Thread: Asteroid's orbit around sun mimics Earth's orbit? How can that be?

This article on ScienceDaily.com tells of an asteroid whose orbit is so similar to Earth's orbit that the asteroid has been "following" Earth for hundreds of thousands of years. This confuses me: given that its mass is so much smaller than Earth's, I would expect that it would have to travel at a much lower speed around the sun, which suggests (to this novice) that we'd be overtaking it periodically, and at a rate far too high to allow it to sit behind us for thousands of years. Also, given its small mass again, if it were moving at the same speed around the sun as the Earth does, then its orbit should be much further out, shouldn't it? Can someone help me to understand this? I thought it might have something to do with Lagrange points, but I would have expected the article to mention that, as it seems like a strange coincidence.

Originally Posted by GreatBigBore

This article on ScienceDaily.com tells of an asteroid whose orbit is so similar to Earth's orbit that the asteroid has been "following" Earth for hundreds of thousands of years. This confuses me: given that its mass is so much smaller than Earth's, I would expect that it would have to travel at a much lower speed around the sun, which suggests (to this novice) that we'd be overtaking it periodically, and at a rate far too high to allow it to sit behind us for thousands of years. Also, given its small mass again, if it were moving at the same speed around the sun as the Earth does, then its orbit should be much further out, shouldn't it? Can someone help me to understand this? I thought it might have something to do with Lagrange points, but I would have expected the article to mention that, as it seems like a strange coincidence.

The mass of the orbiting object has an insignificant effect its orbit. If you replaced the Earth with a 1 kg object, its orbit would be all but indistinguishable from the Earth's present orbit. The "horseshoe" orbit is typical for a small object who's average distance is the same as another.

The process goes something like this: Assume that the object is approaching from behind the Earth in its orbit. As the Earth's gravity pulls forward on the object it gains orbital energy, this pushes further out from the Sun into a slightly higher orbit. But a higher orbit is slower orbit and the object actually loses speed on the Earth and starts to fall behind. Eventually, the Earth begins to catch up to the object from the other direction (kind of like a car lapping another car on a circular track). Again the Earth's gravity pulls on the object, but this time it is pulling the object backward. This causes the object to drop into a lower orbit which is a faster orbit, and it begins to out pace the Earth and pull away. This cycle continues with the Object approaching the Earth from one direction and then the other only to recede again.

Originally Posted by Janus

The mass of the orbiting object has an insignificant effect its orbit. If you replaced the Earth with a 1 kg object, its orbit would be all but indistinguishable from the Earth's present orbit. The "horseshoe" orbit is typical for a small object who's average distance is the same as another.

Thanks, your description of the process helps a lot! But I'm confused. A 1kg object occupying the Earth's orbit? Let's remove Earth from the picture and just have your 1kg object. If it were to stay in the orbit that the Earth had occupied, then the object would have to move a lot more slowly, wouldn't it? I mean, if it were moving at Earth speed, wouldn't it move out to a much larger orbit? Again, your description is awesome. I had no idea that there would be such a complex interplay between Earth and an asteroid. Thanks much.

Originally Posted by GreatBigBore

Originally Posted by Janus

The mass of the orbiting object has an insignificant effect its orbit. If you replaced the Earth with a 1 kg object, its orbit would be all but indistinguishable from the Earth's present orbit. The "horseshoe" orbit is typical for a small object who's average distance is the same as another.

Thanks, your description of the process helps a lot! But I'm confused. A 1kg object occupying the Earth's orbit? Let's remove Earth from the picture and just have your 1kg object. If it were to stay in the orbit that the Earth had occupied, then the object would have to move a lot more slowly, wouldn't it? I mean, if it were moving at Earth speed, wouldn't it move out to a much larger orbit? Again, your description is awesome. I had no idea that there would be such a complex interplay between Earth and an asteroid. Thanks much.

Actually that's the strange thing about gravity. The mass of the object being acted upon has an absolute zero effect on how fast it will fall. Galileo demonstrated this in 1589 by dropping two objects with different mass from the leaning tower of Piza and having them land at the same time. Or... at least according to legend. I may have been merely a thought experiment.

http://en.wikipedia.org/wiki/Galileo's_Leaning_Tower_of_Pisa_experiment
..

The point is that a small object doesn't orbit at a different speed than a heavy one. (The one exception is if the falling object is so heavy that it can move the pulling object, but the Sun is really really big, so I don't think that exception applies in this case. )

Originally Posted by GreatBigBore

Originally Posted by Janus

The mass of the orbiting object has an insignificant effect its orbit. If you replaced the Earth with a 1 kg object, its orbit would be all but indistinguishable from the Earth's present orbit. The "horseshoe" orbit is typical for a small object who's average distance is the same as another.

Thanks, your description of the process helps a lot! But I'm confused. A 1kg object occupying the Earth's orbit? Let's remove Earth from the picture and just have your 1kg object. If it were to stay in the orbit that the Earth had occupied, then the object would have to move a lot more slowly, wouldn't it? I mean, if it were moving at Earth speed, wouldn't it move out to a much larger orbit?

No, in situation where the primary object being orbited (in this case the Sun), is many times more massive than the orbiting object (true for both the Earth and 1 kg object), the mass of the orbital object has almost no effect on the orbit.

Simple case, a circular orbit:

The gravitational force between the object and the Sun is found by



Ms is the mass of the sun, Mo is the mass of the object , d is the distance between them and G is the universal gravitational constant.

The force needed to hold a object in a circular path is



v is the velocity of the object and r is the radius of the circle.

For an orbit, v is the orbital speed and d=r and the gravitational force is what provides the force to maintain the orbit.

thus:

Fg=Fc and



The first thing to notice is that Mo and r appear on both sides of the equation, so we can do some canceling. One r on the right cancels out one r from r² on the left, leaving r, and both "Mo"s
cancel each other out leaving:



Thus the orbital speed depends on the mass of the Sun and the orbital distance alone (G being a constant that does not change).

Thanks again, but now I'm even more confused. I get the math, but what about satellites orbiting Earth? Their mass is tiny in comparison, but I always had the impression that if an Earth satellite were to slow down enough, it would fall to Earth. Is that not the case? Is it really possible for a satellite to slow to a crawl and stay in orbit? Thanks for being patient with this non-specialist.

Originally Posted by GreatBigBore

Thanks again, but now I'm even more confused. I get the math, but what about satellites orbiting Earth? Their mass is tiny in comparison, but I always had the impression that if an Earth satellite were to slow down enough, it would fall to Earth. Is that not the case? Is it really possible for a satellite to slow to a crawl and stay in orbit? Thanks for being patient with this non-specialist.

Orbital mechanics can be tricky to grasp. If you slow a satellite, it will change orbit. It will enter an orbit which has a closer average distance to the Earth. The orbit will also become more elliptical (Assuming that it was circular to begin with. If it was already elliptical, it could either increase or decrease the eccentricity of the orbit, depending on at what point of the orbit you slow it down.)

This new orbit will remain stable unless it becomes low enough for it to brush the atmosphere and lose even more speed through friction. It will also be a faster orbit. If you look at the equation I gave it shows that as the orbital distance decreases, the orbital speed increases. So by slowing down, a satellite can actually gain average speed. It can do this, because as it moves in closer to the Earth it loses gravitational potential , which it converts into kinetic energy in the form of speed.

Wow. I guess celestial mechanics is a bit more complex than I thought, and I already thought it was pretty complex. So I guess if I stopped my satellite dead in its tracks on the major axis, the orbit would become "infinitely eccentric," falling straight to Earth? Thanks so much for the brain exercise!