Cent. Eur. J. Phys. • 11(5) • 2013 • 531-544
DOI: 10.2478/s11534-013-0189-1
Central European Journal of Physics
History of the attempts to measure orbital
frame-dragging with artificial satellites
Review Article
Giacomo Renzetti ∗
Received 16 September 2012; accepted 11 February 2013
Abstract:
An annotated guide to navigate the intricate history of the attempts to measure the Lense-Thirring orbital
precessions with artificial satellites is offered to the reader.
PACS (2008): 04.80.-y, 04.80.Cc, 91.10.Sp, 91.10.Qm
Keywords:
experimental studies of gravity • experimental tests of gravitational theories • satellite orbits • harmonics of
the gravity potential field
© Versita Sp. z o.o.
1. Brief remarks on the LenseThirring effect
On an evening in 1918, Einstein, in closing a discussion
with H. Thirring who was telling the father of General
Relativity (GR) about his work with Lense [1], would have
lamented [2] 1 “Wie schade dass wir nicht einen Erdmond
haben, der gerade nur ausserhalb der Erdatmosphere umlauft!”.
The object of the conversation between Einstein and
Thirring was the so-called Lense-Thirring effect [1], a
small cumulative precession of the orbit of a test particle about a slowly rotating central object. Pfister [3] recently made a detailed historical research of the genesis of
the GR prediction which is nowadays commonly known as
∗
E-mail: [email protected]
“What a pity the Earth has no moon in an orbit just
outside its atmosphere!”.
1
Lense-Thirring effect. Actually, it would be more correct
to denominate it as Einstein-Thirring-Lense effect [3]. Indeed, in a letter to Thirring dated August 2, 1917, Einstein
calculated the Coriolis-type field of the rotating Earth and
Sun, and its influence on the orbital elements of planets
and moons [4]; moreover, according to Pfister [3], Lense
would have likely just put the numbers in the formulas to
obtain the numerical results [5] of the paper co-authored
with Thirring [1] who, instead, would have made most of
the theoretical calculation. Nonetheless, the usual denomination entered nowadays into common usage will be
~ is the angular momentum
adopted in the following. If S
of the central body, GR predicts that the longitude of the
ascending node Ω and the argument of pericenter ω of a
satellite’s orbit are not constant, changing with the rates
2GS
Ω̇LT =
c 2 a3
(1 −
e2 )3/2
, ω̇LT = −
6GS cos i
c 2 a3
(1 − e2 )3/2
.
(1)
G is the Newton’s constant of gravitation, c is the speed
of light in vacuum, a is the semimajor axis of the satellite’s orbit, e is its eccentricity, and i is the inclination
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History of the attempts to measure orbital frame-dragging with artificial satellites
of the orbital plane to the equator of the central rotating
mass in such a way that equatorial orbits have i = 0◦ ,
~ if i = 90◦ . The semiwhile the orbital plane contains S
major axis a has dimensions of length setting the size of
the Keplerian ellipse whose shape is determined by the
eccentricity 0 ≤ e < 1 in such a way that a circle corresponds to e = 0, while highly elliptical orbits have e
close to 1. The node Ω and the argument of pericenter
ω are angles characterizing the orientation of the ellipse
in the space and in the orbital plane, respectively. Actually, Lense and Thirring [1] calculated the precessions of
Ω and of the longitude of the pericenter, which is a “broken” angle defined as π = Ω + ω. The precession of the
argument of pericenter ω shown in Eq. (1) was obtained
by Bogorodskii [6] and later by Barker and O’Connel [7].
For other derivations, see also [8–10]. Kalitzin [11] obtained the in-plane precession of the pericenter, i.e. he
considered the rate of the angle Ω cos i + ω. The formulas
of Eq. (1) hold in a particular coordinate system with the
~ the general case for an arbitrary
z axis aligned along S;
orientation of the body’s spin axis was explicitly treated
by Iorio [12].
In this paper, the focus is on the attempts made so far to
measure the Lense-Thirring orbital shifts in the terrestrial
gravitational field with artificial satellites. Thus, the ongoing LARES mission [13–17], previously known also as
Lageos-3 and LARES/WEBER-SAT and finally launched
in early 2012 aiming to measure them at 1% level [16], will
not be considered here. Here, just some key facts about
it are mentioned2 . Lageos-3 was to be a copy of Lageos
with a similar orbit except that its orbit inclination i = 70◦
would have been supplementary to the Lageos’ one to allow a determination of the Lense-Thirring effect free from
certain disturbances arising from the asphericity of the
Earth’s gravity field (see Section 2). About Lageos-2, the
Europeans who built and paid for it chose i = 52◦ as
giving more frequent laser-geodesy passes over Europe.
LARES was planned as a smaller size and hence lower
mass spacecraft to take advantage of recent improvements
in laser-ranging performance which provides an adequate
ranging signal from such a smaller target; the lower mass
means that a lower performance, lower cost launch vehicle could be used to put it in orbit. The LARES designers settled for an orbit only 1450 km high because
they were offered a free launch, the first (demonstration)
launch of the new Italian VEGA launch vehicle. Professor Doug Currie from the University of Maryland (USA),
who had long been involved with the design of retroreflecI thank M. Efroimsky of the US Naval Observatory for
the useful informations provided.
2
tors for satellite and lunar laser ranging, recommended
to have this relativity-testing Lageos follow-on be called
WEBER-SAT in honor of Dr. Joseph Weber who built a
pioneering gravitational wave detector at the University
of Maryland about 1961. The tests with the Mars Global
Surveyor (MGS) spacecraft and Mars [18–21], and the
Sun-planets scenario [22–27] will be left aside as well; a
recent, comprehensive overview can be found in [28].
Another general relativistic orbital effect caused by the rotation of a central body is the so-called gravitomagnetic
clock effect [29–34]. It affects the orbital periods of two
counter-rotating test particles along otherwise identical
trajectories in such a way that if one of them revolves in
the same direction as the primary spins, it takes longer
time to describe a full orbital revolution, whereas the orbital period of the other one gets shorter if it moves oppositely with respect to the body’s rotation. The possibility
of measuring the gravitomagnetic clock effect in space experiments was the subject of several works [35–38]. It will
not be treated further in this paper.
