Editing Absolute magnitude (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"/>