5.6.11 The First Atomic Explosion

In March 1950, Sir Geoffrey Taylor [1950] published two papers which examined the first atomic explosion in 1945 in New Mexico. The author concludes that a similarity analysis of the experiment is in excellent agreement with the theory and can be used to calculate the energy release during the explosion. The information on the total release of energy was a well guarded secret of the U.S. government in these days. The paper by Taylor was therefore classified when the theoretical investigations were made. However, the publication 5 years later resolved this secret and made the results on energy release public contrary to the intention of the U.S. government. The results on the energy release caused much embarrassment in American government circles. The flaw of the government was that motion pictures recorded by Mack [1947] became unclassified while the energy release was considered top secret. These pictures contained not only the explosion but also a time record which allowed an estimation of the physical quantities.

How such an estimation can be carried out is the subject of the present example. First, let us recall the sequence of pictures which were used by Sir Geoffrey Taylor to carry out the calculations. We took these pictures from the work of Taylor [1950]. They demonstrate the evolution of the blast in the first 2 ms. The animation capabilities of Mathematica empower us to follow the explosion at the desk.

The main observation we make is that the thermal wave expands from a point to a spherical object. The motion of the gas was assumed spherically symmetric. This simplifying assumption received excellent confirmation in the first atomic test.

The picture 15 after the explosion looks like a very large mushroom. This photograph of the fire ball of the atomic explosion in New Mexico confirms the spherical symmetry of the gas motion in an excellent way.

The development of the gas after 127 ms is still nearly spherical. However, the region near the ground is more disturbed.

The pictures above were taken from G.I. Taylor and originally recorded by Mack in 1945. Sir Geoffrey Taylor used in his theoretical considerations three basic equations of motion describing the evolution of the pressure , the density , and the radial velocity . The three equations are the equation of continuity, the Euler equation for the velocity, and the equation of state for a polytropic medium. The third equation expresses the fact that the entropy is constant along the path of a particle which generally is not the case, as Courant and Friederichs [1948] remark. According to the spherical symmetry of the problem Taylor used only the radial component of these quantities. The three equations of motion for the density ρ, the velocity field , and the pressure are

In Mathematica we first define the three variables by

The left-hand side of the three equations of motion are then collected in a list:

The equations contain two parameters, γ and , describing the ratio of the specific heats and the pressure of the undisturbed atmosphere. A symmetry analysis of these equations gives us the result

The three equations permit a four-dimensional discrete symmetry group. The main symmetry is a scaling symmetry for all variables. In addition, there is a symmetry of translation in time. For the moment, we concentrate on the scaling symmetry with infinitesimals for the independent and dependent variables:

The group parameters are chosen in such a way that we can find the parameters α and c in accordance with the measurements carried out by Taylor. The reduction of the equations of motion follows by

Thus, the original equations reduce to a coupled system of first-order ODEs. This set of equations contains the new variable , where is a constant. This constant is determined by the radial coordinate and the time . To determine the exponent α in our analysis, we use the measurements of Taylor for the radius of the explosion front. Since is a constant, we can determine the exponent α in a double logarithmic plot of the radius versus time . The slope in the log-log plot is directly related to 1/α. The data we need for this kind of analysis are tabulated in Taylor's paper and are reproduced here. The first figure of the data set denotes time in seconds and the second the radius in meters:

The double logarithmic plot demonstrates the linear relation between the fire-ball radius and the time elapsed since the ignition.

The scaling exponent α can be estimated by fitting a linear relation on the logarithmic data:

The fit of the logData follows by

The result is a relation connecting time with the radius in a logarithmic representation. The slope of the straight line is given by . The exponent α thus takes approximately the value α = 5/2 within an error of 1.4%.

Combining both the measurement and the fit in a common picture shows the excellent agreement between experiment and theory:

The plot shows that the scaling relation is satisfied over a range of about three decades in time.

The exponent α was calculated by Taylor by different reasonings. He used a dimensional analysis of the problem and ended up with the value for the scaling exponent. The idea behind a dimensional analysis is that a physical quantity is expressed by other physical quantities which mainly govern the process. The atomic explosion is mainly determined by the total energy released at ignition, the density of the surrounding air , and the time . The dimensions of the governing quantities in the length, mass, and time () system are , , and . The dimension of the radius expressed by these quantities is . Since and are constants, we find . Comparing this result with the formula derived from the similarity analysis, we can identify . This relation also suggests that

is a constant in time. Assuming that the density and that , depending on the pressure, is near unity, we can estimate the energy from Taylor's data. The following plot shows an overview of the energy calculated by (5.87) for all data points:

Despite the first point, the energy values oscillate around a mean energy value of about

The value calculated in joules corresponds to a T.N.T. equivalent of about 19,488 tons. The other information contained in Taylor's model is the dynamic behavior of the density ρ, the velocity , and the pressure . Inserting the value of α into the similarity reduction, we find the governing equations for these quantities:

The set of equations contain common factors which are eliminated in the following representation of the first-order coupled ODEs:

This set of equations contains parameters describing the pressure of air , the ratio of the specific heats γ, and a group parameter . Inserting numerical values for these three quantities allows us to integrate the ODEs numerically:

For a numerical integration, we need initial conditions for the density ρ, the velocity field , and the pressure :

The integration for in the range delivers the solution by an interpolated function:

The functions are represented by Plot[] for the three variables , , and :

The plot shows that all three quantities decay in . This behavior is expected since the total amount of energy is released into free space. In conclusion, we not only estimated the released thermal energy of the explosion but have also the spatial and temporal decay of the physical quantities available.

Created by Mathematica  (September 15, 2003)