Relativistic effects and dark matter in the Solar system from observations of planets and spacecraft
aa r X i v : . [ a s t r o - ph . E P ] J un Relativistic effects and dark matter in the solar systemfrom observations of planets and spacecrafts
E.V. Pitjeva ∗ , N.P. Pitjev ∗∗ Institute of Applied Astronomy of RAS, St. Petersburg, Russia St. Petersburg State University, St. Petersburg, Russia
Accepted 2013
The high precision of the latest version of the planetary ephemeris EPM2011 enables oneto explore more accurately a variety of small effects in the solar system. The processing ofabout 678 thousand of position observations of planets and spacecrafts for 1913–2011 withthe predominance of modern radar measurements resulted in improving the PPN parameters,dynamic oblateness of the Sun, secular variation of the heliocentric gravitational constant GM ⊙ , and variation range of the gravitational constant G . This processing made it possibleto estimate the potential additional gravitational influence of dark matter on the motion ofthe solar system bodies. The density of dark matter ρ dm , if any, turned out to be substantiallybelow the accuracy achieved by the present determination of such parameters. At the distanceof the orbit of Saturn the density ρ dm is estimated to be under . · − g/cm , and themass of dark matter in the area inside the orbit of Saturn is less than . · − M ⊙ eventaking into account its possible tendency to concentrate in the center. Key words : solar system: ephemerides, relativistic effects, heliocentric gravitationalconstant, dark matter ∗ E-mail: [email protected] ∗∗ E-mail: [email protected] . INTRODUCTION
The possibility to test and refine various relativistic and cosmological effects from theanalysis of the motion of the solar system bodies is due to the present meter accuracyradio techniques (Standish, 2008) and millimeter accuracy laser techniques (Murphy et al.,2008) for the distance measurements. Just these techniques have provided an observationalfoundation of the contemporary high–precision theories of planetary motions.The numerical theories of planetary motions have been improved and developed by severalgroups in different countries and their accuracy is constantly growing. The progress is relatedwith the increase of the number of high–precision radio observations and the inclusionof a number of small effects (perturbations from a set of asteroids, the solar oblatenessperturbations, etc.) in constructing a dynamic model of the solar system. The radio technicalobservations, having much higher accuracy as compared with the optical ones, are commonlyused now in astrometric practice. These high–precision measurements covering more than50 year time interval allow us to find the orbital elements, masses and other parametersdetermining the motion of the bodies. Moreover, they also give a possibility to check somerelativistic parameters, to estimate the secular change of the heliocentric gravitation constantand to examine the presence of dark matter in the solar system. The last point is of particularimportance for the contemporary cosmological theories. A more accurate and extensive setof observations permits us not only to determine the relativistic perihelion precession ofplanets, but also to estimate the oblateness of the Sun with the corresponding contributioninto the drift of the perihelia. Moreover, these observations provide a means for finding thesecular variation of the heliocentric gravitational constant GM ⊙ and the constraint on thesecular variation of the gravitational constant G (Pitjeva & Pitjev, 2012). In addition, theseprecise observations enable us to consider the assumption of the presence of dark matter inthe solar system and to estimate the upper limits of its mass and density.The present research has been performed on the basis of the current version of EPM2011(the numerical Ephemerides of the Planets and the Moon) of IAA RAS.
