The Statistical Drake Equation:

Abstract

The creation of the Drake Equation by Frank Drake was viewed with skeptical criticism by fellow scientists because the numbers which he used for the variables in the formula were far too optimistic to be true. If they were true, we should have already established contact with extraterrestrial life a long time ago. But when we look out into the endless expanse above us, filled with vast resources and beautiful galaxies, stars, constellations and nebulae, we can see nothing that would suggest a space faring, communicative civilization. No evidence for them at all to ever exist. Of course, this brings into question the Fermi Paradox, formulated by Enrico Fermi. According to Frank Drake’s calculations, there should be at least 36 technologically advanced alien civilizations in the Milky Way Galaxy capable of establishing contact with us. However, this number was calculated using incredibly optimistic numbers.

This article aims to statistically estimate the actual number of extraterrestrial civilizations capable of receiving and sending out signals into space, in other words, discovering how many civilizations are out there in our own Milky Way Galaxy that we can establish contact with.

This will be achieved by replacing the variables in the Drake Equation with contrasting numbers and comparing the results. For example, the original Drake Equation estimates that only about 1 new star is formed every year, yet recent studies by NASA and the European Space Agency show that there are about 3 new stars born every year in the Milky Way Galaxy, and extraterrestrial life forms do not necessarily need to be carbon-based. These two points raise the numbers, and thus, the eventual probability of intelligent life occurring dramatically.

Introduction

The Drake Equation is a rather famous equation and a statistical argument used to predict the number of active, communicative and space faring extraterrestrial civilizations in the Milky Way Galaxy. It was formulated in 1961 by Frank Drake, not specifically for the purposes of evaluating the number of civilizations, but rather to introduce the topic at the first scientific meeting on the Search for Extraterrestrial Intelligence, (SETI). The equation summarizes the main concepts and probabilities that scientists must understand and contemplate when considering the question of other intelligent life. However, although it is often misunderstood as a serious attempt to determine a precise number, it should be thought of as an approximation of what that number could be. Therefore, the numbers used in the original approximation by Frank Drake are much more optimistic than modern science leads us to believe, even though the science and mathematics behind the equation is understandable and plausible. The Drake Equation formula is written out as such:

N=R*fp*ne*fl*fi*fc*L

Where,

N represents the total number of civilisations with which humanity could communicate with, in the Milky Way Galaxy alone.

R represents the estimated number of stars birthed in our galaxy every year.

fp is the fraction (percentage) of stars that have at least one planet orbiting in a system.

ne is the number of Earth-type, habitable planets around the given star. In other words, it is the number of planets in a given star system that house the correct chemical conditions needed for life to start. They are “ready for life”.

fl is the fraction (= percentage) of such habitable, “ready for life” planets on which life actually starts and develops, but not yet to the intelligence such as ourselves.

fi represents the fraction of such planets with basic life forms that actually develop and prosper until some form of intelligent civilization emerges, (like the first humans several tens of thousands of years ago)

fc is the fraction of such ‘planets with civilizations’ whose civilizations develop to the point in which they have access to and can readily communicate interstellar distances with other (at least) similarly advanced civilizations. This means that, at this stage in their development, they must be aware of Maxwell’s equations governing radio waves, as well as of computers and radio astronomy.

L represents the fraction, or percentage of advanced galactic civilizations alive and able to receive and send out signals at the time when we, humanity, attempt to pick up their signals. (These signals don’t have to be intentional, just any signal thrown off by their civilization into space from when they first invented radios, just as we have done since the early 1900s)

The original numbers that Frank Drake used in his formula in 1961 were;

R= 1yr-1(1 star formed per year, on average over the lifespan of the galaxy)

fp= 0.2 to 0.5 (one fifth to one half of all stars formed in the galaxy will have planets)

ne= 1-5 (stars with planets will have between 1 and 5 planets capable of developing life)

fl= 1 (100% of these planets will develop life)

fi= 1 (100% of which will develop intelligent life)

fc= 0.1 to 0.2 (10-20% of which will be able to communicate)

L= 1000 to 100,000,000 communicative civilizations (which will last somewhere between 1000 to 100,000,000 years)

Therefore, when we substitute these numbers which Frank Drake used himself into his formula, we receive a wide range of values for the answer, depending on the assumptions and numbers used. Hence, the result can be N<<1, meaning we are likely alone in the galaxy, or N>>1, implying that the galaxy is teeming with life waiting to be discovered and contacted. By inserting the minimum numbers Frank Drake gave into the equation, we acquire:

N=1*0.2*1*1*1*0.1*1000

Using these numbers gives us a minimum answer of 20. Twenty intelligent civilizations in our galaxy alone that are capable of receiving and transmitting signals and establishing contact with other civilizations.

