Basics
This is edited from an email discussion around 20100630 .I find it useful to simply express the basic physics in finite , executable "APL" notation as presented on my website . StefanBoltzmann shows that objects radiate energy per unit time , power , as the fourth power of their temperature Computations in energy are linear . Thus the rate of energy transfer between two objects is the simple difference of each of their individual rates , ie , {[
T0 ; T1 ] / PatT'
( T0 ; T1 ) } . The
net heat flow is from whichever is warmer to whichever is cooler . No
mystery ( except that { sb * T ^ 4 }
bit solved
more than a century ago , itself ) .In classical physics fashion , one can determine the power density , and therefore temperature , at a point by simply integrating ( summing ) the energy flux over its surrounding sphere ,
{[
T ; s ] +/ PatT T
@ s } , where s
is
a partition of the sphere . As a Gaussian surface I'm sure
tons of
related implications can immediately be made for spheres in place of
points . For a sun with the following radius , distance , and temperature : we compute the following :
Converting to Proportion of total Sphere
and
appending
complement :
We can now make a table :
The table Q with
labels : Summing the last column and taking the fourth root , we have : ( +/ Q[ 3 ] ) ^ % 4 />/
278.6791 or ( +/ PofS * T ^ 4 ) ^ % 4 />/
278.6791
~ 280 THIS is the crucial value difference from which must be explained . That's about 8 or 9 Kelvin ( = Celsius ) , NOT 30+ !! It is hard to over emphasize what a Bull Shit computation leads to that number and the damage it has done to the understanding of planetary temperature . How the hell it ever came to be the foundational computation in too many textbooks is a thesis for a History of Science degree . So how can a body deviate from this temperature without being , essentially by definition , a sink or a source ? By throwing different reflective filters into the different partitions . Filters are symmetric ; they don't know which direction the the energy is flowing , whether absorbing or emitting . We can add a column , which I'll label K for Kirchhoff , for this parameter . where K[ 0 ] ... are scalars , single numbers . Thus they define a gray scale . This gets up to the case I have implemented on my http://CoSy.com which computes a number of interesting cases .
You will find there that I am offering a $300 "prize" for any student who extends the implementation of each K[ i ] to full spectra rather than gray values so actual quantitative values can be calculated for the actual spectra of interest . I feel a number of recent blog debates have given me some clues to quantitatively understanding the vertical temperature structure of atmospheres , but simply to get the fundamental algorithms presented here widely replicated would be a massive step towards broader understanding of this most basic , nonoptional , physics . 
I reserve the right to post all communications I receive or generate to CoSy website for further reflection .
