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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.

Write the initial mass density as $\rho =mn$. The ideal gas law gives pressure $p=n{k}_{B}T$, and the sphere has $M=4\pi \rho R^3/3$. Here $k_B$ is the Boltzmann constant and $G$ is the gravitational constant. First obtain the gravitational potential energy by adding concentric shells. The mass inside radius $r$ is $M(r)=M{r}^3/R^3$, soIn a small contraction $dR<0$ at fixed total mass, the gravitational energy released is $-dW=3G{M}^2|dR|/(5R^2)$. The positive work required against the gas pressure is $-pdV=4\pi R^{2}p|dR|$. Equating these givesUse $R=(3M/(4\pi mn){)}^{1/3}$ to eliminate the radius. The uniform-sphere pressure-work collapse criterion isThe units are kilograms: $(k_{B}T/(Gm){)}^{3/2}$ has units $\mathrm{k}{\mathrm{g}}^{3/2}{\mathrm{m}}^{-3/2}$, and ${\rho }^{-1/2}$ supplies $\mathrm{k}{\mathrm{g}}^{-1/2}{\mathrm{m}}^{3/2}$. At fixed initial $n$ and $T$, $M>M_{\mathrm{crit}}$ makes gravity stronger than the pressure-work requirement in this model, allowing contraction.

This is a Jeans mass estimate for homologous contraction of an initially uniform sphere. It is not an exact stability limit for a hydrostatic gas sphere: uniform mass density and uniform temperature would give uniform pressure, rather than the gradient required by hydrostatic equilibrium. The numerical coefficient therefore differs from conventions based on a Jeans instability wavelength or a pressure-confined isothermal sphere. Subsequent contraction also depends on heat exchange; isothermal pressure support during gravitational collapse weakens relative to gravity as the radius decreases.