Analysis of a nuclear explosion

Question P6.4.1

The expansion of the spherical ball of fire generated in an explosion may be analysed to deduce the initial energy, $E$, released by a nuclear weapon. The British physicist Geoffrey Taylor used dimensional analysis to demonstrate that the radius of this sphere, $R(t)$ should be related to $E$, the air density, $\rho_\mathrm{air}$, and time, $t$, through $$ R(t) = CE^{\frac{1}{5}}\rho_\mathrm{air}^{-\frac{1}{5}}t^{\frac{2}{5}}, $$ where, using model-shock wave problems, Taylor estimated the dimensionless constant $C \approx 1$. Using the data obtained from declassified timed images of the first New Mexico atomic explosion, Taylor confirmed this law and produced an estimate of the (then unknown) value of $E$. Use a log-log plot to fit the data in the table below (after G. I. Taylor, Proc. Roy. Soc. London A201, 159 (1950)) to the model and confirm the time-dependence of $R$. Taking $\rho_\mathrm{air} = 1.25\;\mathrm{kg\,m^{-3}}$ deduce $E$ and express its value in Joules and in 'kilotons of TNT' where the explosive energy released by 1 ton of TNT is arbitrarily defined to be $4.184\times 10^9\;\mathrm{J}$.

$t\;/\mathrm{ms}$$R\;/\mathrm{m}$ $t\;/\mathrm{ms}$$R\;/\mathrm{m}$ $t\;/\mathrm{ms}$$R\;/\mathrm{m}$

Note: this data can be downloaded as new-mexico-blast-data.txt.

Solution P6.4.1