Jansky

The spectral flux density or monochromatic flux, S, of a source is the integral of the spectral radiance, B, over the source solid angle: $${\displaystyle S=\underset{\text{source}}{\iint }B(\theta ,\varphi )\mathrm{d}\mathrm{\Omega }.}$$

The unit is named after pioneering US radio astronomer Karl Guthe Jansky and is defined as

• ${\displaystyle 1\mathrm{J}\mathrm{y}={10}^{-26}\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot \mathrm{H}{\mathrm{z}}^{-1}}$ (SI)
• ${\displaystyle 1\mathrm{J}\mathrm{y}={10}^{-23}\mathrm{e}\mathrm{r}\mathrm{g}\cdot {\mathrm{s}}^{-1}\cdot \mathrm{c}{\mathrm{m}}^{-2}\cdot \mathrm{H}{\mathrm{z}}^{-1}}$ (CGS).

Since the jansky is obtained by integrating over the whole source solid angle, it is most simply used to describe point sources; for example, the Third Cambridge Catalogue of Radio Sources (3C) reports results in janskys.

• For extended sources, the surface brightness is often described with units of janskys per solid angle; for example, far-infrared (FIR) maps from the IRAS satellite are in megajanskys per steradian (MJy⋅sr⁻¹).
• Although extended sources at all wavelengths can be reported with these units, for radio-frequency maps, extended sources have traditionally been described in terms of a brightness temperature; for example the Haslam et al. 408 MHz all-sky continuum survey is reported in terms of a brightness temperature in kelvin.

Unit conversions

Jansky units are not a standard SI unit, so it may be necessary to convert the measurements made in the unit to the SI equivalent in terms of watts per square metre per hertz (W·m⁻²·Hz⁻¹). However, other unit conversions are possible with respect to measuring this unit.

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: $${\displaystyle S_v[\mu \text{Jy}]={10}^6\cdot {10}^{23}\cdot {10}^{-\frac{\text{AB}+48.6}{2.5}}={10}^{\frac{23.9-\text{AB}}{2.5}}.}$$

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: $${\displaystyle \begin{array}{cc}{P}_{\text{dBW}\cdot {\text{m}}^{-2}\cdot {\text{Hz}}^{-1}}& =10{\log }_{10}\left(P_{\text{Jy}}\right)-260,\\ P_{\text{dBm}\cdot {\text{m}}^{-2}\cdot {\text{Hz}}^{-1}}& =10{\log }_{10}\left(P_{\text{Jy}}\right)-230.\end{array}}$$

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