Drake Equation

Also see: Fermi Paradox

The Drake Equation is a mathematical equation, created in 1961 by the astronomer Frank Drake, used to estimate the number of extraterrestrial civilizations in the Milky Way. In the original form, this number is given as the product of seven factors:

 N = R* × fp × ne× fl × fi × fc × L

R*: star formation
The first factor represents the rate of star formation in the Milky Way. This is the only well-known factor in the equation: about 7 (as an average, seven new stars are formed each year). Though only a fraction of these stars will be similar to the Sun, perhaps as low as 2%, this will be accounted of with the third factor. (Actually, given the time that intelligence takes to develop, we should consider the rate of star formation a few billion years ago - but there shouldn't be much difference).


 * Value used by Drake: 1
 * Modern value: 7
 * Value used by Cosmos [1]: 0.14 (only Sun-like stars)

fp: stars with planets
The second factor represents the fraction of stars that has a planetary system (also see: Habitable solar systems). At Drake's time, some scientists thought that planets were formed by a nebula around the star, and therefore they were probably common, while others thought they were captured by the Sun when passing near other stars, and therefore they should be extremely rare. Today, though it's not as well known as the first, but there are a number of solid estimates: infrared surveys suggest that at least 20% of stars, more likely up to 60%, has rocky planets ; a new study based on gravity lensing implies that virtually every star in the Milky Way has planets, with an average of 1.6 planets per star.
 * Value used by Drake: 0.5
 * Value used by Sagan: 0.25
 * Value used by Cosmos: 0.7
 * Pessimistic value: 0.2
 * Optimistic value: 0.99

ne: Earth-like planets
Also see: Extrasolar planets, Earth Similarity Index

The third factor is the average number of Earth-like planets (or, at least, planets able to support life) that orbit around each planet-bearing star. Drake thought that each star could reasonably have two planets in its habitable zone; in fact, besides the Earth, Mars would have good chance of containing liquid water, if it had a stronger tectonic activity and a thicker atmosphere; even early Venus (before its extreme greenhouse effect), Europa and Titan are celestial bodies that might have (or have had) enough water for life.

According to the Rare Earth Hypothesis, the occurrence of life requires a large number of improbable and independent factors: right distance from the galactic Core, roughly circular orbit around the galaxy, right distance from a metal-rich star of the right class and size, stable orbits that block asteroids, plate tectonics, a large moon, right biochemistry, etc., which would imply that the number of planets suitable for life might be very low even in the whole Milky Way.

However, these requirements are believed by some to be too restrictive: extraterrestrial life might not need oxygen, large moons and external jovian planets might be unnecessary for life and/or common; and anyway, likely Earth-like planets have already been observed, such as Gliese 581 d and Kepler-22b. An estimate from NASA gives Earth-like planets for up to 2.7% of sun-like stars, or about 2 billions of Earth-like planets in all the Milky Way - two of these stars, Gliese 581 and HD 10180 are among the 100 Sun-like stars closer to earth (though closer stars could have still unobserved planets). This suggest that, as an average, there is roughly an Earth-like planet for every hundred planet-bearing star.

The number of planets suitable for life could increase further if the possibility of life on jovian planets were taken into account, and if an array of alternative biochemistries were allowed.
 * Value used by Drake: 2
 * Value used by Sagan: 2
 * Value used by Cosmos: 1-2
 * Pessimistic value: negligible, perhaps 0.0001
 * Optimistic value: 2 (0.2 considering only Sun-like stars)

fl: extraterrestrial life
The fourth factor represents the fraction of Earth-like planets where life actually appears. There are no solid data about this factor, or any of the following; we can only extrapolate from Earth. It's true, as noted by Carl Sagan, that life appears very quickly after the stabilization of early Earth, suggesting that it's statistically inevitable, given favourable conditions; therefore, Sagan and Drake posit that virtually every Earth-like planet will have life. On the other hand, this could be a consequence of the anthropic principle: given the long time required by intelligence (see the next section), the very fact that we are here to observe ourselves forces us to exist on one of the few planets where life arises soon, even if it really is unlikely.
 * Value used by Drake: 1
 * Value used by Sagan: 0.5
 * Value used by Cosmos: 0.13-0.5
 * Pessimistic value: 0.001
 * Optimistic value: 1

fi: extraterrestrial intelligence
Also see: Development of intelligence, Intelligence on Earth, Intelligent aliens

