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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 2M🜨 planet with a companion that is 10.4M☽ (.1279M🜨). 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 .64M☉.

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 See also Phys.org's December 3, 2014 Can binary terrestrial planets exist?

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  • $\begingroup$ 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. $\endgroup$
    – David Hammen
    Commented Dec 2, 2022 at 7:46
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    $\begingroup$ 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/27832-binary-earth-size-alien-planets.html $\endgroup$
    – Thoth
    Commented Dec 5, 2022 at 17:12
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    $\begingroup$ @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. 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. $\endgroup$
    – David Hammen
    Commented Dec 5, 2022 at 21:13
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    $\begingroup$ @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. $\endgroup$
    – Thoth
    Commented Dec 5, 2022 at 21:14
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    $\begingroup$ @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. $\endgroup$
    – Thoth
    Commented Dec 5, 2022 at 21:18

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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;

also E M Standish, "Sufficient Conditions for Escape in the Three-Body Problem", Celestial Mechanics, v.4 no.1 (1971), pp.44-48.

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.

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