Effects of standard and modified gravity on interplanetary ranges

0.56 MB·English

by  Lorenzo Iorio

January 19, 2011 1:37 WSPC/INSTRUCTION FILE SSB˙multieffects
1 EFFECTS OF STANDARD AND MODIFIED GRAVITY ON
1
INTERPLANETARY RANGES
0
2

n
a
LORENZOIORIO
J
Ministero dell’Istruzione, dell’Universita`e della Ricerca (M.I.U.R.), Fellow of the Royal
8
Astronomical Society (F.R.A.S.). Permanent address: Viale Unit`a di Italia 68
1
Bari, (BA)70125, Italy

] [email protected]
c
q
Received22May2010
-
r Revised25October 2010
g Accepted5November2010
[ CommunicatedbyJ.Pullin

5
v We numerically investigate the impact on the two-body range of several Newtonian
5 and non-Newtonian dynamical effects for some Earth-planet (Mercury, Venus, Mars,
8 Jupiter,Saturn)pairsinviewoftheexpectedcm-levelaccuracyinsomefutureplanned
5 or proposedinterplanetary rangingoperations. Thegeneral relativisticgravitomagnetic
4 Lense-Thirringeffect shouldbemodeledandsolved-forinfutureaccurate rangingtests
. ofNewtonianandpost-Newtoniangravitybecauseitfallswithintheirmeasurabilitydo-
2
main. It could a-priori “imprint” the determination of some of the target parameters
0
of the tests considered. Moreover, the ring of the minor asteroids, Ceres, Pallas, Vesta
0
(andalsomanyotherasteroidsifMarsisconsidered)andtheTrans-NeptunianObjects
1
(TNOs)actassourcesofnonnegligiblesystematicuncertaintyonthelargergravitoelec-
:
v tric post-Newtonian signals from which it is intended to determine the parameters γ
i and β of the Parameterized Post-Newtonian (PPN)formalismwithveryhigh precision
X (several orders of magnitude better than the current 10−4 −10−5 levels). Also other
r putative,nonconventionalgravitationaleffectslikeaviolationoftheStrongEquivalence
a
Principle(SEP),asecularvariationoftheNewtonianconstantofgravitationG,andthe
Pioneeranomalyareconsidered.Thepresenceofahypothetical,distantplanetary-sized
body X could be detectable with future high-accuracy planetary ranging. Our analysis
can, in principle, be extended also to future interplanetary ranging scenarios in which
oneormorespacecraftinheliocentricorbitsareinvolved.Theimpactoffittingtheinitial
conditions, and of the noise in the observations, on the actual detectability of the dy-
namicalsignaturesinvestigated,whichmaybepartlyabsorbedintheestimationprocess,
shouldbequantitatively addressedinfurtherstudies.
Keywords:Experimentalstudiesofgravity;Experimentaltestsofgravitationaltheories;
Modified theories of gravity; Ephemerides, almanacs, and calendars; Remote observing
techniques
PACS:04.80.-y,04.80.Cc,04.50.Kd,95.10.Km,95.75.Rs
1. Introduction
Recent years have seen increasingefforts towards the implementation of the Plane-
taryLaserRanging(PLR)techniqueaccuratetocm-level1,2,3,4,5,6,7,8.Itwouldallow
1
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2 L. Iorio
toreachmajorimprovementsinthreerelatedfields:solarsystemdynamics,testsof
general relativity and alternative theories of gravity, and physical properties of the
targetplanetitself.Inprinciple,anysolarsystembodyendowedwithasolidsurface
and a transparentatmosphere would be a suitable platform for a PLR system, but
some targets are more accessible than others. Major efforts have been practically
devoted so far to Mercury1 and Mars2,5, although simulations reaching 93 a.u. or
morehavebeenundertakenaswell4,7.In2005twointerplanetarylasertransponder
experiments were successfully demonstrated by the Goddard Geophysical Astro-
nomical Observatory (GGAO). The first utilized the nonoptimized Mercury Laser
Altimeter (MLA) on the Messenger spacecraft1,3, obtaining a formal error in the
laser range solution of 0.2 m, or one part in 1011. The second utilized the Mars
Orbiting Laser Altimeter (MOLA) on the Mars Global Surveyor spacecraft9,3. A
precisemeasureoftheEarth-Marsdistance,measuredbetweentheircentersofmass
andtakenoveranextendedperiod(fiveyearsormore),wouldsupport,amongother
things,abetter determinationofseveralparametersofthesolarsystem.Sensitivity
analyses point towards measurement uncertainties between2 1 mm and 100 mm.
