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Absolute magnitude () is a measure of the
luminosity Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a ...
of a celestial object on an inverse logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the
apparent magnitude Apparent magnitude () is a measure of the brightness of a star or other astronomical object observed from Earth. An object's apparent magnitude depends on its intrinsic luminosity, its distance from Earth, and any extinction of the object's ...
that the object would have if it were viewed from a distance of exactly , without
extinction Extinction is the termination of a kind of organism or of a group of kinds (taxon), usually a species. The moment of extinction is generally considered to be the death of the last individual of the species, although the capacity to breed and ...
(or dimming) of its light due to absorption by interstellar matter and cosmic 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. As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength ranges corresponding to specified filter bands or
passband A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
s; for stars a commonly quoted absolute magnitude is the absolute visual magnitude, which uses the visual (V) band of the spectrum (in the
UBV photometric system The UBV photometric system (from ''Ultraviolet, Blue, Visual''), also called the Johnson system (or Johnson-Morgan system), is a photometric system usually employed for classifying stars according to their colors. It was the first standardized ...
). 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. 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. The
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared r ...
has absolute magnitude MV = +4.83. Highly luminous objects can have negative absolute magnitudes: for example, the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
galaxy has an absolute B magnitude of about −20.8. An object's absolute ''bolometric'' magnitude (Mbol) represents its total
luminosity Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a ...
over all
wavelengths In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, 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, a
bolometric correction In astronomy, the bolometric correction is the correction made to the absolute magnitude of an object in order to convert its visible magnitude to its bolometric magnitude. It is large for stars which radiate most of their energy outside of the v ...
(BC) is applied. For Solar System bodies that shine in reflected light, a different definition of absolute magnitude (H) is used, based on a standard reference distance of one astronomical unit.

# Stars and galaxies

In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light-years, 308.57 petameters or 308.57
trillion ''Trillion'' is a number with two distinct definitions: * 1,000,000,000,000, i.e. one million million, or (ten to the twelfth power), as defined on the short scale. This is now the meaning in both American and British English. * 1,000,000,00 ...
kilometres). A star at 10 parsecs has a
parallax Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects ...
of 0.1″ (100 milli
arcseconds A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...
). Galaxies (and other extended 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 object ''equals'' the apparent magnitude it ''would 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 the
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s and cast shadows if they were at 10 parsecs from the Earth. Examples include
Rigel Rigel is a blue supergiant star in the constellation of Orion. It has the Bayer designation β Orionis, which is Latinized to Beta Orionis and abbreviated Beta Ori or β Ori. Rigel is the brightest and most massive componentand ...
(−7.0),
Deneb Deneb () is a first-magnitude star in the constellation of Cygnus, the swan. Deneb is one of the vertices of the asterism known as the Summer Triangle and the "head" of the Northern Cross. It is the brightest star in Cygnus and the ...
(−7.2), Naos (−6.0), and
Betelgeuse Betelgeuse is a red supergiant of spectral type M1-2 and one of the largest stars visible to the naked eye. It is usually the tenth-brightest star in the night sky and, after Rigel, the second-brightest in the constellation of Orion ...
(−5.6). For comparison,
Sirius Sirius is the brightest star in the night sky. Its name is derived from the Greek word , or , meaning 'glowing' or 'scorching'. The star is designated α Canis Majoris, Latinized to Alpha Canis Majoris, and abbreviated Alpha CMa ...
has an absolute magnitude of only 1.4, which is still brighter than the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared r ...
, whose absolute visual magnitude is 4.83. The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75. Absolute magnitudes of stars generally range from approximately −10 to +20. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giant
elliptical galaxy M87 Messier 87 (also known as Virgo A or NGC 4486, generally abbreviated to M87) is a supergiant elliptical galaxy with several trillion stars in the constellation Virgo. One of the largest and most massive galaxies in the local uni ...
has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10). Some active galactic nuclei (
quasars A quasar is an extremely luminous active galactic nucleus (AGN). It is pronounced , and sometimes known as a quasi-stellar object, abbreviated QSO. This emission from a galaxy nucleus is powered by a supermassive black hole with a mass rangin ...
like
CTA-102 CTA 102, also known by its B1950 coordinates as 2230+114 (QSR B2230+114) and its J2000 coordinates as J2232+1143 (QSO J2232+1143), is a blazar-type quasar discovered in the early 1960s by a radio survey carried out by the California Institute of ...
) 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 burst GRB 080319B reached, according to one paper, an absolute r magnitude brighter than −38 for a few tens of seconds.

