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The Drake Equation

Jan 14, 2022


The Drake Equation was developed by Frank Drake in 1961 as a way to focus on the factors which determine how many intelligent, communicating civilizations there are in our galaxy. It was originally written simply as a means to stimulate discussion in a meeting on radio astronomy, but has since been seen as a useful tool for estimating our likelihood of contacting intelliget alien civilisations. The simplest form of the Drake Equation to understand is as follows: The number of communicating civilisations in our galaxy, N, is:

`$$ N = S \times f_{p} \times n_{e} \times f_{l} \times f_{i} \times f_{c} \times f_{L} $$`

In this equation:

  1. \(S\) represents the number of stars in the Milky Way Galaxy (approx 100 billion, or 109). The fact that the Drake Equation and the Search for Extra Terrestrial Intelligence (SETI) is restricted to the Milky Way galaxy gives an opportunity to teach the scale of space and the speed of light, and thus the reason that no-one is looking outside of our galaxy for intelligent alien life.

  2. \(f_{p}\) is the fraction of stars that have planets around them (estimates range from 20% to 50%).

  3. \(n_{e}\) is the number of planets per star system that are capable of sustaining life (estimates range from 1 to 5).

  4. \(f_{l}\) is the fraction of habitable planets where life actually evolves (here estimates range wildly, from 100%, i.e. if life can evolve then it will, down to almost 0%, i.e. we are very rare in the universe).

  5. \(f_{i}\) is the fraction of inhabited planets where intelligent life evolves (here estimates range from 100%, because intelligence gives such a good survival advantage that it surely must evolve, down to nearly 0%, because in all the billions of years of life and billions of species that have lived on Earth, only one intelligent species has arisen).

  6. \(fc\) is the fraction of the intelligent civilisations which have discovered the means to communicate (e.g. by radio waves) and which actively do so (estimates here are low, 10-20%, and I’m not sure what their basis is!). This includes civilisations which are actively sending out signals, in the hope that others might detect them, and also ones which are using radio waves for their own internal communications, such that we might “eavesdrop” on radio waves which have reached us. (This was the premise for the Carl Sagan novel, and later film, Contact, in which aliens detect early 20th century television transmissions, and so decide to make contact with us.)

  7. \(f_{L}\) is fraction of their planet’s life during which the communicating civilizations carry on communicating.

Historical estimates of the parameters

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

Drake states that given the uncertainties, the original meeting concluded that there were probably between 1,000 and 100,000,000 civilizations in the galaxy.

Current estimates

\(R*\) = the rate of star creation in our galaxy

Latest calculations from NASA and the European Space Agency indicate that the current rate of star formation in our galaxy is about 7 per year. 1

\(f_{p}\) = the fraction of those stars that have planets

It is known from modern planet searches that at least 40% of sun-like stars have planets, and the true proportion may be much higher, since only planets considerably larger than Earth can be detected with current technology. The Kepler mission, from its initial data, estimates that about 34% of stars host at least one planet. However, recent analysis of microlensing surveys has found that $f_{p}$ may approach 1—that is, stars are orbited by planets as a rule, rather than the exception; and that there are one or more bound planets per Milky Way star.2

\(n_{e}\) = the average number of planets per star having planets that might support life

The consensus at the Green Bank meeting was that ne had a minimum value between three and five. Dutch astronomer Govert Schilling has opined that this is optimistic. Even if planets are in the habitable zone, the number of planets with the right proportion of elements is difficult to estimate. Brad Gibson, Yeshe Fenner, and Charley Lineweaver determined that about 10% of star systems in the Milky Way galaxy are hospitable to life, by having heavy elements, being far from supernovae and being stable for a sufficient time 3

\(f_{l}\) = the fraction of the above that actually go on to develop life

Geological evidence from the Earth suggests that fl may be very high; life on Earth appears to have begun around the same time as favorable conditions arose, suggesting that abiogenesis may be relatively common once conditions are right. However, this evidence only looks at the Earth (a single model planet), and contains anthropic bias, as the planet of study was not chosen randomly, but by the living organisms that already inhabit it (ourselves). Also countering this argument is that there is no evidence for abiogenesis occurring more than once on the Earth—that is, all terrestrial life stems from a common origin. If abiogenesis were more common it would be speculated to have occurred more than once on the Earth. In addition, from a classical hypothesis testing standpoint, there are zero degrees of freedom, permitting no valid estimates to be made. If life were to be found on Mars that developed independently from life on Earth it would imply a value for fl close to one. While this would improve the degrees of freedom from zero to one, there would remain a great deal of uncertainty on any estimate due to the small sample size, and the chance they are not really independent. In 2002, Charles H. Lineweaver and Tamara M. Davis (at the University of New South Wales and the Australian Centre for Astrobiology) estimated \(f_{l}\) as > 0.13 on planets that have existed for at least one billion years using a statistical argument based on the length of time life took to evolve on Earth.

