I’m in a Complex Systems and Data Science PhD program. In one of my classes we’ve been performing dimensional analysis, used to derive equations describing how different parts of a system relate to one another. This is a pretty empowering technique, and I wanted to walk through an example here.
G. I. Taylor was a British physicist tasked with estimating the atomic bomb’s kinetic energy during World War 2. For bureaucratic security clearance reasons he wasn’t given access to research data from the Manhattan Project, and needed to make his estimates based on declassified video footage of the Trinity bomb test. This video was sufficient to estimate the size of the mushroom cloud and the time elapsed since detonation. From this mushroom cloud footage alone, with almost no understanding of explosives or nuclear physics, we’ll try to estimate the kinetic energy of the blast.
(Details in this story are mythologized; in reality it appears that Taylor did have access to classified research data before the Trinity footage was made public, but the academic re-telling of the story usually stresses that you could derive all the necessary details from the Trinity film alone)
We know that the size of the mushroom cloud will be related to both the energy in the bomb and the time since the explosion. Taylor assumed the density of air would also be relevant. What are those variables measured in?
The radius of the mushroom cloud,
r, is measured in some length units (meters, feet, etc), which we’ll refer to as
Time since explosion,
t, is measured in time units (seconds, minutes), or
The density of air,
d, is measured as “mass over volume” (like “grams per cubic centimeter”), or
[M/V], but volume itself is measured as a length cubed, so we can simplify to
The kinetic energy of the blast,
E, is measured as “force across distance”, where “force” is “mass times acceleration”, and “acceleration” is “distance over time squared”. Therefore energy is measured in the dimensions
Note that the exact units don’t matter: We don’t care whether the radius is measured in meters or feet, we care that “as the energy increases, so does the radius.” If we switch from kilometers to centimeters, or switch from metric to imperial units, the scale of this relationship should stay the same.
We want to solve for the energy of the bomb, so we’ll put energy on one side of the equation, and set it equal to our other terms, all raised to unknown exponents, times a constant. In other words, all we’re saying is “all these terms are related somehow, but we don’t know how yet.”
Now let’s write that same equation, substituting in the dimensions each term is measured in:
Now we just need to find exponents that satisfy this equation, so that all the units match. Let’s assume
x1 = 1, because we want to solve for “energy”, not something like “energy squared”. This means mass is only raised to the first power on the left side of the equation. The only way to get mass to the first power on the right side of the equation is if
x2 = 1. Alright, great, but that means we have length squared on the left, and length to the negative third on the right. To compensate, we’ll need to set
x3 = 5. Finally, we have time to the negative two on the left, so we must set
x4 = -2 on the right. Plugging these derived exponents in, we have:
Switching back from dimensions to variables, we now find:
Neat! The last difficulty is calculating the constant, which needs to be determined experimentally. That’s out of scope for this post, and for dimensional analysis, but Taylor basically estimated it based on the heat capacity ratio of air, which is related to how much pressure air exerts when heated. At the altitude of the mushroom cloud, this value is around 1.036. This constant has no units (or in other words, it’s “dimensionless”), because it’s a ratio representing a property of the air.
Using the estimates from the Trinity film (in one frame, the mushroom cloud had a radius of roughly 100 meters 0.016 seconds after the explosion, at an altitude where the density of air is about 1.1 kilograms per cubic meter), we can enter all values into our equation, and estimate that the Trinity bomb had a blast energy of
4 x 10^13 Joules, or about 10 kilotons of TNT. The established blast energy of the bomb is 18-22 kilotons of TNT, so our estimate is the right order of magnitude, which isn’t half bad for just a couple lines of math.
We got lucky with this example, but sometimes the math isn’t as obvious. If several terms are measured in the same dimensions, then we can only find correct values for exponents by balancing the dimensions like a series of equations. Here’s the same problem, starting from our dimensions, solved with linear algebra:
Next, we transform the matrix into reduced echelon form, which makes it easier to isolate individual values:
Remember that this matrix represents three linear equations:
We can therefore see how each variable must be related:
Since we only have one independent variable (
x4) we can simply set
x4 = 1 and solve for the other variables.
x1 = 2,
x2 = -2,
x3 = -10,
x4 = 1.
If we had multiple independent variables (which would represent many dimensions but only a few terms), we’d have to set each independent variable to
1 one at a time, setting the other independent variables to
0, and then would need to set our partial solutions equal to one another to combine into a complete answer. Fortunately this is a simpler case!
We’ll now plug those exponents back into our dimensional equation, and painstakingly work our way back to parameters:
Certainly a longer path, but a more robust one.
Dimensional analysis allows us to examine arbitrary systems, moving from “these different components are probably related” to “this is how different attributes of the system scale with one another”, based only on what units each component is measured in. This is an awfully powerful tool that can give us a much deeper understanding of how a system works - limited by a constant we’ll need to experimentally solve for. All we’ve left out is one of the most challenging aspects of dimensional analysis: knowing which variables are relevant to the behavior of the system. Answering this question often requires a lot of experience with the system in question, so we can best think of dimensional analysis as “taking an expert’s intuition about a system, and solving for behavior analytically.”