Using the largest estimates within Frank Drake’s original numbers, however, grants us a total number of 50,000,000 advanced extraterrestrial civilizations in our galaxy alone:

N=1*0.5*5*1*1*0.2*100,000,000

→N=50,000,000

Therefore, according to Frank Drake’s original estimations in 1961, there could be between 20-50,000,000 alien civilizations capable of communicating with us in the Milky Way Galaxy, yet we haven’t made contact with a single one.

Theory

As already mentioned before, the minimum number of intelligent civilizations in the Milky Way Galaxy alone is 20, according to the original assumptions and estimates of the Drake Equation. However, humanity’s understanding of the universe and astronomy has changed dramatically over the past century, and thus many of Frank Drake’s original estimations are essentially obsolete. Because the numbers representing the different variables in the formula have changed, the number of civilizations must change too, and there should be a wider range of answers to the Equation, considering many optimistic and pessimistic views of the Universe. Furthermore, the Fermi Paradox and the possibility of past intelligent civilizations that are now extinct must be taken into account.

Therefore, I propose a new set of figures used for the variables in the Drake Equation that will bring it into accordance with modern knowledge and understanding of the universe and astrobiology.

Analysis, Results & Discussion

The original Drake Equation consists of:

N=R*fp*ne*fl*fi*fc*L

Where,

N=1*0.2 to 0.5*1 to 5*1*1*0.1 to 0.2*1000 to 100,000,000

↑ This is using Frank Drake’s own estimates in 1961.

For the variable, R, Frank estimates that 1 star is birthed each year in the Milky Way galaxy. Thanks to expansive research conducted by NASA and the European Space Agency, we know that 275 million stars are born every day throughout the entire observable universe. Multiply that by 365, because there are 365 days in a year, and divide it by 100 billion. This is the estimated number of galaxies in the universe. When you put this equation in a calculator, you will get: 19712000, which equals 0.9855. Rounded up, that means that on average, one new star is born in every galaxy across the entire universe in a year. The reason we know (or can accurately predict) the number of stars in celestial regions is because red, white and blue stars give off different amounts of light. By measuring that starlight, specifically, its color and brightness, astronomers can estimate how many stars our galaxy, or even universe, holds. Astronomers also take pictures of tiny zoomed ‘squares’ of the night sky, in which they then count the number of stars in that picture. Thus, they can predict the number of stars across the universe.

For the variable, fp, Frank Drake predicted that 0.2 to 0.5 of all stars in the galaxy will contain planets and star systems. However, that prediction was made even before the Moon Landings, and way before the first exoplanets were discovered. As more and more exoplanets were being discovered, it became understood that essentially all stars in the galaxy have their own star system, with very rare exceptions (Exoplanet Exploration: Planets Beyond our Solar System, n.d.). Thus, the number that Drake used should be revised and replaced with a higher number, such as between 0.5-0.99, as a pessimistic choice.

The variable, ne, in the Drake Equation represents the number of planets which can host life in a star system which has planets. Drake used the figures, 1-5, which seems incredibly optimistic considering the now revised number of total planet-bearing stars. A more realistic scenario is that, at most, there are a handful of habitable planets in a single system, but the overall majority of systems only have one or none. And, rather than classifying the actual worlds as inhabitable or uninhabitable, astronomers have started to classify regions of orbits around stars, as any rocky worlds found in such regions will likely be habitable and good enough to live in.

Thus, as the likelihood of finding more than one habitable planet in any particular star system decreases exponentially as the number of planets increase, the variable ne must be altered to include these points of interest.

A better variable would be the one below, where the quantity, t represents time, and ƛ represents the exponential decay constant:

ne=dNdt=-ƛN

Where ne is equal to exponential decay (ƛ) This represents the exponentially decreasing likelihood of you finding star systems with more habitable planets (Up to 5)

This formula can be graphed as:

We can deduce from the graph above that the probability of finding star systems with habitable planets decreases/decays along with the number of planets that are habitable in each system. This means that it is more likely that you will find a stellar system with only one habitable planet rather than 4 or 5 habitable planets. The probability will decrease by 50% for every added habitable planet in one star system you try to find.

Again, with the variable of fl, Drake seems far too optimistic, in view of the fact that he assumes that 100% of all the planets that can possibly bear life, will inevitably bear life. However, the only life we know of so far is located right here on planet Earth, and it is extremely difficult to predict how life started in the first place. We don’t even know for sure how life on Earth started. This is why it is incredibly optimistic and unscientific to assume that every single planet that has the right conditions for life will inevitably develop life. Even though we know that because of humanity’s existence, the number cannot be <0, we do not know exactly how life starts, and because the only place we can study life is on Earth, it would be wise to assume a wide range of numbers for this variable, from 0.1 to 1. This would give the variable a range of 10% to 100%.

