Past exam of the physics B course of the University of Cambridge / 2011 / Paper 1 / Section B / B8 / b

Kepler's laws of planetary motion state that, in the relative two-body description, bound trajectories are ellipses with the attracting body at a focus, the radius vector sweeps equal areas in equal times, and the square of the period is proportional to the cube of the semimajor axis for a fixed total mass. Their Newtonian derivation is especially informative. With relative coordinate $\mathbf{r}$ and ${\mu }_g=G(M+m)$, the inverse-square acceleration is $-{\mu }_g\mathbf{r}/r^3$. Its zero torque conserves $\mathbf{h}=\mathbf{r}\times \dot{\mathbf{r}}$, fixing a plane and the areal velocity $h/2$.

For $u=1/r$, the orbit equation is $u^{\prime \prime }+u={\mu }_g/h^2$ as a function of polar angle. ThereforeBound negative-energy motion has $0\le e<1$ and semilatus rectum $h^2/{\mu }_g=a(1-e^2)$. Dividing ellipse area by areal velocity givesFor a planet much lighter than its star, replace $M+m$ by $M$. Zero-energy and positive-energy encounters are parabolic and hyperbolic rather than periodic ellipses. Perturbations from other bodies and departures from point-mass Newtonian gravity can precess or otherwise modify the orbit; Kepler's laws are a two-body approximation, not an exact rule for every observed planetary path.