A SETI thought experiment
I've made my opinion of the Drake equation clear several times in the past. Instead of revisiting that, I want to look at SETI from another angle.
Consider if you will what we can "look at" with SETI. We have been conducting a SETI program for - let's call it an even fifty years. So, if we handwave for the moment the question of whether we've been looking in the right places or using the right means, and just assume that if a signal existed we would have picked it up (a generous assumption), let's say we have examined a sphere of fifty light years radius for intelligent life. Let's be even more generous and say that our ability to detect such signals has been constant throughout the lifetime of the SETI project. (It has not; it's pretty fair to say that it has increased continuously throughout the life of the project.)
In actual fact, it's a little better than that. You see, if a signal had been sent from fifty-one years away, anywhere from one year to fifty-one years before SETI began, we would (under the generous parameters above) have received it. Likewise if a signal were sent from 60LY anywhere from ten to sixty years before the beginning of SETI. In a sense, we can say that we have studied a fifty-year/50LY thick surface layer of a "light cone" stretching back into the galaxy's past and spreading out as it goes. Any signal sent more than fifty years inside that cone, we missed. Any signal sent outside it, we haven't received yet. Any signal sent more than, to be generous again, a hundred years or 100LY outside that cone will probably not be received during the lifetime of anyone now working on SETI (even if only to the extent of running SETI@Home).
OK?
Now. The galaxy is a hundred thousand light years across. At any given instant in the past, the intersection of that "light cone" with the galaxy that we have so far examined is fifty light years wide. That's one two-thousandth of the diameter of the galaxy. Looks about as thick as a pencil line, doesn't it? And if we assume that we're "not special", then the further back into the past that light cone propagates, the less likely it is to intersect a technological civilization.
So let's forget about the deep past for now. Let's instead imagine, for the moment, that the galaxy is seeded with a thousand civilizations comparable and contemporary to our own. (If they're more than about a hundred years behind us, they can neither send any signals we can detect, nor could they detect any of ours. If they're more than about a hundred years ahead of us ... who knows what they use for communication. For all we know, they use phase-modulated artificial gravity waves or quantum-entanglement ansibles.) Each of them, like ours; is surrounded by an expanding 100LY-radius bubble of radio emissions bearing the signature of intelligent generation.
Pretend, for the moment, that the galaxy has zero thickness. (It doesn't. It's about sixteen thousand light-years thick in its central bulge, and about five thousand out here in our region. But we'll ignore that for now.) That makes the plane area of the galaxy roughly 7.85 billion square light years (pi times the square of fifty thousand light years). Let's assume that, on average, civilizations are more or less evenly scattered¹, and divide that area into a thousand roughly equal domains. Then each of these domains is 7.85 million square light years, from which we can work backwards to find that each of those domains - including our own - has a radius of ... oh dear. About 707 light years. Which puts our hypothetical thousand Earthlike civilizations about fourteen hundred light-years apart. Uh-oh. The odds are we're not going to hear anything from any of them for a LONG time.
So let's improve the odds. Let's say there's ten thousand civilizations. Now those domains are about 785,000 square light years, and their radius is therefore ... um. A hair under five hundred light years. They still average a thousand light years apart. That doesn't help us much, does it?
OK, what if there's fifty thousand civilizations? Well, that about halves the separation. Domains are now about 225LY radius, making them about 450LY apart ... no. Still doesn't help us. To have roughly a 50/50 chance of a detection in the next fifty years, we probably need them to be no more than about 150LY apart on average. (We'll ignore developmental-stage offsets, because lacking any information to the contrary or any reason to believe that we are somehow "special", the most reasonable guess we can make is that the number of civilizations N years ahead of us is roughly comparable to the number of civilizations N years behind us, for any arbitrarily chosen value of N. So, without even worrying about distribution, we'll assume they roughly cancel each other out. Remember, we're working with averages here.) That means a domain radius of about 75LY.
So, how many civilizations do we need to get the domain radius down to about 75LY?
Turns out we need about half a million. And that, remember, is not to have a detection now, it's to have a roughly fifty-fifty chance of a detection in the next fifty years.
And we're still ignoring the thickness of the galaxy, which is five thousand light years - call it about thirty-three domains deep - even out here. So if we pretend the galaxy is an even five thousand light years thick all the way across, then the odds are on the order of thirty times worse than we just calculated. And at that, I've been VERY generous in SETI's favor throughout.
So ... how's the odds on that SETI program looking, huh...?
[1] We're not going to try to correct for what the habitable and non-habitable regions of the galaxy might be, because bluntly we don't know enough about the answer to make the effort any more than pure guesswork and handwaving. We can reasonably assume that the area in the immediate vicinity - let's be generous and say a thousand light years or so - of the supermassive black hole we believe to lie at the heart of the galaxy is probably pretty hostile to life; but then, that central bulge is sixteen thousand light years thick. And we have found life in some pretty damned hostile environments here on earth, from ocean-floor vents to the cores of nuclear reactors.
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