AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

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UNCLASSIFIED/ /fOR OFFI&Ial.k Wlili QPU,¥
Equation, of the corresponding probability density
function fET_Distance(r) : here r is the distance
between us and the nearest ET civilization
assumed as the independent variable of its own
prob~bility density function. The ensuing sections
provide more mathematical details about this
.firr_oistana, (r) such as its mean value, variance,
standard deviation, all central moments, mode,
median, cumulants, skewness and kurtosis.
CLASSICAL, NON-PROBABILISTIC
DERIVATION OF THE DISTANCE OF THE
NEAREST ET CIVILIZATION
Consider the Galactic Disk and assume that:
1) The diameter of the Galaxy is (about) 100,000
light years, (abbreviated ly) i.e. its radius,
R calaxy• is about 50,000 ly.
2) The thickness of the Galactic Disk at half-way
from its center, hcalaxy , is about 16,000 ly.
Then
3) The
volume
of the
Galaxy
may
be
approximated
as
the
volume
of
the
corresponding cylinder, i.e.
V
-
R1
h
(100)
Gal«ry - 1l Ga/,uy
4) Now consider the sphere around us having a
radius r. The volume of such as sphere is
4 (Er Distance)3
VOw·_ Sphere =31l'
2
(101)
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 space 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 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
aware that this is weak point in the reasoning.
Having thus assumed that ET Civilizations
are UNIFORMLY SPACED IN THE GALAXY'
we can write down this proportion:
Vcal<Lry = Vo11r_ Sphere
(102)
N
That is, upon replacing both (100) and (101) into
(102):
4 (Er Distance) 3
1l RJalaxyh = 3" ---=- 2-­
(103)
N
I
The only unknown in the last equation is
ET _Distance, and so we may solve for it, thus
getting the:
(A VERA GE) DISTANCE BETWEEN ANY PAIR
OF NEIGHBOURING CIVILIZATIONS IN
THE GALAXY
116 R 2
h
_ V
Galaxy
C
Er D.
(104)
istance -
1 ,.--;
= ­
-
tN
ifii
where the positive constant C is defined by
C =3 6 Ria1a.,y h calcuy ""28845 light years . ( l 05)
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 DE SITY FUNCTION FOR
ET_DISTA CE
~he probability density function (pdt) yielding
the distance of the ET Civilization nearest to us in
the Galaxy and pre ented in this section, was
discovered by this author on September 5th' 2007.
He did not disclose it to 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 February 5-6-7-8, 2008.
This meeting was also attended by SETI Institute
experts Jill Tarter, Seth Shostak, Doug Vakoch,
Tom Pierson and others. During this author's talk,
Paul Davies suggested to call "the Maccone
distribution" the new probability density function
that yields the ET_Distance and is derived in this
section.
UNCLASSIFIED/ ffOR OFFI&Ial.L Wlili QPII..¥
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