a. fienga[1,2], v. viswanathan[1,2], l. bernus[1], n. rambaux

Rencontres de Moriond, 23-30 March 2019, La Thuile Session: Gravity on Earth and in the Solar System

Results obtained with the new lunar ephemeris INPOP17a and its

application to fundamental physics A. Fienga[1,2], V. Viswanathan[1,2], L. Bernus[1], N. Rambaux[1], O.Minazzoli[3,4]

M. Gastineau[1], H. Manche[1], J. Laskar[1]

[1] ASD-IMCCE, Observatoire de Paris – 75014 Paris[2] Laboratoire Géoazur, Observatoire de la Cote d’Azur – 06560 Valbonne[3] Centre Scientifique de Monaco, 8 Quai Antoine 1er, 98000 Monaco [4] Laboratoire Artemis, Université Côte d’Azur, CNRS

Outline1. Dynamical model

- Planetary and Lunar Ephemeris (INPOP)

2. Observations and reduction procedures - Lunar Laser Ranging (LLR) data

3. Model parameter constraints and estimations

4. Results- LLR Post-fit residuals, estimates (INPOP17a)

- SEP

5. Discussion and future work

2

INPOP planetary and lunar ephemeris

Modelparameters

Orbits and orientation

ReductionmodelGINS

LLRdata

Regression

Intégrateur Numérique Planétaire de l'Observatoire de Paris - INPOP

• Numerical integration of the equations of motion (EIH, c-2 PPN approximation)

• Adams-Cowell integrator in extended precision

• 8 planets + Pluto + Moon + asteroids (point-mass, ring), GR, J2Sun, Earth rotation

• Moon: orbit and librations (now includes a lunar fluid core)

• Constant updates: Cassini re-analysis, Juno, Mars orbiters…

• Additional LLR dataset (including new IR data from Grasse)

3


Modelparameters


ReductionmodelGINS

LLRdata

Regression

4

• 2 layer lunar interior structure - using Euler-Liouville equation => solid mantle (shape described up to degree-6) => liquid core (axisymmetric)

Earth-Moon dynamical model INPOP17a

Modelparameters

ReductionmodelGINS

LLRdata

Regression

Weber et al. 2011 INPOP planetary and lunar ephemeris


• Perturbations - point mass mutual interactions: => Sun, planets and asteroids- extended bodies mutual interactions: => figure-point mass (up to degree-6); => figure-figure interactions (up to degree-3)- Earth tides* : - orbital and rotation time delays —> complex Love numbers

• Dissipation* at CMB- Viscous friction at fluid core-solid mantle boundary

5

Earth-Moon dynamical model


Modelparameters


ReductionmodelGINS

LLRdata

Regression

Fienga et al. 2011, 2015; Folkner et al. 2014;Williams et al. 2014; Manche, 2010; Pavlov et al. 2016; Viswanathan, 2017

Observation: Lunar Laser Ranging (LLR)

• 2-way Time of flight measurement to lunar retroreflectors

• Observed since 1969 till date

• Accuracy of LLR observations from APOLLO station (New Mexico, USA) at few millimetersand Grasse station (Calern, France) at few centimeters over E-M distances

• Contains intrinsic E-M distance information;as well as geophysical and relativistic effects

6


Modelparameters


ReductionmodelGINS

LLRdata

Regression

Faller et al. 1969; Bender et al. 1973Samain et al. 1998; Murphy et al. 2008Courde et al. 2017

Murphy 2010

3.5 meter

2.5 meter

laser

people

2010.12.10 5LLR Analysis Workshop

1.5 meter

Grasse

APOLLO

Since 2016, IR LLR data from Grasse

7

Viswanathan et al. 2018 (MNRAS)Courde et al. 2017 (A&A)

• Better atmospheric transmission—> more observations

• Observations round the clock

• Diversification of observed reflectors

• Observations during new and full moon

Bettertransmissionefficiency

Maximum sensitivity for tests of EP

8http://www.geoazur.fr/astrogeo/?href=observations/donnees/luneRG/statistiques_new

Up to 12/2018

LLR reduction model• Light-time computation (Moyer, 2003)

- IERS 2010 recommendations => geophysical and relativistic corrections- EOPs from JPL KEOF (Ratcliff and Gross, 2017)- Lunar solid tides similar to the Earth (only degree-2)

• Developed within GINS software ( OCA-GRGS-CNES )- Reduction model step-wise comparison (sub-mm)- Earth orientation bug fixes- Improved precision- Available within the last GINS version ( >v14.2 )

• —> geodesy: multi-technics (LLR/SLR/GPS/Doris) constraints on station positions (Mémin et al. 2017)

9


Modelparameters


ReductionmodelGINS

LLRdata

Regression

• GRAIL-derived gravity field for the Moon (up to deg-6 - Konopliv et al. 2013, Lemoine et al. 2013)

• Combined GGM05C gravity field for the Earth (up to deg-6 - Ries et al. 2016)

• Earth-Moon mass ratio (planetary fit of INPOP - Fienga et al. 2011 )

• Polar moment of inertia ratio of fluid core v total Moon(Cf/CT - Folkner et al. 2014)

• Orbital time delay for Earth(phase lag induced by tidal components - Williams & Boggs, 2016)

Constraints on model parameters

10


Modelparameters


ReductionmodelGINS

LLRdata

Regression

Fixed parameters

estimates

Viswanathan, 2017 (Ph.D thesis, Observatoire de Paris) Fienga et al. 2011 (CMDA)

Constraints

• GRAIL-derived gravity field for the Moon (see discussion at the end)

• fc, flattening of the fluid core

• Rotational time delays for Earth and Moon

• h2 Love number

• friction coefficient at the CMB

• Total moment of inertia of the Moon

• Initial conditions for Moon’s orbit and libration angles

• observational bias + station positions

Estimated model parameters

11


Modelparameters


ReductionmodelGINS

LLRdata

Regression

Fixed parameters

estimates

Viswanathan, 2017 (Ph.D thesis, Observatoire de Paris) Fienga et al. 2011 (CMDA)

Constraints

12

x 20moreL2andx10moreL1

Results: LLR Post-fit Residuals: INPOP17a

from 1.2 to 1.8 cm

13

Results: INPOP17a versus INPOP13c

14

6 V. Viswanathan et al.

Table 2. Grasse LLR data retroreflector statistics computed using post-fitresiduals obtained with INPOPG and INPOPG + IR, within the fit intervals01/01/2015 to 01/01/2017 (with a 3σ filter), with the WRMS in m (rmsweighted by the number of normal points from each reflector).

