The Drake equation is an attempt to roughly calculate the number or percentage of stars having panets with intelligent life.
Another equation that would be of interest to science ficiton fans in general and
Star Trek fans in particular is one to calculate the percentage of stars having planets naturally habitable for humans and for beings with the same environmental requirements.
Most of the planets visited by starship crews in TOS and other
Star Trek productions are class M planets, which seem to roughly correspond with planets where Earth Humans can survive for hours at a time with no obvious environmental protection gear, planets which are naturally habitable for humans.
As astronomers came to understand the fusion processes in stars, they began to make calculations about the destinies of stars of various types. And thus it became possilbe for astrophysicists who were science fiction fans to calculate which stars are most suitable for having planets naturally suitable for humans to live on.
Characters in
Starman Jones (1953) and
Time for the Stars (1956) by Robert A.Heinlein commented that main sequence spectral class G stars were most suitable for having habitable planets. But a lot of other science fiction writers continued to depict habitable planets orbiting quite different types of stars.
As far as I know the only scientific study of the possibility of exoplanets of other stars being naturally habitable for humans (and also of course for beings with the same environmental requirements) was
Habitable Planets for Man, Stephen H. Dole, 1964.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
As far as I can tell scientific discussions of planetary habitability in recent decades discuss habitability for liquid water using life forms in general, and not the more restricted case of habitability for humans in particular. I note that even on Earth, which is habitable for humans in many places, the vast majority of the biosphere would be swiftly fatal to unprotected and unprepared humans teleported there. Human habitable planets should be a minority of planets habitable for liquid water using lifeforms.
Reading
Habitable Panets for Man I was disappointed by the many scientific restrictions on habitability discussed, making habitable planets seem much rarer than I desired. On pages 82 to 101 Dole disussed the probability of each of a long list of factors necessary for an individual star to have habitable planets and came up with a total probability that an indivudual star will have habitable planets.
In table 18 on page 104, Dole concludes that in our section of the galaxy there should be about 4.03 times ten to the minus 4th power stars with habitable planets per cubic parsec, or 0.000403 stars with habitable planets per cubic parsec. Since a parsec is 3.26156 light years, a cubic parsec is 10.637773 cubic light years, and thus Dole calculates that there are 0.0000378 stars with habitable planets per cubic light year.
Stellar density is the average number of
stars within a unit volume. It is similar to the stellar mass density, which is the total
solar masses (MSun) found within a unit volume. Typically, the volume used by astronomers to describe the stellar density is a cubic
parsec (pc3).
In the
solar neighborhood, this value can be determined from surveys of nearby stars, combined with estimates of the number of faint stars that may have been missed. The true stellar density near the Sun is estimated as 0.004 stars per cubic
light year, or 0.14 stars pc−3.
This indicates that Dole believed that about one star out of a few hundred would have a human habitable planet.
Table 19 on page 105 gives the expected number of habitable planets within spheres with specifed radii around the Sun.
A spere with a radius of 27.2 light years should have 1 habitable planet.
A spere with a radius of 34.3 light years should have 2 habitable planets.
A spere with a radius of 46.5 light years should have 5 habitable planets.
A spere with a radius of 58.5 light years should have 10 habitable planets.
A spere with a radius of 100 light years should have 50 habitable planets.
A sphere with a radius of 100 lightyears would have avolume of 4,188, 790.2 cubic lightyears and thus it should contain about 16,755.1608 stars, thus indicating that Dole's calculations found that about 1 star in every 335.1 would have a habitable planet.
On page 103 Dole said that there should be about 600 million stars with habitable planets in the Milky Way Galaxy. Since the Milky Way Galaxy is now believed to contain about 100 billion to 400 billion stars, there could be about 298 million to 1.1936 billion human habitable planets in the galaxy, if Dole's calculations are correct.
In
Star Trek human habitable planets seem much more common than Dole estimated.
