Just a curiosity:
Let $g \in \mathbb{Z}_{>0}$. Is it possible for a planet of topological genus $g$ to exist? For example, is there any contradiction (from the point of view of physics) in assuming the existence of a torus planet ($g=1$)?
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1$\begingroup$ Related: physics.stackexchange.com/q/428986/123208 $\endgroup$– PM 2RingCommented Jun 19 at 12:44
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1$\begingroup$ Is there a reason you complicate your question with introducing concepts of topology/differential geometry instead of just asking "can there exists planets with holes" (or so)? Note that questions should be formulated as easy as it can be, since we want questions/answers to be useful and interesting to a broader audience. $\endgroup$– Tobias FünkeCommented Jun 19 at 12:48
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3$\begingroup$ On a small scale the Earth has a genus greater than 0, e.g. Arches National Park, cave systems with multiple entrances, etc. $\endgroup$– Michael SeifertCommented Jun 19 at 12:49
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3$\begingroup$ Effectively a dupe of physics.stackexchange.com/q/101301/25301; See also physics.stackexchange.com/q/616421/25301 and probably a few others? $\endgroup$– Kyle KanosCommented Jun 19 at 12:54
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1$\begingroup$ In the 1970's, Larry Niven wrote a series of science fiction books about a ribbon shaped planet. It was manufactured from a ridiculously strong material and spun fast enough that centrifugal force held the inhabitants to the inside. The science is good, aside from a few made up things that make it all possible. The first is Ringworld $\endgroup$– mmesser314Commented Jun 19 at 14:13
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2 Answers
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There is no problem from the point of view of physics for a toroidal planet to exist. There are two problems which are different to existence however
- Formation
- Stability
There is no known natural process by which a toroidal planet would form, so to get them you probably need some deliberate "manufacturing" of them by aliens.
Then if you perturb a toroidal planet enough it will gravitationally collapse into a spherical planet (probably rather dramatically). Quite a lot of people have examined the stability of toroidal planets, for example see this paper
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Celestial objects such as planets, moons and stars are typically only held together by gravity, which is why they dissolve past the Roche limit, where another nearby gravitational field is stronger. An object held together only by gravity will be a spheroid, since its gravitational field is approximately spherically symmetric.
Smaller objects such as asteroids can consist of one large chunk of ice or rock, and can have any topological genus.
