Notes from Under Sky

The Drake Equation

Some thoughts on what it really says

The idea of life in outer space is still recent enough for it to seem modern to me. So it's with some surprise that I find that the Drake Equation is moving on toward its 50th birthday. Why, it's even older than I am.

The Drake Equation is a formula, put together by Cornell astronomer and astrophysicist Frank Drake (1930-), as a way to focus discussion on the prevalence of extraterrestrial life. It has appeared in a number of different forms, but it was originally written in 1961 as follows:

N = R* × fp × ne × fl × fi × fc × L

What I'd like to do now is say a little about what each of those terms means, and what the entire equation says to me as a whole. To begin with, R* is the average rate of star formation in the Milky Way. Drake and his colleagues estimated this rate at 10 per year, a value supported by considerable evidence. Even after 50 years, the estimate of this value has not changed much: The latest figures from NASA and ESA (European Space Agency) suggest a value of 7 new stars per year. So Drake made a pretty good guess.

Next, fp is the fraction of stars that have planets. Drake estimated this at 1/2, well before any extrasolar planets were even discovered. This turned out to be an accurate guess: Based on the current findings, at least 30 percent of Sun-like stars have planets about them. Since our current techniques are not sensitive enough to detect small planets (Earth-sized or smaller), especially in longer orbits, this figure is probably larger.

The remaining factors are much less reliably known. Drake estimated the mean number of potentially life-supporting planets per viable star, ne, as 2. Estimating ne is difficult because the kind of planets that we know from experience is capable of supporting life are Earth-sized or smaller, and again, our techniques are not sufficiently sensitive to detect these. At this point, we know of only one planet—the Earth itself—out of a total of about 200 planetary systems. However, it might be that other kinds of planets are also capable of supporting life, kinds that we're better able to detect.

Perhaps the most startling estimate made by Drake was that of the fraction of potentially life-supporting planets that actually go on to develop life. Drake supposed that this probability was 1: all such planets go on to develop life. He may have based this in part on the relative speed with which life developed on the Earth. Using a somewhat more sophisticated statistical argument, an Australian team arrived at a minimum value of fl > 0.13.

Personally, I'm not sure I buy this estimate (especially to two significant digits!), even as a minimum. Such statistical arguments invariably require some assumption about the a priori probabilities of life developing. Intuitively, I feel that the speed with which life developed is a powerful argument, but it seems almost impossible to quantify that without knowing more about what processes were present at the start that led to life.

Although Drake felt that life itself was almost inevitable, whenever possible, he apparently felt that the development of intelligence was not. Again, he may have been swayed by how long it took intelligence to develop on the Earth: Life developed only a few hundred million years (at most) after the Earth cooled down enough to support it, yet it took another three billion years for recognizably advanced organisms to take hold, and another half a billion years after that for us to evolve.

This delay is all the more poignant if you consider that recent studies of stellar evolution suggest that the Earth will be able to sustain liquid water (a prime prerequisite for life as we know it) for about a billion years more. In other words, if intelligence had been just 25 percent slower in developing, it would never have gotten off the ground.

Besides, intelligence is not an obvious evolutionary advantage, considering that (at least in our case) it's being carried around in a relatively fragile case. It's a common misconception that evolution tends toward advancement of species; rather, evolution tends toward reproductive success. On the Earth, at least, animal species have two basic strategies toward ensuring reproductive success: fecundity and robustness, and it should come as no surprise that fecundity has it, by a wide margin. Even fish persist primarily through fecundity. It's only when you get to larger animals, where having a litter of hundreds or even dozens becomes impractical, that robustness begins to have the upper hand.

And if you were trying to engineer something robust (as opposed to having it just happen), where on the list of features would intelligence go? First? Second? Fifth? Personally, I would think that most engineers would conclude that an intelligent animal would just go and get itself into trouble more often than not (perhaps a perceptive conclusion). It might be better to have the animal be a bit more predictable in its behavior, and build in robustness with a tougher hide, faster healing, more menacing defense mechanisms, and so on.

At any rate, Drake estimated that fi = 0.01: Only one percent of all life-sustaining planets go on to develop intelligence.

Drake was similarly pessimistic about the development of technical communicative abilities, which in our case means radio communication; he also estimated fc = 0.01. Frankly, I think that might be a bit too pessimistic, assuming that by intelligence, we mean something at least as smart as a reptile. It seems to me that although intelligence is an unlikely thing to develop, once it does develop, it's pretty persistent. Intuitively, I feel that it's almost certain that this intelligence would develop to the point where it would be able to destroy itself, and it's hard for me to see how that would happen without the concommittant ability to communicate via radio (or similar electromagnetic means).

Probably the hardest parameter to estimate of all in the Drake Equation is L, the mean lifetime of a communicative civilization. Drake felt that it was about 10,000 years, but that was surely a shot in the dark. More recently, noted skeptic Michael Shermer (1954-) has suggested, based on the survivability of historical civilizations, that a closer estimate would be on the order of hundreds of years. However, few of these civilizations left no intellectual descendants at all (a la the probably mythical Atlantis), so they probably shouldn't be taken in isolation. On the other hand, none of those civilizations had the technical wherewithal to wipe out all human life on the planet, either.

