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# Absolute magnitude

Absolute magnitude (M) 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 were viewed from a distance of exactly 10 parsecs (32.6 light-years), 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 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 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.

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 100(n/5). For example, a star of absolute magnitude MV=3.0 would be 100 times more luminous than a star of absolute magnitude MV=8.0 as measured in the V filter band. The Sun has absolute magnitude MV=+4.83.[1] Highly luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute B magnitude of about −20.8.[2]

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.[3]

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.

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• ✪ Stargazing Basics 2: Understanding star magnitude in astronomy
• ✪ Astronomy - Measuring Distance, Size, and Luminosity (18 of 30) Absolute Magnitude
• ✪ Introductory Astronomy: Magnitudes of Stars
• ✪ Top 10 Brightest Stars (Absolute Magnitude)
• ✪ Astronomy - Ch. 17: The Nature of Stars (4 of 37) What is Absolute Magnitude?

#### Transcription

Hi I’m David Fuller from the “Eyes on the Sky” video series. Let’s look at another aspects of Stargazing Basics, this time, magnitudes of various objects. Magnitude is just a fancy way to describe the difference in brightness of objects we see in the night sky. And it’s not too hard to learn how it works, but let’s start with some history. The Greek astronomer Hipparchus developed the magnitude scale back in the 2-nd century BC, when he assigned the brightest stars a magnitude of “one” and the dimmest stars that of “six,” the in-between stars of 2, 3 , 4 and 5 magnitude assigned according to brightness. He assumed that the difference in brightness of stars was 2.512 times brighter than the next dimmest one. Taken across that full 6 magnitude scale, this meant that the brightest stars – the first magnitude ones - were 100 times brighter than the dimmest, 6-th magnitude ones. It also allows for a fairly easy way to determine other brightness differences: The difference from first to third magnitude is 6.3 times; first to fourth, 15.8, first to fifth , about 40 times. This worked just fine until Galileo turned a telescope towards the heavens, and humans discovered there were a LOT more stars out there than just the first through sixth magnitude ones they could see naked eye. But what they did was just extrapolate the scale further. So much like golf or ERA in baseball, the lower the number, the brighter the star, and the higher the number, the dimmer the star. 11-th magnitude is 100 times dimmer than 6-th magnitude, and that 11-th magnitude star would also be 10,000 times dimmer than a first magnitude star. In the other direction, we have some objects that are brighter than first magnitude stars. Hipparchus actually fudged the numbers a bit; the star Sirius in the winter sky is definitely brighter than most other bright stars, and it shines at magnitude negative one point four. The planet Jupiter is brighter than that, often appearing at around magnitude negative 2, and Venus brighter still, typically around negative four. When objects are bright, they are denoted on star charts with larger dots; dimmer stars are usually given smaller dots. This helps us locate brighter objects in the sky more easily. But beyond the stars’ magnitudes is that of the Moon and Sun. The full Moon in the night sky shines at magnitude negative 12.7, and the Sun at magnitude negative 26.7! Those are both bright compared to starlight, but also a huge difference, even between themselves, as the Sun is 400,000 times brighter than the full Moon! Now everything we have discussed has been “visual magnitude,” meaning how bright or dim objects appear to our location on Earth. Another concept is “absolute magnitude,” used by astronomers to compare the relative brightness of objects when placed the same distance. But that is more complex than what we need to know for simple stargazing purposes. However, for visual magnitude, it helps to understand how it relates to other objects besides stars. Stars are point-like objects in the sky, so it is easy to assess their magnitude and brightness. But galaxies, star clusters and nebula are not quite as easy. So we look at their “integrated magnitude.” That’s a way of saying that the same magnitude is now spread out over a larger area. For small objects, that can mean it is something that will look brighter to us in telescopes. If the area if very large though – such as the galaxy M33 or the nebula M1, those objects may look very dim, relatively speaking. If possible, try to look for the surface brightness of an object; objects above a surface brightness of 11 or 12 can be very difficult to find from light polluted areas. That’s a quick overview of the magnitude scale, and how to understand it. Just remember the Sun and Moon have negative magnitudes and are the brightest visual magnitude objects, and lower numbers are dimmer stars and objects, and you’ll be in good shape! In the next video, I’ll explain how to easily measure distance in the sky, moving FROM bright objects to the dimmer ones, so we can better utilize optical aids like binoculars and telescopes to find interesting objects. Thanks for watching; I’m David Fuller. Keep your eyes on the sky and your outdoor lights aimed down by using dark sky friendly lighting fixtures, so we can all see, what’s up.

