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Are there a direct mathematical relationship between Quantum tunnelling and Gravitational potential energy of stars?

The true source of the Sun's energy was shown by Hans Bethe to be nuclear fusion (through Quantum tunnelling) caused by gravitational potential energy from the contraction of the Sun.

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    $\begingroup$ Uh...why should there be such a relationship? $\endgroup$ – ACuriousMind Sep 22 '14 at 15:57
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    $\begingroup$ Not sure why this question is gathering some close votes, as it is via quantum tunneling that nuclear reactions take place in the core of a star. $\endgroup$ – Kyle Kanos Sep 22 '14 at 16:39
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There isn't exactly a mathematical relationship, but there is a physical one.

It is the gravitational compression that causes the increase in temperature in the core of the gas cloud that becomes a star. When the temperature reaches a critical value (in the millions of Kelvin range), hydrogen fusion can occur. This is because the temperature is great enough that the Coulomb barrier, $$ U_{Coulomb}\sim\frac{q_1q_2}{r^2} $$ is less than the kinetic energy of a pair of protons colliding.

What the discovery of quantum tunneling did was allow this fusion temperature to decrease from the classical value, so that nuclear fusion would occur before the classical temperature (which I think is in the billions of Kelvins).

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