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If giant planets have a rocky core. Stars don't. Why not? BD have a core of what?
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Stars don't have rocky cores because their cores are completely ionised, even though many have a substantial amount of heavy elements present. BD's would normally have cores much like stars, except not ionised to the same extent (if at all). Some though, may have a substantial amount of solids present, depending on how they formed and the conditions in their core regions.
Planets are a weird bunch...you can have planets with weak cores or well developed cores, planets with mostly ice and metallic gases and a smattering of scattered heavy elements, small to medium sized rocky cores surrounded by mantles of ice and liquid/metallic gases, or massive rocky cores with little or no ices/metallic gases...and everything in between. You can find numerous examples of all possible internal structures present in the planets they've found.