Gokul43201 said:
Dave, do you have a proof for this? With sufficient angular velocity and tensile strength, I don't see any obvious reason that a toroid will not be stable.
I've heard a rough estimate for the oblateness of an object to be the centripetal acceleration divided by the acceleration due to gravity at the surface. (It at least gets you in the correct order of magnitude, so a planet like Jupiter is much flatter than a planet like the Earth.)
If you slowly spun the planet up, you'd reach a point where the centripetal acceleration at the surface were equal to the gravitational acceleration. That means the equatorial matter could achieve orbit - even though that orbit might only be an inch above the surface. That's as fast as the matter should ever get, since as soon as it's in orbit, the planet isn't sweeping it along anymore. The spinning up would be a non-conservative process.
The eventual oblateness of a planet spinning this fast should be 1. In other words, the planet should wind up being a flat platter. The planet would continue to flatten, supplying new matter at the critical surface velocity, throwing new matter into orbit. Additionally, as the mass of the old planet decreased, the velocity required to throw matter into orbit would decrease, meaning new matter would be thrown into orbit faster and faster.
The newer matter tossed into orbit would collide with older matter in orbit and you'd wind up with an unstable ring. It would flow along in the shape of a ring. The outer most portion of the ring would be held in place by gravity, but chunks of the inside could break off and drift weightlessly across the inside of the ring, striking the ring in some random place, breaking off new chunks, etc. (Matter would still be held inside the plane of the ring - the weightlessness would only occur inside the ring).
The outer portion of the ring would even have a tenuous structure. The new matter colliding with the earlier matter would knock the earlier matter into a higher elliptical orbit, but the earlier matter would never reach perigee since it would hit the newer matter on the way down. That way, the ring slows as it expands, hopefully enough to hold the outer portion of the ring in contact with the inner portion. I'm not sure whether that would happen or if you'd be stuck with a ring of debris instead of a "planet". You'd have a planet plagued by constant earthquakes from the collisions on the inside if it did hold. The formation into a ring would have to be a conservative process.
But, if the planet's rotation then slowed some more - enough that the outer most objects should find an orbit lower than the ring - then the weight of the outside of the ring would start compressing the inside of the ring, just like the arches of an old Roman bridge. The force of gravity compressing the ring would finally give you the tensile strength you needed. The additional slowing down would have to be a non-conservative process since you're reducing angular velocity and the radius simultaneously.
It would be a strange place. You'd have gravity on the outside of the ring and be weightless on the inside of the ring.
It would also raise the question about what caused the planet to start spinning so fast and what caused its rotation to slow down - all without breaking the laws of conservation of energy and momentum. And whatever caused the planet to spin up would have to cease almost as soon as the ring started forming, since it would have the same effect on the matter in the ring as it did on the matter in old spherical planet. (Actually, a slight delay and a higher orbit for the outer portion of the ring would give you more room to cram all of the mass of your sphere into a toroid.) And, if the planet kept slowing down, it's weight would become so heavy that it would only be a matter of time before one of those compressed chunks on the inside of the ring were just destroyed, starting the decomposition of the entire planet. It would eventually decompose back into a sphere. So whatever was slowing the planet down would have to cease at some point if the planet were to remain stable.
So, while a toroid planet could be stable if it existed, it would take a whole lot of PFM to create it.
(Could a planet actually form in a toroid during its initial formation? I don't think so since part of the matter it accumulates comes from sweeping up debris in its orbital path.)
