The Drake Equation

This page was copied from Nick Strobel's Astronomy Notes. Go to his site at www.astronomynotes.com for the updated and corrected version.

The Drake Equation is a way to estimate the number of communicating advanced civilizations (N) inhabiting the Galaxy. It is named after Frank Drake who first summarized the things we need to know to answer the question, "how many of them are out there?" The equation breaks this big unknown, complex question into several smaller (hopefully manageable) parts. Once you know how to deal with each of the pieces, you can put them together to come up with a decent guess.

\[N = R_* \times f_p \times n_E \times f_l \times f_i \times f_c \times L\]

R_* =

average star formation rate (number of stars formed each year). Roughly 200 billion stars in the Galaxy / 10 billion years of Galaxy's lifetime = 20 stars/year.

f_p =

average fraction of stars with planets. Astronomers are currently focussing on single star systems so planets would more likely have stable orbits. With our current technology it is also easier to find planets around single star systems, though we have found planets in binary and multiple star systems. The recent discovery of planets in stable orbits in binary and multiple star systems was certainly a surprise for planetary dynamicists who model planet orbits and boosts the f_p term from what we originally thought. Astronomers are looking at stars where the star is not too hot (hence, short life) nor too cold (hence, narrow habitable zone and tidal locking of rotation). Also, we should look at stars that have signatures of "metals" (elements heavier than helium) in their spectra---stars in the galactic disk and bulge. Leftover "metal" material from the gas/dust cloud that formed the star may have formed Earth-like planets. The census of stars by the Kepler mission has shown that at least 70% of ordinary stars including those hotter than the Sun and the great majority cooler than the Sun have a planet of some size orbiting them (the percentage will only increase as the Kepler team continues processing the data).

n_E =

average number of Earth-like planets per suitable star system. The planet has a solid surface and liquid medium on top to get the chemical elements together for biochemical reactions. The planet has strong enough gravity to hold onto an atmosphere. Current statistics from the Kepler mission show that 23% of sun-like stars have a planet less than 3 Earth diameters in size. However, the dividing line of size between a rocky planet and a more gaseous one like Neptune is about 1.5 Earth diameters.

f_l =

average fraction of Earth-like planets with life. Extrasolar life will probably be carbon-based because carbon can bond in so many different ways and even with itself. Therefore, carbon can make the large and complex molecules needed for any sort of biological processes. Also, carbon is common in the galaxy. Many complex organic molecules are naturally made in the depths of space and are found in molecular clouds throughout the Galaxy. The rarer element silicon is often quoted as another possible base, but there are problems with its chemical reactions. When silicon reacts with oxygen, it forms a solid called silica. Carbon oxidizes to form a gas. Silicon has a much lesser ability to form the complex molecules needed to store and release energy. See the previous section and Raymond Dessy's article at Scientific American's "Ask the Experts -- Space" web site for further discussion of the limitations of silicon chemistry. The habitable zone idea would be contained either in this term or put in the n_E above.

f_i =

average fraction of life-bearing planets evolving at least one intelligent species. Is intelligence necessary for survival? Will life on a planet naturally develop toward more complexity and intelligence? Those are questions that must be answered before "reasonable" guesses can be put in for f_i. Take note that on the Earth, there is only one intelligent (self-aware) species among millions of other species. (Perhaps, whales, dolphins, and some apes should be considered intelligent too, but even still, the number of intelligent species is extremely small among the other inhabitants of our planet.) Sharks have done very well for hundreds of millions of years and they are stupid enough to eat tires! Bacteria have thrived on the Earth for billions of years. Being intelligent enough to read an astronomy textbook is very nice but it is not essential for the mandates of life.

Bacteria and other simple forms of life have been found in some very extreme conditions on the Earth. Simple forms of life can even survive long passages through space. However, we will not be talking with such simple forms of life. We are more interested in complex life---multi-cellular animal and plant life. Complex life is more fragile than simple life, so while new research seems to increase the f_l term, the f_i term might be smaller than was initially thought. If the rise of intelligence is accidental, then the f_i term will be nearly zero. If intelligence is an emergent property of any biological system, then intelligence would be an expected result of complex life, boosting the f_i term. An emergent property is a property that arises when a simple system with a sufficient number of interacting parts spontaneously becomes more complex. The property cannot be seen in individual members of a system but can be seen only in large assemblages of them---complex behaviors of a collective that are more than just the sum of the individual members.

f_c =

average fraction of intelligent-bearing planets capable of interstellar communication. The intelligent life we will be able to talk to will have to use some sort of symbolic language. Will intelligent life want to communicate to beings of a different species? The anthropologists, psychologists, philosophers, and theologians will have a lot of input on this term in the Drake Equation.

