# How would I make my Quintuple Star System Stable?

So I've been working on a star system that I want to have inhabitable planets, so by stable I mean stable on the timescale of the stars themselves; billions of years at least.

So my initial plan was a quintuple star system with two pairs of binaries and one lone star, all orbiting a common barycenter. The single star is a main sequence K-type star, an orange dwarf I think is the term. For now I've named it Shapash for the Canaanite sun goddess. The binary stars are (1) the "Majestic" pair, another K-type star with an M-type (red dwarf); and (2), the "Tempest" pair, a K-type star and a brown dwarf - I suppose this pair is more like the brown dwarf orbiting the orange dwarf.

But, I am worried that my system will run into the three-body problem (because its three "points" orbiting a common barycenter). Now, I don't know much about orbital mechanics, and I'm not sure how to make this system more stable. I have seen the article here about a quintuple system, but the single star is orbiting one of the binary star pairs. That's not quite what I want, but I'd settle with it if there is no other way for a quintuple system to be stable.

tl;dr how do I make a quintuple star system stable over billions of years?

• I was gonna say "Kemplerer rosette!" just for the joy of a Larry Niven reference, but then I looked up the wikipedia article and it corrected a lot of my notions (and my spelling) about this kind of formation. And it turns out they are not stable. :(
– Qami
Commented Aug 2, 2022 at 23:58
• Yeah; the kemplerer rosette is vaguely like what I'm describing - but yeah as you said its unstable. Commented Aug 3, 2022 at 2:25

For a multiple star system to have long term stable orbits, it should have what is called a hierarchical structure and spacing of the stellar orbits.

https://en.wikipedia.org/wiki/Star_system#Hierarchical_systems

A system where all the stars are about the same distance from the common center of mass would not be stable.

The two binaries should have stars relatively close together compared to the distances between the binaries themselves.

Thus a guadruple system would be easy to vizulize, with each binary consisting of two stars orbiting their center of gravity, and the two binaries orbiting the center of gravity of the two binaries.

And the fifth star should orbit the center of gravity of the whole system.

There are two possibilities.

Maybe the two binaries orbit around their common center of gravity and the single star orbits around the center of the whole system and at a much greater distance than the separation between the two binaries.

Maybe one binary and the single star obit their common center of gravity at a distance at least several times the separation of the stars in the binary, and the other binary orbits the center of gravity of all the stars, and at a much greater distance.

One question is the relative masses of the single star and of each component of the binaries and the total mass of each binary.

There ae three main sequence spectral class K stars in the system, the single star and one in each of the binaries. I know that all of the K type stars are main sequence stars, since you wrote that you want the system to have habitable planets.

Anyway, the masses of main sequence class K stars range from 0.59 the mass of the Sun for a K9V star to 0.88 the mass of the Sun for a K0V star.

https://en.wikipedia.org/wiki/K-type_main-sequence_star

Main sequence class M stars (red dwarfs) have masses ranging from 0.079 the mass of the Sun for M9V stars to 0.57 the mass of the Sun for M0V stars.

https://en.wikipedia.org/wiki/Red_dwarf

Brown dwarfs are neither planets nor stars, so techically your system is more like a quadruple star than a quintiple star. Anyway, brown dwarfs have a mass range of aobut 13 Jupiter masses to about 75 Jupiter masses. Since jUpiter isabout 1/1000 as massive as the Sun the mss range of brown dwarfs is from about 0.013 to 0.075 the mass of the Sun.

Any star which you want to have habitable planets orbiting it will have to be massive enough to have habitable planets orbiting it. And that could result in only the K class stars having habitable planets, and in the k class stars all being the heaviest K0V or K1V stars. That would mean the two binaries and the single would have almost the same mass, since the K class stars would have similar masses and the masses of the M class star and the brown dwarf would be very small compared to them.

There are several reasons why class M stars (and probably still more so brown dwarfs) might not be capable of having habitable planets orbiting them.

