# Can this planetary system remain stable?

Before we begin, this system is created by aliens, and all orbits are on the same plane. These orbits are similar to earth's in shape. So, none of this has to occur naturally it just has to be stable for a period of about as long as our solar system will last. (And I know the sun won't live that long.)

First, the sun is the size of ours, and identical in every way. There is a Mars sized planet just within the habitable zone of this star. On the outer edge of the habitable zone is a binary pair of what you can think of as earth clones orbiting each other at a distance of 3 million miles. Directly between the binary planet pair and the Mars-like planet there is a second binary pair of planets, identical to the ones on the edge of the habitable zone.

Would this system be stable? Would the planets in the binary pairs be too far apart? Would the different bodies in the solar system wreak havoc on each other's orbits? Would life as we know if cease to exist?

• A trillion years? The Lyapunov time of this system won't be anywhere near that, so you most likely wouldn't get any good results from simulations (if anyone was considering doing one). The stability algorithm I gave here (see Mardling (2008) might be a good starting point, if you can analyze the resonances at work. You'd have to do it in pairs, though, not all three at once. Feb 16 '16 at 16:31
• @HDE226868 The trillion years is one of the least important aspects of the question, so I changed it to the length of our solar system. Feb 16 '16 at 23:18

You can separate this into two problems:

1. Treat the binary planets as a single planet with the combined mass of both and see if the planetary configuration is stable - which is probably only true over the length of time you're talking about if there are no external events to perturb any of the planets in your system. Our own solar system has probably only been stable for a few 100 million years or so - the last major impact on the Earth was 65 mya, but there wasn't one (that we know of) for quite a while before that. There have been impacts on other planets up to modern times. (Shoemaker-Levi on Jupiter for example).

Unless your aliens remove all the asteroids and comets having a stable solar system over the length of time you're expecting would be virtually impossible.

On a more basic level, will the binary pairs affect their neighbours enough to perturb their orbits just by existing? The Sun's habitable zone is estimated to be from just within the orbit of Venus to around the orbit of Ceres (source) so it sounds like you have enough space to play with here to have your three sets of planets where you want them.

2. Are the binary pairs stable? Are they too close together to start off with? Given that the Moon is 384,400 km from the Earth and 1/6th of the size, 4.82 million km seems to be a little far apart. You are increasing the distance 12 fold so this will outweigh the 6 fold increase in mass. The planets might not be close enough to stay together over the long term.

The planets will drift apart over time in the same way that the Moon is drifting away from the Earth, so you need them to be close enough together for the "lifetime" of your system. Also they are going to become tidally locked with each other so that they end up presenting one face to each other in the same way that the Moon always presents the same face to the Earth.

Also, exactly circular orbits are likely to be rarer and less stable than slightly elliptical orbits, so I'd drop that requirement.

• Earth's orbit is pretty nearly circular anyway. Feb 16 '16 at 10:06
• @Whelkaholism but not perfectly circular ;) Feb 16 '16 at 10:29
• Given that distance in the binary system, they are going to become tidally locked. See the question related to that worldbuilding.stackexchange.com/q/35784/16186 Feb 16 '16 at 11:52
• @ChrisF, can I change my question a little bit? I was kind of looking for hard science, but I forgot that tag. Also, I do understand that they eventually will become tidal locked, I might change the distance in the question if you allow it. I don't really want to invalidate your answer, seeing as you've gotten upvotes. Feb 16 '16 at 15:02
• @XandarTheZenon your desire for hard science in this seems at odd with the age of the system your proposing. I can't really stop you updating the question, so go ahead. Feb 16 '16 at 15:09

The question about whether a given six-body celestial system in a given configuration is stable is a research-grade question in the subfield of mathematics called dynamical systems. If I were able to exactly answer your question with support from mathematical proofs, I could probably publish a couple of scientific papers based on my work. Perhaps even a book. But I can tell you what the answer would probably be: No.

