Journal of Modern Physics, 2011, 2, 210-218
doi:10.4236/jmp.2011.24029 Published Online April 2011 (http://www.SciRP.org/journal/jmp)
Copyright © 2011 SciRes. JMP
On Some Critical Issues of the LAGEOS-Based Tests of the
Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.)
Fellow of the Royal Astronomical Society (F.R.A.S.) Viale Unità di Italia, Bari (BA), Italy
Received October 6, 2010; revised November 23, 2010; accepted November 27, 2010
We summarize some critical issues pertaining the tests of the general relativistic Lense-Thirring effect per-
formed by I. Ciufolini and coworkers in the gravitational field of the Earth with the geodetic satellites
LAGEOS and LAGEOS II tracked with the Satellite Laser Ranging technique.
Keywords: Experimental Studies of Gravity, Experimental Tests of Gravitational Theories, Satellite Orbits,
Harmonics of the Gravity Potential Field, Geopotential Theory and Determination
In the scalar theory of gravitation by Newton, which
does not fulfill the Lorentz invariance, the gravitational
field of a spherical body does not depend on its state of
motion, being, indeed, determined only by its mass M.
On the contrary, in the tensorial General Theory of Rela-
tivity (GTR) by Einstein, which is a fully relativistic the-
ory of gravitation, non-static distributions of matter-energy
yield their own contributions to the overall gravitational
field in addition to the static ones. Such peculiar terms
are connected with the off-diagonal components 0,
of the spacetime metric tensor, and were
dubbed “gravitomagnetic” [1-3] in analogy with the
magnetic fields generated by the electric currents. Indeed,
in its weak-field and slow-motion approximation, the
fully non-linear field equations of GTR get linearized,
thus resembling those of the linear Maxwellian electro-
magnetism; in particular1, 000iii
. The resulting gravitomagnetic part of the
equations of motion of a test particle is 
where dots denote time derivatives, and c is the speed of
light in vacuum.
At great distances r from a slowly rotating body en-
dowed with proper angular momentum S, it is
with the gravitomagnetic vector potential given by 
where G is the Newtonian constant of gravitation. The
related gravitomagnetic field is [1,5,6]
Among other phenomena, the resulting Lorentz-like,
non-central acceleration due to Equation (4)
causes the secular precession of the spin
of a gyro-
scope (Schiff spin-spin effect, ) with frequency
and the secular precessions of the longitude of the as-
and of the argument of pericenter
are the components of the metric
,,0,1, 2, 3
tensor of the “flat” Minkowskian spacetime.
of the orbit of a test particle (Lense-Th
effect, ), both in geodesic motion ar
body. The parameters a, e, I are the semimajor axis, the
ial and natural satellites in the Solar
d analyses which have not yet been ex-
ons which yielded them have been repeated by different
oncerning GP-B, if, on the one hand, the analysis of the
practice, be repeated by other independent researchers,
ound the spinning
eccentricity and the inclination, respectively, of the par-
ticle’s orbit. Such phenomena are often collectively de-
noted with the catchy denomination “frame-dragging”,
although also the general relativistic gravitoelectric de
Sitter precession  of an orbiting gyroscope in the field
of a static mass is a part of such a category .
Experimental/observational efforts have been dedi-
cated in recent times to obtain empirical corroborations
of the aforementioned predictions of GTR. The
obe B (GP-B) mission [11,12] is an extremely refined,
sophisticated and expensive experiment [13-15], con-
ceived 50 years ago, explicitly aimed to measure, among
other things, the gravitomagnetic Schiff effect with four
gyroscopes in a controlled environment enclosed in an
active spacecraft orbiting the spinning Earth since 2004.
The properly scientific phase of the mission ended in
2005, and the analysis of the data collected during it is
still ongoing [16-18]. The expected accuracy was origi-
nally 1% or better, but it seems that the occurrence of
some unexpected systematic errors [15,19,20] may fi-
nally undermine the actual attainment of such a goal. At
present, according to the official mission’s website2, the
claimed statistical error is 14%, while the systematic
uncertainty is 10%.