The perspectives inmeasuringthe distance betweenanEarth-basedstationandan
activelasertransponderontheMartianmoonPhoboscapableofreachingmm-level
range resolution have recently been investigated in Ref. 10. The authors of Ref. 11
envisage the possibility of using also Deimos, in addition to Phobos, as a target
for an in-situ lander in the framework of the proposed Gravity Experiment with
TimE Metrology on Martian satEllites (GETEMME) missiona. Its goal is to use
laser to measure intermartian distances with an accuracy of a few tenth of mm,
and its nominal durationshould be 3−5 yr. See also Ref. 12 for an earlier,prelim-
inary study on the possibility of using both artificial and natural satellites to test
generalrelativityin the martiansystem. ConcerningMercury,a recentanalysison
the future BepiColombob mission, aimed to accurately determining, among other
things, severalkey parameters of post-Newtonian gravity and the solar quadrupole
moment from Earth-Mercury distance data collected with a multi-frequency ra-
dio link13,14, points toward a maximum uncertainty of 4.5−10 cm in determining
the Earth-Mercury range over a multi-year time span13,15,14 (1-8 yr). A proposed
spacecraft-based mission aimed to accurately measure also the general relativistic
gravitomagneticfield of the Sun and its adimensional quadrupole mass moment J
2
alongwithotherPPNparameterslikeγ andβ bymeansofinterplanetaryrangingis
the Astrodynamical Space Test of Relativity using Optical Devicesc (ASTROD)17.
Another space-based missions proposed to accurately test several aspects of the
gravitationalinteractionvia interplanetary laserranging are the Laser Astrometric
aSeeontheWEBhttp://meetingorganizer.copernicus.org/EPSC2010/EPSC2010-60.pdf.
bIt is an ESA mission,including two spacecraft, one of which provided by Japan, to be put into
orbit around Mercury. The launch is scheduled for 2014. The construction of the instruments is
currentlyongoing.
cItscheaperversionASTRODImakesuseofonespacecraftinaVenus-gravity-assistedsolarorbit,
rangingopticallywithgroundstations16.
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Effects of Standard and Modified Gravity on Interplanetary Ranges 3
Test of Relativity (LATOR)18, and the interplanetary range to Phobos10 one goal
of which is a measurement of γ with an accuracy of 10−7. For a review of the
motivations for accurately determining the parameters of post-Newtonian gravity,
in particular β and γ, see, e.g., Ref. 19, 20 and references therein.
In this paper we study the effects that several Newtonian and non-Newtonian
dynamical features of motion have on the two-body range for the Earth and some
planets of the solar system for which accurate ranging to spacecraft exists or is
plannedin future. Our goalis to inspect the potentialaliasing posedby other com-
peting dynamical forces acting as source of systematic uncertainty. Indeed, it must
be recalled that in the range observables actually used in testing post-Newtonian
gravity there is also a part due to the Earth-planet orbital motions in addition to
the purely post-Newtonian Shapiro delay connected with the propagation of elec-
tromagnetic waves. Thus, reaching unprecedented accuracy in only measuring the
latter effect is useless if the accuracy of the orbital signal is not at a comparable
level.Ontheotherhand,itshouldberemarkedthatwedonotaimtoquantitatively
assesstheactualmeasurabilityofthedynamicaleffectsinvestigated.Itisadifferent
andimportanttaskwhichwoulddeserveadedicatedwork.Indeed,afitoftheinitial
conditionstotherealobservationswouldbeneededinordertorealisticallyevaluate
the level of removal of the effects of interest from the signatures. It is a nontrivial
task which is beyond the scopes of the present analysis which could be fruitfully
used to single out the most relevant dynamical signals and focus future efforts on
them. Anyway, we will use in the following a heuristic rule-of-thumb. The paper
is organized as follows. In Section 2 we outline the strategy followed and mention
theNewtonianandnon-Newtonianeffectsinvestigated.InSection3,Section4,Sec-
tion 5, Section 6, and Section 7 we deal with the ranges of Mercury, Venus, Mars,
Jupiter and Saturn, respectively. Section 8 is devoted to the conclusions.