GrasseLRRR INPOPG INPOPG + IR per cent change NPTs

A15 0.0183 0.0181 1.1 1018A14 0.0203 0.0177 12.8 172A11 0.0267 0.0239 10.5 215L1 0.0215 0.0166 22.8 265L2 0.0246 0.0215 12.6 256WRMS 0.0207 0.0189 9.5 1926

small corrections to the LLR station coordinates help for the im-provement of LLR residuals during the construction of the lunarephemerides. The Earth orientation parameters and the modellingof the Earth rotation are however kept fixed to the IERS convention(see Section 2.3).

The solution INPOPG with an axisymmetric core fitted to LLRobservations serves as a validation of our lunar model and analysisprocedure, against the DE430 Jet Propulsion Laboratory planetaryand lunar ephemeris analysis described in Folkner et al. (2014) andEphemeris of Planets and the Moon (EPM) Institute of AppliedAstronomy Russian Academy of Sciences ephemeris in Pavlov,Williams & Suvorkin (2016). Only 532 nm wavelength LLR data areused for matching with the DE430 and EPM ephemeris. In Folkneret al. (2014), Pavlov et al. (2016), and INPOPG, gravity field coef-ficients up to degree and order 6 are used for the Moon (GL0660bfrom Konopliv et al. 2013) and the Earth [GGM05C from Ries et al.(2016) for INPOP17a ephemeris and EGM2008 from Pavlis et al.(2012, 2013) for DE/EPM ephemerides]. Coefficients C32, S32, andC33 are then included in the fit parameters as they improve theoverall post-fit residuals. For INPOPG, the improvement of the for-mal uncertainty compared to Pavlov et al. (2016), especially in theestimation of parameter kv/CT, indicates a strong dissipation mech-anism within the Moon, through viscous torques at the fluid core–mantle boundary. Overall, INPOP uncertainties are consistent withEPM Pavlov et al. (2016) published values. DE Williams, Boggs& Folkner (2013); Folkner et al. (2014) uncertainties are greaterthan INPOP and EPM, and should therefore be considered as morerealistic.

Differences between GL0660b values and fitted C32, S32, and C33

from Folkner et al. (2014), Pavlov et al. (2016), or in INPOPG areseveral orders of magnitude greater than the mean GRAIL uncer-tainties (see Konopliv et al. 2013). These results suggest that somesignificant effects impacting the LLR observations are absorbed bythe adjustment of the degree 3 of the full Moon gravity field.

The solution INPOPG + IR refers to the addition of 2 yr of IR LLRobservations Courde et al. (2017) described in Section 2.1 and builtin following the same specification as of INPOPG.

This data set is weighted at the same level as the APOLLO stationnormal points within the estimation procedure (see Section 2.4).

The first outcome from the introduction of the IR data sets is theimprovement of the post-fit residuals obtained for L1 reflector asone can see in Tables 2 and 3 and in Figs 5–8. This is due to theincrease of normal points obtained for this reflector as discussed inSection 2.1.1.

The second conclusion is that because of only 2 yr on data, theimprovement brought by the addition of IR data on the estimatedparameters characterizing the Moon and its inner structure is sig-nificant, especially for those quantifying the dissipation mechanism

Table 3. APOLLO LLR data retroreflector statistics computed using post-fit residuals obtained with INPOPG and INPOPG + IR, within the fit intervals01/01/2015 to 01/01/2017 (with a 3σ filter), with the WRMS in m (rmsweighted by the number of normal points from each reflector).

APOLLOLRRR INPOPG INPOPG + IR per cent change NPTs

A15 0.0127 0.0127 0.2 344A14 0.0192 0.0177 7.8 176A11 0.0185 0.0169 8.7 164L1 0.0186 0.0157 15.6 89L2 0.0136 0.0137 −0.7 64WRMS 0.0159 0.0149 6.7 837

Table 4. Fixed parameters for the Earth–Moon system.

Parameter Units INPOP DE430 EPM

(EMRATa − 81.300 570) × 106 1.87 −0.92 −0.92c

(RE − 6378.1366) × 104 km 0.0 −3 0.0(J2E − 2.6 × 10−11) yr−1 0.0 0.0 0.0(k20, E − 0.335) 0.0 0.0 0.0(k21, E − 0.32) 0.0 0.0 0.0(k22, E − 0.301 02) −0.019 02 0.018 98 −0.019 02(τO0, E − 7.8 × 10−2) × 102 d 0.0 −1.4 0.0(τO1, E + 4.4 × 10−2) d 0.0 0.0b 0.0τO2, E + 1.13 × 10−1) × 101 d 0.0 0.13 0.0(RM − 1738.0) km 0.0 0.0 0.0(αC − 7.0 × 10−4) 0.0 0.0 0.0(k2, M − 0.024 059) 0.0 0.0 0.0(l2 − 0.0107) 0.0 0.0 0.0

Notes. a EMRAT is fitted during the joint analysis between the lunar andplanetary part.bτO1, E in Folkner et al. (2014) given as −0.0044 is a typographical error.cEMRAT in the EPM solution Pavlov et al., 2016, is fixed to a value obtainedfrom DE430

such as Q27.212 and τM with a decreasing uncertainty or kvCT

and fc

with a significant change in the fitted value (see Table 5).A significant global improvement is noticeable when one com-

pares post-fit residuals obtained with INPOPG and with INPOPG + IR

with those obtained with INPOP13c as presented in Fienga et al.(2014) or in Tables 2 and 3. Finally, one should notice in Table 1the 1.15 cm obtained for the post-fit weighted rms obtained for the3 yr of the last period of the APOLLO data (group D) as well asthat for the IR Grasse station.

3 T E S T O F TH E E QU I VA L E N C E P R I N C I P L E

3.1 Context

Among all possibilities to test GR, the tests of the motion of massivebodies as well as the propagation of light in the Solar system werehistorically the first ones, and still provide the highest accuraciesfor several aspects of gravity tests (see Berti et al. 2015; Joyceet al. 2015; Yunes, Yagi & Pretorius 2016 for recent overviewsof constraints on alternative theories from many different types ofobservations). This is in part due to the fact that the dynamics of theSolar system is well understood and supported by a long history ofobservational data.

In GR, not only do test particles with different compositionsfall equally in a given gravitational field, but also extended bodies


Table 2. Grasse LLR data retroreflector statistics computed using post-fitresiduals obtained with INPOPG and INPOPG + IR, within the fit intervals01/01/2015 to 01/01/2017 (with a 3σ filter), with the WRMS in m (rmsweighted by the number of normal points from each reflector).

GrasseLRRR INPOPG INPOPG + IR per cent change NPTs

A15 0.0183 0.0181 1.1 1018A14 0.0203 0.0177 12.8 172A11 0.0267 0.0239 10.5 215L1 0.0215 0.0166 22.8 265L2 0.0246 0.0215 12.6 256WRMS 0.0207 0.0189 9.5 1926

small corrections to the LLR station coordinates help for the im-provement of LLR residuals during the construction of the lunarephemerides. The Earth orientation parameters and the modellingof the Earth rotation are however kept fixed to the IERS convention(see Section 2.3).

