# Orbiting through a torus shaped planet.

sonhouse
Posers and Puzzles 21 Feb '07 22:20
1. sonhouse
Fast and Curious
21 Feb '07 22:20
Can you establish a stable orbit through the center of the torus, like a ring within a ring, think of clutching a doughnut through the hole with thumb and index finger, that would be the shape of the orbit. The planet's mass is equal to the earth, but the hole in the torus is maintained by super machinery like super strong diamond threads or something. So can a stable orbit exist through such an object?
2. 23 Feb '07 08:40
Originally posted by sonhouse
Can you establish a stable orbit through the center of the torus, like a ring within a ring, think of clutching a doughnut through the hole with thumb and index finger, that would be the shape of the orbit. The planet's mass is equal to the earth, but the hole in the torus is maintained by super machinery like super strong diamond threads or something. So can a stable orbit exist through such an object?
We have an exact mathematical solutions for a 2-body system but not for a 3-body system. But even for the 2-body the masses are assumed to be point-like. In real life, a satellite orbit around the Earth is not exactly stable, you have always to take account for all kinds of perturbations, like gravitation from the moon and sun, and a lot of other disturbances. In real life, there are no stable orbits.

In the 3-body system, there are specific orbits that can be stable, but they are very exact in it's nature so in reality you can't use them as stable.

When we have a toroidal bodies, and want to find a stable orbit that goes through the hole, well, my experience is not enough, I'm afraid. I just can’t answer the question. You can't treat the body as a point of mass formed the centre, because you are supposed to go near the centre of the orbit.

But the first idea that comes to my mind is – No, this orbit is not stable.
3. 23 Feb '07 10:27
Originally posted by FabianFnas
We have an exact mathematical solutions for a 2-body system but not for a 3-body system. But even for the 2-body the masses are assumed to be point-like. In real life, a satellite orbit around the Earth is not exactly stable, you have always to take account for all kinds of perturbations, like gravitation from the moon and sun, and a lot of other disturbances. In real life, there are no stable orbits.
Just because there are small variations of the kind you describe doesn't mean it isn't stable. If it was an unstable orbit then these disturbances would disrupt it completely - the fact that the orbits can remain pretty predictable over a long period of time suggests they are stable, in the technical definition of the term.
4. 23 Feb '07 10:35
Originally posted by sonhouse
Can you establish a stable orbit through the center of the torus, like a ring within a ring, think of clutching a doughnut through the hole with thumb and index finger, that would be the shape of the orbit. The planet's mass is equal to the earth, but the hole in the torus is maintained by super machinery like super strong diamond threads or something. So can a stable orbit exist through such an object?
This doesn't prove it either way, but it's an argument that suggests orbits would exist in at least some cases.

Imagine an extremely long and thin torus. Now imagine something orbiting close in to the ring, so it doesn't go near the centre of the torus. Close to the ring the torus would actually appear to be a long cylinder, and the other side of the torus would be so far away that its effect would be negligible.

So you've now got the equivalent of an orbit in two-dimensional space. I would expect that such orbits exist, and may well be stable, although I can't be bothered doing the maths at the moment! The main difference would be that gravity is proportional to 1/r instead of 1/r^2, I think.

If this is right, it shows that orbits exist in this extreme case.
5. 23 Feb '07 10:54
Originally posted by mtthw
Just because there are small variations of the kind you describe doesn't mean it isn't stable. If it was an unstable orbit then these disturbances would disrupt it completely - the fact that the orbits can remain pretty predictable over a long period of time suggests they are stable, in the technical definition of the term.
In satellite orbits they are so insignificant so they can easily be disregarded for a while - but nevertheless they are not stable. MIR fell down because you stopped maintain the orbital parameters, it was not simply stable.

The solar system as a whole is chaotic. In the long run the planets will alter its orbits considerable. Only 2-body systems can be stable, and this can be mathematical proven.

Are we talking only technical? All right then. But in reality, not knowing all forces acting on the bodies can lead to disastrous results for astronauts and others.
6. 23 Feb '07 12:03
What about orbits that stay within the toroid centre and rotate in the same axis as the toroid itself. (Like a small doughnut in the centre of the larger doughnut.)

This could be possible because the mass of the closest part of the toroid is much less than the mass of the toroid away from the satellite, although much closer.

I don't think the other type of orbit could exist because it feels (with my gut - truthiness) that the satellite would need to achieve some sort of escape velocity every time it tried to leave the toroid centre.
7. 23 Feb '07 13:18
Originally posted by FabianFnas
The solar system as a whole is chaotic. In the long run the planets will alter its orbits considerable. Only 2-body systems can be stable, and this can be mathematical proven.

