Photometric Quantities

The nomenclature of photometric quantities in use in astronomical literature is far from standard and sometimes ambiguous. Here we therefore give a brief summary of the definitions and units of measure of these quantities as they are used in this study.

• The Luminosity $L$ of a source is the energy radiated by the whole surface of the source per unit time, that is

  $$L=\frac{\mathrm{d} E}{\mathrm{d} t}~\mathrm{[J \: / \: s]}~.$$ (14)

  
  

• The Brightness $F$ of a source is the energy radiated by the whole surface of the source per unit time per unit area (of the receiver), that is

  $$F=\frac{\mathrm{d} L}{\mathrm{d} A}=\frac{\mathrm{d} E}{\mathrm{d} A \, \mathrm{d} t}~\mathrm{[J \: / \: m^2 \: s]}~.$$ (15)

  
  

• The Frequency Specific Brightness $F_{\nu}$ of a source is the energy radiated by the whole surface of the source per unit time per unit area (of the receiver) per unit frequency interval, that is

  $$F_{\nu}=\frac{\mathrm{d} F}{\mathrm{d} \nu}= \frac{\mathrm{d} L}{... ...{d} \nu \, \mathrm{d} A \, \mathrm{d} t} ~\mathrm{[J \: / \: Hz \: m^2 \: s]}~.$$ (16)

  
  
  
  In astronomy, $F_{\nu}$ is generally expressed in Janskys, where
  
  $$1~\textrm{Jansky}=1~\textrm{Jy}=10^{-26}~\mathrm{J \: / \: s \: m^2 \: Hz}=10^{-23}~\mathrm{erg \: / \: s \: cm^2 \: Hz}~.$$
  
  Originally used by radio astronomers and named after one, while a rather perplexing unit itself (the meaning of the $-26$ exponent in particular having being extensively discussed in front of many a pint by discouraged early postgraduates), the Jansky is also important in that it provides the foundation for a magnitude system which, given the idiosyncracies of most astronomy practitioners, is about as absolute as it gets, the socalled AB magnitude system. This is defined, at ANY wavelength, by setting $m_{AB}=0$ for a source of $F_{\nu}=3631~\textrm{Jy}$ (following Oke 1983, hereafter OG83). This translates into the following conversions
  
  $$F_{\nu\,\textrm{[Jy]}}=3631\cdot10^{-0.4\,m_{AB}}~,\textrm{[OG83]}$$
  $$m_{AB}=-2.5\cdot\log\,F_{\nu\,\textrm{[Jy]}}+8.9~,\textrm{[OG83]}$$
  
  and
  
  $$L_{\nu\,\textrm{[W/Hz]}}=4.345\cdot10^{13}\cdot10^{-0.4\,M_{AB}}~,\textrm{[OG83]}$$
  $$M_{AB}=-2.5\cdot\log\,L_{\nu\,\textrm{[W/Hz]}}+34.095,\textrm{[OG83]}$$
  
  for apparent and absolute $AB$ magnitudes, respectively. Note, however, that Tokunaga (2005) revised the absolute flux densities for Vega and thus the zero point of the AB magnitude system so that
  
  $$F_{\nu\,\textrm{[Jy]}}=3720\cdot10^{-0.4\,m_{AB}}~,\textrm{[TV05]}$$
  $$m_{AB}=-2.5\cdot\log\,F_{\nu\,\textrm{[Jy]}}+8.926~,\textrm{[TV05]}$$
  
  and this conversion is now being preferred by a number of projects.
  
  Like in any astronomical magnitude system, magnitude differences are
  
  $$m_1-m_2=-2.5\cdot\log\,(f_1/f_2)~,$$
  
  which applied to the conversion of magnitude and flux errors yields
  
  $$m_{err} = -2.5 \cdot \log( (F-F_{err}) / F )~,$$
  $$F_{err} = F \cdot ( 1 - 10^{-0.4*m_{err}})~.$$
  
  
  Conversely, the suitably-named Vega magnitudes are defined so that the magnitude of Vega is zero in all bands. Somewhat confusingly, absolute fluxes of Vega (and secondary calibrators) are measured with increasing accuracy and thus the very definition of Vega magnitudes changes in time. Having said that, estimates for the conversion factor defined by

  $$m_{AB} = m_{Vega} + conv$$ (17)

