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\preprint{}
\title{Relativistic spin effects in the Earth-Moon system}
\author{David Vokrouhlick\'y\cite{byline}}
\address{Observatoire de la C\^ote d'Azur, D\'epartement
 CERGA, URA CNRS 1360, \\ Avenue N Copernic, 06130 Grasse,
 France}
\date{\today}
\maketitle
\begin{abstract}
 Relativistic precession and nutation effects applied to the
 Earth and Moon coordinate systems are discussed in
 detail. Apart from the effects common to the two reference
 systems (including the well-known geodetic precession),
 the lunar reference frame undergoes an additional
 precession of $28.9\, {\rm mas/cy}$. This value is
 theoretically within the range of the Lunar Laser Ranging
 (LLR) technique in the forthcoming years, but impossible to 
 decorrelate from other secular effects of selenophysical
 origin. Due to a nearly perfect cancellation of the de Sitter
 and Lense-Thirring phenomena, additional nutations of the lunar
 referential are below the sensitivity of the LLR.
 Analogous cancellation occurs also for the secular part
 of the Lense-Thirring precession of the Earth-Moon
 referentials. Terminological ambiguity of the geodetic
 precession constant notion, related to the inclusion
 of some of the Lense-Thirring terms, is pointed out.
\end{abstract}
\pacs{04.25.Nx, 95.10.Jk, 95.30.Sf, 96.20.-n}

\narrowtext

\section{Introduction}
Recently, Damour and co-workers \cite{dam1,dam2,dam3} have 
developed a general scheme for the modeling of post-Newtonian
dynamics (both translational and rotational)
of a system of weakly self-gravitating extended bodies. A detailed
theory of the coordinate systems is a natural part of the
approach.

In this paper we address the problem of the correspondence between
local systems attached to a given moving body and the global system 
used to describe the motion of all bodies. Although that principles 
of general relativity admit arbitrary coordinate systems it appears
that judicious choice make the mathematical problem much simpler.
Such a kind of coordinate system is commonly called {\it dynamically
nonrotating} due to the absence of the relativistic
Coriolis effects \cite{dam3} (sometimes also ``quasi-inertial frame''
\cite{ab,kop,bk}). Another commonly recommended choice for the 
rotational state of the
local coordinate systems, called {\it kinematically nonrotating},
does not relate to the dynamical principles but to the simplicity
of the coordinate transformation between the local and global
systems. In the latter approach, recommended by the IAU Working 
Group on Reference systems \cite{gui,iers}, the direction
of the local frame axes
coincides with that of the global one. In this case one
has to apply the relativistic Coriolis force ``$\propto\;{\bf
\Omega}_r \times {\bf S}$'' in local body systems. In both 
approaches the relativistic spin
effects remain. One naturally faces the question of implementation
and observability of these phenomena in case of actual dynamical
systems.

The Earth-Moon system seems to be promising for detection of the
relativistic effects thanks to the availability of very high
accuracy positional data obtained by the Lunar Laser Ranging
(LLR) technique. The lunar motion is thus the most precisely
known among the natural bodies due to the current LLR technology
precision of about $2-3$ centimeters for the distance.
In the coming decade, with the use of multicolor
lasers, an improvement is expected up to the ultimate limit of 
about $2$ -- $3$ millimeters for normal points of the lunar motion
(Ch. Veillet, personal communication). Corresponding technical 
improvements are being implemented on the French lunar laser 
station at the Observatoire de la C\^ote d'Azur. Translated into
amplitudes of the lunar librations the current best
performances are on the level of $1$ milliarcsecond with an
ultimate precision limit of about $0.3$ milliarcseconds.

Thanks to these high quality data the Earth-Moon dynamics has 
already been used for important relativistic tests. First, one 
has to mention the significant contribution to the equivalence 
principle testing via ``the Nordtvedt effect''
\cite{dnw,mull,sci,wnd}. Second, the principal term of the de Sitter
precession of the Earth-Moon center-of-mass frame has been detected 
with a precision of about $1$ percent \cite{mull,sci,wnd,bcb,sha}.
Finally, a theoretical modeling of a new class of
relativistic lunar librations has been recently presented \cite{bv}.
On the other hand, one must be aware that from the
relativistic point of view the Earth-Moon system is ``polluted'' by a
lot of phenomenologically modeled geophysical and selenophysical
phenomena, namely those related to the anelastic deformation of 
the two bodies.

