Absolute 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 m = 1, and the dimmest stars visible to the naked eye are assigned m = 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 magnitude M and apparent magnitude m from any distance d (in parsecs, with 1 pc = 3.2616 light-years) are related by

$${\displaystyle 100^{\frac {m-M}{5}}={\frac {F_{10}}{F}}=\left({\frac {d}{10\;\mathrm {pc} }}\right)^{2},}$$

where F is the radiant flux measured at distance d (in parsecs), F₁₀ the radiant flux measured at distance 10 pc. Using the common logarithm, the equation can be written as

$${\displaystyle M=m-5\log _{10}(d_{\text{pc}})+5=m-5\left(\log _{10}d_{\text{pc}}-1\right),}$$

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.[8]

For objects at very large distances (outside the Milky Way) the luminosity distance d_L (distance defined using luminosity measurements) must be used instead of 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 M can also be written in terms of the apparent magnitude m and stellar parallax p:

$${\displaystyle M=m+5\left(\log _{10}p+1\right),}$$

or using apparent magnitude m and distance modulus μ:

$${\displaystyle M=m-\mu .}$$

Rigel has a visual magnitude m_V of 0.12 and distance of about 860 light-years:

$${\displaystyle M_{\mathrm {V} }=0.12-5\left(\log _{10}{\frac {860}{3.2616}}-1\right)=-7.0.}$$

Vega has a parallax p of 0.129″, and an apparent magnitude m_V of 0.03:

$${\displaystyle M_{\mathrm {V} }=0.03+5\left(\log _{10}{0.129}+1\right)=+0.6.}$$

The Black Eye Galaxy has a visual magnitude m_V of 9.36 and a distance modulus μ of 31.06:

$${\displaystyle M_{\mathrm {V} }=9.36-31.06=-21.7.}$$

The bolometric absolute magnitude 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:[7]

$${\displaystyle M_{\mathrm {bol,\star } }-M_{\mathrm {bol,\odot } }=-2.5\log _{10}\left({\frac {L_{\star }}{L_{\odot }}}\right)}$$

which makes by inversion:

$${\displaystyle {\frac {L_{\star }}{L_{\odot }}}=10^{0.4\left(M_{\mathrm {bol,\odot } }-M_{\mathrm {bol,\star } }\right)}}$$

where

• L_⊙ is the Sun's luminosity (bolometric luminosity)
• L_★ is the star's luminosity (bolometric luminosity)
• M_bol,⊙ is the bolometric magnitude of the Sun
• M_bol,★ is the bolometric magnitude of the star.

In August 2015, the International Astronomical Union passed Resolution B2[9] 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.[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 where M_bol = 0 corresponds to luminosity L₀ = 3.0128×10²⁸ W, with the zero point luminosity L₀ set such that the Sun (with nominal luminosity 3.828×10²⁶ W) corresponds to absolute bolometric magnitude 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 m_bol = 0 corresponds to irradiance f₀ = 2.518021002×10⁻⁸ W/m². Using the IAU 2015 scale, the nominal total solar irradiance ("solar constant") measured at 1 astronomical unit (1361 W/m²) corresponds to an apparent bolometric magnitude of the Sun of m_bol,⊙ = −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:

$${\displaystyle M_{\mathrm {bol} }=-2.5\log _{10}{\frac {L_{\star }}{L_{0}}}\approx -2.5\log _{10}L_{\star }+71.197425}$$

where

• L_★ is the star's luminosity (bolometric luminosity) in watts
• L₀ is the zero point luminosity 3.0128×10²⁸ W
• 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 M_bol = 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 magnitude M_bol as:

$${\displaystyle L_{\star }=L_{0}10^{-0.4M_{\mathrm {bol} }}}$$

using the variables as defined previously.

The phase angle ${\displaystyle \alpha }$ can be calculated from the distances body-sun, observer-sun and observer-body, using the law of cosines.

