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Jupiter will never (not on any timescale like the lifetime of the Sun anyway) accrete enough mass to begin hydrogen fusion. It would need to accrete 12 times its current mass to undergo a brief period of fusing its interior deuterium and to accrete more than 70 times its current mass to attain a central temperature high enough to sustain hydrogen (pp chain) ...


I'm not entirely sure what you mean about 'pulling hydrogen', all bodies, whether they be planets or literally human bodies, will pull hydrogen via gravity. Earth can lose the H it attracts as H is so light that it can have speeds greater than the escape velocity (just due to random thermal motion). Perhaps Jupiter is sufficiently massive that this happens ...

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