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According to the popular Solar System formation models, what are the bounds on the Gas Giants' cores (note I am only interested in the bounds for our Solar System).

For example according to Wikipedia on Jupiter

Assuming [a core] did exist, it may have shrunk as convection currents of hot liquid metallic hydrogen mixed with the molten core and carried its contents to higher levels in the planetary interior.

This suggests that a core would shrink over time, which would put an upper bound on the size today.

Are there other bounds? I would like numbers together with some reasons for those numbers.

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My advise on this question is to follow the data from the Juno spacecraft, which is now successfully in Jovian orbit. I suspect that a lot of phenomenology about Jupiter and related gas giants might be run through a paper shredder, and maybe a few will be supported. This spacecraft is tasked to address just this and related questions.

I am no expert on this, but as I understand the argument for a solid core in Jupiter and other gas giants it is that given their mass without a solid core they would have smaller volumes.

Lots of planetary models in the past have been thrown out as spacecraft actually reach planets and perform measurements. I suspect the same will happen here. We are fortunate enough to live in an age where we can get the real data on these things.

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