Steve Willner <swillner@cfa.harvard.edu>

There are various forms of it, but basically it is a means of doing
boundary calculations for the prevalence of intelligent life in the
universe.  It might take the form of saying that if there are:

X   stars in the Galaxy, of which
Y % have planets, of which
Z % can support life, on which
A % intelligent life has arisen, with
B   representing the average duration of civilizations

then you fool around with the numbers to figure out how close on average
the nearest civilization is.  There are various mathematical expressions
for this formula (see below), and there are variations on how many terms
the equations include.

The problem, of course, is that some of the variables are easy to pick
(e.g., stars in the Galaxy), some are under study (e.g., how many
stars have terrestrial-like planets), and others are just flat-out
wild guesses (e.g., duration of civilization, where we are currently
running an experiment to test this here on Terra of Sol).


One useful form says the number of detectable civilizations is:
        N  = R * fp * ne * fl * fi * fc * L
 where        
        R  = "the average rate of star formation in the region in question",
        fp = "the fraction of stars that form planets"
        ne = "the average number of planets hospitable to life per star"
        fl = "the fraction of those planets where life actually emerges"
        fi = "the fraction of life-bearing planets where life evolves into
              intelligent beings"
        fc = "the fraction of planets with intelligent creatures capable
              of interstellar communication"
        L = "the length of time that such a civilization remains
              detectable".

(If you want some definition of civilization other than detectability,
just change your definition of fc and L accordingly.)

Can we provide reasonable estimates for any of the above numbers?  The
"social/biological" quantities are at best speculative and aren't
appropriate for this newsgroup anyway.  (For arguments that they are
quite small, see biologist Ernst Mayr's article in _Bioastronomy
News_, Quarter 1995, <URL:http://planetary.org/tps/mayr.html>.)  Even
the "astronomical" numbers, though determinable in principle, have
considerable uncertainty.  Nevertheless, I will attempt to provide
reasonable estimates.  I'll take the "region in question" to be the
Milky Way Galaxy and consider only cases "similar to" our solar
system.

For R, I'm going to use only stars with luminosities between half and
double that of the Sun.  Dimmer stars have a very small zone where
Earth-like temperatures will be found, and more luminous stars have
relatively short lifetimes.  Near the Sun, there are about 4.5E-3 such
stars in a cubic parsec.  I'm only going to consider stars in the
Galactic disk, which I take to have a scale height of 660 pc and scale
length of between 5 and 8 kpc.  (Stars outside the disk either have
lower metallicity than the Sun or live in a very different environment
and may have formed in a different way.)  The Sun is about 8 kpc from
the Galactic center, and thus in a region of lower than maximum star
density.  Putting everything together, there ought to be around 1.4E9
stars in the class defined.  This represents about 1% of the total mass
of the Galaxy.  The age of the Sun is about 4.5E9 years, so the average
rate of formation R is about 0.3 "solar like stars" per year.

Planets are more problematic, since extrasolar planets cannot generally
be detected, but it is thought that their formation is a natural and
indeed inevitable part of star formation.  For stars like the Sun, in
fact, there is either observational evidence or clear theoretical
justification for every stage of the planet formation process as it is
currently understood.  We might therefore be tempted to take fp=1 (for
stars in the luminosity range defined), but we have to consider binary
stars.  A second star may disrupt planetary orbits or may somehow
prevent planets forming in the first place.  Because about 2/3 of the
relevant stars are in binary systems, I'm going to take fp=1/3.

Now we are pretty much out of the range of observation and into
speculation.  It seems reasonable to take ne=1 or even 1.5 on the basis
of the Solar system (Earth and Mars), but a pessimist could surely take
a smaller number.  You can insert your own values for the probabilities,
but if we arbitrarily set all of them equal to one
  N <= 0.1 L
seems consistent with all known data.

A more detailed discussion of interpretation of the Drake equation and
the factors in it can be found in Issue 5 of SETIQuest.