Drake's Equation and Searching for Life in the Milky Way

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Lesson Overview

In this lesson, we'll discuss the prospect of

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life in the Milky Way galaxy beyond the Earth. We'll begin by discussing the speculations made in a paper written by Carl Sagan about the possibility of life in Jupiter's atmosphere. From there, we shall derive a formula which describes the habitable zone of a star. Using this formula and data obtained by the Kepler Space Telescope, we can estimate the total number of "Earth-like" planets in the Milky Way. From there, we discuss the fraction of those planets on which simple and intelligent life evolve; then we'll discuss the fraction of those planets on which advanced communicating civilizations evolve and what fraction of those civilizations are communicating right now.

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Possible strange and exotic lifeforms

"In 2005 I worked on a project to update some of the fx sequences in Carl Sagan's seminal series Cosmos. For this sequence I did all the shots on Jupiter. I started with the original high-res and beautiful matte painting by Adolf Schaller and tried to recreate the creatures as closely as possible."\(^{[1][2]}\)

In this lesson, we'll estimate how many worlds in our home Milky Way galaxy is inhabited with life and, among those worlds which life had arisen, how many of them developed intelligent life. We'll also estimate how many of those extant intelligent civilizations developed high-tech communications systems which would allow them, if they wanted, to communicate across interstellar distances; these advanced civilizations, if they still exist, could be contacted right now if only we knew where in the Milky Way they were hiding.

On our home-planet, the Earth, all life requires liquid water, solar energy from the Sun, and rich and complex carbon-based molecules. The reason for the latter, why all life on Earth requires organic chemistry and carbon-based molecules, is because of the unique and astonishing properties of carbon atoms and, in particular, because of the properties of the outer-most valance electrons of carbon atoms. To fully explain what these properties are would require an entire course on organic chemistry but, to briefly describe them, what makes carbon so special is that it can bond with other carbon atoms and other elements in periodic table to form long chains of organic molecules all of which are necessary for all forms of Earth-based life. These organic molecules are necessary to support the metabolism of all forms of life on this planet.

  "Mr. Spock (Leonard Nimoy), communicating with the Horta through a Vulcan mind meld."\(^{[3]}\)

"Mr. Spock (Leonard Nimoy), communicating with the Horta through a Vulcan mind meld."\(^{[3]}\)

It is only the life here, on Earth, which we understand and know well. But this has not stopped scientists from speculating about what other kinds of life may be possible in the Milky Way. For example, scientists have speculated about non-carbon based lifeforms. They have imagined that silicon-based life might be possible because the chemical properties of silicon are very similar to those of carbon. This scientific possibility has inspired science fiction writers; for example, in one episode in the TV series Star Trek, the crew onboard the Enterprise visited an alien world and encountered silicon-based life. These alien lifeforms looked like interconnected rocks shuffling along the planetary surface.

Despite the possibility of silicon-based life receiving serious scientific attention, such lifeforms arising through spontaneous natural processes would be highly unlikely. This is because, unlike carbon which is one of the most abundant elements in the universe, silicon is comparatively rare in the universe. (Indeed, silicon is 10 times less abundant than carbon.) It would also be much more difficult for long chains of molecules to form and react with one another if they were silicon-based since silicon atoms don't like sharing their electrons as much as carbon atoms do. 

There have also been far more ambitious proposals in scientific literature of how extraterrestrial life might be radically different from the life we see here on Earth. The first ever paper on extraterrestrial life to be published was pioneered by Carl Sagan and this paper speculated on the possibility of life in Jupiter's atmosphere. These possible Jovian lifeforms were called Floaters, Hunters, and Sinkers and are illustrated in an artist's depictions below.


Possible star systems that life might evolve in

We can indeed speculate about what other kinds of life there may be in this universe, but we cannot—at least at our present time—be certain about whether or not it is possible for such life to arise. The only kinds of life which we can estimate the probability of existing on other worlds in the galaxy besides the Earth are lifeforms that are similar to those on Earth. What I mean by lifeforms "similar" to that of the Earth's are any forms of life which require energy, liquid water, an atmosphere, and carbon-based chemistry.

