gr-qc0103088.tex
\documentclass[12pt]{article}
\usepackage[english]{babel}
\usepackage{graphicx,epsfig}
%\columnsep 0.8cm
\makeindex \textwidth 170mm \textheight 220mm \topmargin -5mm
%\addtolength{\textheight}{-\headheight}
%\addtolength{\textheight}{-\headsep}
%\addtolength{\textheight}{-\footheight}
\oddsidemargin 0mm \evensidemargin 0mm
%\setlength{\parskip}{\baselineskip}
\pagestyle{plain} \pagenumbering{arabic}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\nn}{\nonumber}
\newcommand{\bra}[1]{\left\langle #1 \right|}
\newcommand{\ket}[1]{\left| #1 \right\rangle}

% MACRO MIE
\def\nvd{4\b-\ga-3}
\def\ol#1{\overline{#1}}
\def\vass#1{\left\vert\ #1 \right\vert}
\def\somma#1#2{\sum_{\rm #1}^{\rm #2}}
\def\derp#1#2{\rp{\partial{#1}}{\partial{#2}}}
\def\rfr#1{eq. (\ref{#1})}
\def\rfrs#1#2{eqs. (\ref{#1})-(\ref{#2})}
\def\Rfr#1{Eq. (\ref{#1})}
\def\Rfrs#1#2{Eqs. (\ref{#1})-(\ref{#2})}
\def\bb{\bibitem}
\def\eqi{\begin{equation}}
\def\eqf{\end{equation}}
\def\eqia{\begin{eqnarray}}
\def\eqfa{\end{eqnarray}}
\def\ci{\cos{i}}
\def\si{\sin{i}}
\def\rp#1#2{{#1\over#2}}
\def\gr{General Relativity}
\def\ct#1{\cite{#1}}
\def\lb#1{\label{#1}}
\def\lg{LAGEOS}
\def\lgg{LAGEOS II}
\def\lt{Lense-Thirring\ }
\def\grc{gravitomagnetic}
\def\grm{gravitomagnetism}
\def\nd{node}
\def\pg{perigee}
\def\nds{nodes}
\def\aj{Ajisai}
\def\st{Starlette}
\def\etl{ETALON I}
\def\ett{ETALON II}
\def\lr{LARES}
\def\rc{residual combination}
\def\lte{Lense-Thirring effect}
\def\ltd{Lense-Thirring drag}
\def\sat{satellite}
\def\orb{orbit}
\def\keps{Keplerian orbital elements}
\def\sot{solid and ocean tides}
\def\tps{tidal perturbations}
\def\per{perturbation}
\def\srp{solar radiation pressure}
\def\ngp{non-gravitational perturbations}
\def\gp{gravitational perturbations}
\def\gc{general relativistic}
\def\orr{orbital residuals}
\def\iot{in order to}
\def\du{due to}
\def\bcs{because}
\def\nlg{\O^{\rm I}}
\def\nlgg{\O^{\rm II}}
\def\nel{\O^{\rm E1}}
\def\nee{\O^{\rm E2}}
\def\naj{\O^{\rm Aj}}
\def\nst{\O^{\rm St}}
\def\nlr{\O^{\rm LR}}
\def\dsl{\d\dot\O^{\rm Sl}}
\def\dws{\d\dot\O^{\rm WS}}
\def\dlg{\d\dot\O^{\rm I}}
\def\dlgg{\d\dot\O^{\rm II}}
\def\de{\d\dot\O^{\rm E1}}
\def\dee{\d\dot\O^{\rm E2}}
\def\daj{\d\dot\O^{\rm Aj}}
\def\dst{\d\dot\O^{\rm St}}
\def\dlr{\d\dot\O^{\rm LR}}
\def\peri{\d\dot\og^{\rm II}}
\def\per{\d\dot\og^{\rm I}}
\def\perir{\dot\og^{\rm II}_{\rm GR}}
\def\prir{\dot\og^{\rm I}_{\rm GR}}
\def\osp{e a\dot\og}
\def\dnd{\d\O}
\def\cmb#1#2#3{{#1+c_{\rm 1} #2 + c_{\rm 2} #3}}
\def\cmbb#1#2#3#4{{#1+u_1 #2 + u_2 #3 + u_3 #4}}
%-------------------------------------------
\def\a{\alpha}
\def\b{\beta}
\def\ga{\gamma}
\def\d{\delta}
\def\ve{\varepsilon}
\def\e{\epsilon}
\def\z{\zeta}
\def\et{\eta}
\def\th{\theta}
\def\vth{\vartheta}
\def\io{\iota}
\def\k{\kappa}
\def\lm{\lambda}
\def\m{\mu}
\def\n{\nu}
\def\x{\xi}
\def\o{\omicron}
\def\vp{\varpi}
\def\p{\pi}
\def\r{\rho}
\def\vr{\varrho}
\def\s{\sigma}
\def\vs{\varsigma}
\def\t{\tau}
\def\u{\upsilon}
\def\f{\phi}
\def\vf{\varphi}
\def\ch{\chi}
\def\ps{\psi}
\def\og{\omega}
\def\G{\Gamma}
\def\D{\Delta}
\def\TH{\Theta}
\def\L{\Lambda}
\def\X{\Xi}
\def\S{\Sigma}
\def\P{\Pi}
\def\U{\Upsilon}
\def\F{\Phi}
\def\PS{\Psi}
\def\O{\Omega}
\def\to{$T_{\rm obs}$}
\def\ppn{\rp{2+2\ga-\b}{3}}
%\def\cst#1#2{{\eqia c_{\rm 1}&=&{#1},\\ c_{\rm 2}&=&{#2}.\eqfa}}
%\def\cstt#1#2#3{\eqia c_1 &=,\\ c_2 &=& #2, \\ c_3 &=&#3\eqfa}
% FINE MACRO MIE

\begin{document}
\begin{titlepage}
\begin{flushright}
\today\\
BARI-TH/00\\
\end{flushright}
\vspace{.5cm}
\begin{center}
{\LARGE Measuring the Relativistic Perigee Advance with Satellite
Laser Ranging} \vspace{.5cm}
\quad\\
{Lorenzo Iorio$^{\dag}$, Ignazio Ciufolini$^{\ddag}$, Erricos C. Pavlis$^{*}$\\
\vspace{0.3cm}
\quad\\
{\dag}Dipartimento di Fisica dell' Universit{\`{a}} di Bari, via
Amendola 173, 70126, Bari, Italy\\ \vspace{0.5cm}
\quad\\
{\ddag}Dipartimento di Ingegneria dell'Innovazione
dell'Universit{\`{a}} di Lecce, via per Arnesano, 73100, Lecce,
Italy
\\
\vspace{0.3cm}
\quad\\
{*} Joint Center for Earth Systems Technology, JCET/UMBC and NASA
Goddard Space Flight Center, Greenbelt, Maryland, 20771-0001
U.S.A.}\\ \vspace{.5cm}

{\bf Abstract\\}
\end{center}
{\noindent \footnotesize The pericenter advance of a test body by
a central mass is one of the classical tests of General
Relativity. To day this effect is measured with radar ranging by
the perihelion shift of Mercury and other planets, in the
gravitational field of Sun, with a relative accuracy of the order
of $10^{\rm -2}$--$10^{\rm -3}$. In this paper we explore the
possibility of a measurement of the pericenter advance in the
gravitational field of Earth by analyzing the laser-ranged data of
some orbiting, or proposed, laser-ranged geodetic \sat s. Such a
measurement of the perigee advance would place limits on
hypothetical, very weak, Yukawa-type components of the
gravitational interaction with a finite range of the order of
$10^4$ km. Thus, we show that, at the present level of knowledge
of the orbital perturbations, the relative accuracy, achievable
with suitably combined orbital elements of LAGEOS and LAGEOS II,
is of the order of $10^{\rm -3}$. With the corresponding measured
value of $\ppn$, by using $\et=4\beta-\gamma-3$ from Lunar Laser
Ranging, we could get an estimate of the PPN parameters $\gamma$
and $\beta$ with an accuracy of the order of $10^{-2}-10^{-3}$.
Nevertheless, these accuracies would be substantially improved in
the near future with the new Earth gravity field models by the
CHAMP and GRACE missions. The use of the perigee of LARES (LAser
RElativity Satellite), with a suitable combination of orbital
residuals including also the node and the perigee of  LAGEOS II,
would also further improve the accuracy of the proposed
measurement.}

\end{titlepage} \newpage \pagestyle{myheadings} \setcounter{page}{1}
\vspace{0.2cm}
\baselineskip 14pt

\setcounter{footnote}{0}
\setlength{\baselineskip}{1.5\baselineskip}
\renewcommand{\theequation}{\mbox{$\arabic{equation}$}}
\noindent

\section{Introduction}
The well known secular shift of the pericenter
$\omega$ of a test body induced by the Schwarzschild metric generated
by a static, spherically symmetric distribution of
mass [{\it Misner et al.,} 1973; {\it Ciufolini and
Wheeler,} 1995] is one of the classical tests of
Einstein's General Relativity.
\subsection{The Solar System tests}
The advance of pericenter, which can be expressed in terms of
$\ppn$, where $\beta$ and $\gamma$ are the
Eddington--Robertson--Schiff PPN parameters [{\it Will}, 1993],
has been measured in the Solar System, on the Mercury perihelion,
by radar signals transmitted from Earth to Mercury and back to
Earth [{\it Shapiro et al.,} 1972], yielding $\ppn=1.005\pm
7\times 10^{\rm -3}$. The contribution of possible systematic
errors raises the uncertainty to about $2\times 10^{\rm -2}$.
 From the analysis of ten years of data the relative error
published in [{\it Shapiro et al.}, 1976] amounts to 5$\times
10^{-3}$.
%***IORIO CHECK FOR MORE RECENT MEASUREMENTS IN THE PAPER WILL-2001
%AND ALSO CHECK THESE NUMBERS***
The major sources of systematic error in this measurement are the
poorly-known variations in the topography of the planet's surface
and the uncertainties in the radar scattering law [{\it Shapiro,}
1990; {\it Pitjeva}, 1993]. For a review of periastron advance
measurements on binary pulsars see [{\it Will}, 2001] and
references therein.

