View source for Mar 31, 2019 - Brightness temperature

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

For a blackbody, the brightness is given by
<math>
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>
\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>
\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>dI_\nu</math> is in Jy/sr, in agreement with https://lambda.gsfc.nasa.gov/toolbox/tb_sim_info.cfm