1. 1 1 Determination of the PPN parameters, and limiting estimations for the density of dark matter and change G in Solar system 1 E.V. Pitjeva, 2 N. P. Pitjev 1 Institute of Applied Astronomy, Russian Academy of Sciences 2 St. Petersburg State University Workshop on Precision Physics and Fundamental Physical Constants (FFK 2013)

2. 2 2 The EPM ephemerides (Ephemerides of Planets and the Moon) of IAA RAS originated in the seventies of the last century to support space flights and have been developed since that time. All the modern ephemerides (DE – JPL, EPM – IAA RAS, INPOP – are based upon IMCCE) are based upon relativistic equations of motion for astronomical bodies and light rays as well as relativistic time scales. The numerical integration of the equations of celestial bodies motion has been performed in the Parameterized Post-Newtonian metric for General Relativity in the TDB time scale. EPM ephemerides are computed by numerical integration of the equations of motion of planets, the Sun, the Moon, asteroids, TNO and the equations of the lunar physical libration in the barycentric coordinate frame of J2000.0 over the 400 years interval (1800 – 2200).

3. 3 3 The dynamical model of EPM2011takes into account the following: mutual perturbations from the major planets, the Sun, the Moon and 5 more massive asteroids; perturbations from the other 296 asteroids chosen due to their strong perturbations upon Mars and the Earth; perturbation from the massive asteroid ring with the constant mass distribution; perturbations from the TNO; perturbation from a massive ring of TNO in the ecliptic plane with the radius of 43 au; relativistic perturbations; perturbations due to the solar oblateness J 2 =210 -7 ; perturbations due the figures of the Earth and the Moon.

4. 4 Observations 677670 observations are used for fitting EPM2011 Planet RadioOptical Interval of observ. Number of observ. Interval of observ. Number of observ. Mercury1964-2009948—— Venus1961-201040061—— Mars1965-2010578918—— Jupiter +4sat.1973-1997511914-201113364 Saturn+9sat.1979-20091261913-201115956 Uranus+4sat.198631914-201111846 Neptune+1sat.198931913-201111634 Pluto—— 1914-20115660 In total1961-20106201101913-201157560 4

5. 5 Accuracy of astrometric observations 1 mas 1 µas 10 µas 100 µas 10 mas 100 mas 1“ 10” 100” 1000” 1 µas 10 µas 100 µas 1 mas 10 mas 100 mas 1” 10” 100” 1000” 140015001700190020002100016001800 Ulugh Beg Wilhelm IV Tycho Brahe Hevelius Flamsteed Bradley-Bessel FK5 Hipparcos Gaia SIM ICRF GC naked eye telescopes space 140015001700190020002100016001800 Hipparchus 1  as is the thickness of a sheet of paper seen from the other side of the Earth

6. 6 Approximately 270 parameters were determined while improving the of planetary part of EPM2011 ephemerides the orbital elements of planets and 18 satellites of the outer planets; the value of the astronomical unit or GM  ; three angles of orientation of the ephemerides with respect to the ICRF; thirteen parameters of Mars’ rotation and the coordinates of three landers on Mars; the masses of 21 asteroids; the mean densities of asteroids for three taxonomic types (C, S, and M); the mass and radius of the asteroid 1-or 2-dimensional rings; the mass of the TNO belt; the Earth to Moon mass ratio; the Sun’s quadrupole moment (J 2 ) and parameters of the solar corona for different conjunctions of planets with the Sun; eight coefficients of Mercury’s topography and corrections to the level surfaces of Venus and Mars; the constant bias for three runs of planetary radar observations and seven spacecraft; five coefficients for the supplementary phase effect of the outer planets; post - model parameters (β, γ, π advances, ĠM  /GM , change of a i ). The values of some estimated parameters of EPM2011 (with uncertainties 3σ): the heliocentric gravitation constant: GM  = (132712440031.1 ± 0.3) km 3 /s 2, the Earth to Moon mass ratio: M Earth /M Moon = 81.30056763  0.00000005.. 6