L =

average lifetime (in years) that a civilization remains technologically active. How long will the civilization use radio communication? Will they be around long enough to send messages and get a reply? Even if we manage to take better care of the Earth and each other, our technology is changing so we may not use radio communication. It used to be that our television and music/talk broadcasts were "over the air" using radio and microwaves. Some of that radio/microwave energy leaked out into interstellar space (that may be why all of the extraterrestrials are staying away). Those broadcasts are now happening mostly via cable. Now the voice communication that used to be via cable is now happening mostly with radio/microwaves (land lines vs. cellular phones). There is also the fact (yes, I use "fact") that individual species have changed and died out as their environment changed. Humans will be very different in a million years from now (if we survive that long). Because of the huge interstellar distances in the Galaxy, the L term is the most significant constraint on communicating with an extraterrestrial.

Another version of the Drake Equation (used by Carl Sagan, for example) replaces R_* with N_*---the number of stars in the Milky Way Galaxy and L with f_L---the fraction of a planetary lifetime graced by a technological civilization. Once you have found N, the average distance d between each civilization can be found from Nd³ = volume of Galaxy = 5.65 × 10¹² light years³. Solve for the average distance between each civilization = (volume of the Galaxy/N)^1/3 light years.

The certainty we have of the values of the terms in the Drake Equation decreases substantially as you go from R_* to L. Astronomical observations will enable us to get a handle on R_*, f_p, n_E, and f_l. Our knowledge of biology and biochemistry will enable us to make some decent estimates for f_l and some rough estimates for f_i. Our studies in anthropology, social sciences, economics, politics, philosophy, and religion will enable us to make some rough guesses for f_i, f_c, and L but those terms involve sociological factors (behaviors of alien civilizations) and we have a hard enough time trying to understand how our own human societies interact (our predictions often fail miserably). Furthermore, numerical analysis is not used in philosophy and religion.

Some astronomy authors are so bold as to publish their guesses for all of the terms in the Drake equation even though estimates of n_E and f_l are only rough and values quoted for the last three, f_i, f_c, L, are just wild guesses. I will not publish my values for the last few terms because I do not want to bias your efforts in trying come up with a value for N. We do know enough astronomy to make some good estimates for the first two terms. The current star formation rate is about 2 to 3 stars/year, but in the past it was much larger so I quote the average value of 20 stars/year. The fraction of stars that are sun-like medium temperature stars that are single and that would have any chance of life-filled planets orbiting them is about 1/50 = 2%. Proto-planetary disks have been detected around some stars and astronomers are now just beginning to detect planets around solar-type stars, including those in binary or multiple-star systems as well as planets in the habitable zones of the very common cooler stars (so the 2% figure is probably too small for f_p). See the end of the Solar System Fluff chapter for a discussion of finding exoplanets and web links to up-to-date information about them.

A nice interactive to try out your values in the Drake Equation is The Drake Equation interactive from NOVA's Origins series that was broadcast on PBS (selecting the link will bring it up in a new window either in front of or behind this window).

For a sample of the scientific debates over the values in the Drake Equation (and perhaps the need to add more terms!), see the Complex Life Elsewhere in the Universe? debate that includes the "Rare Earth" authors, Don Brownlee and Peter Ward (selecting the link will bring it up in another window). Ward and Brownlee got the astrobiology/SETI community to re-examine its assumptions about extra-terrestrial life when they laid out their case for why complex life (life beyond the microbial level) may be very rare in the universe in their book "Rare Earth". Needless to say there is disagreement, but it is a healthy debate in the determination of what it takes to make a habitable planet that can support complex, intelligent (self-aware) life.

Contributors and Attributions

• Nick Strobel (Bakersfield College)

Is this page a copy of Strobel's Astronomy Notes?

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