The Drake Equation

The Drake Equation is used in our world for the purposes of getting a guesstimate of the likelihood of contact with another civilization. For our purposes, however, it can be used to get a rough estimate of the size of the Galactic Empire.

We can adjust the Drake Equation, eliminating those data points which we do not need, such as those involving another civilization reaching a certain tech level, and so on.  Thus, we might end up with:

Habitable systems = Star-formation-rate * stable-time-factor * stellar-lifespan * fraction-with-planets * likelihood-of-Earth-type-planets 

Or:

HS = R_* * S_l * R_l * f_p * n_e

Translation and Comments:

R_* - Suitable Star Formation Rate -
Right now, our galaxy is forming approximately one new star per year, on average. In the past, this figure was as high as 3-5. For our purposes, F, G, and K stars are the most likely to support life, and these comprise about 10% of all our stars. Let's say that the Star Wars galaxy is (and has been) making one suitable star every ten years, or .1 stars per year.   (This would suggest a stellar formation rate (counting all stars) of one star per year, with a similar percentage being F, G, and K.)  

(Update:  The above value may be extremely low.  Drake himself recently stated that the Milky Way is making 20 stars per year, with up to four of those being of the proper stellar type.  I'll look into the situation and update accordingly.)

S_l - Time factor -
This is a new figure in the Drake Equation, since his was more specific... dealing with life arising and communicating in the galaxy. I define it in reference to a period of time during which a star is stable, putting out just the right amount of heat and light to allow for a stable habitable zone (i.e. a continuous distance from the star at which one could expect the solar radiation to be such as to allow liquid water to exist).  Let's guesstimate that the average star is stable for 70 percent of its lifespan.

R_l - Average Stellar Lifespan - 

A star such as ours has a lifespan of ten billion years.   Some stars last longer . . . some don't.   A star with 90% of the sun's mass and 93% of its diameter would have a lifespan of fifteen billion years . . . a star with 140% of the sun's mass and 120% of the diameter would last a "mere" four billion.  In theory, lifespans of anywhere from millions of years for a large, hot, fast-burning star, to hundreds of billions of years for a small, dim, slow-burning star are possible.  Let's use ten billion, to not only account for the many stars which live and die more quickly, but also to account for those old-timers.

F_p - Fraction with Planets -
Given the number of planetary systems we're finding nearby, a good figure for this might be .2, or 1 in 5. That suggests that 20% of stars have planets, and that means any planet (i.e. so far, we're only finding gas giants, not the terrestrial worlds we'd like to find).   Remember, though, that Qui-Gon said that "most" stars of the SW galaxy had planets.   (This implies a much higher average stellar metallicity (i.e. ratio of a heavy element (like iron) to hydrogen) for the stars of the Republic (or the entire galaxy), which would suggest that terrestrial planets could be extremely common by comparison.)   So, let's bump this up to 55%, or .55 as the fraction with planets.

N_e - Number of planets per solar system which can support life -
Given our recent discoveries (though confined to systems with extra-large planets), this number might be low, or might be high. For now, I'll say that 1 in 10 is a decent figure. This should probably be higher for the Star Wars galaxy, given the way many close-by systems in SW have habitable planets and moons, but I'll keep it for now.

Feel free to argue with any of these figures, since they are rough estimations anyway.  But, I do think they'll lead to a conclusion within a couple of orders of magnitude or so.

So, at the moment, we have:

HS = .1 * .7 * 10,000,000,000 * .55 * .1
HS = 38,500,000

That would be 38.5 million habitable systems in the Star Wars galaxy. Given the million systems of the Empire, it would suggest that about 1 in 40 habitable worlds of the galaxy were within their grip and colonized.  If these habitable systems were evenly distributed throughout the galaxy and the 1 in 40 (i.e. "million systems") were contiguous (i.e. in a group), it would only require 2.5% of the galaxy.   In a galaxy like ours of 100,000 light-years diameter, 2.5% would work out to a diameter of 15,800 light-years.

Remember well, however, the fact that the SW galaxy is canonically stated to be "modest-sized".  If we grant it a diameter of 50,000 light-years (larger than the highest-end local average and thus hardly modest, but workable) we end up with an Empire spanning just under 8,000 light-years.

But, let's try a different perspective.   If we were to assume that the Galactic Empire actually spans the length and breadth of the SW galaxy (suggested by the non-canon, though it's contrary to canon fact), and has 1 in 40 habitable planets colonized, that means the galaxy need only be large enough to support about 10 billion stars.  One of our neighboring satellite galaxies, the Large Magellanic Cloud, harbors about ten billion stars and is 10,000 light years across, albeit in a somewhat less "galactic" shape than our own galaxy (it has a disk and bars like ours, but it's not quite so well ordered).  However, given that the SW galaxy seems to be a rotating, well-ordered one (as per canon maps from AoTC), it could have a moderately larger diameter along the planar axis.

