# Minnaert function

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The '''Minnaert function''' is a [photometric](/source/photometry_(astronomy)) function used to interpret [astronomical](/source/astronomical) observations<ref>{{cite journal | doi = 10.1016/S0019-1035(03)00075-7 | bibcode=2003Icar..163..150C | volume=163 | title=Probing Titan's lower atmosphere with acousto-optic tuning | journal=Icarus | year=2003 | pages=150–163| last1=Chanover | first1=N.J. | last2=Anderson | first2=C.M. | last3=McKay | first3=C.P. | last4=Rannou | first4=P. | last5=Glenar | first5=D.A. | last6=Hillman | first6=J.J. | last7=Blass | first7=W.E. | issue=1 }}</ref><ref>{{cite journal | doi = 10.1016/j.icarus.2006.05.006 | bibcode=2006Icar..184..401S | volume=184 | title=Martian phase function: Modeling the visible to near-infrared surface photometric function using HST-WFPC2 data | journal=Icarus | year=2006 | pages=401–423| last1=Soderblom | first1=J. | last2=Belliii | first2=J. | last3=Hubbard | first3=M. | last4=Wolff | first4=M. | issue=2 }}</ref> and [remote sensing](/source/remote_sensing) data for the [Earth](/source/Earth).<ref>{{cite journal | doi = 10.1080/01431160500104194 | volume=26 | title=The use of the Minnaert correction for land-cover classification in mountainous terrain | journal=International Journal of Remote Sensing | year=2005 | pages=3831–3851| last1=Blesius | first1=L. | last2=Weirich | first2=F. | issue=17 | bibcode=2005IJRS...26.3831B | s2cid=129750287 }}</ref> It was named after the astronomer [Marcel Minnaert](/source/Marcel_Minnaert). This function expresses the [radiance factor](/source/radiance_factor) (RADF) as a function the [phase angle](/source/Phase_angle_(astronomy)) (<math>\alpha</math>), the [photometric latitude](/source/photometric_latitude) (<math>\varphi</math>) and the [photometric longitude](/source/photometric_longitude) (<math>\lambda</math>).

:<math>
   \text{RADF} = \frac{I}{F} = \pi~A_M~\mu_0^k~\mu^{k-1}
 </math>
where <math>A_M</math> is the Minnaert [albedo](/source/albedo), <math>k</math> is an empirical parameter, <math>I</math> is the scattered [radiance](/source/radiance) in the direction <math>(\alpha,\varphi,\lambda)</math>, <math>\pi F</math> is the incident radiance, and
:<math>
   \mu_0 = \cos\varphi~\cos(\alpha-\lambda) ~;~~ \mu = \cos\varphi~\cos\lambda ~.
 </math>
The phase angle is the angle between the light source and the observer with the object as the center.

The assumptions made are:
* the surface is illuminated by a distant point source.
* the surface is isotropic and flat.

Minnaert's contribution is the introduction of the parameter <math>k</math>, having a value between 0 and 1,<ref>{{Cite journal|url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1941ApJ....93..403M&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf|bibcode=1941ApJ....93..403M|title=The reciprocity principle in lunar photometry|last1=Minnaert|first1=M.|journal=The Astrophysical Journal|year=1941|volume=93|page=403|doi=10.1086/144279}}</ref> originally for a better interpretation of observations of the [Moon](/source/Moon). In remote sensing the use of this function is referred to as ''Minnaert topographic correction'', a necessity when interpreting images of rough terrain.

== References ==
<references/>

Category:Observational astronomy
Category:Photometric systems
Category:Equations of astronomy

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Adapted from the Wikipedia article [Minnaert function](https://en.wikipedia.org/wiki/Minnaert_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Minnaert_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.