December 11, 2002
One Hundred Interesting Mathematical Calculations, Number 2

The Drake equation (also known as the Green Bank equation) is a famous result in the speculative fields of xenobiology and the search for extraterrestrial intelligence.

This equation was devised by Dr. Frank Drake in the 1960s in an attempt to estimate the number of extraterrestrial civilizations in our galaxy with which we might come in contact. It seems to suggest that contact with extraterrestrials should not be a remarkably uncommon event. The Drake equation states that

N = R* x fp x ne x fl x fi x fc x L

where:

N is the number of extraterrestrial civilizations in our galaxy with which we might expect to be able to communicate

and

R* is the rate of star creation in our galaxy
fp is the fraction of those stars which have planets
ne is average number of planets which can potentially support life per star that has planets
fl is the fraction of the above which actually go on to develop life
fi is the fraction of the above which actually go on to develop intelligent life
fc is the fraction of the above which are willing and able to communicate
L is the expected lifetime of such a civilisation

Considerable disagreement on the values of most these parameters exists, but the values used by Drake and his colleagues in 1961 are:

R* = 10/year
fp = 0.5
ne = 2
fl = 1
fi = fc = 0.01
L = 10 years.

The value of R* is the least disputed. fp is more uncertain, but is still much firmer than the values following. Confidence in ne was once higher, but the discovery of numerous gas giants in close orbit with their stars has introduced doubt that life-supporting planets commonly survive the creation of their stellar systems. In addition, most stars in our galaxy are red dwarfs which have little of the ultraviolet radiation that causes the types of mutations needed by evolution. Instead they flare violently, mostly in X-rays - a property not conducive to life as we know it (simulations also suggest that these bursts errode planetary atmospheres). The possibility of life on moons of gas giants such as Europa adds further uncertainty to this figure.

What evidence is currently visible to humanity suggests that fl is very high; life on Earth appears to have begun almost immediately after conditions arrived in which it was possible, suggesting that abiogenesis is relatively "easy" once conditions are right. But this evidence is limited in scope, and so this term remains in considerable dispute. fi, fc, and L are obviously little more than guesses. (Note, however, that in the year 2001 a value of 50 for L can be used with exactly the same degree of confidence that Drake had in using 10 in the year 1961.)

The remarkable thing about the Drake equation is that by plugging in apparently fairly plausible values for each of the parameters above, the resultant expectant value of N is generally often >>†1. This has provided considerable motivation for the SETI movement. However, this conflicts with the currently observed value of N†=†1, namely ourselves. This conflict is often called the Fermi paradox, after Enrico Fermi who first publicised the subject, and suggests that our understanding of what is a "conservative" value for some of the parameters may be overly optimistic or that some other factor is involved to suppress the development of intelligent space-faring life....

Some computations of the Drake equation, given different assumptions:

R* = 10/year, fp = 0.5, ne = 2, fl = 1, fi = fc = 0.01, and L = 50 years
N = 10 x 0.5 x 2 x 1 x 0.01 x 0.01 x 50 = 0.05dd>

Alternatively, making some more optimistic assumptions, and assuming that 10% of civilisations become willing and able to communicate, and then spread through their local star systems for 100,000 years (a very short period in geologic time):

R* = 20/year, fp = 0.1, ne = 0.5, fl = 1, fi = 0.5, fc = 0.1, and L = 100,000 years
N = 20 x 0.1 x 0.5 x 1 x 0.5 x 0.1 x 100000 = 5000

## References:

• Charles H. Lineweaver and Tamara M. Davis, Does the Rapid Appearance of Life on Earth Suggest that Life is Common in the Universe?, arXiv:astro-ph/0205014 v1 2 May 2002
• Michael Shermer, Why ET Hasn't Called, Scientific American, August 2002, page 21

External references:

#### Drake equation

The Drake equation (also known as the Green Bank equation) is a famous result in the speculative fields of xenobiology and the search for extraterrestrial intelligence.

