Editing Apparent magnitude (section)

== Calculations ==
[[File:VISTA Magellanic Cloud Survey view of the Tarantula Nebula.jpg|thumb|upright=1.2|Image of [[30 Doradus]] taken by [[ESO]]'s [[VISTA (telescope)|VISTA]]. This [[nebula]] has a visual magnitude of 8.]]
[[File:Apparent magnitude.svg|thumb|Graph of relative brightness versus magnitude]]

The dimmer an object appears, the higher the numerical value given to its magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Therefore, the magnitude {{mvar|m}}, in the [[Photometric system|spectral band]] {{mvar|x}}, would be given by
<math display="block">m_{x}= -5 \log_{100} \left(\frac {F_x}{F_{x,0}}\right),</math>
which is more commonly expressed in terms of [[Common logarithm|common (base-10) logarithms]] as
<math display="block">m_{x} = -2.5 \log_{10} \left(\frac {F_x}{F_{x,0}}\right),</math>
where {{mvar|F<sub>x</sub>}} is the observed [[irradiance]] using spectral filter {{mvar|x}}, and {{math|''F''<sub>''x'',0</sub>}} is the reference flux (zero-point) for that [[Photometric system#Filters used|photometric filter]]. Since an increase of 5 magnitudes corresponds to a decrease in brightness by a factor of exactly 100, each magnitude increase implies a decrease in brightness by the factor <math>\sqrt[5]{100} \approx 2.512</math> (Pogson's ratio). Inverting the above formula, a magnitude difference {{math|''m''<sub>1</sub> − ''m''<sub>2</sub> {{=}} Δ''m''}} implies a brightness factor of
<math display="block"> \frac{F_2}{F_1} = 100^\frac{\Delta m}{5} = 10^{0.4 \Delta m} \approx 2.512^{\Delta m}.</math>

=== Example: Sun and Moon ===
''What is the ratio in brightness between the [[Sun]] and the full [[Moon]]?''

The apparent magnitude of the Sun is −26.832<ref name="IAU2015B2"/> (brighter), and the mean magnitude of the [[full moon]] is −12.74<ref name="moon-fact" /> (dimmer).

Difference in magnitude:
<math display="block"> x = m_1 - m_2 = (-12.74) - (-26.832) = 14.09. </math>

Brightness factor:
<math display="block"> v_b = 10^{0.4 x} = 10^{0.4 \times 14.09} \approx 432\,513. </math>

The Sun appears to be approximately {{val|400000}} times as bright as the full Moon.

===Magnitude addition===
Sometimes one might wish to add brightness. For example, [[Photometry (astronomy)|photometry]] on closely separated [[double star]]s may only be able to produce a measurement of their combined light output. To find the combined magnitude of that double star knowing only the magnitudes of the individual components, this can be done by adding the brightness (in linear units) corresponding to each magnitude.<ref>{{cite web | title=Magnitude Arithmetic|url=http://www.caglow.com/info/wtopic/mag-arith | work=Weekly Topic | publisher=Caglow | access-date=30 January 2012 | archive-url=https://web.archive.org/web/20120201203951/http://www.caglow.com/info/wtopic/mag-arith | archive-date=1 February 2012|url-status=live}}</ref>
<math display="block"> 10^{-m_f \times 0.4} = 10^{-m_1 \times 0.4} + 10^{-m_2 \times 0.4}. </math>

Solving for <math>m_f</math> yields
<math display="block"> m_f = -2.5\log_{10} \left(10^{-m_1 \times 0.4} + 10^{-m_2 \times 0.4} \right), </math>
where {{mvar|m<sub>f</sub>}} is the resulting magnitude after adding the brightnesses referred to by {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}.

===Apparent bolometric magnitude===
While magnitude generally refers to a measurement in a particular filter band corresponding to some range of wavelengths, the apparent or absolute '''bolometric magnitude''' (m<sub>bol</sub>) is a measure of an object's apparent or absolute brightness integrated over all wavelengths of the electromagnetic spectrum (also known as the object's [[irradiance]] or power, respectively). The zero point of the apparent bolometric magnitude scale is based on the definition that an apparent bolometric magnitude of 0 mag is equivalent to a received irradiance of 2.518×10<sup>−8</sup> [[watt]]s per square metre (W·m<sup>−2</sup>).<ref name="IAU2015B2">{{cite journal |author=IAU Inter-Division A-G Working Group on Nominal Units for Stellar & Planetary Astronomy |title=IAU 2015 Resolution B2 on Recommended Zero Points for the Absolute and Apparent Bolometric Magnitude Scales |url=https://www.iau.org/static/resolutions/IAU2015_English.pdf |journal=Resolutions Adopted at the General Assemblies |date=13 August 2015 |arxiv=1510.06262 |bibcode=2015arXiv151006262M |access-date=19 May 2019 |archive-url=https://web.archive.org/web/20160128180606/https://www.iau.org/static/resolutions/IAU2015_English.pdf |archive-date=28 January 2016 |url-status=live }}</ref>