In this paper, we focus on detailed elaboration of the relativistic
rotational effects of the reference frames in the Earth-Moon system. 
A subclass of these effects, common to the two referentials, may be 
attributed to the rotation of the Earth-Moon center-of-mass system
(including the widely known de Sitter term having the magnitude
of about $1.9194\, {\rm as/cy}$). However, due to the mutual 
Earth-Moon dynamics both referentials are also submitted to 
individual precession and nutations. Their principal terms are 
evaluated. It is shown that the additional
precession of the lunar reference frame is within the capabilities
of the LLR technique. However, due to difficulties in decorrelating
it from the other (phenomenologically modeled) effects,
it is hardly seen in the data. A similar situation occurs for
the newly described relativistic nutations of the lunar reference
system due to fine cancellation of the de Sitter and Lense-Thirring
phenomena. Corresponding terms of the Earth reference system
coincide with those computed recently by Brumberg and co-workers 
\cite{bbf1,bbf2}. Also the ambiguity in the precise definition of
the geodetic precession constant is briefly discussed.

Finally, it is to be noticed that the phenomena discussed in this
paper are not to be mistaken for the relativistic perigee
precession of the lunar orbit due to the Earth mass gravity
monopole \cite{wi,no}.

\section{RELATIVISTIC SPIN EFFECTS FOR REFERENCE SYSTEMS}
\subsection{Background}
Formally, the relativistic Coriolis force term in the equations of
rotational motion in a given local coordinate system of body A 
reads\footnote{ As discussed in detail in Ref.~\cite{dam3}, it is 
this term which can take the kinematical interpretations the
body A reference frame relativistic rotation.}
\begin{equation}
 \left({d S_a^{\rm A} \over dT_{\rm A}}\right)_R^{\rm Coriolis} =
 {1 \over 2c^2} \epsilon_{abc} S_b^{\rm A} H_c^{\rm A} \; ,
 \label{one} 
\end{equation}
where $S_a^{\rm A}$ is the (1PN-) spin vector of body A,
$T_{\rm A}$ is the local coordinate time, $\epsilon_{abc}$ is the
usual Levi-Civita fully antisymmetric symbol and $H_a^{\rm A}$ is the
central gravitomagnetic field in the coordinate system of the
A-th body. Detailed analysis shows \cite{dam3} that the latter
quantity
is composed of two parts, one depending on the dynamical state
of the external bodies and another depending only on the
dynamical state of the reference frame of the A-th body (i.e. on
the manner how it is related to the global coordinate system).
If $V_{\rm A}$ and $A_{\rm A}$ denote respectively the velocity and
acceleration of the origin of the coordinate system of the A-th body,
$R^i_{{\rm A} a}$ the orthogonal rotation matrix involved in the
transformation between the local system of body A and the global 
coordinate systems, ${\bar w}^{\rm A}$ and ${\bf {\bar w}}^{\rm A}$
the scalar and vector post-Newtonian potentials of the external
gravitational field measured in the local system of body A, one has
\begin{eqnarray}
 H_a\left(T_{\rm A}\right) &=&\; -\,4\epsilon_{ijk} R^i_{{\rm A} a}
 \left[ \partial_j {\bar w}^{\rm A}_k + v_{\rm A}^j \partial_k
 {\bar w}^{\rm A} \right]_{\cal L_{\rm A}} \nonumber \\
 & & +\;\epsilon_{abc}\left[V^{\rm A}_b A^{\rm A}_c
 + c^2 {dR^i_{{\rm A} b} \over dT_{\rm A}} R^i_{{\rm A} c}\right]
 _{\cal L_{\rm A}} \; . \label{two}
\end{eqnarray}
An approximation of the potentials sufficient for our purposes is
given by
\begin{mathletters}
\begin{eqnarray}
 {\bar w}^{\rm A} &=& G \sum_{{\rm B} \neq {\rm A}} \left[{M_{\rm B}
 \over r_{\rm B}} \right]_{{\cal L}_A}
 + {\cal O}(2) \;, \label{threea} \\
 {\bar w}^{\rm A}_i &=& G \sum_{{\rm B} \neq {\rm A}}
 \left[{M_{\rm B} \over r_{\rm B}} v_{\rm B}^i - s_{\rm B}^{ji}
 \partial_j\left({1 \over r_{\rm B}}
 \right) \right]_{{\cal L}_A} + {\cal O}(2)\; ,  \label{threeb}
\end{eqnarray}
\end{mathletters}
where $M_{\rm B}$'s are the mass monopoles of the individual bodies,
$r_{\rm B}^i = x^i - z_{\rm B}^i$ with $z_{\rm B}^i$ being the 
coordinate, in the global reference system, of the frame with
origin at B and $s_{\rm B}^{ij} = \epsilon_{ijk}s_{\rm B}^k =
\epsilon_{ijk} R^k_{{\rm A} a} S_{\rm B}^a$. Symbol ${\cal L}_A$ means
that the bracket term has to be evaluated at the origin of the
coordinate system of body A and ${\cal O}(2) = {\cal O}(c^{-2})$
points neglected post-Newtonian terms. Notice the absence of spin
contribution to the scalar potential ${\bar w}^{\rm A}$. As for
the rotation matrix $R^i_{{\rm A} a}$,
either it is submitted to the trivial condition $R^i_{{\rm A} a}
\equiv \delta_{ia}$ to express the absence of kinematical rotation
or one adjusts its value in order to cancel the gravitomagnetic 
field $H_a^{\rm A}(T_{\rm A})$ at any instant, which is
equivalent to the absence of dynamical rotation. Although this
choice has been adopted in this paper, our results can be
easily transposed in the other way, with the appearance of the
Coriolis term in Eq.\ (\ref{one}).

