  Newton famously showed that a gravitational force inversely proportional to
the square of the distance, $F \sim 1/r^2$, 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 $F \sim 1/r^{\alpha} (1 \leq \alpha < 2)$. 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.
