<p>Besides the approximations mentioned in the comments, the information available does defines some conditions for which stability can be assured, and other conditions for which instability can be assured. (But the two sets of conditions together are not exhaustive, and the time to escape in an unstable system is not measured.)</p>
<p>So maybe it could be helpful for your questions to test your system against the criteria shown in these two papers:</p>
<p><a href="https://ui.adsabs.harvard.edu/link_gateway/1972CeMec...6..352S/ADS_PDF" rel="nofollow noreferrer">E M Standish, &quot;Sufficient Conditions for Return in the Three-Body Problem&quot;, Celestial Mechanics, v.6 no.3 (1972), p.352-355</a>;</p>
<p>also
<a href="https://ui.adsabs.harvard.edu/link_gateway/1971CeMec...4...44S/ADS_PDF" rel="nofollow noreferrer">E M Standish, &quot;Sufficient Conditions for Escape in the Three-Body Problem&quot;, Celestial Mechanics, v.4 no.1 (1971), pp.44-48</a>.</p>
<p>There's also a further paper from the same author based on a numerical approach, integrating 800 sample systems. He identifies that an  important factor in the fate of the triple is the angular momentum in the system: <a href="https://articles.adsabs.harvard.edu/pdf/1972A%26A....21..185S" rel="nofollow noreferrer">Astron &amp; Astrophys 21 (1972), 185-191</a>.</p>

CC BY-SA 4.0
2022-12-06T22:33:43.610Z
This is *really interesting!* My coffee hasn't kicked in yet but I will try to get these today and dig in. In the mean time you may find this Math SE question interesting [Can there be an energetically unbounded three-body orbit where escape is impossible due to conservation of angular momentum?](https://math.stackexchange.com/q/3657498/284619) I have been [thinking about  posting it to Math Overflow](https://meta.mathoverflow.net/q/4582/96907) but there was one caveat holding me back. It's possible these offer an actual answer! Now I have to remember what math's "sufficient" does/doesn't mean
uhoh
CC BY-SA 4.0
2022-12-08T22:10:39.633Z
I'm about to plug my system's values into these equations. Should Mass C be the orbital parent of both bodies?
Thoth
CC BY-SA 4.0
2022-12-06T19:55:59.510Z
2022-12-06T20:10:03.157Z
2022-12-06T20:10:03.157Z
terry-s
<p>I recently found an article claiming a double planet system needs to be at least .5 AU from its parent star to be stable for billions of years. It was specifically talking about two same-mass bodies, though I'm curious about the stability of differently-sized ones.</p>
<p>My system consists of a 2<em>M</em> planet with a companion that is 10.4<em>M</em>☽ (.1279<em>M</em>). They are tidally locked to each other, with a semi-major axis of 79,250 km and an eccentricity of .00986. They both orbit their parent star at .4 AU, with an eccentricity of .0196. The star is .64<em>M</em>☉.</p>
<p>I would like to know how long this system would last given its orbital distance. The companion body is well within the main one's hill sphere, though I'm not sure if the star's tidal influence would make it destabilize within the first few billion years.</p>
<p><strong>update:</strong> Here's the article I found. Unfortunately, it does not cite the original study from the physicists it interviews. also, the planet's hill sphere is roughly 875192 km, 11x the second body's semi-major axis. Space.com's November 21, 2014 <a href="https://www.space.com/27832-binary-earth-size-alien-planets.html" rel="nofollow noreferrer">Binary Earth-Size Planets Possible Around Distant Stars</a> See also Phys.org's December 3, 2014 <a href="https://phys.org/news/2014-12-binary-terrestrial-planets.html" rel="nofollow noreferrer">Can binary terrestrial planets exist?</a></p>

CC BY-SA 4.0
2022-12-02T07:46:00.450Z
Please add a link to the referenced article. Also, the Hill sphere is an approximation. The general consensus is that orbits remain long-term stable out to about half of the Hill sphere radius. Beyond that, long term stability becomes ever more dubious.
David Hammen
CC BY-SA 4.0
2022-12-05T17:12:46.043Z
Here's the article I found. Unfortunately, it does not cite the original study from the physicists it interviews. also, the planet's hill sphere is roughly 875192 km, 11x the second body's semi-major axis. https://www.space.com/27832-binary-earth-size-alien-planets.html
Thoth
CC BY-SA 4.0
2022-12-05T20:53:11.567Z
@Thoth I've moved your information back into your question where it will be easier for others to see it. Just fyi if you reply to a comment, don't forget the @username which will send them a notification that you've replied. (from [FAQ](https://meta.stackexchange.com/q/7931/303080) see [How do comment `@`replies work?](https://meta.stackexchange.com/q/43019/303080)) Interesting question, *Welcome to Stack Exchange!*
uhoh
CC BY-SA 4.0
2022-12-05T21:02:35.097Z
@DavidHammen I've updated the question with the OP's source  unrelated: any thoughts on [this question](https://astronomy.stackexchange.com/q/51200/7982)?
uhoh
CC BY-SA 4.0
2022-12-05T21:13:13.430Z
@uhoh space.com is notorious for not using references in its articles. (I don't read space.com primarily for that reason.) They did however provide the authors of the study. I assume it is this [American Astronomical Society meeting conference presentation](https://ui.adsabs.harvard.edu/abs/2014DPS....4620102R/abstract). Unfortunately, that's just the abstract; I can't find the full paper. Even more unfortunately, that paper has been cited but once since it was presented eight years ago. I don't give much credence to conference papers that have been cited only one time in eight years.
David Hammen
CC BY-SA 4.0
2022-12-05T21:14:59.730Z
@uhoh Thank you for the edits. In this case I'm less concerned about whether or not this kind of system is common, and more concerned about how long it may persist below .5 AU from its host star.
Thoth
CC BY-SA 4.0
2022-12-05T21:18:10.147Z
@DavidHammen If I'm being honest I did not put much effort into finding a reputable source, I merely found this article scrolling through Google results and the notion of the .5 AU limit had me concerned. The example system is from a worldbuilding project of mine, which I came up with before finding the article, and I want to make sure it's scientifically plausible.
Thoth
<orbital-mechanics><natural-satellites><planetary-formation><planetary-systems><solar-system-evolution>
Orbital Stability of a Double Planet System
Thoth
# Orbital Stability of a Double Planet System

(Asked by: username_0 on 2022-12-02)

I recently found an article claiming a double planet system needs to be at least .5 AU from its parent star to be stable for billions of years. It was specifically talking about two same-mass bodies, though I'm curious about the stability of differently-sized ones.

