# How many co-orbital Earths can we fit and NOT get them tidally locked?

According to Sean Raymond, a co-orbit of six Earth-sized planets, separated by 60 degrees of space, will stay stable for millions of years.

And in his latest blog, he has gone a bit further and doubled the tally, adding a large moon on each world.

Neat to look at, but there is one major obstacle that cannot be overlooked--tidal locking. Our Earth species are adapted to a life run by day-night cycles, so in the event of terraforming an alien world into something like Serina, seeding Earth species on a world in which the sun never sets on one side and never rises on the other would be an instant failure. But if you put two large bodies close enough together, then that is exactly what you'll get.

So can six or a dozen Earth-sized planets orbiting their sun from one same distance rotate without being tidally locked to each other? Or would that require a lower number?

• So the question is “does adding more bodies in the same orbit increase the chance of these bodies becoming tidally locked to their sun?” Oct 30, 2021 at 2:49
• Why would you expect them to become tidally locked, any more than Earth has been tidally locked by the Sun, or the Moon? Tidal forces from those co-orbital planets would be utterly negligible compared to the force from the Moon, given that (for the 6 planet case) they're about 400 times further away. Tidal force is proportional to $1/r^3$. Oct 30, 2021 at 3:04
• @jamesqf Earth is tidally locked to neither. If it were tidally locked, we would never have day-night cycles, just eternal light on one side and eternal darkness on another. Oct 30, 2021 at 4:05
• @JohnWDailey they would start slamming into each other long before any tidal lockingeffects are noticeable. By <s>several</s> many magnitudes longer. Oct 30, 2021 at 7:03
• @JohnWDailey: Yes, THAT'S WHAT I WROTE. If Earth has not been tidally locked to either the Moon or the Sun, why would the OP expect it to become tidally locked by the orders of magnitude smaller forces that would be exerted by those hypothetical co-orbital Earths? Nov 1, 2021 at 1:28

I suppect that the number of Earth mass planets that can be equally spaced in a ring around a star will depend on the distance at which the ring planets orbit and thus the circumference of the orbit.

I believe that the minimum spacing was discussd by Sean Raymond in one of his posts about planets in rings. Those posts are based on a scientific paper which seems to establish that between 7 and 42 equal mass objects equally spaced in a ring around a much more massive object can have stable orbits.

And when discussing such systems of rings of planets Raymond says:

It turns out there is a stability limit for the number of planets that can be spread along the same orbit. The planets must be evenly spaced and there must be at least 7 on one orbit (not a typo: at least 7!). The limit is simple: the planets sharing the same orbit must be separated by at least 12 Hill radii in distance along the orbit. This is different from before, where we were looking at the distance between orbits.

https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/

Imagine there is a star exactly like the Earth, and there is a ring of Earth mass planets equally spaced in an obit with a radius of 1 AU, the same as the semi-major axis of the Earth's orbit.

Since the size of a planet's Hill sphere or Hill radius is determined by the mass of the planet, the mass of the star, and the orbital distance of the planet, the Hill radius of each of those Earth mass planets will be the same as Earth's, which is about 1,471,400 kilometers.

12 times about 1,471,400 kilometers is about 17,656,800 kilometers.

The semi-major axis of Earth's orbit is about 149,598,023 kilometrs, so the circumference of Earth's orbit is approximately 939,951,306.2 kilometers. 17,656,800 kilometers goes about 53.23 times. 7 to 42 equally spaced planets in Earth's orbit would be spaced 22,379,793 to 134,278,758 kilometers apart.

If the Earth mass planets were separated by only the minimum of 17,656,80 kilometers, the Sun would be 8.472544459 times as far away, weakening its tidal braking effect compared to the closer Earth mass planets. On the other hand, the sun has a mass of about 333,000 times the mass of Earth, greatly strengthening its gravitational braking effect relative to those of the other Earth mass planets.

Thus it seems intuative that the Sun's tidal braking effect on the Earth is many times as great as the tidal braking effects of hyppthetical other Earth mass planets sharing the same orbit as the Earth.

Since the Sun's tidal braking effect on the Earth is not strong enough to make the Earth tidally locked to the Sun, and since it should be many times as great as the tidal braking effects of hyppthetical other Earth mass planets sharing the same orbit as the Earth, those hypothetical other Earth mass planets in a planetary ring would probably not tidally lock each planet to its neighbors.

Of course it would be better to actually calculate the strength of the tidal braking force on each planet in a ring of planets and compare that to the tidal braking forces on objects which are known to be tidally locked and on objects known to not be tidally locked.

• Stumbled over the thread of reasoning a bit, but this seems right. Oct 30, 2021 at 6:55