How does the Sun shine if its core is too cold for fusion?

It cheats, using a quantum effect called tunnelling. The Sun shines by fusing hydrogen nuclei, which are single protons, into helium. But protons are all positively charged and repel each other, and to fuse they must get close enough to stick. When Arthur Eddington proposed in the 1920s that the Sun runs on this, the core's temperature, close to 15.7 million kelvin, was thought too low for protons to overcome that repulsion. Quantum mechanics solved it: a proton's wave can tunnel through the barrier it could never climb over, so fusion happens at a lower temperature than classical physics allows.

Even so, it is astonishingly slow. The first step, two protons becoming deuterium, also needs the weak nuclear force to turn one proton into a neutron. More than 99.99% of proton collisions simply bounce, and the average proton in the core waits 9 billion years before it fuses. After that the chain races ahead: the new deuterium lasts about a second before becoming helium-3, which lasts about 400 years before helium-4 is made.

The payoff comes from Einstein's E = mc². The final helium nucleus weighs about 0.7% less than the four protons that made it, and that missing mass becomes energy: the Sun turns 4.26 billion kilograms of mass into energy every second. Yet per cubic metre, the centre of the core makes only about as much power as a compost heap. The Sun is bright because its core is gigantic.

Before fusion, the Sun's power was a real puzzle. Lord Kelvin calculated that if it glowed from slowly shrinking, it could be only about 20 million years old, and Charles Darwin complained that this left far too little time for evolution.