## ERQ · 14 marks · Topics: E.5 Fusion and stars + D.1 Gravitational fields
**Stem.** Astronomers studying the nearby main-sequence star Tau Ceti model it as a uniform sphere of mass M = 1.55 × 10³⁰ kg and radius R = 5.40 × 10⁸ m. The star generates energy by the proton–proton chain, in which the net reaction 4 ¹₁H → ⁴₂He + 2 e⁺ + 2 νₑ releases 26.7 MeV per helium nucleus formed. The measured luminosity of Tau Ceti is L = 1.95 × 10²⁶ W. The mass of a hydrogen atom is 1.673 × 10⁻²⁷ kg and the mass of a helium-4 atom is 6.645 × 10⁻²⁷ kg. For a star in hydrostatic equilibrium, an order-of-magnitude estimate gives a mean core temperature T_c ≈ GMm_p / (k_B R), where m_p is the proton mass.
### Part (a) State [2 marks] · AO1 · Topic: E.5
State the two conditions that must be satisfied at the core of a main-sequence star for hydrogen fusion to occur, and explain why both are required.
### Part (b)(i) Calculate [3 marks] · AO2/AO3 · Topic: E.5
Calculate the rate, in kg s⁻¹, at which Tau Ceti converts hydrogen mass into helium mass.
### Part (b)(ii) Calculate [3 marks] · AO2/AO3 · Topic: E.5
Calculate the rate, in kg s⁻¹, at which the rest mass of the star is being converted to energy.
### Part (c) Show that [3 marks] · AO2/AO3 · Topic: E.5 + D.1
Using the order-of-magnitude expression for core temperature given in the stem (which arises from equating gravitational potential energy per proton to thermal energy), show that the mean core temperature of Tau Ceti is of the order of 10⁷ K.
### Part (d) Suggest [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR
The expression T_c ≈ GMm_p / (k_B R) treats the star as a uniform sphere with a single representative temperature. Suggest three distinct reasons why the actual central temperature of Tau Ceti (observationally ≈ 1.6 × 10⁷ K) differs from the value calculated in part (c), referring to the gravitational field model used.
---
## Mark Scheme
### Part (a) [2 marks]
- M1: Temperature must be sufficiently high (≈ 10⁷ K) so that protons have enough kinetic energy to overcome the Coulomb (electrostatic) repulsion barrier / approach within range of the strong nuclear force [ECF: no]
- M2: Density (or pressure) must be sufficiently high so that the rate of proton–proton collisions is high enough to sustain the reaction / produce net energy output [ECF: no]
### Part (b)(i) [3 marks]
- M1: Mass defect per He formed: Δm = 4(1.673 × 10⁻²⁷) − 6.645 × 10⁻²⁷ = 4.7 × 10⁻²⁹ kg, OR uses E = 26.7 MeV → 4.28 × 10⁻¹² J per reaction [ECF: no]
- M2: Rate of He production = L / E_per_reaction = 1.95 × 10²⁶ / 4.28 × 10⁻¹² = 4.56 × 10³⁷ s⁻¹; hydrogen mass rate = 4 × 1.673 × 10⁻²⁷ × 4.56 × 10³⁷ [ECF: yes]
- M3: ≈ 3.05 × 10¹¹ kg s⁻¹ (accept 3.0–3.1 × 10¹¹ kg s⁻¹) [ECF: yes]
### Part (b)(ii) [3 marks]
- M1: Uses E = mc² → dm/dt = L / c² [ECF: no]
- M2: dm/dt = 1.95 × 10²⁶ / (3.00 × 10⁸)² [ECF: yes]
- M3: = 2.17 × 10⁹ kg s⁻¹ (accept 2.2 × 10⁹ kg s⁻¹) [ECF: yes]
### Part (c) [3 marks]
- M1: Substitution: T_c ≈ (6.67 × 10⁻¹¹)(1.55 × 10³⁰)(1.673 × 10⁻²⁷) / [(1.38 × 10⁻²³)(5.40 × 10⁸)] [ECF: no]
- M2: Numerator ≈ 1.73 × 10⁻⁷; denominator ≈ 7.45 × 10⁻¹⁵ [ECF: yes]
- M3: T_c ≈ 2.3 × 10⁷ K, which is of order 10⁷ K ✓ (must explicitly state order of magnitude conclusion) [ECF: yes]
### Part (d) [3 marks]
Award 1 mark each for any three distinct critiques:
- The star is NOT uniform — density and gravitational field strength increase steeply toward the centre, so the actual core (where fusion occurs) experiences a much stronger local g than the volume-averaged value used here [ECF: no]
- The expression sets thermal energy ≈ classical Coulomb/gravitational energy per proton; actual fusion proceeds via quantum tunnelling so fusion can occur at temperatures lower than the naïve classical estimate [ECF: no]
- Treating the star as a point mass / using R as the relevant length scale ignores that pressure (and hence temperature) is set by integrating hydrostatic equilibrium dP/dr = −ρ(r)g(r) through the interior, not by a single shell value [ECF: no]
- Radiation pressure and electron degeneracy pressure contribute to support, reducing the temperature needed for equilibrium relative to a pure ideal-gas + gravity balance [ECF: no]
- Composition is not pure hydrogen — heavier elements alter the mean molecular mass, so m_p is not the correct mass per particle [ECF: no]
### Marker notes
- Alternative method accepted for (b)(i): work in MeV throughout — L converted to MeV s⁻¹ then divided by 26.7 MeV gives He rate directly.
- (b)(ii) is independent of (b)(i); do NOT penalise twice if student conflates the two — the discriminating point is that mass-to-energy rate (≈ 2 × 10⁹ kg s⁻¹) is ~140× smaller than hydrogen-processing rate (≈ 3 × 10¹¹ kg s⁻¹) because only ~0.7% of rest mass is released.
- (c) accept 1 × 10⁷ to 5 × 10⁷ K; conclusion sentence required for M3.
- (d) discriminator: accept any three from the list; do not award two marks for restatements of the same underlying point (e.g. "g varies with r" and "density varies with r" count as one).