Estimating the Incidence of Intelligent Life in the Galaxy
Bruce L. Gary, Last Updated 2023.03.17

I endeavor to show that if the Moon was produced by a grazing collision of proto-Earth by a Mars-sized body, humans are likely to be the only sentient creatures in our galaxy at this time. My argument will rely upon set theory and sampling theory. Keep in mind that this is a “concept demonstration” so it’s not important to have accurate values for the needed parameters.

Let’s begin by considering the super-set of all rocky planets in our galaxy that are in the habitable zone, HZ. The size of this super-set is large, possibly 1e+10.

Now consider the sub-set of this super-set that have a super-size moon formed by the same process as our moon; namely, a Mars-sized grazing collision of a proto-Earth. The Moon’s mass is 1.23 % of the Earth’s mass. For the other solar system planets the median mass ratio for all satellites of a planet to the planet’s mass is 0.010 %. In other words, the Moon’s mass in relation to Eaaaarth is an outlier, being 118 times what would be expected based the other planets. We Earthlings have a super-sized moon. Here’s a graph that illustrates this.

Figure 1. Ratio of sum of masses of moons to the planet mass for our solar system.

The sub-set of planets with a “super-sized moon” is a small fraction of the super-set, possibly one in a million, or 1e-6, leading to a total number for this sub-set of 1e+4. Keep in mind that the fraction of planets that undergo a Mars-size grazing collision is highly uncertain, but the exact value is not important for illustrating the important role it plays in estimating the incidence of life in the galaxy.

Figure 2. The open circles represent a sequence of 100 rocky planets in the HZ somewhere in the galaxy. Tho filled circles show that two of them have super-sized moons. Imagine a more realistic diagram in which only about one in a million rocky planets have a super-large moon.

Now let’s consider a totally different sub-set: those rocky planets in the HZ that currently have intelligent life existing on them. This sub-set will be a small fraction of the super-set, with a large uncertainty. For example, the fraction may range from 1e-10 to 1e-4. (Note: 1e-10 is a lower limit given that intelligent life exists on Earth. The upper limit of 1e-4 is guided by the fraction of time that an intelligent species exists during the lifespan of the planet, which for Earth is approximately 300,000 years divided by 4.5 billion years, or 1e-4.)

Figure 3. Illustration of currently existing intelligent life among the same set of rocky planets in the HZ shown in the previous figure, assuming no correlation of super-sized moons with the evolution of intelligent life. For this illustration only one planet (filled circle) is assumed to host intelligent life.

We now consider two hypotheses: H1 states that there is no correlation between these two sub-sets (i.e., the presence of a super-sized moon is irrelevant to the evolution of intelligent life), and H2 states that there’s a strong correlation between these two sub-sets (i.e., that the presence of a super-sized moon is an essential condition for the evolution of intelligent life).

Consider that H1 is true. This corresponds to Fig.’s 2 and 3 being uncorrelated. In other words, if we made a list of planets with intelligent life on them almost all of them would have regular-sized moons. Only one-in-a-million would have a super-sized moon. We humans could have been on any one of the life-bearing planets, so why are we also on one of those rare super-sized moon planets? This joint-probability situation is tremendously unlikely. On any of the Fig. 3 planets with intelligent life a sentient could ask “How likely is it that we are residents of both rare and random conditions (we are intelligent beings on a planet that has a super-sized moon)?”

But we humans have such a super-sized moon, so we’re stuck with accounting for the improbable situation that we just happen to be one of those rare intelligent species that also share the rare condition of having a super-sized moon. The answer, of course, is that there’s just one chance in a million that we just happen to be one of those alien species. This is equivalent to stating that there’s just one chance in a million that H1 is true; i.e, H1 is probably not true!

Now consider that H2 is true. This corresponds to a high correlation between Fig.’s 2 and 3. It’s equivalent to stating that a super-sized moon does something, such as preserving habitability for a long time so that evolution can proceed, uninterrupted, to possibly produce an intelligent species. Whatever the reason for the correlation, we are permitted to ask one more sampling theory question. “If a super-sized moon orbiting a rocky planet in the HZ has a significant probability of producing an intelligent species at some time during a few billion years interval, and if the intelligent species lifetime is on the order of 300,000 years, what’s a likely estimate for N, the number of intelligent species that now exist in the galaxy?” The answer is obtained by multiplying the number of rocky planets in the galaxy that are located in the HZ, 1e+10, by the fraction that have siper-sized moons, 1e-6, times the ratio of a typical intelligent species existence to how long it took for that species to evolve, 300,000 years / 4.5 billion years. Doing this yields N = 1e+10 * 1e-6 * 3e+4/4.5e+9 = ~ 0.1. In other words, at the present time we’re likely the only intelligent species in the galaxy!

This derivation for N is of course dependent upon the parameter values that go into it. Probably the most uncertain parameter is the fraction of planets with super-sized moons. The uncertainty for this parameter could easily be 3 orders of magnitude, leading to the same uncertainty for N.

There’s one loose end in the above that deserves comment. I adopted a lifetime for an intelligent human existence to be 300,000 years. That’s how long homo sapiens has existed. What if humans are going to exist for another 10 million years, for example? That would change the previous calculation for N, raising it to 2, and we might then be allowed to believe that we’re not alone because there’s probably another intelligent species now alive in the galaxy. But this new assumption requires a violation of sampling theory. This matter was treated more than 30 years ago, and it sometimes goes by the name “anthropic principle.” (I independently discovered this idea a few years later and was teased at the JPL cafeteria when I explained it to my astronomer friends.) It goes like this: So far there have been ~ 70 billion human lives lived. The total number of humans that will ever live can be estimated by assuming we’re now not at a privileged position on that sequence. So, there’s a 50 % probability, according to sampling theory, that the total number of humans that will ever live is 140 billion (i.e., double 70 billion). But given our past global population, and given reasonable projections for the future (e.g., leveling off at ~ 12 billion), 140 billion lives will be reached in less than 200 years.

Figure 4. Three scenarios for future world population, corresponding to our present time being at the 75th, 50th and 25th percentiles of the entire human sequence.

Therefore, humanity must be near its end, so it's fair to assign humanity a 300,000-year lifespan, and, finally, we are probably alone in the galaxy.

The previous “armchair calculations” show how important it is to obtain a more accurate estimate for super-size moon creation by grazing collisions of Earth-size proto-planets by Mars-size planetesimals.  People engaged in SETI projects might want to consider this.