%yield an accuracy level $<10^{\rm -3}$.

%Unfortunately, the interpretation of the perihelion advance of
%Mercury as a test of general relativity is complicated by the
%uncertainty in the mass quadrupole moment of the Sun which also
%contributes to the perihelion advance. From the expression of the
%perihelion precession and from the 1976 experimental uncertainties
%in $\dot \omega$, one can easily calculate that for Mercury
%orbiting the Sun, any value of $J_{\rm 2 \odot}$ larger than about
%3$\times 10^{\rm -6}$ would disagree with the general relativistic
%prediction of 42.98 arcseconds/century; indeed, according to some
%authors measured values of $J_{\rm 2 \odot}$ may be as large as $
%\sim 5.5\times 10^{\rm -6}$ (for a comprehensive discussion see
%[{\it Ciufolini and Wheeler}, 1995; {\it Pireaux et al.,} 2001]).
%Laboratory experiments have been also proposed [{\it Cacciani et
%al.,} 1989] to measure the solar mode $l = 1$, to get informations
%on the rotation of the core of the Sun from the $l = 1$ rotational
%frequency splitting. Nevertheless, according to more recent
%determinations of $J_{\rm 2 \odot}$ using the analysis of the
%Sun's pressure modes, both from the ground-based network of
%observatories and the space based SOHO, $J_{\rm 2 \odot }=(2.3\pm
%0.1)\times 10^{-7}$ [{\it Shapiro}, 1999]; in [{\it Godier and
%Rozelot,} 2000] a theoretical estimate of $J_{\rm 2 \odot
%}=(2.0\pm 0.4)\times 10^{-7}$ is reported. Thus, according to
%these values of $J_{\rm 2 \odot }$ the observed perihelion advance
%of Mercury is well in agreement with the general relativistic
%predictions.

Several space missions have been proposed in order to measure,
among other effects, this phenomenon by using a variety of techniques:
the most recent and promising are SORT, IPLR and ASTROD [{\it Ni},
2001].
%***IORIO CHECK LATEST NUMBERS OF J2 IN THE
%WILL 2001 PAPER, ADD THE REFERENCES FROM MY BOOK, AND CHECK WITH
%CACCIANI, PHYSICS DEP. UNIV. LA SAPIENZA ROMA, MY FRIEND, TO KNOW
%THE STATUS OF THE ART OF J2 MEASUREMENTS***

More accurate measurements of $\ppn$ might be performed in the
future by means of the ESA BepiColombo Mercury orbiter [{\it
Balogh et al.,} 2000; {\it Milani et al.,} 2001a; {\it Milani et
al.}, 2001b]. In [{\it Milani et al.}, 2001b] it is claimed that
the BepiColombo mission, which should be launched in 2009, should
allow to measure $\gamma$ and $\beta$ with a relative accuracy of
the order of $2\times 10^{-6}$.
%\subsection{The binary pulsar measurement}
%A recent important test of General Relativity, performed in a
%strong--field astrophysical context, is undoubtedly represented,
%among other things, by the detection of the periastron advance in
%the binary pulsar PSR B1534+12: see [{\it Stairs et al.}, 1998]
%and the references therein. The measurements done up to now would
%agree with the values predicted by General Relativity to $2\times
%10^{-5}$. Further improvements should come from the upgrade of the
%Arecibo Radio Telescope and from the possible discovery of new
%binary pulsars.
%
%However, there is not a general consensus on the quoted accuracy
%as representative of a truly test of General Relativity because
%the orbital parameters of the system such as the masses of the
%pulsar and its companion, the semimajor axis of the orbit, the
%eccentricity, etc. are not all measured to the $2\times 10^{-5}$
%precision. So, the least precisely measured parameter among all
%those which enter the formula of the precession do indeed control
%the precision of this experiment as a test of General Relativity.
%Then, the precision, in the end, could boil down to about
%$10^{-2}$ or slightly better.
\subsection{The proposed experiment}
In this paper we propose to perform a measurement of the general
relativistic ``gravitoelectric" pericenter advance in the
gravitational field of the Earth by using some suitable
combinations of the orbital residuals of existing or proposed,
spherical, passive, geodetic laser-ranged satellites [{\it Iorio},
2002a] with particular emphasis to \lg\ and \lgg\, in order to
exploit the relevant experience obtained with the \grc\
Lense-Thirring measurements in [{\it Ciufolini et al.}, 1996;
1997; 1998]. In [{\it Ciufolini and Matzner,} 1992] a
determination of the LAGEOS general relativistic gravitoelectric
perigee shift in the field of the Earth is reported, but the
accuracy amounts only to $2\times 10^{\rm -1}$.
%......................................................................................

%In order to test general relativity versus alternative metric
%theories of gravitation, following a method introduced by
%Eddington, Robertson and Schiff, one can expand the spacetime
%metric (which describes the gravitational interaction) at the
%order beyond Newtonian theory (post-Newtonian) and then multiply
%each post-Newtonian term by a dimensionless parameter to be
%experimentally determined. Using this method, Nordtvedt and Will
%[{\it Will}, 1993] have developed the PPN
%(Parameterized-Post-Newtonian) formalism, a powerful and useful
%tool for testing general relativity and alternative metric
%theories. Therefore, the PPN formalism is a post--Newtonian
%parameterized expansion of the metric tensor ${\bf g}$ and of the
%energy--momentum tensor ${\bf T}$ in terms of small known
%classical potentials.

%In the slow-motion and weak-field approximation, at the so-called
%post-Newtonian limit, the structure of known metric theories of
%gravity is identical apart from the numerical values of the PPN
%parameters which appear in the expansion of the metric
%coefficients [{\it Will,} 1993; {\it Ciufolini and Wheeler,} 1995;
%{\it Will,} 2001]. Especially meaningful are the parameters
%$\beta$ and $\ga$, i.e the usual Eddington-Robertson-Schiff
%parameters used to describe the ``classical" tests of \gr. $\ga$
%measures how much space curvature is produced by unit rest mass
%and in the standard parameterized post-Newtonian gauge $\b$
%accounts for the level of nonlinearity in the superposition law
%for gravity [{\it Misner et al.}, 1973]. In general relativity
%$\beta = 1$ and $\ga = 1$. In [{\it Damour and Nordvedt}, 1993a;
%{\it Damour and Nordvedt}, 1993b] it has been pointed out that
%there are theoretical arguments, related to scalar--tensor
%extensions of general relativity with cosmological implications as
%well, for the necessity of constraining $\gamma$ and $\beta$ well
%beyond the $10^{-4}$ level. However, it must pointed out that, as
%we will show below, the accuracies obtained up to now in various
%experiments using different techniques, including the proposed
%analysis here, are situated in the $10^{-3}-10^{-4}$ region.

%***IORIO controlli che i valori dati in questo paragrafo siano
%aggiornati agli ultimi 2000-2001 esperimenti***
%To-day a very accurate measurement of $\ga$ was performed by using
%the Viking time delay [{\it Reasenberg et al.,} 1979]: the result
%was $\ga=1.000\pm 2\times 10^{\rm -3}$. The quoted uncertainty
%allows for possible systematic errors. A more recent measurement
%based on the time delay of the NEAR spacecraft [{\it Elliott et
%al.}, 1998] claims an uncertainty $\leq 10^{\rm -3}$. Other recent
%measurements exploit the gravitational bending of the
%electromagnetic waves at various wavelengths [{\it Robertson and
%Carter,} 1984; {\it Robertson et al.,} 1991; {\it Fr\oe
%schl{\'{e}} et al.,} 1997] obtaining an uncertainty of the order
%of $10^{\rm -3}$; e.g., according to the VLBI radio analysis by
%[{\it Lebach et al.}, 1995] $\ga=0.9996\pm1.7\times 10^{-3}$,
%while in [{\it Fr\oe schl{\'{e}} et al.,} 1997] $\ga=0.997\pm
%3\times 10^{\rm -3}$ is quoted based on the astrometric
%observations of the electromagnetic waves deflection in the
%visible. An improvement in the accuracy of two orders of magnitude
%[{\it Fr\oe schl{\'{e}} et al.,} 1997] is expected from the future
%GAIA astrometric mission [{\it GAIA,} 2000]. Lunar laser ranging
%(LLR) measurements of the geodetic precession yields for $\ga$ an
%accuracy of $10^{\rm -2}$ [{\it Williams et al.,} 1996]. However a
%careful discussion of the different terms in the lunar motion
%yields eventually $\ga=1.000\pm 5\times 10^{\rm -3}$ as the
%present LLR result [{\it Fr\oe schl{\'{e}} et al.,} 1997]. It must
%pointed out that, as we will show below,
%

%The value of the parameter $\b$, which cannot be measured
%independently, can be obtained from the measured values of $\ga$
%and of some combinations of $\ga$ and $\b$: $\et=\nvd$ and $\ppn$
%are the most accurately determined. General relativity predicts
%$\et=0$ and $\ppn=1$. For $\beta$ in [{\it Anderson and Williams},
%2001] by using LLR data $\delta\beta=4.7\times 10^{\rm -4}$ is
%reported.
%%There
%%exists also other combinations sensitive to features of Moon and
%%Mars motions, but neither the accuracy of the performed or future
%%experiments is so high as for $\et$ and $\ppn$ [{\it Eubanks et
%%al.,} 1999] nor the sensitivity to such combinations is
%theoretically well understood
%***IORIO questa ultima frase e'assolutamente oscura e va chiarita o tolta***
%[{\it Williams et
%al.,} 1996].