2. THE PLANETARY EPHEMERIS EPM2011
Numerical Ephemerides of Planets and the Moon (EPM) had started in the 1970s. Eachsubsequent version is characterized by additional new observations, refined values of the2rbital elements and masses of the bodies, an improved dynamical model of the celestialbodies motion, as well as a more advanced reduction of observational data.All presently used main planetary ephemerides DE (Standish, 1998), EPM (Pitjeva,2005a), and INPOP (Fienga, 2008) are based on General Relativity involving the relativisticequations of celestial bodies motion and light propagation as well as the relativistic timescales. In addition, these ephemerides involve estimating from observations some parameters( β , γ , ˙ G ) to check their compatibility with General Relativity.The current EPM2011 ephemerides were constructed using approximately 680 thousanddata (1913-2011) of different types. The equations of the bodies motion were taken within theparameterized post–Newtonian n–body metric in the barycentric coordinate system – BCRS(Brumberg 1991), the same as that of DE. Integration in TDB (Barycentric DynamicalTime) time scale (see the IAU2006 resolution B3) was performed using the Everhart’smethod over the 400-year interval (1800-2200) with the lunar and planetary integratorof the ERA-7 software package (Krasinsky & Vasilyev, 1997). The EPM ephemeridesincluding also the time differences TT-TDB, and seven additional asteroids, namely,Ceres, Pallas, Vesta, Eris, Haumea, Makemake, Sedna, are available via FTP by meansof ftp://quasar.ipa.nw.ru/incoming/EPM/.Since the basic observational data for producing the next version of the planetaryephemerides EPM2011 were mainly related to the spacecrafts, the control of the orientationof the EPM2011 ephemerides with respect to the ICRF frame has required a particularattention. For this purpose we have used the VLBI observations of the spacecrafts nearplanets at the background of quasars. The coordinates of the quasars are given in the ICRFframe (Table 1), where ( α + δ ) are one–dimensional measurements of the α and δ combination,( α, δ ) being the two–dimensional measurements (the position of the planet is observed to bedisplaced relative to the base ephemeris (DE405) by a correction measured counter-clockwisealong a line at an angle to the right ascension axis, see Folkner, 1992).3 able 1. VLBI observations of near–planet spacecrafts at the ICRF background quasarsPlanet Spacecraft Interval of observations Number of observationsVenus Magellan 1990-1994 18( α + δ )Venus Express 2007-2010 29( α + δ )Mars Phobos 1989 2( α + δ )MGS 2001-2003 15( α + δ )Odyssey 2002-2010 86( α + δ )MRO 2006-2010 41( α + δ )Saturn Cassini 2004-2009 22( α, δ )The accuracy of such observations increased to tenths of mas (1 mas = 0".001) for Marsand Saturn in 2001-2010 (Jones et al., 2011)enabling us to improve the orientation of EPMephemerides (Table 2) in the same way, as it was done by Standish (1998). The angles ofrotation of the Earth-Moon barycenter vector about the x,y,z-axes of the BCRS system wereobtained from VLBI observations described above. Table 2.
The angles of rotation of the EPM2011 ephemerides to ICRF (1 mas = . ′′ )Interval of Number of ε x ε y ε z observations observations mas mas mas1989-2010 213 − . ± . − . ± .
048 0 . ± . More than 270 parameters are estimated in the planetary part of EPM2011 ephemeridesas follows:- the orbital elements of planets and satellites of the outer planets,- the value of the astronomical unit or GM ⊙ ,- the angles of orientation of the EPM ephemerides with respect to the ICRF system,- parameters of the Mars rotation and the coordinates of the three Mars landers,- masses of 21 asteroids, the average density of the taxonomic class of asteroids (C, S,M),- the mass and radius of the asteroid ring and the mass of the TNO ring,- the mass ratio of the Earth and Moon,- the quadrupole moment of the Sun and the solar corona parameters for differentconjunctions of the planets with the Sun,- the coefficients for the Mercury topography and the corrections to the level surfaces of4enus and Mars,- coefficients for the additional phase effect of the outer planets.In the lunar part of EPM ephemerides about 70 parameters are estimated from LLRdata (see for example, Krasinsky, Prokhorenko, Yagudina, 2011). All estimated parametersin both parts are consistent within the frame of the combined theory of motion of the planetsand the Moon given by the EPM ephemerides.The initial parameters of EPM2011 represent the constants adopted by the IAU GA27 (Luzum et al., 2011) as the current best values for ephemeris astronomy. Among themfive constants are resulted from the ephemeris improvement of DE and EPM ephemerides(Pitjeva & Standish, 2009). These five parameters adjusted from processing all observationsfor EPM2011 are as follows:the masses of the largest asteroids, i.e. M Ceres /M ⊙ = . · − , M P allas /M ⊙ = . · − , M V esta /M ⊙ = . · − ;ratio of the masses of the Earth and Moon M Earth /M Moon = 81 . ± . ;the value of the astronomical unit in meters au = (149597870695 . ± . or theheliocentric gravitation constant GM ⊙ = (132712440031 ± km /s .Presently, in accordance with the IAU 2012 resolution B2 the astronomical unit (au) isre-defined by fixing its value. Up to now, both values of au and the heliocentric gravitationconstant ( GM ⊙ ) were in use. It was possible to determine the au value and to calculate thevalue of GM ⊙ from it, or vice versa, to determine GM ⊙ and to calculate the value of aufrom it. Here the values of au and GM ⊙ are given as in the paper Pitjeva & Standish, (2009)published before the IAU 2012 resolution B2. At present, only the value of GM ⊙ is estimatedfrom observations.A serious problem in developing modern planetary ephemerides arises due to the necessityto take into account the perturbations caused by asteroids. The factors affecting the planetarymotions and needed to be included in developing high–precision ephemerides are of particularconsideration in this paper. To start with, the hazard near–Earth asteroids are relativelysmall (D < 5 km) and their perturbations do not affect practically the Earth motion. That’swhy they are not examined in this paper. The main asteroid belt substantially affecting themotion of Mars and some other planets is modeled in the EPM ephemerides by using themotion of 301 large asteroids and a homogeneous material ring representing the influence of5ll other numerous small asteroids (Krasinsky et al., 2002; Pitjeva, 2010a). The parameterscharacterizing the ring of small asteroids (its mass and radius) were determined from theanalysis of observations resulting in the values: M ring = (1 . ± . · − M ⊙ (3 σ ) , R ring = (3 . ± .