Getting back to the Drake equation:
Planets with intelligent beings contemporary with humans are also very common in
Star Trek. Some of those alien species reside on human habitable planets, and some reside on planets where humans would instantly die.
In the TOS episode "Who Mourns for Adonais?" the
Enterprise is exploring the system of Pollux.
CAROLYN: Here's the report on Pollux Five, Captain. This entire system has been almost the same. A strange lack of intelligent life on the planets. It bugs the percentages.
KIRK: Bugs the? Well, carry out the standard procedures on Pollux Four.
So despite the fact that a normal star system should have only a few planets in the habitable zone, Lt. Palomas seems to be think it is very odd that they haven't found any with intelligent life there yet.
It gives the impression that Starfleet has found at least one planet with current intelligent life in every solar system so far.
In the Enterprise episode "Fight or Flight":
T'POL: I'm sure you're aware that only one out of every forty three thousand planets supports intelligent life.
ARCHER: I took exobiology, I know the statistics, but we're travelling at warp five. There's got to be someone out here. Come in. (squeak)
Considering that you might expect the average star to have about 5 to 10 planets, that statistic suggests that only about one star out of every 4,300 to 8,600 supports intelligent life.
And if only one star out of several thousand has a planet with intelligent life, it is certainly an odd coincidence that Vulcan is said to be 16 light years and over 16 light years from Earth in two episodes of Enterprise. A sphere with a radius of 17 light years has a volume of 20,579.52628 cubic light years, and with 0.004 stars per cubic light year, there should be only about 82.3 stars in that volume of space, so having at least two stars with planets with intelligent life is really against the odds used by T'Pol and Archer.
Thus there is a vast discrpancy in how common stars with planets with intelligent life are alleged to be in different versions of
Star Trek. It is like the different series have different Drake equations.
And one one way to reconcile the two series is that in the time of Enterprise a very broad defininition of planet and a very narrrow definition of intelligent life is used, while during the era of TOS a very narrow definiton of planet and a very broad definition of intelligent life is used.
In the TOS era
Star tTek IV: The Voyage Home:
McCOY: Well, dammit? You think this is its way of saying 'Hi there' to the people of the Earth?
SPOCK: There are other forms on intelligence on Earth, Doctor. Only human arrogance would assume the message must be meant for man.
KIRK: You're suggesting the transmission is meant for lifeform other than man?
SPOCK: It is a possibility, Admiral. The President did say that it was directed at Earth's oceans
So in the era of TOS some lifeforms in the oceans of Earth are known or suspected to be intelligent.
On Earth, in addiiton to extinct relatives and ancestors of
Homo sapiens, there at least four ape species, 3 proboscidean species, and 80 or 90 cetacean species, whose intelligence ranges probably greatly overlap with the intelligence range of
Homo sapiens, and which could possibly be considered intelligent beings and people by neutral observers. And there have been now extinct possibly intelligent species of primates, proboscideans, and cetaceans for many millions of years. And that is just considering possibly intelligent species of mammals, and ignoring potentially intelligent species in other classes of animals.
All of those species use tools to a greater or lesser degree depending on their environments and how easy it is for their bodies to make and use tools. But out of possibly hundreds of possibly intelligent species in the history of Earth, as far as we know only one,
Homo sapiens, ever developed high technology and civilization. Note that
Homo sapiens fossils have been found datng back as far as over 300,000 years ago, but members of
Homo sapiens only began to develop higher technology and the beginnings of civilization about 12,000 years ago, during the latest 25th or 0.4 of the existence of
Homo sapiens.
That indicates to me that it may be common for a planet with large muliticelled animals to have tens or hundreds of species of inelligent beings and people at a time, but only a tiny minority of them develop civilization. And if that is the case in TOS, planets with recognized intelligent beings may be very common, but maybe only a fraction of them have civilizations in TOS.
So possibly in the era of TOS,many times as many species are recognized as being intelligent beings and people as are civilized.
And possibly in the era of
Enterprise, only civilized species are recognized as intelligent beings, thus reducing the numbers of recognized species of intelligent beings greatly.
