Fusion & Stars

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Lesson notes

Fusion Reactions in Stars

  • Nuclear fusion is the joining of two small nuclei to produce a larger nucleus, releasing energy.
  • In stars, four hydrogen nuclei (protons) fuse to produce one helium nucleus, releasing a huge amount of energy.
  • The proton-proton chain is a series of reactions that produces helium from hydrogen; it powers stars like the Sun.
  • For fusion to occur, nuclei must have high kinetic energy to overcome electrostatic repulsion between protons.
  • The strong nuclear force binds nucleons together but has a very short range, so nuclei must get very close to fuse.
  • Fusion requires extremely high temperature and pressure/density, such as in the core of a star.
  • On Earth, research focuses on the deuterium-tritium (D-T) reaction: ²₁H + ³₁H → ⁴₂He + ¹₀n.

Hydrogen fusion in a star

Hydrogen fusion in a star

Energy Released in Fusion Reactions

  • The larger nucleus produced by fusion has a higher binding energy per nucleon than the original nuclei.
  • A mass defect occurs because the total mass of the products is less than the total mass of the reactants.
  • The energy released equals the difference in binding energy, according to ΔE = Δmc².
  • For example, fusing two deuterium nuclei into helium releases about 28 − 4 = 24 MeV.
  • In the proton-proton chain, the overall reaction is: 4¹₁H → ⁴₂He + 2e⁺ + 2νₑ + energy.
  • Neutrinos carry away a small fraction of the energy released in the first step of the proton-proton chain.

Star Formation

  • Stars form from a nebula, a giant cloud of hydrogen gas and dust.
  • Gravitational attraction causes denser clumps to form, leading to gravitational collapse.
  • The collapsing gas heats up and glows, forming a protostar.
  • Protostars can be detected by their infrared radiation.
  • When the core reaches millions of kelvin, hydrogen fusion begins, and the star becomes a main sequence star.
  • A star remains stable when outward radiation and gas pressure balance the inward gravitational force.

Star formation stages

Star formation stages

Life Cycle of a Star

  • All stars begin with the same stages: nebula → protostar → main sequence star.
  • Low-mass stars (like the Sun) evolve: red giant → planetary nebula → white dwarf.
  • High-mass stars evolve: red supergiant → supernova → neutron star or black hole.
  • A red giant forms when hydrogen in the core runs out and the outer layers expand and cool.
  • A planetary nebula is formed when the outer layers of a low-mass star are ejected.
  • A white dwarf is the hot, dense remnant core left behind.
  • A supernova occurs when the iron core of a massive star collapses and the outer layers explode.
  • If the remnant core after a supernova exceeds about 3 solar masses, it collapses into a black hole.

Life cycle of a large mass star

Life cycle of a large mass star

The Hertzsprung–Russell (HR) Diagram

  • The HR diagram plots luminosity (y-axis) against surface temperature (x-axis).
  • Temperature increases from right to left on the HR diagram.
  • Most stars lie in a diagonal band called the main sequence.
  • For main sequence stars, luminosity increases with surface temperature.
  • Red giants and red supergiants are above the main sequence, with high luminosity at cooler temperatures.
  • White dwarfs are below and to the left of the main sequence: hot but low luminosity.
  • The HR diagram only shows stars in stable phases; black holes cannot be seen as they emit no light.

Emission & Absorption Spectra in Stars

  • A continuous spectrum contains all wavelengths and is produced by hot, dense sources like stellar cores.
  • An emission line spectrum consists of discrete bright lines on a black background, produced by hot, low-pressure gases.
  • An absorption line spectrum consists of dark lines on a continuous spectrum, produced when white light passes through a cool, low-pressure gas.
  • Stars emit an absorption line spectrum because their outer atmospheres are cooler than the core.
  • Each element produces a unique pattern of spectral lines, acting as a fingerprint for chemical composition.
  • The chemical composition of a star can be determined by comparing its absorption spectrum with known emission spectra.
  • The Sun is predominantly made of hydrogen and helium, confirmed by its absorption spectrum.

