A Bussard ramjet, one of many designs for a true interstellar spacecraft. Ramjets wouldn’t go more than a fraction of the speed of light. By NASA [Public domain], via Wikimedia Commons

This is the third in a series of three posts I did on the Fermi Paradox, starting with some numbers from over at the Planetary Habitability Laboratory. In the first two I wrote about why alien civilizations would be hard to see, either via radio waves or Dyson projects. In this one I will get into the numbers behind planetary colonization and what that says about the odds that there’s anyone out there.

I’ll do two things. First is a look at the mathematics of interstellar colonization – how long could we reasonably expect it to take for a civilization to fill the galaxy? The second is to use a little data on the distribution of stars in the galactic habitable zone to see how likely it is that there’s anyone around currently.

Picture a civilization that sends out starships, and let’s say they have a massive colonization program. They send out a whole fleet, to the nearest 10 habitable planets. That will fill a sphere of about 10 light years, assuming that there’s an average distance of about 10 light years between planets. (I work out the numbers here and the PHL has theirs here). Then, let’s say it takes a century for the colony worlds to send out their own ships. Another 10 each, so you get 100 going out. We will further assume that these ships go at just a hair under the speed of light.

That would get us to a sphere with 110 habitable worlds, and it would be 20 to 22 light years wide. It would take a total of 120 years to fill that up. The next step we have 1,100 ships going out, filling a sphere about 150 or so light years wide. But note how much bigger that sphere is; the ships will take about another 130 years to get to the edges. So in year 450 of the colonization era the next batch of ships goes out but it will take them 1,000 years or so to fill the next spherical shell of planets.

Note that the rate of increase in the size of the colonization sphere starts to slow down, and become linear. Eventually, we see that it takes about 100 million years to colonize the whole galaxy, because it’s 100,000 light years across and nothing can go faster than light. (This assumes that one of the colonization “lines” goes straight across).

Think about that span of time – a 100 million year old society, appearing in our solar system now, would have been starting in the Cretaceous. They’d have been launching  Apollo missions 40 million years before the famous Tyrannosaurus Rex had even evolved.

This also assumes a pretty brisk schedule of launches. It assumes that people keep on building and building and never, ever stop. I’d say it’s likely that any colonization effort would take a lot longer.

But out of millions of civilizations, wouldn’t somebody make it to the 100-million-yar anniversary?

Possibly. Here’s where we can look at an old idea called the Drake Equation. Formulated by Frank Drake, it was more of a thought experiment than anything else. But it’s a useful way to quantify what you don’t know, as they say. It looks like this:

N = R_*ƒ_p n_e ƒ_l ƒ_i ƒ_c L

R_*= the rate of star formation (stars per year)
ƒ_p = the fraction of stars with planets
n_e = the number of planets per star that can support life
ƒ_l = the fraction that develop life to begin with
ƒ_i = the fraction that get to intelligent life
ƒ_c = the fraction that develop a technical civilization that can send radio signals
L = the length of time they send signals or exist, in years

ƒ_l , ƒ_i , ƒ_c , and L are all unknowns. So let’s come up with some reasonable values.

Life showed up on Earth almost as soon as conditions allowed it. To be really generous, we will assume it’s 1. That is, every habitable world will have something alive on it at some time. Maybe it’s bacteria or lichen but it’s alive.

ƒ_i is one that we’d be more conservative on. The reason is there’s no reason to think intelligence is a natural and inevitable consequence of complex life. Yes we are here, but look at how few intelligent species there are, given how many we share the planet with. Then consider how many species have come and gone in the last 3 billion years. It probably amounts to billions – after all there are at least a million species of insect around right now.

And of all those species, only a few got smart enough to make and use tools. Even restricting ourselves to considering families of animals, or orders, the evolution of tool-using and manipulation of the environment in the way humans do seems to be exceedingly rare. The late Stephen Jay Gould might have been wrong that the Burgess Shale fauna represented new phyla of animals, but he was probably right that nothing like a human would be likely if we replayed Earth’s history, given the sort-of-random nature of evolution.

That appears to indicate that ƒ_i is very, very small, on the order of 10^-8 or 10^-7– one in a hundred million, or on in ten million, even assuming every smart animal develops a technical civilization.

Then we get L. This is dicey, we only have our own civilization to go on, and we might destroy it (at least it’s ability to get into space) darn fast. We’ve only had such a civilization (radio and space-capable) for 100 years or so. Heck, the whole human species (Homo Sapiens in this case) has only existed for 100,000 years at the most. As a fraction of planetary lifetime that is very, very small.

Recall the Drake equation. Using the values we guesstimated earlier, and assuming a civilization lasts or 100 million years,

N = (7)(1)(2)(1)(1/10,000,000)(1)(100,000,000) = 140

That isn’t many. It would put civilizations at about 5,000 to 7,500 light years apart on average, depending on whether you exclude a big chunk of the galactic center and its environs. Since the galactic center is probably too chaotic (lots of supernovae and crowded stars that mess up orbits) we can go with the lower figure.

Using the calculation at the beginning of the post, that means about a half a million years for a group of smart aliens to run into another group of smart aliens. Working from that, it would still seem they should be here.

That, I think, provides evidence that civilizations are simply rare. There is, however, another factor we need to think through. Not all civilizations will form at the same time. Some might have come and gone millions of years ago. So maybe intelligence and space travel are common but we just missed them – they showed up back in the age of dinosaurs, or perhaps earlier than that. Or they came when the smartest primate was a lemur.

This gets interesting, though. A study by Charles H. Lineweaver, Yeshe Fenner, and Brad K. Gibson at the University of New South Wales back in 2004 examines the Galactic Habitable Zone, and looks at the distribution of stars that could have habitable planets over time as well as space.

Basically it finds that you get the center of the bell curve about 5 billion years ago for the origins of stars that would harbor life at all, and that gives a rough idea of where you’d expect civilizations to happen in time as well as space. If it takes around 4-5 billion years to get to a civilization, then you’d expect the bulk of intelligent civilizations to just be appearing now, give or take a half a billion years. Note that we got a figure of about 140 civilizations at any given instant; but if we take the bell curve approach we can see that they might spread out over quite a lot of centuries.

But what if most civilizations don’t last that long? This would say that even with a relatively narrow “peak” we’re still talking about time scales of hundreds of millions of years. Civilizations might have even passed through our solar system, and disappeared before we noticed. (Stephen Baxter’s novel Space gets into this possibility).

At this point one might invoke some kind of “great filter,” which destroys civilizations before they get a chance to develop. Baxter’s book posits that gamma-ray bursters sterilize huge chunks of the galaxy every so often in a cosmic “reboot.” Another idea is that civilizations that are prone to colonizing (like ours) tend to destroy themselves thorough overuse of resources. More outlandish theories are that there’s some race of killer robots that gets rid of the competition.

I don’t feel that any of these are particularly helpful, because they rely too much on assumptions that may or may not be right. I would argue that we should go with the principle of parsimony – the simplest idea. And that is that civilizations are simply rare in space, or rare in time.

Even a “tight” grouping of aliens in time has a spread that is as long as multi-celled animals have existed on Earth. Since we don’t see anyone else, that would appear to show aliens are rare enough that in any given span of a century or less we aren’t going to see them.

If civilizations are rare – so rare that we might be the only ones in our galactic neighborhood, then I’d argue it’s doubly important that we think in the long term, that we act to make sure we go on surviving, even into deep time. That’s going to take a lot of work and dare I say it, social change. With luck, though, we’ll do it — if a meteor doesn’t smack us first. I suspect a dinosaur might have something to say about that.