A comment on "A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model", by I. Ciufolini et al

0.19 MB·

by  Lorenzo Iorio

Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
A comment on “A test of general relativity using
the LARES and LAGEOS satellites and a GRACE
Earth gravity model”, by I. Ciufolini et al.
Lorenzo Iorio
Received: 14April2016/Accepted: 8January2017
7
1
0
Abstract Recently, Ciufolini et al. reported on a test of the general rela-
2
tivistic gravitomagnetic Lense-Thirring effect by analyzing about 3.5 years
n
of laser ranging data to the LAGEOS, LAGEOS II, LARES geodetic satel-
a
J lites orbiting the Earth. By using the GRACE-based GGM05S Earth’s global
3 gravity model and a linear combination of the nodes Ω of the three satel-
2 lites designed to remove the impact of errors in the first two even zonal har-
monic coefficients J , J of the multipolar expansion of the Newtonian part
2 4
]
of the Earth’s gravitationalpotential, they claimed an overallaccuracy of 5%
c
q for the Lense-Thirring caused node motion. We show that the scatter in the
- nominal values of the uncancelled even zonals of degree ℓ = 6, 8, 10 from
r
g some of the most recent global gravity models does not yet allow to reach
[ unambiguously and univocally the expected 1% level, being large up to
1 .15% (ℓ=6), 6% (ℓ=8), 36% (ℓ=10) for s≈ome pairs of models.
v
Keywords Experimental studies of gravity Experimental tests of grav-
4
·
7 itational theories Satellite orbits Harmonics of the gravity potential
· ·
4 field
6
0 PACS 04.80.y 04.80.Cc 91.10.Sp 91.10.Qm
. · · ·
1
0
7
1 Introduction
1
:
v InRef.[5],Ciufolinietal.reportedatestofthegeneralrelativisticgravitomag-
i
X neticLense-ThirringeffectinthegravitationalfieldoftheEarthusingalinear
combination of the right ascensions of the ascending nodes Ω of the LARES,
r
a
L.Iorio
Ministerodell’Istruzione,dell’Universit`aedellaRicerca(M.I.U.R.),VialeUnit`adiItalia68
Bari,(BA)70125,Italy
Tel.:[phone]
E-mail:[email protected]
2 L.Iorio
Table1 Relevantorbitalparametersa(km), e, I(deg)ofLAGEOS,LAGEOSII,LARES,
their classical node precession coefficients Ω˙.ℓ for the first five even zonal harmonics (ℓ =
2−10),theirLense-ThirringnodeprecessionsΩ˙LT forS=5.86×1033 kgm2 s−1,andthe
combination coefficients c1, c2. All the rates, in milliarcseconds per year (mas yr−1), are
expressed to an accuracy of 0.1 mas yr−1 which corresponds to about a percent fraction
of the predicted Lense-Thirring combined signature CLT = 50.2 masyr−1. The numerical
values ofthecoefficients c1, c2 arequoted toanaccuracy adequate foracancelation ofJ2
toasub-percentlevel.
LAGEOS LAGEOSII LARES
a 12270 12163 7828
e 0.0045 0.0135 0.0008
I 109.84 52.64 69.5
Ω˙.2 4.159523197035×1011 −7.671024751108×1011 −2.0691803570443×1012
Ω˙.4 1.541082434098×1011 −5.57207688363×1010 −1.8385054326934×1012
Ω˙.6 3.29198354689×1010 4.98585219772×1010 −9.061255341802×1011
Ω˙.8 2.3906795991×109 1.10181009277×1010 −9.43157797573×1010
Ω˙.10 −1.407631461×109 −2.213156639×109 3.04267201897×1011
Ω˙LT 30.7 31.5 118.1
cj, j=1,2 - 0.344281069 0.073388218
LAGEOS and LAGEOS II laser-ranged geodetic satellites and a GRACE-
based Earth gravity model claiming an alleged overallsystematic uncertainty
of 5%. For an overview of past attempts, see Refs. [8,12,3] and references
therein.
Inthispaper,wewillquantitativelyshowthattheclaimsinRef.[5]aretoo
optimistic despite the recent advances in modeling the Earth’s gravity field.
