Apparent magnitude explained

Apparent magnitude () is a measure of the brightness of a star, astronomical object or other celestial objects like artificial satellites. Its value depends on its intrinsic luminosity, its distance, and any extinction of the object's light caused by interstellar dust along the line of sight to the observer.

Unless stated otherwise, the word magnitude in astronomy usually refers to a celestial object's apparent magnitude. The magnitude scale likely dates to before the ancient Roman astronomer Claudius Ptolemy, whose star catalog popularized the system by listing stars from 1st magnitude (brightest) to 6th magnitude (dimmest).[1] The modern scale was mathematically defined to closely match this historical system by Norman Pogson in 1856.

The scale is reverse logarithmic: the brighter an object is, the lower its magnitude number. A difference of 1.0 in magnitude corresponds to the brightness ratio of

\sqrt[5]{100}

, or about 2.512. For example, a magnitude 2.0 star is 2.512 times as bright as a magnitude 3.0 star, 6.31 times as magnitude 4.0, and 100 times magnitude 7.0.

The brightest astronomical objects have negative apparent magnitudes: for example, Venus at −4.2 or Sirius at −1.46. The faintest stars visible with the naked eye on the darkest night have apparent magnitudes of about +6.5, though this varies depending on a person's eyesight and with altitude and atmospheric conditions.[2] The apparent magnitudes of known objects range from the Sun at −26.832 to objects in deep Hubble Space Telescope images of magnitude +31.5.[3]

The measurement of apparent magnitude is called photometry. Photometric measurements are made in the ultraviolet, visible, or infrared wavelength bands using standard passband filters belonging to photometric systems such as the UBV system or the Strömgren uvbyβ system. Measurement in the V-band may be referred to as the apparent visual magnitude.

Absolute magnitude is a related quantity which measures the luminosity that a celestial object emits, rather than its apparent brightness when observed, and is expressed on the same reverse logarithmic scale. Absolute magnitude is defined as the apparent magnitude that a star or object would have if it were observed from a distance of 10abbr=offNaNabbr=off. Therefore, it is of greater use in stellar astrophysics since it refers to a property of a star regardless of how close it is to Earth. But in observational astronomy and popular stargazing, references to "magnitude" are understood to mean apparent magnitude.

Amateur astronomers commonly express the darkness of the sky in terms of limiting magnitude, i.e. the apparent magnitude of the faintest star they can see with the naked eye. This can be useful as a way of monitoring the spread of light pollution.

Apparent magnitude is technically a measure of illuminance, which can also be measured in photometric units such as lux.[4]

History

Visible to
typical
human
eye
Apparent
magnitude
Bright-
ness
relative
to Vega
Number of stars
(other than the Sun)
brighter than
apparent magnitude[5]
in the night sky
Yes−1.0 251% 1 (Sirius)
0.0 100% 4(Vega, Canopus, Alpha Centauri,Arcturus)
1.0 40% 15
2.0 16% 48
3.0 6.3% 171
4.0 2.5% 513
5.0 1.0%
6.0 0.4%
6.5 0.25% [6]
No7.0 0.16%
8.0 0.063%
9.0 0.025%
10.0 0.010%

The scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes. The brightest stars in the night sky were said to be of first magnitude (= 1), whereas the faintest were of sixth magnitude (= 6), which is the limit of human visual perception (without the aid of a telescope). Each grade of magnitude was considered twice the brightness of the following grade (a logarithmic scale), although that ratio was subjective as no photodetectors existed. This rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest and is generally believed to have originated with Hipparchus. This cannot be proved or disproved because Hipparchus's original star catalogue is lost. The only preserved text by Hipparchus himself (a commentary to Aratus) clearly documents that he did not have a system to describe brightness with numbers: He always uses terms like "big" or "small", "bright" or "faint" or even descriptions such as "visible at full moon".[7]

In 1856, Norman Robert Pogson formalized the system by defining a first magnitude star as a star that is 100 times as bright as a sixth-magnitude star, thereby establishing the logarithmic scale still in use today. This implies that a star of magnitude is about 2.512 times as bright as a star of magnitude . This figure, the fifth root of 100, became known as Pogson's Ratio.[8] The 1884 Harvard Photometry and 1886 Potsdamer Duchmusterung star catalogs popularized Pogson's ratio, and eventually it became a de facto standard in modern astronomy to describe differences in brightness.[9]

