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1 E.V. Pitjeva, 2 N . P . Pitjev 1 Institute of Applied Astronomy, Russian Academy of Sciences

This workshop focuses on determining the PPN parameters, estimating the density of dark matter and changes in the gravitational constant G in the solar system.

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1 E.V. Pitjeva, 2 N . P . Pitjev 1 Institute of Applied Astronomy, Russian Academy of Sciences

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

  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 – 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). 2

  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 J2=2•10-7; perturbations due the figures of the Earth and the Moon. 3

  4. Observations677670observations are used for fitting EPM2011 4

  5. Accuracy of astrometric observations naked eye telescopes space 0 1400 1500 1600 1700 1800 1900 2000 2100 Hipparchus Ulugh Beg 1000” 1000” 100” 100” Wilhelm IV Tycho Brahe Flamsteed 10” Hevelius 10” Bradley-Bessel 1“ 1” GC 100 mas 100 mas FK5 10 mas 10 mas Hipparcos 1 mas 1 mas ICRF 100 µas 100 µas 10 µas Gaia 10 µas 1 µas SIM 1 µas 0 1400 1500 1600 1700 1800 1900 2000 2100 1 as is the thickness of a sheet of paper seen from the other side of the Earth

  6. Approximately 270 parametersweredeterminedwhileimprovingtheofplanetarypartof 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 (J2 ) 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 ai). The values of some estimated parameters of EPM2011 (with uncertainties 3σ): the heliocentric gravitation constant: GM = (132712440031.1 ± 0.3)km3/s2 , the Earth to Moon mass ratio: MEarth/MMoon = 81.30056763  0.00000005.. 6

  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. Estimation of the secular changesofGM and G The following relation = Ġ/G +

  9. The decrease of the solar mass The decrease of the solar massdue to radiation is: -6.789• 10-14 per year. The decrease of the solar massdue tothe 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 dustfall is:< < 10-16 ÷ 10-17 per year The fall of asteroids is: < (10-16 ÷ 10-17)•Mper year The fall of comets is: < 3.2•10-14Mper year The total value interval of - 9.8 •10-14 < < -3.6 •10-14 per year 9

  10. The secular change ofGM Taking into account the monotony and smallness of , it was shown (Jeans, 1924) that the invariant holds μ(t)·a(t)= const, whereais 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 planetsMercury, 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. 10

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

  12. Estimations of dark matter in the Solar system

  13. The additional central mass Any planet at distance r from the Sun can be assumed to undergo an additional acceleration from dark matter: (d2r/dt2)dm = - GM(r)dm /r2 , (1) whereM(r)dm is the mass of the additional matter in a sphere of radius r around the Sun. At a uniform density ρdm of the gravitating medium filling the Solar system, the additional acceleration on a body will be proportional to r: (d2r/dt2)dm = - kr.(2) 13

  14. Corrections to the central attractive mass 14

  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/r2 /{ 2[E+U(r)] - (J/r)2 }1/2 .(4) 15

  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): Δθ0 = -4π2ρdm /MSun •a3(1-e2)1/2(5) where Δθ0 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. 16

  17. Precession of a planet orbit 17

  18. Additional perihelion precessions from the observationsof planets and spacecraft 1mas = 0".001 18

  19. Estimates of the density ρdm from the perihelion precessions σΔπ 19

  20. Estimations for a uniform distribution of density If we proceed from the assumption of a uniform ρdm distribution in the Solar system, then the most stringent constraint is obtained from the data for Saturn: ρdm < 1.1•10-20g/cm3. The mass within the spherical volume with the size of Saturn’s orbit is Mdm < 7.1•10-11MSun , (11) which is within the error of the total mass of the main asteroid belt (1). 20

  21. Estimations for an exponential decrease in density to the periphery As a model of the ρdm distribution, we took the expression: ρdm = ρ0• e-cr , (6) where ρ0 is the central density and cis a positive parameter characterizing an exponential decrease in density to the periphery. The expressions for the gravitational potential for an inner point at distance rfor distribution (6) is U(r) = 4πG ρ0 /r •[2- e-cr(cr+2)]/c3(7) The parameters of distribution (6) can be estimated from obtained results. The mass inside a sphere of radius r for distribution (6) is Mdm = 4πρ0 [2/c3 – e-cr (r2/c + 2r/c2 + 2/c3)] (8) 21

  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 (ρdm < 1.1•10-20 g/cm3), Mars (ρdm < 1.4•10-20g/cm3 ) and Earth (ρdm < 1.4•10-19g/cm3) 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 ρ0= 1.47•10-20g/cm3 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 ρ0= 1.17•10-17g/cm3 и c=4.42 ае-1 . 22

  23. 23

  24. The mass in the space between the orbits of Mars and Saturn is Mdm< 7.33•10-11MSun . The mass between the Sun and the orbit of Mars is Mdm<0.55•10-11MSun . Summing masses for both intervals, the upper limit for the total mass of dark matter was estimated as Mdm< 7.88•10-11MSun 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 ±1.13•10-10MSun(3σ) of the total mass of the asteroid belt. 24

  25. Results • The mass of the dark matter, if present, and its density ρdm are much lower than the today's errors of these parameters. • It was foundthe density ρdm at the orbital distances of Saturn is less than ρdm < 1.1•10-20g/cm3 . • The dark matter mass in the sphere within Saturn’s orbit should be less than Mdm <7.9•10-11Msun 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. Thanks ! 26

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