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FERMI PROBLEMS

Simply in passing, and because it was so easy to do, we paused in the last section to mention how to compute the weight of things on the moon from a picture on a television screen. This exercise was an example of a whole category of delightful and (to the unfamiliar) amazing exercises that have come to be known by a special name, in honor of one of their great proponents, Enrico Fermi.

Fermi was uncontestably one of the most important research physicists of this century, and a great many of the working tools of the modern physicist were invented by him. He was also for many years a professor at the University of Chicago, who had a reputation for asking his students outrageous and seemingly impossible questions, and then showing them that they had the necessary knowledge and tools to answer them, which is why this kind of problem came to be named Fermi problems.

The point of Fermi problems is in part to show how much use can be made of commonly available knowledge by the person willing to be resourceful and make approximate simple calculations, but it is more to illustrate the difference between estimation and guessing. The fabled canonical Fermi problem was the question ``how many piano tuners are there in Chicago?''. Faced with such a question without warning in a physics lecture hall, one response would simply be to declare that you don't know and cannot know, and if forced to produce an answer to simply guess at one that is ``plausible''. The primary disadvantage of this guessing is not that it yields imprecise answers, because any question answered with incomplete or not-well measured information necessarily yields imprecise answers. And in fact, even guesses about common experiences are often plausible, precisely because they are constrained by an intuitive touch with reality. One would not even guess that the number of piano tuners in Chicago is something comparable to the number of people living in Chicago, because that would violate the experience that most people one meets are not piano tuners.

The problem with guessing is that one does not know how much confidence to place in the answer, because the constraints from which it follows have not been clearly identified, and so even if estimations have been made at an intuitive level, the degree of imprecision in the estimations has no way to be identified.

To illustrate the difference between an answer based on a guess, but constrained intuitively by the realization that not everyone in a city is a piano tuner, and an answer based on estimation, we sketch the way such an estimation might be constructed. First, one must make some basic assumptions. It is good that they be reasonable, but necessary only that they be made explicitly, so that one can go back and check them if the estimated answers are wildly in error. This is how physicists find mistakes, and it amounts to being aware of the processes of one's thinking. So, for example, one might assume that a piano tuner cannot stay in business unless there are pianos to be tuned, which then limits the number of piano tuners by the number of pianos and the frequency with which they require re-tuning (play on words. Ha.). One can easily estimate, to within a factor of ten and often less, values like the number of people in a city (most people know such numbers for their own cities to accuracies better than ten or fifteen percent, and even for unfamiliar cities one can estimate to within a factor of ten, because the populations of whole contries are usually within less than a thousand times the populations of their largest cities, and one typically knows, to within a factor of ten, how many such cities there are), the number of universities, concert halls and conservatories (places likely to have high concentrations of pianos), and so forth. Then among all the people in a city, some fraction can be expected to have personal pianos, and again such a fraction can be estimated fairly easily to within a factor of ten. (Is it plausible, for example, that All of the families in a city have one piano per family, or that only one tenth of them do, or one one-hundredth, one-thousandth, etc? We would estimate from experience a number between one tenth and one one-hundredth, and estimate the number of families used to produce this number as about one fifth the number of people, given that all people came from somewhere, and the ratio of the number of parents to children is closer to 2/5 (2 parents)/(2+3 total family members) than it is to 4 or to 1/25.)

Proceeding in this way, one can obtain estimates of the number of pianos in a city. Then one must make assumptions about how often they must be tuned. Is it one hundred times a year? That would require that pianos go out of tune in three days, which defies experience with most pianos, though for pianos in concert halls it may be that they require re-tuning every few concerts, to guarantee the quality of the music or even to change the tuning to suit the character of the performance. So from this sector of the piano population one obtains an estimate of how many tunings are required in a year. On the other hand, for the pianos in universities and conservatories, which get a lot of use but are not required to be perfect each time, perhaps twice to ten times a year is as often as they are re-tuned. And finally, many pianos in homes are retuned once every several years or not at all. So from these factor-of-ten estimations and basic multiplication, one obtains a numerical estimate for how many tunings are required in Chicago in a year.

