How to derive "a blob of water in space becomes a sphere" analytically? I'd like to understand better the assertion that planets are spheres because of gravity.
The thing which I believe is true, but don't completely get, is that any arrangement of a sufficient number of molecules, all with zero velocity, in isolation from anything else, will over time form a sphere due to gravitational attraction between the molecules.
If this is true, what is the argument for why this will happen? (i.e. what is the proof that I can't construct a weird initial arrangement of molecules for which it won't happen?) Can we do some math and see it pop out? If this is false, what additional stipulations do I need?
 A: A sphere is the only shape with no "downhill"
Intuitively, things move downhill spontaneously - any matter under the force of gravity will try to minimize its distance to that center of gravity, unless something else prevents it. A sphere is the only possible shape that has all points on the surface equidistant from the center of gravity - there is no uphill or downhill on the surface of a sphere. A sphere is the only shape that gravity doesn't try to rearrange.
If you build a mountain on a sphere, the natural tendency will be for that mountain to erode over time and become flat. If you dig a hole in a sphere, the tendency will be for the surrounding matter to collapse inward, filling the hole. Over time, any deviation from a sphere shape will be "fixed" by gravity, which is why planets are roughly spherical. It's possible to have rigid structures that resist gravity (skyscrapers, rocky mountains), but that requires that the structure is held together by something stronger than gravity. A planet is more like a loose pile of gravel than a solid rock, so those deviations tend to get flattened out, leaving a sphere.
A: The formation of spherical blobs of water in space is because of surface tension. This is only exhibited because it is a liquid. However at the astronomical scale, the gravity of a sufficiently large object will eventually turn it spherical. The object has to be at least the size of a planet because only then will the gravitational forces be large enough to change the shape.
The shape is usually close to a spheroid. This has got to do with the fact that a sphere has the smallest surface area to volume ratio of all 3-dimensional shapes.
The earth is flattened at the ends due to its angular velocity.