Attempts to measure the Lense-Thirring effect were
proposed, and in some cases implemented, with some
stem; for a recent, comprehensive overview see, e.g.,
Ref.  and references therein. Concerning the per-
formed analyses, the first tests date back to the mid of
90s [22-25]; they were conducted in the gravitational
field of the Earth with the non-dedicated LAGEOS and
LAGEOS II geodynamic satellites3 continuously tracked
with the Satellite Laser Ranging (SLR) technique  by
looking at the nodes of both the satellites and the perigee
of LAGEOS II. Such attempts are still ongoing [27-32]
by retaining only the nodes of LAGEOS and LAGEOS II.
The claimed accuracy in such more recent tests is 10% -
15% [27-32], but other evaluations, dealing with certain
sources of systematic errors in a more conservative way,
point towards figures which may be up to 2-3 times lar-
ger: for such critical views, see Refs. [21,33] and refer-
In this paper we want to clearly point out some epis-
temological and physical issues pertaining the performed
icitly addressed in a satisfactorily way. In Section 2,
after a brief review of the status of the GP-B mission and
the perspectives for performing other measurements of
the Schiff precession with artificial and natural probes,
we discuss if independent tests of the Lense-Thirring
effect really exist in literature after about 15 years since
the first attempts were implemented. Section 3 is devoted
to the relation among the general relativistic gravi-
tomagnetic field of the Earth and the spacecraft-based
global solutions for the classical part of the terrestrial
gravitational field produced so far. Some alternative ap-
proaches to process the data of the LAGEOS and
LAGEOS II spacecraft are discussed in Section 4. The
issue of the actual level of cancelation of the corrupting
bias due to the classical quadrupole mass moment of the
rotating Earth in the tests performed so far is tackled in
Section 5; in it the impact of the other multipoles of the
terrestrial gravitational field according to the first models
from GOCE is discussed as well. Section 6 summarizes
2. Do Really Independent Tests of
Physics is an activity whose results are c
ovisionally) established if the
teams of independent researchers in different laboratories
with different methodologies. Actually, this is not (yet?)
the case for gravitomagnetism.
2.1. The Schiff Effect and the GP-B Experiment
data collected in 2004 - 2005 could, both in principle and
on the other hand it will likely not be possible to do that
for the entire experiment in any foreseeable future in
view of its extreme sophistication and cost.
This is certainly not satisfactorily from an epistemo-
logical point of view because GP-B seems destined to
remain a unique empirical check of the Schi
deed, a proposal to use spacecraft orbiting the Sun and
Jupiter  had not sequel so far; on the other hand, its
complexity, cost and technological difficulties would
certainly not have been lower than those of GP-B itself.
Moreover, independent measurements of the Schiff
effect with natural bodies in, e.g., the Solar System are in
all probability unfeasible. To this aim, let us recall tha
2The See http://einstein.stanford.edu/ on the WEB.
3They are dense, spherical targets entirely covered with retroreflectors
for passively bouncing back the laser impulses sent to them by
ground stations. Both the LAGEOS spacecraft orbit at altitudes o
about 6000 km, so that they do not suffer macroscopic orbit decay due
to the atmospheric drag. As a consequence, their lifetime is evaluated
to be of the order of 105 yr.
e maximum value of the gravitomagnetic Schiff pre-
cession occurs when the angular momentum S of the
central source and the precessing spin σ of the gyroscope
Copyright © 2011 SciRes. JMP
he situation is, in principle, more favorable for the
the LAGEOS satel-
AGEOS II are 14-15 years old, dating back to
throughout the world, so that the
either performed so
S satellites only if and when papers published in
are mutually perpendicular, being, instead, zero when
they are aligned . In principle, a natural scenario sat-
isfying such a requirement is the Sun-Uranus system.