GR predicts also a further effect occurring in the neighbourhood of a central rotating body: the precession of the
spin of an orbiting gyroscope discovered independently
by Pugh [39] and Schiff [40, 41] in 1959-1960. It was
recently measured by Everitt et al. [42] with four gyros carried onboard the Gravity Probe-B (GP-B) spacecraft orbiting the Earth at a claimed accuracy of 19%. It
will not be treated in this paper. For further details, see
http://einstein.stanford.edu/ on the WEB.
Testing gravitomagnetism directly in more or less controlled and/or known local scenarios with macroscopic objects is important because it may play important roles
in relatively more uncertain and speculative high-energy
processes occurring in the ergosphere of rotating black
holes [43–45] through the Penrose mechanism [46], and
with slowly moving spin-1/2 elementary particles in weak
fields as well [47].
2. Earlier proposals for measuring
the Lense-Thirring effect with Earth’s
artifical satellites
Around the beginning of the space era, marked by the
launch of the Sputnik satellite on October 4, 1957, the
possibility of measuring the Lense-Thirring effect with
man-made moons orbiting just outside the Earth’s atmosphere, to paraphrase Einstein, started to be seriously
taken into account.
Ginzburg and Bogorodskii [2, 6, 48–50] preliminarily
looked at the node and the perigee of a single Earth satellite in relatively low orbits with a more or less pronounced
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eccentricity to be tracked by radio signals; for example,
Bogorodskii [6] by choosing the values a = 7420 km, e =
0.11, i = 65◦ of the third Soviet artificial satellite obtained a node and perigee precessions of 16.3 arcseconds
per century and −20.6 arcseconds per century, respectively. These authors, although aware of the potentially
corrupting effects exerted by the asphericity of the Earth
on the relativistic precessions sought, did not treat in detail the issue of how to cope with them. For example,
Ginzburg concluded his paper [48] by writing: “It seems to
us that a discussion of the possibility of detecting the relativistic ’rotation effect’ should attract attention”. Ginzburg
[2] observed that “the interpretation of such observations
is a great deal more complicated for a satellite than for
a planet […] the orbit of a satellite is also perturbed by
irregularities in the earth’s shape and density […]”. Bogorodskii [6] acknowledged that “The discovery of such
effects […] would require […] painstaking development of
satellite motion theory, which would enable us to separate out those effects from the multiplicity of perturbations
due to other causes.”.
It is well known that certain departures from spherical
symmetry of the terrestrial gravitational field, usually parameterized in terms of even zonal harmonic coefficients
[51, 52] J` , ` = 2, 4, . . ., causes secular precessions of the
node and the perigee of a satellite [52] which tend to mask
the relativistic effects because of their quite larger magnitude. The largest precessions, caused by the quadrupole
mass moment J2 , were recently calculated for an arbitrary
orientation in space of the Earth’s spin axis by Iorio [53];
for the precessions up to degree ` = 20, calculated by
customarily aligning the z axis to the spin axis of the
central body, see [54]. The even zonals should be known
with sufficient accuracy to allow for an acceptable aliasing of the Lense-Thirring drag in the overall signature, but,
in general, it is not yet possible with a single spacecraft
since, as shown below, the Earth’s J2 should be known to
a level of accuracy not yet reached in the current Earth’s
gravity modelling. Indeed, the classical precessions of the
node and the perigee of degree ` = 2 are
2
R
J2 cos i
3
, ω̇2
Ω̇2 = − n
2
a
(1 − e2 )2
2
J2 5 cos2 i − 1
R
3
= n
,
4
a
(1 − e2 )2
(2)
√
where n = GM/a3 is the mean orbital frequency and
R is the Earth’s mean equatorial radius; for the Lageos
satellites they are as large as ≈ 108 milliarcseconds per
year (mas/y), while the Lense-Thirring precessions of Eq.
(1) are of the order of ≈ 101 mas/y (see Table 1). The
orbital precessions caused by the other even zonals with
degree ` > 2 are about 1000 times smaller than those
due to J2 . Thus, the impact of the asphericity of the Earth
is a major systematic bias in the attempts of measuring
the Lense-Thirring effect with artificial satellites. It can
be seen from Table 1 that the Earth’s J2 should be known
with a ≈ 10−13 uncertainty to allow for a ≈ 1% determination of the Lense-Thirring drag from individual satellites’
nodes; such a level of accuracy is currently far from being
obtained (see the discussion in Section 3.2 and [55, 56]).
The even zonals are nowadays determined as solved-for
parameters of global gravity field solutions by processing large data sets collected by dedicated spacecrafts
such as CHAMP, GRACE and GOCE [57]; also observations of laser-ranged geodetic satellites are combined
with CHAMP/GRACE/GOCE data [58]. Most of the next
efforts by several researchers were devoted just to reduce
as much as possible the unavoidable bias caused by the
even zonals.
In 1959 Yilmaz [59] proposed “to launch an artificial satellite whose plane contains the axis of rotation of the earth”;
see also [60]. In principle, the idea of using a satellite in a strict polar orbit is sound because, while the
Lense-Thirring precession of the node is independent of
the satellite’s inclination i (see Eq. (1)), the competing
much larger precessions caused by the Earth’s quadrupole
and higher moments are all proportional to cos i [54]; thus,
for i = 90◦ they would be exactly canceled out. As we will
see, the polar geometry will be revamped several times in
the forthcoming years.
A step further came in 1974 when Davies [61] noticed:
“The advance of the perigee or apogee of an Earth satellite has several sources such as the irregular shape of the
Earth […] How then do we separate out all these effects?
All of them depend upon the direction of motion of the
satellite except the Lense-Thirring effect which depends
on the direction of rotation of the Earth. Consequently,
with two satellites moving in the same type of orbit, but in
opposite directions one would […] have a differencing type
of experiment.” Davies [61] went on by suggesting to use
drag-free satellites to compensate the unwanted effects
of the atmospheric drag which would be non-negligible
at the small altitudes (h = 122 km) suggested by him.
