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INPOP planetary ephemerides: Recent results intesting gravity

A. Fienga1

J. Laskar2 P. Exertier1 H. Manche2 M. Gastineau2

1GéoAzur, Observatoire de la Côte d’Azur, France

3IMCCE, Observatoire de Paris, France

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The planetary ephemerides today

DE430,EPM2011, INPOP13c: what they have in common ...

Numerical integration of the (Einstein-Imfeld-Hoffmann, c−4 PPN approximation)equations of motion.

ẍPlanet =∑A6=B

µBrAB‖rAB‖3

+ ẍGR(β, γ, c−4) + ẍAST ,300 + ẍJ�2

Adams-Cowell in extended precision

8 planets + Pluto + Moon + asteroids (point-mass, ring),GR, J�2 , Earth rotation (Euler angles, specific INPOP)

Moon: orbit and librations

Simultaneous numerical integration TT-TDB, TCG-TCB

Fit to observations in ICRF over 1 cy (1914-2014)

GAIA ESA planetary ephemerides, Asteroid physics,Tests of gravity

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The Solar system and the tests of gravity

With such accuracy, the solar system is still the ideal lab for testinggravity

0

5

10

15

20

25

30

35

40

1970 1975 1980 1985 1990 1995 2000 2005 2010Year

(cm

)

LLR WRMS based on INPOP10a

5 cm

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and the modified gravity comes ...!

For example,

Theories Phenomenology Object

Standard Model violation of EP Moon-LLR

MOND d$̇supp, dΩ̇supp planets

Scalar field theories Ġ/G Moon-LLR, planetsvariation of β, γ planets

Dark Energy Ġ/G Moon-LLR, planetsAWE/chameleons variation of β, γ planetsDark Matter linear drift of AU planets

asupp Moon-LLR,planets

d$̇supp, dΩ̇supp planetsISL d$̇supp planets,Moon-LLRf(r) asupp planets

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INPOP and gravity tests

In Planetary and Lunar ephemerides (like INPOP), GR plays a role in

∆tSHAP = (1 + γ)GM�(t)lnl0 + l1 + t

l0 + l1 − t

∆$̇PLA =2π(2γ − β + 2)GM�(t)

a(1− e2)c2+

3πJ2R2�

a2(1− e2)c2+ ∆$̇AST

∆$̇Moon =2π(2γ − β + 2)GM�(t)

a(1− e2)c2+ ∆$̇GEO + ∆$̇SEL + ∆$̇S,PLA

GR tests are then limited by

Contributions by J�2 , Asteroids, 2γ − β + 2Lunar and Earth physics

BUTDecorrelation with all the planetsBenefit of PE global fit versus single space mission

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Specific INPOP developments for testing gravity

Variations/Estimations of PPN β, γ, Sun J�2 (Fienga et al. 2011)

Simulation of a Pioneer anomaly type of acceleration → ẍconstant (Fienga et al. 2011)

Supplementary advance of perihelia $̇ and nodes Ω̇ → R($(ti ),Ω(ti )) → MONDconstraints (Fienga et al. 2011)

Equivalence Principle @ astronomical scale

→ ẍj =mGj

mIjF (xi , ẋi ,m

Gi , ...) = (1 + η)F (xi , ẋi ,m

Gi , ...)

With µ� = GM�, µj = GMj for planet j

Ṁ�M�

and ĠG with˙µ�µ�

= ĠG +Ṁ�M�

andµ̇jµj

= ĠG

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more than 1 century of data

Mercury: independent analysis ofMESSENGER data (Verma et al. 2014)

Saturn: Cassini VLBI and radio tracking

Jupiter: Galileo VLBI and 5 s/c flybys

Venus, Mars: VEX, MEX, MGS, MRO,MO, ...

α δ ρS/C VLBI 1990-2010 V, Ma, J, S 1/10 mas 1/10 mas

S/C Flybys 1976-2014 Me, J, S, U, N 0.1/1 mas 0.1/1 mas 1/30 m

S/C Range 1976-2014 Me, V, Ma 2/30 m

Direct range 1965-1997 Me,V 1 km

Optical 1914-2014 J, S, U, N, P 300 mas 300 mas

LLR 1980-2014 Moon 1cm

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INPOP13C accuracy versus to DE, EPM (I)

Name # Perturbers # fitted masses Ring GM�

TNO Main belt TNO Main belt TNO Main belt

INPOP13c 0 150 0 150 0 F F

DE430 0 343 0 343 0 0 F

EPM2011 21 301 0 21 + 3 density classes F F

Differences in dynamical modeling and adjustments

INPOP13c data sets ≈ DE430 data sets BUT

2 specific data analysis for MESSENGER

new JPL analysis for CASSINI → Independent analysis in progress

EPM2011 older data sets without MESSENGER data and recently published

Cassini VLBI

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INPOP13C accuracy versus to DE, EPM (II)

Differences in angles [mas] Differences in range [m]

(INPOP13c - DE430)

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Tests of gravity

ẍPlanet =∑A6=B

µBrAB‖rAB‖3

+ ẍGR(β, γ, c−4) + ẍAST ,300 + ẍJ�2

With µ� = GM�, µj = GMj for planet j

Ṁ�M�

and ĠG with˙µ�µ�

= ĠG +Ṁ�M�

andµ̇jµj

= ĠG

∀ti ,M�(ti ) andG (ti )→µ(ti ), µ�(ti ) → ẍPlanet , ẍAst , ẍMoon

What values of ˙µ�µ� (and thenṀ�M�

or ĠG ) BUT also PPN β, γ and

J�2 are acceptable / INPOP13c accuracy ?

