Drake equation

This article is about Frank Drake's equation. For the Tub Ring music album, see Drake Equation (album).

The Drake equation (sometimes called the Green Bank equation or the Green Bank Formula) is an equation used to estimate the number of detectable extraterrestrial civilizations in the Milky Way galaxy. It is used in the fields of exobiology and the search for extraterrestrial intelligence (SETI). The equation was devised by Frank Drake in 1961.

History

In 1960, Frank Drake conducted the first search for radio signals from extraterrestrial civilizations at the National Radio Astronomy Observatory in Green Bank, West Virginia. Soon thereafter, the National Academy of Sciences asked Drake to convene a meeting on detecting extraterrestrial intelligence. The meeting was held at the Green Bank facility in 1961. The equation that bears Drake's name arose out of his preparations for the meeting:

As I planned the meeting, I realized a few day[s] ahead of time we needed an agenda. And so I wrote down all the things you needed to know to predict how hard it's going to be to detect extraterrestrial life. And looking at them it became pretty evident that if you multiplied all these together, you got a number, N, which is the number of detectable civilizations in our galaxy. This, of course, was aimed at the radio search, and not to search for primordial or primitive life forms.

— Frank Drake

This meeting established SETI as a scientific discipline. The meeting's dozen participants — astronomers, physicists, biologists, social scientists, and industry leaders — became known as the "Order of the Dolphin". The Green Bank meeting has been commemorated by a plaque at the site.

The Drake equation is closely related to the Fermi paradox in that Drake suggested that a large number of extraterrestrial civilizations would form, but that the lack of evidence of such civilizations (the Fermi paradox) suggests that technological civilizations tend to disappear rather quickly. This theory often stimulates an interest in identifying and publicizing ways in which humanity could destroy itself, and then counters with hopes of avoiding such destruction and eventually becoming a space-faring species. A similar argument is The Great Filter, which notes that since there are no observed extraterrestrial civilizations, despite the vast number of stars, then some step in the process must be acting as a filter to reduce the final value. According to this view, either it is very hard for intelligent life to arise, or the lifetime of such civilizations must be relatively short.

Carl Sagan, a great proponent of SETI, quoted the formula often and as a result the formula is sometimes mislabeled as "The Sagan Equation."

The equation

The Drake equation states that:

$N=R^{\ast }\times f_{p}\times n_{e}\times f_{\ell }\times f_{i}\times f_{c}\times L\!$

where:

N = the number of civilizations in our galaxy with which communication might be possible;

and

R^* = the average rate of star formation per year in our galaxy

f_p = the fraction of those stars that have planets

n_e = the average number of planets that can potentially support life per star that has planets

f_ℓ = the fraction of the above that actually go on to develop life at some point

f_i = the fraction of the above that actually go on to develop intelligent life

f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space

L = the length of time such civilizations release detectable signals into space.

Alternative expression

The number of stars in the galaxy now, N^*, is related to the star formation rate R^* by

$N^{\ast }=\int _{0}^{T_{g}}R^{\ast }(t)dt,\,\!$

where T_g = the age of the galaxy. Assuming for simplicity that R^* is constant, then $N^{\ast }=R^{\ast }\times T_{g}$ and the Drake equation can be rewritten into an alternate form phrased in terms of the more easily observable value, N^*.

$N=N^{\ast }\times f_{p}\times n_{e}\times f_{\ell }\times f_{i}\times f_{c}\times L/T_{g}\,\!$

R factor

One can question why the number of civilizations should be proportional to the star formation rate, though this makes technical sense. (The product of all the terms except L tells how many new communicating civilizations are born each year. Then you multiply by the lifetime to get the expected number. For example, if an average of 0.01 new civilizations are born each year, and they each last 500 years on the average, then on the average 5 will exist at any time.) The original Drake Equation can be extended to a more realistic model, where the equation uses not the number of stars that are forming now, but those that were forming several billion years ago. The alternate formulation, in terms of the number of stars in the galaxy, is easier to explain and understand, but implicitly assumes the star formation rate is constant over the life of the galaxy.

Elaborations of the Drake Equation

As many observers have pointed out, the Drake equation is a very simple model that does not include potentially relevant parameters. David Brin states:

[The Drake Equation] merely speaks of the number of sites at which ETIs spontaneously arise. It says nothing directly about the contact cross-section between an ETIS and contemporary human society.

