 Hi, well, I'm Professor Steven Neschabang and I'm here to help you out with some ideas having to do with gray body emission and Kirchhoff's law. So the first idea that I'll talk about is the idea of radiance as a function of wavelength of light. And I've drawn two black body curves here, Planck black body curves. We're using the symbol r for that. So that would be a black body curve, let's say, of the Earth. And this could be the black body curve of the atmosphere. If the atmosphere was really thick and emitted like a black body, I'd kind of shifted it to the right a little bit to higher wavelength because according to Veen's law, you know, that lambda max is inversely proportional to the temperature. And we generally think of the atmosphere as having a lower temperature than the surface of the Earth. So that's why that's been shifted a little bit to the right. So now what happens if we have an atmosphere that's not thick? Okay, so it doesn't emit like a black body. Well, what we say is, well, it's going to have a spectral pattern. It's going to have a distribution of radiant intensities as a function of wavelengths that looks like a black body, but it's just going to be less at all wavelengths. And how much less? Well, what we're going to say is this. We'll just multiply by some number smaller than one. So this would be the black body of radiance for the atmosphere. And as I've drawn it, we just reduced it down here. And now this dash curve here, I've just multiplied this by, say, 0.4 or whatever the emissivity is. That's that constant factor smaller than one that lowers the radiance that comes off the atmosphere because the atmosphere is not completely thick. Okay, so that's the emissivity. And what we say is that a substance like the atmosphere that emits in this way, that is to say like a black body, but multiplied by an emissivity smaller than one, we call that a gray body, and that's why this is called RGB instead of RGB. Okay, so how does this relate to the total radiance that comes off that body? Well, let's suppose, if you recall, that the total energy that comes off a given object is just the area under that curve. And obviously as the temperature gets bigger, that area gets bigger, and therefore the total flux of energy that comes off that object becomes bigger. So just remember this, the temperature of the atmosphere is lower than the temperature of the earth. So we expect less flux to come off the atmosphere, even if it were a black body. So that's what's being graphed here. You see, I have the temperature of the atmosphere a little bit lower than the temperature of the surface of the earth. This curve here is the proportional to sigma times t to the fourth. And as you can see, I've just put the total flux over here. The four pi r squared takes into account the surface area of the earth. There's the sigma and the step in Boltzmann constant, and there's the temperature of the earth raised to the fourth. That never is higher because the temperature there is greater than the temperature of the atmosphere. So what's it going to look like? What is the flux going to look like when I have a gray body? Well, the area under this curve is obviously going to be just the area under the first curve multiplied by that constant emissivity. So in this case, I'm looking at an emissivity of about 0.4. So I would say, oh well, that must have a total flux corresponding to about that right here. You see, I've lined it up with the temperature of the atmosphere, but I've come down to an emissivity of about 0.4. So now in terms of our atmospheric model, what are we talking about? This number that we're talking about is I5. It's the flux coming down from the atmosphere, emitting as a gray body. So of course, we would write I5 as something like this. We would say, oh well, it could be 4 pi r squared sigma T atmosphere to the fourth times epsilon. All right, so that's good. Now of course, I2 here, just filling this out, that would be this term right here because it's the temperature of the earth. We've got I5, we've talked about how I6 is just equal to I5 because the atmosphere can radiate down or radiate up. So the last bit of it that we haven't talked about is how the flux from the earth gets, how much of them that gets absorbed by the atmosphere. And the idea here is that some fraction of it, we'll call it A, gets absorbed by this atmosphere. And some fraction of it, say 1 minus A, gets passed through and lost to space. So I would say something like this. That must be A times I2. That is I3, that's how much gets absorbed. And that must be 1 minus A times I2. So that part, that is there. The last idea here is just conceptually, the thicker the atmosphere, the bigger epsilon is, that is the closest thing is to emitting like a black body, but also the more it's likely to absorb. And in fact, Kirchhoff's law says that the absorptivity is equal to the emissivity, that's Kirchhoff's law, which means in this case, if we thought we had an emissivity in the atmosphere of, say, 0.4, then we would say that the absorptivity is 0.4 as well. And then we have all the terms in our relative isothermal atmosphere picture. Okay.