26) (3 σ ) au . The total mass of the asteroid main belt represented by the sum of the mass of 301 largestasteroids and the homogeneous material ring (involving the main uncertainty) is M belt = (12 . ± . · − M ⊙ (3 σ ) , that is ≈ M Ceres . (1)This value of M belt is close to M belt = (13 . ± . · − M ⊙ ( σ ) , obtained from the Marsranging data in the paper by Kuchynka & Folkner (2013) by means of another method inestimating the masses of 3714 individual asteroids.Hundreds of trans–neptunian objects (TNO) discovered in recent years also affect themotion of the planets, especially outer ones. A dynamic model of EPM ephemerides includesEris (the planet–dwarf found in 2003 and surpassing Pluto by its mass) and other 20 largestTNO into the process of the simultaneous integration. Perturbations from other TNO aremodeled by the perturbation from a homogeneous ring located in the ecliptic plane with theradius of 43 au and the mass estimated in (Pitjeva, 2010a). The mass of the TNO ring foundfrom the analysis of observations amounts to M T NOring = (501 ± · − M ⊙ (3 σ ) . The total mass of all TNOs including the mass of Pluto, 21 largest TNOs and the TNO ringcomes to M T NO = (790 ± · − M ⊙ (3 σ ) , that is ≈ M Ceres or ≈ M Moon . (2)In addition to the mutual perturbations of the major planets and the Moon the EPM2011dynamic model includes- the perturbations of the 301 most massive asteroids,- the perturbations from the remaining minor planets of the main asteroid belt modeledby a homogeneous ring,- the perturbations from the 21 largest TNO,6 the perturbations from the remaining TNOs modeled by a uniform ring at the averagedistance of 43 au,- the relativistic perturbations,- the perturbation due to the oblateness of the Sun estimated in EPM2011 fitting as( J = 2 · − ).
3. OBSERVATION DATA AND THEIR REDUCTIONS
The total amount of the high-precision observations used for fitting EPM2011 has beenincreased due to the recent data. They include 677 670 positional measurements of differenttypes for 1913-2011 from classic meridian measurements to modern spacecraft tracking data(Table 3).
Table 3.
