The Schwarzschild metric for a spherically symmetric black hole is given by

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

where we have taken units in which we set $G=c=1$. Consider a photon moving within the equatorial plane $\theta=\frac{\pi}{2}$, along a path $x^{a}(\lambda)$ with affine parameter $\lambda$. Using a variational principle with Lagrangian

$$L=g_{a b} \frac{d x^{a}}{d \lambda} \frac{d x^{b}}{d \lambda}$$

or otherwise, show that

$$\left(1-\frac{2 M}{r}\right)\left(\frac{d t}{d \lambda}\right)=E \quad \text { and } \quad r^{2}\left(\frac{d \phi}{d \lambda}\right)=h$$

for constants $E$ and $h$. Deduce that

$$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$$

Assume now that the photon approaches from infinity. Show that the impact parameter (distance of closest approach) is given by

$$b=\frac{h}{E}$$

Denote the right hand side of equation $(*)$ as $f(r)$. By sketching $f(r)$ in each of the cases below, or otherwise, show that:

(a) if $b^{2}>27 M^{2}$, the photon is deflected but not captured by the black hole;

(b) if $b^{2}<27 M^{2}$, the photon is captured;

(c) if $b^{2}=27 M^{2}$, the photon orbit has a particular form, which should be described.