I Is a horseshoe orbit a hyperbolic orbit?

[Mike S.]

TL;DR Summary

Can a horseshoe orbit be considered as two moons following hyperbolic orbits around each other?

Epimetheus and Janus switch places periodically, because they follow a horseshoe orbit around Saturn, which is considered a "pseudo-orbit" around each other. I'm thinking that if you look at the conic sections - taking an elliptical orbit of two moons to greater and greater extremes until they don't come back together again - you might end up at a horseshoe orbit, and that you might then view that as a hyperbolic orbit in which, due to the presence of the planet, the moons must inevitably meet up again. Maybe you could represent it as a hyperbolic orbit inside some manifold or curved space?? My intuition is sniffing around and I'm wondering if you can point it in the right direction. :)

 

[Filip Larsen]

If I understand your question right then, yes, you might argue that the two moons may be, for each close (enough) encounter they have, be seen as in a (perturbed) hyperbolic orbit relative to each other while they both are also in a bound orbit around Saturn, but I would also say it would be pointless classification since the configuration really is a type of three body situation that does not model well as a series of two-body patched conics trajectories.

To understand the the configuration and its apparent stability its probably more useful to look at the resonance effect in the full dynamics. By the way, https://www.planetary.org/articles/janus-epimetheus-swap has a nice short explanation.

 

[Drakkith]

I'm with Filip. A hyperbolic orbit is a description of a 2-body interaction, or one in which you can closely approximate a 2-body interaction. To elaborate a bit, a hyperbolic orbit is a type of Keplerian orbit. Per wiki a Keplerian orbit is: the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on.

A horseshoe orbit is inherently a multi-body interaction and does not stay in a single plane like a Keplerian orbit does. Whatever you'd have to do to somehow fit a horseshoe orbit into the classification of a hyperbolic orbit would undoubtedly take things out of classical gravitation and thus away from the simple Keplerian orbits anyways.