Why the Sun needs quantum tunnelling to shine

Two protons at the centre of the Sun are moving far too slowly to touch. They fuse anyway. The way they manage it is the reason the Sun gives off light at all.

Passing through a wall you cannot climb

Roll a ball at a hill with too little energy to reach the top and it comes back. That is the classical rule. A proton does not obey it. In quantum mechanics a particle is described by a wave, and a wave does not stop dead at a barrier. It fades inside it. If the barrier is thin enough, a little of the wave remains on the far side, and that remainder is a real chance of finding the proton past a hill it never had the energy to climb. This is quantum tunnelling.

Why the Sun needs it

Protons carry positive charge, and like charges repel harder the closer they get. To fuse, two protons must come within range of the strong nuclear force, which only reaches about as far as a nucleus is wide. In between sits a hill of electrical repulsion, the Coulomb barrier. The Sun's core is at about 15.5 million kelvin, and at that temperature a typical proton carries a little over a thousand electronvolts of energy. The top of the barrier is hundreds of times higher than that. By the classical rule not one pair of protons in the Sun could ever meet, and a star that cannot fuse anything has no way to keep making heat.

Two steep curves and a narrow window

Two things decide how often protons fuse; both depend sharply on energy.

The first is the chance of tunnelling. A faster proton gets closer before repulsion turns it back, so the wall it has to pass through is thinner, and the leaked wave is larger. The chance grows steeply with energy, but at core energies it is still a small fraction of one.

The second is how many protons have that energy. In a hot gas, most particles sit near the typical energy, and the number with far more falls off exponentially. Fast protons are rare, and the faster, the rarer.

Multiply the two. The product is a narrow bump: on the left, plenty of protons that almost never tunnel; on the right, protons that tunnel easily but barely exist. The bump between is called the Gamow peak, the most probable energy of the pairs that actually react. The Solar Fusion review works it out for each step of the chain; for the first step, proton on proton, it sits near 6 keV, about four and a half times the typical thermal energy and roughly a hundredth of the barrier height. Nearly all the Sun's energy comes from this thin slice of unusually fast nuclei doing something unlikely.

Reaction probabilities in that window are so small that measuring them on Earth is hard. The LUNA collaboration had to run its accelerator under a mountain, shielded from cosmic rays, to follow one pp-chain reaction (helium-3 fusing with helium-3) down to 16.5 keV, the lower edge of its solar Gamow peak, which sits near 22 keV.

In one breath

Protons in the Sun repel one another, and none is moving fast enough to overcome that repulsion. But a proton is a wave, and a wave leaks through a barrier it cannot clear. Rare, fast protons meet a wall thin enough to leak through often enough, and that leak is the fusion that powers the Sun. Without tunnelling the core would be a hot gas doing nothing, and there would be no sunlight to explain.