Luminosity

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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.

Luminosity is generally understood as a measurement of brightness. Each discipline, however, defines the term differently, depending on what is being measured.

In astronomy, luminosity measures the total amount of energy emitted by a star or other astronomical object per unit time. Thus the SI units of this definition of luminosity are joules per second, which are watts. A watt is one unit of power, and just as a light bulb is measured in watts, so too is the Sun, the latter having a total power output of 3.846×1026 W. It is this number which constitutes the basic metric used in astronomy and is known as 1 solar luminosity, the symbol for which is $L_\odot$. Radiant power, however, is not the only way to conceptualize brightness, so other metrics are also used. The most common is apparent magnitude, which is the perceived brightness of an object from an observer on Earth at visible wavelengths. Other metrics are absolute magnitude, which is an object's intrinsic brightness at visible wavelengths, irrespective of distance, while bolometric magnitude is the total power output across all wavelengths.

The field of optical photometry uses a different set of distinctions, the main ones being luminance and illuminance. 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.

Astronomy

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 electromagnetic energy a body radiates per unit of time.[1] It is most frequently measured in two forms: visual (visible light only) and bolometric (total radiant energy),[2] 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 and measurement of heating. When not qualified, the term "luminosity" means bolometric luminosity, which is measured either in the SI units, watts, or in terms of solar luminosities. A star also radiates neutrinos, which carry off some energy, about 2% in case of our Sun, producing a stellar wind and contributing to the star's total luminosity.[3] 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.[1] The former is typically represented in terms of solar radii, $R_{\odot}$, while the latter is represented in kelvins, 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.

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 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 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.[4] 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 Class O stars boasting temperatures in excess of 30,000 K while the less massive, typically older Class M stars exhibit temperatures less than 3,500 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.[5] 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.

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$L_{\odot}$, a spectral type of A2, and an effective temperature around 8,500 K, meaning it has a radius around 203$R_{\odot}$. For comparison, the red supergiant Betelgeuse has a luminosity around 100,000$L_{\odot}$, a spectral type of M2, and a temperature around 3,500 K, meaning its radius is about 1,000$R_{\odot}$. Red supergiants are the largest type of star, but the most luminous are much smaller and hotter, with temperatures up to 50,000 K and more and luminosities of several million$L_{\odot}$, meaning their radii are just a few tens of$R_{\odot}$. An example is R136a1, over 50,000 K and shining at over 8,000,000$L_{\odot}$ (mostly in the UV), it is only 35$R_{\odot}$.

Magnitude

GroupingRangeExampleVMag
First Magnitude< 1.5Vega0.03
Second magnitude1.5 to 2.5Denebola2.14
Third Magnitude2.5 to 3.5Rastaban2.79
Fourth Magnitude3.5 to 4.5Sadalpheretz3.96
Fifth Magnitude4.5 to 5.5Pleione5.05
Sixth Magnitude5.5 to 6.554 Piscium5.88
Seventh Magnitude6.5 to 7.5HD 403077.17
Eighth Magnitude7.5 to 8.5HD 1137667.56
Ninth Magnitude8.5 to 9.5HD 1493828.94
Tenth Magnitude9.5 to 10.5HIP 130449.98

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.[6] 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.[6][7]

In 1856 Norman Pogson, noticing that 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 1001/5 or roughly 2.512 times. Consequently, a first magnitude star is about 2.5 times brighter than a second magnitude star, 2.52 brighter than a third magnitude star, 2.53 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 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.[6]

Artist impression of a transiting planet temporarily diminishing the star's brightness, leading to its discovery.[8]

The Pogson logarithmic scale is used to measure both apparent and absolute magnitudes, the latter corresponding to the apparent brightness in visible light of a star or other celestial body as seen at the interstellar distance of 10 parsecs. By contrast, apparent brightness describes the diminishing intensity of light as a result of distance according to an inverse-square law.[9] In addition to this brightness decrease from increased distance, there is an extra linear decrease of brightness due to extinction from intervening interstellar dust.[7]

By measuring the width of certain absorption lines in the 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.

