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\title{Improved constraint on the $\alpha_1$ PPN
       parameter from lunar motion}
\author{J\"urgen M\"uller}
\address{Institute of Astronomical and Physical Geodesy, TU M\"unchen,
         D-80290 M\"unchen, Germany}
\author{Kenneth Nordtvedt}
\address{Northwest Analysis, 118 Sourdough Ridge Road, Bozeman,
         MT 59715, USA}
\author{David Vokrouhlick\'y}
\address{Institute of Astronomy, Charles University,
         \v Sv\'edsk\'a~8, CZ-15000 Prague, Czech Republic}
\date{\today}
\maketitle
\begin{abstract}
 Some theories of gravity, alternative to general relativity,
 introduce a preferred rest frame for local gravitational physics.
 At the post-Newtonian level, these preferred-frame effects in
 lagrangian-based theories are described by two phenomenological
 parameters $\alpha_1$ and $\alpha_2$, of which only the latter is
 strongly constrained today. Following previous theoretical suggestions
 of Nordtvedt, Damour and Vokrouhlick\'y we show that the
 analysis of the lunar motion provides an improved bound
 of the former parameter, notably $\alpha_1 = (-8 \pm 9)\times 10^{-5}$
 realistic error [and $\alpha_1 = (-8\pm 4)\times 10^{-5}$ (90
 \% C.L.)].
\end{abstract}
\pacs{ 04.80.Cc, 96.20.-n}

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\narrowtext
\section{Introduction}
Modern unification theories suggest that the (long-range gravitational)
interaction between macroscopic bodies may be mediated not only by
the metric tensor field, but also by other scalar, vector or tensor
fields. In the theories which contain vector or other tensor fields,
in addition to the metric field, distribution of matter in the
Universe typically selects a preferred rest frame for local
gravitational physics. At the post-Newtonian level, appropriate for
dynamics of the solar system bodies, such preferred-frame effects are
phenomenologically describable by two parameters $\alpha_1$ and
$\alpha_2$ \cite{WN72,NW72,W81}. While the latter of the two parameters
has been tightly constrained by noting the close alignment of the solar
spin axis with the angular momentum vector of the whole solar system
\cite{N87}, the former is constrained relatively weakly. The best
current solar system value $\alpha_1 = (2.1 \pm 3.1) \times 10^{-4}$
(90 \% C.L.) has been reported in Ref.~\cite{H84}, while analysis of
the binary pulsar data yields a slightly better value of $|{\hat
\alpha}_1| < 1.7 \times 10^{-4}$ (90 \% C.L.) \cite{BCD95}.\footnote{
Note, however, that the strong field parameter ${\hat \alpha}_1$ can
be expressed as a linear function of its weak field counterpart
$\alpha_1$, discussed in this paper, and an additional parameter
$\alpha^\prime_1$ \cite{DEF92,BCD95}. Combining the solar system and
binary pulsar tests of the preferred-frame effects, qualitatively
independent, thus yields the possibility to constrain the two
parameters $\alpha_1$ and $\alpha^\prime_1$.} These constraints are
only slightly better than the corresponding limits on the much more
fundamental parameters $\beta$ and $\gamma$ \cite{Sci,WND96}.

This situation motivated several suggestions for experiments aiming
to obtain better limits on $\alpha_1$. For instance, Damour and
Esposito-Far\`ese \cite{DEF94} considered motion of artificial Earth
satellites on well-tuned orbits as possible probes of the $\alpha_1$
parameter. However, despite of a network of singularly sensitive
satellite orbits, frozen with respect to fixed space, it seems that
none of the currently laser tracked objects suit the task (with
regard to LAGEOS, the geodynamics satellite with the currently
best-measured orbit, see Ref.~\cite{MVRE}).

Another promising candidate of a well-tracked Earth satellite
is the Moon. Nordtvedt considered this possibility soon after the
lunar laser ranging (LLR) project became operational and the PPN
framework classifying gravitational theories was formulated \cite{N73}.
Recently, Damour and Vokrouhlick\'y \cite{DV3} revisited the
problem by using a more involved analytical method (Hill-Brown
technique) and confirmed that the lunar ranging data may include
interesting information on the $\alpha_1$ parameter. The latter
work is the prime motivation for our communication, in which we
aim to use the lunar data to obtain a new bound on the
$\alpha_1$ PPN parameter.