My system consists of a 2*M* planet with a companion that is 10.4*M*☽ (.1279*M*). They are tidally locked to each other, with a semi-major axis of 79,250 km and an eccentricity of .00986. They both orbit their parent star at .4 AU, with an eccentricity of .0196. The star is .64*M*☉.

I would like to know how long this system would last given its orbital distance. The companion body is well within the main one's hill sphere, though I'm not sure if the star's tidal influence would make it destabilize within the first few billion years.

**update:** Here's the article I found. Unfortunately, it does not cite the original study from the physicists it interviews. also, the planet's hill sphere is roughly 875192 km, 11x the second body's semi-major axis. Space.com's November 21, 2014 [Binary Earth-Size Planets Possible Around Distant Stars](https://www.space.com/27832-binary-earth-size-alien-planets.html) See also Phys.org's December 3, 2014 [Can binary terrestrial planets exist?](https://phys.org/news/2014-12-binary-terrestrial-planets.html)

**Tags:** orbital-mechanics, natural-satellites, planetary-formation, planetary-systems, solar-system-evolution

> **Comments:**
> **username_1**: Please add a link to the referenced article. Also, the Hill sphere is an approximation. The general consensus is that orbits remain long-term stable out to about half of the Hill sphere radius. Beyond that, long term stability becomes ever more dubious.  
> **username_0**: Here's the article I found. Unfortunately, it does not cite the original study from the physicists it interviews. also, the planet's hill sphere is roughly 875192 km, 11x the second body's semi-major axis. https://www.space.com/27832-binary-earth-size-alien-planets.html  
> **username_2**: @username_0 I've moved your information back into your question where it will be easier for others to see it. Just fyi if you reply to a comment, don't forget the @username which will send them a notification that you've replied. (from [FAQ](https://meta.stackexchange.com/q/7931/303080) see [How do comment `@`replies work?](https://meta.stackexchange.com/q/43019/303080)) Interesting question, *Welcome to Stack Exchange!*  
> **username_2**: @username_1 I've updated the question with the OP's source  unrelated: any thoughts on [this question](https://astronomy.stackexchange.com/q/51200/7982)?  
> **username_1**: @username_2 space.com is notorious for not using references in its articles. (I don't read space.com primarily for that reason.) They did however provide the authors of the study. I assume it is this [American Astronomical Society meeting conference presentation](https://ui.adsabs.harvard.edu/abs/2014DPS....4620102R/abstract). Unfortunately, that's just the abstract; I can't find the full paper. Even more unfortunately, that paper has been cited but once since it was presented eight years ago. I don't give much credence to conference papers that have been cited only one time in eight years.  
> **username_0**: @username_2 Thank you for the edits. In this case I'm less concerned about whether or not this kind of system is common, and more concerned about how long it may persist below .5 AU from its host star.  
> **username_0**: @username_1 If I'm being honest I did not put much effort into finding a reputable source, I merely found this article scrolling through Google results and the notion of the .5 AU limit had me concerned. The example system is from a worldbuilding project of mine, which I came up with before finding the article, and I want to make sure it's scientifically plausible.  

---

## 1 answers

### Answer by username_3 on 2022-12-06. Score: 1

Besides the approximations mentioned in the comments, the information available does defines some conditions for which stability can be assured, and other conditions for which instability can be assured. (But the two sets of conditions together are not exhaustive, and the time to escape in an unstable system is not measured.)

So maybe it could be helpful for your questions to test your system against the criteria shown in these two papers:

[E M Standish, "Sufficient Conditions for Return in the Three-Body Problem", Celestial Mechanics, v.6 no.3 (1972), p.352-355](https://ui.adsabs.harvard.edu/link_gateway/1972CeMec...6..352S/ADS_PDF);

also
[E M Standish, "Sufficient Conditions for Escape in the Three-Body Problem", Celestial Mechanics, v.4 no.1 (1971), pp.44-48](https://ui.adsabs.harvard.edu/link_gateway/1971CeMec...4...44S/ADS_PDF).

There's also a further paper from the same author based on a numerical approach, integrating 800 sample systems. He identifies that an important factor in the fate of the triple is the angular momentum in the system: [Astron & Astrophys 21 (1972), 185-191](https://articles.adsabs.harvard.edu/pdf/1972A%26A....21..185S).

> **Comments:**
> **username_2**: This is *really interesting!* My coffee hasn't kicked in yet but I will try to get these today and dig in. In the mean time you may find this Math SE question interesting [Can there be an energetically unbounded three-body orbit where escape is impossible due to conservation of angular momentum?](https://math.stackexchange.com/q/3657498/284619) I have been [thinking about  posting it to Math Overflow](https://meta.mathoverflow.net/q/4582/96907) but there was one caveat holding me back. It's possible these offer an actual answer! Now I have to remember what math's "sufficient" does/doesn't mean  
> **username_0**: I'm about to plug my system's values into these equations. Should Mass C be the orbital parent of both bodies?  

---

