Basic Concepts: Estimation and Dimensions

Ages and ages ago, Jennifer Ouellette commented on the start of the Basic Concepts series with a list of topics she'd like to see done. One of these was "Size and Scaling:"

First, let's tackle the jargon problem: Just what the heck is an order of magnitude? I use the phrase all the time now, after years of hanging around physicists, but as a budding science writer, I found the term a bit opaque, and I'd wager the average person on the street is a bit unclear on the specifics, too. Second, this is one of those areas where a picture really can be worth a thousand words -- or, barring that, it helps to paint a word picture. Many science museums use the "powers of ten" approach when discussing various size scales in the universe, from the subatomic level to the farthest reaches of the universe. It's been around since at least the 1960s, and it endures because it's effective. But it's just a start. The laws of physics actually start to change as one approaches the subatomic level, and a clear explication of how size and scaling can change a system's behavior would help with the public's chronic confusion about a number of things

This has been saved in my RSS reader for months, but I keep not getting around to it. This is one of those ideas that exists in a really tricky grey area: It's sufficiently basic that it's actually hard to articuate what's going on, but it's still abstract enough that it's hard to see why it matters. Even when we try to explicitly teach students about order-of-magnitude estimates, they don't really see the point, and I admit, I didn't either, until I spent a few years working in the field.

I was reminded of this topic while reading a book for an upcoming review, and I figured I really ought to take a crack at it. This will be somewhat less focussed than some of the previous Basic Concepts entries, and I'm going to lump together two slightly different ideas: Order-of-Magnitude Estimation and Dimensional Analysis, both of which are an important part of the physicist's mental toolkit.

The term order of magnitude, as Jennifer notes, is casually thrown around by physicists and astronomers all the time. In casual use, it means something like "roughly equal," but it has a fairly precise technical meaning, based on scientific notation. If you recall your middle school math, or have one of those calculators that display in scientific notation, you'll remember that any number can be written as a decimal multiplied by ten to some power. Thus, we can write 3,141.59 as:

3.14159 x 10³

In this notation, "order of magnitude" means "just the 10³ part." 3,141.59 is of the same order of magnitude as 1,500 or 4,567.8. There's a little uncertainty about whether you round up or not-- most people would probably say that 9,000 is the same order of magnitude as 10,000, but it's not clear whether 5,123 would be of order 1,000 or 10,000-- I've seen both, and used both, in different circumstances.

Either way, "order of magnitude" applied to an individual number just means the appropriate power of ten. Two numbers that are of the same order are within a factor of 10 of each other, and two numbers separated by an order of magnitude differ by a factor of ten-- 150 is an order of magnitude smaller than 3,141.59, and 45,678 is an order of magnitude larger.

This is an extremely coarse way of looking at numbers (though it might pass for precision measurement in astronomy), but it's actually more useful than you might think. Physicists frequently do order-of-magnitude calculations as a first step in solving problems.

For example, let's say that you want to find the velocity of a beam of electrons that have been accelerated through some potential difference (something that has come up several times in my class this term). To do this, you set the change in potential energy of the charges equal to the kinetic energy they acquire, and get something like:

qV = 1/2 mv²

where q is the charge, V is the electric potential difference (in volts), m is the mass, and v is the speed we're looking for. If you re-arrange this, you find an equation for the speed:

v = (2qV/m)^1/2)

An electron has a charge of 1.602 x 10^-19 coulombs and a mass of 9.109 x 10^-31 kg, and let's say we're talking about a voltage difference of 250V. Getting an exact answer to this requires a fair bit of button-punching on a calculator, but you can do an order of magntiude estimate in seconds by stripping each number down to just an order of magnitude, and putting that in.

The charge is just 10^-19 C, the mass is 10^-30 kg, and the voltage is 10² V, so this becomes:

v ≈ (10^-19 10³ / 10^-30)^(1/2) = (10¹³)^(1/2) = 10^6.5

So, we expect the speed of the electrons to be somewhere between 10⁶ and 10⁷ m/s, depending on the exact values. In fact, it's 9.377 x 10⁶, using the numbers I gave above.

