= Quadratic-polytrope disk with constant aspect ratio
{title2=$\Sigma=\frac{GM}{K}[\operatorname{arsinh}\epsilon-\epsilon/\sqrt{1+\epsilon^2}]$}
For $p=K\rho^2$, a point-mass <gravitational potential> and a zero-density free surface at $z=\pm\epsilon r$, the <barotropic cylindrical rotation theorem> gives $\Omega^2=GM/(r^3\sqrt{1+\epsilon^2})$. Vertical balance gives $\rho=GM[1/\sqrt{r^2+z^2}-1/(r\sqrt{1+\epsilon^2})]/(2K)$. Integrating over the finite height produces the displayed constant <surface density>. This is a local radial equilibrium family; infinite radial extent would have infinite mass.
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