Then one proceeds similarly to estimate how many such tunings are expected to keep a piano tuner in business. One could assume that most piano tuners also do other work, and that a tuning job once a week, or perhaps even less, is sufficient to maintain the vocation. Again, it is not so important that this number be accurate (compared to what, for instance?) as the it be explicit at which point number has been assumed so that it can be checked again if more precision is required. From this final number of tunings per tuner per year, and our previously generated figures of tunings required per city per year, we divide to obtain the estimate of tuners required per city.

This example is in some ways one of the more obscure of Fermi problems, which is probably why it is popular as an anecdote. Not only is the number of piano tuners in Chicago a figure of marginal interest and usefulness to most people most of the time, but it requires assumptions about rather specific human patterns that are often not commonly experienced, such as the number of tuning jobs an average piano tuner requires per year to make it worth his while to stay in business as a tuner. In this latter sense, it is even a misleading example of the kind of Fermi problems that will concern us. And, at the same time as the piano numbers are harder to estimate based on common experience, they are easier to simply go out and compile statistically than those of many of the problems we will consider. Both of these factors stem from the nature of the piano question as a question about the behavior of people. In our work, we will restrict ourselves to questions about the behavior of somewhat less complex aspects of the natural world than people, though they will often be harder to measure directly, which is why Fermi problems are such an important tool in discussing them.

As a more reasonable and very simple example, we could ask, ``what is the mass of the earth?'' That is a figure available in all sorts of books, which means that someone or even many people have, at least approximately, figured it out. But how did they do it? Clearly they did not find a spring scale and place the earth on it, since there is nowhere to place the scale. In this sense, a figure like the mass of the Earth is much more difficult to access directly than the statistics about piano tuners in Chicago. On the other hand, it is far more concrete and easily estimated from common experience.

For example: The earth is made of rock, almost entirely. And with little effort, we can estimate the mass of a cubic meter of rock (what kind does not matter, since they do not differ from each other by even as much as a factor of five, and on average by much less than a factor of two). Of course, one must have some feel for some unit of mass to have any starting place at all, but most people know that a paperclip has a mass of about a gram, and that a cubic centimeter of water has a mass of (by definition, at 4 degrees C) a gram, so we have a relation between mass and volume for water. Then we can see that rocks sink, so we know they have more mass than water per gram, but by picking up a glass of water and a rock of equal size we can estimate whether it weighs about twice as much, or five times, or ten (typically somewhere between five and ten times as much). So, either by knowing or by doing simple tests to find out, we obtain the mass of rock per unit volume, and we need only the volume of the earth to estimate its mass.

But most people have known from elementary math the volume of a sphere in relation to its radius (and if you forget the details for a sphere, you can estimate it from the value of a cube and be off by less than a factor of ten). So we need only know the radius of the earth. But again, this is easily obtained. Most people have driven from one city to another in a different time zone. Clearly on average such cities are not one hundred miles apart (barely more than twice the distance from edge to edge of some cities, like Los Angeles), or ten thousand miles apart (the whole Unites States is only about three thousand miles across), so a useable estimate is about one thousand miles, center-to-center, of time zones. And there are 24 time zones, because they all switch by one hour to follow the progress of the sun over the surface of the earth in each of the 24 hours of a day, so we can estimate that the circumference of the earth is about 24000 miles. And since the circumference of a circle is about 6 times its radius (if you didn't know this, it is not hard to draw a rough circle on a piece of paper and use string to estimate the relation), so the radius of the earth is estimated at about 4000 miles. From which we can then proceed to estimate the volume and hence the mass, and we are done.

It turns out that this estimate is actually quite good, because while our estimate of the distance between time zones was not accurate to better than a factor of two, the round answer of a thousand miles happens to be much closer than that to the correct answer at the equator. Sometimes in this way one gets lucky and comes closer to the correct answer than the precision of the estimates makes it legitimate to expect, just as other times one will be further away, because all of the mistakes add up.

