Editing Jansky (section)

=== Temperature units ===

The [[spectral radiance]] in janskys per [[steradian]] can be converted to a [[brightness temperature]], useful in radio and microwave astronomy. 

Starting with [[Planck's law]], we see
<math display="block">B_{\nu} = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT}-1}.</math>
This can be solved for temperature, giving
<math display="block">T = \frac{h\nu}{k\ln\left (1+\frac{2h\nu^3}{B_\nu c^2}\right )}.</math>
In the low-frequency, high-temperature regime, when <math>h\nu \ll kT</math>, we can use the [[asymptotic expression]]:
<math display="block">T\sim \frac{h\nu}k\left(\frac{B_\nu c^2}{2h\nu^3}+\frac 12\right).</math>

A less accurate form is
<math display="block">T_b = \frac{B_{\nu}c^2}{2k\nu^2},</math>
which can be derived from the [[Rayleigh–Jeans law]]
<math display="block">B_{\nu} = \frac{2\nu^2 kT}{c^2}.</math>