<p>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.</p>
<p><strong>Conservation of energy</strong></p>
<p>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 <a href="https://en.wikipedia.org/wiki/Tidal_heating" rel="nofollow noreferrer">tidal heating</a>.</p>
<p><strong>Conservation of AM (Angular Momentum)</strong></p>
<p>If we assume AM is conserved, according to <a href="https://www.annualreviews.org/doi/abs/10.1146/annurev-earth-050212-124057?journalCode=earth" rel="nofollow noreferrer">Asphaug</a> 2014:</p>
<blockquote>
<p>If somehow brought together into one place, <span class="math-container">$\vec{L}_{EM}$</span> would equal the
angular momentum of an Earth-mass planet spinning with a period of∼5
h—an anomalously large value that Cameron &amp; Ward (1976) took to
indicate a giant impact.</p>
</blockquote>
<p>Here, <span class="math-container">$\vec{L}_{EM}$</span>  is the combined angular momentum of the Earth/Moon system.</p>
<p>If the Moon formed just outside Earth’s Roche limit of 2.9 <span class="math-container">$R \oplus$</span>, 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:</p>
<ol>
<li>The Earth’s rotation is slowing due to its own tidal interaction with the Sun. <a href="https://en.wikipedia.org/wiki/Tidal_acceleration#Effect_of_the_Sun" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Tidal_acceleration#Effect_of_the_Sun</a></li>
<li>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 <span class="math-container">$R \oplus$</span>.  The Earth/Moon system shed angular momentum as passed through each of the resonances.  Between 6 and 8 <span class="math-container">$R \oplus$</span>, 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 <a href="https://www.annualreviews.org/doi/abs/10.1146/annurev-earth-050212-124057?journalCode=earth" rel="nofollow noreferrer">Asphaug</a>).</li>
<li>Solar mass ejections and other impacts aren’t thought to significantly alter the Earth/Moon AM since formation.</li>
</ol>
<p><strong>Is there a range of Earth rotation speeds possible?</strong></p>
<p><a href="https://arxiv.org/abs/1802.10223" rel="nofollow noreferrer">Lock et. Al</a>  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 <a href="https://en.wikipedia.org/wiki/Synestia" rel="nofollow noreferrer">synestia</a> instead of a planet. Here is a figure from their paper:</p>
<p><a href="https://i.sstatic.net/tyc9Q.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tyc9Q.png" alt="enter image description here" /></a></p>
<p>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.</p>
<p>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: <a href="https://astronomy.stackexchange.com/questions/23489/why-doesnt-the-earths-rotation-throw-us-off-the-surface/23491#23491">Why doesn&#39;t the Earth&#39;s rotation throw us off the surface?</a>.</p>
<p>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.</p>

CC BY-SA 4.0
2021-01-08T04:19:47.687Z
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).
Acccumulation
CC BY-SA 4.0
2021-01-08T04:56:27.923Z
@Acccumulation True, there is some loss to heat as well.  Those losses would also factor in as a faster initial rotation speed.
Connor Garcia
CC BY-SA 4.0
2021-01-21T20:45:31.730Z
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](https://en.wikipedia.org/wiki/Standard_gravitational_parameter).
PM 2Ring
CC BY-SA 4.0
2024-11-22T18:31:27.803Z
@ConnorGarcia It seems you know quite a lot about production of moons by impact. I would be much in your debt if you could answer the question https://astronomy.stackexchange.com/q/58979/30705 that I placed today. Thank you.
Michael_1812
CC BY-SA 4.0
2021-01-08T02:49:43.920Z
2021-01-25T03:54:15.287Z
2021-01-25T03:54:15.287Z
Connor Garcia
<p>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.</p>

CC BY-SA 4.0
2018-05-11T13:12:14.817Z
Part of the difficulty is that there still are competing theories as to the exact formation process.
Carl Witthoft
CC BY-SA 4.0
2018-05-11T15:23:08.517Z
True. I have clarified the question.
Ags1
CC BY-SA 4.0
2018-05-11T17:40:39.793Z
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 :-)
Carl Witthoft
CC BY-SA 4.0
2018-05-11T20:08:34.817Z
Well, it must be short of escape velocity or you wouldn't have a planet... But even at current 24h rotation, it pays to launch from the equator.
Ags1
CC BY-SA 4.0
2018-05-11T21:50:47.090Z
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.
David Hammen
CC BY-SA 4.0
2018-05-12T05:25:38.577Z
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.  :-)
userLTK
CC BY-SA 4.0
2019-11-15T02:23:58.883Z
"Scientists estimate that a day in the life of early Earth was only about 6 hours long." [NASA](https://spaceplace.nasa.gov/review/dr-marc-earth/earth-rotation.html)
Keith McClary
<the-moon>
How fast was the Earth spinning directly after the Moon formed?
Ags1
# How fast was the Earth spinning directly after the Moon formed?

(Asked by: username_0 on 2018-05-11)

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.

**Tags:** the-moon
> **Comments:**
> **username_1**: Part of the difficulty is that there still are competing theories as to the exact formation process.  
> **username_0**: True. I have clarified the question.  
> **username_1**: 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 :-)  
> **username_0**: Well, it must be short of escape velocity or you wouldn't have a planet... But even at current 24h rotation, it pays to launch from the equator.  
> **username_2**: 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.  
> **username_3**: 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.  :-)  
> **username_4**: "Scientists estimate that a day in the life of early Earth was only about 6 hours long." [NASA](https://spaceplace.nasa.gov/review/dr-marc-earth/earth-rotation.html)  

---

## 1 answers

### Answer by username_5 on 2021-01-08. Score: 2 | Accepted answer

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](https://en.wikipedia.org/wiki/Tidal_heating).

**Conservation of AM (Angular Momentum)**

If we assume AM is conserved, according to [Asphaug](https://www.annualreviews.org/doi/abs/10.1146/annurev-earth-050212-124057?journalCode=earth) 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](https://www.annualreviews.org/doi/abs/10.1146/annurev-earth-050212-124057?journalCode=earth)).
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](https://arxiv.org/abs/1802.10223) 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](https://en.wikipedia.org/wiki/Synestia) instead of a planet. Here is a figure from their paper:

[![enter image description here](https://i.sstatic.net/tyc9Q.png)](https://i.sstatic.net/tyc9Q.png)

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?](https://astronomy.stackexchange.com/questions/23489/why-doesnt-the-earths-rotation-throw-us-off-the-surface/23491#23491).

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.

> **Comments:**
> **username_6**: 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).  
> **username_5**: @Acccumulation True, there is some loss to heat as well.  Those losses would also factor in as a faster initial rotation speed.  
> **username_7**: 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](https://en.wikipedia.org/wiki/Standard_gravitational_parameter).  
> **username_8**: @ConnorGarcia It seems you know quite a lot about production of moons by impact. I would be much in your debt if you could answer the question https://astronomy.stackexchange.com/q/58979/30705 that I placed today. Thank you.  

---

