<p>The point you appear to refer to, is called the <a href="https://en.wikipedia.org/wiki/Lagrangian_point" rel="noreferrer">Lagrangian point</a> $L_1$.
This point is a saddle in the field of gravity, hence not to be considered to be stable in the strict sense.
Two other Lagrangian points, called $L_4$ and $L_5$, can be stable, provided the considered orbiting objects are of small mass in comparison to the two main bodies of the system, and if the masses of the binary components are sufficiently different.</p>

<p>According to theorem 4.1 of <a href="http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf" rel="noreferrer">this paper</a>, $L_4$ and $L_5$ are stable in all directions, if and only if the mass ratio of the two main binary components $\frac{m_1}{m_2}\geq\frac{25+3\sqrt{69}}{2}\approx 24.9599$.
According to theorem 3.1 of the same paper all Lagrangian points are stable in z-direction, which is the direction perpendicular to the orbital plane of the binary system.
(Credits for this corrected version go to user DylanSp.)</p>

CC BY-SA 3.0
2016-03-03T18:56:50.660Z
@Gerald According to [these notes from NASA](http://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf), the ratio between $M_1$ and $M_2$ also needs to be sufficiently high for $L_4$ and $L_5$ need to be stable. Your source doesn't go into sufficient detail on the conditions for stability of $L_4$ and $L_5$.
DylanSp
CC BY-SA 3.0
2016-03-03T22:24:18.077Z
@DylanSp That's a good point. However, if the paper you referenced is read with a strict mathematical interpretation, it says "This will
be true if", not "This will be true iff", with "iff" meaning "if and only if". So I'm not sure, whether the conclusion in the other direction holds. The figure on page 2 of the paper I referenced, shows a minimum for L4 and L5 in the equal-mass binary case. Out of the hip, I can't decide which interpretation is correct.
Gerald
CC BY-SA 3.0
2016-03-04T15:10:07.677Z
@Gerald I dug up [this paper](http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf), which confirms it as stable if and _only if_ $M_1/M_2 \geq 24.9559$ (approximately).
DylanSp
CC BY-SA 3.0
2016-03-08T18:38:11.643Z
@DylanSp Thanks a lot! This paper is unambiguous. I've adjusted the answer.
Gerald
CC BY-SA 3.0
2015-12-04T09:57:51.707Z
For a binary star system, L4 and L5 are generally *not* stable: the mass ratio between the two stars isn't high enough.
Mark
CC BY-SA 3.0
2015-12-04T16:00:34.747Z
It's the mass ratio between the third body and the two other bodies, which needs to close to zero for L4 and L5 to be stable, not the ratio between the two stars.
Gerald
CC BY-SA 3.0
2015-12-04T16:12:24.037Z
added equal mass binary case
Gerald
CC BY-SA 3.0
2015-12-03T17:32:49.957Z
2016-03-08T18:35:48.223Z
2016-03-08T18:35:48.223Z
Gerald
<p>I know of three sets of stable orbits in a binary star system: orbiting closely around star A, orbiting closely around star B, or orbiting distantly around both stars (and their mutual center of gravity) at once.</p>

<p>Is there a fourth set of stable orbits, around the mutual center of gravity, but <em>inside</em> both stars' orbits?</p>

<p><a href="https://i.sstatic.net/jcFfG.png" rel="noreferrer"><img src="https://i.sstatic.net/jcFfG.png" alt="Two stars orbiting their mutual center of gravity, with a planet orbiting that same point"></a></p>

CC BY-SA 3.0
2015-07-26T13:25:48.067Z
Can you draw a picture of what this would look like?
HDE 226868
CC BY-SA 3.0
2015-07-26T19:45:17.637Z
I've added an image.
Mark
CC BY-SA 3.0
2015-07-27T08:15:33.777Z
I strongly suspect that's not stable long term, even though the planet does orbit the 2 stars center of gravity.   I find it hard to believe that wouldn't rather quickly destabilize.  There's a handful of calculated 3 body scenarios that work, my favorite is the figure 8 (this one is 3 equal bodies)  http://www.ams.org/featurecolumn/images/simo03.gif and more here:  http://news.sciencemag.org/physics/2013/03/physicists-discover-whopping-13-new-solutions-three-body-problem.  Your question is slightly different, but I still feel pretty confident in saying that wouldn't be stable for long.
userLTK
CC BY-SA 3.0
2015-07-28T18:26:01.210Z
You can't have an orbit like that. There is no force toward the barycenter. This planet would be shot out of the binary system at high speed in no time flat.
David Hammen
<orbit><binary-star>
Orbits in a binary star system
Mark
# Orbits in a binary star system

