Topics

- Make our first models
- To learn how to get rough estimates based on common sense and knowledge
- Gather intuition for the value and limitations of crude estimates

Further reading

Consider a spherical cow: a course in environmental modelling by John Harte

The Art of Approximation in Science and Engineering course notes by Sanjoy Mahajan at MIT.

A collection of Fermi modelling problems posted by the University of Maryland physics education group.

NASA's pages on quantitative thinking

XKCD author Randall Munroe's book What if? is largely a collection of expositions on interesting Fermi models.

Let us start with a problem we will return to repeatedly in the coming chapters -- measuring the earth. We know the earth is big, but how big? 10? 100? A million?

First, we need to agree on how we're measuring "big". Is this big as in what is the volume of the earth? Or what's the mass? Total mass can be useful for astrophysical problems, but from the perspective of our daily lives, volume and mass are not so useful. More useful are distances and areas. Suppose we assume the earth is a perfect sphere. Then, if we can estimate a basic measure like the radius of the earth, then the area, volume, and mass can all be calculated using standard formulas.

To get an estimate of the radius of the earth, we can leverage our knowledge about modern air-travel. A jumbo jet can now fly us between any parts of the world -- journeys that used to take months now take hours. Jets try to fly efficiently, along the shortest paths between any two points, so the longest flights will be no more than 1/2 the circumference of the earth. Well, the longest flights I've taken are 12 hours long, and it seems certain one could reach anyplace on earth within 24 hours of flight time -- a quick classroom survey (wisdom from a crowd) reveals flights around 20 hours long; multiply that by the speed of the airplane, and that will be an estimate of distance. How fast does a commercial jet fly? Well, jets are much faster than cars, so probably more than 100 miles an hour. But commercial jets also fly at speeds below the sound-barrier -- around 350 meters per second or 800 miles an hour. Jumbo jets are probably closer to this high end than the low end, so let's just go with 800mph for the plane's speed.

\[\begin{align} \text{circumference} &\approx 2 \times \text{distance of the longest commercial flight}, \\ 2 \pi r_{\text{earth}} &\approx 2 \times \text{duration} \times \text{speed}, \\ 2 \times 3 \times r_{\text{earth}} &\approx 2 \times 20\, \text{hours} \times 800\, \text{mph}, \\ r _ {\text{earth}} &\approx 5,300\, \text{miles}. \end{align}\]Thus, we are guessing that the radius of the earth is about 5,300 miles (8.5 megameters) with order-of-magnitude confidence intervals between 500 and 50,000 miles, based on algebra, geometry, and a few pieces of knowledge. More expensive measurements of the radius of the earth show it is about 4,000 miles (6.36 megameters), so our estimate is only off by 30% of the true value. Note that we have assumed \(\pi \approx 3\) in these calculations. Our estimates are all very crude, so there's no benefit to using greater precision.

What do you know about the world? At first pass, we might say quite a lot -- we've all been living here for years and observing new and old things over that time. It is certain that we are able to function in day-to-day life much better at age 20 than we could when we were only a year old because of all the things we've learned.

But suppose I rephrase my question. What things in this world can you describe using numbers? At first, we can quickly list a few things that we know by numbers -- our age, the length of a day, month and year, ... But after a bit, most of us start to have difficulty -- an elephant is bigger than a dog, but how much bigger? Philadelphia is a bigger city than State College Pennsylvania, but how many people actually live there? When it snows a foot in winter, how much water actually makes up that snow? How much rain falls in our home towns each year? How much land would you need to feed yourself for a month? Unless we are trained to, we seldom think of the world in terms of such specific numbers. But a lot of questions and arguments can be settled quickly with some basic numbers, appropriately applied.

Approximations based on common knowledge and rules of thumb are often called "Fermi models" in honor of Italian-born physicist Enrico Fermi -- the scientist who built the first nuclear reactor in 1942. Fermi was particularly good at making ballpark estimates of things based on surprisingly little information. In one widely-told story, Fermi attended the test of the first atomic bomb during World War II, after which he pronounced that he believed the explosion released energy equivalent to 10 kilotons of TNT. The actual value was 20 kilotons. That may not seem very accurate until you consider that nobody had ever seen a nuclear explosion before, and the uncertainty in the explosion size ranged from nothing bigger than the box of dynamite used to start the bomb (a.k.a a "fizzle") to enough gigatons to ignite the entire atmosphere and destroy the world.