Defining and calibrating what magnitude 0.0 means is difficult, and different types of measurements which detect different kinds of light (possibly by using filters) have different zero points. Pogson's original 1856 paper defined magnitude 6.0 to be the faintest star the unaided eye can see,[10] but the true limit for faintest possible visible star varies depending on the atmosphere and how high a star is in the sky. The Harvard Photometry used an average of 100 stars close to Polaris to define magnitude 5.0.[11] Later, the Johnson UVB photometric system defined multiple types of photometric measurements with different filters, where magnitude 0.0 for each filter is defined to be the average of six stars with the same spectral type as Vega. This was done so the color index of these stars would be 0.[12] Although this system is often called "Vega normalized", Vega is slightly dimmer than the six-star average used to define magnitude 0.0, meaning Vega's magnitude is normalized to 0.03 by definition.

Limiting Magnitudes for Visual Observation at High Magnification[13]
Telescope
aperture
(mm)
Limiting
Magnitude
3511.3
6012.3
10213.3
15214.1
20314.7
30515.4
40615.7
50816.4
With the modern magnitude systems, brightness is described using Pogson's ratio. In practice magnitude numbers rarely go above 30 before stars become too faint to detect. While Vega is close to magnitude 0, there are four brighter stars in the night sky at visible wavelengths (and more at infrared wavelengths) as well as the bright planets Venus, Mars, and Jupiter, and since brighter means smaller magnitude, these must be described by negative magnitudes. For example, Sirius, the brightest star of the celestial sphere, has a magnitude of −1.4 in the visible. Negative magnitudes for other very bright astronomical objects can be found in the table below.

Astronomers have developed other photometric zero point systems as alternatives to Vega normalized systems. The most widely used is the AB magnitude system,[14] in which photometric zero points are based on a hypothetical reference spectrum having constant flux per unit frequency interval, rather than using a stellar spectrum or blackbody curve as the reference. The AB magnitude zero point is defined such that an object's AB and Vega-based magnitudes will be approximately equal in the V filter band. However, the AB magnitude system is defined assuming an idealized detector measuring only one wavelength of light, while real detectors accept energy from a range of wavelengths.

Measurement

See main article: Photometry (astronomy). Precision measurement of magnitude (photometry) requires calibration of the photographic or (usually) electronic detection apparatus. This generally involves contemporaneous observation, under identical conditions, of standard stars whose magnitude using that spectral filter is accurately known. Moreover, as the amount of light actually received by a telescope is reduced due to transmission through the Earth's atmosphere, the airmasses of the target and calibration stars must be taken into account. Typically one would observe a few different stars of known magnitude which are sufficiently similar. Calibrator stars close in the sky to the target are favoured (to avoid large differences in the atmospheric paths). If those stars have somewhat different zenith angles (altitudes) then a correction factor as a function of airmass can be derived and applied to the airmass at the target's position. Such calibration obtains the brightness as would be observed from above the atmosphere, where apparent magnitude is defined.

The apparent magnitude scale in astronomy reflects the received power of stars and not their amplitude. Newcomers should consider using the relative brightness measure in astrophotography to adjust exposure times between stars. Apparent magnitude also integrates over the entire object, regardless of its focus, and this needs to be taken into account when scaling exposure times for objects with significant apparent size, like the Sun, Moon and planets. For example, directly scaling the exposure time from the Moon to the Sun works because they are approximately the same size in the sky. However, scaling the exposure from the Moon to Saturn would result in an overexposure if the image of Saturn takes up a smaller area on your sensor than the Moon did (at the same magnification, or more generally, f/#).

Calculations

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, in the spectral band, would be given bym_= -5 \log_ \left(\frac \right),which is more commonly expressed in terms of common (base-10) logarithms asm_ = -2.5 \log_ \left(\frac \right),where is the observed irradiance using spectral filter, and is the reference flux (zero-point) for that 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

\sqrt[5]{100}2.512

(Pogson's ratio). Inverting the above formula, a magnitude difference implies a brightness factor of \frac = 100^\frac = 10^ \approx 2.512^.

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 (brighter), and the mean magnitude of the full moon is −12.74 (dimmer).

Difference in magnitude: x = m_1 - m_2 = (-12.74) - (-26.832) = 14.09.

Brightness factor: v_b = 10^ = 10^ \approx 432\,513.

The Sun appears to be approximately times as bright as the full Moon.

Magnitude addition

Sometimes one might wish to add brightness. For example, photometry on closely separated double stars 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.[15] 10^ = 10^ + 10^.