"Earth-like" lifeforms would, for the most part, only be able to survive in star systems with "Sun-like" stars. And by Sun-like stars, I mean G-stars (our Sun is a main sequence G-star) and K-stars. G-stars and K-stars are "Sun-like" because they generally have similar luminosities (luminosity is the measure of a star's total energy output), similar cycles and sizes of stellar flares, and similar lifetimes. Estimates of the total number of stars in the Milky Way range from 100-400 billion. We'll estimate this number to be 300 billion. Out of this immense multitude of stars, about 20% of them are Sun-like G- and K-type stars. Only in the star-systems in the Milky Way containing G- and K-type stars do we think it is possible for what we called "Earth-like" life to survive.

I have said earlier on that, due to our ignorance of what's possible in terms of life, we shall restrict our attention to estimating the possibility of "Earth-like" life elsewhere in the galaxy. But let me just go on one more slight tangent about what other kinds of life might be possible. Lifeforms which are totally different to the life that we see on Earth might be capable of surviving on worlds orbiting types of stars besides G-stars and K-stars.

 Artist's depiction of an exoplanet orbiting a pulsar. Due to the pulsar's radiation, the exoplanet glows.

Artist's depiction of an exoplanet orbiting a pulsar. Due to the pulsar's radiation, the exoplanet glows.

We have discovered exoplanets orbiting pulsars. Such pulsars emit a lot of high-frequency radiation such as x-rays and would drench such exoplanets in x-rays causing them to give off a faint glow. An artist's depiction of such a glowing planet is illustrated in the artist's depiction on the right. Now, we're pretty sure that star systems containing stars like pulsars would be inhospitable to life (it would definitely be inhospitable to Earth-like life), but we cannot be sure of this with a hundred percent confidence. Some very exotic lifeforms might be able to survive in the harsh environment of a planet orbiting a pulsar. After all, we have found strange lifeforms here on Earth which can survive in boiling hot water, in sulphereic acid, and we've even discovered life that can survive in space despite the high amounts of radiation in space which would be lethal to us.

There are also stars known as M-stars which are essentially red dwarf stars. Because M-stars are very similar to our own Sun, we initially thought of them as stars that would be good candidates for "Sun-like stars"—that is, stars suitable for "Earth-like" life. But the luminosity of an M-star is very low, so low in fact that a planet would have to huddle very close to such a star in order to support life which requires liquid water. Also, in order for an exoplanet orbiting an M-star to sustain liquid water, it would have to revolve around a star in a roughly circular orbit. For a planet revolving around an M-star in that star's habitable zone (a range of distances away from a star in which a planet with a given reflectivity and atmospheric greenhouses would be neither too hot nor too cold to sustain liquid water) in a roughly circular orbit, that planet would become tidally locked with the star. This means that one side of the planet would always be facing the star and the other side would always be facing away form the star. Because of this it would be impossible for such a planet to sustain an atmosphere. Here's why. The dark side of the planet would be too cold to keep an atmosphere. Gases on the bright, hot side of the planet would circulate to the dark, cold side of the planet and subsequently freeze out of the atmosphere. It would be very difficult for even simple life to form on a planet without an atmosphere. This is because a planet's atmosphere shield its surface from harmful radiation and, without one, a planet's surface would be constantly bathed in radiation. Thus, it would be impossible for intelligent life to evolve on a planet orbiting around an M-star in the habitable zone.

Now, along the circumference of such a planet which roughly divides the bright hot half and the dark cold half of the planet, there must be a region of the planet which has a temperature that would allow for liquid water. Perhaps then, within these regions, it would be possible for life to arise. But even if this were true, we still wouldn't expect intelligent life to arise on such a planet. This isn't just because of the fact that such a planet wouldn't be able to sustain an atmosphere. The other reason why we think that it would be impossible for intelligent life to arise on such a world is because intelligent life takes very long to evolve and M-stars have very large flares far more often than G-type and K-type stars. We know from our planet's fossil record that simple life arises fairly quickly compared to geological time scales and we therefore think that it might be possible for simple life to develop in a star-system with an M-type star. But we also know from our planet's fossil record that intelligent life takes a very long time to evolve. If we assume that changes in life through evolution by natural selection occur at more or less the same speed on another planet as our own, then we'd expect that intelligent life would also take an enormous amount of time to evolve in a star system with an M-star. But because of how close a planet's orbit must be to an M-star to sustain liquid water and because of the fact that M-star's have flares more often than the Sun and flares which are also much more violent, we'd expect somewhat sophisticated life to get wiped out before it could ever evolve into intelligent beings.


How many habitable planets are out there?

