IB Physics E.5 Fusion and Stars — Notes (SL & HL)

Why does a star neither collapse nor fly apart?

A star is a huge ball of hot ionised gas (plasma) held together by its own gravity. Every layer is pulled towards the centre by the gravitational attraction of the mass inside it. If nothing opposed this, the star would collapse in less than an hour.

• Inward: gravity. Each layer carries the weight of all the layers above it, so the pressure needed to hold it up is greatest at the centre.
• Outward: pressure. The energy released by fusion in the core flows outwards as radiation. Photons carry momentum, so they exert a radiation pressure; the very hot gas also exerts a large gas (thermal) pressure. Use the term radiation pressure, and always name it in an answer; you may add gas pressure.
• Because the two balance, the resultant force on each layer is zero and the star keeps a constant radius. This state is called hydrostatic equilibrium (the name is useful but not required).

Why is the equilibrium stable?

The balance corrects itself. If the fusion rate rises slightly, the extra energy raises the pressure, so the star expands a little. The expanding core cools and becomes less dense, the fusion rate falls, and the star settles back. If the fusion rate drops, gravity wins slightly, the core contracts and heats up, and the fusion rate rises again. This feedback keeps a main-sequence star almost unchanged for most of its life.

The equilibrium ends only when the core runs out of the nuclei it has been fusing. Then the outward pressure from the core falls, gravity wins, and the core contracts — the start of the changes described in section 4.

How does fusion supply a star's energy?

• Fusion of light nuclei releases energy because binding energy per nucleon rises steeply from hydrogen-2 to helium-4 and then, apart from a few dips, generally continues to rise up to the peak near iron-56 (E.3).
• The energy released is \(E=mc^2\) applied to the mass defect of the reaction: (mass of reactants) − (mass of products). In atomic mass units, 1 u corresponds to 931.5 MeV (data booklet).
• Equivalently, energy released = (total binding energy of products) − (total binding energy of reactants).
• Most of the energy appears as kinetic energy of the products and as gamma-ray photons. It heats the core and eventually leaves the surface as the star's luminosity.

Hydrogen burning in the Sun

In the Sun, hydrogen is turned into helium mainly through the proton–proton chain. You do not need to recall every step, but you should know the overall change and that energy is released at each stage.

• Charge is conserved: \(+4e\) before, \(+2e+2e\) after. In stage 1 a proton changes into a neutron, emitting a positron and a neutrino — a β⁺ process (E.3).
• The positrons quickly annihilate with electrons in the plasma, adding to the energy released.
• The neutrinos interact so weakly that they escape the Sun directly, carrying a few per cent of the energy with them.

What conditions does fusion need?

• Temperature. All nuclei are positively charged and repel each other. Only if they approach at very high speed do they get close enough (about \(10^{-15}\ \text{m}\)) for the strong nuclear force to bind them. The mean kinetic energy of a particle is \(\bar E_k=\tfrac32k_BT\) (B.1), so a high kinetic energy needs a high temperature: the Sun's core is about \(1.5\times10^{7}\ \text{K}\).
• Density. At a given temperature, only a tiny fraction of collisions lead to fusion. A very high density (more nuclei per unit volume) makes collisions frequent enough to release energy at the rate needed to support the star.
• Gravity provides both. When a cloud of gas and dust (a nebula) collapses under gravity, gravitational potential energy is transferred to the kinetic energy of the particles, so the gas heats up and becomes denser. When the core of the protostar becomes hot and dense enough, hydrogen fusion begins and the star joins the main sequence.
• Heavier nuclei have more charge, so they repel more strongly: fusing helium, carbon and oxygen needs progressively higher core temperatures. This is why only massive stars can fuse elements beyond carbon.

Quick estimate. At \(1.5\times10^{7}\ \text{K}\), \(\bar E_k=\tfrac32\times1.38\times10^{-23}\times1.5\times10^{7}=3.1\times10^{-16}\ \text{J}\approx1.9\ \text{keV}\). At the Sun's surface (5800 K) it is only 0.75 eV — far too little, which is why fusion happens only in the core.

How does a star's mass decide its evolution?

A star spends about 90 % of its life on the main sequence, fusing hydrogen into helium in its core. Its mass controls everything that follows. A more massive star has stronger gravity, so its core is hotter and denser, its fusion rate is far higher and its luminosity is much greater. It therefore uses up its hydrogen much faster: massive stars have shorter lives, even though they start with more fuel.

Low-mass stars (up to about 8 solar masses), including the Sun

• When the core hydrogen runs out, fusion in the core stops and gravity makes the core contract and heat up. Hydrogen continues to fuse in a shell around the core.
• The outer layers expand enormously and cool: the star becomes a red giant — larger, more luminous and with a lower surface temperature than before.
• The core becomes hot enough to fuse helium into carbon (and some oxygen). In a star like the Sun, the core never becomes hot enough for carbon fusion.
• The outer layers drift away as a planetary nebula, leaving the hot, exposed core: a white dwarf. No fusion occurs in a white dwarf; it is held up against gravity by electron pressure (a quantum effect) and simply cools over billions of years. A white dwarf can be stable only if its mass is below about 1.4 M_☉ (the Chandrasekhar limit).

