Why does the Moon orbit the Earth rather than the Sun?

[Ryker]

I was just wondering why, if the gravitational force exerted on it by the Sun is greater than the one exerted on it by the Earth, the Moon stays in orbit around the Earth? Is it due to angular momentum or are there any other factors?

 

[Janus]

Since both the Earth and Moon are essentially the same distance from the Sun, they both experience the same acceleration towards the Sun. In order for there to be net force pulling them a part, this acceleration would have to be different.

Granted, at times the Moon is closer to or further from the Sun than the Earth is, and at those times the accelerations differ. However, this difference in distance is small compared to the total Earth-Sun distance, and results in only a small tendency for the Moon to pull away from the Earth. Small enough that the Earth has no trouble holding on the Moon.

 

[D H]

The answer is that gravitational force is the wrong metric.

The gravitational force from the Earth is equal to the from the Sun at a distance of about 40.65 Earth radii from the center of the Earth. There is very little difference in behavior of an object orbiting the Earth just inside this distance versus just outside it. They are both orbiting the Earth.

A better metric is mechanical energy. If the total mechanical energy of some object from the perspective of an Earth-centered frame is positive the object is not in an bound orbit about the Earth. If it's negative, it would be in a bound orbit if it was just a question of the Earth and the object. However, there are pesky objects like the Sun, Jupiter, and Venus that can make the object escape the Earth. An even better metric is needed to determine if an object truly is in a bound orbit.

This better metric is the Hill sphere (google that term).

 

[Ryker]

I was actually trying to avoid saying anything about the Hill sphere. A question related to mine is namely also one of the questions for our assignment, and we haven't mentioned the sphere at all (and I doubt we ever will). After looking it up, I also don't quite understand why exactly a Hill sphere emerges and what the reasons for the bodies that are within it stay inside the gravitational influence of the central body.

Also, does angular momentum then have nothing to do with it? I'm having trouble imagining it doesn't, because surely if the Moon wasn't rotating an was instead between the Earth and the Sun, it would start accelerating towards the Sun due to the positive net force in that direction.

 

[Ken G]

Perhaps another way to frame the answer in terms of what you are saying about angular momentum is that the Moon does orbit the Sun, and it also orbits the Earth. If it did not orbit the Sun, it certainly couldn't orbit the Earth either. So we can layer the answer, saying that first and foremost, every planet and every moon in the solar system orbits the Sun, or else it isn't here. Then, on top of that, they might also orbit each other, if they are within each other's Hill sphere. In other words, it's not an either/or question, it's a first one and then the other question. Note this does not contradict the answers you already got, it merely slices the pie in a different way, which seems like what you are looking for.

 

[Ryker]

Alright, thanks everyone. How would saying the following sound, though? During the evolution of the Solar system the Moon has acquired angular momentum in the direction that is causing it to revolve around (orbit) the Earth. Due to conservation of angular momentum, the orbit could only be altered and the Moon torn away from the Earth by applying enough torque so as to "nullify" that angular momentum. But since the Moon is orbiting the Earth, so that its position changes in time, the Sun's gravitational force itself isn't sufficient to cause that change in angular momentum. Is there anything wrong with this explanation?

 

[D H]

I was actually trying to avoid saying anything about the Hill sphere. A question related to mine is namely also one of the questions for our assignment, and we haven't mentioned the sphere at all (and I doubt we ever will).

Then don't use the Hill sphere. Use the term gravitational sphere of influence instead.

The Hill sphere and Laplace sphere are the two leading contenders in determining the gravitational sphere of influence of some body.

Also, does angular momentum then have nothing to do with it?

Not really.

You have missed one obvious explanation, and that is frame of reference.

What is the gravitational force of the Sun on the Moon from the perspective of a reference frame with origin at the center of the Earth? The answer is not GM_sunm_moon/R^2. That is but a part of the answer. There is another part. Hint: The Earth is accelerating toward the Sun.