 The habitable zone is a range of orbital distances (between an exoplanet and the star it orbits) where the planet is neither too hot nor too cold. This is also sometimes called the Goldilocks zone. In order for a planet to be truly "Earth-like," it must also have a mass 1-2 times that of the Earth, have an atmosphere, and orbit in a stable elliptical orbit that is not too essentric.

The habitable zone is a range of orbital distances (between an exoplanet and the star it orbits) where the planet is neither too hot nor too cold. This is also sometimes called the Goldilocks zone. In order for a planet to be truly "Earth-like," it must also have a mass 1-2 times that of the Earth, have an atmosphere, and orbit in a stable elliptical orbit that is not too essentric.

 Artist's depiction of the Kepler space telescope.

Artist's depiction of the Kepler space telescope.

The next question which we must ask ourselves is the following: among those star systems which contain G- and K-type stars, what fraction of those star systems contain "Earth-like" planets? And by Earth-like, I mean planets which are 1-2 times as massive as that of the Earth and which orbit in their stars habitable zones. When the astrophysicist Frank Drake first formulated his Drake equation, no other planet in another star system had ever been observed before. In his time, it would've been impossible to predict the number of other Earth-like planets orbiting Sun-like stars. But in our time, things are much different. As of the time of this writing, the Kepler space telescope has discovered nearly 4,000 exoplanets orbiting other stars. All of the points in the graph in Figure 3 represent some of the planets discovered by the Kepler telescope. The vertical position of each point represents the radius of each exoplanet and the horizontal distance of a point represents the distance that some exoplanet is away from its home-star. As you can see from this graph, most of the exoplanets discovered by the Kepler telescope are large Neptune- to Jupiter-sized planets that orbit very closely next to their home-star.

Why is it that most of the exoplanets which we have discovered using the Kepler telescope are very big and very close to the stars which they orbit? Why is it that most of the exoplanets we've discovered using any technique tend to be large planets orbiting close to their home-stars? To answer this question, let me first tell you how astronomers find exoplanets in the first place. There are basically two different ways of finding an exoplanets. The first technique we discussed in the lesson, Spectroscopy. If an exoplanet is orbiting around a star, the exoplanet will exert gravitational forces on the star which it is orbiting causing that star to wobble. When the star wobbles, its speed increases slightly; thus, the light it emits gets slightly redshifted. The redshifts in the stars spectral lines and "signature" can be analyzed and measured using an instrument called a spectroscope. The larger that the exoplanet is and the closer it is orbiting its home-star, the bigger of a wobble and redshift it causes in the star and the star's light respectively. Smaller exoplanets that orbit farther away cause the star to wobble very slightly and cause very small redshifts in the star's spectral lines which becomes increasingly difficult to detect the smaller and farther away that the exoplanet is. Since large exoplanets orbiting very close are easier to detect, we tend to detect more such exoplanets using this techniques.

Here is the second technique that astronomers like to use to detect exoplanets. The radius of a large, Jupiter-sized planet will be roughly one-tenth the size of the radius of a G-type or K-type star. Let's write this mathematically as

$$r_s≈10(r_p)\tag{2}$$

The cross-sectional area of the star is given by \(πR_s^2\) where \(R_s\) is the star's radius; but when a Jupiter-sized planet crosses our line of sight in front of the star, the planet (whose surface area is \(π(0.1R_s)^2=0.01(πR_s^2)\)) blocks out 0.01=1% of the star's light. By a smaller Earth-like exoplanet has a radius about one hundred times smaller than the radius of the average G-star or K-star; we can write this mathematically as

$$r_s≈100(r_p).\tag{3}$$

Artist's depiction of an exoplanet crossing our line of sight in front of its home-star. 

When an Earth-sized planet crosses our line of sight in front of the star, the planet (with surface area \(0.001(πR_s^2)\)) blocks out only 0.001=0.1% of the star's light. An exoplanet whose size is similar to that of the Earth's is therefore much more difficult to detect than a Jupiter-sized exoplanet. Also, the farther out an exoplanet orbits a star the long is its orbital period and, thus, the less likely we are to observe it. Using either technique, the smaller the exoplanet and the bigger its orbit, the more difficult it is to detect the presence of the exoplanet. This is why most discoveries of exoplanets have been have enormous Jupiter-sized planets that orbit very close to their home-stars.

