E.5 Fusion and starsIB Physics SL: Revision notes

A main-sequence star is stable because two effects balance: the outward radiation pressure (with gas pressure) produced by energy released in core fusion, and the inward gravitational force of the star's own mass. This balance is called hydrostatic equilibrium. If fusion speeds up the star expands slightly; if it slows, gravity makes the core contract.

In the core, hydrogen fuses into helium. Helium-4 has a greater binding energy per nucleon than hydrogen, so the products have less mass than the reactants and the mass defect is released as energy (E = mc²).

Nuclei are positive and repel each other, so fusion needs:

• a very high temperature (about 10⁷ K) so nuclei have enough kinetic energy to overcome electrostatic repulsion;
• a high density so collisions are frequent enough to sustain the reaction rate.

Gravitational compression of a large mass of gas provides both.

More massive stars have more fuel but are far more luminous, so they use their fuel faster and have much shorter main-sequence lifetimes.

• Low-mass stars (like the Sun): core hydrogen runs out → core contracts and heats → red giant (helium fuses to carbon) → outer layers lost as a planetary nebula → white dwarf.
• High-mass stars: → red supergiant, fusing elements up to iron → supernova → neutron star, or black hole for the most massive.

A remnant core below the Chandrasekhar limit (about 1.4 M☉) becomes a white dwarf; above it, it collapses to a neutron star, and above the Oppenheimer–Volkoff limit (about 2–3 M☉) to a black hole.

The HR diagram plots luminosity (vertical, increasing upwards) against surface temperature (horizontal, increasing to the left). Its main regions:

• Main sequence: a band from hot, bright, massive stars (top left) to cool, dim, low-mass stars (bottom right); stars fusing hydrogen in the core.
• Red giants and supergiants: top right; cool but very luminous, so very large radius.
• White dwarfs: bottom left; hot but dim, so very small radius.

Because L = σ4πR²T⁴, a star's position tells you its radius: at fixed temperature, higher luminosity means a larger star.

As the Earth orbits the Sun, a nearby star appears to shift against distant background stars. Half of the total angular shift over six months is the parallax angle p. The distance is d (parsec) = 1 / p (arc-second). The method only works for relatively nearby stars (up to a few hundred parsecs) because more distant stars have parallax angles too small to measure.

Combine three measurements:

1. Distance d from parallax.
2. Luminosity from apparent brightness: L = 4πd²b.
3. Surface temperature from Wien's law: λmax T = 2.9 × 10⁻³ m K.

Then use the Stefan–Boltzmann law, L = σ4πR²T⁴, so R = √(L / 4πσT⁴). For comparing two stars, R ∝ √L / T².