- Split View
-
Views
-
Cite
Cite
E. V. Pitjeva, N. P. Pitjev, Relativistic effects and dark matter in the Solar system from observations of planets and spacecraft, Monthly Notices of the Royal Astronomical Society, Volume 432, Issue 4, 11 July 2013, Pages 3431–3437, https://doi.org/10.1093/mnras/stt695
- Share Icon Share
Abstract
The high precision of the latest version of the planetary ephemeris Ephemerides of the Planets and the Moon (EPM2011) enables one to explore more accurately a variety of small effects in the Solar system. The processing of about 678 thousand of position observations of planets and spacecraft for 1913–2011 with the predominance of modern radar measurements resulted in improving the PPN parameters, dynamic oblateness of the Sun, secular variation of the heliocentric gravitational constant G M⊙, and the stronger limits on variation of the gravitational constant G. This processing made it possible to estimate the potential additional gravitational influence of dark matter on the motion of the Solar system bodies. The density of dark matter ρdm, if any, turned out to be substantially below the accuracy achieved by the present determination of such parameters. At the distance of the orbit of Saturn the density ρdm is estimated to be under 1.1 × 10−20 g cm−3, and the mass of dark matter in the area inside the orbit of Saturn is less than 7.9 × 10−11 M⊙ even taking into account its possible tendency to concentrate in the centre.
INTRODUCTION
The possibility to test and refine various relativistic and cosmological effects from the analysis of the motion of the Solar system bodies is due to the present metre accuracy radio techniques (Standish 2008) and millimeter accuracy laser techniques (Murphy et al. 2008) for the distance measurements. Just these techniques have provided an observational foundation of the contemporary high-precision theories of planetary motions.
The numerical theories of planetary motions have been improved and developed by several groups in different countries and their accuracy is constantly growing. The progress is related with the increase of the number of high-precision radio observations and the inclusion of a number of small effects (perturbations from a set of asteroids, the solar oblateness perturbations, etc.) in constructing a dynamic model of the Solar system. The radio technical observations, having much higher accuracy as compared with the optical ones, are commonly used now in astrometric practice. These high-precision measurements covering more than 50 year time interval allow us to find the orbital elements, masses and other parameters determining the motion of the bodies. Moreover, they also give a possibility to check some relativistic parameters to estimate the secular change of the heliocentric gravitation constant and to examine the presence of dark matter in the Solar system. The last point is of particular importance for the contemporary cosmological theories. A more accurate and extensive set of observations permits us not only to determine the relativistic perihelion precession of planets, but also to estimate the oblateness of the Sun with the corresponding contribution into the drift of the perihelia. Moreover, these observations provide a means for finding the secular variation of the heliocentric gravitational constant G M⊙ and the constraint on the secular variation of the gravitational constant G (Pitjeva & Pitjev 2012). In addition, these precise observations enable us to consider the assumption of the presence of dark matter in the 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 the numerical Ephemerides of the Planets and the Moon (EPM2011) of the Institute of Applied Astronomy of Russian Academy of Sciences (IAA RAS).
THE PLANETARY EPHEMERIS EPM2011
Numerical EPM had started in the 1970s. Each subsequent version is characterized by additional new observations, refined values of the orbital elements and masses of the bodies, an improved dynamical model of the celestial bodies 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 et al. 2008) are based on General Relativity involving the relativistic equations of celestial bodies motion and light propagation as well as the relativistic time-scales. In addition, these ephemerides involve estimating from observations some parameters (β, γ, |$\dot{G}$|) to check their compatibility with General Relativity.
The current EPM2011 ephemerides were constructed using approximately 680 thousand data (1913–2011) of different types. The equations of the bodies motion were taken within the parametrized post-Newtonian N-body metric in the barycentric coordinate system – BCRS (Brumberg 1991), the same as that of DE. Integration in Barycentric Dynamical Time (TDB) time-scale (see the IAU2006 resolution B3) was performed using Everhart's method over the 400 year interval (1800–2200) with the lunar and planetary integrator of the era-7 software package (Krasinsky & Vasilyev 1997). The EPM ephemerides including also the time differences TT–TDB, and seven additional objects, namely, Ceres, Pallas, Vesta, Eris, Haumea, Makemake and Sedna are available via FTP by means of ftp://quasar.ipa.nw.ru/incoming/EPM/.