Let's now get back to answering our original question: what fraction \(f_{HP}\) of those star systems in the Milky Way containing G-stars and K-stars also contain "Earth-like" planets? That is, planets whose masses are 1-2 times that of the Earth's and which orbit in their home-stars habitable zones. According to data obtained by the Kepler telescope, about 10% of G-stars and K-stars contain planets with masses 1-2 times that of the Earth's and which receive a solar radiation flux (amount of luminosity an exoplanet gets from its star per unit area) between 1/4 to 4 times as much as the Earth gets. All of these planets are "Earth-like" in the respect that their masses are 1-2 times that of the Earth's; but some fraction of them are either too close or too far away from their home-stars to support liquid water. We want to find the fraction of these worlds which are within the distance ranges from their stars that are just right for supporting liquid water. These distance ranges are called habitable zones. To find them, we first need to use a little math to determine the relationship between luminosity (and hence solar radiation flux as well) and the separation distance between an exoplanet and the star that it orbits.

All of the energy emitted by the star is in the form of photons and electromagnetic radiation which are emitted radially in all directions away form the star. The luminosity, or total power output of the star, is given by the Stefan–Boltzmann law which states

$$L_s=AσT_s^4,\tag{4}$$

where \(L_s\) is the star's total luminosity, \(σ\) is a constant, \(T_s\) is the temperature of the star, and \(A\) is the surface area of the star. Substituting \(A=4πR_s^2\) into Equation (3) where \(R_s\) is the star's radius, we have

$$L_s=(4πR_s^2)σT_s^4.\tag{5}$$

Now, how much of that luminosity (or power) reaches an exoplanet a distance \(r_p\) away from the star. To answer this question, let's start out by imagining that we draw an imaginary hollow sphere whose radius \(r_p\) is the same as the radius of the exoplanet's orbit. The luminosity/power emitted by the star which passes through this sphere is given by Equation (5). But the fraction of this luminosity which gets to the exoplanet is the ratio \(4πr^2/πR_p^2\) where \(4πr_p^2\) is the surface area of the imaginary hollow sphere and \(πr_p^2\) is the cross-sectional area of the planet (this cross-section is the portion of the planet which receives that star's luminosity). Thus, the luminosity \(L_p\) delivered to the exoplanet is given by

$$L_p=\frac{4πR_s^2}{πr_p^2}σT_s^4.\tag{6}$$

Dividing by some amount of area \(A\) on both sides of Equation (6), we find that the solar radiation flux an exoplanet gets is given by

$$Φ_p=\frac{4πR_s^2}{πr_p^2}\frac{σT_s^4}{A}.\tag{7}$$

Equation (7) gives us the relationship between the stellar radiation flux \(Φ_p\) on an exoplanet and the separation distance \(r_p^2\) between the exoplanet and star. We said earlier that, according to empirical data, 10% of star systems with G-stars and K-stars contain exoplanets that get 1/4 to 4 times the stellar radiation flux that the Earth gets from the Sun. If we let \(r_p\) denote the separation distance between the Earth and the Sun, then in order for a planet to get 4 times as much stellar radiation flux its distance away from its home-star must be \(0.5r_p\)—in other words, it must be twice as close to its home-star as the Earth is from the Sun. In order for a planet to get 1/4 times the stellar radiation flux, its distance away from its home-star must be \(2r_p\). Thus, the empirical fact that 10% of star systems with G-stars and K-stars have "Earth-like" exoplanets getting 1/4 to 4 times as much stellar radiation flux implies that these exoplanets are within a distance range of \(0.5r_p\)-\(2r_p\) away from their home-stars where \(r_p\) is the separation distance between the Earth and the Sun. And since the Earth orbits at a distance of 1 astronomical unit (AU) away form the Sun, it follows that these exoplanets are within a distance range of \(0.5AU\)-\(2AU\). And, according to data obtained by the Kepler telescope plus a little math, it turns out that 45% of these exoplanets also orbit in their stars habitable zones.

To estimate the fraction \(f_{HP}\) of the star systems in the Milky Way galaxy containing Earth-like masses with masses ranging from 1-2 times that of the Earth and which orbit in their stars habitable zones, we must calculate the product \(f_{HP}=\text{0.2 × 0.1 × 0.45}\). Doing so, we find that \(f_{HP}=0.009\). But this fraction of star systems might contain double- or triple-star systems and exoplanets with elliptical orbits which are too eccentric where the exoplanet orbits in and out of its habitable zone causing its water to alternate between being in a gaseous, liquid, and frozen state. Also, it is possible for an exoplanet to have an atmosphere which is not suitable for life. So in reality, the fraction of star systems with truly "Earth-like" exoplanets that can sustain liquid water and life is probably slightly lower. Let's lower our estimate of 0.009 down to 0.006.

