reply to post by Ex_CT2
NASA has a reputation for wanting precision within "millimeters" on it's reports yet when their own data does not fit the model they will actually
change the data. The Lunar co-ordinates have been changed MANY times in the recent past (less than 5 years) and you still need to do some math to make
them fit. The internet is full of data confirming anomalous changes in lunar orbit, here is a memo from the source that is a few years old but you can
work forward from here if you like............
1
JET PROPULSION LABORATORY INTEROFFICE MEMORANDUM
CALIFORNIA INSTITUTE OF TECHNOLOGY IOM 335-JW,DB,WF-20080314-001
March 14, 2008
To: Lunar Distribution
From: J. G. Williams, D. H. Boggs and W. M. Folkner
Subject: DE421 Lunar Orbit, Physical Librations, and Surface Coordinates
Introduction
This memo discusses the DE421 lunar orbit, orientation angles and coordinate frames. The
DE421 orbit is compared with DE403 and DE418. The construction of the lunar part of DE421 is
described. For coordinate frames rotating with the Moon, two choices are described and the
rotation between the two frames is given.
The planetary and lunar ephemeris DE421 (Folkner et al., 2008) was provided for the Phoenix
mission to Mars. Also updated were the orbit and physical librations of the Moon. This memo
concentrates on the lunar aspects of DE421.
DE421 vs. DE418 and DE403 Moon
DE403 was generated in 1995 and, compared to the recent DE418 and DE421 ephemerides, the
lunar position is separating. DE418 is described by Folkner et al. (2007) and lunar aspects are
discussed in Williams and Folkner (2007). The DE418–DE403 difference illustrated in Figure 1
shows the differences in radius as well as ecliptic longitude and latitude from 1990 to 2020.
Figure 2 shows similar plots for DE421–DE403. The longitude difference shows the t2 behavior
from a 0.8% change in secular tidal acceleration, and the growing monthly variation in radius
difference reflects the related acceleration of mean anomaly difference. A half meter separation
in mean radius is due to a difference in the GM of Earth+Moon. The latitude pattern results from
two monthly oscillations beating together while the amplitude of one increases due to the secular
acceleration difference. The position difference is typically about 6 m in 2008, increasing to ~8
m in 2012, ~11 m in 2015, and ~16 m in 2020. The difference is mainly orthogonal to radius so
corresponding angles are about 3, 4, 6, and 9 milliseconds of arc, respectively.
The lunar orbits of DE418 and DE421 are compared over 30 years in Figure 3 and are similar.
The radial coordinates differ by a few centimeters and the two orthogonal components differ by a
few decimeters. The ecliptic latitude and longitude differences are a few milliseconds of arc
(mas). There is a slow drift of ~3 µas/yr in longitude differences and the 18.6 yr periodicity
(node precession period) is due to small differences between diurnal and semidiurnal tides on
Earth. Both DE418 and DE421 are superior to DE403 for navigation purposes. The accuracy of
DE403 is degrading at an accelerating pace and it is no longer recommended for missions.
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Figure 1. Moon differences for ecliptic longitude and latitude, and radius for DE418 – DE403.
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Figure 2. Moon differences for ecliptic longitude and latitude, and radius for DE421 – DE403.
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Figure 3. Moon differences for ecliptic longitude and latitude, and radius for DE421 – DE418.
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Construction of the Ephemeris
The DE421 planets have been discussed by Folkner et al. (2008). Here we address the lunar
ephemeris and rotation. The initial conditions for the lunar ephemeris and three dimensional
lunar orientation (Euler angles and spin rates), lunar laser retroreflector array positions and other
lunar parameters were fit to Lunar Laser Ranging (LLR) data. A total of 16,601 ranges extend
from March 16, 1970 to December 27, 2007. Modern range accuracies are more than an orderof-
magnitude more accurate than the early data. Ranges were processed from McDonald
Observatory, Texas, Observatoire de la Côte d'Azur, France, Haleakala Observatory, Hawaii, and
Apache Point Observatory, New Mexico. A few ranges from Matera, Italy were also processed.