%The combination $\et$ measures the possible violation of the
%strong equivalence principle, i.e. the so-called Nordtvedt effect
%[{\it Nordtvedt,} 1968; 1991]. $\et$ has been measured by analyzing
%the motion of the Moon via LLR data analyses [{\it Shapiro et
%al.,} 1976; {\it Williams et al.,} 1996; {\it Anderson and
%Williams,} 2001]. The most recent determination of $\et$ by LLR is
%$\et=0.0002\pm 8\times 10^{\rm -4}$ [{\it Anderson and Williams,}
%2001].
%***IORIO check***.

This paper is organized as follows: in section 2 we compare the
present experimental accuracy in satellite laser-ranging
measurements with the relativistic expression of the perigee
shift of the two \lg\ satellites. In section 3 we analyze some of
the most important sources of systematic errors. The accuracy
achievable in measuring $\beta$ and $\gamma$ is also discussed.
%***IORIO spiegare queste ultime
%parole meglio con una frase aggiuntiva***.
By detecting the relativistic perigee rate of the proposed LARES
[{\it Ciufolini}, 1998] laser-ranged satellite, in conjunction
with the node and the perigee of LAGEOS II, it might be possible
to further improve the accuracy in the measurement of these
parameters over a long enough time span. This is the subject of
section 4 together with an analysis of the potential role of the
existing laser-ranged satellite Starlette. Section 5 is devoted to
the conclusions.
%----------------------------------
\section{The relativistic perigee precession of LAGEOS type satellites}

As known, in the slow-motion and weak-field approximation, the
Schwarzschild metric generated by a static, spherically symmetric
distribution of mass-energy induces an additional post-Newtonian
``gravitoelectric force" which acts on the orbit of a test body by
shifting its pericenter; in the PPN formalism the pericenter rate
can be written as: \eqi \dot\og_{\rm GE}=\rp{3 n G M}{c^2 a
(1-e^2)}\times\ppn,\lb{peg}\eqf in which $G$ is the Newtonian
gravitational constant, $c$ is the speed of light in vacuum,
$M$ is the mass of the central object, $a$ and $e$ are
semimajor axis and eccentricity, respectively, of the orbit of
the test body and $n=\sqrt{GM/a^{\rm 3}}$ is its mean motion. In
the following we define $\n\equiv \ppn$. General Relativity
predicts that the perigee shifts for \lg\ and \lgg\ amount to
3,312.35 milliarcseconds per year (mas/y in the following) and
3,387.46 mas/y respectively.

The actual experimental precision allows for detecting such rates
for both the \lg\ satellites [{\it Ciufolini,} 1996]. Indeed, for
the perigee the observable quantity is $r \equiv ea\dot\og$; at
present the RMS of the residuals of the best orbital fits amounts
to $\d r_{\rm exp} \sim 1$ cm, or less, for the two LAGEOS
satellites, over several orbits and with a set of force models
(i.e. not including modeling errors). Since the \lg\ eccentricity
is $e_{\rm I}=4.5\times 10^{\rm -3}$ the accuracy in detecting the
perigee is $\d\og^{\rm I}_{\rm exp} \sim \d r_{\rm exp}/(e_{\rm
I}a_{\rm I}) \simeq 37$ mas. So that, over 1 year the relative
accuracy in the measurement of the relativistic perigee shift
would be of the order of $ 1 \times 10^{\rm -2}$. For \lgg\ this
measurement accuracy is better than for \lg. Indeed, the \lgg\
eccentricity is $e_{\rm II}=1.4\times 10^{\rm -2}$, thus
$\d\og^{\rm II}_{\rm exp} \sim 12$ mas; this may yield an accuracy
of the order of $3\times 10^{\rm -3}$ over 1 year. Over 8 years,
by assuming the same fit error as before, it would amount to $
\sim 4\times 10^{-4}$. These considerations rule out the
possibility of directly using the perigee of the other existing,
spherical, geodetic laser-ranged satellites Etalon-1, Etalon-2,
Ajisai, Stella, Westpac-1 because their eccentricities are even
smaller than that of \lg. On the contrary, Starlette has an
eccentricity of the order of $2\times 10^{\rm -2}$; however, since
it orbits at a lower altitude is more sensitive than the \lg\
satellites to atmospheric drag and to Earth's zonal harmonics, so
that would be difficult to process its data at an acceptable level
of accuracy (See also section 4). Accordingly, in order to detect
the gravitoelectric relativistic shift in the gravitational field
of Earth, we will focus on the perigee of \lg\ and especially of
\lgg\ .
%-------------------------------------------------------
\section{The systematic errors}
The perigee of an Earth satellite is a "non-clean" orbital
element in the sense that it is affected by a large number of
aliasing classical forces inducing systematic errors which must
be carefully identified and analyzed.
\subsection{The static geopotential error}
The most important source of systematic errors in such a
measurement is represented by the mismodeling of the classical
perigee precession induced by the errors in even zonal harmonics
of the Earth's gravitational field [{\it Kaula}, 1966]. This error
is really critical because the resulting aliasing trend cannot be
removed from the data and nothing can be done about it apart from
assessing the related error as reliably as possible.

By using the covariance matrix of the EGM96 Earth gravity model
[{\it Lemoine et al.,} 1998] and adding the correlated terms in a
root-sum-square fashion up to degree $l=20$, we obtain for \lg\ a
systematic error $\d\n/\n_{\rm zonals}=8.1\times 10^{\rm -3}$
whereas for \lgg\ we have $\d\n/\n_{\rm zonals}=1.5\times 10^{\rm
-2}$. Since the major source of error lies in the uncertainty of
the first two mismodeled even zonal harmonics $\d J_2$ and $\d
J_4$, in order to eliminate most of the static and dynamical even
zonal terms of the geopotential, following the method in [{\it
Ciufolini,} 1996] for the measurement of the \grc\ \lt\ effect, we
search for suitable combinations of the orbital residuals of the
existing laser-ranged satellites. In Tab.\ref{combi} we report the
most promising combinations: their general form is \eqi
\d\dot\og^{\rm II}+\somma{i=1}{N}c_{\rm i}\d\dot\O^{\rm i}+c_{\rm
N+1}\d\dot\og^{\rm I}=x_{\rm GR}\ \n,\lb{zione}\eqf in where $N$
is the number of the nodes of different laser-ranged satellites
employed, $x_{\rm GR}$ is the slope, in mas/y, of the
relativistic trend to be measured, in Tab.\ref{combi}
$\d\n/\n_{\rm zonals}$ is the systematic error induced by the
even zonal harmonics up to degree $l=20$ calculated with the
EGM96 covariance matrix. The coefficients $c_i$ are determined in
terms of the orbital parameters of the satellites entering the
combinations.

\begin{table}[ht!]
\caption{Combined residuals: numerical values}\label{combi}
\begin{center}\begin{tabular}{lllllll}\noalign{\hrule height 1.5pt}
& $\O^{\rm II}$ & $\O^{\rm I}$ & $\O^{\rm Aj}$ & $\og^{\rm
I}$\\
\hline $\k$ & $c_1$ & $c_2$ & $c_3$ & $c_4$ & $x_{\rm GR}$
(mas/y)
& $\d\n/\n_{\rm zonals}$
\\
\hline

{\sl 1} & $-0.87$ & $-2.86$ & 0 & 0 & 3,387.46 &
$6.59\times 10^{\rm -3}$ \\

{\sl 2} & $-2.51$ & $-4.37$ & 0 & 2.51 & 11,704.92 &
$1.1\times 10^{\rm -3}$\\

{\sl 3} & $-1.96$ & $-3.69$ & $0.037$ & $1.37$ & 7,928.51
&
$8.1\times
10^{\rm -4}$\\

\hline \noalign{\hrule height 1.5pt}
\end{tabular}
\end{center}
\end{table}


It is important to stress that the use of the LAGEOS satellites,
due to their altitude, makes our measurement substantially
insensitive to the errors in the zonal harmonics of degree $l>20$,
so that our estimates of $\d\n/\n_{\rm zonals}$ presented here are
valid even in the case that the EGM96 covariance matrix for
degrees higher than $l>20$ would not be accurate enough.

In addition to \lg\ and \lgg\ we have only considered Ajisai [{\it
Iorio}, 2002a] since it is well tracked, contrary for example to
the Etalon satellites, and it would be less demanding than for the
other satellites to reduce its laser-ranged data to a level of
accuracy comparable to that of \lg\ and \lgg. Moreover, the other
laser-ranged \sat s orbit at lower altitudes, therefore they are
more sensitive to terms of the geopotential of higher degree.
Consequently, as confirmed by numerical calculations, the
inclusion of their data in the combined residuals would increase
$\d\n/\n_{\rm zonals}$\footnote{About combination \textsl{3}, it
should be noted that, in order to obtain more reliable and
accurate estimates of the systematic error due to the even zonal
harmonics of the geopotential, the calculation of the error should
include terms with degree higher than $l=20$ due to the
sensitivity of Ajisai to such higher degree terms. }.

It is important to point out that the uncertainties quoted in
Tab.\ref{combi} for $\d\n/\n_{\rm zonals}$ will be reduced in the
near future when the new gravity models from the CHAMP and GRACE
missions will be released [{\it Pavlis}, 2002a].
%indeed, it has been claimed an improvement in the accuracy of the
%lower even zonal harmonics of Earth field by about 2 orders of
%magnitude.
%***IORIO check that the claimed improvement of about
%100 is I believe only for the lower harmonics, also put a
%reference about it!***
%--------------------------------------
\subsection{The time--dependent systematic errors}
In regard to the evaluation of the impact of the other sources of
systematic errors, the role of the coefficients $c_i$ entering
the combinations has to be taken into account, indeed, according
to their magnitude, they can reduce or emphasize the effects of
the perturbing forces acting on the orbital elements weighted by
them.

