The Drake equation

The Drake equation is formulated by the American astronomer Frank Drake in 1961 to calculate the number of civilizations with which we could communicate, without imagining that this equation would make him famous. In 1961, a handful of researchers, including Carl Sagan, gathered to decipher the results of Drake, who had conducted the first radio observations of the sky the previous year. They try to come up with a scientific approach to searching for signals that might come from an extraterrestrial civilization. To structure the discussion, Drake scribbled the parameters of the problem. These few symbols will serve as a common thread for reflections on the possibilities of extraterrestrial life or, more precisely of intelligence.
The idea can be summarised as follows: "What if we estimated the number of galactic civilisations by multiplying probabilities?".

$$ N_{\text{civ}} = F_{\text{st}} \times P_{\text{pla}} \times N_{\text{pla}} \times P_{\text{life}} \times P_{\text{int}} \times P_{\text{com}} \times T $$

• $F_{\text{st}}$ is the rate of formation of stars in the Galaxy ^? .
• $P_{\cdots }$ is the probability or fraction of the stars meeting a particular condition (from 0 to 100%, i.e. from 0 to 1).
• $P_{\text{pla}}$ is the probability that a star has planets.
• $N_{\text{pla}}$ is the average number of habitable planets per star.
• $P_{\text{life}}$ is the probability that life appears on a habitable planet.
• $P_{\text{int}}$ is the probability that intelligence appears on a planet where life has appeared.
• $P_{\text{com}}$ is the probability that an intelligent life form develops the means of communicating with other worlds.
• $T$ is the time for which such communication can be detected. This is thus the lifespan of a communicative civilisation.

This formula gives us the required factors and thus enables us to study them, but it does not provide an answer. Drake’s equation is a means of measuring our ignorance… and our progress.

Rough estimates

• The number of stars in the Galaxy ^? lies somewhere between 100 and 200 billion. The age of the Galaxy ^? is about 10 billion years. We therefore estimate the star formation rate ($F_{\text{st}}$) at 10 per year. It has been virtually constant for about 5 billion years. This star formation rate was the only term of the equation that was relatively well known up until 1995.

• The discovery of exoplanets, in particular with the Kepler satellite ^? , showed that the probability of a star having planets, $P_{\text{pla}}$, is very high, close to 100%, at least in the galactic disc. This is one of the major results of the discovery of exoplanets. The stars far from the galactic plane ^? , which are old and poor in metals, could have far fewer planets.

• The time factor does not appear in the Drake equation, but it is important, in particular because stars have variable lifespans according to their mass. The most massive become very bright but have a short life. The less massive are neither very hot nor very bright, and can also have highly eruptive activity. Stable Sun-type stars represent about 1% of the total, but the range of acceptable stars (neither too massive nor too small) can be expanded to about 10%.

• The planetary constraints (mass and distance from the star) depend on the conditions required for the appearance of life, and thus on a definition of the concept of “life”. We can however limit the mass in the following way: the most massive planets are gas giants which have no surface (Jupiter, Saturn, Uranus, Neptune), whereas Mercury and dwarf planets such as Pluto have no atmosphere (they are not massive enough to retain one). It should be noted that life could also appear on the satellites ^? of the giant planets: even if they are not actually “planets” as such, they should be included as possibilities in the term $N_{\text{pla}}$. In the Solar System, this chiefly refers to Europa (satellite ^? of Jupiter, which could have liquid water under the ice).

• The number of habitable planets per star is beginning to be known thanks to Kepler’s observations. 20% of the Sun-type stars could have an Earth type planet in the HZ. It is also estimated that there is a 40 to 90% chance that a solar-type star has an Earth-like planet (rocky, approximately the same size and temperature).

• The probability of the appearance of life, $P_{\text{life}}$ is unknown. According to one optimistic viewpoint, life could appear as soon as the physical and chemical conditions on the surface of a planet are “right”. Several clues tell astronomers that life can form easily on a planet. The presence of prebiotic molecules ^? is detected in comets ^? and in the interstellar medium ^? . These molecules are the basic elements which, on Earth, formed the first living cells.
  The early days on planet Earth were highly agitated: Earth, along with the other planets, was permanently bombarded by planetesimals ^? , the descendants of which are comets ^? . When this incessant bombardment ended,
  the temperature on the Earth’s surface had fallen, liquid water then emerged land appeared and first living cells appeared. These earliest traces of life may be up to 3.8 to 4.1 billion years old. The planets closest to Earth, Mars and Venus, experienced the same conditions. It is possible that this process also occurred on these planets.
  But exceptional conditions may also have been needed: for example, it would appear that Jupiter acted as a gravitational shield in relation to Earth, preventing a large number of comets ^? from impacting it. Without this shield, Earth could have had a surface covered with water and would have been less favourable to life. In addition, the Moon, which has the particularity of being relatively massive in relation to Earth, stabilises the Earth’s axis of rotation, and therefore the long-term climate. The Moon also produces significant tides, intensifying liquid/solid exchanges.

• The estimation of $P_{\text{int}}$ is even harder, starting with the definition of intelligence.

• The same goes for $P_{\text{com}}$. A developed civilisation could for example not seek to communicate.

• Finally, $T$ is totally unknown! Our technological civilisation (capable of communicating via radio waves) has been around for about 100 years (even though homo erectus appeared a million years ago). Will we still be here in millions of years or will we disappear in a few centuries following a natural disaster, destruction of the ecosystem by pollution, or even a nuclear war?

Result

An optimistic calculation gives: $N_{\text{civ}} = 10 \times 1 \times 3 \times 0.02 \times 1 \times 0.25 \times {10}^8 = 1.5 \times {10}^7$.
A pessimistic calculation gives: $N_{\text{civ}} = 10 \times 1 \times 3 \times 0.01 \times 0.02 \times 0.1 \times 200 = 0.12$.
In reality $N_{\text{civ}}$ is apparently equal to 1 because we are here, but then we would be alone in the Galaxy ^? . So, there are still significant possible margins.

You can try various parameter values with the Drake equation applet.