# Is it possible for two stars to be in a horseshoe orbit around a much larger star?

I was reading about how Saturn has two moons, Janus and Epimetheus, that swap orbits once every four years. Could something like this happen on a much larger scale, but with stars instead of a planet and moons? Let's say you had a large, massive star with two red dwarfs orbiting around it. Could those two red dwarf stars fall into a horseshoe orbit, regularly swapping orbits every few years or so?

• That's amazing! I didn't know that Janus and Epimetheus are doing that. Very interesting question.
– uhoh
Commented Jan 24, 2018 at 8:55
• @Alchimista "swap" does not appear in the link. It might be a poor choice of words. The two moons are moving in approximately the same orbit. Imagine first that they are opposite each other, both rotating around Saturn. One drifts back slowly while the other advances, until one looks like it's "in front" of the other. Then the one in front speeds up, the one "behind" slows down, they pass through the moment when they are opposite each other and keep going until the one that used to be "in front" is now "behind". If you ask a separate question about how they move, it will allow a better answer.
– uhoh
Commented Jan 24, 2018 at 17:48
• @uhoh thanks for explanation. I will have a look in Celestia as I suspect the two satellite are handled there :) Commented Jan 24, 2018 at 18:14
• I've (kind of) worked out an answer to this but it's ugly. Debating on posting it cause it's pretty raw but I will if nobody else answers. In short, two stars passing each other every few years is impossible. The nature of the horseshoe orbit requires the orbits be close and being close, the orbital velocity is also close so it would take a long time for one star to catch the other. They would swap after hundreds or even thousands of orbits which would mean centuries or millennia. Other than the time-frame, I see no reason why this wouldn't be possible. Unlikely perhaps but possible. Commented Jan 25, 2018 at 14:54
• Janus and Epimethius, for example, are so close to Saturn that they orbit the planet in less than a day, and it takes about 2,000 orbits (about 4 years), for Janus to catch Epimethius or vise versa, in order for them to switch. They also don't exactly switch places as Janus is more massive, they kind of move around a theoretical barycentric orbit between them but closer to Janus. Epimethius moves about 3.6 km for every 1 km Janus moves due to conservation of momentum. Commented Jan 25, 2018 at 14:59

Yes, it is possible, however such orbits are likely to be unstable.

For example in the paper

Ćuk, Matija, Douglas P. Hamilton, and Matthew J. Holman. Long-term stability of horseshoe orbits. Monthly Notices of the Royal Astronomical Society 426.4 (2012): 3051-3056. doi:10.1111/j.1365-2966.2012.21964.x

it was found that if the mass of second body becomes larger than 1/1200 of the first body, horseshoe orbits become unstable. (Third body was assumed to have negligible mass).

Nonzero mass of the third body was investigated recently in a thesis:

Balaji, Bhaskaran. Mass ratios in stellar triple systems that admit horseshoe orbits. Diss. Massachusetts Institute of Technology, 2016. abstract, pdf

and it was found that in addition to constraint of Ćuk et al. nonzero mass of the third body further decreases stability of such systems, so that the lifetime of horseshoe orbit $\tau \propto m_3^{-1}$.

For stars only coupling of more massive giants with lighter red dwarfs would fulfill first constraint and then the third star would never have 'negligible' mass, one could conclude that horseshoe orbits when all three bodies are stars are always unstable. And for larger masses of smaller stars the lifetime of such orbits would be too short to expect such orbits in our vicinity.

However it would probably be possible to have stable horseshoe orbits of two stars (say a blue giant and red dwarf) and a planet.

• "However it would probably be possible to have stable horseshoe orbits of two stars (say a blue giant and red dwarf) and a planet." That would be an interesting system for a planet to be in. Commented Jan 31, 2018 at 23:00