or, with simplified notation:
where C is a constant.
Should the source be a graybody with emittance ε, the received radiation would
consequently be εW
source
.
We are now ready to write the three collected radiation power terms:
1 – Emission from the object = ετW
obj
, where ε is the emittance of the object and τ
is the transmittance of the atmosphere. The object temperature is T
obj
.
2 – Reflected emission from ambient sources = (1 – ε)τW
refl
, where (1 – ε) is the re-
flectance of the object. The ambient sources have the temperature T
refl
.
It has here been assumed that the temperature T
refl
is the same for all emitting
surfaces within the halfsphere seen from a point on the object surface. This is of
course sometimes a simplification of the true situation. It is, however, a necessary
simplification in order to derive a workable formula, and T
refl
can – at least theo-
retically – be given a value that represents an efficient temperature of a complex
surrounding.
Note also that we have assumed that the emittance for the surroundings = 1. This
is correct in accordance with Kirchhoff’s law: All radiation impinging on the sur-
rounding surfaces will eventually be absorbed by the same surfaces. Thus the
emittance = 1. (Note though that the latest discussion requires the complete
sphere around the object to be considered.)
3 – Emission from the atmosphere = (1 – τ)τW
atm
, where (1 – τ) is the emittance of
the atmosphere. The temperature of the atmosphere is T
atm
.
The total received radiation power can now be written (Equation 2):
We multiply each term by the constant C of Equation 1 and replace the CW
products by the corresponding U according to the same equation, and get
(Equation 3):
Solve Equation 3 for U
obj
(Equation 4):
98 Publ. No. 1 557 536 Rev. a35 – ENGLISH (EN) – January 20, 2004
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