Introducing the angular velocity vector $\Omega^{\rm A}_i$
characterizing rotation of the dynamically non-rotating frame of the
body A with respect to the global coordinate system by
\begin{equation}
 \Omega_i^{\rm A} = -{1 \over 2} \epsilon_{ijk} {dR^j_{{\rm A} a}
 \over dT_{\rm A}} R^k_{{\rm A} a} \; , \label{four} 
\end{equation}
we obtain
\begin{equation}
 {\vec \Omega}^{\rm A} = -{1 \over 2c^2} {\bf \vec v}_{\rm A}
 \times{\bf \vec a}_{\rm A}
 + {2 \over c^2} {\bf \vec v}_{\rm A}\times{\nabla {\bar w}^{\rm A}}
 + {2 \over c^2} \nabla \times {\bf \vec {\bar w}}^{\rm A}\; ,
 \label{five} 
\end{equation}
which has to be compared with corresponding terms in
\cite{wi,sof,bru}. Notice, that in the parameterized version of
the post-Newtonian theory factors $2$ in the last two terms of Eq.\
(\ref{five}) are to be replaced by $(1+\gamma)$, where $\gamma$ is 
the traditional Eddingtonian post-Newtonian
parameter. If only gravitational forces are applied one advantageously
combines the first two terms into a single one with factor $(\gamma+
{1 \over 2})$. In agreement with the common terminology \cite{wi} we
shall refer to them as the geodetic terms. The last term in 
Eq.\ (\ref{five}) then expresses gravitomagnetic or Lense-Thirring 
phenomena.

In the following, we shall investigate the influence of all terms in 
Eq.\ (\ref{five}) on the dynamics of the coordinate frames in the 
Earth-Moon(-Sun) system. We adopt the following
notation: index C is reserved for the Sun, index B for the 
Moon (with mass $M_{\rm B}$) and index A for the Earth (with mass
$M_{\rm A}$). Since Eq.\ (\ref{five}) is limited to the terms
in $1/c^2$, we can use for the vectors of position, velocity
and acceleration Newtonian approximation. The center-of-mass
of the Earth-Moon system is assumed to move on an elliptic 
orbit with semimajor axis $a_1$, eccentricity $e_1$ and
mean motion $n_1$ around the Sun, which is supposed to be located
at the center-of-mass of the global coordinate system. We thus neglect
the solar motion with respect to the global system, an approximation
justified by the inverse mass ratio of the Sun and the Earth-Moon 
system of about $328900$. The mutual Earth-Moon dynamics is 
approximated by the elliptical motion with semimajor axis $a_2$,
eccentricity $e_2$, inclination $I$ with respect to the ecliptic plane
and mean motion $n_2$. The following abbreviations will be used:
$M_{\rm AB} \equiv M_{\rm A}+M_{\rm B}$, $\iota = \sin I$. If 
${\bf R}$ denotes the position of Earth-Moon center-of-mass in
the global system and ${\bf r} (= {\bf r}_{\rm BA}) = {\bf z}_{\rm B}
- {\bf z}_{\rm A}$ the Earth-Moon vector, positions of the Earth and
Moon in the global coordinate system are
\begin{mathletters}
\begin{eqnarray}
 {\bf z}_{\rm A} & = & {\bf R} - {M_{\rm B} \over M_{\rm AB}}
 {\bf r} \; , \\
 {\bf z}_{\rm B} & = & {\bf R} + {M_{\rm A} \over M_{\rm AB}}
 {\bf r} \; , 
\end{eqnarray}
\end{mathletters}
and similarly for the velocities. An analogous relation can
also be established for the accelerations provided the solar
tidal terms are neglected in the Earth-Moon center-of-mass system.
Within the accuracy aimed at in this paper, these terms
can be safely omitted.

Our representation of the Earth-Moon(-Sun) dynamics is thus much
simpler than in the case considered by Brumberg and co-workers 
\cite{bbf1,bbf2} who implemented for similar calculations the
analytical planetary theory VSOP87 \cite{bf}. However, simplified
formulas given in the following sections yield a better view of
the structure of the particular terms and allow to localize and discuss
clearly the discrepancy between geodetic precession constant
given by Brumberg et~al. \cite{bbf1,bbf2} and by those of the
IERS Standard \cite{iers} (also e.g. in \cite{willi}).