Indeed, while the solar equator is inclined to the mean
ecliptic at the epoch J2000 by the Carrington angle
deg , the spin σ of Uranus is tilted to the
ecliptic by 97.77 deg . Of course, apart from the dif-
ficulties of devising some effective methods for con-
monitoring the precessional motion of the spin
of Uranus, the magnitude of the Sun-Uranus Schiff pre-
cession would be insignificantly small. In principle, an-
other potential natural laboratory may, perhaps, be the
double pulsar PSR J0737–3039A/B [38,39]. Indeed,
while the spin4 SA of A is perpendicular to the orbital
plane , σB is not aligned with SA [40,41] because of
the de Sitter precession  which has recently been
measured with a 13% accuracy [40,41]. Actually, the
gravitomagnetic Schiff-like spin precession  of σB
caused by SA would be much smaller and quite difficult
2.2. The LAGEOS-Based Tests
Lense-Thirring tests performed with
The first attempts to reveal the existence of the Earth’s
gravitomagnetic field by analyzing the data of LAGEOS
The network of the5 International Laser Ranging Ser-
vice (ILRS)  consists of a large set of laser ranging
R community is quite numerous. The LAGEOS satel-
lites, which are some of the most important SLR targets,
are continuously tracked since long time. The GEODYN
software , developed by NASA, is widely dissemi-
nated throughout the SLR stations also because it is free
of charge. Moreover, some institutions developed their
own orbit analysis systems like UTOPIA by the Center
for Space Research (CSR) of the University of Texas,
and the Earth Parameter and Orbit System (EPOS) by
Despite this situation, potentially favorable for per-
forming several truly independent tests of such a predic-
tion of GTR, none has been actually
r, or published in international peer-reviewed journals.
Indeed, apart from a pair of conference talks given by J.
Ries et al. (CSR) [45,46], all the relatively more accu-
rate6 tests published so far in peer-reviewed papers or
edited books have I. Ciufolini in the authorship as first
author or editor himself [27-32]. Moreover, to date, no
independent works on the Lense-Thirring effect attribut-
able to members of GFZ exist in literature. Thus, al-
though the list of co-authors of the papers by Ciufolini
has often changed and in some of the most recent works
[31,32] unpublished results obtained with UTOPIA and
EPOS are described as well, such tests cannot be consid-
ered as truly independent ones. This is particularly true
also in view of the fact that the methodology adopted is
basically the same, apart from the orbital processors used.
This point will be explained better in Section 3 and Sec-
Thus, it will be possible to speak about genuinely in-
dependent tests of the Lense-Thirring effect with the
er-reviewed journals without Ciufolini in the author-
ship, and authored if possible by different researchers
with respect to those having more or less systematically
co-operated with him, will appear in literature. Moreover,
and, perhaps, most importantly, also the methodology
used should be different from that adopted so far. Such
studies, which should be made publicly available, would
be of great importance even in the case of negative
and/or inconclusive outcomes. Finally, somebody may
likely wonder why the author of the present paper does
not undertake himself the task that he is suggesting to
others. It may be pointed out that if, on the one hand, the
results presented by skillful and experienced researchers
in satellite data processing have raised doubts until now,
on the other hand it is likely that analogous uncertainties
would be even stronger in the case of a work produced
by a scientist not yet actively engaged in such a difficult
art. Moreover, the reliability of such results may, perhaps,
be reduced in the eyes of a part of the community in
view of the fact that the author of the present paper
would difficultly be considered as sufficiently neutral
and detached from the subject considered. Analyses by
really independent third parties may have more chances
to be accepted without some sorts of prejudices.
Concerning the non-negligible role played by such
considerations, it maybe instructive to illustrate the fol-
lowing case. In late 2007 a preprint titled “A
alysis of the GP-B mission. I: on the impossibility of a
reliable measurement of the gravitomagnetic precession
of the GP-B gyroscopes”, authored by G. Forst, was
posted on the arXiv repository . This author never
either posted other preprints on the arXiv website or pub-
lished any peer-reviewed papers. Moreover, there is no
mention at all on the WEB of the organization quoted as
4Since the rotational periods of A and B are 23 ms and 2.8 s, respec-
tively, SA is larger than σB. The latter one describe a full precessional
cycle in 75 yr because of the general relativistic de Sitter precession
5See http://ilrs.gsfc.nasa.gov/ on the WEB.
6One of the major critical points of the earlier tests was the use of the
erigee of LAGEOS II , heavily perturbed by several non-gravita-
tional classical forces.
Copyright © 2011 SciRes. JMP
sobtained so far by several independent
stitutions from the data of the dedicated spacecraft
his affiliation. Finally, the references cited by G. Forst
did not actually show what was attributed to them in the
main text of his preprint, as noted first by7 K. Krogh. In
early 2008, the arXiv moderators removed the preprint
by G. Forst with the following motivation: “This sub-
mission has been removed because ‘G. Forst’ is a pseu-
donym of a physicist based in Italy who is unwilling to
submit articles under his own name. This is in explicit
violation of arXiv policies. Roughly similar content,
contrasting the relative merits of the LAGEOS and GP-B
measurements of the frame-dragging effect, can be found
in pp. 43-45 of .” Even so, in late 2008 Ref.  was
cited by I. Ciufolini in some talks of him [49-51].
