Inastronomy,absolute magnitude (M) is a measure of theluminosity of acelestial object on an inverselogarithmicastronomical magnitude scale; the more luminous (intrinsically bright) an object, the lower its magnitude number. An object's absolute magnitude is defined to be equal to theapparent magnitude that the object would have if it were viewed from a distance of exactly 10parsecs (32.6light-years), withoutextinction (or dimming) of its light due to absorption byinterstellar matter andcosmic dust. By hypothetically placing all objects at a standard reference distance from the observer, their luminosities can be directly compared among each other on a magnitude scale. ForSolar System bodies that shine in reflected light, a different definition ofabsolute magnitude (H) is used, based on a standard reference distance of oneastronomical unit.
Absolute magnitudes of stars generally range from approximately −10 to +20. The absolute magnitudes of galaxies can be much lower (brighter).
The more luminous an object, the smaller the numerical value of its absolute magnitude. A difference of 5 magnitudes between the absolute magnitudes of two objects corresponds to a ratio of 100 in their luminosities, and a difference of n magnitudes in absolute magnitude corresponds to a luminosity ratio of 100n/5. For example, a star of absolute magnitude MV = 3.0 would be 100 times as luminous as a star of absolute magnitude MV = 8.0 as measured in the V filter band. TheSun has absolute magnitude MV = +4.83.[1] Highly luminous objects can have negative absolute magnitudes: for example, theMilky Way galaxy has an absoluteB magnitude of about −20.8.[2]
As with all astronomicalmagnitudes, the absolute magnitude can be specified for differentwavelength ranges corresponding to specifiedfilter bands orpassbands; for stars a commonly quoted absolute magnitude is theabsolute visual magnitude, which uses the visual (V) band of the spectrum (in theUBV photometric system). Absolute magnitudes are denoted by a capital M, with a subscript representing the filter band used for measurement, such as MV for absolute magnitude in the V band.
An object's absolutebolometric magnitude (Mbol) represents its totalluminosity over allwavelengths, rather than in a single filter band, as expressed on a logarithmic magnitude scale. To convert from an absolute magnitude in a specific filter band to absolute bolometric magnitude, abolometric correction (BC) is applied.[3]
In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light-years, 308.57 petameters or 308.57trillion kilometres). A star at 10 parsecs has aparallax of 0.1″ (100milliarcseconds). Galaxies (and otherextended 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 objectequals the apparent magnitude itwould 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 theplanets and cast shadows if they were at 10 parsecs from the Earth. Examples includeRigel (−7.8),Deneb (−8.4),Naos (−6.2), andBetelgeuse (−5.8). For comparison,Sirius has an absolute magnitude of only 1.4, which is still brighter than theSun, whose absolute visual magnitude is 4.83. The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.[4][5]Absolute magnitudes of stars generally range from approximately −10 to +20. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giantelliptical galaxy M87 has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10). Someactive galactic nuclei (quasars likeCTA-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 burstGRB 080319B reached, according to one paper, an absoluter magnitude brighter than −38 for a few tens of seconds.[6]
The Greek astronomerHipparchus 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 magnitudem = 1, and the dimmest stars visible to the naked eye are assignedm = 6.[7] The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitudeM and apparent magnitudem from any distanced (inparsecs, with 1 pc = 3.2616light-years) are related bywhereF is the radiant flux measured at distanced (in parsecs),F10 the radiant flux measured at distance10 pc. Using thecommon logarithm, the equation can be written aswhere it is assumed thatextinction from gas and dust is negligible. Typical extinction rates within theMilky Way galaxy are 1 to 2 magnitudes per kiloparsec, whendark clouds are taken into account.[8]
For objects at very large distances (outside the Milky Way) the luminosity distancedL (distance defined using luminosity measurements) must be used instead ofd, because theEuclidean approximation is invalid for distant objects. Instead,general relativity must be taken into account. Moreover, thecosmological 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, aK correction might have to be applied to the magnitudes of the distant objects.