## 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.

The measurement of absolute magnitude is made with an instrument called a bolometer. When using an absolute magnitude, one must specify the type of electromagnetic radiation being measured. When referring to total energy output, the proper term is bolometric magnitude. The bolometric magnitude usually is computed from the visual magnitude plus a bolometric correction, Mbol = MV + BC. This correction is needed because very hot stars radiate mostly ultraviolet radiation, whereas very cool stars radiate mostly infrared radiation (see Planck's law).

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 (−7.0), Deneb (−7.2), Naos (−6.0), and Betelgeuse (−5.6). For comparison, Sirius has an absolute magnitude of 1.4, which is brighter than the Sun, whose absolute visual magnitude is 4.83 (it actually serves as a reference point). The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75.[4][5] Absolute magnitudes of stars generally range from −10 to +17. 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).

### 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 m = 1, and the dimmest stars visible to the naked eye are assigned m = 6.[6] 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) is 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. The relation can be written in terms of logarithm:

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

where the insignificance of extinction by gas and dust is assumed. Typical extinction rates within the galaxy are 1 to 2 magnitudes per kiloparsec, when dark clouds are taken into account.[7]

For objects at very large distances (outside the Milky Way) the luminosity distance dL (distance defined using luminosity measurements) must be used instead of d (in parsecs), because the Euclidean approximation is invalid for distant objects and general relativity must be taken into account. Moreover, the cosmological redshift complicates the relation 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 approximated using 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:

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

#### Examples

Rigel has a visual magnitude mV of 0.12 and distance 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 mV 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 mV of 9.36 and a distance modulus μ of 31.06

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

### Bolometric magnitude

The bolometric magnitude Mbol, 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:[6]

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

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)
Mbol,⊙ is the bolometric magnitude of the Sun
Mbol,★ is the bolometric magnitude of the star.

In August 2015, the International Astronomical Union passed Resolution B2[8] 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.[9] Combined with incorrect assumed absolute bolometric magnitudes for the Sun could lead to systematic errors in estimated stellar luminosities (and stellar properties calculated which rely on stellar luminosity, such as radii, ages, and so on).

Resolution B2 defines an absolute bolometric magnitude scale where Mbol = 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 Mbol,⊙ = 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 mbol = 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 mbol,⊙ = −26.832.[9]

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
Mbol is the bolometric magnitude of the star

The new IAU absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, the nominal solar luminosity corresponds closely to Mbol = 4.74, a value that was commonly adopted by astronomers before the 2015 IAU resolution.[9]

The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude Mbol as:

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

using the variables as defined previously.

## Solar System bodies (H)

Abs Mag (H)
and Diameter
for asteroids
(albedo=0.15)[10]
H Diameter
10 34 km
12.6 10 km
15 3.4 km
17.6 1 km
19.2 500 meter
20 340 meter
22.6 100 meter
24.2 50 meter
25 34 meter
27.6 10 meter
30 3.4 meter

For planets and asteroids a definition of absolute magnitude that is more meaningful for non-stellar objects is used. The absolute magnitude, commonly called ${\displaystyle H}$, 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).[11] Solar System bodies are illuminated by the Sun, therefore the magnitude 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.