Solving for

mf

yields m_f = -2.5\log_ \left(10^ + 10^ \right), where is the resulting magnitude after adding the brightnesses referred to by and .

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 (mbol) 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 2.518×10−8 watts per square metre (W·m−2).[16]

Absolute magnitude

See main article: 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.

The absolute magnitude, of a star or astronomical object is defined as the apparent magnitude it would have as seen from a distance of 10pc. The absolute magnitude of the Sun is 4.83 in the V band (visual), 4.68 in the Gaia satellite's G band (green) and 5.48 in the B band (blue).[17] [18] [19]

In the case of a planet or asteroid, the absolute magnitude rather means the apparent magnitude it would have if it were 1AU 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).[20]

Standard reference values

Standard apparent magnitudes and fluxes for typical bands[21]
Band
(μm)

(FWHM)
Flux at,
Jy10−20 erg/(s·cm2·Hz)
U 0.36 0.15 1810 1.81
B 0.44 0.22 4260 4.26
V 0.55 0.16 3640 3.64
R 0.64 0.23 3080 3.08
I 0.79 0.19 2550 2.55
J 1.26 0.16 1600 1.60
H 1.60 0.23 1080 1.08
K 2.22 0.23 670 0.67
L 3.50
g 0.52 0.14 3730 3.73
r 0.67 0.14 4490 4.49
i 0.79 0.16 4760 4.76
z 0.91 0.13 4810 4.81

The magnitude scale is a reverse logarithmic scale. A common misconception is that the logarithmic nature of the scale is because 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 .[22]

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 it varies depends on the type of light detector. For this reason, it is necessary to specify how the magnitude is measured 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 human eye. When an apparent magnitude is discussed without further qualification, the V magnitude is generally understood.[23]

Because cooler stars, such as red giants and red dwarfs, emit little energy in the blue and UV regions of the spectrum, their power is often under-represented by the UBV scale. Indeed, some L and T class stars have an estimated magnitude of well over 100, because they emit extremely little visible light, but are strongest in infrared.[24]

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 human eyes perceive, because this archaic film is more sensitive to blue light than it is to red light. Magnitudes obtained from this method are known as photographic magnitudes, and are now considered obsolete.[25]

For objects within the Milky Way with a given absolute magnitude, 5 is added to the apparent magnitude for every tenfold increase in the distance to the object. For objects at very great distances (far beyond the Milky Way), this relationship must be adjusted for redshifts and for non-Euclidean distance measures due to general relativity.[26] [27]

For planets and other Solar System bodies, the apparent magnitude is derived from its phase curve and the distances to the Sun and observer.[28]

List of apparent magnitudes

See also: List of brightest stars.

Some of the listed magnitudes are approximate. Telescope sensitivity depends on observing time, optical bandpass, and interfering light from scattering and airglow.