This equation was devised by Dr. Frank Drake in the 1960s in an attempt to estimate the number of extraterrestrial civilizations in our galaxy with which we might come in contact. It seems to suggest that contact with extraterrestrials should not be a remarkably uncommon event.

The Drake equation is closely related to the Fermi paradox (for which, see below).

The Drake equation states that

N = R* × fp × ne × fl × fi × fc × L

where:

N is the number of extraterrestrial civilizations in our galaxy with which we might expect to be able to communicate

and

R* is the rate of star creation in our galaxy
fp is the fraction of those stars which have planets
ne is average number of planets which can potentially support life per star that has planets
fl is the fraction of the above which actually go on to develop life
fi is the fraction of the above which actually go on to develop intelligent life
fc is the fraction of the above which are willing and able to communicate
L is the expected lifetime of such a civilisation

Considerable disagreement on the values of most these parameters exists, but the values used by Drake and his colleagues in 1961 are: R* = 10/year, fp = 0.5, ne = 2, fl = 1, fi = fc = 0.01, and L = 10 years. The value of R* is the least disputed. fp is more uncertain, but is still much firmer than the values following. Confidence in ne was once higher, but the discovery of numerous gas giants in close orbit with their stars has introduced doubt that life-supporting planets commonly survive the creation of their stellar systems. In addition, most stars in our galaxy are red dwarfs which have little of the ultraviolet radiation that causes the types of mutations needed by evolution. Instead they flare violently, mostly in X-rays - a property not conducive to life as we know it (simulations also suggest that these bursts errode planetary atmospheres). The possibility of life on moons of gas giants such as Europa adds further uncertainty to this figure. What evidence is currently visible to humanity suggests that fl is very high; life on Earth appears to have begun almost immediately after conditions arrived in which it was possible, suggesting that abiogenesis is relatively "easy" once conditions are right. But this evidence is limited in scope, and so this term remains in considerable dispute. fi, fc, and L are obviously little more than guesses. (Note, however, that in the year 2001 a value of 50 for L can be used with exactly the same degree of confidence that Drake had in using 10 in the year 1961.)

The remarkable thing about the Drake equation is that by plugging in apparently fairly plausible values for each of the parameters above, the resultant expectant value of N is generally often >> 1. This has provided considerable motivation for the SETI movement. However, this conflicts with the currently observed value of N = 1, namely ourselves. This conflict is often called the Fermi paradox, after Enrico Fermi who first publicised the subject, and suggests that our understanding of what is a "conservative" value for some of the parameters may be overly optimistic or that some other factor is involved to suppress the development of intelligent space-faring life.

Other assumptions give values of N that are << 1, but some observers believe this is compatible with observations due to the anthropic principle; no matter how low the probability that any given galaxy will have intelligent life in it, the galaxy that we are in must have at least one intelligent species by definition. There could be hundreds of galaxies in our galactic cluster with no intelligent life whatsoever, but of course we would not be present in those galaxies to observe this fact.

Others regard the anthropic principle as controversial, and consider the N << 1 case puzzling from the viewpoint of the Copernican principle.

Some computations of the Drake equation, given different assumptions:

R* = 10/year, fp = 0.5, ne = 2, fl = 1, fi = fc = 0.01, and L = 50 years
N = 10 × 0.5 × 2 × 1 × 0.01 × 0.01 × 50 = 0.05

Alternatively, making some more optimistic assumptions, and assuming that 10% of civilisations become willing and able to communicate, and then spread through their local star systems for 100,000 years (a very short period in geologic time):

R* = 20/year, fp = 0.1, ne = 0.5, fl = 1, fi = 0.5, fc = 0.1, and L = 100,000 years
N = 20 × 0.1 × 0.5 × 1 × 0.5 × 0.1 × 100000 = 5000

## Estimates of the Drake equation parameters

This section attempts to list best current estimates for the parameters of the Drake equation. Please list new estimates for these values here, giving the rationale behind the estimate and a citation to their source.