===Absolute magnitude===
{{Main|Absolute magnitude}}
While apparent magnitude is a measure of the brightness of an object as seen by a particular observer, absolute magnitude is a measure of the ''intrinsic'' brightness of an object.  Flux decreases with distance according to an [[inverse-square law]], so the apparent magnitude of a star depends on both its absolute brightness and its distance (and any extinction). For example, a star at one distance will have the same apparent magnitude as a star four times as bright at twice that distance.  In contrast, the intrinsic brightness of an astronomical object, does not depend on the distance of the observer or any [[Extinction (astronomy)|extinction]].<ref>{{Cite web |title=Lecture 4: Page 3, Properties of the Stars |url=https://homepages.uc.edu/~hansonmm/ASTRO/LECTURENOTES/W03/Lec6/Page3.html |access-date=2024-12-05 |website=homepages.uc.edu}}</ref>

The absolute magnitude {{mvar|M}}, of a star or astronomical object is defined as the apparent magnitude it would have as seen from a distance of {{convert|10|pc|ly|lk=out}}. The absolute magnitude of the Sun is 4.83 in the V band (visual), 4.68 in the [[Gaia (spacecraft)|Gaia satellite's]] G band (green) and 5.48 in the B band (blue).<ref name="Bband">{{cite web | title=Some Useful Astronomical Definitions | publisher=Stony Brook Astronomy Program | first=Aaron | last=Evans | url=http://www.astro.sunysb.edu/aevans/PHY523/classnotes523/useful-definitions-pp.pdf | access-date=12 July 2009 | archive-url=https://web.archive.org/web/20110720052227/http://www.astro.sunysb.edu/aevans/PHY523/classnotes523/useful-definitions-pp.pdf | archive-date=20 July 2011 | url-status=live}}</ref><ref name="Gband">{{cite journal | last1=Čotar | first1=Klemen | last2=Zwitter | first2=Tomaž | last3=Traven | first3=Gregor | last4=Kos | first4=Janez | last5=Asplund | first5=Martin | last6=Bland-Hawthorn | first6=Joss | last7=Buder | first7=Sven | last8=D'Orazi | first8=Valentina | last9=De Silva | first9=Gayandhi M | last10=Lin | first10=Jane | last11=Martell | first11=Sarah L | last12=Sharma | first12=Sanjib | last13=Simpson | first13=Jeffrey D | last14=Zucker | first14=Daniel B | last15=Horner | first15=Jonathan | last16=Lewis | first16=Geraint F | last17=Nordlander | first17=Thomas | last18=Ting | first18=Yuan-Sen | last19=Wittenmyer | first19=Rob A |display-authors=2 | title=The GALAH survey: unresolved triple Sun-like stars discovered by the Gaia mission | journal=Monthly Notices of the Royal Astronomical Society | publisher=Oxford University Press (OUP) | volume=487 | issue=2 | date=21 May 2019 | issn=0035-8711 | doi=10.1093/mnras/stz1397 |arxiv=1904.04841  |doi-access=free | pages=2474–2490}}</ref><ref name="Bessell2005">{{cite journal|last1=Bessell|first1=Michael S.|title=Standard Photometric Systems|journal=Annual Review of Astronomy and Astrophysics|volume=43|issue=1|date=September 2005|pages=293–336|issn=0066-4146|doi=10.1146/annurev.astro.41.082801.100251|bibcode=2005ARA&A..43..293B|url=http://www.mso.anu.edu.au/~bessell/araapaper.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.mso.anu.edu.au/~bessell/araapaper.pdf |archive-date=9 October 2022 |url-status=live}}</ref>

In the case of a planet or asteroid, the absolute magnitude {{mvar|H}} rather means the apparent magnitude it would have if it were {{convert|1|AU|km|lk=in}} from both the observer and the Sun, and fully illuminated at maximum opposition (a configuration that is only theoretically achievable, with the observer situated on the surface of the Sun).<ref name="Luciuk">{{Cite web | url = http://www.asterism.org/tutorials/tut35%20Magnitudes.pdf | author = Luciuk, M. | title = Astronomical Magnitudes | page = 8 | access-date = 11 January 2019}}</ref>