Having background aliens is a matter of immersion. We're told the Federation is an equal alliance of hundreds of planets then it creates a dissonance if everyone is human and everything revolves around Earth.
Are we told that the Federation contains hundreds of planets or millions of planets?
First a few facts about the size and number of stars in our Milky Way Galaxy:
The Milky Way Galaxy has a central bulge, a galactic disc, and a thinly populated spherical halo around both. The galactic disk is about 100,000 light years in diameter. Stars thin out with increasing distance from the central plane of the galactic disk, and it is common to read that the galactic disc is about 1,000 light years thick near Earth. There are an estimated 100,000,000 to 400,000,000 stars in the Milky Way Galaxy.
So as a space traveling society expands from its home world, it would occupy an ever growing spherical volume of space, until the sphere expands to reach the "upper" and "lower"edges of the galactic disc, and then the expanding society should become an ever expanding flat cylinder with a thickness of 1,000 light years and an ever expanding radius. Until of course, it meets othe expanding civilizations, which is very common in space opera stories, and is unable to expand beyound various borders with those other societies, thus distorting the shape of the volume of space it rules.
The true stellar density near the Sun is estimated as 0.004 stars per cubic
light year, or 0.14 stars pc−3.
https://en.wikipedia.org/wiki/Stellar_density
In the ENT 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)
If only one planet out of every 43,000 supports intellligent life, then the percentge of stars with planets with intelligent life should be about 1/43,000, or 0.00002325, multiplied by the number of planets per star. So if there are one hundred billion to four hundred billion stars in the Milky Way galaxxy there should be abou 2,325,581 to 9,302,325 planets with intelligent life, multiplied by the average number of planets per star, in the Milky Way Galaxy.
So the 150 member planets later said to be in the Federation would require about 6,450,000 planets. And also about 6,450,000 star systems, divided by the average number of planets per star system. Assuming an average of one planet per star, and with about 0.004 stars per cubic light year, the Federation would occupy a volume of at least 1,612,500,000 cubic light years, equilvalent to to a sphere with a radius of 727.5 light years, or a diameter of l,455 lgiht years.
In "Who Mourns For Adonais?"
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.
So have they checked 430,000 planets in the Pollux system already without finding intelligent life on10 of them as would be statistically likely?
I find it easier to believe that in Archer's era a very broad definitin of planets, and a very narrow definition of intelligent life, are used. And in Kirk's era a very narrow definition of planets, and broader definition of intelligent life, are used.
Or maybe Palomas means that they have checked a bunch of planets with life in the circumstellar habitable zone of Pollux (and only a small subset of all planets would be in the circumstellar habitable zones of their stars) and found fewer of those planets with intelligent life than would be usual.
In the Tos epsode "Balance of Terror":
MCCOY: But I've got one. Something I seldom say to a customer, Jim. In this galaxy, there's a mathematical probability of three million Earth-type planets. And in all of the universe, three million million galaxies like this. And in all of that, and perhaps more, only one of each of us. Don't destroy the one named Kirk.
Since McCoy doesn't specify how Earth like his Earth-type planets are, it would be hard to compare his estimate with 2020 estimates.
In the TOS epsiode "Mudd's Women" Harry Mudd is arrested at Rigel. In the later episode "I, Mudd" Kirk & co. meet Mudd again.
MUDD: Yes, well, I organised a technical information service bringing modern industrial techniques to backward planets, making available certain valuable patents to struggling young civilisations throughout the galaxy.
KIRK: Did you pay royalties to the owners of those patents?
MUDD: Well, actually, Kirk, as a defender of the free enterprise system, I found myself in a rather ambiguous conflict as a matter of principle.
SPOCK: He did not pay royalties.
MUDD: Knowledge, sir, should be free to all.
KIRK: Who caught you?
MUDD: That, sir, is an outrageous assumption.
KIRK: Yes. Who caught you?
MUDD: I sold the Denebians all the rights to a Vulcan fuel synthesiser.
KIRK: And the Denebians contacted the Vulcans.
So Harry Mudd has been to Deneb between the two eps isodes. Deneb, or Alpha Cygni, is about 2,500 light years from Earth, give or tke a few hundred.. So in the TOs era the zone of exporation raches to dneb and about 2,500 lightyears from Earth.
MUDD: Worse than that. Do know what the penalty for fraud is on Deneb Five?
