Fusion and Stars

Stars shine because nuclear fusion in their cores converts mass into energy. In a main-sequence star such as the Sun, hydrogen is fused into helium through the proton-proton chain: the overall result is that four protons become one helium-4 nucleus plus two positrons and two neutrinos.

The helium-4 nucleus is about $0.7\%$ lighter than the four protons that formed it. This mass defect $\Delta m$ is released as energy, roughly $27\ \text{MeV}$ per helium nucleus produced, according to:

$$E = \Delta m c^2$$

Because the binding energy per nucleon rises steadily up to iron-56, fusing light nuclei always moves the products towards greater stability, so energy is released. Fusion is only possible because core temperatures are high enough for nuclei to approach one another against their mutual electrostatic repulsion (helped by quantum tunnelling), and core densities are high enough to make collisions frequent.

You should be able to describe hydrogen fusion in terms of mass defect and binding energy per nucleon, and explain why fusion, rather than chemical reactions or gravitational contraction alone, can power a star steadily for billions of years.

A main-sequence star is in hydrostatic equilibrium: at every layer, the inward pull of gravity is balanced by the outward push of thermal gas pressure and radiation pressure generated by core fusion. This balance is self-regulating, acting like a thermostat.

If the fusion rate drops, the core cools slightly, pressure falls, and gravity compresses the core; compression raises the temperature, which speeds fusion back up. If the fusion rate rises, the core expands and cools, slowing fusion again.

This feedback is why stars are remarkably stable for most of their lives: the Sun has remained on the main sequence for about $4.6$ billion years with nearly constant output.

Equilibrium fails only when the core runs out of hydrogen fuel. With no energy source, the outward pressure can no longer match gravity, the core contracts and heats, and the star moves into the next stage of its evolution.

You must be able to name both balancing influences (gravity inward versus gas plus radiation pressure outward) and explain the self-correcting feedback argument qualitatively.

Hydrogen fusion requires a core temperature of about $10^7\ \text{K}$ so nuclei have enough kinetic energy to overcome the Coulomb barrier; helium fusion needs roughly $10^8\ \text{K}$, and each heavier fuel needs hotter conditions still.

A star's initial mass therefore controls its whole life: more massive stars have hotter, denser cores, burn fuel far faster, and have much shorter main-sequence lifetimes.

When core hydrogen is exhausted, low-mass stars (below about $8 M_{\odot}$) swell into red giants, fuse helium into carbon, shed their outer layers, and end as white dwarfs. High-mass stars become supergiants, build fusion shells of progressively heavier elements up to iron, and end in a supernova explosion.

When a star exhausts its thermonuclear fuel, it can no longer maintain hydrostatic equilibrium through thermal pressure:

• If the progenitor mass is low (under $8 M_{\odot}$), the collapsing core is halted by electron degeneracy pressure, a quantum mechanical effect stemming from the Pauli exclusion principle, which prevents electrons from occupying the same quantum state. This forms a white dwarf.
• If the core mass exceeds the Chandrasekhar limit ($1.4 M_{\odot}$), gravity overcomes electron degeneracy, forcing electrons to combine with protons to form neutrons (electron capture). The resulting collapse is halted by neutron degeneracy pressure, forming a neutron star.
• If the core exceeds the Oppenheimer-Volkoff limit ($2$ to $3 M_{\odot}$), no known force can prevent complete collapse into a black hole.

The Hertzsprung-Russell (H-R) diagram plots stellar luminosity (vertical axis, usually as $L/L_{\odot}$ on a logarithmic scale) against surface temperature (horizontal axis, in kelvin, decreasing to the right).

• Most stars lie on the main sequence, a diagonal band running from hot, luminous stars at the top left to cool, dim red dwarfs at the bottom right; position along this band is set by mass, with the most massive stars at the top left.
• Red giants and supergiants occupy the upper right: their surfaces are cool but their luminosities are enormous, so they must be very large.
• White dwarfs sit at the lower left: hot but faint, so they must be small.

Luminosity, radius and temperature are linked by:

$$L = 4 \pi R^2 \sigma T^4$$

Lines of constant radius therefore run diagonally across the diagram, letting you compare stellar sizes at a glance. As a star evolves off the main sequence, its position on the diagram traces its changing luminosity and temperature.

You should be able to sketch the diagram with correctly labelled axes, mark the main sequence, red giant and white dwarf regions, and place a star on it given its temperature and luminosity.

For nearby stars, distance is measured by stellar parallax. As Earth orbits the Sun, a nearby star appears to shift against the far more distant background stars; half the total angular shift over six months is the parallax angle $p$.

One parsec is the distance at which $p = 1$ arcsecond ($1\ \text{pc} = 3.09 \times 10^{16}\ \text{m}$). With $d$ in parsecs and $p$ in arcseconds:

$$d = \frac{1}{p}$$

Because parallax angles shrink as distance grows, the method only works for relatively nearby stars.

Luminosity $L$ is a star's total radiated power, but what we actually measure on Earth is its apparent brightness $b$: the power per square metre after the radiation has spread over a sphere of radius $d$, given by:

$$b = \frac{L}{4 \pi d^2}$$

Combining measurements lets you deduce a star's radius. Wien's law, $\lambda_{\text{max}} T = 2.90 \times 10^{-3}\ \text{m K}$, gives the surface temperature from the peak wavelength of the star's spectrum, and the Stefan-Boltzmann law $L = 4 \pi R^2 \sigma T^4$ then yields $R$.

Practise chaining these steps fluently:

• parallax to distance
• brightness to luminosity
• peak wavelength to temperature
• luminosity and temperature to radius