Let us recallthat the gravitomagneticnode precessionof a test particle in
geodesic motion around a central rotating primary is [10]
2GS
Ω˙ = , (1)
LT c2a3(1 e2)3/2
−
whereGandcaretheNewtonianconstantofgravitationandthespeedoflight
invacuum,respectively,S isthebody’sangularmomentum,whileaandeare
the orbit’s semimajor axis and eccentricity, respectively. The magnitudes of
the Lense-Thirringprecessions for the satellites of the LAGEOS family, along
with their relevant orbital parameters, are listed in Table 1.
2 The systematic uncertainty due to the uncancelled even zonal
harmonics
Oneofthemostcriticalissuesofalltheattemptsplannedorperformedsofarto
detect the Lense-Thirringeffect with Earth’sartificialsatellites is represented
bythecorrecthandlingofthecompetingsecularnodeprecessionsΩ˙ induced
Jℓ
Acommenton“Atestofgeneralrelativity[...]” 3
bytheevenzonalharmoniccoefficients1 J = √2ℓ+1C , ℓ=2,4,6,...of
ℓ ℓ,0
−
the multipolar expansion of the Newtonian terrestrialgravitationalpotential.
Indeed, their nominal sizes are several orders of magnitude larger than the
Lense-Thirring node precessions; moreover, their current level of modeling is
not yet accurate enough to allow to use a single satellite’s node to extract
the relativistic signature. Thus, in the past years, some strategies to circum-
vent such an issue were devised. Contrary to what is claimed in Ref. [5], the
combination presumably used in Ref. [5], not displayed there, was explicitly
proposed by the present author2 on the basis of a strategy proposed for the
first time in Ref. [2]. It was suitably designed to cancel out, at least in princi-
ple, the node precessions due to the first two even zonals J , J , being fully
2 4
impacted by all the other ones of higher degree ℓ 6. The hope is that, since
≥
the geopotential is modeled in the data reduction softwares used to process
the satellites’ data, the level of mismodeling in the uncancelled even zonals
is accurate enough to allow for a reliable determination of the Lense-Thirring
effect with a percent accuracy.
Actually, as summarized in Refs. [8,12] and references therein, there are
sounddoubtsthatsuchastrategycouldallowtoreacha 1%overallaccuracy,
≈
as steadily claimed by Ciufolini et al. over the years. In this paper, we will
enforcesuchconcernsbyshowingthatsevereissueslurkeveninthelow-degree
part of the spectrum of the geopotential.
2.1 Review of the method used so far to extract the Lense-Thirring signature
For the convenience of the reader, we will now review the aforementioned
approach, which is a generalization of the one proposed for the first time in
Ref. [2].
By assuming to use the nodes of, say, N different satellites, the following
N linear combinations can be written down
N−1 ∂Ω˙(i)
µ Ω˙(i)+ J2k δJ , i=1,2,...N. (2)
LT LT ∂J 2k
2k !
k=1
X
They involve the Lense-Thirring node precessions as predicted by General
Relativity (see Eq. (1)), scaled by a multiplicative parameter µ , and errors
LT
in the computed secularnode precessionsdue to errorsin the firstN 1 even
−
zonals J , k = 1,2,...N 1, assumed as mismodeled through δJ , k =
2k 2k
−
1 TheparametersCℓ,0determiningJℓarethefullynormalizedStokescoefficientsofdegree
ℓ = 2j, j = 1,2,3,... and order m= 0 of the geopotential’s expansion. For a given value
of ℓ, which can assume odd values as well, m generally runs from 0 to ℓ. The multipolar
expansionofthegravitationalpotentialincludesalsotheStokescoefficientsSℓ,m;itisalways
Sℓ,0 =0, and the remainingones do not induce secular orbital precessions [9]. Thus, for a
givendegreeℓ,thereare2ℓ+1generallynonvanishingnormalizedStokes coefficients.