To conceptualize the range of magnitudes in our own 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.[5]

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.

Luminosity formulae

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 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:[1]

$L= \sigma A T^4$ ,

where A is the area and σ is the Stefan–Boltzmann constant, with a value of 5.67010×10−8 W m−2 K−4.[10]

Imagine a point source of light of luminosity $L$ 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.

$F = \frac{L}{A}$ ,

where

$A$ is the area of the illuminated surface.
$F$ is the flux density of the illuminated surface.

The surface area of a sphere with radius r is $A = 4\pi r^2$, so for stars and other point sources of light:

$F = \frac{L}{4\pi r^2} \,$ ,

where $r$ is the distance from the observer to the light source.

It has been shown that the luminosity of a star $L$ (assuming the star is a black body, which is a good approximation) is also related to temperature $T$ and radius $R$ of the star by the equation:[1]

$L = 4\pi R^2\sigma T^4 \,$ ,

where

σ is the Stefan–Boltzmann constant 5.67×10−8 W·m-2·K-4.

Dividing by the luminosity of the Sun $L_{\odot}$ and cancelling constants, we obtain the relationship:[1]

$\frac{L}{L_{\odot}} = {\left ( \frac{R}{R_{\odot}} \right )}^2 {\left ( \frac{T}{T_{\odot}} \right )}^4$ ,

where $R_{\odot}$ and $T_{\odot}$ are the radius and temperature of the Sun, respectively.

For stars on the main sequence, luminosity is also related to mass:

$\frac{L}{L_{\odot}} \approx {\left ( \frac{M}{M_{\odot}} \right )}^{3.9}$ .

Magnitude formulae

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. Given a visible luminosity (not total luminosity), one can calculate the apparent magnitude of a star from a given distance (ignoring extinction):

$m_{\rm star}=m_{\rm Sun}-2.5\log_{10}\left({ L_{\rm star} \over L_{\odot} } \cdot \left(\frac{ d_{\rm Sun} }{ d_{\rm star} }\right)^2\right)$

where

mstar is the apparent magnitude of the star (a pure number)
mSun is the apparent magnitude of the Sun (also a pure number)
Lstar is the visible luminosity of the star
$L_{\odot}$ is the solar visible luminosity
dstar is the distance to the star
dSun is the distance to the Sun

Or simplified, given mSun = −26.73, distSun = 1.58 × 10−5 lyr:

mstar = − 2.72 − 2.5 · log(Lstar/diststar2), where Lstar is measured in $L_{\odot}$.

Bolometric

The difference in bolometric magnitude is related to the luminosity ratio according to:

$M_{bol_{\rm star}} - M_{bol_{\rm Sun}} = -2.5 \log_{10} {\frac{L_{\rm star}}{L_{\odot}}}$

which makes by inversion:

$\frac{L_{\rm star}}{L_{\odot}} = 10^{\left( Mbol_{\rm Sun} - Mbol_{\rm star} \right) / 2.5}$

where

$L_{\odot}$ is the Sun's (sol) luminosity (bolometric luminosity)
$L_{\rm star}$ is the star's luminosity (bolometric luminosity)
$M_{bol_{\rm Sun}}$ is the bolometric magnitude of the Sun
$M_{bol_{\rm star}}$ is the bolometric magnitude of the star.

Computational challenges

Calculating a star's luminosity and magnitude is sometimes a tremendous astrophysical challenge. Although the formulae are well understood, obtaining accurate data to plug into those formulae is not always easy. This is particularly the case for enigmatic stars like Betelgeuse whose thick circumstellar nebula makes it difficult to identify the size and shape of the star's photosphere, leading to significant error factors in determining its luminosity.

