How do protons fuse in the sun?

Question

I know that Nucelar fusion in the sun requires quantum tunneling to occur, as otherwise the Sun's core wouldn't be hot enough to overcome the Fusion barrier. However while the 2nd and further stage of the Proton-Proton chain I have no problem with, I've noticed an issue with the first stage. The first stage if the Proton-Proton fusion chain involves 1. Two protons fusing to a diproton. 2. One of the protons changing into a neutron and emitting a Positron. The issue is that the first step is endothermic. A diproton has a mass more then a Mev greater then the combinded mass of two protons.

How does quantume tuneling occur when the kinetic energy of the two paritcles is not only smaller then the Cloumb barrier, but also smaller then the difference in mass energy between the Diproton and the two protons?

Answer 1

My thoughts, for what little those are worth, are that you have to look at it as a pair of canceling reactions. Two Hydrogen-1 nuclei come together for an instant in a fusion reaction, then almost immediately decay through proton emmission or spontaneous fission (in this context the exact same thing). This reaction dominate the fusion cycle of the sun.

The energy balances as the masses are not diminishing, as a net reaction it is neither endothermic nor exothermic. 1 out of 10^19 times this fusion occurs the decompostion spits a positron rather than the proton. That positron finds an electron and annihilates, or gets absorbed, or emitted, or whatever else, it doesn't matter in this context.

What does matter is that the newly formed atom is Hydrogen-2 (Deuterium). That deuterium can then fuse with Hydrogen-1 or Hydrogen-2 (Deuterium) to form Helium-3 or Helium-4 both of which are stable, and this is for sure an exothermic reaction.

Now lets think about this on a time scale. If that H1+H1->H2 + p only occurs in 1 out of 10^19 reactions, and for the sake of argument lets say the atoms in that reaction fuse and decompose a 100 times a second (made up just chose a number at random). Then you are looking at the desired reaction to make deuterium occurring randomly at any point in the next 3.17 Billion years [(10^19) / (100 * 3600 * 24 * 365)].

So you have this reaction that is really doing nothing over and over again for billions of years, waiting on the real party to get started.

That was super verbose and upon rereading your question it is not really addressing the question. I just did the math 3 times myself to confirm that what you say is correct. The mass of the diproton particle is in fact higher than the combined masses of two hydrogen-1 atoms. Upon reflection this makes some level of sense. The two proton are held together only briefly by their kinetic energies which must be bound up somewhere (in this case the mass difference). So that would make the H1+H1=>He2 (Diproton) Endothermic and the He2=>H1+H1 Exothermic as that binding energy (mass) would again be released.

Again just my thoughts for what it is worth.

Answer 2

The Coulomb energy of two protons separated by $1.5\times 10^{-15}$ m is about 1 MeV. The modulus of the (negative) diproton binding energy is less than this. There is therefore a need for protons, which have typical energies of keV in the centre of a star, to "tunnel" through the classical Coulomb barrier into the local energy minimum.

Of course, because the diproton energy is still higher than the combined rest mass and initial kinetic energy of the separated protons then the tunneling probability is suppressed (but non-zero). Likewise, it is then very easy for the diproton to decay back into two protons, or very rarely, into a bound deuteron whilst ejecting a positron.

Possibly a way to think about it is that the decay lifetime of the di-proton is so short ($\sim 10^{-21}$ s) that its energy is broadened by the uncertainty principle such that there is some overlap with the initial total energy of its constituents.

Answer 3

When two protons come close, it becomes energetically favorite for one of them to turn into neutron because you lose Coulomb attraction. Sun's core is not vacuum, it's very far form vacuum. It is better to look at it like this: if you're a proton in sun's core you are close to a lot of protons. The distance to your closest proton varies over time, the smaller that distance is the more likely you are to emit a positron (and neutrino) and turn into a neutron. As soon as you turn into neutron you get immediately fused with a proton.