Are we talking only technical? All right then. But in reality, not knowing all forces acting on the bodies can lead to disastrous results for astronauts and others.
OK, if you consider the really long run. But wouldn't you say the orbit of the moon round the Earth is pretty stable?

I'm thinking of stability in terms of what happens if you perturb the orbit slightly? Do the perturbations grow exponentially? (Thinking about it, they probably do, but very slowly, so over a long enough time scale you're right).
8. 23 Feb '07 13:20
Originally posted by mtthw
This doesn't prove it either way, but it's an argument that suggests orbits would exist in at least some cases.

Imagine an extremely long and thin torus. Now imagine something orbiting close in to the ring, so it doesn't go near the centre of the torus. Close to the ring the torus would actually appear to be a long cylinder, and the other side of t ...[text shortened]... /r^2, I think.

If this is right, it shows that orbits exist in this extreme case.
9. 23 Feb '07 16:06
Originally posted by mtthw
OK, if you consider the really long run. But wouldn't you say the orbit of the moon round the Earth is pretty stable?

I'm thinking of stability in terms of what happens if you perturb the orbit slightly? Do the perturbations grow exponentially? (Thinking about it, they probably do, but very slowly, so over a long enough time scale you're right).
Originally posted by mtthw

No you're not, it's not complete rubbish. You did a reasonable fair simplification of a thorus and you stated a special case, and you did it well.

Doesn't change, however, that I don't find the orbit very stable.

But what you think is stable and what I think of a stable orbit is not very much the same. That's is the key that people find us disagreeing. But we're not. The key word is 'stable' here.

A low orbit satellite is not stable for more than years, for higher altitude orbits for decades, and for the geostationary orbits, well I don't know, but they are regularly influenced pretty much from the Moon. For satellites in this very particular orbit you have to quite often stabilize its position so it doesn't drift off location.

The Moon moves steadily away from Earth. Earth itself wiggles its polar axis, changes its apogee and perigee in a cyclic movement. The orbit of Venus can in a few billion of years come so close to Earth that our orbits will be totally disrupted and send either of us into the sun or out of the solar system. If this 'long run' should be disregarded and treat the solar system as stable depends pretty much of the very definition of the word 'stable'.

I have very enjoyable moments when thinking of stable orbits around and within a thorodial bodies in planetary sizes.
10. 23 Feb '07 18:03
Originally posted by FabianFnas
Originally posted by mtthw

No you're not, it's not complete rubbish. You did a reasonable fair simplification of a thorus and you stated a special case, and you did it well.
Oh, it's a plausible argument. I just have a sneaking suspicion that I can't get away with just neglecting the rest of the torus like that (it's a long way away, but there's so much of it!). I'd have to do the calculation to be sure - integrate round the torus. Maybe later π
11. sonhouse
Fast and Curious
23 Feb '07 18:182 edits
Originally posted by mtthw
Oh, it's a plausible argument. I just have a sneaking suspicion that I can't get away with just neglecting the rest of the torus like that (it's a long way away, but there's so much of it!). I'd have to do the calculation to be sure - integrate round the torus. Maybe later π
One thing about MIR and others, it's not gravitational instabilies that kill that kind of orbit, it's dust and upper atmoshperic gasses gradually slowing down the craft, especially big ones with relatively low mass, ISS is in the same boat. If it was in orbit around the moon, the same orbit would last a whole lot longer due to less dust and air. If there was no gas there would be no instability except for the pertubutions of external masses like the sun, etc. Another thing that perturbs orbits is something called "Mascons", concentrations of higher or lower density mass under the surface of a planet or moon. That is being traced now with a pair of satellites called GRACE. Two craft in constant touch by laser where they know their positions with respect to one another within one micron or so. Mass concentrations cause the orbit of the leading craft first encountering it to get slightly closer and that change is position is charted somewhat like an atomic force microscope except this is one humunous microscope, eh.
What led me to think about toroid shaped planets and such was thinking about the idea (scientifically impossible with present day technology) of drilling a hole straight through the earth, evacuating the hole and using it as a transportation system, just drop a craft into the hole which is in vacuum, it gets acellerated starting with one G and dropping to zero at the exact center of the earth, then negative G's until you reach the other end with exactly zero velocity and just clamp on to a stop, get out and new passengers get on and back down the hole you go. Free transport (at least after you spend the trillions of dollars for the tunnelπ Now I was thinking what if you did not stop at the end, just let it cycle down and back up forever. How long would it do that, what forces would act on it to slow it down and then end up like a dampened wave, stopping eventually in the center of the earth.
So thinking about that, I mentally stretched out the tunnel to approximate a toroid shape. I think the same thing would happen. If you had a craft at some altitude, it would acellerate to the center and then decellerate and reach the other side of the toroid with zero velocity. So then I figured, what if it was coming in with a pretty large velocity, could it be a stable orbit? I think it would have to be a very large ellipse because the curved nature of the orbit would mean it would need to intersect the center of the toroid but that leaves the curve favoring one side of the torous. Which probably means instability. That's what I have been struggling with.
12. 23 Feb '07 20:13
Originally posted by mtthw
Oh, it's a plausible argument. I just have a sneaking suspicion that I can't get away with just neglecting the rest of the torus like that (it's a long way away, but there's so much of it!). I'd have to do the calculation to be sure - integrate round the torus. Maybe later π
As I recall, in three dimensions, at least, an object inside a uniform spherical shell feels no gravity from the shell: everything cancels out. I think this would also be the case for an object in the plane of the torus when it is within the "hole."

