Intro to Dimensional Analysis: How Much Energy in an Atomic Bomb? Or a Supernova?

Dimensional Analysis is an extremely useful tool to create mathematical models. It is very simple: Decide what variables your function depends on, and then solve a linear system of equations so you can make at least the dimensions or units of match correctly. This will give you the basic shape, and then with some experimental data you can find the values of the constants. Even though Dimensional Analysis is easy, it is extremely powerful!

This approach was used by Sir G.I. Taylor in 1947 to estimate the energy used in the Trinity test of an atomic bomb. He looked at the video and photos to measure radius and time and decide that the radius would depend on time, energy, and density. Using Dimensional Analysis he would compute the energy correctly within 10%. Amazingly, you can use the same formula to make accurate predictions of the explosions of supernova such as the Tycho Supernova that astronomers saw briefly in 1572.

Tycho Supernova image photo credit: NASA/JPL-Caltech/UCLA

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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.

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[oafysgbk]

Cool story bro, but you clearly never actually read Taylor's papers. The actual analysis was chock full of thermodynamics, gas law considerations, differential equations, and numerical integration. Although this is an entertaining story that also happens to be useful for teaching unit analysis, it bears no resemblance to the actual derivation. In fact, Taylor's derivation has no unit analysis at all. My point in making this comment is not to suggest that the unit analysis story is garbage; in fact it's actually astounding that we can use such a simple strategy to guess at a result that actually takes a lot more thought to derive properly. Rather, I am a science educator myself (in a much smaller scale), and I think that as science educators we have a duty not to oversell the "cool" while underselling the hard work.

(Edit: If you've actually read Taylor's papers and you still tell this story of what he did, then you're being very dishonest.)

[RomanNumural]

Fantastic video on the topic! I know this stuff well but I still feel very entertained and interested. I've been harping to my brother about unit analysis for ages now, it's awesome how you can use it for such accurate predictions.

Thank you for sharing. Extremely useful today, when universities are closed due to the coronavirus outbreak. Such high quality videos are of a very high value in distant learning.

[Achill]

I like dimensional analysis (I'm a physicist), but we need to be aware of its limitations.
@5:25 you go from r = f(t, E, rho) to r = t^a * E^b * rho^c without saying so.
That we can estimate r as product of powers of t, E, and rho is a big ASSUMPTION. It's an assumption that often works surprisingly well, but not always. As a physicist, I would put more time into stating that assumption - the rest is rather simple math, done in a minute.

[דודרנדלמן]

Great video and really interesting. Captivating. Thank you.

[FranFerioli]

Congratulations for tackling such a tricky (for me at least) subject. A small thing to add. There is another characteristic for an explosion in air: pressure. But atmospheric pressure is negligible compared to the pressure behind the shockwave so we can ignore this dimensional parameter. Once the explosion weakens and atmospheric pressure is no more negligible, the problem stop being self-similar and the 1/5 formula is not valid anymore.

[AttilaRZA]

This video is exactly what I looking for. I typed: how the hell they estimated the Trinity power via watching the video!!!

[aashsyed]

thanks! also extra thanks for the little bit of physics! appreciate A lot!

[yongmrchen]

This is interesting. There are two questions that are not clear to me. One: it seems that we assume that we know the function f, even not completely, which is a product of the three variables scaled up with some constant. Are other functions available and even better? Two: What if there are variables that are not considered in the function?

[JR-iuyl]

Thank you so much for this vid, really

[douglas]

Why only multiplication? How do we know there is no sum, or others functions like 'e' or sine in the equation?

[ernestboston]

Question: Is this video part of one of your playlists?

[hrobert]

We can use this to analyze the Beirut blast! Would this formula work? E=(p0/t2)(r/c). Sorry couldn’t represent sub- and super scripts clearly. Where p0=1.25 kg*m-3 and c=1.033. We could use the big white building in the video to estimate size by looking up its dimensions and break down the video frame by frame for time.

[bscutajar]

If he had multiple values of r and t, why did he have to do experiments? From multiple values of r and t you can find the best fit for C and E. Theoretically, just two pairs of values of r and t would be sufficient.

[JR-iuyl]

Maybe I missed something but why did he choose that particular value of C and its still unclear to my as to why the actual "C" fits in with the approximated C.

[justinlumpkin]

Wonder why YouTube is recommending this now...