Thermonuclear Fusion in the Sun and Other Stars — Question 8

Question 8

An intermediate helium abundance adjusts to balance production and destruction

In a uniform stellar region let $^3$He number density n obey $\dot n=S-2Kn^2$, where two $^3$He nuclei are consumed per fusion event. Take $S=2.00\times10^{10}\,\mathrm{m^{-3}s^{-1}}$ and $K=10^{-20}\,\mathrm{m^3/s}$.

1. Find equilibrium abundance and the rate of final helium-producing events there.

2. Linearize about equilibrium to find the relaxation time and determine stability.

Question 8 - Solution

An intermediate helium abundance adjusts to balance production and destruction

(a) Method and calculation

$$S-2Kn_{\mathrm{eq}}^2=0\Rightarrow\boxed{n_{\mathrm{eq}}=\sqrt{S/(2K)}=10^{15}\,\mathrm{m^{-3}}}.$$$$\boxed{R_{\mathrm{eq}}=Kn_{\mathrm{eq}}^2=S/2=10^{10}\,\mathrm{m^{-3}s^{-1}}}.$$

(b) Interpretation and further calculation

$$n=n_{\mathrm{eq}}+\delta n\Rightarrow\dot{\delta n}\simeq-4Kn_{\mathrm{eq}}\delta n.$$$$\boxed{\tau=(4Kn_{\mathrm{eq}})^{-1}=2.50\times10^4\,\mathrm s,\quad\delta n\propto e^{-t/\tau}}.$$

Physical check

The negative linear coefficient makes equilibrium stable. The factor two in abundance removal is essential even though each event produces only one final helium nucleus.