Simple useful calculations

Brightness temperature from flux

Let $S_\nu$ be the measured flux of a source unresolved with a beam having FWHM of $\theta_m$ and $\theta_m$. The birghtness of the source is given by:

$$B=\frac{S_\nu}{\Omega} = \frac{2kT}{\lambda^2} \quad\rm {mJy asec^{-2}}$$ (3.1)

where

$$\Omega = \frac{3\pi\theta_M\theta_m}{8\ln 2}\approx 1.7\theta_M\theta_m$$ (3.2)

is the solid angle covered by an equivalent elliptical source of uniform brightness distribution. Using appropiate units, the brightness temperature can then be expressed as:

$$T \approx \left(\frac{S_\nu}{\rm 1 mJy}\right) \left(\frac{\lambda}{\rm 1 cm}\right)^2 \left(\frac{\theta_M\theta_m}{\rm asec^2}\right)^{-1}$$ (3.3)

Note that this is only a lower limit because the source can be much smaller than the beam implying a higher brightness.