In [[astronomy]], '''absolute magnitude''' is the [[apparent magnitude]], capital ''M'', an object would have if it were at a standard [[luminosity distance]] away from us, in the absence of [[Extinction (astronomy)|interstellar extinction]]. It allows the overall brightnesses of objects to be compared without regard to distance.

The absolute magnitude uses the same convention as the visual magnitude, with a ~2.512<!--- exact = 100**(1/5) ---> difference in [[brightness]] between steps in magnitude (because 2.512⁵ ≈ 100). The [[Milky Way]], for example, has an absolute magnitude of about −20.5. So a [[quasar]] at an absolute magnitude of −25.5 is 100 times brighter than our [[galaxy]]. If this particular quasar and our galaxy could be seen side by side at the same distance, the quasar would be 5 magnitudes (or 100 times) brighter than our galaxy.

== Absolute magnitude for stars and galaxies ('''''M''''') ==
In stellar and galactic astronomy, the standard [[distance]] is 10 [[parsec]]s (about 32.616 [[light year]]s, or 3 × 10¹⁴ [[kilometre]]s). A star at ten parsecs has a [[parallax]] of 0.1" (100 milli arc seconds).

In defining absolute magnitude it is necessary to specify the type of [[electromagnetic radiation]] being [[measurement|measured]]. When referring to total [[energy]] output, the proper term is '''[[bolometer|bolometric]] magnitude'''. The bolometric magnitude can be computed from the visual magnitude plus a bolometric correction, <math>M_{bol}=M_V+BC</math>. This correction is needed because very hot stars radiate mostly ultraviolet radiation, while very cool stars radiate mostly infrared radiation (see [[Planck's law]]). The dimmer an object (at a distance of 10 parsecs) would appear, the higher its absolute magnitude. The lower an object's absolute magnitude, the higher its [[luminosity]]. A [[mathematics|mathematical]] [[equation]] [[Relation (mathematics)|relates]] apparent magnitude with absolute magnitude, via parallax.

Many stars visible to the naked eye have an absolute magnitude which is capable of casting [[shadow]]s from a distance of 10 parsecs; [[Rigel]] (−7.0), [[Deneb]] (−7.2), [[Zeta Puppis|Naos]] (−6.0), and [[Betelgeuse]] (−5.6).

For comparison, [[Sirius]] has an absolute magnitude of 1.4 and the [[Sun]] has an absolute visual magnitude of 4.83 (it actually serves as a reference point). The Sun's absolute bolometric magnitude is 4.75.

Absolute magnitudes for stars generally [[Range (statistics)|range]] from −10 to +17. The absolute magnitude for galaxies can be much lower (brighter). For example, the giant [[elliptical galaxy]] [[Elliptical Galaxy M87|M87]] has an absolute magnitude of −22.

=== Computation ===
One can compute the absolute magnitude <math>M\!\,</math> of an object given its [[apparent magnitude]] <math>m\!\,</math> and [[luminosity distance]] <math>D_L\!\,</math>:
:<math> M = m - 5 ((\log_{10}{D_L}) - 1)\!\,</math>

where <math>D_L\!\,</math> is the star's luminosity distance in [[parsecs]], which are (≈ 3.2616 [[light-year]]s)

For nearby astronomical objects (such as stars in our galaxy) the [[luminosity distance]] ''D_L'' is almost identical to the real [[distance]] to the object, because spacetime within our galaxy is almost Euclidean. For much more distant objects the Euclidean approximation is not valid, and [[General Relativity]] must be taken into account when calculating the luminosity distance of an object.

In the Euclidean approximation for nearby objects, the absolute magnitude <math>M\!\,</math> of a star can be calculated from its [[apparent magnitude]] and [[parallax]]:
:<math> M = m + 5 (\log_{10}{\pi} + 1)\!\,</math>

where π is the star's parallax in arcseconds.

You can also compute the absolute magnitude <math>M\!\,</math> of an object given its apparent magnitude <math>m\!\,</math> and [[distance modulus]] <math>\mu\!\,</math>:
:<math> M = m - {\mu}\!\,</math>

==== Example ====
: [[Rigel]] has a visual magnitude of m_V=0.18 and distance about 773 light-years.
:: M_{V_Rigel} = 0.18 + 5*(1 + log₁₀(3.2616/773)) = −6.7
: [[Vega]] has a parallax of 0.133", and an apparent magnitude of +0.03
:: M_{V_Vega} = 0.03 + 5*(1 + log₁₀(0.133)) = +0.65
: [[Alpha Centauri]] has a [[parallax]] of 10.750" and an apparent magnitude of −0.01
:: M_{V_{α Cen}} = −0.01 + 5*(1 + log₁₀(10.750)) = +4.37
: [[Black Eye Galaxy]] has a visual magnitude of m_V=+9.36 and a distance modulus of 31.06.
:: M_{V_{Black Eye Galaxy}} = 9.36 − 31.06 = −21.7

=== Apparent magnitude ===
Given the absolute magnitude <math>M\!\,</math>, for objects within our galaxy you can also calculate the apparent magnitude <math>m\!\,</math> from any distance <math>d\!\,</math>:

:<math> m = M + 5 (\log_{10}{d} - 1)\!\,</math>

For objects at very great distances (outside our galaxy) the [[luminosity distance]] ''D_L'' must be used instead of ''d''.