SPOCK: The guilty party has his choice. Death by electrocution, death by gas, death by phaser, death by hanging.
MUDD: The key word in your entire peroration, Mister Spock, was, death. Barbarians. Well, of course, I left.
Someone who thinks that a Federation planet can have the death penalty can accept that deneb V could be a member of the Federation. Someone who thinks that no planet with the death penalty can be admitted to the Federation would think that Deneb V can not be a Federation member.
Is Deneb part of the Federation?
In the TOS episode "Journey To Babel" KIrk tells McCoy that the delegation from Vulcan will be the last one they pick up.
KIRK: The Vulcans are the last delegates we have to pick up. As soon as we get them aboard, we'll be able to relax.
And:
MCCOY: Sure. A formal reception tonight, a hundred and fourteen delegates aboard for two weeks, thirty two of them ambassadors, (Spock joins them) half of them mad at the other half, and the whole lot touchier than a raw antimatter pile over this Coridan question.
Possibly McCoy means that there are only 32 Federation members, and each planet's delegation consists of one ambassador and an average of about 2.5 other persons. Or possibly Mccoy means that there are 114 Federation members, and 114 delegations, 32 of them commanded by ambassadors and 82 of them by persons of other ranks, and that there are a few assistants in each delegation, thus making a total of about 200 to 400 guests on the Enterprise.
Later:
Captain's log, Stardate 3842.3. We have departed Vulcan for the neutral planetoid code-named Babel. Since it is in our sector, the Enterprise has been assigned to transport ambassadors of Federation planets to this vitally important council. The issues of the council are politically complex, the passengers explosive.
.
This implies that the Enterprise is the only starship carrying delegations, and that the only representatives of Federation planets going to Babel are those aboard the Enterprise.
In the TOS episode "Methamorphosis" Kirk tells Cochrane:
KIRK: We're on a thousand planets and spreading out. We cross fantastic distances and everything's alive, Cochrane. Life everywhere. We estimate there are millions of planets with intelligent life. We haven't begun to map them. Interesting?
It is uncertain where Kirk's "we" means Earth humans or Federation members in genral, nor how large a population a planet has to have to be counted.
If Kirk means that the estimated millions of planets with Intelligent life are scattered across the entire universe, then it would be statiscially very improable for a single galaxy to have more than one planet with intelligent life. That is not true in
tSar Trek So Kirk probably meant millions of planets with intelligent life in our Milky Way Galaxy. If "millions" is somewhere between 1,000,000 and 1,000,000,000, and if there are between 100,000,000,000 and 400,000,000,000 stars in our galaxy, the average ratio of stars per planet with intelligent life would be between one in four hundred thousand and one in a hundred.
And later Cochrane tells Nancy/the Companion:
COCHRANE: Everything will be an eye-opener to you. There's a thousand planets out there, a thousand races, and I'll show everything to you, soon as I learn my way around again. Maybe I can make up a little for everything you've done for me.
And I don't know how accurately Cochrane interpreted Kirk's statemens.
In the TNG episode "Where No One Has Gone before":
KOSINSKI: As an explorer. In three centuries of space flight, we've charted just eleven percent of our galaxy. And then we accomplish this.
in the TNG epsisode "The Dauphin":
WESLEY: For both of us. This is all just beginning. We've only charted nineteen percent of our galaxy. The rest is out there, just waiting. Look what we've already discovered.
So apparently 8 percent of of the galaxy was charted within about a year Unless, of course, Kosinsky and Wesley used different standards of charting.
Obviously every planet within the Federation, and probably many times as many stars and planets which are not within the Federation, would be within the percentage of the galaxy which has been charted.
Assuming that the percentages of the galaxy refer to percentages of the volume, and that the volume is restricted to the glactic disc, and that the galactic disc in 100,000 light years and diameter and uniformly 1,000 light years thick, which is a bunch of assumptions, the size of the charted regions of the galaxy can be calculated.
Eleven percent of the galaxy would be a cylinder with a thickness of 1,000 light years, a radius of 16,583 light years, and a diameter of 33,166 light years.
Nineteen percent of the galaxy would be a cylinder with a thickness of 1,000 light years, a radius of 21,794 light years, and a diameter of 43,589 light years.