For example, for the combination \textsl{2} of Table 1, over 1
year, the impact of the error in measuring the perigee rate of
LAGEOS amounts to about $2.5\times(3.1\times 10^{\rm
-3})=7.7\times 10^{\rm -3}$ while for combination \textsl{3} it
is about $1.37\times(4.7\times 10^{\rm -3})=6.4\times 10^{\rm
-3}$.

In this paper we only consider in detail the combination {\sl 1}
of Table 1
  \eqi\d\dot\og^{\rm II}-0.87\times
\d\dot\O^{\rm II}-2.86\times\d\dot\O^{\rm I}=3,387.46\times\n
,\lb{rsd}\eqf so that we exploit the insight acquired with the \lt\
\lg\ experiment [{\it Ciufolini et al.,} 1996; 1997; 1998].

The long-period harmonic perturbations, according to their
periods $P$ and to the adopted observational time span $T_{\rm
obs}$, may turn out to be less insidious than the mismodeled
secular perturbations due to the zonal harmonics of the
geopotential since, if $P<T_{\rm obs}$ and $T_{\rm obs}=nP,\
n=1,2,...$ they average out; if their periods are shorter than the
time span they can be viewed as empirically fitted quantities
which can be subsequently removed from the signal.

The results and estimates recently obtained for the LAGEOS
gravitomagnetic experiment can be applied to our proposed
measurement. Indeed, the solid Earth and ocean tidal perturbations
on LAGEOS and LAGEOS II and their impact on the Lense--Thirring
effect have been studied in [{\it Iorio}, 2001; {\it Iorio and
Pavlis}, 2001; {\it Pavlis and Iorio}, 2002], while the role
played by the non--gravitational perturbations has been analyzed
in [{\it Lucchesi}, 2001; 2002]. Regarding the atmospheric drag,
whose impact might be thought of as a serious drawback, especially
for the perigee of LAGEOS II, in [{\it Ciufolini et al.}, 1997] it
is shown that it essentially averages out over an orbital
revolution.

In Tab.\ref{8anni} we summarize the results obtained with the
proposed combination using LAGEOS and LAGEOS II for a 8-year
time span with orbital fits, arcs, each of 7 days. In assessing the total
systematic error we have accounted for the fact that the
gravitational errors are not independent, indeed we have simply summed them
up; then we have added these uncertainties and the other independent
systematic errors in a root-sum-square fashion.
%In the first
%column we quoted the kind of error. In the second column there are
%the sources, in this paper, of the quoted numbers in the third
%column.
%The formal statistical error has been quoted independently.
%Moreover, the
%estimate of the statistical formal error has been obtained by
%considering the noise level in the LAGEOS data of 1998
%Lense-Thirring experiment.
In Tab.\ref{8anni} we can observe that the error budget is
mainly dominated by the even zonal harmonics and by the LAGEOS II
perigee measurement error.
%which has not a time--dependent
%signature (***???*** but, of course, depends on the data set chosen at a
%given time span***???***).
In regard to the non--gravitational errors, according to the
results of [{\it Lucchesi}, 2001] for the direct solar radiation
pressure and the Earth's albedo and to the results of [{\it
Lucchesi}, 2002] for the thermal thrust perturbations and the
asymmetric reflectivity, the corresponding uncertainty would
amount to almost $1\times 10^{-2}$ over 7 years. In obtaining this
result a very pessimistic approach has been adopted by assuming a
mismodeling of 20$\%$ for all the perturbing effects except for
the direct solar radiation pressure which has been assumed to have
an uncertainty at the 0.5$\%$ level. However, we stress that only
the Earth's thermal thrust, or Yarkovski-Rubincam effect, induces
a mismodeled linear trend whose impact would amount to about
$1\times 10^{-4}$: the other non--gravitational forces are
time--varying with known periodicities and can be fitted and
removed from the signal as done for the tidal perturbations.

\begin{table}[ht!]
\caption[PPN preliminar error budget: $T_{obs}=8$ years, $\D t=7$
days.]{Preliminary error budget: $T_{obs}=8$ years, $\D t=7$
days.}\label{8anni}
\begin{center}
\begin{tabular}{ll}\noalign{\hrule height 1.5pt}

Even zonal harmonics (par. 3.2) & $6.59\times 10^{-3}$  \\

$J_3$ geopotential [{\it Pavlis and Iorio}, 2002]& $3.2\times 10^{-4}$ \\

Tides [{\it Iorio}, 2001; {\it Pavlis and Iorio}, 2002] & $4.4\times
10^{-4}$ \\

Non-gravitational effects [{\it Lucchesi}, 2001; 2002] & 1$\times
10^{-4}$
\\

%Inclination & $7.6\times 10^{-4}$ \\

Measurement error in LAGEOS II perigee (sect. 2) & $4\times 10^{-4}$ \\

\hline

Total systematic error & $7.3\times 10^{-3}$\\

%\hline

%Formal statistical error & $5.7\times 10^{-4}$ \\

\hline \noalign{\hrule height 1.5pt}
\end{tabular}
\end{center}
\end{table}

Forthcoming improvements in the Earth gravity field modeling
achievable with the CHAMP and GRACE data have been recently
extensively studied and reported in [{\it Pavlis}, 2002a]. They will yield
important improvements in the accuracy of our experiment. We
finally observe that we already have the required laser-ranging
data since, at present, we have almost 10 years of LAGEOS and
LAGEOS II data.

\subsection{The parameters $\beta$ and $\gamma$}
The results obtained for the combination {\sl 1} examined here,
together with $\delta\eta=8\times 10^{-4}$ reported in [{\it
Anderson and Williams,} 2001] for the combination
$\et=4\beta-\gamma -3$ [{\it Nordtvedt,} 1968; 1991], would allow
to measure $\b=\rp{2}{7}\et+\rp{3}{7}\n+\rp{4}{7}$ at the level $
\sim 3\times 10^{\rm
-3}$ independently
of other measurements of $\ga$; this result, which is of the same
order of magnitude of that
obtained with radar ranging [{\it Shapiro}, 1990],
should be compared with the most recent $\d\b=4.7\times 10^{\rm
-4}$, obtained with LLR data [{\it Anderson and Williams,}
2001].

The parameter $\gamma$, that may be written as
$\gamma=\frac{1}{7}\eta+\frac{12}{7}\nu-\frac{5}{7}$, would be
measured less precisely: $\delta\gamma=1.2\times 10^{\rm -2}$.
However, $\gamma$ can be directly measured via light deflection or
radar time delay. For example, in [{\it Fr\oe schl{\'{e}} et al.,}
1997] the result $\ga=0.997\pm 3\times 10^{\rm -3}$ is based on
the astrometric observations of electromagnetic waves deflection
in the visible frequency. By using the Viking time delay [{\it
Reasenberg et al.,} 1979], $\ga=1.000\pm 2\times 10^{\rm -3}$ was
obtained; the quoted uncertainty allows for possible systematic
errors. A more recent measurement based on the time delay with
the NEAR spacecraft [{\it Elliott et al.}, 1998] claims an
uncertainty $\leq 10^{\rm -3}$.

For an updated overview of the present status of the measurements
of the PPN parameters, see [{\it Will}, 2001]

%***IORIO IN THIS FIGURE IS NOT CLEAR THE DIFFERENCE BETWEEN THE 2
%SOLID LINES***
%\begin{figure}[ht!]
%\begin{center} %\mbox{\epsfig{file=ckm.eps,width=8cm,height=8cm}}
%\includegraphics*[width=15cm,height=12cm]{errppn.eps}
%\end{center}
%\caption{\footnotesize Systematic errors induced by the major
%harmonic perturbations. The horizontal solid line is the error due
%to the mismodeled even zonal harmonics of the geopotential. }
%\label{figura1}
%\end{figure}

%For different time spans $T_{\rm obs}$, we have simulated with
%MATLAB the time series of the mismodeled harmonic part of the
%signal by including the solid Earth and ocean tides
%[\textit{Iorio,} 2001], the $l=3$ Earth's zonal harmonic
%perturbation of the perigee of \lgg\ and the solar radiation
%pressure [{\it Lucchesi}, 2001]\footnote{In the present error
%budget we have neglected other subtle non-gravitational
%perturbations acting on the LAGEOS orbits (Earth's albedo, Earth's
%direct IR radiation pressure, thermal effects) whose effects on
%the perigee of LAGEOS II are currently under detailed
%consideration. Anyway, their impact on our measurement should be
%at the level of $10^{-3}$ or less.}. Subsequently, at fixed
%$T_{\rm obs}$, we have performed 5,000 runs by varying the initial
%phases so to account for the different cases occurring in the real
%world, and for each run at fixed $T_{\rm obs}$ we have calculated
%$\left\langle\d \O^{\rm I}\right\rangle_{ T_{\rm obs}}$,
%$\left\langle\d \O^{\rm II}\right\rangle_{ T_{\rm obs}}$ and
%$\left\langle\d \og^{\rm II}\right\rangle_{ T_{\rm obs}}$ in mas.
%We have then inserted them in \rfr{rsd} and we have taken the
%ratios of the averages of the long-periodic systematic errors
%obtained in this way with the general relativistic shift for the
%given time span. In this way we have obtained a vector of 5000
%figures, for any given $T_{\rm obs}$, depending on the initial
%phases. Such values turn out to be normally distributed, so that
%we have chosen their standard deviation to represent $\d\n/\n_{\rm
%harmonics}$ for the various $T_{\rm obs}$ chosen. Note that over 5
%years we have $\d\n/\n_{\rm harmonics}=1.4\times 10^{\rm -3}$
%while over 8 years we have $\d\n/\n_{\rm harmonics}=6.7\times
%10^{\rm -4}$.
%---------------------------------
\section{The role of LARES and Starlette}
The proposed LARES satellite [{\it Ciufolini}, 1998] will enable us to
perform several general relativistic tests and geophysical
measurements. In this section we investigate its role for the
measurement of the gravitoelectric perigee advance that would be
well detectable on LARES thanks to its proposed relatively large
eccentricity: $e_{\rm LR}\simeq0.04$.