High-mass stars (more than about 8 solar masses)

• After the main sequence the star becomes a red supergiant. Its core reaches temperatures high enough to fuse successively heavier nuclei: helium → carbon and oxygen → neon and magnesium → silicon → iron, each stage in a shell surrounding the next.
• Iron sits at the peak of the binding-energy curve: fusing it would absorb energy. Once the core is iron, no energy is released to support it, so it collapses in a fraction of a second.
• The collapse rebounds as a supernova explosion that ejects the outer layers. Many nuclei heavier than iron are made by neutron capture during this explosion, and the products are scattered into space to form new stars and planets.
• The remnant core becomes a neutron star if its mass is between about 1.4 M_☉ and about 3 M_☉, or a black hole if it is larger.

What does the Hertzsprung–Russell diagram show?

• Luminosity \(L\) is the total power radiated by a star (W). It is often given in units of the Sun's luminosity, \(L_{\odot}\). Both axes are usually logarithmic because the values range over many orders of magnitude.
• Surface temperature runs from about 40 000 K on the left to about 2500 K on the right. Hot stars appear blue-white; cool stars appear red.

Lines of constant radius

From the Stefan–Boltzmann law, \(L=\sigma4\pi R^2T^4\). For stars of the same radius, \(L\propto T^4\), so on log–log axes every line of constant radius is a straight line, sloping down from upper left to lower right. Moving up and to the right on the diagram means a larger star: cool but very luminous stars must be giants. Moving down and to the left means a smaller star: hot but dim stars must be tiny.

A star's track across the diagram shows its evolution. The Sun will move up and to the right as it becomes a red giant, then down and to the left to become a white dwarf.

How is stellar parallax used to measure distance?

As the Earth orbits the Sun, a nearby star appears to move against the background of much more distant stars. Measuring this shift six months apart gives the star's distance by simple geometry.

• \(d\) = distance to the star (pc), \(p\) = parallax angle (arc-seconds). In the data booklet. The equation works only with these units.
• 1 arc-second \(=\dfrac{1}{3600}\) of a degree. Because \(p\) is tiny, \(\tan p\approx p\) (in radians), so \(d=\dfrac{1\ \text{AU}}{p}\) — this is what defines the parsec.
• Smaller parallax angle means a more distant star. Ground-based telescopes can measure angles down to roughly 0.01″ (about 100 pc); space telescopes above the atmosphere measure far smaller angles and reach thousands of parsecs.

How do we find a star's temperature, composition and radius?

Information from the spectrum

• Surface temperature. A star radiates approximately as a black body (B.1). The wavelength at which the intensity peaks gives the surface temperature through Wien's displacement law, \(\lambda_{\max}T=2.9\times10^{-3}\ \text{m K}\) (data booklet). Hotter stars peak at shorter wavelengths and look bluer.
• Composition. The continuous spectrum is crossed by dark absorption lines. Atoms in the cooler outer layers absorb photons whose energies match differences between their energy levels (E.1). Each element has its own pattern of lines, so the lines identify the elements present in the star's outer layers.

Luminosity, apparent brightness and radius

• \(L\) = luminosity (W), \(\sigma\) = Stefan–Boltzmann constant \(5.67\times10^{-8}\ \text{W m}^{-2}\ \text{K}^{-4}\), \(A\) = surface area (m²), \(R\) = radius of the star (m), \(T\) = surface temperature (K), \(b\) = apparent brightness (W m⁻²), \(d\) = distance (m).
• \(L=\sigma AT^4\) and \(b=\dfrac{L}{4\pi d^2}\) are in the data booklet (B.1). The surface area of a sphere, \(A=4\pi R^2\), is not, so learn it.
• Apparent brightness is what we measure at the Earth; luminosity is the power the star emits. A dim-looking star may be very luminous but far away.

Comparing with the Sun avoids the constants. Dividing \(L=\sigma4\pi R^2T^4\) for the star by the same equation for the Sun:

This ratio form is not in the data booklet; it follows directly from \(L=\sigma AT^4\). Solar values (\(L_{\odot}\), \(R_{\odot}\), \(T_{\odot}\)) are given in a question when they are needed.

Exam tips for E.5

Common misconceptions

How this topic connects

Fusion and fission side by side: shared principles, opposite directions

Alike: in both, the products have a greater binding energy per nucleon than the reactants, so the mass decreases and energy \(E=\Delta mc^2\) is released, mostly as kinetic energy of the products (E.4). Unlike: fusion joins light nuclei (left of the iron peak) whereas fission splits very heavy nuclei (right of the peak). Fusion releases more energy per unit mass of fuel (about 0.7 % of the mass, compared with about 0.1 % for uranium fission), needs extremely high temperatures and densities rather than a critical mass, and its main product, helium, is not radioactive. Fission is triggered by neutron absorption and can run as a chain reaction at ordinary temperatures, which is why it, not fusion, powers today's nuclear power stations.

Stars as black bodies: temperature and radius from their radiation

Treating a star as a black body (B.1) lets us read its surface temperature from the peak wavelength of its spectrum using Wien's law. Combined with its luminosity (from apparent brightness and distance), the Stefan–Boltzmann law \(L=\sigma4\pi R^2T^4\) then gives its radius, which no telescope could measure directly. The same laws produce the lines of constant radius on the HR diagram and explain why the colour of a star indicates its temperature.

A star as an ideal gas: modelling stellar stability with the gas laws

The plasma in a star behaves approximately as an ideal gas (B.3), \(PV=nRT\): the pressure depends on the temperature and the density. This simple model explains the self-regulation of a star — if the core contracts, its temperature and density rise, the pressure rises and pushes back. It also explains why a collapsing nebula heats up (work done on the gas by gravity increases its internal energy). The model is approximate: at the very high densities of white dwarfs and neutron stars the ideal-gas model fails and quantum effects provide the pressure — a reminder that models have limits of validity.

Key equations

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