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\centerline{\bf The Fermi Paradox:}
\centerline{\bf Timescale for the Colonization of the Galaxy}
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The search for extraterrestrial intelligence (SETI) operates under a
few key assumptions. One of the most stringent is that other life is
``like us,'' that is, with similar biochemistry and preferring to live
on roughly earth-like planets. Our Galaxy, the Milky Way, contains
roughly 400 billion stars, the vast majority of which are long-lived
like our sun. It is about 100,000 light years across, and about 10
billion years old. Credited to physicist Enrico Fermi is the observation
that if a civilization had arisen on any single extrasolar planet at
any time in the past, they would have colonized the whole galaxy quite
quickly in comparison with the Galaxy's age. Such a civilization
should therefore already have a presence here in the solar system, and
``Fermi's paradox'' is the question ``So, where are they?''

Let us estimate the timescale for colonization of the galaxy in two
ways.

\pointbegin {\bf Population Growth}

One way to think about this is to look at population pressure as a
motivating force for Galactic expansion. Nobody likes overcrowding,
right? Let us start with the earth. We have 6 billion people and the
population doubles every 30 years. How many people can earth hold
before serious resource depletion occurs? I have no idea, but maybe it
is 100 billion = $10^{11}$ people. At that point, people are going to
want to go into space pretty badly. Lets assume there are 100 billion
habitable planets in the Galaxy. How long before all those planets get
filled up, too?

Procedure: Make a table of two columns, ``year'' and ``population.''
Start with year 2000, and population $5\times 10^9$. Then increment 30
years into the future, doubling the population each 30 years. How long
before we overpopulate the earth at $10^{11}$ souls? How long before
we overpopulate the Galaxy at ($10^{11}$ people)$\times$( $10^{11}$
planets)?


\point {\bf Spacecraft + Infrastructure Speed}

In the last example we may have exceeded the speed of light in
populating the Galaxy, which, after all, is 100,000 light years
across. Using the attached picture of a spiral galaxy (really M100 in
the Virgo galaxy cluster) let's chart out how long it would take a
``colonization wave'' to reach our sun from various points in the
galaxy. We will use the handy formula $d=vt$, or (distance) =
(velocity)$\times$(time).

At what velocity will the colonization wave travel? Well, it cannot
exceed the speed of light, as we will learn when we tackle special
relativity. However, experienced space travelers will always want to
travel VERY near the speed of light so that time dilation causes their
trip to be as short as possible. We expect that a technically advanced
civilization will build spaceships that travel very near the speed of
light. However, it may take time to build the ships and it may take
time for individual planets to build up the infrastructure needed for
colonization efforts. So an average speed may be only 0.1 or even 0.01
of the speed of light. Let us consider both cases.

Procedure: First we estimate the typical distance the little green men
will have to travel in order to arrive here at earth. On your picture
of M100 (representing the Milky Way seen from above) the scale is
approximately 1 cm = 5000 light years. Draw a dot representing the sun
about 2 cm down from top center. Label it ``sun.'' Draw a square 4 cm
on a side centered on the center of the galaxy. The stars are more
dense in this region, so there should be a better chance of having a
civilization evolve there. Draw 10 dots inside the square, in no particular
pattern, but filling the square uniformly. Number them. Draw 10 more
spots outside the square (but still in the Galaxy) again trying to
uniformly spread the points around. Pay no attention to where the sun
is while you do this. Number these, too. These dots represent possible
origin locations for extraterrestrial civilizations.

Next, list the 20 distances from each dot to the sun, in cm, and then
take the average. Convert to light years (1 cm = 5000 light years).
Now we want a slow-velocity (0.01c) time estimate and a fast-velocity
(0.1c) time estimate. For the time estimates, invert $d=vt$ to read
$t=d/v$. If you use distance in light-years, no other units
conversions are needed. Find the fast and slow colonization times.

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\centerline{\bf Analysis}

You now have three estimates for the time it takes for an alien
civilization to reach planet earth (one for population growth
pressure, two for colonization wave speed). How do they compare with the 10
billion-year age of the Galaxy? (Divide 10 billion by your estimates
to get $N_c$, the number of times the Milky Way could have been
colonized again and again --- given your 3 different colonization times.)

Do you agree that there is a ``Fermi paradox?'' Think of at least 5
ways to explain why we are NOT knee-deep in aliens right now.


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