As the investigated system is close to the planar three body
problem, remind that $\iota \approx 0.087$, the main
relativistic precession and nutations contribute to the rotation
around the ecliptic normal. It is thus natural to adopt
an ecliptic global coordinate system with the z-axis normal to the 
ecliptic plane. The following two sections are devoted to the analysis
of the geodetic and Lense-Thirring terms related to the rotations
around the ecliptic normal for both the terrestrial and lunar 
reference systems. In order to simplify notations, we 
omit the index ``z'' in the computed quantities. Final section D
is devoted to the computation of new nutations of the lunar
coordinate system around the in-ecliptic plane axes. It will be
demonstrated that their amplitudes are surprisingly greater than 
for the corresponding nutations about the third axis as a result
of: (i) a fine tuning and
cancellation of the geodetic and Lense-Thirring terms
contributing to the out-of-ecliptic ``z'' component, and (ii)
an associated long period, given by the lunar node rate, with the
in-ecliptic components.

\subsection{Application to the Earth-Moon(-Sun) system -- 
geodetic terms}
First, we focus on the secular geodetic phenomena resulting from the 
first two terms in Eq.\ (\ref{five}). Inserting global frame velocity
and accelerations of the investigated bodies A and B we obtain the
secular rates  
\begin{eqnarray}
 \Omega_{\rm A} &=& \left(\gamma+{1 \over 2}\right)\biggl[\left({n_1
  a_1 \over c}\right)^2 {n_1 \over 1-e_1^2} \nonumber \\
 & & \quad + \left({M_{\rm B} \over M_{\rm AB}}\right)^2
 \left({n_2 a_2 \over c}\right)^2 n_2\, {\sqrt{1-\iota^2} \over
 1-e_2^2} \biggr] \; , \label{seven} 
\end{eqnarray}
and
\begin{eqnarray}
 \Omega_{\rm B} &=& \left(\gamma+{1 \over 2}\right)\biggl[\left({n_1
 a_1 \over c}\right)^2 {n_1 \over 1-e_1^2} \nonumber \\
 & & \quad+ \left({M_{\rm A} \over M_{\rm AB}}\right)^2
 \left({n_2 a_2 \over c}\right)^2 n_2\, {\sqrt{1-\iota^2} \over
 1-e_2^2} \biggr] \; , \label{eight} 
\end{eqnarray}
characterizing the relativistic precession of the corresponding 
referentials. After an additional integration of 
Eqs.\ (\ref{seven})--(\ref{eight}) one sets the angles of precession
of the two referentials (to be denoted $\theta_{\rm A}$, resp.
$\theta_{\rm B}$)
\begin{eqnarray}
 \theta_{\rm A} &=& \left(\gamma+{1 \over 2}\right)\biggl[\left({n_1
 a_1 \over c}\right)^2 {l' \over 1-e_1^2} \nonumber \\
 & &\quad+ \left({M_{\rm B} \over M_{\rm AB}}\right)^2
 \left({n_2 a_2 \over c}\right)^2 l\, {\sqrt{1-\iota^2} \over
 1-e_2^2} \biggr] \; , \label{nine} 
\end{eqnarray}
and
\begin{eqnarray}
 \theta_{\rm B} & =& \left(\gamma+{1 \over 2}\right)\biggl[\left({n_1
 a_1 \over c}\right)^2 {l' \over 1-e_1^2} \nonumber \\
 & & \quad+ \left({M_{\rm A} \over M_{\rm AB}}\right)^2
 \left({n_2 a_2 \over c}\right)^2 l\, {\sqrt{1-\iota^2} \over
 1-e_2^2} \biggr] \; . \label{ten} 
\end{eqnarray}
Here, $l$ and $l'$ are the mean anomalies of the Moon and the
Sun. The first part of the previous formulas, which is common to the
two coordinate systems, is the usual de Sitter precession which has
been known since 1916 \cite{ds}. Its rate is of about $1.9194\, {\rm
as/cy}$ \cite{willi,fu} and it is due to the solar gravity field.
The second terms in Eqs.\ (\ref{nine})--(\ref{ten}) are specific
to each reference frame and follow from the mutual Earth-Moon 
gravitational interaction. Their difference reads
\begin{equation}
 \delta\Omega_{\rm AB} = \left(\gamma+{1 \over 2}\right) \left({n_2
  a_2
 \over c}\right)^2 n_2\, {\sqrt{1-\iota^2} \over 1-e_2^2} {M_{\rm A}
 - M_{\rm B} \over M_{\rm AB}} \; . \label{eleven} 
\end{equation}
If the Earth-Moon system parameters are substituted one
obtains $\delta\Omega_{\rm AB} \approx 28.9\, {\rm mas/cy}$. 
Naturally, most of this is due to a secular advance of the 
lunar reference frame. Due to the small value of
$(M_{\rm B}/M_{\rm AB})^2$ the Earth reference frame is subjected
to an additional relativistic precession of about $0.004\, {\rm
mas/cy}$.

Notice, that previous formulas for the secular rates of the two local
coordinate systems relating to the orbiting bodies A and B are given
in closed form in small parameters ($\iota$, $e_1$ and $e_2$).

