Thermonuclear Fusion in the Sun and Other Stars — Question 6

Question 6

A steeper temperature dependence lets CNO burning overtake the pp chain

At fixed density and composition, use supplied local laws $\epsilon_{pp}\propto T^4$ and $\epsilon_{\mathrm{CNO}}\propto T^{17}$. At $T_0$, their ratio is .0100. Treat the laws as valid over the interval considered.

1. Find the ratio at $1.10T_0$.

2. Find the temperature ratio at which the contributions become equal and identify the key extrapolation assumption.

Question 6 - Solution

A steeper temperature dependence lets CNO burning overtake the pp chain

(a) Method and calculation

$$\boxed{\frac{\epsilon_{\mathrm{CNO}}}{\epsilon_{pp}}=.0100(T/T_0)^{13}}.$$$$\boxed{\left.\epsilon_{\mathrm{CNO}}/\epsilon_{pp}\right|_{1.10T_0}=.0100(1.10)^{13}=.0345227}.$$

(b) Interpretation and further calculation

$$.0100(T/T_0)^{13}=1\Rightarrow\boxed{T/T_0=100^{1/13}=1.42510}.$$$$\frac{d\ln(\epsilon_{\mathrm{CNO}}/\epsilon_{pp})}{d\ln T}=17-4=13.$$

Physical check

The powers are local approximations and composition includes catalyst abundance. A real crossover requires temperature-dependent reaction rates and stellar structure.