G.I. Taylor’s Dimensional Analysis of the Atomic Bomb

In 1938 two chemists and two physicists discovered a scientific phenomenon which made the world’s problems both a lot smaller and a lot bigger. Like every group project since the dawn of time, one person got stiffed in the credits - in this case, Lise Meitner, who wasn’t recognized until much later. They discovered a process called fission [2], which generally goes like this:

An atom is made up of electrons, protons, and neutrons [1]. Protons and neutrons make up the nucleus of an atom, and are heavier than electrons. Logically, the more of these ingredients you have in an atom, the heavier the atom will be. Helium, for example, has two protons, and gold has seventy-nine. Douglas Adams

(creator of “The Hitchhiker’s Guide to the Galaxy”) would strongly consider the possibility that a gold-filled balloon would float away. The rest of us, probably, would not.

In many ways, physicists behave like the toddler who rampages about the playground with an insatiable desire to smash, throw, and set things on fire. We similarly drive our poor, sleep-deprived mothers to need another glass of wine at night. Sometimes, science is simply expensive target practice. When we learned the

structure of atoms, we began to shoot neutrons at them to see if we could make heavier elements.

The nucleus of uranium-235 is like a dinner party that’s one guest away from chaos - and that stray neutron is the drunk friend who shows up after midnight asking what’s for dinner. Instead of welcoming in that last neutron and becoming a heavier element or isotope, civil war breaks out at the dinner party, and the atom

breaks apart into two or more smaller atoms [2]. Splitting an atom like this releases a lot of energy compared to the size of the atom, so splitting a lot of atoms all at once releases an enormous amount of energy.

If you’re thinking, “1938... Isn’t that...”, yes, it is. The year before WWII began.

The discovery of fission one year before a world war broke out was, to put it mildly, terrifying. It was widely understood that a fission bomb would be more powerful than any weapon thus created, and whichever country was the first to create the weapon held humanity in their hands. Thus began the Manhattan Project.

Before I ever read anything about the Manhattan Project, I imagined it was a smoky nightclub where bedazzled movie stars clinked glasses of champagne and discussed the stock market. After more than a decade of curating this mental image, you may imagine how shocked I was to discover that “Manhattan”

was actually a laboratory in the middle of the New Mexican desert, where exactly (1) bongo-playing, safe-cracking, nut-case terrorized his colleagues as they toiled to create an atomic bomb.

After successfully building the weapon, in 1945 the United States government tested it by detonating the bomb in what was called the Trinity test [3]. Time-stamped pictures were taken of the bomb as it exploded, and were overlaid with a length scale. For all the genius that went into the Manhattan Project, when the government declassified these pictures, they made a major slip-up that can be caught with the simplest of physics tools: Dimensional Analysis.

Basic Dimensional Analysis Rules

1. Terms of different dimensions cannot be added or subtracted. For example, 10 meters plus 15 seconds means... nothing.

2. Terms of different dimensions can be multiplied and divided. For example, 10 meters per (divided by) 15 seconds is a speed. Speed qualifies as a something.

These two rules are useful when setting up equations. Consider the equation

where r is a radius and F (t) is a function of time. We know that the dimension of the radius, [r], is length, and therefore that the dimension of the function of time, [F (t)], must also be length.

To speak plainly, consistency is the issue. We can’t have one side of the equation have one dimension and the other side of the equation a different dimension. How much sense does it make for me to say, “Behold, an orange with a radius of 15 kilograms” ? Now, the radius of the orange did increase as a function of time while it was growing on the tree. That function F (t) does have time in there somewhere. Ultimately, however, the units of time need to cancel in some way and leave us with a final dimension of length, or else the equation is nonsensical.

Generally, dimensional analysis is used as a means of verifying a set of calculations. A check. I can immediately tell that something isn’t correct in an equation if I am adding something like energy (which has dimensions M*L^2/T^2) to something like volume (which has dimensions L^3). Sometimes, however, dimensional analysis can be used to solve a problem from scratch, giving a pretty good estimation of the true solution. This is precisely what happened in the case of G.I. Taylor and the atomic bomb.

