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{{About|the measurement of brightness|the Cedar Point show|Luminosity — Ignite the Night!}}
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[[File:Phot-27e-07.jpg|thumb|230px|right|Image of galaxy [[NGC 4945]] showing the huge luminosity of the central few star clusters, suggesting there are 10 to 100 supergiant stars in each of these, packed into regions just a few [[parsecs]] across.]]
{{Wiktionary}}
'''Luminosity''' is generally understood as a [[measurement]] of [[brightness]].{{Citation needed|date=December 2013}}  Each discipline, however, defines the term differently, depending on what is being measured.{{Citation needed|date=December 2013}}


In astronomy, luminosity measures the total amount of energy emitted by a [[star]] or other [[astronomical object]] per unit time. In [[SI]] units  this is expressed as [[joules]] per second or [[watt]]s. I.e. The [[Sun]] has a total power output of {{val|3.846|e=26|u=[[Watt|W]]}} or 1.00 [[solar luminosity]], the symbol for which is <math>L_\odot</math>.   
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The field of [[Photometry (optics)|optical photometry]] uses a different set of distinctions, the main ones being [[luminance]] and [[illuminance]]. [[Photometry (astronomy)|Astronomical photometry]], by contrast, is concerned with measuring the ''[[flux]]'', or intensity of an astronomical object's [[electromagnetic radiation]]. In the field of [[computer graphics]] the concept of luminosity is different altogether, a synonym in fact for the concept of [[lightness]], otherwise known as the value or tone component of a color.
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[[File:Hertzsprung-Russel StarData.png|thumb|230px|right|[[Hertzsprung–Russell diagram]] identifying stellar luminosity as a function of temperature for many stars in our solar neighborhood.]]In [[astronomy]], ''luminosity'' is the amount of [[Radiant energy|electromagnetic energy]] a body radiates per unit of time.<ref name="AUSTRALIA2004">{{cite web
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|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
|accessdate=2 July 2012}}</ref> It is most frequently measured in two forms: ''visual'' (visible light only) and ''bolometric'' (total radiant energy),<ref name="COSMOS1">{{cite web
|title=Luminosity
|publisher=[[Swinburne University of Technology]]
|url=http://astronomy.swin.edu.au/cosmos/L/Luminosity
|accessdate=2 July 2012}}</ref> although luminosities at other wavelengths are increasingly being used as instruments become available to measure them. A [[bolometer]] is the instrument used to measure [[radiant energy]] over a wide band by [[Absorption (electromagnetic radiation)|absorption]] and measurement of heating.  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]]. A star also radiates [[neutrino]]s, which carry off some energy, about 2% in the case of our Sun, producing a [[stellar wind]] and contributing to the star's total luminosity.<ref name="BAHCALL1">
{{cite web
|first=John |last=Bahcall |authorlink=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
|accessdate=2012-07-03
}}</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 infra-red.


A star's luminosity can be determined from two stellar characteristics: size and [[effective temperature]].<ref name="AUSTRALIA2004" /> The former is typically represented in terms of solar [[radius|radii]], <math>R_{\odot}</math>, 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, often calculated using [[parallax]].  Both can be measured with great accuracy in certain cases, with cool supergiants often having large angular diameters, and some cool evolved stars having masers 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.
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An alternate 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
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| last1 = Karttunen
| first1 = Hannu
| title = Fundamental astronomy
| work = Physics and Astronomy Online Library
| publisher = Springer
| year = 2003
| page = 289
| accessdate = 2012-06-02
| 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
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| title=The Real Starry Sky
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| 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 0 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.
 
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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<math>L_{\odot}</math>, a spectral type of A2, and an effective temperature around 8,500&nbsp;K, meaning it has a radius around 203<math>R_{\odot}</math>.  For comparison, the red supergiant Betelgeuse has a luminosity around 100,000<math>L_{\odot}</math>, a spectral type of M2, and a temperature around 3,500&nbsp;K, meaning its radius is about 1,000<math>R_{\odot}</math>.  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<math>L_{\odot}</math>, meaning their radii are just a few tens of<math>R_{\odot}</math>.  An example is [[R136a1]], over 50,000&nbsp;K and shining at over 8,000,000<math>L_{\odot}</math> (mostly in the UV), it is only 35<math>R_{\odot}</math>.
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{{Main|Magnitude (astronomy)}}
 
