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Fig. 3 for the source element P. The efficiency e of the telescope will generally be more easily expressed in terms of the these latter coordinates because it is usually symmetric about the forward direction at 9' = 0. For the purpose of the integration (30), one must settle on one coordinate system for all the variables. Since I(9,<,v) is expressed in terms of the sky-based system, the sky system is used in (30). It is possible to convert from e(9',<') to e(9,<) with appropriate transformation expressions.

We have shown in (30) how the power P observed with a real antenna is derived from the intrinsic true specific intensity I(9,<,v) of the sky. We next discuss the converse operation wherein a real and imperfect antenna detects a power P but it is the specific intensity I(9,<, v) we wish to know.

Average specific intensity

The specific intensity can be measured precisely only with an ideal narrow-beam and narrow-bandwidth antenna. The ideal measure of I (9,<,v) with perfect angular and frequency resolution can not be attained from measurements with a real antenna. Our knowledge of the sky can be no better than the resolution of our telescopes.

A real measurement yields the average specific intensity Iav over the real antenna angles and frequencies. From (30) and the definition of an average,

, fv¡9f< 1 (9<v A e(0,<,v) sm9 d9 d< dv m v = -f f f A m A \AC>A- (Av. specific (8.31)

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