It’s not widely appreciated how often physicists can guess the answer to a problem before they even start calculating. By combining a basic consistency requirement with scientific reasoning, they can often use a heuristic approach to solving problems that allows them to derive most of a formula without doing any work at all. This week I want to introduce this to you, and show you some of its power.

The trick, called “dimensional analysis” or “unit analysis” or “dimensional reasoning”, involves requiring consistency among units, sometimes called “dimensions.” For instance, the distance from the Earth to the Sun is, obviously, a **length**. We can state the length in kilometers, or in miles, or in inches; each is a * unit* of length. But for today’s purposes, it’s irrelevant which one we use. What’s important is this: the Earth-Sun distance has to be expressed in

**unit of length, because, well, it’s a length! Or in physics-speak, it has the “dimensions of**

*some***length**.”

For any equation in physics of the form X = Y, **the two sides of the equation have to be consistent with one another.** If X has dimensions of length, then Y must also have dimensions of length. If X has dimensions of mass, then Y must also. Just as you can’t meaningfully say “I weigh twelve meters” or “I am seventy kilograms old”, physics equations ** have to make sense**, relating weights to weights, or lengths to lengths, or energies to energies. If you see an equation X=Y where X is in meters and Y is in Joules (a measure of energy), then you know there’s a typo or a conceptual mistake in the equation.

*In fact, looking for this type of inconsistency is a powerful tool, used by students and professionals alike, in checking calculations for errors. I use it both in my own research and when trying to figure out, when grading, where a student went wrong.*

That’s nice, but why is it useful beyond checking for mistakes?

Sometimes, when you have a problem to solve involving a few physical quantities, **there might be only one consistent equation relating them** — only one way to set an X equal to a Y. And you can guess that equation without doing any work.

Well, that’s pretty abstract; let’s see how it works in a couple of examples.

### Simple Example: Velocity, Radius and Period

First, a super-easy one just to illustrate the point. Suppose we want to find the velocity v of the Earth as it travels round the Sun. If we call the average radius of the Earth’s roughly circular orbit R, and we call T the time it takes to orbit the Sun, then what formula should we write down? Well, there’s only one possibility that’s consistent. Velocity v is a length traveled per time; it has dimensions of length over time. R is a length. T is a time. And so the equation that relates them must be of the form

- v = # R / T

where “#” is an unknown number that this argument doesn’t specify.

Since we don’t know this number, have we really learned anything? Yes we have! The formula cannot set v equal to R^{2}/T, or T^{2}/R, or R T, or R^{1.4} /T^{2.6}. **Any formula other than v = # R/T would relate a length per time to something that isn’t a length per time**… and would therefore be nonsensical. ** Just by demanding sense, we have mostly solved the problem without doing any work at all**, except for one unknown #.

If we want to be precise, we’ll still have to calculate the unknown #. If the orbit were circular that would be easy; # = 2π. For a realistic, elliptical orbit, you have to actually calculate it. Still, for a nearly-circular orbit like the Earth’s, this # it’s not going to be a billion for a nearly-circular orbit, nor is it going to be a billionth. It will be a number that’s not far from 2π, which in turn is not too, too far from 1. (2π is approximately 6.) **So we can make an estimate without doing much, if any, calculation.**

### Interesting Example: Kepler’s Law, In Detail

Now let’s take a less trivial example, though still easy to do using other methods. Recently, using do-it-yourself techniques, I showed you how you yourself could derive Kepler’s third law, which relates the radius of a planet’s orbit R to its orbital period T, specifically that R^{3} is proportional to T^{2}. We found this was true for objects that orbit the Sun. We also found it was true for objects that orbit the Earth, but with details that were different. Can we find a formula which is true *both* for the Sun *and* the Earth — one that explains the difference?

Well, **under an assumption** — that Newton’s gravity is involved somehow — we can. This is where physics reasoning and some experience comes in.

First, if gravity is at work, an experienced physicist knows that Newton’s constant G **always** appears, because this constant characterizes the overall strength of gravity. The dimensions of G have to be consistent with Newton’s gravitational force equation

- F = G M m / r
^{2}

which gives the force of gravity between two objects of mass M and m that are separated by a distance r. Rearranging for convenience, we can write this as

- G = F r
^{2}/(M m)

In first-year physics we learn that force has dimensions of **mass times length divided by time squared**. M and m have dimensions of **mass**, and r has dimensions of **length**. From the above equation G = F r^{2} /(M m), we find the dimensions of G itself:

- dimensions of G = dimensions of F r
^{2}/(M m) [for consistency!]- = (dimensions of F) * (dimensions of r
^{2})/(dimensions of M m)

- =
**(mass * length / time**^{2}) * (length^{2})/(mass^{2}) = (length^{3}/time^{2}/mass)

- = (dimensions of F) * (dimensions of r

Moreover, under our assumption that gravity is at work, and since we are considering objects that all orbit the same central body, such as the Sun, we can guess that the mass of that central body comes in somehow. Let’s refer to that mass as “M”.