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2 approaches based on INPOP13c

1 - Global fit including ˙µ�µ� , PPN β, γ and J�2

Planet CI, 290 GMast ,GMring , GM�, EMRAT + GRFull data samples including Messenger and Cassini data

Correlations between parameters

Correlated datasets

2 - Monte Carlo simulation

about 36000 runs

optimized by a genetic algorithm

With Ṁ�M� = (−0.92± 0.46)× 10−13 yr−1 → ĠG (Pinto et al. 2013),(Pitjeva et al. 2012)

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GR Global fit

Correlations between parameters (Ashby et al. 2007), (Fienga et al. 2009), (Iorio et al.2011)

∆$̇ = 2π(2γ−β+2)µ�(t)a(1−e2)c2 +

3πJ�2 R2�

a2(1−e2)c2 + ∆$̇AST

→ Full modeling with 290 GMast and Limited modeling with 60 GMast

Correlated datasets (Konopliv et al. 2011), (Kuchynka et al. 2012)

S/C electronic delay → bias by s/cJPL data analysis (MO,MRO) →Mis-calibration per station → bias bystation DSS

Independant analysis (MEX, VEX,MGS, MESSENGER): mean rangeper station and per data arc

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GR Global fit: results

Method PPN β − 1 PPN γ − 1 Ġ/G J�2× 105 × 105 × 1013 yr−1 × 107

Limited -12.8 ± 5.4 10.2 ± 6.6 1.12 ± 0.40 2.23 ± 0.2Limited + SC + DSS -4.45 ± 9.1 10.1 ± 2.4 1.12 ± 0.34 2.23 ± 0.2Full -4.9 ± 5.1 -2.0 ± 5.1 -0.54± 0.51 2.27 ± 0.3Full + SC + DSS -4.4 ± 5.3 -0.81 ± 4.5 0.42 ± 0.53 2.27 ± 0.25INPOP13a (Verma et al. 2014) 0.2 ± 2.5 -0.3 ± 2.5 0.0 2.40 ± 0.20

First INPOP Full fit of the 4 GR parameters all together

Stable estimation for J�2

Greater uncertainties for β and γ vs (Verma et al. 2014)

Sensivity of 3 GR β, γ and Ġ/G to dynamical modeling

∆χ2 ≈ 1%

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MC simulations”Real” uncertainty/LS estimations + ”my theory proposes this violation of GR. Is it compatible with INPOP ?”

Grid of sensitivity for GRP determinations(Fienga et al. 2009, 2011), (Verma et al. 2014)

GRP: PPN β,γ, $̇, Ω̇, asupp, Ġ/G , J�2

Construction of different INPOP for different values of GRP

For each value of GRP , all parameters (IC planets, GMAst ,GM�) of INPOP are fitted.

Iteration = all correlations are taken into account2 selection criteria

1 C1 = Postfit residuals /INPOP → GRP intervals with ∆residuals < 50%

2 C2 = ∆χ2 < 3%

What values of GRP are acceptable at the level of data accuracy ?

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INPOP13a and tests of GR: PPN β and γ

INPOP13a (Verma et al. 2014):

(β − 1)× 105 = 0.2± 2.5 , (γ − 1)× 105 = −0.3± 2.5 , J�2 × 107 = 2.40± 0.20

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Direct Monte Carlo of µ̇/µ, J�2 , β, γ

4000 INPOP runs with random

selection of (µ̇/µ, J�2 , β, γ)

1 run = 4 iterations (1hr/iteration@ 16 itanium processors)

Selection of INPOP(µ̇/µ, J�2 , β, γ)

inducing differences to INPOP13a

residuals < 50 %

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Direct Monte Carlo of µ̇/µ, J�2 , β, γ

Only 15 % INPOP(µ̇/µ,J�2 , β, γ) < 50 %1.4% for INPOP() < 25 %

No clear gaussiandistribution especiallyfor β and γ

→ Optimisation of the MCby a genetic algorithm

< J�2 > (2.21 ± 0.29) × 10−7

W-test = 0.984

< β − 1 > (-0.8 ± 8.2)× 10−5 ?0.969

< γ − 1 > (0.2± 8.2)× 10−5 ?0.968

< Ġ/G > (0.04± 2.46)*× 1013 yr−1 (0.27± 1.66)**

0.987

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Simple Genetic Algorithm with mutation (SGAM)

