The <Toomre stability criterion> for a razor-thin isothermal gas disk uses
$$
\boxed{Q=\frac{c_s\kappa}{\pi G\Sigma}}.
$$
Self-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 $Q\geq1$ and unstable for $Q<1$.

For a Keplerian disk, $\kappa=\Omega$, $H\simeq c_s/\Omega$, $\Omega^2\simeq GM_*/r^3$, and $\Sigma\sim M_D/(\pi r^2)$. Therefore
$$
Q\sim\frac{H\Omega^2r^2}{GM_D}
\sim\frac Hr\frac{M_*}{M_D}.
$$
The instability condition $Q\lesssim1$ is equivalently
$$
\boxed{\frac{M_D}{M_*}\gtrsim\frac Hr}.
$$

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-321/3/a/solution