But I'm not sure this will happen if the object is not in the plane of the torus, so looping around one part of the torus might still be possible because the object is only in that plane for an instant during each revolution...
13. 23 Feb '07 21:33
Originally posted by GregM
As I recall, in three dimensions, at least, an object inside a uniform spherical shell feels no gravity from the shell: everything cancels out. I think this would also be the case for an object in the plane of the torus when it is within the "hole."

But I'm not sure this will happen if the object is not in the plane of the torus, so looping around one part ...[text shortened]... be possible because the object is only in that plane for an instant during each revolution...
Yes, you are right. In the middle of the torus hole, at the torus mass centre, the gravitation is zero. In the rotational axis the gravitation is pointed towards the mass centre. But off the rotational centre the gravitation is pointed towards the nearest surface. Correct me if I'm wrong.

In the example of a sphere, only in the mass centre, gravitation is zero, and everywhere else the gravitation is pointed towards this mass centre.
14. sonhouse
Fast and Curious
24 Feb '07 00:38
Originally posted by FabianFnas
Yes, you are right. In the middle of the torus hole, at the torus mass centre, the gravitation is zero. In the rotational axis the gravitation is pointed towards the mass centre. But off the rotational centre the gravitation is pointed towards the nearest surface. Correct me if I'm wrong.

In the example of a sphere, only in the mass centre, gravitation is zero, and everywhere else the gravitation is pointed towards this mass centre.
That is correct but you are not going to be attracted to the nearest wall when you venture from dead center, the vector of gravitation is pointed down or up, not sideways. If you for instance, go up from the center, say 100 Km, and drop, that's all you do, drop, you don't go careening off the nearest wall because the mass sideways speaking is canceled out, equal mass all around you. Think of a stack of vinyl records, with the hole in the center, the stack starts out with small records representing the poles, gradually getting bigger till the biggest one is at the position of the equator, then getting smaller again, till you get to the poles. So if you blow that image up you see a stack of records separated by some distance. At the center of the hole of each record you have equal mass so you stay in the center.
As you go away from dead center of the planet, you see more and more records underneath you. That represents the actual gravitation at each point along the path.
15. 24 Feb '07 16:321 edit
Originally posted by FabianFnas
Yes, you are right. In the middle of the torus hole, at the torus mass centre, the gravitation is zero. In the rotational axis the gravitation is pointed towards the mass centre. But off the rotational centre the gravitation is pointed towards the nearest surface. Correct me if I'm wrong.

In the example of a sphere, only in the mass centre, gravitation is zero, and everywhere else the gravitation is pointed towards this mass centre.
No: inside a spherical shell, an object feels no gravity at all from the shell. You might think you would feel a force toward the closest wall, but it turns out that the pull of the extra mass of the farther wall exactly cancels out the pull of the closer wall.

see http://www.merlyn.demon.co.uk/gravity1.htm#FiSSh

Edit: You are right that in a sphere, there is always a pull towards the center; I am talking about hollow shells, since the torus has a big hole in the center.

That site states something I didn't know before: outside a uniform spherical shell, the gravity of the shell acts like it is from a point of equal mass at the center of the sphere.

These two statements should apply for objects in the plane of a large torus as well.

It is clear that at least one type of orbit around a torus is possible, though it's not clear that it is stable: a circular orbit outside of and "parallel" to the torus, which so that the orbit and the torus are like two concentric circles.

I guess there is also an orbit that consists of a straight line: position an object on the line that goes through center of the torus and is perpendicular to the plane of the torus. If you let go of it the object will slide back and forth on this line. I'm not sure if this is stable, either.