Drake Equation

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See also: Fermi Paradox

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

$N = R^{*} \times f_{p} \times n_{e} \times f_{l} \times f_{i} \times f_{c} \times L$

Factors

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: 0.14 (only Sun-like stars)

f_p: Stars with Planets

The second factor represents the fraction of stars that have a planetary system. 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 (as a fraction of Sun-like stars)
• Pessimistic value: 0.2
• Optimistic value: 0.99

n_e: 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.

Artistic rendition of Kepler-22, a possibly Earth-like planet 600 light years from here.

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. For example, if all of Irwin's and Schulze-Makuch's planetary models were allowed for, we'd need to count along with Earth a dry planet with possible pockets of water mixed with peroxide (Mars), an extreme-greenhouse planet with sulfuric acid clouds (Venus), four gas giants with ammonia and methane clouds (Jupiter, Saturn, Uranus and Neptune), two moons with a vast water ocean covered in ice (Europa and Enceladus), a sulfur-rich moon with strong volcanic activity (Io), a moon with a thick atmosphere and liquid hydrocarbons (Titan) and three worlds with likely pockets of liquid nitrogen (Triton, Pluto and Charon), for a total of fourteen hypothetical abodes for life.

• 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)

f_l: 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.

This value might rise dramatically if we take into account the possibility on panspermia, a phenomenon in which life is flung in space by a meteorite impact and travels between planets and even solar systems aboard asteroids. If this were a common occurrence, life just has to start once in a galaxy before spreading to many worlds, light years away, "fertilizing" all the potentially hospitable planets on its way.

• 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

f_i: 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

f_c: Communication

Also see: SETI, Fermi Paradox, Interstellar Communication

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: 1,000,000,000?

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 $L = 10000$, that gives 1000 active civilizations in every given year.

	Moderate estimate	Drake's estimate	Sagan's estimate	Cosmos estimate (lower)	Cosmos estimate (upper)	Pessimistic estimate	Optimistic estimate
$R^*$	7	1	40	0.14	0.14	7	7
$f_{p}$	0.8	0.5	0.25	0.7	0.7	0.2	0.99
$n_e$	0.1	2	2	1	2	0.0001	0.2
$f_l$	0.1	1	0.5	0.13	0.5	0.001	1
$f_i$	0.1	1	0.1	0.01	0.025	0.001	1
$f_c$	0.1	0.1	0.1	0.001	0.01	0.001	1
$N^*$	0.00056	0.1	0.1	10-7	0.0000245	10-13	1.4

The average distance between civilizations can be calculated as $d = \sqrt[3]{(V \div N)}$, 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 $N^* = 0.1$ and $L = 1000000$, $N = 100000$ and therefore $d = \sqrt[3]{(V \div N)} = \sqrt[3]{5 \times 10^9} = 1700$; there are 1700 light years between each civilization; with an extremely optimistic estimate ($N^* = 10$ and $L = 1000000000$) the distance is roughly 37 light years (assuming, of course, a random distribution in all the galaxy).

Average distance between civilizations

Number	Distance	Number	Distance
3	55 000 l.y.	1 million	794 l.y.
10	36 800 l.y.	3 millions	550 l.y.
30	25 500 l.y.	10 millions	368 l.y.
100	17 100 l.y.	30 millions	255 l.y.
300	11 900 l.y.	100 millions	171 l.y.
1000	7940 l.y.	300 millions	119 l.y.
3000	5500 l.y.	1 billion	79 l.y.
10 000	3680 l.y.	3 billions	55 l.y.
30 000	2550 l.y.	10 billions	37 l.y.
100 000	1710 l.y.	30 billions	26 l.y.
300 000	1190 l.y.	100 billions	17 l.y.

Variations and Alternative Uses

A common variation of the original Drake Equation, also used in Xenology, replaces the first factor with $N_g$ (the total number of solar systems suitable for life in the Milky Way, perhaps ten billion) and the last with $f_L$ (the fraction of the total lifespan of a star in which the civilization exists, 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 develop (and employ) interstellar travel, and then they establish on 1000 star systems as an average; $N$ becomes 100 times greater. Thus, $f_m = 100$.

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:

Rank	Energy consumption	Average stars controlled	Actual stars controlled (?)
≤ 2.0	≤ 1026 W	(1)	(1)
2.1	1027 W	5	14
2.2	1028 W	50	180
2.3	1029 W	500	2500
2.4	1030 W	5000	33 000
2.5	1031 W	50 000	450 000
2.6	1032 W	500 000	6.0 million
2.7	1033 W	5 millions	81 millions
2.8	1034 W	50 million	1.1 billion
2.9	1035 W	500 million	15 billion
3.0	1036 W	5 billion	200 billion

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 billion.

The BBC version linked under References also includes $n_r$, the number of times civilization develops on a particular planet. This allows for the possibility of a technological civilization arising again from the destruction of the previous one, whether it's due to the latest sapient species rebuilding or an entirely new one evolving. Civilizations that last for a brief time would also have more chances of arising again (unless their end is something utterly destructive on a planetary scale). This factor can be included by calculating $L$ as the total lifetime of all technological civilizations on a planet.

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 $R^* \times f_p \times n_e \times f_l \times f_i = 0.056$) and let's interpret $L$ as the lifespan of all the biosphere (perhaps five billions years); in this case, we get 280 million life-bearing systems in the Milky Way, with an average distance of 120 light years.

References

• Beyond the Drake Equation, an exploration of further implications and details of the equation
• A Review of the Drake Equation
• [1] [2] Two visual calculators of the Drake Equation