Absolute magnitude

{{Short description|Measure of the luminosity of celestial objects}} {{About|the brightness of stars|the science fiction magazine|Absolute Magnitude (magazine)}} {{Use dmy dates|date=April 2020}}

In astronomy, '''absolute magnitude''' ('''{{mvar|M}}''') is a measure of the luminosity of a celestial object on an inverse logarithmic astronomical 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 the apparent magnitude that the object would have if it were viewed from a distance of exactly {{convert|10|pc|ly|1|abbr=off|lk=on}}, without extinction (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. 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.

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. The Sun has absolute magnitude MV = +4.83.<ref name="SunAbs"/> Highly luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute B magnitude of about −20.8.<ref name="Karachentsev"/>

As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength ranges corresponding to specified filter bands or passbands; 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). 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 absolute ''bolometric'' magnitude (Mbol) represents its total luminosity over all wavelengths, 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 (BC) is applied.<ref name="Flower1996"/>

==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 kilometres). A star at 10 parsecs has a parallax of 0.1″ (100 milliarcseconds). 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 planets and cast shadows if they were at 10 parsecs from the Earth. Examples include Rigel (−7.8), Deneb (−8.4), Naos (−6.2), and Betelgeuse (−5.8). For comparison, Sirius has an absolute magnitude of only 1.4, which is still brighter than the Sun, whose absolute visual magnitude is 4.83. The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.<ref name="Strobel"/><ref name="Casagrande"/> 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 has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10). Some active galactic nuclei (quasars like CTA-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 burst GRB 080319B reached, according to one paper, an absolute r magnitude brighter than −38 for a few tens of seconds.<ref name=bloom2009/>

=== Apparent magnitude === {{Main|Apparent magnitude}}

The Greek astronomer Hipparchus 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 {{math|1=''m'' = 1}}, and the dimmest stars visible to the naked eye are assigned {{math|1=''m'' = 6}}.<ref name="Carroll"/> The difference between them corresponds to a factor of 100 in brightness. For objects within the immediate neighborhood of the Sun, the absolute magnitude {{mvar|M}} and apparent magnitude {{mvar|m}} from any distance {{mvar|d}} (in parsecs, with 1 pc = 3.2616 light-years) are related by <math display="block"> 100^{\frac{m-M}{5}}=\frac{F_{10}}{F} = \left(\frac{d}{10\;\mathrm{pc}}\right)^{2}, </math> where {{mvar|F}} is the radiant flux measured at distance {{mvar|d}} (in parsecs), {{math|''F''10}} the radiant flux measured at distance {{math|10 pc}}. Using the common logarithm, the equation can be written as <math display="block"> M = m - 5 \log_{10}(d_\text{pc})+5 = m - 5 \left(\log_{10}d_\text{pc}-1\right),</math> where it is assumed that extinction from gas and dust is negligible. Typical extinction rates within the Milky Way galaxy are 1 to 2 magnitudes per kiloparsec, when dark clouds are taken into account.<ref name="Unsoeld2013"/>

For objects at very large distances (outside the Milky Way) the luminosity distance {{math|''d''L}} (distance defined using luminosity measurements) must be used instead of {{mvar|d}}, because the Euclidean approximation is invalid for distant objects. Instead, general relativity must be taken into account. Moreover, the cosmological 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, a K correction might have to be applied to the magnitudes of the distant objects.

The absolute magnitude {{mvar|M}} can also be written in terms of the apparent magnitude {{mvar|m}} and stellar parallax {{mvar|p}}: <math display="block"> M = m + 5 \left(\log_{10}p+1\right),</math> or using apparent magnitude {{mvar|m}} and distance modulus {{mvar|μ}}: <math display="block"> M = m - \mu.</math>

==== Examples ==== Rigel has a visual magnitude {{math|''m''V}} of 0.12 and distance of about 860 light-years: <math display="block">M_\mathrm{V} = 0.12 - 5 \left(\log_{10} \frac{860}{3.2616} - 1 \right) = -7.0.</math>

Vega has a parallax {{mvar|p}} of 0.129″, and an apparent magnitude {{math|''m''V}} of 0.03: <math display="block">M_\mathrm{V} = 0.03 + 5 \left(\log_{10}{0.129} + 1\right) = +0.6.</math>

