Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 4 / 10B / ii / Solution

Choose the gravitational potential energy to vanish at infinity. The total mechanical energy isPut $\gamma =GM/h^2$ and $\phi =\theta -{\theta }_0$. From part (i), $u=\gamma (1+e\cos \phi )$ and $u^{\prime }=-\gamma e\sin \phi$. Consequentlyand the angular terms cancel against the gravitational potential energy:For a nonradial Kepler orbit, negative, zero and positive mechanical energy correspond respectively to $e<1$, $e=1$ and $e>1$.