Editing Absolute magnitude (section)

==== 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.