To estimate the total number of Earth-like, habitable exoplanets in the Milky Way galaxy, we need to calculate the product \(N_sf_{HP}=\text{300 billion × 0.006}\). Doing so, we find that there are roughly 1.8 billion Earth-like, habitable planets in the Milky Way—an astonishing fact!


Planning an interstellar voyage to a habitable planet

The total number of stars in the Milky Way galaxy is given by, roughly, \(N_s=3\text{ × }10^{11}\). Of this immense multitude of stars, the fraction of those stars which contain Earth-like planets orbiting in their home-stars habitable zone is given by roughly \(f_{HP}=0.006\). By taking the product \(N_sf_{HP}\), we obtain the following estimate of the total number of Earth-like, habitable planets within our Milky Way galaxy:

$$N_sf_{HP}=(3\text{ × }10^{11})(0.006)=\text{1.8 billion}.$$

There are therefore approximately 1.8 billion habitable planets within our galaxy. It is astonishing to live in a time when we are able to use part of the Drake equation and data obtained by the Kepler space telescope to infer how many other planets like our own are out there in our galaxy.

We are also able to use these calculations to estimate the number of "nearby" habitable planets. Within a 40 light-year radius of the Earth, there are roughly 1,000 other stars besides our Sun. The approximate fraction of these stars which are orbited by habitable planets is 0.006. Taking the product of 0.006 and 1,000, we find that there are roughly 6 star systems within a 40 light-year radius of the Earth which contain habitable planets. Indeed, astronomers have already found habitable planets within this 40 light-year radius of our home planet. (In fact, a new habitable planet named Ross 128 which is just 11 light-years away from us was discovered on the day I was writing this!)

 An artist's depiction of the starship proposed by Project Daedalus.\(^{[5]}\)

An artist's depiction of the starship proposed by Project Daedalus.\(^{[5]}\)

The first interstellar voyage undertaken by humanity to the stars will most likely be to a star named Proxima Centuari within the triple star system, Alpha Centuari. But certainly some of the subsequent interstellar journeys will be to these nearby habitable planets. An interstellar voyage using a nuclear propulsion starship proposed by Project Daedalus (see image above) would involve using an immense starship powered by fusing together deuterium and helium-3. Such a starship could reach Ross 128 in about 92 years. But a far more efficient matter–anti-matter starship—powered by the fusion of matter and anti-matter—could approach speeds about 98% the speed of light. Due to the effects of Einstein's theory of special relativity, the crew on board such a ship would arrive at Ross 128 in just 3 years ship time while everyone on Earth would have aged 12 years into the future


Estimating the number of intelligent communicating civilizations

Whenever we estimate the probability of life evolving on other habitable planets—whether its simple lifeforms such as microbes or intelligent thinking beings such as we capable of inventing radio communication—we, at least for the time being, always assume as true something known as the principle of mediocrity. This principle states that there is nothing special about the history of events which unfolded here on Earth and that if life or intelligent beings evolved on other exoplanets, the history of events which lead to their development would be more or less the same as the history of events which occurred here on Earth. Let me give an example of what I mean by this statement. All forms of life on this planet, down to even the simplest of microbes, required energy, liquid water, and carbon atoms to evolve. Carbon is a very special element and it has properties unlike any other element in the periodic table. In particular, carbon atoms are able to bond with many other atoms (including other carbon atoms) to form very large molecules such as those comprising DNA, ATP, or the amino acids in proteins. These molecules are the stuff of life. Energy is necessary for those molecules to react with each other in such a way as to allow the metabolisms of living creatures to function properly. And liquid water is the medium in which these chemical reactions most easily occur. And, as a final note, we know of no other atom besides carbon which has the right chemical properties to form very large, complex molecules. And this is a crucial point because it are those large, complex molecules which all forms of life—at least as we know of—need in order to exist. Indeed, what I have said about the importance of energy, liquid water and carbon-based chemistry applies to all lifeforms on the planet Earth.