Ranges to four retroreflector arrays on the Moon were used. They are located at the Apollo 11,
14, 15 and Lunokhod 2 sites. A majority of the ranges are to the largest array at the Apollo 15
site (77.9%), while Lunokhod 2 gets the fewest number of ranges (2.7%). Apollo 11 and 14
make up 9.8% and 9.6% of the total data set, respectively. Ranges to multiple arrays are
important for determining the physical librations and lunar geophysical parameters.
The construction of a new ephemeris involves a series of choices and sometimes compromises.
The models for the computation of both the acceleration of the Moon in its orbit and the torques
about its center of mass for the numerical integration, and the computation of range for the range
data fits, depend on geophysical processes at the Earth and Moon. Some of the geophysical
parameters are input and held constant while others are fit to the lunar ranges. Linear constraints
between parameters can be applied. Many more lunar parameters were adjusted during the
DE421 fit than was the case for DE418 increasing confidence in the final product.
For the Earth's gravity field, J3 and J4 were taken from the GGM02C gravity field and the
equatorial Earth radius used with gravity was set to 6378.1363 km. The J2 coefficient was based
on the GGM02C "tide free" value, but the J2 value was adjusted for a different Love number k20
used here. The constant part of the zonal ocean tide is part of the tidal acceleration computation
here, while that contribution is included in the GGM02C J2 value, so it is necessary to account
for the difference. The tidal gravity model here uses three Love numbers k20, k21, and k22 with
three tidal time delays for the long period (zonal), diurnal, and semidiurnal tides, respectively.
The three Love numbers and the zonal (long period) time delay were combined from separate
Earth and ocean tides. Tidal response changes with frequency and values were chosen to
approximately match the Mf, O1, and M2 tides, which are the most important tides in each of the
three frequency bands for the tidal secular acceleration of the Moon. The Earth tides came from
the IERS Conventions (McCarthy and Petit, 2003) and the ocean tides were based on the
FES2004 result (Lyard et al., 2006; reformatted by Richard Ray on a web site, 2007). The zonal
time delay (∆t20) was based on the Mf ocean tide, but the diurnal (∆t21) and semidiurnal (∆t22)
time delays were solution parameters. Combined Earth tide parameters are summarized in Table
1. “Type” indicates whether a parameter was fixed or free to change during the solution, or
derived from other parameters after the solution. Tidal secular acceleration in orbital longitude
and semimajor axis rate were derived from the tide values using a theory after the solution was
complete. The computation of the range depends on the coordinates of the ranging stations which
are fit. Station motion was fit when the station’s data span was years. Two rotation angles at
J2000 (X rotation about equinox direction and Y about axis at 90° right ascension and 0
declination) are fit to orient the Earth’s equator in space with respect to its orbit. The alignment
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of the inner four planets with the international celestial reference frame (ICRF) is established
mainly through planetary VLBI data to Mars and Venus. The precession and obliquity rates with
respect to space and also several nutation coefficients were fit. The X and Y rotation values have
some dependence on the secular and long-period variations. The diurnal and semidiurnal UT1
terms were based on the IERS conventions. GM of the Earth+Moon was fixed during solutions,
so the value of the Sun/(Earth+Moon) mass ratio and the derived GMearth, using TDB seconds,
are not new results.
Table 1. Geophysical and orientation parameters for Earth.
Parameter Type Unit Value
Sun/(Earth+Moon) fixed 1 328900.55915
GMearth derived km3/sec2 398600.4362
equatorial radius fixed km 6378.1363
k20 fixed 1 0.335
k21 fixed 1 0.320
k22 fixed 1 0.320
∆t20 fixed day 0.0640
∆t21 fit day 0.01114245
∆t22 fit day 0.00657429
zonal dn/dt derived “/cent2 0.12
diurnal dn/dt derived “/cent2 -3.31
semidiurnal dn/dt derived “/cent2 -22.88
zonal da/dt derived mm/yr -0.18
diurnal da/dt derived mm/yr 4.89
semidiurnal da/dt derived mm/yr 33.75
X axis rotation fit mas 5.8
Y axis rotation fit mas -16.9
obliquity rate fit mas/yr -0.26
luni-solar precession fit “/yr 50.38490
Table 2 gives the lunar geophysical parameters. In Tables 1 and 2 extended numbers of digits are
given for internal consistency; they do not imply accuracy. Some of the lunar gravity field
coefficients were fixed to LP150Q values (Konopliv, website) and some were fit, as described in
the next section. The radius used with the gravity field was 1738 km, a reasonable value for the
equatorial radius, though the mean radius is smaller (Smith et al., 1997). The mass of the Moon
depends on the Earth/Moon mass ratio and the GM of the Earth-Moon system. The mass ratio
was fit to planetary data. The lunar mantle orientation (physical libration) initial conditions and
retroreflector coordinates for Apollo 11, 14, 15 and Lunokhod 2 were solution parameters. These
results are given in later sections. Lunar Love numbers h2 and k2 were fit, but the ratio was
constrained to a model value of 1.75 during the fit. Love number l2 was fixed to a model value.
