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I have read online in numerous places about the possible existence of toroidal planets, and I most people seem to believe that they could exist, but they also have no evidence to support this claim. I recently did a project in which I calculated the gravitational field around one-such planet, and I would like to continue working on it to either prove or disprove the possibility of these planets.

Before I continue, I would like to know if any mathematically rigorous investigations into this question have already been conducted. I have searched, but I haven't found anything on planets, only black holes.

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You may want to read this article: io9.com/… –  Kyle Kanos Feb 28 at 17:13
    
It's a good article, thanks, but most of the math isn't shown, just a lot of graphics. I have no way of knowing if it's legitimate or not. For example, he clams that "a moon bobbing up and down through the hole" is a stable orbit, but my results show that it definitely is not. –  Platatat Feb 28 at 17:18
    
Then you didn't actually read it (which makes sense since it took you 5 minutes to respond). –  Kyle Kanos Feb 28 at 17:21
    
I read it a week ago and considered it non-mathematically rigorous by a long shot. And there's no need to be rude by telling me that I'm too lazy to read a response to my own question. –  Platatat Feb 28 at 17:24
    
Oh, so you saw all the links (at the top and scattered in the text) to relevant articles/papers but judged them as not mathematically rigorous? Could you then define what is mathematically rigorous to you? –  Kyle Kanos Feb 28 at 17:25

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up vote 7 down vote accepted

I posted an answer to a similar question here:

http://physics.stackexchange.com/a/97978/1255

The wording there was pretty much asking for a broad description of the physics (which come down to bifurcation possibilities) up to, and including, the topology changes of the planet.

The paper I linked to seems to be pretty much the state of the art on this problem. Calculations producing some of the shapes within this set were reproduced by the io9 link someone gave:

torus diagram

As you can see, it's not exactly a torus (although upon later inspection, it looks like this might be a simple rotated ellipse, in which case I'd file it away in "wrong approaches"). This is expected, because the field distribution for a single ring of wire or gravitational mass in Newtonian gravity is extremely well understood. With certain limit-case assumptions, you can also divide the problem up into 2 components - one analogous to an infinite line, and one for the exclusively radial field from the global dynamics. But that method is approximate by definition. An infinite line of mass would obvious be a perfect infinite cylinder under the assumptions for how planetary mass behaves. It's that radial field that causes the shape to be somewhat egg shape.

The author says that he used a Monte Carlo method of randomly placing ring sources of mass within the interior of the thing. The equation for the ring is well-know, so this is a fairly obvious method. What's not obvious is how you perform an iteration between the surface and the field. How do you even define the curve of the planet's surface in the first place? I worked on this problem a little bit myself, and I've seen multiple wrong solutions online, so I'm quite skeptical if there isn't obvious voluminous work behind it. I have asked the author of that article about these details. So far, I have not received any response.

There's no better metric for the scope of the problem than the question on this site, Why is the Earth so fat? This was only concerned with a tiny tiny limit case problem. It involved a lot of numerical attempts, and the solutions weren't actually all that good in terms of accuracy.

In fact, the Monte Carlo approach has some serious issues. If you're testing the field/potential close to the surface of the planet, you'll almost certainly blow up numerically. The problem is that the field/potential could easily be infinite if a point just happens to be randomly selected very very close to the test-point on the surface. Maybe you'll just de-tune it a little bit. But then you need this very value in order to get to do the iteration on the surface profile!

I recently did a project in which I calculated the gravitational field around one-such planet, and I would like to continue working on it to either prove or disprove the possibility of these planets.

At the risk of being condescending, I'll venture a guess that you did a numerical approximation of the thin ring case? We have an explicit solution for that, and it's the bread-and-butter of the more advanced investigations. Maybe I'm wrong, I honestly have no idea what you've done. But in terms of your interests:

have read online in numerous places about the possible existence of toroidal planets, and I most people seem to believe that they could exist, but they also have no evidence to support this claim.

