Closed periodic orbits in anomalous gravitation

TL;DR

This paper explores the dynamics of two-body systems under gravitational forces with different inverse power laws, revealing conditions for closed periodic orbits and connecting these findings to modified gravity theories.

Contribution

It provides a detailed geometric analysis of orbits under anomalous gravitation laws with exponents between 1 and 2, identifying conditions for closed periodic orbits.

Findings

Existence of closed self-intersecting orbits resembling hypotrochoids.

Periodic orbits occur for specific initial conditions under anomalous gravity.

Connections made between the case $ extalpha=1$ and Modified Newtonian Dynamics.

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 $F \sim 1/r^{\alpha} (1 \leq \alpha < 2)$. 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.