Periodic terms called nutations (following the terminology of
Fukushima \cite{fu}), are superimposed to the geodetic precession.
Analogously to the case of the secular effects
mentioned above, the main relativistic nutation terms have their
origin in the solar gravity (and are thus of ``the tidal
origin''). Annual and semiannual terms derived by
Fukushima \cite{fu} have amplitudes of $0.153$ mas and $0.002$ mas
(see also \cite{bv,biz}). They are again common both to the
Earth and
Moon reference frames. In the following, we shall derive another
class of relativistic nutations arising from the mutual
Earth-Moon dynamics. As a consequence, their period equals one
synodic month.
Following the classical theories of the Moon, we shall denote the
mean longitudinal elongation of the Moon and the Sun by the Delaunay
variable $D$ and the ratio of the mean motion of the Earth-Moon 
center-of-mass around the Sun and the lunar synodic mean motion by
$m$ [thus $m = n_1/(n_2-n_1)$], and we have $m \approx 0.081$.

Inserting the global frame velocity and accelerations of the
involved bodies into the first two terms of Eq.\ (\ref{five})
and retaining only terms of the zeroth order in the power series
in eccentricity $e_2$ and inclination $\iota$ we obtain
(apart for the previously mentioned Fukushima terms)
\begin{mathletters}
\begin{eqnarray}
 \delta\Omega_{\rm A} &= & -\left(\gamma+{1 \over 2}\right)
 {M_{\rm B} \over M_{\rm AB}}{a_2 \over a_1} \left({n_1 a_1 \over c}
 \right)^2 n_2
 \left(1+{n_2 \over n_1}\right) \cos D \nonumber \\
 & & \quad + {\cal O}(e_2,\iota^2) \; ,  \label{twelvea} \\
 \delta\Omega_{\rm B} &= & \left(\gamma+{1 \over 2}\right)
 {M_{\rm A} \over M_{\rm AB}} {a_2 \over a_1}\left({n_1 a_1 \over
 c}\right)^2 n_2 \left(1+{n_2 \over n_1}\right) \cos D\nonumber \\
 & & \quad + {\cal O}(e_2,\iota^2) \; , \label{twelveb}
\end{eqnarray}
\end{mathletters}
and the corresponding solution for the local system nutations
\widetext
\begin{mathletters}
\begin{eqnarray}
 \delta\theta_{\rm A} &= & -\left(\gamma+{1 \over 2}\right)
 {M_{\rm B} \over M_{\rm AB}} {a_2 \over a_1}\left({n_1 a_1 \over c}
 \right)^2 {(1+m)(1+2 m) \over  m} \sin D 
 + {\cal O}(e_2,\iota^2) \; , \label{thirteena} \\
 \delta\theta_{\rm B} &= & \left(\gamma+{1 \over 2}\right)
 {M_{\rm A} \over M_{\rm AB}} {a_2 \over a_1}\left({n_1 a_1 \over c}
 \right)^2 {(1+m)(1+2m) \over m} \sin D + {\cal O}(e_2,
 \iota^2) \; . \label{thirteenb}
\end{eqnarray}
\end{mathletters}
\narrowtext
\noindent In the case of the lunar referential these
synodic terms have an amplitude comparable to the first Fukushima
nutation term (approximately $0.1$ mas). We shall however see in 
the next section, that a great part of the effect is cancelled by the
corresponding synodic term due to the Lense-Thirring effect.

\subsection{Application to the Earth-Moon(-Sun) system -- 
Lense-Thirring terms}
Next, we evaluate the principal effects due to the last 
-- Lense-Thirring -- term of Eq.\ (\ref{five}). By restricting
to the secular effects and substituting the vector potential
(\ref{threeb}) into formula (\ref{five}) we obtain two kinds
of effects. The first is due to the solar angular momentum and reads
\begin{equation}
 \Omega_{\rm A} = -{\gamma+1 \over c^2} {G S_{\rm C} \over a_1^3
 (1-e_1^2)^{3/2}} \; , \label{fourteen} 
\end{equation}
while the second, linked to the total angular momentum of
the Earth-Moon system, reads
\begin{eqnarray}
 \Omega_{\rm A} &=& \left(\gamma+1\right) {M_{\rm A} M_{\rm B} \over
  M_{\rm AB}^2}
 \left({n_2 a_2 \over c}\right)^2 n_2\, {\sqrt{1-\iota^2} \over
 1-e_2^2} \nonumber \\ & =& {\gamma+1
 \over c^2} {G L_{\rm AB}^z \over a_2^3 (1-e_2^2)^{3/2}}
 \label{fifteen}
\end{eqnarray}
[$G$ is the gravitational constant]. The same contributions appear 
in the Lense-Thirring precession of the body B frame.
Interestingly, both secular Lense-Thirring phenomena nearly cancel out
for the Earth-Moon system. The Earth-Moon angular momentum
term (\ref{fifteen}) yields a value of about $0.47\,
{\rm mas/cy}$ and the solar induced term (\ref{fourteen}) amounts
to $-0.28\, {\rm mas/cy}$. One must notice, however, that
the latter depends on the total solar angular momentum $S_{\rm C}$,
which is not well known \cite{ul}. 

