Brightness temperature is the temperature an object would be if it were a black body radiating light at a specific frequency. It is often used to determine whether a source of radiation is nonthermal, usually in the radio region of the electromagnetic spectrum.

If an object is a true black body, the spectrum of light it emits is only a function of its temperature. This function is The Planck Law

Bν(T) = 2hν3c-2 × (exp(hν/kT) - 1)-1 (eq. 1)

When we observe in the radio region of the spectrum, the quantity is very small relative to kT, and we can assume the Rayleigh-Jeans Law,

Bν(T) = 2ν2c-2kT (eq. 2)

which you can then solve for the brightness temperature

Tν = c2Bν/(2νk) (eq. 3)

By equation 3, if we specify the frequency, ν, we want to measure, and measure the brightness of the source at this frequency, we then obtain the brightness temperature.

If we are observing a perfect black body, the brightness temperature is exactly the temperature of the body. However, true black bodies do not exist: some manmade objects like incandescent lights are almost black bodies, as are some natural sources like stars and the microwave background radiation. So, if we measure the brightness temperature of these objects, they will be close to the true temperature of the source.

However, many astrophysical objects are not thermal sources of radiation. For example, emission line radiation is confined to a very narrow range of frequencies, and the brightness temperature measured at the line frequency will be much higher than the true temperature. Even worse are synchrotron radiation sources like supernova remnants, active galaxies, and quasars. They can have brightness temperatures of billions of degrees when their spectra are measured. No stable object can have a surface temperature of a billion degrees because radiation pressure would blow it apart. Thus a high brightness temperature will immediately tell an observer that the source of light is nonthermal and/or strongly beamed, as in the case of extragalactic radio jets.

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brightness temperature -- color temperature -- effective temperature
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Sources:
Radiative Processes In Astrophysics, G. Rybicki and A. Lightman, Wiley Interscience
Class lecture notes

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