### 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? http://adsabs.harvard.edu/full/1991PASP..103.1033K may also be helpful?

• 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.