Fusion and stars

📚The physics

Fusion needs heat because nuclei repel. Two positively charged nuclei must be brought within about \( 10^{-15} \) m for the strong nuclear force to take over, and to get that close they must overcome the electric repulsion — the Coulomb barrier. Only at temperatures of order \( 10^{7} \) K do nuclei move fast enough. This is why fusion happens in stellar cores and is so difficult to sustain on Earth.

In the Sun the proton–proton chain converts four hydrogen nuclei into one helium-4 nucleus, releasing about 26 MeV. Referring back to E.4: helium-4 sits higher on the binding energy per nucleon curve than hydrogen, so the reaction runs downhill in energy terms.

Stellar equilibrium is the balance a main-sequence star holds for most of its life: gravity pulling inward against radiation and gas pressure from fusion pushing outward. The balance is self-correcting. If the core cools, gravity wins slightly, the core compresses, the temperature rises and fusion speeds up, pushing back out. A star is a thermostat, and this is why a star’s life is stable rather than explosive.

Luminosity and apparent brightness are different quantities and the distinction is examined. Luminosity \(L\) is the total power the star radiates, from Stefan’s law \( L = \sigma A T^{4} \). Apparent brightness \(b\) is the power we receive per square metre:

which falls off as the inverse square of distance. A dim-looking star may be an intrinsically bright one that is very far away — the reason apparent brightness alone tells you almost nothing about a star.

The Hertzsprung–Russell diagram plots luminosity against surface temperature, with temperature increasing to the left, which catches people out. Most stars lie on a diagonal band, the main sequence, running from hot bright stars at the top left to cool dim ones at the bottom right. Off that band sit the red giants (cool but luminous, so they must be enormous) and the white dwarfs (hot but faint, so they must be tiny). Reading a star’s size off its position is exactly the reasoning \( L = \sigma A T^{4} \) supports.

The mass–luminosity relationship for main-sequence stars is roughly \( L \propto M^{3.5} \). That steep power has a consequence worth stating: a star ten times the Sun’s mass is around three thousand times more luminous, so it burns through its fuel far faster and lives a far shorter life. Massive stars die young.

Stellar evolution. When core hydrogen runs out, the core contracts and heats while the outer layers expand and cool — a red giant. What follows depends on mass. A star like the Sun sheds its outer layers as a planetary nebula and leaves a white dwarf, supported by electron degeneracy pressure, but only if its remnant mass is below the Chandrasekhar limit of about 1.4 solar masses. Above that the collapse continues to a neutron star, and beyond roughly 3 solar masses of remnant, to a black hole. Fusion in massive stars builds elements only up to iron, because iron sits at the peak of the binding energy curve and fusing beyond it absorbs energy rather than releasing it — which is why the core collapse happens at all.