Drake Equation Tutorial

By Jim Plaxco

In November 2006, I was a participant in a panel discussion Defining the Drake Equation at the Windycon Science Fiction Convention. My co-panelists were Seth Shostak of the SETI (Search for Extraterrestrial Intelligence) Institute; Bill Higgins, a physicist at Fermi National Accelerator Laboratory (Fermilab); and Bill Thomasson. You can see a picture of our panel at MidAmerican Fan Photo Archive Windycon 33 Saturday Panels. I have decided to turn the preparation that I did for that panel, and notes taken during the panel discussion, into a tutorial on the Drake Equation.

Drake Equation History

The year is 1960 and Frank Drake of the National Radio Astronomy Observatory (NRAO) in Green Bank, West Virginia undertakes the first attempt to find extraterrestrial civilizations. Dubbed Project Ozma, for a period of 6 hours a day for four months the NRAO radio telescope listens for radio signals of intelligent origin. None are found.

Within a year a meeting is hosted in Green Bank to explore the issue of extraterrestrial intelligence. Frank Drake needed to come up with an agenda for the meeting in order to provide some structure to the discussion. To serve as an agenda, he devises the Drake Equation. Sometimes known as the Sagan-Drake Equation in the past, the meeting was attended by approximately a dozen interested parties.

Drake Equation Overview

The Drake Equation is an attempt to encapsulate all the variables that would be relevant to establishing the number of intelligent civilizations that existed in the Milky Way galaxy and which were broadcasting radio signals at this particular point in time. The Drake Equation is composed of seven terms. The first six are used to compute the rate at which intelligent civilizations are being created and the final term identifies how long each lasts on average as a broadcasting civilization. It is worth stressing that the Drake Equation applies only to intelligent civilizations in the Milky Way galaxy. It does not apply to civilizations in other galaxies because they are too distant to be able to detect their radio signals.

The Drake Equation is:
N = R * f_p * n_e * f_l * f_i * f_c * L

where:
N = The number of broadcasting civilizations.
R = Average rate of formation of suitable stars (stars/year) in the Milky Way galaxy
f_p = Fraction of stars that form planets
n_e = Average number of habitable planets per star
f_l = Fraction of habitable planets (n_e) where life emerges
f_i = Fraction of habitable planets with life where intelligent evolves
f_c = Fraction of planets with intelligent life capable of interstellar communication
L = Years a civilization remains detectable

According to the Wikipedia entry for the Drake Equation, the following values were those used in the original formulation of the Drake Equation:
R = 10
f_p = 0.5
n_e = 2.0
f_l = 1.0
f_i = 0.01
f_c = 0.01
L = 10000

Plugging Drake's original numbers into the Drake Equation produces a value of 10 for the number of broadcasting civilizations in our galaxy. Now lets go through each of the terms in detail.

R - The rate of formation of Suitable Stars in the Milky Way Galaxy

Estimates for the number of stars in the Milky Way vary from a low of 100 billion to a high of 400 billion. Estimates for the age of the Milky Way also vary from a low of 800 million years to a high of 13 billion years. If we go with the lowest star count and the oldest age for the galaxy, the average rate of star formation works out to 7.7 new stars per year. If we go with the highest star count and the youngest age for the galaxy, the average rate of star formation becomes 500 new stars per year.

An important caveat to the above values is that the rate of star formation in the galaxy is not constant over time. In the galaxy's younger days, stars were being formed at a much higher rate. Today, estimates for the overall star formation rate range from 5 to 20.

Another caveat is that not all stars are created equal. For example, very massive stars are not considered suitable. Some versions of the Drake Equation use the R term for the overall rate of star formation and then add a second term to estimate the fraction of these stars that are like our own Sun. A suitable star would be one that has a reasonably long life (approximately 10 billion years for our Sun which is now in midlife) and sized so that the fusion process that powers the star produces the right amount of energy to sufficiently warm the planets but not turn them into toast. Estimates are that the rate of formation of Sun sized stars is on the order of 1 per year.

f_p - The Fraction of Stars with Planets

At the time the Drake Equation was created, the only planets that were known were those of our own solar system. Since that time approximately 200 extrasolar planets have been discovered.

When the Drake Equation was created, it was thought that planets would only be found in single star systems. It was believed that gravitational disruptions in multiple star systems would prevent planets from forming. This hypothesis removed approximately 50 percent of the stars from consideration. It has now been shown theoretically that these multiple star systems can have planets. For example, if a planet is in orbit around a star that is X units of distance away, then the planet's orbit can be stable if the companion star is more than 5X units away. Alternatively, if two stars are X units away from one another, then a planet that orbits these stars from a distance of more than 5X units should have a stable orbit.

So what fraction of stars have planets? Estimates range from a low of 5% to a high of 90%. If you use a value of 0.1 you are saying that you believe that 1 in 10 stars will have planets. Alternatively if you use a value of 1.0 you are saying that every single star will have planets.

n_e - The Average Number of Habitable Planets per Star

In his original equation, Drake optimistically assigned a value of 2 to this parameter meaning that there are on average two Earth-like planets per star for those stars with planets. Factors that must be considered in arriving at a value for this parameter are the chemical composition of the solar nebula from which the planets were created (the presence of sufficient quantities of the necessary elements) and the idea of a star's habitable zone (the range of orbital distances within which liquid water can exist)

Something else to consider is that our idea of habitable may be too restrictive. Does life require an Earth-like planet? This is a question of life as we know it versus life as we don't know it. However, from a biochemical standpoint, it is hard for us to imagine life that does not require liquid water.

