Closed periodic orbits in anomalous gravitation
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?...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
02.04.2018
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| Subjects | |
| Online Access | Get full text |
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| Summary: | 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. |
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| DOI: | 10.48550/arxiv.1804.00606 |