🚀 go-pugleaf

OK, here is the theory of radiative transfer in a grey atmosphere which
might be useful is producng accurate planets:

(See 'The Physics of Atmospheres' by John T. Houghton (1986) for more details,
but this is my interpretation of what he is trying to say :)

-Garrett


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The intensity I absorbed as a function of path length z is the

'optical density' X:

	X = integral(0, z) k rho dz

k = absorption coefficient, rho = molecular density.

X starts at the top of the atmosphere and increases towards the surface in

the following model.

                       -X
So we have that  I = I
                      0

The intensity emitted by a layer of atmosphere at optical density X is

B(T) dX where B(T) is the blackbody function at temperature T:

                   4                             -8
	B = sigma T             sigma = 5.6703x10   W
                                                   ----
                                                    2 4
                                                   m K

Now, consider a layer of atmosphere dX, let the outward flow of radiation
    +                                        -
be I  and the downward flow of radiation be I . With a little though it can

be shown that
           -     -
         dI  = -I dX + B dX

and that
           +     +
         dI  =  I dX - B dX (going against dX!  dX < 0 in this case)

We get two differential equations

         +                            -
       dI     +                     dI        -
(1)    --  = I  - B        and (2)  -- = B - I
       dX                           dX

                       +    -
Define:		phi = I  - I     (net flux outwards)

                       +    -
      and       psi = I  + I     (total intensity radiated by the layer!)

         d    +    -           dT
Now:     -- (I  - I ) = pho C  -- . In equilibrium dT/dt = 0 hence
         dz                  p dt

 +    -
I  - I  = constant = phi.   (The net flux is the same at each atmospheric level).
                         d psi
So, (1) + (2) gives:     ----- = phi
                         d  X

and
                         d phi
(1) - (2) gives:         ----- = psi - 2B
                         d  X

However since phi is constant then  psi = 2B.

           dB   phi
So, then   -- = --- which gives the solution for B:
           dX    2

                    phi
                B = --- X  + constant
                     2

Now we apply boundary conditions at the top of the atmosphere (X = 0) where
 -
I  = 0, and there we have that
          +
   phi = I  = I               (I  is the necessary outward radiation flux to
               0                0        balance the effect of solar heating,
                                         I will discuss this below).
           +
and psi = I  = I .
                0


Therefore for any X we have that the local blackbody function B is

given by:

         I
          0
     B = --- (X + 1)
          2

Just above ground level we have that

           I
            0
     B  = --- (X  + 1)
      g    2    g


Note that this *does not* determine the ground temperature, and in fact

it is easy to show that there is a discontinuity at the ground, where the

black body of the surface B   is actually
                           E

          I
           0
    B  = --- (X  + 2)
     E    2    g

Note that in the case of no absorption that B   is simply I , that this
                                             E             0

blackbody radiates the correct amount to maintain thermal equilibrium.



   How to determine 'decent' surface temperatures (based on this simple

radiative model) is then straightforward:
                 4
     B  = sigma T
      E          E

so we have
        	 4
	2 sigma T  = I  (X  + 2)
	         E    0   g

or

	   2         4
	 ---- sigma T  = I
	 2+X         E    0
	    g

           1
Since I  = - (1-r) S  
       0   4

(where r is the planet's albedo and S is the solar intensity)

Then it is easy to see that the planet's surface temperature is simply

         4    (1-r) S
	T  = --------  
             4e sigma

where the planet's emissivity e is related to the optical depth of the

atmosphere X by:
           2
     e = -----
         2 + X
              g


So, based on knowledge of the optical depth of the planet, its albedo r

and the solar constant it should be quite easy to determine an accurate

value for the planet's surface temperature!


-Garrett