The basic philosophy of Fermi models is to make crude ballpark estimates of the quantities needed based on information readily at hand. A single significant digit of precision is good-enough. There should be no need to resort to reference works, libraries, or network searches. The mathematics should involve nothing more than highschool algebra. Even such crude estimates can be very useful in distinguishing the important and the likely from the negligible and rare. Thinking in this way might inspired one to consider Sherlock Holmes's limited but practical view of knowledge:

... a person's brain originally is like a little empty attic, and you have to stock it with such furniture as you choose. ... the skillful workman is very careful indeed as to what they take into their brain-attic. -- paraphrased from

A Study in Scarletby A. C. Doyle

Here are two instructive examples.

One of the problems Fermi used to challenge his students was estimating how many piano tuners lived in Chicago, a city of about 3.5 million people in 1950. This seems un-answerable at first glance -- how can we count all the piano tuners? Unless you own a piano, you've probably never even met a piano tuner! Counting all of them in Chicago would definitely be a close kin of counting needles in a haystack.

Never-the-less, there is a sensible way to proceed. We can apply one of the simple principles of economics: supply matches demand. In Chicago, the supply of piano tuner hours each year should equal the demand for piano tuner hours each year. On the supply side, if \(T\) is the number of piano tuners and each tuner works a \(w=40\) hour week for \(y=50\) weeks a year, then there is a supply of \(T \times w \times y\) hours of piano tuning per year. The demand side is a little trickier. It probably takes about \(h = 3\) hours, to tune a piano, including travel, and a good piano is tuned probably once a year. But how many pianos are there to tune each year? Well, how many pianos do WE know about in typical households. We know most households don't (and probably didn't) have pianos. But most of us probably know a household that does have a piano (my grandmother had one), and they were probably more common back in 1950. So perhaps \(f _ h = 1/20\) of households has a piano. If the average household had \(1/f _ p = 4\) people, and there were \(N = 3.5\) million people in Chicago, then Chicago needed \(h \times f_h \times f_p \times N\) hours of piano tuning each year. By setting supply equal to demand, \[T \times w \times y = N \times f _ h \times f _ p \times h.\] Solving for the number of tuners, \[T = \frac{N \times f _ h \times f _ p \times h}{w \times y} = \frac{3.5 \times 10^6 \times 0.25 \times 0.05 \times 3}{40 \times 50} \approx 65\] Thus, we estimate that there were around 65 piano tuners in Chicago in 1950. This number, of courses, is imprecise, but we can be relatively confident there were more than 6 tuners, and there were probably less than 600 tuners.

Our final example of a Fermi model is the infamous Drake equation. It is an excellent illustration of both the power and the limitation of Fermi models. The Drake equation was proposed in 1961 by Frank Drake as a minimal model for predicting the number of civilizations in our galaxy whose radio signals might be detected using radio astronomy. When \(N\) is the number of civilizations communicating by radio, \(R_*\) is the rate of star formation, \(f_p\) is the fraction of stars with planets, \(n_e\) is the number of habitable planets per planeted star, \(f_l\) is the fraction of habitable planets with life, \(f_i\) is the fraction of planets with life where intelligent life takes root, \(f_c\) is the fraction of planets populated by intelligent life where radio signals are released, and \(L\) is length of the window of time a civilization transmits, the equation reads \[N = R_* f_p n_e f_l f_i f_c L.\]

Although a little long, the Drake equation is simple enough to be understood. Just find estimates of these 7 numbers and multiply them together. Yet the physical, biological, and societal processes it summarizes -- coalescence of planets from dust clouds, spontaneous emergence of life, invention of radio ... -- are complicated and chaotic. How can such a simple equation capture all of them at once?

The conceptual justification of the Drake equation is summarized in the pictures below. There are many different paths history can take from star formation to a planetary civilization communicating by radio signal. There are also many histories that do not lead to stable detectable civilizations. Estimating the number of such civilizations requires approximating the "path integral" over the histories -- for each history path, estimate the chance of that path and transmission window \(L\) occurring and then sum up all the paths.

While intuitively simple, this can be very hard to do because there are so many different paths possible. But the path integral can be approximated by identifying a sequence of check-points that all histories have to pass through on their way to a detectable civilization. Then, the number of civilizations is approximated by estimating the path integrals of each segment between successive pairs of successive checkpoints and multiply all of these together.

So, to estimate the number of detectable civilizations in our galaxy, we need only estimate these seven parameters. Drake suggested numbers like \(R_* \approx 1\), \(f_p \approx 0.35\), \(n_e \approx 2.5\), \(f_l \approx 1\), \(f_i \approx 1\), \(f_c \approx 0.15\), and \(L > 1,000\). Together these imply \(N > 130\) detectable civilizations in our galaxy currently.