The point is that this number, which Nobody can just go out and ``measure'', can be estimated by almost anyone from simple simple parts of common experience. There is freedom in knowing and practicing this. One no longer views ``facts'', like the mass of the earth written in textbooks, as remote pieces of dogma handed down from untestable authorities. Rather, they are accessible features of the experiences all of us share, and we can test them for plausibility, guard ourselves agains the inevitable mistakes we will encounter, and perhaps most importantly, generate such numbers for ourselves when there are not textbooks handy to do it for us. All of these represent steps away from a passive intellectual dependence on the reputation and authority of others.

The importance of this point is very hard to emphasize adequately. We use such numbers, and with the increasing division and interdependence of our societies, we have come to rely on them, as on the other contributions from physics to our base of knowledge and methodology. We have become this way because the use of such numbers allows us to make our own lives better. Yet the collection of such useful numbers, ``how big'', ``how much'', ``how often'', ``how strong'', etc, available to us in other people's books is far smaller than the set we could generate for ourselves as we need them from everyday experiences. And, unlike the number written in books, the numbers we generate for ourselves do not have to be looked up, which means that we can use ever more of them without being paralyzed by the inability to find them in increasingly complex and bloated libraries.

This last point is one of the potentially greatest short-term solutions to the ``information crisis'' that we are experiencing as a society. We have the ability to generate staggering quantities of useful numbers, and we have the desire to use more of them all the time. Yet, as long as the person who uses the numbers is dependent on someone else to generate them, he is encumberd by the need to find them among the archives. This lookup problem in fact rapidly becomes more difficult than the generation of the information itself, and as we generate more and more useful numbers, we get progressively less use from the simplest of them, which are buried and obscured by a bigger bulk of other data.

While this problem of finding things once we have created them is a fascinating one that merits solutions for its own sake, an increadible fraction of the problem is not necessary at this time, and stems entirely from our own gross and ingrained intellectual dependence, our conviction that we are incapable of generating our own answers as we need them, and that we must rely on someone else to generate them for us. While this may inescapably be true in certain areas of life, for many of the most basic numbers of physics that could make our lives better it is not true, because the same source of our need for the numbers provides a source for their creation, the fact that they describe the experiences with nature that we share.

There is an excellent book on the mathematical aspects of estimation, Innumeracy, by J. A. Paulos, which should probably be considered a required adjunct to this text. Most of the volume of that work deals with probablility and risk, which though fascinating and relevant to living, are slightly different from the main emphasis of the Fermi problems we will develop here. Nonetheless, the approach to the use of numbers and the attitude are identical in both.

In this chapter and all that follow, a large bulk of the problems will be Fermi problems, sometimes dealing with the particular subject material newly introduced in the chapters, and sometimes dealing just with life in general. As the reader does more of these from the text, he should begin to notice that these are exactly the kinds of questions that occur to curious minds throughout the course of normal life. And hopefully, as he begins to appreciate the efficacy with which he can answer them, he will develop the habit of asking them about more aspects of what he sees, because such questions do not need to simply be dismissed as unknowable, but should be parts of the active process of thinking independently.

As teasers and examples, we conclude with some that the reader can answer either in this or the next two chapters: What is the mass of the sun? (easier after two more chapters). As a population, who walks faster, tall people or short people, and by how much relative to their heights? Upon seeing a movie; How could large dinosaurs have been shaped, and how could they have moved? Would they look more like elephants or giraffes in their motions, if they were built in a certain way? How much of your food goes into doing a day's physical work, and how much just to keeping you alive? From science fiction; Could ANYthing at all live at the center of the sun? How about in the emptiest parts of empty space? (Maybe some more `culturally responsible' type questions, if they seem relevant.)



Next: WHY ISN'T DIMENSIONAL Up: DIMENSIONAL ANALYSIS: THE Previous: THE TECHNIQUE OF


desmith@
Thu Aug 31 12:01:42 CDT 1995