Possibly in the era of TOS the definition of a planet is very similar to that used by the International Astronioical Union (IAU) since 2006 for planets in our star system.. In "The Changeling" the diagram of the solar system shows 9 planets, so either Pluto is considered a planet in TOS or a ninth planet has been discovered by the era of TOS. The IAU has not yet made a formal definition of what is an exoplanet in another star system.
In the era of
Enterpise the definition of a planet might be much broader.
Here is a link to a list of objects in our solar system which have been considered to be "planets" at some time during the last few thousand years or so but are no longer counted as planets.
https://en.wikipedia.org/wiki/List_of_former_planets
It lists 26 former planets. But all of the first 15 asteroids or minor palnets to be discovered were at least occassionally considered to be planets until 1855, so that adds 8 more for a total of 34 former planets. Andit would be fairly reasonable to reclassify some of them as planets again.
A planetary mass object or planemo or planetary body or geophysical planet is an object less massive than a brown dwarf or a star and massive enough for its gravity to pull and squeeze its matter into a spheroidal or ellipsoidal shape.
Here is a link to a list of gravitationally rounded objects in our solar system. Except for the Sun, they are all geophyical planets.
https://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_System
They include 8 planets and 5 dwarf planets which allorbit directly around the Sun. It also lists 5 objects which ae good candidates to be classified as dwarf planets. That makes a total of 13 to 18 planetary mass objects orbiting the Sun which could be redefined as planets. Many other Trans Neptuntian Objects (TNOS) are not known well enough to judge how well they would qualify as geophysical planets. And of course it is posssible, depending on the process of solar system formation, that mny hudnreds or thousands of planetary mass TNOs might be discovered in the future.
A antural satellite or moon is as astronomical planet which orbits around a planet or other object which orbits around a star instead of directly orbiting that star. Most of the hundreds of known moons in the solar system are small objects, but a few are planetary mass objects which could be reclassified as planets.
At the present ttime 19 known moons in our solar sytem are geophysicla planets and could be reclassified as planets in the future. So in the era of
Enterprise, between 8 and 37 presently known solar system objects might be classifed as planets, and more objects might be discovered and/or classifed as planets by then.
Planetary mass objects are more massive than the minor irregularly shaped objects, and range in mass up to about roughly 13 times the mass of jupiter. Objects between aproximtely 13 times the mass of jupiter and roughly 75 to 80 times the mass of jupiter are classified as brown dwarfs. Objects more massive than about 75 to 80 times the mass of Jupiter are classified as stars.
Since brown dwarfs are not classified as planets or stars, objects which orbit brown dwarfs are technicallny not either moons or planets. Maybe they could be called planoons or moonets. And probably there are a lot of planoons or moonets or whatever orbiting brown dwarfs, and probably a lot of them are planetary mass objects.
Posssibly by the era of
Enterprise, some or all brown dwarfs migh be reclassified as planets, which would significantly increase the numbrof planets. Since many scientists consider life on brown dwarfs impossible, that might increase the ratio of panets to planets with intelligent beings.
The astronomical objects in the galaxy are not limited to stars and the various objects which orbit those stars. Many objects less massive than stars exist in interstellar space, orbiting the center of mass of the galaxy like the stars do. They include brown dwarfs, "rogue planets" includng all planetary mass objects in interstellar space, and countless smaller, irregularly shaped objects. There are three confimed examples of interstellar objects entering our solar system.
https://en.wikipedia.org/wiki/Rogue_planet
https://en.wikipedia.org/wiki/Interstellar_object
And possibly the numbers of planetary mass objects wandering in interstellar space could be enough to have hundreds or thousands of objects classified as "planets" in the era of
Enterprise for every planet which has intelligent life as defined in the era of
Enterprise.
So I suspect that diffferent eras of
Star trek use different versions of the Drake equation, based more on observation than on theory, but differing because of differing definitions of of planets and of intelligent life.