Facilities > VLA > Proposing > Flux Density - Brightness Temperature Conversion

Flux Density - Brightness Temperature Conversion

by Gustaaf Van Moorsel last modified Dec 18, 2014 by Juergen Ott

Start with the Rayleigh-Jeans law:

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

where T is the brightness temperature, [inline]\lambda[/inline] the wavelength, k the Boltzmann constant, [inline]\Omega[/inline] the beam solid angle, and S the flux density, all in mks units.  For a Gaussian beam, its solid angle [inline]\Omega[/inline] is related to the HPBW [inline]\theta[/inline] by:

[display]\Omega = \frac{\pi \theta^2}{4 \ln 2}[/display]

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

[display]T = 0.32 \times 10^{23} \frac{\lambda^2}{\theta^2} S[/display]

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

[display]T = 1.36 \frac{\lambda^2}{\theta^2} S[/display]