R*, the rate of star creation in our galaxy

Estimated by Drake as 10/year

fp, the fraction of those stars which have planets

Estimated by Drake as 0.5

ne, the average number of planets which can potentially support life per star that has planets

Estimated by Drake as 2

fl, the fraction of the above which actually go on to develop life

Estimated by Drake as 1

In 2002, Charles H. Lineweaver and Tamara M. Davis (at the University of New South Wales and the Australian Centre for Astrobiology) estimated fl as > 0.33 usng a statistical argument based on the length of time life took to evolve on Earth.

fi, the fraction of the above which actually go on to develop intelligent life

Estimated by Drake as 0.01

fc, the fraction of the above which are willing and able to communicate

Estimated by Drake as 0.01

L, the expected lifetime of such a civilisation

Estimated by Drake as 10 years.

A lower bound on L can be estimated from the lifetime of our current civilization from the advent of radio astronomy in 1938 (dated from Grote Reber's parabolic dish radio telescope) to the current date. In 2002, this gives a lower bound on L of 64 years.

In an article in Scientific American, Michael Shermer estimated L as 420 years, based on compiling the durations of sixty historical civilizations. Using twenty-eight civilizations more recent than the Roman Empire he calculates a figure of 304 years for "modern" civilizations. Note, however, that the fall of most of these civilizations did not destroy their technology, and they were succeeded by later civilizations which carried on those technologies, so Shermer's estimates should be regarded as pessimistic.

References:

• Charles H. Lineweaver and Tamara M. Davis, Does the Rapid Appearance of Life on Earth Suggest that Life is Common in the Universe?, arXiv:astro-ph/0205014 v1 2 May 2002
• Michael Shermer, Why ET Hasn't Called, Scientific American, August 2002, page 21

External references:

Posted by DeLong at December 11, 2002 02:34 PM | Trackback

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I'm a multi-year participant in Seti-at-home. I've always wondered why we haven't ALREADY heard from ETs. But now I see it's not such a sure thing as I've expected...expected without taking the time to work through the Drake equation.

Plugging in numbers that *I* would use:

R=10/year,

fp=0.5,

ne=0.1 (I'm very pessimistic here...I think only 1 solar system in 10 that has planets has earthlike planets),

fl=1,

fi=0.001 (I'll assume that only 1 in 1000 develops intelligent life, rather than 1 in 100),

fc=0.01,

L=10000 (I think it's probably even UNDERestimating that a civilization that can communicate will be able to do so for 10,000 years).

So I would get:

N = 10 x 0.5 x 0.1 x 1 x 0.001 x 0.01 x 10,000

N = 0.05

That's only a 1-in-20 chance of intelligent life trying to communicate with us in this galaxy. Hmmmm...interesting. And a bit depressing.

Posted by: Mark Bahner on December 11, 2002 04:00 PM

>> I'm a multi-year participant in Seti-at-home. I've always wondered why we haven't ALREADY heard from ETs ... <<

Posted by: Jim Glass on December 11, 2002 04:16 PM

Gosh, somehow hit the send button too soon.

The Fermi Paradox wasn't really an answer/response to the Drake equation, as Wikipedia indicates, since it preceded it, IIRC.

Fermi figured that the time it would take for a civilization to colonize the whole galaxy would be slight compared to the life of the galaxy, and so wondered "if they are out there why aren't they here?"

Posted by: Jim Glass on December 11, 2002 04:34 PM

I might add that there was an interesting debate between Carl Sagan, who of course championed the idea of intelligent life being out there among those "billions and billions" of stars, and Ernst Mayr, a preeminent evolutionary biologist who took a more skeptical view.

Sagan and astromer types tend to be impressed by the very big numbers of stars and to apply the probabilites that they use all the time in their kind of work to come straight to their conlcusion. The Drake equation being just a number of probabilities compounded.

A lot of evolutionary biologists think that's rather naive, and think instead that it's entirely possible that evolution selects *against* intelligence. In which case the apparent lack of technolocigal life out there, as far as we've been able to see, is not just the result of Drake-type probabilities but of systematic evolutionary force at work.

Mayr and Sagan go around this here:

Of course, they both agree in the end that with only our one first-hand example from which to try to deduce probabilities and the systematic forces at work, and our very limited experience at looking around us, the rigorously best answer to all these questions today is "Who knows?"