The Black Eye Galaxy has a visual magnitude {{math|''m''V}} of 9.36 and a distance modulus {{mvar|μ}} of 31.06: <math display="block">M_\mathrm{V} = 9.36 - 31.06 = -21.7.</math>

=== Bolometric magnitude === {{See also|Apparent bolometric magnitude}}

The absolute bolometric magnitude ({{math|''M''bol}}) takes into account electromagnetic radiation at all wavelengths. It includes those unobserved due to instrumental passband, the Earth's atmospheric absorption, and extinction by interstellar dust. It is defined based on the luminosity 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:<ref name="Carroll"/> <math display="block">M_\mathrm{bol,\star} - M_\mathrm{bol,\odot} = -2.5 \log_{10} \left(\frac{L_\star}{L_\odot}\right)</math> which makes by inversion: <math display="block">\frac{L_\star}{L_\odot} = 10^{0.4\left(M_\mathrm{bol,\odot} - M_\mathrm{bol,\star}\right)}</math> where *{{math|''L''☉}} is the Sun's luminosity (bolometric luminosity) *{{math|''L''★}} is the star's luminosity (bolometric luminosity) *{{math|''M''bol,☉}} is the bolometric magnitude of the Sun *{{math|''M''bol,★}} is the bolometric magnitude of the star.

In August 2015, the International Astronomical Union passed Resolution B2<ref name="IAU_XXIX"/> defining the zero points of the absolute and apparent bolometric 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.<ref name="IAU2015B2"/> 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 {{math|1=''M''bol = 0}} corresponds to luminosity {{math|1=''L''0 = {{val|3.0128|e=28|u=W}}}}, with the zero point luminosity {{math|''L''0}} set such that the Sun (with nominal luminosity {{val|3.828|e=26|u=W}}) corresponds to absolute bolometric magnitude {{math|1=''M''bol,☉ = 4.74}}. Placing a radiation source (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale {{math|''m''bol {{=}} 0}} corresponds to irradiance {{math|1=''f''0 = {{val|2.518021002|e=-8|u=W/m2}}}}. Using the IAU 2015 scale, the nominal total solar irradiance ("solar constant") measured at 1 astronomical unit ({{val|1361|u=W/m2}}) corresponds to an apparent bolometric magnitude of the Sun of {{math|1=''m''bol,☉ = −26.832}}.<ref name="IAU2015B2" />

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: <math display="block">M_\mathrm{bol} = -2.5 \log_{10} \frac{L_\star}{L_0} \approx -2.5 \log_{10} L_\star + 71.197425</math> where *{{math|''L''★}} is the star's luminosity (bolometric luminosity) in watts *{{math|''L''0}} is the zero point luminosity {{val|3.0128|e=28|u=W}} *{{math|''M''bol}} 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 {{math|1=''M''bol = 4.74}}, a value that was commonly adopted by astronomers before the 2015 IAU resolution.<ref name="IAU2015B2" />

The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude {{math|''M''bol}} as: <math display="block">L_\star = L_0 10^{-0.4 M_\mathrm{bol}}</math> using the variables as defined previously.

{{anchor|Solar System|Solar system|Solar System bodies|Solar system bodies}}

== Solar System bodies ({{mvar|''H''}}) == {{anchor|Solar System bodies (H)}} {{For|an introduction|Magnitude (astronomy)}}

{| class="wikitable floatright" style=" text-align:center; font-size:0.9em;" |+ Abs Mag (H) and Diameter for asteroids (albedo=0.14)<ref>[https://cneos.jpl.nasa.gov/tools/ast_size_est.html CNEOS Asteroid Size Estimator]</ref> ! H !! Diameter |- | 10 || 36 km |- | 12.7 || 10 km |- | 15 || 3.6 km |-id=1km | 17.7 || 1 km |- | 19.2 || 510 m |- | 20 || 360 m |-id=PHA | 22 || 140 m |- | 22.7 || 100 m |- | 24.2 || 51 m |- | 25 || 36 m |-id=Chelyabinsk | 26.6 || 17 m |- | 27.7 || 10 m |- | 30 || 3.6 m |-id=1m | 32.7 || 1 m |}

For planets and asteroids, a definition of absolute magnitude that is more meaningful for non-stellar objects is used. The absolute magnitude, commonly called <math>H</math>, is defined as the apparent magnitude that the object would have if it were one astronomical unit (AU) from both the Sun and the observer, and in conditions of ideal solar opposition (an arrangement that is impossible in practice).<ref name="Luciuk"/> 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. Newly discovered comets with a weakly active coma can masquerade as an asteroid resulting in unrealistic size estimates as a comet's absolute magnitude (M1) can not be directly compared to an asteroid's absolute magnitude (H).