The absolute magnitudeM can also be written in terms of the apparent magnitudem andstellar parallaxp:or using apparent magnitudem anddistance modulusμ:
Classically, the difference in bolometric magnitude is related to the luminosity ratio according to:[7]which makes by inversion:where
L⊙ is the Sun's luminosity (bolometric luminosity)
L★ is the star's luminosity (bolometric luminosity)
Mbol,⊙ is the bolometric magnitude of the Sun
Mbol,★ is the bolometric magnitude of the star.
In August 2015, theInternational Astronomical Union passed Resolution B2[9] defining thezero points of the absolute and apparentbolometric magnitude scales in SI units for power (watts) and irradiance (W/m2), 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.[10] 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 whereMbol = 0 corresponds to luminosityL0 =3.0128×1028 W, with the zero pointluminosityL0 set such that the Sun (with nominal luminosity3.828×1026 W) corresponds to absolutebolometric magnitudeMbol,⊙ = 4.74. Placing aradiation source (e.g. star) at the standard distance of 10parsecs, it follows that the zero point of the apparent bolometric magnitude scalembol = 0 corresponds toirradiancef0 =2.518021002×10−8 W/m2. Using the IAU 2015 scale, the nominal totalsolar irradiance ("solar constant") measured at 1astronomical unit (1361 W/m2) corresponds to an apparent bolometric magnitude of theSun ofmbol,⊙ = −26.832.[10]
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:where
L★ is the star's luminosity (bolometric luminosity) inwatts
L0 is the zero point luminosity3.0128×1028 W
Mbol 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 nominalsolar luminosity corresponds closely toMbol = 4.74, a value that was commonly adopted by astronomers before the 2015 IAU resolution.[10]
The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitudeMbol as:using the variables as defined previously.
Abs Mag (H) and Diameter for asteroids (albedo=0.14)[11]
H
Diameter
10
36 km
12.7
10 km
15
3.6 km
17.7
1 km
19.2
510 m
20
360 m
22
140 m
22.7
100 m
24.2
51 m
25
36 m
26.6
17 m
27.7
10 m
30
3.6 m
32.7
1 m
Forplanets andasteroids, a definition of absolute magnitude that is more meaningful for non-stellar objects is used. The absolute magnitude, commonly called, is defined as theapparent magnitude that the object would have if it were oneastronomical unit (AU) from both theSun and the observer, and in conditions of ideal solar opposition (an arrangement that is impossible in practice).[12] Because Solar System bodies are illuminated by the Sun, their brightness varies as a function of illumination conditions, described by thephase angle. This relationship is referred to as thephase curve. The absolute magnitude is the brightness at phase angle zero, an arrangement known asopposition, from a distance of one AU.
The phase angle can be calculated from the distances body-sun, observer-sun and observer-body, using thelaw of cosines.
The absolute magnitude can be used to calculate the apparent magnitude of a body. For an objectreflecting sunlight, and are connected by the relationwhere is thephase angle, the angle between the body-Sun and body–observer lines. is thephase integral (theintegration of reflected light; a number in the 0 to 1 range).[13]
The value of depends on the properties of the reflecting surface, in particular on itsroughness. In practice, different approximations are used based on the known or assumed properties of the surface. The surfaces of terrestrial planets are generally more difficult to model than those of gaseous planets, the latter of which have smoother visible surfaces.[13]
Diffuse reflection on sphere and flat diskBrightness with phase for diffuse reflection models. The sphere is 2/3 as bright at zero phase, while the disk can't be seen beyond 90 degrees.
Planetary bodies can be approximated reasonably well asideal diffuse reflectingspheres. Let be the phase angle indegrees, then[14]A full-phase diffuse sphere reflects two-thirds as much light as a diffuse flat disk of the same diameter. A quarter phase () has as much light as full phase ().
By contrast, adiffuse disk reflector model is simply, which isn't realistic, but it does represent theopposition surge for rough surfaces that reflect more uniform light back at low phase angles.
