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}$
0.111.11.3642.84.3465.6
0.2419.91.5044.44.6167.3
0.3825.41.6546.015.0106.5
0.5228.81.7946.925.0130.0
0.6631.91.9348.734.0145.0
0.8034.23.2659.053.0175.0
0.9436.33.5361.162.0185.0
1.0838.93.8062.9
1.2241.04.0764.3

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


Solution

To access solutions, please obtain an access code from Cambridge University Press at the Lecturer Resources page for my book (registration required) and then sign up to scipython.com providing this code.