The '''apparent magnitude''' ('''''m''''') of a celestial body is a [[measurement|measure]] of its [[brightness]] as seen by an observer on [[Earth]], normalized to the value it would have in the absence of the [[Earth's atmosphere|atmosphere]]. The brighter the object appears, the lower the numerical value of its '''magnitude'''.

== Explanation ==
The scale upon which magnitude is now measured has its origin in the [[Hellenistic Greece|Hellenistic]] practice of dividing those stars visible to the naked eye into six ''magnitudes''. The brightest stars were said to be of first magnitude (''m'' = 1), while the faintest were of sixth magnitude (''m'' = 6), the limit of [[human]] [[visual perception]] (without the aid of a [[telescope]]). Each grade of magnitude was considered to be twice the brightness of the following grade (a [[logarithmic scale]]). This somewhat crude method of indicating the brightness of stars was popularized by [[Ptolemy]] in his ''[[Almagest]]'', and is generally believed to have originated with [[Hipparchus]]. This original system did not measure the magnitude of the [[Sun]].

In 1856, [[Norman Robert Pogson|Pogson]] formalized the system by defining a typical first magnitude star as a star that is 100 times as bright as a typical sixth magnitude star; thus, a first magnitude star is about 2.512 times as bright as a second magnitude star. The fifth root of 100 is known as ''Pogson's Ratio''<ref>[http://articles.adsabs.harvard.edu//full/seri/MNRAS/0017//0000012.000.html Magnitudes of Thirty-six of the Minor Planets for the first day of each month of the year 1857], [[Norman Robert Pogson|N. Pogson]], [[Monthly Notices of the Royal Astronomical Society|MNRAS]] Vol. 17, pg. 12 (1856)</ref>. Pogson's scale was originally fixed by assigning [[Polaris]] a magnitude of 2. Astronomers later discovered that Polaris is slightly variable, so they first switched to [[Vega]] as the standard reference star, and then switched to using tabulated zero points for the measured fluxes<ref>[http://ukads.nottingham.ac.uk/cgi-bin/nph-bib_query?bibcode=1982lbor.book.....A&amp;db_key=AST Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology - New Series " Gruppe/Group 6 Astronomy and Astrophysics " Volume 2 Schaifers/Voigt: Astronomy and Astrophysics / Astronomie und Astrophysik " Stars and Star Clusters / Sterne und Sternhaufen] [[Lawrence H. Aller|L. H. Aller]] ''et al.'', ISBN # 3-540-10976-5; 0-387-10976-5 (1982)</ref>. The magnitude depends on the wavelength band (see below).

The modern system is no longer limited to 6 magnitudes or only to visible light. Very bright objects have ''negative'' magnitudes. For example, [[Sirius]], the brightest star of the [[celestial sphere]], has an apparent magnitude of &minus;1.47. The modern scale includes the [[Moon]] and the [[Sun]]; the full Moon has an apparent magnitude of &minus;12.6 and the Sun has an apparent magnitude of &minus;26.73. The [[Hubble Space Telescope]] has located stars with magnitudes of 30 at visible wavelengths and the [[Keck telescopes]] have located similarly faint stars in the infrared.

{| class="wikitable" style="float: right; margin-left: 1em;"
|+Scale of Apparent Magnitude
!Apparent<br />Magnitude
!Range&nbsp;<br />of&nbsp;Magnitude<ref>{{cite web
| url = http://www.stargazing.net/David/constel/howmanystars.html
| title = How Many Stars You Can Observe
| publisher = David Haworth
| language = English
| accessdate = 2007-09-15
}}</ref>
|-
|style="text-align: center;"|−5
|style="text-align: center;"|−4.51 to −5.50
|-
|style="text-align: center;"|−4
|style="text-align: center;"|−3.51 to −4.50
|-
|style="text-align: center;"|−3
|style="text-align: center;"|−2.51 to −3.50
|-
|style="text-align: center;"|−2
|style="text-align: center;"|−1.51 to −2.50
|-
|style="text-align: center;"|−1
|style="text-align: center;"|−0.51 to −1.50
|-
|style="text-align: center;"|0
|style="text-align: center;"|−0.50 to 0.49
|-
|style="text-align: center;"|1
|style="text-align: center;"|0.50 to 1.49
|-
|style="text-align: center;"|2
|style="text-align: center;"|1.50 to 2.49
|-
|style="text-align: center;"|3
|style="text-align: center;"|2.50 to 3.49
|-
|style="text-align: center;"|4
|style="text-align: center;"|3.50 to 4.49
|-
|style="text-align: center;"|5
|style="text-align: center;"|4.50 to 5.49
|-
|style="text-align: center;"|6
|style="text-align: center;"|5.50 to 6.49
|-
|style="text-align: center;"|7
|style="text-align: center;"|6.50 to 7.49
|}