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![[Apparent magnitude|VMag]]
 
|-bgcolor="#e0e080"
  <li>[http://www.importia.com/index.php/forum/newtopic http://www.importia.com/index.php/forum/newtopic]</li>
|First Magnitude
 
|      < 1.5
  <li>[http://emr4u.net/index.php?option=com_blog&view=blog http://emr4u.net/index.php?option=com_blog&view=blog]</li>
|[[Vega]]
 
|0.03
  </ul>
|-bgcolor="#d2d275"
|Second magnitude
|1.5 to 2.5
|[[Denebola]]
|2.14
|-bgcolor="#c4c46a"
|Third Magnitude
|2.5 to 3.5
|[[Rastaban]]
|2.79
|-bgcolor="#b6b65f"
|Fourth Magnitude
|3.5 to 4.5
|[[Sadalpheretz]]
|3.96
|-bgcolor="#a8a854"
|Fifth Magnitude
|4.5 to 5.5
|[[Pleione (star)|Pleione]]
|5.05
|-bgcolor="#9a9a49"
|Sixth Magnitude
|5.5 to 6.5
|[[54 Piscium]]
|5.88
|-bgcolor="#8c8c3e"
|Seventh Magnitude
|6.5 to 7.5
|[[HD 40307]]
|7.17
|-bgcolor="#7e7e33"
|Eighth Magnitude
|7.5 to 8.5
|[[HD 113766]]
|7.56
|-bgcolor="#707028"
|Ninth Magnitude
|8.5 to 9.5
|[[HD 149382]]
|8.94
|-bgcolor="#62621d"
|Tenth Magnitude
|9.5 to 10.5
|[[HIP 13044]]
|9.98
|}
Luminosity is an intrinsic measurable property of a star independent of distance. The concept of magnitude, on the other hand, incorporates distance. First conceived by the Greek astronomer [[Hipparchus]] in the second century BC, the original concept of magnitude grouped stars into six discrete categories depending on how bright they appeared.  The brightest first magnitude stars were twice as bright as the next brightest stars, which were second magnitude; second was twice as bright as third, third twice as bright as fourth and so on down to the faintest stars, which Hipparchus categorized as sixth magnitude.<ref name="KNOWLE2001"/> The system was but a simple delineation of stellar brightness into six distinct groups and made no allowance for the variations in brightness within a group.  With the invention of the [[telescope]] at the beginning of the seventeenth century, researchers soon realized that there were subtle variations among stars and millions fainter than the sixth magnitude—hence the need for a more sophisticated system to describe a continuous range of values beyond what the naked eye could see.<ref name="KNOWLE2001"/><ref name="ASTRONOTES1">{{cite web
|title=Magnitude System
|publisher=Astronomy Notes
|url=http://www.astronomynotes.com/starprop/s4.htm
|date=2 November 2010
  |accessdate=2 July 2012}}</ref>


In 1856 [[Norman Pogson]], noticing that [[Photometry (astronomy)|photometric]] measurements had established first magnitude stars as being about 100 times brighter than sixth magnitude stars, formalized the Hipparchus system by creating a [[logarithmic scale]], with every interval of one magnitude equating to a variation in brightness of 100<sup>1/5</sup> or roughly 2.512 times. Consequently, a first magnitude star is about 2.5 times brighter than a second magnitude star, 2.5<sup>2</sup> brighter than a third magnitude star, 2.5<sup>3</sup> brighter than a fourth magnitude star, et cetera. Based on this continuous scale, any star with a magnitude between 5.5 and 6.5 is now considered to be sixth magnitude, a star with a magnitude between 4.5 and 5.5 is fifth magnitude and so on.  With this new mathematical rigor, a first magnitude star should then have a magnitude in the range 0.5 to 1.5, thus excluding the [[List of brightest stars|nine brightest stars]] with magnitudes lower than 0.5, as well as the four brightest with negative values. It is customary therefore to extend the definition of a first magnitude star to any star with a magnitude less than 0.5, as can be seen in accompanying table.<ref name="KNOWLE2001">{{cite web
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|title=THE FIRST MAGNITUDE STARS
|author=Nigel Foster
|publisher=Knowle Astronomical Society
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[[File:Heic0612b H.jpg|thumb|200px|left|Artist impression of a transiting planet temporarily diminishing the star's brightness, leading to its discovery.<ref name="ESA1">{{cite web
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|title=COROT discovers its first exoplanet and catches scientists by surprise
 
|publisher=[[European Space Agency]]
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|accessdate=2 July 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 10 [[parsec]]s. 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
 
|title=The Inverse-Square Law
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|publisher=Institute for Astronomy - University of Hawaii
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|accessdate=26 September 2012}}</ref>  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"/>
 