So now to Kepler’s law: *might there be an equation that relates gravity’s ever-present constant, G, the mass of a central body, M, the period T of an object orbiting that central body, and the radius R of that orbit?* Well, how about

- G / M = # R
^{4}/T^{2}

or

- G M
^{5}= # R^{9}T^{2}

or

or

- G M
^{3/2}= # T^{5}/R^{7/3}?

No Way! Each of these possible equations is nonsense! because the dimensions of the left hand side are not equal to the dimensions of the right hand side!

But there is in fact (as I’ll convince you in a moment) **one and only one possible answer that could make sense! **That’s this one:

**G M = # R**^{3}/T^{2}

And this confirms that for a particular central object of mass M, all objects that orbit it have the same ratio for R^{3} to T^{2}. In other words, **you can guess Kepler’s third law of orbits simply by using dimensional analysis.** No complicated equations are required.

Again, # is an unknown number that we would have to calculate. But even though we don’t know it yet, we’re most of the way to finding the formula we want, and we haven’t done any work other than checking dimensions!

*Why is this the only possible formula? One can be systematic about it, but here’s a quick way to see it. R and T have no units of mass, but G and M do. So to relate G and M to R and T, there must be some combination of G and M in which the dimensions of mass cancel. Since, as we just saw above, G has dimensions of something divided by mass, G M is the only combination where the dimensions of mass cancel, leaving only dimensions of length and time! In fact GM has dimensions of length^{3} divided by time^{2} — and that means G M can only be related to R^{3}/T^{2}. That’s all there is to it*!

A little calculation shows that the unknown # is (2π)** ^{2}**, approximately 39.5, which is not too, too far from 1. (It’s not that close, admittedly. But remember that this unknown # could have been 483,248,342,198 or 0.000000000000932 — and so, relative to what it could have been, it’s still pretty close to 1.) This tendency for these unknown #’s to be not to so far from 1 is one we need to keep an eye on.

Yet we don’t even need to know the unknown # to learn something extremely important! Suppose we have studied **R ^{3}/T^{2}** for the Moon and satellites moving around the Earth, and

**R**for the planets orbiting the Sun. We have

^{3}/T^{2}- G M
_{sun}= #**R**for objects orbiting the Sun^{3}/T^{2}

For instance we could focus on the Earth’s orbit, so we would take R to be the Earth-Sun distance R_{ES} and T to be one Earth year T_{E}. Meanwhile,

- G M
_{earth}= #**R**for objects orbiting the Earth^{3}/T^{2}

Here we could focus on the Moon’s orbit, and take R to be the Moon-Earth distance R_{ME} and T to be one Moon month T_{M}. Now we can take the ratio of these two expressions! **The G cancels, and so does the unknown #**! That leaves us with

- M
_{sun}/M_{earth}= (**R**)^{3}/T^{2}_{earth_around_sun}/(**R**)^{3}/T^{2}_{moon_around_earth}= (R_{ES}^{3}/R_{ME}^{3})(T_{M}^{2}/T_{E}^{2})

Since the Earth-Sun distance R_{ES} is about 388 times the Moon-Earth distance, and the Earth-Year is about 13.4 times the Moon-Month, we find

- M
_{sun}/M_{earth}= (388)/(13.4)^{3}= 325,000^{2}

which is correct, to within a few percent. Look at that! **We calculated the ratio of the Sun’s mass to the Earth’s mass just using dimensional analysis**! All we needed to know was the distances and times relevant to the orbits of the Earth and Moon.

** We never had to solve a gravity equation to figure this out! We just had to assume that gravity was involved somehow**.

I hope that this convinces you that if you use this reasoning well, it can be immensely powerful, with the potential to simplify difficult problems dramatically. Next time we’ll look at how dimensional analysis can be (and was) used to learn things about relativity and black holes, and then we’ll look at atomic physics and beyond.

What does “this constant [G] characterizes the overall strength of gravity” mean? Do you mean *unit* strength of gravity?