Or maybe the percentages mentioned were eleven and nineteen percent of the total number of stars in the galaxy. Since there are an estimated 100,000,000,000 to 400,000,000,000 stars in the galaxy, eleven percent could be as few as 11,000,000,000 stars, and nineteen percent could be as many as 76,000,000,000 stars. If there are about 0.004 stars per cubic light years in the region of the galaxy near Earth, the charted part of the ggalxy could be a cylinder 1,000 light years thick and between 935.6 and 245* light years in radius, between 1,871.2 and 4,918 light years in diameter, if my calculations are correct.
And the volume of space ruled by the Federation would probably be a small portion of those charted areas.
In
Star Trek: First Contact Picard describes the Federation to Lily
LILY: How many planets are in this Federation?
PICARD: Over one hundred and fifty ...spread across eight thousand light years.
The Milky Way Galaxy has a central bulge, a galactic disc, and a thin spherical halo around both. The galactic disk is about 100,000 light years in diameter. Stars thin out with increasing distance from the central plane of the galactic disk, and it is common to read that the galactic disc is about 1,000 light years thick near Earth.
So as a space travelling society expands from its home world, it would occupy an ever growing spherical volume of space, until the sphere expands to reach the "upper" and "lower"edges of the galactic disc, and then the expanding society should become an ever expanding flat cylinder with a thickness of 1,000 light years and an ever expanding radius. Until of course, it meets othe expanding civilizations, which is very common in space opera stories, and is unable to expand before various bordrs with those other societies, thus distorting the shape of the volume of space it rules.
So one could assume that Picard's statement means that the Federation occupies a flat cylinder with a thickness of about 1,000 light years, a radius of about 4,000 light years and a diameter of about 8,000 light years. It would have a volume of about 50,265,482,457 cubic light years.
The true stellar density near the Sun is estimated as 0.004 stars per cubic
light year, or 0.14 stars pc−3.
https://en.wikipedia.org/wiki/Stellar_density
So there should be approximately 201,061,928 stars in the cylinder of space ruled by the Federation. And it has been estiamted that the number of planets in the Milky Way Galaxy should be greater than the number of stars. So the Federation should contain at least 200,000,000 planets. With only 150 member planets in the Federation, that would mean that only one star out of about 1,333,333.333 in Federation space has a planet which is a Federation member.
And many people might say that probably a much smaller proportion of stars than one in a million would have a planet with intellligent life in real life.
But in
Star Trek a number of stars close to Earth are memtioned as having intelligent life on some of their planets. If the stars near to Earth are a typical sample, the proportion of stars with planets with intelligent life should be more like one in ten or one in a hundred So there should be about 2,000,000 to 20 000,000 stars with intelligent life in the space of the Federation, not a mere 150.
Maybe Picard said that the Federation with 150 member planets was spread over 8,000 cubic light years. That would be a cube with sides 20 light years long, or a sphere with a radius of 12.407 light years and a diameter of 24.814 light years.In any case, a volume of space with 8,000 cubic light years should have about 32 stars, so the average star in that sphere would have to have about 5 planets which were Federation members.
Maybe the Federation covers 8,000,000 cubic light years, and Picard said that 8,000 by mistake. In that case the Federation could be a cube 200 light years on a side, or a sphere with a radius of 124.07 light years and a diameter of 248.14 light years There would be about 32,000 stars in Federation space, and about one star out of every 213.33 would have a Federation member planet.
Or maybe Federation space consists of hundreds or thousnds of disconnected spheres, cubes,and other shaped volumes of space around Federation member planets and other star systems ruled by the Federation, and the greatest distance between two volumes of space claimed by the Federation is about 8,000 light years.
And of course there is other evidence of the size of the United Federation of Planets. Many maps of the
Star Trek galaxy depict the Federation, the Klingons, the Romulans, and other realms as big blobs of space spreading over hundreds or maybe thousands of light years, and possibly speading through the entire thickness of the galactic disc.
If such borders are correct every well known realm in
Star Trek should rule over thousands or millions of star systems within the volume of space it rules. Or possibly billions of star systems. Any space government depicted as covering a noticeable area, instead of being a dimensionless little dot, on maps of the galactic disk of the Milky Way Galaxy must rule a vast and unimaginably large number of star ssytems.
And thus it should rule a vast number of star systems, some of them with intelligent life.
Does she wear her hair in blonde pigtails and is named "Elsa" too?