Because it is the contact cross-section that is of interest to the SETI community, many additional factors and modifications of the Drake equation have been proposed. These include the number of times a civilization might re-appear on the same planet, the number of nearby stars that might be colonized and form sites of their own, and other factors.

Colonization

Brin has proposed generalizing the Drake Equation to include additional effects of alien civilizations colonizing other star systems. Each original site expands with an expansion velocity v, and establishes additional sites that survive for a lifetime L'. The result is a more complex set of 3 equations.

Reappearance number

The Drake equation may furthermore be multiplied by how many times an intelligent civilization may occur on planets where it has happened once. Even if an intelligent civilization reaches the end of its lifetime after, for example, 10,000 years, life may still prevail on the planet for billions of years, permitting the next civilization to evolve. Thus, several civilizations may come and go during the lifespan of one and the same planet. Thus, if n_r is the average number of times a new civilization reappears on the same planet where a previous civilization once has appeared and ended, then the total number of civilizations on such a planet would be (1+n_r), which is the actual reappearance factor added to the equation.

The factor depends on what generally is the cause of civilization extinction. If it is generally by temporary uninhabitability, for example a nuclear winter, then n_r may be relatively high. On the other hand, if it is generally by permanent uninhabitability, such as stellar evolution, then n_r may be almost zero.

In the case of total life extinction, a similar factor may be applicable for f_ℓ, that is, how many times life may appear on a planet where it has appeared once.

METI factor

Alexander Zaitsev said that to be in a communicative phase and emit dedicated messages are not the same. For example, humans, although being in a communicative phase, are not a communicative civilization; we do not practice such activities as the purposeful and regular transmission of interstellar messages. For this reason, he suggested introducing the METI factor (Messaging to Extra-Terrestrial Intelligence) to the classical Drake Equation. The factor is defined as "The fraction of communicative civilizations with clear and non-paranoid planetary consciousness", or alternatively expressed, the fraction of communicative civilizations that actually engage in deliberate interstellar transmission.

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:

• R* = 10/year (10 stars formed per year, on the average over the life of the galaxy)
• f_p = 0.5 (half of all stars formed will have planets)
• n_e = 2 (stars with planets will have 2 planets capable of developing life)
• f_l = 1 (100% of these planets will develop life)
• f_i = 0.01 (1% of which will be intelligent life)
• f_c = 0.01 (1% of which will be able to communicate)
• L = 10,000 years (which will last 10,000 years)

Drake's values give N = 10 × 0.5 × 2 × 1 × 0.01 × 0.01 × 10,000 = 10.

The value of R* is determined from considerable astronomical data, and is the least disputed term of the equation; f_p is less certain, but is still much firmer than the values following. The value of n_e is based on our own solar system, and assumes that two planets had the possibility of having life. This not only has problems with anthropic bias, but also is inconsistent with a f_l of one unless we do find life on Mars. Also, 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 flare violently, mostly in X-rays—a property not conducive to life as we know it (simulations also suggest that these bursts erode planetary atmospheres). The possibility of life on moons of gas giants (such as Jupiter's moon Europa, or Saturn's moon Titan) adds further uncertainty to this figure.

Geological evidence from the Earth suggests that f_l 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 f_l 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.

Similar arguments of bias can be made regarding f_i and f_c by considering the Earth as a model: intelligence with the capacity of extraterrestrial communication occurs only in one species in the 4 billion year history of life on Earth. If generalized, this means only relatively old planets may have intelligent life capable of extraterrestrial communication. 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.

f_i, f_c and L, like f_l, are also guesses. Estimates of f_i have been affected by discoveries that the solar system's orbit is circular in the galaxy, at such a distance that it remains out of the spiral arms for hundreds of millions of years (evading radiation from novae). Also, Earth's large moon may aid the evolution of life by stabilizing the planet's axis of rotation. 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 terms.

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.

By plugging in apparently "plausible" 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.

Some computations of the Drake equation, given different assumptions:

Current estimates (see below):

R* = 7/year, f_p = 0.5, n_e = 2, f_l = 0.33, f_i = 0.01, f_c = 0.01, and L = 10,000 years

N = 7 × 0.5 × 2 × 0.33 × 0.01 × 0.01 × 10,000 = 2.31 (so two communicative civilizations exist in our galaxy at any given time, on average, plus two hundred more that are not trying to communicate).

But a pessimist might equally well believe that suitable planets are rare, life seldom becomes intelligent, and intelligent civilizations do not last very long:

R* = 10/year, f_p = 0.5, n_e = 0.01, f_l = 0.13, f_i = 0.001, f_c = 0.01, and L = 1000 years

N = 10 × 0.5 × 0.01 × 0.13 × 0.001 × 0.01 × 1000 = 0.000065 (we are almost surely alone in our galaxy).

Alternatively, making some more optimistic assumptions, assuming that planets are common, life always arises when planets are favorable, 10% of civilizations 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, f_p = 0.5, n_e = 2, f_l = 1, f_i = 0.1, f_c = 0.1, and L = 100,000 years

N = 20 × 0.5 × 2 × 1 × 0.1 × 0.1 × 100,000 = 20,000 (there's quite a few civilizations, although the closest one would still be about 1500 light years away).

Current estimates of the parameters

This section attempts to list best current estimates for the parameters of the Drake equation.

R* = the rate of star creation in our galaxy

Estimated by Drake as 10/year. 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.

f_p = the fraction of those stars that have planets

Estimated by Drake as 0.5. It is now 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. Infra-red surveys of dust discs around young stars imply that 20-60% of sun-like stars may form terrestrial planets. Microlensing surveys, sensitive to planets further from their star, see planets in about 1/3 of systems examined–a lower limit since not all planets are seen. A mid-2010 estimate by Dimitar Sasselov, of the Kepler planet-hunting team estimates the number of terrestrial planets in the Milky Way to be as much as 100 million.

n_e = the average number of planets (satellites may perhaps sometimes be just as good candidates) that can potentially support life per star that has planets

Estimated by Drake as 2. The combination of this with f_l=1 implies that two planets per star develop life, which not only assumes that our solar system is typical but that there is (or was) life on Mars. Marcy et al. note that most of the observed planets have very eccentric orbits, or orbit very close to the sun where the temperature is too high for earth-like life. However, several planetary systems that look more solar-system-like are known, such as HD 70642, HD 154345, Gliese 849 or Gliese 581. There may well be other, as yet unseen, earth-sized planets in the habitable zones of these stars. Also, the variety of solar systems that might have habitable zones is not just limited to solar-type stars and earth-sized planets; it is now believed that even tidally locked planets close to red dwarfs might have habitable zones, and some of the large planets detected so far could potentially support life

In early 2008, two different research groups concluded that Gliese 581 d may possibly be habitable. Since about 200 planetary systems are known, this very roughly estimates $n_{e}>0.005$. In 2010, researchers announced the discovery of Gliese 581 g, a 3.1 Earth-mass planet in near the middle of the habitable zone of Gliese 581, and a strong candidate for being the first known Earth-like habitable planet. Given the closeness of the planet's star, and the number of stars examined to the level of detail needed to find such planets, they estimate ε_Earth, or the fraction of stars with Earth-like planets, as 10-20%.

Using different criteria, Lineweaver has also determined that about 10% of star systems in the Galaxy are hospitable to life, by having heavy elements, being far from supernovae and being stable for a sufficient time.

NASA's Kepler mission was launched on March 6, 2009. Unlike previous searches, it is sensitive to planets as small as Earth, and with orbital periods as long as a year. If successful, Kepler should provide a much better estimate of the number of planets per star that are found in the habitable zone.

Even if planets are in the habitable zone, however, the number of planets with the right proportion of elements may be difficult to estimate. Also, the Rare Earth hypothesis, which posits that conditions for intelligent life are quite rare, has advanced a set of arguments based on the Drake equation that the number of planets or satellites that could support life is small, and quite possibly limited to Earth alone; in this case, the estimate of n_e would be infinitesimal.

f_l = the fraction of the above that 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 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

Estimated by Drake as 0.01 based on little or no evidence. This value remains particularly controversial. The famous biologist Ernst Mayr pointed out that of the billions of species that have existed on Earth, only one has become intelligent and infer a tiny value for f_i.Skeptics point out that the large spread of values in this term and others make all estimates unreliable. (See criticism).

f_c = the fraction of the above that are willing and able to communicate

Estimated by Drake as 0.01. There is considerable speculation why a civilization might exist but choose not to communicate, but there is no hard data.

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

Estimated by Drake as 10,000 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. 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.

Values based on the above estimates,

R* = 7/year, f_p = 0.5, n_e = 2, f_l = 0.33, f_i = 0.01, f_c = 0.01, and L = 10000 years

result in

N = 7 × 0.5 × 2 × 0.33 × 0.01 × 0.01 × 10000 = 2.31

James Kasting, in his book "How To Find A Habitable Planet", gives the equation as $N=N_{g}\times f_{p}\times n_{e}\times f_{l}\times f_{i}\times f_{c}\times f_{L}$, where the first term on the right hand side of the equation is the number of stars in the galaxy. He estimates the first three terms at 4 * 10 to the ninth power. He then uses Carl Sagan's figures for the next three terms, disclaiming responsibility, and arrives at approximately 10 to the seventh power as an estimate, not considering the final term, $f_{L}$, which is the fraction of a planet's lifetime during which it supports a technical civilization. He notes that this is the most uncertain factor in the equation.

Criticism

Criticism of the Drake equation follows mostly from the observation that several terms in the equation are largely or entirely based on conjecture. Thus the equation cannot be used to draw firm conclusions of any kind. As T.J. Nelson states:

The Drake equation consists of a large number of probabilities multiplied together. Since each factor is guaranteed to be somewhere between 0 and 1, the result is also guaranteed to be a reasonable-looking number between 0 and 1. Unfortunately, all the probabilities are completely unknown, making the result worse than useless.

Likewise, in a 2003 lecture at Caltech, Michael Crichton, a science fiction author, stated:

The problem, of course, is that none of the terms can

be known, and most cannot even be estimated. The only way to work the equation is to fill in with guesses. [...] As a result, the Drake equation can have any value from "billions and billions" to zero. An expression that can mean anything means nothing.

Speaking precisely, the Drake equation is literally meaningless...

Another objection is that the very form of the Drake equation assumes that civilizations arise and then die out within their original solar systems. If interstellar colonization is possible, then this assumption is invalid, and the equations of population dynamics would apply instead.

One reply to such criticisms is that even though the Drake equation currently involves speculation about unmeasured parameters, it was not meant to be science, but intended as a way to stimulate dialog on these topics. Then the focus becomes how to proceed experimentally. Indeed, Drake originally formulated the equation merely as an agenda for discussion at the Green Bank conference.

Another reply to such criticisms is that the Drake Equation is a Fermi problem which involves the multiplication of several estimated factors, and such calculations (e.g. the number of piano tuners in Chicago) will probably be more accurate than might be first supposed (assuming that there is no consistent bias in the estimated factors). This is because if there is no consistent bias, then there will probably (with a binomial distribution) be some factors that are estimated too high and other factors that are estimated too low, and such errors will partially cancel each other out.

In fiction

The Drake equation and the Fermi paradox have been discussed many times in science fiction, including both serious takes in stories such as Frederik Pohl's Hugo award-winning "Fermi and Frost", which cites the paradox as evidence for the short lifetime of technical civilizations — that is, the possibility that once a civilization develops the power to destroy itself (perhaps by nuclear winter), it does, and humorous commentary in stories such as Terry Bisson's classic short story "They're Made Out of Meat."

• The equation was cited by Gene Roddenberry as supporting the multiplicity of inhabited planets shown in Star Trek, the television show he created. However, Roddenberry didn't have the equation with him, and he was forced to "invent" it for his original proposal. The invented equation created by Roddenberry is:

$Ff^{2}(MgE)-C^{1}Ri^{1}~\times ~M=L/So.\$

Drake has gently pointed out, however, that a number raised to the first power is merely the number itself. A poster with both versions of the equation was seen in the Star Trek: Voyager episode "Future's End."

• The formula is also cited in Michael Crichton's Sphere.
• In the Evolution-based game Spore, after eventually coming into contact with living beings on other planets, a picture is shown, along with the comment, "Drake's Equation was right...a living alien race!"
• George Alec Effinger's short story "One" uses an expedition confident in the Drake Equation as a backdrop to explore the psychological implications of a lonely humanity.
• Alastair Reynolds' Revelation Space trilogy and short stories focus very much on the Drake Equation and the Fermi Paradox, using genocidal self-replicating machines as a great filter.
• Stephen Baxter's Manifold Trilogy explores the Drake Equation and the Fermi Paradox in three distinct perspectives.
• Ian R. MacLeod's 2001 novella "New Light On The Drake Equation" concerns a man who is obsessed by the Drake Equation.
• The Ultimate Marvel comic book mini-series Ultimate Secret has Reed Richards examining the Drake Equation and considering the Fermi Paradox. He claims that when Drake plugged in his numbers, he came up with 10,000 alien races that would have a civilization advanced enough to contact Earth, but only one that Richards knew of had done it (actually 10 others already had, and two more were about to, but Richards did not know that at the time). In discussion with Sue Storm, it is revealed that he believes that advanced civilizations destroy themselves. In the story it turns out that they are also destroyed by Gah Lak Tus.
• Eleanor Ann Arroway paraphrases the Drake equation several times in the film Contact, using the magnitude of N^* and its implications on the output value to justify the SETI program. However, in one scene of the 1997 motion picture based upon the Carl Sagan novel, she misstates her Drake Equation result by several billion.
• This equation was mentioned by Howard and detailed by Sheldon in the 20th episode of the second season of the television series The Big Bang Theory. Howard goes on to modify the terms in the equation to project the likelihood of a member of the group hooking up with a member of the opposite sex. He uses the "Wallowitz Coefficient" which equals neediness times dress size squared to achieve this.

See also

• Big Bounce
• Black swan theory
• Doomsday argument
• Final anthropic principle
• Goldilocks Principle
• Great Filter
• Inverse gambler's fallacy
• Kardashev scale
• Metaphysical naturalism
• Neocatastrophism
• Nick Bostrom
• Principle of mediocrity
• Rare Earth hypothesis
• Selection bias
• Zoo hypothesis

References

• Drake, Frank (1992). Is Anyone Out There? The Scientific Search for Extraterrestrial Intelligence. New York: Delacorte Pr. ISBN 0-385-30532-X. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |unused_data= ignored (help)
• Rood, Robert T. (1981). Are We Alone? The Possibility of Extraterrestrial Civilizations. New York: Scribner. ISBN 0-684-16826-X. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |unused_data= ignored (help)
• Lineweaver, Charles H. (2 May 2002). "Does the Rapid Appearance of Life on Earth Suggest that Life is Common in the Universe?". arXiv:astro-ph/0205014. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Shermer, Michael (2002). "Why ET Hasn't Called". Scientific American: 21. {{cite journal}}: Unknown parameter |month= ignored (help)
• Bates, Gary (2004). Alien Intrusion. Master books. ISBN 0-89051-435-6. {{cite book}}: Unknown parameter |unused_data= ignored (help)
• Morton, Oliver (2002). Graham Formelo (ed.). It Must Be Beautiful. Granta Books. ISBN 0-86207-555-7. {{cite book}}: Check |isbn= value: checksum (help); Unknown parameter |unused_data= ignored (help) Chapter A Mirror in the Sky is dedicated to Drake equation

External links

• "Only a matter of time, says Frank Drake" A Q&A with Frank Drake about his famous equation and the meaning of SETI, from an interview in February 2010, leading up to the 50th birthday of SETI.
• Frank Drake (2004). "The E.T. Equation, Recalculated". Wired. {{cite web}}: Unknown parameter |month= ignored (help)
• Drake Calculator Calculates the Drake equation with each parameter a variable that can be changed and is explained.
• http://www.skypub.com/news/special/9812seti_aliens.html
• Beyond the Drake Equation
• January 2002 space.com article about estimated prevalence of extrasolar planets
• Preprint by Lineweaver and Davis estimating f_l as > 0.13
• July 2003 discovery of a planetary system similar to our solar system
• Macromedia Flash page allowing the user to modify Drake's values from PBS Nova
• The Drake Equation Astronomy Cast episode #23, includes full transcript.
• The Drake Equation A critical examination of the Drake Equation
• The Drake Equation Animated simulation of the Drake Equation