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arXiv · 1804.00606

Closed periodic orbits in anomalous gravitation

Bjorn A. Vermeersch
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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 two-body `sun/planet' system under anomalous gravitation $F \sim 1/r^α (1 \leq α< 2)$. 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 $α$, and discuss conceptual connections of the case $α= 1$ to Modified Newtonian Dynamics and galactic rotation curves.

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Bjorn A. Vermeersch. 2018-08-15. Closed periodic orbits in anomalous gravitation. https://arxiv.org/abs/1804.00606

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