=== Apparent magnitude === [[File:Phase angle explanation.png|thumb|right|250px|The phase angle <math>\alpha</math> can be calculated from the distances body-sun, observer-sun and observer-body, using the law of cosines.]] The absolute magnitude <math>H</math> can be used to calculate the apparent magnitude <math>m</math> of a body. For an object reflecting sunlight, <math>H</math> and <math>m</math> are connected by the relation <math display="block">m = H + 5 \log_{10}{\left(\frac{d_{BS} d_{BO}}{d_0^2}\right)} - 2.5 \log_{10}{q(\alpha)},</math> where <math>\alpha</math> is the phase angle, the angle between the body-Sun and body–observer lines. <math>q(\alpha)</math> is the phase integral (the integration of reflected light; a number in the 0 to 1 range).<ref name="Karttunen2016"/>

By the law of cosines, we have: <math display="block">\cos{\alpha} = \frac{ d_\mathrm{BO}^2 + d_\mathrm{BS}^2 - d_\mathrm{OS}^2 } {2 d_\mathrm{BO} d_\mathrm{BS}}.</math>

Distances: * {{math|''d''BO}} is the distance between the body and the observer * {{math|''d''BS}} is the distance between the body and the Sun * {{math|''d''OS}} is the distance between the observer and the Sun * {{math|''d''0}}, a unit conversion factor, is the constant 1 AU, the average distance between the Earth and the Sun

=== Approximations for phase integral {{serif|''q''(''α'')}} === The value of <math>q(\alpha)</math> 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.<ref name="Karttunen2016"/>

==== Planets as diffuse spheres ==== thumb|right|240px|Diffuse reflection on sphere and flat disk thumb|240px|Brightness 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 as ideal diffuse reflecting spheres. Let <math>\alpha</math> be the phase angle in degrees, then<ref name="Whitmell1907"/> <math display="block">q(\alpha) = \frac23 \left(\left(1-\frac{\alpha}{180^{\circ}}\right)\cos{\alpha}+\frac{1}{\pi}\sin{\alpha}\right).</math> A full-phase diffuse sphere reflects two-thirds as much light as a diffuse flat disk of the same diameter. A quarter phase (<math>\alpha = 90^{\circ}</math>) has <math display="inline">\frac{1}{\pi}</math> as much light as full phase (<math>\alpha = 0^{\circ}</math>).

By contrast, a ''diffuse disk reflector model'' is simply <math>q(\alpha) = \cos{\alpha}</math>, 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 <math>p</math>, a measure for the reflectivity of planetary surfaces, is based on the diffuse disk reflector model. The absolute magnitude <math>H</math>, diameter <math>D</math> (in kilometers) and geometric albedo <math>p</math> of a body are related by<ref name="sizemagnitude"/>{{efn|name="Mag_formula"}}<ref name="H_derivation"/> <math display="block">D = \frac{1329}{\sqrt{p}} \times 10^{-0.2H} \mathrm{km},</math> or equivalently, <math display="block">H = 5\log_{10}{\frac{1329\text{ km}}{D\sqrt{p}}}.</math>

Example: The Moon's absolute magnitude <math>H</math> can be calculated from its diameter <math>D=3474\text{ km}</math> and geometric albedo <math>p = 0.113</math>:<ref name="Albedo"/> <math display="block">H = 5\log_{10}{\frac{1329\text{ km}}{3474\text{ km}\cdot\sqrt{0.113}}} = +0.28.</math> Cox gives a value close to that (<math>+0.21</math>).<ref name="cox2000"/>

We have <math>d_{BS}=1\text{ AU}</math>, <math>d_{BO}=384400\text{ km}=0.00257\text{ AU}.</math> At quarter phase, <math display="inline">q(\alpha)\approx \frac{2}{3\pi}</math> (according to the diffuse reflector model), this yields an apparent magnitude of <math display="block">m = +0.28+5\log_{10}{\left(1\cdot0.00257\right)} - 2.5\log_{10}{\left(\frac{2}{3\pi}\right)} = -10.99.</math> The actual value is somewhat lower than that, <math>m=-10.0.</math> This is not a good approximation, because the phase curve of the Moon is too complicated for the diffuse reflector model.<ref name="Luciuk2"/> A more accurate formula is given in the following section.