Davies [61] argued that “equatorial orbits are optimal for
the experiment itself but are not as practical as inclined
orbits from a launching viewpoint”. Finally, he suggested
to use laser-ranging to track the satellites. Although proposed preliminarily and without quantitative details, the
ideas by Davies [61] are rather remarkable since they anticipate certain further studies, seemingly independent of
it, which can be substantially traced to more or less faithful variations of what was present in [61].
In 1976 van Patten and Everitt [62, 63] proposed to con533
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History of the attempts to measure orbital frame-dragging with artificial satellites
Table 1.
Key orbital parameters of the satellites of the Lageos family and of Sputnik-3, and their node precessions. Ω̇LT is the Lense-Thirring rate,
while Ω̇2 is the classical rate caused by the Earth’s even zonal J2 = 1.08 × 10−3 . The accuracy with which J2 should be known to allow
for a ≈ 1% determination of the Lense-Thirring effect is of the order of ≈ 10−13 .
i (◦ )
S/C
a (km) e
Lageos
12, 270 0.0045 109.9 30.7
Lageos-2 12, 163 0.014
LARES
7, 820
Sputnik-3 7, 420
Ω̇LT (mas/y) Ω̇2 (mas/y)
52.65 31.5
4 × 108
−8 × 108
0.0007 69.5
118
−2 × 109
0.11
141
−3 × 109
sider the sum of the node precessions of two counterorbiting drag-free satellites in nearly polar orbits. The
concept of the experiment implied that, in addition to precision Doppler tracking data from existing ground stations,
satellite-to-satellite Doppler data should have been taken
at points of passing near the poles to yield an accurate
measurement of the separation distance between the two
satellites. Precautions to avoid collisions were studied in
[64].
Later, in 1978 Cugusi and Proverbio [22] proposed to
use the Satellite Laser Ranging (SLR) technique [65] and
geodetic satellites such as Starlette and Lageos, which
are passive being entirely covered by retroreflectors to
bounce back the laser pulses sent to them from groundbased stations, to measure the Lense-Thirring effect. Cugusi and Proverbio [22] were aware of the problems posed
by the even zonals, but did not deal with it in details.
They wrote: “Therefore the possibility of showing the relativistic effects by means of the analysis of laser ranging
data should not be disregarded, provided this analysis
also takes into account all the secular or periodic perturbations […] Satellites almost drag-free, as e.g. Starlette, are the most suitable to this purpose”. Cugusi and
Proverbio [22] explored also the possibility of launching a
hypothetical new satellite with the purpose of maximizing
the relativistic effects; nonetheless, they were pessimistic
since [22] “no significant increase of relativistic perturbation values seem possible not even by means of a particular choice of the orbital elements, without also increasing
air-dragging and effects arising from spatial and temporal
variations of the Earth’s gravitational field.”.
A variant of the idea by Davies [61], and of the proposal
by van Patten and Everitt [62, 63] itself, was put forth
by Ciufolini [13] in 1986. After writing about the zonals issue that “A way of overcoming this problem would
be to measure accurately J2 , J4 , J6 , . . ., by orbiting several
high-altitude, laser-ranged satellites, plus Lageos to measure the Lense-Thirring effect”, Ciufolini [13] proposed to
launch a new geodetic satellite Lageos X in a supplemen-
65
tary orbit with respect to Lageos3 aLageos = aLageos X =
12270 km; iLageos = 110◦ , iLageos X = 70◦ in such a way
that the sum of the nodes would have canceled out the
precessions due to the Earth’s multipoles. Putting aside
technological considerations, the conceptual relationship
of the proposal by Ciufolini [13] with that by van Patten and Everitt [62, 63] is evident since it can easily be
shown that an orbital configuration having two distinct orbital planes with supplementary inclinations is equivalent
to one with a pair of counter-revolving satellites in the
same plane.
In 1997 Peterson [66] studied various constellations of
new SLR satellites to be launched to measure the LenseThirring drag along with the then existing Lageos and
Lageos-2. In particular, he first analyzed the sum of
the nodes of the Lageos/Lageos-3 configuration originally
proposed by Ciufolini [13] and of an analogous supplementary configuration with the existing Lageos-2 and a newly
proposed Lageos-6. Then, a constellation with Lageos
and two new satellites, dubbed Lageos-4 and Lageos-5,
to be placed in quasi-supplementary orbits with respect
to Lageos and their longitudes of perigee π was considered. Using the JGM3 model [67], and explicitly solving for
a dedicated Lense-Thirring parameter from the simulated
data, Peterson [66] concluded that the optimal choice was
the Lageos/Lageos-3 constellation, followed by the analogous Lageos-2/Lageos-6 one at about the same level of
accuracy (5 − 8%); the Lageos/Lageos-4/Lageos-5 combination was not competitive with its 22% level of accuracy.
The impact of certain non-gravitational disturbing thermal accelerations connected to the time evolution of the
satellites’ spin vectors [68, 69] was 5% (Lageos/Lageos-3),
12% (Lageos-2/Lageos-6), 30% (Lageos/Lageos-4/LageosIt
was
launched
on
May
4,
1976
with
a
Delta-2913
rocket;
see
http://space.jpl.nasa.gov/msl/QuickLooks/lageosQL.html
on the WEB.
3
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5). For a better comprehension of the overall discussion, the reader must be aware that non-gravitational
perturbations like atmospheric drag, direct Sun’s radiation pressure, Earth’s albedo, etc. [70] are, in general,
much stronger on the perigee than the node of an Earth’s
satellite such as Lageos. Rubincam [68, 69] originally
introduced thermal thrust (the Yarkovsky effect) into the
Lageos analysis to explain the observed anomalous alongtrack acceleration. Farinella et al. [71] pointed out that
thermal thrust can also produce long-period and secular
perturbations to the Lageos node; the perturbations produced depend on the satellite’s spin axis orientation in
the inertial space; it changes with time because of naturally occurring torques. These perturbations introduce
a significant systematic error into determinations of the
Lense-Thirring effect.
The error from thermal thrust cannot be eliminated by just
taking certain combinations of node rates for two satellites, a procedure discussed above for gravitational field
determination errors. Indeed, each satellite’s spin axis has
a different orientation and evolution with time, so thermal thrust errors for different satellites are not correlated.
