VLA > Flux Density - Brightness Temperature Conversion

# Flux Density - Brightness Temperature Conversion

$$T = \frac{\lambda^2}{2 \, k \, \Omega} S$$
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 with peak 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
$$\Omega = \frac{\pi \, \theta_{maj} \, \theta_{min}}{4 \, \ln 2}$$
(see the unnumbered equation following Eq 3G4 in the NRAO Interferometers II course).
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:
$$T = 1.36 \frac{\lambda^2}{\theta_{maj} \, \theta_{min}}\, I$$
or in frequencies $\nu$ in units of GHz:
$$T = 1.222\times10^{3} \frac{I}{\nu^2 \, \theta_{maj} \, \theta_{min}}$$ 