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Analyses of Lunar Laser ranges show a displacement in direction of the Moon's pole of rotation which indicates that strong dissipation is acting on the rotation. Two possible sources of dissipation are monthly solid-body tides raised by the Earth (and Sun) and a fluid core with a rotation distinct from the solid body. Both effects have been introduced into a numerical integration of the lunar rotation. Theoretical consequences of tides and core on rotation and orbit are also calculated analytically. These computations indicate that the tide and core dissipation signatures are separable. They also allow unrestricted laws for tidal specific dissipation Q versus frequency to be applied. Fits of Lunar Laser ranges detect three small dissipation terms in addition to the dominant pole-displacement term. Tidal dissipation alone does not give a good match to all four amplitudes. Dissipation from tides plus fluid core accounts for them. The best match indicates a tidal Q which increases slowly with period plus a small fluid core. The core size depends on imperfectly known properties of the fluid and core-mantle interface. The radius of a core could be as much as 352 km if iron and 374 km for the Fe-FeS eutectic composition. If tidal Q versus frequency is assumed to be represented by a power law, then the exponent is -0.19+/-0.13. The monthly tidal Q is 37 (-4, +6), and the annual Q is 60 (-15, +30). The power presently dissipated by solid body and core is small, but it may have been dramatic for the early Moon. The outwardly evolving Moon passed through a change of spin state which caused a burst of dissipated power in the mantle and at the core-mantle boundary. The energy deposited at the boundary plausibly drove convection in the core and temporarily powered a dynamo. The remanent magnetism in lunar rocks may result from these events, and the peak field may mark the passage of the Moon through the spin transition.
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Lunar rotational dissipation in solid body and molten core
James G. Williams, Dale H. Boggs, Charles F. Yoder,
J. Todd Ratcliff, and Jean O. Dickey
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA
Abstract. Analyses of Lunar Laser ranges show a displacement in direction of the Moon's pole of
rotation which indicates that strong dissipation is acting on the rotation. Two possible sources of
dissipation are monthly solid-body tides raised by the Earth (and Sun) and a fluid core with a
rotation distinct from the solid body. Both effects have been introduced into a numerical
integration of the lunar rotation. Theoretical consequences of tides and core on rotation and orbit
are also calculated analytically. These computations indicate that the tide and core dissipation
signatures are separable. They also allow unrestricted laws for tidal specific dissipation Q versus
frequency to be applied. Fits of Lunar Laser ranges detect three small dissipation terms in
addition to the dominant pole-displacement term. Tidal dissipation alone does not give a good
match to all four amplitudes. Dissipation from tides plus fluid core accounts for them. The best
match indicates a tidal Q which increases slowly with period plus a small fluid core. The core
size depends on imperfectly known properties of the fluid and core-mantle interface. The radius
of a core could be as much as 352 km if iron and 374 km for the Fe-FeS eutectic composition. If
tidal Q versus frequency is assumed to be represented by a power law, then the exponent is
-0.19_+0.13. The monthly tidal Q is 37 (-4,+6), and the annual Q is 60 (-15,+30). The power
presently dissipated by solid body and core is small, but it may have been dramatic for the early
Moon. The outwardly evolving Moon passed through a change of spin state which caused a burst
of dissipated power in the mantle and at the core-mantle boundary. The energy deposited at the
boundary plausibly drove convection in the core and temporarily powered a dynamo. The
remanent magnetism in lunar rocks may result from these events, and the peak field may mark the
passage of the Moon through the spin transition.
1. Introduction
The Moon keeps one face toward the Earth. This simple
statement of the equality of the rotational and orbital periods has
a deeper implication. Since there is no reason to expect that the
Moon formed in such a special rotational state, there must have
been one or more mechanisms for changing the lunar rotational
angular momentum and energy.
Laser ranges from the Earth to the Moon started in 1969. The
analyses of laser ranges discovered active lunar rotational
dissipation nearly a decade later, and during the past 2 decades
the detection has improved [Yoder et al., 1978; Ferrari et al.,
1980; Cappallo et al., 1981; Dickey et al., 1982; Williams et al.,
1987; Dickey et al., 1994]. The Mooifs rotation is locked in a
spin state (Cassini state) such that the 18.6 year retrograde
precession of the lunar equator plane along the ecliptic plane
matches the precession of the lunar orbit plane. In the absence of
dissipation the equator's average descending node aligns with the
orbit's average ascending node. Laser range analysis finds an
average shift between the two nodes which indicates ongoing
dissipation. The presently measured shift is -9.8" in the node of
the equator on the ecliptic equivalent to an arc length shift of
0.263" in the pole direction. The arrangement and precession of
spin and orbit poles is shown in Figure 1. Over the past
2 decades the significance of the pole shift has improved from the
first detection to the present 1% uncertainty.
Copyright 2001 by the American Geophysical Union.
Paper number 2000JE001396.
There are two proposed mechanisms for the lunar rotational
dissipation: solid-body tidal dissipation [Yoder, 1979; Cappallo
et al., 1981] and dissipation at a liquid-core/solid-body boundary
[Yoder, 1981 ]. Tidal dissipation must exist for the Moon at some
strength. Core dissipation requires a fluid lunar core. While
there are several reasons to suspect that a core is present (see
section 19), and the recent Lunar Prospector mission has
strengthened the evidence, the consequences of a small core are
subtle, and it has remained unclear whether it is solid or liquid.
Both tidal and core dissipation can displace the equator plane
in the observed manner. In the past it has not been possible to
distinguish between them. Improvements in the range accuracy
and increasing data span now make it possible to use small
additional signatures to discriminate.
This paper explores the two dissipation models used for
numerical or analytical computation of the lunar rotation
(sections 2 and 3 for tides and section 9 for core). It presents
analytical developments for the effect of each model on the
rotation (tides: sections 4, 5, 7, core: sections 10, 12-14) and
orbit (tides: section 8, core: section 15). Results from fits to the
Lunar Laser Ranging (LLR) data using the two dissipation
models are presented ( section 18). Results are discussed and
compared with other evidence on the lunar interior (sections 19
and 20).
2. Rotational Dynamics
The attraction of the Earth and Sun on the nonspherical figure
of the Moon applies torques. The Earth dominates the torques.
As a consequence, the lunar equator plane precesses along the
The torque T includes the gravitational interaction of the lunar
figure with external bodies. In the integration model these are
Earth, Sun, Venus, and Jupiter. For a spherical attracting body,
the second-degree torques depend on I and take the form
T 2 = r5 rxIr. (2)
M is the mass of the attracting body, and r is its position with
respect to the Moon's center. G is the gravitational constant. In
the Jet Propulsion Laboratory (JPL) model, additional torques
come from third- and fourth-degree lunar gravitational harmonics
and figure-figure interactions (triaxial Moon with oblate Earth).
Since the orbits used for torque computation include the
influence of gravitational harmonics, planetary perturbations, and
relativity, the torques include indirect effects due to those
perturbations. The lunar orientation is required to compute the
torques, and the body-referenced angular velocities depend on the
Euler angles and their rates.
/-q/sin0 sintp - {3 cøstP /
ß ß
to= -• sin0 costp + 0 sintp . (3)
cos 0 + (p
Figure 1. The spin axis and orbit normal precess in 18.6 years
about the ecliptic pole in a retrograde direction. Without
dissipation the three poles would be coplanar. Dissipation in the
Moon causes a small displacement of the spin pole orthogonal to
that plane.
ecliptic plane in 18.6 years (tilt 1.54 ø) with a superimposed
sequence of periodic variations in pole direction, and the rotation
is synchronous with variations in rotation about the polar axis.
Much of the sensitivity of the LLR data to lunar science
information comes through this time-varying three-dimensional
rotation of the Moon called physical libration. These parameters
include the moment of inertia combinations [•=(C-A)/B and
¾=(B-A)/C, seven third-degree gravitational harmonics,
dissipation due to solid-body tides and core, and Love number k 2.
Dickey et al. [1994] review the Lunar Laser-Ranging technique
and results.
The range accuracy has improved with time, and the most
recent data are fit with a 2 cm rms residual. A highly accurate
model for the orbit and rotation of the Moon is needed to fit the
lunar ranges. The orbits of the Moon and planets and the rotation
of the Moon are simultaneously numerically integrated. The
lunar initial conditions for these integrations and the parameters
of the previous paragraph come from least squares fits to the
lunar range data.
The numerical integration of the lunar rotation requires the
equations of motion and a model for torques. The orientation of
the Moon is specified by three Euler angles. The angular
velocities are computed from the Euler angles and their rates.
The lunar rotation is computed from differential equations for the
angular momentum. The vector differential equation is the Euler
equation when expressed in a frame rotating with the body
d(Ito) + taxIt0 = T. (1)
I is the moment of inertia matrix, ta is the angular velocity vector,
and t is time. The angular momentum vector is the product Ito.
In the JPL numerical integration model the Euler aiqgles consist
of a node-like angle from the J2000 equinox along the J2000
Earth's equator to the descending node of the lunar equator, a tilt
0 between the two equators, and an angle (p from the node along
the lunar equator to the lunar zero meridian. For analytical
calculations it is more useful to give Euler angles defined so that
the Earth's equator plane replaces the ecliptic plane in the
foregoing sequence of three angles. Equation (1) is equivalent to
three second-order, nonlinear differential equations for the Euler
Tidal effects cause I and the gravitational harmonics tO be time
varying. This will be described in the next section. If there is a
fluid core, then in addition to (1) a vector differential equation is
needed to describe the core rotation. There would be torques
from interactions at a core-mantle interface which must be
applied with equal magnitude and opposite sign to the mantle and
core (section 9).
3. Computational Model for Tidal
and Rotational Deformation
In addition to causing torques, the attraction of the Earth and
Sun also raises tides on the Moon. The time-varying tidal
distortion of the Moon changes both the moments of inertia and
the torques, thereby modifying the rotation. Spin also distorts the
Moon, and that time-varying deformation can be treated along
with tides.
The Moon must be distorted by solid-body tides. The elastic
tidal response of the Moon is modeled with Love numbers. The
amount of anelastic tidal dissipation is not known a priori, but
dissipation must be present. Consequently, for 2 decades a tidal
dissipation model has been used to fit the observed lunar
dissipation for Lunar Laser range data analysis. A time-varying
expression for the lunar moments of inertia is used ii• the
program which numerically integrates the rotation of the Moon
and the orbits of the Moon and planets.
An early theoretical investigation by Peale [1973] of elastic
tidal effects on rotation about the pole concluded that the effects
were small, but he did not find the larger effect in pole direction.
Analytical theories for both elastic tides and tidal dissipation
have been presented by Yoder [ 1979] and Eckhardt [ 1981 ]. Bois
and Journet [1993] attempted a numerical approach. An
equation for time-delayed lunar moments of inertia is used by
Newhall and Williams [1997] for numerical LLR data analysis.
The moment-of-inertia expression can be split up into a fixed
part, a part for tidal deformation, and a part for spin-related
I = Irigid + Itide + Ispin. (4)
In the principal axis system the rigid-body principal moments of
inertia are A<B<C. The first axis, associated with A, is
approximately toward the Earth, and the third axis, associated
with C, is nearly in the direction of the spin vector:
Irigid = 0 B 0 . (5)
0 0 C
The rigid-body moments are used to define ½z=(C-B)/A,
[•=(C-A)/B and ¾=(B-A)/C. Only two are independent with
o•=([3-¾)/(1-[3¾). Those relative differences and the ratios
A/C=(1-[3¾)/(1+[3) and B/C=(1+¾)/(1+[3) can be determined
much more accurately than the moments of inertia.
The tides affect the moments. The second-degree tide-raising
potential at a point on the lunar surface (Moon-centered unit
vector u') is
Vtide = r3 P2(u.u'). (6)
For the tide-raising body, M is the mass, and r is its Moon-
centered position vector (components r i, distance r, unit vector
u=r/r, components ui). For the Moon, R is the radius 1738 km.
P2(u.u') = (3/2)[(u.u')2-1/3] is the second-degree Legendre
polynomial. To calculate forces, the positive gradient of (6) is
taken with respect to the position Ru' (potential sign convention
is plus for the point mass potential). Along the Earth-Moon line
the acceleration is outward from the Moon. For the tidal part of
the moment the nine matrix components (indices i, j) are
k 2 M R s •i[ ) (7)
/tide, ij = - r• ( u i ttj - 3 '
where k 2 is the second-degree potential lunar Love number and
the delta function 15ij modifies the diagonal components.
An elastic body will also distort from rotation. In a rotating
Irame the additional potential at the surface is separated into two
parts: one spherically symmetric and the other multiplying a
second-degree spherical harmonic.
R 2 032
Vspin = -•• [ 1 -P2(u"•0) ] . (8)
The an•gular velocity vector is to (components 03i' scalar 03, unit
vector to). Distortion from both parts of the potential contributes
to the moment of inertia components.
_ R s 032
/spin, g- 3G [k2 (03i03J---•80 ' )+s03280' ]' (9)
The Love number k 2 and the spherical parameter s depend on the
elastic properties of the Moon. See Appendix A for a discussion
of the spherical term. Rotational acceleration can also distort the
Moon. These distortions are shown to be small in Appendix B.
Since 033/03 = 1 and rl/r = 1, there are static-deformation
contributions to both the spin and tidal parts of the moments. It is
a matter of definition whether such constant parts are left in the
tidal and spin parts of the moments or moved to the "rigid" part.
In the work by Newhall and Williams [1997] the average values
of the three diagonal terms of the spin part were nearly nulled by
ignoring the s term and adding to the diagonal n2/3, n2/3,-2n2/3,
respectively, inside the parenthesis of (9). Here n is the sidereal
mean motion. This is a wise choice for a rapidly spinning object
like the Earth, where significant oblateness is caused by spin, but
for the slowly rotating Moon the spin-induced oblateness is
smaller than the permanent figure and either choice is reasonable
(see section 6).
