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Detectable civilizations in our galaxy 3

Reconciling with the traditional Drake Equation. Created by Sal Khan.

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Video transcript

What I want to do in this video is reconcile the more traditional Drake equation with the stuff that we derived, or that we came up with. We kind of thought through it in the last several videos. So the more known Drake equation is this. The number of detectable civilizations in the galaxy is equal to-- and they'll have this. And this is not the number of stars in the galaxy. This is the average rate of star formation per year in the galaxy. So star-- so let me write this down. Average rate of star formation, which seems kind of unintuitive, and frankly, it is, but hopefully we'll reconcile to show you that this, and what we're going to show with the traditional Drake equation are actually the same thing. So that's the average rate of star formation. So I don't know what it is. Maybe it's 10 stars a year, or something on that order. And then the rest of it looks pretty similar. So times the fraction of stars that have planets. So this product would give you the average stars with planet formations per year. You multiply that times the average number of planets capable of sustaining life for a star that has planets, for a solar system that has planets. So essentially, if you multiply this, this is the average new planets per year in our galaxy capable of sustaining life. You multiply that times this, which is the same exact fraction. The fraction of those planets that are capable of sustaining life, that actually, this is capable of sustaining life, now we're saying that the fraction that actually do develop life. And then of the life, we care about the fraction that actually does become intelligent. And of the fraction that actually does become intelligent, we care about the fraction that eventually becomes detectable. That can actually communicate. And then in the traditional Drake equation, we multiply that times this L over here. Times the detectable life of the civilization. So how long is that civilization detectable? Are they releasing radio waves, or something like it that a civilization like ours can detect? Maybe there are other ways to communicate, and we're just not advanced enough. Maybe in a few years, we'll discover in a few decades, or a few hundreds of years, we'll discover that all the other advanced civilizations are using a much more sophisticated way of communicating that doesn't involve electromagnetic waves. Who knows? But this is what we're thinking right now. But anyway, the whole point here is to reconcile this thing which is less intuitive-- for me at least-- than with this thing. Because I started up here with the total number of stars in the galaxy. The traditional Drake equation starts with the average rate of star formation. So it's like, well, how does the average rate of star formation gel with the total number of stars, or civilizations that are now detectable? What I want to do is diagram that out a little bit, and I'm going to make a few assumptions. I'm going to assume that this is kind of constant, that we're in a steady state. So this is constant, and we are in a steady state. The reality is that what would matter is the rate of star formation maybe 4, 5, 6 billion years ago, I don't know how long it has to be ago, so that now it starts to become realistic for real intelligence and real detectable intelligence to exist. But let's just assume that this number is constant for most of the life of the galaxy. Obviously we're making all sorts of crazy assumptions here, so why not make another one? But what I want you to show is that this is equivalent to the number of stars in the galaxy divided by the average life of a star, or the average life of a solar system. And if n divided by this t sub s, if that's the same thing as our star, then essentially we have the same formulas. And to see that they're the same, imagine this. Imagine this. That this year, so this is this year, so this is-- well, let me say this year. This year. Let's say that we have our star. Let's say that this number is 10. We have 10 new stars in the galaxy. So this is-- I'll just say it's 10. So our star is equal to 10. So this height over here is 10. That's what I'm depicting. So if I were to slide it, I could show that this is 10 units high, or whatever. And then last year, there was also 10, so on and so forth. Now, let's go to whatever-- let's say that this number, this number right over here is 10 billion years. The average star life is 10 billion years. So let's go back 10 billion years into the past. So the average life of a star is equal to 10 billion years. And we're assuming that this is constant. So 10 billion years ago this year, there were also 10 new stars came about. And every year in between you had 10 stars come about. Now, how many total stars would there be in our galaxy? Well, any star that came about-- so we could go beyond that. We could go to stars that were born more than 10 billion years ago, more than this t sub s years ago. So you could have a star that was born 10 billion and 1 years ago, on average. We're talking about on averages here. On average, that star will not exist anymore, so that star is not in existence. The stars that are in existence, once again, on average, are the ones that were born 10 billion years ago, all the way to the ones that were born this year. So you have 10 billion years of star birth, the ones that are still around. Each year there's 10 of those years, so the total number of stars should be equal to the number of stars that are born each year-- assuming that that is constant-- times the average lifespan of the stars. Times the average lifespan of the stars. And once again, this works because the stars that were born before this lifespan don't exist anymore. They've died out, on average. We care about kind of this area right over here. 10 stars per year times 10 billion years. And now if you manipulate this a little bit, you'll see that we'll get the result we want. Let's solve for r. So we could just divide both sides by this t. So you get n star, so the number of stars in our galaxy now, making a bunch of assumptions, divided by the average life of the stars is equal to the average number of new stars per year. Is equal to the average number of new stars per year, and we get our result. If you replace this with the total number of stars divided by t, you get the exact same result that we had before. You just change the order a bit. We can take this divided by t, put it under this n, take it out up here, and then you get the exact same thing. So hopefully that reconciles it a little bit for you.