Editing Luminosity

{{Short description|Measurement of radiant electromagnetic power emitted by an object}}
{{Other uses}}
{{Use dmy dates|date=October 2019}}
[[File:The Sun in white light.jpg|thumb|The [[Sun]] has an intrinsic luminosity of {{val|3.83|e=26|u=[[watt]]s}}. In astronomy, this amount is equal to one [[solar luminosity]], represented by the symbol ''L''<sub>⊙</sub>. A star with four times the radiative power of the Sun has a luminosity of {{val|4|u=''L''<sub>⊙</sub>}}.]]
'''Luminosity''' is an absolute measure of radiated [[electromagnetic radiation|electromagnetic energy]] per unit time, and is synonymous with the [[radiant power]] emitted by a light-emitting object.<ref>{{Cite news|url=https://www.britannica.com/science/luminosity|title=Luminosity {{!}} astronomy|work=Encyclopedia Britannica|access-date=24 June 2018|language=en}}</ref><ref>{{Cite web|url=https://en.mimi.hu/astronomy/luminosity.html|title=* Luminosity (Astronomy) - Definition, meaning - Online Encyclopedia|website=en.mimi.hu|access-date=24 June 2018}}</ref> In [[astronomy]], luminosity is the total amount of electromagnetic [[energy]] emitted per unit of [[time]] by a [[star]], [[galaxy]], or other [[astronomical object|astronomical objects]].<ref name=Hopkins1980>{{cite book |last=Hopkins |first=Jeanne |title=Glossary of Astronomy and Astrophysics |edition=2nd |publisher=[[The University of Chicago Press]] |date=1980 |isbn=978-0-226-35171-1}}</ref><ref>{{cite book |last1=Morison |first1=Ian |title=Introduction to Astronomy and Cosmology |date=2013 |publisher=Wiley |isbn=978-1-118-68152-7|url=https://books.google.com/books?id=Fh_yo8Jv7t8C&pg=PT193|page=193}}</ref>

In [[SI]] units, luminosity is measured in [[joules]] per second, or [[watt]]s. In astronomy, values for luminosity are often given in the terms of the [[Solar luminosity|luminosity of the Sun]], ''L''<sub>⊙</sub>. Luminosity can also be given in terms of the astronomical [[Magnitude (astronomy)|magnitude]] system: the [[Absolute magnitude#Bolometric magnitude|absolute bolometric magnitude]] (''M''<sub>bol</sub>) of an object is a logarithmic measure of its total energy emission rate, while [[absolute magnitude]] is a logarithmic measure of the luminosity within some specific [[wavelength]] range or [[Passband|filter band]].

In contrast, the term ''brightness'' in astronomy is generally used to refer to an object's apparent brightness: that is, how bright an object appears to an observer. Apparent brightness depends on both the luminosity of the object and the distance between the object and observer, and also on any [[Extinction (astronomy)|absorption]] of light along the path from object to observer. [[Apparent magnitude]] is a logarithmic measure of apparent brightness. The distance determined by luminosity measures can be somewhat ambiguous, and is thus sometimes called the [[luminosity distance]].

== Measurement ==
When not qualified, the term "luminosity" means bolometric luminosity, which is measured either in the [[SI]] units, [[watt]]s, or in terms of [[solar luminosity|solar luminosities]] ({{solar luminosity}}).  A [[bolometer]] is the instrument used to measure [[radiant energy]] over a wide band by [[absorption (electromagnetic radiation)|absorption]] and measurement of heating.  A star also radiates [[neutrino]]s, which carry off some energy (about 2% in the case of the Sun), contributing to the star's total luminosity.<ref name="BAHCALL1">{{cite web |first=John |last=Bahcall |author-link=John N. Bahcall |url=http://www.sns.ias.edu/~jnb/SNviewgraphs/snviewgraphs.html |title=Solar Neutrino Viewgraphs |publisher=[[Institute for Advanced Study]] School of Natural Science |access-date=3 July 2012
}}</ref> The IAU has defined a nominal solar luminosity of {{val|3.828|e=26|u=W}} to promote publication of consistent and comparable values in units of the solar luminosity.<ref name=iau>{{cite arXiv |eprint=1510.07674|class=astro-ph.SR|title=IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties |last1=Mamajek|first1=E. E.|last2=Prsa|first2=A.|last3=Torres|first3=G.|last4=Harmanec|first4=P.|last5=Asplund|first5=M.|last6=Bennett|first6=P. D. |last7=Capitaine|first7=N. |last8=Christensen-Dalsgaard|first8=J.|last9=Depagne|first9=E.|last10=Folkner|first10=W. M.|last11=Haberreiter|first11=M. |last12=Hekker|first12=S. |last13=Hilton|first13=J. L.|last14=Kostov|first14=V.|last15=Kurtz|first15=D. W.|last16=Laskar|first16=J.|last17=Mason|first17=B. D.| last18=Milone|first18=E. F. |last19=Montgomery|first19=M. M.|last20=Richards|first20=M. T.|last21=Schou|first21=J.|last22=Stewart|first22=S. G.|year=2015}}</ref>