In the tidal and spin parts of I, the position r and spin rate to
are functions of time. If the moments Itide and Ispin are evaluated
using r(t) and to(t), respectively, then the elastic response of the
Moon will be accounted for in the resulting rotation. The
sensitivity of the LLR analysis to the Love number k 2 comes
through these terms. Tidal and spin dissipation effects arise if the
distortion is not an instantaneous response. In the program which
numerically integrates the rotation and orbits the tidal dissipation
is introduced with a time delay At by using r(t-At) and tO(t-At)
when computing the distorted moments. In the differential
equations (1) and the torque (2) it is I which is time delayed. The
time-delayed position and spin rate appear only in the moments
and not in the tO explicit in (1) or the r explicit in (2). With an
analytical expansion more generality can be introduced through a
separate time delay, or, equivalently, a separate phase shift, for
each periodic term in the moments. Such an analytical solution
will be developed in the next section.
Some numerical values can be assigned to the above effects.
The model used for the lunar and planetary ephemeris DE403
included tidal dissipation but not core dissipation, so the DE403
solution generated in 1995 represents a limiting possibility with
the Love number k 2 = 0.0300, the time delay At = 0.1673 day,
and the polar moment normalized with the lunar mass and radius
C/mR 2 = 0.3944. With these values the ratio of the tidal moment
factor to C is (k2MR5/Ca 3 ) = 5.7x10 -7, where a = 384,399 km
is the semimajor axis of the lunar orbit. Similarly, take the
common factor in (9) with 03=n (for constant part) and
normalize by C to get (k2n2RS/3GC)= 1.9x10 -7. The time
variation is even smaller than these values. The direction of the
Earth as seen in the lunar principal axis frame varies 0.1 radian in
both the north-south and east-west directions. The eccentricity e
of the lunar orbit is 0.055, so that the (a/r)) tidal factor varies by
3e. The spin rate direction varies <0.001 radian with respect to
the principal axes, and the spin rate relative magnitude varies
about 10 -4. Thus the relative time variation of the moments is of
order 10 -7 for tides and 10 -lø for spin. The relative variation due
to time delay is smaller yet since it involves the factor nat, which
is 0.039 = 1/26 for DE403 values.
4. Tidal and Rotational Dissipation:
Analytical Development
What are the dynamical consequences for the rotation angles
of the tidal and rotational deformation and dissipation? Series
solutions with numerical coefficients have previously been given
by Yoder [1979] and Eckhardt [1981]. The results depend on
how the specific dissipation Q varies with deformation
frequency. The specific dissipation used here is a whole-body Q,
and just as k 2 depends on elastic properties of lunar material as a
function of radius, k2/Q is a function of the distribution of
internal dissipation. The numerical model with constant time
delay is equivalent to Q proportional to 1/frequency. For the
values of time delay and k 2 given in the previous paragraph,
Q = 26, which, as will be seen below, is for a 1 month period.
Yoder gives series for the inverse frequency case, and he gives
the difference between series for that case and a constant Q case
(no frequency dependence). Eckhardt gives series for the
constant Q case. The solution in this and the next section will
have a separate Q for each deformation frequency. Thus the
coefficients of each periodic term in the rotation series can be
functions of more than one deformation frequency.
The torque expression (2) involves uxIu/r 3, where the unit
vector u=r/r. The Euler equation (1) involves toxIto.
Restricting the following development to the second-degree
torques and tides yields
d(Ito) 3 G M
dt = - toxlto + r3 uxlu. (10)
The tidal and spin pans of I depend on r, u, and to. With a dyad
form for products of components the moment matrices can be
written as
Itide =- r3 ( uu-•- i ), (11)
R s 03 2
Isp in= 3G [ k2(ø•ø•---• - i) +s032i ]' (12)
where i is the identity matrix. It is immediately evident that the
parts involving the identity matrix will disappear in the cross
products. With dissipation the tidal and spin deformation parts of
I have delayed responses. An asterisk is used to distinguish the
parameters which originate from I. These include parameters of
the tide-raising body, which may be different from the torquing
body, and time-delayed quantities. Then (10) becomes
d(Ico) 3 G m
dt + •XIrig id • - r 3 UXlrigid U =
k2RS [ 3GMM* M
- r3 r* 3 uxu* u.u* + 7 ux•* u.•*
M* 1 ]
q- {0xu* {0-u*- •--• {0x•* (0'{0' . (13)
When the tide-raising body and the torquing body are the same,
the asterisk indicates the time-delayed parameters and M*=M.
When the tide-raising and torquing bodies are different, the
asterisk indicates the time-delayed parameters of the tide-raising
body and the right-hand side requires sums over the bodies (two
sums for the first term and one sum for each of the second and
third terms).
Note that if there is no dissipation (r=-r*, u=u*, and
and the tide-raising and torquing bodies are the same (M=M*),
then the first and fourth terms on the right-hand side of (13) are
zero because of the cross products and the second and third terms
cancel. Without dissipation, not only does a bulge directly under
the attracting body exert no torque (first term on fight-hand side),
and not only is the apparent torque (-t0xIt0) from working in a
rotating frame unable to interact with the spin-induced
deformation (fourth term), but the torque from the spin
deformation (second term) and the apparent torque from the tides
(third term) cancel one another. In the rotating frame the same
tide-plus-spin forces which elastically distort the Moon cannot
also apply torque on that deformation since they are aligned. In
an inertial frame the attracting body does apply torque on the
rotation-caused bulge. The time variation of the angular
momentum It0 in the rotating frame is not altered by the elastic
deformations, but the rotation rates and Euler angles are still
influenced because of the time variation of I in that product.
Another piece of information can be gleaned from (13). For
multiple bodies raising tides and causing torques, there would be
sums over the bodies (briefly use a subscript for the body): two
sums in the first term on the right-hand side and one in each of
the second and third terms. Without dissipation, for every term
M n UnXt0 there is a term M n t0xu n which cancels it, and for every
M n Mtn UnXU m there is a MmM n UmXU n. For a constant Love
number, multiple attracting bodies cannot alter the angular
momentum in the rotating frame through deformations without
With dissipation the four deformation terms on the right-hand
side of (13) are nonzero. The important torque terms arise from
the Earth interacting with Earth-raised tides, while the Sun is
only a minor influence. In component form the functions
Uij = (a/r) 3 tx i txj and 03i 03j In2 are needed. The diagonals of the
functions give (a/r) 3 and 032 which occur in I in the derivative
on the left-hand side of (13). (The radius r is conveniently
normalized by the semimajor axis a, and the spin is normalized
by the mean motion n.) The series for these functions were
developed using the lunar orbit theory of Chapront-Touzd and
Chapront [1988, 1991] and the physical libration series by
J. G. Williams et al. (manuscript in preparation, 2001)
(hereinafter referred to as Williams et al., 2001). The functions
with and without phase shifts/time lags are multiplied together to
represent the four terms on the right-hand side of (13). When
written out in component form, each of the three vector
components of the differential equation has 24 terms on the right
side, and each term has a series expansion. Economy of effort is
achieved by combining the second-degree functions from Earth,
Sun, and spin into one matrix. The coefficients are in proportion
to the -M/a 3 and n2/3G that can be deduced from (7) and (9).
Then the 24 terms for each component (54 if the Sun is included)
can be replaced with six.
Since u 1--1 and 033/n -- 1, the larger deformation terms involve
these components. As an example, the most important pair of
terms on the right-hand side of the third component of the vector
differential equation (13) is
_ k2 Rs 3 G M 2 , ,
aS [UiiUi2-Ui2Ull ]. (14)
Without dissipation this pair of terms will cancel, but with
dissipation a component multiplying a phase shift is selected for
each periodicity. The u 2 depends on orbit and physical libration
variations, with the dominant periodic terms from the longitude
variations of the lunar orbit. The largest of these is the monthly
(27.555 days) eccentricity-caused term depending on mean
anomaly g, approximately 2esin •. With this term as an
example, the brackets in (14) plus a smaller contribution (indices
2212) yield
22,000" ( sin •* - sin • )-- 22,000" ( •* - g ) cos g. (15)
For a positive frequency a positive time delay corresponds to
a negative phase shift and a positive specific dissipation Q
so ( •* - g )-- -1/Qtj. Terms of the form of (15) arise from a
constant torque coefficient multiplying a periodic deformation,
minus a periodic torque times a constant deformation. Other
terms result when a periodic torque multiplies a periodic
deformation, and a constant results when the periods are equal.
The phase for constant terms enters directly as a difference, e.g.,
sin(g*-e), while mixes of different periods give arguments with
angles mixed together, e.g., sin(2F*-e*-e)- sin(2F-e-e*).
The factor GM/a 3 is ubiquitous, and for analytical computation
it is useful to relate it to sidereal mean motion n. Kepler's third
law is modified for solar attraction [Brouwer and Clemence,
1961, chap. 12], and 1/a is set equal to the time-averaged 1/r for
the perturbed orbit:
GM_ n2 1+ (16a)
a 3 - 2 n 2 (M+m) '
a 3 -- 0.9906 n 2 . (16b)
terms which depend on the mean anomaly result from the radial
variation and the variation in orbit longitude. Consequently,
forcing terms proportional to e sin i have arguments
(1/2 month period) and F-e (2190 days = 6.0 years). The
strongest forcing functions for rigid or deformed motion of the
lunar pole have arguments F, F-e, and F+e.
The influence of deformation on the pole direction, the latitude
physical librations, is calculated using two orientation
parameters. The P l and P2 parameters are the x and y coordinates
of the ecliptic pole, respectively, using the lunar principal axis
P 1 = -sin 0 sin {p, (18a)
where n' is the mean motion of the Earth-Moon center of mass
about the Sun. The Earth/Moon mass ratio is M/m = 81.3006,
and for R = 1738 km the ratio R/a = 1/221.17.
The third component of the differential equation (13) describes
the rotation about the polar axis. This rotation angle nearly
follows the mean Earth as seen from the Moon, the Moon's
orbital mean longitude L plus 180 ø. The small remaining part,
the "longitude" angle of physical librations, is called •:. For the
ecliptic definition of Euler angles in section 2, ½+•t = L+'r+180 ø.
The theory of the lunar rotation with torques on the lunar figure is
a classic problem [Eckhardt, 1981; Moons, 1982a, 1982b;
Petrova, 1996]. While the differential equations for rotation are
nonlinear, a linearized form gives a good first approximation.
For the present purpose, use 03 3 = n + :r, ignore the small 031032
term, extract a linear q: term from the rigid-body torque, and treat
the remainder of that torque as a forcing function. Then the polar
component of the differential equation becomes
C(':r + 3 ¾n 2 'r ) + •3 n =f: (17)
The forcing term f_ includes both the rigid-body forcing (without
linear 'r term) and the right-hand side of (13). The solution from
the rigid-body forcing is not an objective here but is treated in the
above three references (also see section 13). The resonant
frequency n ( 3 ¾ )1/2 for the longitude variable has a period of
1056.1 days (including a correction factor S 3 =0.9759 and
adjustment for third-degree harmonics discussed by Williams et
al. (2001)). As an example, the resulting solution with the
forcing term proportional to (15) is-1.3" (k2/Q 0 cos e, but there
is a small correction from the derivative of •33, and the final
contribution to 'r is -1.1" (k2/Qt,) cos g. With the DE403 solution
values the coefficient is-0.0012" or -1 cm at the lunar equator,
which projects into a few millimeters in range.
The solution of the differential equation (17) for a periodic
forcing term amplifies longer periodicities more than monthly
terms. Libration amplitudes larger than the monthly example
occur for annual, 206 day, and 1095 day periods. The latter
requires the most care since it is near the resonance. Dissipation
also induces a constant offset of 'r which is larger than any of the
periodic terms. Solar influences decrease the constant coefficient
by 0.2%. The derivative of I plays only a minor role for
longitude librations because it favors fast terms, while the
solution of the differential equation favors slow terms.
The mean lunar orbit plane is inclined 5.145 ø to the ecliptic
plane. The resulting ecliptic latitude motion of the Moon
depends on the angle measured from the node, with period
27.212 days, and the polynomial representation of the angle is
denoted F (=L-• or mean argument of latitude). The leading
term for ecliptic latitude is 5.13 ø sin F, and this gives the
strongest forcing term for the lunar pole. Additional forcing
P2 = -sin 0 cos {p. (18b)
The differential equations for Pl and P2 are coupled together [see
Eckhardt, 1981 ]. The linear approximation to (13) comes from
taking 033 constant, expressing the first two angular velocity
components as functions of Pl and P2 and their derivatives, and
extracting a linear term in P l from the rigid-body torque term
(second component):
A ( •2 + n (1-ct)Pl + ctn2p2 ) +)13 n =fx, (19a)
B(-fil +n(1-•)ib2-4[•n2pl ) +/23 n =fy (19b)
The forcing functions about the x and y axes have been multiplied
by the cosine of the equator's 1.54 ø tilt to the ecliptic plane
to give fx and fy, respectively. Resonance frequencies are
27.29638 days and 74.63 years (Williams et al., 2001). The rigid
or deformed forcing terms at 27.212 days (F) and 6 years (F-g)
cause significant responses in the pole direction, but the
1/2 month response is weak. The first three terms on the right-
hand side of (13) are important for the pole. The derivative of I
plays a major role for the F term. The Sun increases the F term
magnitude by 0.3%.
For the linear part of the rotational dissipation solution, six
elements Uij are considered for each of the constant plus 52
periodicities of the Earth-induced torque/tide functions. These
include the largest functions plus smaller periodicities selected to
give longer periods or near resonant terms. To these are added
the Sun-induced functions for the constant and 13 periodicities
plus the larger spin terms. The appropriate combination of
elements for the right-hand side of (13) and the moment rate on
the left-hand side are computed for 52 (constant times periodic)
plus 2x522 (periodic times periodic, giving sum and difference
frequencies) combinations. Rotational coefficients are retained
above a threshold size.