The principle of mediocrity is the assumption that the same must also be true for the evolution of life on other planets. Now, the assumption that all lifeforms in the galaxy and beyond (and not just here on Earth) require these three main ingredients in order to evolve actually isn't too bad of a conjecture to make. Also, evidence suggests that given the right conditions (that is, liquid water, carbon-based chemistry, and energy) on any habitable planet, the transitions from atoms to organic molecules to simple lifeforms is practically inevitable. Indeed, some scientists even think that these transitions occurred cyclically and multiple times during a time in early Earth history known as Late Heavy Bombardment. They argued that in Earth's primordial oceans, spontaneous and entirely natural processes caused atoms to form complex molecules and complex molecules to form life; but, after that initial life arose, it was short-lived and eventually got wiped out by the scores of asteroids colliding with the Earth during this chaotic time in Earth's history. And then, this process would occur all over again: chemistry would, for a second time, transition to life. And then a third time, and so on. This cyclical process, some argue, occurred multiple times. If this did indeed occur, then this would be compelling evidence that life does indeed inevitably occur, in a comparatively short period of time, given the right conditions. It is because of evidence such as this that we, with a great deal of confidence, assert that the fraction of habitable planets on which simple life evolves must be very close to one. We therefore let this fraction, \(f_l\), equal one:

$$f_l=1.$$

By taking the product \(N_sf_{HP}f_l\), we can estimate that the total number of habitable planets on which at least simple, single-celled organisms evolve on is given by roughly

$$N_sf_{HP}f_l=(3\text{ × }10^{11})(0.006)(1)=\text{1.8 billion}.$$

Thus, it is not at all an exaggeration to assert with great confidence, on the basis of current scientific understanding, that on well over one billion exoplanets simple, single-celled lifeforms must have had evolved. A considerable fraction of those exoplanets could be called "pond scum planets." That is, planets with liquid water oceans on its surface; and on the surface of those oceans reside enormous colonies of single-celled microbes such as cyanobacteria. In other words, the surface of these planets would resemble what the surface of the Earth looked like roughly 3.5 billion years ago.

To estimate the total number of habitable planets on which intelligent beings lived at some point in that planet's history, we also rely heavily on the principle of mediocrity. To put things bluntly right here from the get-go, despite the fact that the evolution of simple life is probably very likely to occur on a habitable planet, the evolution of intelligent beings and advanced civilizations seems to be very unlikely. Even on the planet Earth, we think that the probability of intelligent beings (which is to say, us!) evolving was very unlikely. (Indeed, it is estimated that at one time in our early history, the total number of humans was perhaps less than one thousand and we came very close to extinction.) The main argument as to why this is is because intelligence does not seem to be an important attribute for survival; if it was, then we would expect to see intelligence evolve multiple times in the history of life on this planet. And this is, of course, not the case—an intelligent species evolved only once in Earth's history. Attributes such as locomotion and eyes do, however, to the contrary, appear to be very important when it comes to species survival. This is due to the simple fact that these two characteristics evolved multiple times during the history of life on Earth. There are also other arguments such as the Great Filters Hypothesis which attempt to explain why the evolution of intelligence must be profoundly rare, but we shall save such discussion for future lessons. 

To represent the fact that the fraction of habitable planets on which intelligence arose during some period of time in that planet's history (we'll represent this fraction by writing \(f_i\)) is very unlikely, we can write

$$f_i<<1.$$

But the tricky question to answer is, how unlikely? The simple answer is that we do not yet know. We'll guestimate that \(f_i<0.1\) knowing in the back of our minds that this number could be much lower. The next fraction in Drake's equation, \(f_c\), represents the fraction of those intelligent beings which go on to invent radio communication and advanced technology. This number is likely very close to one and, thus, we'll let \(f_c=1\). We're actually very confident in the value of \(f_c\) unlike the value of \(f_i\) for which, the best we can presently do, is guess. The reason why we think that \(f_c≈1\) is due to the fact that once we intelligent beings finally evolved, the development of many kinds of technology and scientific knowledge occurred independently multiple times. For example, calculus was developed twice independently by Isaac Newton and Leibniz. Also, if Einstein did not formulate his general theory of relativity then David Hilbert certainly would have developed the equivalent of that theory. Any study would reveal that this same pattern applies to the development of many technologies and scientific theories. This seems to imply that given the existence of intelligent beings, the invention of technologies and the discovery of scientific theories is not some rare and unlikely occurrence but, rather, nearly inevitable. By taking the product \(N_sf_{HP}f_lf_i\), we can estimate that the total number of planets on which intelligent lifeforms evolved is given by