==== More advanced models ====

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.<ref name="Karttunen2016"/> For planets, approximations for the correction term <math>-2.5\log_{10}{q(\alpha)}</math> in the formula for {{mvar|m}} have been derived empirically, to match observations at different phase angles. The approximations recommended by the Astronomical Almanac<ref name="Mallama_and_Hilton"/> are (with <math>\alpha</math> in degrees):

{| class="wikitable" |- ! Planet ! Referenced calculation<ref name="IMCCE"/> ! <math>H</math> ! Approximation for <math>-2.5\log_{10}{q(\alpha)}</math> |- | Mercury | −0.4 | −0.613 | <math>+6.328\times10^{-2}\alpha - 1.6336\times10^{-3}\alpha^{2}+3.3644\times10^{-5}\alpha^{3}-3.4265\times10^{-7}\alpha^{4}+1.6893\times10^{-9}\alpha^{5}-3.0334\times10^{-12}\alpha^{6}</math> |- | Venus | −4.4 | −4.384 | * <math>-1.044\times10^{-3}\alpha+3.687\times10^{-4}\alpha^{2}-2.814\times10^{-6}\alpha^{3}+8.938\times10^{-9}\alpha^{4}</math> (for <math>0^{\circ}<\alpha \le 163.7^{\circ}</math>) * <math>+236.05828-2.81914\alpha+8.39034\times10^{-3}\alpha^{2}</math> (for <math>163.7^{\circ}<\alpha<179^{\circ}</math>) |- | Earth | − | −3.99 |<math>-1.060\times10^{-3}\alpha+2.054\times10^{-4}\alpha^{2}</math> |- | Moon<ref name=cox2000/> | 0.2 | +0.21{{efn|1=Other sources may give slightly different values.}} | * <math>+2.9994\times10^{-2}\alpha-1.6057\times10^{-4}\alpha^{2}+3.1543\times10^{-6}\alpha^{3}-2.0667\times10^{-8}\alpha^{4}+6.2553\times10^{-11}\alpha^{5}</math> (for <math>\alpha\le150^{\circ}</math>, before full Moon) * <math>+3.3234\times10^{-2}\alpha-3.0725\times10^{-4}\alpha^{2}+6.1575\times10^{-6}\alpha^{3}-4.7723\times10^{-8}\alpha^{4}+1.4681\times10^{-10}\alpha^{5}</math> (for <math>\alpha\le150^{\circ}</math>, after full Moon) |- | Mars | −1.5 | −1.601 | * <math>+2.267\times10^{-2}\alpha-1.302\times10^{-4}\alpha^{2}</math> (for <math>0^{\circ}<\alpha\le50^{\circ}</math>) * <math>+1.234-2.573\times10^{-2}\alpha+3.445\times10^{-4}\alpha^{2}</math> (for <math>50^{\circ}<\alpha\le120^{\circ}</math>) |- | Jupiter | −9.4 | −9.395 | * <math>-3.7\times10^{-4}\alpha+6.16\times10^{-4}\alpha^{2}</math> (for <math>\alpha\le12^{\circ}</math>) * <math>-0.033-2.5\log_{10}{\left(1-1.507\left(\frac{\alpha}{180^{\circ}}\right)-0.363\left(\frac{\alpha}{180^{\circ}}\right)^{2}-0.062\left(\frac{\alpha}{180^{\circ}}\right)^{3}+2.809\left(\frac{\alpha}{180^{\circ}}\right)^{4}-1.876\left(\frac{\alpha}{180^{\circ}}\right)^{5}\right)}</math> (for <math>\alpha>12^{\circ}</math>) |- | Saturn | −9.7 | −8.914 | * <math>-1.825\sin{\left(\beta\right)}+2.6\times10^{-2}\alpha-0.378\sin{\left(\beta\right)}e^{-2.25\alpha}</math> (for planet and rings, <math>\alpha<6.5^{\circ}</math> and <math>\beta<27^{\circ}</math>) * <math>-0.036-3.7\times10^{-4}\alpha+6.16\times10^{-4}\alpha^{2}</math> (for the globe alone, <math>\alpha\le6^{\circ}</math>) * <math>+0.026+2.446\times10^{-4}\alpha+2.672\times10^{-4}\alpha^{2}-1.505\times10^{-6}\alpha^{3}+4.767\times10^{-9}\alpha^{4}</math> (for the globe alone, <math>6^{\circ}<\alpha<150^{\circ}</math>) |- | Uranus | −7.2 | −7.110 |<math>-8.4\times10^{-4}\phi'+6.587\times10^{-3}\alpha+1.045\times10^{-4}\alpha^{2}</math> (for <math>\alpha < 3.1^{\circ}</math>) |- | Neptune | −6.9 | −7.00 |<math>+7.944\times10^{-3}\alpha+9.617\times10^{-5}\alpha^{2}</math> (for <math>\alpha < 133^{\circ}</math> and <math>t > 2000.0</math>) |} {{Multiple image | header = The different halves of the Moon, as seen from Earth | image1 = Daniel Hershman - march moon (by).jpg | caption1 = Moon at first quarter | image2 = Waning gibbous moon near last quarter - 23 Sept. 2016.png | caption2 = Moon at last quarter }} Here <math>\beta</math> is the effective inclination of Saturn'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 <math>\phi'</math> is a small correction term depending on Uranus' sub-Earth and sub-solar latitudes. <math>t</math> is the Common 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 <math>\alpha \ge 179^{\circ}</math> 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 was <math>d_{BS}=0.719\text{ AU}</math> from the Sun, and <math>d_{BO} = 0.645\text{ AU}</math> from Earth, at a phase angle of <math>\alpha=93.0^{\circ}</math> (near quarter phase). Under full-phase conditions, Venus would have been visible at <math>m=-4.384+5\log_{10}{\left(0.719 \cdot 0.645\right)}=-6.09.</math> Accounting for the high phase angle, the correction term above yields an actual apparent magnitude of <math display="block">m = -6.09 + \left(-1.044 \times 10^{-3} \cdot 93.0 + 3.687\times10^{-4} \cdot 93.0^{2} - 2.814 \times 10^{-6} \cdot 93.0^{3} + 8.938 \times 10^{-9} \cdot 93.0^{4}\right) = -4.59.</math> This is close to the value of <math>m=-4.62</math> predicted by the Jet Propulsion Laboratory.<ref name="JPLHorizonsVenus"/>

