## Brightness and Color Temperature

Suppose we have an instrument that can measure radiant power over some range of frequencies anywhere in the electromagnetic spectrum. For simplicity we assume a narrow field of view for the instrument, but this is not necessary. If we were to point the instrument in a particular direction at a source of radiation, which could, but need not, be a measurably emitting body, the instrument would dutifully measure a radiant power. Now we can ask, What temperature must a blackbody have in order for the instrument reading to be the same? This temperature is called the brightness temperature of the source, not to be confused with the ordinary (or thermodynamic) temperature. Even if the radiation measured is mostly or entirely emitted (as opposed to reflected) by a body, its brightness temperature is not the same as its temperature unless we happen to choose a frequency range over which the emissivity of the body is almost 1.

To show that a brightness temperature always exists consider the integral of the Planck function Eq. (1.11) over any range of frequencies:

This integral approaches 0 as T ^ 0 and to as T ^to, and its derivative with respect to T is always positive. Thus whatever our instrument reads, we can always find one and only one temperature such that Eq. (1.60) matches it. But keep in mind that this temperature depends on the frequency interval and possibly the direction (unless the source is isotropic). And if the instrument is equipped with a polarizing filter, and we were to rotate it, the brightness temperature might change (unless the source is unpolarized.)

Although the concept of brightness temperature is not restricted, color temperature is. The color temperature of a source of (necessarily) visible radiation is the temperature of a blackbody with the same perceived color. As we show in Section 4.3, the gamut of colors accessible to the human observer can be represented as a set of points in a two-dimensional space, whereas the possible colors of blackbodies lie on a curve in this space. Thus blackbod-ies of all temperatures can match only a small sample of possible colors. Nevertheless, color temperatures can be useful as long as we recognize their limitations. The color temperature of average daylight (sunlight plus skylight) is around 6500 K; that of an ordinary incandescent (tungsten) lamp around 3000 K. Color temperatures of skylight are 10,000-40,000 K. Perhaps it is fortunate that we can't touch the sky. If we could, we'd surely burn our fingers.

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