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 in 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.

Contents

Astronomy

Luminosity

In astronomy, luminosity is the amount of electromagnetic energy a body radiates per unit of time.[1] It is measured in two forms: apparent (visible light only) and bolometric (total radiant energy).[2] 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]

A star's luminosity is determined primarily by two stellar characteristics: size and temperature.[1] The former is typically represented in terms of solar radii, R_{\odot}, while the latter is represented in degrees Kelvin. To determine a star's radius, however, two other metrics are needed: the star's angular diameter and its distance from Earth, usually calculated using parallax. A third component needed to derive stellar 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]

Hertzsprung–Russell diagram identifying stellar luminosity as a function of temperature for many stars in our solar neighborhood.

In the current system of stellar classification, stars are grouped according to temperature, with the very young and energetic Class O stars boasting temperatures in excess of 30,000K while the older Class M stars exhibit temperatures less than 3,700K—a vast difference that has a huge impact on the star's luminosity.[5] As a result, the most luminous stars in the galaxy are usually the youngest; as stars evolve, their stellar wind causes them to lose mass, a phenomenon which varies greatly with each stellar class but which nevertheless causes a diminution in the star's luminosity over its lifetime. In the Hertzsprung-Russel diagram, the x-axis represents temperature while the y-axis represents luminosity; 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 right. The reason certain stars like Deneb and Betelgeuse are found off the main sequence is because of their extremely high luminosity resulting from their enormous size. A star like Deneb, for instance, has a radius that is 203\begin{smallmatrix}R_{\odot} \end{smallmatrix}, yielding a mass of 19\begin{smallmatrix}M_{\odot} \end{smallmatrix} and luminosity of 196,000\begin{smallmatrix}L_{\odot} \end{smallmatrix}, which means that this blue-white supergiant radiates one hundred and ninety-six thousand times as much energy as the Sun.[6] By contrast, the much cooler Betelgeuse has a luminosity of approximately 120,000\begin{smallmatrix}L_{\odot} \end{smallmatrix}, a figure which is only possible because it is considerably larger than Deneb; with a radius of 995\begin{smallmatrix}R_{\odot} \end{smallmatrix}, Betelgeuse is about 15 times its size.[7] The most brilliant star ever discovered is a Wolf-Rayet star known as R136a1, spotted in the Large Magellanic Cloud in 2010. At 265\begin{smallmatrix}M_{\odot} \end{smallmatrix} it is one of the most massive and most luminous stars yet identified, boasting a total bolometric luminosity 8,700,000 times that of the Sun.[8]

Magnitude

Artist impression of a transiting planet temporarily diminishing the star's luminosity, leading to its discovery.[9]

Luminosity is an intrinsic measurable property independent of distance, and is appraised as absolute magnitude, corresponding to the apparent luminosity in visible light of a star or other celestial body as seen at the interstellar distance of 10 parsecs. In contrast, apparent brightness is related to the distance by an inverse square law.[citation needed] In addition to this brightness decrease from increased distance, there is an extra linear decrease of brightness due to extinction from intervening interstellar dust. Visible brightness is usually measured by apparent magnitude. Both absolute and apparent magnitudes are on an inverse logarithmic scale, where a 5 magnitude increase counterparts a 100th part decrease in nonlogarithmic luminosity.[10]

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 formulas

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 blackbody gives the value for luminosity for a blackbody:[1]

L= σAT4.

A is the area (for 4πr2 for a star and σ is the Stefan–Boltzmann constant, with a value of 5.67010×10−8 W m−2 K−4.[11]

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

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 formulas

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:

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).

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^{((Mbol_{\rm Sun} - Mbol_{\rm star})/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 formulas are well understood, obtaining accurate data to plug into those formulas 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 the red supergiant's luminosity.

As discussed in the Luminosity section above, 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".[12] Recently, reported angular diameters have ranged from 42.05 to 56.60 arcseconds,[13][14] distances from 152 ± 20pc to 197 ± 45pc (520 ± 73ly to 643 ± 146ly),[15][16] and temperatures from 3,100 to 3,660 Kelvin,[13][14] 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 [13]Perrin 2004: 43.33 ± 0.04 mas [14]
DistanceHarper 2008: 197 ± 45pc [16]van Leeuwen 2007: 152 ± 20pc [15]
TemperatureSmith 2009: 3,300K [13]Perrin 2004: 3,641K [14]

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 luminosityL_{\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 = 3.4 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 ( 3.4 AU \right )} = \quad 730 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)


Note: 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%

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.[17][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, usually expressed in either the cgs units cm−2 s−1 or b−1 s−1. The integrated luminosity is the integral of the luminosity with respect to time. The luminosity is an important value to characterize the performance of an accelerator.