Progress in the Lense-Thirring determination requires that
the Lageos spin axis be determined as a function of time
[72, 73].
With the the spin axis known, the actual thermal thrust
must still be modeled, the actual spacecraft construction
introducing extra complexity into the computation [74].
A further development of the mission concept originated
by Davies [61] came with the 2003 papers by Iorio [75–
77] in which he proposed to look at the difference of the
perigee precessions of two drag-free satellites orbiting
in supplementary orbital planes. In this case, the LenseThirring perigee precessions are opposite and sum up (see
Eq. (1)), while the even zonal perigee perturbations are
identical [54] and cancel out. [68, 69, 72, 78–92]. Iorio and
Lucchesi [76] used the covariance matrix of the EGM96
gravity model [93] up to ` = 20 and a pair of hypothetical
satellites S1/S2 with a = 12000 km, iS1 = 63.4◦ , iS2 =
116.6◦ by obtaining an overall accuracy of ∼ 5% over a 6
yr mission duration.
On February 13, 2012 LARES [17], the heir of the original proposal by Ciufolini [13], was launched, although at
a much lower altitude than Lageos (and Lageos-2). Its
proponent aims to reach a 1% accuracy [94] by combining its data with those of the other two existing satellites
of the Lageos family. For a debate about the feasibility
of such an ambitious goal, see [16, 28, 94–99]. However,
some years will be needed to obtain the first results.
3. The tests with the Lageos satellites
Lageos was followed by its twin Lageos-2, put into orbit
on4 October 22, 1992 by the Space Shuttle. Its orbital
elements (a = 12163 km, i = 52.65◦ ) are different from
those required in [13].
3.1.
The tests with the perigee of Lageos-2
Nonetheless, the laser pulses from both Lageos and
Lageos-2 were soon used for the first attempts to measure the Lense-Thirring effect in the Earth’s gravity field.
In 1996 Ciufolini [100] proposed to linearly combine the
numerically integrated residuals of the nodes of Lageos
and Lageos-2 and the perigee of Lageos-2 with numerical
coefficients c1 , c2 theoretically computed from the standard formulas [54, 100] of the secular precessions of these
orbital elements caused by J2 and J4 . Thus, the resulting
linear combination
δΩLageos + c1 δΩLageos−2 + c2 δωLageos−2
(3)
was, by construction, ideally not impacted by the first two
even zonals causing the largest competing precessions.
Indeed, at that time the mismodeling in the estimated values of J2 and J4 did not allow the use of a single orbital element to extract the Lense-Thirring precession because the
overall systematic uncertainty in the Earth’s multipoles
would have been overwhelming. The Lense-Thirring effect
would have impacted Eq. (3) in full as a linear trend since
it was planned neither to explicitly model it nor to solve
for it in the data reduction. Ciufolini [100] argued that the
accuracy of the existing Earth gravity models at that epoch
such as JGM3 [67] and EGM96 [93] would have been sufficient to constrain the impact of the other uncancelled even
zonals J6 , J8 , . . . to an acceptable level (25% or less). This
strategy was implemented in the subsequent tests [101–
104]. As a measure of the realistic uncertainty in the even
zonals, Ciufolini [100, 102, 103] took
the absolute value
.
of the differences |∆J` | = J`A − J`B , ` = 2, 4, . . . among
the estimated coefficients of two different Earth gravity
models A and B whose intrinsic accuracies were quite different; for example, the models JGM3 [67] and GEMT-3S
[105] were used in [101–103], where the estimated errors
σJ` , ` = 2, 4, . . . of the individual coefficients of JGM3 [67]
were more accurate than those of GEMT-3S [105] by about
See http://space.jpl.nasa.gov/msl/QuickLooks/lageosQL.html
on the WEB.
4
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one order of magnitude. More precisely, Ciufolini [100]
wrote: “Now, there is a basic problem to evaluate if these
estimated errors in the spherical harmonic coefficients of
the Earth’s gravity field solution are consistent with the
true errors in the value of these coefficients. To see where
the main errors are likely to be concentrated […], one might
take the difference between two different gravity field solutions. This method has been, for example, applied by
Lerch et al. [106] to the solution GEML-2.” About the
differences |∆J` | , ` = 2, 4, . . ., Ciufolini [100] wrote that
they “should provide upper limits to the real errors” of the
even zonals. The same concepts are repeated in [103] as
well. It turned out that the differences |∆J` | , ` = 2, 4 were
larger than the individual estimated errors σJ` , ` = 2, 4 of
the more accurate model JGM3 [67], being almost equal
or smaller for ` ≥ 6. The impact of other long-period
gravitational perturbations such as the tides was studied
in [107–109]. In their latest test, Ciufolini et al. [104]
used the EGM96 Earth gravity model [93] by reporting
a successful confirmation of the relativistic prediction for
the Lense-Thirring effect with a 20% total uncertainty; the
systematic bias caused by the imperfect knowledge of the
Earth’s multipoles was evaluated as large as [104] 13%
from the covariance matrix of EGM96. The systematic bias
due to the non-conservative accelerations was treated in
several independent analyses [85–87, 90, 110, 111] because of their potentially corrupting effect on the perigee
of Lageos-2. Also the systematics due to the even zonals
was reanalyzed by Iorio [54] and Ries et al. [110, 111].