While bolometers do exist, they cannot be used to measure even the apparent brightness of a star because they are insufficiently sensitive across the [[electromagnetic spectrum]] and because most wavelengths do not reach the surface of the Earth.  In practice bolometric magnitudes are measured by taking measurements at certain wavelengths and constructing a model of the total spectrum that is most likely to match those measurements.  In some cases, the process of estimation is extreme, with luminosities being calculated when less than 1% of the energy output is observed, for example with a hot [[Wolf-Rayet star]] observed only in the infrared. Bolometric luminosities can also be calculated using a [[bolometric correction]] to a luminosity in a particular passband.<ref name=nieva>{{cite journal|bibcode=2013A&A...550A..26N| arxiv=1212.0928| title=Temperature, gravity, and bolometric correction scales for non-supergiant OB stars|journal=Astronomy & Astrophysics|volume=550|pages=A26|last1=Nieva|first1=M.-F|year=2013| doi=10.1051/0004-6361/201219677|s2cid=119275940}}</ref><ref name=buzzoni>{{cite journal|bibcode=2010MNRAS.403.1592B|arxiv=1002.1972|title=Bolometric correction and spectral energy distribution of cool stars in Galactic clusters|journal=Monthly Notices of the Royal Astronomical Society| volume=403|issue=3|pages=1592 |last1=Buzzoni|first1=A |last2=Patelli|first2=L|last3=Bellazzini|first3=M|last4=Pecci|first4=F. Fusi|last5=Oliva|first5=E| year=2010| doi=10.1111/j.1365-2966.2009.16223.x|doi-access=free |s2cid=119181086}}</ref>

The term luminosity is also used in relation to particular [[passband]]s such as a visual luminosity of [[k band (infrared)|K-band]] luminosity.<ref>{{Cite web|url=http://www.faculty.virginia.edu/ASTR5610/lectures/LECTURE2/lec2a.html|title=ASTR 5610, Majewski [SPRING 2016]. Lecture Notes|website=www.faculty.virginia.edu|access-date=3 February 2019|archive-date=24 April 2021|archive-url=https://web.archive.org/web/20210424171700/https://faculty.virginia.edu/ASTR5610/lectures/LECTURE2/lec2a.html|url-status=dead}}</ref> These are not generally luminosities in the strict sense of an absolute measure of radiated power, but absolute magnitudes defined for a given filter in a [[photometric system]].  Several different photometric systems exist.  Some such as the UBV or [[UBV photometric system|Johnson]] system are defined against photometric standard stars, while others such as the [[AB magnitude|AB system]] are defined in terms of a [[spectral flux density]].<ref name=delfosse>{{citation |bibcode=2000A&A...364..217D |display-authors=1 |last1=Delfosse |first1=Xavier |last2=Forveille |first2=Thierry |last3=Ségransan |first3=Damien |last4=Beuzit |first4=Jean-Luc |last5=Udry |first5=Stéphane |last6=Perrier |first6=Christian |last7=Mayor |first7=Michel |title=Accurate masses of very low mass stars. IV. Improved mass-luminosity relations |journal=Astronomy and Astrophysics |volume=364 |pages=217–224 |date=December 2000 |arxiv = astro-ph/0010586 }}</ref>

