# How would Earth mass loss theoretically affect its orbital mechanics?

1. Aug 20, 2012

### mofobro

Firstly, what happens to earth's orbit around the sun should it gain or lose mass?

I would instinctively guess that a less massive earth would have less gravitational attraction to the sun, therefore widening the orbit. However, I understand that orbital mechanics can run counter-intuitively, and of course there are relativistic issues involved.

Here is an article that claims an estimated loss of 50,000 tons per year.
http://gizmodo.com/5882517/did-you-know-that-earth-is-getting-lighter-every-day

I will attempt to perform the calculation myself of the change in orbital length per millenium. Anyone have a ballpark estimate? Perhaps a kilometer longer? That's assuming a lighter earth moves away from the sun.

I'd also like to take into account the effect of a lighter earth on the distance to the moon. At this rate of Earth mass loss, how many millenia does is take for the moon to be moving away in kilometers per year, not cm?

2. Aug 20, 2012

### Chronos

3. Aug 20, 2012

### mofobro

Thank you.

I can't seem to find anything one way or another about mass in relation to orbit from the link you provided.

I just want to make sure, you're saying that if, say, an asteroid knocked away the moon (thereby reducing the mass of the earth-moon system) there wouldn't be any change in the distance from the earth to the sun, or the period of its orbit?

Last edited: Aug 20, 2012
4. Aug 20, 2012

### mofobro

5. Sep 21, 2012

### Buckleymanor

Very much doubt it would have no effect. We know asteroid orbital paths are effected by the sun's radiation via orbital outgassing you would imagine a planet would act in a similar manner.
Increasing or decreasing the earths mass would have a consiquence of enlarging it's diameter or vice versa. Resulting in more or less radiation outgassing and change in orbital speed and path.

6. Sep 24, 2012

### mofobro

According to the thread 1722283, increasing the earth's mass would only change the "barycenter" around which the earth orbits slightly. I tried using Newton's equations for gravitational force and centripetal force, and found that the mass of the earth cancels out (under the crude assumption that earth's orbit is circular.)

It makes me wonder, though, if we were to send off 90% of earth's mass on rockets (ok, ok, disregard the fact that we haven't that much hydrogen fuel) - we would still orbit more or less the same way? Intuition screams "no," but the math says otherwise.

7. Sep 24, 2012

### HallsofIvy

Do you remember Gallileos' experiment of dropping a two balls of different mass of the 'leaning tower of Pisa'? Mass is not relevant to acceleration because gravitational force is directly proportional to mass and acceleration is inversely proportional to mass. The "mass" cancels out. Mass has no effect on any gravitational motion.

8. Sep 24, 2012

### Buckleymanor

Agreed though the OP question was with regards to theoreticaly affecting it's orbital mechanics.You therefore would have to take into account all possible effects as well as gravity to be thorough.http://earthsky.org/astronomy-essentials/the-yarkovsky-effect-pushing-asteroids-around-with-sunlight

9. Sep 24, 2012

### TurtleMeister

I think you have a couple of errors in this post. The second sentence contains a contradiction (mixing free fall with Newton's second law?) and the last sentence is just plain wrong.

Edit:
Ok, I get your second sentence now, and I agree. At first it looked like a contradiction because you say "Mass is not relevant to acceleration" and then in the same sentence you say "acceleration is inversely proportional to mass". But I was taking those two things out of context.

And, I was probably taking your last sentence out of context. But you did use the word "any" gravitational motion, which would not exclude:
$$A^{rel}=\frac{G(M_1+M_2)}{R^2}$$

Last edited: Sep 24, 2012
10. Oct 13, 2012

### Buckleymanor

Sorry for restarting this thread but just remembered something as you do.
Keplers Laws keep being mentioned as far as any insistence is made on any change mass having an effect on the orbital period or distance from the sun.The only change being the barycentre.
If you were to take an extreme case of adding mass to the Earth to the point where it is larger than the sun won't the sun end up orbitiing the earth would there still be no change of the orbital period and distance. Won't this also have an effect on the other planets orbital speeds and distances within the solar system.

11. Oct 13, 2012

### TurtleMeister

In the extreme example you gave, both the barycentre and the orbital period would change. And yes, it would affect other planets as well. All bodies in the solar system affect each other gravitationally. You may find it interesting that the barycentre and the orbital period are determined by two different properties of mass:

The barycentre is determined by the inertial mass of the bodies. In the two body problem:
$$r_1=a\frac{m_2}{m_1+m_2}$$
$$r_2=a\frac{m_1}{m_1+m_2}$$
where:
$r_1$ is the distance from $m_1$ to the barycentre
$r_2$ is the distance from $m_2$ to the barycentre
$a$ is the distance between the two bodies

The orbital period in the two body problem is determined by the sum of the active gravitational mass of the two bodies and their separation (this is derived from Kepler's third law):
$$T=2\pi\sqrt{\frac{a^3}{\mu_1+\mu_2}}$$
where:
$T$ is the orbital period
$a$ is the sum of the simi-major axes of the two bodies
$\mu_1$ is the standard gravitational parameter (active gravitational mass) of $m_1$.
$\mu_2$ is the standard gravitational parameter (active gravitational mass) of $m_2$.

However, since inertial mass and active gravitational mass are proportionally equivalent, their effects are also proportional.

12. Oct 14, 2012

### Buckleymanor

Now I have to ask the question at which point the barycentre and orbital period would change does it have to be extreme.If adding mass by any amount changes the barycentre then by extension this will affect all bodies in the solar system including the Earth.So you could end up with both an alteration of the Earth's orbital period and a barycentre change between it and the Sun for small or large exchanges of mass.

13. Oct 14, 2012

### TurtleMeister

No, it does not have to be extreme. Any change has an effect. But it can be so small that it can be ignored, depending on your application. Small changes can make a difference over a very long period of time. Just plug the numbers into the equations in post #11 and see for yourself. The Sun is more than 300,000 times the mass of the Earth. It is probably safe to say that there is never going to be enough change in the earths mass that would cause a detectable change in it's orbital period. Therefore, the mass of the earth can be ignored for most all applications.