Since the basic observational data for producing the next version of the planetary ephemerides EPM2011 were mainly related to the spacecraft, the control of the orientation of the EPM2011 ephemerides with respect to the ICRF frame has required a particular attention. For this purpose, we have used the very long baseline interferometry (VLBI) observations of spacecraft near planets at the background of quasars whose coordinates are given in the ICRF frame (Table 1), where (α, δ) are two-dimensional measurements, (α + δ) being one-dimensional measurements of the α and δ combination (the position of the planet is observed to be displaced from the base ephemeris (DE405) by a correction measured counter-clockwise along a line at an angle to the right ascension axis, see Folkner 1992).
Planet . | Spacecraft . | Interval of observations . | Number of observations . |
---|---|---|---|
Venus | Magellan | 1990–1994 | 18(α + δ) |
VEX | 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) |
Planet . | Spacecraft . | Interval of observations . | Number of observations . |
---|---|---|---|
Venus | Magellan | 1990–1994 | 18(α + δ) |
VEX | 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) |
Planet . | Spacecraft . | Interval of observations . | Number of observations . |
---|---|---|---|
Venus | Magellan | 1990–1994 | 18(α + δ) |
VEX | 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) |
Planet . | Spacecraft . | Interval of observations . | Number of observations . |
---|---|---|---|
Venus | Magellan | 1990–1994 | 18(α + δ) |
VEX | 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 arcsec) for Mars and Saturn in 2001–2010 (Jones et al. 2011) enabling us to improve the orientation of EPM ephemerides (Table 2) in the same way, as it was done by Standish (1998). The angles of rotation of the Earth–Moon barycentre vector about the x-, y-, z-axes of the BCRS system were obtained from VLBI observations described above.
Interval of . | Number of . | εx . | εy . | εz . |
---|---|---|---|---|
observations . | observations . | (mas) . | (mas) . | (mas) . |
1989–2010 | 213 | −0.000 ± 0.042 | −0.025 ± 0.048 | 0.004 ± 0.028 |
Interval of . | Number of . | εx . | εy . | εz . |
---|---|---|---|---|
observations . | observations . | (mas) . | (mas) . | (mas) . |
1989–2010 | 213 | −0.000 ± 0.042 | −0.025 ± 0.048 | 0.004 ± 0.028 |
Interval of . | Number of . | εx . | εy . | εz . |
---|---|---|---|---|
observations . | observations . | (mas) . | (mas) . | (mas) . |
1989–2010 | 213 | −0.000 ± 0.042 | −0.025 ± 0.048 | 0.004 ± 0.028 |
Interval of . | Number of . | εx . | εy . | εz . |
---|---|---|---|---|
observations . | observations . | (mas) . | (mas) . | (mas) . |
1989–2010 | 213 | −0.000 ± 0.042 | −0.025 ± 0.048 | 0.004 ± 0.028 |
More than 270 parameters are estimated in the planetary part of EPM2011 ephemerides as follows:
the orbital elements of planets and satellites of the outer planets,
the value of the astronomical unit (au) or G M⊙,
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 trans-neptunian object (TNO) ring,
the mass ratio of the Earth and Moon,
the quadrupole moment of the Sun and the solar corona parameters for different conjunctions of the planets with the Sun,
the coefficients for the Mercury topography and the corrections to the level surfaces of Venus 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 LLR data (see for example, Krasinsky, Prokhorenko & Yagudina 2011). All estimated parameters in both parts are consistent within the frame of the combined theory of motion of the planets and the Moon given by the EPM ephemerides.
The initial parameters of EPM2011 represented the constants adopted by the IAU GA 27 (Luzum et al. 2011) as the current best values for ephemeris astronomy. Among them five constants were resulted from the ephemeris improvement of DE and EPM ephemerides (Pitjeva & Standish 2009). At present, these five parameters adjusted from processing all observations for EPM2011 are as follows: the masses of the largest asteroids, i.e. MCeres/M⊙ = 4.722(8) × 10−10, MPallas/M⊙ = 1.047(9) × 10−10, MVesta/M⊙ = 1.297(5) × 10−10; ratio of the masses of the Earth and Moon MEarth/MMoon = 81.300 567 63 ± 0.000 000 05; the value of the au in metres au = (149 597 870 695.88 ± 0.14) or the heliocentric gravitation constant G M⊙ = (132 712 440 031 ± 1) km3 s−2.
