Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2015 / Paper 1 / 4 / a / Solution

For a uniform sphere, $M=4\pi \rho R^3/3$ and its gravitational potential energy is $\Omega =-3G{M}^2/(5R)$. The ideal-gas kinetic energy is $E_{\mathrm{kin}}=3M{k}_{B}T/(2\bar{m})$. Equating these magnitudes using the virial theorem gives $GM/R=5k_{B}T/\bar{m}$. Eliminating $R$ yields the uniform-sphere pressure-work collapse criterionIn particular, for fixed composition,The same Jeans mass scaling follows from $c_s^2{k}^2\sim 4\pi G\rho$ in the Jeans instability dispersion relation and a characteristic size $k^{-1}$. Pressure communicates support in a sound-crossing time; sufficiently strong self-gravity wins when the gravitational time is shorter.

The numerical coefficient depends on the equilibrium, boundary and wavelength-to-cloud conventions. A sphere with both exactly uniform density and uniform temperature is not an exact hydrostatic self-gravitating ideal-gas equilibrium without additional stresses. The uniform-sphere argument establishes the characteristic mass scaling; it does not prove an exact stability boundary for that nonexistent equilibrium.