Lunar Phase Function

A precise determination of the lunar phase function, defined as the change of the moonshine intensity as a function of the lunar phase, is essential in deriving the accurate measurement of the Earth's albedo. We have undertaken several steps of calibration in order to achieve the lunar phase function to the accuracy of half a percent. For this purpose, the data have been collected on the basis of daily observation for 2 years so far covering more than 300 useful nights.

## Atmospheric Extinction

To eliminate the effect of the atmospheric extinction, observations are carried out for as long as possible during the night so that a measurement of the intensity at varying airmass can be obtained which should follow the Beer's law. The airmass, as determined from the altitude of the moon in the sky at different times, is incorperated into the Beer's law fitting so that the intensity at zero airmass intensity for each night can be extrapolated as an output from the fitting. Fig.1a and Fig.1b illustrate the result of the airmass fit in a sample night. From top panel to bottom panel, the figures show the moonshine and earthshine intensities read out from corresponding fiducial patches and the intergral intensity of the whole crescent. Experiencs from observations show that the data usually follow the Beer's law very well and the fitting accuracy is often better than 1%.

 Beer's law fitting Beer's law fitting Fig.1a Intensity against time Fig.1b Intensity against airmass (click here for PostScript file) (click here for PostScript file)

We also note that in some nights, evolution of the eartshine intensity throughout a night does not well follow the Beer's law, because apart from the atmospheric transmission, the evolution of the earthshine is also influenced by changing of the earth during a given night. We study this problem by investigating the relationship between the earthshine atmospheric absorption coefficient $\alpha_e$ and the moonshine coefficient $\alpha_m$, and find that the absorptivity of earthshine $\alpha_e$ shows a linear correlation to $\alpha_m$. From the linear scaling law that we find between $\alpha_e$ and $\alpha_m$, we therefore can make a better determination of $\alpha_e$ from $\alpha_m$ for the nights when earthshine variation shows a mixture of the local airmass change and global change hence does not follow exactly Beer's law.

## Night Variation

The measurement of the moonshine intensity at different nights, even after correction of airmass so that the input numbers from each night are now at zero airmass, is subject to the change of local atmospheric condition. To correct this, we use the overall intensity of the bright side of the moon as the standard star. Good correlation is found between the change of the moonshine intensity and the crescent intensity (Fig.2). The cross-correlation between the moonshine scattering and the integrated crescent intensity scattering is 0.73 for the morning observations, and 0.77 for the evening observations. A simple least-square linear fit is made between the moonshine scattering and crescent scattering, and the correlated part of the scattering is subtracted from the moonshine data as a correction for the nightly variation. The scattering of the data points is considerably reduced after such correction (see second panel of Fig.3).

Fig.2 Intensity against phase

## Topocentric Position of Moon and Sun

The second step correction is made by taking into consideration of the relative position of the moon and the sun with respect to the earth. For different nights at the same lunar phase, because the relative Sun-Moon position can be different, the intensity measured may vary. Such variation is more evident towards the zero lunar phase and hence correction is done for lunar phases in the range of -15 to 15 degree (see third panel of Fig.3).

## Libration Effect

The third step correction accounts for the libration effect. Because of the longitudinal, latitudinal and dynamical librations, in different cycles of lunar orbit, at the same lunar phase, we would expect the change of the position of the fiducial patches in the lunar disk. The read out intensity thus changes as a function of the geometric postion of the fiducial patch in the lunar disk. The description of such variation as a function of the lunar libration is derived empirically and the effect is corrected, which gives the final result of the determination of the lunar phase function (see last panel of Fig.3).

## Opposition Peak

The final lunar phase function for each fiducial patch needs to be normalized to the full moon opposition peak. For this purpose, we analyzed the lunar eclipse data observed on November 29, 1993. Toward the full moon when the lunar phase is close to zero, the evolution of the moonshine intensity is controlled not only by the changing airmass but also by the changing phase angle. We developed a simple solution to this problem under the reasonable assumption that the opposition effect is a linear effect within a small range of phase angles, and made non-linear least-square fit which incorporates the atmospheric attenuation and the opposition effect. The retrieved opposition effect parameters are used to normalize the phase function. The plots in Fig.3 show the normalized lunar phase function. The results from the analysis of the lunar eclipse data also give us the ratio of the reflectivity of the two fiducial patches.