As already discussed, the calculation of stellar brightness requires 3 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 × 108 km (2.58 AU) based on the then-current parallax value of 0.018".[11] Recently, reported angular diameters have ranged from 42.05 to 56.60 milliarcseconds,[12][13] distances from 152 ± 20pc to 197 ± 45pc (520 ± 73ly to 643 ± 146ly),[14][15] and temperatures from 3,100 to 3,660 kelvin,[12][13] variables that have produced wide discrepancies.

To understand these computational challenges, let's explore two distinct scenarios which are currently being debated:

ParameterScenario IScenario II
Angular DiameterBester 1996: 56.6 ± 1.0 mas [12]Perrin 2004: 43.33 ± 0.04 mas [13]
DistanceHarper 2008: 197 ± 45pc [15]van Leeuwen 2007: 152 ± 20pc [14]
TemperatureSmith 2009: 3,300 K [12]Perrin 2004: 3,641 K [13]

To determine the star's luminosity, there are 3 computational steps:

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 $R_{\odot}$; and finally
3) convert its solar radius $R_{\odot}$ and temperature (kelvin) into solar luminosity$L_{\odot}$.

Arcseconds to AU

The calculations begin with the formula for a star's angular diameter, as follows:

$If: {\delta} = \frac{d_B}{D_B} \qquad Then:d_B = \delta \cdot D_B \quad And: R_B ={\left ( {\frac {\delta \cdot D_B}{2}} \right )}$

where ${\delta}$ represents Betelgeuse's angular diameter in arcseconds, ${D_B}$ the Distance from Earth in parsecs
${d_B}$ Betelgeuse's diameter in AU, and ${R_B}$ Betelgeuse's Radius in AU. Therefore:

$Scenario \quad I: \qquad R_B = {\left ( {\frac {{0.05660} \cdot 197.0}{2}} \right )} = 5.582 AU = 5.6 AU$ (rounded)
$Scenario \ II: \qquad R_B = {\left ( {\frac {{0.04333} \cdot 152.0}{2}} \right )} = 3.308 AU = 3.3 AU$ (rounded)

AU to R☉

To convert the above into solar units, the math is straightforward. Since 1 AU = 149,597,871 km and the mean diameter of the Sun = 1,392,000 km (hence a mean radius of 696,000 km), the calculation is as follows:

$Scenario \quad I: \qquad d_B = {\left ( {\frac {149,597,871 km}{696,000 km}} \right )} {\left ( 5.6 AU \right )} = 1,204 R_{\odot}$ (rounded).
$Scenario \ II: \qquad d_B = {\left ( {\frac {149,597,871 km}{696,000 km}} \right )} {\left ( 3.3 AU \right )} = \quad 710 R_{\odot}$ (rounded).

R☉ to L☉

Incorporating the R 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:

$Scenario \quad I: \qquad \frac{L_{\rm B}}{L_{\odot}} = {\left ( {\frac{1,204}{1}} \right )}^2 {\left ( {\frac{3,300}{5,778}} \right )}^4 = 154,000 L_{\odot} (rounded)$
$Scenario \ II: \qquad \frac{L_{\rm B}}{L_{\odot}} = \quad {\left ( {\frac{730}{1}} \right )}^2 {\left ( {\frac{3,641}{5,778}} \right )}^4 = \ 84,000 L_{\odot} (rounded)$

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, and hence their size, cannot be determined directly while quantities such as the luminosity and temperature can be estimated from other observable quantities.

Optical photometry

In 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.[16][not in citation given]

Computer graphics

In Adobe Photoshop's imaging operations, luminosity is the term used incorrectly to refer to the 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

In scattering theory and accelerator physics, luminosity is the number of particles per unit area per unit time times the opacity of the target. It is an important value to characterize the performance of an accelerator.

