Planetary orbit in a two-star system

My daughter asked me the following question and I couldn’t answer it. Help please.

In a stellar system with two (approximately equal mass) stars, and one planet, could the planet orbit around both stars in a figure-8 path? (with one star centered in each of the top and bottom part of the ‘8’)


Anything’s possible, I guess–ever seen Saturn’s “braided” ring?–but a likelier orbit would be a huge ellipse around the both of them.

I attended a lecture tonight on searching for extra-solar planets. Coincidence happens.

The speaker - an astronomy prof involved in this research - was asked if any of the 200+ systems with known planets had multiple stars. She said no - that planetary orbits would likely be too unstable.

However, she also said, in effect, that because of the greatly reduced likelihood of findings planets around binary star systems, nobody is currently wasting their time searching around binary star systems. Too many better targets.

There may be a theoretical stable orbit for some odd configuration.

Yes, it’s possible -

Asimov wrote a short story, “Sucker Bait”, that postulated a planet orbiting in a binary system. The orbit was stable because the planet was in one of the “Trojan points” of the smaller star, which in turn orbited a much larger star. Not a figure 8, but I guess in theory it’s possible.

Wait a minute. Remember we’re talking about two stars – same size. IF their not the same exact size than over time wouldn’t all heck break lose as far as stability of the planetary orbits? Wouldn’t that be infinitely more complex thann a figure eight? So the question becomes - in our universe how many two-star systems contain stars with the same mass? None - so says chance. That it IMHO at least.

How about a planet in orbit around a lagrange point?

Forgive me, but would that be circular orbits rather than figure eights? Mayybe I don’t understand but the orbit paths wouldn’t cross in a figure eight in what you described - right?

I suppose that if the planet were as massive as the stars, one could indeed have a figure eight orbit. But that’s clearly never going to be the case.

Re DarrenS’s link: Is the author saying that figure eight orbits are possible although unstable, or are they too unstable to ever come to be (even for a short period)?

If you look at what happens when the planet passes right inbetween the suns in the figure eight pattern, you can see that it is highly unstable. If the planet does not move exactly through the midpoint, it would be pulled closer to one star and ejected from the system. Or at least moved into a different wider orbit.

I woud think the odds of a planet managing to find the exact center of mass on each orbit would be so low as to be negligable.

How about that! I’m not a big SF reader, but I’ve read many of Asimov’s non-SF books and he is the one that taught me that biary systems don’t have planetary systems. Then along comes DarrenS to prove ol’ Isaac wrong. :frowning:

Don’t know about figure-eight orbits; that would probably require a lot of things going right, all at the same time. Regarding planets in binary stars in general; current best wisdom seems to be that stable planetary orbits might well be possible, but it’s not at all clear a planet could form in such an orbit. That is, if you’ve got one you can keep it, but how do you get it?
On the other hand: right up till the discovery of planets larger than Jupiter in close-in orbits, current best wisdom was pretty sure gas giants would have to be further out (or lose their hydrogen to solar wind).
Since we’ve only seen one solar system close-up, I’d think the best way to answer some of these questions would be to go look at a few more.

Are we assuming that the stars in questions would remain static? At the same mass, each star would most likely have seperate elliptical orbits around a fixed point. So not only would the planets be moving relative to some frame of reference but also one, or both, of the stars depending on whether you have one fixed in you frame of reference.

I thopught at first that it’s possible, but thought better of it. Follow along with me. – imagine the potential energy surface. If you make a model in that shape you can roll a ball aeround it and it will mimic an orbit (a lot of planetariums and science museums have these. Heck – I’ve seen shopping malls with similar things that encoiurage you to spin coins into them to collect for charities.) The difference is that this well would have two pits in it, one for each star. You’d also want to spin it, to mimic the effect of rotation (those two stars better be rotating about a common center, or gravity will suck them together.)

Now, if that rotation wasn’t there, I could easily see a figure-8 orbit developing. You just have to finesse rolling that imaginary ball at the right speed and angle to get it to go through the exact center of the system.

But with the system rotating, you’re outaluck. The ball (planet) has to be rotating about the center, too, so it has to maintain constant angular momentum. The closer it gets to the center of the system, the faster it has to go. For zero distance (passing through the center) you have to go to infinite velocity. Ooops.

Another way of looking at that it that you can, instead of rotating your model, simply add a “centrifugal potential”, in the form of an upward-pointing funnel in the middle of your model. With a single well and a cebtrifugal potential you end up with a big “spike” sticking out of the middle, dropping to a ring-shaped trough arounbd that, then climbing on the outside. Your planet will sit in the trough that defines the orbit (you don’t have to spin the ball around the potential well – your cetrifugal potential does that for you). You can see how a stable radius is defined by the radius of the trough. If you “rock” the ball back and forth in the trough, you get an elliptical orbit (the trough is shallower away from the center of the centrifugal potential).

With a two-mass system having two pgravity wells and one centrifugal anti-well, you can see how you’d never get to the center. Hence, no figure-8 orbits. The height of that centrifugal spike depends on your velocity/angular momentum. If it’s high enough, all your orbits are distorted circles or ellipses around both stars. If it’s low enough, however, you can get some eccentric shapes. But that spike in the center of the system forbids figure 8’s.

By the Way, Lagrage Points are only stable if most of the mass is in one of the two gravity wells. Two equal masses don’t fulfill that requirement, so the “Lagrage Points” of a two-equal-star system aren’t stable.

Actually, “all heck” will usually break loose even if they have the same mass.

