DOW-UAP-D127, AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

This document is a Defense Intelligence Reference Document (DIRD), a technical reference format used by the Defense Intelligence Agency (DIA) to capture baseline knowledge on a specific topic for later analytic use. DIRDs are best understood as reference and synthesis products rather than as original research. It is one of 38 DIRDs produced under the Advanced Aerospace Weapon System Applications Program (AAWSAP) between 2009 and 2011. Because AAWSAP’s scope permitted a broad range of supporting topics, not every DIRD in the series directly concerns aerospace systems or future threat assessment. The following summary reflects the DIRD’s scope and framing at the time of writing and should not be read as implying current validation of the concepts discussed. This DIRD introduces the Drake Equation, a well-known thought framework for estimating how many communicative extraterrestrial civilizations might exist in the galaxy. It reformulates the equation in statistical terms, arguing that the usual approach of assigning fixed values to its variables is too simplistic because major inputs are uncertain and are better modeled as probability distributions. Using that approach, it concludes that, if one accepts the underlying logic of the Drake Equation, the estimated number of communicating civilizations should be treated as a range of possible values, and that the likely distance between neighboring civilizations can likewise be expressed statistically rather than as a single figure. The document is primarily a mathematical and methodological exercise, and its worked examples rely on assumed values to illustrate the framework rather than to establish a firm astrophysical estimate. Overall, it is an attempt to formalize uncertainty within the Drake framework rather than an attempt to bound the actual likelihood, prevalence, or proximity of extraterrestrial civilizations.

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Defense
Intelligence
Reference
Document
Acquisition Threat Support
11 March 2010
!COD: 1 December 2009
An Introduction to the
Statistical Drake Equation
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An Introduction to the Statistical Drake Equation
Prepared by:
Acquisition Support Division (DW0-3)
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.
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Contents
1. Introduction .......................................................................................................iv
2. The Key Question: How Far are They ? .............................................................. 4
3. Computing N By Virtue of the Drake Equation (1961) ........................................ 7
4. The Drake Equation is Over-Simplified ............................................................. 10
5. The Statistical Drake Equation ......................................................................... 11
6. Solving the Statistical Drake Equation By Virtue of the Central Limit Theorem
(CLT) of Statistics .......................................................... , .................................... 13
7. An Example Explaining the Statistical Drake Equation ..................................... 15
8. Finding the Probability Distribution of the Et-Distance By Virtue of the Statistical
Drake Equation................................................................................................. 18
9. The "Data Enrichment Principle" as the Best CLT Consequence Upon the
Statistical Drake Equation (Any Number of Factors Allowed) ........................... 23
10. Conclusions .................................................................................................... 23
Appendix A: Proof of Shannon's 1948 Theorem Stating That the Uniform
Distribution is the "Most Uncertain" One Over a Finite Range of
Values ............................................................................................. 25
Appendix B: Original Text of the Author's Paper #IAC-08-A4.1.4 Entitled the
Statistical Drake Equation ............................................................... 28
References ........................................................................................................... 55
iii
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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, is about 50,000 ly.
R Gala.1y ,
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The thickness of the Galactic Disk at half-way from its center, is about 16,000 ly.
h a " 1t'-'Y '
The volume of the galaxy may then be approximated as the volume of the
corresponding cylinder, i.e.
(1)
Now consider the sphere around us having a radius r. The volume of such a sphere is
3
Vo ur _Sphere =
34
n
( Ef
-
Di 2s tance) (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:
-Va a -/a.,y =-Vo , ~,r_Sph -ere (3)
N
That is, upon replacing both (1) and (2) into (3):
3
4 ( Ef Dis lance )
2 - i'l" _ -___
i'l" RGa/axyh = 3 2 (4)
N 1
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)
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Ef _ ff , stance (N) = 3 6R VC2 N a!a ,y h C (5)
VN
where the positive constant C is defined by
C =V6 R~ala.,y hca/axy ,., 28845 light years . (6)
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=l. 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=l000, 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=l00000O, 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 t he following key diagram of the average ET distance vs. the
assumed number of communicating civilizations, N, in the galaxy right now (Figure 1):
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Average DIST A CE of the nearest ET civilization vs. the ASSUMED NUMBffi of ET civilizations in the Gah
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0
:i
150
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125
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100 \
750 '
"--
500
~
250
0
0 I 00000 200000 300000 400000 500000 600000 700000 800000 900000 I 000000
ASS MED NUMBER of civiliz ations in the Galaxy (that is, Nin the Drake equation)
Figure 1. DISTANCE LAW; i.e., the Average Distance (plot along the ver-1:ical 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.
"But is there anyone out there to talk to? With a third or a half a trillion stars in our
Milky Way galaxy alone, could ours be the only one accompanied by an inhabited
planet? How much more likely it is that technical civilizations are a cosmic
commonplace, that the galaxy is pulsing and humming with advanced societies, and,
therefore, that the nearest such culture is not so very far away - perhaps transmitting
from antennas established on a planet of a naked-eye star just next door. Perhaps
when we look up at the sky at night, near one of those faint pinpoints of light is a world
on which someone quite different from us is then glancing id ly at a star we call the Sun
and entertaining, for just a moment, an outrageous speculation.
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It is very hard to be sure. There may be several impediments to the evolution of a
technical civilization. Planets may be rarer than we think. Perhaps the origin of life is
not so easy as our laboratory experiments suggest. Perhaps the evolution of advanced
life forms is improbable. Or it may be that complex life forms evolve more readily, but
intelligence and technical societies require an unlikely set of coincidences - just as the
evolution of the human species depended on the demise of the dinosaurs and the ice­
age recession of the forests in whose trees our ancestors screeched and dimly
wondered. Or perhaps civilizations arise repeatedly, inexorably, on innumerable planets
in the Milky Way, but are generally unstable; so all but a tiny fraction are unable to
survive their technology and succumb to greed and ignorance, pollution and nuclear
war.
It is possible to explore this great issue further and make a crude estimate of N, the
number of advanced civilizations in the galaxy. We define an advanced civilization as
one capable of radio astronomy. This is, of course, a parochial if essential definition.
There may be countless worlds on which the inhabitants are accomplished linguists or
superb poets but indifferent radio astronomers. We will not hear from them. N can be
written as the product or multiplication of a number of factors, each a kind of filter,
every one of which must be sizable for there to be a large number of civilizations:
• Ns, the number of stars in the Milky Way galaxy.
• fp, the fraction of stars that have planetary systems.
• ne, the number of planets in a given system that are ecologically suitable for life.
• fl, the fraction of otherwise suitable planets on which life actually arises.
• fi, the fraction of inhabited planets on which an intelligent form of life evolves.
• fc, the fraction of planets inhabited by intelligent beings on which a communicative
technical civilization develops.
• fl, the fraction of planetary lifetime graced by a technical civilization.
Written out, the equation reads
N = Ns •jjJ •ne •fl ·ft ·Jc· fL (7)
All of the f's are fractions, having values between Oand 1; they will pare down the
large value of Ns.
To derive N we must estimate each of these quantities. We know a fair amount about
the early factors in the equation, the number of stars and planetary systems. We know
very little about the later factors, concerning the evolution of intelligence or the lifetime
of technical societies. In these cases our estimates will be little better than guesses. I
invite you, if you disagree with my estimates below, make your own choices and see
what implications your alternative suggestions have for the number of advanced
civilizations in