(Asked by: username_0 on 2015-07-26)

I know of three sets of stable orbits in a binary star system: orbiting closely around star A, orbiting closely around star B, or orbiting distantly around both stars (and their mutual center of gravity) at once.

Is there a fourth set of stable orbits, around the mutual center of gravity, but *inside* both stars' orbits?

[![Two stars orbiting their mutual center of gravity, with a planet orbiting that same point](https://i.sstatic.net/jcFfG.png)](https://i.sstatic.net/jcFfG.png)

**Tags:** orbit, binary-star

> **Comments:**
> **username_1**: Can you draw a picture of what this would look like?  
> **username_0**: I've added an image.  
> **username_2**: I strongly suspect that's not stable long term, even though the planet does orbit the 2 stars center of gravity.   I find it hard to believe that wouldn't rather quickly destabilize.  There's a handful of calculated 3 body scenarios that work, my favorite is the figure 8 (this one is 3 equal bodies)  http://www.ams.org/featurecolumn/images/simo03.gif and more here:  http://news.sciencemag.org/physics/2013/03/physicists-discover-whopping-13-new-solutions-three-body-problem.  Your question is slightly different, but I still feel pretty confident in saying that wouldn't be stable for long.  
> **username_3**: You can't have an orbit like that. There is no force toward the barycenter. This planet would be shot out of the binary system at high speed in no time flat.  

---

## 1 answers

### Answer by username_4 on 2015-12-03. Score: 6 | Accepted answer

The point you appear to refer to, is called the [Lagrangian point](https://en.wikipedia.org/wiki/Lagrangian_point) $L\_1$.
This point is a saddle in the field of gravity, hence not to be considered to be stable in the strict sense.
Two other Lagrangian points, called $L\_4$ and $L\_5$, can be stable, provided the considered orbiting objects are of small mass in comparison to the two main bodies of the system, and if the masses of the binary components are sufficiently different.

According to theorem 4.1 of [this paper](http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf), $L\_4$ and $L\_5$ are stable in all directions, if and only if the mass ratio of the two main binary components $\frac{m\_1}{m\_2}\geq\frac{25+3\sqrt{69}}{2}\approx 24.9599$.
According to theorem 3.1 of the same paper all Lagrangian points are stable in z-direction, which is the direction perpendicular to the orbital plane of the binary system.
(Credits for this corrected version go to user username_5.)

> **Comments:**
> **username_0**: For a binary star system, L4 and L5 are generally *not* stable: the mass ratio between the two stars isn't high enough.  
> **username_4**: It's the mass ratio between the third body and the two other bodies, which needs to close to zero for L4 and L5 to be stable, not the ratio between the two stars.  
> **username_4**: added equal mass binary case  
> **username_5**: @username_4 According to [these notes from NASA](http://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf), the ratio between $M_1$ and $M_2$ also needs to be sufficiently high for $L_4$ and $L_5$ need to be stable. Your source doesn't go into sufficient detail on the conditions for stability of $L_4$ and $L_5$.  
> **username_4**: @username_5 That's a good point. However, if the paper you referenced is read with a strict mathematical interpretation, it says "This will be true if", not "This will be true iff", with "iff" meaning "if and only if". So I'm not sure, whether the conclusion in the other direction holds. The figure on page 2 of the paper I referenced, shows a minimum for L4 and L5 in the equal-mass binary case. Out of the hip, I can't decide which interpretation is correct.  
> **username_5**: @username_4 I dug up [this paper](http://www.math.cornell.edu/~templier/junior/final_paper/Thomas_Greenspan-Stability_of_Lagrange_points.pdf), which confirms it as stable if and _only if_ $M_1/M_2 \geq 24.9559$ (approximately).  
> **username_4**: @username_5 Thanks a lot! This paper is unambiguous. I've adjusted the answer.  

---

