Why is fusion hard? Protons have the same charge, so they repel. Two protons would have to get within about a femtometre (10⁻¹⁵ m), where the strong force can bind them, but at that distance the repulsion is about 0.7 MeV. The core of the Sun is at 15 million K, where a typical proton has only about 1.3 keV of thermal energy, around 500 times too little. A proton with that energy can only get as close as about 1100 fm in a head-on collision. The Sun works because of quantum tunnelling: the proton has a small chance of appearing on the other side of the barrier. The chance is about e^{−√(E_G/E)}, with the Gamow energy E_G = 493 keV for two protons.

kT = 0.0862 keV per million K, closest approach = 1440 keV fm / kT, tunnelling chance ≈ exp(−√(493 keV / E))

The temperature. A hotter core means more protons with enough energy to tunnel, and the tunnelling chance rises steeply. Near the Sun’s core the energy output of the p-p chain is roughly proportional to T⁴: 20 million K gives about three times the power of 15 million K, and 10 million K gives only a fifth. The slider changes the speed of the protons, and the graph shows the rate.

Step 1 is very slow. Two protons have to fuse, and one of them has to change into a neutron by the weak interaction: p → n + e⁺ + ν_e (an up quark becomes a down quark). Even after the protons tunnel, this almost never happens: a given proton waits about ten billion years on average. This is why the Sun lasts so long. The positron then meets an electron and they annihilate, giving two gamma rays (1.022 MeV). The neutrino hardly interacts with anything and leaves the Sun, taking about 0.26 MeV with it.

Steps 2 and 3 are fast. A deuteron captures a proton in about a second to make helium-3 and a gamma ray (5.49 MeV). Two helium-3 nuclei then fuse into helium-4 and release two protons (12.86 MeV). The chain needs step 1 and step 2 twice each, to make the two helium-3 nuclei for step 3.

The energy ledger. 2 × (0.42 + 1.02) + 2 × 5.49 + 12.86 = 26.72 MeV. This is the mass defect of four hydrogen atoms turning into one helium-4 atom: (4 × 1.007825 − 4.002603) u × 931.494 MeV/u = 26.73 MeV. This is 0.7% of the mass: E = mc². The energy that warms the Sun is about 26.2 MeV, since the two neutrinos take away about 0.5 MeV.

Checks. In each reaction the electric charge, the number of nucleons (baryon number) and the lepton number are conserved. In step 1: charge +2 = +1 (deuteron) + 1 (positron); baryon number 2 = 2; lepton number 0 = (+1 for the neutrino) + (−1 for the positron).

The picture is not to scale. The nuclei are really about a hundred thousand times smaller than the space they move in.