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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.  Let's chart out how long it would take a ``colonization
wave'' to reach all the way across the Milky Way. 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, but 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 to expand colonization efforts to stars even
further away from the home world. So an average speed may be only 0.1
or even 0.01 of the speed of light. Let us consider both cases.

Procedure: Assume an (overestimated) distance through which a
colonization wave would travel: 100,000 light-years. Why is this an
overestimate?
For colonization timescales in this method, let us do it twice. 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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\hrule
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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 these 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.)  Make a little table of two columns:
``Milky Way colonization time'' and ``$N_c$''. Give your results for
the three cases in three rows below the headings.

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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