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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.

Rock tends to be stronger than ice. Rocky objects are able to withstand their own gravity for longer than icy ones. Pallas is a reasonable cut of point. The next smaller asteroids (Vesta, Hygiea etc) are round-ish, but not in hydrostatic equilibrium. On the other hand Small, icy moons such as Miranda and Mimas are in, or close to equilibrium. Mimas has a diameter of just under 400km.

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Asking for clarification may be against policy, but I feel necessary in this exchange. James Kilfiger states, "Pallas, at 572km clearly has an irregular, non spherical shape. All rocky objects larger than Pallas (and there aren't many) are spherical." However, is this diameter, 572 km, an approximate threshold for spherical bodies, or are there rocky, spherical bodies with a smaller diameter? – GS1969 Jul 17 at 7:24
    
I;ve edited my answer. – James K Jul 17 at 9:31

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