$$N_sf_{HP}f_lf_i=(3\text{ × }10^{11})(0.006)(1)(<0.1)<\text{180 million}.$$

The final term in Drake's equation (which we shall denote by \(f_{cn}\) where "cn" denotes "communicating now") represents the fraction of intelligent civilizations which invent radio communication and which are communicating "now." \(f_{cn}\) can be calculated by dividing the age of the galaxy (which is \(∽10^{10}\) years) by the average lifetime \(L\) of an intelligent, communicating civilization. We'll represent this by writing

$$f_{cn}=\frac{L}{10^{10}years},$$

where \(L\) is the average lifetime of an intelligent, communicating civilization. Frankly put we have no idea what the value of \(L\) actually is. Estimates of this value range from 100 years (roughly how long we have been communicating) to 100 million years—the latter being a geological amount of time that represents roughly the time interval in between major meteorite and asteroid impacts on the Earth. Indeed, when the Drake equation was first formulated in 1961 by Frank Drake with the help of Carl Sagan and John Lilly, they estimated that these were the lower and upper bounds of the value of \(L\). If it is indeed possible that the value of \(L\) could potentially be as high as 100 million years then, according to the Drake equation, it should also be a possibility that there millions of advanced civilizations in our galaxy communicated right now. But due to something known as Fermi's Paradox (which we'll discuss in greater detail in a future article or video), this seems very unlikely. Most modern estimates for \(L\) seem to range between 100 and 10,000 years. These estimates reduce the upper bound of the lifetime of an intelligent communicating civilization based on the social and institutional forces we have seen in human history. No human civilization has ever survived longer than 10,000 years before collapsing and, if other alien civilizations in the galaxy are subject to similar social and institutional forces, we'd expect that this same upper bound should apply to them too. Plugging in our estimated value of \(f_{cn}\), we have

$$N_sf_{HP}f_lf_if_{cn}=(3\text{ × }10^{11})(0.006)(1)(<0.1)\biggl(\frac{100<L<10,000}{10^{10}}\biggr).$$

The lower and upper bounds of \(f_{cn}\) are given by the following inequalities:

$$\frac{1}{10^8}<f_{cn}<\frac{1}{10^6}.$$

Plugging this result into Drake's equation, we have

$$N_sf_{HP}f_lf_if_{cn}=(3\text{ × }10^{11})(0.006)(1)(<0.1)()10^{-8}<f_{cn}<10^{-6}.$$

Assuming that \(f_i=.1\), we can evaluate the expressions above to find that the total number \(N_{cn}\) of intelligent civilizations communicating now in our galaxy must be given by the inequalities

$$1.8<N_{cn}<180\text{ where }f_i=0.1$$

This reduces the number of other intelligent communicating civilizations in our galactic neighborhood down from a little over a hundred to less than one (meaning, that not every galaxy has an intelligent communicating civilization) if we consider values of \(f_i\) which are less than 0.1. This is a much less exciting prospect than the estimates made in 1961; however, it is one which seems much more likely.


This article is licensed under a CC BY-NC-SA 4.0 license.

References

1. Brian Emerson. "Carl Sagan's Cosmos: Life on Jupiter". Online video clip. YouTube. YouTube, 10 September 2007. Web. 16 November 2017.

2. Sagan, Carl, et al. “Cosmos: A Personal Voyage: One Voice in the Cosmic Fugue.” Cosmos: A Personal Voyage, PBS, 1980.                         

3. Malinconico, MaryAnn. (2015, March). "to seek out new life" even if not carbon-based. . . . Retrieved from http://carbonacea.blogspot.com/2015/03/to-seek-out-new-life-even-if-not-carbon.html.

4. Schaller, Adolf. (2013, November). Creating Life on a Gas Giant. Retrieved from http://www.planetary.org/blogs/guest-blogs/2013/20131023-on-hunters-floaters-and-sinkers-from-cosmos.html.

5. Chen, Qiao. “Project Daedalus.” ArtStation. Retrieved from http://bridge.artstation.com/projects/kX116.

6. Tyson, Neil deGrasse. Welcome to the Universe: An Astrophysical Tour. Princeton University Press, 2017.

7. SUMMERS, M. (2018). EXOPLANETS: Diamond worlds, super earths, pulsar planets, and the new search for life beyond our... solar system. S.l.: SMITHSONIAN BOOKS.