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At radio frequencies, cosmic source strengths are usually measured in Janskys3.6 (Jy). Consider a plane wave from a distant point source falling on the Earth. If the energy per unit frequency passing through an area of 1 square meter held perpendicular to the line of sight to the source is $10^{-26}$ watts then the source is said to have a brightness of 1 Jy, i.e.

\begin{displaymath}1\ Jy = 10^{-26}\ W/m^2/Hz,\end{displaymath}

For an extended source, there is no longer a unique direction to hold the square meter, such sources are hence characterized by a sky brightness B, the energy flow at Earth per unit area, per unit time, per unit solid angle, per unit Frequency, i.e. the units of brightness are $W/m^2/Hz/sr$.

Very often the sky brightness is also measured in temperature units. To motivate these units, consider a black body at temperature $T$. The radiation from the black body is described by the Planck spectrum

\begin{displaymath}B(\nu) = {2 h \nu^3 \over c^2} {1\over e^{h\nu/kT} -1}
~~~~~~~~~~ W/m^2/Hz/sr\end{displaymath}

i.e. the same units as the brightness. For a typical radio frequency of 1000 MHz, $h\nu/k = 0.048$, hence

\begin{displaymath}e^{h\nu/kT} \sim 1 +h \nu/kT\end{displaymath}


\begin{displaymath}B(\nu) \simeq {2 \nu^2 \over c^2}kT = {2kT/\lambda^2}\end{displaymath}

This approximation to the Planck spectrum is called the Rayleigh-Jeans approximation, and is valid through most of the radio regime. From the R-J approximation,

\begin{displaymath}T = {\lambda^2 \over 2k} B(\nu)\end{displaymath}

In analogy, the brightness temperature $T_B$ of an extended source is defined as

\begin{displaymath}T_B = {\lambda^2 \over 2k} B(\nu).\end{displaymath}

where $B(\nu)$ is the sky brightness of the source. Note that in general the brightness temperature $T_B$ has no relation to the physical temperature of the source.

For certain sources, like the quiet sun and HII regions, the emission mechanism is thermal bremstrahlung, and for these sources, provided the optical depth is large enough, the observed spectrum will be the Rayleigh-Jeans tail of the black body spectrum. In this case, the brightness temperature is a directly related to the physical temperature of the electrons in the source. Sources for which the synchrotron emission mechanism dominates, the spectrum is not black-body, but is usually what is called steep spectrum3.7, i.e. the flux increases sharply with increasing wavelength. At low frequencies, the most prominent such source is the Galactic non-thermal continuum, for which the flux $S \propto \nu^{-\alpha},\ \alpha \sim 1 $. At low frequencies hence, the sky brightness temperature dominates the system temperature3.8. Pulsars and extended extra-galactic radio sources too in general have steep spectra and are brightest at low frequencies. At the extreme end of the brightness temperature are masers where a lot of energy is pumped out in a narrow collimated molecular line, the brightness temperatures could reach $\sim 10^{12}$ K. This could certainly not be the physical temperature of the source since the molecules disintegrate at temperatures well below $10^{12}$ K.


As befitting its relative youth, this is a linear, MKS based scale. At most other wavelengths, the brightness is traditionally measured in units far too idiosyncratic to be described in this footnote.
... spectrum3.7
provided that the source is optically thin
... temperature3.8
See the discussion on system temperature later in this section

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