another, each with approximately the same area. The surface of the Earth radiates energy upward only (i.e. to the atmosphere and space), while the atmosphere radiates energy in both directions (i.e. back to the surface and to space). So far, recall, we’ve been treating the part of the Earth absorbing energy from the sun as a uniform disk with an area equal to the “shadow” of the sun (that is, ¼ the area of the entire Earth’s surface); this is a fairly good approximation, since we’re already disregarding variations in albedo and emissivity across different latitudes and longitudes (that’s part of what it means to be a zero-dimensional model). We can think of the atmosphere, then, as consisting of another slab with approximately the same surface area as the surface itself. This is not quite right, but it is also a fairly good approximation. Since the atmosphere, as we’ve seen, absorbs energy only from the surface of the Earth, but emits energy both back toward the Earth and to space, we have to adjust its surface area accordingly in our model. For the purposes of absorption, we can treat the atmosphere as having twice the area of the surface, since it radiates along both the inside and outside. Just as with the surface of the Earth, the atmosphere radiates energy in accord with the Stefan-Boltzmann law. That is, it radiates energy as a function of its surface area and temperature.
We also stipulate that (since this is an energy balance model), the atmosphere emits exactly as much as it absorbs. We’ve already noted that the atmosphere isn’t entirely transparent from the perspective of the Earth: it absorbs some (but not all) of the outgoing radiation. Let us add a term to our model to reflect the opacity of absorbing surfaces. Call this term γ. A surface that is totally opaque has γ = 1 (it absorbs all the energy that actually reaches it), and a surface that is totally transparent to incoming radiation has γ = 0. Note that this term is independent of α: a surface’s opacity only comes into play with regard to the energy that isn’t just reflected outright.