Object! Seen from...!Notes
−67.57gamma-ray burst GRB 080319Bseen from 1 AU awaywould be over (20 quadrillion) times as bright as the Sun when seen from the Earth
−41.39star Cygnus OB2-12seen from 1 AU away
−40.67star M33-013406.63seen from 1 AU away
−40.17star Eta Carinae Aseen from 1 AU away
−40.07 seen from 1 AU away
−39.66 seen from 1 AU away
−39.47star P Cygniseen from 1 AU away
−38.00 seen from 1 AU awaywould be seen as a large, very bright bluish disk of 35° apparent diameter
−30.30 star Sirius A seen from 1 AU away
−29.30 seen from Mercury at perihelion
−27.40 star Sun seen from Venus at perihelion
−26.832star Sun seen from Earthabout 400,000 times as bright as mean full Moon
−25.60 star Sun seen from Mars at aphelion
−25.00 Minimum brightness that causes the typical eye slight pain to look at
−23.00 star Sun seen from Jupiter at aphelion
−21.70 star Sun seen from Saturn at aphelion
−21.00 star Sun seen from Earth on an overcast middaymeasuring about 1000 lux
−20.20 star Sun seen from Uranus at aphelion
−19.30 star Sun seen from Neptune
−19.00 star Sun seen from Earth on a very strongly overcast middaymeasuring about 100 lux
−18.20 star Sun seen from Pluto at aphelion
−17.70 planet Earth seen fully illuminated as earthlight from the Moon[29]
−16.70 star Sun seen from Eris at aphelion
−16.00 star Sun as twilight on Earthmeasuring about 10 lux[30]
−14.20 An illumination level of 1 lux[31] [32]
−12.60 seen from Earth at perihelionmaximum brightness of perigee + perihelion + full Moon (~0.267 lux; mean distance value is −12.74, though values are about 0.18 magnitude brighter when including the opposition effect)
−12.40 Betelgeuse (when supernova)seen from Earth when it goes supernova[33]
−11.20 star Sun seen from Sedna at aphelion
−10.00 Comet Ikeya–Seki (1965) seen from Earthwhich was the brightest Kreutz Sungrazer of modern times[34]
−9.50seen from Earthmaximum brightness
−9 to −10 Phobos (moon)seen from Marsmaximum brightness
−7.50seen from Earththe brightest stellar event in recorded history (7200 light-years away)
−6.80seen from Proxima Centauri b[35]
−6.00The total integrated magnitude of the night sky (incl. airglow) seen from Earthmeasuring about 0.002 lux
−6.00seen from Earth(6500 light-years away)
−5.90seen from Earthwhen the ISS is at its perigee and fully lit by the Sun[36]
−4.92planet Venus seen from Earthmaximum brightness when illuminated as a crescent
−4.14planet Venus seen from Earthmean brightness
−4 Faintest objects observable during the day with naked eye when Sun is high. An astronomical object casts human-visible shadows when its apparent magnitude is equal to or lower than −4 [37]
−3.99seen from Earthmaximum brightness of 4.7 million years ago, the historical brightest star of the last and next five million years.[38]
−3.69 Moon lit by earthlight, reflecting earthshine seen from Earth (maximum)
−2.98planet Venus seen from Earthminimum brightness when it is on the far side of the Sun
−2.94planet Jupiter seen from Earthmaximum brightness
−2.94planet Mars seen from Earthmaximum brightness
−2.5 Faintest objects visible during the day with naked eye when Sun is less than 10° above the horizon
−2.50seen from Earthminimum brightness
−2.50 planet Earth seen from Marsmaximum brightness
−2.48planet Mercury seen from Earthmaximum brightness at superior conjunction (unlike Venus, Mercury is at its brightest when on the far side of the Sun, the reason being their different phase curves)
−2.20planet Jupiter seen from Earthmean brightness
−1.66planet Jupiter seen from Earthminimum brightness
−1.47star system Sirius seen from EarthBrightest star except for the Sun at visible wavelengths
−0.83seen from Earthapparent brightness as a supernova impostor in April 1843
−0.72seen from Earth2nd brightest star in night sky
−0.55planet Saturn seen from Earthmaximum brightness near opposition and perihelion when the rings are angled toward Earth
−0.3seen from EarthExpected apparent magnitude at 2061 passage
−0.27star system Alpha Centauri AB seen from EarthCombined magnitude (3rd brightest star in night sky)
−0.04seen from Earth4th brightest star to the naked eye
−0.01star Alpha Centauri A seen from Earth4th brightest individual star visible telescopically in the night sky
+0.03seen from Earthoriginally chosen as a definition of the zero point
+0.23planet Mercury seen from Earthmean brightness
+0.46star Sun seen from Alpha Centauri
+0.46planet Saturn seen from Earthmean brightness
+0.71planet Mars seen from Earthmean brightness
+0.90Moon seen from Marsmaximum brightness
+1.17planet Saturn seen from Earthminimum brightness
+1.33star Alpha Centauri B seen from Earth
+1.86planet Mars seen from Earthminimum brightness
+1.98seen from Earthmean brightness[39]
+3.03seen from Earthin the Large Magellanic Cloud (160,000 light-years away)
+3 to +4Faintest stars visible in an urban neighborhood with naked eye
+3.44seen from EarthM31
+4Orion Nebulaseen from EarthM42
+4.38seen from Earthmaximum brightness (moon of Jupiter and the largest moon in the Solar System)