{| class="wikitable"
|+'''Apparent magnitudes of known celestial objects'''
|-
! App. Mag.
! Celestial object
|-
| &minus;26.73 || [[Sun]] (449,000 times brighter than full moon)
|-
| &minus;12.6 || Full [[Moon]]
|-
| &minus;4.7
| Maximum brightness of [[Venus (planet)|Venus]] and the [[International Space Station]] (when the ISS is at its [[perigee]] and fully lit by the sun)<ref>{{cite web
| url = http://www.heavens-above.com/satinfo.asp?SatID=25544
| title = ISS Information - Heavens-above.com
| publisher = Heavens-above
| language = English
| accessdate = 2007-12-22}}</ref>
|-
| &minus;3.9
| Faintest objects observable during the day with naked eye
|-
| &minus;3.7
| Minimum brightness of [[Venus]]
|-
| &minus;3.0
| Maximum brightness of [[Mars]]
|-
| &minus;2.8
| Maximum brightness of [[Jupiter]]
|-
| &minus;1.9
| Maximum brightness of [[Mercury (planet)|Mercury]]
|-
| &minus;1.47 || Brightest star (except for the sun) at visible wavelengths: [[Sirius]]
|-
| &minus;0.7 || Second-brightest star: [[Canopus (star)|Canopus]]
|-
| &minus;0.24
| Maximum brightness of [[Saturn]]
|-
| 0
| The zero point by definition: This used to be [[Vega]]<br/> (see [[Apparent magnitude#References|references]] for modern zero point)
|-
| 3
| Faintest stars visible in an urban neighborhood with naked eye
|-
| 4.6
| Maximum brightness of [[Ganymede (moon)|Ganymede]]
|-
| 5.5
| Maximum brightness of [[Uranus]]
|-
| 6.5
| Faintest [[star]]s observable with [[naked eye]] under perfect conditions
|-
| 6.7
| Maximum brightness of [[Ceres (dwarf planet)|Ceres]]
|-
| 7.7
| Maximum brightness of [[Neptune (planet)|Neptune]]
|-
| 9.1
| Maximum brightness of [[10 Hygiea]]
|-
| 9.5
| Faintest objects visible with [[binoculars]]
|-
| 10.2
| Maximum brightness of [[Iapetus (moon)|Iapetus]]
|-
| 12.6 || Brightest [[quasar]]
|-
| 13.65
| Maximum brightness of [[Pluto#Physical characteristics|Pluto]] (1,148 times fainter than naked-eye visibility)
|-
| 18.7
| Maximum brightness of [[Eris (dwarf planet)|Eris]]
|-
| 23
| Maximum brightness of Pluto's smallest moons [[Hydra (moon)|Hydra]] and [[Nix (moon)|Nix]]
|-
| 27
| Faintest objects observable in visible light with 8m ground-based telescopes
|-
| 30
| Faintest objects observable in visible light with [[Hubble Space Telescope]]
|-
|-
| 38
| Faintest objects observable in visible light with planned [[Overwhelmingly Large Telescope|OWL]] (2020)
|-
| colspan=2 | (see also [[List of brightest stars]])
|}
These are only approximate values at visible wavelengths (in reality the values depend on the precise bandpass used) &mdash; see [[Airglow#How to calculate the effects of airglow|airglow]] for more details of telescope sensitivity.

As the amount of light received actually depends on the thickness of the Earth's atmosphere in the line of sight to the object, the apparent magnitudes are normalized to the value it would have in the absence of the atmosphere. The dimmer an object appears, the higher its apparent magnitude. Note that brightness varies with distance; an extremely bright object may appear quite dim, if it is far away. Brightness varies [[inverse-square law|indirectly with the square]] of the distance. (This is an approximation at cosmological distance scales due to the [[general relativity|curvature of spacetime]]). The [[absolute magnitude]], ''M'', of a celestial body (outside of the solar system) is the apparent magnitude it would have if it were 10 [[parsec]]s (~32 [[light years]]) away; that of a planet (or other solar system body) is the apparent magnitude it would have if it were 1 [[astronomical unit]] away from both the [[Sun]] and [[Earth]]. The absolute magnitude of the Sun is 4.83 in the V band (yellow) and 5.48 in the B band (blue).{{Fact|date=October 2007}}

The apparent magnitude in the band x can be defined as (noting that <math>\log_{\sqrt[5]{100}} F = \frac{\log_{10} F }{\log_{10} 100^{1/5}} = 2.5\log_{10} F</math>)

:<math>m_{x}= -2.5 \log_{10} (F_x) + C\!\,</math>

where <math>F_x\!\,</math> is the observed [[flux]] in the band x,
and <math>C\!\,</math> is a constant that depends on the units of the flux and the band. The constant <math>C\!\,</math> is defined in [[Apparent magnitude#References|Aller et al 1982]] for the most commonly used system.

The variation in brightness between two luminous objects can be calculated another way by subtracting the magnitude number of the brighter object from the magnitude number of the fainter object, then using the difference as an exponent for the base number 2.512; that is to say (<math> m_f - m_b = x </math>; and <math> 2.512^x = </math>''variation in brightness'').