\(f_{i}\) = the fraction of the above that actually go on to develop intelligent life

This value remains particularly controversial. Those who favor a low value, such as the biologist Ernst Mayr, point out that of the billions of species that have existed on Earth, only one has become intelligent and from this infer a tiny value for fi. Those who favor higher values note the generally increasing complexity of life and conclude that the eventual appearance of intelligence might be inevitable, implying an fi approaching 1. Skeptics point out that the large spread of values in this factor and others make all estimates unreliable. In addition, while it appears that life developed soon after the formation of Earth, the Cambrian explosion, in which a large variety of multicellular life forms came into being, occurred a considerable amount of time after the formation of Earth, which suggests the possibility that special conditions were necessary. Some scenarios such as the Snowball Earth or research into the extinction events have raised the possibility that life on Earth is relatively fragile. Again, the controversy over life on Mars is relevant since a discovery that life did form on Mars but ceased to exist would affect estimates of these factors. Again this model has a large anthropic bias and there are still zero degrees of freedom. Note that the capacity and willingness to participate in extraterrestrial communication has come relatively “quickly”, with the Earth having only an estimated 100,000 year history of intelligent human life, and less than a century of technological ability.

For deliberate communication, the one example we have (the Earth) does not do much explicit communication, though there are some efforts covering only a tiny fraction of the stars that might look for our presence. (The Arecibo message, for example). For other civilizations, there is considerable speculation why a civilization might exist but choose not to communicate, but no hard data. However, deliberate communication is not required, and calculations indicate that current or near-future Earth-level technology might well be visible to civilizations not too much in advance of our own. By this standard the Earth is a communicating civilization.

\(f_{c}\) = the fraction of the above that release detectable signs of their existence into space

\(L =\) the expected lifetime of such a civilization for the period that it can communicate across interstellar space

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. It could also be argued from Michael Shermer’s results that the fall of most of these civilizations was followed by later civilizations that carried on the technologies, so it’s doubtful that they are separate civilizations in the context of the Drake equation. In the expanded version, including reappearance number, this lack of specificity in defining single civilizations doesn’t matter for the end result, since such a civilization turnover could be described as an increase in the reappearance number rather than increase in L, stating that a civilization reappears in the form of the succeeding cultures. Furthermore, since none could communicate over interstellar space, the method of comparing with historical civilizations could be regarded as invalid. David Grinspoon has argued that once a civilization has developed it might overcome all threats to its survival. It will then last for an indefinite period of time, making the value for L potentially billions of years. If this is the case, then the galaxy has been steadily accumulating advanced civilizations since it formed. He proposes that the last factor L be replaced with \(f_{IC} \times T\), where \(f_{IC}\) is the fraction of communicating civilizations become “immortal” (in the sense that they simply don’t die out), and T representing the length of time during which this process has been going on. This has the advantage that T would be a relatively easy to discover number, as it would simply be some fraction of the age of the universe. It has also been pointed out that, once a civilization has learned of a more advanced one, its longevity could increase because it can learn from the experiences of the other. The astronomer Carl Sagan speculated that all of the terms, except for the lifetime of a civilization, are relatively high and the determining factor in whether there are large or small numbers of civilizations in the universe is the civilization lifetime, or in other words, the ability of technological civilizations to avoid self-destruction. In Sagan’s case, the Drake equation was a strong motivating factor for his interest in environmental issues and his efforts to warn against the dangers of nuclear warfare.

Equation results

As many skeptics have pointed out, the Drake equation can give a very wide range of values, depending on the assumptions. The following examples use numbers within the ranges suggested for each of the parameters. Using lowest values in the above estimates \(N = 8 \times 10^{-20}\) (not only are we probably alone in the galaxy, but maybe in the whole universe) On the other hand, plugging in other proposed values for each of the parameters above, the resultant value of N can be made greater than 1. This has provided considerable motivation for the SETI movement. However, we have no evidence for extraterrestrial civilizations. This conflict is often called the Fermi paradox, after Enrico Fermi who first asked about our lack of observation of extraterrestrials, and motivates advocates of SETI to continually expand the volume of space in which another civilization could be observed. The highest values in the above estimates, assuming 1 for parameters that have no backup for any estimate result in \(N = 182\) million

  1. https://www.nasa.gov/centers/goddard/news/topstory/2006/milkyway_seven.html 

  2. Cassan, A.; et al. (11 January 2012). “One or more bound planets per Milky Way star from microlensing observations”. Nature. 481 (7380): 167–169. arXiv:1202.0903 

  3. Lineweaver, C. H.; Fenner, Y.; Gibson, B. K. (2004). “The Galactic Habitable Zone and the Age Distribution of Complex Life in the Milky Way”. Science. 303 (5654): 59–62. arXiv:astro-ph/0401024 

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