Closed periodic orbits in anomalous gravitation
Abstract
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 twobody `sun/planet' system under anomalous gravitation $F \sim 1/r^{\alpha} (1 \leq \alpha < 2)$. A subset of initial conditions induce closed selfintersecting periodic orbits resembling hypotrochoids with perihelia and aphelia forming regular polygons. We provide timeresolved trajectories for a variety of exponents $\alpha$, and discuss conceptual connections of the case $\alpha = 1$ to Modified Newtonian Dynamics and galactic rotation curves.
 Publication:

arXiv eprints
 Pub Date:
 April 2018
 arXiv:
 arXiv:1804.00606
 Bibcode:
 2018arXiv180400606V
 Keywords:

 Physics  Popular Physics
 EPrint:
 This version discusses two additional references and corrects a previous error in Fig. 6. Animated orbits are available as mp4 video in submission files