## Apparent magnitude

The Greek astronomer
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equ ...
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 magnitude , and the dimmest stars visible to the naked eye are assigned . The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude and apparent magnitude from any distance (in
parsec The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, and ...
s, with 1 pc = 3.2616
light-year A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distances and is equivalent to about 9.46 trillion kilometers (), or 5.88 trillion miles ().One trillion here is taken to be 1012 ...
s) are related by $100^=\frac = \left(\frac\right)^,$ where is the radiant flux measured at distance (in parsecs), the radiant flux measured at distance . Using the common logarithm, the equation can be written as $M = m - 5 \log_(d_\text)+5 = m - 5 \left(\log_d_\text-1\right),$ where it is assumed that extinction from gas and dust is negligible. Typical extinction rates within the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
galaxy are 1 to 2 magnitudes per kiloparsec, when dark clouds are taken into account. For objects at very large distances (outside the Milky Way) the luminosity distance (distance defined using luminosity measurements) must be used instead of , because the Euclidean approximation is invalid for distant objects. Instead, general relativity must be taken into account. Moreover, the
cosmological redshift Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
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, a K correction might have to be applied to the magnitudes of the distant objects. The absolute magnitude can also be written in terms of the apparent magnitude and
stellar parallax Stellar parallax is the apparent shift of position of any nearby star (or other object) against the background of distant objects, and a basis for determining (through trigonometry) the distance of the object. Created by the different orbital p ...
: $M = m + 5 \left(\log_p+1\right),$ or using apparent magnitude and distance modulus : $M = m - \mu.$

### Examples

Rigel Rigel is a blue supergiant star in the constellation of Orion. It has the Bayer designation β Orionis, which is Latinized to Beta Orionis and abbreviated Beta Ori or β Ori. Rigel is the brightest and most massive componentand ...
has a visual magnitude of 0.12 and distance of about 860 light-years: $M_\mathrm = 0.12 - 5 \left(\log_ \frac - 1 \right) = -7.0.$
Vega Vega is the brightest star in the northern constellation of Lyra. It has the Bayer designation α Lyrae, which is Latinised to Alpha Lyrae and abbreviated Alpha Lyr or α Lyr. This star is relatively close at only from the Sun, ...
has a parallax of 0.129″, and an apparent magnitude of 0.03: $M_\mathrm = 0.03 + 5 \left(\log_ + 1\right) = +0.6.$ The Black Eye Galaxy has a visual magnitude of 9.36 and a distance modulus of 31.06: $M_\mathrm = 9.36 - 31.06 = -21.7.$

## Bolometric magnitude

The bolometric absolute magnitude , takes into account
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
at all
wavelengths In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
. It includes those unobserved due to instrumental
passband A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenn ...
, the Earth's atmospheric absorption, and extinction by interstellar dust. It is defined based on the
luminosity Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a ...
of the stars. In the case of stars with few observations, it must be computed assuming an effective temperature. Classically, the difference in bolometric magnitude is related to the luminosity ratio according to: $M_\mathrm - M_\mathrm = -2.5 \log_ \left(\frac\right)$ which makes by inversion: $\frac = 10^$ where * is the Sun's luminosity (bolometric luminosity) * is the star's luminosity (bolometric luminosity) * is the bolometric magnitude of the Sun * is the bolometric magnitude of the star. In August 2015, the
International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreach ...
passed Resolution B2 defining the zero points of the absolute and apparent
bolometric magnitude Absolute magnitude () is a measure of the luminosity of a celestial object on an inverse logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if it ...
scales in SI units for power (
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wat ...
s) 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. 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 where corresponds to luminosity , with the zero point
luminosity Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a ...
set such that the Sun (with nominal luminosity ) corresponds to absolute
bolometric magnitude Absolute magnitude () is a measure of the luminosity of a celestial object on an inverse logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if it ...
. Placing a
radiation In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes: * ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
source (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale corresponds to
irradiance In radiometry, irradiance is the radiant flux ''received'' by a ''surface'' per unit area. The SI unit of irradiance is the watt per square metre (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used ...
. Using the IAU 2015 scale, the nominal total
solar irradiance Solar irradiance is the power per unit area ( surface power density) received from the Sun in the form of electromagnetic radiation in the wavelength range of the measuring instrument. Solar irradiance is measured in watts per square metre ...
(" solar constant") measured at 1 astronomical unit () corresponds to an apparent bolometric magnitude of the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared r ...
of . 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: $M_\mathrm = -2.5 \log_ \frac \approx -2.5 \log_ L_\star + 71.197425$ where * is the star's luminosity (bolometric luminosity) in
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wat ...
s * is the zero point luminosity * 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 nominal solar luminosity corresponds closely to , a value that was commonly adopted by astronomers before the 2015 IAU resolution. The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude as: $L_\star = L_0 10^$ using the variables as defined previously.

