Radiative gradient in a uniformly rotating star (${\nabla }_{\mathrm{rad}}=\frac{3\kappa PL}{16\pi acGm{T}^4}{\left(1-\frac{{\Omega }^{2}V}{2\pi Gm}\right)}^{-1}$)

The divergence theorem and Poisson equation give ${\int }_S\nabla \varphi \cdot d\mathbf{S}=4\pi Gm-2{\Omega }^{2}V$ on a bounding equipotential surface. Equipotential luminosity conservation with stellar circulation then fixes $f=L/(4\pi Gm-2{\Omega }^{2}V)$. Combining $dT/d\varphi =-f/\chi$, $dP/d\varphi =-\rho$ and radiative conductivity gives the displayed logarithmic temperature-pressure gradient. Both appearances of mass mean the enclosed mass $m$; only on the outer boundary can both be replaced by the total mass.