<p>I will answer the specific case of the Moon disappearing since the more general question, is a bit open-ended. </p>

<p>Of course the Moon cannot just "disappear." However, there are very real circumstances where a planet or other stellar object might be disrupted. The best example I can think of is if a sufficiently large asteroid or moon from another planet somehow make their way into Earth's orbit and collide with the Moon. Such collisions between Solar System objects are rare, but occasionally occur in models of the future evolution of the Solar System (see work by <a href="https://en.wikipedia.org/wiki/Jacques_Laskar" rel="nofollow noreferrer">Jacques Laskar</a>). </p>

<p>However, one must be careful to first consider whether that mass is actually lost from the system. For instance, in the Giant Impact Scenario a Mars-sized object is theorized to have slammed into the Earth, but the material never left the system. If a similar collision were to disintegrate the Moon and the mass remained as rocks and dust in orbit around the Earth, the effect on Earth's orbit would be minor. This is because it is the Earth-Moon <em>system</em> that orbits around the Sun. Note that there is a small change since the Earth-Moon barycenter has shifted position, but the effect is tiny so let's ignore it for now.</p>

<p>Instead, let's consider a case where the collision disrupts the Moon, and its material leaves Earth's orbit. If this occurs sufficiently quickly, (faster than the orbital period, i.e., 1 year) then from an orbital dynamics standpoint, the mass is effectively lost instantaneously. The Dutch astrophysicist Adriaan Blaauw, <a href="https://ui.adsabs.harvard.edu/abs/1961BAN....15..265B/abstract" rel="nofollow noreferrer">first calculated the relevant physics</a>. Earth's orbit will expand, but by an almost miniscule amount. The reason is that the dynamics of the Earth's orbit around the Sun is dictated by the <em>total</em> mass of the system (i.e. Earth's mass plus the Sun's mass). To lowest order, the new orbit's orbital separation (<span class="math-container">$a$</span>) will be:</p>

<p><span class="math-container">$$ a \approx 1 + \frac{\Delta M}{M} {\rm AU} \approx 1 + \frac{1/81 M_{\oplus}}{M_{\odot}} {\rm AU} \approx 1 + 4\times10^{-8} {\rm AU} $$</span></p>

<p>So, the orbit will expand by roughly 40 parts in a billion. A quick calculation using Kepler's third law (<span class="math-container">$a^3\sim P^2$</span>) shows that the length of a year would increase by about 2 seconds. A similar change in the Earth's orbital eccentricity will occur (<span class="math-container">$\Delta e \sim \frac{\Delta M}{M}$</span>), but since this change is smaller than the precision to which we have measured the Earth's orbital eccentricity, the effect is negligible.</p>

CC BY-SA 4.0
2020-02-10T12:27:36.773Z
Okay, but I was expecting more of a Keplerian equation based on stable orbits.
Tyler Durden
CC BY-SA 4.0
2020-02-10T19:52:35.757Z
Surely something needs to be said about angular momentum loss?
ProfRob
CC BY-SA 4.0
2020-02-11T02:42:43.537Z
I added something about orbital eccentricity, which should help address angular momentum loss. Or maybe I'm missing something? Happy to add more, @TylerDurden if you can tell me more about what you're looking for.
Jeff
CC BY-SA 4.0
2020-02-10T04:55:42.470Z
2020-02-11T02:39:59.390Z
2020-02-11T02:39:59.390Z
Jeff
<p>If a planet becomes lighter, then I believe its orbit will change.</p>

<p>For example, the moon is 1/81 of the mass of the earth. So, if the moon disappeared, then the earth-moon system would lose 1/81th of its mass. How would the orbit of the earth change in that circumstance?</p>