A possible observable is the following combination of
orbital residuals \eqi\delta\dot\omega^{\rm
II}+c_1\delta\dot\omega^{\rm LR}+c_2\delta\dot\Omega^{\rm
II}=x_{\rm GR}\nu,\lb{lares}\eqf with \bea
c_1 & = & -4.71,\\
c_2 & = & 2.26,\\
  x_{\rm GR} & = & -12,117.4\ {\rm mas/y}. \eea The coefficients
$c_i$ depend on the orbital parameters of the selected satellites
and have been calculated corresponding to the proposed orbital
parameters of LARES: $a_{\rm LR}=12,270$ km, $i_{\rm LR }= 70$
deg and $e_{\rm LR}=0.04$. Let us now calculate the systematic
errors induced by the even zonal harmonics of the geopotential,
according to the EGM96 gravity model. \Rfr{lares} would allow to
cancel out the errors due to the first four even zonal harmonics;
the impact of the remaining ones, with degree higher than four,
amounts to an uncertainty of about $6\times 10^{-3}$. This error
will further reduce when the new gravity models from the CHAMP
and GRACE missions will be available. An interesting feature of
\rfr{lares} is that the impact of the non--gravitational
perturbations would produce an uncertainty of about $3.9\times
10^{-3}$ over a time span of 7 years, according to the very
conservative estimates discussed above; in particular the Earth
thermal thrust, i.e. the so-called Yarkovski--Rubincam effect,
would induce a mismodeled secular trend with an uncertainty of
only $6.4\times 10^{-5}$. The inclusion of LARES would thus
represent an improvement with respect to the LAGEOS and LAGEOS II
scenario previously outlined, especially when, in the near future,
the impact of the mismodeled non--gravitational perturbations will
increase relatively to the gravitational perturbations thanks to
the more accurate Earth's gravity field models.

   Among the existing laser-ranged satellites, Starlette, with its
   relatively large eccentricity of 0.0204, seems to be a natural candidate
   for the measurement of the perigee advance. Unfortunately,
   it orbits at an altitude much lower than the LAGEOS
   satellites ($a_{\rm Str}=7,331$ km) and thus the mismodeling of the
even zonal harmonics
   of the geopotential would induce large systematic errors in
   the measurement of the perigee rate. It turns out that the inclusion of the
   perigee of Starlette in the combined residuals if, on the one hand,
provides a
   further observable and thus would allow to
   cancel out the error of an additional mismodeled even zonal harmonic,
   on the other hand it raises the total error due to
   the geopotential, even with the LARES data.
   Indeed, in a combination including the nodes of LAGEOS and LAGEOS
   II and the perigees of LAGEOS II and Starlette, the relative error
   due to the geopotential amounts to only $6.2\times 10^{-2}$. This
rules out the possibility
   of using the Starlette data, at present level of knowledge of the Earth's gravitational field,
   to improve the measurement
of the relativistic perigee  advance.

