Editing Luminosity (section)

== Radio luminosity ==
The luminosity of a [[Astronomical radio source|radio source]] is measured in {{math|W Hz<sup>−1</sup>}}, to avoid having to specify a [[bandwidth (signal processing)|bandwidth]] over which it is measured. The observed strength, or [[flux density]], of a radio source is measured in [[Jansky]] where  {{math|1 Jy {{=}} 10<sup>−26</sup> W m<sup>−2</sup> Hz<sup>−1</sup>}}.

For example, consider a 10{{nbsp}}W transmitter at a distance of 1 million metres, radiating over a bandwidth of 1&nbsp;MHz. By the time that power has reached the observer, the power is spread over the surface of a sphere with area {{math|4''πr''<sup>2</sup>}} or about {{math|1.26×10<sup>13</sup> m<sup>2</sup>}}, so its flux density is {{math|1=10 / 10<sup>6</sup> / (1.26×10<sup>13</sup>) W m<sup>−2</sup> Hz<sup>−1</sup> = 8×10<sup>7</sup> Jy}}.

More generally, for sources at cosmological distances, a [[k-correction]] must be made for the spectral index α of the source, and a relativistic correction must be made for the fact that the frequency scale in the emitted [[rest frame]] is different from that in the observer's [[rest frame]]. So the full expression for radio luminosity, assuming [[isotropic]] emission, is
<math display="block">L_{\nu} = \frac{S_{\mathrm{obs}} 4 \pi {D_{L}}^{2}}{(1+z)^{1+\alpha}}</math>
where ''L''<sub>ν</sub> is the luminosity in {{math|W Hz<sup>−1</sup>}}, ''S''<sub>obs</sub> is the observed [[flux density]] in {{math|W m<sup>−2</sup> Hz<sup>−1</sup>}}, ''D<sub>L</sub>'' is the [[luminosity distance]] in metres, ''z'' is the redshift, ''&alpha;'' is the [[spectral index]] (in the sense <math>I \propto {\nu}^{\alpha}</math>, and in radio astronomy, assuming thermal emission the spectral index is typically [[Spectral index|equal to 2.]])<ref>{{cite journal |last1=Singal |first1=J. |last2=Petrosian |first2=V. |last3=Lawrence |first3=A. |last4=Stawarz |first4=Ł. |title=On the Radio and Optical Luminosity Evolution of Quasars |journal=The Astrophysical Journal |date=20 December 2011 |volume=743 |issue=2 |pages=104 |doi=10.1088/0004-637X/743/2/104|arxiv=1101.2930 |bibcode=2011ApJ...743..104S |s2cid=10579880 }}</ref>

For example, consider a 1 Jy signal from a radio source at a [[redshift]] of 1, at a frequency of 1.4&nbsp;GHz.
[http://www.astro.ucla.edu/~wright/CosmoCalc.html Ned Wright's cosmology calculator] calculates a [[luminosity distance]] for a redshift of 1 to be 6701 Mpc = 2×10<sup>26</sup> m giving a radio luminosity of {{math|1=10<sup>−26</sup> × 4{{pi}}(2×10<sup>26</sup>)<sup>2</sup> / (1 + 1)<sup>(1 + 2)</sup> = 6×10<sup>26</sup> W Hz<sup>−1</sup>}}.

To calculate the total radio power, this luminosity must be integrated over the bandwidth of the emission. A common assumption is to set the bandwidth to the observing frequency, which effectively assumes the power radiated has uniform intensity from zero frequency up to the observing frequency. In the case above, the total power is {{math|1=4×10<sup>27</sup> × 1.4×10<sup>9</sup> = 5.7×10<sup>36</sup> W}}. This is sometimes expressed in terms of the total (i.e. integrated over all wavelengths) luminosity of the [[Sun]] which is {{math|3.86×10<sup>26</sup> W}}, giving a radio power of {{math|1.5×10<sup>10</sup> ''L''<sub>⊙</sub>}}.