Editing Absolute magnitude (section)

=== 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 [[Surface roughness|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 ====
[[File:Diffuse reflector sphere disk.png|thumb|right|240px|Diffuse reflection on sphere and flat disk]]
[[File:Diffuse reflection model phase functions.svg|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 [[Lambertian diffuse lighting model|ideal diffuse reflecting]] [[sphere]]s. Let <math>\alpha</math> be the phase angle in [[Degree (angle)|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 [[kilometer]]s) and geometric albedo <math>p</math> of a body are related by<ref name="sizemagnitude"/><ref 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}{D\sqrt{p}}}.</math>

Example: The [[Moon|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}{3474\sqrt{0.113}}} = +0.28.</math>
We have <math>d_{BS}=1\text{ AU}</math>, <math>d_{BO}=384400\text{ km}=0.00257\text{ AU}.</math>
At [[lunar phases|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 [[phase curve (astronomy)|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">{{cite web | title=Encyclopedia - the brightest bodies | website=IMCCE | url=https://promenade.imcce.fr/en/pages5/572.html | access-date=2023-05-29}}</ref>
! <math>H</math>
! Approximation for <math>-2.5\log_{10}{q(\alpha)}</math>
|-
| [[Mercury (planet)|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 (planet)|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>+240.44228-2.81914\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>{{Cite book|first=A.N.|last=Cox|year=2000|title=Allen's Astrophysical Quantities, fourth edition|publisher=Springer-Verlag|pages=310}}</ref>
| 0.2
| +0.28
|
* <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 (planet)|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 (planet)|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 (planet)|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 (planet)|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 (planet)|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 (planet)|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|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 = +0.28+5\log_{10}{\left(1\cdot0.00257\right)}+2.71= -9.96,</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.

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 clouds|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)|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]].]]
[[File:Asteroid HG phase integrals.svg|thumb|240px|Phase integrals for various values of G]]
[[File:Slope parameter G.png|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 [[International Astronomical Union|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 [[ephemeris|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=-0.08</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 [[Light curve|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"/>