So, you think you know what a planet looks like? A sphere, perhaps flattened at the poles? But if you've been following recent discoveries of dwarf planets, you may know that rapidly spinning dwarf planets like Haumea typically are rugby ball shaped (triaxial spheroids), rather surprisingly perhaps. So, what other shapes can a planet have? Theory, and experiment with droplets in simulated zero g suggest several exotic possibilities.

In Hal Clement's classic hard science fiction novel Mission of Gravity his explorers discover a rapidly spinning planet, and naturally enough he imagines it as flattened at the poles, in the same way as the Earth but more so

He imagined it as an oblate spheroid

Spheroid

When he wrote that nobody had ever discovered a rapidly spinning planet. But now we have found very rapidly spinning dwarf planets in the outer reaches of the solar system, and it turns out, that they look like this (at least the ones found so far):

Haumea: Rugby Ball Planet - artist's impression

And here is an artist's impression of Haumea spinning, with its red spot.

Another example 20000 Varuna - artist's impression

They may look similar to Hal Clemens planet, but are actually a little different. These are triaxial ellipsoids

Ellipsoid

They rotate around their minor axis, the shortest of the three axes.

It turns out that both these shapes are right, because there are two possible solutions, as the spin rate increases. You can get an oblate spheroid, or a triaxial ellipsoid - the solution "bifurcates". But the triaxial ellipsoid is the most stable of the two as Jacobi dramatically predicted, over a century ago, in his paper published in 1834 - overturning the previous prediction that rapidly spinning planets could only be oblate spheroids.

The mathematician Jacobi, who predicted that rapidly spinning planets could take the form of triaxial ellipsoids in 1834

For more about the history of mathematical equilibrium solutions for rapidly spinning planets, see this Figures of Equilibrium - Historical Account - by Chandrasekhar. 

Subrahmanyan Chandrasekhar, author of Figures of Equilibrium - Historical Account - best known for his work on collapse of stars to black holes and the Chandrasekhar limit.

In practice, this triaxial ellipsoid is the shape that the rapidly spinning dwarf planets take, preferentially.

Now if that seems bizarre, it's not the most unusual planet shape possible.

Planets can actually be triangular, or square, or even pentagonal in shape in theory. And in principle they can also be donut shaped too, with a hole in the middle - a stable configuration though one that is easily disturbed so there probably aren't that many donut shaped planets in the universe :).

For that to happen they have to be spinning so fast they are on the verge of breaking apart.

There are cool videos of triangular and square droplets in simulated zero g here.

LiveLeak.com - Spinning water droplets behave like black holes

Here the droplets are held together by surface tension. But apparently theoretically droplets held together by surface tension in zero g have the same shapes as droplets that form from a wide range of processes including a planet or star held together by gravity. So this is experimental work that also confirms theoretical predictions for possible shapes of planets too.

So these are also possible shapes for rapidly rotating planets. Paper here: Nonaxisymmetric Shapes of a Magnetically Levitated and Spinning Water Droplet

Then as it spins faster, it can turn into two spheres joined together. This is the scenario of a binary planet.

GRAVITATIONALLY LOCKED BINARY PLANET

Normally planets can't get too close to each other, or they will tear each other apart through tidal forces. But if the two planets are gravitationally locked to each other, in the same way as Pluto and Charon, then tidal effects no longer matter.

Binary planets like that are just permanently distorted, and so can be as close together as you like. You could in principle even have an Earth sized planet almost touching our Earth - with a shared atmosphere and ocean, and the Earth would be in no risk of tidal disruption. They can even join together in an "overcontact binary" in principle. 

Indeed recently some scientists suggested that we may find habitable double worlds like this as we continue to search for exoplanets.

(Image NASA) 'Double Earths' Could Be Fun Exoplanets To Hunt For -- If They Exist see: Can binary terrestrial planets exist?

This idea is explored fictionally in Robert Forward's "Rocheworld". In his case it is an "overcontact binary" - well at least the atmosphere is shared between the two planets, and sometimes the ocean

Rocheworld

We haven't found any Rocheworld planets yet, but there are many contact binary asteroids. These are too small to be rounded under their own gravity - still, they are approximately the shape of an overcontact binary, and are probably "loose rubble piles" mainly held together by gravitational effects like a contact binary planet.

This is 216 Kleopatra

And comets also, this is 67P/Churyumov–Gerasimenko which is also approximately the shape of an overcontact binary

And we now know of a contact binary moon, at least it seems so, Pluto's tiny moon Kerberos.

New Horizons image of Kerberos.

Also many contact binary stars are known, again including "overcontact binaries" also like these asteroids and comets, so close together that their atmospheres overlap - the first discovered W Ursae Majoris.

figure from: Chapter 19-7 and for another image see also Contact Binaries


As it spins even faster then in theory anyway, a planet or star can turn into three or even four spheres joined together to make an "overcontact ternary" or "overcontact quarternary" planet or star, or you could have a donut shaped planet too, though so far we haven't found anything like this:

See Viewpoint: The many shapes of spinning drops

This is an animation someone did of the last possibility the "hoopworld"

But thin hoops like this are unstable and easily break up into droplets.

Thicker hoops like this visualization from io9 though can be stable.

I shouldn't think there are many of these in our universe :). Needs very special conditions to form. But - not impossible apparently.

Here is a youtube video of a Moon of a donut Earth:

You can also get donut shaped solutions for end state of a collapsing gas cloud, which are stable, in principle. So, you could have a huge toroidal “planet” with a young star at the center, as the author suggests in this paper : General Relativisitic Sturcture of Star Toroidal Systems (1992).

Anders Sandberg's fun article explores many aspects of life on a donut shaped planet - and the orbits of satellites and moons around such a planet.

What would the Earth be like if it was the shape of a donut?

There are many other possibilities not shown in that diagram, including the rounded triangles, squares and pentagons, cross shaped, and also three, four or five bodies in contact with each other in a circle with hole in the middle, like this.

See page 494 of this paper.

TIDALLY STRETCHED PLANETS

This is another way that planets may vary in shape from the usual round or oblate spheroid. Gravitationally locked tidally stretched planets can be triaxial ellipsoids as well, with the long axis towards their sun.

Again we don't know of any planets like this yet. But we do have moons. They are too small to be in gravitational stasis, but the innermost moons of Jupiter do seem seriously tidally squished - or at least - with their long axis aligned towards Jupiter - as you can see from these photographs of them taken by Galileo:

In this photograph then Jupiter is to the right. So they are stretched out towards Jupiter. Catalog Page for PIA01624

Here are more images of the largest of these moons, Amalthea

And in colour, an image taken by Voyager (it is very red, probably because of sulfur from Io)

Amalthea (moon)