In addition to the first-order solution, selected nonlinear
corrections from the rigid-body torques are added as second- and
third-order corrections. This has the effect of increasing the 'r
constant by 3% and increasing the magnitude of the F
coefficients for the pole by 2%. The pole response at 6 years is
made larger.
5. Tidal and Rotational Dissipation'
Series Solution
This section presents and discusses the lunar physical libration
series solution for tidal and spin dissipation. Comparisons are
made with the previous computations of Yoder [1979] and
Eckhardt [ 1981 ].
Table 1. Longitude Libration Tidal and Spin Dissipation Terms are Given as a Function of Deformation Q Parameters a
Argument Period, Coefficients for Deformation Q Parameters
2D+œ 2F F+œ 2D œ+œ' 2F-œ F œ 2œ-F 3œ-2F D œ-œ' 2D-œ 2œ-2D œ'
9.6 13.6 13.7 13.8 14.8 25.6 26.9 27.2 27.6 27.9 28.3 29.5 29.8 31.8 206 365
œ-D 2F-2œ F-œ
412 1095 2190
0 oo 0.5 -0.3
œ 27.55
2D-œ 31.81
2œ-2D 205.89
œ' 365.26
œ-D 411.78
2F-2œ 1095.18 -0.5
F-œ-79 ø 2190.35
1.9 5.5 7.6 8.4 305.4 10.8 -0.2
-0.3 -0.3 -2.2 -2.3 0.9
0.4 -0.4 8.5
0.3 0.3 -0.3
-0.5 -14.9 -5.9 -14.8 -5.3 -0.3 17.9
aEach libration term is the product of a cosine of the argument at the left, with its period in days, times the sum of the coefficients (in arcseconds) to the
right. Each coefficient is multiplied by the Love number k 2 and divided by the Q for the deformation period (days) and deformation argument at the top.
The arguments of the series solution depend on polynomial
expressions for four angles. The polynomials are denoted g for
lunar mean anomaly (period 27.555 days), g' for the mean
anomaly of the Earth-Moon center of mass about the Sun
(1 year), F for argument of latitude (27.212 days), and D for
mean elongation of the Moon from the Sun (29.531 days). Also
useful is the polynomial for the lunar orbit node f2 measured
from the precessing equinox. It is also convenient to use L and L'
for the polynomial expressions for the mean longitudes of the
Moon and Sun, respectively, both measured from the precessing
equinox, where L = F + f2 and D = L- L'.
By subtracting the uniform rotation and precession motion
from the Euler angles, there results a set of small libration
parameters q:, p, and (5. For the ecliptic definition of Euler angles
(section 2) the conversions between Euler angles and the libration
parameters are = f2 + (5, 0 = I + p, and (p = F + 180 ø + q: - (5.
Equations (18a) and (18b) provide the connection to Pl and P2'
The angle I (not to be confused with the moment of inertia) is the
1.54 ø mean tilt of the precessing equator to the ecliptic plane.
The product 1(5 is convenient because it is comparable in size to p
The analytical dissipation series for the longitude libration ('[)
is tabulated in Table 1, and the latitude librations (pl and P2) are
in Table 2. Coefficients down to 0.2 are presented (a borderline
188 day term was not included in Table 2). In Table 1 the 6 year
term with phase is orthogonal to the rigid-body term owing to
third-degree harmonics. The amplitude of each periodic term in
the rotation depends on one or more of the Q parameters for the
deformation frequencies. For example, in arcseconds the
monthly p• term in Table 2 is
k, (217.4 8.0 4.7 1.8 0.7 )
+ ¾ + 0-75 + + +'"
The coefficient is dominated by the Q for the 27.212 day month
(north-south motion), but the Q for deformation at the 27.555 day
anomalistic month and the Q at 1/2 month contribute a few
percent. Most of the p• and P2 coefficients for argument F are
equivalent to a constant, negative shift of the equator's precessing
node. The constant 1(5 shift is given in arcseconds by
/(sconst k• ( 216.4 0.2 4.7
=" - QF + Qt - QF+t
1.8 0.7
- Q2F - QF-• + '" ) ' (21)
Compared to the monthly p• and P2 coefficients, the Qe
dependence has virtually disappeared, and the sensitivity to the
three principal frequencies of latitude forcing remains. In
Table 2. Latitude Libration Tidal and Spin Dissipation Terms a
Argument Period, Libration Function
2F F+(
days 13.6 13.7
Coefficient for Each Deformation Q
2D F 2D-t F-t
14.8 27.2 27.6 31.8 2190
F 27.212 Pl cos 1.8 4.7
F 27.212 P2 sin -1.8 -4.7
F-t 2190.350 Pl cos -0.2
F-œ 2190.350 P2 sin 0.3
2F 13.606 1(5 cos
2F 13.606 p sin
2F-t 26.877 1(5 cos
2F-œ 26.877 p sin
27.555 1(5 cos
27.555 p sin
0 oo 1(5 1 -1.8 -4.7
217.4 8.0 0.3 0.7
-216.0 -8.0 -0.3 -0.7
-6.9 -5.8 -1.9
8.3 7.5 2.6
0.3 -0.8 -0.3
-0.3 0.8 0.3
3.7 6.5 2.3
-3.6 -6.5 -2.3
-216.4 -0.7
aLatitude libration parameters are p• and P2 and, equivalently, p and 1•. Each libration term is the specified trigonometric
function of the argument at the left (with its period) times the sum of the coefficients to the right. Each coefficient is multiplied
by the Love number k 2 and divided by the Q for the deformation period (in days) and associated argument at the top.
addition to the P l and P2 parameters in Table 2, an approximate
conversion to p and Ic• is given. The latter pair is less complete
since it omits some smaller combinations including differently
phased mixes with F arid fl.
The physical libration Pl is approximately the tilt of the lunar
pole away from the Earth, and the monthly term is the largest
observable dissipation periodicity. (The constant in longitude
libration is not directly measurable since a change is
compensated by a shift of reflector longitude coordinates during a
solution.) The 27.212 day periodicity is the dissipation signature
that has been seen by LLR for 2 decades. With Q proportional to
1/frequency and the DE403 value of k2/Q = 0.030/25.9 =
1.16x10 -3, the coefficient of the Pl term is 0.276". Since the
coefficient in (20) is dominated by the monthly Q F, the Q
determined by the DE403 fit of LLR data to a time-delay tidal
dissipation model effectively corresponds to a monthly period of
27.212 days. A different dependence of Q on frequency will
change the Q inferred from observations by only a few percent.
The Ic• shift is-0.265" and the node shift is-9.8"
For the DE403 value of k2/Q a unit value in Tables 1 and 2
corresponds to a rotational displacement of 9.7 mm at the lunar
radius. It is interesting to compare the tidal sensitivities for
periodic rotation terms in the tables with tide heights. For the
largest tides of-0.1 m, with arguments • and F, the Q• and QF
are well represented among major rotation terms. Of the tides
from 1 cm to several centimeters, namely, 2D-g, 2D, 2g, F+g, the
latter is most important in the rotation. Of the many tides from
1 mm to several millimeters, the rotation is sensitive to Q
parameters for F-e, •' 2•-2D. The 2F-2• tide is only -0.1 mm
but is selected by the near resonant period. The phase-shifted
part of the tide height is proportional to 1/Q. So the larger
sensitivities in Tables 1 and 2 correspond to phase-shifted tidal
displacements of a few millimeters down to a few micrometers.
For selected tidal frequencies the influence on the rotation
exceeds the tide height in size.
The dissipation terms have been evaluated for two
dependences of Q on frequency using the expressions in Tables 1
and 2 augmented with smaller coefficients. Table 3 evaluates the
coefficients for Q independent of frequency, and Table 4 uses Q
proportional to F/frequency. The latter corresponds to the time
delay tidal model used for the numerical integration of the
rotation. For the 6 year longitude term, only the cosine
component is shown, but most of that term is in the sine
component (see Table 1). Most noteworthy are the monthly and
6 year terms for (Pl and P2) latitude librations and the 3 year,
1 year, and 206 day terms for longitude libration. The most
interesting terms for testing frequency dependence of Q are the
3 year and annual terms in longitude libration. Table 1 shows
that the annual term is sensitive to the annual tidal Q, while the
3 year term is most sensitive to monthly Q and 3 year tidal Q.
The series of Tables 1-4 scale inversely with C/mR 2, here taken
as 0.3932 with an uncertainty of 0.0002 [Konopliv et al., 1998].
Table 3 can be compared with Eckhardt's [1981]
computations, and Table 4 can be compared with Yoder[1979].
For the constant in longitude, Eckhardt (multiply his tabulated
differences by-2000) gave 342, and Yoder gave 350.4.
Eckhardt's values should be -1/2% larger owing to his smaller
value of C/mR 2, so the constant term here is slightly less than the
two published calculations. For the 3 year longitude term,
Eckhardt has -24, in good agreement with Table 3. Yoder's
value for this near-resonant term is off by an order of magnitude.
For the 206 day term, Yoder has the right magnitude (5.0), but
the reversed sign, while for the difference between the annual
terms of Tables 3 and 4 he gives 8.4. Eckhardt does not give
terms smaller than 10. For the large term in latitude libration,
Eckhardt gives 210 and -208 for the monthly P l and p2
coefficients, respectively, and-208 for the I•J constant.
Compared with Table 3, his monthly magnitudes are 10% smaller
and the lc• magnitude is 7% smaller. The magnitude of the Ic•
constant should be less (qJconst sin I = 9) than the average of the
two monthly magnitudes, so there is a 4% internal inconsistency
in Eckhardt. Yoder defines his latitude results as though a
rotation of the p and l•J variables, and the 229.6 value for the
latter parameter (there is a sign ambiguity due to an apparently
misplaced n in his definitions) is a good match with Table 4. The
second term in latitude librations is elliptical in p l and P2 and
splits into g and 2F-g terms in l•J and p. Eckhardt gives -14 for
P l and 20 for P2, in reasonable agreement with Table 3, while
Table 3. Evaluation of the Coefficients of the Physical Libration Theory for Tidal
Dissipation Using Q Independent of Frequency a
Argument Period, 'c Pl P. 2 Io p
cos, cos, sin, cos, sin,
days ..........
0 oo 339.95
F 27.21
F-• 2190.35 -0.14
œ 27.56 -1.12
2•-2D 205.89 -4.14
365.26 8.20
2œ-F 27.91
2F-2œ 1095.18 -24.30
œ-D 411.78 0.36
F+œ-2D 188.20
2D-œ 31.81 -0.39
2D-F 32.28
2F-2D 173.31 0.19
F+f•-81 ø 27.32
81ø-f• 6798.38
2F-œ 26.88
2F 13.61
13.78 -0.03
233.73 -232.41
-15.02 18.96
-0.01 0.03
-0.34 0.46
-0.16 0.22
-0.18 0.18
aEach coefficient (units arcseconds) should be multiplied by k2/Q.
Table 4. Evaluation of the Coefficients of the Physical Libration Theory for Tidal
Dissipation Using Q = QF •v / Frequencya
Argument Period, •: P 1 P. 2 Io p
cos, cos, sin, cos, sin,
days ..........
0 oo 349.30
F 27.21
F-/• 2190.35
P. 27.56 -1.13
2P.-2D 205.89 -5.03
g 365.26 0.34
2g-F 27.91
2F-2g 1095.18 -43.31
œ-D 411.78 0.64
F+œ-2D 188.20
2D-•t 31.81 -0.38
2D-F 32.28
2F-2D 173.31 0.10
F+f•-81 ø 27.32
81 ø-f• 6798.38
2F-•e 26.88
2F 13.61
2P. 13.78 -0.06
240.30 -238.98
-13.39 16.86
-0.21 0.23
-0.31 0.44
-0.18 0.26
-0.18 0.18
aEach coefficient should be multiplied by k2/Q F. Units are arcseconds.
Yoder gives 12.5 by 15.2, which is similar to Table 4's entries.
The numerical results of Bois and Journet [1993] are much
smaller than the analytical results and are in error.
The most important dissipation terms are at monthly, 206 day,
annual, 3 year, and 6 year periods. The series of this section will
be used for interpretation of LLR data fits (section 18).
6. Average Values and Definitions
Section 3 pointed out that the tidal deformation of (7) and the
spin deformation of (9) have constant parts. With deformations,
the "rigid-body" moments of inertia of (5) are not the time-
averaged moments. Since the second-degree harmonics J2 and
C22 depend on the moments, careful definitions must be given.
The rigid-body moments A, B, and C are used to define
ct=(C-B)/A, [•=(C-A)/B, and T=(B-A)/C. J2 is taken as an
independent parameter, while C22 and C/mR 2 are derived
J2 rigid (1 + •)
C22rigid- 2(2•-¾+•¾) (22)
C 4 C22 rigid (23)
mR 2 -
The constant part of the functions (a/r) 3 U i Uj and 03 i 03j / n 2 are
used to compute the averages. For accurate time-averaged values
of the moments normalized by mR 2 and the second-degree
harmonics, add the corrections from the appropriate columns of
Table 5 to the rigid-body values. There are very small tidal
contributions to the off-diagonal moments, and two second-
degree harmonics because two of the principal axes are not quite
aligned with the mean Earth and mean spin directions. The
principal axes of the rigid body and average deformed body do
not quite match.
In the JPL LLR software, [•, ¾, k 2, and J2 are the independent
parameters, while C22 and C/mR 2 are derived. In the numerical
integrator the mean spin values have been virtually nulled out of
/spin' which forces the mean spin effects into the "rigid-body"
quantities. Only the average tidal contributions from the Earth
(no Sun) should be added to rigid-body quantities to get averages.
Thus the LLR-derived values of [• and ¾ reported in this and past
JPL papers depend on the rigid-body part without mean tides.
Ferrari et al. [1980] gave expressions to link values of J2 and
C22 which include average Earth-raised tides with k 2 and rigid
values of [•, ¾, and C/mR 2. Those expressions were used to report
numerical values there and by Dickey et al. [1994]. The original
rationale was that spacecraft-derived harmonics were generated
without a tidal or spin deformation model, while LLR analyses
did use a tidal model and a nulled average spin deformation.