The planet can’t orbit around a Lagrange point. All five L-points in the 3-body system are either attractive or repelling, which means that you’ll stay put at the point, but that small disturbances push you away from the repelling ones. The attracting points pull you back in.

Quick and dirty restricted 3-body problem (generally following V.I. Arnold’s Mathematical Methods of Classical Mechanics):
We assume one of the three bodies is far smaller than the other two, so its effect on their orbit is negligible. The other two are essentially a 2-body system, for which we can pick rotating coordinates fixing one (generally the larger, if there is one) at the origin and the second moving periodically along the x-axis. Now it’s possible to explicitly write down the (time-dependant) gravitational potential V, set up the Hamiltonian

H(x,p,t) = p[sup]2[/sup]/2m + V(x,t)

where (x,p) are coordinates on the cotangent bundle (position and momentum for Newtonians). Hamilton’s equations give

dp[sub]i[/sub]/dt = -?H/?x[sub]i[/sub] = -?V/?x[sub]i[/sub]
dq[sub]i[/sub]/dt = ?H/?p[sub]i[/sub] = p/m

This system, even for circular choices of the orbit of the two large bodies is not generally integrable. There are some known solutions, but in general the system behaves chaotically. In fact, this was (in a sense) the first recognized chaotic dynamical system.

Wait a minute.

Let’s look at our closest star system, Alpha Centauri. If I am not mistaken, it is a 3 star system, right? Alpha Centauri A and Alpha Centauri B are both sun-like stars that orbit each other closely, and Proxima Centauri is a red dwarf that orbits both at a much greater distance (farther than Pluto orbits our sun, IIRC).

If a planet cannot stably orbit a two-star system, then how could Proxima orbit Alpha Centuari A+B?

I understand that the closer a body gets to the stars the more chaotic the orbit will be…but those perturbations are going to diminish the farther you get from the stars. At some point you can pretty much treat them as one mass. A planet orbiting Alpha Centauri A+B at the distance of Earth’s orbit might not be possible, but why not one orbiting a Jupiter, or Saturn, or Neptune’s distance?

Or we could look at situations where one star is very massive and the other is much smaller. Let’s say Jupiter or Neptune was a red dwarf star. Would that preclude planets from orbiting the sun, either inside or outside the orbit of the red dwarf? Or a situation where a star with planets orbits a more massive star. How is that different than a planet with moons orbiting a star?

What I’m getting at, is that I agree that a “three body problem” is chaotic…but our solar system is also a multi-body problem. It’s just that the gravity of Jupiter and the other planets can be treated as neglible for most purposes when figuring the orbit of Earth. I am not a physicist, is there something I’m missing here?

Try another example – Algol (Beta Persei) is actually three stars. Two stars of nearly equal mass (just as in our case!) are orbited by a third, smaller star that’s much farther away. The orbits are pretty close to stable – Algol is an eclipsing variable with a very precise three-day period (Read my book! – shameless plug), but you can treat them all as “sort of” 2-body problems. Algol C is so small and so far out that you can treat Algol A and Algol B as rotating about a common center, and ignore C. C is so far out relative to the separation between A and B that they act very much like a single gravity well, so the orbit is nearly circular.
But, as the OP asks, I don’t think you could get C in a figure-8 orbit running between A and B.

I did not say there were no stable orbits (those that neither escape to infinity nor crash into one of the large bodies). I said that the system was chaotic. Chaotic systems have periodic orbits, though they usually repel nearby trajectories. Chaotic systems have many (if not most) of their trajectories contained in a compact subset of phase space. What chaotic systems do not have is easy predictability. For most choices of initial position and initial momentum in the restricted 3-body system, small changes in the initial conditions leads to enormous changes in observed behavior.

In short, yes there exist Lagrange points (though the planet sits at one rather than orbiting around it), but usually the planet follows a bizarre nonperiodic (and not even nearly periodic, like the precessing orbit of Mercury) path which may or may not (very difficult to predict) crash into one of the stars.

Yes, but the OP was concerned with “figure-eight” orbits, which preclude being at that sort of a distance.

You’re still being too generous with the difference in mass of the large bodies. The restricted 3-body problem encompasses the Sun-Earth-satellite system; neglecting contributions from the moon, Venus, and Mercury of course.

For making first approximations, yes they can be neglected. However, the solar system is definitely chaotic and has been accepted as such since Poincaré won King Oscar II’s prize for determining the stability of the solar system with his work on the restricted 3-body problem. If anything, what you’re missing is a good handle on time scales in celestial mechanics. Just because we can predict pretty well for the next century doesn’t mean the system is stable. A century is nothing in celestial mechanics.

Orbital resonances create many of the features of our own solar system.

In the asteroid belt, areas with integral orbital periods with Jupiter (5:2, i.e. an orbit in which every five revolutions matches two of Jupiter’s, 3:1, 7:3, etc.) are nearly free of planetesimals because they get kicked out of orbit. But closer to Jupiter, the reverse occurs. At 4:3 the orbits become more stable than the surroundings, so planetesimals gather there.

Same with Pluto and Neptune, which are in a 2:3 resonance. That resonance also has large numbers of other bodies at various eccentricities.

Without integral resonances, the orbits become chaotic and the bodies get kicked out of the system.

At least, that’s my memory of the prof’s rather chaotic speech.

It seems that all of the research on the three-body problem assumes 2D orbits (i.e., all bodies orbit in the same plane. Can anything be predicted about 3D orbits? Inherently unstable? Possible stable configurations? Too difficult to determine?