**Becca Tanner**

So far we have seen a whole bunch of problems that can be solved using Fermi Estimation, but what about one of the biggest questions of them all: is there intelligent life somewhere else in the Milky Way galaxy? In 1961, Frank Drake tried to answer exactly that. Although written only to stimulate discussion, rather than to actually quantify the number of extraterrestrial civilizations, the Drake Equation is perhaps one of the most famous Fermi Problems ever written. The equation is as follows:

Where:**N =*** the number of civilizations in our galaxy with which communication may be possible***R _{*} =**

*the average rate of star formation in our galaxy*

**f**

_{p}=*the fraction of those stars which have planets*

**n**

_{e}=*the average number of planets that can potentially support life per star that has planets*

**f**

_{1}=*the fraction of planets that could support life which actually develop life at some point*

**f**

_{i}=*the fraction of planets with life that actually go on to develop intelligent life*

**f**

_{c}=*the fraction of intelligent life which go on to develop communications that could send out detectable signs of their existence*

**L =**

*the length of time for which these detectable signs are sent out into space*

There is a fairly straightforward thought process behind this equation. Essentially, if you take all the planets forming in our galaxy and then systematically estimate the probability, or the fraction, of these planets which have the potential to sustain life, which then do sustain life, and which sustain intelligent life, you arrive at an estimate for the total number of intelligent civilizations. All seems pretty easy right? Well, the problems arise when you look at the value, or rather the range of values, that can be calculated for N.

When estimates were first made for each of the parameters, the difference between the upper and lower bounds spanned multiple orders of magnitude. Therefore, it was of little surprise that the values calculated for N were much the same; ranging from just 20 civilizations, all the way up to 50 million civilizations. Below is a summary of these original estimates:

**R _{*} =**

**1 star per year**

**f**

_{p}=

**0.2-0.5**

*(one fifth to one half of stars formed will have planets)*

**n**

_{e}=

**1-5**

*(stars with planets will have between 1 and 5 planets capable of developing life)*

**f**

_{1}=

**1**

*(100% of these planets will develop life)*

**f**

_{i}=

**1**

*(100% of which will develop intelligent life)*

**f**

_{c}=

**0.1-0.2**

*(10%-20% of which will be able to communicate)*

**L =**

**1,000 – 100,000,000 years**

*(timespan of communication)*

Drake stated that, due to the uncertainties, it was likely that N~L so the true value of N fell somewhere between 1,000 and 100 million (that spans five orders of magnitude!). Unfortunately, that’s not the only problem. A decade before the Drake Equation was written, similarly large estimates for the number of extraterrestrial civilizations were being discussed, and our old friend Enrico Fermi was a part of those discussions. Allegedly, Fermi suddenly exclaimed “But where is everybody?” and the Drake Equation has resulted in the same question being asked ever since.

This rather unavoidable contradiction between the supposedly large estimates for the number of intelligent civilizations, but the complete lack of evidence for any such in existence has been termed the Fermi Paradox.

There are two aspects to the Fermi Paradox: the first is a matter of scale. The Milky Way galaxy is vast with many millions of stars. The Drake Equation predicts that a relatively small percentage of these stars will have planets that are able to sustain intelligent life, but due to the large scale of our galaxy, this should still translate to many thousands or even millions of extraterrestrial civilizations – yet we have no evidence for any. The second aspect is a matter of probability. On Earth, we observe the continuous ability of intelligent life to overcome scarcity and colonize new habitats. So why wouldn’t we expect the same of other intelligent civilizations that may exist around stars much older than our sun and therefore have had much more time to develop the necessary technology to colonize beyond their solar system? We would likely expect the probability of interstellar travel to be high. Yet, despite the universe existing for 13.8 billion years, there is zero evidence for any such galactic colonization.

There are many potential solutions to the Fermi Paradox – a favourite of mine is *The Zoo Hypothesis, *developed by John Ball in 1973. He argued that due to the age of the universe, multiple civilizations, capable of harnessing the energy of an entire galaxy, would have formed by the time Earth came into existence. Much like humans compared to other species on Earth, their advanced intelligence would leave them with the ability to control or destroy less advanced forms of life. This leaves our aliens with an ethical dilemma; intervene, or observe quietly from a distance. If they decided on the latter, Earth is simply a cosmic zoo, completely oblivious to an array of onlookers from a distant corner of the galaxy.

There are many more mind-bending solutions to the Fermi Paradox, some of which are summarised in this article by Kunal Jasty.

A final thought to consider is the reliability of using the Drake Equation when considering questions such as the existence of extraterrestrial life. As we’ve discussed before, Fermi Problems are effective when estimates can be made confidently within one order of magnitude. Of course, it is expected that these estimates will not be exact and will therefore have some variance (or spread) from the true value. This variance will be positive or negative depending on whether it is an overestimate or underestimate. In addition to this, the number of steps within our Fermi Problem will impact the accuracy of our overall answer. The following idea has many comparisons to that of a Random Walk (a link is provided if you wish to read into this concept further).

We start by supposing a problem has n steps, with each step involving an estimate with a variance 𝜎^{2} from the true value. Because the variance of random variables is additive, the overall variance of the Fermi solution from the true answer will be ~n𝜎^{2}, with overestimates and underestimates roughly canceling each other out. Therefore, the standard deviation 𝜎 (square root of variance), increases with the square root of n. This is useful as the standard deviation essentially describes the overall deviation from the true value – the lower the standard deviation, the closer you are to the true value.

Applying this to the Drake Equation, we have seven steps, some of which have a very small variance, and others, such as the final step in estimating L, which have a variance of multiple orders of magnitude. Therefore, the overall standard deviation will be large.

However, if we think back to our example in calculating the mass of the ocean in article 2, we only had four steps (the overall equation being mass = density x depth x surface area of Earth x percentage of ocean coverage). Each of these estimates had a much smaller variance which remained within one order of magnitude, and so the overall standard deviation would be smaller. We know this is the case as our value for the mass of the ocean was well within the same order of magnitude as the accepted values.

So, in summary, whilst Fermi Problems can be both incredibly versatile and useful, they also have limits which we need to be aware of. I hope you’ve enjoyed this short series of articles and will now feel more confident if you ever have to approach an estimation problem in the future. And remember, the next time you find yourself contemplating some seemingly impossible questions, know that you’re probably being watched by an array of extraterrestrial civilizations on a trip to their interstellar zoo…

[…] calculate the number of intelligent extraterrestrial civilizations in the Milky Way galaxy. In the final article of this series we will look further into this infamous equation, before delving deeper into the mathematics behind […]

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[…] Fermi Problems Part 3: Where is everybody? […]

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[…] Fermi Problems Part 3: Where is everybody? […]

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Great article overall – just one small note. Variance is only additive when adding random variables. The majority of Fermi problems (including the examples here) break the problem down into the multiplication of various estimations (treated as random variables so that referring to their variance makes sense). The formula for variance of a product is:

Var(XY) = Var(X)Var(Y) + Var(Y)E(X)^2 + Var(X)E(Y)^2

where X and Y are random variables and E is expectation. This means that a relatively small error multiplied by a very large number can be come a large error very quickly, and in fact, Fermi estimation often strongly depends on accurate initial estimations – estimating the number of US states as 5 instead of 50 will likely result in the final answer being 1/10 of what it should be (a significant error) as opposed to 45 less than what it should be (likely insignificant on the scale of Fermi problems)

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