Imagine asking, à la a Jay Leno “Jaywalking” segment, some random people this question: “Why is our Moon round?” Or, “Why are planets always round (including the Earth) – like spheres – and not like a cube or a boulder or potato or other irregular solid?”
Science communicator Ethan Siegel answered this question in a recent article.
His article gets into some physics, using examples of significant objects in our solar system. But first, let’s consider our earthly experience and common sense. Like for snowballs and Play-Doh objects.
Snowballs. Scoop up some snow from the ground, perhaps a mash-up of snow, ice, and water. A texture from powdery to slush. Maybe with some grit or pebbles mixed in. Using your hands, form that mass into sort of a ball. Pressing and compressing. Maybe even adding more snow until the desired size; then shaping the mass again.
Play-Doh can be even more interesting. Making small objects. Then pressing them together. Perhaps adding some solid plastic objects into the shape. But then mashing the whole thing into a ball-like object to start over. Fun.
Anyway, whether snowy or putty-like stuff, you use force on these objects – pushing (and maybe pulling), compressing – to shape them. An outside force. External pressure.
What do we know about forces inside objects? Might internal forces also be in play for astronomical objects? Internal pressure.
• Big Think > “Ask Ethan: Why are planets always round?” by Ethan Siegel (November 24, 2021) – The stars, planets, and many moons are extremely round. Why don’t they take other shapes?
While generally spherical, planets vary in roundness. Stars are round as well. Are all planetary moons in our solar system round? Are any asteroids or Kuiper belt objects round?
So, does size matter? Is there a critical size – a size below which roundness is unlikely?
In fact, how do objects get to be the size of planets? Especially rocky ones.
Siegel notes that normal matter clumps naturally. Well beyond the atomic scale, we’re talking about gravitational aggregation.
If you get enough normal matter together in one place — regardless of the type, phase, origin, or nature of matter that you have — it will contract until it’s a single, gravitationally bound object.
[Perhaps some experiments on the ISS might demonstrate this behavior?]
At a scale of “more than a ‘grain’ of matter but not more than a few million tons,” Siegel distinguishes between rubble piles, objects with inner denser cores and outer less dense layers, and merged bodies. With shapes due to “luck-of-the draw.”
Like a mash-up of rocky boulders or chunks of ice, these objects aren’t round. Two types of forces (using a classical context) are at play: electromagnetic force and gravity.
The solidity and enduring shape of everyday objects – floors, walls, desks, chairs, coffee cups, cows, etc. – depend on the electromagnetic force. For the molecules and atoms of solid matter to stay in place, hold their arrangment.
But gravity becomes dominant at a certain scale, a threshold – strong enough to break the arrangement of molecules and atoms.
Siegel notes one of the most powerful concepts in physics, namely, center of mass. A mash-up of objects acts as if every bit is attracted to its center of mass (which may or may not be at the “center” of that mash-up). Self-gravitation.
So, when the pressure of that attraction overcomes the electromagnetic force, it’s shape-shifting time. Hulk smash (quite slowly). 
If the pressure is great enough, it will override whatever initial conditions or shapes an object possesses to start with, and compel it to reshape itself into a more energetically stable configuration.
Reaching that point – “behave in a plastic fashion” – depends on the type of stuff in the mash-up. Rock, ice, grit, metal, etc. Crushing (deforming) ice is easier than crushing concrete, for example.
If you’re below about 10^18 kilograms (a quadrillion tons or so), you’ll be below about 100 kilometers in radius, and that is always too small, or low in mass, to pull yourself into a round shape.
[Wish list: A visualization simulating this shape-shifting over time.]
Saturn’s moon Mimas, … is slightly under 200 kilometers in radius, but is undoubtedly rounded. In fact, it’s the smallest astronomical body presently known that’s in a round shape owing to self-gravitation … [Minas] … is very low in density, only barely denser than water-ice, …
A rocky or metal rich Mimas would need to be two or more times larger to be round.
Siegel points out the shape of Mimas also illustrates whether an object has the gravitational pressure to preserve its spherical shape when repeatedly impacted. He mentions the role of rotation in achieving “hydrostatic equilibrium.” 
… once you get up to about 800 kilometers in radius, everything known above that size is not only round [“with only small imperfections superimposed atop that”], it’s in hydrostatic equilibrium as well.
 In his latest Thanksgiving blog post, Sean Carroll notes:
The simplest thing that lines of force can do is just to extend away from a source, traveling forever through space until they hit some other source. … That corresponds to field being in the Coulomb phase. … The magnitude of the force therefore goes as the inverse of the square — the famous inverse square law. In the real world, both gravity and electromagnetism are in the Coulomb phase, and exhibit inverse-square laws.
• Preposterous Universe > “Thanksgiving – electromagnetism” by Sean Carroll (November 25, 2021)
 Siegel also points out that rotation alters the shapes of many planets and stars from being strictly spherical to oblate spheroids. Like the Earth. But not the Moon.
Gravitational pressure is so extreme on neutron stars, that their “mountains” are mere millimeters.
• Science Daily > “Millimeter-tall ‘mountains’ on neutron stars” by Royal Astronomical Society (July 18, 2021)
New models of neutron stars show that their tallest mountains may be only fractions of millimetres high, due to the huge gravity on the ultra-dense objects.
Because of their compactness, neutron stars have an enormous gravitational pull around a billion times stronger than the Earth. This squashes every feature on the surface to miniscule dimensions, and means that the stellar remnant is an almost perfect sphere.