%-------------------------------------------------------------
\section{Conclusions} The main features of our proposed
measurement of perigee advance using laser-ranged satellites are
the following:
\begin{itemize}
   \item It will provide a test of a general
   relativistic effect, i.e. the pericenter advance, {\it in the gravitational
   field of Earth}, complementary to similar tests in the field of Sun.
   Indeed, such a measurement in the gravitational
   field of Earth would place stronger limits
   on possible, very weak, Yukawa-type components of the
   gravitational interaction with a finite range of the order of $10^4$ km [{\it Iorio}, 2002b].
   \item For a suitable combination of the orbital residuals of the
   nodes of LAGEOS and LAGEOS II and perigee of LAGEOS II,
   the relative error due to the mismodeled classical precessions by
the static even zonal
   harmonics of the geopotential,
   which is the most insidious source of systematic
   error, amounts, according to the EGM96 Earth gravity
   model, to about $6\times 10^{-3}$. The total error in our proposed
combination of orbital
   elements, also due to the other time--dependent systematic errors,
   amounts, over an observational time span of eight years, to almost $7\times
   10^{-3}$.
   \item In the near  future the knowledge of the Earth gravitational
field will increase   thanks to the
   data from the CHAMP and GRACE missions. Moreover, the rms accuracy
   in measuring the position of the LAGEOS satellites should reach the few
   mm level. Consequently, there will be an increase in the
   accuracy of our measurement.
   %\item In regard to the precision of the measurements of $\gamma$ and
   %$\beta$, other future space missions, like GAIA and BepiColombo,
   %claim accuracies almost two orders of magnitude larger than thatavailable with the proposed experiment and
   %the CHAMP and GRACE data.
   %However,
   %such missions will be implemented in many years (for example, the
   %BepiColmbo Mercury orbiter is planned to be launched
   %not before than
   %2009), are expensive and complex. The proposed analysis,
   %instead, would be comparatively low--cost and low--time
   %consuming. Moreover, with the data from CHAMP and GRACE it soon
   %would improve of almost one order of magnitude the present--day
   %available accuracy in $\beta$ and $\gamma$
   \item  In regard to the terrestrial space
    environment, which induces a large number of orbital perturbations
  on the satellites,
  it is very well studied and the action of these perturbations,
    especially on the LAGEOS satellites, is very well modeled. The
data processing would be reliably based on
   the large experience collected in several years of analysis of
   the LAGEOS orbits [{\it Pavlis}, 2002b].
    Moreover,
  it would be relatively fast and easy to include the future
improvements in the force modeling
    in our analysis, e.g. the improvements due to a better
experimental and theoretical
    knowledge
    of the rotation rate and spin axis orientation of LAGEOS type satellites.
    \item The proposed measurement, as an additional outcome, from $\ppn$ in
    conjunction with $\et=4\beta-\gamma-3$ from Lunar Laser Ranging,
would allow to obtain
    the PPN
    parameters $\gamma$ and $\beta$ in the gravitational field of
Earth (see above) with an accuracy at the level of $10^{-2}-10^{-3}$.
    This accuracy would  improve with
    the new Earth's gravitational models from CHAMP and GRACE.
    \item Starlette data, although this satellite has a relatively
eccentric orbit, should not
   be included in the combined residuals. Indeed, Starlette
   would raise the systematic error in the geopotential due to its
lower altitude with respect to the
   LAGEOS satellites
   and thus due to its higher sensitivity to the even zonal
   harmonics of the terrestrial gravitational field.
   \item The use of the perigee of the proposed LARES satellite, for a
suitable combination
   of observables including also the node
   and the perigee of LAGEOS II, would
   allow to reduce the systematic relative error due to the mismodeled even
   zonal harmonics of the geopotential to about 6$ \times 10^{-3}$,
according to EGM96.
   This proposed measurement would also take advantage of the new
Earth gravity models from CHAMP and GRACE.
   According to conservative
   estimates, the impact of the non--gravitational perturbations would
amount to about $4\times 10^{-3}$
   over 7 years; in particular the impact of the mismodeled Earth
thermal thrust, or Yarkovski-Rubincam
   effect,
   would amount to only about $6\times 10^{-5}$.
   \end{itemize}
%-----------------------------------------
\section*{Acknowledgements}L. I. would like to thank L. Guerriero for
his encouragement and support, C.M. Will and I.I. Shapiro for the
useful information kindly supplied to him.
E. C. Pavlis gratefully acknowledges NIMA support through the NURI
grant NMA201-01-1-2008.
%-----------------------------------------
\begin{thebibliography}{Dillo83}
\addcontentsline{toc}{chapter}{References}
%-------------------------------------
\bb[{\it Anderson and Williams}, 2001]{andw} Anderson, J.D., and
J.G. Williams, Long-range tests of the equivalence principle, {\it
Class. and Quantum Grav., 18}, 2447-2456, 2001.
%--------------------------------------------------------------
\bb[{\it Balogh et al.,} 2000]{bal} Balogh, et al., BepiColombo An
Interdisciplinary Cornerstone Mission to the Planet Mercury,
\textit{ESA-SCI, 1,} pp. 21-24 and pp. 35-38, 2000.
%--------------------------------------------------------------
\bb[{\it Ciufolini and Matzner}, 1992]{ciufmat} Ciufolini, I., and
R. Matzner, Non-Riemannian theories of gravity and lunar and
satellite laser ranging, {\it Int. J. of Mod. Phys. A, 7, 4},
843-852, 1992.
%--------------------------------------------------------------
\bb[{\it Ciufolini and Wheeler}, 1995]{ciuf1} Ciufolini, I., and
J. A. Wheeler, {\it Gravitation and Inertia}, 498 pp.,
Princeton University Press, New York, 1995.
%------------------------------------
\bb[{\it Ciufolini}, 1996]{ciuff} Ciufolini, I., On a new method to
measure the
gravitomagnetic field using two orbiting satellites,
{\it Il Nuovo Cimento, 109A, 12}, 1709-1720, 1996.
%------------------------------------------
\bb[{\it Ciufolini et al.}, 1996]{ciufn} Ciufolini
I., D. Lucchesi, F. Vespe and A. Mandiello,
Measurement of
dragging of inertial
frames and gravitomagnetic field using laser-ranged satellites,
{\it Il Nuovo Cimento 109A, 5}, 575-590, 1996.
%------------------------------------------
\bb[{\it Ciufolini et al.}, 1997]{superciuf} Ciufolini, I., F.
Chieppa, D. Lucchesi, and F. Vespe, Test of Lense-Thirring orbital
shift due to spin, {\it Class. and Quantum Grav., 14}, 2701-2726,
1997.
%-------------------------------------------
\bb[{\it Ciufolini et al.}, 1998]{ciu98} Ciufolini, I., E. Pavlis,
F. Chieppa, E. Fernandes-Vieira, and J. P{\'{e}}rez-Mercader, Test
of General Relativity and Measurement of the Lense-Thirring Effect
with Two Earth Satellites, {\it Science, 279}, 2100-2103, 1998.
%---------------------------------------------
\bb[{\it Ciufolini}, 1998]{ciuA} Ciufolini, I., Accurate Test of
the Relativistic Perigee Precession and Measurement of the
Relativistic Parameters $\b$ and $\ga$ using LARES, in {\it LARES
Phase-A Study}, pp. 39-40, Rome, 1998.
%-------------------------------------------
%\bb[{\it Damour and Nordvedt}, 1993a]{damn} Damour, T., and K.
%Nordvedt, General Relativity as a Cosmological Attractor of
%Tensor--Scalar Theories, {\it Phys. Rev. Lett., 70}, 2217--2219,
%1993.
%-------------------------------------------
%\bb[{\it Damour and Nordvedt}, 1993b]{damnb} Damour, T., and K.
%Nordvedt, Tensor--scalar cosmological models and their relaxation
%toward general relativity , {\it Phys. Rev. D, 48}, 3436--3450,
%1993.
%-------------------------------------------
\bb[{\it Elliott et al.}, 1998]{elli} Elliott, A. A., R. W.
Hellings, and J. K. Miller, PPN Gamma Results from the NEAR
Conjunction, AAS 191st Meeting, Washington, DC, Session 48, no.
09, January 1998.
%---------------------------------------------
%\bb[{\it ESA}, 1995]{esa} ESA collaboration, Horizon 2000 Plus-European Space
%Science in the 21st Century, ESA
%sp-1180, Noordwijk, August 1995.
%-------------------------------------
%\bb[{\it Eubanks and Klioner}, 1999]{eub} Eubanks, T. M., and S. A.Klioner, New
%Test of Gravitation using Planetary
%VLBI and Ranging, abstract submitted for the APR99 Meeting of The American
%Physical Society, March 20-26, Atlanta, GA, 1999.
%---------------------------------------
\bb[{\it Fr\oe schl{\'{e}} et al.}, 1997]{frosc} Fr\oe
schl{\'{e}}, M., F. Mignard, and F. Arenou, Determination of the
PPN parameter $\ga$ with the Hipparcos data, ESA SP-402, 1997.
%--------------------------------------
%\bb[{\it GAIA}, 2000]{GAIA} GAIA Science Advisory Group. GAIA:
%Composition, formation and evolution of the galaxy. Results of the
%Concept and Technology Study. Draft version available on as
%\verb"http://astro.estec.esa.nl/GAIA" march-report.pdf,version
%1.6, 2000.
%--------------------------------------
%\bb[{\it Grard et al.}, 1994]{merc} Grard, R., G. Scoon, and M. Coradini,
%Mercury Orbiter-An
%Interdisciplinary Mission, {\it ESA Journal, 18,} 197-205, 1994.