== Stellar luminosity ==
A star's luminosity can be determined from two stellar characteristics: size and [[effective temperature]].<ref name="AUSTRALIA2004">{{cite web |title=Luminosity of Stars |publisher=[[Australia Telescope National Facility]]|url=http://outreach.atnf.csiro.au/education/senior/astrophysics/photometry_luminosity.html |date=12 July 2004 |archive-url=https://web.archive.org/web/20140809144429/http://www.atnf.csiro.au/outreach//education/senior/astrophysics/photometry_luminosity.html |archive-date=9 August 2014}}</ref> The former is typically represented in terms of solar [[radius|radii]], ''R''<sub>⊙</sub>, while the latter is represented in [[kelvin]]s, but in most cases neither can be measured directly.  To determine a star's radius, two other metrics are needed: the star's [[angular diameter]] and its distance from Earth.  Both can be measured with great accuracy in certain cases, with cool supergiants often having large angular diameters, and some cool evolved stars having [[astrophysical maser|maser]]s in their atmospheres that can be used to measure the parallax using [[VLBI]].  However, for most stars the angular diameter or parallax, or both, are far below our ability to measure with any certainty.  Since the effective temperature is merely a number that represents the temperature of a black body that would reproduce the luminosity, it obviously cannot be measured directly, but it can be estimated from the spectrum.

An alternative way to measure stellar luminosity is to measure the star's apparent brightness and distance.  A third component needed to derive the luminosity is the degree of [[Extinction (astronomy)|interstellar extinction]] that is present, a condition that usually arises because of gas and dust present in the [[interstellar medium]] (ISM), the [[Earth's atmosphere]], and [[circumstellar dust|circumstellar matter]].  Consequently, one of astronomy's central challenges in determining a star's luminosity is to derive accurate measurements for each of these components, without which an accurate luminosity figure remains elusive.<ref name="KARTTUNEN1">{{cite book  | last1 = Karttunen  | first1 = Hannu | title = Fundamental Astronomy  | publisher = [[Springer-Verlag]]  | date = 2003  | page = 289  | url=https://books.google.com/books?id=OEhHqwW-kgQC | isbn = 978-3-540-00179-9}}</ref> Extinction can only be measured directly if the actual and observed luminosities are both known, but it can be estimated from the observed colour of a star, using models of the expected level of reddening from the interstellar medium.

In the current system of [[stellar classification]], stars are grouped according to temperature, with the massive, very young and energetic [[O-type main sequence star|Class O]] stars boasting temperatures in excess of 30,000&nbsp;[[kelvin|K]] while the less massive, typically older [[Stellar classification#Class M|Class M]] stars exhibit temperatures less than 3,500&nbsp;K. Because luminosity is proportional to temperature to the fourth power, the large variation in stellar temperatures produces an even vaster variation in stellar luminosity.<ref name="LEDREW1">{{cite journal | author=Ledrew, Glenn | title=The Real Starry Sky | journal=Journal of the Royal Astronomical Society of Canada |date=February 2001 | volume=95 | pages=32–33 | url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?2001JRASC..95...32L&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf | access-date=2 July 2012 | bibcode=2001JRASC..95...32L}}</ref> Because the luminosity depends on a high power of the stellar mass, high mass luminous stars have much shorter lifetimes.  The most luminous stars are always young stars, no more than a few million years for the most extreme.  In the [[Hertzsprung–Russell diagram]], the x-axis represents temperature or spectral type while the y-axis represents luminosity or magnitude.  The vast majority of stars are found along the [[main sequence]] with blue Class O stars found at the top left of the chart while red Class M stars fall to the bottom right.  Certain stars like [[Deneb]] and [[Betelgeuse]] are found above and to the right of the main sequence, more luminous or cooler than their equivalents on the main sequence.  Increased luminosity at the same temperature, or alternatively cooler temperature at the same luminosity, indicates that these stars are larger than those on the main sequence and they are called giants or supergiants.