The observational materialPlanet Radio observations Optical observationsTime interval Number Time interval NumberMercury 1964-2009 948 – –Venus 1961-2010 40061 – –Mars 1965-2010 578918 – –Jupiter+4 sat. 1973-1997 51 1914-2011 13364Saturn+9 sat. 1979-2009 126 1913-2011 15956Uran+4 sat. 1986 3 1914-2011 11846Neptun+1 sat. 1989 3 1913-2011 11634Pluton – – 1914-2011 5660Total 620110 57560Radar measurements (the detailed description of them is given in Pitjeva 2005a, 2013)have a high accuracy. At present, the relative accuracy ∼ − for the spacecraft trajectorymeasurements became usual, exceeding the accuracy of classical optical measurements byfive orders of magnitude. However, in general only Mercury, Venus, and Mars are providedwith radio observations. Initially, the surfaces of these planets were radio located from 1961to 1995. Later on many spacecrafts passed by, orbited or landed to these planets. A largeportion of the spacecraft data was used to get the astrometric positions. There are muchless radio observations for Jupiter and Saturn, and only one set of the three–dimensional7ormal points ( α , δ , R ) obtained from the Voyager-2 spacecraft are available for Uranusand Neptune. Therefore, the optical observations are still of great importance for the outerplanets. Thereby, the varied data of 19 spacecrafts were used for constructing the EPM2011ephemerides and estimating the relevant parameters, in particular, the additional perihelionprecessions of the planets (see Table 4 below).The recent data from the spacecrafts have been added to the previous ones for thelatest version of the EPM ephemerides. It involves data related to Odyssey, MRO (MarsReconnaissance Orbiter) (Konopliv et al., 2011), Mars Express (MEX), Venus Express(VEX), and, more specifically, VLBI observations of Odyssey and MRO, three–dimensionalnormal points of Cassini and Messenger observations, along with the CCD observations ofthe outer planets and their satellites obtained at Flagstaff and Table Mountain observatories.The most part of observations was taken from the database of JPL/Caltech created by Dr.Standish and continued by Dr. Folkner. MEX and VEX data provided by ESA becameavailable thanks to a kindness of Dr. Fienga (private communications of T.Morlay toA.Fienga).The detailed description of methods for all reductions of planetary observations (bothoptical and radar ones) was given by Standish (1990). This is a basic paper in the field ofplanetary observations discussion. In the EPM ephemerides some reductions changed slightlyare described in Pitjeva 2005a, 2013. All necessary reductions listed therein were introducedinto actual observation data as follows:Reductions of the radar observations • the reduction of moments of observations to a uniform time scale; • the relativistic corrections – the time–delay of propagation of radio signals in thegravitational field of the Sun, Jupiter and Saturn (Shapiro effect) and the reduction ofTDB time (ephemeris argument) to the observer’s proper time; • the delay of radio signals in the troposphere of the Earth; • the delay of radio signals in the plasma of the solar corona; • the correction for the topography of the planetary surfaces (Mercury, Venus, Mars).8eductions of the optical observations • the transformation of observations to the ICRF framecatalogue differences => FK4 => FK5 => ICRF; • the relativistic correction for the light bending of the Sun; • the correction for the additional phase effect.
4. RESULTS4.1 Estimates of relativistic effects
Some small parameters are determined (in addition to the orbital elements of the planets)while constructing the EPM ephemerides using new observations and the method similar to(Pitjeva, 2005a,b). In most cases the parameters can be found from the analysis of the secularchanges of the orbital elements. Therefore, the uncertainties of their determination decreasewith increasing the time interval of observations.The simplified relativistic equations of the planetary motion were derived more than30 years ago in different coordinate systems of the Schwarzschild metric supplementedwith coordinate parameter alpha to specify standard, harmonic, isotropic, or any othercoordinates. These equations were described in (Brumberg, 1972, 1991). For example, theintegration exposed in (Oesterwinter & Cohen, 1972) was made in the standard coordinates(alpha=1). However, planetary coordinates turned out to be essentially different for thestandard and harmonic systems. It was shown in (Brumberg, 1979) that the ephemerisconstruction and processing of observations should be done in the same coordinate systemresulting to the relativistic effects not dependent on the coordinate system (effacing ofparameter alpha). Later on, the resolutions of IAU (1991, 2000) recommended to use theharmonic