I mean that if you doubled G, all gravitational effects would double in strength; if you made it 1/10 of what it is now, all gravitational effects would become 10 times weaker.

“A little calculation shows that the unknown # is (2π)2…” How do we get this 4PI^2?

Just from replacing R with circumference 2 Pi R in the places that it is needed.

G is interesting because it is not a dimensionless constant. So fine, having defined it, then it is reasonable to use in similar equations (ie “under an assumption — that Newton’s gravity is involved somehow”).

But would you like to comment on the process/jump of defining such constants in the first place, and whether dimensioned or dimensionless gives you insight?

Well, that will become a pressing question in later posts, so let’s get back to it later.

You might mention that the equations for Planck length. Planck time and Planck mass in terms of hbar, c and G also can be readily found by applying dimensional analysis. In fact this may be the only way to derive equations for these three quantities(?). I do not know if Planck used dimensional analysis for this purpose.

This will come later. Planck certainly knew about dimensional analysis, I don’t know its origin, but it goes back hundreds of years. Bohr used it directly; that will be a subject of a later post.

The best example is the c^2 in E = m * c^2. It had to be c^2, no other solution was possible. Gilbert Strang (MIT) mentioned this.

See the next post, https://profmattstrassler.com/2022/06/23/e-m-c-squared-the-simple-dimensions-of-a-discovery/. It is an obvious point to any physicist, but little appreciated outside of physics. Einstein’s real advance wasn’t the equation but the realization that it could reasonably apply to every object in the universe (and calculating the # out front.)

Dimensional analysis arguments are scaling arguments in disguise. To see this, note that given physical variables X_1, X,2, X_3,… ,X_n and Y, you can write down a dimensionally correct equation of the form:

Y = Yp f[X_1/Xp_1, X_2/Xp_2, X_3/Xp_3,…,X_n/Xp_N]

where f(x1,x2,x3,…,x_n) is any arbitrary dimensionless function of n variables, and Yp, and the Xp_j are the Planck units for Y and the X_j, respectively. So, this means that you can always make any arbitrary relationship between physical variables dimensionally correct. But this then introduces constants hbar, G and c.

So, what makes dimensional analysis work are constraints such as e.g. demanding that hbar and c do not appear in a relation. Suppose then that we work in natural units where hbar = c = G = 1. Then we’re free to put back hbar and c in the way they would appear in SI units as they are equal to 1 anyway. We can then consider these variables to be dimensionless scaling constants and consider the limit of hbar to zero and c to infinity. We then demand that in that limit the relation between the variables yields a limiting relation between the variables. This is then formally identical to dimensional analysis where you would demand that hbar and c do not appear.

But if we start out with natural variables where everything is dimensionless, we have complete freedom in how we are going to rescale the variables and therefore which scaling limit we want to consider. There is then a particular choice that corresponds to a classical limit of restoring hbar and c and sending them to zero and infinity, respectively, but there are other choices that can be useful which can be incompatible with conventional dimensional analysis.

An example of such an alternative scaling argument is to derive the formula for a ground state energy of the hydrogen atom as follows. We can restore c in the fine structure constant and keep that independent from the electric charge, so alpha become de-facto dimensional becoming proportional to 1/c. We then say that restoring c in E = m yields E = m c^2. and to get to a classical limit we must cancel that c^2, so we must multiply this by alpha^2.

Yes, this is the way professionals often make these arguments. But it isn’t easy for non-experts to follow.

I’ve learned quite a bit over the past week thinking about the justification for dimensional analysis, which initially looked obvious but turned out to be more profoundly subtle. However, I believe it’s based upon the basic principle that physics shouldn’t depend upon how we label physical quantities; and that assigning numbers to physical quantities involves their adding and scaling within functions, which leads me to this question: is the justification for dimensional analysis a particular case of general covariance?

I don’t think that’s quite right, though I should give it a bit more thought when less sleepy. General covariance says that how we put coordinates all across space-time doesn’t matter. Here we say something both more and less general: we consider units of all sorts (not just lengths and times) but we don’t allow general space-time-dependent choices. It’s true that both are examples of focusing on the question of distinguishing physical processes from description of physical processes.

I suggest you think about what it means to change from one coordinate to another. Any equation can be written in a dimensionless form, i.e. # = f(quantities), and then requiring that f be labelling-independent is dimensional analysis; in this language, relabeling is a symmetry of physics, and any measurable physical quantity such as f should be invariant under that symmetry. This is related to the previous commenter’s remark.

Note, however, that the argument that # should be of order 1 is logically independent of this, and deserves a separate discussion.