+4.50seen from Earthan open cluster that may have been seen by Aristotle[40]
+4.5Sagittarius Dwarf Spheroidal Galaxyseen from Earth
+5.20seen from Earthmaximum brightness
+5.38[41] planet Uranus seen from Earthmaximum brightness (Uranus comes to perihelion in 2050)
+5.68planet Uranus seen from Earthmean brightness
+5.72seen from Earthwhich is used as a test for naked eye seeing under dark skies[42] [43]
+5.8 seen from EarthPeak visual magnitude (the "Clarke Event") seen on Earth on 19 March 2008 from a distance of 7.5 billion light-years.
+6.03planet Uranus seen from Earthminimum brightness
+6.49seen from Earthmaximum brightness
+6.5 Approximate limit of stars observed by a mean naked eye observer under very good conditions. There are about 9,500 stars visible to mag 6.5.
+6.64seen from Earthmaximum brightness
+6.75seen from Earthmaximum brightness
+6.90seen from EarthThis is an extreme naked-eye target that pushes human eyesight and the Bortle scale to the limit[44]
+7.25planet Mercury seen from Earthminimum brightness
+7.67[45] planet Neptune seen from Earthmaximum brightness (Neptune comes to perihelion in 2042)
+7.78planet Neptune seen from Earthmean brightness
+8.00planet Neptune seen from Earthminimum brightness
+8Extreme naked-eye limit, Class 1 on Bortle scale, the darkest skies available on Earth.[46]
+8.10seen from Earthmaximum brightness; largest moon of Saturn; mean opposition magnitude 8.4
+8.29star UY Scutiseen from EarthMaximum brightness; one of largest known stars by radius
+8.94seen from Earthmaximum brightness
+9.50Faintest objects visible using common 7×50 binoculars under typical conditions
+10Apollo 8 CSM in orbit around the Moon seen from Earthcalculated (Liemohn)[47]
+10.20seen from Earthmaximum brightness, brightest when west of Saturn and takes 40 days to switch sides
+11.05star Proxima Centauriseen from Earthclosest star
+11.8moon Phobosseen from EarthMaximum brightness; brighter moon of Mars
+12.23star R136a1seen from EarthMost luminous and massive star known[48]
+12.89moon Deimosseen from EarthMaximum brightness
+12.91seen from Earthbrightest (luminosity distance of 2.4 billion light-years)
+13.42seen from EarthMaximum brightness
+13.65seen from Earthmaximum brightness, 725 times fainter than magnitude 6.5 naked eye skies
+13.9moon Titaniaseen from EarthMaximum brightness; brightest moon of Uranus
+14.1star WR 102seen from EarthHottest known star
+15.4 seen from Earthmaximum brightness
+15.55seen from Earthmaximum brightness (the largest moon of Pluto)
+16.8 seen from EarthCurrent opposition brightness
+17.27seen from EarthCurrent opposition brightness
+18.7 seen from EarthCurrent opposition brightness
+19.5 Faintest objects observable with the Catalina Sky Survey 0.7-meter telescope using a 30-second exposure[49] and also the approximate limiting magnitude of Asteroid Terrestrial-impact Last Alert System (ATLAS)
+20.7 seen from Earth(small ≈8 km satellite of Jupiter)
+22 Faintest objects observable in visible light with a 600 mm (24″) Ritchey-Chrétien telescope with 30 minutes of stacked images (6 subframes at 5 minutes each) using a CCD detector[50]
+22.8Luhman 16seen from EarthClosest brown dwarfs (Luhman 16A=23.25, Luhman 16B=24.07)
+22.91seen from Earthmaximum brightness of Pluto's moon
+23.38seen from Earthmaximum brightness of Pluto's moon
+24 Faintest objects observable with the Pan-STARRS 1.8-meter telescope using a 60-second exposure[51] This is currently the limiting magnitude of automated allsky astronomical surveys.
+25.0 seen from Earth(small ≈4 km satellite of Saturn)
+25.3 Trans-Neptunian object seen from EarthFurthest known observable object in the Solar System about 132abbr=unitNaNabbr=unit from the Sun
+26.2 Trans-Neptunian object seen from Earth200 km sized object about 90abbr=unitNaNabbr=unit from the Sun and about 75 million times fainter than what can be seen with the naked eye.
+27.7 Faintest objects observable with a single 8-meter class ground-based telescope such as the Subaru Telescope in a 10-hour image[52]
+28.2 seen from Earth (2003)in 2003 when it was 28abbr=unitNaNabbr=unit from the Sun, imaged using 3 of 4 synchronised individual scopes in the ESO's Very Large Telescope array using a total exposure time of about 9 hours[53]
+28.4 asteroid seen from Earth orbitobserved magnitude of ≈15-kilometer Kuiper belt object seen by the Hubble Space Telescope (HST) in 2003, dimmest known directly observed asteroid.
+29.4 seen from Earth Discovered by the James Webb Space Telescope. One of the furthest objects discovered.[54]
+31.5 Faintest objects observable in visible light with Hubble Space Telescope via the EXtreme Deep Field with ≈23 days of exposure time collected over 10 years[55]
+34 Faintest objects observable in visible light with James Webb Space Telescope[56]
+35 unnamed asteroid seen from Earth orbitexpected magnitude of dimmest known asteroid, a 950-meter Kuiper belt object discovered (by the HST) passing in front of a star in 2009.[57]
+35 seen from Eartha luminous blue variable star, expected magnitude at visible wavelengths due to interstellar extinction

