Closed periodic orbits in anomalous gravitation

Newton famously showed that a gravitational force inversely proportional to the square of the distance, $F\sim 1\mathrm{/}{r}^2$, formally explains Kepler’s three laws of planetary motion. But what happens to the familiar elliptical orbits if the force were to taper off with a different spatial exponent? Here we expand generic textbook treatments by a detailed geometric characterisation of the general solution to the equation of motion for a two-body `sun/planet’ system under anomalous gravitation $F\sim 1\mathrm{/}{r}^{\alpha }(1\le \alpha <2)$. A subset of initial conditions induce closed self-intersecting periodic orbits resembling hypotrochoids with perihelia and aphelia forming regular polygons. We provide time-resolved trajectories for a variety of exponents $\alpha$, and discuss conceptual connections of the case $\alpha =1$ to Modified Newtonian Dynamics and galactic rotation curves.