<p>The magnitude of the Moon without the <em>opposition surge</em> is given by a simple equation:</p>
<p><span class="math-container">$m = -12.73 + 1.49 \cdot |\psi| + 0.043 \cdot \psi^4,$</span></p>
<p>where <span class="math-container">$\psi$</span> is the <em>phase angle</em> in radians <a href="http://esoads.eso.org/abs/1976asqu.book.....A" rel="nofollow noreferrer">(Allen 1976)</a>.  This can be converted to flux using <span class="math-container">$m \propto -2.5 \cdot ^{10}\!\log F$</span>,
where <span class="math-container">$^{10}\!\log$</span> denotes the 10-base logarithm (as opposed to e.g. the natural log).</p>
<p>For the increase in flux due to the variable distance and opposition surge, I multiply F with the factor</p>
<p><span class="math-container">$\left(\frac{\Delta_0}{\Delta}\right)^2  \cdot  \max\left(1, 1.35 - 2.865 \cdot |\psi| \right),$</span></p>
<p>where <span class="math-container">$\Delta_0$</span> and <span class="math-container">$\Delta$</span> are the mean and current distance to the Moon, respectively.  However, I can't find a reference for the last term at the moment, and I can't remember where I obtained it.  I'll post that information if I find it.</p>

CC BY-SA 4.0
2023-07-02T16:18:45.010Z
The reference can be taken as [Krisciunas, Schaefer, _"A model of the Brightness of Moonlight"_](https://iopscience.iop.org/article/10.1086/132921), which says: _"For $|\alpha| < 7°$ the lunar opposition effect should be taken into account, in which case the derived value of $B_{\text{moon}}$ must be multiplied by a factor in between 1.00 and
1.35."_
Ruslan
CC BY-SA 4.0
2023-07-05T10:01:21.270Z
I *think* the correction can be found in Schaefer B., "To the visual limits", Sky & Telescope vol. 95, No. 5, p 57 (1998), but I can't get hold of a copy and I'm not sure it is the original reference.
AstroFloyd
CC BY-SA 4.0
2023-07-05T12:22:17.617Z
This article (which is easily googlable BTW) only shows a program with a magical addition `CM` to the lunar magnitude, it doesn't discuss the meaning.
Ruslan
CC BY-SA 4.0
2023-07-06T14:52:20.083Z
@Ruslan CM is the magnitude correction for the different optical bands (U,B,V,R,I).  I got the article from a colleague; I haven't discovered an opposition surge in it.  I was mistaken...
AstroFloyd
CC BY-SA 4.0
2016-07-17T15:10:17.270Z
2020-11-06T16:49:13.107Z
2020-11-06T16:49:13.107Z
AstroFloyd
<p>Brightness of the lunar disk varies as a function of phase angle. This phenomenon is very well established, as is the increase in brightness at low phase angles.</p>

<p>Is there a simple analytical formula that describes this function? It looks like brightness could be modeled as an exponential as a function of phase angle, modulo the opposition surge.</p>

<p>Thanks.</p>

<p>EDIT: I realize that these effects depend on the wavelength of the reflected light. I'm looking for an average curve, though.</p>

<p><img src="https://i.sstatic.net/CNcmj.gif" alt="Lunar brightness curve, pointing out the opposition surge"></p>

<the-moon><light><moon-phases>
Is there a simple, analytical formula for the lunar phase brightness curve?
qpo2
# Is there a simple, analytical formula for the lunar phase brightness curve?

(Asked by: username_0 on 2015-03-24)

Brightness of the lunar disk varies as a function of phase angle. This phenomenon is very well established, as is the increase in brightness at low phase angles.

Is there a simple analytical formula that describes this function? It looks like brightness could be modeled as an exponential as a function of phase angle, modulo the opposition surge.

Thanks.

EDIT: I realize that these effects depend on the wavelength of the reflected light. I'm looking for an average curve, though.

![Lunar brightness curve, pointing out the opposition surge](https://i.sstatic.net/CNcmj.gif)

**Tags:** the-moon, light, moon-phases

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## 1 answers

### Answer by username_1 on 2016-07-17. Score: 4

The magnitude of the Moon without the *opposition surge* is given by a simple equation:

$m = -12.73 + 1.49 \cdot |\psi| + 0.043 \cdot \psi^4,$

where $\psi$ is the *phase angle* in radians [(Allen 1976)](http://esoads.eso.org/abs/1976asqu.book.....A). This can be converted to flux using $m \propto -2.5 \cdot ^{10}\!\log F$,
where $^{10}\!\log$ denotes the 10-base logarithm (as opposed to e.g. the natural log).

For the increase in flux due to the variable distance and opposition surge, I multiply F with the factor

$\left(\frac{\Delta\_0}{\Delta}\right)^2 \cdot \max\left(1, 1.35 - 2.865 \cdot |\psi| \right),$

where $\Delta\_0$ and $\Delta$ are the mean and current distance to the Moon, respectively. However, I can't find a reference for the last term at the moment, and I can't remember where I obtained it. I'll post that information if I find it.

> **Comments:**
> **username_2**: The reference can be taken as [Krisciunas, Schaefer, _"A model of the Brightness of Moonlight"_](https://iopscience.iop.org/article/10.1086/132921), which says: _"For $|\alpha| < 7°$ the lunar opposition effect should be taken into account, in which case the derived value of $B_{\text{moon}}$ must be multiplied by a factor in between 1.00 and 1.35."_  
> **username_1**: I *think* the correction can be found in Schaefer B., "To the visual limits", Sky & Telescope vol. 95, No. 5, p 57 (1998), but I can't get hold of a copy and I'm not sure it is the original reference.  
> **username_2**: This article (which is easily googlable BTW) only shows a program with a magical addition `CM` to the lunar magnitude, it doesn't discuss the meaning.  
> **username_1**: @Ruslan CM is the magnitude correction for the different optical bands (U,B,V,R,I).  I got the article from a colleague; I haven't discovered an opposition surge in it.  I was mistaken...  

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