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.
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
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%.
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.
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
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).
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).
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.