Editing Absolute magnitude (section)

==== 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"/>