Posted by: Jim Glass on December 11, 2002 05:24 PM

The latest stuff I've seen (without remembering where) has suggested that the combination of moderate temperature and presence of water is not as likely or possible as theoretical math might suggest.

Right or wrong, isn't part of the problem possibly related to the distance from earth of the intelligent beings? Light years of distance, and all that.

Posted by: Anarchus on December 11, 2002 05:26 PM

Posted by: Ben Hyde on December 11, 2002 05:42 PM

No no no no no. Anything which talks about us "communicating with" anyone else surely has to have some term reflecting the fact that the speed of communication is bounded by the speed of light; if any such civilisation happened to exist 5000 light years away, it might as well not bother.

Posted by: dsquared on December 11, 2002 10:50 PM

Once a sentient race really gets going, it's easy to show, as Fermi did, that their expansion should proceed at a significant fraction of the speed of light.

The solar system has been around 4.5 billion years; the universe, what, 10? Compared to that, once sentience appears it should colonize the galaxy in a relative eyeblink (a few million years, tops) on those timescales. So the fundamental question of the Fermi paradox isn't "why haven't they contacted us," it's "why didn't they colonize our planet before we ever arose from the muck?" After all, if they've had 10 billion years (half that if you assume generation 2 stars are a requirement), at least *one* should have appeared and overran everything.

The only semi-convincing resolution I've seen to this is that the galaxy is undergoing a transition in frequency of gamma-ray bursts events. Previously, they were frequent enough to wipe out sentience before it really got going; now they're becoming too infrequent to stop the process.

That said, I hope we develop serious electromagnetic shielding before the next burst, lest all sentience be scoured from the galaxy again. I give it 1,000 years until we're ready. :D

Posted by: Jason McCullough on December 12, 2002 12:41 AM

Perhaps it would help to give the result of Fermi's estimate of the time it would take a space-travelling civilization to colonize the galaxy: 100 million years. Fermi had the reputation in physics of being extraordinary skilled at these kinds of estimates (we even call them "Fermi calculations"), which is saying something because all physicists would like to think they are good at this. So, taking the 4-5 billion years of the Milky Way galaxy's existence, we're talking about 40-50 times the amount of time necessary for colonization. Sagan (and many others) resolve the Fermi paradox by arguing that we might be among the first civilizations, that we're somehow still special. Interestingly, this puts Sagan somewhat near Mayr in the "evolution is hard" camp. Others argue that something may be fundamentally wrong with Fermi's calculation, and that not all parts of the galaxy are equally accessible.

Posted by: Ben Vollmayr-Lee on December 12, 2002 06:45 AM

I should clarify, when I say that something may be fundamentally wrong with Fermi's calcluation, this is not at all discrediting Fermi. He most likely believed something was wrong with the assumptions in the calculation also. The interesting question, which the calculation poses, is which assumption is wrong, and why.

Posted by: Ben Vollmayr-Lee on December 12, 2002 12:25 PM

I think I see an answer to Fermi's paradox, and it is somewhat depressing. It just isn't technically possible to colonize another star system. This site, Warp Drive When?, does a pretty good job of explaining the problem, while keeping up some hope.

It explains that even using a hyperefficient fusion rocket of some sort, if you wanted to take a schoolbus sized payload past our nearest neighboring star, you would need a thousand supertankers' volume of propellant, and it would still take 900 years to get there. And that doesn't include the amount of propellant it would take to stop once the schoolbus has arrived. And our nearest star is only a little more than 4 lightyears away. The nearest potentially habitable planet might be much farther away than that.

With the laws of physics as we currently know them (no exotic technologies such as warp drives or wormholes), it does not seem possible to colonize another star system. That's the answer to Fermi's paradox.

Posted by: Mitch on December 12, 2002 02:41 PM

Whenever people raise the Fermi paradox, I always like to mention the most insightful paper I ever read on the subject, Robin Hanson's "The Great Filter."

