# Is there a simple, analytical formula for the lunar phase brightness curve?

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.

• Could you source that image and/or data? Presumably, curve fitting methods could find an arbitrarily good match? adsabs.harvard.edu/full/1991PASP..103.1033K may also be helpful?
– user21
Mar 26, 2015 at 14:12

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). 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.

• The reference can be taken as Krisciunas, Schaefer, "A model of the Brightness of Moonlight", 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." Jul 2, 2023 at 16:18
• 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. Jul 5, 2023 at 10:01
• 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. Jul 5, 2023 at 12:22
• @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... Jul 6, 2023 at 14:52