The solution INPOPG with an axisymmetric core fitted to LLRobservations serves as a validation of our lunar model and analysisprocedure, against the DE430 Jet Propulsion Laboratory planetaryand lunar ephemeris analysis described in Folkner et al. (2014) andEphemeris of Planets and the Moon (EPM) Institute of AppliedAstronomy Russian Academy of Sciences ephemeris in Pavlov,Williams & Suvorkin (2016). Only 532 nm wavelength LLR data areused for matching with the DE430 and EPM ephemeris. In Folkneret al. (2014), Pavlov et al. (2016), and INPOPG, gravity field coef-ficients up to degree and order 6 are used for the Moon (GL0660bfrom Konopliv et al. 2013) and the Earth [GGM05C from Ries et al.(2016) for INPOP17a ephemeris and EGM2008 from Pavlis et al.(2012, 2013) for DE/EPM ephemerides]. Coefficients C32, S32, andC33 are then included in the fit parameters as they improve theoverall post-fit residuals. For INPOPG, the improvement of the for-mal uncertainty compared to Pavlov et al. (2016), especially in theestimation of parameter kv/CT, indicates a strong dissipation mech-anism within the Moon, through viscous torques at the fluid core–mantle boundary. Overall, INPOP uncertainties are consistent withEPM Pavlov et al. (2016) published values. DE Williams, Boggs& Folkner (2013); Folkner et al. (2014) uncertainties are greaterthan INPOP and EPM, and should therefore be considered as morerealistic.

Differences between GL0660b values and fitted C32, S32, and C33

from Folkner et al. (2014), Pavlov et al. (2016), or in INPOPG areseveral orders of magnitude greater than the mean GRAIL uncer-tainties (see Konopliv et al. 2013). These results suggest that somesignificant effects impacting the LLR observations are absorbed bythe adjustment of the degree 3 of the full Moon gravity field.

The solution INPOPG + IR refers to the addition of 2 yr of IR LLRobservations Courde et al. (2017) described in Section 2.1 and builtin following the same specification as of INPOPG.

This data set is weighted at the same level as the APOLLO stationnormal points within the estimation procedure (see Section 2.4).

The first outcome from the introduction of the IR data sets is theimprovement of the post-fit residuals obtained for L1 reflector asone can see in Tables 2 and 3 and in Figs 5–8. This is due to theincrease of normal points obtained for this reflector as discussed inSection 2.1.1.

The second conclusion is that because of only 2 yr on data, theimprovement brought by the addition of IR data on the estimatedparameters characterizing the Moon and its inner structure is sig-nificant, especially for those quantifying the dissipation mechanism

Table 3. APOLLO LLR data retroreflector statistics computed using post-fit residuals obtained with INPOPG and INPOPG + IR, within the fit intervals01/01/2015 to 01/01/2017 (with a 3σ filter), with the WRMS in m (rmsweighted by the number of normal points from each reflector).

APOLLOLRRR INPOPG INPOPG + IR per cent change NPTs

A15 0.0127 0.0127 0.2 344A14 0.0192 0.0177 7.8 176A11 0.0185 0.0169 8.7 164L1 0.0186 0.0157 15.6 89L2 0.0136 0.0137 −0.7 64WRMS 0.0159 0.0149 6.7 837

Table 4. Fixed parameters for the Earth–Moon system.

Parameter Units INPOP DE430 EPM

(EMRATa − 81.300 570) × 106 1.87 −0.92 −0.92c

(RE − 6378.1366) × 104 km 0.0 −3 0.0(J2E − 2.6 × 10−11) yr−1 0.0 0.0 0.0(k20, E − 0.335) 0.0 0.0 0.0(k21, E − 0.32) 0.0 0.0 0.0(k22, E − 0.301 02) −0.019 02 0.018 98 −0.019 02(τO0, E − 7.8 × 10−2) × 102 d 0.0 −1.4 0.0(τO1, E + 4.4 × 10−2) d 0.0 0.0b 0.0τO2, E + 1.13 × 10−1) × 101 d 0.0 0.13 0.0(RM − 1738.0) km 0.0 0.0 0.0(αC − 7.0 × 10−4) 0.0 0.0 0.0(k2, M − 0.024 059) 0.0 0.0 0.0(l2 − 0.0107) 0.0 0.0 0.0

Notes. a EMRAT is fitted during the joint analysis between the lunar andplanetary part.bτO1, E in Folkner et al. (2014) given as −0.0044 is a typographical error.cEMRAT in the EPM solution Pavlov et al., 2016, is fixed to a value obtainedfrom DE430

such as Q27.212 and τM with a decreasing uncertainty or kvCT

and fc

with a significant change in the fitted value (see Table 5).A significant global improvement is noticeable when one com-

pares post-fit residuals obtained with INPOPG and with INPOPG + IR

with those obtained with INPOP13c as presented in Fienga et al.(2014) or in Tables 2 and 3. Finally, one should notice in Table 1the 1.15 cm obtained for the post-fit weighted rms obtained for the3 yr of the last period of the APOLLO data (group D) as well asthat for the IR Grasse station.

3 T E S T O F TH E E QU I VA L E N C E P R I N C I P L E

3.1 Context

Among all possibilities to test GR, the tests of the motion of massivebodies as well as the propagation of light in the Solar system werehistorically the first ones, and still provide the highest accuraciesfor several aspects of gravity tests (see Berti et al. 2015; Joyceet al. 2015; Yunes, Yagi & Pretorius 2016 for recent overviewsof constraints on alternative theories from many different types ofobservations). This is in part due to the fact that the dynamics of theSolar system is well understood and supported by a long history ofobservational data.

In GR, not only do test particles with different compositionsfall equally in a given gravitational field, but also extended bodies

Results: LLR Post-fit Residuals: INPOP17a

[m]

•For Grasse, NPT sigma > WRMS•For APOLLO, WRMS > NPT sigma

—> Still some room for improvement at the level of the APOLLO accuracy

[m]

Grasse

APOLLO

LLR 532nm LLR 532nm + 1064nm

Results: Test of fundamental physics using LLR

• Accurate model of planetary and lunar ephemeris~400,000 km (E-M distance) below 2 cm accuracy

• Tool to test equivalence principle violation at astronomical scales

15


with µ ≡ mGM + mG

E +!!

mG

mI

"

E− 1

"mG

M +!!

mG

mI

"

M− 1

"mG

E .!

mG

mI

"

Eand

!mG

mI

"

Mare the ratios between the gravitational and

the inertial masses of the Earth and Moon, respectively.With ephemerides, the first term of equation (7) does not lead

to a sensitive test of the UFF, because it is absorbed in the fit ofthe parameter mG

M + mGE (e.g. Williams et al. 2012), while the last

term does. At leading order, one can approximate both distancesappearing in this last term as being approximately equal. One gets

!aUFF ≡ (aM − aE)UFF

≈ GmGS

#rSE

r3SE

$$mG

mI

%

E− 1

%− rSM

r3SM

$$mG

mI

%

M− 1

%&

≈ aE

#$$mG

mI

%

E− 1

%−

$$mG

mI

%

M− 1

%&

≡ aE!ESM (8)

with

!ESM =#$

mG

mI

%

E−

$mG

mI

%

M

&. (9)

One recovers equation (3). Therefore, in this context, constraintson !ESM can be interpreted as constraints on the difference ofthe gravitational-to-inertial mass ratios between the Earth and theMoon.