About this issue, Ries et al. [111] remarked that a “serious
problem is the use of a very favorable negative correlation
between zonals in EGM96 (the result of poor separation
of the zonals in the gravity solution) to reduce the error
introduced by the gravity model from approximately 50%
to 13%. The EGM96 covariance, like any gravity solution covariance, is only an approximate estimate of the
errors in the gravity solution; it cannot be considered to
be an exact representation of the magnitude or correlation
of the error in the individual coefficients. Further, there is
no reason to expect that the errors in the EGM96 gravity model (which is a multi-decade mean gravity solution)
are representative of the actual errors in the gravity model
during the period of the Lense-Thirring analysis, in light
of known secular, seasonal and decadal variations in the
Earth’s gravity field. A more realistic error assessment
would not rely on the cancellation of the errors due to
a fortunate correlation, and it probably would treat the
magnitude of the errors in the higher degree zonals given
by the EGM96 covariance with some caution. This would
lead to an estimated error in the current determination of
the Lense-Thirring precession of at least 50 to 100%, if
not larger.” Iorio [54] obtained a 46.5% bias by using the
published sigmas σJ` , ` = 6, 8, . . . , 20 of the variance matrix of EGM96; he neglected the correlations among the
estimated multipoles. About the general strategy adopted,
Ries et al. [110] stated that “It is risky, however, to put too
much weight on the fact that the estimate is close to the
general relativity prediction, because the current models
for the gravity field and the thermal forces have been developed using the same data and models that already have
assumed that general relativity is correct. […] A simulatenous recovery of the gravity field, the LT precession and
all the other adjusted parameters would be a more reliable
experiment, and […] would provide a much more rigorous
error estimate.” Ciufolini et al. [104] evaluated the overall
impact of the non-conservative accelerations to be ∼ 13%
or less, but Ries et al. [110], in dealing with the thermal
forces, wrote that “the influence of these unknown forces
on the perigee cannot be confidently quantified. […] It appears to be questionable to expect that the LT perigee
signal can be extracted from the nongravitational noise
with any reliability”. Also Vespe [85] recognized the perturbations of Earth penumbra on the perigee of Lageos-2
as a further source of systematic uncertainty not properly accounted for in [104], thus increasing the total error
budget.
3.1.1.
Looking for alternative orbital combinations
In 2002 Iorio [112], extending the 1996 approach by Ciufolini [100], looked at the possibility of getting rid of more
even zonals with linear combinations including the nodes
of other SLR satellites such as Ajisai, Starlette, Stella
and Westpac-1 in addition to the nodes of both the Lageos satellites and the perigee of Lageos-2. By using the
covariance matrix of EGM96 [93] up to ` = 20, a slight
improvement was obtained from the inclusion of the node
of Ajisai which would allow to cancel out J6 as well [112].
In [113] the same calculation was repeated with EGM96
[93] up to ` = 20 neglecting the reciprocal correlations
among the estimated even zonals: a systematic error
as large as 64% was obtained for that combination with
EGM96 [93] up to ` = 20.
An earlier analysis about the use of the other existing
geodetic satellites was done in 1990 by Casotto et al.
[114], but the nodes of each spacecraft were treated separately.
3.2. Eliminating the perigee of Lageos-2 with
the GRACE models
The launch of CHAMP (July 15, 2000) and, especially,
GRACE (March 17, 2002), two dedicated missions for the
precise measurement of the Earth’s multipoles, opened
a new era for the accurate determination of geopoten-
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tial with space-based techniques [57]. Several global
gravity field solutions, produced by different worldwide
institutions, started soon to be released: most of them
are freely available at the ICGEM [115] WEB portal
http://icgem.gfz-potsdam.de/ICGEM/
The opportunities offered by the new, more accurate models to drastically improve the overall accuracy of the
Lense-Thirring tests with the Lageos satellites were readily investigated by several researchers [110, 111, 116–
121]. Basically, it was acknowledged, at different levels
of detail, that it became possible to discard the perigee
of Lageos-2 by combining just the nodes of Lageos and
Lageos-2 to cancel out J2 . Actually, Peterson [116] looked
at the sum of the nodes of Lageos and Lageos-2 in view of
the forthcoming models from GRACE. About the expected
improvements by the GRACE models, Ries et al. [110]
wrote that “A significant improvement in the knowledge of
the Earth’s gravity field will be provided by the upcoming
GRACE mission. This will remove the dependence on the
perigee signal […]”. Later, Ries et al. [111] added that
“the only major source of uncertainty in the static gravity
field remaining would be in the dominant J2 coefficient.
This problem is eliminated by using the two node rates
to recover the LT parameter and J2 .” In April 2003 [118],
August 2003 [121] and September 2003 [120] Iorio posted
on the ArXiv database some papers, later published in
journals and conference proceedings, where he explicitly
worked out a linear combination of the nodes of Lageos
and Lageos-25
δΩLageos + k1 δΩLageos−2 , k1 ∼ 0.546
(4)
impacted by all the even zonals apart from J2 . The numerical value of k1 was obtained by the known standard
formula [54, 101] for the J2 node precession, and it allows, in principle, to exactly remove the bias due to the
Earth’s quadrupole. Moreover, Iorio [118–121] quantitatively calculated the magnitude of the systematic error
due to the terrestrial multipoles with some of the recently
released CHAMP/GRACE models and Eq. (4). Iorio and
Morea [118] used Eq. (4) and the EIGEN2 (CHAMP)
model [122] by using its covariance matrix, and its sigmas
σJ` , ` = 4, 6, . . . 70 in both a Root-Sum-Square (RSS) and
Sum of the Absolute Values (SAV) way getting figures as
large as 18 − 37%. The sigmas σJ` , ` = 4, 6, . . . 70 of the
GGM01C (GRACE) model [123] yielded 14% (RSS) and
18% (SAV) [118]. The same results are reported in [120].
In [121] also the EIGEN-GRACE01S model was used with
5
Iorio [120, 121] acknowledged [111].
Eq. (4) by finding a 21% (RSS) error. As reported later by
Iorio [124], Ciufolini was aware of such findings as a result
of several private communications6 between Iorio and Ciufolini himself on March, 2003 [124], and between Iorio and
Pavlis on September 2003 [124]. All the models EIGEN2,
EIGEN-GRACE01S and GGM01 were quite preliminary,
being based on a limited amount of data from CHAMP
and GRACE. Later, in August 2004 Iorio7 [119] applied the
more trustable EIGEN-GRACE02S model [125] to Eq. (4)
in a calculation up to ` = 70 by finding a total systematic
bias of 3% (RSS) and 4% (SAV). As a consequence, Iorio
[119] wrote: “Then, with a little time–consuming reanalysis of the nodes only of the existing Lageos and Lageos2 satellites with the EIGEN-GRACE02S data it would
at once be possible to obtain a more accurate and reliable measurement of the Lense Thirring effect, avoiding
the problem of the uncertainties related to the use of the
perigee of Lageos II.” In [119] there are also the first quantitative evaluations of the potentially corrupting bias due
to the secular variations J̇` , ` = 4, 6 of the even zonals
on a measurement of the Lense-Thirring effect with Eq.
