Editing Absolute magnitude (section)

==Stars and galaxies==
In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light-years, 308.57 petameters or 308.57 [[Orders of magnitude (numbers)#1012|trillion]] kilometres). A star at 10 parsecs has a [[parallax]] of 0.1″ (100 [[minute of arc|milliarcseconds]]). Galaxies (and other [[nebula|extended objects]]) are much larger than 10 parsecs; their light is radiated over an extended patch of sky, and their overall brightness cannot be directly observed from relatively short distances, but the same convention is used. A galaxy's magnitude is defined by measuring all the light radiated over the entire object, treating that integrated brightness as the brightness of a single point-like or star-like source, and computing the magnitude of that point-like source as it would appear if observed at the standard 10 parsecs distance. Consequently, the absolute magnitude of any object ''equals'' the apparent magnitude it ''would have'' if it were 10 parsecs away.

Some stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to outshine the [[planet]]s and cast shadows if they were at 10 parsecs from the Earth. Examples include [[Rigel]] (−7.8), [[Deneb]] (−8.4), [[Zeta Puppis|Naos]] (−6.2), and [[Betelgeuse]] (−5.8). For comparison, [[Sirius]] has an absolute magnitude of only 1.4, which is still brighter than the [[Sun]], whose absolute visual magnitude is 4.83. The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.<ref name="Strobel"/><ref name="Casagrande"/>
Absolute magnitudes of stars generally range from approximately −10 to +20. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giant [[elliptical galaxy M87]] has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10). Some [[active galactic nuclei]] ([[quasars]] like [[CTA-102]]) can reach absolute magnitudes in excess of −32, making them the most luminous persistent objects in the observable universe, although these objects can vary in brightness over astronomically short timescales. At the extreme end, the optical afterglow of the gamma ray burst [[GRB 080319B]] reached, according to one paper, an absolute [[Photometric system|r magnitude]] brighter than −38 for a few tens of seconds.<ref>{{Cite journal|last1=Bloom|first1=J. S.|last2=Perley|first2=D. A.|last3=Li|first3=W.|last4=Butler|first4=N. R.| last5=Miller|first5=A. A.|last6=Kocevski|first6=D.|last7=Kann|first7=D. A.|last8=Foley|first8=R. J.|last9=Chen|first9=H.-W.| last10=Filippenko|first10=A. V.|last11=Starr|first11=D. L.|title=Observations of the Naked-Eye GRB 080319B: Implications of Nature's Brightest Explosion|date=2009-01-19|journal=The Astrophysical Journal|language=en| volume=691|issue=1| pages=723–737| doi=10.1088/0004-637x/691/1/723|arxiv=0803.3215|bibcode=2009ApJ...691..723B|issn=0004-637X|doi-access=free}}</ref>

=== Apparent magnitude ===
{{Main|Apparent magnitude}}

The Greek astronomer [[Hipparchus]] established a numerical scale to describe the brightness of each star appearing in the sky. The brightest stars in the sky were assigned an apparent magnitude {{math|1=''m'' = 1}}, and the dimmest stars visible to the naked eye are assigned {{math|1=''m'' = 6}}.<ref name="Carroll"/> The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude {{mvar|M}} and apparent magnitude {{mvar|m}} from any distance {{mvar|d}} (in [[parsec]]s, with 1 pc = 3.2616 [[light-year]]s) are related by
<math display="block"> 100^{\frac{m-M}{5}}=\frac{F_{10}}{F} = \left(\frac{d}{10\;\mathrm{pc}}\right)^{2}, </math>
where {{mvar|F}} is the radiant flux measured at distance {{mvar|d}} (in parsecs), {{math|''F''<sub>10</sub>}} the radiant flux measured at distance {{math|10 pc}}. Using the [[common logarithm]], the equation can be written as
<math display="block"> M = m - 5 \log_{10}(d_\text{pc})+5 = m - 5 \left(\log_{10}d_\text{pc}-1\right),</math>
where it is assumed that [[Extinction (astronomy)|extinction from gas and dust]] is negligible. Typical extinction rates within the [[Milky Way]] galaxy are 1 to 2 magnitudes per kiloparsec, when [[Dark nebula|dark clouds]] are taken into account.<ref name="Unsoeld2013"/>

For objects at very large distances (outside the Milky Way) the luminosity distance {{math|''d''<sub>L</sub>}} (distance defined using luminosity measurements) must be used instead of {{mvar|d}}, because the [[Euclidean space|Euclidean]] approximation is invalid for distant objects. Instead, [[general relativity]] must be taken into account. Moreover, the [[cosmological redshift]] complicates the relationship between absolute and apparent magnitude, because the radiation observed was shifted into the red range of the spectrum. To compare the magnitudes of very distant objects with those of local objects, a [[K correction]] might have to be applied to the magnitudes of the distant objects.