At a different level of relevance, we also mention the8
editing-war which involved the voice “frame-dragging”
on Wikipedia in 2006-2007. It mainly consisted of
stematic and reiterated removal by Italian IPs followed
by their almost immediate reinstatement, of all and only
the references by the author of the present paper on some
critical aspects of the Lense-Thirring tests with the
LAGEOS satellites. On the contrary, the references by I.
Ciufolini were never removed.
3. New Global Earth’s Gravity Field
A distinctive feature of all the global Earth’s grav
lutions  o
CHAMP , GRACE  and GOCE  is that GTR
was never explicitly solved for along with, say, the even
zonal harmonic coefficients ,0 ,2,4,6,C
geopotential. The first global solution, obtained from
CHAMP, dates back to 2001 . Both the previous
ones and all those of the CHACE era
are publicly available on the Internet at the official web-
site of the International Centre for Global Earth Models
(ICGEM), http://icgem.gfz-potsdam.de/ICGEM/. This
fact implies, among other things, that the even zonals
may retain a sort of a-priori “imprinting” of the Lense-
Thirring effect itself; similar arguments were put forth in
the pre-CHAMP/GRACE/GOCE era in Refs.[47,57]. In
Ref.  it was explicitly shown that this may actually
be just the case for GRACE, given the present-day level
of accuracy in estimating the even zonal coefficients and
the size of the gravitomagnetic effect on the orbit of
GRACE. Although likely time consuming, producing
new global Earth’s gravity field solutions by explicitly
solving for relativity as well would be a really important
and independent test of the general relativistic gravi-
tomagnetic component of the field of the Earth, also be-
cause it would allow to inspect the correlations among
the estimated solve-for relativity parameter(s) and the
even zonals in the covariance matrix . It would be
important to judge if it made sense to employ that par-
ticular gravity model in processing the data of the
LAGEOS satellites to try to safely extract the Lense-
Thirring effect. The inquiries of the author of the present
paper to some scientists presently involved in the genera-
tion of the global gravity field solutions have, in general,
received rather evasive answers, if any, mainly centered
on the issue of the great computational and time efforts
which would be required to re-process all the data sets
from, say, GRACE.
4. A Different Approach in Processing the
lowt the Lense-Thirring effect from
e data of the LAGEOS satellites, common to all the
y some correc-
An issue related to the previous one is the approach f
ed to directly extrac
analyses performed so far. The directly observable quan-
tity in SLR is the station-spacecraft range computed in
terms of the two-way time-of-flight recorded by a
ground-based clock. Actually, in all the tests implemented
so far the gravitomagnetic effect on the LAGEOS ranges
was never explicitly modeled in terms of one or more
dedicated solve-for parameters to be estimated in the
usually least-square sense , as done, instead, for a
host of other parameters pertaining certain physical
properties of the spacecraft, their orbital motions and the
Earth-fixed stations. Note that in the pre-CHAMP/
GRACE/GOCE era the global Earth’s gravity field solu-
tions were produced just in such a way, i.e. by globally
fitting long data records from a constellation of SLR tar-
gets, among which LAGEOS and LAGEOS II always
played a dominant role, and estimating the even zonal
harmonics as solve-for parameters. Incidentally, let us
note that even in such circumstances the Lense-Thirring
effect was never modeled and solved-for.
Another approach which may be followed may consist
of not modeling the Lense-Thirring effect at all, and es-
timating in a purely phenomenological wa
to the node precessions. This may typically
be done for each orbital arcs. A similar approach was
followed in the determination of the corrections
the stard perihelion precessions of some planets of
the Solar System , which, in principle, account for
any unmodeled/mismodeled dynamical effect ljust
the Lense-Thirring one. Also in the case of the timing of
the binary systems hosting one or more pulsars a
post-Keplerian periastron precession PK
7See on the WEB http://www.physicsforums.com/showthread.php?t=
troversy on the WEB.