2. Method
In order to numerically obtain the effect of a given gravitational acceleration, con-
sidered as a relatively small perturbation P of the Newtonian Sun’s monopole, on
the range |ρ~| between the Earth-Moon Barycenter (EMB) and a planet we used
MATHEMATICA to simultaneously integrate with the Runge-Kutta method the
equations of motion in Cartesian coordinates of EMB and the planet considered
with and without the perturbation P investigated by using the same set of initial
conditions. We adopted the ICRF/J2000.0 reference frame, with the ecliptic and
mean equinox of the reference epoch, i.e. J2000.0, centered at the Solar System
Barycenter (SSB); the initial conditions at the epoch J2000.0 were retrieved with
theHORIZONSWEBinterfacebyJPL,NASA.Thetemporalintervalofthenumer-
icalintegrationforMercuryandVenushasbeentakenequalto∆t=2yrinviewof
thefactthatthe typicaloperationaltime spansenvisagedforfuture PLRtechnique
are similar. For Mars and Saturn, for which ranging to spacecraft is currently on-
going, we adopted ∆t=5 yr. Also for Jupiter we used ∆t=5 yr. The basic model
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4 L. Iorio
adopted consists of the barycentric equations of motion of the Sun, the eight plan-
ets,theMoon,Ceres,Pallas,Vesta,PlutoandEris,tobesimultaneouslyintegrated;
the forces acting onthem include the mutual NewtonianN−body interactions,the
perturbation due to the solar quadrupolar mass moment J , the effect of two rings
2
modeling the actions of the minor asteroids and of the Trans-Neptunian Objects
(TNOs), and the general relativistic gravitoelectric Schwarzschild and gravitomag-
netic Lense-Thirringfields ofthe Sun. Asadditionalperturbations,we modeledthe
action of a distant, planet-like body kept fixed in a given spatial position (planet
X), secular rate G˙/G of the Newtonian gravitational constant G, a violation of
the Strong Equivalence Principle (SEP) by means of the Nordtvedt parameter η, a
constantanduniformaccelerationradiallydirected towardsthe Sun, and actingon
Uranus, Neptune, Pluto (and Eris), to account for the Pioneer anomaly.
Inordertopreliminarilygiveanideaofthepotentialmeasurabilityoftheeffects
.
considered,thecomputeddifferences∆|ρ~|=|ρ~ |−|ρ~ |,whereRreferstoareference
P R
orbit which does not contain the perturbation P of interest, were subsequently
comparedtotheavailabletimeseriesoftherangeresidualsfortheinnerplanetsand
Saturnwhichsetthepresent-dayaccuracylevelinrangingtoplanets21,22.Whenthe
possibility that a given, unmodeled dynamical effect may show or not its signature
in the range residuals it must be considered that the magnitude of such an effect
shouldroughlybe one orderofmagnitude largerthan the rangeresiduals accuracy.
Thistoavoidtheriskthatitmaybeabsorbedandpartiallyortotallyremovedfrom
the signature in the process of estimation of the initial conditions and of the other
numerous solve-for parameters in the real data reduction. As outlined before, it is
justa sortofrule ofthumb; fully quantitative,realisticanalyses,outside the scopes
ofthepresentpaper,shouldrequiretheactualfittingoftheinitialconditionstothe
observations.
Dependingonthedynamicaleffectoneisinterestedin,someoftheperturbations
examined here are to be considered as sources of noise inducing systematic bias on
the targetsignal.For example,if the goalofthe analysisis,say,the Lense-Thirring
effect, then the rangeperturbationdue to the TNOs is clearly a sourceof potential
systematic error which has to be evaluated. Thus, our plots are useful to assess
the level of aliasing of severalpotential sources of aliasing for some non-Newtonian
effects and the correlations that may occur in estimating them. Dynamical effects
whichareviewedasnoiseinagivencontextcanalsoberegardedasmaintargetsin
another one; see, e.g., the proposed determination of asteroid masses through the
ASTROD mission23.