Furthermore, the LLR test of UFF captures a combined effectof the SEP, from the differences in the gravitational self-energies,and the WEP due to compositional differences, of the Earth–Moonsystem. In general, one has

!ESM = !WEPESM + !SEP

ESM. (10)

In order to separate the effects of WEP, we rely on results fromlaboratory experiments that simulate the composition of the coreand the mantle materials of the Earth–Moon system. One suchestimate is provided by Adelberger (2001), which translates to thefollowing mass ratio difference:

!WEPESM =

#$mG

mI

%

E−

$mG

mI

%

M

&

WEP(11)

= (1.0 ± 1.4) × 10−13. (12)

It is also possible to deduce the Nordtvedt parameter (η) defined as

!SEPESM = ηSEP

#$|#|mc2

%

E−

$|#|mc2

%

M

&(13)

≈ ηSEP × (−4.45 × 10−10), (14)

where # and mc2 are the gravitational binding and rest mass en-ergies, respectively, for the Earth and the Moon (subscripts E andM, respectively). The value of −4.45 × 10−10 is obtained fromWilliams et al. (2009, equation 7).

However, all metric theories lead to a violation of the SEP only.Therefore, for metric theories, it is irrelevant to try to separateviolation effects of the WEP and SEP, as the WEP is intrinsicallyrespected.

3.4.2 Dilaton theory and a generalization of the Nordtvedtinterpretation

Starting from a general dilaton theory, a more general equationgoverning celestial mechanics than equation (6) has been found to

be Hees & Minazzoli (2015); Minazzoli & Hees (2016)

aT = −'

A=T

GmGA

r3AT

rAT (1 + δT + δAT ) . (15)

The coefficients δT and δAT parametrize the violation of theUFF. In this expression, the inertial mass mI

A writes in termsof the gravitational mass mG

A as mGA = (1 + δA)mI

A Hees & Mi-nazzoli (2015); Minazzoli & Hees (2016). Of course, sincemG

A/mIA = 1 + δA, one recovers equation (6) when δAB = 0 for

all A and B. From equation (15), one can check that the gravi-tational force in this context still satisfies Newton’s third law ofmotion:

mIAaA = GmI

AmIB

r3AB

rAB (1 + δA + δB + δAB ) = −mIB aB. (16)

In the dilaton theory, the δ coefficients are functions of ‘dila-tonic charges’ and of the fundamental parameters of the theoryDamour & Donoghue (2010); Hees & Minazzoli (2015); Minazzoli& Hees (2016). However, in what follows, we will consider thephenomenology based on the δ parameters independently of its the-oretical origin, as a similar phenomenology may occur in a differenttheoretical framework.

In general, δT can be decomposed into two contributions:one from a violation of the WEP and one from a violation ofthe SEP:

δT = δWEPT + δSEP

T , with δSEPT = η

|#T|mTc2

. (17)

The quantity δSEPT depends only on the gravitational energy content

of the body T. On the other hand, δWEPT depends on the composition

of the falling body T (Damour & Donoghue 2010; Hees & Minazzoli2015; Minazzoli & Hees 2016). In some theoretical situations (seee.g. Damour & Donoghue 2010), if δWEP

T = 0, then δWEPT ≫ δSEP

T ,such that one can have either a clean WEP violation or a clean SEPviolation.

Like the parameter δWEPT , δAT depends on the composition of

the falling bodies. However, unlike δWEPT , it also depends on the

composition of the body A that is the source of the gravita-tional field in which the body T is falling (Hees & Minazzoli2015; Minazzoli & Hees 2016). As a consequence, the relativeacceleration of two test particles with different compositions can-not be related to the ratios between their gravitational-to-inertialmasses in general (i.e. mG

A/mIA = 1 + δA). This contrasts with

the usual interpretation (see for instance Williams et al. 2012).However, with some theoretical models, δWEP

T is much greaterthan δAT (Damour & Donoghue 2010; Hees & Minazzoli 2015;Minazzoli & Hees 2016).

At the Newtonian level, the relative acceleration between theEarth and the Moon reads

aM − aE = −Gµ

r3EM

rEM + GmGS

#rSE

r3SE

− rSM

r3SM

&

+ GmGS

#rSE

r3SE

(δE + δSE) − rSM

r3SM

(δM + δSM)&

, (18)

with µ ≡ mGM + mG

E + (δE + δEM)mGM + (δM + δEM)mG

E . As dis-cussed already in the previous subsection, the first term of equation(18) does not lead to a sensitive test of the UFF, because it canbe absorbed in the fit of the parameter mG

M + mGE (e.g. Williams

et al. 2012), while the last term does. At leading order, one canapproximate both distances appearing in this last term as being

Interpretation 1: Nordtvedt’s interpretation : inertial versus gravitational mass

16

All metric theories can be reduced to a SEP violation, WEP is always respected.In terms of self-gravitation SEP violation, we have

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S

h?2Q`2iB+HBMi2`T`2i@iBQMb

*QKT`BbQMrBi?JA*_P@a*PS1

*QM+HmbBQM

LQ`/ip2/iǶb BMi2`T`2iiBQM ,BM2`iBH p2`bmb ;`pBiiBQMH Kbb

HH K2i`B+ i?2Q`B2b +M #2 `2/m+2/ iQ a1S pBQHiBQM- q1SBb Hrvb `2bT2+i2/XAM i2`Kb Q7 miQ;`pBiiBQM a1S pBQHiBQM- r2 ?p2

mG

mI= ηN

Ω

mc2U8V

r?2`2 Ω Bb i?2 b2H7@;`pBiiBQM 2M2`;vX q2 ;2i i?2M

∆ESM ≈ ηN × (−4.45× 10−10) UeV

qBi? Qm` `2bmHi- Bi BKTHB2b ,

ηN = (0.85± 1.59)× 10−4 UdV

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM



mG

mI= ηN

Ω

mc2U8V


∆ESM ≈ ηN × (−4.45× 10−10) UeV


ηN = (0.85± 1.59)× 10−4 UdV

where Ω is the self-gravitation energy. We get then

LLR

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM



mG

mI= ηN

Ω

mc2U8V


∆ESM ≈ ηN × (−4.45× 10−10) UeV


ηN = (0.85± 1.59)× 10−4 UdV

17

Interpretation 2: DilatonIn the dilation frame where a scalar interacts with matter, the equation of motion becomes

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM

.BHiQM T?2MQK2MQHQ;v

HA = mAv2A2

− G!

