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Consider lots of mass in isolated 3D space, close to each other. Consider that only the gravitational force (Newtonian) exists. Also consider that there is no rotational motion.

It is evident that a sphere has the geometry/configuration that allows for the least possible gravitational potential in such a configuration. Hence, most of our planets/stars are spheres.

This post Is it possible to prove that planets should be approximately spherical using the calculus of variations? gives a mathematical proof of this fact.

What would be the most stable shape if gravity was proportional to various interesting $r^\alpha$ instead for various $\alpha$, rather than $r^{-2}$? Can we tweak the linked proof to get that shape?

Qmechanic
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whoisit
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  • "What would be the most stable shape if gravity was proportional to various interesting $r^\alpha$ instead for various α, rather than $r^{-2}$?" Look at the linked duplicate and think about what would have to change (if anything) in the final results. – hft Nov 17 '22 at 17:52
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    We could do several pages of calculations like @Qmechanic, but if the force is spherically symmetric, thus unable to specify an orientation, why wouldn't the stable shape be? If it's e.g. a cube, where will the vertices go? – J.G. Nov 17 '22 at 20:28

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