Posted by: Tom on December 12, 2002 05:20 PM

Actually, I think the F sub i component is the truly hard bit. How can we extrapolate about intelligence given one example? Searle, among others, treats intelligence as an essentially biological process inherent to human anatomy. We don't recognise anything other than people as intelligent, and we hardly could expect aliens to share our biology to any significant degree. The universe could well be full of life without anything that we could ever consider intelligent, because the only meter we have for evaluating intelligence is ourselves. Even if we might evaluate an alien organism's cognitive abilities as comparable to our own, the very alien nature of it might make us incapable of meaningful interaction. Elephants probably posess greater computational power in their bodies than we do, yet we would hardly class them as having a civilisation.

Why would aliens communicate with us? Would they even communicate amongst themselves? Would it occur to them to want to? Why would they colonise the stars? None of this stuff ever occurs to cats, and they're more like us than any alien would be. No assumption we make about ourselves, no matter how trivial or obvious, necessarily holds true of an alien.

This set of ideas is advanced in part by Stanislaw Lem in Solaris (now a film in the US, starring George Clooney, but rent Tartakovsky's 1972 version if you can), where the alien's capacity for cognition appears immense, but its actions are incomprehensible, and no real communication is possible with it.

Mitch, your case against interstellar colonisation puts me in mind of arguments made in the 19th century against heavier than air flight. Using stored antimatter, or some form of high efficiency fusion, or even generating small black holes on demand provides enough energy to reduce fuel to payload ratios for interstellar travel to reasonable levels, and can raise speeds to a large part of the speed of light. While we can't do any of those things now, at least we already know they aren't impossible. No new physics required.

Posted by: Scott Martens on December 12, 2002 10:45 PM

Actually, I think the F sub i component is the truly hard bit. How can we extrapolate about intelligence given one example? Searle, among others, treats intelligence as an essentially biological process inherent to human anatomy. We don't recognise anything other than people as intelligent, and we hardly could expect aliens to share our biology to any significant degree. The universe could well be full of life without anything that we could ever consider intelligent, because the only meter we have for evaluating intelligence is ourselves. Even if we might evaluate an alien organism's cognitive abilities as comparable to our own, the very alien nature of it might make us incapable of meaningful interaction. Elephants probably posess greater computational power in their bodies than we do, yet we would hardly class them as having a civilisation.

Why would aliens communicate with us? Would they even communicate amongst themselves? Would it occur to them to want to? Why would they colonise the stars? None of this stuff ever occurs to cats, and they're more like us than any alien would be. No assumption we make about ourselves, no matter how trivial or obvious, necessarily holds true of an alien.

This set of ideas is advanced in part by Stanislaw Lem in Solaris (now a film in the US, starring George Clooney, but rent Tartakovsky's 1972 version if you can), where the alien's capacity for cognition appears immense, but its actions are incomprehensible, and no real communication is possible with it.

Mitch, your case against interstellar colonisation puts me in mind of arguments made in the 19th century against heavier than air flight. Using stored antimatter, or some form of high efficiency fusion, or even generating small black holes on demand provides enough energy to reduce fuel to payload ratios for interstellar travel to reasonable levels, and can raise speeds to a large part of the speed of light. While we can't do any of those things now, at least we already know they aren't impossible. No new physics required.

Posted by: Scott Martens on December 12, 2002 10:45 PM

Actually, I think the F sub i component is the truly hard bit. How can we extrapolate about intelligence given one example? Searle, among others, treats intelligence as an essentially biological process inherent to human anatomy. We don't recognise anything other than people as intelligent, and we hardly could expect aliens to share our biology to any significant degree. The universe could well be full of life without anything that we could ever consider intelligent, because the only meter we have for evaluating intelligence is ourselves. Even if we might evaluate an alien organism's cognitive abilities as comparable to our own, the very alien nature of it might make us incapable of meaningful interaction. Elephants probably posess greater computational power in their bodies than we do, yet we would hardly class them as having a civilisation.

