1H1 +1H1 →1H2 +e+ + νe
1H1 + 1H2 → 2He3
2He3 +2He3 →2He4 + 21H1 .
Although I use here a notation similar to the notation of chemical reactions, I am referring to the reactions of bare nuclei. Enough electrons are present to maintain electrical neutrality, but, because of the high temperature, they are not bound to nuclei, nor do they participate in the reaction.
In the first of these three reactions, two protons (pp) unite to form a deuteron (pn) while a positron and neutrino are created and fly away. In effect, one of the two protons is transformed into a neutron in a beta decay process. Normally, it is a neutron that undergoes beta decay as it transforms into a proton while emitting an electron and antineutrino. According to the theory of weak interactions, a proton is equally favorably disposed to undergo beta decay but is prohibited from doing so by energy conservation (since the proton is less massive than the neutron). In this solar fusion reaction, the binding energy of the deuteron (2.2 MeV) is more than sufficient to provide the neutron-proton mass difference plus the mass of the emitted positron.
In the second of these three reaction, a deuteron formed in the first reaction fuses with a proton to make a helium 3 nucleus. Finally, in the third reaction, two helium 3 nuclei react to form an alpha particle and two protons. The net result of two of the first reactions, two of the second, and one of the third can be summarized by a single reaction equation,
41H1 →2He4 +2e+ +2νe ,
or, in somewhat simpler notation,
4p → α + 2e+ + 2νe .
Note, in this net reaction, the conservation of charge, of baryon number, and of electron family number.
In this so-called proton-proton cycle in the Sun, 6.2 MeV of energy per proton is released. A few percent of this energy is given to neutrinos, which escape at once from the center of the Sun without impediment. Nearly 1015 (one quadrillion) of these solar neutrinos strike each square meter of the Earth each second—most of them passing right on through the Earth, a few reacting in the detectors of physicists well below Earth’s surface. The rest of the energy goes at first into kinetic energy of charged particles, eventually into electromagnetic energy—photons. In addition, by annihilating with electrons, the positrons add another increment of energy to the total. In each annihilation event,
e+ + e– → 2γ ,
the complete conversion of mass to energy yields about 1 MeV. Although positron annihilation involves neither nuclei nor nuclear force, it is another exothermic (energy-generating) contributor to the proton-proton cycle, raising the total energy release from 6.2 to 6.7 MeV per nucleon. Including positron annihilation, the net reaction is
2e– + 4p → α + 2νe + nγ .
The notation nγ on the right designates an unknown number of photons. Eventually, for each proton burned in the solar furnace, millions of photons are radiated at the surface.
The net energy release in the Sun’s proton-proton cycle may be calculated readily from the masses of the participating particles.
Before
Mass of 4 protons = (4) (1.007276) = 4.029104 amu
Mass of 2 electrons = (2) (0.000549) = 0.001098 amu
Total mass of reacting particles = 4.03020 amu
After
Mass of alpha particle = 4.00150 amu
Net loss of mass = 0.02870 amu
Net loss of energy = (0.02870 amu) (931 MeV/amu) = 26.7 MeV.
In the last line I have used the mass-to-energy conversion factor that is especially useful in the nuclear domain, 931 MeV/amu. Dividing 26.7 MeV by 4, we get 6.7 MeV, the energy release per nucleon stated above. The fractional conversion of mass to energy in the Sun’s fusion reactions is 0.7%, or one part in 140. This is about eight times the fractional conversion in uranium fission, and more than 100 million times the fractional conversion in a typical chemical explosion. Each day the Sun burns 5.3 × 1016 kg of its hydrogen into alpha-particle ashes, thereby transforming about 3.7 × 1014 kg of its mass into energy, a rate of output equivalent to the explosion of billions of multi-megaton weapons each second.