Uniformly rotating stellar hydrostatic equilibrium ($\nabla P=-\rho \nabla \varphi ,{\nabla }^2\varphi =4\pi G\rho -2{\Omega }^2$)

Uniform solid-body rotation adds the centrifugal potential $-{\Omega }^2{\varpi }^2/2$ to the gravitational potential. In the corotating frame, hydrostatic equilibrium becomes $\nabla P=-\rho \nabla \varphi$. The Poisson equation gives ${\nabla }^2\varphi =4\pi G\rho -2{\Omega }^2$. 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 $|\nabla \varphi |$ to be constant there.

This approximation represents the gravitational potential by a central point mass while retaining the centrifugal potential. A surface through polar radius $R_p$ satisfies $R_p/r+q(r/R_p{)}^2{\sin }^2\theta /2=1$, where $q={\Omega }^2{R}_p^3/(GM)$. The inner closed surface is regular below $q=8/27$; at the limiting value its equatorial radius is $3R_p/2$ and equatorial effective gravity vanishes. The approximation is useful for illustrating oblateness and gravity darkening, rather than modeling the density distribution of every rotating star.

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.

Slow circulation in planes containing the rotation axis can transport entropy through a rotating star while the leading mechanical balance remains nearly hydrostatic equilibrium. Its steady mass flux satisfies the continuity equation. It is distinct from assuming an exactly motionless radiative interior, and from turbulent convection.

The thermal imbalance of a slowly rotating radiative star can drive stellar meridional circulation. Its fractional centrifugal distortion is ${ϵ}_{\Omega }\sim {\Omega }^2{R}^3/(GM)$, so balancing an order-${ϵ}_{\Omega }$ fraction of the luminosity by thermal advection estimates $t_{\mathrm{ES}}\sim t_{\mathrm{KH}}/{ϵ}_{\Omega }$, where $t_{\mathrm{KH}}\sim G{M}^2/(RL)$. Stable composition gradients can further impede the circulation. This radiative thermal estimate is different from rapid torque-driven circulation in a convective envelope.

For steady stellar meridional circulation, $T\nabla s=\nabla (h+\varphi )$ and $\nabla \cdot (\rho \mathbf{v})=0$. Thus $\rho \mathbf{v}\cdot T\nabla s=\nabla \cdot [\rho \mathbf{v}(h+\varphi )]$. In the leading uniformly rotating stellar hydrostatic equilibrium, $h+\varphi$ is constant on an equipotential surface; its total advected flux vanishes because the net mass flux vanishes. Integrating the energy equation then identifies the outward luminosity with enclosed nuclear energy generation, even though local heat advection is nonzero.

On a regular equipotential surface, $f,f^{\prime },\rho$ and ${\nabla }^2\varphi$ are constant, but $|\nabla \varphi {|}^2$ generally varies. Consequently a flux $\mathbf{F}=f(\varphi )\nabla \varphi$ cannot ordinarily satisfy local radiative equilibrium $\nabla \cdot \mathbf{F}=\rho ϵ$ at all latitudes. Special cases such as $f^{\prime }=0$ or spherical symmetry evade this obstruction. Stellar meridional circulation supplies heat advection in the leading hydrostatic treatment.

Gravity darkening is the variation of surface brightness and effective temperature with effective gravity over a rotating or tidally distorted star. Lower equatorial gravity generally corresponds to cooler equatorial regions. The radiative gravity-darkening law applies to the idealized uniformly rotating, radiatively transporting, composition-uniform case; its exponent is not universal for every stellar envelope.

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.