Hydrostatic self-gravitating incompressible cylinder ($p(r)=p_{\mathrm{ext}}+\pi G{\rho }^2(R^2-r^2)$)

For an infinitely long circular cylinder of uniform mass density $\rho$, the gravitational Poisson equation and the divergence theorem give $d\Phi /dr=2\pi G\rho r$ internally. The hydrostatic equilibrium equation $dp/dr=-\rho d\Phi /dr$ and boundary value $p(R)=p_{\mathrm{ext}}$ then give the title formula. This model neglects rotation, surface tension and end effects.