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
(exp(hν/kT) - 1)-1 (eq. 1)
When we observe in the radio region of the spectrum, the quantity hν
is very small relative to kT, and we can assume the
2ν2c-2kT (eq. 2)
which you can then solve for the brightness temperature
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.
brightness temperature -- color temperature -- effective temperature
Radiative Processes In Astrophysics, G. Rybicki and A. Lightman,
Class lecture notes