I was reading a couple of Earth-Moon related questions (Mars just collided with Earth! A question of eccentricity, Could the earth have another moon?) and they got me thinking about planet-moon systems in general.
Binary star systems are pretty common. The types of the two stars in the binary can vary pretty widely (main sequence, puslar/neutron star, black hole, white dwarf, giant phases, etc, etc), but some are formed of a pair of roughly equal mass (within a factor of <10).
I can't say I've ever heard of a binary planet system, though. Of course a planet with a moon is sort of a binary, but I've never heard of an equal mass binary planet. I think the closest thing in the solar system would be the Pluto-Charon system, with a mass ratio of about 10:1.
Is there any reason a binary planet would be unstable? Obviously this is a three-body system, which has some inherent instability, but Earth-Moon-Sun seems pretty stable. Would increasing the mass of the Moon to match that of Earth make the system unstable?
How about gas-giants? I think a Jupiter-Jupiter binary close to a star would be short lived because of three-body interactions, but what about further out? Would, for instance, a double-Jupiter or double-Saturn be stable in our solar system? Or is there some tidal effect that would cause the orbit to decay and the binary planet to merge?
As an aside, it seems that binary asteroids aren't terribly difficult to find... perhaps we just haven't seen any binary planets yet because they're only stable relatively far away from their star, making them difficult to detect outside our own solar system?
Answer
They are sometimes also called double planets and they're more widespread in fiction than in observations. I don't think that there is any new instability that would appear for the system of double planets orbiting a star and that wouldn't be present for other, more asymmetric pairs of planets. Obviously, the tidal forces would be really large if the planets were close enough to each other. But because the tidal forces go like $1/r^3$, it's enough to choose the distance that is 5 times larger than the Earth-Moon distance and the tidal forces from the other Earth would be weakened 125 times and would already be as weak as they are actually from the Moon now (with a lower frequency).
One must realize that the systems with two or several planets are rather rare and the condition that the mass of the leading two planets is comparable is even more constraining.
Imagine that each of the two planets has a mass that is uniformly distributed between 1/20 of Earth's mass (like Mercury) and 300 Earth masses (like Jupiter) on the log scale. The interval goes from the minimum to the maximum that is 6,000 times heavier. That's more than 12 doublings, $2^{12}=4,096$. So if you pick the first planet to be at a random place on that interval of masses (uniformly at the log scale), the probability that the second planet's mass (which is independent) differs by less than the factor of $\sqrt{2}$ from the first mass is about $1/12$.
Only $1/12$ of systems that look like a pair of planets will be this symmetric. And the number of pairs of planets - even asymmetric ones – is rather low, indeed. The reason is really that during the violent eras when Solar-like systems were created, rocks had large enough velocities and they flew in pretty random directions so they were unbound at the end. It's just unlikely to find two large rocks in the same small volume of space: compare this statement with some high-temperature, high-entropy configurations of molecules in statistical physics.
It's also rather unlikely that a collision with another object creates two objects that will orbit one another. After all, the two-body orbits are periodic so if the two parts were in contact during the collision, they will collide again after one period (or earlier). Equivalently, the eccentricity of the orbit is likely to be too extreme which will lead to a fast reunification of the two new planets. Moreover, even if something would place the two newly created planets from a "divorce" on a near-circular orbit, perhaps a collision with a second external object (good luck), it's very unlikely that such an orbit will have the right radius, like the 1 million km I was suggesting in the case of the hypothetical double Earth above. If the two objects are too close, the tidal forces will be huge and (at least for some signs of the internal angular momentum) they will gradually make the planets collapse into one object again. And if the energy with which the planets are ejected from one another is too high, no bound state will be created at all. So the initial kinetic energy of the newborn 2 planets would have to be almost exactly tuned to their gravitational potential energy (without the minus sign) and that's generally unlikely, too.
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