# The Drake equation

## How to Estimate the Number of Aliens in the Milky Way

Oct 6, 2019 · 8 min read

The so-called Drake equation is a probabilistic argument purposed to estimate the number of active, communicative extraterrestrial civilizations in our galaxy. The equation is:

where

• R* represents the average rate of star formation in our galaxy
• fp is the fraction of stars that have planets
• ne is the fraction of planets that are in the circumstellar habitable zone, i.e. can potentially support life
• fl is the fraction of planets that could support life that actually do develop life at some point
• fi is the fraction of planets with life that develop (intelligent) civilizations
• fc is the fraction of civilizations that develop technology which emit detectable signals of their existance into space
• L is the length of time for which extra terrestrial civilizations release detectable signals into space

Subject to the assumptions that give rise to these variables, estimates of the number of extraterrestrial civilizations in our galaxy have ranged widely from as low as 0.000000000091 (we are probably alone in our galaxy) to as high as 15,600,000 (our galaxy is populated by millions of intelligent civilizations). The following animation eloquently explains the factors that go into the estimation:

# The Drake Equation

The Drake equation in other words summarizes the factors which are thought to affect the likelihood that humans will be able to detect radio-communication from intelligent extraterrestrial life (Burchell, 2006). The argument was first proposed in 1961 by American astronomer Frank Drake, for the purpose of stimulating scientific dialogue at the meeting of the Search for Extra-Terrestrial Intelligence (SETI) community:

As I planned the meeting, I realized a few day[s] ahead of time we needed an agenda. And so I wrote down all the things you needed to know to predict how hard it’s going to be to detect extraterrestrial life. And looking at them it became pretty evident that if you multiplied all these together, you got a number, N, which is the number of detectable civilizations in our galaxy. This was aimed at the radio search, and not to search for primordial or primitive life forms. — Frank Drake

The 1961 meeting of the SETI community was arranged following a surge of interest among various well-known physicists and astronomers’ in searching for extra terrestrial communications. Evidence of this interest includes Giuseppe Cocconi and Philip Morrison’s Nature article Searching for Interstellar Communications which first proposed the idea of using radio telescopes to try to detect transmissions broadcast by extraterrestrial civilizations. It also includes a newspaper article entitled Life on other Planets? in the Sydney Morning Harald (1959) in which Harlow Shapley (father of Nobel laureate mathematician Lloyd Shapley) and William Howells’ work for the Darwin Centennial celebration is highlighted.

# Original estimates

Depending on the assumptions, as many skeptics have pointed out, the Drake equation can give a very wide range of values. Original “educated guesses” for each parameter proposed by Drake and his colleagues in 1961 were:

• R* ≈ 1, the yearly average rate of star formation in our galaxy
• fp ≈ 0.2– 0.5, one fifth to one half of all stars will have planets
• ne ≈ 1–5, stars with planets will have between one and five planets in the habitable zone
• fl ≈ 1, every planet in the habitable zone will develop life
• fi ≈ 1, every planet will life will develop intelligent life
• fc ≈ 0.1–0.2, between 10% – 20% of planets with intelligent life will be able to communicate
• L ≈ 1000–100 000 000, the length of time for which extra terrestrials will send detectable signals into space and these signals’ duration

If we assume the lower estimates, the solution to the Drake equation then yields:

If we instead assume the higher estimates, the solution to the Drake equation then yields:

Suggesting that intelligent extra terrestrial civilizations in the Milky Way number somewhere between 20 (two zero) and 50 million. In other words, if one agrees with the 1961 assumptions of Drake and his colleagues’, it is likely that there currently exists intelligent extra terrestrial civilizations capable of communicating with Earth, as we speak. However, as should be clear, a few of these assumptions seem implausibly optimistic.

# Better informed estimates

Although we have a fairly good idea of the rate of stellar formation, a dearth of data for the other components means that calculations are often reduced to the creative speculations of quixotic astronomers — Michael Shermer

Since the above first estimate was proposed in 1961, each factor of the Drake equation have been further researched, altering the original estimation drastically. The only factor we can claim to have empirically valid, statistically significant estimates for, as Michael Shermer points out, is the first, the rate of star formation in the Milky Way (Shermer, 2002):

## The star formation rate

The first factor (R*) representing the rate of star formation in the Milky Way was in 2010 estimated by researchers Thomas Robitaille and Barbara A. Whitney to be approximately equal to between R* = 0.68 M☉ and 1.45 M☉, where M☉ represents a solar mass. Using the data from the GLIMPSE infrared survey conducted with NASA’s Spitzer infrared telescope, the study derived the galactic star formation rate (SFR) by comparing the number of young stellar objects (YSO) with a refined stellar-population synthesis model (Wanjek, 2015). The approximation yields a star formation rate of about 1.5–3 stars per year.

