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.  For a Gaussian beam, its solid angle $\Omega$ is related to the HPBW $\theta$ by:

$$\Omega = \frac{\pi \theta^2}{4 \ln 2}$$

(see the unnumbered equation following Eq 3G4 in the NRAO Interferometers II course).  Substituting $\Omega$ in the first equation leads to:

$$T = 0.32 \times 10^{23} \frac{\lambda^2}{\theta^2} S$$

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

$$T = 1.36 \frac{\lambda^2}{\theta^2} S$$