The infamy of the Drake equation comes in part from the observation that while there are now data we can use to estimate the rate of formation of habitable worlds in our galaxy (encompassing the first three parameters), the last four parameters \(f _ l\), \(f _ i\), \(f _ c\), and \(L\) all must be based on one biased data point -- our own planet. We have not yet found any other life in our galaxy. Drake proposed that life was certain to evolve on all habitable planets. But is that reasonable? If water is a prerequisite for the creation of life, and many of these worlds lack water, \(f _ l\) may actually deserve to be a small fraction, not unity. The parameters \(f _ i\), \(f _ c\), and \(L\) involve similar uncertainties, and thus their product can span practically the whole range of possibilities from a single civilization right now to millions of civilizations.

The other issue with the Drake equation is its limited scope -- it assumes that the appearance of civilizations requires exactly these seven steps. But other stages and events might be necessary and are not accounted for. For example, how important is it that the earth is a tilted planet rather than an upright planet. Without a tilt, a planet would not have slow seasonality, and without slow seasonality, there would not be evolutionary selection for long-term planning and thus intelligence.

We can dive into the details of the Drake equation ever more deeply. There are lots more criticisms one can heap on the Drake equation, and lots of ideas for improving it. In the Drake equation's defense, it was always intended to be a conversation starter, and that it does very well.

In spite of their simplicity, Fermi models illustrate a principle characteristic of the modelling process -- the creative re-organization of knowledge to answer a specific question. In estimating the size of the earth, we combined geometric knowledge about a sphere, elementary physics of sound, and observations of aircraft travel. To determine the number of piano tuners, we combined the economic theory of supply-and-demand with common observation and knowledge. In building the Drake equation, we assembled ideas from astronomy, biology, history, and philosophy. The models are fundamentally mathematical, but the math itself provides the most modest of scaffoldings for the creative process. The models are useful not just for their estimates but also for giving a starting point to organize the important ingredients going into an answer. As our studies progress, we will learn to use more sophisticated math and richer data sets, but our emphasis will remain on the creation rather than the mathematical nuances of solution.

And as we gain more experience, we'll see that there are many reasons why you might want to build a model.

Models help us reorganize information to more easily retrieve more useful answers.

Models can help us guess which things we will need to measure to answer our questions.

Models can help us explore alternatives.

Models confront our imaginations with new possibilities.

Models create new language and metaphors to compress complexity and facilitate communication.

Models can turn vague uncertainties into precise questions.

Models predict.

Models can help us plan.

Models provide guides for optimizing.

Each reason will bring its own viewpoint and implies a different mathematical framework.

Answer each of these questions with 80% confidence bounds our your error.

- Thomas Edison started out life as a telegraph operator. How long would it take Edison to transmit the text of an 1865 front page of the New York Times to San Francisco by Morse code?
- How tall is the elm tree outside McAllister Hall?
- How many grooves are there in a LP record?
- How much does a baby elephant weight?
- How many icecream parlors are there in the USA?
- How much does it cost to charge 2 AA batteries?
- How many gallons of gas does it take to drive from Pittsburgh to Philadelphia?
- What is worth more, 10 pounds of gold or 10 pounds of $100 bills?
- How much would a 18-wheel trailer worth of styrofoam peanuts weight?
- Could you carpet all of the united states with a mole of mole pelts?
- How for is the moon from the earth?
- How much does an aircraft carrier weight?
- What fraction of the incident sunlight does a window screen block out?
- It has been estimated that as of 2018, sea level is rising 3 millimeters each year. One possible explanation for this is erosion -- sediment washed off of islands and continents fills up the ocean basis and raises sea level like dropping a rock into a class of water. Assuming erosion occurs equally across all land masses, how much would the height of our land be decreasing by each year to create the observed sea level rise?
- (James Jean's puzzle) What's the probability that the next breath you take will contain the same air that was exhaled in the last breath of Nicoli Copernicus?
- At the beginning of the movie "Raiders of the Lost Ark", Indians Jones eyes a golden idol on a pedestal. He pulls out a bag of sand, removes a handful, and them replaces the idol with the bag of sand. We can safely presume Indiana wanted to bag of sand to weight the same as the idol, but he guesses wrong. How close was he?
- In
*Return of the Jedi*, when Lando Calrissian is piloting the Millenium Falcon through the superstructure of the death star to get to its reactor core, how fast is the ship flying? (see this clip) - Make your own Fermi model to show some scene in an adventure movie is actually unrealistic.
- Assuming the fence below is made of steel, how much would it cost to build it across the length of the US-Mexico border?