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Apart from thermo-nuclear reactions in the Sun, (and in hydrogen bombs and other methods developed by scientists) does nuclear fusion occur naturally in the planets? If not, why?

Also, do the Sun and the Outer planets - Jupiter through Neptune - have rocky cores similar to the Earth (consisting of iron, nickel, silicon etc) in addition to the gases? I have read that there is not much of hydrogen or helium in the inner planets because of their low escape velocities and the solar winds from the Sun. Hence only the heavier elements remained.

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Related: physics.stackexchange.com/questions/8558/… . Fusion can occur by tunneling across the potential barrier, but the rate is sufficiently mindbogglingly suppressed that you can ignore it for almost all purposes. –  dmckee Aug 9 '13 at 18:15
However Earth had a natural fission reactor –  babou Aug 9 '13 at 22:14
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The only planet in the Solar system where nuclear fusion occurs is Earth. And that is only because we have the means to achieve the combination of high pressure and high temperature to overcome the Coulomb barrier. Even the heaviest of the planets, Jupiter, is about ten times too small to achieve the pressure required to sustain fusion.

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