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As it continues to compress in size is there any reason it could not begin burning hydrogen and eventually initiate a nuclear reaction, and hence become a star?
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Jupiter isn't compressing, it's quite stable. It would need to be about 80 times larger (mass wise) to begin burning hydrogen as a red dwarf. I think (??) to burn deuterium as a brown dwarf the minimum size is about 15 Jupiter masses....don't hold me to that one though.