If we multiply out all of Drake's estimates, we get

N = 10 × 0.5 × 2 × 1 × 0.01 × 0.01 × 10,000 = 10

In other words, at any point in time, there are just 10 communicative civilizations in the galaxy. If this were so, and we assumed that these civilizations were distributed randomly around the galaxy, the closest one would be perhaps 10,000 light-years away, so it comes perhaps as little surprise that we haven't heard from one yet, and assuming that they haven't found some way to travel 10,000 light-years, it would be pretty difficult for us to carry on a conversation by radio, given that each line in the dialogue would have a 10,000-year delay on it.

However, one of the problems with the Drake Equation is that the values are so uncertain. Aside from maybe the first two parameters, our estimates are mostly ill-informed, based primarily on our knowledge of the Earth and our societies. If we were to change these estimates even a little, the number of currently communicative civilizations could jump as high as the millions, or it could drop down to a tiny fraction. To a degree, the results of the Drake Equation say more about our own feelings about civilization than about their actual prevalence around the galaxy. What, then, is the point of the Drake Equation? Is there any point at all? Or is it just a mindless exercise?

My own feeling is that there is a point to it, but to get into that, let's consider how the Drake Equation is put together.

Although Drake never put it in such terms, the equation can be stated in terms of a result from queueing theory (the study of waiting in lines, essentially) called Little's Law. That's hardly surprising, since Little actually proved his law after Drake put his equation together. However, Little's Law is intuitively compelling, and it was probably used countless times before it was in fact proved.

Little's Law states that for any stable system (such as the galaxy, perhaps), the mean number of objects in that system equals the rate at which those objects enter the system, times the mean time that they stay in the system. Algebraically,

N = ß × L

Little's Law holds for any system whatsoever, as long as it is stable (neither growing nor shrinking in the long run). So, for example, if 5,000 people enter Disneyland per hour, and they stay at Disneyland for 8 hours on average, then the average number of people in Disneyland, at any time, is 5,000 times 8, or 40,000. It's a remarkable property of Little's Law that it doesn't depend on whether the people enter one at a time, or in big clumps of 100, or whether they all stay for 8 hours, or some stay for 4 and others stay for 10; you still just multiply the average values.

Drake might even have employed Little's Law, subconsciously, to derive R*, the rate of star birth. If a star like the Sun lasts for 12 billion years, and there are (let's say) 120 billion stars in the galaxy, then the rate of star birth must be 120 billion stars, divided by 12 billion years, or 10 stars per year.

What does Little's Law state about galactic civilizations?

Well, if Rc is the rate at which civilizations come into being in the galaxy, and L is the time they spend before perishing, on average, then Little's Law clearly states that the average number of them at any time must be

N = Rc × L

That looks a little like the Drake Equation: After all, there's an L in each formula. However, where we have just a single Rc in this expression of Little's Law, there are lots of parameters in the Drake Equation. How do they all relate?

Consider that there are lots of things that go into making a communicative civilization. First of all, as far as we know, we need a planet, and then that planet needs to develop a communicative civilization. We might write that as the following:

N = Rp × P(communicative civilization develops, given a suitable planet) × L

where P means "probability." We might decide, however, that it's just too hard to estimate the probability of radio communication developing, given a suitable planet; we might want to break that down:

N = Rp × P(life develops, given a suitable planet)
× P
(intelligence develops, given life)
× P
(communicative civilization develops, given intelligence) × L

We could even apply the same kind of reasoning to Rp, relating it to R*:

N = R* × P(planetary system develops, given a star)
× E
(suitable planets, given a planetary system)
× P
(life develops, given a suitable planet)
× P
(intelligence develops, given life)
× P
(communicative civilization develops, given intelligence) × L

where E represents average, or expected, value. And what is this last equation, lo and behold, except the Drake Equation itself?

What this process illustrates is the logical interconnection between each of the different parameters. They are not scattered here and there at random. Rather, each of them is a conditional probability—building on the conditions set up by the previous one. They represent a ladder of steps, all of which must be satisfied if the result is to be a communicative civilization. That, to me, is the value of the Drake Equation: its implicit statement of what specific factors go into the development of such a civilization, not the actual value that comes out in the end, which must be so uncertain, after all.

At the same time, there's something unsatisfying about the Drake Equation. A basic rule concerning probabilities is that when all of a set of things have to happen for the end result to occur, you multiply the probabilities together; when any single one of a mutually exclusive set of things can happen, you add the probabilities.

But there are no additions in the Drake Equation, only multiplications. The very structure of the Drake Equation suggests only a single path toward a communicative civilization. But do we really know that? We are biased by our own history and pre-history. We see only our own development, and cannot conceive of any other way for matters to proceed. Is there any way for us to get out of our own preconceptions?

At some level, of course, the multiplication rule must apply. We know that civilizations must arise, and they must persist for a certain length of time, in order for them to be around to communicate with. That is the message of Little's Law. But how many different paths to civilization there must be, if we look at the parameters deeply enough! Is an industrial revolution necessary, or under different circumstances, under different models of inter-species competition, would a different path be plausible? Such a reconstruction of the Drake Equation would require a much more advanced understanding of society than we currently possess, but it would also yield more significant results. Maybe, as the Drake Equation enters its second half-century, it's time to revisit its terms, and consider in what other lights we might see them.

Copyright (c) 2006, 2008 Brian Tung