Example 2: At first quarter phase, the approximation for the Moon gives <math display="inline">-2.5\log_{10}{q(90^{\circ})}=2.71.</math> With that, the apparent magnitude of the Moon is <math display="inline">m = +0.21+5\log_{10}{\left(1\cdot0.00257\right)}+2.71= -10.03,</math> close to the expected value of about <math>-10.0</math>. 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.<ref name="cox2000"/>{{efn|1=Calculation: Using the phase function at <math>\alpha=90^{\circ}</math> in Cox, table 12.15, gives <math>2.5\cdot\log_{10}{0.0824/0.0780}=0.0596\text{ mag}</math>.}}

Earth's albedo 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, Earth's apparent magnitude cannot be predicted as accurately as that of most other planets.<ref name="Mallama_and_Hilton"/>

==== Asteroids ==== [[File:Ceres opposition effect.png|thumb|right|240px|Asteroid 1 Ceres, imaged by the Dawn 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 the opposition effect.]] thumb|240px|Phase integrals for various values of G thumb|right|240px|Relationship between the slope parameter <math>G</math> and the opposition surge. Larger values of <math>G</math> correspond to a less pronounced opposition effect. For most asteroids, a value of <math>G = 0.15</math> is assumed, corresponding to an opposition surge of <math>0.3\text{ mag}</math>. 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 <math>0^{\circ}</math>. 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.<ref name="Karttunen2016"/>

In 1985, the IAU adopted the semi-empirical <math>HG</math>-system, based on two parameters <math>H</math> and <math>G</math> called ''absolute magnitude'' and ''slope'', to model the opposition effect for the ephemerides published by the Minor Planet Center.<ref name="MPC1985"/> <math display="block">m = H + 5\log_{10}{\left(\frac{d_{BS}d_{BO}}{d_{0}^{2}}\right)}-2.5\log_{10}{q(\alpha)},</math>

where *the phase integral is <math>q(\alpha)=\left(1-G\right)\phi_{1}\left(\alpha\right)+G\phi_{2}\left(\alpha\right)</math> and *<math display="inline">\phi_{i}\left(\alpha\right) = \exp{\left(-A_i \left(\tan{\frac{\alpha}{2}}\right)^{B_i}\right)}</math> for <math>i = 1</math> or <math>2</math>, <math>A_{1}=3.332</math>, <math>A_{2}=1.862</math>, <math>B_{1}=0.631</math> and <math>B_2 = 1.218</math>.<ref name="Lagerkvist"/>

