# Can a planet be on a stable orbit at a Lagrangian point of a binary star system?

Suppose we have a binary star system and a planet at the L4 or L5 point of the orbit of one of the stars around the other. The planet is thus illuminated by the two stars from the same distance and the angle between the stars is about 60 degrees. If the planet is tidally locked, it will be constantly illuminated by two non-moving stars. Is this possible?

• Short answer, yes it is possible, but the planet would likely be too hot for life as we know it. For the Lagrange point to be in the "goldilocks zone", the stars would have to be orbiting each other at a distance similar to that of our sun and Jupiter, and even then with the planet constantly illuminated it would be a desert world with most of the liquid water buried underground. A real answer's going to need more details about the classes of stars in the binary system and what climate you want. – KeithS Oct 29 '15 at 14:22
• KeithS, go ahead and submit that as an answer. I agree with the initial point (it would be stable) but disagree on your conclusion (I think it might or might not be too hot depending upon the specifics of the system). – Jim2B Oct 29 '15 at 17:26

## 3 Answers

I saw a reference once that indicated the magic ratio for the L4 / L5 points to remain stable, is 9 / 1 (primary / secondary). As long as your world was less than 1/9 of the mass of the smaller star, it could remain in a stable orbit in the secondary's L4 / L5 points.

Whether this allows the planet a habitable temperature depends upon the specifics of the orbital parameters. For instance, use a G2 start (like our Sun) and a barely fusing M9 dwarf as the secondary. Place the secondary a little further out than Earth's orbit and Voila, you have a habitable planet.

Tidal forces between your planet and the primary at those distances would less than the Solar tides the Earth experiences. The tidal forces between the Earth and the secondary would be even smaller. So the planet would not be tidally locked.

Of course many different configurations could be used. However, you wouldn't want the tidal forces too get too strong or you'd end up with a planet tidally locked to not just one star but too. Or maybe that is what you want :)

• According to a world building book I have and wikipedia the L4 and L5 points aren't stable unless the difference in mass between the two suns in this case is 25. If smaller sun must have a mass 25 times smaller than the larger one then I kind of assume one would be brighter than the other since they are at the same distance. I don't know that much about suns. I do know that having one 25 times larger than another is not problem, but I don't know if somehow a smaller sun can give off the same amount of light as a larger one. Anixx didn't say if that was important. – ozone Oct 31 '15 at 11:12
• A dim red star with $M_{\text{red dwarf}} = 0.08 \times M_{\text{Sun}}$ would have a luminosity $L_{\text{red dwarf}} = 0.0006 \times L_{\text{Sun}}$. A body with mass $= M_{\text{Sun}} / 25$ would not ignite. – Jim2B Nov 2 '15 at 4:19
• So minimum star mass to get fusion ignition is about $0.08 \times M_{Sun}$. Which means the primary must be at least $25 \times 0.08 = 2.0 \times M_{Sun}$ – Jim2B Nov 29 '15 at 5:22
• @ozone, a follow-up to our original comments. Later I saw something that said size difference between primary and secondary of 25:1 (as you said) but size difference between secondary and tertiary of 9:1. Which means the minimum sizes are Primary > $2 \cdot M_{Sun}$, Secondary ~$0.08 \cdot M_{Sun}$, Tertiary (planet) < $0.009 \cdot M_{Sun}$ (which is about $8 \cdot M_{Jupiter}$ I think). – Jim2B Mar 12 '16 at 2:22

Sorry, due to only being able to edit comments for 5 minutes (it took me longer than that to even complete the comment) when I decided to split the comment into an answer they got messed up.

If smaller sun must have a mass 25 times smaller than the larger one then I kind of assume one would be brighter than the other since they are at the same distance. I don't know that much about suns. I do know that having one 25 times larger than another is not problem, but I don't know if somehow a smaller sun can give off the same amount of light as a larger one. Anixx didn't say if that was important. If it were me, I would find two suns that would work, then place them so that the planet had the right climate and calculate what kind of orbit they had. Lather rinse repeat.

It might also be possible to have two suns that orbit each other so closely that they almost exchange atmospheres and have the planet orbit both of them. I'm not sure how the tidal lock would work, but intuitively, I figure the planet could lock to the barycenter. Again, these things are fairly easily calculated and it gives more freedom for the types of suns.

Hope this helps.

• stackexchange hint: when you want to change a comment after 5 minutes, you can just delete it and post a new one. – Philipp Oct 31 '15 at 12:53

The Langrangian points L4 and L5 are stable, but not very stable. When the two binary stars and the planet* are the only non-negligible masses in the vicinity and the mass of the planet is very small compared to that of the stars, then the configuration would be stable.

But add any disturbing factors, like additional planets and moons, and the whole configuration will become unstable.

In our solar system, many planets have small asteroids at their L4 and L5 points, but the majority of them are only temporary because the gravity influence by other planets soon knocks them out of their orbits.

For further reading, I recommend the Wikipedia article Trojan (astronomy).

Regarding tidal locking: It's unlikely. For tidal locking to happen, the satellite must be not perfectly spherical. It only works because different points of the satellite experience different gravity and due to the uneven form this generates torque forcing the satellite to face the gravity source. The necessary gravity gradient only appears when the gravity source is very close compared to the irregularity of the planet. In the given configuration, the suns must be quite far away or they will bake the satellite (in our solar system, the only planet tidally locked to the sun is Mercury) or the satellite must be very irregular in shape (unlikely when it is large enough to have non-negligible gravity itself).

For further reading, I recommend the Wikipedia article on tidal locking.

*It's technically not a planet when it shares its orbit with a much larger mass

• "For tidal locking to happen, the satellite must be not perfectly spherical." - the body changes its shape due to tides. – Anixx Oct 31 '15 at 18:26
• "(in our solar system, the only planet tidally locked to the sun is Mercury" - it is not – Anixx Oct 31 '15 at 18:27
• It is. 3:2 resonance is locked as any small change will cause negative feedback. – JDługosz Nov 1 '15 at 0:42