CC BY-SA 4.0
2020-02-10T01:39:44.103Z
So should we move the Earth to the location of the original Earth-Moon center of mass position, and give it the center of mass' velocity?
uhoh
CC BY-SA 4.0
2020-02-10T02:15:00.683Z
This question has been asked many times, in many forms. The problem is that the Earth cannot instantaneously lose 1/81th of its mass, in either Newtonian mechanics or general relativity. You essentially are asking what the laws of physics say will happen when the laws of physics are violated. Such questions do not make sense.
David Hammen
CC BY-SA 4.0
2020-02-10T02:36:43.213Z
@DavidHammen you're not a big Space 1999 fan? (humor) https://youtu.be/Y6BXaGEuqxo?t=2393 lower volume before playing!
uhoh
CC BY-SA 4.0
2020-02-10T02:52:40.093Z
I'm voting to close this question as off-topic because the question has no basis in science.
David Hammen
CC BY-SA 4.0
2020-02-10T02:52:46.760Z
@uhoh - Such questions might be appropriate for the sci-fi or world building sister sites, where random violations of science are fair game. But not here. Voting to close.
David Hammen
CC BY-SA 4.0
2020-02-13T21:26:14.250Z
More to the point, the question says "if a planet becomes lighter I believe its orbit will change".  But if we replaced the Moon with, say, a baseball moving at the same speed and at the same distance, the baseball would still take a month to orbit the Earth and the Earth would still take a year to orbit the Sun. Can you say why you believe this seemingly false thing?  By understanding why you believe something false, or what subtle thing you're not telling us that makes it true, we can better address your real question.
Eric Lippert
CC BY-SA 4.0
2020-02-13T22:09:35.813Z
@EricLippert If the orbiting body looses mass (like a comet for example), then what happens is that the gravitational force decreases, but velocity remains the same, so the orbit expands and the period of the orbit increases.
Tyler Durden
CC BY-SA 4.0
2020-02-13T22:15:02.817Z
@TylerDurden: And now we're getting to the crux of your false beliefs, which is good; that will let you replace them with true beliefs. If the gravitational force decreases because the mass decreases, **what happens to the acceleration**? Can you state Newton's laws?
Eric Lippert
CC BY-SA 4.0
2020-02-13T22:19:44.920Z
@EricLippert I think it is pretty well established that the orbit will expand due both to the theory of stable orbits and studies of comets (see this question for example: https://astronomy.stackexchange.com/questions/18539/does-the-loss-of-mass-create-an-observable-change-in-a-comets-orbit)
Tyler Durden
CC BY-SA 4.0
2020-02-13T22:41:34.463Z
@TylerDurden: Comets are particularly interesting because they're melting, because the rate of melting is a function of distance, and because the momentum of the gas that is produced needs to be taken into account. If you're proposing that we melt the Moon then I submit to you that the critique above that the question is unphysical is germane. By what process do you propose that we get rid of the Moon?
Eric Lippert
CC BY-SA 4.0
2020-02-13T23:04:46.793Z
Consider the following thought experiment. Suppose we have two identical bowling balls side by side in identical orbits around the Sun, attached together by a very light, strong thread. The balls are small enough that we can neglect their gravitational attraction to each other. The total mass of the object is 2m, where m is the mass of one ball. Do you propose that when we cut the thread, and now have two independently orbiting objects of mass m each, that their orbits will significantly change?
Eric Lippert
CC BY-SA 4.0
2020-12-22T19:38:52.513Z
It *could* be a valid question if “disappearance” of Moon was elucidated. Moon hit by energy weapon and slowly vaporized ⇒ one scenario, Moon kicked away by an impact ⇒ another scenario. Even within the vaporization scenario the result would depend on details: temperature, spacial asymmetry, etc.
Incnis Mrsi
<the-moon><earth><astrophysics>
If the earth lost 1/81th of its mass how would it orbit change?
Tyler Durden
# If the earth lost 1/81th of its mass how would it orbit change?

(Asked by: username_0 on 2020-02-10)

If a planet becomes lighter, then I believe its orbit will change.

For example, the moon is 1/81 of the mass of the earth. So, if the moon disappeared, then the earth-moon system would lose 1/81th of its mass. How would the orbit of the earth change in that circumstance?