(4) which may be as large as 13% (RSS)-42% (SAV). The
role of temporal changes in the Earth’s multipoles on the
Lageos tests was qualitatively addressed by Ries et al.
[110, 111].
In July 2004 Lucchesi [126] gave a talk at the 35th
COSPAR Scientific Assembly in which he presented an
analysis of the laser data of Lageos and Lageos-2 combined according to Eq. (4) and the EIGEN2 model [122]
which confirmed the error budget by Iorio [118, 120, 121].
Later, in November 2004 Lucchesi8 [127] submitted a paper to the journal Advances in Space Research, published
in 2007, in which he expanded and refined the COSPAR
analysis based on Eq. (4) and the EIGEN2 model [122].
In June 2004 Ciufolini and Pavlis [128] submitted a paper to the journal Nature, published in October 2004,
in which they used Eq. (4) and the EIGEN-GRACE02S
model [125] by reporting a successful measurement of the
Lense-Thirring effect with a total accuracy of 5−10%. Ciufolini and Pavlis [128] acknowledged none of the previous
works by other researchers on this topic. In particular,
they acknowledged neither [111, 126] nor [118, 120, 121].
Moreover, they explicitly attributed Eq. (4) to Ciufolini
himself by citing [13].
Iorio [129] soon criticized some aspects of [130]. He noted
that the total error may be as large as 15 − 45% (1 − 3σ )
In particular, Iorio sent the draft of [118] to Ciufolini on
March 26, 2003 [124].
7
Iorio [119] acknowledged [111].
8
Lucchesi [127] acknowledged [118].
6
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History of the attempts to measure orbital frame-dragging with artificial satellites
because of the uncertainties in the long-term temporal
variations J̇` , ` = 4, 6, . . . of the even zonals impacting
Eq. (4). Also the a-priori evaluation of the error due to
the time-independent part of the geopotential by Ciufolini
and Pavlis [128] would be too optimistic since based on
ad-hoc choices in the computational strategies aimed to
get a too small figure with just a RSS calculation. Moreover, the needs of using also other Earth’s gravity field
solutions and varying the data sets were pointed out in
[129]. Iorio [129] also preliminarily remarked that the test
may be plagued by a sort of a-priori “memory” effect of GR
itself in the CHAMP/GRACE models. This point was further developed quantitatively in [131] in which the impact
of frame-dragging on the orbits of GRACE spacecrafts is
assessed, and in [132] where it was demonstrated how GR
globally does affect also the GRACE intersatellite tracking in a non-negligible way. Iorio [129] blamed Ciufolini
and Pavlis [128] because they did not acknowledge his
contribution to Eq. (4).
Ciufolini and Pavlis [130] replied to Iorio [129] with arguments sometimes weak. For example, they blamed Iorio
[129] for having allegedly proposed to include the mean
anomaly in future Lense-Thirring experiments, but such
claims are not correct. Moreover, Ciufolini and Pavlis [130]
defended themselves against the accusation of plagiarism
by Iorio [129] about Eq. (4) in an objectively unconvincing
way. Indeed, they claimed that Eq. (4) was substantially
contained in what Ciufolini wrote on pag. 279 of his 1986
paper [13] (see Section 2 for a quotation of his words). To
date, Ciufolini has never changed his mind, still refusing
to cite most of the works of his former collaborator. Iorio,
in turn, replied to [130] with [124]. Among other things, he
[124] discovered a further 9% systematic gravitational error
likely plaguing Eq. (4), caused by mixed effects between
the J2 node precessions and the inclination. Indeed, the
contribution of the uncertainty in the inclination to the
oblateness-driven node precession can be computed by
taking the derivative of Eq. (2) with respect to i. It turns
out that i should be known with an accuracy of9 ≈ 0.05
mas to cause a mismodelled classical node precession as
little as ≈ 1% of the Lense-Thirring rate. For further criticisms by Ciufolini and Pavlis [130] to other points of [129],
see Section 3.2.1. About the “imprinting” issue, Ciufolini
and Pavlis [130] rejected the warnings in [129] by erroneously comparing the node rates of GRACE due to the
Earth’s even zonals with the Lense-Thirring node rates of
It would roughly correspond to ≈ 2 − 3 mm in determining the satellite’s trajectory in a root-mean-square sense,
which is one of the goals, not yet completely reached by
all the ranging stations, of the laser-ranging community.
9
the Lageos satellites. Moreover, they [130] incorrectly referred to pre-GRACE era simulations of the Lageos orbits,
but did not give any quantitative details concerning them.
In [133] there is the first comparative study of the impact of different Earth gravity models on the Lageos node
test of the Lense-Thirring drag. The SAV approach was
followed with the sigmas σJ` , ` = 4, 6, . . . 20 of EIGENCG03C [134] (3.9%), EIGEN-CG01C [135] (6%), EIGENGRACE02S [125] (4%) and GGM02S [136] (9%) by finding
a non-negligible scatter as far as the static part of the
geopotential is concerned. The consequences of the uncertainties in the secular variations J̇` , ` = 4, 6 of the
even zonals were investigated with numerical simulations
in [137] finding a 10−20% bias at 1−σ level. By combining
these results with those for the static part of the geopotential in [133], Iorio [137] obtained the range 22 − 25%
for the total error at 1 − σ level.
In a series of further works [28, 138, 139], Iorio et al.
took a step forward in the assessment of the systematic
error caused by the static part of geopotential with new
quantitative arguments. Other potential weak points of
the Lageos tests were elucidated in [28, 138, 139] as
well. Summarizing, Iorio et al. [28, 138, 139], having at disposal several Earth’s gravity models released
by different
institutions
worldwide, took the differences
. ∆J` = J`A − J`B , ` = 2, 4, 6 . . . 20 between the estimated
multipoles for several pairs (A, B) of models as representative of the real uncertainties of the even zonals.