Tidal models are now used to analyze spacecraft data [Lemoine et
al., 1997; A. S. Konopliv, private communication, 1996] as well
as LLR data. Table 5 can be used to recover average values for a
variety of definitions.
A fluid or strengthless Moon would relax to the shape of the
tidal plus synchronously rotating spin potential. To calculate the
equilibrium moment differences or second-degree gravitational
harmonics for the Moon, the fluid Love number kf = 1.44 is
appropriate rather than the smaller quantity from elastic theory.
Such a calculation shows that J2 is 22 times larger, [• is 17 times
larger, and ¾ and C22 are 8 times larger than the equilibrium
figure for the present distance. The Moon is strong enough to
support the stress elastically. It is appealing to conjecture that the
tidal plus spin figure was frozen into an earlier Moon closer to
the Earth [Jeffreys, 1915, 1937; Kopal, 1969; Larnbeck and
Pullan, 1980]. The spread of factors from 8 to 22, corresponding
to distances of 0.50 to 0.36 times the present Moon, does not
make it easy to embrace the hypothesis. Lambeck and Pullan
invoke noise in the gravity field, the spectrum of power in the
higher-degree field extrapolated to second degree, to explain the
spread. Here the spectrum of Konopliv et al. [1998] is adopted
for the extrapolation, and a linear combination, which would be
zero for an equilibrium figure, is formed. The linear combination
of harmonics is J2 - 10 C22 / 3 = ( 1.3+ 1.1 )x 10 -4, or the equivalent
expression [• - 4 ¾ / 3 = (3.3+2.7)xl 0 -4, and the departure from
Table 5. Mean Values of Deformations for Moments and Harmonics a
Parameter Rigid Tide by Earth Tide by Sun Oblate Spin Spherical Spin
Ill/mR2 A/mR 2 -4.935x10 -6 -7x10 -9 -0.843x10 -6 2.529x10 -6
122/mR 2 B/mR 2 2.469x10 -6 -7x10 -9 -0.843x10 -6 2.529x10 -6
133/mR 2 C/mR 2 2.466x 10 -6 1.4x 10 -8 1.686x 10 -6 2.529x 10 -6
ll2/mR 2 0 2.3x10 -9 0 0 0
I• 3/mR2 0 -2.8x 10 -9 0 0 0
/23/mR 2 0 0 0 0 0
J2 J2 rigid 3.698X10 -6 2.1X10 -8 2.529X10 -6 0
C2! 0 2.8x10 -9 0 0 0
S2• 0 0 0 0 0
022 022 rigid 1'851X10-6 0 0 0
S22 0 - 1.2x 10 -9 0 0 0
aThe tidal and spin deformations of the moments of inertia and the second-degree
harmonics have mean values (columns 3-6). The symbol (or zero value) for the rigid-body
quantity is given in the second column. The numerical values in columns 3-5 should be
multiplied by the Love number k 2. The last column should be multiplied by s.
equilibrium is comparable to the extrapolated power. The frozen
figure hypothesis is viable.
7. Frequency Shifts and Damping
From Deformation
The forced lunar physical librations have three resonances: one
in longitude libration and two for pole direction. The resonance
periods are the same as the periods of the three free libration
modes. The free librations are analogous to the solutions of the
reduced equations for linear differential equations, and the
unpredictable amplitude and phase must be established by
observation. See Williams et al. (2001) for a study of free
librations. Elastic deformation will shift the resonance periods
from the rigid-body values, and dissipation will damp the free
librations in addition to causing the forced terms of sections 4
and 5.
Elastic deformation without dissipation does not contribute
forced terms from the right-hand side of (13). It does influence
the rotation through the derivative of I in the Ico term. The
largest modification comes from the i-1, j=3 tidal term in (7).
The u 3 component is a function of Pl, and its derivative is
introduced into the differential equations. The square of the
monthly resonance frequency for pole direction
(precession/nutation mode) in the rotating frame is modified to
2=n 211_ •2 sin2 I+3(S lct+S 2[•')+k 2•cosl] (24)
¾p '•- ,
where S2=0.9778, Sl=0.0018, and [•'=629.978x10 -6 is a
modification of [• to include effects of third-degree harmonics
(see Williams et al., 2001). The tidal part depends on the
mR2 M ( aR__)3
- = 1.91 x 10 -5 . (25)
•- C m
For the DE403 k 2 value, the tidal part shortens the monthly
resonance period by 8x10 -6 day. The equivalent 81 year period
in the nonrotating frame is shortened by 9 days, and the 24 year
period in the 18.6 year precessing frame is lengthened by 0.8 day.
Other elastic effects on the three resonance frequencies multiply
ct, [•, or ¾ and so are less important than the contribution in (24).
While elasticity causes a dramatic increase in the wobble period
for the Earth, this, as Peale [ 1973] realized, is not the case for the
The free libration in longitude has a 1056 day period
(Williams et al., 2001). A variation of 'r causes an east-west
motion of the tidal bulge, and a delayed response in the bulge
causes damping from the tidal torque term. A linear term for 'r
comes through u 2 in (14) and this is the source of most of the
damping in (17). For damping like exp(-Dt) the damping time is
1/D. The damping for the longitude mode is
3 k2
D/_? 0.497 -•- n Q/_,, (26a)
k 2
D L - 0.091 •LL Yr-l' (26b)
The QL is at the 1056 day period, and ¾ ', with value 228.6x 10 -6,
is a modification of ¾ for third-degree harmonics (Williams et al.,
2001). The expression (26a) is similar to that given by Eckhardt
[1993], and (26b) is 4% different from the numerical expression
of Peale [ 1976].
The motion of the pole direction moves the tidal bulge in a
north-south direction. The tidal torque term (first on right-hand
side of (13)) is the main influence on damping the 27.296 day
monthly mode. Terms from the derivative of the moment and the
spin acting on the tidal bulge (third term) cancel. The spin on
spin and torque on spin bulge terms are ineffective because the
spin axis stays near the principal axis for the monthly mode. The
damping is given by
k 2
Dp = 1.47 n Qp, (27a)
k 2
Dp = 2.35x10 -3 •pp yr -1. (27b)
The Q is at 1 month (27.296 days). The agreement with Peale's
numerical value is excellent. For the DE403 value of k2/Q the
damping time is 3.67xl 05 years.
For the wobble mode the spin axis is displaced from the
principal axis. The bulges from tides and spin are both effective
in damping the 74.6 year wobble. The expression for the
damping of the elliptical wobble depends on the ratio E (=2.474)
of the axis of the ellipse, where E 2 = (
Dw ( 2.62 k2
= + 0.168 E) n Qw' (28a)
k 2
D w = 1.47 n Qw' (28b)
D w = 2.36x 10 -3 k2
-•w yr-l' (28c)
The wobble Q is at 74.6 years. The numerical expression is 17%
different from Peale's. The similarity of numerical coefficients
for the damping of the two pole modes is coincidence.
Fits of the LLR data will be used to estimate Q as a function of
frequency (section 18). Damping times will then be calculated
(section 20).
8. Orbit Perturbations From Tidal Dissipation
The tidal and spin deformations not only affect the lunar
rotation but also perturb the orbit. There are both elastic and
dissipation effects, but only the latter are considered in this
section. Dissipation causes the exchange of energy and angular
momentum between the rotation and orbit. This section first
presents the potentials for deformations and then gives numerical
and analytical expressions for secular orbit changes.
An external body raises tides on the Moon, and those tides
generate forces on the tide-raising and any other external bodies.
The tidal distortion from a tide-raising body (denoted by *) has a
potential energy at an external body of
R 5
Vtide =k 2 G MM* r3 r, 3 P2(u'u*). (29)
The potential energy at the external body from second-degree
spin distortion is
R 5
Vspin---k 2 M to '2 -- P2(u'•)*). (30)
3 r 3
P2 is the second Legendre polynomial, and •o is the unit spin
vector. The remaining notation is as before. For dissipation the
phase-shifted or time-delayed variables (except M) indicated with
an asterisk are displaced as seen from the frame of the rotating
body. To calculate forces, the positive gradients of (29) and (30)
are taken with respect to the position coordinates without an
asterisk (sign convention for the point mass potential is plus).
Along the Earth-Moon line the acceleration is inward toward the
A rotating frame is natural for computing time-delayed lunar
deformation. Both the orbit motion and rotation are time
delayed. For orbit computations it can be convenient to expand
the vector and scalar radius through first order in the time delay
At using a space-fixed frame
r* -- r-( i'- toxr ) At, (31a)
r* • r-/' At. (3lb)
The expression in parentheses is the conversion from space- to
body-referenced velocity.
As seen from the rotating Moon, the Earth's angular and
distance variations cause tides. Here secular orbit changes from
energy and angular momentum exchange are considered. The
orbit is perturbed in two ways by the deformations' directly from
the forces calculated from the gradients of (29) and (30) and from
forces due to the rigid figure of the Moon through the rotational
displacements of its principal axes. To compute the power going
into the orbit, calculate i'.V V, where V is the sum of the rigid
figure, tide, and spin potentials. With manipulation the equation
for power is derived.
dV d(Ito)
i-. VV = -•- - to .-•--. (32)
Since the Euler equation (1) permits the derivative of the angular
momentum to be replaced with the torque, this equation may
seem self-evident, but the right-hand side is evaluated in the
frame rotating with the Moon, which is computationally
convenient, and the left-hand side is in the nonrotating frame, as
needed for orbit perturbations. For the time derivative of V one
differentiates the u and r variables but not the parameters with an
asterisk. Simplifications can be made. Owing to the
synchronous rotation, the power flowing into the rotation rate is
only C/ma 2 -- 10 -5 of the dissipated power, so the spin potential
and the second term on the right-hand side can be ignored. The
trigonometric series for Uij = u i uj (a/r) 3 were developed for the
computations of section 4, and these series appear in the rigid
figure and tide potentials. The rigid figure potential is linear in
the U i. and its time derivative gives periodic terms, but the tidal
•J.' .
potential contains products Uij U i j, and its derivative contains
periodic and secular terms. For Earth-raised tides acting back on
the Earth the average power, Pave' depends on the tidal potential
through the constant part of
k 2 G M 2 ,
Pave= 2a (-ff-)5(3•bijUij- Z /-]ii Z U•jj )' (33)
ij i j
This power is drawn from the lunar orbit and dissipated in the
Moon. The average power depends on squared tidal amplitudes
times the frequency. Note that Uii = (a/r) 3 The average
power from solar tides is three orders of magnitude smaller than
the power from Earth-raised tides.
The power is related to the semimajor axis change through the
derivative of the total energy -GMm/2a. The secular semimajor
axis and mean motion changes ( 3 Ad/a =-2 Ati/n ) are given in
Table 6. The dependence on each tidal Q is explicit. In
calculating the table, power is converted to semimajor axis
change using a mean semimajor axis, rather than an osculating
one. To convert/xti in mm yr -l to average power in ergs yr -•,
multiply by 0.99x 10 24.
For dissipative effects the torques on the lunar rotation and
orbit, due to displaced second-degree figure and deformation, are
equal in magnitude and opposite in sign (there are ignored figure-
figure effects which are effectively fourth degree). About the
polar axis the constant part of the torque due to tides is balanced
against the constant part due to the rigid figure being displaced
by tides. The average torque about the polar axis is zero. The
tide-caused displacement of the pole direction is a dynamical
rather than static response, and the sum of torques about the body
y axis is not zero. This time-varying torque has a constant
component projected along the line of the equator/ecliptic
intersection. This component causes the Moon's equator to
precess, but the dissipation-induced shift in the direction of the
constant torque by o from the orbit node on the ecliptic (section
5) causes secular orbit perturbations. Since the torque vector
does not quite lie in the orbit plane, the orbital angular
momentum is perturbed, and since it is not quite aligned with the
Table 6. Secular Orbit Changes From Periodic Tides a
Argument Period, Ah, Ah, Ap, Ak, A•, A/•, A•,
days "cent -2 mm yr -1 mm yr -1 10 -• yr -I •as yr -I "cent -2 "cent -2
œ 27.555 205 -302 -4 -705 15 -1.71 2.36
F 27.212 136 -201 -201 2 -601 -0.89 0.34
2D-• e 31.812 6 -10 0 -22 0 -0.06 0.08
2D 14.765 10 -14 0 -33 0 -0.08 0.11
13.777 7 -10 0 -23 0 -0.05 0.08
F+• 13.691 6 -9 -4 -10 -13 -0.04 0.04
2F 13.606 1 -2 -2 0 -5 -0.01 0
F-œ 2190.350 0 0 - 1 2 -2 0 0
2D+• e 9.614 1 -1 0 -3 0 -0.01 0.01
Sum for constant Q 373 -550 -212 -795 -606 -2.86 3.02
Sum for Q-l/frequency 394 -580 -218 -854 -623 -3.02 3.22
aTidal argument and period are at left. The remaining columns are to be multiplied by k2/Q, with Q appropriate to the
tidal frequency. The last two lines give the sum of terms for Q constant and Q proportional to inverse frequency (multiply
last line by k2/Qr).
node, the inclination is perturbed. The angular momentum
component normal to the ecliptic plane is preserved.
For angular momentum exchange between rotation and orbit
the torque rxVV is required. In section 4 the tidal torques were
developed for physical libration calculations but must be rotated
from body-referenced coordinates into the orbit frame. For the
computations of Table 6 the total orbital angular momentum is
proportional to the square root of the semilatus rectum p=a(1-e2),
and the torque normal to the orbit plane gives the change in p.
The eccentricity rate comes from the change in p and a. The
torque component in the orbit plane directed 90 ø from the node
gives the secular orbit inclination rate.
There are indirect effects of the above a, e, and i rates which
cause the perigee and node precession rates to change. The solar-
induced precession rates depend strongly on the mean motion and
more weakly on eccentricity and inclination. Like the mean
longitude, the node and perigee angles experience tidal
accelerations. The partial derivatives of the longitude of perigee
(•) and node (f2) precession rates [Chapront-Touzd and
Chapront, 1988], with the tabulated tidal rates for a, e, and i, give
the accelerations '• and in Table 6.