**Tags:** the-moon, earth, astrophysics
> **Comments:**
> **username_1**: So should we move the Earth to the location of the original Earth-Moon center of mass position, and give it the center of mass' velocity?  
> **username_2**: This question has been asked many times, in many forms. The problem is that the Earth cannot instantaneously lose 1/81th of its mass, in either Newtonian mechanics or general relativity. You essentially are asking what the laws of physics say will happen when the laws of physics are violated. Such questions do not make sense.  
> **username_1**: @DavidHammen you're not a big Space 1999 fan? (humor) https://youtu.be/Y6BXaGEuqxo?t=2393 lower volume before playing!  
> **username_2**: I'm voting to close this question as off-topic because the question has no basis in science.  
> **username_2**: @uhoh - Such questions might be appropriate for the sci-fi or world building sister sites, where random violations of science are fair game. But not here. Voting to close.  
> **username_3**: More to the point, the question says "if a planet becomes lighter I believe its orbit will change".  But if we replaced the Moon with, say, a baseball moving at the same speed and at the same distance, the baseball would still take a month to orbit the Earth and the Earth would still take a year to orbit the Sun. Can you say why you believe this seemingly false thing?  By understanding why you believe something false, or what subtle thing you're not telling us that makes it true, we can better address your real question.  
> **username_0**: @EricLippert If the orbiting body looses mass (like a comet for example), then what happens is that the gravitational force decreases, but velocity remains the same, so the orbit expands and the period of the orbit increases.  
> **username_3**: @TylerDurden: And now we're getting to the crux of your false beliefs, which is good; that will let you replace them with true beliefs. If the gravitational force decreases because the mass decreases, **what happens to the acceleration**? Can you state Newton's laws?  
> **username_0**: @EricLippert I think it is pretty well established that the orbit will expand due both to the theory of stable orbits and studies of comets (see this question for example: https://astronomy.stackexchange.com/questions/18539/does-the-loss-of-mass-create-an-observable-change-in-a-comets-orbit)  
> **username_3**: @TylerDurden: Comets are particularly interesting because they're melting, because the rate of melting is a function of distance, and because the momentum of the gas that is produced needs to be taken into account. If you're proposing that we melt the Moon then I submit to you that the critique above that the question is unphysical is germane. By what process do you propose that we get rid of the Moon?  
> **username_3**: Consider the following thought experiment. Suppose we have two identical bowling balls side by side in identical orbits around the Sun, attached together by a very light, strong thread. The balls are small enough that we can neglect their gravitational attraction to each other. The total mass of the object is 2m, where m is the mass of one ball. Do you propose that when we cut the thread, and now have two independently orbiting objects of mass m each, that their orbits will significantly change?  
> **username_4**: It *could* be a valid question if “disappearance” of Moon was elucidated. Moon hit by energy weapon and slowly vaporized ⇒ one scenario, Moon kicked away by an impact ⇒ another scenario. Even within the vaporization scenario the result would depend on details: temperature, spacial asymmetry, etc.  

---

## 1 answers

### Answer by username_5 on 2020-02-10. Score: 3

I will answer the specific case of the Moon disappearing since the more general question, is a bit open-ended.

Of course the Moon cannot just "disappear." However, there are very real circumstances where a planet or other stellar object might be disrupted. The best example I can think of is if a sufficiently large asteroid or moon from another planet somehow make their way into Earth's orbit and collide with the Moon. Such collisions between Solar System objects are rare, but occasionally occur in models of the future evolution of the Solar System (see work by [Jacques Laskar](https://en.wikipedia.org/wiki/Jacques_Laskar)).

However, one must be careful to first consider whether that mass is actually lost from the system. For instance, in the Giant Impact Scenario a Mars-sized object is theorized to have slammed into the Earth, but the material never left the system. If a similar collision were to disintegrate the Moon and the mass remained as rocks and dust in orbit around the Earth, the effect on Earth's orbit would be minor. This is because it is the Earth-Moon *system* that orbits around the Sun. Note that there is a small change since the Earth-Moon barycenter has shifted position, but the effect is tiny so let's ignore it for now.

Instead, let's consider a case where the collision disrupts the Moon, and its material leaves Earth's orbit. If this occurs sufficiently quickly, (faster than the orbital period, i.e., 1 year) then from an orbital dynamics standpoint, the mass is effectively lost instantaneously. The Dutch astrophysicist Adriaan Blaauw, [first calculated the relevant physics](https://ui.adsabs.harvard.edu/abs/1961BAN....15..265B/abstract). Earth's orbit will expand, but by an almost miniscule amount. The reason is that the dynamics of the Earth's orbit around the Sun is dictated by the *total* mass of the system (i.e. Earth's mass plus the Sun's mass). To lowest order, the new orbit's orbital separation ($a$) will be:

$$ a \approx 1 + \frac{\Delta M}{M} {\rm AU} \approx 1 + \frac{1/81 M\_{\oplus}}{M\_{\odot}} {\rm AU} \approx 1 + 4\times10^{-8} {\rm AU} $$

So, the orbit will expand by roughly 40 parts in a billion. A quick calculation using Kepler's third law ($a^3\sim P^2$) shows that the length of a year would increase by about 2 seconds. A similar change in the Earth's orbital eccentricity will occur ($\Delta e \sim \frac{\Delta M}{M}$), but since this change is smaller than the precision to which we have measured the Earth's orbital eccentricity, the effect is negligible.

> **Comments:**
> **username_0**: Okay, but I was expecting more of a Keplerian equation based on stable orbits.  
> **username_6**: Surely something needs to be said about angular momentum loss?  
> **username_5**: I added something about orbital eccentricity, which should help address angular momentum loss. Or maybe I'm missing something? Happy to add more, @TylerDurden if you can tell me more about what you're looking for.  

---