Copyright © 2011 SciRes. JMP
ns which will be intentionally produced without
AGEOS II purposely designed to cancel out the impact
nomenologically estimated as a solve-for parameter
along with other ones . Subsequently, it was identi-
fied with the gravitoelectric precessioredicted by
Future independent LAGEOS-based tests should try to
implement such strategies. In doing that, those global
taining a-priori “imprinting” of relativity itself should
be used as reference gravity field models (see Section 3).
5. Issues Pertaining the Bias Due to the
Newtonian Multipoles of the Terrestrial
. The Cancelation of the First Even Zonal
Harmonic of the Geo
All the most recent tests performed so far rely upo
r combination of the nodes of
of the first even zonal harmonic coefficient 22,0
of the multipolar expansion of the Newtonian gravita-
tional potential of the Earth, which is a major source of
systematic uncertainty. Indeed, it turns
nominal values of the competing secular node preces-
sions of LAGEOS and LAGEOS II caused by 2
out that the
orders of magnitude larger than the gravitomagnetic ones.
Such a combination was explicitly worked out in Ref.
, following a strategy put forth in Ref. ee also
Actually, the coefficient c1 of such a combination is a
function of the semimajor axes a, the eccentricities e and
the inclinations I t
o the Earth’s equator of characterizing
the orbits of both the LAGEOS satellites. This implies
that the unavoidable uncertainties in the computation of
such Keplerian orbital elements from the estimated state
vectors of the satellites yield an overall uncertainty in
itself which, thus, can be known only with a certain
numbers of significant digits . In turn, this fact in-
troduces a further source of systematic bias becaus,
given a certain uncertainty 89
pending on the level of accuracy with which one assumes
in determining the inclinations I of LAGEOS and
LAGEOS II, the resulting can2uced
node precessions is necessarily not perfect, contrary to
what implicitly assumed so far. It turns out that the re-
sidual J2 signature would amount to 14% - 23% of the
Lense-Thirring one . It should be remarked that a
value of c1 known up to the 8-9th decimal digits should
be used to obtain just the aforementioned level of accu-
racy in the cancelation of the effect of J2. Instead, c1 has
always been treated so far with a very limited number of
decimal digits; in, e.g., Ref.  they are just 3 (c1 =
5.2. The Bias Due to the Other Even Zonal
celation of the J-ind
rmonics of Higher Degree
Concerning the other even zonals of higher d
, which J
are not canceled
ut by the aforementioned linear combination, their
sulting signal which may am
tial produced by different institutions with
from GRACE and GOCE, still
mismodeling induces a systematic uncertainty in the re-
ount to a non-negligible
fraction of the Lense-Thirring signal. The realistic
evaluation of such a systematic alias was evaluated in
several papers; see, e.g., Refs. [33,21] and references
Let us briefly recall that nowadays we have at our dis-
posal several estimated values of the even zonals of the
riable approaches and techniques from the data col-
lected by the CHAMP, GRACE and GOCE dedicated
missions. In evaluating the aliasing of the even zonals on
the Lense-Thirring signal it should be clear that, since we
are dealing with the same physical quantities simply
measured with different techniques, there are a-priori no
reasons, in principle, to prefer just one specific solution
instead of other ones, unless objective and quantitative
arguments are provided for trusting just it. Certainly, in
the framework of a test of fundamental physics, it is not
acceptable to pick-up just this or that particular Earth’
gravity models that, for some reasons, yield the best re-
sult in terms of fitted straight line9, and evaluating the
systematic error on the Lense-Thirring effect by only
using such particular solutions; it would be a sort of se-
lection bias towards that solution just yielding the closest
outcome to the one expected in advance. Instead, it is
much more realistic and conservative to take into ac-
count a large number of gravity models, provided that
they are roughly of comparable accuracy, and adopt the
differences among their estimated values for each even
zonals as representative of the realistic uncertainty in
them. In any case, quantitative, statistical arguments
should be used to discard one or more determinations of
a given even zonal, as discussed in Ref. . Stated sim-
ply, it is not admissible to play with the various gravity
models by retaining only those convenient to the a-priori,
desired outcome and discarding, instead, those yielding
less favorable results.
In this respect, here we point out that the first, pre-
liminary global solutions from GOCE like10 GOCO01S,
which combines data
9The issues previously discussed in Section 3 and Section 4 should, at
this point, not be forgotten.