The model for the DE403 integration is based on tidal
dissipation, but no core. The DE403 solution effectively sets a
limit to the tidal contribution' Ar• = 0.46 "cent -2 and
Aft = -0.67 mm yr -1. Additional rates are Ap = -0.25 mm yr -l,
Ag = -0.99x10 -li yr -l, and Adi/dt- -0.72 •as yr -1. The
accelerations are A/• = -0.0035 and A• = 0.0037" cent -2. The
inclination rate and the last two accelerations are too small to
detect with the present data set. The secular acceleration Ah is
positive. Tides on the Earth cause a negative secular acceleration
of-26" cent -2. Tidal dissipation in the Moon contributes <2%
of the total tidal secular acceleration. The above eccentricity rate
is 70% of that from the Earth. The product aAk =-3.8 mm yr -l.
With the above Ad, lunar tides cause the perigee to increase
3.2 mm yr -I and the apogee to decrease 4.5 mm yr -l. These
changes, along with the secular acceleration, are large enough to
detect with the Lunar Laser data analysis, but other masking
influences on these rates must be considered (see section 16).
Analytical approximations for the orbit changes are useful,
e.g., for evolutionary calculations. For the effects due to the
displaced figure axes the dissipation-induced constant •: and
terms are needed. Analytical approximations are
k2 M mR2 (_•)3 [ 6e2
A'C=Q m C sin(i+/) sin I ] , (34a)
k 2 M tnR 2 (_•_)3 sin(i+/)sin/
AIO=-Q rn C [3 sini (34b)
The Q is for a 1 month tidal period.
The analytical approximations correspond to the •? and F tides
in Table 6. The leading terms in the U0. series are
U•l--l+3ecosg , Ul2--2esing, and Ul3--sin(i+/)sinF.
These may be used with the power equation (33) and converted
to the secular acceleration in orbital mean longitude •h:
9 k2 M (_•__)5 n 2 [7 e 2 + sin2(i+/) ]. (35)
Ah=2 Qm
The orbit eccentricity is e (0.0549), the semimajor axis is a
(384,399 km), and the mean motion is n (13.3685 rev yr-•). The
inclinations of the orbit and equator planes to the ecliptic plane
are i = 5.145 ø and I = 1.543 ø, respectively. The numerical
evaluation 348 k2/Q" cent -2 may be compared with Table 6. The
semimajor axis perturbation follows from Aft =-2 a Ah/3 n. The
numerical evaluation is •fi =-515 k2/Q mm yF 1.
Analytical approximations for eccentricity and inclination
rates follow from angular momentum transfer as before:
•b: 21 k2 M
-• -- -- ne (36)
2Qm '
di 3 k2 M (_•__)5 sin2(i+/)
= -- -- n (37)
dt 2 QF rn sin i
The Q is for a month. The numerical evaluations are
Ak = -7.4x10 -9 k2/Q yr -• and di/dt = -6.0x10 -4 k2/Q F" yr -•.
Lunar tidal dissipation extracts energy from the orbit and
deposits it in the Moon. Angular momentum from the orbit keeps
the lunar pole direction offset but does not change the spin rate
(apart from the small secular acceleration Ah). This is quite
different from the Earth, where the spin energy and angular
momentum power the orbit changes. (Zonal tides on the Earth do
extract their energy from the orbit rather than the spin, but they
affect tidal h by only - 1%.)
How does the Mooifs spin rate follow the slowly increasing
orbit period from dissipation on Earth and Moon? The rigid-
body axis displaces slightly east of the mean Earth direction, so
torques decrease the lunar spin. This is a rigid-body dynamical
balance of deceleration against torque. The expression comes
from solving the equivalent of (17) with a quadratic time term in
the polynomial for mean longitude L:
:• + h + 3 ¾ n 2 'c = 0. (38)
Assuming fourth and higher derivatives of L are zero, the
displacement in q: is
A'c = --- (39)
3¾n 2 ß
To follow the tidal deceleration of-26 "cent -2 requires a
displacement of only 0.0006". The quadratic (t 2) term in L
depends on the changing eccentricity of the Earth-Moon orbit
around the Sun as well as the tidal acceleration. The total
acceleration is -13 "cent -2 [Simon et al., 1994], and it requires
only 0.0003" shift of the axis for the lunar spin to follow the orbit
change. The longitude libration follows slow orbital longitude
accelerations as assumed in analytical theories and experienced in
numerical integrations [Bois et al., 1996].
The lunar tidal forces which give rise to the above secular
orbit effects are part of the JPL numerical integration program for
orbits and rotation. The numerical orbit integration does not use
this section's approximations. The time-varying moments of
inertia are converted to the five second-degree gravitational
harmonics, and the orbit perturbations are computed from the
harmonics. This is convenient because perturbations from the
large rigid-body parts of the lunar J2 and C22 must also be
calculated. The detectability of these orbit effects will be
considered further in section 16.
9. Computational Model for Core Dissipation
If a liquid lunar core exists, then dissipation at the core-mantle
boundary is expected when the fluid moves at a different rate
than the overlying mantle. This section presents the core model
used in the numerical orbit and rotation integrations and
theoretical computations.
Though motions in the fluid may be complex, we adopt a
simplified model based on the average fluid rotation m'. The
differential angular velocity between the core and mantle is
Ato = to'-to. At a point on the surface of a spherical core-mantle
boundary (radius R') the relative velocity of the fluid is AtoxR',
and a viscous force proportional to the relative velocity gives a
torque proportional to R'x(AtoxR')= R '2 Am- (R'.Ato) R'. When
integrated over the spherical surface, the total torque is
proportional to Am.
A core dissipation model is implemented in the LLR analysis
software. The equations of sections 2 and 3 are now interpreted
as applying to the mantle. To the large gravitational torques
acting on the mantle in T on the fight-hand side of (1) is added
the small additional torque T c
T c = K ( •o'-•o ), (40)
where K is a dissipation parameter which couples mantle and
core. The ratio of K to the mantle moment C is a parameter to be
fit to data. The core-mantle boundary is taken as spherical, so the
only torque on the core is -Tc. The Euler equation governing the
overall rotation of the core is then
d-•• + •o'xi'•o' = -Tc. (41 )
For a spherically symmetric core, the core moment matrix I' has
equal diagonal elements C' (tidal distortions are ignored), and the
above cross product is zero.
do)' K
dt= C' ( to- to'). (42)
The moment ratio C7C is an input parameter. For the Euler
equations the torque on the core is in the core's rotating frame,
while the opposite core torque on the mantle is expressed in the
mantle's rotating frame.
If the (laminar) viscous force is replaced with a turbulent force
proportional to the square of the relative velocity, then the total
torque integrated over the sphere is proportional to IX,,,I •x,,, and
the counterpart to (40) would require an additional factor of IXol.
Yoder [1981] concludes that a lunar core-mantle interaction
would be turbulent. There is further discussion in section 11.
The core-mantle coupling is weak, and m' shows less variation
than •o. The magnitude of the difference •o'-to is nearly
constant, and the direction is mostly uniform precession (the
mantle rate varies <10 -4 n, and the direction varies <10 -3 radians
from uniform precession). The difference between turbulent and
viscous interactions is subtle, and (40) is used in this paper for
data analysis.
The equations of rotation for the mantle and core are
numerically integrated along with the equations of motion for the
orbits of the Moon and planets. The initial time is 1969. Partial
derivatives of the lunar Euler angles and orbit with respect to
K/C, the two initial angular velocity vectors, two sets of initial
Euler angles, two mantle moment differences (C-A)/B and
(B-A)/C, gravitational harmonics, k 2, and lunar tidal dissipation
are also integrated so that solutions can be made.
10. Precession of Core
The equator of the observed solid Moon is tilted 1.54 ø to the
ecliptic plane, and its retrograde precession is locked to the 18.6
year precession of the orbit plane. It can be guessed that any core
will exhibit some analogous precession. The core tilt angle is
unknown. Goldreich [1967] considered viscous, turbulent, and
shape effects and concluded that the coupling of the core to the
_nantle is too weak to align the rotation axes of solid and fluid
parts. Thus the core's equator is likely to lie closer to the ecliptic
plane than to the mantle's equator, but it should exhibit some
precession-induced motion.
To compute the precession of core and mantle, a coordinate
system rotating at the 18.6 year node rate is chosen. For the
torques and angular velocities in the mantle system, the x axis
points toward the intersection of the equator and ecliptic planes,
and the z axis is normal to the equator plane; y completes the
triad. There is an analogous set of axes for the core. The Euler
angles are (1) the angle from the equinox along the ecliptic
plane to the descending equator plane, (2) the angle 0 between
the equator and ecliptic planes, and (3) the angle {p from the
intersection to the lunar zero meridian. Primed quantities are for
the core, and unprimed are for the mantle. For uniform
precession of core and mantle plus uniform rotations of mantle
about the z axis and core about the z' axis {•= •, •= •'= fi,
ß o
0 = 0' = 0, and 0 = I. Then the core/mantle angular velocity
difference in the mantle xyz frame is
-•0' sinO' sin(xl/'-XlD
•o'- •o = (p' [ cos 0 sin 0' cos(•'-•) - sin 0 cos 0' . (43)
(p' [ sin 0 sin 0' cos(•t'-•t) + cos 0 cos 0' ] -
To get the angular velocity difference in the core frame,
interchange primed and unprimed quantities.
For steady state precession the differential equations for the
mantle in the xyz frame are
-C (l t to., sin 0 + 2 (1 t2 sin 0 cos 0: Tg x + K (to'-to) x , (448)
0: Tgy + K (to'-tO)y, (44b)
0 = Tgz + K (to'-to) z . (44c)
A, B, and C are now the mantle moments, not the total lunar
moments. The gravitational torque on the mantle is T g. The
differential equations for the core in the primed frame are
-C' (0' •' sin 0' = K (co-to)x,, (458)
0 = K (co-tO')y,, (45b)
0 = K (co-to')z,. (45c)
There are no gravitational torques on a spherical core.
The core equations are solved first. The second and third
components are combined to derive (0 cos 0 = (p' cos 0'. Since the
precession rates of coi'e and mantle are the same, their angular
velocity components normal to the ecliptic plane, 4)'cos 0 + and
(0' cos 0' + (l t', are equal. However, the angular velocity normal
to the mantle's equator to z = •p + (l t cos 0 is different from
that norma! to the core's equator to•,= 4)'+ (l t' cos 0'. Define
• =-(K/C'f2), which is positive since the node rate is negative.
Then the solution for the core is
cot(•'-•): •, (46)
tan 0
tan 0' = , (47)
N/1 +•2
cos 0 N/l+tan20 '. (48)
Since 0 is expected to be bigger than 0', the core must spin at a
rate of •-99.96% of the mantle rate.
To develope the gravitational torques T g on the mantle in the
xyz frame, analytical expressions for Uij = (a/r) 3 u i uj were first
written in the body-fixed frame and then rotated by q). Here the
notation of libration theory is used for the mantle's uniform
precession and rotation, so (p = F+x-o+180 ø, 'qt = fl+o, and 0 = I.
The largest terms are linear in sin i and sin I, but third-degree
terms which multiply sin i and sin I by sin2i, sin i sin/, sin21, and
e 2 were included. These small third-degree terms, plus periodic
librations multiplying the torque functions, were evaluated and
combined with the numerical factors of the linear terms. Solar
torques make a small contribution. Only the constant part is
retained below. The best accuracy is needed for the first of the
three components.
Tg x = { [ 0.9758 (C-A) + 0.0048 (C-B) ] sin I
+ [ 0.9872 (C-A) + 0.0041 (C-B) ] sin i cos(o-z) }. (498)
Tgy = •' { -[ 0.9833 (C-A) + 0.0059 (C-B) ] sin i sin(o-z)
+ x [ (B-A) sin I- (C-B) sin i ] } . (49b)
Tg z =-3 S 3 (B-A) n 2 ( x + T sin i ). (49c)
Here x and o are constant, and S 3 = 0.9759.
For the mantle prece.ssion s.olution the notation of libration
theory is used with (I) = F, qt = f•, and 0 = I. The three constant
torques cause a tilt I, a shift in the equator's node o, and a
constant offset in longitude x:
K 2/5 sin I cos I
sin(o-z) = - • . (51)
C (1+• 2) 3 n 2 sin i ( 0.9840 [3 + 0.0059 ct )
An upper limit can be put on K/C(I+• 2) by iising the constant
Io =-0.265" found from the DE403 pure tidal solution. The
K/C(I+• 2) limit is 3.4x10 -8 d -l, while the x limit is -0.021".
Note that the x offset has a sign opposite that for tidal dissipation.
The combination sin I sin F enters the range observations in a
direct manner (see section 17), and the tilt ! may be considered a
well-observed quantity. The following relation from the first
component of (43), (448), (498), and the core solution links I to
physical parameters:
G t = -3 n 2 sin i cos(o-z) ( 0.9865 [3 + 0.0041 ct + E ),
G b = 2.0002 to z + 3 n 2 ( 0.9754 [3 + 0.0048 ct + E )
- 1.9982 •2_ 2 P K
C (1+•2) ß
sin I = (52c)
G b '
The inclination i = 5.145 ø, and the elastic combination
E=k2•/3, where • is defined by (25). The combination
[3=(C-A)/B is the solution parameter which most strongly
adjusts the mantle's tilt when analyzing data, but there are weaker
dependences on Love number, third-degree harmonics, and
•K/C(I+•2). To account for the influence of C31 and C33,
replace [3 and ct with the primed quantities defined by Williams et
al. (2001). Also, Williams et al. used a Fourier analysis to extract
i = 5553.63" from the DE403 numerical integration of physical
librations. The physical parameters for the numerical integration
were fit to the Lunar Laser data. The above expression is within
1" of the numerical result.
