For a timelike geodesic in the equatorial plane $\left(\theta=\frac{1}{2} \pi\right)$ of the Schwarzschild spacetime with line element

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

derive the equation

$$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2}(E / c)^{2}$$

where

$$\frac{2 V(r)}{c^{2}}=1-\frac{r_{s}}{r}+\frac{h^{2}}{c^{2} r^{2}}-\frac{h^{2} r_{s}}{c^{2} r^{3}}$$

and $h$ and $E$ are constants. The dot denotes the derivative with respect to an affine parameter $\tau$ satisfying $c^{2} d \tau^{2}=-d s^{2}$.

Given that there is a stable circular orbit at $r=R$, show that

$$\frac{h^{2}}{c^{2}}=\frac{R^{2} \epsilon}{2-3 \epsilon}$$

where $\epsilon=r_{s} / R$.

Compute $\Omega$, the orbital angular frequency (with respect to $\tau$ ).

Show that the angular frequency $\omega$ of small radial perturbations is given by

$$\frac{\omega^{2} R^{2}}{c^{2}}=\frac{\epsilon(1-3 \epsilon)}{2-3 \epsilon}$$

Deduce that the rate of precession of the perihelion of the Earth's orbit, $\Omega-\omega$, is approximately $3 \Omega^{3} T^{2}$, where $T$ is the time taken for light to travel from the Sun to the Earth. [You should assume that the Earth's orbit is approximately circular, with $r_{s} / R \ll 1$ and $\left.E \simeq c^{2} .\right]$