The absolute magnitude ${\displaystyle H}$ can be used to calculate the apparent magnitude ${\displaystyle m}$ of a body. For an object reflecting sunlight, ${\displaystyle H}$ and ${\displaystyle m}$ are connected by the relation

$${\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}$$

where ${\displaystyle \alpha }$ is the phase angle, the angle between the body-Sun and body–observer lines. ${\displaystyle q(\alpha )}$ is the phase integral (the integration of reflected light; a number in the 0 to 1 range).[13]

By the law of cosines, we have:

$${\displaystyle \cos {\alpha }={\frac {d_{\mathrm {BO} }^{2}+d_{\mathrm {BS} }^{2}-d_{\mathrm {OS} }^{2}}{2d_{\mathrm {BO} }d_{\mathrm {BS} }}}.}$$

Distances:

• d_BO is the distance between the body and the observer
• d_BS is the distance between the body and the Sun
• d_OS is the distance between the observer and the Sun
• d₀, a unit conversion factor, is the constant 1 AU, the average distance between the Earth and the Sun

The value of ${\displaystyle q(\alpha )}$ 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.[13]

Diffuse reflection on sphere and flat disk

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 ${\displaystyle \alpha }$ be the phase angle in degrees, then[14]

$${\displaystyle q(\alpha )={\frac {2}{3}}\left(\left(1-{\frac {\alpha }{180^{\circ }}}\right)\cos {\alpha }+{\frac {1}{\pi }}\sin {\alpha }\right).}$$

A full-phase diffuse sphere reflects two-thirds as much light as a diffuse flat disk of the same diameter. A quarter phase (${\displaystyle \alpha =90^{\circ }}$) has ${\textstyle {\frac {1}{\pi }}}$ as much light as full phase (${\displaystyle \alpha =0^{\circ }}$).

By contrast, a diffuse disk reflector model is simply ${\displaystyle q(\alpha )=\cos {\alpha }}$, 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 ${\displaystyle p}$, a measure for the reflectivity of planetary surfaces, is based on the diffuse disk reflector model. The absolute magnitude ${\displaystyle H}$, diameter ${\displaystyle D}$ (in kilometers) and geometric albedo ${\displaystyle p}$ of a body are related by[15][16][17]

$${\displaystyle D={\frac {1329}{\sqrt {p}}}\times 10^{-0.2H}\mathrm {km} .}$$

Example: The Moon's absolute magnitude ${\displaystyle H}$ can be calculated from its diameter ${\displaystyle D=3474{\text{ km}}}$ and geometric albedo ${\displaystyle p=0.113}$:[18]

$${\displaystyle H=5\log _{10}{\frac {1329}{3474{\sqrt {0.113}}}}=+0.28.}$$

We have ${\displaystyle d_{BS}=1{\text{ AU}}}$, ${\displaystyle d_{BO}=384400{\text{ km}}=0.00257{\text{ AU}}.}$ At quarter phase, ${\textstyle q(\alpha )\approx {\frac {2}{3\pi }}}$ (according to the diffuse reflector model), this yields an apparent magnitude of ${\textstyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}-2.5\log _{10}{\left({\frac {2}{3\pi }}\right)}=-10.99.}$ The actual value is somewhat lower than that, ${\displaystyle m=-10.0.}$ The phase curve of the Moon is too complicated for the diffuse reflector model.[19]

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 ${\displaystyle -2.5\log _{10}{q(\alpha )}}$ in the formula for m have been derived empirically, to match observations at different phase angles. The approximations recommended by the Astronomical Almanac[20] are (with ${\displaystyle \alpha }$ in degrees):

