Fundamentals of Thermal Radiation - Brightness Temperature

In the previous chapter it was stated that if a part of the universe completely "intercepted" the reception pattern of the horn antenna then the measured antenna temperature would equal the temperature of that part of the universe. This assumes that the part of the universe viewed by the horn antenna is at a uniform temperature. We also assumed that the part of the universe being viewed is a blackbody, meaning that no photons incident upon it are reflected (from either direction, i.e., all photons coming from within the material are able to penetrate the material surface without being reflected). When these conceptually simple conditions are met then it can be said that the measured antenna temperature equals the "physical temperature" of the emitting material. When these assumptions cannot be made we use the concept "brightness temperature," TB. A surface emitting 95% of the photons incident upon it (from either direction) appewars the same to a radiomter as a surface with 100% emissivity at a temperature that is 0.95 times its actual physical temperature. Therefore, it is useful to define brightness temperature as physical temperature times emissivity.

It follows that a material's brightness temperature equals its physical temperature when it is 100% emissive. The atmosphere is ~100% emissive at microwaves, so this greatly simplifies analysis of MTP observers. It is typical for dry ground to have an emissivity of ~90%, which means that an MTP that views only the ground will be viewing a target that has a brightness temperature that is ~90% of the ground's physical (Kelvin) temperature plus 10% of whatever emission is incident upon the ground (and being partially reflected). The reason for this lower emissivity for the ground is that there's an abrupt change in the real part of dielectric constant experienced by a photon incident upon the ground-to-air boundary.

Let's consider another of the assumptions in the first paragraph. It was stated that we assumed that the entirety of the antenna's reception pattern was "intercepted" by a part of the universe whose brightness temperature was under discussion. The word "intercepted" deserves comment, and that's what this paragraph is about. It is very useful to sometimes think of a passive radiometer, attached to a horn antenna, to be "radiating" photons instead of receiving them (which was alluded to in the previous chapter). With this direction reversal we can then ask "what percentage of the radiated photons are intercepted by an extended target of interest?" If 99% of the radiated photons are intercepted by the target of interest, then it can also be stated that this target fills only 99% of the antenna pattern of the radiometer. The radiation pattern of a horn antenna consists of a main beam, Gaussian in shape, and surrounding sidelobes. It is typical for the main beam to contain ~98% of the radiated photons. Hence, a target that has a solid angle that exactly matches the main beam will fill only ~98% of a horn antenna's reception pattern. The rest of the 4-pi solid angle fills the remaining ~2% of the reception pattern. In order to reduce the influence of sidelobes it is common to arrange for the horn antenna to "under-illuminate" the reflector, or to employ an over-sized reflector (either flat or curved). The over-sized reflector can serve to intercept some of the sidelobe pattern and "direct it" onto the same part of the sky viewed by the main beam.

As a practical matter only ~99% of a horn antenna's antenna pattern can be directed to a small patch of sky, with ~1% directed at locations that are difficult to determine. This means that measured antenna temperature is a weighted-average of the intended patch of sky and an unknown part of the rest of the 4-pi sphere. For example, consider that the MTP main beam and nearby sidelobes are directed at the atmosphere having a brightness temperature of TB_atmos, and a stray 1% of the antenna pattern is directed at the fairing that shields the MTP from the airstream and that the fairing has a brightness temperature TB_fairing. The measured antenna temperature will be 0.99 * TB_atmos + 0.01 * TB_fairing. The closer together the brightness temperatures of the fairing and atmosphere are to each other the smaller will be the unwanted effect.

Returning to the meaning of brightness temperature, note that the fairing's brightness temperature will be slightly different from its physical temperature: TB_fairing = T_physical_fairing * Emissivity_fairing, where Emissivity_fairing ~90%. In other words, the fairing will emit only ~90% of the radio photons that would be emitted by a blackbody at the same temperature as the fairing. This also means that the fairing reflects ~10% of the radio photons incident upon it, and these will be received by the horn antenna. Let's not get involved with a complete equation for brightness temperature here; rather, the goal of this chapter is to state that:

    Brightness Temperature = Physical Temperature times Emissivity (plus smaller terms related to reflected photons)

which states that when emissivity = 100%, as it does for the atmosphere,

Brightness Temperature = Physical Temperature (when viewing the atmosphere)

For the MTP case, where most of the antenna pattern intercepts the atmosphere and a small fraction (such as 1%) intercepts nearby material,

    Antenna Temperature = Atmosphere Physical Temperature (plus small corrections related to sidelobes)

This discussion of "brightness temperature" and its relationship to the thing that can be measured, antenna temperature, prepares the way for understanding how to interpret observations of a real atmosphere with a temperature that changes with depth along the line-of-sight. This will be discussed in the next chapter.

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