This relation is valid for phase angles <math>\alpha < 120^{\circ}</math>, and works best when <math>\alpha < 20^{\circ}</math>.<ref name="dymock"/>

The slope parameter <math>G</math> relates to the surge in brightness, typically {{val|0.3|u=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 <math>G=0.15</math> is assumed.<ref name="dymock"/> In rare cases, <math>G</math> can be negative.<ref name="Lagerkvist"/><ref name="JPLdoc"/> An example is 101955 Bennu, with <math>G=-0.03</math>.<ref name="Bennu"/>

In 2012, the <math>HG</math>-system was officially replaced by an improved system with three parameters <math>H</math>, <math>G_1</math> and <math>G_2</math>, which produces more satisfactory results if the opposition effect is very small or restricted to very small phase angles. However, as of 2022, this <math>H G_1 G_2</math>-system has not been adopted by either the Minor Planet Center nor Jet Propulsion Laboratory.<ref name="Karttunen2016"/><ref name="Shevchenko2016"/>

The apparent magnitude of asteroids varies as they rotate, on time scales of seconds to weeks depending on their rotation period, by up to <math>2\text{ mag}</math> or more.<ref name="lc"/> 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.<ref name="dymock"/><ref name="Karttunen2016"/>

=== Cometary magnitudes === The brightness of comets is given separately as ''total magnitude'' (<math>m_{1}</math>, the brightness integrated over the entire visible extent of the coma) and ''nuclear magnitude'' (<math>m_{2}</math>, the brightness of the core region alone).<ref name="MPES"/> 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 {{mvar|H}}.

The activity of comets varies with their distance from the Sun. Their brightness can be approximated as <math display="block">m_{1} = M_{1} + 2.5\cdot K_{1}\log_{10}{\left(\frac{d_{BS}}{d_0}\right)} + 5\log_{10}{\left(\frac{d_{BO}}{d_0}\right)}</math> <math display="block">m_{2} = M_{2} + 2.5\cdot K_{2}\log_{10}{\left(\frac{d_{BS}}{d_0}\right)} + 5\log_{10}{\left(\frac{d_{BO}}{d_0}\right)},</math> where <math>m_{1,2}</math> are the total and nuclear apparent magnitudes of the comet, respectively, <math>M_{1,2}</math> are its "absolute" total and nuclear magnitudes, <math>d_{BS}</math> and <math>d_{BO}</math> are the body-sun and body-observer distances, <math>d_{0}</math> is the Astronomical Unit, and <math>K_{1,2}</math> are the slope parameters characterising the comet's activity. For <math>K=2</math>, this reduces to the formula for a purely reflecting body (showing no cometary activity).<ref name="Meisel1976"/>

For example, the lightcurve of comet C/2011 L4 (PANSTARRS) can be approximated by <math>M_{1}=5.5\text{, }K_{1}=3.7</math>.<ref name="COBS 2011L4"/> On the day of its perihelion passage, 10 March 2013, comet PANSTARRS was <math>0.302\text{ AU}</math> from the Sun and <math>1.109\text{ AU}</math> from Earth. The total apparent magnitude <math>m_{1}</math> is predicted to have been <math>m_1 = 5.5 + 2.5\cdot3.7\cdot\log_{10}{\left(0.302\right)}+5\log_{10}{\left(1.109\right)} = +0.9</math> at that time. The Minor Planet Center gives a value close to that, <math>m_{1} = +0.5</math>.<ref name="MPC2011L4"/>

