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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. Ell, PAGES 27,933-27,968, NOVEMBER 25, 2001

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.

0148-0227/01/2000JE001396509.00

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

27,933

27,934 WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE

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

3GM

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

(Moon):

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

angles.

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

WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND CORE 27,935

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

distortion:

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

GMR 2

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

27,936 WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE

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

1

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

dissipation.

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.,

WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND CORE 27,937

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) '

GM

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

frame:

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

ß

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 ].

27,938 WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND CORE

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

days

2D+œ 2F F+œ 2œ 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

-1.1

-0.3

-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

and

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

0.3

-0.3

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

-1.1

1.1

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.

WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND CORE 27,939

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

ß

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

2œ 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

-223.88

12.81

-0.11

0.22

-0.61

0.01

0.39

0.19

0.18

-0.94

-1.19

0.30

-12.75

-0.02

-0.40

-0.19

-0.18

0.92

1.13

-0.15

aEach coefficient (units arcseconds) should be multiplied by k2/Q.

27,940 WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE

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

-230.20

10.83

-0.14

0.01

-0.93

0.02

0.36

0.22

0.18

-0.68

-1.19

0.26

-10.76

-0.22

-0.37

-0.22

-0.18

0.66

1.14

-0.12

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

parameters:

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

WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE 27,941

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

combination

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

Moon.

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

27,942 WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE

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

Moon.

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

WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND CORE 27,943

Table 6. Secular Orbit Changes From Periodic Tides a

di

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

2œ 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%.)

27,944 WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE

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

h

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(r½o')

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

WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE 27,945

-•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

(A+B)

-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

simpler:

-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.

3n2

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)

3n2

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

(528)

(52b)

t

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

27,946 WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND CORE

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]

gives

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)

<p'

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

offset.

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)

cos?

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

WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND CORE 27,947

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' •-•]

K K

+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

coupling.

The damping for the mantle free libration mode is

K

Dœ = , (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'

(67)

The linearized differential equations for mantle and core rotation

are

K

•J2 + 0)3 (1-•x)pl + {x 0)3 2 P2 + • cos I ( 0)1 - 0)i ) =fx, (68a)

K

-•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

27,948 WILLIAMS ET AL.' LUNAR DISSIPATION IN MANTLE AND 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

discarded)

Am = V4-V2 0332 ( 1 + 3 [•+ a •) +4a•033 4

K

- 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

infinity.

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

K

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

large.

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

WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE 27,949

as--

-v 2 K K cos 21 } (74a)

C C' cos 2 I' '

3¾n2H

as-- (3S 37n2-v2) (74b)

The cosine mantle coefficient is

ac---I K 3¾n2Hv 3 (•,)2 sin21

(75a)

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*

K

cos/ [ K

cosI' (3 S 3 ¾n 2-v 2 ) C'

(76a)

as

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

v

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

terms.

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

COS,

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).

a •

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)

K

-[ 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

27,950 WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE

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

0 •

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

g/c'(Ivl-n).

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

equator.

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.

WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE 27,951

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 '

K

Ah = 1.1 lx106 "cent -2 . (82b)

c

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)

K

Aa = -1.64x10 3 m yr -I (83b)

c

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.

27,952 WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE

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

used.

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

solution.

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

WILLIAMS ET AL.: LUNAR DISSIPATION IN MANTLE AND CORE 27,953

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 <