2 See[8,12]andreferencestherein;inparticular,[7].
4 L.Iorio
1,2,...N 1. In the following, we will use the shorthand
−
Ω˙ =. ∂Ω˙Jℓ (3)
.ℓ
∂J
ℓ
for the partial derivative of the classical node precession with respect to the
generic even zonal J of degree ℓ. Then, the N combinations of Eq. (2) are
ℓ
equatedtotheexperimentalresiduals∆Ω˙(i), i=1,2,...N ofeachnodeofthe
N satellites considered. In principle, such residuals account for the purposely
unmodelled Lense-Thirring effect, the mismodelling of the static and time-
varying parts of the geopotential, and the non-gravitationalforces. Thus, one
gets
N−1
∆Ω˙(i) =µ Ω˙(i)+ Ω˙(i)δJ , i=1,2,...N. (4)
LT LT .2k 2k
k=1
X
If we look at the Lense-Thirring scaling parameter µ and the mismodeling
LT
in the even zonals δJ , k = 1,2,...N 1 as unknowns, we can interpret
2k
−
Eq. (4) as an inhomogenous linear system of N algebraic equations in the N
unknowns
µLT, δJ2, δJ4...δJ2(N−1), (5)
N
whose coefficients are | {z }
Ω˙(i), Ω˙(i), i=1,2,...N, k =1,2,...N 1, (6)
LT .2k −
while the constant terms are the N node residuals
∆Ω˙(i), i=1,2,...N. (7)
Itturnsoutthat,aftersomealgebraicmanipulations,thedimensionlessLense-
Thirringscalingparameter,whichis1 inGeneralRelativity,canbe expressed
as
∆
µ = C . (8)
LT
LT
C
In Eq. (8), the combination of the N node residuals
N−1
.
=∆Ω˙(1)+ c ∆Ω˙(j+1) (9)
∆ j
C
j=1
X
is,byconstruction,independentofthe firstN 1evenzonals,beingimpacted
−
bytheotheronesofdegreeℓ>2(N 1)alongwiththe non-gravitationalper-
−
turbations and other possible orbital perturbations which cannot be reduced
to the same formal expressions of the first N 1 even zonal rates. Instead,
−
N−1
=. Ω˙(1)+ c Ω˙(j+1) (10)
CLT LT j LT
j=1
X
Acommenton“Atestofgeneralrelativity[...]” 5
combines the N gravitomagnetic node precessions as predicted by General
Relativity. The dimensionless coefficients c , j = 1,2,...N 1 in Eq. (9)-
j
−
Eq. (10) depend only on some of the orbital parameters of the N satellites
involved in such a way that, by construction, = 0 if Eq. (9) is calculated
∆
C
by posing
∆Ω˙(i) =Ω˙(i)δJ , i=1,2,...N (11)
.ℓ ℓ
for any of the first N 1 even zonals,independently of the value assumed for
−
its uncertainty δJ .
ℓ
In our case [7], it is N = 3; the satellites employed are LAGEOS, LA-
GEOS II, LARES. The resulting combinationcoefficients c , c yielding µ ,
1 2 LT
worked out from [1]
3 R 2 cosI
Ω˙ = n , (12)
.2 −2 b a (1 e2)2
(cid:18) (cid:19) −
5 R 2 1+ 3e2
Ω˙ =Ω˙ 2 7sin2I 4 , (13)
.4 .2"8(cid:18)a(cid:19) (cid:0)(1−e2)2(cid:1) − #
(cid:0) (cid:1)
calculated for LAGEOS, LAGEOS II, LARES as in Table 1, turn out to be
[7,8]
ΩLR ΩL ΩL ΩLR
c = .2 .4− .2 .4 0.344281069, (14)
1 ΩLII ΩLR ΩLR ΩLII≃
.2 .4 − .2 .4
ΩL ΩLII ΩLII ΩL
c = .2 .4 − .2 .4 0.073388218. (15)
2 ΩLII ΩLR ΩLR ΩLII≃
.2 .4 − .2 .4
In Eqs. (12)-(13), R is the primary’s equatorial radius, while n = GM/a3
b
and I are the satellite’s Keplerianmean motion and orbital inclination to the
p
reference x,y plane, respectively; M is the mass of the central body. The
{ }
combinedLense-Thirringsignal,calculatedfrom Eq.(1), Eq.(10), Eqs.(14)-
(15) and Eqs. (12)-(13), is
=50.2 mas yr−1. (16)
LT
C
We point out that the numerical values of c , c in Eqs. (14)-(15) are quoted
1 2
with nine decimal digits in order to assure a cancelation of J accurate to
2
better than 1% level. If the only concern were the evaluation of the impact of
theerrorsintheevenzonalsofdegreeℓ 6discussedinSection2.2,muchless
≥
accurate values as c 0.344, c 0.0733 would be adequate; nonetheless,
1 2
≈ ≈
the first even zonal harmonic would nominally impact the combined Lense-
Thirring trend at a 860% level. The authors of Ref. [5] did not quote any
numerical values for c , c .
