This article is kind of neat, not least for the charming way it uses "potato" (and its adjectival variant "potatoid") as a technical term. The point of it is to try to work out how big an object has to be for it to be round.
One particularly nice part of this paper is their description of why astronomical objects are the shape they are. In something like the astronomical periodic table (hydrogen, helium, metals), they classify all objects into "irregular", potatoes, spheres, and disks. Small objects can be irregular because they're held in shape by electronic forces, but as they get bigger they normally turn into spheres. When you get to really large objects, electronic forces (or even neutron degeneracy pressure) are not enough to support objects against gravity, so really large objects are either black holes or supported by the motion of their component elements (like the Solar system is). In this largest category, the shape is (they say) either spherical or disk-like, and which it is depends on how easily the system can get rid of energy and angular momentum. If the system can get rid of energy and angular momentum easily, it collapses down into a star, planet, or black hole. If it can get rid of energy easily, but not angular momentum — which is generally harder — the system settles itself down into a disk. If the system can't get rid of either energy or angular momentum, it swirls around randomly and is spherical.
While I like this classification, it seems to me that it's a little more subtle than that; for example, elliptical galaxies are very old and not spherical, and it turns out that globular clusters remain the way they are only because they get energy kicked into them from the binaries they contain. But in any case, the paper is aimed at the transition between irregular, electronically-supported objects and round gravitationally-flattened objects.
The paper bases its conclusion in part on observation of Solar System objects, in particular asteroids and icy moons for which we know the shape, but there really aren't all that many such objects to work from. So the authors try to work out a theoretical prediction for what they call the "potato radius". The basic ingredients are the density of the material and the "yield strength", at which the material squishes and flows. (A demonstration of the importance of the plasticity of solid material on planetary scales is given by the fact that earthquakes never originate any deeper than 30 km in the Earth's crust: below this depth material just squashes rather than getting stressed and cracking.)
As often happens in astronomy, their firmest prediction is a scaling law: the potato radius increases with the square root of the yield strength and decreases inversely with the density. The constant out front is a little more complicated to get, but in the end they come up with a model of a nearly-spherical object with a "bump" and ask what shear strength is needed to support the bump. The result is reasonably close to the observed transition; they also point out that what matters is not so much the shear strength now as the shear strength when the object was forming, which is fairly difficult to pin down.
In any case, they argue that the "potato radius" for rocky bodies is about 300 km, and that for icy bodies it is 200 km. This is lower than has been thought, which has the interesting consequence that many small trans-Neptunian objects we have found ought to be reclassified from small bodies to dwarf planets because they are probably round. What I find myself wondering is, for objects that are well below the potato radius — say rocky asteroids of 100 km in radius or less — just how irregular can they be? If they formed from the slow accumulation of stray rocks and they're not massive enough to really squash those rocks together, does that mean they're full of voids? Cave systems? Could future astronauts go spelunking deep inside large asteroids?
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