The Schwarzschild metric is

$$d s^{2}=\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)-\left(1-\frac{2 M}{r}\right) d t^{2}$$

Writing $u=1 / r$, obtain the equation

$$\frac{d^{2} u}{d \phi^{2}}+u=3 M u^{2}$$

determining the spatial orbit of a null (massless) particle moving in the equatorial plane $\theta=\pi / 2$.

Verify that two solutions of $(*)$ are

$$\begin{aligned} \text { (i) } u &=\frac{1}{3 M}, \quad \text { and } \\ \text { (ii) } \quad u &=\frac{1}{3 M}-\frac{1}{M} \frac{1}{\cosh \phi+1} . \end{aligned}$$

What is the significance of solution (i)? Sketch solution (ii) and describe its relation to solution (i).

Show that, near $\phi=\cosh ^{-1} 2$, one may approximate the solution (ii) by

$$r \sin \left(\phi-\cosh ^{-1} 2\right) \approx \sqrt{27} M$$

and hence obtain the impact parameter.