Paper 4, Section I, B

The motion of a planet in the gravitational field of a star of mass $M$ obeys

$\frac{\mathrm{d}^{2} r}{\mathrm{~d} t^{2}}-\frac{h^{2}}{r^{3}}=-\frac{G M}{r^{2}}, \quad r^{2} \frac{\mathrm{d} \theta}{\mathrm{d} t}=h$

where $r(t)$ and $\theta(t)$ are polar coordinates in a plane and $h$ is a constant. Explain one of Kepler's Laws by giving a geometrical interpretation of $h$.

Show that circular orbits are possible, and derive another of Kepler's Laws relating the radius $a$ and the period $T$ of such an orbit. Show that any circular orbit is stable under small perturbations that leave $h$ unchanged.

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