> For now we shall stick with the simple model where the average population of a civilisation increases with the planet’s surface area.
This is exactly what keeps it as an amusing thought experiment a la the doomsday argument—it's an unintuitive result from treating our population growth as simple as possible and extrapolating it to a general law.
I accept the doomsday argument, but I had heard that it is controversial, so I expected more debate here.
[1] https://en.wikipedia.org/wiki/Self-indication_assumption [2] https://en.wikipedia.org/wiki/Self-sampling_assumption [3] https://en.wikipedia.org/wiki/Self-Indication_Assumption_Doo...
Or another way, you at least need to argue AGAINST rational rationing of existing resources to make calculating potential populations over all time quite difficult and run up against heat death calculations.
That said, I love both these thought experiments because they highlight how hard it is to figure out population growth with (theoretically) rational populations.
Of note, if fusion get's cheap enough planets far from stars may be more useful as they get to radiate more energy into space.
Have you heard of the German Tank Problem? https://en.wikipedia.org/wiki/German_tank_problem
You can make estimates from a sample size of one. That's literally the point of the entire article. If you want their justification, it's here:
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A similar connection between area and population is seen among countries on Earth. Those with a larger population also tend to have a larger area. This effect is not quite as strong as you might think, due to the way countries are formed. A tiny region of land is less likely to declare its independence if only a handful of people were living in that region to begin with. The smallest countries therefore tend to have slightly higher population densities. Overall though the trend is clear, larger countries do hold significantly larger populations. Most countries are smaller than sixty thousand square kilometres, yet most individuals live in a country of over one million square kilometres.
If we are to estimate the size of an ordinary alien planet - one that hosts intelligent organisms - we first need to make two decisions. The first is the connection between population size and planet size. The simplest approach is to suppose that, on average, the population density will not change with planet size. For small changes in planetary radius this ought to be a good approximation. For planets much larger than Earth one could imagine that a larger proportion of the planet’s surface - for example near the poles and the equator - tend to become uninhabitable. A more detailed study of planetary atmospheres, and the prevalence of water, is needed to give a better answer here. For now we shall stick with the simple model where the average population of a civilisation increases with the planet’s surface area.