The definition of thegeometric albedo, a measure for the reflectivity of planetary surfaces, is based on the diffuse disk reflector model. The absolute magnitude, diameter (inkilometers) and geometric albedo of a body are related by[15][16][17] or equivalently,
Example: TheMoon's absolute magnitude can be calculated from its diameter andgeometric albedo:[18]We have,Atquarter phase, (according to the diffuse reflector model), this yields an apparent magnitude of The actual value is somewhat lower than that, This is not a good approximation, because the phase curve of the Moon is too complicated for the diffuse reflector model.[19] A more accurate formula is given in the following section.
Because Solar System bodies are never perfect diffuse reflectors, astronomers use different models to predict apparent magnitudes based on known or assumed properties of the body.[13] For planets, approximations for the correction term in the formula form have been derived empirically, to matchobservations at different phase angles. The approximations recommended by theAstronomical Almanac[20] are (with in degrees):
The different halves of the Moon, as seen from Earth
Moon at first quarter
Moon at last quarter
Here is the effective inclination ofSaturn's rings (their tilt relative to the observer), which as seen from Earth varies between 0° and 27° over the course of one Saturn orbit, and is a small correction term depending on Uranus' sub-Earth and sub-solar latitudes. is theCommon Era year. Neptune's absolute magnitude is changing slowly due to seasonal effects as the planet moves along its 165-year orbit around the Sun, and the approximation above is only valid after the year 2000. For some circumstances, like for Venus, no observations are available, and the phase curve is unknown in those cases. The formula for the Moon is only applicable to thenear side of the Moon, the portion that is visible from the Earth.
Example 1: On 1 January 2019,Venus was from the Sun, and from Earth, at a phase angle of (near quarter phase). Under full-phase conditions, Venus would have been visible at Accounting for the high phase angle, the correction term above yields an actual apparent magnitude of This is close to the value of predicted by the Jet Propulsion Laboratory.[23]
Example 2: Atfirst quarter phase, the approximation for the Moon gives With that, the apparent magnitude of the Moon is close to the expected value of about. Atlast quarter, the Moon is about 0.06 mag fainter than at first quarter, because that part of its surface has a lower albedo.
Earth'salbedo varies by a factor of 6, from 0.12 in the cloud-free case to 0.76 in the case ofaltostratus cloud. The absolute magnitude in the table corresponds to an albedo of 0.434. Due to the variability of theweather, Earth's apparent magnitude cannot be predicted as accurately as that of most other planets.[20]
Asteroid1 Ceres, imaged by theDawn spacecraft at phase angles of 0°, 7° and 33°. The strong difference in brightness between the three is real. The left image at 0° phase angle shows the brightness surge due to theopposition effect.Phase integrals for various values of GRelationship between the slope parameter and the opposition surge. Larger values of correspond to a less pronounced opposition effect. For most asteroids, a value of is assumed, corresponding to an opposition surge of.
If an object has an atmosphere, it reflects light more or less isotropically in all directions, and its brightness can be modelled as a diffuse reflector. Bodies with no atmosphere, like asteroids or moons, tend to reflect light more strongly to the direction of the incident light, and their brightness increases rapidly as the phase angle approaches. This rapid brightening near opposition is called theopposition effect. Its strength depends on the physical properties of the body's surface, and hence it differs from asteroid to asteroid.[13]
This relation is valid for phase angles, and works best when.[26]
The slope parameter relates to the surge in brightness, typically0.3 mag, when the object is near opposition. It is known accurately only for a small number of asteroids, hence for most asteroids a value of is assumed.[26] In rare cases, can be negative.[25][27] An example is101955 Bennu, with.[28]
In 2012, the-system was officially replaced by an improved system with three parameters, and, which produces more satisfactory results if the opposition effect is very small or restricted to very small phase angles. However, as of 2022, this-system has not been adopted by either the Minor Planet Center norJet Propulsion Laboratory.[13][29]
The apparent magnitude of asteroidsvaries as they rotate, on time scales of seconds to weeks depending on theirrotation period, by up to or more.[30] In addition, their absolute magnitude can vary with the viewing direction, depending on theiraxial tilt. In many cases, neither the rotation period nor the axial tilt are known, limiting the predictability. The models presented here do not capture those effects.[26][13]
The brightness ofcomets is given separately astotal magnitude (, the brightness integrated over the entire visible extend of thecoma) andnuclear magnitude (, the brightness of the core region alone).[31] Both are different scales than the magnitude scale used for planets and asteroids, and can not be used for a size comparison with an asteroid's absolute magnitudeH.
