<p>The best we can do is estimate.</p>
<p>Using <span class="math-container">$\frac{1}{S}=\frac{1}{P_{moon}}+\frac{1}{P_{earth}}$</span> (sign is positive as Sun and Moon moves opposite in the geocentric frame) you can get the corresponding sidereal period for the ancient 30 day synodal month to be 27.72300 days. Then using Kepler's Third Law <span class="math-container">$P^2=\frac{4\pi^2}{G(M+m)}a^3$</span> you will get the ancient semi-major axis to be <span class="math-container">$3.8850\times 10^8\text{ m}$</span>.</p>
<p>Let's assume direct impact against direction of velocity. We can use vis-viva to calculate the velocities. <span class="math-container">$v^2=GM(\frac{2}{r}-\frac{1}{a})$</span> and assume the final orbit (current) to be circular (so the ancient one is slightly elliptical, and <span class="math-container">$r$</span> is constant), to get <span class="math-container">$v_i=1.2886\text{ km/s}$</span> and <span class="math-container">$v_f=1.2748\text{ km/s}$</span>.</p>
<p>For elastic collision, there are infinite solutions that can cause such a change using <span class="math-container">$\Delta v_{moon}=\frac{2m_{asteroid}}{m_{moon}+m_{asteroid}}u_{asteroid}$</span>. Two are <span class="math-container">$m=7\times10^{18}\text{ kg}$</span>, <span class="math-container">$u=70\text{ km s}^{-1}$</span> and <span class="math-container">$m=10^{19}\text{ kg}$</span>, <span class="math-container">$u=50\text{ km s}^{-1}$</span> with respect to the Moon, but this is just a <em>very</em> rough estimation because we assumed the asteroid collided with the Moon like a billiard ball when in reality the Moon could be blasted into chunks. Also <span class="math-container">$10^{18}$</span> means 1 billion objects of 1 million tons.</p>
<p><em>Corollary: Should we expect the moon also to orbit more closely to the earth in result?</em></p>
<p>Yes, from Kepler's Third Law, a shorter period means a shorter orbit.</p>
<p><strong>Edit:</strong></p>
<p>For inelastic collision (which ProfRob pointed out to be closer to the actual scenario), we can use <span class="math-container">$\Delta v_{moon}=\frac{m_{asteroid}}{m_{moon}+m_{asteroid}}u_{asteroid}$</span> and get infinite solutions including <span class="math-container">$m=2
\times10^{19}\text{ kg}$</span>, <span class="math-container">$u=50\text{ km s}^{-1}$</span>. Your answer should be roughly bounded by these two cases.</p>
<p>Also answering the question in the title &quot;How much energy to shorten a synodic month by about 1.56%?&quot;, we can use <span class="math-container">$\Delta E=-\frac{GM_{earth}m_{moon}}{2}(\frac{1}{a_f}-\frac{1}{a_i})$</span> and get <span class="math-container">$\Delta E=-8.0\times10^{26}\text{ J}$</span></p>