B =A

mAmB

rAB(1 + δA + δAB) +O

"1

c2

#

UkyV

r?2`2

δA =α0αA

1 + α20

− ηNΩA

mAc2, δAB =

αAαB

1 + α20

UkRV

Ç α0 , mMBp2`bH +QmTHBM; +QMbiMiXÇ αA , MQM mMBp2`bH +QmTHBM; +QMbiMi- /2T2M/b QM

+?2KB+H +QKTQbBiBQM Q7 AXÇ a1S pBQHiBQM Bb biBHH ?2`2 5 Hees & Minazzoli, arXiv:1512.05233 (2015)

18

Interpretation 2: DilatonIn the dilation frame where a scalar interacts with matter, the equation of motion becomeCa :_J

kyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM

AM i?Bb 7`K2rQ`F- i?2 BMi2`T`2iiBQM Bb

∆ESM = [(δ1`i? + δamM@1`i?)− (δJQQM + δamM@JQQM)] UkkV

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM

.BHiQM T?2MQK2MQHQ;v

HA = mAv2A2

− G!

B =A

mAmB

rAB(1 + δA + δAB) +O

"1

c2

#

UkyV

r?2`2

δA =α0αA

1 + α20

− ηNΩA

mAc2, δAB =

αAαB

1 + α20

UkRV

Ç α0 , mMBp2`bH +QmTHBM; +QMbiMiXÇ αA , MQM mMBp2`bH +QmTHBM; +QMbiMi- /2T2M/b QM

+?2KB+H +QKTQbBiBQM Q7 AXÇ a1S pBQHiBQM Bb biBHH ?2`2 5

V Viswanathan, A Fienga, O Minazzoli, L Bernus, J Laskar, M Gastineau; The new lunar ephemeris INPOP17a and its application to fundamental physics, Monthly Notices of the Royal Astronomical Society, Volume 476, Issue 2, 11 May 2018, Pages 1877–1888, https://doi.org/10.1093/mnras/sty096

https://doi.org/10.1093/mnras/sty096

19

INPOP17a and fundamental physics tests 9

Table 6. Comparison of results for the value of !ESM (column 4) estimated with the solution INPOP17A fitted to LLR data set between (1) 1969–2011(for comparison with Muller et al. 2012; Williams et al. 2012); (2) 1969–2017 with data obtained only in green wavelength, (3) 1969–2017 with dataobtained with both green and IR wavelength. Column 5 empirically corrects the radial perturbation from effects related to solar radiation pressure andthermal expansion of retroreflectors using equation (4), with a value !r = 3.0 ± 0.5 mm Williams et al. (2012). Column 6 contains the value of !ESM afterapplying the corrections of column 5. Column 7 contains the parameter η obtained using equation (13). See discussion in Section 4.

Reference Data Uncertainty Estimated Corrected Corrected Parametertime span !ESM cos D !ESM ηc

(Year) (× 10−14) (mm) (× 10−14) (× 10−4)

Williams et al. (2009)a 1969–2004 N/A 3.0 ± 14.2 2.8 ± 4.1 −9.6 ± 14.2 2.24 ± 3.14Williams et al. (2012) 1969–2011 N/A 0.3 ± 12.8 2.9 ± 3.8 −9.9 ± 12.9 2.25 ± 2.90Muller et al. (2012)a, b 1969–2011 3σ −14 ± 16 – – –INPOP17A (limited data) 1969–2011 3σ −3.3 ± 17.7 4.0 ± 5.2 −13.5 ± 17.8 3.03 ± 4.00Hofmann & Muller (2016)a 1969–2016 3σ – – −3.0 ± 6.6 0.67 ± 1.48INPOP17A (green only) 1969–2017 3σ 5.2 ± 8.7 1.5 ± 2.6 −5.0 ± 8.9 1.12 ± 2.00INPOP17A (green and IR) 1969–2017 3σ 6.4 ± 6.9 1.1 ± 2.1 −3.8 ± 7.1 0.85 ± 1.59

Notes. aThermal expansion correction not applied.bSRP correction not applied.cDerived using |$E|

mEc2 − |$M|mMc2 = −4.45× 10−10 (Williams et al. 2012, equation 6).

and the Earth–Moon mass ratio (EMRAT), respectively. In all thesolutions w.r.t. LLR EP estimation, the gravitational mass of theEarth–Moon barycentre (GMEMB) remains as a fit parameter dueits high correlation with the EP parameter (!ESM). EMRAT wasestimated from a joint planetary solution and kept fixed duringLLR EP tests (for all INPOP solutions in Table 6) due to its weakdetermination from LLR.

A test solution that fitted EMRAT, with GMEMB as a fixed param-eter, gives an estimate of !ESM = (8 ± 7.0) × 10−14. However, thevalue of EMRAT estimated from an LLR-only solution has an un-certainty of one order of magnitude greater than that obtained fromthe joint planetary fit. This is also consistent with a similar resultby Williams et al. (2009). As a result, EMRAT was not included asa fit parameter for the estimates provided in Table 6, as it resultedin a degraded fit of the overall solution.

Williams et al. (2012) show that including annual nutation com-ponents of the Earth pole direction in space, to the list of fit-ted parameters during the estimation of LLR EP solution, in-creases the uncertainty of the estimated UFF violation parame-ter (!ESM) by 2.5 times. Moreover, it is to be noted that withinTable 6, the solutions by Williams, Turyshev & Boggs (2009,2012) and Muller, Hofmann & Biskupek (2012) use the IERS2003 McCarthy & Petit (2004) recommendations within the re-duction model, while all INPOP17 solutions use IERS 2010 Pe-tit & Luzum (2010) recommendations. The notable difference be-tween the two IERS models impacting the LLR EP estimation isexpected to be from the precession nutation of the celestial in-termediate pole within the ITRS–GCRS transformation (Petit &Luzum 2010, p. 8).

Equation (4) shows the dependence of !ESM w.r.t. the cosine ofthe lunar orbit synodic angle, synonymous with the illuminationcycle of the lunar phases. Due to the difficulties involved withranging to the Moon during the lunar phases with the extreme valuesof cos D (new and full Moon) as described in Section 2.1.2, the LLRobservations during these phases remain scarce. The availability ofIR LLR observations from Grasse contributes to the improvement ofthis situation, as shown in Fig. 2. This is reflected in the improvementof the uncertainty of the estimated value of !ESM by 14 per cent,with solutions including the IR LLR data.

