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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 Sciences2St. Petersburg State University

Workshop on Precision Physics and Fundamental Physical Constants (FFK 2013)

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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 uponare 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).

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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.

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Observations 677670 observations are used for fitting EPM2011

Planet Radio Optical

Interval of observ.

Number of observ.

Interval of observ.

Number of observ.

Mercury 1964-2009 948 —— ——

Venus 1961-2010 40061 —— ——

Mars 1965-2010 578918 —— ——

Jupiter +4sat. 1973-1997 51 1914-2011 13364

Saturn+9sat. 1979-2009 126 1913-2011 15956

Uranus+4sat. 1986 3 1914-2011 11846

Neptune+1sat. 1989 3 1913-2011 11634

Pluto —— —— 1914-2011 5660

In total 1961-2010 620110 1913-2011 575604

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Accuracy of astrometric observations

1 mas

1 µas10 µas

100 µas

10 mas

100 mas

1“

10”

100”

1000”

1 µas10 µas

100 µas

1 mas

10 mas

100 mas

1”

10”

100”

1000”

1400 1500 1700 1900 2000 21000 1600 1800

Ulugh Beg

Wilhelm IVTycho Brahe

HeveliusFlamsteed

Bradley-Bessel

FK5

Hipparcos

Gaia

SIM

ICRF

GC

naked eye telescopes space

1400 1500 1700 1900 2000 21000 1600 1800

Hipparchus

1 as is the thickness of a sheet of paper seen from the other side of the Earth

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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 (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..

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PPN parameters and (General Relativity: = =1)

-1 = 0.000020.00003, -1 = +0.000040.00006 => 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

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Estimation of the secular changes ofof GM and G

= Ġ/G +

The following relation

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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 SunThe 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: < 3.2•10-14M per year

The total value interval of

- 9.8 • 10-14 < < -3.6 • 10-14 per year

The decrease of the solar mass

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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 ofThe secular change of GM

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

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Estimations of dark matter in the Solar system

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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)

where M(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

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Planets ΔMSun [•10-10MSun ] |σΔMSun / ΔMSun |

Mercury -0.5 ± 117.7 235.4

Venus -0.67 ± 5.86 8.7

Mars 0.20 ± 2.65 13.3

Jupiter 0.4 ± 1671.4 4178.5

Saturn -0.27 ± 15.16 56.1

Corrections to the central attractive mass

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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)

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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.

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Precession of a planet orbit

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Planets π |σπ / π|mas/yr

Mercury -0.020 ± 0.030 1.5

Venus 0.026 ± 0.016 0.62

Earth 0.0019 ± 0.0019 1.0

Mars -0.00020 ± 0.00037 1.9

Jupiter 0.587 ± 0.283 0.48

Saturn -0.0032 ± 0.0047 1.5

Additional perihelion precessions from the observationsof planets and spacecraft

1 mas = 0".001

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Planets σΔπ ["/yr] ρ [г/см3]

Mercury 0.000030 < 9.3•10-18

Venus 0.000016 < 1.9•10-18

Earth 0.00000190 < 1.4•10-19

Mars 0.00000037 ≤ 1.40•10-20

Jupiter 0.000283 ≤ 1.7•10-18

Saturn 0.0000047 ≤ 1.1•10-20

Estimates of the density ρdm from the

perihelion precessions σΔπ

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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•10-20 g/cm3.

The mass within the spherical volume with the size of Saturn’s orbit is

Mdm < 7.1•10-11 MSun , (11)

which is within the error of the total mass of the main asteroid belt (1).

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As a model of the ρdm distribution, we took the expression: ρdm = ρ0 • e-cr , (6)

where ρ0 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 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 = 4π ρ0 [2/c3 – e-cr (r2/c + 2r/c2 + 2/c3)] (8)

Estimations for an exponential decrease in

density to the periphery

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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•10-20 g/cm3 )

and Earth (ρdm < 1.4•10-19 g/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•10-20 g/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•10-17 g/cm3 и c =4.42 ае-1 .

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-20,5-20

-19,5-19

-18,5-18

-17,5-17

-16,5

0 2 4 6 8 10 12

r (kpc)

lg (r

)

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The mass in the space between the orbits of Mars and Saturn is Mdm < 7.33•10-11 MSun .

The mass between the Sun and the orbit of Mars is Mdm < 0.55•10-11 MSun .

Summing masses for both intervals, the upper limit for the total mass of dark matter was estimated as

Mdm < 7.88•10-11 MSun

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•10-10 MSun (3σ)

of the total mass of the asteroid belt. 24

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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 found the density ρdm at the orbital distances of Saturn is less than

ρdm < 1.1•10-20 g/cm3 . • The dark matter mass in the sphere within Saturn’s

orbit should be less than

Mdm < 7.9•10-11 Msun

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

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Thanks !