The absolute magnitude {{mvar|M}} can also be written in terms of the apparent magnitude {{mvar|m}} and [[stellar parallax]] {{mvar|p}}:
<math display="block"> M = m + 5 \left(\log_{10}p+1\right),</math>
or using apparent magnitude {{mvar|m}} and [[distance modulus]] {{mvar|μ}}:
<math display="block"> M = m - \mu.</math>

==== Examples ====
[[Rigel]] has a visual magnitude {{math|''m''<sub>V</sub>}} of 0.12 and distance of about 860 light-years:
<math display="block">M_\mathrm{V} = 0.12 - 5 \left(\log_{10} \frac{860}{3.2616} - 1 \right) = -7.0.</math>

[[Vega]] has a parallax {{mvar|p}} of 0.129″, and an apparent magnitude {{math|''m''<sub>V</sub>}} of 0.03:
<math display="block">M_\mathrm{V} = 0.03 + 5 \left(\log_{10}{0.129} + 1\right) = +0.6.</math>

The [[Black Eye Galaxy]] has a visual magnitude {{math|''m''<sub>V</sub>}} of 9.36 and a distance modulus {{mvar|μ}} of 31.06:
<math display="block">M_\mathrm{V} = 9.36 - 31.06 = -21.7.</math>

=== Bolometric magnitude ===
{{See also|Apparent bolometric magnitude}}

The absolute [[Bolometer|bolometric]] magnitude ({{math|''M''<sub>bol</sub>}}) takes into account [[electromagnetic radiation]] at all [[wavelengths]]. It includes those unobserved due to instrumental [[passband]], the Earth's atmospheric absorption, and [[Extinction (astronomy)|extinction by interstellar dust]]. It is defined based on the [[luminosity]] of the stars. In the case of stars with few observations, it must be computed assuming an [[effective temperature]].

Classically, the difference in bolometric magnitude is related to the luminosity ratio according to:<ref name="Carroll"/>
<math display="block">M_\mathrm{bol,\star} - M_\mathrm{bol,\odot} = -2.5 \log_{10} \left(\frac{L_\star}{L_\odot}\right)</math>
which makes by inversion:
<math display="block">\frac{L_\star}{L_\odot} = 10^{0.4\left(M_\mathrm{bol,\odot} - M_\mathrm{bol,\star}\right)}</math>
where
*{{math|''L''<sub>⊙</sub>}} is the Sun's luminosity (bolometric luminosity)
*{{math|''L''<sub>★</sub>}} is the star's luminosity (bolometric luminosity)
*{{math|''M''<sub>bol,⊙</sub>}} is the bolometric magnitude of the Sun
*{{math|''M''<sub>bol,★</sub>}} is the bolometric magnitude of the star.

In August 2015, the [[International Astronomical Union]] passed Resolution B2<ref name="IAU_XXIX"/> defining the [[Zero Point (photometry)|zero points]] of the absolute and apparent [[bolometric magnitude]] scales in SI units for power ([[watt]]s) and irradiance (W/m<sup>2</sup>), respectively. Although bolometric magnitudes had been used by astronomers for many decades, there had been systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references, and no international standardization. This led to systematic differences in bolometric corrections scales.<ref name="IAU2015B2"/> Combined with incorrect assumed absolute bolometric magnitudes for the Sun, this could lead to systematic errors in estimated stellar luminosities (and other stellar properties, such as radii or ages, which rely on stellar luminosity to be calculated).

Resolution B2 defines an absolute bolometric magnitude scale where {{math|1=''M''<sub>bol</sub> = 0}} corresponds to luminosity {{math|1=''L''<sub>0</sub> = {{val|3.0128|e=28|u=W}}}}, with the zero point [[luminosity]] {{math|''L''<sub>0</sub>}} set such that the Sun (with nominal luminosity {{val|3.828|e=26|u=W}}) corresponds to absolute [[bolometric magnitude]] {{math|1=''M''<sub>bol,⊙</sub> = 4.74}}. Placing a [[radiation]] source (e.g. star) at the standard distance of 10 [[parsecs]], it follows that the zero point of the apparent bolometric magnitude scale {{math|''m''<sub>bol</sub> {{=}} 0}} corresponds to [[irradiance]] {{math|1=''f''<sub>0</sub> = {{val|2.518021002|e=-8|u=W/m<sup>2</sup>}}}}. Using the IAU 2015 scale, the nominal total [[solar irradiance]] ("[[solar constant]]") measured at 1 [[astronomical unit]] ({{val|1361|u=W/m<sup>2</sup>}}) corresponds to an apparent bolometric magnitude of the [[Sun]] of {{math|1=''m''<sub>bol,⊙</sub> = −26.832}}.<ref name="IAU2015B2" />

Following Resolution B2, the relation between a star's absolute bolometric magnitude and its luminosity is no longer directly tied to the Sun's (variable) luminosity:
<math display="block">M_\mathrm{bol} = -2.5 \log_{10} \frac{L_\star}{L_0} \approx -2.5 \log_{10} L_\star + 71.197425</math>
where
*{{math|''L''<sub>★</sub>}} is the star's luminosity (bolometric luminosity) in [[watt]]s
*{{math|''L''<sub>0</sub>}} is the zero point luminosity {{val|3.0128|e=28|u=W}}
*{{math|''M''<sub>bol</sub>}} is the bolometric magnitude of the star

The new IAU absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, the nominal [[solar luminosity]] corresponds closely to {{math|1=''M''<sub>bol</sub> = 4.74}}, a value that was commonly adopted by astronomers before the 2015 IAU resolution.<ref name="IAU2015B2" />

The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude {{math|''M''<sub>bol</sub>}} as:
<math display="block">L_\star = L_0 10^{-0.4 M_\mathrm{bol}}</math>
using the variables as defined previously.

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