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 location of radiative sources and sinks; we know that all the radiation that comes in has to go out eventually (we're still assuming things are in equilibrium, or rather close to it). So here's what we have.

Incoming solar radiation reaches the Earth, passing mostly unimpeded through the atmosphere. It reaches the surface of the Earth, where some of it is immediately reflected, which we've accounted for already by building in a term for albedo. The remainder is absorbed by the Earth. Later, it is reradiated, but at a very different wavelength than it was when it came in. On its way out, some of this radiation is absorbed by greenhouse gas molecules in the atmosphere, and the rest of it passes back out into space. The radiation that is absorbed by the atmosphere creates (in effect) a new source of radiation, which radiates energy both back toward the surface and out to space. Our picture, then, consists of three sources: the sun (which radiates energy to the surface), the surface (which radiates energy to the atmosphere and space), and the atmosphere (which radiates energy to the surface and space). The true temperature of the surface $$T_s$$, then, is a function of both the radiation that reaches it from the sun and the radiation that reaches it from the atmosphere after being absorbed and re-emitted. Let's see how to go about formalizing that. Recall that before we had the radiation balance of the planet, which predicts the effective temperature of the planet as seen from the outside:

OK, so how shall we find the actual surface temperature of the planet? To start, let's note that we can model the atmosphere and the surface of the Earth as two "slabs" that sit on top of one

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