This is part 3 of the series of posts on the odds that life might have arisen spontaneously, as some people believe. This time I plug the result of the previous calculations into the famous Drake equation, with rather scary consequences.

The Drake equation, as you undoubtedly know, estimates the number of alien civilizations capable of communicating with us earthlings. It was created in 1961 at a historic conference where the search for extraterrestrial intelligence (SETI) was born as well. This is the equation, taken from this source:

N = R_{*} • f_{p} • n_{e} • f_{l} • f_{i} • f_{c} • L

where the dot means multiplication and the different factors are:

N = number of civilizations in our galaxy we could communicate with

R_{*} = average rate of star formation in our galaxy

f_{p} = fraction of stars with planetary systems

n_{e} = number of planets that can support life

f_{l} = fraction of those planets that will develop life

f_{i} = fraction of those planets that will develop intelligent life

f_{c} = fraction of civilizations that might develop transmission technologies

L = amount of time that these civilizations would have to transmit their signals into space.

The Drake equation has encouraged much debate on the existence of intelligent aliens out there, with estimates for N ranging from zero to a hundred thousand, depending on the estimates used for the different terms. Back in 1961 we didn’t know much about many of the terms, but we do now. For instance, f_{p} could have been a small fraction if planetary formation is mostly the result of catastrophic events such as star collisions. Now that we have observed a couple thousand extrasolar planets we know that f_{p} is close to unity. Other factors are still very hard to estimate but, since many of them are fractions that must be between zero and one, we can come up with bounds. If you have followed my two previous posts calculating the odds of spontaneous life (here and here), you know I have a value for f_{l} that I’m itching to use in the equation. Because the value, although based on quite optimistic assumptions, is quite small, I will use the most optimistic assumptions for all the other terms, and see what happens. Let’s go!

R_{*} = average rate of star formation in our galaxy

The Wikipedia article on the Drake equation gives bounds between 1.5 and 3 stars per year, based on 2010 data. So let’s take the upper bound of 3.

f_{p} = fraction of stars with planetary systems

Every time we look at a star with powerful enough telescopes, we find planets around it. So a value of 1 seems accurate.

n_{e} = number of planets that can support life

This depends on the definition of habitable zone, which typically means that you can find liquid water. Its location will vary with the size of the star, being closer to the star for small ones, farther for large ones. A current consensus is that the number should be 3 to 5, but since I want to be optimistic, let’s make n_{e} equal to 10.

f_{l} = fraction of those planets that will develop life

This is the monkey wrench. This number varies enormously depending on whether or not you admit a theistic explanation for the origin of earthly life, or instead you believe that it must have arisen spontaneously from the laws of physics (including the 2nd law of thermodynamics). The first option gives us an upper bound of 1, and a lower bound of 0. The second option, which I worked out in some detail in my previous posts, gives us an optimistic estimate of 1.91788 * 10ˆ(-323168) Since SETI by and large does not have theistic roots, let’s take the latter number for our calculation. But wait! My calculation was for terrestrial life. What if there is an entirely different kind of life out there? What if it’s based on silicon, or some other element? How many kinds of life are talking about? For lack of a better estimate, let’s say each element can give rise to an entirely different kind of life, or a factor of about 100. I don’t expect life that eventually leads to intelligence and communication through space to be any less complex than terrestrial life, so I’ll just multiply my initial number by 100, giving 1.91788 * 10ˆ(-323166)

f_{i} = fraction of those planets that will develop intelligent life

This is another potential monkey wrench, dealing with the problem of the emergence of consciousness. It is hoped that our current efforts with exceedingly powerful computers will eventually shed light on whether a sufficiently powerful set of neurons will eventually develop consciousness or rather the consciousness requires a “first program” as hard to arise spontaneously as the DNA of the first living cell. Meanwhile, I’m going to be optimistic and give this factor a value of 1.

f_{c} = fraction of civilizations that might develop transmission technologies

It’s taken us a while to achieve this, but we finally did it, so why not be optimistic and expect all other intelligent beings to be able to do it as well? I’ll give it a 1.

L = amount of time that these civilizations would have to transmit their signals into space.

Drake added this to the equation for two reasons: 1, it needs a quantity having the units of time in order to produce a pure number as a result, and 2, back in 1961 there was a strong fear that we humans would blow ourselves up into tiny radioactive fragments, which might make L rather small. But with the fall of the Iron Curtain we feel less apprehensive now, and I’m ready to say, why not, that L could be as long as the age of the universe itself, that is, 13.77 billion years by the latest estimate.

So here’s how all our factors line up, as a table:

R* (1/years) | f_{p} | n_{e} | f_{l} | f_{i} | f_{c} | L (years) |

3 | 1 | 10 | 1.91788 * 10^{-323166} | 1 | 1 | 13.77* 10^{9} |

I’m sure you see where this is going. When we multiply all these factors, our optimistic guess for N is 1.03 * 10ˆ(-323154), that is, very, very close to zero. Even when we multiply by the number of galaxies in the universe (now estimated as 2 trillion), this only raises the number to 2.06* 10ˆ(-323142)

In other words, unless you accept theist answers for things like the origin of life, we are alone in the universe. If you believe SETI is worthwhile, you have to be a theist. The math doesn’t work otherwise.

Think about all the consequences for Science Fiction in movies, books, etc. Many famous books would have to be rewritten, movies remade. Doesn’t that send a tingle down your spine?

Interesting exercise(s), Francisco. Elon Musk et.al. are working on populating nearby satellites. Boy, that’s a big project! But hey why not? We’ll need to find a way to get lots of air, water, etc. there first. Then, we’ll need to grow food there. Then, we’ll need an army of robots to parent the conception, delivery, growth, education, etc. of the beings born there. AND, they will probably not be the same size as us. Wow, this would be a huge project.

Maybe you could blog that.