The force model included the third body perturbations the Earth, Sun, and the planets, the
solar radiation pressure, and the Earth-induced and solar-induced lunar tides. We applied
a k2 Love number of 0.027 as derived by Williams et al. [1987]. The DSN station
coordinates are derived from Folkner [1993], a solution obtained for the Mars Observer
project, and are mapped back to the 1960's using station velocities derived by the
International Earth Rotation Service [IERS, 1994]. Stations 12 (at Goldstone) and 61 (at
Madrid) were enlarged in the late 1970's from 26 meters to 34 meters. Based on surveys
taken prior to and after the reconstruction, the coordinates of these stations prior to the
move can be redetermined and mapped back to the epoch of the Lunar Orbiter tracking data.
We have also applied the effect of direct oblation on the lunar satellite's orbit (J2 of the
Earth acting on a lunar satellite), and indirect oblation (acceleration of the Moon caused by
the Moon's 12 interacting with the point mass of the Earth) [Moyer, 1971]. We applied the
1991 IAU model, with the appropriate corrections for the typographical errors for some of
the librational terms [Davies et al., 1992]. We used the DE200 set of lunar and planetary
ephemerides [Standish, 1990].
The GEODYN orbit determination program was used to process the lunar tracking data and
create the normal equations [Pavlis et al., 1995]. A companion program to GEODYN,
SOLVE [Ullman, 1994] was used to combine the normal equations, and obtain the least
squares solutions.
Calibration of GLGM-2
The GLGM-2 gravity model was calibrated using the method of Lerch [1991]. The data
were assembled into ten sets of normal equations, based on orbit geometry. The nominal,
or a priori, gravity model for GLGM-2 was the Konopliv et al. [1993] model. The
tracking data were processed with an a priori sigma of 1 cm/s. This sigma is typically 3 to
5 times the RMS of fit of the tracking data, depending on the arc. Then, the a priori
weights were refined following extensive analysis, including orbit tests, recalculation of
the RMS of fit to the tracking data using interim models, and inspection of anomaly maps.
The data attaining the lowest periapse altitudes (30 km or less) were assigned a priori
sigmas of 3 cm/s. This included the Apollo 16 subsatellite data, and several batches of
data from Lunar Orbiters 2 and 3, that descended to as low as 27 to 38 km above the
reference ellipsoid (radius of 1738 Ian) as they disappeared over the western limb.
Ten solutions complete to degree and order 70 were computed, and calibration factors
were obtained for each of the ten subset solutions. As indicated in Table 2, all the
calibration factors are approximately unity. Only the Lunar Orbiter 1 data needed to be
down weighted more than the other sets of data. No valid calibration factors were obtained
for the Lunar Orbiter 4 data, set 1. This set of data is extremely weak, because of the high
periapse height (2700 km), long orbital period (12 hours), and numerous attitude
maneuvers. We applied the calibration factors from Lunar Orbiter 4 set 2 for these data.
The fmal calibrated solution (GLGM-2) was derived using the calibration factors obtained
from all ten of the subset solutions. We also attempted to iterate on the calibration process,
by obtaining a refinement of the weights. This step did not involve recalculating the
normal equations, but rather required the calculation of ten more subset solutions. We
began with the calibrated solution (GLGM-2 or LGM-309a), and repeated the calibration
process, to obtain a second calibrated solution (LGM-320). However, this final solution
did not perform satisfactorily on the orbit and RMS of fit tests compared with GLGM-2, so
we rejected this solution. The calibration factors are summarized in Table 2.
180
solar radiation pressure, and the Earth-induced and solar-induced lunar tides. We applied
a k2 Love number of 0.027 as derived by Williams et al. [1987]. The DSN station
coordinates are derived from Folkner [1993], a solution obtained for the Mars Observer
project, and are mapped back to the 1960's using station velocities derived by the
International Earth Rotation Service [IERS, 1994]. Stations 12 (at Goldstone) and 61 (at
Madrid) were enlarged in the late 1970's from 26 meters to 34 meters. Based on surveys
taken prior to and after the reconstruction, the coordinates of these stations prior to the
move can be redetermined and mapped back to the epoch of the Lunar Orbiter tracking data.
We have also applied the effect of direct oblation on the lunar satellite's orbit (J2 of the
Earth acting on a lunar satellite), and indirect oblation (acceleration of the Moon caused by
the Moon's 12 interacting with the point mass of the Earth) [Moyer, 1971]. We applied the
1991 IAU model, with the appropriate corrections for the typographical errors for some of
the librational terms [Davies et al., 1992]. We used the DE200 set of lunar and planetary
ephemerides [Standish, 1990].
The GEODYN orbit determination program was used to process the lunar tracking data and
create the normal equations [Pavlis et al., 1995]. A companion program to GEODYN,
SOLVE [Ullman, 1994] was used to combine the normal equations, and obtain the least
squares solutions.
Calibration of GLGM-2
The GLGM-2 gravity model was calibrated using the method of Lerch [1991]. The data
were assembled into ten sets of normal equations, based on orbit geometry. The nominal,
or a priori, gravity model for GLGM-2 was the Konopliv et al. [1993] model. The
tracking data were processed with an a priori sigma of 1 cm/s. This sigma is typically 3 to
5 times the RMS of fit of the tracking data, depending on the arc. Then, the a priori
weights were refined following extensive analysis, including orbit tests, recalculation of
the RMS of fit to the tracking data using interim models, and inspection of anomaly maps.
The data attaining the lowest periapse altitudes (30 km or less) were assigned a priori
sigmas of 3 cm/s. This included the Apollo 16 subsatellite data, and several batches of
data from Lunar Orbiters 2 and 3, that descended to as low as 27 to 38 km above the
reference ellipsoid (radius of 1738 Ian) as they disappeared over the western limb.
Ten solutions complete to degree and order 70 were computed, and calibration factors
were obtained for each of the ten subset solutions. As indicated in Table 2, all the
calibration factors are approximately unity. Only the Lunar Orbiter 1 data needed to be
down weighted more than the other sets of data. No valid calibration factors were obtained
for the Lunar Orbiter 4 data, set 1. This set of data is extremely weak, because of the high
periapse height (2700 km), long orbital period (12 hours), and numerous attitude
maneuvers. We applied the calibration factors from Lunar Orbiter 4 set 2 for these data.
The fmal calibrated solution (GLGM-2) was derived using the calibration factors obtained
from all ten of the subset solutions. We also attempted to iterate on the calibration process,
by obtaining a refinement of the weights. This step did not involve recalculating the
normal equations, but rather required the calculation of ten more subset solutions. We
began with the calibrated solution (GLGM-2 or LGM-309a), and repeated the calibration
process, to obtain a second calibrated solution (LGM-320). However, this final solution
did not perform satisfactorily on the orbit and RMS of fit tests compared with GLGM-2, so
we rejected this solution. The calibration factors are summarized in Table 2.
180