Consider the following information.
Albedo of the Earth:
Emissivity of the Earth: (Blackbody)
Part 1: Calculate the surface temperature of the Earth with the absence of the atmosphere. Note that the surface temperature is labelled in the above diagram.
Part 2: Calculate the surface temperature of the Earth with the atmosphere present.
Without an atmosphere the power of solar radiation absorbed by the surface of the Earth is equal to the power radiated by the Earth.
This temperature is , which is an ice world!
Add an atmosphere. The power of solar radiation absorbed by the atmosphere of the Earth is equal to the power radiated by the atmosphere. Thus,
This is the same calculation as in part 1, so .
Now, the power of the solar radiation still reaches the surface of the Earth alongside and the power radiated by the atmosphere. Therefore, we tweak the equation from part 1 by adding an atmospheric radiation term.
This temperature is around , which is too hot! Sure, some places on Earth can be hotter than this temperature, but the average surface temperature is only around . One possible source of error in our calculation in part 2 is the amount of solar radiation that reaches the surface of the Earth, but there could be a plethora of other causes such as other thermal transfer effects.