3. Earth-Mercury range
At present, the 1-way range residuals of Mercury from radar-ranging span 30 yr
(1967-1997) and are at a few km-level (Figure B-2 of Ref. 21); the same holds for
the1-wayMercuryradarclosureresidualscovering8yr(1989-1997,FigureB-3a)of
Ref.21).TherearealsoapairofMariner10rangeresidualsinthe70satMercuryat
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Effects of Standard and Modified Gravity on Interplanetary Ranges 5
0.2kmlevel(FigureB-3b)ofRef.21).RangingtoBepiColomboshouldbeaccurate
to15,14 4.5−10 cm over a few years.
3.1. The Schwarzschild field of the Sun
In Figure 1 we plot the effect of the gravitoelectric Schwarzschild field of the Sun
on the Earth-Mercury range. We modeled its acceleration as24
GM GM
a = 2(β+γ) −γv·v r+2(1+γ)(r·v)v , (1)
SS c2r3 r
(cid:26)(cid:20) (cid:21) (cid:27)
where we inserted the PPN parameters β and γ: they are equal to 1 in general
relativity and we used such values. Figure 1 can be compared with Figure 1 of
Ref. 14, obtained for unspecified initial conditionsd: they are quite similar. The
maximum variation of the signal is of the order of 4 ×105 m, corresponding to
a measurement accuracy of about 2.5×10−7. The expected realistic accuracy in
determining β and γ is 2×10−6 in BepiColombo13.
3.2. The oblateness of the Sun
Figure 2 shows the nominal effect of the Sun’s quadrupolar mass moment on the
Mercury range for J =2×10−7. Its action has been modeled as25
2
3J R2GM
a =− 2 1−5(rˆ·k)2 rˆ+2(rˆ·k)k , (2)
J2 2r4
where R is the Sun’s mean equatorniahl radius and ki is the unit veoctor of the z axis
directedalongthe body’s rotationaxis.Since eq.(2)holds ina frame withits {xy}
plane coinciding with the body’s equator, we rotated the the mean ecliptic at the
epochtotheSun’sequatorwhichisinclinedtoitbytheCarringtonangle26i=7.15
deg.The signalof Figure 2 has a maximumspan of 300m, correspondingto an ac-
dItalsoincludestheShapirodelaycontribution.
EMB-Mercuryrange:SSBnumericalcalculation
100000
Lm 0
®ÈÈHDΡ-100000
-200000
0 0.5 1 1.5 2
tHyL
.
Fig. 1. Difference ∆|ρ~| = |ρ~P|−|ρ~R| in the numerically integrated EMB-Mercury ranges with
and without the perturbation due to the Sun’s Schwarzschild field over ∆t = 2 yr. The same
initial conditions (J2000.0) have been used for both the integrations. The state vectors at the
referenceepochhavebeenretrievedfromtheNASAJPLHorizonssystem.Theintegrations have
been performedinthe ICRF/J2000.0 reference frame, withthe ecliptic and mean equinox of the
referenceepoch, centeredattheSolarSystem Barycenter(SSB).
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6 L. Iorio
curacy measurementof 3×10−4.A measure of the solarJ accurateto 10−2 is one
2
ofthe goalsofBepiColombo13;knowingpreciselyJ wouldyieldimportantinsights
2
on the internal rotation of the Sun. At present, it is known with an uncertainty of
about29 10%. The solar quadrupole mass moment may play the role of source of
systematic bias with respect to, e.g., some non-Newtonian dynamical effects. Con-
cerning the gravitoelectricsignalpreviously analyzed,the mismodeled J signature
2
would impact it a 7.5×10−5 level. It is important to note that the patterns of the
two signals are rather different. Conversely, as we will see, the determination of J
2
at the desired level of accuracy may be affected by other unmodeled/mismdeled
dynamical effects acting as systematic sources of aliasing on it.