Given the absolute magnitude <math>M\!\,</math>, you can also compute apparent magnitude <math>m\!\,</math> from its [[parallax]] <math>p\!\,</math>:

:<math> m = M - 5 (\log_{10}p + 1)\!\,</math>

Also calculating absolute magnitude <math>M\!\,</math> from [[distance modulus]] <math>\mu\!\,</math>:

:<math> m = M + {\mu}\!\,</math>

== Absolute magnitude for planets ('''''H''''') ==
For [[planets]], [[comet]]s and [[asteroid]]s a different definition of absolute magnitude is used which is more meaningful for nonstellar objects.

In this case, the absolute magnitude is defined as the apparent magnitude that the object would have if it were one [[astronomical unit]] (au) from both the [[Sun]] and the [[Earth]] and at a [[phase angle (astronomy)|phase angle]] of zero degrees. This is a physical impossibility, as it requires the observing telescope to be at the centre of the Sun, but it is convenient for purposes of calculation.

To convert a stellar or galactic absolute magnitude into a planetary one, subtract 31.57. This factor also corresponds to the difference between the Sun's [[visual magnitude]] of −26.8 and its (stellar) absolute magnitude of +4.8. Thus, the Milky Way (galactic absolute magnitude −20.5) would have a planetary absolute magnitude of −52.

=== Apparent magnitude ===
The absolute magnitude can be used to help calculate the apparent magnitude of a body under different conditions.

:<math>m = H + 2.5 \log_{10}{(\frac{d_{BS}^2 d_{BO}^2}{p(\chi) d_0^4})}\!\,</math>

where

<math>d_0\!\,</math> is 1 au, <math>\chi\!\,</math> is the [[phase angle (astronomy)|phase angle]], the angle between the Sun-Body and Body-Observer lines; by the [[law of cosines]], we have:
:<math>\cos{\chi} = \frac{ d_{BO}^2 + d_{BS}^2 - d_{OS}^2 } {2 d_{BO} d_{BS}}\!\,</math>

<math>p(\chi)\!\,</math> is the [[phase integral]] (integration of reflected light; a number in the 0 to 1 range)
:Example: (An [[Lambertian diffuse lighting model|ideal diffuse reflecting]] [[sphere]]) - A reasonable first approximation for planetary bodies

<math>p(\chi) = \frac{2}{3} ( (1 - \frac{\chi}{\pi}) \cos{\chi} + (1/\pi) \sin{\chi} )\!\,</math>
: A full-phase diffuse sphere reflects ⅔ as much light as a diffuse disc of the same diameter
: Distances:
:: <math>d_{BO}\!\,</math> is the distance between the observer and the body
:: <math>d_{BS}\!\,</math> is the distance between the Sun and the body
:: <math>d_{OS}\!\,</math> is the distance between the observer and the Sun

==== Example ====
Moon
: <math>H_{Moon}\!\,</math> = +0.25
: <math>d_{OS}\!\,</math> = <math>d_{BS}\!\,</math> = 1 au
: <math>d_{BO}\!\,</math> = 384.5 Mm = 2.57 mau

: How bright is the Moon from Earth?
:: Full Moon: <math>\chi\!\,</math> = 0, (<math>p(\chi)\!\,</math> ≈ 2/3)
::: <math>m_{Moon} = 0.25 + 2.5 \log_{10}{(\frac{3}{2} 0.00257^2)} = -12.26\!\,</math>
::: (Actual -12.7) A full Moon reflects 30% more light at full phase than a perfect diffuse reflector predicts.
:: Quarter Moon: <math>\chi\!\,</math> = 90°, <math>p(\chi) \approx \frac{2}{3\pi}\!\,</math> (if diffuse reflector)
::: <math>m_{Moon} = 0.25 + 2.5 \log_{10}{(\frac{3\pi}{2} 0.00257^2)} = -11.02\!\,</math>
::: (Actual approximately -11.0) The diffuse reflector formula does better for smaller phases.

== See also ==
* [[H-R diagram|Hertzsprung-Russell diagram]] - Relates absolute magnitude or [[luminosity]] versus spectral color or surface [[temperature]].
* [[Jansky]] radio astronomer's preferred unit - linear in power/unit area
* [[Surface Brightness]]- The ''magnitude'' for extended objects

==External links==
* [http://astro.pas.rochester.edu/~aquillen/ast142/costanti.html Reference zero-magnitude fluxes]
* [http://www.astronomynotes.com/starprop/s4.htm The Magnitude system]
* [http://csep10.phys.utk.edu/astr162/lect/stars/magnitudes.html About stellar magnitudes]
* [http://simbad.u-strasbg.fr/sim-fid.pl Obtain the magnitude of any star] - [[SIMBAD]]
* [http://www.cfa.harvard.edu/iau/lists/Sizes.html Converting magnitude of minor planets to diameter]
* [http://www.dnapublications.com/absmag/ DNA Publications website]

[[Category:Observational astronomy]]

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