## ERQ · 14 marks · Topics: E.5 Fusion and stars + D.1 Gravitational fields
**Stem.** Astronomers studying a newly discovered main-sequence star, designated HD-2410, obtain the following data from spectroscopic and parallax measurements:
- Mass: M = 3.8 × 10³⁰ kg
- Radius: R = 1.4 × 10⁹ m
- Surface temperature: T_s = 7200 K
- Core temperature: T_c = 1.5 × 10⁷ K
- Composition (by mass): 71% hydrogen, 27% helium, 2% heavier elements
The star generates its luminosity through the proton–proton (pp) chain, in which four protons fuse to form one helium-4 nucleus with a total energy release of 26.7 MeV per cycle (of which 0.5 MeV is lost to neutrinos). Hydrostatic equilibrium is maintained by the balance between outward radiation/gas pressure and inward gravitational attraction. Assume the star radiates as a black body and that the proton number density at the core is n_p = 6.0 × 10³¹ m⁻³.
### Part (a) State [2 marks] · AO1 · Topic: E.5
State the conditions required at the core of HD-2410 for the pp chain fusion to occur.
### Part (b)(i) Calculate [3 marks] · AO2/AO3 · Topic: E.5
Calculate the luminosity of HD-2410, treating it as a black-body emitter.
### Part (b)(ii) Determine [3 marks] · AO2/AO3 · Topic: E.5
Using your answer to (b)(i), determine the rate at which hydrogen nuclei are consumed in the core of HD-2410, in protons per second. (Useful: 1 MeV = 1.60 × 10⁻¹³ J.)
### Part (c) Derive [3 marks] · AO3 · Topic: E.5 + D.1
For the star to remain in hydrostatic equilibrium, the inward gravitational pressure at the core must be balanced by the outward gas pressure. By equating the gravitational pressure P_g ≈ GM²/R⁴ to the ideal-gas pressure of the core protons P = n_p k_B T_c, derive an expression for the core temperature T_c in terms of M, R, n_p and fundamental constants, and evaluate it numerically for HD-2410. Compare with the measured value 1.5 × 10⁷ K.
### Part (d) Suggest [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR
Suggest three distinct reasons why the value of T_c derived in (c) differs from the measured core temperature.
(d)(i) [1 mark] — Identify one limitation of modelling the gravitational pressure as a single bulk value GM²/R⁴ rather than as a radially varying quantity.
(d)(ii) [1 mark] — Identify one limitation of using only proton number density n_p in the ideal-gas pressure expression.
(d)(iii) [1 mark] — Identify one physical pressure contribution that the model neglects entirely.
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## Mark Scheme
### Part (a) [2 marks]
- Mark 1: Sufficiently high temperature (≈ 10⁷ K) so that protons have enough kinetic energy to overcome the Coulomb repulsion / barrier [ECF: no]
- Mark 2: Sufficiently high density / pressure of protons to give a high enough collision rate (OR explicit reference to quantum tunnelling enabling fusion below classical Coulomb barrier) [ECF: no]
### Part (b)(i) [3 marks]
- Mark 1: Use of Stefan–Boltzmann law L = 4πR²σT_s⁴ [ECF: no]
- Mark 2: Correct substitution: L = 4π(1.4 × 10⁹)² × (5.67 × 10⁻⁸) × (7200)⁴ [ECF: yes]
- Mark 3: L ≈ 3.0 × 10²⁶ W (accept 2.9–3.1 × 10²⁶ W), 2 s.f., correct units [ECF: yes]
### Part (b)(ii) [3 marks]
- Mark 1: Energy released per 4 protons consumed = (26.7 − 0.5) MeV = 26.2 MeV → 26.2 × 1.60 × 10⁻¹³ = 4.19 × 10⁻¹² J [ECF: no]
- Mark 2: Energy per proton consumed = 4.19 × 10⁻¹² / 4 = 1.05 × 10⁻¹² J; rate = L / (energy per proton) using (b)(i) [ECF: yes]
- Mark 3: Rate ≈ 3.0 × 10²⁶ / 1.05 × 10⁻¹² ≈ 2.9 × 10³⁸ protons s⁻¹ (accept 2.7–3.0 × 10³⁸), correct units, 2 s.f. [ECF: yes]
### Part (c) [3 marks]
- Mark 1: Equate pressures GM²/R⁴ = n_p k_B T_c and rearrange to T_c = GM² / (n_p k_B R⁴) [ECF: no]
- Mark 2: Correct substitution: T_c = (6.67 × 10⁻¹¹)(3.8 × 10³⁰)² / [(6.0 × 10³¹)(1.38 × 10⁻²³)(1.4 × 10⁹)⁴] [ECF: yes]
- Mark 3: T_c ≈ 3.2 × 10⁷ K (accept 3 × 10⁷ K to within ×2), with explicit comparison statement that the derived value is roughly twice the measured 1.5 × 10⁷ K [ECF: yes]
### Part (d) [3 marks]
#### (d)(i) [1 mark]
- Mark 1: Gravitational pressure varies with radius (is much larger near the centre and zero at the surface); using a single bulk value GM²/R⁴ either over- or under-estimates the central pressure that the core gas must actually support [ECF: no]
#### (d)(ii) [1 mark]
- Mark 1: The pressure-supporting particles are not just protons — electrons (and helium nuclei) also contribute to the gas pressure; the total particle number density is larger than n_p, so the required temperature for equilibrium is lower than the derived value [ECF: no]
#### (d)(iii) [1 mark]
- Mark 1: The model neglects radiation pressure (P_rad = (1/3)aT⁴), which is significant at core temperatures of 10⁷ K and contributes to supporting the star against gravity, reducing the gas-pressure (and hence temperature) requirement [ECF: no]
### Marker notes
- Alternative method accepted for (b)(ii): candidates who include the neutrino energy (using full 26.7 MeV) gain Marks 1–2 by ECF but lose Mark 1; accept rate ≈ 2.8 × 10³⁸ s⁻¹.
- (c) Accept derivations that retain a numerical factor of order unity (e.g. T_c = GMm_p/(k_B R) form from hydrostatic scaling) provided the equating step is shown; numerical answer must still be within ×2 of the derived expression used.
- (d)(iii) alternative accepted contributions: electron degeneracy pressure (acceptable but less significant for main-sequence stars — award if justified), turbulent/convective pressure, magnetic pressure.
- (d)(i) accept equivalent statements that the star is not uniform-density / that hydrostatic equilibrium is a local (dP/dr) condition, not a global one.