1 2
In principle, the strategy reviewed above allows one to extract also δJ or
2
δJ , provided the substitution Ω˙(i) or Ω˙(i) Ω˙(i) is performed in Eqs. (14)-
4 .2 .4 → LT
(15).
6 L.Iorio
2.2 The impact of the uncancelled even zonals of degree higher than ℓ=4
and the need of using several Earth gravity models
A major source of systematic uncertainty, to be reliably and realistically as-
sessed, is represented by the impact of the mismodeling in the even zonals of
degree ℓ 6 affecting the classical node precessions which are not cancelled
≥
by the adopted linear combination.
Ciufolini et al. [5] limited themselves to use only a single Earth’s global
gravity field solution, GGM05S [13], without any motivation for such a seem-
inglyarbitrarychoice.Moreover,inRef.[5],itisclaimedthat,afterarbitrarily
tripling the releasedsigmas of the even zonal coefficients of such a model, the
overallsystematicuncertaintywouldbeaslittle as4%,withoutprovidingany
detailsoneitherthecomputationalstrategyadoptedorthemaximumdegreeℓ
considered.Furthermore,inRef.[5],itisalsoclaimed,withoutanyexplicitand
quantitativejustifications,thatdifferentEarthgravitymodelswouldallegedly
lead just to slightly different results.
Actually, in view of the nowadays ever increasing number of different
Earth’s global gravity models released by several independent institutions
worldwide3, a fair practice should consist of taking into account an ensem-
ble of various models and look into the resulting scatter of the a-priori eval-
uations of the overall systematic uncertainties with respect to the expected
Lense-Thirring trend. Moreover, correctly assessing the realistic errors in the
evenzonalsrequiresgreatcare,followingquantitativeandunambiguousproce-
duressuchasthoseoutlinedinRef.[14];itshouldnotbedoneinahand-wavy
fashion.
All such critical issues, ignored by the authors of Ref. [5], are still valid
despitethemostrecentadvancesinthefieldofthedeterminationoftheEarth’s
global gravity field models. Below, we will demonstrate it by looking at some
of the most recently released geopotential models.
In general, a straightforward way to analytically calculate the long-term
orbitalperturbationsexperiencedbyatestparticleorbitingaprimaryofmass
M duetoadisturbingadditionalpotentialU withrespecttotheNewtonian
pert
monopole consists of taking its average over one orbital period P
b
1 Pb
U = U dt, (17)
pert pert
h i P
b Z0
and,then,insertingitinthe Lagrangeequationsforthe ratesofchangeofthe
Keplerian orbital elements. In the case of the node, it is
dΩ 1 ∂ U
pert
= h i. (18)
dt −n a2√1 e2sinI ∂I
(cid:28) (cid:29) b −
3 Seehttp://icgem.gfz-potsdam.de/ICGEM/ ontheInternet.
Acommenton“Atestofgeneralrelativity[...]” 7
The temporal average of Eq. (17) has to be calculated onto the unperturbed
Keplerian ellipse characterizedby
1 e2 3/2
dt = − df, (19)
n (1+ecosf)2
b(cid:0) (cid:1)
a(1 e2)
r = − , (20)
1+ecosf
x= r(cosΩcosu cosIsinΩsinu), (21)
−
y = r(sinΩcosu+cosIcosΩsinu), (22)
z = rsinIsinu, (23)
.
where ω, f, u=ω+f arethe argumentofpericenter,the true anomaly,and
the argument of latitude, respectively.