For a third time, with the variable of fi, Frank Drake seems awfully optimistic. We can already tell from the multitudes of animal species and those that are already extinct that we are the first intelligent species to exist on this planet since its dawn. (Letzter, 2020). In fact, considering all the variables, we are lucky to exist at all. Thus, considering that the dinosaurs and many species before us had much more time than us to evolve into an intelligent species, they did not, because they didn’t need to. But, essentially, this means that the number Drake used for this variable just doesn’t make sense. It should be closer to 0 than 1. But since we do not have access to information about this from anywhere other than Earth, it would also be wise to assume a wide range of possibilities, from 0.1-0.5. This would give a 10-50% chance of life developing into an intelligent civilization.

The variable, fc, however, at the range of 0.1-0.2 proposed by Frank Drake, is perhaps one of the more accurate variable estimations proposed in the Drake Equation, as we do not have an accurate source of information to propose a different range of numbers for this figure, I believe that it is perhaps a good estimation for the number of intelligent civilizations which would be able to communicate (sending and receiving signals in space) You also have to take into account the fact that not all intelligent civilizations would develop advanced communication methods capable of transmitting signals across the deep, dark expanse of space.

The variable L represents the amount of time such civilizations have been transmitting detectable signals across space. This is measured in years, Frank Drake personally using the figures, between 1,000 and 100,000,000. This means that the lowest estimate he used in the Drake Equation was that civilizations have been transmitting signals for 1,000 years, and the highest, at 100 million years. Considering the fact that intelligent civilizations, such as ourselves, could have been transmitting signals into space for much less than that, ourselves only having been dishing out radio waves for two centuries, the minimum number that is used in this variable should be changed to 1 year, and the maximum should really be 13 billion, as the age of the Milky Way galaxy, our testing ground, is 13.51 billion years old. Setting the maximum figure to 13 billion leaves us with 510,000,000 years of planetary and solar development, allowing incredibly early star systems to form.

With the new figures and numbers to substitute our variables, we can now experiment with our theory.

We have chosen these numbers because they are all the medians of the minimum and maximum numbers we have chosen to use. When inserting the formula in a calculator, the answer we receive is 170,100,000, so according to our new formula, throughout the past 7 billion years in the Milky Way Galaxy, there could have been over 170 billion different civilizations that were capable of establishing contact with us. A number as large as that is vastly different from the feeble 20 minimum that Frank Drake predicted. However, in our new equation, we were using numbers that were in the middle between the highest and lowest, the median. If we were to use the minimum figures for each number, like Frank Drake, we would get this formula:

N = 1*0.5*1*0.1*0.1*0.1*1

Where,

R (1) = 1 New star born every year in the galaxy

fp(0.5) = The fraction of stars which have planets orbiting them.

ne(1) = The number of habitable planets in each star system with planets

fl(0.1) = The fraction of habitable planets where life actually starts

fi(0.1) = The fraction of all life bearing planets which actually develop intelligent species

fc(0.1) = The fraction of intelligent species that develop the capabilities to communicate through space

L(1) = The number of years which an alien civilization could have been transmitting detectable signals into space. The lowest number is 1, meaning that a civilization has only just been transmitting signals for one year.

Using these minimum figures gives us an answer of 5*10-4 This essentially means that there is an incredibly small chance that even one intelligent civilization exists. Although, these predictions take into account the number of years for which they have been transmitting signals into space, so what the answer is actually saying is that there is a 0.0005% chance that an intelligent species somewhere in the Milky Way galaxy has incredibly recently (a year ago) developed radio and has only started unintentionally beaming detectable radio signals into space. 0.0005 is a far cry from the potential 170 billion civilizations all through the past 7 billion years in the Milky Way galaxy. This means that we conclude with a wide range of answers, much like Frank Drake and his colleagues in 1961:

0<x<170,100,000,000

Conclusion

The updated variables that we have used in the Drake Equation to conclude with the range of numbers as an answer are perhaps more vindictive, realistic and accurate than the original figures proposed by Frank Drake. The numbers we have used support new research and studies in the field and offer an updated view on the actual and more accurately estimated number of communicative and space faring civilizations in the Milky Way Galaxy. The original variables that Frank Drake proposed himself when he first formulated the equation were inaccurate and unrealistic, especially in light of new studies and research projects conducted recently by professional organizations like NASA and the European Space Agency, and private foundations and individuals. Evidence of these inaccuracies include the fact that Drake estimated the birthing of new stars to be 1 in the Milky Way galaxy per year, when the actual number is closer to 3 or 4. Therefore, in conclusion, Frank Drake’s equation and variables make sense and should be kept, but the numbers they represent, although being an accurate guess at the time, should be changed to better fit society’s modern knowledge of the universe, astrobiology and astrophysics.

References

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