7. 7 PPN parameters  and  (General Relativity:  =  =1)  -1 =  0.00002  0.00003,  -1 = +0.00004  0.00006 => a correspondence of the planetary motions and the propagation of light to General Relativity and narrow significantly the range of possible values for alternative theories of gravitation Pitjeva, Proc. IAU Symp. No. 261, 2010, 170-178; Pitjeva, Pitjev, MNRAS, 432, 2013, 3431-3437 7

8. 8 of Estimation of the secular changes of GM  and G = Ġ/G + The following relation

9. 9 The decrease of the solar mass due to radiation is: - 6.789 10 -14 per year. The decrease of the solar mass due to the solar wind is: (2-3)10 - 14 per year (Hundhausen, 1997; Meyer-Vernet N., 2007). The total effect of the solar mass loss due to radiation and the solar wind is: The fall (increase) of the matter on the Sun The dust fall is: < < 10 -16 ÷ 10 -17 per year The fall of asteroids is: < (10 -16 ÷ 10 -17 ) M  per year The fall of comets is: < 3.210 -14 M  per year The total value interval of - 9.8 10 -14 < < -3.6 10 -14 per year The decrease of the solar mass 9

10. 10 Taking into account the monotony and smallness of, it was shown (Jeans, 1924) that the invariant holds μ(t)·a(t) = const, where a is the orbital semi-major axis and μ(t)=G(M+m), then = –. The change of the geliocentic gravitation constant GM  is determined for certain – the accuracy increases as the square of the time interval of observations as: = (-6.3±4.24)10 -14 per year (2σ) being with the century changes of semi-major axes of planets determined simultaneously. The positive values for the planets Mercury, Venus, Mars, Jupiter, Saturn provided with the high-accuracy observations confirm the decrease of GM . Perhaps, loss of the mass of the Sun M  the produces change of GM  due to the solar radiation and the solar wind compensated partially by the matter dropping on the Sun. The secular change of The secular change of GM  10

11. 11 Pitjeva, Pitjev, Solar System Research, 2012, 46, 78-87; Pitjeva, Pitjev, MNRAS, 432, 2013, 3431-3437 11

12. 12 Estimations of dark matter in the Solar system

13. 13 The additional central mass Any planet at distance r from the Sun can be assumed to undergo an additional acceleration from dark matter: (d 2 r/dt 2 ) dm = - GM(r) dm /r 2, (1) where M(r) dm is the mass of the additional matter in a sphere of radius r around the Sun. At a uniform density ρ dm of the gravitating medium filling the Solar system, the additional acceleration on a body will be proportional to r: (d 2 r/dt 2 ) dm = - kr. (2) 13

14. 14 PlanetsΔM Sun [10 -10 M Sun ] | σ ΔMSun / ΔM Sun | Mercury-0.5 ± 117.7235.4 Venus-0.67 ± 5.868.7 Mars 0.20 ± 2.6513.3 Jupiter 0.4 ± 1671.44178.5 Saturn-0.27 ± 15.1656.1 Corrections to the central attractive mass

15. 15 Additional perihelion precessions If we denote the energy and area integrals per unit mass by E and J and a spherically symmetric potential by U(r), then (Landau and Lifshitz, 1988) the equation of motion along the radius r can be written as dr/dt = { 2[E+U(r)] - (J/r) 2 } 1/2. (3) The equation along the azimuth θ is dθ/dr = J/r 2 /{ 2[E+U(r)] - (J/r) 2 } 1/2. (4)

16. 16 The presence of the additional gravitating medium leads to a shorter radial period and a negative drift of the pericenter and apocenter positions (in a direction opposite to the planetary motion) : Δθ 0 = -4π 2 ρ dm /M Sun a 3 (1-e 2 ) 1/2 (5) where Δθ 0 is the perihelion drift in one complete radial oscillation. Khriplovich I. B., Pitjeva E. V., International Journal of Modern Physics D, 2006, V.15, 4, 615-618.

