Dimensions and Units and Atomic Bombs, Oh My!

Dimensional analysis is the art and science of doing algebra with units.  When I say units, I mean feet, pounds, meters, kilograms, femtoseconds, milliliters, etc.  And there is definitely an art to dimensional analysis: despite its humdrum-sounding name, dimensional analysis is a subtle and sometimes beautiful subject that requires creative thinking to do well.  When done effectively, dimensional analysis can give profound insight into the inner workings of a physical phenomenon at a fraction of the effort that one normally expends by doing many tedious experiments or calculations.  It's used all the time in science and engineering.

What's the difference between dimensions and units?

Consider velocity.  Velocity, of course, is expressed as a distance divided by a time.  It can be measured in miles per hour, for example.  In somewhat-technical language, scientists say that velocity has "fundamental dimensions of length divided by time."  Notice I didn't say "dimensions of miles per hour" or "dimensions of meters per second."  That's because dimensions aren't units.  Units (like kilograms, meters, seconds) are arbitrary constructs of the human mind - what we call a kilogram is based on the mass of an arbitrarily sized piece of platinum/iridium alloy - while dimensions (like mass, length, or time) have a physical meaning.  This turns out to be a very important distinction.

Dimensional Consistency

The key to dimensional analysis is that any physically meaningful equation has to be dimensionally consistent.  For example, if I wrote an equation for the distance traveled by an object moving at a constant speed, it might look like this:

distance = velocity x time

Both sides of this equation have dimensions of length.  Obviously, "distance" has dimensions of length, and on the right side we have (length / time) x (time), so the "time"s cancel out, leaving length.  This is what I mean by doing algebra with units.  And if I were to add any additional terms to this equation, they'd better also have dimensions of length.  For example, I could write

total distance = initial distance + velocity x time

The point of dimensional consistency is that every additive term must have the same dimensions.  If I tried to add another term that had dimensions of time / length, then we would know that there is something fundamentally wrong with the equation; we would lose all confidence in its ability to describe reality.  This turns out to be true even in more complicated situations where the formulas are not so simple.

Dimensionless Numbers

If you were to ask me how fast a 747 airplane can fly, and I replied "987," that answer wouldn't mean very much.  The obvious question is: what units am I using?  I could be expressing my answer in miles per hour, meters per second, micrometers per nanosecond, or any other combination.  Again, units are arbitrary.

But if you asked me "what is the Mach number of a Boeing 747?" and I replied "0.85," I've still just given you a number, but now that number has physical meaning.  I've just told you that a 747 can fly at 85% the speed of sound.  But, of course, it doesn't matter whether I calculated both speeds in meters per second or miles per hour - the result I get for the Mach number is independent of the untis system I use, which is precisely why the number is physically meaningful - the only reference point is something physical (the speed of sound).

The Mach number is perhaps the most familiar example of a dimensionless number.  These are numbers that have no dimensions, and thus no units.  It's easy to see this with the Mach number - we are dividing one speed (the speed of the airplane) by another speed (the speed of sound), so as long as we use the same units for both speeds, we will get the same Mach number no matter what unit system we choose.  This turns out to have important consequences.  An important result in dimensional analysis, known as the Buckingham Pi Theorem, describes how I can take a physically meaningful equation (like the one for distance above), and express it in dimensionless form.  For the case of the distance equation, the dimensionless result is very, very simple:

distance / (length x time) = 1.

Both sides of this equation are dimensionless.  On the left I have a distance divided by (length x time), so a distance divided by a distance.  On the right I just have a number.  This example is pretty trivial, but it illustrates the point.  Both sides of the equation are now dimensionless.  OK, you say, that's great.  When is this actually useful?

On July 16, 1945, about one month before the bombings of Hiroshima and Nagasaki, the United States Government conducted the Trinity nuclear test, which was the first detonation of a nuclear weapon in history.  A photographer captured the explosion on camera:

The rapidly expanding fireball from the Trinity nuclear test, 16/1000 of a second after the explosion.  SOURCE: http://nuclearweaponarchive.org/Usa/Tests/Trinity.html

The U.S. Government published 25 of these photos, each at different times, in Life Magazine in 1947.   One of the most sensitive pieces of information was the yield of the bomb, or how much energy it released.  Much to the government's chagrin, however, it turns out that we can use dimensional analysis to determine how the growth of the explosion is related to the yield of the bomb!

The Buckingham Pi theorem is essentially a recipe for arranging the relevant parameters into dimensionless form.  I'll skip the mathematical details (some examples can be found on Wikipedia), but it turns out that the parameters that matter the most here are the yield of the bomb, which we'll call E, the radius R of the fireball, and time t, since the radius obviously grows with time.  It also turns out to depend on the density of the atmosphere, which we'll call d.  (see note 1).  If we arrange these parameters into a dimensionless equation, we will get

Formula for the radius of the fireball as a function of the yield of the bomb E, density of the surrounding atmosphere d, and time t.  K is a constant which turns out to have a value of approximately 1.

The exponents might look a little funny, but they have to be there in order to be dimensionally consistent.  It turns out that, if these are the parameters that matter, then this is the ONLY way to express the radius in terms of the other parameters in a dimensionally consistent fashion!  

This equation predicts that the radius grows as time to the 2/5ths power.  Is it accurate?  Well, the photos (rather helpfully) have a length scale and the elapsed time indicated in each image, so we can actually use the 25 images provided to put the formula provided by dimensional analysis (above) to the test:

The predictions of the formula above, compared with experiment.  The line is predicted by the theory; the + signs are data from the images.

I hope you're blown away by the agreement between theory and experiment here, because if not, I haven't conveyed clearly enough just how easily we came up with that formula.  The above plot was produced by the famous fluid dynamicist Sir Geoffrey Ingram Taylor in a 1950 paper.  He subsequently used the formula above, along with some insights from the mathematical equations governing the behavior of gases, to estimate the yield of the bomb to be 16,000 tons of TNT.  This turns out to be in the correct range, which was given by the head of the Manhattan Project as between 15,000 and 20,000 tons!  (see note 2)

Dimensional analysis isn't easy.  It took real insight for us to know the physical parameters that matter in the equation above.  But when dimensional analysis is done right, it is a cheap way to learn a lot in a short period of time.

NOTES!

1) One way to interpret this is that the denser the atmosphere is, the more mass the fireball needs to push out of the way in order to continue expanding.

2) Taylor's one-time student and biographer, George Batchelor (himself a preeminent fluid dynamicist), notes that Taylor was “…mildly admonished by the US Army for publishing his deductions from their photographs.”  Suffice it to say, when they added the distance and time data, the government did not appreciate the power of dimensional analysis!