A.4 Entire Sphere to Any Area on its Surface or to its Entire Surface

From Fig. A-4, since θk = θj let them be simply θ; then S = 2R cos θ. Starting with (CD-A4), dAj cos θ/S2 is the solid angle by which dAj is viewed from dAk. The intersection of the solid angle with a unit hemisphere shows that this equals 2π sin θ . Then

 

A006x004

FIGURE A-4 Geometry for exchange from surface of gas-filled sphere to itself.

 

Integrating gives

 

                      (A-11)

 

which has a single parameter 2kλR, the sphere optical diameter.

 

Equation (A-11) is integrated over any finite area Ak to give  from the entire sphere to Ak as .
Since Fj–k = Ak/Aj, from (4-17),

 

                                  (A-12)

 

which also applies for the entire sphere to its entire surface.