See also [140]. The models considered in [28, 138, 139]
were EIGEN-GRACE02S [125], GGM02S [136], GGM03S
[141], EIGEN-CG03C [134], ITG-Grace02s [142], ITGGrace03s [143], ITG-Grace2010s [144], AIUB-GRACE01S
[145], AIUB-GRACE02S [146], JEM01-RL03B from JPL
(NASA, USA), EGM2008 [147]. For each pair of models,
both SAV and RSS calculation were performed. The resulting scatter was rather large, ranging from ∼ 10% to
∼ 25 − 30%. The differencing method to assess the realistic errors in the Earth’s gravity field multipoles is widely
used in the space geodesists community, irrespectively of
their relative accuracy: see the discussion in [28] and the
recent [55]. Ciufolini himself [101–103] used it in the earlier tests including the perigee of Lageos-2; see Section
3.1. Nonetheless, he [16] now changed his mind by criticizing the choice of Iorio [138] with arguments which can
be judged incorrect. Indeed, Ciufolini et al. [16] asserted
that one should not compare models with different intrinsic accuracies. Actually, a recent analysis by Wagner and
McAdoo [55] described extensively how it is necessary to
use models with quite different intrinsic accuracies. Later,
Iorio followed their method in [56], although in a different context. Incidentally, it can be remarked that, as a
consequence of [56], the current uncertainty in J2 is up to
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Giacomo Renzetti
2 − 3 orders of magnitude worse than the level required to
get a ≈ 1% determination of the Lense-Thirring effect from
individual node rates (see Section 2 and Table 1). Iorio et
al. replied to Ciufolini et al. [16] in [28] by remarking that
the rejection of some particular values of an experimentally determined quantity has to be based on quantitative,
statistical criteria. Iorio et al. [28] applied some of them
to the values of J4 determined in several models available at that time by showing that, actually, there are no
quantitative reasons to discard any of them. Iorio [139]
used the sigmas σJ` , ` = 4, 6, . . . 20 for the uncertainties
in the zonals, and obtained a 15% error from an average
over all the models considered. Iorio et al. [28, 138] followed also another strategy consisting of computing the
sum up to ` = 20 of all the nominal even zonals node
precessions combined with Eq. (4) for each pair (A, B) of
models and taking the absolute values of their differences.
This particular method yielded, on average, a 17% bias.
As previously noticed by Nordtvedt [148] and Ries et al.
[110], Iorio et al. [28, 138, 139] stressed once more that a
possibly more robust determination of the Lense-Thirring
effect implies its explicit inclusion in the Lageos models in
such a way that a dedicated parameter can be solved-for
in the data reduction process. Even better, it should be
explicitly modeled and simultaneously determined along
with the Earth’s multipoles in a new generation of Earth’s
global gravity field solutions. Ries et al. [110] wrote that
“It is unlikely that the error estimates would be the same
if the LT precession were estimated simultaneously with
the even zonals, which highlights what may be one of the
most problematic aspect of the error analysis”. To date, it
has not yet been done.
Another problematic issue, neglected so far in all the aprori error budgets proposed in literature, was treated in
[149] where it is basically remarked that the cancelation
of the even zonal of lowest degree by Eq. (4) can be exact
only in principle. Since the coefficient k1 entering Eq. (4)
is theoretically computed so that its analytical expression
contains some orbital elements of both the Lageos satellites, the necessarily limited accuracy with which they can
be determined from real data reduction translates into a
certain level of uncertainty in the value of k1 itself. Thus,
the combined impact of J2 on the Lageos/Lageos-2 nodes
is removed from Eq. (4) only imperfectly; Iorio [149] suggested a further contribution of 14 − 23% to the total error
budget depending on the errors assumed in the inclinations of the satellites.
In [150], two further weak points of the Lageos tests were
stressed. First, it was noticed that after the first attempts
about 15 years ago no really independent tests have been
published so far in peer-reviewed journals by authors dif-
ferent from I. Ciufolini, apart from10 a handful of conference
talks by a group led by Ries et al. [151–154] who even
joined later the team of Ciufolini himself. Their results
were included in [16, 98, 155–157]. Although different orbital processors were used by Ries et al., the strategy
followed so far was always the same as in the earlier
works by Ciufolini et al. in the sense that frame-dragging
was neither explicitly modeled nor solved-for in the Lageos data. This is a little bit surprising in view of the
statements and extensive numerical simulations [158] by
Ries et al. prior to their enrollment with the Ciufolini
team. Second, Iorio [150] applied the well established
method of assessing the real errors in the low-degree even
zonals of different models by comparing each to the first
Earth gravity models produced from GOCE data. The pair
AIUB-GRACE02S [146] and GOCO01S [159] provided a
23% error for ` = 4, 6, . . ., while EIGEN51C [160] -AIUBGRACE02S [146] yielded 27%.
Lately, Iorio [161] remarked that a further 20% bias might
occur over multi-decadal time spans comparable to those
used in the data analyses performed so far with Eq. (4)
because of the the uncertainties in the spatial orientation
of the terrestrial spin axis. Indeed, it changes in time due
to a variety of reasons, known with necessarily limited accuracy, while the value employed so far for k1 holds just
for a fixed spin aligned with the z axis of the coordinate
system employed. Thus, also for this reason the removal
of J2 from Eq. (4) would not be perfect. Moreover, Iorio
[161] noticed that a partial/total cancelation of the relativistic signal itself may occur in the estimation of, say,
the satellites’ state vectors at the beginning of each arc
used in actual data reduction. The need of explicitly modeling and estimating a frame-dragging parameter is, thus,
stressed. Finally, another form of a-priori “imprinting” of
relativity itself may actually lurk in the Lageos tests since
the data are analyzed within the framework of a coordinate system whose materialization is largely based just
on Lageos/Lageos-2 SLR observations [161].
Funkhouser et al. [162] criticized the overall accuracy
of the error budgets in both the Lageos experiments by
Ciufolini et al. [104, 128].