In the case of an axially symmetric primary with equatorial radius R and
symmetry axis kˆ, the resulting perturbing potential of degree ℓ and order
m=0 is
ℓ
U = GMJ R P (ψ), ψ =. rˆkˆ, (24)
Jℓ r ℓ r ℓ ·
(cid:18) (cid:19)
whereJ istheevenzonalharmonicofdegreeℓ,andrˆ= r/ristheunitvector
ℓ
pointingfromtheprimarytothetestparticle.Iftheprimary’sequatorialplane
is assumed as reference x,y plane, as in the case of the Earth’s artificial
{ }
satellites, then
ψ =sinIsin(ω+f). (25)
In fact, also the uncancelled even zonals of degree as low as ℓ = 6, 8, 10
may have a non-negligible impact on the linear combination used because of
their contribution to the nodal precessionrate for LARES, as we will show in
this Section.
8 L.Iorio
Table 2 Estimated values Cℓ,0 and formal errors σCℓ,0 of the even zonal harmon-
ics of degree ℓ = 6, 8, 10 according to the Earth’s global gravity field solutions
GOCO05S,ITU GRACE16,ITSG-Grace2014s, JYY GOCE04SretrievableontheInternet
athttp://icgem.gfz-potsdam.de/ICGEM/. Onlythesignificantdigitsaredisplayed.
GOCO05S ITU GRACE16 ITSG-Grace2014s JYY GOCE04S
C6,0 −1.499663×10−7 −1.4999827×10−7 −1.499746×10−7 −1.4998×10−7
σ 1×10−13 6×10−14 2×10−13 2×10−11
C6,0
C8,0 4.94816×10−8 4.948113×10−8 4.94779×10−8 4.938×10−8
σ 1×10−13 5×10−14 1×10−13 2×10−11
C8,0
C10,0 5.334319×10−8 5.335971×10−8 5.334268×10−8 5.352×10−8
σ 8×10−14 4×10−14 9×10−14 3×10−11
C10,0
Toorderzerointheeccentricity,whichisfullyadequateforaquasi-circular
orbit like that of LARES, Eqs. (17)-(18) and Eq. (24) straightforwardlyyield
6
105 R
Ω˙ = n cosI(19+12cos2I+33cos4I), (26)
.6 b
−1024 a
(cid:18) (cid:19)
8
315 R
Ω˙ = n cosI(178+869cos2I+286cos4I+715cos6I),
.8 b
65536 a
(cid:18) (cid:19)
(27)
10
3465 R
Ω˙ = n cosI(2773+2392cos2I+5252cos4I+
.10 b
−4194304 a
(cid:18) (cid:19)
+ 1768cos6I+4199cos8I) (28)
for ℓ = 6, 8, 10. In the limit e 0, Equations (12) to (14) of Ref. [6],
→
obtained with the formalism of the inclination and eccentricity functions by
[9], agree with Eqs. (26)-(28). In the case of LAGEOS, LAGEOS II, LARES,
Eqs. (26)-(28) return the values reported in Table 1.
On the other hand, the remaining scatter in the estimated values of the
correspondingStokesgeopotentialcoefficientsC fromseveralglobalgravity
ℓ,0
models does not yet allow for an unequivocal and unambiguous mismodeling
below the 1% level in the resulting node precessions of LARES.
This can be quickly shown by considering the simple differences ∆C
ℓ,0
amongtheestimatesforC forvariouspairsofEarth’sglobalgravitymodels
ℓ,0
as a naive measure of the realistic uncertainties in the even zonals of interest.
In the following, we will demonstrate that also more refined calculation yield
essentially the same conclusions. Table 2 displays the determined values C
ℓ,0
along with their sigmas σ of some recent field solutions for the degrees
Cℓ,0
ℓ=6, 8, 10.
Acommenton“Atestofgeneralrelativity[...]” 9
Table 3 Absolutevaluesofthe differences ∆C6,0 amongtheestimatedvalues C6,0 ofthe
modelsofTable2.Theyareveryclosetotheuncertaintiesretrievablefromtheapplicationof
Eq.(A11)byWagnerandMcAdoo[14]tothesigmaofthemodeltobetestedbycontrasting
ittoaformallysuperioroneaccordingtothemethodofsuchauthors.