\section{The method and data set}
% METHOD:
Because we use primarily lunar ranging data, it is appropriate to
focus on the dynamical influence of the preferred-frame effects
on the three-body system consisting of the Moon (hereafter
indexed by 1), the Earth (indexed by 2), and the Sun (indexed by
3). The associated lagrangian which enters the gravitational dynamics
of $N$-bodies, reads \cite{DEF94} ($A,B = 1,2,3$)
\begin{equation}
 L_{\alpha_1} = - {\alpha_1\over 4}\sum_{A \neq B} {G M_A M_B\over
  c^2 r_{AB}} \left({\bf v}_A^0 . {\bf v}_B^0\right)\; , \label{2one}
\end{equation}
where $M_A$ is the mass of body A, ${\bf r}_A$ its position coordinate
vector in the solar system center-of-mass frame ($r_{AB} \equiv
|{\bf r}_A-{\bf r}_B|$), and ${\bf v}_A^0$ its ``cosmic velocity''
which splits into the velocity ${\bf v}_A$ of the body relative to
solar system center-of-mass, and the global motion ${\bf w}$ of the
solar system with respect to the gravitationally preferred frame.
A standard assumption \cite{W81} links the latter frame to the rest
system of the cosmic microwave radiation. In this case $w \equiv |{\bf
w}| = 368.9\pm 2.5\,{\rm km/s}$ in the direction $(\alpha,\delta) =
(168^0\pm 0.07^0,-7.23^0\pm 0.07^0)$ \cite{Pee93,Fi94}.

Because of the smallness of the investigated effects (see the
existing limits on $\alpha_1$ from Sec~I), we adopt the following
simplifications: (i) the solar motion with respect to the solar system
center-of-mass is neglected (hence, ${\bf r}_3
\equiv 0$, ${\bf v}_3^0 \equiv {\bf w}$), (ii) in deriving
formulae for the ``$\alpha_1$--accelerations'', resulting from the
lagrangian (\ref{2one}), we substitute only the leading newtonian
parts for the accelerations (neglecting tidal influence
of the solar gravitational field in the Earth-Moon vicinity). The
geocentric coordinate position of the Moon is denoted by ${\bf r}
= {\bf r}_1 - {\bf r}_2$ and the corresponding velocity ${\bf v} =
d{\bf r}/dt$. The barycentric accelerations corresponding to the
(hypothetical) preferred-frame effect modification of the solar
system dynamics of the Earth and Moon then read
\begin{mathletters}
 \label{2two}
\begin{eqnarray}
 \left({d^2 {\bf r}_1\over dt^2}\right)_{\alpha_1} &=& {\alpha_1
  \over 2}{GM_2\over c^2 r^2} \left[\left({\bf v}^0_1.{\bf v}^0_2
  \right){\bf n} + r {d {\bf v}_2\over dt} - \left({\bf n}.{\bf v}
  \right){\bf v}^0_2\right] \nonumber \\
 & & + {\alpha_1\over 2}{GM_3\over c^2 r_1^2}\left[\left({\bf
  v}^0_1.{\bf w}\right) {\bf n}_1 - \left({\bf n}_1.{\bf v}_1\right)
  {\bf w}\right] \; , \nonumber\\ & & \label{2twoa} \\
 \left({d^2 {\bf r}_2\over dt^2}\right)_{\alpha_1} &=& -{\alpha_1
  \over 2}{GM_1\over c^2 r^2} \left[\left({\bf v}^0_1.{\bf v}^0_2
  \right){\bf n} - r {d {\bf v}_1\over dt} + \left({\bf n}.{\bf v}
  \right){\bf v}^0_1\right] \nonumber \\
 & & + {\alpha_1\over 2}{GM_3\over c^2 r_2^2}\left[\left({\bf
  v}^0_2.{\bf w}\right) {\bf n}_2 - \left({\bf n}_2.{\bf v}_2\right)
  {\bf w}\right] \; , \nonumber \\ & & \label{2twob}
\end{eqnarray}
\end{mathletters}
\noindent (Here, we denote ${\bf n}={\bf r}/r$, ${\bf n}_1={\bf r}_1
/r_1$, ${\bf n}_2={\bf r}_2/r_2$, and $d{\bf v}_1/dt$, $d{\bf v}_2/dt$
indicate the newtonian accelerations.) One must be careful when
using the previous formulas. For instance, it is
necessary to subtract the $w^2$ part from the first terms in square
brackets, it being simply a recalibration of the local gravitational
coupling factor $G$.

The above formulation of the preferred-frame perturbation has been
introduced into our numerical model for integration of the orbits of
Moon, Earth and Sun, as well as the other bodies of the solar system
(including several asteroids). The force terms (\ref{2two}) generate
an $\alpha_1$ ``partial'' which as a correction to the nominal orbits
allows a least squares fit of $\alpha_1$ along with other LLR
model parameters.