  </ul>


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, allowing astronomers to estimate a star's distance and extinction without  [[parallax]] calculations. Since the [[stellar parallax]] is usually too small to be measured for many distant stars, this is a common method of determining such distances.
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To conceptualize the range of magnitudes in our own [[Milky Way|galaxy]], the smallest star to be identified has about 8% of the Sun’s mass and glows feebly at absolute magnitude +19. Compared to the Sun, which has an absolute of +4.8, this faint star is 14 magnitudes or 400,000 times dimmer than our Sun. Our galaxy's most massive stars begin their lives with masses of roughly 100 times solar, radiating at upwards of absolute magnitude –8, over 160,000 times the solar luminosity. The total range of stellar luminosities, then, occupies a range of 27 magnitudes, or a factor of 60 billion.<ref name="LEDREW1" />
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In measuring star brightnesses, absolute magnitude, apparent magnitude, and distance are interrelated parameters. If you know two, you can determine the third. 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.
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[[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"/>
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:<math>L= \sigma A T^4</math> ,
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where A is the area and σ is the [[Stefan–Boltzmann]] constant, with a value of {{physconst|sigma|after=.}}
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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.
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:<math>F = \frac{L}{A}</math> ,
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where
 
:<math>A</math> is the area of the illuminated surface.
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:<math>F</math> is the [[flux density]] of the illuminated surface.
 
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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:
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:<math>F = \frac{L}{4\pi r^2} \,</math> ,
where <math>r</math> is the distance from the observer to the light source.


It has been shown that the luminosity of a star <math>L</math> (assuming the star is a [[black body]], which is a good approximation) is also related to temperature <math>T</math> and radius <math>R</math> of the star by the equation:<ref name="AUSTRALIA2004"/>
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:<math>L = 4\pi R^2\sigma T^4 \,</math> ,
 
where
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:&sigma; is the [[Stefan–Boltzmann constant]] 5.67{{e|&minus;8}} [[Watt|W]]·m<sup>-2</sup>·K<sup>-4</sup>.
 
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Dividing by the [[Solar luminosity|luminosity of the Sun]] <math>L_{\odot}</math> and cancelling constants, we obtain the relationship:<ref name="AUSTRALIA2004"/>
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:<math>\frac{L}{L_{\odot}} = {\left ( \frac{R}{R_{\odot}} \right )}^2 {\left ( \frac{T}{T_{\odot}} \right )}^4</math> ,
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where <math>R_{\odot}</math> and <math>T_{\odot}</math> are the radius and temperature of the Sun, respectively.
  <li>[http://www.9ggmm.com/forum.php?mod=viewthread&tid=53399 http://www.9ggmm.com/forum.php?mod=viewthread&tid=53399]</li>
 
 
For stars on the [[main sequence]], luminosity is also [[mass–luminosity relation|related to mass]]:
  <li>[http://kinishao.free.fr/read.php?tid=75244 http://kinishao.free.fr/read.php?tid=75244]</li>
:<math>\frac{L}{L_{\odot}} \approx {\left ( \frac{M}{M_{\odot}} \right )}^{3.9}</math> .
 
 
  <li>[http://wikidiplo.com/Main_Page http://wikidiplo.com/Main_Page]</li>
=== Magnitude formulae ===
 
 
  <li>[http://mercaves.com/aviariosylinks/index.php?page=item&id=147171 http://mercaves.com/aviariosylinks/index.php?page=item&id=147171]</li>
==== Apparent ====
 
The magnitude of a star is a logarithmic scale of observed visible brightness. The apparent magnitude is the observed visible brightness from [[Earth]], and the absolute magnitude is the apparent magnitude at a distance of 10 parsecs.
  <li>[http://enseignement-lsf.com/spip.php?article65#forum17548706 http://enseignement-lsf.com/spip.php?article65#forum17548706]</li>
Given a visible luminosity (not total luminosity), one can calculate the apparent magnitude of a star from a given distance (ignoring extinction):
 
 
  </ul>
:<math>m_{\rm star}=m_\odot-2.5\log_{10}\left[\frac{L_{\rm star}}{L_\odot}\left(\frac{d_\odot}{d_{\rm star}}\right)^2\right]</math>
 
where
:<math>m_\text{star}</math> is the apparent magnitude of the star (a pure number)
:<math>m_\odot</math> is the apparent magnitude of the Sun (also a pure number)
:<math>L_\text{star}</math> is the visible luminosity of the star
:<math>L_\odot</math> is the solar visible luminosity
:<math>d_\text{star}</math> is the distance to the star
:<math>d_\odot</math> is the distance to the Sun
 
Or simplified, given <math>m_\odot=-26.73</math>, <math>d_\odot = 1.58{\times}10^{-5}\,\text{lyr}</math>:
: <math>m_\text{star}=-2.72-2.5\,\lg(L_\text{star}/d_\text{star}^2)</math>, where <math>L_\text{star}</math> is measured in <math>L_\odot</math>.
 