# Solar System bodies ()

For
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s and
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
s, a definition of absolute magnitude that is more meaningful for non-stellar objects is used. The absolute magnitude, commonly called $H$, is defined as the
apparent magnitude Apparent magnitude () is a measure of the brightness of a star or other astronomical object observed from Earth. An object's apparent magnitude depends on its intrinsic luminosity, its distance from Earth, and any extinction of the object's ...
that the object would have if it were one astronomical unit (AU) from both the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared r ...
and the observer, and in conditions of ideal solar opposition (an arrangement that is impossible in practice). Because Solar System bodies are illuminated by the Sun, their brightness varies as a function of illumination conditions, described by the phase angle. This relationship is referred to as the phase curve. The absolute magnitude is the brightness at phase angle zero, an arrangement known as opposition, from a distance of one AU.

## Apparent magnitude

The absolute magnitude $H$ can be used to calculate the apparent magnitude $m$ of a body. For an object reflecting sunlight, $H$ and $m$ are connected by the relation $m = H + 5 \log_ - 2.5 \log_,$ where $\alpha$ is the phase angle, the angle between the body-Sun and body–observer lines. $q\left(\alpha\right)$ is the phase integral (the
integration Integration may refer to: Biology *Multisensory integration * Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technolog ...
of reflected light; a number in the 0 to 1 range). By the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
, we have: $\cos = \frac .$ Distances: * is the distance between the body and the observer * is the distance between the body and the Sun * is the distance between the observer and the Sun * , a unit conversion factor, is the constant 1  AU, the average distance between the Earth and the Sun

## Approximations for phase integral

The value of $q\left(\alpha\right)$ depends on the properties of the reflecting surface, in particular on its roughness. 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.