GOCO05S ITU GRACE16 ITSG-Grace2014s JYY GOCE04S
GOCO05S - 3.197×10−11 8.3×10−12 1.37×10−11
ITU GRACE16 - 2.367×10−11 1.827×10−11
ITSG-Grace2014s - 5.4×10−12
JYY GOCE04S -
Table 4 PercentuncertaintyδµLT (%)duetheerrorsinC6,0 accordingtoTable3.
GOCO05S ITU GRACE16 ITSG-Grace2014s JYY GOCE04S
GOCO05S - 15 4 7
ITU GRACE16 - 11 9
ITSG-Grace2014s - 3
JYY GOCE04S -
2.2.1 The impact of the mismodeling in the third even zonal (ℓ=6)
It turns out that, even by restricting to ℓ =6 alone, whose differences ∆C
6,0
for some gravity models are displayed in Table 3, the corresponding mismod-
elled node precession of LARES δΩ˙LR, calculated with the value in Table 1,
is as large as 104 mas yr−1 for the Jp6air ITU GRACE16/GOCO05S, while it
amountsto77mas yr−1forthepairITU GRACE16/ITSG-Grace2014s,corre-
spondingto15%, 11%,respectively,ofthecombinedLense-Thirringsignature
of Eq. (16). A smaller value can be obtained by considering ITU GRACE16
and JYY GOCE04S; indeed, it is δΩ˙LR =60 mas yr−1, corresponding to 9%
J6
of the predicted relativistic trend. The complete list of percent uncertainty
due to the errors in J is quoted in Table 4.
6
Let us, now, adopt the more advanced approach to realistically assess the
uncertainty in the even zonals proposed by Wagner and McAdoo in Ref. [14].
Eq. (A11) and Eq. (A13) of Ref. [14] yield the resulting (squared) scale coef-
10 L.Iorio
ficients
Cref Ctest 2 σref 2
f2 = ℓ,0− ℓ,0 − Cℓ,0 , (29)
test, ℓ (cid:16) (cid:17) 2(cid:16) (cid:17)
σtest
Cℓ,0
(cid:16) (cid:17)
ℓ Cref Ctest 2 σref 2
g2 = 1 ℓ,m− ℓ,m − Cℓ,m +
test, ℓ 2ℓ+1 (cid:16) (cid:17) 2(cid:16) (cid:17) 
mX=0 σCteℓs,tm 
 (cid:16) (cid:17) 

ℓ Sref Stest 2 σref 2
+ ℓ,m− ℓ,m − Sℓ,m (30)
(cid:16) (cid:17) 2(cid:16) (cid:17) 
mX=1 σSteℓs,tm 
 (cid:16) (cid:17) 
to be applied to the sigma of the Stokes coefficient of degree ℓ of the field
labeled with “test” in order to have a realistic evaluation of its uncertainty;
the model labeled with “ref” has a superior formal accuracy, and is used as
a reference (calibrating) gravity field [14]. By adopting ITU GRACE16 as
formally superior reference model and GOCO05S as test model, it turns out
that the sigma of GOCO05S, quoted in Table 2, has to be scaled by
f =235.97, (31)
test, 6
g =96.81. (32)
test, 6
It can be noted that Eq. (31) yields a realistic uncertainty for C very
6,0
close to the simple difference ∆C between the estimated coefficients of
6,0
ITU GRACE16 and GGM05S. Eq. (32) returns a scaling factor which is less
thanhalfthepreviousone;itcorrespondstoanoveralluncertaintyof6%ofthe
combined Lense-Thirring signature. If, instead, ITSG-Grace2014s is assumed
as test model by keeping ITU GRACE16 as reference model, we have
f =125.65, (33)
test, 6
g =73.21. (34)
test, 6
In view of the sigma of ITSG-Grace2014s reported in Table 2, Eqs. (33)-(34)
imply a mismodeled LARES node precession δΩ˙LR as large as 78 mas yr−1
and 45 mas yr−1, respectively, corresponding to aJn6overall uncertainty in the
combinedLense-Thirringtrendof11%and7%,respectively.Alsointhiscase,
Eq.(29)yieldsessentiallythe sameresultasthesimpledifferencebetweenthe
nominal coefficients of the models considered. If the JYY GOCE04S global