Roche model of a uniformly rotating star ($\varphi =-GM/r-{\Omega }^2{\varpi }^2/2$)

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.