DIA, 2010: estudo do AAWSAP aplica estatística à equação de Drake e estima a distância da civilização extraterrestre mais próxima

An Introduction to the Statistical Drake Equation. Prepared by: Acquisition Support Division (DWO-3) Defense Warning Office Directorate for Analysis Defense Intelligence Agency. Author: [AAP Person 80]. Administrative Note. COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized. This product is one in a series of advanced technology reports produced in FY 2009 under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapon System Applications (AAWSA) Program. Comments or questions pertaining to this document should be addressed to [AAP Person 1], AAWSA Program Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington, DC 20340-5100.

An Introduction to the Statistical Drake Equation. 1. Introduction. SETI (an acronym for “Search for Extraterrestrial Intelligence”) is a relatively new branch of scientific research, having begun only in 1959. Its goal is to ascertain whether alien civilizations exist in the universe, how far from us they exist, and possibly how much more advanced than us they may be. 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. This forces us to use the largest radiotelescopes on Earth for SETI research, because the higher our collecting area of electromagnetic radiation is, the higher our sensitivity is (that is, the farther in space we can probe). Yet, even by using the largest radiotelescopes on Earth (the 310-meter dish at Arecibo, for instance), we cannot search for aliens beyond, say, a few hundred light years away. This is a very, very small amount of space around us within our galaxy, the Milky Way, that is about 100,000 light years in diameter. 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! This demands the construction of much more powerful and radically new radiotelescopes. Rather than big and heavy metal dishes, whose mechanical problems hamper SETI research too much, we are now turning to “software radiotelescopes,” where a large number of small dishes (ATA = Allen Telescope Array, and ALMA = Atacama Large Millimeter/submillimeter Array) or even just of simple dipoles (LOFAR = Low Frequency Array) using state-of-the-art electronics and very-high-speed computing can outperform the classical radiotelescopes in many regards. The final dream in this field is the SKA (= Square Kilometer Array), currently being designed and expected to be completed around 2020. 2. The Key Question: How Far are They? But still, the key question remains: how far are they? Or, more correctly, how far do we expect the NEAREST extraterrestrial civilization to be from the Solar System in the galaxy? This question was first faced in a scientific manner back in 1961 by the same scientist who also was the first experimental SETI radio astronomer ever: the American, Frank Donald Drake (born 1930). He first considered the shape and size of the galaxy where we are living: the Milky Way. This is a spiral galaxy measuring some 100,000 light years in diameter and some 16,000 light years in thickness of the Galactic Disk at half-way from its center. That is: The diameter of the galaxy is (about) 100,000 light years, (abbreviated ly) i.e., its radius, R Galaxy, is about 50,000 ly.

The thickness of the Galactic Disk at half-way from its center, h Galaxy, is about 16,000 ly. The volume of the galaxy may then be approximated as the volume of the corresponding cylinder, i.e. [equation (1)]. Now consider the sphere around us having a radius r. The volume of such a sphere is [equation (2)]. In the last equation, we had to divide the distance “ET_Distance” between ourselves and the nearest ET civilization by 2 because we are now going to make the unwarranted assumption that all ET civilizations are equally spaced from each other in the galaxy! 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. At the moment, however, this is the best guess that we can make, and so we shall take it for granted, although we are aware that this is a weak point in the reasoning. Furthermore, let us denote by N the total number of civilizations now living in the galaxy, including ourselves. Of course, this number N is unknown. We only know that N ≥ 1 since one civilization does at least exist! Having thus assumed that ET civilizations are UNIFORMLY SPACED IN THE GALAXY, we can then write down the proportion: [equation (3)]. That is, upon replacing both (1) and (2) into (3): [equation (4)]. The last equation contains two unknowns: N and ET_Distance, and so we don't know which one it is better to solve for. However, we may suppose that, by resorting to the (rather uncertain) knowledge that we have about the Evolution of the galaxy through the last 10 billion years or so, we might somehow compute an approximate value for N. Then, we may solve (4) for ET_Distance thus obtaining the (AVERAGE) DISTANCE BETWEEN ANY PAIR OF NEIGHBORING CIVILIZATIONS IN THE GALAXY (DISTANCE LAW)