Using both IR and green wavelength data, and empirically cor-recting for the radial perturbation for effects related to solar radi-ation pressure and thermal expansion, our final result on the UFF

violation parameter is given by (see also, Table 6)

!ESM = (−3.8 ± 7.1) × 10−14. (5)

The continuation of the IR observational sessions at Grasse willhelp to continue the improvement in the !ESM estimations.

An observable bias in the differential radial perturbation of thelunar orbit w.r.t. the Earth, towards the direction of the Sun, if sig-nificant and not accounted for within the dynamical model, wouldresult in a false indication of the violation of the principle of equiv-alence estimated with the LLR observations. Oberst et al. (2012)show the distribution of meteoroid impacts with the lunar phase.Peaks within the histogram in Oberst et al. (2012, p. 186) indicate anon-uniform temporal distribution with a non-negligible increase inboth small and large impacts during the new and full Moon phase.Future improvements to the LLR EP estimation must consider theimpact of such a bias that could potentially be absorbed during thefit by the LLR UFF violation parameter !ESM.

3.4 Theoretical interpretations

3.4.1 Nordtvedt’s interpretation: gravitational versus inertialmasses

Although equations of motion are developed at the pN level inINPOP Moyer (2003), violations of the UFF can be cast entirelyin the Newtonian equation of motion with sufficient accuracy. Asdescribed by Nordtvedt (1968b), a difference of the inertial (mI)and gravitational (mG) masses would lead to an alteration of bodytrajectories in celestial mechanics according to the following equa-tion:

aT = −!

mG

mI

"

T

#

A=T

GmGA

r3AT

rAT , (6)

where rAT = xT − xA and G is the constant of Newton.Following Williams et al. (2012), the relative acceleration at the

Newtonian level between the Earth and the Moon due to the attrac-tion of the Sun reads

aM − aE = −Gµ

r3EM

rEM + GmGS

$rSE

r3SE

− rSM

r3SM

%

+ GmGS

$rSE

r3SE

!!mG

mI

"

E− 1

"− rSM

r3SM

!!mG

mI

"

M− 1

"%, (7)


∆ESM = (-3.8 ± 7.1) ×10−14

new interpretation: dilaton theory

Results: Test of fundamental physics using LLRCa :_J

kyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM



mG

mI= ηN

Ω

mc2U8V


∆ESM ≈ ηN × (−4.45× 10−10) UeV


ηN = (0.85± 1.59)× 10−4 UdV

Thermal expansion term


20

INPOP17a and fundamental physics tests 9

Table 6. Comparison of results for the value of !ESM (column 4) estimated with the solution INPOP17A fitted to LLR data set between (1) 1969–2011(for comparison with Muller et al. 2012; Williams et al. 2012); (2) 1969–2017 with data obtained only in green wavelength, (3) 1969–2017 with dataobtained with both green and IR wavelength. Column 5 empirically corrects the radial perturbation from effects related to solar radiation pressure andthermal expansion of retroreflectors using equation (4), with a value !r = 3.0 ± 0.5 mm Williams et al. (2012). Column 6 contains the value of !ESM afterapplying the corrections of column 5. Column 7 contains the parameter η obtained using equation (13). See discussion in Section 4.

Reference Data Uncertainty Estimated Corrected Corrected Parametertime span !ESM cos D !ESM ηc

(Year) (× 10−14) (mm) (× 10−14) (× 10−4)

Williams et al. (2009)a 1969–2004 N/A 3.0 ± 14.2 2.8 ± 4.1 −9.6 ± 14.2 2.24 ± 3.14Williams et al. (2012) 1969–2011 N/A 0.3 ± 12.8 2.9 ± 3.8 −9.9 ± 12.9 2.25 ± 2.90Muller et al. (2012)a, b 1969–2011 3σ −14 ± 16 – – –INPOP17A (limited data) 1969–2011 3σ −3.3 ± 17.7 4.0 ± 5.2 −13.5 ± 17.8 3.03 ± 4.00Hofmann & Muller (2016)a 1969–2016 3σ – – −3.0 ± 6.6 0.67 ± 1.48INPOP17A (green only) 1969–2017 3σ 5.2 ± 8.7 1.5 ± 2.6 −5.0 ± 8.9 1.12 ± 2.00INPOP17A (green and IR) 1969–2017 3σ 6.4 ± 6.9 1.1 ± 2.1 −3.8 ± 7.1 0.85 ± 1.59

Notes. aThermal expansion correction not applied.bSRP correction not applied.cDerived using |$E|

mEc2 − |$M|mMc2 = −4.45× 10−10 (Williams et al. 2012, equation 6).

and the Earth–Moon mass ratio (EMRAT), respectively. In all thesolutions w.r.t. LLR EP estimation, the gravitational mass of theEarth–Moon barycentre (GMEMB) remains as a fit parameter dueits high correlation with the EP parameter (!ESM). EMRAT wasestimated from a joint planetary solution and kept fixed duringLLR EP tests (for all INPOP solutions in Table 6) due to its weakdetermination from LLR.

A test solution that fitted EMRAT, with GMEMB as a fixed param-eter, gives an estimate of !ESM = (8 ± 7.0) × 10−14. However, thevalue of EMRAT estimated from an LLR-only solution has an un-certainty of one order of magnitude greater than that obtained fromthe joint planetary fit. This is also consistent with a similar resultby Williams et al. (2009). As a result, EMRAT was not included asa fit parameter for the estimates provided in Table 6, as it resultedin a degraded fit of the overall solution.

Williams et al. (2012) show that including annual nutation com-ponents of the Earth pole direction in space, to the list of fit-ted parameters during the estimation of LLR EP solution, in-creases the uncertainty of the estimated UFF violation parame-ter (!ESM) by 2.5 times. Moreover, it is to be noted that withinTable 6, the solutions by Williams, Turyshev & Boggs (2009,2012) and Muller, Hofmann & Biskupek (2012) use the IERS2003 McCarthy & Petit (2004) recommendations within the re-duction model, while all INPOP17 solutions use IERS 2010 Pe-tit & Luzum (2010) recommendations. The notable difference be-tween the two IERS models impacting the LLR EP estimation isexpected to be from the precession nutation of the celestial in-termediate pole within the ITRS–GCRS transformation (Petit &Luzum 2010, p. 8).

Equation (4) shows the dependence of !ESM w.r.t. the cosine ofthe lunar orbit synodic angle, synonymous with the illuminationcycle of the lunar phases. Due to the difficulties involved withranging to the Moon during the lunar phases with the extreme valuesof cos D (new and full Moon) as described in Section 2.1.2, the LLRobservations during these phases remain scarce. The availability ofIR LLR observations from Grasse contributes to the improvement ofthis situation, as shown in Fig. 2. This is reflected in the improvementof the uncertainty of the estimated value of !ESM by 14 per cent,with solutions including the IR LLR data.

