Brian , finally some substance . Couldn't justify spending time on this yesterday .  I assume you don't have an income stream from this conflict either .

I don't have anything to edit PDFs which makes contextual commenting clumsy . Your spreadsheet motivates me to try to "read" one ( using OpenOffice ) for about the first time , and I haven't yet figured out how to find the formulas for the graphs .

Is that your your derivation  ?  If so , I take back my questioning of your pedigre . In any case , it's one of the cleanest I have seen . You ought to go get the Wikipedia pages on Steran-Boltzmann and Black-Body fixed . I like the implicit recognition that the Kirchhoff parameter is a single parameter , ε .

I see no difference between us on the physics .  You are wrong that I assume a temperature of 6000k . Take another look at the table K modeling of basic black body planetary temps . You will see I also come to a temperature of 278.7 for a solar temperature of 5778 . An important point to note is that you see that the essential parameter is the ratio between the values of ε for radiation to and from earth . That ratio is what I label AE . You recognize that a default ratio is 1 which is the ratio for a uniform flat-spectrum ( gray ) body . The overall albedo does not matter .  So you don't start with the absurd assumption , which Martin Herzberg calls the cold earth hypothesis , that the earth absorbs acording to its measured albedo , but radiates like a black body .  Pierrehumbert starts from this absurdity which is where the notion that the earth would be , by my implementation :

AE : .7 1 1  ;  Tcs : 5778 3 3 ; ?[ Tdif ; 0.0 ; 6000 ]   /  />/ 257.06

leaving 9 degrees to be explained instead of 31 .

The main differences in approach is that your formula is one dimensional implementation of the earh-sun geometry ; mine models a point surrounded by a sphere partitioned into areas at various temperatures , each with its particular AE