Flux Density - Brightness Temperature Conversion

Start with the Rayleigh-Jeans law:
\[T = rac{\lambda^2}{2 \, k \, \Omega} S \]
where T is the brightness temperature, \(\lambda\) the wavelength, k the Boltzmann constant, \(\Omega\) the beam solid angle, and S the flux density, all in mks units.  
Interferometric maps are measured in brightness units \(I\) of Jy/beam, where “beam” is a nominal area over which the brightness is defined.  
We replace \(rac{S}{\Omega}\) by \(rac{S}{beam}rac{beam}{\Omega} = I \,rac{beam}{\Omega} \), where the second term is the conversion of “beam” to a solid angle in units of steradian.
For a "dirty map", the “beam” is the "dirty beam" which has a complicated structure that depends on the number and orientation of baselines as a function of time. Once the image is deconvolved, however, the "dirty beam" is replaced by a "clean beam", a Gaussian with peak of unity and \( heta_{maj}\) and \( heta_{min}\) as the half-power beam widths along the major and minor axes, respectively. 
The area of a Gaussian beam is defined by its 2-dimensional integral 
\[\Omega = rac{\pi \, heta_{maj} \, heta_{min}}{4 \, \ln 2}\]
Substituting \(\Omega\) in the first equation and converting all constants to a pre-factor leads to:
\[T = 0.32 imes 10^{23} rac{\lambda^2}{ heta_{maj} \, heta_{min}}\, I\]
Converting to units to cm, seconds of arc, and mJy/beam results in:
\[T = 1.36 rac{\lambda^2}{ heta_{maj} \, heta_{min}}\, I\] 
or in frequencies \( u\) in units of GHz: 
\[T = 1.222 imes10^{3} rac{I}{ u^2 \, heta_{maj} \, heta_{min}}\] 
 

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