Using both IR and green wavelength data, and empirically cor-recting for the radial perturbation for effects related to solar radi-ation pressure and thermal expansion, our final result on the UFF

violation parameter is given by (see also, Table 6)

!ESM = (−3.8 ± 7.1) × 10−14. (5)

The continuation of the IR observational sessions at Grasse willhelp to continue the improvement in the !ESM estimations.

An observable bias in the differential radial perturbation of thelunar orbit w.r.t. the Earth, towards the direction of the Sun, if sig-nificant and not accounted for within the dynamical model, wouldresult in a false indication of the violation of the principle of equiv-alence estimated with the LLR observations. Oberst et al. (2012)show the distribution of meteoroid impacts with the lunar phase.Peaks within the histogram in Oberst et al. (2012, p. 186) indicate anon-uniform temporal distribution with a non-negligible increase inboth small and large impacts during the new and full Moon phase.Future improvements to the LLR EP estimation must consider theimpact of such a bias that could potentially be absorbed during thefit by the LLR UFF violation parameter !ESM.

3.4 Theoretical interpretations

3.4.1 Nordtvedt’s interpretation: gravitational versus inertialmasses

Although equations of motion are developed at the pN level inINPOP Moyer (2003), violations of the UFF can be cast entirelyin the Newtonian equation of motion with sufficient accuracy. Asdescribed by Nordtvedt (1968b), a difference of the inertial (mI)and gravitational (mG) masses would lead to an alteration of bodytrajectories in celestial mechanics according to the following equa-tion:

aT = −!

mG

mI

"

T

#

A=T

GmGA

r3AT

rAT , (6)

where rAT = xT − xA and G is the constant of Newton.Following Williams et al. (2012), the relative acceleration at the

Newtonian level between the Earth and the Moon due to the attrac-tion of the Sun reads

aM − aE = −Gµ

r3EM

rEM + GmGS

$rSE

r3SE

− rSM

r3SM

%

+ GmGS

$rSE

r3SE

!!mG

mI

"

E− 1

"− rSM

r3SM

!!mG

mI

"

M− 1

"%, (7)


∆ESM = (-3.8 ± 7.1) ×10−14

Results: Test of fundamental physics using LLRCa :_J

kyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM



mG

mI= ηN

Ω

mc2U8V


∆ESM ≈ ηN × (−4.45× 10−10) UeV


ηN = (0.85± 1.59)× 10−4 UdV

Thermal expansion term

40% of uncertainty comes from poor knowledge of

fluid core size


• GRAIL-derived gravity field for the Moon (see discussion at the end)

• fc, flattening of the fluid core for a given FC radius (seismology)

• Rotational time delays for Earth and Moon

• h2 Love number

• friction coefficient at the CMB

• Total moment of inertia of the Moon

• Initial conditions for Moon’s orbit and libration angles

• observational bias + station positions

Sources of possible improvement: Estimated model parameters

21


Modelparameters


ReductionmodelGINS

LLRdata

Regression

Fixed parameters

estimates

Constraints

Sonnemann et al. 2005

22

Example 1: LLR as a Lunar Interior Probe• Test the sensitivity of LLR to RFC and fc

• Estimates of core oblateness intersect with corresponding theoretical values of a hydrostatic core.

• The accuracy of oblateness and radii of a presently-relaxedlunar core is improved by a factor of 3.

• Direct impact on EP tests usingLLR data

23

V Viswanathan, N Rambaux, A Fienga, J Laskar, M Gastineau;Observational constraint on the radius and oblateness of the lunar core-mantle boundary, 2019, submitted to GRL

Example 1: LLR as a Lunar Interior Probe• Test the sensitivity of LLR to RFC and fc

• Estimates of core oblateness intersect with corresponding theoretical values of a hydrostatic core.

• The accuracy of oblateness and radii of a presently-relaxedlunar core is improved by a factor of 3.

• Direct impact on EP tests usingLLR data

Example 2: Current state-of-the-art Planetary and Lunar Ephemeris

24

• Full list of parameters available through the ephemeris files • See documentation for comparison of planetary orbits with JPL ephemerides

96CHAPTER

5.CONSTRUCTIO

NOFA

LUNAR

EPHEMERIS:IN

POP17A

Table 5.5: Extended body parameters for the Earth and the Moon. Uncertainties for INPOPG and INPOP17a(1-) are obtained from a 5% jackknife (JK). DE430 uncertainties seem to be inflated (unknown scaling) formaluncertainties and EPM solutions provide the 1- formal uncertainties. †: C32, S32 and C33 are reference values fromthe GRAIL analysis by Konopliv et al. (2013). ‡: h2 reference value from LRO-LOLA analysis by Mazarico et al.(2014). ∗ : derived quantity. Refer to Section 5.3.2 for the description of the solution INPOPG

Parameter Units INPOPG INPOP17a DE430 EPM

(GMEMB − 8.997011400 × 10−10) × 1019 AU3/day2 4 ± 2 4 ± 2 -10 10 ± 5(R1,E − 7.3 × 10−3) × 105 day 0 ± 4 6 ± 3 6 ± 30 57 ± 5(R2,E − 2.8 × 10−3) × 105 day 9.2 ± 0.4 8.7 ± 0.3 −27 ± 2 5.5 ± 0.4(CT (mMR2) − 0.393140) × 106 6.9 ± 0.2 8.2 ± 0.2 2∗ 2∗(C32 − 4.8404981 × 10−6†) × 109 4.1 ± 0.3 3.9 ± 0.3 4.4 4.4 ± 0.1(S32 − 1.6661414 × 10−6†) × 108 1.707 ± 0.006 1.666 ± 0.006 1.84 1.84 ± 0.02(C33 − 1.7116596 × 10−6†) × 108 −1.19 ± 0.04 -2.40 ± 0.04 −3.6 −4.2 ± 0.2(M − 9 × 10−2) × 104 day −14 ± 5 -35 ± 3 58.0 ± 100 60 ± 10( kvCT− 1.6 × 10−8) × 1010 day−1 12.7 ± 0.4 15.3 ± 0.5 4.0 ± 10.0 3.0 ± 2.0(fc − 2.1 × 10−4) × 106 37 ± 3 42 ± 3 36 ± 28 37 ± 4(h2 − 3.71 × 10−2‡) × 103 6.3 ± 0.2 6.8± 0.2 11.0 ± 6 6 ± 1

Q27.212 − 45 (derived) 3.9 ± 0.5 5.0 ± 0.2 0 ± 5 0 ± 1

25

GRAIL gravity field ~ 20km resolution

Current state-of-the-art for lunar geodesy: GRAIL

• Asymmetry of the crustal thickness (Wieczorek et al. 2013): difference of about 40km between nearside and farside • Gravity field with < 30 km resolution (Lemoine et al. 2014, Konopliv et al. 2013) • topography, altimetry (first measurement of h2 with LRO and LOLA, Mazarico et al. 2012)

Some examples of GRAIL results:

• From March to December 2012• Altitude : 15km to 50 km and up to 8 km• Mutual distances: from 65 to 225 km

NASA/JPL-Caltech/MIT/GSFC

NASA/JPL-Caltech/MIT/GSFC

Example 2:Comparison of LLR estimates with GRAIL

26

ratio of LLR-GRAIL difference to GRAIL “scaled” uncertainty ≈ 103

27

Open question: Interior signature or systematics ?1-

way

LT

[cm

]

To be investigated withGRAIL team..

