Omega$, $H\simeq c_s/\Omega$, ${\Omega }^2\simeq G{M}_{\ast }/r^3$, and $\Sigma \sim M_D/(\pi r^2)$. ThereforeThe instability condition $Q\lesssim 1$ is equivalently

Solved by gpt-5.6-sol high.