3.3. The Lense-Thirring effect of the Sun
Figure3 depicts the rangeperturbationdue to the Sun’s Lense-Thirringeffect,nei-
ther considered so far in the dynamical force models of the planetary ephemerides
nor in the BepiColombo analyses. It is a general relativistic feature of motion in-
duced by the rotation of the Sun which acts upon a test particle moving with
velocity v with a noncentral acceleration24
(1+γ)G 3
a = (r×v)(r·S)+(v×S) , (3)
LT c2r3 r2
(cid:20) (cid:21)
whereS istheSun’s properangularmomentum.Accordingtohelioseismology27,28,
its magnitude is S = (190.0±1.5)×1039 kg m2 s−1. Note that such a value does
not come from planetary orbital dynamics, so that there is no risk of a-priori “im-
printing” ofgeneralrelativityitself onrange tests ofthe solarLense-Thirringeffect
which could, thus, be regarded as genuine and unbiased. Also in this case we ro-
tated the reference frame to the mean ecliptic at the epoch to the Sun’s equator
by the Carrington angle because eq. (3) holds in a frame with its z axis aligned
with S. The peak-to-peak amplitude of the Lense-Thirring signal is up to 17.5 m
EMB-Mercuryrange:SSBnumericalcalculation
150
100
50
Lm
®ÈÈHDΡ -500
-100
-150
0 0.5 1 1.5 2
tHyL
.
Fig.2. Difference∆|ρ~|=|ρ~P|−|ρ~R|inthenumericallyintegratedEMB-Mercuryrangeswithand
withoutthenominalperturbationduetotheSun’squadrupolemassmomentJ2=2.0×10−7 over
∆t = 2 yr. The same initial conditions (J2000.0) have been used for both the integrations. The
statevectorsatthereferenceepochhavebeenretrievedfromtheNASAJPLHorizonssystem.The
integrationshavebeenperformedintheICRF/J2000.0referenceframe,withthemeanequinoxof
the reference epoch and the reference {xy} plane rotated from the mean ecliptic of the epoch to
theSun’sequator, centeredattheSolarSystem Barycenter(SSB).
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Effects of Standard and Modified Gravity on Interplanetary Ranges 7
over2yr,which,ifontheonehandisunmeasurablefromcurrentlyavailableradar-
rangingto Mercury,onthe other handcorrespondsto a potential relative accuracy
in measuringitwith BepiColomboof 2−5×10−3;this clearlyshowsthat the solar
gravitomagneticfieldshouldbe takenintoaccountinfuture analysesanddatapro-
cessing.Otherwise,it wouldalias the recoveryof other effects. For example,it may
affectthedeterminationofJ at12%level.Ontheotherhand,inordertoallowfor
2
a determination of the Lense-Thirring effect, the Sun’s quadrupole mass moment
should be known with an accuracy better than the present-day one by at least one
order of magnitude; this is just one of the goals of BepiColombo. Moreover, since
the Lense-Thirring effect depends one γ, neglecting it may alias the determination
of γ through the larger gravitoelectric signal at 4×10−5 level.
3.4. The ring of the minor asteroids and Ceres, Pallas and Vesta
In Figure 4 we depict one potential source of systematic bias, i.e. the action of the
ring of minor asteroids29. We modeled it following Ref. 30. For those planets for
.
which r>R , by posing α=R /r, we obtained
ring ring
Gm 3 45
a ≃− ring 2+ α2+ α4 r, (4)
inner ring 2r3 2 32
(cid:18) (cid:19)
from
Gm
a = ring αb(1)(α)−b(0)(α) r. (5)
inner ring 2r3 32 23
. h i
For r <R , by posing α=r/R , we obtained
ring ring
Gm 9 75
a ≃ ring α+ α3+ α5 r, (6)
outer ring 2rR2 8 64
ring (cid:18) (cid:19)
eThemultiplicativefactor2infrontofeq.(3)comesfrom24 1+γ.
EMB-Mercuryrange:SSBnumericalcalculation
10
7.5
5
Lm 2.5
®ÈÈHDΡ 0
-2.5
-5
-7.5
0 0.5 1 1.5 2
tHyL
.