{| class="wikitable sortable" style="font-size: 0.9em;" |+Absolute magnitudes and sizes of comet nuclei ! Comet ! Absolute magnitude <math>M_{1}</math><ref name="kidger"/> ! Nucleus diameter |- |Comet Sarabat || −3.0 || ≈100 km? |- |Comet Hale-Bopp || −1.3 || 60 ± 20 km |- |Comet Halley || 4.0 || 14.9 x 8.2 km |- |average new comet || 6.5 || ≈2 km<ref name="Hughes"/> |- |C/2014 UN271 (Bernardinelli-Bernstein) || {{sort|6.2|{{val|6.2|0.9}}}}<ref name=Bernardinelli/> || 119 to 137 km<ref name="Lellouch2022"/> |- |289P/Blanpain (during 1819 outburst) || 8.5<ref name="Yoshida"/> || 320 m<ref name="Jewitt"/> |- |289P/Blanpain (normal activity) || 22.9<ref name="JPL_289"/> || 320 m |}

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 <math>M_{1} = 8.5</math>.<ref name="Yoshida"/> It was subsequently lost and was only rediscovered in 2003. At that time, its absolute magnitude had decreased to <math>M_{1} = 22.9</math>,<ref name="JPL_289"/> 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.<ref name="Yoshida"/><ref name="Jewitt"/>

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.<ref name="Lamy2004"/>

== Meteors == For a meteor, the standard distance for measurement of magnitudes is at an altitude of {{convert|100|km|0|abbr=on}} at the observer's zenith.<ref name="IMO"/><ref name="SSD_Glossary"/>

== See also == * Araucaria Project * Hertzsprung–Russell diagram – relates absolute magnitude or luminosity versus spectral color or surface temperature. * Jansky - the preferred unit for radio astronomy – linear in power/unit area * List of most luminous stars * Photographic magnitude * Surface brightness – the ''magnitude'' for extended objects * Zero point (photometry) – the typical calibration point for star flux

==Notes== {{notelist|refs=

{{efn|name="Mag_formula"|The <math>1,329\text{ km}</math> factor can be computed as <math>2\text{ AU}\cdot10^{H_{\text{Sun}}/5}</math>, where <math>H_{\text{Sun}}=-26.76</math>, the absolute magnitude of the Sun, and <math>1\text{ AU}=1.4959787\times10^{8}\text{ km}.</math>}}

}}

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<ref name="COBS 2011L4">{{Cite web | url = https://cobs.si/analysis/?comet=654&from_date=2013-01-01+00%3A00&to_date=2013-06-01+00%3A00&observation_type=V&plot_x_value=1&plot_y_value=1&fit_option=1&fit_option=3&exclude_faint=on&exclude_not_accurate=on&exclude_issue=on&observer=&association=&country=&compare_values=compare | title = Comet C/2011 L4 (PANSTARRS) | publisher = COBS | access-date = 22 March 2026 }} Simple fit: H0=5.5, n=3.7</ref>

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<ref name="IMCCE">{{cite web | title=Encyclopedia - the brightest bodies | website=IMCCE | url=https://promenade.imcce.fr/en/pages5/572.html | access-date=2023-05-29}}</ref>

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<ref name=Bernardinelli>{{cite web|type = 2021-08-08 last obs.|title = JPL Small-Body Database Browser: (2014 UN271)|url = https://ssd.jpl.nasa.gov/sbdb.cgi?sstr=54161348|publisher = Jet Propulsion Laboratory|accessdate = 15 September 2021}}</ref>

}}

== External links == * [http://astro.pas.rochester.edu/~aquillen/ast142/costanti.html Reference zero-magnitude fluxes] {{Webarchive|url=https://web.archive.org/web/20030222000548/http://astro.pas.rochester.edu/~aquillen/ast142/costanti.html |date=22 February 2003 }} * [http://www.iau.org/ International Astronomical Union] * [http://www.fxsolver.com/solve/share/knc8CRLY7WMX324G2I7GXg==/ Absolute Magnitude of a Star calculator] * [http://www.astronomynotes.com/starprop/s4.htm The Magnitude system] * [http://webcitation.org/query.php About stellar magnitudes] {{Webarchive|url=https://web.archive.org/web/20211027163807/https://webcitation.org/query.php |date=27 October 2021 }} * [http://simbad.u-strasbg.fr/sim-fid.pl Obtain the magnitude of any star] – SIMBAD * [http://www.minorplanetcenter.org/iau/lists/Sizes.html Converting magnitude of minor planets to diameter] * [https://web.archive.org/web/20010302182040/http://neo.jpl.nasa.gov/glossary/h.html Another table for converting asteroid magnitude to estimated diameter]

Category:Observational astronomy