# Binary planets that are not tidally locked

The story takes place in Alpha Centauri, with this binary orbiting just over 1 AU from Rigil Kentaurus. The star system will tightly packed, but following the 10 Mutual Hill Radii separation setup. The primary planet in this pair will be 1.2 Earth masses and second planet will be 0.85 Earth masses. I understand that it is more common for binary pairs like this to form a tidal locking situation (such as Earth-Moon), but is it possible for these to orbit at around 800,000km and have their own spins/days.

Edit: both planets still orbit eachother within an acceptable range of their hill sphere with Rigil Kentaurus.

Bonus question, would it then be possible for one (or both) planet to have a natural satellite orbit it around half to a third of Moon's mass?

• How old are these planets? – Joe Bloggs Feb 21 at 13:55
• The Earth is not tidally locked to the Moon. – AlexP Feb 21 at 14:21
• From what I can find, the stars are between 5 and 8 billion years old. So I would assume the planets would be between 3 and 6. Kind of like Earth. – Markitect Feb 21 at 14:49
• @user72655 The planets will be the same age as the star, not two billion years younger. A few million years younger, yes, but that’s not significant. – Mike Scott Feb 21 at 15:00
• Hard science tag. I don't know that this kind of question is really answerable under that tag. – Zeiss Ikon Feb 21 at 15:16

Wikipedia offers a formula (from a paid-for source, here) for estimating the time to tidal locking for two given planets. (See Tidal Locking Timescale.)

$$t_{lock}=6(a^{6}R\mu)/(m_sm_{p}^{2})×10^{10}$$ years.

Masses in kilograms and distances in meters.

• For your earthlike worlds, $$\mu$$ can roughly be $$3×10^{10} N·m^{−2}$$.

• $$m_{p}$$ is the mass of the planet

• $$m_{s}$$ is the mass of the satellite

• $$R$$ is the mean radius of the satellite

• $$a$$ is the semi-major axis of the satellite around the planet. (The square of the orbital period of a body is proportional to the cube of the semimajor axis of its orbit.)

The formula assumes an initial 12-hour rotation period. (However, a more cumbersome formula exists on the linked page for calculating tidal lock time for different rates of rotation. 12 hours seems like a reasonable value, however, primordial Earth probably had a rotational period of about 6 hours, so there's room to push things.)

Plugging the values you've given for masses and orbital distance, we can estimate:

... that the double planet will tidally-lock perhaps 12 million years after formation. (The planet loses on average 0.0036 seconds of its day per year.) There are many factors that can contribute to error in the formulas. Wikipedia states that in some cases, they may be off by orders of magnitude.

It's important to note that larger satellites tidally lock themselves faster than smaller ones. Mass $$m_{s}$$ grows with the cube of radius, and so does the mutual attraction.

A possible example of this is in the Saturn system, where Hyperion is not tidally locked, whereas the larger Iapetus, which orbits at a greater distance, is.

• Thanks, I guess that pretty much sets it if these two planets started out together. I think now I need to rephrase my question to ask in what way would it be possible for these two planets to orbit eachother like I want them too. Maybe, just like the Earth and the moon, they were formed by a large planetary collision and are now habitable. If the collision could be done recently enough that the planets have cooled and grown vegetation that would be perfect. – Markitect Feb 22 at 22:56
• @user72655 I would suggest that. The Proxima system is very old, so even if the 12 million year factor is off by an order of magnitude or two, I think it's still reasonable that any binary planets that formed in the protoplanetary disk of that star would be locked by now. Collisions with planetoids I think are less likely billions of years into formation, but I don't see any reason why it couldn't happen despite the odds. – BMF Feb 22 at 23:21
• I'd suggest a collision with a large planetoid (maybe Mars size) a few hundred million years ago to resurface one of the planets, set it off axis, and eject some material so that the planet increases it's rate of spin. Because the planets are not axially-aligned, I think it would take longer to make them tidally locked again. The ejected material would probably rain down on the companion world as well. You'd have at least one rotating planet (the other would probably be fixed on the binary barycenter). – BMF Feb 22 at 23:26
• Would it be unreasonable to allow for the same event to cause the second world to gain a spin? Let's say the planetoid had another similar sized objects with it, maybe it was destroyed and sent out of orbit from the other star. Then both planets would be hit. And maybe, depending on the impact angles, the second planet has a retro rotation. – Markitect Feb 25 at 3:01
• @user72655 I think it might be. Taking our solar system to be "average" (it may not be, but still...), large collisions like that tapered off soon after planet formation and migration. The odds are astronomical. I think the first collision would be best explained by an encounter with another star system, which isn't too uncommon. Perturbations could send a planetoid astray. – BMF Feb 25 at 3:10