Dissipation parameters are fit for lunar tides (time delay ∆tm) and fluid-core/solid-mantle
interface (CMB) interaction (Kv); see Williams et al. (2001) for definitions. Oblateness of the
CMB and initial values for the fluid core orientation were fit. The moment of inertia of the fluid
core was set to 7x10-4 of the total moment, consistent with recent solutions (Williams et al.,
2008). The static J2 coefficient was based on the LP150Q value by applying a constraint which
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adjusts for the difference in the constant J2 tide contribution from the LP150Q k2 value and the
solution value here. The constraint, based on Williams et al. (2001), is
J2 + 3.6987x10-6 k2 = 2.033532597x10-4 . (1)
C22 is calculated from J2 and the two lunar moment of inertia differences (C-A)/B and (B-A)/C.
All four of these values do not include the tidal contributions.
Table 2. Moon geophysical parameters.
Parameter type Unit Value
mass ratio Earth/Moon fit 1 81.30056907
GMmoon derived km3/sec2 4902.80008
equatorial radius fixed km 1738.0
(C-A)/B fit 1 631.0022x10-6
(B-A)/C fit 1 227.7305x10-6
J2 constrained 1 203.27326x10-6
C22 derived 1 22.38977x10-6
k2 constrained fit 1 0.02163
h2 constrained fit 1 0.03786
l2 fixed to model 1 0.01050
∆tm fit day 0.10786
Kv fit 1/day 1.49376x10-8
tidal dn/dt derived “/cent2 0.20
CMB dn/dt derived “/cent2 0.02
tidal da/dt derived mm/yr -0.30
CMB da/dt derived mm/yr -0.02
core moment/total moment fixed 1 7x10-4
CMB flattening fit 1 3.7977x10-4
When the dissipation effects from Earth and Moon are added together, the resulting acceleration
in longitude is -25.85 “/cent2 and the semimajor axis rate is 38.14 mm/yr. These derived values
depend on a theory which is not accurate to the number of digits given. The conversion for Earth
tides is thought to be accurate to ~1/2% of the total, but that is a rough estimate. Differences
between distinct ephemerides are useful, e.g. the computed DE421–DE418 tidal acceleration
difference is only 0.01 “/cent2 and is not evident in Figure 3 while the DE421–DE403 difference
of -0.21 “/cent2 is prominent in Figure 2 as the 4 cm/yr2 acceleration. Dissipation effects also
cause an eccentricity rate, but there is an anomalous rate as well (Williams et al., 2001). Because
we do not know the source of this anomalous rate, we cannot include it in the numerical
integration model. Consequently, an anomalous eccentricity rate is not solved for here and that is
a compromise. Compared to solutions with an independent rate (Williams et al., 2008), the
solution here responds by decreasing the dissipation in lunar tides and increasing the dissipation
from the CMB interaction and this distorted fit has a slightly larger rms postfit residual. Another
compromise is the dependence of the Moon’s tidal damping on frequency. The integration model
uses a time delay model, corresponding to a tidal Q proportional to 1/frequency, but fits with
parameters sensitive to the frequency dependence support only a weak dependence on frequency
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(Williams et al., 2001; Williams et al., 2008). The different tidal frequency dependences cause
physical libration effects at the few milliseconds of arc level which are not present in the
integration; these are detectable with the LLR fits but are not troublesome to most users.
