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 $4.184\times 10^9\;\mathrm{J}$.

 $t\;/\mathrm{ms}$ $R\;/\mathrm{m}$ $t\;/\mathrm{ms}$ $R\;/\mathrm{m}$ $t\;/\mathrm{ms}$ $R\;/\mathrm{m}$ 0.1 11.1 1.36 42.8 4.34 65.6 0.24 19.9 1.50 44.4 4.61 67.3 0.38 25.4 1.65 46.0 15.0 106.5 0.52 28.8 1.79 46.9 25.0 130.0 0.66 31.9 1.93 48.7 34.0 145.0 0.80 34.2 3.26 59.0 53.0 175.0 0.94 36.3 3.53 61.1 62.0 185.0 1.08 38.9 3.80 62.9 1.22 41.0 4.07 64.3