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  •  Bob Armstrong
    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 .
     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 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

       


    ~ 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 .
    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 .

    --
    -- Bob Armstrong -- CoSy.com -- 719-337-2733 -- SKYPE :  cosybob
          TURNING THOUGHTS INTO COMPUTATIONS
    I reserve the right to post all communications I receive or generate to CoSy website for further reflection .


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