The magnitude of the spin rate difference between core and
mantle is
sin I
Io'- o1= X/l+ 2 (53)
If the core couples strongly to the mantle (•>>1), then its spin
pole nearly lines up with the mantle's pole. For weak coupling
(•<<1), the core's spin pole is nearly normal to the ecliptic plane.
11. Core-Coupling Parameter K
The ratio K/C will be fit to data. The core-coupling constant K
depends on fluid dynamics. In this section, interactions from two
possibilities, laminar and turbulent flow, are investigated. For
these cases, K is a function of physical parameters, including core
radius R' fluid density p', and kinematic viscosity v,
At the core-mantle boundary a viscous interaction in a laminar
boundary layer gives a stress proportional to v p' v, where the
core-mantle relative velocity v = AtoxR '. Yoder [1981, 1995]
C,= 2.6 R' (54)
By assuming a core of uniform density, K/C' can be converted to
K/C. From the maximum value given in the preceding section,
set the numerical value of K/C = fc (1 .•2) 3.4x 10 -8 d -1, where
fc is the fraction of the observed lo offset which comes from
the core. The core radius in kilometers is then
R'= 837 [fc(l+•2)/p']l/4/vl/8 with p' in gm cm -3 and v in
cm 2 s -1. For the limiting case Offc=l a liquid iron core density of
7 gm cm -3 and a viscosity of 0.01 cm 2 s -1 give a 900 km core,
which other lunar interior data indicate is unacceptably large (see
discussion in section 19). As Yoder [1981] concluded, the
viscous laminar interpretation fails for the Moon, and an
alternative must be considered.
At a point on the core-mantle boundary the turbulent stress for
relative velocity v=Ac. oxR' is equal to <p']v]v, where p' is the
fluid density and < is a dimensionless parameter which depends
on viscosity. (Topographic irregularities on the core-mantle
boundary can give an additional stress.) Integrating the stress
over the surface and computing the torque gives
3 •2 ,5
K t: < p'R A00. (55)
Concerned about the oscillating direction of the relative velocity,
Yoder [1995] replaced the scalar speed Ivl with its maximum
value divided by x/2, but that is not done here. With the mean
density of the Moon p and Ao0 from (53) one gets
(•_•') 5 16 C K p •]1+• 2 (56)
= 9It mR 2 C,k <p' sin/
Using the limiting case for K/C scaled by fc, the numerical
expression for core Size is then
R'= 145.2 km [fc (!_+•))3/2] 1/5 (57)
Yoder [ 1981 ] used < = 0.002. It is stated by Dickey et al. [ 1994]
that < is within a factor of 2 of 0.001. Yoder [1995] gives an
approximate boundary layer theory. With some rearrangement
(the < symbol here and that used by Yoder are not the same
parameter) and the addition of •, the functional and numerical
(cgs units) forms for < are
•= 0.4 , (586)
In[ 0.4 •R' 2 ,k sin 2 1 ] - In[ v (1 -• 2) ]
= 0.4 (58b)
2 In R' + In •d-•-< - In[ v (1+• 2) ] - 21.0
The Karman constant is set to 0.4. This equation is solved
iteratively if the radius is known. The < and R' equations are
solved iteratively iffc is known; < depends logarithmically on the
core size, kinematic viscosity, and •, so those uncertainties have
modest effects. For a viscosity of 0.01 cm 2 s -1, a 400 km core
gives <=0.00071, while a 300 km core gives <=0.00076. For the
limiting case 0Cc=l) with the density of liquid iron (7 gm cm-3),
the core radius is 421 km. Topography on the boundary would
decrease this core size. For reasonable core sizes the theoretical
K from turbulent interactions exceeds that from laminar flow, so
turbulence is expected as Yoder [1981] concluded. The limiting
core size differs from Yoder's 330 km limit mainly owing to the
smaller value of < and slightly because of his 13% smaller pole
For core radii between 300 and 400 km the peak monthly
velocity difference between core and mantle is 2 to 3 cm s -1
(R' n sin/). Since C' is proportional to mean core density times
R '5, the turbulent K/C' depends mainly on <, which is weakly
dependent on core radius and viscosity. The dynamics of the
core depend on K/C'. For the above values of <, the is 0.02 and
the core tilt to the ecliptic plane is 2', much smaller than the 93'
mantle tilt. For dissipative effects, Goldreich's [1967] assertion
is upheld. The core's equator intersects the ecliptic plane 89 ø
ahead of the mantle's equator intersection. The core changes the
mantle tilt by-0.006", which will be compensated during LLR
data fits by changing [3 and other parameters.
12. Core Differential Equations, Free Modes,
and Damping
Torque on the Moon from the Earth's gravitational attraction
drives the forced librations and causes the mantle's free librations
to oscillate about the forced state. The dissipative core-mantle
interaction causes slow damping of the three periodic free
librations, just as damping is also caused by tidal dissipation
(section 7). Moreover, the core is capable of its own rotational
motion, so there are additional free modes. These are damping
modes, not oscillatory motion. The development of the core and
mantle differential equations for rotation, the free modes, and the
damping are this section's subjects.
First, the coupled differential equations for the longitude
librations are written for mantle and core. The uniform
precession of mantle and core introduces functions of I, I' (mean
0'), and •. Small nonlinear terms are dropped. The mantle
equation is
K cos 1 sin21
:{;+3S3¾n2'r+ (:r- :t' + /[' )=fz' (59)
The core longitude libration •:' contains the periodic terms in
•' + q0'. The mantle moment C is used for ¾= (B-A)/C. C' is
roughly three orders of magnitude smaller than C. The •'sin21
term gives rise to the linear contribution in the constant offset.
This was previously computed (equation (50)) and will not be
considered further here. Small nonlinear terms are also dropped
in the core differential equation.
K( cos/)
:i:' + •; :r'- i: =0
COS ' (6O)
Since the core is assumed spherical without any gravitational
torque, there are x' derivatives but no x' term. Mantle
periodicities are driven by core periodicities through terms
factored by K/C'. Since C'/C is small, the coupling terms will
influence the core more than the mantle. The m'-m component
appears different in the two differential equations because two
frames are used. The ratio cos 1 / cos I' is computed from
cos 2 I sin 2 1
= 1- (61)
cos 2 I' 1+•2 ß
The • =-(K/C'•), defined in section 10, depends on the
(negative) node rate.
The forcing term for the mantle comes from the
U12=(a/r)3u 1 u 2 function factored by 3¾n 2 and the 0.9906
numerical factor of (16), but the forcing function on the fight-
hand side of (59) has the linear 'r term removed to give the
3 S 3 ¾ n2'r on the left-hand side. The free librations are solutions
of the mantle and core differential equations when the fight-hand
side of (59) is zero.
To investigate the free libration modes, substitute
'r = a exp(iv t) and 'r' = a' exp(iv t) into the linearized differential
equations. Two linear equations for a and a' result. The complex
determinant of the coefficients of a and a' is
K K sin2I
At=-V2 [(3S3¾n2-v2)+ C C' •-•]
+iv[(3S 3¾n 2-v2)•;-v2•]. (62)
The inverse 1 /A t is A• /A t A•, where the asterisk denotes the
complex conjugate:
K K sin2I 2
AtA•= V4 [(3S3¾n2-v2)+ C C, 1-•]
K K 2
+ v 2 [(3 S 3 ¾ n 2- v2)• - v2• ] (63)
To find the free libration frequencies (real part of v) and
damping (imaginary part) for the longitude modes, find the roots
with the determinant (62) set to zero. The zero root means that
the spherical core can be rotated by an arbitrary angle. While an
exact solution of the remaining cubic is possible, approximate
solutions are presented here. To guide the approximations, the
sizes of parameter combinations are needed. The combination
(3¾)1/2= 0.026 is well determined. For a small core,
K/C' > K/C. For turbulent coupling K/C'n = 10 -4, which may be
increased by boundary topography. From the limiting case,
K/Cn _<l.5x10 -7. So for the lunar case the combinations n ( 3¾
)1/2 >>K/C'>>K/C are well separated.
One of the roots of the cubic is near iK/C'. If the core rotation
rate is not at the steady state value of (48) plus forced librations,
it will damp very nearly as exp(-Kt/C'). This could have been
guessed from the form of (42) and (60). For a homogeneous iron
core, damping times of 140 years are expected for turbulent
coupling. Topography would decrease the damping time.
The (mantle) free libration frequency for longitude, with
period 1056 days, comes from the square root of 3 S 3 ¾ n 2. For
the Moon the free libration frequency is much larger than K/C',
so the first bracket in (62) dominates the frequency. If the
reverse were true, the free libration frequency would be
determined most strongly by the second bracket and the
¾ =(B-A)/C would be replaced by (B-A)/(C+C'). The core would
rotate with the mantle. in general, there is a slight dependence of
the free libration frequency on the strength of the core-mantle
The damping for the mantle free libration mode is
= , (64)
2 C (l+•z, 2)
where 5,L = K/C'n •3S3¾ is the ratio of core damping constant
to free libration frequency. For turbulent coupling, 5, L = 0.003
(weak coupling). Then from the DE403 limiting case the core-
induced damping time (1/DL) must be >l.6x105 years. The
above damping expression agrees with Peale [1976].
The effect of the core on the latitude librations is more
difficult. The Euler equations for the mantle (equations (1) and
(40)) and core (equation (42)) are not in the same reference
frame. The core differential equation can be expressed in the
mantle body frame
d(rto') K
d• + toxI'to' = ( to- to') , (65)
where I'm' and the angular velocity difference are also in the
mantle frame.
The differential equations for mantle and core rotation are
nonlinear owing to the fox operation as well as terms in the
forcing torques. Except for the precession term of section 10,
nonlinearities are small. A linear treatment suffices in most
cases, but nonlinearities can be treated as additional forcing terms
during an iteration. Analogous to the P l and P2 which describe
the motion of the mantle's pole, the core parameters pl and p• are
defined as
-sin 0' sin( {p + •t- •r), (66a)
p• = -sin 0' cos ({p + •t- •t'). (66b)
This definition removes the rate difference between the core and
mantle systems from the argument.
The difference in angular velocities is needed in the mantle
coordinate frame. Some small nonlinear terms are discarded.
_pl •b sin 2_•/ P 2 sin 2 1 1 +cos I
cos I- cos I + P lt½ 2 cos I' +/5• 2 cos I'
-P2 •b sin2 1 P l sin 2 1 l+cos I
Cos I + -p• b•- Pl
cos I 2 cos I' 2 cos I'
ß sin 2 1 cos/
-F •-•- + i' -i
1 + cos I'
The linearized differential equations for mantle and core rotation
•J2 + 0)3 (1-•x)pl + {x 0)3 2 P2 + • cos I ( 0)1 - 0)i ) =fx, (68a)
-•Jl + 0)3 (1-[•)P2 -4 [3 0)3 2 p• + •cos I( 0)2 -0)• )=fy' (68b)
j3• +[ 0)3 + (1-COS/) PlPi -P• (1-cos/) P0)3
K 2 cos I'
+ C' l+cos I ( 0)• - 0)• ) = 0, (69a)
-•Jl +[ 0)3 +(1-COS/) Plp• +Pl (•-cos I) P0)3
K 2 cos I'
+ C' l+cos I ( 0)• - 0)2 ) = 0. (69b)
The mean spin rate component 0)3 -- • + COS I-- n. Terms of
order sin 21 have been retained in the core differential equations
since the core rotation rate, 0),2= 0)2_/•2 sin2I/(l+•2) from
section 10's steady state rotation, is slower by such an amount.
There is some conflict between the objectives of linearity,
retaining sin2I terms, and the wish to simplify the core
differential equations by removing small terms. In (69a) and
(69b), terms of order •2 sin 21 have been eliminated, so terms of
order sin 2 I' are not complete.
To get the free libration frequehcies and damping, zero the
forcing functions on the right-hand sides and substitute four
unknowns multiplying exp(ivt) for the mantle and core p
parameters. The matrix multiplying the four unknowns is 4x4,
and setting its determinant equal to zero gives an eighth degree
polynomial for the free frequencies and damping. So
approximations are in order (free libration frequencies
>K/C'>>K/C). A first approximation is to solve the core and
mantle differential equations separately, eliminating core
variables in the mantle equations and vice versa. In this
approximation the motion of the mantle's pole causes interaction
with the core, but the mantle does not sense any response of the
core (in the longitude damping, the response of the core shows as
the 1 +•œ2 in the denominator). Similarly, the core does not sense
the mantle's response.
The complex 2x2 core determinant may be written as
Ac [V 2 (•)2 K 1) 2
= -0332 - - 2 i v •; ] [ v 2 -(1-cos •b2 ]. (70)
Setting it e. qual to zero gives four roots: -+033 + iK/C' and
ß +(1-cos/) F. The first pair of roots means the core's pole of
rotation could be tilted differently in space from that computed
for core precession plus forced libration, but damping will move
it toward the latter state. The K/C' damping parameter applies.
The second pair of roots reflects the slower core rotation rate
through the arguments in the definitions (66a) and (66b) based on
the uniform solution of (48) and (61). A sphere does not have a
unique principal axis, and there is no damping.
The 2x2 mantle determinant is approximately (smallest terms
Am = V4-V2 0332 ( 1 + 3 [•+ a •) +4a•033 4
- i v ( 2 v 2 - 2 ]v03 3 sin21 - tx 033 2 -4 03 2
3 )' (71)
Dw= 2'19x10- 3 K
. (73b)
From the limiting case the damping time is _>3.7x 107 years. The
above wobble damping does not agree with Peale's [1976]
stronger result. The difference appears to arise from the toxI'tff
term needed to express the core differential equation in the
mantle frame. While there is a •w = K/C'n, it is very small.
Yoder [1981] gives numerical values for damping time but not
analytical expressions. For all three free modes the values are
four to five times larger than this paper's values.
To compare damping from turbulent core dissipation and tidal
dissipation, consider cases with equal pole offsets. The core is
more efficient than tides for damping the free precession. For the
other two modes the core damping lies between the tidal cases for
constant Q and Q-I/frequency.