## Fraction of stars with planets

The current best estimate of how many stars have planets (fp), perhaps surprisingly, argues that the fraction may approach 1, suggesting that not only do most stars have planets, every star does.

## Fraction of planets in the habitable zone of sun-like stars

A 2013 paper in the Proceedings of the National Academy of Sciences purported that based on Kepler space mission data, there could be as many as 40 billion Earth-sized planets orbiting in the habitable zones of sun-like stars and red-dwarf stars in the Milky Way (Petigura et al, 2013). Considering that there are an estimated 100 billion stars in our galaxy, this implies that the fraction of planets in the habitable zone of sun-like stars (ne) is equal to 0.4.

## Fraction of habitable planets that develop life

Considering no other planet than Earth have as of yet shown signs of life, estimations of the fraction of habitable planets that go on to develop life (fl) are speculative at best. If signs of life were to be found on e.g. Mars, Europa or Titan (which developed independently of life on earth), it could imply a value of fl approaching 1.

## Fraction of habituated planets that develop intelligent life

Estimations of the fraction of habituated planets that go on to develop not only life, but intelligent life (fi), expectedly, varies widely. Among those who argue for low values, it is pointed out that among the billions of species on earth, only one (that we know of) has become intelligent, and so 1 divided by billions appears to argue for a small fraction. Among those who argue for high values, it is commonly pointed out that as the complexity of life increases over time, intelligence appears almost inevitable, given long enough time (Bonner, 1988).

## Fraction of habituated planets with intelligent life that send signals into space

The motivation and incentives for the discovery of the usefulness of electromagnetic waves for the purposes of long-distance communications appear strong enough to argue for the inevitability of a fraction of intelligent civilizations eventually emitting signals into space (fc), revealing their existence. This would argue for a high fraction. Arguments for a low fraction vary widely from those who claim that civilizations broadcast detectable radio signals only for a brief period of time before superior technology takes over, to those who argue that civilizations tend to isolate themselves. Either estimate of course, varies widely and is essentially purely speculative.

## Lifetime of an intelligent, communicative civilization

Although we do have empirical data to support estimations of the final factor, how long an intelligent, communicative civilization stays communicating (L), we lack an ability to extrapolate this data to other planets. Shermer (2002) argues that there have been 60 such civilizations thus far on earth (going back to Mesopotamia, Babylonia, Ancient Egypt and Greece, the Roman Empire, Chinese, Japanese, African, Indian and South American dynasties, and including six modern states of Europe and America) which survived a total of 25,234 years, giving an estimated average lifetime of an intelligent, communicative civilization to be L = 420.6 years. Others have added that at a certain level of complexity, civilizations overcome threats to their existence and go on forever. It has also been noted that as more civilizations arise and are extinct, future civilizations become increasingly sophisticated by learning from history.

# A modern estimation

Inputing the estimates found above and using fractions of 0.5 where there is no data to rely on, we obtain:

Suggesting, somewhat nonsensically, that there are approximately 46 extra terrestrial civilizations in the Milky Way currently capable of communicating with earth.

# Relationship to the Fermi paradox

The Drake equation has a relationship with the famous Fermi paradox which argues that there exists a contradiction between:

1. The lack of empirical evidence of extraterrestrial civilizations; and
2. The high estimates of the probability of extraterrestrial life (such as those arrived at by choosing high values for the factors in the Drake equation);

The argument for the paradox was formalized by Michael H. Hart in 1975, but bears Fermi’s name due to a well-known conversation about recent UFO reports between Enrico Fermi, Edward Teller, Herbert York and Emil Konopinski at the end of which Fermi supposedly proclaimed “But where is everybody?”.

Hart’s paper Explanation for the Absence of Extraterrestrials on Earth highlights the following axioms for the claim:

• There are billons of other stars in our galaxy that are similar to our sun;
• It is highly probable that some of these stars have Earth-like planets;
• If the earth is typical, some of these planets may have developed intelligent life and interstellar travel;
• Even at the slow velocity of envisioned interstellar travel, one could travel across the Milky Way in a few million years;

Following this reasoning, Hart argues, earth should already have been visited by extra terrestrial aliens or their probes, if they existed.

`Howard: Are you not familiar with the Drake equation?Sheldon: The one that estimates the odds of making contact with extraterrestrials, by calculating the product of an increasingly restrictive series of fractional values, such as those stars with planets and those planets likely to develop life, N = R* × fₚ × nₑ × fₗ × fᵢ × f𝒸 × LHoward: Yeah, that one.`

This essay is part of a series of stories on math-related topics, published in Cantor’s Paradise, a weekly Medium publication. Thank you for reading!

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