The Mathematics of Drake's Equation - Planets, stars, and life elsewhere? (ppt presentation)
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The Drake Equation
Does life existelsewhere
in the universe?
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And might there be other advanced civilizations out there?
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What are the chances of technologically-advanced
civilizations elsewhere in the universe?
And how many such civilizations, if any,
might there be?
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We don’t know yet……
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But we can conduct a preliminary analysis using “The Drake Equation”
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The math that we will use is known as The Drake Equation
N = ( R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
The equation was originally developed by
Dr. Frank Drake
When he was professor of physicsand astrophysics at the
University of California, Santa Cruz
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
What possibilities can its mathematics suggest?
Drake’s Equation
We would like to estimate “N” – the
potential numbers oftechnologically
advanced civilizations
elsewhere in the universe
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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Drake’s Equation
We would like to estimate “N” – the
potential numbers oftechnologically
advanced civilizations
elsewhere in the universe
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
The number will vary, of course, with different starting assumptions
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Drake’s Equation
We would like to estimate “N” – the
potential numbers oftechnologically
advanced civilizations
elsewhere in the universe
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s equation allows us to test alternateassumptions in a methodical and analytic way
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The good news is that the math itself will be done by this presentation
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
We start with an estimateof the number of stars
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Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Footnote: After completing this introductory presentation , we
could use Drake’s equation to test other estimates
such as the “fraction of stars with suitable characteristics” (not all stars are sun-like, for example)
We start with an estimateof the number of stars
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Fractionof stars that have
planets
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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We can employ different estimates here to testthe effects if planets turn out to be extremely
common - or if they are comparatively rare
Fractionof stars that have
planets
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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What fraction of planets areHABITABLE
(earth-like, for example)
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Not all planets , for example, are likely to be suitable for life
We want only ‘earth-like’ planets or others whose conditions allow life to exist
What fraction of planets areHABITABLE
(earth-like, for example)
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What portion of habitable planets are actually inhabited
by LIFE-FORMS of any sort?
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Marine plankton
Yeast cells
Anything like these?
Artw
ork courtesy of R. Fem
mer
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
What portion of INHABITED planetsinclude ‘intelligent’ life?
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Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Tool-making? Mathematical? Technological?
On earth, there are multiple ‘degrees’ of
intelligence
Which organisms would satisfy the definition we would use? Chimps?
Dolphins? Only humans?
What portion of INHABITED planetsinclude ‘intelligent’ life?
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*
What fraction of planets with intelligent beings will also have
CIVILIZATIONS?
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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What fraction of planets with intelligent beings will also have
CIVILIZATIONS?
And must they be technologically-advanced civilizations or not?
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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We will be saving this factor for later
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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Part Two
Let’s insert some numerical estimates and see what results we obtain
Early analyses using Drake’s equation oftenemployed estimates of the number of stars
in the Milky Way galaxy
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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Early analyses using Drake’s equation oftenemployed estimates of the number of stars
in the Milky Way galaxy
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
For this presentation, however, assume that an approximate number of stars in the entire universe is something like 1 x 10
This would mean that the value of R* would be100,000,000,000,000,000,000,000
total stars
23
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23If the number of stars present in the universe is 1 x 10
What if PLANETS are RARE and only
1/10th of 1% have planets?
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
23If the number of stars present in the universe is 1 x 10
What if PLANETS are RARE and only
1/10th of 1% have planets?
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
31 out of 1 x 10
…1 out of 1,000…
100 000 000 000 000 000 000 000
1000
Do the calculation1. x 10 23
100 000 000 000 000 000 000 000
1000
Do the calculation
100 000 000 000 000 000 000
1. x 10 23
100 000 000 000 000 000 000 000
1000
Do the calculation
100 000 000 000 000 000 000
1. x 10 23
1 x 10 divided by 1 x 10 = 1 x 10 23 203
So if there are approximately 100,000,000,000,000,000,000 planets
What if EARTH-LIKE planets are rareand only 1/10th of 1% of planets are HABITABLE?
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
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So if there are approximately 100,000,000,000,000,000,000 planets
What if EARTH-LIKE planets are rareand only 1/10th of 1% of planets are HABITABLE?
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
1 out of 1 x 10
…1 out of 1,000…
3
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100 000 000 000 000 000 000
1000
Do the calculation1. x 10 20
100 000 000 000 000 000 000
1000
Do the calculation1. x 10 20
100 000 000 000 000 000
100 000 000 000 000 000 000
1000
Do the calculation1. x 10 20
1 x 10 divided by 1 x 10 = 1 x 10
100 000 000 000 000 000
20 173
1 x 10 divided by 1 x 10 = 1 x 10
100 000 000 000 000 000
20 17
100 000 000 000 000 000 000
1000
Do the calculation1. x 10 20
So this would suggest approximately 100,000,000,000,000,000
planets with conditions suitable for life
3
Even if, however, there were approximately100 000 000 000 000 000
habitable earth-like planets
What if development of LIFE on habitable planets is also RAREand only 1/10th of 1% of habitable planets are hosts to life ?
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Even if, however, there were approximately100 000 000 000 000 000
habitable earth-like planets
What if development of LIFE on habitable planets is also RAREand only 1/10th of 1% of habitable planets are hosts to life ?