In the most complete investigation of the relativistic spin effects
of the Earth reference frame by Brumberg and co-workers
\cite{bbf1,bbf2}, the authors seem to include arbitrarily the 
second Lense-Thirring
precession (\ref{fifteen}) in the definition of the geodetic 
precession constant (in the broader sense following their 
terminology). As a result, their corrected value of the geodetic
precession constant is $1.9199\, {\rm as/cy}$ instead of the
value $1.9194\, {\rm as/cy}$ deduced from the first two terms of
Eq.\ (\ref{five}). In our opinion it might not be a favourable 
route, as the first Lense-Thirring term (\ref{fourteen}) 
investigated previously is of the same order of magnitude as the 
second one. Moreover, not only it has an opposite
sign but its value is uncertain, at least by a factor of two, due 
to the uncertainty of the total solar angular momentum
\cite{ul}. It may thus happen by chance that the Brumberg et~al.
additional term in the geodetic precession constant will be cancelled
by an opposite contribution due to the solar rotation Lense-Thirring
term. However, it should be acknowledged that the discussed
Lense-Thirring originated
terms are by a factor of $30$ smaller than the current precision
level of the dynamical determination of the geodetic precession
\cite{dnw,mull,sci,wnd}. The dominant cause of this uncertainty
is due to the poorly known second zonal harmonic $J_2$ of the Moon
\cite{sci,wnd}. Of course, an improvement in the determination of this
parameter by future lunar missions combined with a larger time
span of the LLR data will make the determination of the geodetic
precession more accurate.
However, with current state of the art, it is unlikely that the
dynamics of the Earth-Moon system could contribute significantly
to a precise determination of the Lense-Thirring effect, and
other opportunities are discussed, for instance, in Ref.~\cite{wi}.

As for the periodic terms in the Lense-Thirring effect (LT
nutations), they are given by\footnote{ We again omit all
eccentricity and inclination $\iota$ dependent terms.}
\begin{mathletters}
\begin{eqnarray}
 \delta\Omega_{\rm A} &= & \left(\gamma+1\right)
 {M_{\rm B} \over M_{\rm AB}} {a_2 \over a_1} {n_2 \over n_1}
 \left({n_1 a_1 \over c}\right)^2 n_2 \cos D \nonumber \\
 & &\; + {\cal O}(e_2,\iota^2) \; , \label{sixteena} \\
 \delta\Omega_{\rm B} &= & -\left(\gamma+1\right)
 {M_{\rm A} \over M_{\rm AB}} {a_2 \over a_1} {n_2 \over n_1}
 \left({n_1 a_1 \over c}\right)^2 n_2 \cos D \nonumber \\
 & & \;+ {\cal O}(e_2,\iota^2) \; , \label{sixteenb}
\end{eqnarray}
\end{mathletters}
which give after integration the nutation angles, as
\begin{mathletters}
\begin{eqnarray}
 \delta\theta_{\rm A} &= & \left(\gamma+1\right)
 {M_{\rm B} \over M_{\rm AB}} {a_2 \over a_1} \left({n_1 a_1 \over
 c}\right)^2 {(1+m)^2 \over m} \sin D \nonumber \\
 & & \;+ {\cal O}(e_2,\iota^2) \; , \label{seventeena} \\
 \delta\theta_{\rm B}&= & -\left(\gamma+1\right)
 {M_{\rm A} \over M_{\rm AB}} {a_2 \over a_1} \left({n_1 a_1 \over
 c}\right)^2  {(1+m)^2 \over m}\sin D \nonumber \\
 & &\;+ {\cal O}(e_2,\iota^2) \; . \label{seventeenb}
\end{eqnarray}
\end{mathletters}

Collecting the synodic nutations given previously, both of the
geodetic and Lense-Thirring origin, we observe that a significant 
part of them cancels out. The final relativistic synodic 
signal reads
\begin{mathletters}
\label{eighteen}
\begin{eqnarray}
 \delta\theta_{\rm A} &= & {1 \over 2} {M_{\rm B} \over M_{\rm AB}}
 {a_2 \over
 a_1} \left({n_1 a_1 \over c}\right)^2 {1+ m \over m}
 \left(1-2\gamma m\right) \sin D \nonumber \\
 & & \;+ {\cal O}(e_2,\iota^2) \; , \label{eighteena} \\
 \delta\theta_{\rm B}&= & -{1 \over 2} {M_{\rm A} \over M_{\rm AB}}
 {a_2 \over a_1}
 \left({n_1 a_1 \over c}\right)^2  {1+m \over m}
 \left(1-2\gamma m\right) \sin D \nonumber \\
 & & + {\cal O}(e_2,\iota^2) \; . \label{eighteenb}
\end{eqnarray}
\end{mathletters}
Substituting the relevant numerical values into formulas
(\ref{eighteen}) we obtain an amplitude of $0.03$ mas for the lunar
coordinate system, and $4\times10^{-4}$ mas for the terrestrial
coordinate system. The latter value related to the Earth 
referential coincides with the result of Brumberg and co-workers 
[16, Eq.~(4.3)].

