Thermonuclear Fusion in the Sun and Other Stars — Question 5

Question 5

Only accessible hydrogen contributes to a simple stellar fuel lifetime

A star has mass $2.00\times10^{30}$ kg, hydrogen mass fraction .700, and only .100 of its mass accessible to core burning. Hydrogen processing releases .00700 of its rest energy. Use luminosity $3.84\times10^{26}$ W, $c=3.00\times10^8$ m/s and $1\,\mathrm{yr}=3.156\times10^7$ s. Neglect neutrino losses here.

1. Estimate accessible nuclear energy and constant-luminosity lifetime.

2. Find the error factor from assuming all stellar hydrogen is available, and state why the result is only a timescale.

Question 5 - Solution

Only accessible hydrogen contributes to a simple stellar fuel lifetime

(a) Method and calculation

$$\boxed{M_{H,\mathrm{usable}}=.100(.700)(2.00\times10^{30})=1.40\times10^{29}\,\mathrm{kg}}.$$$$\boxed{E_{\mathrm{usable}}=.00700M_{H,\mathrm{usable}}c^2=8.82\times10^{43}\,\mathrm J}.$$

(b) Interpretation and further calculation

$$\boxed{t=E_{\mathrm{usable}}/L=2.29688\times10^{17}\,\mathrm s=7.27780\times10^9\,\mathrm{yr}}.$$$$\boxed{t_{\mathrm{all}}/t=1/.100=10}.$$

Physical check

Stars evolve: luminosity, core size and composition change. The accessible-fuel fraction is a supplied simplification rather than a universal constant.