==== Bolometric ====
The difference in bolometric magnitude is related to the luminosity ratio according to:
 
:<math>M_\text{bol,star} - M_{\text{bol},\odot} = -2.5 \log_{10}\frac{L_\text{star}}{L_\odot}</math>
 
which makes by inversion:
 
:<math>\frac{L_\text{star}}{L_\odot} = 10^{(M_{\text{bol},\odot} - M_\text{bol,star})/2.5}</math>
 
where
:<math>L_\odot</math> is the Sun's (sol) luminosity (bolometric luminosity)
:<math>L_\text{star}</math> is the star's luminosity (bolometric luminosity)
:<math>M_{\text{bol},\odot}</math> is the bolometric magnitude of the Sun
:<math>M_\text{bol,star}</math> is the bolometric magnitude of the star.
 
=== Computational challenges ===
<!----------This section has been referenced by the Betelgeuse article--------->
Calculating a star's luminosity and magnitude is sometimes a tremendous astrophysical challenge.{{Citation needed|date=December 2013}} Although the formulae are well understood, obtaining accurate data to plug into those formulae is not always easy.{{Citation needed|date=December 2013}}  This is particularly the case for enigmatic stars like [[Betelgeuse]] whose thick circumstellar nebula {{Citation needed|date=December 2013}} makes it difficult to identify the size and shape of the star's [[photosphere]] {{Citation needed|date=December 2013}}, leading to significant error factors in determining its luminosity.{{Citation needed|date=December 2013}}
 
As already discussed, the calculation of stellar brightness requires three variables: angular diameter, distance and temperature. A wide variance in any of these components will lead to significant error factors in the star's luminosity.  In the last century, there has been a noticeable variance in all 3 components, leading to much debate on the star's actual brightness. In 1920 when the photosphere was first measured, the published angular diameter was 0.047 [[arcseconds]], a measurement which resulted in a diameter of 3.84&nbsp;×&nbsp;10<sup>8</sup>&nbsp;km (2.58 [[Astronomical Unit|AU]]) based on the then-current [[stellar parallax|parallax]] value of 0.018[[arcsecond|"]].<ref name="MICHELSON">{{cite journal
| author=Michelson, Albert Abraham; Pease, Francis G.
| title=Measurement of the diameter of alpha Orionis with the interferometer
| journal=Astrophysical Journal
| year=1921
| volume=53
| pages=249–59
| doi = 10.1086/142603
| bibcode = 1921ApJ....53..249M
| quote=The 0.047 arcsecond measurement was for a uniform disk. In the article Michelson notes that [[limb darkening]] would increase the angular diameter by about 17%, hence 0.055 arcseconds}}</ref>  Recently, reported angular diameters have ranged from 42.05 to 56.60 milliarcseconds,<ref name="SMITH2009">{{cite journal
| author=Smith, Nathan; Hinkle, Kenneth H.; Ryde, Nils
| title=Red Supergiants as Potential Type IIn Supernova Progenitors: Spatially Resolved 4.6 μm CO Emission Around VY CMa and Betelgeuse
| journal=The Astronomical Journal
|date=March 2009
| volume=137
| issue=3
| pages=3558–3573
| url=http://arxiv.org/pdf/0811.3037v1.pdf
| arxiv=0811.3037
| accessdate=15 September 2012
| doi=10.1088/0004-6256/137/3/3558
| bibcode=2009AJ....137.3558S
| quote=For a table of angular diameters see the [[Betelgeuse#Notes|Note section]] of the [[Betelgeuse]] article.}}</ref><ref name="PERRIN2004">{{cite journal
| title=Interferometric observations of the supergiant stars α Orionis and α Herculis with FLUOR at IOTA
| author=Perrin, G.; Ridgway, S. T.; Coudé du Foresto, V.; Mennesson, B.; Traub, W. A.; Lacasse, M. G.
| journal=Astronomy and Astrophysics
| volume=418
| issue= 2
| pages=675–85
| year=2004
| arxiv = astro-ph/0402099
| url=http://arxiv.org/pdf/astro-ph/0402099v1.pdf
| doi=10.1051/0004-6361:20040052
| bibcode=2004A&A...418..675P
| accessdate=15 September 2012}}</ref> distances from 152&nbsp;±&nbsp;20[[parsecs|pc]] to 197&nbsp;±&nbsp;45pc (520&nbsp;±&nbsp;73[[light-year|ly]] to 643&nbsp;±&nbsp;146ly),<ref name="VANLEEUWEN2007">{{cite journal
  | title=Hipparcos, the New Reduction
   | last1=van Leeuwen
  | first1=Floor
  | journal=Astronomy and Astrophysics
  | work=[[VizieR]]
  | publisher=[[Centre de Données astronomiques de Strasbourg]]
  |date=November 2007
  | volume=474
  | issue=2
  | pages=653
  | arxiv = 0708.1752
  | accessdate=15 September 2012
  | doi=10.1051/0004-6361:20078357
  | bibcode=2007A&A...474..653V}}</ref><ref name="HARPER2008">{{cite journal
| author=Harper, Graham M.; Brown, Alexander; Guinan, Edward F.
| title=A New VLA-Hipparcos Distance to Betelgeuse and its Implications
| journal=The Astronomical Journal
|date=April 2008
| volume=135
| issue=4
| pages=1430–40
| url=http://iopscience.iop.org/1538-3881/135/4/1430/pdf/aj_135_4_1430.pdf
| format=PDF
| accessdate=15 September 2012
| bibcode=2008AJ....135.1430H
| doi=10.1088/0004-6256/135/4/1430}}</ref> and temperatures from 3,100 to 3,660 [[kelvin]],<ref name="SMITH2009"/><ref name="PERRIN2004"/> variables that have produced wide discrepancies.
 
