# How fast was the Earth spinning directly after the Moon formed?

I understand the Moon was perhaps five Earth radii away when it first formed (assuming it was formed by a giant impact of a Mars-sized body), and Earth has since transferred its rotational energy into the moon's orbital energy. How fast would Earth have been spinning at the point the Moon was formed? Assuming it was spinning very fast (day under 10 hours) would this have caused a noticeable decrease in effective "gravity" at the equator? Presumably the Precambrian microbes could have reached equatorial orbit really easily, assuming they had a space program.

• Part of the difficulty is that there still are competing theories as to the exact formation process. May 11, 2018 at 13:12
• True. I have clarified the question.
– Ags1
May 11, 2018 at 15:23
• Hmmm.. 10 hours vs. 24 hours is only a factor of 2.5 in angular speed. I think you'd still be a bit short of escape velocity :-) May 11, 2018 at 17:40
• IIRC, the Earth's rotation rate shortly after the collision that resulted in the formation of the Moon was one rev per four to six hours. I don't have a citation on hand. Also, IIRC, a rotation rate of about twice that rate would have been needed to give something at the surface on the equator orbital velocity. (A high rotation rate would have made the Earth pancake quite a bit, so this is a bit fuzzy.) Finally, yet another IIRC, the rotation rate would have slowed down very quickly because the rate at which angular momentum transfers from the Earth to the Moon is a $1/R^{5/2}$ relationship. May 11, 2018 at 21:50
• I'd prefer to avoid the math, but an estimate could be made by simply adding the Moons angular momentum of it's orbit between 3-5 radii distance and it's current distance to the Earth. Granted the Earth wasn't the same mass back then and some energy was lost over time into friction and heat, but a mathematical estimate could be worked out without too much trouble, that would vary depending on how distant the Moon was when it formed. I think 4-6 hours is about right. Pancaking excluded ofcourse. :-) May 12, 2018 at 5:25

We aren't positive that the Giant Impact hypothesis of Moon formation is correct. If we assume it is, we still don’t know how fast the Earth was rotating just after the collision.

Conservation of energy

If we could assume that energy was conserved from then to now, we could calculate the current energy of the Earth/Moon system and then estimate the rotation speed of the Earth when the Moon was in a near orbit. A previous attempt at this can be seen in the edit revisions for this question. Unfortunately, we can’t assume that energy was conserved since some was certainly lost to tidal heating.

Conservation of AM (Angular Momentum)

If we assume AM is conserved, according to Asphaug 2014:

If somehow brought together into one place, $$\vec{L}_{EM}$$ would equal the angular momentum of an Earth-mass planet spinning with a period of∼5 h—an anomalously large value that Cameron & Ward (1976) took to indicate a giant impact.

Here, $$\vec{L}_{EM}$$ is the combined angular momentum of the Earth/Moon system.

If the Moon formed just outside Earth’s Roche limit of 2.9 $$R \oplus$$, presumably Earth’s rotation rate would be slightly slower than 5 hours. Unfortunately, we can’t assume that AM of the Earth/Moon system is conserved. There are several ways the Earth/Moon system has shed AM:

1. The Earth’s rotation is slowing due to its own tidal interaction with the Sun. https://en.wikipedia.org/wiki/Tidal_acceleration#Effect_of_the_Sun
2. As the size of the Moon’s orbit around the Earth increased, it went through several resonances with other bodies in the Solar System, including Venus at about 49 $$R \oplus$$. The Earth/Moon system shed angular momentum as passed through each of the resonances. Between 6 and 8 $$R \oplus$$, the lunar perigee was precessing synchronous to the Earth’s orbit around the Sun, which is known as the evection resonance. Estimates for the amount of AM shed during the evection vary from 10% to 50% (again referenced by Asphaug).
3. Solar mass ejections and other impacts aren’t thought to significantly alter the Earth/Moon AM since formation.

Is there a range of Earth rotation speeds possible?

Lock et. Al propose the possibility that the impact that created the Moon imparted so much energy that the resultant system exceeded the corotation limit (CoRoL), which is the hottest, highest AM system possible. They theorize the result would be a rapidly spinning vaporized torus of gas called a synestia instead of a planet. Here is a figure from their paper:

They talk about the synestia contracting and shedding angular momentum similar to how the accretion disk of our solar system is thought to have shed momentum as the Sun and planets were formed.

It may not make sense to think about the length of a day for a synestia. However, if Earth coalesced out of a synestia, StephenG calculated the fastest possible rotation rate for the Earth would yield a corresponding fastest day at a little under 1.5 hours: Why doesn't the Earth's rotation throw us off the surface?.

In conclusion, if the Moon was formed from a giant impact, an Earth day was probably somewhere between 1.5 hours and a bit more than 5 hours after the Earth had re-coalesced.

• It seems to me that the angular momentum is more conserved than energy. Rotational energy can be converted to heat (e.g. through tidal dragging). Jan 8, 2021 at 4:19
• @Acccumulation True, there is some loss to heat as well. Those losses would also factor in as a faster initial rotation speed. Jan 8, 2021 at 4:56
• I hereby encourage you in your plans to improve this answer. :) As Accc said, energy conservation is only approximate because the bodies are only approximately rigid, especially just after the Moon's formation. BTW, Wikipedia has a nice table of standard gravitational parameters. Jan 21, 2021 at 20:45