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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 in
Solar system
1E.V.
1Institute
Pitjeva, 2N. P. Pitjev
of Applied Astronomy, Russian Academy of Sciences
2St. Petersburg State University
1
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).
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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
Mercury
Radio
Optical
Interval of Number of Interval of Number of
observ.
observ.
observ.
observ.
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
57560 4
Accuracy of astrometric observations
naked eye
0
1400
Hipparchus
1000”
100”
1500
1600
telescopes
1700
1900
2000
2100
Ulugh Beg
1000”
Wilhelm IV
Tycho Brahe
Hevelius
10”
1800
space
100”
Flamsteed
Bradley-Bessel
1“
10”
1”
GC
100 mas
100 mas
FK5
10 mas
10 mas
Hipparcos
1 mas
1 mas
ICRF
100 µas
Gaia
10 µas
1 µas
SIM
0
1400
1500
1600
1700
1800
1900
2000
100 µas
10 µas
1 µas
2100
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1 as is the thickness of a sheet of paper seen from the other side of the Earth
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 , 6
the Earth to Moon mass ratio: MEarth/MMoon = 81.30056763  0.00000005..
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
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Estimation of the secular changes
of GM and G
The following relation
= Ġ/G +
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The decrease of the solar mass
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:
< 3.2•10-14M per year
The total value interval of
- 9.8 • 10-14 <
< -3.6 • 10-14 per year
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The secular change of GM
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.
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Pitjeva, 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
Corrections to the central attractive mass
Planets
ΔMSun [•10-10MSun ]
|σΔMSun / ΔMSun |
Mercury
Venus
-0.5 ± 117.7
-0.67 ± 5.86
235.4
8.7
Mars
Jupiter
Saturn
0.20 ± 2.65
0.4 ± 1671.4
-0.27 ± 15.16
13.3
4178.5
56.1
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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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Additional perihelion precessions from the
observationsof planets and spacecraft
1 mas = 0".001
Planets
π
|σπ / π|
Mercury
mas/yr
-0.020 ± 0.030
1.5
Venus
Earth
Mars
Jupiter
0.026 ± 0.016
0.0019 ± 0.0019
-0.00020 ± 0.00037
0.587 ± 0.283
0.62
1.0
1.9
0.48
Saturn
-0.0032 ± 0.0047
1.5
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Estimates of the density ρdm from the
perihelion precessions σΔπ
σΔπ ["/yr]
ρ [г/см3]
0.000030
0.000016
< 9.3•10-18
< 1.9•10-18
Earth
Mars
0.00000190
0.00000037
< 1.4•10-19
≤ 1.40•10-20
Jupiter
Saturn
0.000283
0.0000047
≤ 1.7•10-18
≤ 1.1•10-20
Planets
Mercury
Venus
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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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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 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)
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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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lg (r )
-16,5
-17 0
-17,5
-18
-18,5
-19
-19,5
-20
-20,5
2
4
6
r
8
10
12
(kpc)
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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.
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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 !
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