E.5 Fusion and stars

Balance inward and outward effects

A stable main-sequence star is in hydrostatic equilibrium: inward gravitational force is balanced by outward thermal or radiation pressure. The star’s radius remains approximately stable while the balance holds.

Link the balance to fusion

Fusion in the core releases energy. The energy transported outward produces thermal and radiation pressure that resists gravitational collapse.

Predict imbalance

If the outward pressure falls, gravity compresses the star and raises core temperature; if pressure grows, the star expands until a new balance is reached.

Common trap

Do not write only “gravity balances pressure” without directions. State that gravity acts inward and thermal/radiation pressure acts outward, with fusion supplying the energy.

Use fusion as a stellar source

In stellar fusion, light nuclei combine to form more tightly bound nuclei. The mass difference is released as energy, which powers the star and supports its pressure balance.

Worked example — deuterium–tritium fusion

For $^2\mathrm{H}+{}^3\mathrm{H}\rightarrow{}^4\mathrm{He}+n$, the binding energies are $2(1.11)=2.22\,\mathrm{MeV}$, $3(2.83)=8.49\,\mathrm{MeV}$ and $4(7.07)=28.28\,\mathrm{MeV}$. Therefore $E=28.28-2.22-8.49=17.57\,\mathrm{MeV}$. The product is more tightly bound, so energy is released.

Follow nucleosynthesis

Fusion in stars can build elements up to iron through successive reactions. Elements heavier than iron are mainly formed in explosive environments and neutron-capture processes rather than ordinary core fusion.

Compare fusion and fission

Fusion can offer high energy per mass and potentially fewer long-lived waste products, but it requires extreme temperature and confinement conditions.

Common trap

Do not say all heavy elements are made by fusion in ordinary stars. The pathway changes around iron.

Use high temperature

Nuclei are positively charged and repel electrically. High core temperature gives them enough kinetic energy and speed to approach despite this repulsion.

Use high density

High density puts more nuclei into a given volume, increasing the collision frequency and the probability of close encounters.

Reach the strong-force range

Fusion requires nuclei to approach closely enough for the attractive strong interaction to act and for a bound product to form.

Common trap

Do not use surface temperature or star size as the direct fusion condition. The relevant evidence is high core temperature and density.

Start when core hydrogen is depleted

Reduced core fusion lowers outward pressure, so gravity contracts and heats the core. Hydrogen shell fusion then expands the outer layers: lower-mass stars become red giants, while high-mass stars become red supergiants.

Connect mass to lifetime

A more massive main-sequence star has more fuel, but its fusion rate and luminosity rise much more strongly. It therefore uses core hydrogen faster and has a shorter main-sequence lifetime.

Keep the path conditional

Mass controls the pathway; not every star becomes a supernova, and not every supernova leaves a black hole. A planetary nebula is expelled gas from a red giant, not a planet-forming stage.

Read the axes

An HR diagram compares luminosity with surface temperature. Temperature usually decreases from left to right, so the hottest stars are on the left.

Use constant-radius lines

From $L=4\pi R^2\sigma T^4$, a constant-radius line links luminosity and temperature. At the same temperature, greater luminosity means greater radius; at the same luminosity, the hotter star has the smaller radius.

Common trap

Do not read the temperature axis as increasing to the right, and do not identify a white dwarf from temperature alone; its low luminosity is also essential.

Use the Earth’s orbit

Observe a nearby star from opposite sides of Earth’s orbit at different times of year. Its apparent position shifts against distant background stars; half of the total angular shift is the parallax angle $p$.

Calculate distance

When $p$ is measured in arcseconds, distance in parsecs is $d=1/p$. For $p=0.25$ arcsec, $d=4$ pc, which can then be converted to light-years.

Know the range

Parallax is a geometric distance method and is most useful for relatively nearby stars. It does not use a star’s spectrum or brightness directly.

Common trap

Do not use the full annual position shift as p if the diagram shows the total displacement from one side of Earth’s orbit to the other.

Model a star as a spherical black body

Its luminosity $L$ is the total power radiated from surface area $4\pi R^2$ at absolute surface temperature $T$. This approximation connects observable luminosity and spectrum-derived temperature to radius.

Form the ratio

$\frac{R_1}{R_2}=\sqrt{\frac{L_1}{L_2}\left(\frac{T_2}{T_1}\right)^4}$. A hotter star can have a smaller radius at the same luminosity, while a very luminous cool star must be large.

Worked example — Canopus

For $L=10700L_\odot=4.12\times10^{30}\,\mathrm{W}$, $T=7400\,\mathrm{K}$ and $\sigma=5.67\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$, $R=\sqrt{L/(4\pi\sigma T^4)}=4.4\times10^{10}\,\mathrm{m}$. The large radius explains high luminosity despite a moderate surface temperature.

Check powers

Temperature enters to the fourth power and radius enters squared. Keep the temperature ratio in the inverse order shown before taking the square root.

Common trap

Do not use $R\propto L/T$; the correct scaling is $R\propto\sqrt L/T^2$.