A Natural Nuclear Reactor — Question 7

Question 7

A retained daughter ratio can encode a delay before mineral closure

At the end of a hypothetical short episode, two radioactive parents have ratio $P_1(0)/P_2(0)=1$ and decay constants $.400\,\mathrm{h^{-1}}$ and $.200\,\mathrm{h^{-1}}$. Daughters produced before mineral closure escape; after closure all daughters are retained. The eventual retained daughter ratio is .500.

1. Derive the retained daughter ratio in terms of closure delay t.

2. Infer t and explain why the result requires a short initial production episode.

Question 7 - Solution

A retained daughter ratio can encode a delay before mineral closure

(a) Method and calculation

$$D_i(\infty)=P_i(t)=P_i(0)e^{-\lambda_i t}.$$$$\boxed{D_1/D_2=e^{-(\lambda_1-\lambda_2)t}=e^{-.200t}}\quad(t\text{ in h}).$$

(b) Interpretation and further calculation

$$\boxed{t=-\ln(.500)/.200=3.46574\,\mathrm h}.$$$$P_i(t)=\int S_i(t\prime)e^{-\lambda_i(t-t\prime)}dt\prime\quad\text{if production is extended}.$$

Physical check

This generic isotope-clock model requires complete daughter escape before closure and complete retention afterward. Partial retention needs additional parameters.