[equation (5)] where the positive constant C is defined by [equation (6)] ≈ 28845 light years. Equations (5) and (6) are the starting point to understand the origin of the Drake equation that we discuss in detail in Section 3 of this paper. Let us just complete this section by pointing out three different numerical cases of the distance law (5): • We know that we exist, so N may not be smaller than 1, i.e., N ≥ 1. Suppose then that we are alone in the galaxy, i.e., that N=1. Then the distance law (5) yields as distance to the nearest civilization from us just the constant C, i.e., 28,845 light years. This is about the distance in between ourselves and the center of the galaxy (i.e. the Galactic Bulge). Thus, this result seems to suggest that, if we do not find any extraterrestrial civilization around us in these outskirts of the galaxy where we live, we should look around the Galactic Center first. And this is indeed what is happening, i.e., many SETI searches are actually pointing the antennas towards the Galactic Center, looking for beacons (see, for instance ref. [1]). • Suppose next that N=1000, i.e. there are about a thousand extraterrestrial communicating civilizations in the whole galaxy right now. Then the distance law (5) yields an average distance of 2,885 light years. This is a distance that most radiotelescopes in Earth may not reach for SETI searches right now: hence the need to build larger radiotelescopes, like ALMA, LOFAR and the SKA. • Suppose finally that N=1000000, i.e., there are a million communicating civilizations now in the galaxy. Then the distance law (5) yields an average distance of 288 light years. This is within the (upper) range of distances that our current radiotelescopes may reach for SETI searches, and that justifies all SETI searches that have been done so far in the first fifty years of SETI (1960-2010). In conclusion, interpolating the above three special cases of N, we may say that the distance law (5) yields the following key diagram of the average ET distance vs. the assumed number of communicating civilizations, N, in the galaxy right now (Figure 1):

Figure 1. DISTANCE LAW; i.e., the Average Distance (plot along the vertical axis in light years) Versus the NUMBER of Communicating Civilizations ASSUMED to Exist in the Galaxy Right Now. 3. Computing N By Virtue of the Drake Equation (1961). In the previous section, the problem of finding how close the nearest ET civilization may be was “solved” by reducing it to the computation of N, the total number of extraterrestrial civilizations now existing in this galaxy. In this section the famous Drake equation is described, that was proposed back in 1961 by Frank Donald Drake (born 1930) to estimate the numerical value of N. We believe that no better introductory description of the Drake equations exists other than the one given by Carl Sagan in his 1983 book “Cosmos” (ref. [2]), in its turn based on the famous TV series “Cosmos.” So, in this paragraph we report Carl Sagan's description of the Drake equation unabridged. [quotation from Carl Sagan follows]

It is clarifying to draw the graph of the ET_Distance probability density (9): [Figure 3] Figure 3. The Probability of Finding the Nearest Extraterrestrial Civilization at the distance r From Earth (in light years) if the Values Assumed in the Drake Equation are Those Shown in Input Table 1. The relevant probability density function f ET_Distance (r) is given by equation (9). Its mode (peak abscissa) equals 1933 light years, but its mean value is higher since the curve has a long tail on the right: the mean value equals in fact 2670 light years. Finally, the standard deviation equals 1309 light years: THIS IS GOOD NEWS FOR SETI, inasmuch as the nearest ET galaxy civilization might lie at just 1 sigma = 2670-1309 = 1361 light years from us. From Figure 3 we see that the probability of finding extraterrestrials is practically zero up to a distance of about 500 light years from Earth. Then it starts increasing with the increasing distance from Earth, and reaches its maximum at r mode = r peak ≈ 1,933 light years. (14) This is the MOST LIKELY VALUE of the distance at which we can expect to find the nearest extraterrestrial civilization. It is not the mean value of the probability distribution (9) for f ET_Distance (r). In fact, the probability density (9) has an infinite tail on the right, as clearly shown in Figure 3, and hence its mean value must be higher than its peak value. As given by (10) and (12), its mean value is r mean_value ≈ 2670 light years. This is the MEAN (value of the) DISTANCE at which we can expect to find extraterrestrials.

After having found the above two distances (1933 and 2670 light years, respectively), the next natural question that arises is: “what is the range, back and forth around the mean value of the distance, within which we can expect to find extraterrestrials with “the highest hopes?” The answer to this question is given by the notion of standard deviation that we already found to be given by (11) and (13), σ ET_Distance ≈ 1309 light years. More precisely, this is the so-called 1-sigma (distance) level. Probability theory then shows that the nearest extraterrestrial civilization is expected to be located within this range, i.e. within the two distances of (2670-1309) = 1361 light years and (2670+1309) = 3979 light years, with probability given by the integral of f ET_Distance (r) taken in between these two lower and upper limits, that is: ≈ 0.75 = 75% (15). 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. 9. The “Data Enrichment Principle” as the Best CLT Consequence Upon the Statistical Drake Equation (Any Number of Factors Allowed). As a fitting climax to all the statistical equations developed so far, let us now state our “DATA ENRICHMENT PRINCIPLE.” It simply states that “The Higher the Number of Factors in the Statistical Drake equation, The Better.” Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y approach the normal distribution when the number of terms in the sum (4) approaches infinity. And this is the case, indeed. 10. Conclusions. We have sought to extend the classical Drake equation to let it encompass Statistics and Probability. This approach appears to pave the way to future, more profound investigations intended not only to associate “error bars” to each factor in the Drake equation, but especially to increase the number of factors themselves. In fact, this seems to be the only way to incorporate into the Drake equation more and more new scientific information as soon as it becomes available. 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.” [p. 24] Unfortunately, to extend the Drake equation to Statistics, it was necessary to use a mathematical apparatus that is more sophisticated than just the simple product of seven numbers.