Equations I: Drake's equation

People with an interest in speculative biology will probably know Drake's equation well. It describes how many civilisations in our galaxy are at present broadcasting their existence by emitting electromagnetic radiation into the universe. If you are only interested in the purely biological side of speculative biology, then alien intelligence might not appeal to you very much. Still, it makes sense to think that any biological intelligence will be deeply shaped by the specific biological background, so alien intelligence is a part of speculative biology (probably until that in turn gives rise to machine intelligence; would that reflect its maker too?). I aim to write two or three posts on the biological evolution in our galaxy, starting with Drake's equation.

I do not have that much affinity with speculative intelligence. I once started to evolve an intelligent species on Furaha. The creature was derived from hexapod predatory stock, so its forelimbs were not used for locomotion, as an example of centaurism. Most such 'neopredators' evolved their front limbs into clubs or spears, as can be seen on the Furaha website. These modified front legs lost all their toes in becoming spears or clubs, but the proto-intelligent species belonged to a group of small neopredators that had in fact developed the grasping ability of toes on their front leg. This allowed them to radiate into a number of interesting shapes.

Click to enlarge; copyright Gert van Dijk

This old and rather poorsketch shows this putative proto-intelligent species. It evolved on an isolated island and was supposed to have gone extinct shortly before humans came to Furaha, say only 30,000 years before. There would be some evidence of shaped clay or other things suggesting that the use of purposely shaped objects. at the time of human discovery, the island's ecology was supposed to be devoid of large species and to have remarkably little diversity. The idea was based on the presumed history of Easter Island, as described by Jared Diamond in his book Collapse. The story holds that overpopulation caused the inhabitants of Easter Island to cut down all trees and to destroy their environment, and through that their civilisation. Easter Island, seen in this way, holds a mirror to all of Earth, telling us to stop and think what we are doing. While reading up on Easter Island, I found that  these depressing ideas have later been questioned, and the case of Easter Island has even been labelled as a story of efficient adaptation. The trees are still all gone, so I find this rather depressing as successes go. 

I had tucked these proto-intelligent species away on a remote island where they would provide the Furahan equivalent of Easter Island, providing a lesson without ruining the entire planet. I later felt that I did not need such a heavy-handed approach so I erased the story entirely.

Back to Drake's equation. Below is a text taken directly from Wikipedia. The equation describes the number of civilisations, N, with which communication might be possible. It is assumed to be equal to the mathematical product of the following parameters:

R, the average rate of star formations, in our galaxy,
fp, the fraction of formed stars that have planets,
ne, for stars that have planets, the average number of planets that can potentially support life,
fl, the fraction of those planets that actually develop life,
fi, the fraction of planets bearing life on which intelligent, civilized life, has developed,
fc, the fraction of these civilizations that have developed communications, i.e., technologies that release detectable signs into space, and
L, the length of time over which such civilizations release detectable signals.

N = R   fp  ne  fl   fi  fc  L

I confess that I always had trouble understanding why this product represents the number of civilizations that are transmitting signals now. In an interview posted here Frank Drake said he started with the rate of new stars being produced because the equation was based on a continuous production of new planetary systems. As a result, the number of detectable civilizations is proportional to the rate of star formation. That makes sense, but still... Say that 10 stars are formed in the galaxy over one year. The equation ends with the average number of years that a civilisation actually transmit signals, say 300. The product would be 3000, modified by the other parameters. This suggests that the equation results in the total number of 'transmission years' resulting from one year's batch of new stars, and I do not quite understand why that would equal the number of civilisations that are transmitting right now. It seems more logical to start such an equation with the total number of stars in the galaxy and to modify that number. In fact, there are equations out there that do just that, and I found that there are several variants that are also called "Drake's equation".

Regardless of the different versions of Drake's equation, the message is clear enough: any estimate of the number of transmitting civilizations depends on a fairly large number of parameters, most of which rely more on guesses than on facts. The Wikipedia paper discusses that nicely, stating that N can vary from less than one to over 15 million. Drake himself arrived at about 20 civilizations in our galaxy. The more there are, the more you have to wonder why we never heard from them, which is of course well-known as Fermi's paradox. Here is a very thorough and entertaining book discussing 75 possible solutions to Fermi's paradox.    

Something like 20 civilizations distributed over one galaxy is not much. The average distance between such civilizations would be enormous, giving us little chance of hearing them. Note that Drake's equation is about how many civilizations are out there, not about our chances are of detecting them. Any considerations on actually detecting them must take the size of the galaxy into consideration. I could not resist playing with these ideas a bit.

Click to enlarge; copyright Gert van Dijk

The figure above shows a solar system containing a transmitting civilization. This civilisation started transmitting at some point in time, here just 30 years ago. From that moment on the signal travelled into space with the speed of light. For every year of time it obviously travels over a distance of one lightyear. After just 10 years the civilisation stopped transmitting. Perhaps the inhabitants found more efficient ways to contact people on their own planet than wasting energy by blasting a signal in all directions. Perhaps they went the Easter Island way, by cutting down all their trees, by nuking themselves to oblivion, by using creative biological weapons, or perhaps their successors, machine intelligences, decided they did not want pets. Whatever happened, a shell of transmissions with a thickness of 10 lightyears is still expanding outwards at the speed of light. The signal strength will decrease quickly, as it is governed by the square of the distance (see here for an explanation, on sound rather than electromagnetic radiuation, but the principle is the same). I tried to find information about how far the type of unfocused signals earth sends out might be received with current equipment; here is one source saying that 21 light years is optimistic, which is not much at all. Another source, from a senior SETI astronomer, states that detecting Earth from 'a few hundred light years' require an antenna the size of Chicago. That's impractically large...

Here is a simple model of a galaxy with most stars in the middle. The image spans 1,500,000 lightyears horizontally and vertically. Over a span of 100,000 years, civilizations evolve and transmit for a while, in this case for any duration between 0 and 5000 years (as longer as all of human history). The thickness of the expanding rings show the duration the civilisation was transmitting. I assumed their signal could just still be detected at a distance of 25,000 lightyears, requiring fantastically sensitive devices. The brightness of the colour indicates signal strength: at 25,000 lightyears it fades to nothing. Note that signals can only be detected in the coloured rings themselves, not in their blank interiors. The result is clear: the total area of the galaxy that lies in a ring is very small, and those are the only areas where transmissions can be picked up.

 
Here it the same scheme, but with a shorter duration of transmission and a smaller distance over which a signal can be detected. There are thin small shells here and there, but you have to look carefully or you will miss them altogether. This is still a very optimistic vision, I think. If there are just 20 transmitting civilisations that require a ridiculously large antenna to be heard seems to mean that it's not surprising we haven't heard anything yet. But there is Seager's equation:  looks at the problem in another way, so that's one I will have a look at in later post.