Flux Density - Brightness Temperature Conversion

Start with the Rayleigh-Jeans law:

\[T = \frac{\lambda^2}{2 k \Omega} S \]

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 \(\frac{S}{\Omega}\) by \(\frac{S}{beam}\frac{beam}{\Omega} = I \\,\frac{beam}{\Omega} \), 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 withpeak of unity and \(\theta_{maj}\) and \(\theta_{min}\) 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

\[\Omega = \frac{\pi \ \theta_{maj} \ \theta_{min}}{4 \ \\ln 2}\]

Substituting \(\\Omega\) in the first equation and converting all constants to a pre-factor leads to:

\[T = 0.32 \times 10^{23} \frac{\lambda^2}{\theta_{maj} \ \theta_{min}}\ I\]

Converting to units to cm, seconds of arc, and mJy/beam results in:

\[T = 1.36 \frac{\lambda^2}{\theta_{maj} \ \theta_{min}}\ I\]