Tables Ia and Ib show briefly various phenomena studied throughout
the text. Phenomena common to the two referentials are distinguished
{}from the individual terms. 

\subsection{Application to the Earth-Moon(-Sun) system --
 ecliptic terms}
Finally, we discuss briefly relativistic nutations around
the in-ecliptic axes (${\bf e}_x,{\bf e}_y$). The two main
motivations in performing this work are as follows: (i) a fine 
cancellation of the lunar nutations discussed previously, (ii)
possible appearance of small divisors related to the small rate
angles (e.g. longitude of the lunar orbit node $\Omega$). Careful
inspection of the Brumberg et~al. \cite{bbf1} results strongly 
supports this possibility.

Substituting necessary components of the global system velocities
and accelerations into Eq.\ (\ref{five}) we find dominant nutation
terms in the following form
\begin{mathletters}
\label{nineteen}
\begin{eqnarray}
 \theta_{\rm B}^x &=& -\left(\gamma+{1 \over 2}\right) \left(
 {M_{\rm A} \over M_{\rm AB}}\right)^2 \left({n_2 a_2 \over c}
 \right)^2 \iota\,
 {n_2 \over {\dot \Omega}} \cos\Omega \; . \label{nineteena} \\
 \theta_{\rm B}^y &=& -\left(\gamma+{1 \over 2}\right) \left(
 {M_{\rm A} \over M_{\rm AB}}\right)^2 \left({n_2 a_2 \over c}
 \right)^2 \iota\,
 {n_2 \over {\dot \Omega}} \sin\Omega \; . \label{nineteenb}
\end{eqnarray}
\end{mathletters}
The amplitude of both terms in Eqs.\ (\ref{nineteen}) is approximately
$0.09$ mas, greater than that of the relativistic nutations 
around the axis normal to the ecliptic plane. Considering a rotating
axis ${\bf n}(\Omega) = \cos\Omega\, {\bf e}_x + \sin\Omega\, 
{\bf e}_y$ in the ecliptic plane, one can also represent nutations 
(\ref{nineteen}) as a precession around ${\bf n}(\Omega)$ with 
the rate of $2.58 \, {\rm mas/cy}$. Finally, one must bear in mind
that excitations associated with the long-period planetary 
perturbations are likely to exist (see \cite{bbf1}).

Similar search for the dominant terms in the case of the terrestrial
reference frame allows to recover $18.6$ year terms in \cite{bbf1}.
Interestingly, these Earth system nutations are of Lense-Thirring
origin. Inversely, corresponding lunar nutation terms given in
formulas (\ref{nineteen}) originate in the geodetic part of
Eq.\ (\ref{five}).

\section{CONCLUSION}
The relativistic precession and nutation of the lunar and 
terrestrial coordinate systems has been examined. The main
terms connected to the Earth-Moon center-of-mass
referential have been identified together with
individual terms of the two local coordinate systems related
to the mutual Earth-Moon dynamics. In particular, those related
to the relativistic effects of the terrestrial reference system
coincide with the principal terms given by Brumberg et~al. 
\cite{bbf1}.

New terms include mainly additional relativistic precession
of the lunar reference system with the rate of $28.9\, {\rm mas/cy}$.
This effect is in principle ``measurable'' using the best LLR
data, but due to the secular character can be hardly separated
{}from the tidal secular effects (e.g. \cite{fabi}). But,
it might be wise to subtract this value from
the observed lunar data to achieve a better fitting of the 
tidal terms to the observed ranges. The corresponding secular
Lense-Thirring term of the common Earth-Moon center-of-mass system
is very small, largely because of the cancellation of the term
linked respectively to the solar and Earth-Moon angular momentum.
Even with an improved LLR technology, its measurement will be
hardly feasible. Important experimental confirmation of the 
Lense-Thirring effect thus requires the use of another techniques
(e.g. satellite born). The amplitudes of the new relativistic lunar
nutations are unfortunately very small, in part as a result again
of a nearly cancellation of the the geodetic and Lense-Thirring
terms.

Corresponding results for the Earth reference frame are also small 
if compared with the capabilities of the current
astrometric techniques reaching maximally the level of a few $0.01$
mas \cite{ksa,kso}. However, future astrometric projects 
\cite{rea,kk} at the microarcsecond level will need to allow for
the Earth frame precession and nutations given in the previous
sections and in Ref.~\cite{bbf1}.