Lunar Surface Coordinate Frames — Principal Axes and Mean Earth/Mean Rotation Axes
The solutions for lunar orientation Euler angles and lunar laser retroreflector array coordinates
use the principal axes of the lunar moment of inertia matrix for the X, Y and Z directions. These
are principal axes before tidal distortions are applied. It is natural to write and integrate the
equations of motion for the lunar Euler angles using the principal axes. The center of mass is
used for the origin of Moon-fixed frames.
The other frame of interest uses the mean direction toward the Earth for the X axis and the mean
rotation direction for the Z axis. Y completes the right-handed triad. In Lunar Laser Ranging
(LLR) papers we have called this the mean Earth/mean rotation axis frame or more tersely the
mean Earth/rotation axis frame. In papers by Mert Davies and the IAU/IAG working group on
coordinates and rotations (Seidelmann et al., 2007, and earlier papers in this sequence) the
designation mean Earth/polar axis frame has been used but this is only a name change and these
frames are the same. An ellipsoidal Moon with only a second-degree figure (gravity) would have
the mean axis and principal axis frames coincident. Third- and higher-degree representations of
the gravity field cause a constant rotation between the two frames (Williams et al, 2006). There
is also a small constant rotation due to dissipation effects in the Moon.
A constant three-angle rotation relates the two frames, but our knowledge of the three constant
angles depends on the gravity field coefficients and it requires a theory relating them. Since
gravity field coefficients can change for different ephemerides, the rotation angles must be
compatible with the ephemeris. The gravity harmonic coefficients have improved over the years
(Konopliv et al., 1998, 2001). Currently, the LP150Q gravity field (Konopliv, website) is
recommended for spacecraft orbit calculations. For DE421, the three third-degree coefficients
C31, S31 and S33 and all of the fourth-degree coefficients match the LP150Q values. In addition,
the J2 value is based on LP150Q after accounting for a difference in Love number k2 using eq.
(1). The four remaining third-degree coefficients, the moment of inertia differences which adjust
the relation between C22 and J2, and k2 were fit during the LLR data analysis. Prior to the
application of tidal distortions, C21, S21 and S22 are zero consistent with principal axis coordinates.
The fits to lunar rotation and orbit are done simultaneously and the numerical integration for
orbit and Euler angle evolution is simultaneous. Thus, the lunar Euler angles (physical librations)
and orbit on the DE file are compatible and the Euler angles use the principal axis frame. Note
also that the LP150Q gravity field (Konopliv, website) uses the principal axis frame, albeit the
frame for DE403 with an orientation 4" different from DE421. Use of the JPL DE421 ephemeris
file with the LP150Q gravity field is recommended for high accuracy navigation purposes.
There are no equations of motions for Euler angles referenced to mean Earth/mean rotation axes.
The numerically integrated Euler angles rotating from space to principal axes that are provided
with DE files are inherently more accurate than knowledge of the mean axes. For a given gravity
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field, the constant three-angle rotation from principal axes to mean axes is less accurately known
than the integrated Euler angles. The constant rotation is typically computed after the fit and
integration. If one wishes the orientation of the lunar mean Earth/mean rotation axes, we
recommend first extracting the principal axis angles from the file and then rotating by three
constant angles given below.
Note that the semianalytical series expressions for angle W along with pole right ascension and
declination, given in the IAU/IAG working group documents (Seidelmann et al, 2007, and
earlier), are approximations of much lower accuracy than the numerical integrations. By
comparing with integrated values, Konopliv et al. (2001) showed that the series expressions for
orientation lead to position errors which can exceed 100 m. One of us (JGW) derived the series
expressions and that uncertainty is consistent with the level of truncation of the series.
Moon-Centered Coordinates
Rotations between the two lunar coordinate frames were first developed for the Lunar Laser
Ranging (LLR) coordinates of retroreflector arrays. The fits to the LLR data use the integrated
lunar orbit and the physical libration Euler angles, so the derived retroreflector array coordinates
are in the principal axis frame. Techniques have been developed for determining the constant
rotation between principal axis coordinates and mean Earth/mean rotation axis coordinates.