As an alternative method, and cross-check, to the fully numerical
procedure described above, we used the analytical formulas for the
radial orbit perturbation derived in Refs.~\cite{N73,DV3}. They yield an
estimate of the amplitudes for selected principal spectral lines,
namely those with frequencies $n$ (sidereal period), $n'$ (annual
period), and $n-2n'$. In the course of our work we found that this set
of analytic terms is not sufficiently complete for obtaining comparably
precise results as the direct numerical treatment. ``Missing'' terms are
due to eccentricity of the lunar orbit (or, in the terminology of
Ref.~\cite{DV3}, to coupling between free and forced perturbations
of the lunar motion). This problem has been addressed by Nordtvedt,
using procedures analogous to those employed in Ref.~\cite{N96}, and he
has shown that the annual lunar
perturbation produces, apart from the principal effect mentioned above,
important sidebands at frequencies $(n_0-n')$ and $(n_0+n')$
($n_0$ being anomalistic frequency). The resulting oscillations of the
lunar range are given by
\widetext
\begin{mathletters}
 \label{2three}
\begin{eqnarray}
 \delta r &\simeq& -\alpha_1\, {wV\over 2c^2}\, r e\, \left(1+2 n/n'
  \right)\cos\left[\left(n_0-n'\right)\left(t-t_0\right)-\phi_-\right]
  \; ,  \label{2threea} \\
 \delta r &\simeq& -\alpha_1\, {wV\over 2c^2}\, re\, \left(1-2n/n'
  \right)\cos\left[\left(n_0+n'\right)\left(t-t_0\right)-\phi_+\right]
  \; , \label{2threeb}
\end{eqnarray}
\end{mathletters}
\narrowtext\noindent
with $V$ expressing mean velocity of the Earth-Moon center-of-mass
motion around the Sun. Phases $\phi_\pm$ are given by addition and
subtraction of a longitude angle of perigee occurrence $\phi_p$ and
a longitude angle $\phi_a$ of ${\bf w}$ measured in ecliptic from the
lunar (and solar) position at time $t_0$ corresponding to an arbitrary
new-moon phase. Expressed in physical quantities related to the lunar
orbit, we obtain amplitudes for the terms (\ref{2three}) of about $38\,
\alpha_1$ and $31\,\alpha_1$ meters, respectively, which is comparable
to the direct effects. Principal effects, supplemented by terms
(\ref{2three}), are used for the analytical estimate of $\alpha_1$.
It should be noted that there exist a number of additional eccentricity
and inclination induced sidebands with smaller amplitude. Although they
are of less interest for analytical understanding of the results, they
may help in decorrelating different types of preferred direction
effects (see below).

% DATA SET:
The data set involved in our analysis consist of the LLR measurements
collected by the McDonald Observatory (USA), Haleakala station (USA),
and OCA/CERGA (France) in the period March 1970 to February 1996.
Dates of operation of the three stations and discussion of the
continually increasing data precision can be found in
Refs.~\cite{Sci,WND96}. Apart from estimating the $\alpha_1$ parameter
our code solves for about $160$ parameters, such as body initial
positions and velocities, the Earth station
and lunar retroreflector locations, the Earth rotation parameters,
geophysical and selenophysical constants, other PPN parameters etc.
General information can be found in Refs.~\cite{Mphd,Scha89,S89},
some previous estimates of the relativistic parameters are in
Ref.~\cite{M91}.

\section{Results and discussion}
Along with the $\alpha_1$ parameter, we estimate a minimal set of
other PPN parameters, namely Eddington's $\beta$ and $\gamma$.
Time variation of the gravitational constant ${\dot G}/G$ has
been discarded from our tests because most physically well-motivated
gravitation theories with appropriately constrained parameter values
for $\beta$, $\gamma$ and $\alpha_1$ have negligibly small value of
${\dot G}/G$. De Sitter precession has been included in our analysis.

The Eddington parameters do not show significant sensitivity to the
frequencies listed in the previous section \cite{BI85,N95,WND96}, and
their fit is well decorrelated from the estimation
of $\alpha_1$. The following realistic errors\footnote{The precisions
with which $\gamma$ and $\beta$ are explicitly determined from LLR data
can
not be understood theoretically at this time. This has also been noted
by Williams et~al. \cite{WND96} who obtain LLR fits for $\gamma$ and
$\beta$ comparable to our fits. We present these results in hope of
stimulating interest in this problem.} have been obtained: $\beta-1 =
(0.9\pm 5.0)\times 10^{-3}$, $\gamma-1 = (-0.8\pm 6.0)\times 10^{-3}$,
and an error of $0.15$ milliarcseconds/year for de Sitter precession 
of the lunar orbit.
For $\alpha_1$ we obtained an estimate $\alpha_1 = (-8\pm 4)\times
10^{-5}$ (90 \% C.L.).