Blue and white supergiants are high luminosity stars somewhat cooler than the most luminous main sequence stars.  A star like [[Deneb]], for example, has a luminosity around 200,000 ''L''<sub>⊙</sub>, a spectral type of A2, and an effective temperature around 8,500&nbsp;K, meaning it has a radius around {{convert|203|solar radius|m|abbr=on|lk=on}}. For comparison, the red supergiant [[Betelgeuse]] has a luminosity around 100,000 ''L''<sub>⊙</sub>, a spectral type of M2, and a temperature around 3,500&nbsp;K, meaning its radius is about {{convert|1000|solar radius|m|abbr=on|lk=on}}. Red supergiants are the largest type of star, but the most luminous are much smaller and hotter, with temperatures up to 50,000&nbsp;K and more and luminosities of several million ''L''<sub>⊙</sub>, meaning their radii are just a few tens of ''R''<sub>⊙</sub>.  For example, [[R136a1]] has a temperature over 46,000&nbsp;K and a luminosity of more than 6,100,000 ''L''<sub>⊙</sub><ref name="census">{{cite journal|last1=Doran|first1=E. I.|last2=Crowther|first2=P. A.| last3=de Koter|first3=A.|last4=Evans|first4=C. J.|last5=McEvoy|first5=C.|last6=Walborn|first6=N. R.|last7=Bastian|first7=N.|last8=Bestenlehner|first8=J. M.| last9=Gräfener|first9=G.| last10=Herrero|first10=A.|last11=Kohler|first11=K.|last12=Maiz Apellaniz|first12=J.|last13=Najarro|first13=F.| last14=Puls|first14=J.| last15=Sana|first15=H.| last16=Schneider|first16=F. R. N.|last17=Taylor|first17=W. D.|last18=van Loon|first18=J. Th.|last19=Vink|first19=J. S.| title=The VLT-FLAMES Tarantula Survey - XI. A census of the hot luminous stars and their feedback in 30 Doradus|journal=Astronomy & Astrophysics| volume=558| pages=A134| arxiv=1308.3412v1| date=2013| doi=10.1051/0004-6361/201321824|bibcode=2013A&A...558A.134D|s2cid=118510909}}</ref> (mostly in the UV), it is only {{convert|39|solar radius|m|abbr=on|lk=on}}.

== 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>}}.

== Luminosity formulae ==
{{More citations needed section|date=July 2023}}
[[File:Inverse square law.svg|right|thumb|Point source ''S'' is radiating light equally in all directions. The amount passing through an area ''A'' varies with the distance of the surface from the light.]]

The [[Stefan–Boltzmann law|Stefan–Boltzmann]] equation applied to a [[black body]] gives the value for luminosity for a black body, an idealized object which is perfectly opaque and non-reflecting:<ref name="AUSTRALIA2004"/>
<math display="block">L = \sigma A T^4,</math>
where ''A'' is the surface area, ''T'' is the temperature (in kelvins) and {{math|''σ''}} is the [[Stefan–Boltzmann constant]], with a value of {{physconst|sigma|after=.}}

Imagine a point source of light of luminosity <math>L</math> that radiates equally in all directions. A hollow [[sphere]] centered on the point would have its entire interior surface illuminated. As the radius increases, the surface area will also increase, and the constant luminosity has more surface area to illuminate, leading to a decrease in observed brightness.

<math display="block">F = \frac{L}{A},</math>
where
*<math>A</math> is the area of the illuminated surface.
*<math>F</math> is the [[flux density]] of the illuminated surface.

The surface area of a sphere with radius ''r'' is <math>A = 4\pi r^2</math>, so for stars and other point sources of light: 
<math display="block">F = \frac{L}{4\pi r^2} \,,</math>
where <math>r</math> is the distance from the observer to the light source.

For stars on the [[main sequence]], luminosity is also [[mass–luminosity relation|related to mass]] approximately as below:
<math display="block">\frac{L}{L_{\odot}} \approx {\left ( \frac{M}{M_{\odot}} \right )}^{3.5}.</math>

== Relationship to magnitude ==
{{Main|Bolometric magnitude}}
Luminosity is an intrinsic measurable property of a star independent of distance. The concept of magnitude, on the other hand, incorporates distance. The [[apparent magnitude]] is a measure of the diminishing flux of light as a result of distance according to the [[inverse-square law]].<ref name="HAWAII2003">{{cite web |url=http://www.ifa.hawaii.edu/~barnes/ASTR110L_S03/inversesquare.html|title=The Inverse-Square Law| author=Joshua E. Barnes| date=18 February 2003 |publisher=Institute for Astronomy - University of Hawaii |access-date=26 September 2012}}</ref> The Pogson logarithmic scale is used to measure both apparent and absolute magnitudes, the latter corresponding to the brightness of a star or other [[celestial body]] as seen if it would be located at an interstellar distance of {{convert|10|parsec|m|abbr=off|lk=on}}. In addition to this brightness decrease from increased distance, there is an extra decrease of brightness due to extinction from intervening interstellar dust.<ref name="ASTRONOTES1">{{cite web|url=http://www.astronomynotes.com/starprop/s4.htm|title=Magnitude System|date=2 November 2010|publisher=Astronomy Notes|access-date=2 July 2012}}</ref>