References

1. "Luminosity of Stars". Australia Telescope National Facility. 12 July 2004. Retrieved 2 July 2012.
2. ^ "Luminosity". Swinburne University of Technology. Retrieved 2 July 2012.
3. ^ Bahcall, John. "Solar Neutrino Viewgraphs". Institute for Advanced Study School of Natural Science. Retrieved 2012-07-03.
4. ^ Karttunen, Hannu (2003). "Fundamental astronomy". Physics and Astronomy Online Library (Springer). p. 289. ISBN 978-3-540-00179-9. |accessdate= requires |url= (help)
5. ^ a b Ledrew, Glenn (February 2001). "The Real Starry Sky". Journal of the Royal Astronomical Society of Canada 95: 32–33. Bibcode:2001JRASC..95...32L. Retrieved 2 July 2012.
6. ^ a b c Nigel Foster. "THE FIRST MAGNITUDE STARS". Knowle Astronomical Society. Retrieved 25 September 2012.
7. ^ a b "Magnitude System". Astronomy Notes. 2 November 2010. Retrieved 2 July 2012.
8. ^ "COROT discovers its first exoplanet and catches scientists by surprise". European Space Agency. 3 May 2007. Retrieved 2 July 2012.
9. ^ Joshua E. Barnes (February 18, 2003). "The Inverse-Square Law". Institute for Astronomy - University of Hawaii. Retrieved 26 September 2012.
10. ^ "The NIST Reference on Constants, Units and Uncertainty - Stefan-Boltzmann constant". National Institute of Standards and Technology. 2 June 2011. Retrieved 2 July 2012.
11. ^ Michelson, Albert Abraham; Pease, Francis G. (1921). "Measurement of the diameter of alpha Orionis with the interferometer". Astrophysical Journal 53: 249–59. Bibcode:1921ApJ....53..249M. doi:10.1086/142603. "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"
12. ^ a b c d Smith, Nathan; Hinkle, Kenneth H.; Ryde, Nils (March 2009). "Red Supergiants as Potential Type IIn Supernova Progenitors: Spatially Resolved 4.6 μm CO Emission Around VY CMa and Betelgeuse". The Astronomical Journal 137 (3): 3558–3573. arXiv:0811.3037. Bibcode:2009AJ....137.3558S. doi:10.1088/0004-6256/137/3/3558. Retrieved 15 September 2012. "For a table of angular diameters see the Note section of the Betelgeuse article."
13. ^ a b c d Perrin, G.; Ridgway, S. T.; Coudé du Foresto, V.; Mennesson, B.; Traub, W. A.; Lacasse, M. G. (2004). "Interferometric observations of the supergiant stars α Orionis and α Herculis with FLUOR at IOTA". Astronomy and Astrophysics 418 (2): 675–85. arXiv:astro-ph/0402099. Bibcode:2004A&A...418..675P. doi:10.1051/0004-6361:20040052. Retrieved 15 September 2012.
14. ^ a b van Leeuwen, Floor (November 2007). "Hipparcos, the New Reduction". Astronomy and Astrophysics (Centre de Données astronomiques de Strasbourg) 474 (2): 653. arXiv:0708.1752. Bibcode:2007A&A...474..653V. doi:10.1051/0004-6361:20078357. More than one of |work= and |journal= specified (help); |accessdate= requires |url= (help)
15. ^ a b Harper, Graham M.; Brown, Alexander; Guinan, Edward F. (April 2008). "A New VLA-Hipparcos Distance to Betelgeuse and its Implications" (PDF). The Astronomical Journal 135 (4): 1430–40. Bibcode:2008AJ....135.1430H. doi:10.1088/0004-6256/135/4/1430. Retrieved 15 September 2012.
16. ^ Schmidt, Maarten; Schneider, Donald P.; and Gunn, James E. (July 1995). "Spectroscopic CCD Surveys for Quasars at Large Redshift. IV.Evolution of the Luminosity Function from Quasars Detected by Their Lyman-Alpha Emission". Astronomical Journal 110: 68. Bibcode:1995AJ....110...68S. doi:10.1086/117497.