17. 17 Precession of a planet orbit

18. 18 Planets π|σ π / π| mas/yr Mercury-0.020 ± 0.0301.5 Venus 0.026 ± 0.0160.62 Earth 0.0019 ± 0.00191.0 Mars-0.00020 ± 0.000371.9 Jupiter 0.587 ± 0.2830.48 Saturn-0.0032 ± 0.00471.5 Additional perihelion precessions from the observationsof planets and spacecraft 1 mas = 0".001 18

19. 19 Planetsσ Δπ ["/yr]ρ [г/см 3 ] Mercury0.000030< 9.310 -18 Venus0.000016< 1.910 -18 Earth0.00000190< 1.410 -19 Mars0.00000037≤ 1.4010 -20 Jupiter0.000283≤ 1.710 -18 Saturn0.0000047≤ 1.110 -20 Estimates of the density ρ dm from the perihelion precessions σ Δπ 19

20. 20 Estimations for a uniform distribution of density If we proceed from the assumption of a uniform ρ dm distribution in the Solar system, then the most stringent constraint is obtained from the data for Saturn: ρ dm < 1.110 -20 g/cm 3. The mass within the spherical volume with the size of Saturn’s orbit is M dm < 7.110 -11 M Sun, (11) which is within the error of the total mass of the main asteroid belt (1). 20

21. 21 As a model of the ρ dm distribution, we took the expression: ρ dm = ρ 0 e -cr, (6) where ρ 0 is the central density and c is a positive parameter characterizing an exponential decrease in density to the periphery. The expressions for the gravitational potential for an inner point at distance r for distribution (6) is U(r) = 4πG ρ 0 /r [2- e -cr (cr+2)]/c 3 (7) The parameters of distribution (6) can be estimated from obtained results. The mass inside a sphere of radius r for distribution (6) is M dm = 4π ρ 0 [2/c 3 – e -cr (r 2 /c + 2r/c 2 + 2/c 3 )] (8) Estimations for an exponential decrease in density to the periphery 21

22. 22 The estimate of the mass of dark matter within the orbit of Saturn was determined from the evaluation of the masses within the two intervals, i.e. from Saturn to Mars and from Mars to the Sun. For this purpose, the most reliable data for Saturn (ρ dm < 1.110 -20 g/cm 3 ), Mars (ρ dm < 1.410 -20 g/cm 3 ) and Earth (ρ dm < 1.410 -19 g/cm 3 ) were used. Based on the data for Saturn and Mars a very flat trend of the density curve (12) between Mars and Saturn was obtained with ρ 0 = 1.4710 -20 g/cm 3 и c =0.0299 ае -1. The obtained trend of the density curve (12) in the interval between Mars and the Sun gives a steep climb to the Sun according to the data for Earth and Mars with the parameters ρ 0 = 1.1710 -17 g/cm 3 и c =4.42 ае -1. 22

23. 23

24. 24 The mass in the space between the orbits of Mars and Saturn is M dm < 7.3310 -11 M Sun. The mass between the Sun and the orbit of Mars is M dm < 0.5510 -11 M Sun. Summing masses for both intervals, the upper limit for the total mass of dark matter was estimated as M dm < 7.8810 -11 M Sun between the Sun and the orbit of Saturn, taking into account its possible tendency to concentrate in the center. This value is less than the uncertainty ± 1.1310 -10 M Sun (3σ) of the total mass of the asteroid belt. 24

25. 25 Results The mass of the dark matter, if present, and its density ρ dm are much lower than the today's errors of these parameters. It was found the density ρ dm at the orbital distances of Saturn is less than ρ dm < 1.110 -20 g/cm 3. The dark matter mass in the sphere within Saturn’s orbit should be less than M dm < 7.910 -11 M sun even if its possible concentration to the center is taken into account. Pitjev N.P., Pitjeva E.V., Astronomy Letters, 2013, V.39, 3, 141-149; Pitjeva E.V., Pitjev N.P., MNRAS, 432, 2013, 3431-3437 25

26. 26 Thanks !