By measuring the width of certain absorption lines in the [[stellar classification|stellar spectrum]], it is often possible to assign a certain luminosity class to a star without knowing its distance. Thus a fair measure of its absolute magnitude can be determined without knowing its distance nor the interstellar extinction.

In measuring star brightnesses, absolute magnitude, apparent magnitude, and distance are interrelated parameters—if two are known, the third can be determined. Since the Sun's luminosity is the standard, comparing these parameters with the Sun's apparent magnitude and distance is the easiest way to remember how to convert between them, although officially, zero point values are defined by the IAU.

The magnitude of a star, a [[unitless]] measure, is a logarithmic scale of observed visible brightness.  The apparent magnitude is the observed visible brightness from [[Earth]] which depends on the distance of the object.  The absolute magnitude is the apparent magnitude at a distance of {{convert|10|parsec|m|abbr=on|lk=on}}, therefore the bolometric absolute magnitude is a logarithmic measure of the bolometric luminosity.

The difference in bolometric magnitude between two objects is related to their luminosity ratio according to:<ref>{{Cite web| url=http://csep10.phys.utk.edu/OJTA2dev/ojta/c2c/ordinary_stars/magnitudes/absolute_tl.html| title=Absolute Magnitude| website=csep10.phys.utk.edu|access-date=2 February 2019}}</ref>
<math display="block">M_\text{bol1} - M_\text{bol2} = -2.5 \log_{10}\frac{L_\text{1}}{L_\text{2}}</math>

where:
*<math>M_{\text{bol1}}</math> is the bolometric magnitude of the first object
*<math>M_\text{bol2}</math> is the bolometric magnitude of the second object.
*<math>L_\text{1}</math> is the first object's bolometric luminosity
*<math>L_\text{2}</math> is the second object's bolometric luminosity

The zero point of the absolute magnitude scale is actually defined as a fixed luminosity of {{val|3.0128|e=28|u=W}}. Therefore, the absolute magnitude can be calculated from a luminosity in watts:
<math display="block">M_\mathrm{bol} = -2.5 \log_{10} \frac{L_{*}}{L_0} \approx -2.5 \log_{10} L_{*} + 71.1974</math>
where {{math|''L''<sub>0</sub>}} is the zero point luminosity {{val|3.0128|e=28|u=W}}

and the luminosity in watts can be calculated from an absolute magnitude (although absolute magnitudes are often not measured relative to an absolute flux):
<math display="block">L_{*} = L_0 \times 10^{-0.4 M_\mathrm{bol}}</math>

==See also==
* [[Glossary of astronomy]]
* [[List of brightest stars]]
* [[List of most luminous stars]]
* [[Orders of magnitude (power)]]
* [[Solar luminosity]]

==References==
{{Reflist|25em}}

==Further reading==
* {{cite book|last=Böhm-Vitense |first=Erika |title=Introduction to Stellar Astrophysics: Volume 1, Basic Stellar Observations and Data |chapter-url=https://books.google.com/books?id=JWrtilsCycQC&pg=PA41 |year=1989|publisher=[[Cambridge University Press]] |isbn=978-0-521-34869-0 |pages=41–48 |chapter=Chapter 6. The luminosities of the stars}}

==External links==
{{Wiktionary}}
*[https://www.fxsolver.com/browse/formulas/Luminosity+of+a+Star Luminosity calculator]
*[http://www.astro.ucla.edu/~wright/CosmoCalc.html Ned Wright's cosmology calculator]
*{{webarchive |url=https://web.archive.org/web/20150508152746/http://www.astro.soton.ac.uk/~td/flux_convert.html |date=8 May 2015|title=University of Southampton radio luminosity calculator}}

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