Are you interested in:

  • Possible natural formation of the very flat shapes, or the dimpled flat shapes, or a full hole in the middle?
  • The possible artificial construction of such bodies?

The first case hasn't been clearly ruled out yet. It's like the search for life on Mars. They might be out there, but I wouldn't hold my breath for discovery. Within the next 30 years, with the way that planet hunting has been progressing, we'll probably settle the question. So that's a good problem to get involved in. Maybe you could use it toward a degree, but it's certainly not a career.

But for the latter case, it's obviously possible. The stability requirements aren't nearly as restrictive as what some people make them out to be. Large torus plants are unstable in the most obvious sense. They're still locally stable, or at least that's what my money is on. But it's extremely tenuous. Even if you could construct it, it wouldn't be smart.

For small torus planets, the outlook is good, as long as we're not talking about serendipitous natural occurrence. Nature will probably find some other way to shed the angular momentum before it takes on the extreme shapes, because planet forming environments are extremely nasty. But if you're intentionally spinning up a planet, things should go exactly according to the evolution of shapes that the research already illustrates. The real question is - if it breaks up, what does it break up into? For a large torus, this is obvious (lots of tiny planets), but if the mass and angular momentum can barely support 2 separate bodies, then you should have a very robust global stability condition. Note that you can establish some conservative limits easily, but in reality, 2 closely orbiting tidally locked bodies is a very very difficult problem, since the tidal forces deform them. That becomes 3D, none of these nice 2D simplifications.

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My own research was based on a toroid of major radius R=2, and minor radius r=1. I iterated over the volume of the torus to find the gravitational acceleration vector at any point: $\vec{g}(a,b,c)=\int\!\!\int\!\!\int_T[(x-a)^2+(y-b)^2+(z-c)^2]^{\frac{-3}{2}}[\‌​rho*(2+\rho*cos(\phi))]<(x-1),(y-b),(z-c)>du*dv*dw$ where: $x=(2+\rho*cos(\phi))cos(\theta)$ $y=(2+\rho*cos(\phi))sin(\theta)$ $z=\rho*sin(\phi)$ And yes, I did numerically integrate, but this is still pretty much an exact solution. –  Platatat Feb 28 at 19:19
    
@Platatat Ah, I see. That triple integral does go beyond what can be written explicitly. Looking at the io9 link again, I'm thinking that his method might not have been very different from yours. It looks like he might have just used an ellipse, rotated about the z-axis. You can see in the image that the surface isn't equipotential. This is still far from giving any meaningful information near the transition points. If you're interested in a large torus, your field is most likely sufficient unless you're interested in something like rotational period. –  Alan Rominger Feb 28 at 19:33
    
@AlanSE: Huh? Are you even reading the same article I'm reading? The io9 article I read says "The cross-section is neither circular nor elliptic but rather egg-shaped, with a slightly sharper inside curvature than on the outside. ... having a low gravity equator and high gravity poles does not mean stuff will roll or drift towards the poles: ... the surface is an equipotential surface, so gravity (plus the centrifugal correction) is always perpendicular to it." –  David Cary Mar 1 at 17:41
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@DavidCary I read article a while ago, and I agree you have a good point, but he's inconsistent. We can quickly come to agreement on the behavior of apparently gravity to an observer on the surface (what it should be). Gravity feels normal to the surface, but of varying intensity. This makes the surface equipotential. But the plots show a surface that is not equipotential. I have not seen a resolution to this. In some other plots, it shows a local maximum potential above the outer equator. This shows that the plots include the 1/2(omega r)^2 term. That means he's wrong. –  Alan Rominger Mar 1 at 18:07
    
@DavidCary I think I might see it now. The plots are not potential (which would otherwise seem to make the most sense), but represent acceleration. Magnitude of acceleration, to be precise. That likely indicates that he has the correct model. –  Alan Rominger Mar 1 at 18:19

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