Finally, it is useful to draw attention to the question of
terminology when speaking of the geodetic precession. Its 
``broader sense'' pioneered by Brumberg and co-workers \cite{bbf1}
(later also included in \cite{bret,sim}) includes in fact a
Lense-Thirring term resulting from a coupling
between the solar gravity field and the angular momentum of the
Earth-Moon system. Another Lense-Thirring term due to the coupling
between the solar angular momentum and the translational motion of the
Earth-Moon center-of-mass, ignored in the analysis of Brumberg et~al.
is, however, of the same order of magnitude but of opposite sign.
Moreover, the latter term has an uncertainty of nearly $100$~\%
due to the badly determined solar angular momentum.
Terminological unification among the astronomical community
is desirable in the future, as the IERS Standard recommended
value still follows geodetic precession in the ``narrow sense''
based only on the first two terms of the fundamental formula 
(\ref{five}).

\acknowledgments
I am grateful to E. Bois, F. Mignard and Ch. Veillet for informative
discussions. The study has been performed during a stay at
the OCA/CERGA, Grasse (France) thanks to the H. Poincar\'e 
research fellowship.

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\caption{Schematic description of the relativistic
 spin effects related to the definition of the local (dynamically
 non-rotating) lunar (LS) and terrestrial (TS) coordinate systems.
 Secular effects -- denoted as ``precession'' -- are given in part
 a), periodic effects -- denoted as ``nutations'' -- are given in
 part b) (P in the fourth column indicates the period). Common
 terms which  can be equally assigned to the Earth-Moon
 center-of-mass coordinate system (including the principal de Sitter
 term) appear between the columns of the two reference systems.  
 Common Lense-Thirring precession is split into two pieces:
 (i) LT--1 originating in the coupling between the solar gravity
 field and the angular momentum of the Earth-Moon system (the
 term added to the newly proposed value of ``the geodetic precession
 constant''), (ii) LT--2 originating in the coupling between the
 solar angular momentum and the Earth-Moon translational motion.
 The individual geodetic and Lense-Thirring nutations of both
 reference systems are of opposite sign and of comparable
 magnitude.}
\begin{tabular}{ccc}
 \multicolumn{3}{c}{ \rule[-1.5ex]{0mm}{3ex} {\bf PRECESSION} ({\rm 
  mas/cy}) \rule[0.75ex]{0mm}{3ex}} \\
\hline\hline
  \rule[0.5ex]{0mm}{3ex} Source terms & TS & LS\rule[-1.5ex]{0mm}{3ex}
 \\ \hline
 principal geodetic \rule[0.5ex]{0mm}{3ex} \rule[-1.5ex]{0mm}{3ex} &
 \multicolumn{2}{c}{1919.36}\\
\hline
 individual geodetic & 0.004 & 28.86 \rule[0.5ex]{0mm}{3ex}
 \rule[-1.5ex]{0mm}{3ex}\\
\hline
 LT -- 1  \rule[0.5ex]{0mm}{3ex} \rule[-1.5ex]{0mm}{3ex} &
 \multicolumn{2}{c}{0.47}\\
\hline
 LT -- 2 \rule[0.5ex]{0mm}{3ex} \rule[-1.5ex]{0mm}{3ex} &
 \multicolumn{2}{c}{-0.28} \\
  & \multicolumn{2}{c}{(\hbox{\rm rem. error $\approx$
 $100$ \%})} \\
\end{tabular}
\medskip
\noindent {\bf a)}
\bigskip

\begin{tabular}{cccc}
 \multicolumn{4}{c}{ \rule[-1.5ex]{0mm}{3ex} {\bf NUTATIONS} ({\rm 
  mas}) \rule[0.75ex]{0mm}{3ex}} \\
\hline\hline
  \rule[0.5ex]{0mm}{3ex} Source terms & TS  & LS
 & P \rule[-1.5ex]{0mm}{3ex} \\
\hline
 Fukushima -- 1  \rule[0.5ex]{0mm}{3ex}
 \rule[-1.5ex]{0mm}{3ex} & \multicolumn{2}{c}{0.15313} & 1 {\rm year}
 \\
\hline
 Fukushima -- 2 \rule[0.5ex]{0mm}{3ex} \rule[-1.5ex]{0mm}{3ex} &
 \multicolumn{2}{c}{0.00192}  & 0.5 {\rm year} \\
\hline
 individual geodetic & -0.00147 & 0.11964 \rule[0.5ex]{0mm}{3ex}
 \rule[-1.5ex]{0mm}{3ex}& 29.5 {\rm days} \\
\hline
 individual LT & 0.00182 & -0.14832 \rule[0.5ex]{0mm}{3ex}
 \rule[-1.5ex]{0mm}{3ex} & 29.5 {\rm days} \\
\end{tabular}
\medskip
\noindent {\bf b)}
\bigskip
\end{table}

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