While it is convenient to refer to core and mantle modes, there
is a small influence of the classical free librations on the core,
and there is a small reflection of the core damping modes in the
mantle rotation. For the mantle modes the • parameters
determine the core/mantle amplitude ratio. For the precession
mode, with the largest coupling, that ratio is ( - i •p )/(1 +• p2).
So the core response is nearly orthogonal in phase when • is
small, but the core and mantle rotate together as •p approaches
The core mode damping is very fast compared to the mantle
damping. The damping of the three mantle free modes is too
slow to allow K/C to be determined. In principle, the core-
damping modes have a small influence on the mantle and if
observed would be sensitive to K/C'. The expected mantle/core
amplitude ratios are very small, and the short damping time
(140 years for turbulent coupling) would make these effects more
transient than the mantle modes. To be observable in the mantle
rotation, the core modes would need strong stimulation in the
recent past.
The real part corresponds to the classical solid-body dynamics,
and the imaginary part contains the dissipative terms. There are
two free modes for the mantle pole. One is an 81 year free
precession in space (frequency-- 313n/2 ), and the other is a
75 year wobble of the pole as seen in the rotating frame
(frequency = 2n (ct [3) 1/2 ). Dissipation affects these periods very
little. A coupling-dependent shift of frequency analogous to the
longitude mode is expected but does not come from the 2x2
approximation. The damping of the mantle's free precession is
Dp = C (l+•p2) (72)
The parameter •p = 2 K/3 [3 n C' is the ratio of the core damping
to the free precession frequency. The dependence on •p does not
come out of the 2x2 treatment. It requires additional terms from
the 4x4 matrix. For turbulent dissipation, % = 0.1 is the strongest
coupling of the three mantle modes and the 18.6 year forced
precession. Topography at the core-mantle boundary could
strengther• the coupling. The core-caused damping time is
_>8.1x104 years. Peale's [1976] analytical expression is very
complicated, and his numerical damping time is several times as
The damping parameter for the wobble is
K ct sin2 I
Dw= [ 213+-•-+ 1-• ]' (73a)
13. Core Forced Terms
Gravitational attraction acting on the mantle's figure ultimately
drives all forced terms. The feeble interaction between the core
and mantle induces weak mantle periodicities, orthogonal in
phase to the main terms, and small core rotation terms. These
small forced terms are computed in this section.
In differential equation (59) periodic orbit terms and nonlinear
terms (orbit times libration and libration times libration) force the
system. For the longitude librations the nonlinear effects are
small except for the constant offset ([3 term in (50)). The forcing
function depends on a sine series for the largest terms. Here a
periodic forcing function with frequency v is represented as
3yn 2Hexp[i(vt+phase)]. The solution for the libration
amplitudes for mantle, x=aexp[i(vt+phase)], and core,
x'=a' exp[i(vt+phase)], gives complex functions. For a sine
forcing function, the real and imaginary parts of a and a'
correspond to a sine and cosine, respectively.
Presented below are both the full solutions and the
approximate solutions to (59) with the foregoing periodic form
for the forcing function and solutions. As with the free libration
calculations, the inequality n (3y) 1/2 >>K/C'>>K/C guides the
approximations. The solution for the sine (in-phase) mantle
libration includes both the conventional solid-body response and
the core effects (with K). It is very close to the solution without
dissipation, and the coefficient of a periodic sine term is
-v 2 K K cos 21 } (74a)
C C' cos 2 I' '
as-- (3S 37n2-v2) (74b)
The cosine mantle coefficient is
ac---I K 3¾n2Hv 3 (•,)2 sin21
K a s v
ac=-• (3S 3¾n 2-v 2) (1+•) (75b)
The ratio •v = K/C'v measures the strength of the coupling
between core and mantle at the forcing frequency. Cosine terms
which have frequencies either much lower or much higher than
the resonance frequency are suppressed, but a response is favored
near the resonance. The core-caused cosine terms, factored by
the small quantity K/C, are very much smaller than the
conventional solid-body sine terms (equation (74b)), but they are
larger than the small change in the sine terms due to the core.
The mantle longitude series for the core effects is given in
Table 7. The two largest planetary terms are too close to the
resonance to separate from the free librations when fitting data.
The remaining periodic terms are too small to detect. All of the
periodic terms in Table 7 have weak coupling between core and
mantle for the turbulent value of K/C'. For the annual term, the
largest conventional longitude term, •v=0.001.
The core's sine and cosine forced longitude coefficients are
K 37n2Hv 2
a's = C' A t At*
cos/ [ K
cosI' (3 S 3 ¾n 2-v 2 ) C'
2 '
1+• v (76b)
K 37n2Hv 3 cosI [
a'c = - •; A t At* cos ]; ( 3 S 3 7 n2 - v2 )
K K sin21 ]
+ C C' 1+•2 ' (77a)
•v as
a' c = - -- (77b)
1+•2 ß
For I•vl < 1 the cosine term is larger than the sine term. For
increasingly larger •v the amplitude grows and the phase rotates
until, as Ivl approaches infinity, the core couples strongly to the
mantle and they rotate together. Lower-frequency forced terms
couple core and mantle more strongly than higher-frequency
In the conventional longitude librations there is a 14" Venus-
induced term with a 273 year period. The turbulent •v is
estimated to be 0.3, so the core should have a long-period term of
at least 4". Unfortunately, the influence of this term on the
mantle librations is unobservable. For turbulent coupling the
annual core term should be -0.1 ", and an 18.6 year term is -0.2".
Table 7. Maximum Terms in Longitude Libration Due
to Dissipation From a Weakly Coupled Fluid Core a
Argument Period, z
days mas
? 365.260 0.2
2F-2œ 1095.175 1.3
3E-5M-59 ø 1069.313 -0.2
23E-21 V+2D-œ+ 15 ø 1056.415 3.0
V-2E-D+ 2 •-F+25 7 ø 1056.345 3.2
0 oo -21.1
aAll terms use cosines of arguments. Angular units are
milliarcseconds (mas). Planetary mean longitudes for Venus,
Earth, and Mars are denoted V, E, and M. Core parameters are
K/C(I+• 2) = 3.4x10 -8 rad d -l and C7C = 1.7x10 -3, with • = 0.022.
Since the coupling is weak for all of the significant mantle
longitude terms, and the LLR data analysis detects the resonant
frequency through the coefficient a s, the ¾ defined with the
mantle moment C is much closer to the measurable quantity than
if it had been defined with the total moment C+C' (the difference
in the numerator is the same with a spherical core). Holding the
mantle C constant makes the differences of sine terms too small
to list in Table 7. For the tidal acceleration, and the exceedingly
long period (> 10,000 years) "secular" terms in longitude, the core
should couple strongly to the mantle. The ¾ in (39) should use
the total moment, but the induced displacement of longitude
libration is small and not directly observable. For secular terms
in longitude, the core acceleration matches the mantle
acceleration, but the core rate is different by -hC'/K. There is no
obvious way to use the secular terms to learn about the core.
The more complicated latitude terms are done as
approximations. From the 2x2 mantle matrix (71) one gets
forced terms for Pl (complex coefficient a) and P2 (complex b).
The forcing functions on the right-hand sides of differential
equations (68a) and (68b) have been set to Xexp[i(vt+phase)]
and -iY exp[i(vt+phase)]. This choice makes X and Y real for the
largest forcing terms (X with a cosine and Y with a sine), and it
associates the real part of a and b with a cosine and the negative
imaginary part with a sine. The X forcing function comes from
3 0in 2 0.9906 U23 cos/, and the Y function comes from
-313n 2 0.9906 U13 cos/with the linear 313n2pl moved to the
left-hand side of (68b).
g .
i[v 0) 3 ( 1 -[3)-i • Fsin 2 I]X
A m
i [v2-0t 0) 2-iv ] Y
3 Z
A m (78a)
-[ v 2- 4 [3 0)3 2 - i v ] X
b-- A m
K .
[ V 0)3 ( 1 - tx ) - i • F sin 2 I ] Y
+ Am (78b)
Both numerator and denominator are complex. The main
dissipation terms are factored by K/C, analogous to the longitude
case. From the experiences with forced longitude librations, free
Table 8. Maximum Terms in Latitude Librations Due to
Dissipation From a Weakly Coupled Fluid Core a
Argument Period, Pl P2 1(• p
COS, Slrl, COS, sin,
days mas mas mas mas
F 27.212
F-œ 2190.350
g 27.555
2F-œ 26.877
2F 13.606
265.2 -265.0
-1.4 0.8
-265.6 -5.9
-3.6 3.7
1.5 -1.5
-0.6 0.6
aThe latitude physical libration parameters are pl, P2, P, and 1o.
Angular units are milliarcseconds (mas). Core parameters are
K/C(I+• 2) = 3.4x10 -8 rad d -i and C7C = 1.7x10 -3, with = 0.022.
librations, and the solution in section 10 it can be guessed that
core response would put 1 + •v 2 in the denominator, where
Table 8 gives the core-induced latitude series. It is dominated
by the term for pole offset (the more elaborate solution of section
10 is used for this term). Most of the 2190 day term is from a
nonlinear contribution. Table 8 also gives the approximate
conversion to p and 1(5 parameters.
Of the forced terms in Tables 7 and 8, only the large pole
offset term is easily observable. The forced physical librations
are mainly sensitive to K/C, and the sensitivity to K/C' (or •) is
very small in the tables.
14. Sidereal Terms
The Moon's orbit precesses along a plane which nearly
coincides with the ecliptic plane, but this mean plane of
precession is tilted by two causes. The oblateness of the Earth
induces an 8" tilt toward the equator, and the resulting plane is
commonly referred to as the Laplacian plane. The second cause
is the motion of the ecliptic plane. This induces a 1.5" tilt
because the orbit does not quite follow the ecliptic motion. The
two tilts are oriented differently. The •v in the latitude solution
of the preceding section is infinite for a term at the sidereal
period (27.322 days in the rotating frame or zero rate in the
inertial frame), and the solution there should not be used for such
calculations. Both tilt effects are very close to the sidereal rate;
the first case differs by the 26,000 year precession of the Earth's
The effect on librations of a fixed plane for orbital precession
is intuitive. The rotating mantle and core precess along the same
plane as the Moon's orbit whether that plane is the ecliptic plane
or not. There are several reasons that this is not quite true for the
Moon: the Sun is still in the ecliptic, there are figure-figure
torques on the Moon from the Earth's oblateness, and the ecliptic
plane is moving. The torques from the Sun will be ignored
compared to the Earth's, and the figure-figure effect is 1% of the
8". As Eckhardt [ 1981 ] showed, the effect of the ecliptic motion
is sizeable, 6" in addition to the 1.5", because the differential
equations must be modified.
The differential equations for core and mantle can be written
and solved in an inertial frame. The solution has a simple
explanation. The pole of the ecliptic plane moves 0.470 "yr --l,
and the axis of that rotation is at ecliptic longitude H = 174.87 ø at
J2000 and moves slowly (-8.7 "yr-l). Both mantle and core
precession nearly follow this motion. The solid-body rotation
fails to follow by an angle given by the 0.470" yr -1 rate divided
by the free precession frequency (0.47 "yr -• /2n/81 yr- 6.0").
For the steady state solution both spin axes move by the
0.47 "yr -•, but there is a separation between the two axes such
that the turbulent torque causes the core's axis to follow the
motion. The core rotation axis is pulled along by the mantle
owing to the core-mantle interaction. The core is fully coupled to
the mantle, and the appropriate expression for the 6" term is
0.47 (B+C')/1.5n(C-A). The phase is L- H + 90 ø, where the
orbital mean longitude is L=F+f2. The classical latitude libration
terms have weak coupling between core and mantle and are very
sensitive to [3 =(C-A)/B, so the sidereal term associated with
ecliptic motion has independent information on the core moment
C'. The core-sensitive terms are
APl = 6.0" sin( L- 84.87 ø ), (79a)
AP2 = 6.0" -• cos( L- 84.87 ø ). (79b)
The expression for the ecliptic-motion-induced separation
between the core and mantle spin axes is 0.47 "yr -1 C'/K. For
turbulent coupling the spin axis of the core lags the secular
motion of ecliptic and mantle poles by •-1 ', while it also precesses
with a 2' angle.
For turbulent coupling, section 1 l's limiting case of a 421 km
iron core gives C'/C = 1.7x10 -3. This gives an upper limit of
0.010" for the sidereal core signature. The two closest terms (in
frequency) are the forced precession, with an 18.6 year beat
period, and the free precession, with an 81 year beat. There are
solution parameters corresponding to all three frequencies, and
the 81 year beat period will weaken the determination of C'. So
the term is large enough to be useful, but the separation of
parameters will be a challenge. Increasing data span will very
much improve the direct determination of the core moment. All
of the terms in Tables 7 and 8 are orthogonal to the major (solid-
body) terms of the same period. This can be an advantage when
solving for K/C. The core-induced sidereal term does not have
this advantage.
The tidal dissipation Tables 3 and 4 have a sidereal term, but it
was too small to include in Table 2. Split into the two phases and
expressed in arc seconds, the two components are
Ap• = -• [0.01 cosL+0.18 cos(L-84.87ø)], (80a)
AP2 = • [ 0.01 sin L + 0.18 sin( L- 84.87 ø ) ]. (80b)
The Q is monthly. The maximum for the tidal dissipation terms
is 0.2 milliarcsecond (mas). This is much smaller than the
maximum core effect, has different phase, and should be
calculable from a monthly Q. The tidal elastic effect proportional
to k 2 is orthogonal to the tidal dissipation, is several mas in size,
and is more likely to correlate with C'.
An additional effect, core-mantle boundary oblateness, has not
yet been investigated. Given this unknown, the two sources of
sidereal terms with two phases, and the 81 year beat period, the
sidereal terms are not pursued further in this paper. They offer a
very interesting future opportunity for direct determination of
core moment.