In any case, we have a Star Wars Galactic Empire that tops out at a rough maximum of 16,000 light years, and is more likely to be in the 10,000-12,000 light year range as per the LMC example.  If more than 100% of habitable planets are colonized (i.e. uninhabitable worlds like Bespin with technologically-assisted colonies), the figure becomes even smaller.  And let's not even consider those races who are part of the Empire and find worlds unlike ours to be habitable.

One final consideration would be the canon maps. In Attack of the Clones, a point near or just outside the core of the SW galaxy is identified as being "beyond the outer rim", and Tatooine, an outer rim world, is shown to be well within the galaxy's spiral arms. Besides the fact that this disproves non-canon notions of a galaxy-spanning Republic and Empire, the position given implies that the Republic is somewhere far closer to the core of the SW galaxy. This implies far higher stellar population densities and stellar metallicities (i.e. more stars with more likelihood of planets) than one would find, for example, way out toward the edge of the Milky Way, where Earth is . . . and a smaller necessary space for the Empire, accordingly.

Comments:

Those, of course, are just the spatial considerations. Something else to consider would be the longevity of the civilization. One could argue that, because the Republic is much older than the Federation, it would presumably have had more time to explore and grow, suggesting that it would be larger.

On the other hand, one could argue that the Federation is young and more concerned with exploration of the far reaches than colonizing what they have within their grasp. In other words, the Federation would be likely to have a much lower colony density over its territories than an old Republic (in a manner similar the history of the United States and the population clusters within it). Given the ~200,000,000 stars of the Federation, mixed with a maximum canonically-stated world-count of 1150, this argument seems likely.

Update:

Ted Collins of Babtech on the Net pointed out some additional details regarding the arguments of Imperial size and habitation density.

"A quick analysis of your figures for the territories and holdings of the Federation and Empire indicate that the "member world density" (inhabited planets/cubic light-year) of the Empire is roughly 160 times that of the Federation."

"Supporting calculations:

Federation Territory
Radius 4,000 ly
Area 5.03E+07 ly^2
Thickness 1000 ly
Volume 5.03E+10 ly^3
Inhabited planets 1,000
Density 1.99E-08 planets/ly^3"

"Empire Territory
Radius 10,000 ly
Area 3.14E+08 ly^2
Thickness 1000 ly
Volume 3.14E+11 ly^3
Inhabited planets 1,000,000
Density 3.18E-06 planets/ly^3"

(Note:  The Federation values are based on an 8,000 light-year diameter Federation as per Picard's "spread across 8,000 light-years" comment from First Contact, not the 8,000 by 6,000 light-year Federation I use based on canon maps.  Meanwhile, the inhabited planet count is from another tale featuring Cochrane from a century earlier, "Metamorphosis"[TOS].

Treating the Federation as an ellipse shape of 8,000 by 6,000 by 1,000 light-years gives us a volume of 3.8e10ly³, with the final density changing to 2.65e-8 planets/ly³.  Further, if we were to assume that the Federation had expanded to a mere 2,000 worlds by the time First Contact's size data was given, then the value would increase to 5.31e-8 planets/ly³.  These would imply habitation density differences compared to the Empire of 120 and 60 times, respectively.)

In general, the figures makes perfect sense, for a variety of reasons:

1.  According to current knowledge, it certainly cannot be said that most stars have planets.  Within 323 light years, we've only found 107 planets in 87 star systems . . . compare this to the 250,000 total stars within 250 light-years of Earth.   It should also be noted that in our galaxy, about 80% of stars are red dwarfs, as opposed to Sol-type stars.   Of course, the stars we now know of that do have planets all involve gas giants (current detection methods do not allow us to locate smaller bodies) . . . though gas giants would be expected in a solar system. Most of these solar systems which we are aware of are unlikely to have habitable worlds, due to the orbits and qualities of the respective gas giants (as per our current understanding of solar system development).

Compare that to Qui-Gon's statement in TPM, where he said that most of the stars of the SW galaxy had planets.  That alone will drive up the fraction in the SW galaxy significantly.

2.  McCoy, in "Balance of Terror"[TOS], remarks that there is a mathematical probability of 3,000,000 Earth-type planets in the galaxy.  (One wonders if that simply means "Class-M" or specifically "Earth-like", assuming there is a difference.)  Given a galactic volume of about 7,850,000,000,000 ly³ (7.85e12 ly³), that works out to a density of 3.82e-7 planets/ly³.   Using the elliptical TNG Federation and the 1000 planets figure from TOS, that would suggest, going by density, that only 1 in 14 Earth-type habitable worlds within their space was inhabited with a Federation colony, affiliate, protectorate, or member.  Thus, going by volume, the number of "Earth-type" habitable worlds within the Federation should be approximately 14,522.   Some of these, of course, would be inhabited by native species with Prime Directive protection.

