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How the Sun Cheats the Barrier

Nuclear fusion and the strong interaction

In the core of the Sun (temperature

T \approx 1.5 \times 10^{7}\ \text{K}

) hydrogen nuclei (protons) fuse. Two protons can only fuse if they come close enough for the strong nuclear interaction to act, about

r_s = 2 \times 10^{-15}\ \text{m}

apart. Treat protons as point charges. The overall result of the proton–proton chain is

4\,^{1}\text{H} \rightarrow\ ^{4}\text{He} + 2e^{+} + 2\nu

Atomic masses:

^{1}\text{H}

1.007825\ \text{u}

;

^{4}\text{He}

4.002603\ \text{u}

. The average kinetic energy of a particle in a gas at temperature

T

\tfrac{3}{2}k_BT

Data: h = 6.63 × 10⁻³⁴ J s · e = 1.60 × 10⁻¹⁹ C · k = 8.99 × 10⁹ N m² C⁻² · k_B = 1.38 × 10⁻²³ J K⁻¹ · mₚ = 1.673 × 10⁻²⁷ kg · 1 u = 1.661 × 10⁻²⁷ kg = 931.5 MeV/c²

(a)

calculate1 mark

Show that the electric potential energy of two protons separated by

r_s

is about

0.7\ \text{MeV}

, and calculate the average kinetic energy of a proton in the Sun's core, in keV.

(b)

derive2 marks

In the core neither proton is fixed. Two protons, each with kinetic energy

K

, approach each other head-on. Using conservation of momentum and energy, show that their distance of closest approach is

r_{\min} = \dfrac{ke^2}{2K}

. Hence show that, on this classical picture, protons of average energy would just reach

r_s

at a temperature

T = \dfrac{ke^2}{3k_Br_s}

, and compare this temperature with the Sun's core temperature.

(c)

calculate2 marks

For a proton with the average kinetic energy in the core, calculate (i)

r_{\min}

from (b) and (ii) its de Broglie wavelength. (iii) Use the comparison of these two lengths to explain why the classical estimate in (b) should not be trusted.

(d)

calculate1 mark

Calculate the energy released in the overall proton–proton chain, and the energy released per nucleon. Explain why calculating it from the atomic masses given automatically includes the energy released when the two positrons annihilate.

(e)

discuss2 marks

Contrast the strong interaction and the Coulomb interaction between two protons, and use the contrast to explain why fusion needs such extreme temperatures. Then use the shape of the binding-energy-per-nucleon curve to explain why fusing light nuclei releases energy, but stars cannot release energy by fusing nuclei heavier than iron.

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