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What's the minimum possible size of a spherical shaped rocky planet or satellite? Is it possible to occur a spherical shaped planet without having a molten core?

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The minimum radius of a celestial body that allows gravity to transform the body into a globe is informally called the potato radius. Empirical evidence suggests it is about 200-300 km, depending on the composition of the body - the higher the proportion of rock to ice, the larger the potato radius.

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This may depend on the source of heat: the kinetic energy of in-falling material or radioactivity. It has to get hot enough to flow under the action of gravity.

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  • $\begingroup$ For that matter, molten ejecta could conceivably go near-spherical before cooling to a solid object in a zero-g environment. $\endgroup$ – Carl Witthoft Jun 15 at 15:27

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