# Fermi’s Ghost

31 December 2021

It is extremely difficult to create a scientific model that is accurate and precise. One need only look at the complexity of cosmological models or the mathematical description of black holes to see just how complex things can be. On the other hand, it can be very easy to make a prediction that is “roughly correct.”

Suppose you want to calculate the internal temperature of the Sun. The interior of the Sun is a complex dance of nuclear fusion, photon scattering, gravity, and thermodynamics. There are aspects of the Sun’s core we are still trying to understand. Calculating core temperature is hard. But you can close by doing a simple back of the envelope calculation. Assume the Sun is a ball of ideal gas, and just use the ideal gas law and Newtonian gravity to estimate the pressure and temperature of the core. The math is just simple algebra, and you get a temperature of about 4 million Kelvin. The actual value is about 15 million Kelvin.

These kinds of rough calculations are often known as Fermi problems. They are named after Enrico Fermi, who famously estimated the power of the first nuclear explosion at Trinity by dropping bits of paper and measuring how far the blast moved them. His rough answer was 10 kilotons. The actual value was about twice that.

The real power of Fermi problems is that they provide a reasonable answer to even the most complex problems. A simple calculation tells you that the Sun’s core is millions of Kelvin, not billions or thousands. The Trinity blast was kilotons, not megatons. They are so simple and based on such obvious data, that you can absolutely rely on the answer. And we ignore their results at our peril.

Back in March of 2020, I did a simple Fermi calculation for Covid 19. Given the rate of infection and observed death rate, estimate how many people in the United States will die. The answer I got was a few million. About 1% of the U.S. population. We are currently on track to have more than a million deaths by the Spring of 2022. Obviously, small factors could have shifted that number significantly. A more deadly variant without a vaccine and the number could have been 10 million. Higher vaccination rates and more consistent masking and isolation policy and the number might have been under 100,000.

When used as a prediction, Fermi problems are like the ghostly harbingers of A Christmas Carol:

If these shadows remain unaltered by the future, If these shadows remain unaltered by the Future, the child will die.

The data was clear. We knew that masking and isolation would prevent deaths, and by not implementing them more seriously, the blood of many dead is on our hands. As a society, we have killed them. We failed to act to avoid the obvious outcome the ghost of Fermi told us was to come.

When I was an undergraduate physics student my department had weekly lectures where a guest physicist would come to talk about their research. One particular speaker was a climate scientist. An atmospheric physicist to be exact, and he talked about how the rise of carbon dioxide in the atmosphere led to rising global temperatures. This was around the time when the Senate was holding hearings about the potential impacts of global warming. Scientists such as James Hansen were sounding alarm bells, while others condemned the idea as fear-mongering.

During the talk, one of my fellow students asked the speaker about this. After all, these models are so complex and rely on possibly uncertain data. How can we possibly know for sure that global warming poses an existential threat to humanity? The speaker replied with a Fermi calculation. Global temperature is just a matter of heat captured by the Sun vs heat emitted. Take the carbon dioxide levels, a bit of algebra, and you get an answer. If the CO2 level doubles in a century, then global temperature rises by about 3°C. That pretty much agrees with what we’ve observed over the past several decades. A clear and roughly accurate prediction from a simple calculation any physics undergraduate can do.