 Focal length of a rotating liquid mirror

A liquid in steady rigid rotation has a free surface $z-z_0={\Omega }^2{r}^2/(2g)$. Comparing with a paraboloid $z-z_0=r^2/(4f)$ gives $f=g/(2{\Omega }^2)$. The result follows from hydrostatic equilibrium in the rotating frame and is independent of density.