CC BY-SA 4.0
2022-04-24T14:30:00.237Z
what is $v_A$; what is $u_B$, what are $m_A$ and $m_B$? Asteroids don't move at 500 km/s, I think the maximum speed you could asume is around 70 km/s unless hypothesising an interstellar asteroid. I'm also troubled by the assumption of an elastic collision. I think assuming the collision is inelastic would be much more accurate.
ProfRob
CC BY-SA 4.0
2022-04-25T03:14:26.963Z
I agree. What would really happen in the case of the velocities and masses involved here would be *profoundly* inelastic collision, not to mention mass transfer and/or loss. It's a great answer as far as it goes, but I think it can be further improved by avoiding some simplifications. "The best we can do is estimate." Yes, but this is not yet the best that we can do to estimate :-)
uhoh
CC BY-SA 4.0
2022-05-08T04:39:22.020Z
Thanks, Mr. Qin.  I conclude it is impossible.
Peter Heffner
CC BY-SA 4.0
2022-04-24T09:18:15.813Z
2022-04-25T01:29:05.453Z
2022-04-25T01:29:05.453Z
Chooi Je Qin
<p>Suppose the moon underwent a single, massive, large-object bombardment event.  About what number (or range) of about what mean mass of objects could shorten the synodic month by about 1.5633%?</p>
<p>(Assume the moon is not broken up.)</p>
<p>Corollary:  Should we expect the moon also to orbit more closely to the earth in result?</p>
<p>By the way the question is historical.  There is at least one ancient text that states that rocks “hurled to the earth” from the skies, whereupon the synodal month shortened from 30 days to the current value.  That being “outside my field,” literary scholars don’t generally tackle it.  So, is there a general way to approach the problem for antiquarians lacking a technical background in physics?</p>
<p>Hoping for an answer such as, say, ‘If 100 objects of a mean mass of 100 tons each struck the moon at a typical velocity of __, over a period of three weeks, then the synodal period could shorten by 1.56%.’ Or, ‘It would be impossible to shorten the synodal period of the moon 1.56% in an asteroid bombardment, because’…</p>
<p>An answer might also help in interpreting the ancient astronomical vocabulary, which so far is translated word for word as is.</p>
<p>Also, if relevant, I date the text’s putative asteroid bombardment to around 3044 BCE.</p>
<p>Thank you!</p>

CC BY-SA 4.0
2022-04-17T01:59:27.133Z
Does "decelerate" the synodic month mean to make it shorter? So instead of 29 days 12 hours 44 minutes 2.9 sec you want it to be 29 days 1 hour 39 minutes 16 seconds?
JohnHoltz
CC BY-SA 4.0
2022-04-17T03:12:32.230Z
Yes. Thank you. Or, suppose the synodal month were shortened from 30 days to the physical value.
Peter Heffner
CC BY-SA 4.0
2022-04-17T05:17:45.623Z
By the way I think it would be okay to ask what kind of kinematics could have shortened a 30 day orbit to its current value, and how much more circular the original orbit could have been as well. A big "delta-v" will affect eccentricity as well as semi-major axis (period), see the [vis-viva equation](https://en.wikipedia.org/wiki/Vis-viva_equation).
uhoh
CC BY-SA 4.0
2022-04-17T05:17:48.610Z
[The vis-viva equation gives everything we need.](https://space.stackexchange.com/a/23104/12102) and [Many quantitative questions about orbits can be answered using the vis-viva equation](https://space.stackexchange.com/a/45622/12102) and [The vis-viva equation is the go-to equation for a lot of things](https://space.stackexchange.com/a/39696/12102)
uhoh
CC BY-SA 4.0
2022-04-24T11:08:18.543Z
You have to extract energy from the system to shorten the period.
ProfRob
<the-moon><asteroids><history><size><hypothetical>
How much energy to shorten a synodic month by about 1.56%?
Peter Heffner
# How much energy to shorten a synodic month by about 1.56%?

(Asked by: username_0 on 2022-04-17)

Suppose the moon underwent a single, massive, large-object bombardment event. About what number (or range) of about what mean mass of objects could shorten the synodic month by about 1.5633%?

(Assume the moon is not broken up.)

Corollary: Should we expect the moon also to orbit more closely to the earth in result?

By the way the question is historical. There is at least one ancient text that states that rocks “hurled to the earth” from the skies, whereupon the synodal month shortened from 30 days to the current value. That being “outside my field,” literary scholars don’t generally tackle it. So, is there a general way to approach the problem for antiquarians lacking a technical background in physics?

Hoping for an answer such as, say, ‘If 100 objects of a mean mass of 100 tons each struck the moon at a typical velocity of \_\_, over a period of three weeks, then the synodal period could shorten by 1.56%.’ Or, ‘It would be impossible to shorten the synodal period of the moon 1.56% in an asteroid bombardment, because’…

An answer might also help in interpreting the ancient astronomical vocabulary, which so far is translated word for word as is.

Also, if relevant, I date the text’s putative asteroid bombardment to around 3044 BCE.

Thank you!

