Equatorial Kerr null geodesic radial equation ($r^4{\dot{r}}^2=[E(r^2+a^2)-aL{]}^2-\Delta (L-aE{)}^2$)

For an equatorial null geodesic of the Kerr metric, $E=-p_t$ and $L=p_{\varphi }$ are geodesic conserved quantities from Killing vectors. In units $G=c=1$, $\Delta =r^2-2Mr+a^2$, and an affine parameter gives the displayed equation. It follows by solving the stationary and axial momentum equations for $\dot{t},\dot{\varphi }$ and substituting them into the null condition. A zero of its radial polynomial is only a turning point; an exterior circular null geodesic also requires the derivative of that polynomial to vanish with $E,L$ held fixed.