von Zeipel theorem

Remarks

Hydrostatic equilibrium reads $\nabla P = -\rho \nabla \Psi$, which forces $P$ and $\rho$ to be functions of $\Psi$ alone, and then so are $T$ and $\kappa$. So $\nabla T = T'(\Psi) \nabla \Psi$, and the flux is $F = -f(\Psi) \nabla \Psi$ for a function $f$ that is constant on each level surface, which gives $\abs{F} = f(\Psi) \, g_{\mathrm{eff}}$. Since $\sigma T_{\mathrm{eff}}^4 = \abs{F}$ at the surface, with $\sigma$ the Stefan–Boltzmann constant, $T_{\mathrm{eff}} \propto g_{\mathrm{eff}}^{1/4}$. Example: a rapidly rotating star is flattened, and its effective gravity is larger at the poles than at the equator. So its poles are hotter and brighter than its equator; this is called gravity darkening. A consequence, known as von Zeipel's paradox, is that a uniformly rotating star cannot be in strict radiative equilibrium: $\nabla \cdot F$ does not vanish in general, which drives slow meridional circulation.

Going deeper

For a student

Gravity darkening has been observed by interferometry on fast rotators such as Altair and Vega. The law $T_{\mathrm{eff}} \propto g_{\mathrm{eff}}^{1/4}$ holds for radiative envelopes; for convective envelopes a smaller exponent is used in practice. The theorem was stated by Hugo von Zeipel in 1924.