The reply by Ciufolini et al. [98] contains no real advances
from the scientific point of view. Ciufolini et al. [98] repeated once more some points previously exposed elsewhere without providing new quantitative elements supporting them, and leaving many of the remarks by Iorio et
al. [28] essentially unaddressed. For example, according
Strictly speaking, Iorio [150] was wrong because of the
work by Lucchesi [127], but it does not seem it was further
pursued.
10
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History of the attempts to measure orbital frame-dragging with artificial satellites
to Ciufolini et al. [98], none of the claims by Iorio et al.
[28] could be reproduced by any of their independent analyses. Such a statement is, in fact, incorrect since it may
induce the readership to believe that, after the publication
of [28], Ciufolini et al. [98] made new data analyses coping
with the remarks in [28]. Actually, it is not so. Moreover,
several quotations by Ciufolini et al. [98] from [28] appear
as inexact. Some of the claims by Ciufolini et al. [98]
about the a-priori relativistic “imprinting” in the GRACE
data were dealt with in [132].
3.2.1. Alternative
combinations
CHAMP/GRACE models
and
the
Iorio [163, 164] studied the impact of the new
CHAMP/GRACE-based Earth gravity models on alternative combinations including the nodes of Ajisai and Jason1 as well able to remove, in principle, J2 , J4 , J6 . A calculation with the EIGEN-CG01C (CHAMP+GRACE) model
[135] up to ` = 20 yielded [163] 0.7% (RSS) 1.6% (SAV)
for the gravity modeling error; the larger non-gravitational
perturbations on Ajisai and, especially, Jason-1 would
have a 4% impact on the Lense-Thirring trend [163]. A similar analysis was investigated by Vespe and Rutigliano in
[165]; in addition to Lageos, Lageos-2, Ajisai and Jason-1,
they looked also at Starlette, Stella, Etalon and the GPS
satellites. Iorio [129] urged the scientific community to
look at actual feasibility of his proposal of using the node
of Jason-1 with quantitative tests backed by real data. To
date, his suggestion has not yet been practically implemented by anyone. A SAV calculation [164] extended to
` = 40 with EIGEN-CG03C [134], EIGEN-CG01C [135],
EIGEN-GRACE02S [125] and GGM02S [136] provided an
upper bound of 1.0 − 2.6% of the systematic bias due to
the even zonals. Iorio [164] concluded that a test with a
total accuracy of 4 − 5% over at least 3 years would be
possible.
Ciufolini and Pavlis [130] claimed that the proposal in
[163] is unfeasible. Actually, they did not discuss any
specific point of the content of [163]. Ciufolini and Pavlis
[130] neither presented data analysis nor numerical simulations to cogently support their conviction, contrary to
what asked in [129]. Ciufolini and Pavlis [130] wondered
why not using the GRACE satellites themselves, whose
orbit can be reconstructed with a much better accuracy
than Jason-1. Iorio [124] replied showing that the inclusion of their nodes, along with those of CHAMP, Starlette
and Stella as well, would be unfeasible because of their
much lower altitude and, thus, higher sensitivity to a wider
range of even zonal harmonics and of other time-varying
gravitational perturbations like, e.g., the tesseral K1 tide.
Indeed, by using EIGEN-CG01C [135] it was shown [54]
that computational instabilities in the secular node pre-
cessions occur around degree ` ∼ 40, making difficult a
reliable evaluation of the total error budget.
4.
Conclusions
It has been shown how the history of the first attempts to
measure the Lense-Thirring effect with terrestrial artificial
satellites is intricate, complex and branched, being in fact
rooted in the independent efforts by several researchers
aimed to explore different routes with mixed success.
Using Lageos and Lageos-2 proved itself so far the most
promising way to detect the orbital Lense-Thirring drag
in the Earth’s surrounding. The inclusion of other existing
SLR satellites such as Starlette, Stella, etc. turned out
to be a somewhat dead end for a variety of reasons such
as their lower altitude enhancing the perturbations caused
by a wider range of even zonal multipoles of the terrestrial
gravitational field.
The relevant attempts already made with Lageos and
Lageos-2 should be complemented by further analyses
based on a different methodology with respect to that
followed until now. In particular, as remarked by various researchers, the gravitomagentic force should be explicitly modeled and solved-for in the data reductions
of Lageos/Lageos-2 observations. They should be analyzed by varying the data sets and the force models, especially the gravitational ones which should be carefully
selected in order to avoid any possible a-priori “contamination” of GR itself. This requirements necessarily restricts the choice of the suitable Earth’s gravity models
by excluding the latest ones which are currently going
to be produced by using Lageos itself. It would also be
desirable that a new generation of global gravity models will be released, in which GR is simultaneously estimated along with the classical multipoles of the geopotential. The possibility that the existing models based only
on CHAMP/GRACE/GOCE themselves have a relativistic
“imprint” cannot be excluded, as indeed quite plausible.
Similar considerations hold also for the future experiments
to be performed with LARES as well.
The a-priori evaluation of the systematic uncertainty due
to the even zonals should not be restricted to just some
models purposely chosen to yield the smallest figure; the
same holds also for the choice of the computational approach adopted for its quantitative assessment. After following such prescriptions, it turns out that the impact
of our imperfect knowledge of the geopotential on the
expected Lense-Thirring signature is likely as large as
∼ 15 − 30% or, perhaps, even larger, strongly depending
on the Earth gravity models adopted, on the type of calculation of their effects (RSS, SAV), and on the gravitational
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Giacomo Renzetti
disturbances themselves taken into account (necessarily
imperfect cancelation of J2 , other uncanceled even zonals
J4 , J6 , J8 , . . ., etc.).
After 16 years since the first important attempts reported
by Ciufolini et al., it is unsatisfactory that no other really
independent groups worldwide decided either to independently perform their own analyses or, if it was done, not
to publish them in peer-reviewed journals whatever their
outcome may have been. Few conference presentations
spread over the years have so far represented only one
school of thought, especially in view of the fact that the
scientific backgrounds and the methodologies followed are
essentially the same.
Acknowledgments
I thank the referees for their pertinent and important comments which improved the manuscript.
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History of the attempts to measure orbital frame-dragging with artificial satellites
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History of the attempts to measure orbital frame-dragging with artificial satellites
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