The constant rotation between principal axis coordinates and mean Earth/mean rotation axis
coordinates depends on the lunar gravity field coefficients and the calculation ultimately rests on
a theory. Originally these theories were by Eckhardt, most recently (1981), and the relevant
terms are the constant parts of p1, p2, and ! . The first two of these are Moon-fixed x and y
coordinates of the unit vector normal to the ecliptic plane and the ! refers to the longitude
rotation. The constant parts of p1, p2, and ! depend mainly on the gravity field and slightly on
dissipation effects; knowledge of the gravity field was poor in earlier times and consequently
Eckhardt's constant part of the ! angle is a factor of about three larger than the actual value of
the angle! Eckhardt provided partial derivatives with respect to the third-degree gravity
coefficients, but since there are nonlinearities in the differential corrections the accuracy is
limited at some level. In rotating the Moon-centered coordinates of LLR retroreflector arrays
from principal axis to mean Earth/mean rotation axis coordinates, LLR first used Eckhardt's
constant parts of p1, p2, and ! , but later opted for consistency by deriving empirically the
rotations between sets of LLR coordinates determined with different integrated ephemerides.
Even with the latter procedure, Eckhardt’s theory had been used to rotate the earlier set of
coordinates which later empirical rotations depended on.
The last LLR array coordinates which used Eckhardt’s theory for the rotation were in Williams,
Newhall and Dickey (1987). Subsequent LLR coordinates during the past two decades were
aligned with those mean Earth/mean rotation axis coordinates: 1) An internal memo (Williams,
Newhall and Standish, 1993) gave a set for DE245 prepared at the time of the Clementine
mission. 2) Williams, Newhall and Dickey (1996) published a set compatible with the gravity
coefficients in Dickey et al. (1994). The 1996 rotations were also given by in a paper by Davies
and Colvin (2000), but the coordinates there are slightly different from our 1996 paper. 3) LLR
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coordinates using DE403 and the three associated rotations have been distributed by email.
These coordinates were used and published while the associated angles for the DE403-related
rotations are given in Konopliv et al. (2001) and Seidelmann et al. (2007).
In the DE418 lunar memo (Williams and Folkner, 2007) and in this memo the procedure is
changed again. The LLR principal axis coordinates determined during the solution leading to
DE421 are given in Table 3. For the rotation to the mean Earth/mean rotation axis frame, the
rotation in longitude, which is very sensitive to the gravity field coefficients, was computed from
a new theory in order to overcome the less accurate gravity field that Eckhardt (1981) used. The
unpublished new theory (from JGW) balances the C22 torque from the displaced X and Y axes
against the torques from the third- and fourth-degree coefficients. A smaller contribution from
tide and core dissipation (Williams et al, 2001) is also added. For the DE418 coordinates, the
pole direction from a fit of periodic terms to the DE403 Euler angle orientations by Newhall and
Williams (1997) was used with corrections for the difference between DE403 and DE418. Here,
the pole direction has used a fit to the DE418 Euler angles by Rambaux (private communication,
2008) converted to p1 and p2 and modified for the slight difference between the DE418 and
DE421 rotations.
If M is a vector of Cartesian coordinates in the mean Earth/mean rotation (MER) axis frame and
P is a coordinate vector in the principal axis (PA) frame, then the derived rotation follows the
form
M = Rx(-p2) Ry(p1) Rz(- ! ) P (2)
where p1 and p2 and ! are the constant parts of those parameters. Though p1 and p2 are
coordinates, they are small and are expressed as angles. The three rotations are small and eq. (2)
is only first order in the rotations. A more exact expression might differ by a few milliseconds of
arc, a few centimeters in position. For DE421 the rotation between frames is
M = Rx(-0.30") Ry(-78.56") Rz(-67.92") P (3)
where the angles are in seconds of arc and the rotations are around the body X, Y and Z axes.
The inverse rotation is
P = Rz(67.92") Ry(78.56") Rx(0.30") M (4)
Note that rotation of coordinates of eqs. (2) - (4) above and the corresponding rotation of the
frames have the opposite sense. Rotation of the LLR principal axis array coordinates of Table 3
to the mean Earth/mean rotation axis frame gives the LLR array coordinates in Table 4.