See also

External links

Notes and References

  1. Book: Toomer . G. J. . Ptolemy's Almagest . 1984 . Springer-Verlag . New York . 0-387-91220-7 . 16 .
  2. Curtis . Heber Doust . 1903 . 1901-03-27 . On the Limits of Unaided Vision . . . 2 . 67–69 . 1903LicOB...2...67C . 10.5479/ADS/bib/1903LicOB.2.67C . 38.
  3. Web site: Magnitudes: Measuring the Brightness of Stars. Templeton. Matthew. 21 October 2011. American Association of Variable Stars (AAVSO). 19 May 2019. https://web.archive.org/web/20190518084414/https://www.aavso.org/magnitude. 18 May 2019. live.
  4. Crumey . A.. Human Contrast Threshold and Astronomical Visibility.. Monthly Notices of the Royal Astronomical Society. October 2006. 442. 3. 2600–2619 . 10.1093/mnras/stu992 . free. 1405.4209. 2014MNRAS.442.2600C .
  5. Web site: Magnitude . https://web.archive.org/web/20080206074842/http://www.nso.edu/PR/answerbook/magnitude.html . 6 February 2008 . National Solar Observatory—Sacramento Peak . 23 August 2006.
  6. [Bright Star Catalogue]
  7. Hoffmann, S., Hipparchs Himmelsglobus, Springer, Wiesbaden/ New York, 2017
  8. Magnitudes of Thirty-six of the Minor Planets for the first day of each month of the year 1857 . Norman Robert Pogson . N. . Pogson . . 17 . 12 . 1856 . 1856MNRAS..17...12P . 10.1093/mnras/17.1.12 . free .
  9. Book: Hearnshaw, John B. . The measurement of starlight: two centuries of astronomical photometry . 1996 . Cambridge Univ. Press . 978-0-521-40393-1 . 1. publ . Cambridge.
  10. Pogson . N. . 1856-11-14 . Magnitudes of Thirty-six of the Minor Planets for the First Day of each Month of the Year 1857 . Monthly Notices of the Royal Astronomical Society . en . 17 . 1 . 12–15 . 10.1093/mnras/17.1.12 . free . 1856MNRAS..17...12P . 0035-8711.
  11. Book: Hearnshaw, J. B. . The measurement of starlight: two centuries of astronomical photometry . 1996 . Cambridge University Press . 978-0-521-40393-1 . Cambridge [England] ; New York, NY, USA.
  12. Johnson . H. L. . Morgan . W. W. . May 1953 . Fundamental stellar photometry for standards of spectral type on the revised system of the Yerkes spectral atlas . The Astrophysical Journal . en . 117 . 313 . 10.1086/145697 . 1953ApJ...117..313J . 0004-637X.
  13. Book: Observing Variable Stars, Novae and Supernovae . Gerald . North . Nick . James . Cambridge University Press . 2014 . 978-1-107-63612-5 . 24 .
  14. Oke. J. B.. Gunn. J. E.. Secondary standard stars for absolute spectrophotometry. The Astrophysical Journal . 15 March 1983 . 266. 713–717. 10.1086/160817. 1983ApJ...266..713O.
  15. Web site: Magnitude Arithmetic. Weekly Topic . Caglow . 30 January 2012 . https://web.archive.org/web/20120201203951/http://www.caglow.com/info/wtopic/mag-arith . 1 February 2012. live.
  16. IAU Inter-Division A-G Working Group on Nominal Units for Stellar & Planetary Astronomy . IAU 2015 Resolution B2 on Recommended Zero Points for the Absolute and Apparent Bolometric Magnitude Scales . Resolutions Adopted at the General Assemblies . 13 August 2015 . 1510.06262 . 2015arXiv151006262M . 19 May 2019 . https://web.archive.org/web/20160128180606/https://www.iau.org/static/resolutions/IAU2015_English.pdf . 28 January 2016 . live .
  17. Web site: Some Useful Astronomical Definitions . Stony Brook Astronomy Program . Aaron . Evans . 12 July 2009 . https://web.archive.org/web/20110720052227/http://www.astro.sunysb.edu/aevans/PHY523/classnotes523/useful-definitions-pp.pdf . 20 July 2011 . live.
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