coordinates for BCRS. In accordance with the IAU 2000 resolution B1.3 modernplanetary ephemerides should be constructed in the harmonic coordinates for BCRS – thebarycentric (for the solar system) coordinate system.The parameters of the PPN formalism β, γ used to describe the metric theories of gravitymust be equal to 1 in General Relativity. The values of parameters β, γ were obtainedsimultaneously by using the EPM2011 ephemerides and the updated database of high–precision observations (Table 3) to get the relativistic periodic and secular variations of9he orbital elements, as well as the Shapiro effect. Certainly, the periodic variations of theorbital elements are smaller than the secular ones but they are of importance to compute theplanetary motion. We derived expressions for the partial derivatives of the orbital elementswith respect to β and γ using the analytical formulas for the relativistic perturbations ofthe elements, including the secular and principal periodic terms given in (Brumberg, 1972).This technique enabled us to get actually the values for β and γ . Moreover, in the eightiesof the last century we tested relativistic effects by processing the observations available atthat time. It turned out that the relativistic ephemeris for any observed planet provided aconsiderably better fit of observations (by 10%) than the Newtonian theory even if the latterincorporated the observed perihelion secular motion (Krasinsky et al., 1986).The obtained values of β and γ read β − − . ± . , γ − . ± . σ ) . (3)The uncertainties in (3) significantly decreased as compared with the results for the EPM2004(Pitjeva, 2005b) and EPM2008 (Pitjeva, 2010b) | β − | < . , | γ − | < . . In (Fienga et al., 2011) based on the INPOP10a planetary ephemeris these parameterswere determined separately fixing one of these two values, either β = 1 , or γ = 1 . Yet theephemeris fitting results in β − − . ± . , γ − . ± . . For comparison we also quote the new γ value obtained by using Very Long BaselineArray measurements of radio sources by Fomalont et al., 2009, i.e. γ = 0 . ± . .All the obtained values of β, γ are in the close vicinity of 1 within the limits of theiruncertainties. As the uncertainties of these parameters decrease, the range of possiblevalues of the PPN parameters narrows, imposing increasingly stringent constraints on thegravitation theories alternative to General Relativity. The solar oblateness produces the secular trends in all elements of the planets with theexception of their semimajor axes and eccentricities (see, for example, Brumberg, 1972).10herefore, the dynamic solar oblateness can be determined together with other parametersfrom observations in constructing a theory of planetary motion. The quadrupole moment ofthe Sun characterizing the solar oblateness was found in EPM2011 to be J = (2 . ± . · − (3 σ ) , (4)that is close to the previous result (Pitjeva, 2005a,b) for EPM2004 J = (1 . ± . · − and the result of INPOP10a (Fienga et al., 2011) J = (2 . ± . · − (1 σ ) . GM ⊙ and G The value of the secular change of the heliocentric gravitational constant GM ⊙ hasbeen updated for the expanded database and the improved dynamical model of planetarymotions (EPM2011). The determination of secular variation GM ⊙ was carried out by themethod exposed in detail in (Pitjeva & Pitjev, 2012) dealing with the EPM2010 planetaryephemerides.The GM ⊙ change was determined by the weighted method of the least squares withall the basic parameters of the EPM2011 ephemerides. In determining ˙ GM ⊙ , it was takeninto account that the acceleration between the Sun and any planet varies with time when GM ⊙ is changing, but the acceleration between any two planets remains unchanged. Thisis different from the situation when one looks for the G change involving the correspondingchange of the accelerations of all bodies. It should be noted that when we determine ˙ G using the planetary motions (Pitjeva & Pitjev, 2012), it is the Sun that contributes most ofall. Indeed, the equations of planetary motion include the products of the masses of bodiesand the gravitational constant, the main term exceeding other terms by several orders ofmagnitude is that for the Sun ( GM ⊙ ). Therefore, as the GM ⊙ term dominates, it is impossibleto separate the change of G from the change of GM ⊙ considering only the motion of theplanets (Pitjeva & Pitjev, 2012). However, if the change of the solar mass ( M ⊙ ) may beestimated from the independent astrophysical data, then based on the change of the GM ⊙ and the limits of the M ⊙ change, the limits of the gravitation constant (G) change can beobtained taking into account the following relation ˙ GM ⊙ /GM ⊙ = ˙ G/G + ˙ M ⊙ /M ⊙ . (5)(see details in Pitjeva & Pitjev, 2012). 