The fifth factor represents the fraction of life-bearing planets in which an intelligent species appears. This value is unknown, too: optimists such as Drake and Sagan argue that intelligence is a valuable evolutionary strategy and that it's bound to appear in every biosphere, provided enough time. On the other hand, of the four billions years since the origin of life only 500 millions (13%) saw species with a central nervous system, and barely 2 millions (0.05%) saw the advanced tool-making of human ancestors. While intelligence seems to be quite common on earth, at least among mammals and birds, it seems to derive from very recent adaptations - and since four fifths of the time in which liquid water can exist on Earth, life-bearing planets where life disappears before developing intelligence might even be the majority.
 * Value used by Drake: 1
 * Value used by Sagan: 0.1
 * Value used by Cosmos: 0.01-0.025
 * Pessimistic value: 0.001
 * Optimistic value: 0.9

fc: communication
Also see: SETI, Fermi Paradox, Interstellar comunication

The sixth factor respresents the fraction of intelligent species that become able to communicate on interstellar scale. One could say that every intelligent species that survives long enough will eventually develop radio communication or higher technology, though it's entirely possible that they're wiped out during their early development, that they don't elaborate science (thus greatly hindering their technological progress) or that they simply decide not to communicate on great distance. After all, radio waves are a highly wasteful method, which could be replaced by more direct forms of communication less easy to intercept. In fact, civilizations that don't specifically choose to communicate might be detectble only for a very thin slice of their lifespan.
 * Value used by Drake: 0.1
 * Value used by Sagan: 0.1
 * Value used by Cosmos: 0.001-0.01
 * Pessimistic value: 0.001
 * Optimistic value: 1

L: longevity of a civilization
The seventh and last factor represents the timespan in which a civilization survives and remains able to communicate on interstellar scale. While we know that the average lifespan of a species is roughly 2-4 millions years, this is hardly significant for a species endowed with complex technology. Drake fixed a value of 10 000 years, while Sagan, worried by the eventuality of nuclear war and resource depletion, feared that technological species are inherently self-destructive, and set L=100.

For another low estimate, Michael Shermer calculated L as the average longevity of 60 human civilizations, and obtained L=420, but it's likely that highly technological civilization would be much more resistant to environmental modifications and dominant enough on their planet not to worry about conquest by other cultures. A different line of thought (es. Peter Ward) argues that such a civilization is virtually immune from most causes of extinction, and sets L as over a million or even a billion years.
 * Value used by Drake: 10 000
 * Value used by Sagan: 100
 * Value used by Cosmos: 10 000-200 000
 * Pessimistic value: 100?
 * Optimistic value: 1000 000 000?

N: number of civilizations
Let's try to give an estimate of the factors in the equation, and compare the result with Drake's original estimates and with modern ones. Since the estimates on the last factor are so wildly different, we can consider only the first six and obtain N*, the rate with which new civilizations appear in the Milky Way: Drake's estimate gives N*=0.1, that is, a new civilization arises (as an average) every ten years. With L=10 000 years, that gives 1000 active civilizations in every given year.

The average distance between civilizations can be calculated as d = 3√(V/N), where V is the volume of the galaxy (roughly 5×1014 cubic light years for the Milky Way) and N is the number of civilizations. For example, given N* = 0.1 civilizations per year and L = 1 million years, N=100 000 and therefore d = 3√(V/N) = 3√5×109 = 1700 light years between each civilization; with an extremely optimistic estimate (N*=10 and L=1 billion years) the distance is roughly 37 light years (assuming, of course, a random distribution in all the galaxy).

Variations and alternative uses
A common variation of the original Drake Equation, also used in Xenology, replaces the first factor with Ng (the total number of solar systems suitable for life in the Milky Way, perhaps ten billions) and the last with fL (the fraction of the total lifespan of a star in which the civilization exist, about 10 billion years for a Sun-like star - thus 0.1 if the civilization lasts a billion years, 0.0001 if a million years, 10-7 if thousand years, and so on). The procedure and the result is the same.

A common modification adds to the equation an eighth factor, which considers the possibility of civilizations expanding in other stellar systems, thus increasing the number of systems that host communicating civilizations (however, since they'd probably expand in the stars closest to them, this wouldn't affect much the average distance between civilizations). For example, let's say that 10% of all communicating civilizations develops (and employs) interstellar travel, and then they establish on 1000 star systems as an average; N becomes 100 times greater.

Cultures under Type 2 would obviously control only one stellar system, though they still could have colonies in others; and it's safe to say that there aren't any above Type 3 in the Milky Way, or the whole galaxy would, by definition, belong to them. Stellar Civilizations seem therefore the only ones that could reasonably expand in the Milky Way. This is the amount of stars they'd need, at the very least, according to their Kardashev rank: ''Note: the final number of "average stars" is much smaller than the number of stars in the whole Milky Way (about 200 billions) because the great majority of the actual stars are much less luminous than the average (roughly 2×1026 W), being red dwarves. The "actual stars" count is based on a logarithmic scale going from 1 to 200 billions.''

If we ignore communicative civilizations, we can simplify the equation to search any civilization, any intelligent species or any life-bearing stellar systems in the galaxy, simply removing factors. For example, let's ignore the last three (with the moderate estimates, we get R* × fp × ne× fl × fi = 0.056) and let's interpret L as the lifespan of all the biosphere (perhaps five billions years); in this case, we get 280 millions of life-bearing systems in the Milky Way, with an average distance of 120 light years.