===Example 1===

''What is the difference in brightness between the Sun and the full moon?''

<math> m_f - m_b = x \!\ </math>

<math> 2.512^x = </math>''variation in brightness''

''The apparent magnitude of the Sun is -26.73, and the apparent magnitude of the full moon is -12.6. The full moon is the fainter of the two objects, while the Sun is the brighter''.

'''Difference in brightness'''

<math> x = m_f - m_b \!\ </math>

<math> x = -12.6 - -26.73 = 14.13 \!\ </math>

<math> x = 14.13 \!\ </math>

'''Variation in Brightness'''

<math> v_b = 2.512^x \!\ </math>

<math> v_b = 2.512^{14.13} \!\ </math>

<math> v_b = 449,032.16 \!\ </math>

''variation in brightness'' = 449,032.16

''In terms of apparent magnitude, the Sun is more than 449,032 times brighter than the full moon. This is a good reason to avoid looking directly at the Sun, even during the ''non-total'' phases of a solar eclipse. (Viewing the ''completely'' eclipsed Sun is safe, but it only stays completely eclipsed for a very short period of time.)''

===Example 2===

''What is the difference in brightness between Sirius and Polaris?''

<math> m_f - m_b = x \!\ </math>

<math> 2.512^x = \!\ </math> ''variation in brightness''

''The apparent magnitude of Sirius is -1.44, and the apparent magnitude of Polaris is 1.97. Polaris is the fainter of the two stars, while Sirius is the brighter''.

'''Difference in brightness'''

<math> x = m_f - m_b \!\ </math>

<math> x = 1.97 - -1.44 = 3.41 \!\ </math>

<math> x = 3.41 \!\ </math>

'''Variation in brightness'''

<math> v_b = 2.512^x \!\ </math>

<math> v_b = 2.512^{3.41} \!\ </math>

<math> v_b = 23.124 \!\ </math>

''In terms of apparent magnitude, Sirius is 23.124 times brighter than Polaris the North Star''.

The second thing to notice is that the scale is [[logarithm]]ic: the relative brightness of two objects is determined by the difference of their magnitudes. For example, a difference of 3.2 means that one object is about 19 times as bright as the other, because Pogson's ratio raised to the power 3.2 is 19.054607...
A common misconception is that the logarithmic nature of the scale is due to the fact that the human [[eye]] itself has a logarithmic response. In Pogson's time this was thought to be true (see [[Weber-Fechner law]]), but it is now believed that the response is a [[power law]] (see [[Stevens' power law]])<ref>"Misconceptions About Astronomical Magnitudes," [[Eric Schulman|E. Schulman]] and C. V. Cox, ''[[American Journal of Physics]]'', Vol. 65, pg. 1003 (1997).</ref>.

Magnitude is complicated by the fact that light is not [[monochromatic]]. The sensitivity of a light detector varies according to the wavelength of the light, and the way in which it varies depends on the type of light detector. For this reason, it is necessary to specify how the magnitude is measured in order for the value to be meaningful. For this purpose the [[UBV system]] is widely used, in which the magnitude is measured in three different wavelength bands: U (centred at about 350 nm, in the near [[ultraviolet]]), B (about 435 nm, in the blue region) and V (about 555 nm, in the middle of the human visual range in daylight). The V band was chosen for spectral purposes and gives magnitudes closely corresponding to those seen by the light-adapted human eye, and when an apparent magnitude is given without any further qualification, it is usually the V magnitude that is meant, more or less the same as '''visual magnitude'''.

Since cooler stars, such as [[red giant]]s and [[red dwarf]]s, emit little energy in the blue and UV regions of the spectrum their power is often under-represented by the UBV scale. Indeed, some [[stellar classification|L and T class]] stars have an estimated magnitude of well over 100, since they emit extremely little visible light, but are strongest in [[infrared]].

Measures of magnitude need cautious treatment and it is extremely important to measure like with like. On early 20th century and older orthochromatic (blue-sensitive) [[photographic film]], the relative brightnesses of the blue [[supergiant]] [[Rigel]] and the red supergiant [[Betelgeuse]] irregular variable star (at maximum) are reversed compared to what our eyes see since this archaic film is more sensitive to blue light than it is to red light. Magnitudes obtained from this method are known as [[photographic magnitude]]s, and are now considered obsolete.

For objects within our Galaxy with a given [[absolute magnitude]], 5 is added to the apparent magnitude for every tenfold increase in the distance to the object. This relationship does not apply for objects at very great distances (far beyond our galaxy), since a correction for [[General Relativity]] must then be taken into account due to the non-Euclidean nature of space.

==See also==
* [[Absolute magnitude]]
* [[Luminosity#In astronomy|Luminosity in astronomy]]
* [[List of brightest stars]]
* [[List of nearest bright stars]]
* [[List of nearest stars]]
* [[Surface Brightness]]

== References ==
{{reflist}}

[[Category:observational astronomy]]

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