= Uniform-sphere pressure-work collapse criterion
{title2=$M_{\rm crit}=[5k_BT/(Gm)]^{3/2}[3/(4\pi\rho)]^{1/2}$}
For a uniform <ideal gas> sphere, set $\rho=mn$, $p=nk_BT$ and $M=4\pi\rho R^3/3$. Building the sphere by concentric shells gives <gravitational potential energy> $W=-3GM^2/(5R)$. During a small homologous contraction, released gravitational <energy> balances the <work> against gas <pressure> when $3GM^2|dR|/(5R^2)=4\pi R^2p|dR|$, or $GMm/R=5k_BT$. Eliminating $R$ gives the displayed mass. At the same initial <mass density> and <temperature>, larger masses have stronger gravitational contraction than pressure support in this estimate. Its numerical coefficient belongs to the uniform-sphere pressure-work model; the infinite-medium <Jeans instability> and a <pressure-confined isothermal sphere> use different equilibrium and boundary assumptions.
Back to article page
