Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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Equation, of the corresponding probability density
function .f~. 1 Disim,"'(r) here r is the distance
hetwccn u~ and the nearest ET civilintion
assumed as the independent variahle of its own
probability demity function. The emuing sediom
provide more mathematical details about this
fr:T_Di,1.rn"'(r) such as its mean value, variance,
standard deviation, all central moments, mode,
median, cumulant\, skewness and kurtosis.
CLASSICAL, NON-PROBABILISTIC
DERIVATION OF THE DISTANCE OF THE
NEAREST ET CIVILIZATION
Cnmidcr the Galactic Disk and a~sumc that:
1) The diameter of the Galaxy is (about) 100,000
light years. (abbreviated ly) i.e. its radius,
Rr.-ai<i.". is about 50,000 ly.
2) The thickne~s of the Galactic Disk at half-way
from its center, lieu/an , is ahout 16,000 ly.
Then
3) The volume of the
approximated as the
corre~ponding cylinder, i.e.
Galaxy
volume
may
of
be
the
(1001
4) Now consider the sphere around us having a
radim r. The volume of such a~ ~phere is
4 (Er Diqancel'
Vo,,,_s1,11,•1,· = 31r 2 (Ill l)
In the laq equation, \Ve had to divide the di~tance
"ET Di~tancc'' between ourselves am! the nearest
ET Civilization by 2 because we are now going to
make the unwarranted assumption that all ET
Civilization.~ are equally space from each other in
the Galaxy! This is a crazy assumption, clearly,
and should he replaced by more ~cientifically-
grounded assumptions as soon as we kmnv more
about our Galactic Neighbourhood. At the moment,
however, this is the best guess that we can make,
and so we shall take it for granted, although we are
a\vare that this is weak point in the reasoning.
Having thns assumed that ET Civilizations
are UNIFORMLY SPACED IN THE GAUXY,
we can write down this proportion:
46
( I02)
N
That i\, upon replacing hoth (100) and (IOI) into
(102J:
( I03)
The only 1111know11 m the last equation is
ET_Distance, and JO 11'e ma_v solve fbr it, thus
:,?elfin:,? 1he:
(AVERAGE) DISTANCE BETWEEN ANY PAIR
OF NEIGHBOURING CIVILIZATIONS IN
THEGAUXY
C
\IN (104)
where the positive constant C is defined by
C = ~ 6 RL,1"''" hGalrl\\ ""28845 light years. ( 105)
Equations (104) and (105) are the starting point for
our first application of the Statistical Drake
equation, that we discuss in detail in the coming
sections of this paper.
PROBABILISTIC DERIVATION OF THE
PROBABILITY DENSITY FUNCTION FOR
ET_DISTANCE
The probability density function (pdf) yielding
the distance of the ET Civilization nearest to us in
the Galaxy and presented in this section, was
discovered by this author on Septemher 5th , 2007.
He did no! disclose it lo other scientists until the
SETI meeting run by the famous mathematical
physicist and popular science author, Paul Davies,
at the "Beyond" Center of the University of
Arizona at Phoenix, on 1-iebruary 5-6-7-8, 2(1(18.
Thi~ meeting was also attended hy SETI Institute
experts Jill Tarter, Seth Shostak, Doug Vakoch,
Tom Pierson and others. During this author's talk,
Paul Davies suggested to call "the Macrnne
distribution'' the new probability density function
that yields the ET_Distance and i~ derived in thi~
section.
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