*
Drake’s Equation
N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
1 out of 1000
100 000 000 000 000 000
1000
Do the calculation1. x 10 17
100 000 000 000 000 000
1000
Do the calculation1. x 10 17
100 000 000 000 000
100 000 000 000 000 000
1000
Do the calculation1. x 10 17
100 000 000 000 000
1 x 10 divided by 1 x 10 = 1 x 10 17 143
100 000 000 000 000 000
1000
Do the calculation1. x 10 17
100 000 000 000 000
1 x 10 divided by 1 x 10 = 1 x 10 17 143
So this would suggest approximately100,000,000,000,000
planets with some sort of life
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s Equation
So if there are approximately 100 000 000 000 000
planets with life-forms of some sort,
What if INTELLIGENT life is a rare occurrence and only 1/10th of 1% of planets develop intelligent beings?
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s Equation
So if there are approximately 100 000 000 000 000
planets with life-forms of some sort,
What if INTELLIGENT life is a rare occurrence and only 1/10th of 1% of planets develop intelligent beings?
…1 out of 1,000…
1 out of 1 x 10 3
100 000 000 000 000
1000
Do the calculation1. x 10 14
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100 000 000 000 000
1000
Do the calculation1. x 10 14
100 000 000 000
1 x 10 divided by 1 x 10 = 1 x 10 14 113
If correct, this would mean approximately100 000 000 000
planets with intelligent life
ADVANCED CIVILIZATIONS
Photo courtesy of N
AS
A
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
So even if there might exist approximately 100 000 000 000
planets that are home to some form of intelligent life,
What if ADVANCED CIVILIZATIONS rarely develop and only 1/10th of 1% of planets develop advanced civilizations
Drake’s Equation P
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*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
So even if there might exist approximately 100 000 000 000
planets that are home to some form of intelligent life,
What if ADVANCED CIVILIZATIONS rarely develop and only 1/10th of 1% of planets develop advanced civilizations
Drake’s Equation
…1 out of 1,000…
1 out of 1 x 10 3
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100 000 000 000
1000
Do the calculation1. x 10 11
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100 000 000 000
1000
Do the calculation1. x 10 11
100 000 000
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100 000 000 000
1000
Do the calculation1. x 10 11
100 000 000
1 x 10 divided by 1 x 10 = 1 x 10 11 83
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100 000 000 000
1000
Do the calculation1. x 10 11
100 000 000
1 x 10 divided by 1 x 10 = 1 x 10 11 83
So this would suggest the possibility of100 000 000
planets with technological civilizations
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Think how amazing it would beif 100,000,000 planets
with civilizations actually exist
This seems very impressive
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s Equation
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s Equation
Recall, however, this factor , which we deferred earlier
Can you guess what it is?
Recall, however, this factor , which we deferred earlier
It is
…. time ….
Can you guess what it is?
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s Equation
because our own planet has had dozens of great civilizations,but only over the last century do we meet a definition
of “technologically advanced” communicative civilizations
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s Equation
This factor is
…. time ….
and it is very sobering
For example, radio telescopes
Thus, this factor ,
L , represents thepercentage
of a planet’s lifetime
that is marked by the presence of intelligent beingswith a technologically-advanced communicative civilization
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Drake’s Equation
Images courtesy of R
. Femm
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If civilizations do not begin instantly,
take a long time to appear and develop,
and do not last forever
and only exist FOR TINY FRACTIONS of their planet’s total lifetime
or for only a tiny portion of the totalelapsed time of the
universe itself
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Then we must divide once again
Suppose that somehow 1 x 10 advanced civilizations manage to develop
If, however, they only exist for a 1 x 10 portion of their planet’s lifetime **
8
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Then
1 x 10 8
1 x 10 71 x 10 1= = 10
Ear
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illi
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ears
old
1 x 10 8
1 x 10 71 x 10 1= = 10
Thus, given the estimates suppositions, and
assumptions that we haveused in this sample analysis
Just ten planets with technologically advanced civilizations
might exist ata particular moment in time
Employing the estimates and mathematics used in
our example, there may be only TEN
other advanced civilizations
out there somewhere at this moment in time
or there could be NONE at all
we may be it…
It makes you think - doesn’t it ?
What responsibility does thisplace upon our shoulders?
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Footnotes
For convenience, this presentation assumed a 1/10 th of 1% probability for each factor in its discussion
But the percentages that one chooses to assign to each factor can and should be modified on the basis of humankind’s ever-increasing knowledge and understandings
For example, solar systems with multiple planets may not be rare at all, but may be very common so that the equation could be run again to reflect a much higher number of planets
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
On the other hand, many stars are very different than our sun and may be unsuitable for sustaining life as we know it
In that case, the value that that we assign to factor R* should probably be adjusted
We could adjust R* downward, for example, by adding a factor fs to the equation to incorporate a “fraction of suitable stars” into our estimates
Footnotes
*N = (R ) ( fp ) (ne) ( fl ) ( fi ) ( fc ) ( L)*
Footnotes
Many scholars and authors have utilized and discussed Drake’s equation
A web search of books and other resources will reward viewers of this presentation with many additional insights concerning its implications and applications
Particular credit should go to Frank Drake, however, and his fellow astronomer Carl Sagan
Made available courtesy of
The Wecskaop ProjectWhat Every Citizen Should Know About Our Planet
Images courtesy of R
. Femm
er
Images courtesy of R
. Femm
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Images courtesy of R
. Femm
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Images courtesy of R
. Femm
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