Thermonuclear Fusion in the Sun and Other Stars — Question 2

Question 2

The solar luminosity measures completed helium-production cycles

Take total energy per completed hydrogen-to-helium cycle as 26.7 MeV, of which .600 MeV escapes in neutrinos. A steady star has photon luminosity $L_\gamma=3.84\times10^{26}$ W. Use $1\,\mathrm{MeV}=1.602\times10^{-13}$ J.

1. Find deposited energy per cycle and the cycle rate needed to supply the luminosity.

2. Find the proton-consumption rate and neutrino power, assuming steady energy balance.

Question 2 - Solution

The solar luminosity measures completed helium-production cycles

(a) Method and calculation

$$\boxed{q_{\mathrm{heat}}=26.7-.600=26.1\,\mathrm{MeV}=4.18122\times10^{-12}\,\mathrm J}.$$$$\boxed{\dot N_{\mathrm{cyc}}=L_\gamma/q_{\mathrm{heat}}=9.18392\times10^{37}\,\mathrm{s^{-1}}}.$$

(b) Interpretation and further calculation

$$\boxed{|\dot N_p|=4\dot N_{\mathrm{cyc}}=3.67357\times10^{38}\,\mathrm{s^{-1}}}.$$$$\boxed{L_\nu=\dot N_{\mathrm{cyc}}(.600)(1.602\times10^{-13})=8.82759\times10^{24}\,\mathrm W}.$$

Physical check

Photon luminosity corresponds to deposited heat, not the full Q value. This comparison assumes long-term equilibrium between energy generation and escape.