Choosing a value of 1.0 for this parameter means that you think that every star with planets will have one habitable planet. A value of 0.5 means that there will be one habitable planet for every two stars with planets.

f_l - The Fraction of Habitable Planets Where Life Emerges

This parameter is something of a wildcard in that we only have one example of life. It is difficult for us to say how easy or hard it is for life to start given suitable environmental conditions. One interesting point to consider is this:

the Earth is approximately 4.5 billion years old;
the period of heavy bombardment during which the planets were pummeled by debris left over from the birth of the solar system ended about 3.8 billion years ago;
the oldest known sedimentary rocks and deposits, found in northwestern Australia, are estimated to be 3.5 - 3.8 billion years old;
the oldest known fossil evidence of life is of cyanobacteria found in these deposits dated at 3.5 billion years old.

The implication of this is that life got started rather quickly on Earth. The big unknown is just how common are the conditions which resulted in life. This is one reason why the search for evidence of past life on Mars is so important. Finding or not finding evidence of past and/or present life on Mars will help us to better answer the question of the likelihood of life elsewhere in the galaxy and universe.

Choosing a value of 0.01 for this parameter means that you think that life develops on only 1 of every 100 habitable planets whereas a value of 1.0 means that life develops on every habitable planet.

f_i - The Fraction of Planets With Life Where Intelligence Life Evolves

Given that life evolves on a planet, how likely is it that intelligent life will appear? This is another big unknown. Of all the millions of species that have ever existed on Earth, only one has evolved the level of intelligence necessary to develop technology.

Further, while very simple life appeared very quickly on Earth, complex life took far longer to develop. Given that there is not a parameter to distinguish microscopic life (which lacks the complexity to develop intelligence) from the development of complex macroscopic life, this aspect must be taken into account in the context of this parameter.

Whereas Drake believed that life would develop on every planet that had habitable conditions, he estimated that intelligent life would emerge on only 1 of every 100 of these planets

Choosing a value of 0.001 for this parameter means that you think that intelligent life will appear on only 1 of every 1000 planets with life. A value of 1.0 means that the development of intelligent life is a certainty on those planets where life develops

f_c - The Fraction of Intelligent Civilizations with Interstellar Communication

So what if aliens have no equivalent of a Maxwell or a Morse or a Marconi or an Edison? They may be smart enough to construct towns and transportation but do they ever invent radio? Drake was of the opinion that 1 out of every 100 civilizations would discover radio. What do you think?

A value of 1.0 means that every civilization develops radio and a value of 0.001 means that only one in a thousand civilizations develop radio.

L - The Number of Years an Intelligent Civilization Remains Detectable

The L parameter turns the equation from a rate into a number. It is also a number for which there is no real basis for the assignment of a value. We are the only intelligent civilization we know of and we do not know how long we will remain detectable. A conservative estimate for this value would be 50 years based on our own experience to date. Drake felt that 10,000 years was a good guess.

N - The Answer is the Number of Detectable Civilizations at this Time

And the answer is N - the number of intelligent civilizations that are broadcasting their presence to the Universe.

Experiment with the Drake Equation

To facilitate your own experimentation with the Drake Equation, I have created an OpenOffice Calc spreadsheet and a Microsoft Excel spreadsheet. If you do not have OpenOffice, I strongly encourage you to get it. OpenOffice is the free, open source alternative to Microsoft Office. You can learn more at the OpenOffice web site

Download the OpenOffice DrakeEquation.ods spreadsheet

Download the Microsoft Excel DrakeEquation.xls spreadsheet

In the spreadsheet you will find that I have inserted my own values for the seven parameters. Following is an explanation for the values I used.

R = 2 which is double the estimated rate of formation of Sun-like stars but well below the maximum estimate of 20 new stars per year in the galaxy.

f_p = 0.45 which is 1/2 the high estimate of 90% of these stars having planets.

n_e = 0.50 because I do not believe that every star that has planets will have habitable planets. Recall that Drake assigned a number of 2 for this parameter. My optimistic estimate is that for every two stars with planets, there will be one habitable planet.

f_l = 0.2 with no sound basis, I decided that life will emerge on only 1 in 5 habitable planets.

f_i = 0.05 again guessing that intelligent life will develop on only 5 out of every 100 planets with life.

f_c = 0.5 because I am optimistic that if there is intelligent life, there is at least a 50-50 chance that they will develop the technology necessary for interstellar communication.

L = 500 because I am not as optimistic as Frank Drake about the number of years for which an intelligent civilization will be broadcasting its presence by way of radio transmissions.

I was very much surprised to see that the combination of values that I used yielded a result of 1.13 currently broadcasting civilizations. That makes us the one. Going back and changing only the L parameter to Drake's value of 10,000 yields 22.5 broadcasting civilizations. If we were to assume that the Milky Way is a cylinder with a radius of 50,000 light years and a thickness of 1,000 light years, then there would be one broadcasting civilization for every 349 billion cubic light years of space.

Now consider this. Let's make the following assumptions:

the radius of the Milky Way is 50,000 light years
there are currently 22.5 broadcasting civilizations
all civilizations lie on the galactic equator in a 2 dimensional distribution

Given these assumptions, this means that on average each of these civilizations are separated by a distance of just over 21,000 light years. That means that any civilization that began broadcasting less than 21,000 years ago, like us for example, would not yet be detectable.

Conclusion

The Drake Equation must be one of the swaggiest (SWAG being the acronym for Scientific Wild-A** Guess) equations ever created because of the uncertainty associated with its parameters. The Drake Equation does do a great job of identifying and categorizing the relevant parameters. It also accomplishes the task of providing structure to the ongoing debate about the search for extraterrestrial intelligence and the likelihood of its existence. The large degree of uncertainty associated with so many of its parameters does tell us one important thing: that we have a lot more to learn.