# Would this simplex ternary system be stable?

I'm trying to make a trinary star system but I'm not sure how it would work. The primary star is an M0Ve star (red dwarf flare star). The orbits of the other two stars are slightly more complicated. Both other stars (T0 and Y9e) orbit around each other,and a planet orbits around the same barycenter, outside of the brown dwarfs (T0 and Y9e).

M0Ve:

• Mass: .47 solar (9.4*10^29 kg)
• Radius: .60 solar (420000 km)
• Luminosity: .041 solar
• Temperature: 3335 K

T0:

• Mass: .051 solar (1.0*10^29 kg)
• Radius: .111 solar (77000 km)
• Luminosity: .00010 solar
• Temperature: 1722 K

Y9e:

• Mass: .0135 solar (2.7*10^28 kg)
• Radius: .109 solar (76000 km)
• Luminosity: .00000005 solar
• Temperature: 260 K

Planet:

• Mass: .79 terrestrial (4.7*10^24 kg)
• Radius: .94 terrestrial (5989 km)
• Albedo: .42
• Insolation: 1450 W/m^2
• Atmosphere: 565 millibars
• Temperature: 310 K
• Geography: Global desert near equator, P4S3 oceans near poles
• Sidereal Day: 34.75 hours

The barycenter of Y9e and T0 is about .41 AU (61300000 km) from M0e, while the planet is about .072 AU (10800000 km) from the barycenter of the brown dwarfs. The brown dwarfs are very close to being ripped apart by each other, 500000 km away from each other (Roche limit is 290000 km).

Is this system stable on the scale of hundreds of billions of years?

• Have you confirmed that this particular arrangement is stable? It doesn't look like any of the particular stable arrangements that I've memorized. – Cort Ammon Aug 30 '18 at 18:15
• I'm not sure. That was one of the reasons why I asked. – Pyrania Aug 30 '18 at 18:17
• Can you be more specific about the sort of problem you are interested in? – L.Dutch - Reinstate Monica Aug 30 '18 at 18:18
• To the comment by Cort Ammom, I think it's a simplex trinary system. To the comment by L.Dutch, things like high radiation levels of flare stars, low levels of ionizing radiation for mutations, atmosphere reactivity, things like that. – Pyrania Aug 30 '18 at 18:21
• I'm not sure about the greenhouse effect, either. I'm going to calculate that. – Pyrania Aug 30 '18 at 18:27

First order approximation:

Distances and mass ratios have to be large. E.g. The mass of the sun compared to any of the planets is large. This makes the sun the dominant gravitational actor in our solar system.

Jupiter has a raft of moons, some of them quite large, but all are tiny compared to Jupiter.

The earth and moon are unusual:

• Mass ratio is only 80:1

• Moon is far enough from the earth that it's orbit around the sun is constantly concave toward the sun -- the orbit around the earth just changes how concave it is.

So if you want stability: The orbit of the binary pair around their barycentre has to be small compared to their orbit around their primary.

The orbit of the planet has to be large compared to orbit of the binary pair.

Despite being far from the binary pair, the planet should not be appreciably closer to the primary star over the course of its orbit.

Second order approximation: Look for resonances. Look for them in various coordinate systems.

Third order effects: A pair of stars, even small ones, that are orbiting just out of Roche limit I suspect will have some relativistic frame dragging.

I think you are going to have to model it and see.