%-------------------------------------------
%\bb[{\it Godier and Rozelot}, 2000]{groz} Godier, S., and J.-P.
%Rozelot, The solar oblateness and its relationship with the
%structure of the tachocline and of the Sun's subsurface, {\it
%Astron. and Astrophys., 355,} 365-374, 2000.
%-------------------------------------------
\bb[{\it Iorio}, 2001]{iorio} Iorio, L., Earth tides and
Lense-Thirring effect, {\it Celest. Mech., 79} 201-230, 2001.
%-------------------------------------------
\bb[{\it Iorio}, 2002a]{iorio2} Iorio, L., Testing General
Relativity with LAGEOS, LAGEOS II and Ajisai laser--ranged
satellites, {\it J. of the Geodetic Soc. of Japan, 48 }(1),
13--20, 2002a.
%--------------------------------------------
\bb[{\it Iorio}, 2002b]{ioriob} Iorio, L., Constraints to a Yukawa
gravitational potential from laser data to LAGEOS satellites, {\it
Physics Lett. A, 298}(5--6), 315--318, 2002b.
%--------------------------------------------
\bb[{\it Iorio and Pavlis}, 2001]{iorio2} Iorio, L., and E. C.
Pavlis, Tidal satellite perturbations and the Lense-Thirring
effect, {\it J. of the Geodetic Soc. of Japan, 47}, 169-173, 2001.
%------------------------------------------
\bb[{\it Kaula}, 1966]{kau} Kaula, W. M., {\it Theory of Satellite
Geodesy}, 124 pp., Blaisdell Publishing Company, Waltham, 1966.
%--------------------------------------
%\bb[{\it Lebach et al.}, 1995]{leb} Lebach, D. E., et al.,
%Measurement of the solar Gravitational Deflection of Radio Waves
%using Very-Long-Baseline Interferometry, {\it Phys. Rev. Lett.,
%75,} 1439-1442, 1995.
%--------------------------------------
\bb[{\it Lemoine et al.}, 1998]{egm96} Lemoine, F. G., et al., The
Development of the Joint NASA GSFC and the National Imagery
Mapping Agency (NIMA) Geopotential Model EGM96,
NASA/TP-1998-206861, 1998.
%-------------------------------------------------
\bb[{\it Lucchesi,} 2001]{luc} Lucchesi, D., Reassessment of the
error modelling of non-gravitational perturbations on LAGEOS II
and their impact in the Lense-Thirring determination. Part I, {\it
Plan. and Space Sci., 49}, 447-463, 2001.
%--------------------------------------------------
\bb[{\it Lucchesi,} 2002]{luc2} Lucchesi, D., Reassessment of the
error modelling of non-gravitational perturbations on LAGEOS II
and their impact in the Lense--Thirring determination. Part II. To
appear in {\it Plan. and Space Sci.}, 2002.
%---------------------------------------------
\bb[{\it Milani et al.,} 2001a]{mil} Milani, A., A. Rossi, D.
Vokrouhlick$\acute{{\rm y}}$, D. Villani, and C. Bonanno, Gravity
field and rotation state of Mercury from the BepiColombo Radio
Science Experiment, \textit{Plan. and Space Sci.,} {\it 49},
1579-1596, 2001.
%--------------------------------------------------
\bb[{\it Milani et al.,} 2001b]{mil} Milani, A., D.
Vokrouhlick$\acute{{\rm y}}$, D. Villani, C. Bonanno, and  A.
Rossi, Testing general relativity with the BepiColombo radio
science experiment, {\it Quaderni del Dipartimento di Matematica,
Universit$\grave{a}$ di Pisa, 4.89.1348, } 2001.
%--------------------------------------------------
\bb[{\it Misner et al.,} 1973]{misn} C. W. Misner, K. S. Thorne
and J. A. Wheeler, {\it Gravitation}, 1279 pp., Freeman, San
Francisco, 1973.
%------------------------------------
\bb[{\it Ni}, 2001]{wei} Ni, Wei-Tou, Testing Relativistic Gravity
and Measuring Solar System Parameters via Optical Space Missions,
in: L\"{a}mmerzahl, C., C.W.F. Everitt, and F.W. Hehl (Eds.),
\textit{Gyros, Clocks, Interferometers...:Testing Relativistic
Gravity in Space,} Lecture Note in Physics 562,  507 pp., Springer
Verlag, Berlin, 2001.
%-----------------------------------------
%\bb[{\it Nordvedt}, 1968a]{nor1} Nordvedt, K., Equivalence Principlefor Massive
%Bodies. I. Phenomenology, {\it Phys. Rev., 169}, 1014-1016, 1968.
%------------------------------------------------
%\bb[{\it Nordvedt}, 1968b]{nor2} Nordvedt, K., Equivalence Principlefor Massive
%Bodies. II. Theory, {\it Phys. Rev., 169}, 1017-1025, 1968.
%------------------------------------------------
\bb[{\it Nordvedt}, 1968]{nor3} Nordvedt, K., Testing Relativity
with Laser Ranging to the Moon, {\it Phys. Rev., 170}, 1186-1187,
1968.
%------------------------------------------------
\bb[{\it Nordvedt}, 1991]{nor4} Nordvedt, K., Lunar laser ranging
reexamined. The non-null relativistic contribution, {\it Phys.
Rev. D, 43}, 3131-3135, 1991.
%------------------------------------------------
\bb[{\it Pavlis}, 2002a]{pav1} Pavlis, E. C., Geodetic
Contributions to Gravitational Experiments in Space, in: Cianci,
R.(Ed.), {\it Recent Developments in General Relativity,} Springer
Verlag, Milan, 2002.
%------------------------------------------------
\bb[{\it Pavlis}, 2002b]{pav2} Pavlis, E. C., Dynamic
Determination of Origin and Scale the Earth System from Satellite
Laser Ranging, in: {\it Proceedings of the 2001 IAG Scientific
Assembly: Vistas for Geodesy in the New Millennium,} the
International Association of Geodesy, Budapest, Hungary, September
2-7, 2001, electronic publication (CD), 2002.
%------------------------------------------------
\bb[{\it Pavlis and Iorio}, 2002]{pavlor} Pavlis, E. C., and L.
Iorio, The impact of tidal errors on the determination of the
Lense-Thirring effect from satellite laser ranging, {\it Int. J.
of Mod. Phys. D, 11}(4), 599--618, 2002.
%------------------------------------------------
%\bb[{\it Pireaux et al.,} 2001]{pir} Pireaux, S., J.-P. Rozelot,
%and S. Godier, Solar quadrupole moment and purely relativistic
%gravitation contributions to Mercury's perihelion advance,
%astro-ph/0109032, submitted to {\it Astrophysics and Space
%Sciences}, 2001.
%------------------------------------------------
\bb[{\it Pitjeva}, 1993]{pit} Pitjeva, E.V., Experimental testing
of relativistic effects, variability of the gravitational constant
and topography of Mercury surface from radar observations
1964-1989, {\it Celest. Mech., 55,} 313-321, 1993.
%------------------------------------------------
\bb[{\it Reasenberg}, 1979]{reas} Reasenberg, R. D., {\it Viking}
Relativity Experiment: Verification of Signal Retardation by Solar
Gravity, {\it Astrophys. J., 234}, 124-B5, L219-L221, 1979.
%------------------------------------------------
%\bb[{\it Robertson and Carter}, 1984]{rob1} Robertson, D. S., and W.E. Carter,
%Relativistic deflection of radio signals in the solar
%Gravitational field measured with VLBI, {\it Nature,
%310,} 572-574, 1984.
%--------------------------------------------------
%\bb[{\it Robertson et al.}, 1991]{rob2} Robertson, D. S., W. E. Carter, and W.
%H. Dillinger, New measurement of solar gravitational
%deflection of radio signals using VLBI,
%{\it Nature, 349,} 768-770, 1991.
%----------------------------------------
%\bb[{\it Rubincam}, 1977]{rub} Rubincam, D. P.,
%General relativity and satellite orbits: the motion of a test particle in the
%Schwarzschild metric,
%{\it Celest. Mech., 15,} 21-33, 1977.
%------------------------------------------------
\bb[{\it Shapiro et al.}, 1972]{shap1} Shapiro, I. et al.,
Mercury' s Perihelion Advance: Determination By Radar, {\it Phys
Rev. Lett., 28}, 1594-1597, 1972.
%------------------------------------------------
\bb[{\it Shapiro et al.}, 1976]{shap2} Shapiro, I., C. Counselman,
and R. King, Verification of the Principle of Equivalence for
Massive Bodies, {\it Phys Rev. Lett., 36}, 555-558, 1976.
%--------------------------------------------
\bb[{\it Shapiro}, 1990]{shap3} Shapiro, I., Solar system tests of
general relativity: recent results and present plans, in {\it
Proceedings of the 12th International Conference on General
Relativity and Gravitation,} University of Colorado at Boulder,
1989, edited by N. Ashby, D. Bartlett, and W. Wyss, Cambridge
University Press, Cambridge, 313-330, 1990.
%---------------------------------
%\bb[{\it Shapiro}, 1999]{shap4} Shapiro, I., A century of
%relativity, {\it Rev. Mod. Phys., 71}, S41-S53, 1999.
%--------------------------------------
% Nordvedt effect lunar laser ranging
%-----------------------------------------
%\bb[{\it Stairs et al.}, 1998]{stairs} Stairs, I.H., {\it et al.},
%Measurement of Relativistic Orbital Decay in the PSR B1534+12
%Binary System, {\it Astroph. J., 505,} 352--357, 1998.
%-----------------------------------------
\bb[{\it Will}, 1993]{will1} Will, C M., {\it Theory and
Experiment in Gravitational Physics}, 2nd edition, Cambridge
University Press, Cambridge, 1993.
%------------------------------------------------
\bb[{\it Will}, 2001]{will2} Will, C M., The Confrontation between
General Relativity and Experiment, {\it Living Rev. Relativity
2001-4} (January, 2001), [Article in Online Journal] cited on: 09
Aug 2001, http://www.livingreviews.org/Articles/Volume4/2001-4will
%: a 1998 update,
%gr-qc/9811036, lecture given at {\it XXVI SLAC Summer Institute on Particle
%Physics}, August 3-14, Stanford Linear
%Accelerator Center, Stanford, California, USA, 1998.
%-----------------------------------------------
%\bb[{\it Williams et al.,} 1996]{wil} Williams, J. G., X.X. Newhall, and
%J. O. Dickey, Relativity parameters determined from lunar laser-ranging, {\it
%Phys. Rev. D., 53},
%6730-6738, 1996.
% beta and gamma from Nordvedt effect and from other combination lunar laser
%ranging
%------------------------------------
\end{thebibliography}
\end{document}