1 individual = INPOP (µ̇/µ, J�2 , β, γ)

1 chromosome = a set of (µ̇/µ, J�2 , β, γ)

2 crossovers + 1/10 mutation (= new random value each over 10)

2 selection criteria : (O-C) < 50% and ∆χ2 < 3%

35 800 runs with 4000 MC simulation as population 0

set i [(µ̇/µ)i , (J�2 )i , βi , γi ]

set j [(µ̇/µ)j , (J�2 )j , βj , γj ]

1 crossover [(µ̇/µ)i , (J�2 )i ,βj , γj ]

[(µ̇/µ)j , (J�2 )j ,βi , γi ]

2 crossovers [(µ̇/µ)i , (J�2 )j ,βi , γj ]

[(µ̇/µ)j , (J�2 )i ,βj , γi ]

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35 800 runs with SGAM

@ PSL mesocentre : NEC 1472 kernels on 92 nodes2 nodes allocated for INPOP12 runs (= 12 x 4 iterations) @ 1hr / node4000 MC simulation = population 0 @ SGAM30 generationsC1 convergency reached @ 18th (12125 runs)C2 convergency reached @ 25th

C1 C2

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Improvement of gaussianity

C1 C2

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Improvement of gaussianity

→ Proper definitionof mean and 1-σ

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Decrease of σ of the GRP distribution with thenumber of runs

C1 C2

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PPN β, γ, µ̇/µ, J�2

Method PPN β − 1 PPN γ − 1 Ġ/G J�2× 10−5 × 10−5 × 1013 yr−1 × 107

LS -4.4 ± 5.5 -0.81 ± 4.5 0.42 ± 0.53 2.27 ± 0.3MC -0.8 ± 8.2 0.2 ± 8.2 0.04 ± 2.46 2.21 ± 0.29MC + SGAM C1 -0.5 ± 6.3 -1.2 ± 4.4 0.4 ± 1.4 2.26 ± 0.11MC + SGAM C2 -0.01 ± 7.1 -1.7 ± 5.2 0.5 ± 1.6 2.22 ± 0.14MC + SGAM C1,C2 -0.25 ± 6.7 -1.5 ± 4.8 0.5 ± 1.6 2.24 ± 0.125

Ġ/G ≈ 2 ×10−13 yr−1 β − 1 ≈ 7 ×10−5 γ − 1 ≈ 5 ×10−5EP η = 4β − γ − 3≈ 2×10−4

J�2 = (2.24± 0.15)× 10−7

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PPN β, γ, µ̇/µ, J�2Method PPN β − 1 PPN γ − 1 Ġ/G J�2

× 10−5 × 10−5 × 1013 yr−1 × 107LS -4.4 ± 5.5 -0.81 ± 4.5 0.42 ± 0.53 2.27 ± 0.3MC + SGAM C1 -0.5 ± 6.3 -1.2 ± 4.4 0.4 ± 1.4 2.26 ± 0.11MC + SGAM C2 -0.01 ± 7.1 -1.7 ± 5.2 0.5 ± 1.6 2.22 ± 0.14MC + SGAM C1,C2 -0.25 ± 6.7 -1.5 ± 4.8 0.5 ± 1.6 2.24 ± 0.125K11-DE 4 ± 24 18 ± 26 1.0 ± 2.4 fixed to 1.8P13-EMP -2 ± 3 4 ± 6 0.29 ± 1.25 2.0 ± 0.2INPOP13a 0.2 ± 2.5 -0.3 ± 2.5 0.0 2.4 ± 0.2F13-DE 0.0 0.0 0.0 2.1 ± 0.70W09-LLR 12 ± 11 fixed 0.0 fixedM05-LLR 15 ± 18 fixed 6 ± 8 fixedB03-Cass 0.0 2.1 ± 2.3 0.0L11-VLB 0.0 -8 ± 12 0.0 fixedPlanck +WP+BAO -1.42± 2.48Heliosismo. 2.206 ± 0.05

Ġ/G ≈ 2 ×10−13 yr−1 β − 1 ≈ 7 ×10−5 γ − 1 ≈ 5 ×10−5J�2 = (2.24± 0.15)× 10−7 EP η = 4β − γ − 3≈ 2×10−4

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Conclusions

β − 1 ≈ 7 ×10−5 γ − 1 ≈ 5 ×10−5EP η = 4β − γ − 3≈ 2×10−4

Ġ/G ≈ 2 ×10−13 yr−1 J�2 = (2.24± 0.15)× 10−7

Short term improvement:

Saturn: Full reprocessing of Cassini 10 years data

Jupiter: JUNO mission in 2015/2016

Mars: MO,MEX,MRO

In the coming years: GAIA, Bepi-Colombo, JUICE

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