Because of the changed procedure for calculating the constant rotations, the mean axis
coordinates of Table 4 are rotated when compared with those previously distributed for the Euler
angles of DE403 and earlier ephemerides. Compared to our DE403 MER axis orientation, the
differences (ignoring signs) are 0.53" in longitude (rotation about z axis), 0.25" about the y axis
and 0.18” about the x axis. At the lunar surface 1" corresponds to 8.4 m, so the displacement
from the DE403 MER axis frame is 5 meters. The DE418 and DE421 MER axis frames are
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close. In the future it should be possible to improve upon the three angles for the DE421 rotation
between frames in eqs. (2) and (3) further, but the values here should be close enough for meter
level accuracy. A separate question is how much the rotation between frames will change for
future ephemerides. Gravity field uncertainties for S31 and S33 are most important for the
longitude rotation uncertainty. For the LP150Q coefficients that uncertainty is about 1.0” and
future changes in the principal axis direction due to gravity field changes can be of that order.
Comparing mean axis coordinates from DE421 and DE403, there is also a shift of 0.7 m along
the X axis due partly to changed orbital semimajor axis and GM(Earth+Moon) and partly due to
different tidal Love numbers.
Table 3. Lunar laser retroreflector array coordinates using principal axis frame.
Array X Y Z R E Longitude Latitude
meters meters meters meters degrees degrees
Apollo 11 1591967.522 690698.106 21003.309 1735472.732 23.4543075 0.6934309
Apollo 14 1652689.359 -520999.194 -109731.018 1736336.135 -17.4971065 -3.6233288
Apollo 15 1554678.949 98094.117 765004.907 1735477.340 3.6103520 26.1551743
Lunokhod 2 1339364.624 801870.788 756358.470 1734639.009 30.9087426 25.8509928
Table 4. Lunar laser retroreflector array coordinates using mean Earth/mean rotation axis frame.
Array X Y Z R E Longitude Latitude
meters meters meters meters degrees degrees
Apollo 11 1591747.845 691222.345 20397.830 1735472.732 23.4730729 0.6734398
Apollo 14 1652818.934 -520454.721 -110361.346 1736336.135 -17.4786483 -3.6441703
Apollo 15 1554937.875 98605.140 764412.735 1735477.340 3.6285073 26.1333959
Lunokhod 2 1339388.500 802310.872 755849.325 1734639.009 30.9221489 25.8323070
The LLR range model includes solid-body tides on the Moon. That calculation includes a
constant displacement in addition to time variations. The constant part of tidal displacements,
different for each retroreflector array, are not included in Tables 3 and 4 but they are given in
Table 5. These can be added to the Table 3 and 4 locations if precise LLR positions are to be
used without a tide model. The lunar displacement Love numbers used during the fit were h2 =
0.03786 and l2 = 0.01050, giving few decimeter constant tidal displacements as shown.
Table 5. Constant part of tide displacements.
Array ∆R ∆East ∆North
meters meters meters
Apollo 11 0.373 - 0.148 -0.004
Apollo 14 0.420 0.116 0.023
Apollo 15 0.344 -0.023 -0.160
Lunokhod 2 0.192 -0.161 -0.117
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Files
The DE421 ephemeris may be downloaded in an ascii version from
ssd.jpl.nasa.gov... .
The complete set of input parameters for the solar system integration, more extensive than
Tables 1 and 2, is part of the file. The SPICE kernal version of DE421 is available at
ssd.jpl.nasa.gov... .
Summary
The DE421 lunar position should be an improvement over DE403 at the present time and for
years into the future. DE421 and DE418 are much more similar to one another than to the much
older DE403. For high accuracy purposes such as lunar navigation, we recommend use of the
DE421 JPL ephemeris file with the LP150Q gravity field. Tables 1 and 2 give geophysical
parameters used for the Earth and Moon, respectively. Rotation of Moon-centered coordinates
between the mean Earth/mean rotation axis frame and the principal axis frame can be achieved
with eqs. (3) and (4). Coordinates of the lunar laser retroreflector arrays are given in Tables 3
and 4.
Acknowledgments. We thank Nicolas Rambaux for the physical libration rotation information.
The research described in this paper was carried out at the Jet Propulsion Laboratory of the
California Institute of Technology, under a contract with the National Aeronautics and Space
Administration.
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