**Tags:** the-moon, asteroids, history, size, hypothetical
> **Comments:**
> **username_1**: Does "decelerate" the synodic month mean to make it shorter? So instead of 29 days 12 hours 44 minutes 2.9 sec you want it to be 29 days 1 hour 39 minutes 16 seconds?  
> **username_0**: Yes. Thank you. Or, suppose the synodal month were shortened from 30 days to the physical value.  
> **username_2**: By the way I think it would be okay to ask what kind of kinematics could have shortened a 30 day orbit to its current value, and how much more circular the original orbit could have been as well. A big "delta-v" will affect eccentricity as well as semi-major axis (period), see the [vis-viva equation](https://en.wikipedia.org/wiki/Vis-viva_equation).  
> **username_2**: [The vis-viva equation gives everything we need.](https://space.stackexchange.com/a/23104/12102) and [Many quantitative questions about orbits can be answered using the vis-viva equation](https://space.stackexchange.com/a/45622/12102) and [The vis-viva equation is the go-to equation for a lot of things](https://space.stackexchange.com/a/39696/12102)  
> **username_3**: You have to extract energy from the system to shorten the period.  

---

## 1 answers

### Answer by username_4 on 2022-04-24. Score: 1 | Accepted answer

The best we can do is estimate.

Using $\frac{1}{S}=\frac{1}{P\_{moon}}+\frac{1}{P\_{earth}}$ (sign is positive as Sun and Moon moves opposite in the geocentric frame) you can get the corresponding sidereal period for the ancient 30 day synodal month to be 27.72300 days. Then using Kepler's Third Law $P^2=\frac{4\pi^2}{G(M+m)}a^3$ you will get the ancient semi-major axis to be $3.8850\times 10^8\text{ m}$.

Let's assume direct impact against direction of velocity. We can use vis-viva to calculate the velocities. $v^2=GM(\frac{2}{r}-\frac{1}{a})$ and assume the final orbit (current) to be circular (so the ancient one is slightly elliptical, and $r$ is constant), to get $v\_i=1.2886\text{ km/s}$ and $v\_f=1.2748\text{ km/s}$.

For elastic collision, there are infinite solutions that can cause such a change using $\Delta v\_{moon}=\frac{2m\_{asteroid}}{m\_{moon}+m\_{asteroid}}u\_{asteroid}$. Two are $m=7\times10^{18}\text{ kg}$, $u=70\text{ km s}^{-1}$ and $m=10^{19}\text{ kg}$, $u=50\text{ km s}^{-1}$ with respect to the Moon, but this is just a *very* rough estimation because we assumed the asteroid collided with the Moon like a billiard ball when in reality the Moon could be blasted into chunks. Also $10^{18}$ means 1 billion objects of 1 million tons.

*Corollary: Should we expect the moon also to orbit more closely to the earth in result?*

Yes, from Kepler's Third Law, a shorter period means a shorter orbit.

**Edit:**

For inelastic collision (which ProfRob pointed out to be closer to the actual scenario), we can use $\Delta v\_{moon}=\frac{m\_{asteroid}}{m\_{moon}+m\_{asteroid}}u\_{asteroid}$ and get infinite solutions including $m=2
\times10^{19}\text{ kg}$, $u=50\text{ km s}^{-1}$. Your answer should be roughly bounded by these two cases.

Also answering the question in the title "How much energy to shorten a synodic month by about 1.56%?", we can use $\Delta E=-\frac{GM\_{earth}m\_{moon}}{2}(\frac{1}{a\_f}-\frac{1}{a\_i})$ and get $\Delta E=-8.0\times10^{26}\text{ J}$

> **Comments:**
> **username_3**: what is $v_A$; what is $u_B$, what are $m_A$ and $m_B$? Asteroids don't move at 500 km/s, I think the maximum speed you could asume is around 70 km/s unless hypothesising an interstellar asteroid. I'm also troubled by the assumption of an elastic collision. I think assuming the collision is inelastic would be much more accurate.  
> **username_2**: I agree. What would really happen in the case of the velocities and masses involved here would be *profoundly* inelastic collision, not to mention mass transfer and/or loss. It's a great answer as far as it goes, but I think it can be further improved by avoiding some simplifications. "The best we can do is estimate." Yes, but this is not yet the best that we can do to estimate :-)  
> **username_0**: Thanks, Mr. Qin.  I conclude it is impossible.  

---