Stellar Parallax

  • Stellar parallax is the apparent shift in position of a nearby star against distant background stars as Earth orbits the Sun.
  • The parallax angle p is measured in arcseconds (").
  • The distance d to a star in parsecs is given by p = 1/d, where p is in arcseconds.
  • This equation is accurate for distances up to about 100 pc.
  • 1 parsec (pc) is the distance at which 1 AU subtends an angle of 1 arcsecond.
  • 1 pc ≈ 3.1 × 10¹⁶ m ≈ 3.26 light-years.
  • 1 light-year (ly) is the distance light travels in one year ≈ 9.5 × 10¹⁵ m.
  • 1 astronomical unit (AU) is the mean Earth-Sun distance ≈ 1.5 × 10¹¹ m.

Determination of Stellar Radii

  • The radius of a star can be estimated using Wien's displacement law and the Stefan-Boltzmann law.
  • Wien's displacement law: λmax T = 2.9 × 10⁻³ m K, gives the surface temperature from the peak wavelength.
  • The Stefan-Boltzmann law: L = 4πr²σT⁴, relates luminosity, radius, and temperature.
  • By combining these laws, the stellar radius r can be calculated if luminosity and temperature are known.
  • The inverse square law of flux can be used to find luminosity from radiant flux and distance.
  • For example, Betelgeuse has a radius about 1000 times larger than the Sun's.

Slides

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Practice questions

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  1. 1.Which statement about nuclear fusion is correct?

    Easy
    • AIt is the joining of two small nuclei to produce a larger nucleus
    • BIt is the splitting of a large nucleus into smaller nuclei
    • CIt absorbs energy from the surroundings
    • DIt only occurs in particle accelerators

  2. 2.Which row shows the conditions required for fusion to be sustained in the core of a star?

    Easy
    • ADensity: moderate, Temperature: very high
    • BDensity: moderate, Temperature: moderate
    • CDensity: very high, Temperature: very high
    • DDensity: very high, Temperature: moderate

  3. 3.Which of the following is part of a possible evolutionary path of a star?

    Easy
    • AW → Y
    • BX → Y
    • CY → W
    • DY → Z

  4. 4.Which of the following describes the sequence for the evolution of a star of about 10 solar masses?

    Easy
    • Anebula → supernova → protostar → main sequence star → red supergiant → neutron star
    • Bnebula → planetary nebula → main sequence star → red giant → supernova → white dwarf
    • Cnebula → protostar → main sequence star → red giant → planetary nebula → white dwarf
    • Dnebula → protostar → main sequence star → red supergiant → supernova → neutron star

  5. 5.Astronomers measure the parallax angle of two nearby stars. The parallax angle of star X is 3.9 × 10⁻⁶ rad and the parallax angle of star Y is 1.6 × 10⁻⁷ rad. What can be deduced about the relative distances of the two stars from these measurements?

    Medium
    • AStar X is closer to Earth than Star Y.
    • BStar Y is closer to Earth than Star X.
    • CStar X and Star Y are both at a similar distance from Earth.
    • DNothing can be deduced from these measurements alone.

  6. 6.The following fusion reaction occurs in stars: ²₁H + ³₁H → ⁴₂He + ¹₀n. The binding energies are: deuterium ²₁H = 2.2 MeV, tritium ³₁H = 8.5 MeV, helium-4 ⁴₂He = 28.3 MeV. How much energy is released in this fusion process?

    Medium
    • A2.2 MeV
    • B10.7 MeV
    • C17.6 MeV
    • D28.3 MeV

  7. 7.Five regions are labelled on the Hertzsprung-Russell diagram shown. Which sequence could show part of the evolution of a star like the Sun?

    Medium
    • A1 → 5 → 3
    • B2 → 3 → 4
    • C3 → 5 → 1
    • D4 → 3 → 2

  8. 8.Which of the following statements about stars in positions P, Q, R and S is correct?

    Medium
    • AP and R have the same mass.
    • BP has a lower surface temperature than S.
    • CR is more massive than P.
    • DR has a lower surface temperature than Q.

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