Presently, in accordance with the IAU 2012 resolution B2 the au is re-defined by fixing its value. Up to now, both values of au and the heliocentric gravitation constant (G M⊙) were connected. It was possible to determine the au value and to calculate the value of G M⊙ from it, or vice versa, to determine G M⊙ and to calculate the value of au from it. Here, the values of au and G M⊙ are given as in the paper Pitjeva & Standish (2009) published before the IAU 2012 resolution B2. At present, only the value of G M⊙ is estimated from observations.
the perturbations of the 301 most massive asteroids,
the perturbations from the remaining minor planets of the main asteroid belt modelled by a homogeneous ring,
the perturbations from the 21 largest TNO,
the perturbations from the remaining TNO modelled by a uniform ring at the average distance of 43 au,
the relativistic perturbations,
the perturbation due to the oblateness of the Sun estimated in EPM2011 fitting as (J2 = 2 × 10−7).
OBSERVATION DATA AND THEIR REDUCTIONS
The total amount of the high-precision observations used for fitting EPM2011 has been increased due to the recent data. They include 677 670 positional measurements of different types for 1913–2011 from classic meridian measurements to modern spacecraft tracking data (Table 3).
Planet . | Radio observations . | Optical observations . | ||
---|---|---|---|---|
. | Time interval . | Number . | Time interval . | Number . |
Mercury | 1964–2009 | 948 | – | – |
Venus | 1961–2010 | 40 061 | – | – |
Mars | 1965–2010 | 578 918 | – | – |
Jupiter+4 sat. | 1973–1997 | 51 | 1914–2011 | 13 364 |
Saturn+9 sat. | 1979–2009 | 126 | 1913–2011 | 15 956 |
Uran+4 sat. | 1986 | 3 | 1914–2011 | 11 846 |
Neptun+1 sat. | 1989 | 3 | 1913–2011 | 11 634 |
Pluton | – | – | 1914–2011 | 5660 |
Total | – | 620 110 | – | 57 560 |
Planet . | Radio observations . | Optical observations . | ||
---|---|---|---|---|
. | Time interval . | Number . | Time interval . | Number . |
Mercury | 1964–2009 | 948 | – | – |
Venus | 1961–2010 | 40 061 | – | – |
Mars | 1965–2010 | 578 918 | – | – |
Jupiter+4 sat. | 1973–1997 | 51 | 1914–2011 | 13 364 |
Saturn+9 sat. | 1979–2009 | 126 | 1913–2011 | 15 956 |
Uran+4 sat. | 1986 | 3 | 1914–2011 | 11 846 |
Neptun+1 sat. | 1989 | 3 | 1913–2011 | 11 634 |
Pluton | – | – | 1914–2011 | 5660 |
Total | – | 620 110 | – | 57 560 |
Planet . | Radio observations . | Optical observations . | ||
---|---|---|---|---|
. | Time interval . | Number . | Time interval . | Number . |
Mercury | 1964–2009 | 948 | – | – |
Venus | 1961–2010 | 40 061 | – | – |
Mars | 1965–2010 | 578 918 | – | – |
Jupiter+4 sat. | 1973–1997 | 51 | 1914–2011 | 13 364 |
Saturn+9 sat. | 1979–2009 | 126 | 1913–2011 | 15 956 |
Uran+4 sat. | 1986 | 3 | 1914–2011 | 11 846 |
Neptun+1 sat. | 1989 | 3 | 1913–2011 | 11 634 |
Pluton | – | – | 1914–2011 | 5660 |
Total | – | 620 110 | – | 57 560 |
Planet . | Radio observations . | Optical observations . | ||
---|---|---|---|---|
. | Time interval . | Number . | Time interval . | Number . |
Mercury | 1964–2009 | 948 | – | – |
Venus | 1961–2010 | 40 061 | – | – |
Mars | 1965–2010 | 578 918 | – | – |
Jupiter+4 sat. | 1973–1997 | 51 | 1914–2011 | 13 364 |
Saturn+9 sat. | 1979–2009 | 126 | 1913–2011 | 15 956 |
Uran+4 sat. | 1986 | 3 | 1914–2011 | 11 846 |
Neptun+1 sat. | 1989 | 3 | 1913–2011 | 11 634 |
Pluton | – | – | 1914–2011 | 5660 |
Total | – | 620 110 | – | 57 560 |
Radar measurements (the detailed description of them is given in Pitjeva 2005a, 2013) have a high accuracy. At present, the relative accuracy ∼10−12 for the spacecraft trajectory measurements became usual, exceeding the accuracy of classical optical measurements by five orders of magnitude. However, in general only Mercury, Venus and Mars are provided with radio observations. Initially, the surfaces of these planets were radio located from 1961 to 1995. Later on many spacecraft passed by, orbited or landed to these planets. A large portion of the spacecraft data were used to get the astrometric positions. There are much less radio observations for Jupiter and Saturn, and only one set of the three-dimensional normal points (α, δ, R) obtained from the Voyager-2 spacecraft are available for Uranus and Neptune. Therefore, the optical observations are still of great importance for the outer planets. Thereby, the varied data of 19 spacecraft were used for constructing the EPM2011 ephemerides and estimating the relevant parameters, in particular, the additional perihelion precessions of the planets (see Table 4).
