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 dθ. Then
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
.
(A-12)
which also applies for the entire sphere to its entire
surface. |