Estimating the Energy Released by an Atomic Bomb

G.I. Taylor was a British physicist who saw the declassified pictures of the Trinity test in Life magazine and essentially said, “Hold my tea while I calculate information not yet released by the United States government”.

Though his information was limited, G.I. Taylor was able to make an astoundingly accurate estimate of how much energy the bomb released using only dimensional analysis [4]. We will walk through this math. I promise it will be less painful than the bomb.

The pictures were rather minimal, as they were published for their visual shock and aesthetic appeal rather than scientific work. In the pictures, there was an image of a largely spherical fireball, a timestamp indicating how much time had elapsed since the detonation of the device, and a length scale [3].

Figure 1: The “Gadget” was detonated during the Trinity Nuclear Test [3]. Pictures of its detonation were taken in rapid succession and timestamped so that scientists could study how the explosion evolved over time. Some of these pictures were declassified and published in Life magazine, which inadvertently provided casual readers a means of estimating the energy released in the atomic explosion (some seriously classified information).

The first thing he did to solve this problem was suggest that the radius of the fireball was related to a few simple quantities. Clearly, the fireball grew and evolved with respect to time. The size of the blast should also depend on the amount of energy initially released when the Gadget was detonated (a stick of dynamite releases more energy than a fire-cracker, and would presumably create a larger fireball too). Finally, because the fireball was pushing air out of the way so that it could take up room on its own, the density of the air must also be a factor (the blast radius probably wouldn’t have been so large if the bomb had been detonated inside a tank of water, for instance). With this information, we can write the following equation:

where r is the fireball radius, t is time, E is energy, and ρair is the density of the air. Calling L = length, T = time, and M = mass, we can put the variables of our function in terms of their dimensions as,

By the rules of dimensions, somehow we need to multiply these three terms such that they all simplify to length (L). Putting them in an equation, we have,

where I have given the yet-to-be-determined exponents x, y, and z.

If we distribute the exponents and group the terms with common bases, we have

Which we know must all be equal to L. Because of this, time and mass must ultimately have exponents that cause them to vanish, such as T^0 = 1, M^0 = 1, and length must boil down to L^1. We use this to create a system of algebraic equations for the exponent values:

By rearranging and substituting these values, you can find that (10) gives y = −z, which plugs into (11) such that −2z − 3z = 1 becomes z = −1/5 . Finally, plugging in the value for y in (9), we find that x = 2/5 . With all of these values, (x = 2/5 , y = 1/5, z = −1/5), we can create a formula for the radius:

where I have included the constant C which has no dimensions. Now it is a matter of rearranging this and plugging in known values.

The density of air is approximately 1.2 kg/m^3, and according to one of the pictures published in LIFE magazine, the radius of the fireball is approximately 77 meters .006 seconds after detonation. C, as it turns out, is approximately equal to 1. We can estimate, therefore, that the energy released by the Gadget in the Trinity test was

Which is approximately equivalent to 21.6 kilotons of TNT.

G.I. Taylor published these estimations in [4], giving the public a full view of some seriously classified information. We physicists often have the lovable quality of being, let us say, lost in the clouds. The poor chap fancied himself as doing some clever physics, but his publication supposedly stirred up suspicion of his being a spy and ruffled many feathers.

Posting a picture only to realize later that something unfortunate is happening in the background can happen to the best of us; all we can hope is that the embarrassing pose your friend struck is not as mission-critical as a length-scale on Trinity Test photos.

References

[1] May 1932: Chadwick reports the discovery of the neutron. American Physical Society, 2007. URL: https://www.aps.org/apsnews/2007/05/may-1932-chadwick-reports-the-discovery-of-the-neutron.

[2] Alan Chodos. Discovery of nuclear fission. American Physical Society, 2007. URL: https://www.aps.org/apsnews/2007/12/december-1938-discovery-nuclear-fission.

[3] Atomic Heritage Foundation. Trinity test -1945. Nuclear Museum, 2014. URL: https://ahf.

nuclearmuseum.org/ahf/history/trinity-test-1945/.

[4] Geoffrey Ingram Taylor. The formation of a blast wave by a very intense explosion. - ii. the atomic

explosion of 1945. The Royal Society, 1950. URL: https://royalsocietypublishing.org/doi/10.1098/rspa.1950.0050.