Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 35A / Solution

In the equatorial plane, the Killing symmetries give conserved quantities $E=(1-2M/r)\dot{t}$ and $L=r^2\dot{\varphi }$. The null normalization givesFor $L\ne 0$, put $u=1/r$; then $du/d\varphi =-\dot{r}/L$. ConsequentlyDifferentiating gives $u^{\prime \prime }+u=3M{u}^2$, initially where $u^{\prime }\ne 0$ and then at turning points by continuity. The circular solution follows directly from the radial equation as well. The ratio $|L/E|$ is the impact parameter of the asymptotic null orbit.