### Planets as diffuse spheres

Planetary bodies can be approximated reasonably well as ideal diffuse reflecting
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
s. Let $\alpha$ be the phase angle in degrees, then $q(\alpha) = \frac23 \left(\left(1-\frac\right)\cos+\frac\sin\right).$ A full-phase diffuse sphere reflects two-thirds as much light as a diffuse flat disk of the same diameter. A quarter phase ($\alpha = 90^$) has $\frac$ as much light as full phase ($\alpha = 0^$). By contrast, a ''diffuse disk reflector model'' is simply $q\left(\alpha\right) = \cos$, which isn't realistic, but it does represent the opposition surge for rough surfaces that reflect more uniform light back at low phase angles. The definition of the
geometric albedo In astronomy, the geometric albedo of a celestial body is the ratio of its actual brightness as seen from the light source (i.e. at zero phase angle) to that of an ''idealized'' flat, fully reflecting, diffusively scattering ( Lambertian) disk w ...
$p$, a measure for the reflectivity of planetary surfaces, is based on the diffuse disk reflector model. The absolute magnitude $H$, diameter $D$ (in
kilometer The kilometre ( SI symbol: km; or ), spelt kilometer in American English, is a unit of length in the International System of Units (SI), equal to one thousand metres ( kilo- being the SI prefix for ). It is now the measurement unit used f ...
s) and geometric albedo $p$ of a body are related by $D = \frac \times 10^ \mathrm.$ Example: The Moon's absolute magnitude $H$ can be calculated from its diameter $D=3474\text$ and
geometric albedo In astronomy, the geometric albedo of a celestial body is the ratio of its actual brightness as seen from the light source (i.e. at zero phase angle) to that of an ''idealized'' flat, fully reflecting, diffusively scattering ( Lambertian) disk w ...
$p = 0.113$: $H = 5\log_ = +0.28.$ We have $d_=1\text$, $d_=384400\text=0.00257\text.$ At quarter phase, $q(\alpha)\approx \frac$ (according to the diffuse reflector model), this yields an apparent magnitude of $m = +0.28+5\log_ - 2.5\log_ = -10.99.$ The actual value is somewhat lower than that, $m=-10.0.$ The phase curve of the Moon is too complicated for the diffuse reflector model. 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. For planets, approximations for the correction term $-2.5\log_$ in the formula for have been derived empirically, to match observations at different phase angles. The approximations recommended by the Astronomical Almanac are (with $\alpha$ in degrees): Here $\beta$ is the effective inclination of
Saturn's rings The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entire ...
(their tilt relative to the observer), which as seen from Earth varies between 0° and 27° over the course of one Saturn orbit, and $\phi\text{'}$ is a small correction term depending on Uranus' sub-Earth and sub-solar latitudes. $t$ is the
Common Era Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
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 $\alpha \ge 179^$ for Venus, no observations are available, and the phase curve is unknown in those cases. The formula for the Moon is only applicable to the near side of the Moon, the portion that is visible from the Earth. Example 1: On 1 January 2019,
Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never far ...
was $d_=0.719\text$ from the Sun, and $d_ = 0.645\text$ from Earth, at a phase angle of $\alpha=93.0^$ (near quarter phase). Under full-phase conditions, Venus would have been visible at $m=-4.384+5\log_=-6.09.$ Accounting for the high phase angle, the correction term above yields an actual apparent magnitude of $m=-6.09+\left\left(-1.044\times 10^ \cdot93.0+3.687\times10^\cdot93.0^-2.814\times10^\cdot93.0^+8.938\times10^\cdot93.0^\right\right)=-4.59.$ This is close to the value of $m=-4.62$ predicted by the Jet Propulsion Laboratory. Example 2: At first quarter phase, the approximation for the Moon gives $-2.5\log_=2.71.$ With that, the apparent magnitude of the Moon is $m = +0.28+5\log_+2.71= -9.96,$ close to the expected value of about $-10.0$. At last quarter, the Moon is about 0.06 mag fainter than at first quarter, because that part of its surface has a lower albedo. Earth's
albedo Albedo (; ) is the measure of the diffuse reflection of solar radiation out of the total solar radiation and measured on a scale from 0, corresponding to a black body that absorbs all incident radiation, to 1, corresponding to a body that refle ...
varies by a factor of 6, from 0.12 in the cloud-free case to 0.76 in the case of altostratus cloud. The absolute magnitude in the table corresponds to an albedo of 0.434. Due to the variability of the
weather Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloudy. On Earth, most weather phenomena occur in the lowest layer of the planet's atmosphere, the t ...
, Earth's apparent magnitude cannot be predicted as accurately as that of most other planets.