11ontrary to ˙ G , it is the change of GM ⊙ that can be determined more accurately andreliably using the planetary motions. To control the stability of the solution for ˙ GM ⊙ and toobtain the more reliable error, we considered various fitting versions with different numbersof the parameters (the number of the adjusted masses of asteroids, perihelion precessions,etc.). The time–decrease of GM ⊙ was found to be ˙ GM ⊙ /GM ⊙ = ( − . ± . · − per year (2 σ ) . (6)This decrease is caused by the loss of the solar mass M ⊙ through radiation and the solarwind. The estimate of the uncertainty for this value is more reliable and larger than in(Pitjeva & Pitjev, 2012). Analysis of versions with different numbers of the fitting parametersdemonstrates that the value of ˙ GM ⊙ and its uncertainty are the most sensitive to theparameters related to the main asteroid belt, i.e. the amount of the adjusted masses of theselected large asteroids and the estimated characteristics of the ring representing the effectof the small asteroids. The value obtained by Folkner (Konopliv et al., 2011) for DE423ephemerides from the Mars ranging data is ˙ GM ⊙ /GM ⊙ = (1 ± · − per year . The uncertainty of this value is larger (and may be more reliable) due to taking into accountthe uncertainties of many other asteroid masses remained unestimated.Estimation of ˙ M ⊙ has been made by means of the astrophysical data using the valuesfor the average solar radiation and solar wind, and amount of comet and asteroid matterfalling on the Sun. The obtained limits of the possible change of M ⊙ can be bounded by theinequality (Pitjeva & Pitjev, 2012) − . · − < ˙ M ⊙ /M ⊙ < − . · − per year . (7)This interval may be narrowed due to the more accurate estimation of matter falling on theSun. From (6) and taking into account (5) and (7), the ˙ G/G value is found to be within theinterval (with the 95 % probability) − . · − < ˙ G/G < +7 . · − per year . (8)The interval (8) imposes the more rigid limits on the possible change of G than the resultsof the determination ˙ G obtained from processing lunar laser observations by Turyshev &12illiams (2007) ˙ G/G = (6 ± · − per yearand Hofmann, Muller & Biskupek (2010) ˙ G/G = ( − ± · − per year . It is proposed in the modern cosmological theories that the bulk of the average density ofthe universe falls on dark energy (about 73%) and the dark matter 23%, whereas the baryonmatter contains about 4% (Kowalski et al., 2008). The nature of dark matter is non–baryonand its properties are hypothetical (Bertone, Hooper & Silk, 2005; Peter, 2012).Despite the possible absence or the very weak interaction of dark matter with ordinarymatter, it must possess the capacity of gravity, and its presence in the solar system canbe manifested through its gravitational influence on the body motion. Attempts to detectthe possible influence of dark matter on the motion of objects in the solar system havealready been made (Nordtvedt, Mueller & Soffel, 1995; Anderson et al., 1989; Anderson etal., 1995; Sereno & Jetzer, 2006; Khriplovich & Pitjeva, 2006; Khriplovich, 2007; Frere, Ling& Vertongen, 2008).The additional gravitational influence may depend on the density of dark matter, itsdistribution in space, etc. We assume, as it is usually done (Anderson et al., 1989; Andersonet al., 1995; Gron & Soleng, 1996; Khriplovich & Pitjeva, 2006; Frere et al., 2008), thatdark matter is distributed in the solar system spherically symmetric relative to the Sun.Then we may suppose that any planet at distance r from the Sun can be undergone anadditional acceleration from invisible matter along with the accelerations from the Sun,planets, asteroids, trans–neptunian objects ¨r dm = − GM ( r ) dm r r , where M ( r ) dm is the mass of the additional matter in a sphere of radius r around the Sun.Testing the presence of the additional gravitational environment can be carried out eitherby finding the additional acceleration, as was made, for example, in (Nordtvedt et al., 1995;Anderson et al., 1989), or the additional perihelion drift (for example, Gron & Soleng, 1996).13he first method determines actually if there is any extra mass inside the sphericallysymmetric volume, in addition to the masses of the Sun, planets and asteroids already takeninto account. Any detected correction to the central attracting mass (or to the heliocentricgravitational constant GM ⊙ ) from the observational data separately for each planet wouldresult in its increased value in accordance with the additional mass within the sphere withthe mean radius of the planetary orbit.The second way is related with an unclosed trajectory of motion in the presence ofthe additional gravitational medium and the drift of the positions of the pericenters andapocenters from revolution to revolution in contrast to the purely Keplerian case of thetwo-body problem. Denoting the