The recent data from the spacecraft have been added to the previous ones for the latest version of the EPM ephemerides. It involves data related to Odyssey, Mars Reconnaissance Orbiter (MRO; Konopliv et al. 2011), Mars Express (MEX), Venus Express (VEX) and, more specifically, VLBI observations of Odyssey and MRO, three-dimensional normal points of Cassini and Messenger observations, along with the CCD observations of the outer planets and their satellites obtained at Flagstaff and Table Mountain observatories. The most part of observations were taken from the data base of JPL/Caltech created by Dr Standish and continued by Dr Folkner. MEX and VEX data provided by ESA became available thanks to Dr. Fienga's kindness (private communications of T. Morlay to A. Fienga).
The detailed description of methods for all reductions of planetary observations (both optical and radar ones) was given by Standish (1990). This is a basic paper in the field of planetary observations discussion. In the EPM ephemerides some reductions changed slightly are described in Pitjeva (2005a, 2013). All necessary reductions listed therein were introduced into 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 the gravitational field of the Sun, Jupiter and Saturn (Shapiro effect) and the reduction of TDB 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).
Reductions of the optical observations:
the transformation of observations to the ICRF frame catalogue differences ⇒ FK4 ⇒ FK5 ⇒ ICRF;
the relativistic correction for the light bending of the Sun;
the correction for the additional phase effect.
RESULTS
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 secular changes of the orbital elements. Therefore, the uncertainties of their determination decrease with increasing the time interval of observations.
The simplified relativistic equations of the planetary motion were derived more than 30 years ago in different coordinate systems of the Schwarzschild metric supplemented with coordinate parameter α to specify standard, harmonic, isotropic or any other coordinates. These equations were described in Brumberg (1972, 1991). For example, the integration exposed in Oesterwinter & Cohen (1972) was made in the standard coordinates (α = 1). However, planetary coordinates turned out to be essentially different for the standard and harmonic systems. It was shown in Brumberg (1979) that the ephemeris construction and processing of observations should be done in the same coordinate system resulting to the relativistic effects not dependent on the coordinate system (effacing of parameter α). Later on, the resolutions of IAU (1991, 2000) recommended to use the harmonic coordinates for BCRS. In accordance with the IAU 2000 resolution B1.3, all modern planetary ephemerides are constructed in the harmonic coordinates for BCRS – the barycentric (for the Solar system) coordinate system.
The parameters of the PPN formalism β, γ used to describe the metric theories of gravity must be equal to 1 in General Relativity. The values of parameters β, γ were obtained simultaneously by using the EPM2011 ephemerides and the updated data base of high-precision observations (Table 3) from the relativistic periodic and secular variations of the orbital elements, as well as the Shapiro effect. Certainly, the periodic variations of the orbital elements are smaller than the secular ones but they are of importance to compute the planetary motion. We derived expressions for the partial derivatives of the orbital elements with respect to β and γ using the analytical formulas for the relativistic perturbations of the 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 80s of the last century, we tested relativistic effects by processing the observations available at that time. It turned out that the relativistic ephemeris for any observed planet provided a considerably better fit of observations (by 10 per cent) than the Newtonian theory even if the latter incorporated the observed perihelion secular motion (Krasinsky et al. 1986).
