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 .
 Stefan-Boltzmann shows that objects radiate energy per unit time ,  power , as the fourth power of their temperature

PatT : {[ T ] sb * T ^ 4 }
 / where  sb  is the conversion factor from degrees Kelvin to Watts % Meter^2 . It falls out in the examples here where we make the round trip from temperature to 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 :

r	Dist	temp
6.96e+008	1.495979e+011	5778

we compute the following :

 SAd : {[ RD ] ( .R `pi ) * %/ RD ^ 2 }    / Solid Angle from radius and distance .
 : SAsun : SAd Sun[ `r `Dist ] />/ 6.800133e-005    / solid angle subtended by sun

Converting to Proportion of total Sphere and appending complement :

 : SfrPrt : { ( :: ; 1 - ) @\: x % pi4 } SAsun />/ 5.411374e-006 0.9999946

We can now make a table :

 Q : ( SphPrt ; 5778 3 )    / appending a 3k cosmic background .
 Q ,: , Q[ 1 ] ^ 4
 Q ,: , */ Q[ 0 2 ]


The table Q  with labels :

Proportion of Sphere	Temperature	T ^ 4	PofS  *  T ^ 4
5.411374e-006	5778	1.114577e+015	6.031394e+009
0.9999946	3	81	80.99956

Summing the last column and taking the fourth root , we have :
  ( +/ Q[ 3 ] ) ^ % 4 />/ 278.6791
or
  ( +/ PofS * T ^ 4 ) ^ % 4 />/ 278.6791

   

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 .

K

K[ 0 ]

K[ 1 ]

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 , non-optional , physics .