Conclusions

• INPOP17a lunar ephemeris files and documentation available at:

www.imcce.fr/inpop

•

28

∆ESM = (-3.8 ± 7.1) ×10−14

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM



mG

mI= ηN

Ω

mc2U8V


∆ESM ≈ ηN × (−4.45× 10−10) UeV


ηN = (0.85± 1.59)× 10−4 UdV

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM

AM i?Bb 7`K2rQ`F- i?2 BMi2`T`2iiBQM Bb

∆ESM = [(δ1ì? + δamM@1ì?)− (δJQQM + δamM@JQQM)] UkkV

• SEP:

• SEPLLR / SEPMicroscope

• LLR detects SEP when MICROSCOPE detects WEP

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM

GG_ M/ JA*_Pa*PS1

δA =α0αA

1 + α20

− ηNΩA

mAc2UkjV

"mi r2 ?p2

ΩA ≈!

A

!

A

Gρ(r)ρ(r′)|r − r′| /3r/3r′ ≈ 3

5

Gm2A

RAUk9V

U7Q` bT?2`B+H #Q/B2bV bm+? i?i

ηNΩA

mAc2≈ ηN

3

5

mA

RAc2∼ ρAR

2A Uk8V

Ca :_JkyR3

GûQ "2`Mmb

AMi`Q/m+iBQM

:2M2`H i2biQ7 1S



*QM+HmbBQM

Ç JA*_Pa*PS1 2tT2`BK2Mi , bKHH +vHBM/2`b BM bi2HHBi2U6Q` /2iBHb #Qmi JA*_Pa*PS1 , `2K2K#2` :BHH2bJûi`BbǶ M/ JìBM S2`MQi@"Q``¨bǶ T`2b2MiiBQMb Dmbi#27Q`2XV

Ç 1ì?@JQQM 2tT2`BK2Mi , bi`QMQKB+H #Q/B2bXÇ ìBQ Q7 a1S pBQHiBQM ∼ (RMoon/RMicroscope)2

Ç *QM+HmbBQM , B7 a1S Bb pBQHi2/ M/ Bb /2i2+i#H2- Bi rBHH#2 /2i2+i2/ rBi? GG_X JB+`Qb+QT2 i2bib ì?2` q1Si?M a1SX

_27 ,hQm#QmH 2i HX S_G RRN- kjRRyR- kyRd

Future improvements:

• Improvement of the dissipation modeling: High order Lover numbers, viscoelastic behaviour, asymmetry of the crust…

• Better modelisation of the thermal expansion

General comment:

V Viswanathan,et al; 2018, The new lunar ephemeris INPOP17a and its application to fundamental physics, MNRAS, Vol 476, Issue 2, Pages 1877–1888,

http://www.imcce.fr/inpop

29

Acknowledgements:Sincere gratitude to all the observers and engineers at Grasse,

APOLLO, McDonald, Matera and Haleakala LLR stations for timely and accurate observations. CNES-GRGS GINS support and

Labex ESEP—Post-doctoral grant acknowledged.

๏ Courde, C., Torre, J.M., Samain, E., Martinot-Lagarde, G., Aimar, M., Albanese, D., Exertier, P., Fienga, A., Mariey, H., Metris, G., Viot, H., Viswanathan, V.: Lunar laser ranging in infrared at the Grasse laser station. Astron. Astrophys. 602, A90 (2017).

๏ Folkner, W.M., Williams, J.G., Boggs, D.H., Park, R.S., Kuchynka, P.: The Planetary and Lunar Ephemerides DE430 and DE431. (2014).

๏ Konopliv, A.S., Park, R.S., Yuan, D.-N.N., Asmar, S.W., Watkins, M.M., Williams, J.G., Fahnestock, E., Kruizinga, G., Paik, M., Strekalov, D., Harvey, N., Smith, D.E., Zuber, M.T.: The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission. J. Geophys. Res. Planets. 118, 1415–1434 (2013).

๏ Lemoine, F.G., Goossens, S., Sabaka, T.J., Nicholas, J.B., Mazarico, E., Rowlands, D.D., Loomis, B.D., Chinn, D.S., Caprette, D.S., Neumann, G.A., Smith, D.E., Zuber, M.T.: High-degree gravity models from GRAIL primary mission data. Journal of Geophysical Research: Planets. 118, 1676–1698 (2013).

๏ Mémin, A., Viswanathan, V., Fienga, A., Santamarìa-Gómez, A., Boy, J.-P., Cavalié, O., Deleflie, F., Exertier, P., Bernard, J.-D., Hinderer, J.: Multi-geodetic characterization of the seasonal signal at the CERGA geodetic reference station, France. In: EGU General Assembly Conference Abstracts. p. 7450 (2017).

๏ Müller, J., Hofmann, F., Biskupek, L.: Testing various facets of the equivalence principle using lunar laser ranging. Classical and Quantum Gravity. 29, 184006 (2012).

๏ Ries, J., Bettadpur, S., Eanes, R., Kang, Z., Ko, U., McCullough, C., Nagel, P., Pie, N., Poole, S., Richter, T., Save, H., Tapley, B.: The Combined Gravity Model GGM05C. (2016).

๏ Viswanathan, V., Fienga, A., Gastineau, M., Laskar, J.: INPOP17a planetary ephemerides. Notes Sci. Tech. l’Institut Mec. Celeste. 108, (2017).

๏ Viswanathan, V., Fienga, A., Minazzoli, O., Bernus, L., Laskar, J., Gastineau, M.: The new lunar ephemeris INPOP17a and its application to fundamental physics. Mon. Not. R. Astron. Soc. 476, 1877–1888 (2018).

๏ Viswanathan, V.: Improving the dynamical model of the Moon using lunar laser ranging and spacecraft data, Ph.D thesis - Observatoire de Paris, (2017).

๏ Williams, J.G., Boggs, D.H.: Secular tidal changes in lunar orbit and Earth rotation. Celest. Mech. Dyn. Astron. 126, 89–129 (2016)

Thank you !

a. fienga[1,2], v. viswanathan[1,2], l. bernus[1], n. rambaux

Documents