Why would aliens communicate with us? Would they even communicate amongst themselves? Would it occur to them to want to? Why would they colonise the stars? None of this stuff ever occurs to cats, and they're more like us than any alien would be. No assumption we make about ourselves, no matter how trivial or obvious, necessarily holds true of an alien.

This set of ideas is advanced in part by Stanislaw Lem in Solaris (now a film in the US, starring George Clooney, but rent Tartakovsky's 1972 version if you can), where the alien's capacity for cognition appears immense, but its actions are incomprehensible, and no real communication is possible with it.

Mitch, your case against interstellar colonisation puts me in mind of arguments made in the 19th century against heavier than air flight. Using stored antimatter, or some form of high efficiency fusion, or even generating small black holes on demand provides enough energy to reduce fuel to payload ratios for interstellar travel to reasonable levels, and can raise speeds to a large part of the speed of light. While we can't do any of those things now, at least we already know they aren't impossible. No new physics required.

Posted by: Scott Martens on December 12, 2002 10:46 PM

Actually, I think the F sub i component is the truly hard bit. How can we extrapolate about intelligence given one example? Searle, among others, treats intelligence as an essentially biological process inherent to human anatomy. We don't recognise anything other than people as intelligent, and we hardly could expect aliens to share our biology to any significant degree. The universe could well be full of life without anything that we could ever consider intelligent, because the only meter we have for evaluating intelligence is ourselves. Even if we might evaluate an alien organism's cognitive abilities as comparable to our own, the very alien nature of it might make us incapable of meaningful interaction. Elephants probably posess greater computational power in their bodies than we do, yet we would hardly class them as having a civilisation.

Why would aliens communicate with us? Would they even communicate amongst themselves? Would it occur to them to want to? Why would they colonise the stars? None of this stuff ever occurs to cats, and they're more like us than any alien would be. No assumption we make about ourselves, no matter how trivial or obvious, necessarily holds true of an alien.

This set of ideas is advanced in part by Stanislaw Lem in Solaris (now a film in the US, starring George Clooney, but rent Tartakovsky's 1972 version if you can), where the alien's capacity for cognition appears immense, but its actions are incomprehensible, and no real communication is possible with it.

Mitch, your case against interstellar colonisation puts me in mind of arguments made in the 19th century against heavier than air flight. Using stored antimatter, or some form of high efficiency fusion, or even generating small black holes on demand provides enough energy to reduce fuel to payload ratios for interstellar travel to reasonable levels, and can raise speeds to a large part of the speed of light. While we can't do any of those things now, at least we already know they aren't impossible. No new physics required.

Posted by: Scott Martens on December 12, 2002 10:46 PM

Sorry. My browser farted. I don't know why it does this on Moveable Type blog's sometimes, but I'm not trying to multiply submit.

Posted by: Scott Martens on December 12, 2002 10:48 PM

I wrote: "I'm a multi-year participant in Seti-at-home. I've always wondered why we haven't ALREADY heard from ETs ..."

Jim Glass responded: "The Fermi Paradox."

No, I don't see my puzzlement as necessarily falling within the Fermi Paradox.

My understanding is that Fermi postulated that any intelligent life would be able to colonize an entire galaxy fairly rapidly. I don't necessarily agree with that.

But even if intelligent life CAN'T make it over here, surely they already know we exist. Not necessarily from our electromagnetic signals, which haven't moved out very far...but because they could look at our planet and see things were happening. ("Oh, look! Someone on that planet has discovered fire!" ;-))

So why haven't other intelligent life forms been flashing laser beams, or some such, at us?

I think the answer may be that we're alone in this galaxy. But then I wonder why intelligent life in *other* galaxies haven't been sending signals.

Posted by: Mark Bahner on December 13, 2002 09:26 AM

Hi Brad! The only (ahem) "intelligent" species I know of habitually fights intraspecies wars of such limitless ferocity that it's quite unlikely to survive the next century. It seems like it would only take one interstellar interspecies war to run f sub c all the way down to zero for any species that practices history.

Yours WDK - WKiernan@concentric.net

Posted by: W. Kiernan on December 15, 2002 07:25 PM