Planet	${\displaystyle H}$	Approximation for ${\displaystyle -2.5\log _{10}{q(\alpha )}}$
Mercury	−0.613	${\displaystyle +6.328\times 10^{-2}\alpha -1.6336\times 10^{-3}\alpha ^{2}+3.3644\times 10^{-5}\alpha ^{3}-3.4265\times 10^{-7}\alpha ^{4}+1.6893\times 10^{-9}\alpha ^{5}-3.0334\times 10^{-12}\alpha ^{6}}$
Venus	−4.384	${\displaystyle -1.044\times 10^{-3}\alpha +3.687\times 10^{-4}\alpha ^{2}-2.814\times 10^{-6}\alpha ^{3}+8.938\times 10^{-9}\alpha ^{4}}$ (for ${\displaystyle 0^{\circ }<\alpha \leq 163.7^{\circ }}$) ${\displaystyle +240.44228-2.81914\alpha +8.39034\times 10^{-3}\alpha ^{2}}$ (for ${\displaystyle 163.7^{\circ }<\alpha <179^{\circ }}$)
Earth	−3.99	${\displaystyle -1.060\times 10^{-3}\alpha +2.054\times 10^{-4}\alpha ^{2}}$
Mars	−1.601	${\displaystyle +2.267\times 10^{-2}\alpha -1.302\times 10^{-4}\alpha ^{2}}$ (for ${\displaystyle 0^{\circ }<\alpha \leq 50^{\circ }}$) ${\displaystyle +1.234-2.573\times 10^{-2}\alpha +3.445\times 10^{-4}\alpha ^{2}}$ (for ${\displaystyle 50^{\circ }<\alpha \leq 120^{\circ }}$)
Jupiter	−9.395	${\displaystyle -3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}$ (for ${\displaystyle \alpha \leq 12^{\circ }}$) ${\displaystyle -0.033-2.5\log _{10}{\left(1-1.507\left({\frac {\alpha }{180^{\circ }}}\right)-0.363\left({\frac {\alpha }{180^{\circ }}}\right)^{2}-0.062\left({\frac {\alpha }{180^{\circ }}}\right)^{3}+2.809\left({\frac {\alpha }{180^{\circ }}}\right)^{4}-1.876\left({\frac {\alpha }{180^{\circ }}}\right)^{5}\right)}}$ (for ${\displaystyle \alpha >12^{\circ }}$)
Saturn	−8.914	${\displaystyle -1.825\sin {\left(\beta \right)}+2.6\times 10^{-2}\alpha -0.378\sin {\left(\beta \right)}e^{-2.25\alpha }}$ (for planet and rings, ${\displaystyle \alpha <6.5^{\circ }}$ and ${\displaystyle \beta <27^{\circ }}$) ${\displaystyle -0.036-3.7\times 10^{-4}\alpha +6.16\times 10^{-4}\alpha ^{2}}$ (for the globe alone, ${\displaystyle \alpha \leq 6^{\circ }}$) ${\displaystyle +0.026+2.446\times 10^{-4}\alpha +2.672\times 10^{-4}\alpha ^{2}-1.505\times 10^{-6}\alpha ^{3}+4.767\times 10^{-9}\alpha ^{4}}$ (for the globe alone, ${\displaystyle 6^{\circ }<\alpha <150^{\circ }}$)
Uranus	−7.110	${\displaystyle -8.4\times 10^{-4}\phi '+6.587\times 10^{-3}\alpha +1.045\times 10^{-4}\alpha ^{2}}$ (for ${\displaystyle \alpha <3.1^{\circ }}$)
Neptune	−7.00	${\displaystyle +7.944\times 10^{-3}\alpha +9.617\times 10^{-5}\alpha ^{2}}$ (for ${\displaystyle \alpha <133^{\circ }}$ and ${\displaystyle t>2000.0}$)

Here ${\displaystyle \beta }$ 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 ${\displaystyle \phi '}$ is a small correction term depending on Uranus' sub-Earth and sub-solar latitudes. ${\displaystyle t}$ 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 ${\displaystyle \alpha \geq 179^{\circ }}$ for Venus, no observations are available, and the phase curve is unknown in those cases.

Example: On 1 January 2019, Venus was ${\displaystyle d_{BS}=0.719{\text{ AU}}}$ from the Sun, and ${\displaystyle d_{BO}=0.645{\text{ AU}}}$ from Earth, at a phase angle of ${\displaystyle \alpha =93.0^{\circ }}$ (near quarter phase). Under full-phase conditions, Venus would have been visible at ${\displaystyle m=-4.384+5\log _{10}{\left(0.719\cdot 0.645\right)}=-6.09.}$ Accounting for the high phase angle, the correction term above yields an actual apparent magnitude of ${\displaystyle m=-6.09+\left(-1.044\times 10^{-3}\cdot 93.0+3.687\times 10^{-4}\cdot 93.0^{2}-2.814\times 10^{-6}\cdot 93.0^{3}+8.938\times 10^{-9}\cdot 93.0^{4}\right)=-4.59.}$ This is close to the value of ${\displaystyle m=-4.62}$ predicted by the Jet Propulsion Laboratory.[21]

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 here 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.[20]

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.

Phase integrals for various values of G

Relation between the slope parameter ${\displaystyle G}$ and the opposition surge. Larger values of ${\displaystyle G}$ correspond to a less pronounced opposition effect. For most asteroids, a value of ${\displaystyle G=0.15}$ is assumed, corresponding to an opposition surge of ${\displaystyle 0.3{\text{ mag}}}$.

