Editing Absolute magnitude (section)

=== 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 [[reflection (physics)|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 (astronomy)|phase angle]], the angle between the body-Sun and body–observer lines. <math>q(\alpha)</math> is the [[Bond albedo#Phase integral|phase integral]] (the [[integral|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''<sub>BO</sub>}} is the distance between the body and the observer
* {{math|''d''<sub>BS</sub>}} is the distance between the body and the Sun
* {{math|''d''<sub>OS</sub>}} is the distance between the observer and the Sun
* {{math|''d''<sub>0</sub>}}, a [[unit conversion]] factor, is the constant 1&nbsp;[[Astronomical Unit|AU]], the average distance between the Earth and the Sun