### Asteroids

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 $0^$. This rapid brightening near opposition is called the opposition effect. Its strength depends on the physical properties of the body's surface, and hence it differs from asteroid to asteroid. In 1985, the IAU adopted the semi-empirical $HG$-system, based on two parameters $H$ and $G$ called ''absolute magnitude'' and ''slope'', to model the opposition effect for the
ephemerides In astronomy and celestial navigation, an ephemeris (pl. ephemerides; ) is a book with tables that gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly ...
Minor Planet Center The Minor Planet Center (MPC) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory. Function ...
. $m = H + 5\log_-2.5\log_,$ where *the phase integral is $q\left(\alpha\right)=\left\left(1-G\right\right)\phi_\left\left(\alpha\right\right)+G\phi_\left\left(\alpha\right\right)$ and *$\phi_\left(\alpha\right) = \exp$ for $i = 1$ or $2$, $A_=3.332$, $A_=1.862$, $B_=0.631$ and $B_2 = 1.218$. This relation is valid for phase angles $\alpha < 120^$, and works best when $\alpha < 20^$. The slope parameter $G$ relates to the surge in brightness, typically , when the object is near opposition. It is known accurately only for a small number of asteroids, hence for most asteroids a value of $G=0.15$ is assumed. In rare cases, $G$ can be negative. An example is 101955 Bennu, with $G=-0.08$. In 2012, the $HG$-system was officially replaced by an improved system with three parameters $H$, $G_1$ and $G_2$, which produces more satisfactory results if the opposition effect is very small or restricted to very small phase angles. However, as of 2022, this $H G_1 G_2$-system has not been adopted by either the Minor Planet Center nor
Jet Propulsion Laboratory The Jet Propulsion Laboratory (JPL) is a federally funded research and development center and NASA field center in the City of La Cañada Flintridge, California, United States. Founded in the 1930s by Caltech researchers, JPL is owned by NASA ...
. The apparent magnitude of asteroids varies as they rotate, on time scales of seconds to weeks depending on their
rotation period The rotation period of a celestial object (e.g., star, gas giant, planet, moon, asteroid) may refer to its sidereal rotation period, i.e. the time that the object takes to complete a single revolution around its axis of rotation relative to the ...
, by up to $2\text$ or more. In addition, their absolute magnitude can vary with the viewing direction, depending on their axial 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.

## Cometary magnitudes

The brightness of
comet A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena are ...
s is given separately as ''total magnitude'' ($m_$, the brightness integrated over the entire visible extend of the
coma A coma is a deep state of prolonged unconsciousness in which a person cannot be awakened, fails to respond normally to painful stimuli, light, or sound, lacks a normal wake-sleep cycle and does not initiate voluntary actions. Coma patients exhi ...
) and ''nuclear magnitude'' ($m_$, the brightness of the core region alone). 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 magnitude . The activity of comets varies with their distance from the Sun. Their brightness can be approximated as $m_ = M_ + 2.5\cdot K_\log_ + 5\log_$ $m_ = M_ + 2.5\cdot K_\log_ + 5\log_,$ where $m_$ are the total and nuclear apparent magnitudes of the comet, respectively, $M_$ are its "absolute" total and nuclear magnitudes, $d_$ and $d_$ are the body-sun and body-observer distances, $d_$ is the Astronomical Unit, and $K_$ are the slope parameters characterising the comet's activity. For $K=2$, this reduces to the formula for a purely reflecting body (showing no cometary activity). For example, the lightcurve of comet C/2011 L4 (PANSTARRS) can be approximated by $M_=5.41\textK_=3.69.$ On the day of its perihelion passage, 10 March 2013, comet PANSTARRS was $0.302\text$ from the Sun and $1.109\text$ from Earth. The total apparent magnitude $m_$ is predicted to have been $m_1 = 5.41 + 2.5\cdot3.69\cdot\log_+5\log_ = +0.8$ at that time. The Minor Planet Center gives a value close to that, $m_ = +0.5$. 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 comet 289P/Blanpain was discovered in 1819, its absolute magnitude was estimated as $M_ = 8.5$. It was subsequently lost and was only rediscovered in 2003. At that time, its absolute magnitude had decreased to $M_ = 22.9$, 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. 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.

# Meteors

For a meteor, the standard distance for measurement of magnitudes is at an altitude of at the observer's
zenith The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "highest ...
.

* Araucaria Project * Hertzsprung–Russell diagram – relates absolute magnitude or
luminosity Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a ...
versus spectral color or surface temperature. *
Jansky The jansky (symbol Jy, plural ''janskys'') is a non- SI unit of spectral flux density, or spectral irradiance, used especially in radio astronomy. It is equivalent to 10−26 watts per square metre per hertz. The ''flux density'' or ''mono ...
radio astronomer's preferred unit – linear in power/unit area * List of most luminous stars *
Photographic magnitude Photographic magnitude ( or ) is a measure of the relative brightness of a star or other astronomical object as imaged on a photographic film emulsion with a camera attached to a telescope. An object's apparent photographic magnitude depends on i ...
* Surface brightness – the ''magnitude'' for extended objects * Zero point (photometry) – the typical calibration point for star flux

# References

Reference zero-magnitude fluxes

International Astronomical Union

Absolute Magnitude of a Star calculator