Well, let's examine this.
In 2.1 and earlier, the "standard" assumption was that a Pop was similar to 1 billion people. This meant that Earth started with a population of 8 billion in 2200, and could sustain a constant population of 16-17 billion people. Not bad, if a bit small to start with.
However, now with 2.2 around the corner, I feel that the 1 Pop = 1 Billion citizens equation falls apart. 24 billion people on Earth in 2200?? Not to mention I never really liked it too much in the beginning. Populations don't grow at a constant rate of 1 billion people per 33 months, no matter what their biology is.
To make matters harder, the Commonwealth of Man's biography mentions that the UNS Chrysanthemum carried with it a population of 250,000 colonists. This leads me to assume that (without the "A New Life" tradition) the first pop "generated" by a colony is close to 250,000 people, give or take a few hundred thousand, let's say.
Civilization V has a statistic that tells you the total population of your cities based on their population, a system very similar to pops in Stellaris. In that game, The population of your cities is equal to 1000•Population2.4. This exponential growth is much more similar to the population models in Stellaris, I think, rather than a flat rate of 1 billion people per Pop.
In that equation, the 1000 is the minimum population of a city. That is, when a city is a founded it has a population of 1000. In Stellaris, that number is the aforementioned 250,000. All we need is a Stellaris version of the "2.4" that the Population number is raised to.
The United Nations report that the estimated population of Earth in the year 2100 will be 11 billion people. Noting the fact that the Earth can only support a limited population until we invent post-scarcity utopia like is present Stellaris, I think an estimate of Earth's population being ~12.5 billion by the year 2200 is fair. Knowing that the starting amount of Pops on Earth is 24, we can solve for the growth rate, which according to my calculator gives me a result of approximately 3.4.
So, I think that we can approximate the population of your planets and empire using that exponential equation:
Population = 250,000 • Pops3.4
Interestingly, dividing 12.5 billion by 24 Pops gives a value of 1 Pop = 500 million people, which isn't actually that bad a metric either. I purport adding a ±0.1 difference to the growth rate (3.4) depending on whether your race has Rapid or Slow Breeders, respectively. In the case of large multiracial empires, just use the average rate.
If you wanted to be more accurate on an empire level, you could sum the results of this equation for each of your planets. Even if you didn't, this gives that new "We Are Legion" achievement for having 1000 pops as a hive mind a total population of 3.7 quadrillion drones in that hive minds empire. Yeesh.
Let me know what you think, and if I made any mistakes.
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