An important part of this work consists of estimating the ``realistic''
error of the $\alpha_1$ parameter fit. Although the nature of an error
estimation in case of very complex problems (such as that of
lunar orbit determination) is to some degree subjective, we tried to
follow some systematic steps. First, we checked whether presence of
the $\alpha_1$ parameter in other sectors of the gravity theory could
modify the results. We considered two such known contributions: (i)
modification of the gravitational to inertial mass ratio of bodies such
as Earth and Moon in proportion to $\alpha_1$ \cite{N73,DEF92},
(ii) an annual term in relativistic geodetic precession
\cite{N73,W81,DEF94}. Neither of these effects altered estimation of
$\alpha_1$.

There are a number of candidates among non-relativistic or geometrical
effects which may show aliasing with the $\alpha_1$ signal. Covariance
analysis represents an approximate measure of this. The largest
correlation with other LLR parameters is smaller than 0.4 (e.g.\ tidal
lag angle and velocity of the Earth about the Sun). Such a low
correlation coefficient means that $\alpha_1$ can be well determined
within the LLR model. However our LLR model is only complete up to a
certain level. Some physical effects are just modeled with an accuracy
corresponding to about $1$ cm in the Earth-Moon distance,
e.g.\ solid Earth tides, atmospheric correction, Earth rotation. Other
effects are not considered at all, e.g.\ ocean loading, atmospheric
loading. These error sources can lead to signals which influence
the determination of $\alpha_1$. To check the magnitude (worst case) of
these effects, we analyzed the post-fit residuals, and obtained values
between 0.5 cm and 0.8 cm for the amplitudes of annual, semi-annual
and sidereal periods. Therefore a value of $8\times 10^{-5}$ has been
added to the formal $\alpha_1$ error in consideration for the influence
of unmodeled or insufficiently modeled effects in our
computer program. A further error may be introduced by using quantities
in the LLR analysis which are of limited accuracy but can not be
improved during the fit, e.g.\ nutation coefficients, coefficients of
the lunar gravity field of higher degree and order or the solar mass.
For such error sources we have added a value of about $4\times 10^{-5}$
to the error of $\alpha_1$.

In conclusion, we arrive at $(8\pm 9)\times 10^{-5}$ for a realistic
error bar of the $\alpha_1$ PPN parameter.

Analytical estimation of the preferred frame effects mentioned in the
previous section offers the possibility to cross-check numerical
results given above. Five principal spectral lines [three terms from
Ref.~\cite{DV3} and those from Eq.~(\ref{2three})] have been
introduced
in our program and led to the following estimate of $\alpha_1$:
$\alpha_1 = (-9\pm 6) \times 10^{-5}$ (90 \% C.L.).
The weak disagreement between the numerically and analytically
estimated values of $\alpha_1$ probably results from the rich
spectrum of the complete preferred-frame perturbation.

Particular care is paid to possible correlation between the
preferred-frame lunar orbit perturbation with that due to cosmic
polarization \cite{N94,NMS95,DV3}. The latter acts principally
at the sidereal period, competing at this frequency with
the preferred-frame effects, but having negligible sidebands. The
phases of the two effects at
this frequency are close, because the apex direction of the solar
system motion with respect to the cosmic microwave rest frame is
nearly perpendicular to the direction towards the galactic center.
However, simultaneous solution for $\alpha_1$ and a (hypothetical)
differential cosmic acceleration $g_c$ of the Earth and Moon suggests
satisfactory decorrelation of the two effects [correlation
coefficient of about 0.2]. The rich spectrum of the preferred
frame perturbations of the lunar orbit, discussed previously,
appears to be the essential factor in this respect.

In conclusion we note that the LLR data analysis contains high
quality information about possible anisotropy of the ``gravitational
constant''. Exploiting this possibility, we obtained here a new limit
on the $\alpha_1$ parameter. Nordtvedt \cite{N96} has recently
suggested that an interesting LLR measurement bound can be obtained
for the $\alpha_2$ parameter, as well. This task remains for the
future.

\acknowledgements
 We are grateful to Forschungseinrichtung Satellitengeod\"asie
 for financial support, Profs. M.~Schneider and M.H.~Soffel for
 discussions, and the referee for helpful suggestions. K.N. was
 supported by the A. von~Humboldt Foundation
 through the 1995/6 grant, and by the IHES, Bures sur Yvette,
 France.

%%%%%%%%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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