= Uniformly rotating stellar hydrostatic equilibrium
{title2=$\nabla P=-\rho\nabla\phi,\ \nabla^2\phi=4\pi G\rho-2\Omega^2$}
Uniform <solid-body rotation> adds the <centrifugal potential> $-\Omega^2\varpi^2/2$ to the gravitational potential. In the corotating frame, <hydrostatic equilibrium> becomes $\nabla P=-\rho\nabla\phi$. The <Poisson equation> gives $\nabla^2\phi=4\pi G\rho-2\Omega^2$. On regular connected <equipotential surfaces>, $P$ is constant; taking the curl shows that $\rho$ is constant too. Uniform composition and an invertible equation of state then make $T$ constant on each such surface. These statements do not require $|\nabla\phi|$ to be constant there.
Back to article page