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 ${\displaystyle 0^{\circ }}$. 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.[13]

In 1985, the IAU adopted the semi-empirical ${\displaystyle HG}$-system, based on two parameters ${\displaystyle H}$ and ${\displaystyle G}$ called absolute magnitude and slope, to model the opposition effect for the ephemerides published by the Minor Planet Center.[22]

$${\displaystyle m=H+5\log _{10}{\left({\frac {d_{BS}d_{BO}}{d_{0}^{2}}}\right)}-2.5\log _{10}{q(\alpha )},}$$

where

• the phase integral is ${\displaystyle q(\alpha )=\left(1-G\right)\phi _{1}\left(\alpha \right)+G\phi _{2}\left(\alpha \right)}$ and
• ${\textstyle \phi _{i}\left(\alpha \right)=\exp {\left(-A_{i}\left(\tan {\frac {\alpha }{2}}\right)^{B_{i}}\right)}}$ for ${\displaystyle i=1}$ or ${\displaystyle 2}$, ${\displaystyle A_{1}=3.332}$, ${\displaystyle A_{2}=1.862}$, ${\displaystyle B_{1}=0.631}$ and ${\displaystyle B_{2}=1.218}$.[23]

This relation is valid for phase angles ${\displaystyle \alpha <120^{\circ }}$, and works best when ${\displaystyle \alpha <20^{\circ }}$.[24]

The slope parameter ${\displaystyle G}$ relates to the surge in brightness, typically 0.3 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 ${\displaystyle G=0.15}$ is assumed.[24] In rare cases, ${\displaystyle G}$ can be negative.[23][25] An example is 101955 Bennu, with ${\displaystyle G=-0.08}$.[26]

In 2012, the ${\displaystyle HG}$-system was officially replaced by an improved system with three parameters ${\displaystyle H}$, ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$, which produces more satisfactory results if the opposition effect is very small or restricted to very small phase angles. However, as of 2021, this ${\displaystyle HG_{1}G_{2}}$-system has not been adopted by either the Minor Planet Center nor Jet Propulsion Laboratory.[13][27]

The apparent magnitude of asteroids varies as they rotate, on time scales of seconds to weeks depending on their rotation period, by up to ${\displaystyle 2{\text{ mag}}}$ or more.[28] 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.[24][13]

The brightness of comets is given separately as total magnitude (${\displaystyle m_{1}}$, the brightness integrated over the entire visible extend of the coma) and nuclear magnitude (${\displaystyle m_{2}}$, the brightness of the core region alone).[29] 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 H.

The activity of comets varies with their distance from the Sun. Their brightness can be approximated as

$${\displaystyle m_{1}=M_{1}+2.5\cdot K_{1}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)}}$$

$${\displaystyle m_{2}=M_{2}+2.5\cdot K_{2}\log _{10}{\left({\frac {d_{BS}}{d_{0}}}\right)}+5\log _{10}{\left({\frac {d_{BO}}{d_{0}}}\right)},}$$

where ${\displaystyle m_{1,2}}$ are the total and nuclear apparent magnitudes of the comet, respectively, ${\displaystyle M_{1,2}}$ are its "absolute" total and nuclear magnitudes, ${\displaystyle d_{BS}}$ and ${\displaystyle d_{BO}}$ are the body-sun and body-observer distances, ${\displaystyle d_{0}}$ is the Astronomical Unit, and ${\displaystyle K_{1,2}}$ are the slope parameters characterising the comet's activity. For ${\displaystyle K=2}$, this reduces to the formula for a purely reflecting body (showing no cometary activity).[30]

For example, the lightcurve of comet C/2011 L4 (PANSTARRS) can be approximated by ${\displaystyle M_{1}=5.41{\text{, }}K_{1}=3.69.}$[31] On the day of its perihelion passage, 10 March 2013, comet PANSTARRS was ${\displaystyle 0.302{\text{ AU}}}$ from the Sun and ${\displaystyle 1.109{\text{ AU}}}$ from Earth. The total apparent magnitude ${\displaystyle m_{1}}$ is predicted to have been ${\displaystyle m_{1}=5.41+2.5\cdot 3.69\cdot \log _{10}{\left(0.302\right)}+5\log _{10}{\left(1.109\right)}=+0.8}$ at that time. The Minor Planet Center gives a value close to that, ${\displaystyle m_{1}=+0.5}$.[32]

Absolute magnitudes and sizes of comet nuclei

Comet	Absolute magnitude ${\displaystyle M_{1}}$[33]	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[34]
C/2014 UN271 (Bernardinelli-Bernstein)	6.7[35]	60–200 km?[36][37]
289P/Blanpain (during 1819 outburst)	8.5[38]	320 m[39]
289P/Blanpain (normal activity)	22.9[40]	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 ${\displaystyle M_{1}=8.5}$.[38] It was subsequently lost and was only rediscovered in 2003. At that time, its absolute magnitude had decreased to ${\displaystyle M_{1}=22.9}$,[40] 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.[38][39]

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.[41]