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...are you really complaining that they didn't write out and evaluate the surface integral over the daylit hemisphere ∫∫S Isuncos θincidencedS? I assure you that it simplifies to Isunπr²
I assume you are referring to the second law of thermodynamics? The greenhouse effect doesn't violate the second law. The net heat flow is still surface>atmosphere>space. The fact that infrared-absorbing gasses reflect some of this heat back to the surface would not cause the surface to "heat up" in the absence of external solar radiation. The net effect is to slow down the rate of energy emission from the surface out into space. Net heat flows from hot to cold in every step.
I appreciate putting someone in their place with the power of calculus, but a simpler demonstration would be a diagram of light hitting the earth. The portion of light captured by earth will be a circle, not the surface area of a half sphere, let alone the surface area of a full sphere.
The last (which is @Sky is assuming) would be attained if, with a paper earth, we cut earth open and lay it flat on the table to capture sunlight on all its surface.
Those are really fancy symbols there @popocatepetl
Are you sure you know what they mean?
I’ve got a strange hunch that you do not. The jig is up.
phrase of jig INFORMAL•NORTH AMERICAN
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Oh sure, that's the simplifying assumption which is presumed to be understood. Unfortunately, OP did not understand it. He may be confused by how the locally-valid rule I = Isuncos θincidence can give rise to the much simpler rule PEarth = Isunπr² when summed over all localities. The way to resolve such confusion is to evaluate the integral yourself and see that it all cancels out.
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