There is a discussion of the habitability of red dwarf systems at:

https://en.wikipedia.org/wiki/Habitability_of_red_dwarf_systems

The most important problem with red dwarfs is tidal locking to the star, which would result in one side of the planet having eternal day and heat and the other side having eternal night and cold. It has long been been believed that would make a planet uninhabitable. Fortunately recent studies suggest that a decent planetary atmosphere and hydrosphere should be sufficient to keep both hemispheres at habitable temperatures.

So if you accept that tidally locked planets can still be habitable, then you can have tidally locked planets even orbiting the red dwarf, if the other problems with red dwarfs aren't too bad. But if you don't want to risk depicting habitable tidally locked planets you have a problem, since tidal locking will also affect most K class stars. For example, in Habitable Planets for Man, 1964, pages 68 to 72, Stephen H. Dole calculated that the inner part of star's habitable zone would become close enough for tidal locking if a star had a mass of 0.88, the mass of the Sun,and the entire habitable zone would be close enough for tidal locking at a mass of 0.72 times the mass of the Sun.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

According to the Wikpedia table, a K0V star would have a mass of 0.88 Sun, while a star with 0.72 the mass of the Sun would be between a K5V and a K4V. So if you don't want to use habitable tidally locked planets, your class K stars have to be between K4V and K0V.

Unless the planets are tidally locked to a companion world. If one side always faces the companin world as the planet orbits the companion world, then each side o fhe planet would have alternating day and night as it faced toward and away from the star. Thus scientists have speculated a lot about potentially habitable giant exomoons orbiting giant exoplanets inthe habitable zones of K type and M type stars.

On pages 72 to 75 Dole decided that the lower mass limit for a star in such a situation would be about 0.35 the mass of the Sun, or between about M4V and M3V.

Obviously the singe class K star must orbit far enough from the two binaries to have planets in stable orbits around it.

What about the two binaries? If the two stars orbit each other closely enough, they can have a combined circumbinary habitable zone, and planets can orbit in that zone in what is called circumbinary or P-type orbit. If the two stars in a binary orbit far enough from each other, they can have separate circumstellar habitable zones, and planets in those zones can have stable orbits. An orbit around one of the stars in a binary is called a noncircumbinary or S-type orbit.

So you might want to try for having habitable planets in S-type orbits around the single K star, and in P-type orbits around each binary.

Or you might want to try for having habitable planets in S-type orbits around all four of the stars and the brown dwarf. There are problems with having habitable planets in orbits around class M stars, and presumably also with having habitable planets around brown dwarfs.

This study suggests that only the more massive brown dwarfs would be capable of having habitable planets orbiting them, but there still might be as many potentially habitable planets orbiting brown dwarfs as orbit normal stars.

https://iopscience.iop.org/article/10.3847/1538-4357/ab5b13#:~:text=When%20the%20brown%20dwarf%20has,habitable%20over%20geologically%20significant%20timescales.

Thus a writer could depict a star and a brown dwarf orbiting each other so closely that they can have habitable planets in P type orbits around both.

Or they can depict the star and the brown dwarf separated far enough that they stable S-type orbits are possible around both of them, but only the star canhave habitable planets.

Or they can depict the star and the brown dwarf separated far enough that they stable S-type orbits are possible around both of them, and both the star and the brown dwarf can (and maybe do) have habitable planets orbiting them. This third alternative would be the most daring.

"will run into the three-body problem" - never. It is only "problem" in mathematical sense, their orbits generally can't be expressed as a single mathematical expression because the system changes all the time. Not really a problem in real world. It is too unstable to be expressed as mathematical expression, it doesn't mean that everything collides and explodes.

Your five stars rotate around common center of mass, pairs of binaries additionally around their own centers of mass. Entire system easily can be stable at astronomic time scale. Depending on specifics individual stars or even pairs of binaries can have habitable worlds.

"how do I make a quintuple star system stable over billions of years?" - what makes you think it would be unstable?