  
  
  are given in Table 2. Clearly, $conv$ is simply the magnitude of the Vega star in the AB system (in a given band), so that
  $$F_{zp\,[\mathrm{Vega}]}=F_{zp\,[AB]}\cdot10^{-0.4\,conv}~,$$
  or equivalently
  $$conv=2.5\,\log(F_{zp\,[AB]}/F_{zp\,[Vega]})~,$$
  where
  $$~~~\mathrm{where}~~~F_{zp\,[AB]}=3631~\mathrm{Jy}~.$$
  The closely related Wavelength Specific Brightness $F_{\lambda}$ of a source is the energy radiated by the whole surface of the source per unit time per unit area (of the receiver) per unit wavelength interval, that is

  $$F_{\lambda}=\frac{\mathrm{d} F}{\mathrm{d} \lambda}= \frac{\mathr... ... \lambda \, \mathrm{d} A \, \mathrm{d} t} ~\mathrm{[J \: / \: m \: m^2 \: s]}~.$$ (18)

  
  
  
  
  
  Table 2: AB vs Vega Magnitudes. Band, $conv = m_{AB} - m_{Vega}$, Vega zero-points (or zero-Vega-magnitude fluxes in Jy) and source.

  band	$conv$	$F_{zp}$	source
  GALEX FUV	2.223	468.6	Bianchi 2011
  GALEX NUV	1.699	759.3	Bianchi 2011
  SDSS $u$	0.927	1545	Hewett et al. 2006
  SDSS $g$	-0.103	3991	Hewett et al. 2006
  SDSS $r$	0.146	3174	Hewett et al. 2006
  SDSS $i$	0.366	2593	Hewett et al. 2006
  SDSS $z$	0.533	2222	Hewett et al. 2006
  2MASS $J$	0.89	1594	2MASS Expl. Supp. 2006
  2MASS $H$	1.37	1024	2MASS Expl. Supp. 2006
  2MASS $K_s$	1.84	666.7	2MASS Expl. Supp. 2006
  UKIRT/WFCAM $Z$	0.528	2232	Hewett et al. 2006
  UKIRT/WFCAM $Y$	0.634	2026	Hewett et al. 2006
  UKIRT/WFCAM $J$	0.938	1530	Hewett et al. 2006
  UKIRT/WFCAM $H$	1.379	1019	Hewett et al. 2006
  UKIRT/WFCAM $K$	1.900	631	Hewett et al. 2006
  VISTA/VIRCAM $Z$	0.502	2287	Gonzalez-Fernandez et al. 2017
  VISTA/VIRCAM $Y$	0.600	2089	Gonzalez-Fernandez et al. 2017
  VISTA/VIRCAM $J$	0.916	1562	Gonzalez-Fernandez et al. 2017
  VISTA/VIRCAM $H$	1.366	1032	Gonzalez-Fernandez et al. 2017
  VISTA/VIRCAM $Ks$	1.827	675	Gonzalez-Fernandez et al. 2017
  IRAC-1	2.78	280.9	IRAC Data HandBook 3.0 2006
  IRAC-2	3.26	179.7	IRAC Data HandBook 3.0 2006
  IRAC-3	3.75	115.0	IRAC Data HandBook 3.0 2006
  IRAC-4	4.38	64.1	IRAC Data HandBook 3.0 2006
  WISE-1	2.683	306.682	AllWISE Expl. Supp. 2013
  WISE-2	3.319	170.663	AllWISE Expl. Supp. 2013
  WISE-3	5.242	29.045	AllWISE Expl. Supp. 2013
  WISE-4	6.604	8.284	AllWISE Expl. Supp. 2013
  MIPS-24	6.77	7.14	MIPS Data HandBook 3.3.1 2008
  MIPS-70	9.18	0.775	MIPS Data HandBook 3.3.1 2008
  MIPS-160	10.90	0.159	MIPS Data HandBook 3.3.1 2008

  
  
  
  
  Table 3: AB vs Vega Magnitudes in Johnson-Counsins-Glass System. Band, Effective Wavelength and Width in nm, $conv$ factors defined in Equation 17 and Vega zero-points (or zero-magnitude fluxes, in Jy). From http://www.euro-vo.org/internal/Avo/WorkPackageTwoTwo/mag_to_flux_conversions and to be taken with a bit of skepticism.

  band	$\lambda_{eff}$	$\Delta\lambda$	$conv$	$F_{zp}$
  $U$	367	66	0.767933	1790
  $B$	436	94	-0.122051	4063
  $V$	545	85	-0.001494	3636
  $R$	638	160	0.184344	3064
  $I$	797	149	0.442323	2416
  $J$	1220	213	0.897256	1589
  $H$	1630	307	1.37857	1020
  $K$	2190	390	1.88462	640
  $L$	3450	472	2.76295	285
  $M$	4750	460	3.43126	154

  
  
• The Frequency Specific Reduced Brightness $F_{\nu,red}$ of a source is the product of its Frequency Specific Brightness and the frequency at which it is measured, namely

  $$F_{\nu,red}=\nu\:F_{\nu}~\mathrm{[J \: / \: m^2 \: s]}~,$$ (19)

