AB magnitude

The flux density in janskys can be converted to a magnitude basis, for suitable assumptions about the spectrum. For instance, converting an AB magnitude to a flux density in microjanskys is straightforward:^[4]

${\displaystyle S_{v}~[\mathrm {\mu } {\text{Jy}}]=10^{6}\cdot 10^{23}\cdot 10^{-{\tfrac {{\text{AB}}+48.6}{2.5}}}=10^{\tfrac {23.9-{\text{AB}}}{2.5}}.}$

dBW·m⁻²·Hz⁻¹

The linear flux density in janskys can be converted to a decibel basis, suitable for use in fields of telecommunication and radio engineering.

1 jansky is equal to −260 dBW·m⁻²·Hz⁻¹, or −230 dBm·m⁻²·Hz⁻¹:^[5]

${\displaystyle P_{{\text{dBW}}\cdot {\text{m}}^{-2}\cdot {\text{Hz}}^{-1}}=10\log _{10}\left(P_{\text{Jy}}\right)-260,}$

${\displaystyle P_{{\text{dBm}}\cdot {\text{m}}^{-2}\cdot {\text{Hz}}^{-1}}=10\log _{10}\left(P_{\text{Jy}}\right)-230.}$

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

${\displaystyle B_{\nu }={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}}.}$

This can be solved for temperature, giving

${\displaystyle T={\frac {h\nu }{k\ln \left(1+{\frac {2h\nu ^{3}}{B_{\nu }c^{2}}}\right)}}.}$

In the low-frequency, high-temperature regime, when ${\displaystyle h\nu \ll kT}$, we can use the asymptotic expression:

${\displaystyle T\sim {\frac {h\nu }{k}}\left({\frac {B_{\nu }c^{2}}{2h\nu ^{3}}}+{\frac {1}{2}}\right).}$

A less accurate form is

${\displaystyle T_{b}={\frac {B_{\nu }c^{2}}{2k\nu ^{2}}},}$

which can be derived from the Rayleigh–Jeans law

${\displaystyle B_{\nu }={\frac {2\nu ^{2}kT}{c^{2}}}.}$