Jeans instability

Remarks

Linearise the continuity equation, the momentum equation and Poisson's equation, and eliminate the velocity and the potential: the density perturbation $\rho_1$ satisfies $\partial_t^2 \rho_1 = c_s^2 \nabla^2 \rho_1 + 4\pi G \rho_0 \rho_1$. A plane wave then gives the dispersion relation. For $\abs{k} > k_J = \sqrt{4\pi G \rho_0}/c_s$, $\omega$ is real and the perturbation is a sound wave modified by gravity. For $\abs{k} < k_J$, $\omega$ is imaginary and one solution grows like $e^{\abs{\omega} t}$. The growth rate approaches $\sqrt{4\pi G \rho_0}$ for long waves. Example: take a molecular cloud core at $10$ K with about $10^4$ hydrogen molecules per cubic centimetre. Its Jeans length is about $0.2$ parsec, so larger regions tend to collapse. The assumption about the background is not consistent, since a uniform static medium cannot be in equilibrium under its own gravity. Consistent treatments confirm the conclusion, for example the Jeans instability in an expanding universe.

Going deeper

For a student

The criterion compares the time a sound wave needs to cross a region with the time gravity needs to make it collapse, the free-fall time. The same dispersion relation, with a correction for rotation, gives the stability criteria for galactic disks. The Jeans mass is, in one common convention, the mass inside a sphere of diameter $\lambda_J$. It is the minimum mass that can collapse; it decreases as the density grows, which suggests fragmentation of collapsing clouds. For a cloud of finite size bounded by external pressure, the corresponding threshold is the Bonnor–Ebert mass.