integrals of energy and area by E, J , and the sphericallysymmetric potential by U ( r ) the equations of motion of a unit mass along the radius r andalong the azimuthal coordinate θ read, respectively, (Landau & Lifshitz, 1969) ˙ r = (2[ E + U ( r )] − J /r ) / , (9) dθdr = J/r (2[ E + U ( r )] − J /r ) / . (10)In the Keplerian two-body problem the oscillation periods along the radius r (from theperihelion to the apocenter and back) and along azimuth θ around the center coincide, andthe positions of the pericenter and apocenter are not displaced from revolution to revolution.The additional gravitating medium leads to a shorter radial period and a negative drift of theposition of the pericenter and apocenter (in a direction opposite to the planetary motion).The perihelion precession for the uniformly distributed matter ( ρ dm =const) depends on theorbital semimajor axis a and eccentricity e of the planetary orbit (Khriplovich & Pitjeva,2006) ∆ θ = − π ρ dm a (1 − e ) / /M ⊙ , (11)where ∆ θ is the perihelion drift for one complete radial oscillation.Estimations of the density and mass of dark matter are produced often under theassumption that it changes very slowly or is constant within the solar system, i.e. underthe assumption of the uniform distribution of dark matter. A number of papers (Lundberg& Edsjo, 2004; Peter, 2009; Iorio, 2010) assume the concentration of dark matter to thecenter and even its capture and dropping on the Sun. The latter assumption should be made14ith caution. In the item 4.3 (as well as in Pitjeva & Pitjev, 2012), it was found that theheliocentric gravitational constant GM ⊙ decreases, so there is a stringent limitation on theamount of possible dark matter dropping on the Sun. The constraint on the possible presenceof dark matter inside the Sun (no more than 2-5% of the solar mass) was also obtained in(Kardashev, Tutukov & Fedorova, 2005), where the physical characteristics of the Sun havebeen carefully analyzed.Both approaches have been applied in the present work. The more sophisticatedconsideration is given in (Pitjev & Pitjeva, 2013).The corrections to the additional perihelion precession and to the central mass wereobtained by fitting the EPM2011 ephemerides to about 780 thousand of observations of theplanets and spacecrafts (Table 3). The fitting was done by the weighted method of the leastsquares. The various test solutions differing from one another by the sets of the adjustedparameters were considered for obtaining the reliable values of these parameters and theiruncertainties ( σ i ) in the same manner as for getting the ˙ GM ⊙ estimation.The resulting values are exceeded by their uncertainties ( σ ) indicating that the darkmatter density ρ dm , if any, is very small being lower than the accuracy of these parametersachieved by the modern determination. The obtained opposite signs for the values ∆ π and ∆ M for the various planets also show the smallness of such effects.The relative uncertainties in the corrections to the central mass from the observationsseparately for each planet were significantly greater than that for the additional perihelionprecessions exceeding the corrections to the central mass themselves in several times or evenby several orders of magnitude. It should be remembered that the integral estimation of thedark matter mass falling into a spherically symmetric (relative to the Sun) volume dependson the accuracy of knowledge of all body masses into this volume. Basically, it is due to theinaccurate knowledge of the masses of asteroids.More accurate results were obtained for estimates of the perihelion precessions (Table4) allowing to estimate the local density of dark matter at the mean orbital distance of aplanet. Here, the uncertainties of determination of the corrections are comparable with thevalues themselves. Therefore, the estimates from Table 4 were actually used.The investigation of the additional perihelion precession of the planets was carried outtaking into account all other known effects affecting the perihelion drift. Indeed, if there15s an additional gravitating medium, then a negative drift of the perihelion and aphelionoccurs from revolution to revolution in accordance with the formula (11). Since the growthof the perihelion drift is accumulated, this criterion can be sensitive enough for verifying thepresence of additional matter. Table 4.
Additional perihelion precessions from observations of planets and 19spacecraftsPlanets ˙ π | σ ˙ π / ˙ π |mas/yrMercury − . ± . . ± . . ± . − . ± . . ± . − . ± . σ ˙ π may be treated as the upper limits for thepossible additional drifts of the secular motion of the perihelia, and can give the upper limitfor the density of the distributed matter by using (11). The resulting estimates ρ dm are shownin Table 5. Table 5.