Fig.3. Difference∆|ρ~|=|ρ~P|−|ρ~R|inthenumericallyintegratedEMB-Mercuryrangeswithand
without the perturbation due to the Sun’s Lense-Thirringfield over ∆t=2 yr. The same initial
conditions (J2000.0) have been usedfor both the integrations. Thestate vectors atthe reference
epoch have been retrieved from the NASA JPL Horizons system. The integrations have been
performed in the ICRF/J2000.0 reference frame, with the mean equinox of the reference epoch
and the reference {xy} plane rotated from the mean ecliptic of the epoch to the Sun’s equator,
centeredattheSolarSystemBarycenter (SSB).
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8 L. Iorio
from
Gm
a = ring b(1)(α)−αb(0)(α) r. (7)
outer ring 2rRr2ing 32 23
h i
Recall that the Laplace coefficients are defined as
. 1 2π cosjψdψ
b(j)(α)= , (8)
s π (1−2αcosψ+α2)s
Z0
where s is a half-integer; a useful approximate expression in terms of a series is31
s(s+1)...(s+j−1) s(s+j) s(s+1)(s+j)(s+j+1)
b(j) ≃ αj 1+ α2+ α4 .
s 1·3···j (1+j) 1·2(j+1)(j+2)
(cid:20) (cid:21)
(9)
By assuming for the ring of the minor asteroids a nominal mass of29 m =
ring
1×10−10M anda radius29 R =3.14au,itwouldimpact the Mercuryrangeat
⊙ ring
4mlevel(peak-to-peakamplitude),whichis,infact,measurable.Itsnominalbiason
theSchwarzschild,J andLense-Thirringsignalswouldbe1×10−5,1.3×10−2,2.3×
2
10−1, respectively. Anyway, the present-day level of uncertainty in the mass of the
ring is29 δm = 0.3×10−10M . Thus, the impact of such a mismodeling would
ring ⊙
be, 3×10−6,4×10−3,7×10−2, respectively; it cannot be considered negligible.
The effect of Ceres, Pallas and Vesta on the determination of some Newtonian
and non-Newtonian parameters with BepiColombo has been preliminarily inves-
tigated in Ref. 15. Here in Figure 5 we show the nominal perturbation on the
Earth-Mercury range due to the combined actions of Ceres, Pallas and Vesta; the
valuesfortheirmasseshavebeenretrievedfromRef.32.Itspeak-to-peakamplitude
amounts to 80 m; thus, their signature would be measurable at a 0.6−1×10−3
level. Anyway, the mismodeled solar quadrupole mass moment would bias their
signal at 4×10−1 level. The Lense-Thirring effect, if unmodeled, would have an
impact at 2.2×10−1 level. The present-day relative uncertainties in their masses
are 6×10−3,3×10−2,2×10−2 respectively32. This implies a mismodeled signal
EMB-Mercuryrange:SSBnumericalcalculation
2
1
®ÈÈHLDΡm 0
-1
-2
0 0.5 1 1.5 2
tHyL
.
Fig.4. Difference∆|ρ~|=|ρ~P|−|ρ~R|inthenumericallyintegratedEMB-Mercuryrangeswithand
withoutthenominalperturbationduetotheringofminorasteroidswith29 mring=1×10−10M⊙
and Rring = 3.14 a.u. over ∆t = 2 yr. The same initial conditions (J2000.0) have been used for
boththeintegrations.ThestatevectorsatthereferenceepochhavebeenretrievedfromtheNASA
JPLHorizonssystem.TheintegrationshavebeenperformedintheICRF/J2000.0referenceframe,
withtheeclipticandmeanequinoxofthereferenceepoch,centeredattheSolarSystemBarycenter
(SSB).
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Effects of Standard and Modified Gravity on Interplanetary Ranges 9
withapeak-to-peakamplitudeof50cm.ItwouldimpacttheSchwarzschild,J and
2
Lense-Thirringrangeperturbationsat1×10−6,2×10−3,3×10−2level,respectively.
3.5. The Trans-Neptunian Objects
The situationis differentforanotherpotentialsourceofsystematicuncertainty,i.e.
the Trans-Neptunian Objects (TNOs). Figure 6, obtained by modeling them as a
ringwith22 m =5.26×10−8M and22 R =43au,showsthattheirmaximum
ring ⊙ ring
effect would amount to 80 cm. We used the same formulas as for the asteroid ring.