(Update:  As of 2151, the Vulcans (as related by T'Pol) had calculated that only 1 in every 43,000 planets supported intelligent life.  The phrasing and context suggests she referred to indigenous life, not "could support" such life ("Fight or Flight"[ENT1]).  Unfortunately, it isn't clear if this referred to all planets (i.e. our solar system would show 1 out of 9), planets-planets (i.e. the big dirtballs, not the big gasballs), or actual Class-M worlds (which would just be confusing, since it would require way more than 14,522 Class M worlds for us to have seen all those aliens).)  

Other Details from the canon:

1. There are several instances where more than one planet (or moon) per system are habitable or inhabited, though not all of these are Federation worlds. Off the top of my head, I can think of the multiple planets of the Rigel system (various instances), the two worlds Eminiar and Vendikar ("Taste of Armageddon"[TOS1]), Troyius and Elas from "Elaan of Troyius"[TOS3], Romulus and Remus (ST:Nem), Bajor and Jeraddo, its fifth moon ("Progress"[DS9-1]), and Ekos and Zeon from "Patterns of Force"[TOS2]. These would suggest that solar systems that are habitable can often be very habitable, and could arguably be taken to imply that habitable worlds are quite common in the galaxy.

2. On the other hand, mankind's first colony (depicted in the otherwise-forgettable "Terra Nova"[ENT]) was 20 light-years from Earth. There are over 100 stars within 20 light-years of Earth (as per http://www.anzwers.org/free/universe/nearstar.html), which would imply at least some decent spread between habitable worlds. Also, up until "Dead Stop"[ENT2], at which point it is suggested that the Enterprise is about 130 light-years from Earth, only about 7 Class M worlds have been seen by Enterprise. (And only one or two was uninhabited, as I recall.) T'Pol, as of "Strange New World"[ENT1], had visited only 36 Class M worlds in her entire career.  

3. Over the course of the entirety of Star Trek, I'd guesstimate that we've seen or heard of perhaps 500 different species or individual populated/colonized planets within or near Federation space. That suggests a low density of habitable systems.  

4. The Federation has devoted resources toward terraforming (ST2, "Home Soil"[TNG]), and has starships dedicated to this task (the "geoterraforming" vessel USS Strata from "Field of Fire"[DS9]). This also suggests a low density of habitable systems.

5. Bajor had colonized a world on the other side of the wormhole early in the DS9 series. This colony, New Bajor, was destroyed by the Jem'Hadar. Given the many problems of Bajor, this might suggest that a colony isn't hard to make. (Alternately, given the many problems of Bajor, it might mean that there were some people who simply wanted to get the hell out of Dodge.)

6. The Federation colonies we have seen (Delta Rana IV from "Survivors"[TNG], Jouret IV from BoBW1, Caldos IV from "Sub Rosa"[TNG], the new colony being planted in "Silicon Avatar"[TNG], Terra Nova, Deneva from "Operation: Annihilate!"[TOS], the Berthold radiation planet from "This Side of Paradise"[TOS], et cetera) all seem to be on extremely Earth-like worlds (or what are thought to be such worlds). This may suggest a helluva lot of pickiness by those who want to colonize other planets.

Those, I think, would be the only real clues available for habitation density within the Federation. There are a few other interesting facts . . . the uninhabited (di?)lithium cracking station close to the edge of the galaxy, the comment by Spock in "Doomsday Machine"[TOS] about "the most populated" region of the Trek galaxy, and so on . . . but overall it would be hard to get any useful, non-contradictory estimate from the canon.

My own personal opinion (a rough synthesis of the various data points) is that there are probably a fair number of easily-habitable planets in the Federation that haven't been colonized, and a fair number that have. Overall, though, most of the Federation population would be centered in a core area, with lesser numbers of people living on distant colonies (such as Jouret IV, said to be one of the most distant colonies, destroyed by the Borg in BoBW1). (However, there were a substantial number of colonies at or near the Cardassian border.)  This would make some sense, since I imagine that most colonials would prefer to be relatively close to Federation protection . . . though a few intrepid souls might not mind being way out on the frontier. The existence of terraforming would suggest that many of these habitable core worlds are inhabited already, and thus there is pressure to increase the number of colonizable worlds in that area. On the other hand, one would think that most terraforming efforts would be geared toward improving the habitability of already-habitable worlds, as opposed to bothering with Mars-like planets like the one from "Home Soil" . . . I'm thinking here of worlds that are "barely livable" like Galorndon Core (a dark and stormy world seen in "The Enemy"), Nelvana III from the ep that introduced Thomas Riker, and Mab-Bu-whatever from "Power Play"[TNG]. But, as I said . . . this is a rough synthesis . . . because one can hardly create a better one given the peculiarities of the various data bits.

In any event, I think it safe to say that the Federation certainly hasn't got people on every conceivably-livable world, whether it's an easy living there or not. Meanwhile, we see that the Empire even has people on planets like Tatooine.

In short, the Empire has more habitable worlds at its disposal, and after all the centuries of the Republic, it's colonized a huge number of them.