The activity of comets varies with their distance from the Sun. Their brightness can be approximated aswhere are the total and nuclear apparent magnitudes of the comet, respectively, are its "absolute" total and nuclear magnitudes, and are the body-sun and body-observer distances, is theAstronomical Unit, and are the slope parameters characterising the comet's activity. For, this reduces to the formula for a purely reflecting body (showing no cometary activity).[32]
For example, the lightcurve of cometC/2011 L4 (PANSTARRS) can be approximated by[33] On the day of its perihelion passage, 10 March 2013, comet PANSTARRS was from the Sun and from Earth. The total apparent magnitude is predicted to have been at that time. The Minor Planet Center gives a value close to that,.[34]
The absolute magnitude of any given comet can vary dramatically. It can change as the comet becomes more or less active over time or if it undergoes an outburst. This makes it difficult to use the absolute magnitude for a size estimate. When comet289P/Blanpain was discovered in 1819, its absolute magnitude was estimated as.[40] It was subsequently lost and was only rediscovered in 2003. At that time, its absolute magnitude had decreased to,[42] and it was realised that the 1819 apparition coincided with an outburst. 289P/Blanpain reached naked eye brightness (5–8 mag) in 1819, even though it is the comet with the smallest nucleus that has ever been physically characterised, and usually doesn't become brighter than 18 mag.[40][41]
For some comets that have been observed at heliocentric distances large enough to distinguish between light reflected from the coma, and light from the nucleus itself, an absolute magnitude analogous to that used for asteroids has been calculated, allowing to estimate the sizes of their nuclei.[43]
^abcMamajek, E. E.; Torres, G.; Prsa, A.; Harmanec, P.; Asplund, M.; Bennett, P. D.; Capitaine, N.; Christensen-Dalsgaard, J.; Depagne, E.; Folkner, W. M.; Haberreiter, M.; Hekker, S.; Hilton, J. L.; Kostov, V.; Kurtz, D. W.; Laskar, J.; Mason, B. D.; Milone, E. F.; Montgomery, M. M.; Richards, M. T.; Schou, J.; Stewart, S. G. (13 August 2015),"IAU 2015 Resolution B2 on Recommended Zero Points for the Absolute and Apparent Bolometric Magnitude Scales"(PDF),Resolutions Adopted at the General Assemblies, IAU Inter-Division A-G Working Group on Nominal Units for Stellar & Planetary Astronomy,arXiv:1510.06262,Bibcode:2015arXiv151006262M
^Harris, A. W.; Warner, B. D.; Pravec, P. (2016). "Asteroid Lightcurve Derived Data V16.0".NASA Planetary Data System.246: EAR-A-5-DDR-DERIVED-LIGHTCURVE-V16.0.Bibcode:2016PDSS..246.....H.
^Guide to the MPES(PDF), Minor Planet Center, p. 11, retrieved11 January 2019
^Meisel, D. D.; Morris, C. S. (1976), "Comet brightness parameters: Definition, determination, and correlations",NASA. Goddard Space Flight Center the Study of Comets, Part 1,393:410–444,Bibcode:1976NASSP.393..410M
^Hughes, D. W. (16 June 1989). "Cometary Absolute Magnitudes, their Significance and Distribution".Asteroids, Comets, Meteors III, Proceedings of a Meeting (AMC 89) Held at the Astronomical Observatory of the Uppsala University. Uppsala: 337.Bibcode:1990acm..proc..327H.
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