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IB DP Physics (2025 syllabus)

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E.5 Fusion and stars

Practice Worksheet — name: ______________________ date: ____________

Formulas for this topic

Stellar parallax d(parsec)=1p(arc-second)d(\text{parsec}) = \frac{1}{p(\text{arc-second})}
Stellar luminosity L=4πR2σT4L = 4\pi R^{2}\sigma T^{4}
Apparent brightness b=L4πd2b = \frac{L}{4\pi d^{2}}
Wien's law λmaxT=2.9×10−3 m K\lambda_{\text{max}} T = 2.9 \times 10^{-3}\ \text{m K}

Section A — Multiple choice

A1. Fusion in stellar cores requires extremely high temperatures because:

  1. A Nuclei must overcome their mutual electrostatic repulsion to get within range of the strong force
  2. B The strong force only operates at high temperatures
  3. C Photons must have enough energy to split nuclei
  4. D Electrons must be removed from the atoms first

A2. A star has parallax 0.040 arc-seconds. Its distance is:

  1. A 25 pc
  2. B 40 pc
  3. C 0.04 pc
  4. D 2.5 pc

A3. On the HR diagram, white dwarfs lie below the main sequence because they are:

  1. A Cool and large
  2. B Hot but very small, hence dim
  3. C Cool and very small
  4. D Hot and very large

Section B — Short answer

B1. Describe the equilibrium that keeps a main-sequence star stable, and what happens when core hydrogen is exhausted. [4 marks]

B2. A star's spectrum peaks at 500 nm and its apparent brightness is 1.4×10−9 W m−21.4\times10^{-9}\ \text{W m}^{-2} at a distance of 10 pc (3.1×10173.1\times10^{17} m). Determine its surface temperature and luminosity. [4 marks]

B3. Explain why fusion, despite releasing less energy per reaction than fission, releases more energy per unit mass of fuel. [2 marks]

Answer key

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Section A

A1: Nuclei must overcome their mutual electrostatic repulsion to get within range of the strong force — Positive nuclei repel; only at enormous kinetic energies (temperature) do they approach within ~10−1510^{-15} m, where the attractive strong force can bind them. High density raises the collision rate — both conditions are needed.

A2: 25 pc — d=1/p=1/0.040=25d = 1/p = 1/0.040 = 25 parsecs (about 82 light-years). Smaller parallax means greater distance — the method fails beyond ~100 pc where the angle becomes unmeasurably small.

A3: Hot but very small, hence dim — Luminosity L=4πR2σT4L = 4\pi R^2\sigma T^4: white dwarfs have high surface temperatures (left side of diagram) but Earth-sized radii, so their luminosity is thousands of times below main-sequence stars of the same temperature.

Section B

B1: Outward radiation pressure from core fusion balances the inward gravitational pressure of the star's own mass. When hydrogen fuel runs out, radiation pressure falls and gravity wins: the core contracts and heats, igniting hydrogen in a surrounding shell (and later helium in the core); the envelope expands and cools — the star becomes a red giant, moving up-right on the HR diagram.

B2: Wien: T=2.9×10−3/5.0×10−7=5800 KT = 2.9\times10^{-3}/5.0\times10^{-7} = 5800\ \text{K} — Sun-like. Luminosity: L=4πd2b=4π(3.1×1017)2(1.4×10−9)≈1.7×1027 WL = 4\pi d^2 b = 4\pi(3.1\times10^{17})^2(1.4\times10^{-9}) \approx 1.7\times10^{27}\ \text{W}, roughly four times the Sun's.

B3: The binding-energy-per-nucleon curve is steepest at low mass numbers: fusing hydrogen to helium gains ~7 MeV per nucleon, whereas fission gains under 1 MeV per nucleon. Per kilogram of fuel, fusion therefore releases several times more energy.