Estimates of the density ρ dm obtained from σ ∆ π for the perihelion precessionsPlanets σ ˙ π ρ dm ′′ /yr g/cm Mercury . < . · − Venus . < . · − Earth . < . · − Mars . < . · − Jupiter . < . · − Saturn . < . · − The data based on the estimates for the Earth, Mars and Saturn yield the most stringentconstraints on the density ρ dm . The high–precision series of observations of Saturn appeared16hen the Cassini spacecraft arrived to it in 2004. There is the large and long set ofobservations of Mars associated with many spacecrafts on its surface and around it. TheEarth orbit improvement is based on all observations starting with the observations madefrom the Earth. Assuming the homogeneous distribution ρ dm in the solar system the moststringent constraint ρ dm < . · − g/cm is obtained from the data for Saturn. Then themass M dm within the spherical volume with the size of Saturn’s orbit is M dm < . · − M ⊙ . (12)This value is about 2 times smaller than the uncertainty of the obtained total mass of themain asteroid belt (1).Another version can be considered when a continuous medium has some concentration tothe center of the solar system. Investigations under the assumption of density concentrationto the center have already been carried out, for example, by Frere et al. (2008). We havetaken the model for ρ dm with the exponential dependence on the distance rρ dm = ρ · e − cr , (13)where ρ is the central density and c is a positive parameter characterizing an exponentialdecrease of the density to the periphery. The value of c = 0 corresponds to a uniform density.Function (13) is everywhere finite and has no singularities at the center and on the periphery.The mass inside a sphere of radius r for distribution (13) is M dm = 4 πρ · − e − cr ( c r + 2 cr + 2) c . (14)In spite of the presence of c in the denominator this expression does not have singularitiesfor c → . The formula (14) transforms therewith into the expression for the mass of ahomogeneous sphere.The values in Table 5 may be considered as the limits of the density ρ dm at variousdistances. In a relatively narrow interval of the radial distances caused by the eccentricityof the planetary orbit the density of dark matter can be considered to be approximatelyconstant. The potential existence of the dark matter M dm distributed between the Sun andthe orbit of a planet gives very small contribution (the tenths or elevenths fraction of themagnitude) to the total attractive central mass determined by the solar mass. Therefore,17ne can use the formula (11) and obtain the local restrictive estimations for ρ dm in theneighborhood of the planet orbit (Table 5).With the assumption of the concentration to the center the estimate of the mass of darkmatter within the orbit of Saturn was determined from the evaluation of the masses withinthe two intervals, i.e. from Saturn to Mars and from Mars to the Sun. For this purpose themost reliable data of Table 5 for Saturn ( ρ dm < . · − g/cm ), Mars ( ρ dm < . · − g/cm ) and Earth ( ρ dm < . · − g/cm ) were used. Based on the data for Saturn andMars a very flat trend of the density curve (13) between Mars and Saturn was obtained with ρ = 1 . · − g/cm and c = 0 . au − . From these parameters the mass in the spacebetween the orbits of Mars and Saturn is M dm < . · − M ⊙ . The obtained trend of thedensity curve (13) in the interval between Mars and the Sun gives a steep climb to the Sunaccording to the data for Earth and Mars with the parameters ρ = 1 . · − g/cm and c = 4 . au − . For these parameters the mass (14) between the Sun and the orbit of Mars is M dm < . · − M ⊙ .Summing masses for both intervals the upper limit for the total mass of dark matterwas estimated as M dm < . · − M ⊙ between the Sun and the orbit of Saturn, takinginto account its possible tendency to concentrate in the center. This value is less than theuncertainty ± . · − M ⊙ (3 σ ) of the total mass of the asteroid belt. The value M dm doesnot change perceptibly compared to the hypothesis of a uniform density (12), although thetrend of the density curve in the second case provides the significant (by three orders ofmagnitude) increase to the center. CONCLUSION
The estimations of the gravitational PPN parameters, the solar oblateness, the secularchange of the heliocentric gravitation constant GM ⊙ and the gravitation constant G , as wellas the possible gravitational influence of dark matter on the motion of the planets in thesolar system have been made on the basis of the EPM2011 planetary ephemerides of IAARAS using about 678,000 positional observations of planets and spacecrafts, mostly radioand laser ranging ones.The PPN parameters turned out to be β − − . ± . , γ − . ± . σ ) . Our estimation for the change of the heliocentric gravitational constant is18 GM ⊙ /GM ⊙ = ( − . ± . · − per year (2 σ ) . It was found also that the limits for thetime variation of the gravitational constant G are − . · − < ˙ G/G < +7 . · − (2 σ ) per year.The mass and the level of dark matter density in the solar system, if any, was obtainedto be substantially lower than the modern uncertainties of these parameters. The densityof dark matter was found to be lower than ρ dm < . · − g/cm at the distance of theSaturn orbit, and the mass of dark matter in the area inside the orbit of Saturn is less than . · − M ⊙ , even taking into account its possible tendency to concentrate in the center. REFERENCES
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