Mar 31, 2019 - Brightness temperature

Jansky is a unit that is equal to [math]\displaystyle{ 1 \mathrm{Jy} = 10^{-26} \mathrm{\frac{W}{m^2 Hz}} }[/math]

For a blackbody, the brightness is given by [math]\displaystyle{ I_\nu = \frac{2 h \nu^3}{c^2}\frac{1}{e^{h\nu/kT}-1} . }[/math] Its units are Jy/sr.

If we change temperature by a small amount, we have [math]\displaystyle{ \frac{dI_\nu}{dT} = \frac{2 e^{h\nu/kT} \nu^4 h^2}{c^2\left(e^{h\nu/kT}-1\right)^2 kT^2}, }[/math] which can be rewritten as [math]\displaystyle{ \frac{dT}{T} = \frac{c^2\left(e^{h\nu/kT}-1\right)^2 kT}{2 e^{h\nu/kT} \nu^4 h^2} \times dI_\nu . }[/math]

For 30 GHz, the factor on the right is 1.35817e-8 if [math]\displaystyle{ dI_\nu }[/math] is in Jy/sr, in agreement with https://lambda.gsfc.nasa.gov/toolbox/tb_sim_info.cfm