Uniform-density stellar model ($\rho =3M/(4\pi R^3)$)

A uniform-density spherical hydrostatic equilibrium model with zero surface pressure has $P(r)=2\pi G{\rho }^2(R^2-r^2)/3$, $M=4\pi \rho R^3/3$ and gravitational energy $-3G{M}^2/(5R)$. A perfect gas can have this mechanical structure with varying temperature and entropy; the imposed equilibrium density does not require incompressible perturbations.