gr-qc0103088.extracted_bib
\begin{thebibliography}{Dillo83}
\addcontentsline{toc}{chapter}{References}
%-------------------------------------
\bb[{\it Anderson and Williams}, 2001]{andw} Anderson, J.D., and
J.G. Williams, Long-range tests of the equivalence principle, {\it
Class. and Quantum Grav., 18}, 2447-2456, 2001.
%--------------------------------------------------------------
\bb[{\it Balogh et al.,} 2000]{bal} Balogh, et al., BepiColombo An
Interdisciplinary Cornerstone Mission to the Planet Mercury,
\textit{ESA-SCI, 1,} pp. 21-24 and pp. 35-38, 2000.
%--------------------------------------------------------------
\bb[{\it Ciufolini and Matzner}, 1992]{ciufmat} Ciufolini, I., and
R. Matzner, Non-Riemannian theories of gravity and lunar and
satellite laser ranging, {\it Int. J. of Mod. Phys. A, 7, 4},
843-852, 1992.
%--------------------------------------------------------------
\bb[{\it Ciufolini and Wheeler}, 1995]{ciuf1} Ciufolini, I., and
J. A. Wheeler, {\it Gravitation and Inertia}, 498 pp.,
Princeton University Press, New York, 1995.
%------------------------------------
\bb[{\it Ciufolini}, 1996]{ciuff} Ciufolini, I., On a new method to
measure the
gravitomagnetic field using two orbiting satellites,
{\it Il Nuovo Cimento, 109A, 12}, 1709-1720, 1996.
%------------------------------------------
\bb[{\it Ciufolini et al.}, 1996]{ciufn} Ciufolini
I., D. Lucchesi, F. Vespe and A. Mandiello,
Measurement of
dragging of inertial
frames and gravitomagnetic field using laser-ranged satellites,
{\it Il Nuovo Cimento 109A, 5}, 575-590, 1996.
%------------------------------------------
\bb[{\it Ciufolini et al.}, 1997]{superciuf} Ciufolini, I., F.
Chieppa, D. Lucchesi, and F. Vespe, Test of Lense-Thirring orbital
shift due to spin, {\it Class. and Quantum Grav., 14}, 2701-2726,
1997.
%-------------------------------------------
\bb[{\it Ciufolini et al.}, 1998]{ciu98} Ciufolini, I., E. Pavlis,
F. Chieppa, E. Fernandes-Vieira, and J. P{\'{e}}rez-Mercader, Test
of General Relativity and Measurement of the Lense-Thirring Effect
with Two Earth Satellites, {\it Science, 279}, 2100-2103, 1998.
%---------------------------------------------
\bb[{\it Ciufolini}, 1998]{ciuA} Ciufolini, I., Accurate Test of
the Relativistic Perigee Precession and Measurement of the
Relativistic Parameters $\b$ and $\ga$ using LARES, in {\it LARES
Phase-A Study}, pp. 39-40, Rome, 1998.
%-------------------------------------------
%\bb[{\it Damour and Nordvedt}, 1993a]{damn} Damour, T., and K.
%Nordvedt, General Relativity as a Cosmological Attractor of
%Tensor--Scalar Theories, {\it Phys. Rev. Lett., 70}, 2217--2219,
%1993.
%-------------------------------------------
%\bb[{\it Damour and Nordvedt}, 1993b]{damnb} Damour, T., and K.
%Nordvedt, Tensor--scalar cosmological models and their relaxation
%toward general relativity , {\it Phys. Rev. D, 48}, 3436--3450,
%1993.
%-------------------------------------------
\bb[{\it Elliott et al.}, 1998]{elli} Elliott, A. A., R. W.
Hellings, and J. K. Miller, PPN Gamma Results from the NEAR
Conjunction, AAS 191st Meeting, Washington, DC, Session 48, no.
09, January 1998.
%---------------------------------------------
%\bb[{\it ESA}, 1995]{esa} ESA collaboration, Horizon 2000 Plus-European Space
%Science in the 21st Century, ESA
%sp-1180, Noordwijk, August 1995.
%-------------------------------------
%\bb[{\it Eubanks and Klioner}, 1999]{eub} Eubanks, T. M., and S. A.Klioner, New
%Test of Gravitation using Planetary
%VLBI and Ranging, abstract submitted for the APR99 Meeting of The American
%Physical Society, March 20-26, Atlanta, GA, 1999.
%---------------------------------------
\bb[{\it Fr\oe schl{\'{e}} et al.}, 1997]{frosc} Fr\oe
schl{\'{e}}, M., F. Mignard, and F. Arenou, Determination of the
PPN parameter $\ga$ with the Hipparcos data, ESA SP-402, 1997.
%--------------------------------------
%\bb[{\it GAIA}, 2000]{GAIA} GAIA Science Advisory Group. GAIA:
%Composition, formation and evolution of the galaxy. Results of the
%Concept and Technology Study. Draft version available on as
%\verb"http://astro.estec.esa.nl/GAIA" march-report.pdf,version
%1.6, 2000.
%--------------------------------------
%\bb[{\it Grard et al.}, 1994]{merc} Grard, R., G. Scoon, and M. Coradini,
%Mercury Orbiter-An
%Interdisciplinary Mission, {\it ESA Journal, 18,} 197-205, 1994.
%-------------------------------------------
%\bb[{\it Godier and Rozelot}, 2000]{groz} Godier, S., and J.-P.
%Rozelot, The solar oblateness and its relationship with the
%structure of the tachocline and of the Sun's subsurface, {\it
%Astron. and Astrophys., 355,} 365-374, 2000.
%-------------------------------------------
\bb[{\it Iorio}, 2001]{iorio} Iorio, L., Earth tides and
Lense-Thirring effect, {\it Celest. Mech., 79} 201-230, 2001.
%-------------------------------------------
\bb[{\it Iorio}, 2002a]{iorio2} Iorio, L., Testing General
Relativity with LAGEOS, LAGEOS II and Ajisai laser--ranged
satellites, {\it J. of the Geodetic Soc. of Japan, 48 }(1),
13--20, 2002a.
%--------------------------------------------
\bb[{\it Iorio}, 2002b]{ioriob} Iorio, L., Constraints to a Yukawa
gravitational potential from laser data to LAGEOS satellites, {\it
Physics Lett. A, 298}(5--6), 315--318, 2002b.
%--------------------------------------------
\bb[{\it Iorio and Pavlis}, 2001]{iorio2} Iorio, L., and E. C.
Pavlis, Tidal satellite perturbations and the Lense-Thirring
effect, {\it J. of the Geodetic Soc. of Japan, 47}, 169-173, 2001.
%------------------------------------------
\bb[{\it Kaula}, 1966]{kau} Kaula, W. M., {\it Theory of Satellite
Geodesy}, 124 pp., Blaisdell Publishing Company, Waltham, 1966.
%--------------------------------------
%\bb[{\it Lebach et al.}, 1995]{leb} Lebach, D. E., et al.,
%Measurement of the solar Gravitational Deflection of Radio Waves
%using Very-Long-Baseline Interferometry, {\it Phys. Rev. Lett.,
%75,} 1439-1442, 1995.
%--------------------------------------
\bb[{\it Lemoine et al.}, 1998]{egm96} Lemoine, F. G., et al., The
Development of the Joint NASA GSFC and the National Imagery
Mapping Agency (NIMA) Geopotential Model EGM96,
NASA/TP-1998-206861, 1998.
%-------------------------------------------------
\bb[{\it Lucchesi,} 2001]{luc} Lucchesi, D., Reassessment of the
error modelling of non-gravitational perturbations on LAGEOS II
and their impact in the Lense-Thirring determination. Part I, {\it
Plan. and Space Sci., 49}, 447-463, 2001.
%--------------------------------------------------
\bb[{\it Lucchesi,} 2002]{luc2} Lucchesi, D., Reassessment of the
error modelling of non-gravitational perturbations on LAGEOS II
and their impact in the Lense--Thirring determination. Part II. To
appear in {\it Plan. and Space Sci.}, 2002.
%---------------------------------------------
\bb[{\it Milani et al.,} 2001a]{mil} Milani, A., A. Rossi, D.
Vokrouhlick$\acute{{\rm y}}$, D. Villani, and C. Bonanno, Gravity
field and rotation state of Mercury from the BepiColombo Radio
Science Experiment, \textit{Plan. and Space Sci.,} {\it 49},
1579-1596, 2001.
%--------------------------------------------------
\bb[{\it Milani et al.,} 2001b]{mil} Milani, A., D.
Vokrouhlick$\acute{{\rm y}}$, D. Villani, C. Bonanno, and  A.
Rossi, Testing general relativity with the BepiColombo radio
science experiment, {\it Quaderni del Dipartimento di Matematica,
Universit$\grave{a}$ di Pisa, 4.89.1348, } 2001.
%--------------------------------------------------
\bb[{\it Misner et al.,} 1973]{misn} C. W. Misner, K. S. Thorne
and J. A. Wheeler, {\it Gravitation}, 1279 pp., Freeman, San
Francisco, 1973.
%------------------------------------
\bb[{\it Ni}, 2001]{wei} Ni, Wei-Tou, Testing Relativistic Gravity
and Measuring Solar System Parameters via Optical Space Missions,
in: L\"{a}mmerzahl, C., C.W.F. Everitt, and F.W. Hehl (Eds.),
\textit{Gyros, Clocks, Interferometers...:Testing Relativistic
Gravity in Space,} Lecture Note in Physics 562,  507 pp., Springer
Verlag, Berlin, 2001.
%-----------------------------------------
%\bb[{\it Nordvedt}, 1968a]{nor1} Nordvedt, K., Equivalence Principlefor Massive
%Bodies. I. Phenomenology, {\it Phys. Rev., 169}, 1014-1016, 1968.
%------------------------------------------------
%\bb[{\it Nordvedt}, 1968b]{nor2} Nordvedt, K., Equivalence Principlefor Massive
%Bodies. II. Theory, {\it Phys. Rev., 169}, 1017-1025, 1968.
%------------------------------------------------
\bb[{\it Nordvedt}, 1968]{nor3} Nordvedt, K., Testing Relativity
with Laser Ranging to the Moon, {\it Phys. Rev., 170}, 1186-1187,
1968.
%------------------------------------------------
\bb[{\it Nordvedt}, 1991]{nor4} Nordvedt, K., Lunar laser ranging
reexamined. The non-null relativistic contribution, {\it Phys.
Rev. D, 43}, 3131-3135, 1991.
%------------------------------------------------
\bb[{\it Pavlis}, 2002a]{pav1} Pavlis, E. C., Geodetic
Contributions to Gravitational Experiments in Space, in: Cianci,
R.(Ed.), {\it Recent Developments in General Relativity,} Springer
Verlag, Milan, 2002.
%------------------------------------------------
\bb[{\it Pavlis}, 2002b]{pav2} Pavlis, E. C., Dynamic
Determination of Origin and Scale the Earth System from Satellite
Laser Ranging, in: {\it Proceedings of the 2001 IAG Scientific
Assembly: Vistas for Geodesy in the New Millennium,} the
International Association of Geodesy, Budapest, Hungary, September
2-7, 2001, electronic publication (CD), 2002.
%------------------------------------------------
\bb[{\it Pavlis and Iorio}, 2002]{pavlor} Pavlis, E. C., and L.
Iorio, The impact of tidal errors on the determination of the
Lense-Thirring effect from satellite laser ranging, {\it Int. J.
of Mod. Phys. D, 11}(4), 599--618, 2002.
%------------------------------------------------
%\bb[{\it Pireaux et al.,} 2001]{pir} Pireaux, S., J.-P. Rozelot,
%and S. Godier, Solar quadrupole moment and purely relativistic
%gravitation contributions to Mercury's perihelion advance,
%astro-ph/0109032, submitted to {\it Astrophysics and Space
%Sciences}, 2001.
%------------------------------------------------
\bb[{\it Pitjeva}, 1993]{pit} Pitjeva, E.V., Experimental testing
of relativistic effects, variability of the gravitational constant
and topography of Mercury surface from radar observations
1964-1989, {\it Celest. Mech., 55,} 313-321, 1993.
%------------------------------------------------
\bb[{\it Reasenberg}, 1979]{reas} Reasenberg, R. D., {\it Viking}
Relativity Experiment: Verification of Signal Retardation by Solar
Gravity, {\it Astrophys. J., 234}, 124-B5, L219-L221, 1979.
%------------------------------------------------
%\bb[{\it Robertson and Carter}, 1984]{rob1} Robertson, D. S., and W.E. Carter,
%Relativistic deflection of radio signals in the solar
%Gravitational field measured with VLBI, {\it Nature,
%310,} 572-574, 1984.
%--------------------------------------------------
%\bb[{\it Robertson et al.}, 1991]{rob2} Robertson, D. S., W. E. Carter, and W.
%H. Dillinger, New measurement of solar gravitational
%deflection of radio signals using VLBI,
%{\it Nature, 349,} 768-770, 1991.
%----------------------------------------
%\bb[{\it Rubincam}, 1977]{rub} Rubincam, D. P.,
%General relativity and satellite orbits: the motion of a test particle in the
%Schwarzschild metric,
%{\it Celest. Mech., 15,} 21-33, 1977.
%------------------------------------------------
\bb[{\it Shapiro et al.}, 1972]{shap1} Shapiro, I. et al.,
Mercury' s Perihelion Advance: Determination By Radar, {\it Phys
Rev. Lett., 28}, 1594-1597, 1972.
%------------------------------------------------
\bb[{\it Shapiro et al.}, 1976]{shap2} Shapiro, I., C. Counselman,
and R. King, Verification of the Principle of Equivalence for
Massive Bodies, {\it Phys Rev. Lett., 36}, 555-558, 1976.
%--------------------------------------------
\bb[{\it Shapiro}, 1990]{shap3} Shapiro, I., Solar system tests of
general relativity: recent results and present plans, in {\it
Proceedings of the 12th International Conference on General
Relativity and Gravitation,} University of Colorado at Boulder,
1989, edited by N. Ashby, D. Bartlett, and W. Wyss, Cambridge
University Press, Cambridge, 313-330, 1990.
%---------------------------------
%\bb[{\it Shapiro}, 1999]{shap4} Shapiro, I., A century of
%relativity, {\it Rev. Mod. Phys., 71}, S41-S53, 1999.
%--------------------------------------
% Nordvedt effect lunar laser ranging
%-----------------------------------------
%\bb[{\it Stairs et al.}, 1998]{stairs} Stairs, I.H., {\it et al.},
%Measurement of Relativistic Orbital Decay in the PSR B1534+12
%Binary System, {\it Astroph. J., 505,} 352--357, 1998.
%-----------------------------------------
\bb[{\it Will}, 1993]{will1} Will, C M., {\it Theory and
Experiment in Gravitational Physics}, 2nd edition, Cambridge
University Press, Cambridge, 1993.
%------------------------------------------------
\bb[{\it Will}, 2001]{will2} Will, C M., The Confrontation between
General Relativity and Experiment, {\it Living Rev. Relativity
2001-4} (January, 2001), [Article in Online Journal] cited on: 09
Aug 2001, http://www.livingreviews.org/Articles/Volume4/2001-4will
%: a 1998 update,
%gr-qc/9811036, lecture given at {\it XXVI SLAC Summer Institute on Particle
%Physics}, August 3-14, Stanford Linear
%Accelerator Center, Stanford, California, USA, 1998.
%-----------------------------------------------
%\bb[{\it Williams et al.,} 1996]{wil} Williams, J. G., X.X. Newhall, and
%J. O. Dickey, Relativity parameters determined from lunar laser-ranging, {\it
%Phys. Rev. D., 53},
%6730-6738, 1996.
% beta and gamma from Nordvedt effect and from other combination lunar laser
%ranging
%------------------------------------
\end{thebibliography}