 Radiative gravity-darkening law

In uniformly rotating stellar hydrostatic equilibrium, composition-uniform thermodynamic quantities are functions of the total potential. Radiative conductivity gives $\mathbf{F}=f(\varphi )\nabla \varphi$, with $f=-\chi dT/d\varphi$. On one surface, $f$ is constant, so the outward flux is $F_n=fg$. The Stefan–Boltzmann law therefore gives $T_e\propto g^{1/4}$. This is a surface-flux law: the material temperature constant on an interior equipotential is not the latitude-dependent emergent effective temperature.