Elementary relations for luminosity

The following relations hold

\mathcal{L} = \rho v \, (if the target is perfectly opaque)
\frac{dN}{dt} = \mathcal{L} \sigma
 N = \sigma \int \mathcal{L}\ dt
\frac{d\sigma}{d\Omega} = \frac{1}{\mathcal{L}} \frac{d^{2}N}{d\Omega \, dt}

where

 \mathcal{L}\ is the (instantaneous) luminosity.
\int \mathcal{L}\ dt is the integrated luminosity.
 N\ is the number of interactions.
 \rho\ is the number density of a particle beam.
\sigma\ is the total cross section.
d\Omega\ is the differential solid angle.
 \frac{d\sigma}{d\Omega} is the differential cross section.

For an intersecting storage ring collider:

\mathcal{L} = f n \frac{N_{1} N_{2}}{A}

where

f\ is the revolution frequency
n\ is the number of bunches in one beam in the storage ring.
N_{i}\ is the number of particles in each bunch
A\ is the cross section of the beam.

References

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  2. ^ "Luminosity". Swinburne University of Technology. http://astronomy.swin.edu.au/cosmos/L/Luminosity. Retrieved 2 July 2012. 
  3. ^ Bahcall, John. "Solar Neutrino Viewgraphs". Institute for Advanced Study School of Natural Science. http://www.sns.ias.edu/~jnb/SNviewgraphs/snviewgraphs.html. Retrieved 2012-07-03. 
  4. ^ Karttunen, Hannu (2003). Fundamental astronomy. Springer. p. 289. ISBN 978-3-540-00179-9. 
  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. http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?2001JRASC..95...32L&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf. Retrieved 2 July 2012. 
  6. ^ Schiller, F.; Przybilla, N. (March 2008). "Quantitative spectroscopy of Deneb". Astronomy & Astrophysics 479 (3): 849-858. arXiv:0712.0040. Bibcode 2008A&A...479..849S. doi:10.1051/0004-6361:20078590. http://arxiv.org/pdf/0712.0040v1.pdf. Retrieved 30 June 2012. "Earlier data had yielded a luminosity of 54,000L with a radius of 108R" 
  7. ^ Mohamed, S.; Mackey, J.; Langer, N. (May 2012). "3D simulations of Betelgeuse's bow shock". Astronomy & Astrophysics Volume 541, id.A1: A1. arXiv:1109.1555v2.pdf. Bibcode 2012A&A...541A...1M. doi:10.1051/0004-6361/201118002. 
  8. ^ Crowther, Paul A.; Schnurr, Olivier; Hirschi, Raphael; Yusof, Norhasliza; et al.. "The R136 star cluster hosts several stars whose individual masses greatly exceed the accepted 150 M stellar mass limit". Monthly Notices of the Royal Astronomical Society 408 (2): 731–751. arXiv:1007.3284. Bibcode 2010MNRAS.408..731C. doi:10.1111/j.1365-2966.2010.17167.x. http://arxiv.org/pdf/1007.3284v1.pdf. Retrieved 2 July 2012. 
  9. ^ "COROT discovers its first exoplanet and catches scientists by surprise". European Space Agency. 3 May 2007. http://www.esa.int/esaCP/SEMCKNU681F_index_0.html#subhead2. Retrieved 2 July 2012. 
  10. ^ "Magnitude System". Astronomy Notes. 2 November 2010. http://www.astronomynotes.com/starprop/s4.htm. Retrieved 2 July 2012. 
  11. ^ "The NIST Reference on Constants, Units and Uncertainty - Stefan-Boltzmann constant". National Institute of Standards and Technology. 2 June 2011. http://physics.nist.gov/cgi-bin/cuu/Value?sigma. Retrieved 2 July 2012. 
  12. ^ 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" 
  13. ^ 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. http://arxiv.org/pdf/0811.3037v1.pdf. Retrieved 15 September 2012. "For a table of angular diameters see the Note section of the Betelgeuse article." 
  14. ^ 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. http://arxiv.org/pdf/astro-ph/0402099v1.pdf. Retrieved 15 September 2012. 
  15. ^ a b van Leeuwen, Floor (November 2007). "Hipparcos, the New Reduction". VizieR (Centre de Données astronomiques de Strasbourg) 474 (2): 653. arXiv:0708.1752. Bibcode 2007A&A...474..653V. doi:10.1051/0004-6361:20078357. 
  16. ^ 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. http://iopscience.iop.org/1538-3881/135/4/1430/pdf/aj_135_4_1430.pdf. Retrieved 15 September 2012. 
  17. ^ 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.