15. Orbit Perturbations From Core Dissipation
The gravitational attraction from a spherical core acts like a
point mass and does not directly perturb the lunar orbit, but there
is an indirect effect. The core-induced constant shifts in libration
'r and c• (section 10) displace the mantle's principal axes from
what would otherwise be their equilibrium orientations. The
displaced figure of the Moon then perturbs the orbit. The effects
are small, and leading-term approximations are used in this
section. As is the case with tidal dissipation, the orbital
perturbations are computed by the numerical integration
programs from the accelerations. The approximations of this
section do not enter those programs.
Orbit perturbations from a displaced figure were also
considered for tidal perturbations (section 8). The important
effects are in semimajor axis a, mean motion n, and inclination i.
The computation can proceed in a manner similar to section 8
using the 'r and c• offsets of section 10. Changes in a and n are
also related to the power drawn from the orbit and deposited in
the core:
P =-K ( •o' - •o )2, (81a)
K/•2 sin 2 I
Pave = - 1 + •2 (81 b)
The secular mean motion and semimajor axis changes are
calculated (approximately) from the mean power. The mean
motion change is
Ati = -- /?2 1+ -•- 3 sin 2 I (82a)
C (1+• 2) m '
Ah = 1.1 lx106 "cent -2 . (82b)
The • is based on the node rate. The latter equation uses K/C in
radians d -1 to give "cent -2. The limiting case gives an upper
limit of 0.038 "cent -2 from the fluid core. The influence on the
semimajor axis comes from Aft = -2 a Ah/3 n, so the relation is
K• C (m) (R)2
Aft =-C(I+• 2) m R 2 1+• •- 2 a sin 21, (83a)
Aa = -1.64x10 3 m yr -I (83b)
Again, K/C is in radians d -1 to give m yr -•. For the limiting case
this is-0.056 mm yr -l.
In the first approximation there are no torques perpendicular to
the ecliptic plane, but there are torques normal to the orbit. The
semimajor axis and semilatus rectum expand at the same rate so
the eccentricity rate is zero. There is also a torque in the orbit
plane 90 ø from the node which gives rise to an inclination rate
di K C (m)(3)2 sin21
d•=-C(l+• 2) mR 2 1+• sini ' (84a)
di K
d• = - 4.9 "yr -• (84b)
C (1 +•2) ß
The last equation uses K/C in radians d -1 to give inclination rate
in "yr -•. The rate for the limiting case is-1.7x10 -7 "yr -I. This
is too small to detect. The core influence on node and longitude
of perihelion acceleration is about an order of magnitude smaller
than for tidal dissipation for the limiting cases.
For the same pole offset, tidal dissipation in the Moon
provides an order-of-magnitude larger secular change of
semimajor axis and mean motion than does core dissipation.
Also, the tides change eccentricity, while the core does not. As
with the lunar tides, the changes are opposite in sign to those
from tidal dissipation on the Earth. The fluid-core-caused
changes in a and n are three orders of magnitude smaller than
rates caused by tides on the Earth. The differences in orbit
perturbations from the three offer an opportunity to distinguish
between them. This will be discussed further in the next section.
16. Separation of Orbit Perturbations
Can the secular rates of orbital semimajor axis, mean motion,
and eccentricity be used to separate the contribution from lunar
tidal and core dissipation? For semimajor axis and mean motion
rates, tidal dissipation on the Earth is two orders of magnitude
more important than lunar tides and three orders of magnitude
more important than lunar core effects. In principle, one can
subtract the Earth influence from the measured orbit changes to
get the lunar effect. The measured pole offset gives a linear
combination of the two lunar influences, and the total orbital
effect depends on their proportion.
To the secular acceleration h, the Moon contributes between
0.038 "cent -2 (all dissipation in core) and 0.46 "cent -2 (all
dissipation tidal). Table 9 gives the secular acceleration and
eccentricity rates computed from tides on Earth. Tidal
components are deduced from artificial satellite and Lunar Laser
Ranging. The LLR model has Love numbers and tidal time
delays for three frequency bands: semidiurnal, diurnal, and long
period. The semidiurnal and diurnal time delays are LLR fit
parameters. The DE403 lunar ephemeris was generated in 1995,
and its secular acceleration from Earth and Moon dissipation is
-25.64+0.4" cent -2. The predictions of tidal acceleration from
the artificial satellite laser ranging (SLR) deduced tides are
systematically -1 "cent -2 lower (in magnitude) than the LLR
values. Half of this difference is understood. The SLR
calculations of lunar acceleration do not correctly account for the
finite mass of the Moon [Williams et al., 1978], which requires a
correction factor of l+m/M = 1.0123. A modified Kepler's third
law (used in (16a)) contributes an additional factor of 1.0028
(using a=384,399 km from the average inverse distance). These
two corrections increase the magnitude of the SLR values by
0.4" cent -2. A review of the conversion of the LLR Earth and
Moon tidal time delays to h shows that the published (negative)
Table 9. Mean Motion and Eccentricity Rates Computed
From Four Models of Earth Tides a
Tide Model h, k, Reference
,, cent-2 10-• yr-l
GEM-T1 -25.27 1.83 Christodoulidis et al. [1988]
GEM-T2 -24.94 1.68 Marsh et al. [ 1990];
Dickman [ 1994]
Cartwright-Ray -24.88 1.59 Ray [ 1994]
LLR DE403 -26.10 1.35 this paper
aThe first three models depend in whole or in part on multiple tidal
components deduced from artificial satellite laser range data analysis.
The last corresponds to the model used in the lunar and planetary
integrator with two adjustable tidal parameters fit to LLR data.
values of h need to be corrected by +0.15 "cent -2. Earth tides
account for +0.10 "cent -2, and lunar semimonthly tides in
Table 6 add 0.05 "cent -2. (The Dickey et al. [1994] value of
h =-25.88_+0.5 "cent -2 becomes -25.73_+0.5 "cent-2.) Adding
dissipation in the Moon to Table 9 does not improve the
SLR/LLR disparity. Because LLR is sensitive to the total secular
acceleration while SLR senses only Earth tides, lunar tides
increase the SLR/LLR spread more than core dissipation. At
present, knowledge of tides on the Earth is not sufficiently
accurate to extract the lunar contribution to the observed secular
acceleration from the difference between SLR and LLR values.
The situation for eccentricity rate is more hopeful. The Moon
contributes between-1.0x10 -ll yr -1 (all dissipation from tides)
and 0 (all dissipation in core). The contributions from Earth and
Moon are close enough in size that eccentricity rate is useful for
learning about the Moon's interior. An eccentricity rate of
-1.0x10 -ll yr -1 changes the perihelion distance by 3.2 mm yr -1.
The LLR determination of eccentricity rate should improve with
increasing data span.
The internal accuracy of the determination of the dissipation-
induced h is good. However, range perturbation exceeds 15 m
during the data span! But the present uncertainty of tides on
Earth does not permit this to be used for the lunar problem.
Eccentricity rate is a much weaker signal, accumulating a few
centimeters in range during the data span, but is easier to correct
for tides on Earth. At present, the lunar rotation provides a direct
test of lunar dissipation without corruption from external
influences. Since the rotation effects are bounded while the orbit
effects are secular, the orbit perturbations may assume greater
importance in the future.
17. Determination and Separation
of Lunar Variables
This section discusses how the lunar rotation terms affect the
Lunar Laser ranges. It also discusses how the solution
parameters separate from one another. The data analysis program
uses rigorously derived partial derivatives of range with respect
to the solution parameters, but for illustration, approximations are
The range vector R from an observatory on the Earth to a
retroreflector on the Moon is
R = r- R s + R r. (85)
The three position vectors are geocentric Moon r, the geocentric
ranging station R s, and the selenocentric retroreflector position
R r. Orientation matrices for the Earth and Moon are used to
transform between space-fixed coordinates and body-fixed
coordinates. When accurately calculating the round-trip time
delay, two R vectors are needed. One "leg" uses the transmit
time and the lunar bounce time, while the other uses the bounce
time and receive time. Since Rs/r--I/60 and Rr/r--1/221, a first
approximation for the range projects the two smaller vectors
along the Moon to Earth unit vector u =-r/r:
R = r+ u'(R s- R r). (86)
At a given time, the difference in range to different
retroreflectors depends on the reflector coordinates and the lunar
orientation with respect to the Earth-Moon vector. In the lunar
body-referenced frame, u is approximated by
1 1
U 1 = 1- •' U22 - •- U32 , (87a)
u 2 -- sin[ ( 2 e sin • )- 'c ] , (87b)
u 3 =-sin i sin F- sin(I+ p) sin(F-o). (87c)
The direction of this vector is composed of the optical librations,
due to the orbit (eccentricity e and inclination i terms), and the
physical librations, due to rotation (I, 'r, p, and o). The e and i
terms are leading terms of series for ecliptic longitude and sine
latitude, respectively. See Eckhardt [1981] for the exact
expressions. The selenocentric coordinates of a retroreflector
project into the range direction as -U'Rr, where R r = (X, Y, Z) in
the body frame. The main sensitivity of the range to the
longitude libration comes from Y u 2, and the sensitivity to
latitude librations comes from Z %. For the four retroreflectors,
1339<X<1653 km, -521<Y<803 km, and -111<Z<765 km
[Williams et al., 1996]. Figure 1 shows the retroreflector
locations. At the lunar surface a selenocentric angle of 1" is
equivalent to 8.4 m, but the projection into the range direction is
<4 m for the retroreflector positions. Thus a few centimeter
range accuracy is sensitive to physical librations at the =0.005"
level, and numerous observations will improve on this during a
In the range data analysis program a partial derivative of the
range (time delay) is required with respect to each solution
parameter (P) for each leg of the round trip. For lunar parameters
these partials are f•'(3r/3P+3R/3P), in the space-fixed system.
The orbit is separate from the orientation of and location on the
Moon. For illustration, in lunar body-fixed coordinates the
partial of the -u.R r term is -u'3Rr/3P - Rr'•)u/•)P. The •)Rr/•)P
includes partials with respect to the three selenocentric
coordinates for each of the four retroreflectors plus partials for
two Love numbers h 2 and l 2 for tidal displacements. The partials
3Rr/3P come from the geometry and are not integrated. They are
generated and projected into the range direction while processing
data. The sensitivity to the reflector coordinates comes through
the orientation of the Moon with respect to the Earth-Moon line.
The tides vary with time, depend on location, and project
according to variable orientation. A numerical integration
program generates the partials of orientation 3u/3P and orbit
3r/3P with respect to dynamical parameters. These dynamical
parameters include •3, ¾, seven third-degree gravitational
harmonics, Love number k 2, tidal time delay At equivalent to a Q
inversely proportional to frequency, K/C, rotation initial
conditions for solid body and core, and lunar J2' The projection
into the range direction at the observation time is done when the
range data is analyzed. Except for J2, these dynamical
parameters are most sensitive through the orientation. To
distinguish Q values at different frequencies, analytical partials
3u/3P are generated and projected at the time of data analysis.
On the basis of the series solutions of section 5 and Tables 1 and
2, analytical partials are included for coefficients of five out-of-
phase terms: 27.2 days and 2190 days for latitude librations, plus
annual, 1095 days and 206 days for longitude librations. Since
the p• and P2 parameters are coordinates rather than angles, the
analytical latitude partials are implemented using their equivalent
terms for constant o and 27.555 day variations in p and o.
During solutions, how detectable and separable are the
dissipation effects through lunar orientation? Except for the
sidereal term, the dissipation terms are orthogonal in phase to the
terms produced by the second-degree figure (triaxiality). There is
little difficulty in separating orthogonal terms, even when they
have identical periods, provided that the data span is long
enough. Of the seven third-degree harmonics, three produce
Figure 2. Location of the three Apollo retroreflectors and the
two French reflectors on Soviet Lunakhods. The spread of
locations aids separation of parameters during solutions.
terms orthogonal to the dissipation terms, and four (C30, C32, S31,
and S33) produce terms that are phased like dissipation. The
spherical harmonic functions for C30 and C32 are even in
longitude and odd in latitude, while those for S31 and S33 are odd
in longitude and even in latitude. The resulting libration series
are dissimilar for the two pairs [Eckhardt, 1981; Moons, 1982b],
but the paired members will correlate with each other. It is the
separation of S 31, S33, K/C, and At (n k 2 At = k2lQ for monthly Q)
that needs further discussion.
Because of the good geometric spread of retroreflectors
(Figure 2), the physical libration latitude and longitude
components are distinguishable from each other and from the
orbit. Table 10 displays the larger partial derivatives for S 31, S33'
K/C, and two tidal dissipation models (Q constant and
Q-l/frequency). Eckhardt [1981] and Moons [1982b] are the
sources for the two harmonic columns; this paper provides the
dissipation columns. The constant in x is not shown because it
contributes nothing to the separation when reflector longitude (or
X and D is adjusted during the solution. The x partials are
tabulated because they are the physical libration part of u 2 in
(87b). Instead of u 3, the similar, but simpler, Pl - x sin I cos F is
used (Moons tabulates Pl and P2 rather than p and Io). The
columns are normalized like unit vectors.
Table l 0 may be used to understand what happens during the
numerical solutions. Similarity down each column's series of
argumentsffrequencies causes correlation, while dissimilarity
promotes separation. First, notice that the dissipation columns
are dominated by the precession pole offset (cos F latitude term),
but this offset is zero for the harmonics. Only dissipative effects
contribute to the observed 0.26" pole offset. Separation during
solutions depends on the largest dissimilar coefficients, provided
the data span is comparable to or larger than their periods •and
beat periods with other major terms. The number of parameters
in the fit must at least be matched by the number of detectable
periodicities in the partials. In the simplest case the partials
would be considered in decreasing order of size, but there are
complications since of the three free libration modes one is near
the 27.2 day F term (24 year beat period) and another is near the
1095 day term (81 year beat). Though the LLR data span
exceeds 24 years, the earliest data is an order of magnitude less
accurate than the recent data. While the determination of the F
term is weakened <