Why is this useful? I mean, to get the correct answer, you're going to need to go back and plug all these numbers into the calculator anyway, so why not just do that from the beginning?

Well, in this case, it's useful because it gives us a quick check on the validity of our assumptions. When we wrote down that first equation, we used the classical expression for the kinetic energy. If the electrons turn out to be moving close to the speed of light, that expression will give results that are badly wrong, and would need to be replaced with the full relativistic expression. Of course, the relativistic formula is a pain in the ass to work with, and we'd rather not use it if we don't have to.

The order-of-magntiude calculation tells us quickly that we're ok using the classical expression. The speed of light is of order 10⁸ m/s, so our answer of 10⁶-10⁷ m/s is between 1% and 10% of the speed of light. The relativistic formula gives only a 1% correction when v is 10% of the speed of light (a result I know from another easy order-of-magnitude calculation), so for the precision we've got here, we're not going to notice the errors caused by using the classical formula.

"Well, fine," you say, "But I still have to put the full numbers in, so I haven't really saved any work. But it's easy to check an order-of-magntiude calculation, where it's hard to know if you've gone wrong with a full calculation. And with the order-of-magnitude estimate in hand, you can also know when you've done something horribly wrong in the full calculation.

Order of magntiude estimates are more important in physics than most other sciences because physics necessarily spans a hunge number of orders of magnitude. You can see that from this example-- the numbers in this problem range from 10^-30 on up to 10², a span of 32 orders of magntiude. Even if you restrict the possible operations to addition, subtraction, multiplication, and division, that means the correct answer might be anywhere between 10^-30 and 10⁺³⁰, which gives you absolutely no way to judge whether the number that comes out of the calculator is right. An order-of-magnitude calculation lets you restrict the possibilities to the point where you can hope to say something sensible about whether you've done things correctly.

These estimates are also key to understanding the scaling problem that Jennifer mentions. You can use order-of-magnitude calculations to show when odd effects start to have a significant effect-- for example, the statement above that relativistic effects only start to become significant when the speed gets above about 10% of the speed of light. Orders of magnitude are crucial to keeping straight what physical effects we need to consider-- you wouldn't use string theory to calculate the force needed to accelerate a car, because the order of magnitude is all wrong.

Closely related to the idea of order-of-magnitude estimation is the idea of dimensional analysis. It works pretty much the same way that order-of-magnitude estimation does, but in this case, you throw away the numbers entirely, and just work with the units. So, our speed equation becomes:

v ≈ (C V/ kg)^(1/2)

That doesn't seem to have helped, but I know from basic E&M that one volt per meter is equal to one newton per coulomb, and from basic mechanics I know that one newton is one kilogram meter per second squared, so:

v ≈ (C V/ kg)^(1/2) = ((C (kg m/s²)/C)m/kg)^(1/2) = (m²/s²)^(1/2)

This means that the big bundle of stuff on the right hand side of that equation ends up giving you a number with the units of velocity. Which, in turn, means that we most likely did the original calculation correctly, and can have some faith that the quantity we've calculated is the thing we're looking for.

Neither of these methods will tell you whether you've really gotten the right answer in the full calculation, but they'll help catch the most egregious sorts of mistakes that get made in the course of doing basic physics problems. It's hard to get undergraduates to understand this, but I spent hours as a graduate student making a crude simulation of some atomic collision processes, and dimensional analysis saved me more than once.

Dimensional analysis is not just about checking the answers to problems you already know how to do, though. If you understand dimensional analysis well enough, you can actually use it to make new discoveries.

A nice example of this comes from my recent bedtime reading, Götz Hoeppe's Why the Sky Is Blue, which describes how Lord Rayleigh determined the wavelength dependence of the scattering process that bears his name using nothing but dimensional analysis. He recognized that there are only a handful of things that can possibly affect the intensity of the light scattered from particles in the air: the wavelength of the light, the size of the particles, the density of the particles, the density of the surrounding medium, and the distance between the scattering particles and the observer. Using simple dimensional considerations, he was able to deduce that the scattering intensity has to be inversely proportional to the fourth power of the wavelength, which is the correct dependence. And, incidentally, explains why the sky is blue.