Suchaneffect,nottakenintoaccountsofar,wouldbe better measurablethanthat
by the minor asteroids.This implies a bias of 2×10−6 onthe Schwarzschildsignal,
3×10−3forJ and4.5×10−2fortheLense-Thirringeffect.Amajorconcernisthat
2
the mass of the TNOs is far from being accurately known, so that an uncertainty
as large as 100% should be applied.
EMB-Mercuryrange:SSBnumericalcalculation
40
20
Lm
®ÈÈHDΡ 0
-20
-40
0 0.5 1 1.5 2
tHyL
.
Fig. 5. Difference ∆|ρ~| = |ρ~P|−|ρ~R| in the numerically integrated EMB-Mercury ranges with
and without the nominal perturbation due to32 Ceres, Pallas, Vesta over ∆t = 2 yr. The same
initial conditions (J2000.0) have been used for both the integrations. The state vectors at the
referenceepochhavebeenretrievedfromtheNASAJPLHorizonssystem.Theintegrations have
been performedinthe ICRF/J2000.0 reference frame, withthe ecliptic and mean equinox of the
referenceepoch, centeredattheSolarSystem Barycenter(SSB).
EMB-Mercuryrange:SSBnumericalcalculation
0.4
0.2
®ÈÈHLDΡm 0
-0.2
-0.4
0 0.5 1 1.5 2
tHyL
.
Fig. 6. Difference ∆|ρ~| = |ρ~P|−|ρ~R| in the numerically integrated EMB-Mercury ranges with
andwithoutthenominalperturbationduetotheringofTrans-NeptunianObjectswith22mring=
5.26×10−8M⊙ and Rring = 43 a.u. over ∆t=2 yr. The same initial conditions (J2000.0) have
beenusedforboththeintegrations. Thestate vectors atthereferenceepochhavebeenretrieved
fromtheNASAJPLHorizonssystem.TheintegrationshavebeenperformedintheICRF/J2000.0
referenceframe,withtheeclipticandmeanequinox ofthereferenceepoch, centered attheSolar
SystemBarycenter (SSB).
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10 L. Iorio
3.6. Violation of the Strong Equivalence Principle
Let us, now, focus our attention to other nonstandard effects like a SEP violation,
a variation of the Newtonian gravitational constant, the Pioneer anomaly and a
putative planet X.
Concerning the SEP violation, we modeled it in the accelerationa of a body i
i
as
m(G) Gm
a(SEP) = i jr ,i=1,2,...N. (10)
i "m(iI)#SEPjX,i=6 j ri3j ij
 
In it m(G) and m(I) are the gravitational and inertial masses, respectively, of the
i i
body i. Their ratio is
m(G)
i =1+ηΩ (11)
"m(I)# i
i SEP
in which η is the Nordtvedt dimensionless constant 33,34 accounting for SEP viola-
tionf, and
. Ei
Ω = , (12)
i m c2
i
where E is the (negative) gravitational self-energy of the i−th body, and m c2 is
i i
its total mass-energy. For a spherical body of radius R 19,
i
3Gm
i
Ω =− . (13)
i 5R c2
i
Figure 7 shows its effect on the Mercury range for η = 10−5; at present, the most
accurate constraints on η come from Lunar Laser Ranging (LLR) amounting to35
η =(4.0±4.3)×10−4. The peak-to-peak amplitude of the signature of Figure 7 is
6 mm, which is practically impossible to detect. Moreover, the SEP signal would
be totally swamped by other dynamical effects like the ones by the minor asteroid
ring, even if modeled at the present-day level of accuracy, and the TNOs. Also the
Lense-Thirring effect, modeled or not, would be of concern.
3.7. Secular variation of the Newtonian constant of gravitation
The case of a possible variation of the Newtonian gravitational constant is inter-
esting because recently E.V. Pitjeva in Ref. 22 preliminarily reported a secular
variation for it G˙/G = (−5.9±4.4)×10−14 yr−1, statistically significant at 3−σ
level.Otherresearchersgetresultsstatisticallycompatiblewith0.FolknerinRef.36
getsanupperboundof2×10−13yr−1fromplanetaryephemeridesaswell.Williams
fIntermsofthePPNparametersβ andγ itisη=4β−γ−3,sothatη=0ingeneralrelativity.