Equatorial circular Kerr photon orbit ($r^2-3Mr\pm 2a\sqrt{Mr}=0$)

For $0<a<M$, the two exterior equatorial circular null geodesics of the Kerr metric have radii solving the signed equation above. The plus sign gives the prograde branch and the minus sign gives the retrograde branch. To derive it put $b=L/E$ in the double-root conditions of the equatorial Kerr null geodesic radial equation: $r^2=(b^2-a^2)/3$ and $r^3=M(b-a{)}^2$. Both exterior radii exceed $r_{+}=M+\sqrt{M^2-a^2}$ and approach $3M$ when $a\to 0$. The horizon generators are a separate case with $E=L=0$; dividing by $E$ excludes them. Constant-radius null directions at other exterior radii generally have nonzero acceleration and are not geodesics.