An example if two distinct debated scenarios are:
<br/>
{|class="wikitable" cellpadding="1" width=90% style="border:darkgrey; text-align:center; background:none;
! width=20% height: 200px; align="left"|Parameter
! width=40% height: 200px; align="left"|Scenario I
! width=40% height: 200px; align="left"|Scenario II
|-
| '''Angular Diameter'''
| Bester 1996: 56.6 ± 1.0 mas <ref name="SMITH2009"/>
| Perrin 2004: 43.33 ± 0.04 mas <ref name="PERRIN2004"/>
|-
| '''Distance'''
| Harper 2008: 197 ± 45pc <ref name="HARPER2008"/>
| van Leeuwen 2007: 152 ± 20pc <ref name="VANLEEUWEN2007"/>
|-
| '''Temperature'''
| Smith 2009: 3,300&nbsp;K <ref name="SMITH2009"/>
| Perrin 2004: 3,641&nbsp;K <ref name="PERRIN2004"/>
|}
 
To determine the star's luminosity, there are 3 computational steps:<br/>
:1) convert the star's [[angular diameter]] in arcseconds into its radius in [[astronomical units]] ('''AU''');
:2) convert the star's radius in '''AU''' into its solar radius <math>R_{\odot}</math>; and finally
:3) convert its solar radius <math>R_{\odot}</math> and temperature (kelvin) into solar luminosity<math>L_{\odot}</math>.
 
==== Arcseconds to AU ====
 
The calculations begin with the formula for a star's angular diameter, as follows:
<br/>
:<math>If: {\delta} = \frac{d_B}{D_B} \qquad Then:d_B = \delta \cdot D_B \quad And: R_B ={\left ( {\frac {\delta \cdot D_B}{2}} \right )}</math>
where
'''<math>{\delta}</math>''' represents Betelgeuse's angular diameter in arcseconds,
'''<math>{D_B}</math>''' the <big>'''D'''</big>istance from Earth in parsecs <br />
'''<math>{d_B}</math>''' Betelgeuse's <big>'''d'''</big>iameter in AU, and
'''<math>{R_B}</math>''' Betelgeuse's <big>'''R'''</big>adius in AU.  Therefore:
 
:<math>Scenario \quad I: \qquad R_B = {\left ( {\frac {{0.05660} \cdot 197.0}{2}} \right )} = 5.582 AU = 5.6 AU</math> (rounded)
:<math>Scenario \ II: \qquad R_B = {\left ( {\frac {{0.04333} \cdot 152.0}{2}} \right )} = 3.308 AU = 3.3 AU</math> (rounded)
 
==== AU to R<sub>☉</sub> ====
 
To convert the above into solar units, the math is straightforward. Since 1 AU = 149,597,871&nbsp;km and the mean diameter of the Sun = 1,392,000&nbsp;km (hence a mean radius of 696,000&nbsp;km), the calculation is as follows:
 
:<math>Scenario \quad I: \qquad d_B = {\left ( {\frac {149,597,871 km}{696,000 km}} \right )} {\left ( 5.6 AU \right )} = 1,204 R_{\odot}</math> (rounded).
:<math>Scenario \ II: \qquad d_B = {\left ( {\frac {149,597,871 km}{696,000 km}} \right )} {\left ( 3.3 AU \right )} = \quad 710 R_{\odot}</math> (rounded).
 
