What were some historically important Fermi estimates in the history of science?

Question

What were some back-of-the-envelope estimates that scientists made where the result of the estimate was important to either the intellectual development of the field or had significant social consequences?

For example, Helmholtz made some simple estimates to show that chemical energy could not possibly fuel the sun for its known lifespan, then argued that the only remaining energy source was gravitational, and used this to estimate an (incorrect) upper bound for the age of the sun. A detailed model or extensive calculation wasn't necessary; his points could be made just as well with order of magnitude numbers.

This is different from, for example, Fermi's famous estimation of the yield of the first atomic weapon test by dropping slips of paper. While his estimation was impressive, there was (I presume) a team of scientists collecting far more data and crunching through more detailed models to calculate the yield to much greater accuracy than Fermi a short time later, so while his estimation was interesting and amusing it had little real consequence. On the other hand, Taylor's estimate of the Trinity test yield based on photos from a magazine might count because he was able to reveal important classified information with this estimate, and he came to with a very simple method, not needing the more detailed approaches he had been considering.

Answer 1

A challenge is that many scientific projects start with a Fermi estimate to see if an idea is plausible, but if the answer is "yes", much work and analysis is done before the work appears in public where the original Fermi estimate is at best buried in the details.

I will note that Fermi's famous quick determination of the Trinity test yield was more of a "torn up envelope" experiment that a "back of the envelope" estimate. He likely used the best hydrodynamic theory from Hans Bethe to calculate the yield, and he did all the calculations ahead of time so that soon as his bits of paper hit the ground he could use his pre-calculated table of values to give an immediate estimate of the yield. (See this answer over on Physics Stack Exchange.) Taylor's estimate is similar in that he had worked out the relevant physics years earlier. (See "G.I. Taylor and the Trinity test".) Both Taylor and Fermi's genius was in recognizing how serious theory could be applied to simple experimental data - fluttering paper or a series of photos - to extract the yield. Maybe we should have a separate category for "Fermi experiments", where the theory is complex but the measurements simple.

Getting back to the question, here are some ideas, although none of them may be as clear-cut as you'd like.

Weren't geologists' and biologists' crude estimates for lower bounds on the the age of the earth in the 19th century essentially Fermi estimates that created a famous tension with Kelvin's estimate for the age of the Earth that drove research in all three fields. And wasn't even Kelvin's initial value essentially a Fermi estimate based on a crude gravitational collapse model?

The original Drake equation and the Fermi Paradox are both Fermi estimates that are foundational for SETI research.

I would argue that John Ioannidis original paper on "Why Most Published Research Findings Are False" was also a dressed-up Fermi Estimate that helped trigger the Replication Crisis in science.

Although it isn't a specific Fermi Estimate, dimensional analysis has been foundational in fluid dynamics. Faced with intractable Navier–Stokes equations, scientists focused on identifying dimensionless ratios that characterized systems, e.g. the Reynolds Number, Rayleigh Number, Nusselt Number, Grashof number, ….

The Cosmological Constant Problem is another case where the problem is obvious from simple dimensional analysis and the value of Newton's gravitational constant, and this has driven huge theoretical efforts in particle physics.

I think Bohr was using Fermi estimation as he worked towards his Bohr atom. See, for example, his recollection that "I just felt that one knows the order of magnitude of the binding of any electron." or this description that "Bohr, we learn, troubled year after year about the problem of atomic stability, made many an order-of-magnitude estimate of factors likely to be relevant to atomic stability; and reflected deeply about the effect of radiation upon stability - so that ten days after he saw Balmer's formula for the wave lengths in the hydrogen spectrum he had won his way through to the quantum theory of the atom."

Finally, this paper on my "to read" list on "The Value of Imprecise Prediction" may have some relevant examples.