Institute/Homepages/i5080/forschung/GOCO/ on the Internet.
Copyright © 2011 SciRes. JMP
hatever the final outcome of its data analysis will be,
citly dedicated to
easure the general relativistic gravitomagnetic Schiff
ients of the classical
observable quantities in Satellite Laser Ranging
itational potential, which is of degree 2, from
from GOCE and the earlier models from
ank M. Cerdonio for insightful and inspiring corre-
] K. S. Thorne, “Gravitomagnetism, Jets in Quasars, and
yroscope Experiment,” In: J. D. Fairbank,
veritt and P. F. Michelson, Eds.,
A Brief Re-
ald and R. H. Price, Eds.,
sity Press, Yale, 1986.
esent significative discrepancies with respect to earlier
GRACE/CHAMP-only global solutions. Indeed, it can
be shown that the pair11 GOCO01S-EIGEN51C, where
EIGEN51C  is a global solution consisting of 6 years
of CHAMP and GRACE data and the DNSC08 global
gravity anomaly data set, yields an uncertainty of 27% of
the Lense-Thirring signal. Another similar example is
given by the pair GOCO01S-AIUB-GRACE02S yielding
an uncertainty as large as 23% of the Lense-Thirring
signature; AIUB-GRACE02S  is a tide-free GRACE-
only based solution obtained from almost 2 yr of
6. Summary and Conclusions
the Gravity Probe B mission, expli
spin-spin effect in an extremely sophisticated and expen-
sive controlled experiment carried onboard a spacecraft
orbiting the Earth, will remain the only empirical check
of this specific prediction of the General Theory of Rela-
tivity because of the practical impossibility of repeating
it in any foreseeable future. Moreover, no other natural
laboratories in astronomical scenarios can likely be used
to put on the test the Schiff’s gyroscope precession.
In principle, the situation for the tests of the Lense-
Thirring spin-orbit effect performed so far in the gravita-
tional field of the Earth with the non-dedicated LAG
d LAGEOS II satellites tracked with the Satellite Laser
Ranging technique is more favorable. Indeed, the life-
time of such orbiting laser targets is of the order of 105 yr,
they are totally passive not requiring active instrumenta-
tion carried onboard, and the laser ranging community is
made of several teams disseminated in a wide network of
ground stations mainly using an orbital processor system
which is freely available. Instead, despite the first at-
tempts were made about 15 years ago by a group led by I.
Ciufolini, no really independent tests have been pub-
lished so far in peer-reviewed journals by authors differ-
ent from the aforementioned Italian scientist, apart from
a couple of conference talks by a group led by J. Ries.
On the contrary, fake Internet-based attempts to under-
mine the credibility of the Gravity Probe B mission were
undertaken by an Italian physicist.
New Earth’s global gravity field solutions in which the
General Theory of Relativity is explicitly solved for
along with the multipolar coeffic
rt of the geopotential should be produced. On the con-
trary, all the models obtained so far from the dedicated
CHAMP, GRACE and GOCE missions by several inde-
pendent institutions since 2001 may be a-priori “im-
printed” by the General Theory of Relativity itself since
no relativistic effects were ever explicitly estimated in
A closer connection between the gravitomagnetic ef-
fects on the orbits of LAGEOS and LAGEOS II and the
ould be elucidated. In this regard, the Lense-Thirring
effect should be explicitly modeled in the dynamical
force models of the LAGEOS satellites and a dedicated
parameter should be solved-for, as it is common practice
in all other areas of space science and gravitational
The cancelation of the first even zonal harmonic coef-
ficient of the classical multipolar expansion of the terres-
e linear combination of the nodes of LAGEOS and
LAGEOS II used so far is not perfect because of the un-
certainties in their orbital parameters. It turns out that the
uncanceled effect of the Earth’s centrifugal oblateness is
as large as 14% - 23% of the Lense-Thirring combined
Significative discrepancies among the estimated val-
ues of the even zonal harmonics in the first, preliminary
AMP and GRACE exist; according to them, the sys-
tematic uncertainty caused by the mismodeling in the
even zonals of degree higher than 2 on the Lense-
Thirring signature of LAGEOS and LAGEOS II is still
as large as about 20%.
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