==== R<sub>☉</sub> to L<sub>☉</sub> ====
 
Incorporating the R<sub>☉</sub> results into the luminosity formula outlined earlier where B = Betelgeuse, L = Luminosity, R = Radius and T = Temperature, we can calculate Betelgeuse's luminosity per each scenario, as follows:
<br/>
:<math>Scenario \quad I: \qquad \frac{L_{\rm B}}{L_{\odot}} = {\left ( {\frac{1,204}{1}} \right )}^2 {\left ( {\frac{3,300}{5,778}} \right )}^4 = 154,000 L_{\odot} (rounded)</math>
:<math>Scenario \ II: \qquad \frac{L_{\rm B}}{L_{\odot}} = \quad {\left ( {\frac{730}{1}} \right )}^2 {\left ( {\frac{3,641}{5,778}} \right )}^4 = \ 84,000 L_{\odot} (rounded)</math>
 
These luminosity calculations do not take into consideration error factors relating to angular diameter or distance measurements nor any diminution caused by extinction, which in the case of Betelgeuse has been estimated at around 3.1%. Also, while the calculations are correct and useful, in practice they are often performed in reverse because the distance to most stars{{Citation needed|date=December 2013}}, and hence their size, cannot be determined directly while quantities such as the luminosity and temperature can be estimated from other observable quantities.{{Citation needed|date=December 2013}}
 
==Optical photometry==
{{main|Luminance}}
In [[photometry (optics)|photometry]], ''luminosity'' is sometimes incorrectly used to refer to [[luminance]], which is the density of [[luminous intensity]] in a given direction. The [[SI]] unit for luminance is [[candela]] per [[square metre]].
 
The [[luminosity function]] a.k.a. ''luminous efficiency function'' describes the average visual sensitivity of the human eye to light of different wavelengths. There are two luminosity functions in common use. For everyday light levels, the photopic luminosity function best approximates the response of the human eye. For low light levels, the response of the human eye changes, and the scotopic curve applies.<ref>
{{cite journal
| title = Spectroscopic CCD Surveys for Quasars at Large Redshift. IV.Evolution of the Luminosity Function from Quasars Detected by Their Lyman-Alpha Emission
  | author = Schmidt, Maarten; Schneider, Donald P.; and Gunn, James E.
| date = July 1995
| journal = Astronomical Journal
| volume = 110
| page = 68
| bibcode = 1995AJ....110...68S
|doi = 10.1086/117497 }}</ref>{{failed verification|date=January 2011}}
 
==Computer graphics==
{{main|Luma (video)|HSL color space}}
In [[Adobe Photoshop]]'s imaging operations, ''luminosity'' is the term used incorrectly to refer to the [[luma (video)|luma]] component of a color image signal; that is, a weighted sum of the nonlinear red, green, and blue signals.  It seems to be calculated with the Rec. 601 luma co-efficients (Rec. 601: Luma (Y’) = 0.299 R’ + 0.587 G’ + 0.114 B’).
 
The "L" in [[HSL color space]] is sometimes said incorrectly to stand for luminosity.  "L" in this case is calculated as 1/2 (MAX + MIN), where MAX and MIN refer to the highest and lowest of the R'G'B' components to be converted into HSL color space.
 
==Scattering theory==
 
{{Main|Luminosity (scattering theory)}}
 
In [[scattering theory]] and [[Particle accelerator|accelerator]] physics, '''luminosity''' is the number of particles per unit [[area]] per unit [[time]] times the [[Opacity (optics)|opacity]] of the target. It is an important value to characterize the performance of an accelerator.
 
==References==
{{Reflist|colwidth=25em}}
 
[[Category:Astrophysics]]
[[Category:Physical quantities]]
[[Category:Photometry]]
[[Category:Accelerator physics]]
[[Category:Scattering theory]]

Latest revision as of 13:23, 5 May 2014

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