  
  
  whereas the Wavelength Specific Reduced Brightness $F_{\lambda,red}$ of a source is the product of its Wavelength Specific Brightness and the wavelength at which it is measured, namely

  $$F_{\lambda,red}=\lambda\:F_{\lambda}~\mathrm{[J \: / \: m^2 \: s]}~.$$ (20)

  
  
  The fact that these two quantities are expressed in the same units is not a coincidence but is due to their actually being the same quantity

  $$\nu\:F_{\nu}=F_{\nu,red}=F_{\lambda,red}=\lambda\:F_{\lambda}~,$$ (21)

  
  
  expressing the source brightness contained in a given (logarithmic) spectral range. In other words, if we plot $F_{\nu,red}=F_{\lambda,red}$ using a logarithmic scale along the abscissae, the brightness emitted over different wavelength ranges can be directly compared by comparing the areas under the relevant regions of the curve, making it a powerful tool in graphically illustrating the emission processes of astronomical sources.
• Most galaxies, unlike most stars, are resolved objects, so that in addition to measuring their total energy flux, we can in principle measure the energy flux per unit solid angle of the source coming from different regions. The Surface Brightness $\Sigma$ of a region of a diffuse source is the energy radiated by the region per unit time, per unit area (of the receiver) and per unit solid angle (of the source), that is

  $$\Sigma=\frac{\mathrm{d} F}{\mathrm{d} \Omega}=\frac{\mathrm{d} L}... ... \Omega \, \mathrm{d} A \, \mathrm{d} t} ~\mathrm{[J \: / \: sr \: m^2 \: s]}~.$$ (22)

  
  

• The Frequency Specific Surface Brightness $\Sigma_{\nu}$ of a region of a diffuse source is the energy radiated by the region per unit time, per unit area (of the receiver) per unit solid angle (of the source) per unit frequency interval, that is

  $$\Sigma_{\nu}=\frac{\mathrm{d} F}{\mathrm{d} \nu \, \mathrm{d} \Om... ...ga \, \mathrm{d} A \, \mathrm{d} t}~\mathrm{[J \: / \: Hz \: sr \: m^2 \: s]}~.$$ (23)

  
  
  
  The closely related Wavelength Specific Surface Brightness $\Sigma_{\lambda}$ of a region of a diffuse source is the energy radiated by the region per unit time, per unit area (of the receiver) per unit solid angle (of the source) per unit wavelength interval, that is

  $$\Sigma_{\lambda}=\frac{\mathrm{d} F}{\mathrm{d} \lambda \, \mathr... ...ega \, \mathrm{d} A \, \mathrm{d} t}~\mathrm{[J \: / \: m \: sr \: m^2 \: s]}~.$$ (24)

  
  

• When Luminosities rather than Brightnesses are being measured, $L_{\nu}$ and $L_{\lambda}$ are defined rather than $F_{\nu}$ and $F_{\lambda}$.

Note that since

$$\mathrm{d}\nu=\mathrm{d}\left(\frac{c}{\lambda}\right)=\frac{c}{\... ...gleftrightarrow~~~\frac{\mathrm{d}\nu}{\nu}=\frac{\mathrm{d}\lambda}{\lambda}~,$$ (25)

the following relations hold

$$F_{\nu}=\frac{\lambda^2}{c}\,F_{\lambda}~~~\Longleftrightarrow~~~ F_{\nu}\,\nu=F_{\lambda}\,\lambda~,$$ (26)

$$\Sigma_{\nu}=\frac{\lambda^2}{c}\,\Sigma_{\lambda}~~~\Longleftrightarrow~~~ \Sigma_{\nu}\,\nu=\Sigma_{\lambda}\,\lambda~.$$ (27)

Astronomers, however, generally express brightness and surface brightness in logarithmic units, i.e. in magnitudes (mag) and magnitudes per square second of arc (mag/arcsec$^2$), respectively. To define a magnitude scale, one has to arbitrarily choose a reference brightness $F_{zp}$, and the corresponding reference surface brightness $\Sigma_{zp}$ of $F_{zp}$ per square second of arc. The brightness of a source expressed in magnitudes is then

$$m=-2.5\,\log\frac{F}{F_{zp}}~\mathrm{[mag]}~,$$ (28)

while the surface brightness of a region of a diffuse source in magnitudes per square second of arc is

$$\mu=-2.5\,\log\frac{\Sigma}{\Sigma_{zp}}~\mathrm{[mag/arcsec^2]}~.$$ (29)

$F_{zp}$ is called the zero-point of the adopted magnitude scale since $m=0$ for $F=F_{zp}$ (and thus $\mu=0$ for $\Sigma=\Sigma_{zp}$).
Note that these definitions equally apply to bolometric measurements and to measurements in a given photometric band. One then simply has to take into account only the radiation within a given wavelength range weighted by the profile of the photometric band.
Note also that the sky background is often expressed in different units, such as those described by Leinert et al. (1998).