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Is this minimum mass known? or maybe, is it given in terms of density? If so, how much density is the minimum to have an spherical object due to its own gravity?

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You would have to define "objects." Icy bodies start to become round under their own gravity at a certain mass. Rocky planets will take more. Liquids would form a sphere with miniscule mass as I assume gas proto-planets would be round as soon as they have enough gravity to be considered an "object." – Robert Cartaino Apr 1 '14 at 20:10
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This question is more complicated than it seems like it should be!

There is no threshold mass or density beyond which an object becomes perfectly spherical; even supermassive stars are slightly oblong. The only exception is black holes, which are perfectly round up until you reach the quantum level. If we want a simple answer, most guesses are somewhere around $\frac{1}{10000}$ the mass of earth, or $6\cdot10^{20}$ kg, but that is very approximate and depends on the composition of the object.

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Icy objects, such as most in the Kuiper belt can reach an equilibrium if they are about 400km across, whereas the rocky asteroid Pallas, at 572km clearly has an irregular, non spherical shape. All rocky objects larger than Pallas (and there aren't many) are spherical.

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