### Apparent magnitude

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).[12]

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:

• dBO is the distance between the body and the observer
• dBS is the distance between the body and the Sun
• dOS is the distance between the observer and the Sun
• d0 is 1 AU, the average distance between the Earth and the Sun

### Approximations for phase integral ${\displaystyle q(\alpha )}$

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.[12]

#### Planets

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[13]

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

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 ${\displaystyle {\frac {1}{\pi }}}$ as much light as full phase (${\displaystyle \alpha =0^{\circ }}$).

For 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[14][15]

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

Example: The Moon's absolute magnitude ${\displaystyle H}$ can be calculated from its diameter ${\displaystyle D=3474{\text{ km}}}$ and geometric albedo ${\displaystyle p=0.113}$:[16]

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

We have ${\displaystyle d_{BS}=1{\text{ AU}}}$, ${\displaystyle d_{BO}=384400{\text{ km}}=0.00257{\text{ AU}}.}$ At quarter phase, ${\displaystyle q(\alpha )\approx {\frac {2}{3\pi }}}$ (according to the diffuse reflector model), this yields an apparent magnitude of ${\displaystyle m=+0.28+5\log _{10}{\left(1\cdot 0.00257\right)}-2.5\log _{10}{\left({\frac {2}{3\pi }}\right)}=-10.99.}$ 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.[17]

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.[12] 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[18] 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.[19]

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. Earth's apparent magnitude cannot be predicted as accurately as that of most other planets.[18]

#### Asteroids

Asteroid 1 Ceres, imaged by the Dawn spacecraft at phase angles of 0°, 7° and 33°. 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. Atmosphereless bodies, 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.[12]

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.[20]

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

where

the phase integral is ${\displaystyle q(\alpha )=\left(1-G\right)\phi _{1}\left(\alpha \right)+G\phi _{2}\left(\alpha \right)}$

and

${\displaystyle \phi _{i}\left(\alpha \right)=\exp {\left(-A_{i}\left(\tan {\frac {\alpha }{2}}\right)^{B_{i}}\right)}}$ for ${\displaystyle i=1}$ or ${\displaystyle 2}$, ${\displaystyle A_{1}=3.332}$, ${\displaystyle A_{2}=1.862}$, ${\displaystyle B_{1}=0.631}$ and ${\displaystyle B_{2}=1.218}$.[21]

This relation is valid for phase angles ${\displaystyle \alpha <120^{\circ }}$, and works best when ${\displaystyle \alpha <20^{\circ }}$.[22]

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.[22] In rare cases, ${\displaystyle G}$ can be negative.[21][23] An example is 101955 Bennu, with ${\displaystyle G=-0.08}$.[24]

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 2019, this ${\displaystyle HG_{1}G_{2}}$-system has not been adopted by either the Minor Planet Center nor Jet Propulsion Laboratory.[12][25]

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.[26] 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.[22][12]

### Cometary magnitudes

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).[27] 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.[28]

For example, the lightcurve of comet C/2011 L4 (PANSTARRS) can be approximated by ${\displaystyle M_{1}=5.41{\text{, }}K_{1}=3.69.}$[29] 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, ${\displaystyle m_{1}=+0.5}$.[30]

Absolute magnitudes and sizes of comet nuclei
Comet Absolute
magnitude ${\displaystyle M_{1}}$[31]
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[32]
289P/Blanpain (during 1819 outburst) 8.5[33] 320 m[34]
289P/Blanpain (normal activity) 22.9[35] 320 m

At the same distance, Comet Hale-Bopp is about 130 times brighter than Comet Halley.[31]

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 ${\displaystyle M_{1}=8.5}$.[33] It was subsequently lost, and was only rediscovered in 2003. At that time, its absolute magnitude had decreased to ${\displaystyle M_{1}=22.9}$,[35] 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.[33][34]

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.[36]

## Meteors

For a meteor, the standard distance for measurement of magnitudes is at an altitude of 100 km (62 mi) at the observer's zenith.[37][38]

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