The reason for this is that gravitational dynamics are a chaotic system. This means that tiny changes in the initial conditions of the system lead to major changes in the outcome. Any system with more than two bodies is chaotic. Even our own solar system is chaotic; see this page. In the span of a human lifetime, or even the span of all of humanity, it looks like our solar system is unchanging. The planets maintain their respective orbits, and it seems they will forever. But in reality, this time span is tiny compared to the age of the solar system, and even tinier compared to one trillion years. Even with the most precise measurements and careful calculations, we cannot predict what the solar system will look like even one billion years in the future, .1% of your time span. It is almost impossible to find a stable three body orbit, and even harder with six bodies. Even if your system is stable, the tiniest change (like star passing within a few lightyears of the system) would throw off the stability on the time scale of a trillion years.

EDIT: Since you have altered the time scale of the problem, I will augment my answer, which is still no. Our own solar system is not stable, and so I doubt that your somewhat more complicated system would be stable. That being said, an orbital system being stable is a very stringent and rigorously defined condition, and is not what most people mean when they say stable. If you mean "the description of the orbits I gave remains valid for several billion years," this is a much less stringent condition which changes the answer to "probably not." Several billion years is still a long time. The Lyapunov time is the the time scale on which a system behaves in a predictable way. The Lyapunov time for our solar system is about 50 million years. This means that we have absolutely no idea what our solar system will look like in 50 million years. And there is no reason to expect that it will look anything like it does now. Your system likely has a similar Lyapunov time, so while it may be have nicely for a few million years, it will probably devolve into chaos long before you hit the billion year mark.

• This is the correct answer. Quasi-stable dynamic systems of n-bodies over a trillion years are probably quite unusual. IIRC when looking at planetary system we state that a configuration like our Solar System is stable if it remains roughly the same over billions of years. A trillion is 1000x longer. Feb 16 '16 at 19:40
• The time it lasts is not as important to me, so I changed it to the longevity of our solar system. Feb 16 '16 at 23:22
• Great answer, like Jim2B said, +1. I do have issue with two things, though: The assertion that the Solar System is not stable (we don't know one way or the other) and the statement that "we have absolutely no idea what our Solar System will look like in 50 million years". The Lyapunov time does describe the chaos of the system, but it doesn't mean that anything is possible after those 50 million years. We have reason to believe that the Solar System will look pretty much the same after that time - we just can't accurately calculate what the positions (and orbits, to some extent) will be like. Feb 17 '16 at 0:49
• @HDE226868 I may have been a bit hasty in saying that the solar system is unstable, but given that stable configurations constitute a set of measure zero in configuration space, I feel comfortable in believing that it is unstable. You are also right about Lyapunov time. I was confusing this with system time-the amount of time it takes for error to propagate to the size of the system. Given that the Lyapunov time is about 50 million years, I would guess that the system time is still less that a billion years. Feb 17 '16 at 1:10
• Just a quick note that you guys are on the wrong track here. There is plenty of space within the habitable zone for many orbits (up to 6-8, and each orbit can host multiple planets). See my answer or here for more information: planetplanet.net/2014/05/21/… Dec 15 '17 at 11:18

This setup is totally stable in terms of the orbits around the star. The habitable zone itself (which is not super well-defined) extends between roughly 0.9 and somewhere between 1.5-2.5 AU. Planetary spacing is determined by the so-called Hill radius: RH = a (mp/Mstar)^1/3, where a is the orbital radius, mp is the planet mass and Mstar is the star's mass. Planet orbits must be at least ~8-10 RH apart to be stable. That means you can fit about 6-8 stable orbits of Earth-mass planets in the habitable zone. So, for your setup with 3 orbits it's perfectly stable.

As for the binary planet, it is not stable. Binary planets in general are totally stable (see Earth-Moon, PLuto-Charon as examples). But their orbits must be close enough to avoid being torn apart. The criterion for stability is that their orbits must be smaller than about half of the Hill radius (for prograde orbits; it's larger for retrograde). Earth's Hill radius is about 0.01 AU. Your binary Earths are a bit farther out, say at around 2 AU. That means RH is 0.02 AU. However, 3 million miles is about 0.03 AU. You need to chop that separation by a factor of 3 or more to keep it stable -- unless you want to invoke a retrograde orbit for the binary planets, in which case you just need to chop the separation in half.

FYI, I'm an astrophysicist studying orbital dynamics. I have also covered this topic extensively on my blog. Here are a few relevant posts: