DOW-UAP-D127: An Introduction to the Statistical Drake Equation - How Many Communicating Civilizations the Galaxy Might Hold, and How Far Away

Source file: DOW-UAP-D127_AAWSAP-DIRD-An-Introduction-to-the-Statistical-Drake-Equation-March-11-2010.pdf Originating agency: Defense Intelligence Agency (DIA), Defense Warning Office, under the AAWSAP program Document type: Defense Intelligence Reference Document (DIRD), Acquisition Threat Support series; no DIA control number appears on the cover Date: 11 March 2010 (ICOD, information cutoff date: 1 December 2009) Classification: UNCLASSIFIED//FOR OFFICIAL USE ONLY (the FOUO caveat is struck through; released to the public in 2026) Page count: 55 VIRIN: 260918-D-D0360-1116 PURSUE Release: 6

Summary

This is an AAWSAP Defense Intelligence Reference Document (DIRD) dated 11 March 2010 and titled "An Introduction to the Statistical Drake Equation". The title page states that it was prepared by the Acquisition Support Division (DWO-3) of the Defense Warning Office, Directorate for Analysis, DIA. The author's name is redacted and replaced by the pseudonym AAP Person 80. As in other documents of the series, the administrative note describes it as one of a series of "advanced technology reports produced in FY 2009" and directs questions to AAP Person 1, the AAWSA Program Manager.

The subject is the Drake equation, the formula proposed by the astronomer Frank Drake in 1961 to estimate N, the number of civilizations in the galaxy capable of communicating by radio. The author argues that the classical equation is "too simplistic" because it multiplies seven sharp numbers when each of them is known only approximately. In its place the author proposes a "Statistical Drake Equation" in which every factor is a random variable, and concludes that N follows a lognormal distribution. From this the paper also derives a probability distribution for the distance to the nearest civilization.

The paper does not mention UFOs, UAP, alien visitation or aircraft of any kind. The only channel of contact it considers is radio: "As of 2009, the only physical tools we know that could help us get in touch with aliens are the electromagnetic waves an alien civilization could emit and we could detect" (page 4). The author also stresses that the input values in the paper's examples were chosen purely for illustration.

Research Article

Structure: a 24-page introduction and a reprinted conference paper

The document consists of ten short sections (pages 4 to 24), two appendices and a reference list. Appendix A (pages 25 to 27) proves a corollary of a 1948 theorem by Claude Shannon. Appendix B (pages 28 to 54), about half the document, is the "Original Text of the Author's Paper": a paper the author presented on 1 October 2008 at the 59th International Astronautical Congress (IAC) in Glasgow. The body of the DIRD is therefore an accessible explanation, "by speaking easily" in the author's words, of original research already presented elsewhere, rather than a synthesis of existing literature. Another three pages or so (7 to 10) are an "unabridged" quotation of Carl Sagan's description of the Drake equation from "Cosmos".

The cover carries the label Acquisition Threat Support, but the paper contains no reference to acquisition, threats or aviation.

"How far are they?" and the distance law

The introduction places the subject in the search for extraterrestrial intelligence (SETI), which the paper dates to 1959. Even the largest telescopes, such as the Arecibo dish, cannot search beyond a few hundred light years in a galaxy about 100,000 light years across, so, the author says, it is no surprise that fifty years of SETI have found nothing (see quotes). The hope lies in new telescope arrays: the ATA, ALMA, LOFAR and the SKA, expected to be completed "around 2020".

Section 2 presents the "distance law". The galaxy is approximated as a cylinder with a radius of 50,000 light years and a thickness of 16,000 light years, with all N civilizations evenly spaced within it. The average distance between neighbours is then C divided by the cube root of N, where C equals 28,845 light years. Even the author calls the equal-spacing assumption "a crazy assumption" (page 5). Three numerical cases follow: if we are alone (N=1), the distance is 28,845 light years, roughly the distance to the galactic centre; with a thousand civilizations, 2,885 light years; with a million, 288 light years, within the upper range of current telescopes (page 6, Figure 1).

The Drake equation through Sagan's eyes

Section 3 sets out the seven factors: the number of stars in the galaxy (Ns), the fraction of stars with planets (fp), the number of planets per system suitable for life (ne), and the fractions of those planets on which life (fl), intelligence (fi) and a communicating technical civilization (fc) arise. The last factor (fL) is the fraction of a planet's lifetime graced by such a civilization. In the Sagan passage, Ns is about 4×10¹¹, fp about a third, ne two, fl about a third and fi×fc one percent, so that about a billion planets have produced a technical civilization. If civilizations destroy themselves quickly, N is about ten; if one percent survive "technological adolescence", N reaches ten million. A footnote updates the picture: Sagan wrote in the 1970s, before the first extrasolar planet was found in 1995, and by April 2009 some 347 such planets were listed (page 9).

A sharp number versus a distribution

Section 4 states that published estimates of N range "from a few tens" to millions or billions, but always as a single number. The author's remedy (Sections 5 and 6) is to replace each factor with a random variable that has a mean and a standard deviation. The Gaussian is rejected because it allows negative values, and the uniform distribution is chosen instead, because Shannon showed that over a finite range it is the distribution with maximum entropy, that is, maximum uncertainty. The limits of the range are then the mean minus and plus √3 times the standard deviation.

A product of uniform variables has no analytic solution, and the problem, the author writes, "occupied the author's mind for no less than about ten years (1997-2007)" (page 13). The solution, found in September 2007, is to take logarithms: the product becomes a sum, and by the Central Limit Theorem a sum of independent variables tends to a Gaussian. So ln(N) is Gaussian and N is lognormal. From this follows the "Data Enrichment Principle": "The Higher the Number of Factors in the Statistical Drake equation, The Better". An example of a missing factor is asteroid impacts, whose role in the extinction of the dinosaurs was understood only in 1980 (page 14).

The worked example: 3,500 or 4,590?

The input table (page 16) sets 350 billion stars with a standard deviation of one billion, fp and fl at 50 percent, fi and fc at 20 percent, ne at 1, and fL at 10,000 divided by 10¹⁰. The classical product is N = 3,500. The lognormal distribution gives a mode of about 250, a median of about 1,740, a mean of about 4,590 and a standard deviation of 11,195. From this the author concludes that the statistical extension "INCREASES OUR HOPES" (page 17), and in Appendix B calls it "GOOD NEWS FOR SETI".

A caveat is needed here, and the paper itself supplies the basis for it. In Appendix B (page 38) the author computes the exact distribution numerically, without the Central Limit Theorem approximation, and obtains a mean of 3,499.9988, that is, 3,500, and a standard deviation of about 3,953, so that "N might range in between 0 and 7453". This is what basic probability predicts: for independent factors, the mean of a product equals the product of the means. The abstract on page 28 likewise states that the lognormal mean "is the ordinary N in the Drake equation". The figures of 4,590 and 11,195, and the "hopes" drawn from them, therefore come from applying a lognormal approximation to just seven factors, not from new information about the galaxy. According to page 44, the two curves coincide only above N of about 1,500, while the mode of 250 lies precisely in the range where they differ.

The author acknowledges the basic limitation: the input values were chosen "for the sake of making up a numerical example", and in the paper's words "we just care about its mathematics" (page 34). There are also editorial inconsistencies: the body (page 11) says "350 millions stars" with a standard deviation of 50 million, against 350 billion in the appendix; two different tables are both numbered "Table 2"; and page 18 refers to "Appendix A" where Appendix B is meant.

The distance to the nearest civilization

Section 8 turns the distance law into a random quantity. In the worked example, the probability of finding a civilization closer than 500 light years is practically zero; the most likely distance is 1,933 light years, the mean 2,670 light years and the standard deviation 1,309 light years. With 75 percent probability, the nearest civilization lies between 1,361 and 3,979 light years away (page 23). According to Appendix B, the author discovered this distribution on 5 September 2007 and first presented it in February 2008 at a SETI meeting run by the physicist Paul Davies at the "Beyond" Center in Arizona, attended by Jill Tarter, Seth Shostak and others. The paper says Davies suggested naming the distribution after the author (page 46).

The conclusions imagine the Statistical Drake Equation growing into "a huge computer code", which would be "Humanity's first 'Encyclopaedia Galactica'". Appendix B closes with an apology the author reports making at a talk at the SETI Institute on 11 April 2008, with Frank Drake himself in the audience (see quotes).

Significance

The paper does not directly serve any of the 12 technical areas in AAWSAP's Statement of Objectives (D110), such as propulsion, power generation or materials; it belongs among the supporting topics. It may have been meant as background on whether other technological civilizations exist, but the paper itself draws no such link: it contains no mention of UFOs, UAP, interstellar travel or threat assessment, and the only contact it contemplates is receiving radio signals across thousands of light years.

Its archival value lies mainly in showing the breadth of AAWSAP: alongside papers on propulsion and materials, the program also received a mathematical paper on the probability of neighbours in the galaxy, about half of it a reprinted conference paper. That also bears on the official description of the series ("reference and synthesis products rather than original research"): here the author mainly presents their own original work. The war.gov summary is accurate on the other points: this is a methodological exercise that formalises uncertainty within the Drake framework, not an estimate of how many civilizations actually exist. The Drake equation itself is generally regarded in the scientific community as a framework for organising uncertainty, and the result of any version of it, statistical or classical, depends entirely on its inputs.

Key People

Locations

Key Concepts

Notable Quotes

"Thus, current SETI can cover only a very tiny fraction of the galaxy, and it is not surprising that in the past 50 years of SETI searches, NO extraterrestrial civilization was discovered. Quite simply, we did not get far enough!" -- page 4

"This is a crazy assumption, clearly, and should be replaced by more scientifically-grounded assumptions as soon as we know more about our Galactic Neighborhood." -- page 5

"It occupied the author's mind for no less than about ten years (1997-2007)." -- page 13

"Thus, in conclusion, THE STATISTICAL EXTENSION of the classical Drake equation INCREASES OUR HOPES to find an extraterrestrial civilization!" -- page 17

"In plain words: with 75 percent probability, the nearest extraterrestrial civilization is located in between the distances of 1361 and 3979 light years from us, having assumed the input values to the Drake Equation given by table 1. If we change those input values, then all the numbers change again, of course." -- page 23

"In the long run, the Statistical Drake equation might just become a huge computer code, growing in size and especially in the depth of the scientific information it contains. It would thus be Humanity's first 'Encyclopaedia Galactica.'" -- page 23

"So, we really make no assumption about the astronomy, or the biology, or the sociology of the Drake equation: we just care about its mathematics." -- page 34

"My apologies, Frank, for disrupting the beautiful simplicity of your equation." -- page 54