The Big Bang — Question 5

Question 5

Helium abundance follows from counting the surviving neutrons

Immediately before nucleosynthesis, let n/p=1/7. Assume all neutrons enter helium-4, each nucleus using two neutrons and two protons. Ignore binding-mass corrections and other nuclei.

1. Derive the helium baryon mass fraction in terms of r=n/p and evaluate it.

2. Find the helium-to-hydrogen number ratio and the fraction of initial protons incorporated into helium.

Question 5 - Solution

Helium abundance follows from counting the surviving neutrons

(a) Method and calculation

$$N_{\mathrm{He}}=n/2,\quad Y=\frac{4N_{\mathrm{He}}}{n+p}=\boxed{\frac{2r}{1+r}=.25}.$$$$N_H=p-n\quad\text{because helium consumes }n\text{ protons}.$$

(b) Interpretation and further calculation

$$\boxed{N_{\mathrm{He}}/N_H=\frac{r}{2(1-r)}=\frac1{12}=.0833333}.$$$$\boxed{N_{p,\mathrm{He}}/p=n/p=1/7}.$$

Physical check

Helium is about one quarter by mass in this model but about one helium nucleus per twelve hydrogen nuclei; mass and number fractions differ.