For comparison, we also quote the new γ value obtained by using Very Long Baseline Array measurements of radio sources by Fomalont et al. (2009), i.e. γ = 0.9998 ± 0.0003.
All the obtained values of β, γ are in the close vicinity of 1 within the limits of their uncertainties. As the uncertainties of these parameters decrease, the range of possible values of the PPN parameters narrows, imposing increasingly stringent constraints on the gravitation theories alternative to General Relativity.
Estimations of the solar dynamic oblateness
Estimations of the secular changes of G M⊙ and G
The value of the secular change of the heliocentric gravitational constant G M⊙ has been updated for the expanded data base and the improved dynamical model of planetary motions (EPM2011). The determination of secular variation G M⊙ was carried out by the method exposed in detail in (Pitjeva & Pitjev 2012) dealing with the EPM2010 planetary ephemerides.
Estimations of dark matter in the Solar system
It is proposed in the modern cosmological theories that the bulk of the average density of the Universe falls on dark energy (about 73 per cent) and the dark matter 23 per cent, whereas the baryon matter contains about 4 per cent (Kowalski et al. 2008). The nature of dark matter is non-baryon and its properties are hypothetical (Bertone, Hooper & Silk 2005; Peter 2012).
Despite the possible absence or the very weak interaction of dark matter with ordinary matter, it must possess the capacity of gravity, and its presence in the Solar system can be manifested through its gravitational influence on the body motion. Attempts to detect the possible influence of dark matter on the motion of objects in the Solar system have already been made (Anderson et al. 1989, 1995; Nordtvedt, Mueller & Soffel 1995; Khriplovich & Pitjeva 2006; Sereno & Jetzer 2006; Khriplovich 2007; Frere, Ling & Vertongen 2008).
Testing the presence of the additional gravitational environment can be carried out either by finding the additional acceleration, as was made, for example, in Nordtvedt et al. (1995) and Anderson et al. (1989) or the additional perihelion drift (for example, Gron & Soleng 1996).
The first method determines actually if there is any extra mass inside the spherically symmetric volume, in addition to the masses of the Sun, planets and asteroids already taken into account. Any detected correction to the central attracting mass (or to the heliocentric gravitational constant G M⊙) from the observational data separately for each planet would result in its increased value in accordance with the additional mass within the sphere with the mean radius of the planetary orbit.
Estimations of the density and mass of dark matter are produced often under the assumption that it changes very slowly or is constant within the Solar system, i.e. under the 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 the centre and even its capture and dropping on the Sun. The latter assumption should be made with caution. In the Section 4.3 (as well as in Pitjeva & Pitjev 2012), it was found that the heliocentric gravitational constant G M⊙ decreases, so there is a stringent limitation on the amount of possible dark matter dropping on the Sun. The constraint on the possible presence of dark matter inside the Sun (no more than 2–5 per cent of the solar mass) was also obtained in Kardashev, Tutukov & Fedorova (2005), where the physical characteristics of the Sun have been carefully analysed.
Both approaches have been applied in this work. The more sophisticated consideration is given in Pitjev & Pitjeva (2013).
The corrections to the additional perihelion precession and to the central mass were obtained by fitting the EPM2011 ephemerides to about 780 thousand of observations of the planets and spacecraft (Table 3). The fitting was done by the weighted method of the least squares. The various test solutions differing from one another by the sets of the adjusted parameters were considered for obtaining the reliable values of these parameters and their uncertainties (σi) in the same manner as for getting the |$\dot{G{{\rm M}}}_{{\odot }}$| estimation.
The resulting values are exceeded by their uncertainties (σ) indicating that the dark matter density ρdm, if any, is very small being lower than the accuracy of these parameters achieved by the modern determination. The obtained opposite signs for the values Δπ and ΔM0 for the various planets also show the smallness of such effects.
The relative uncertainties in the corrections to the central mass from the observations separately for each planet were significantly greater than that for the additional perihelion precessions exceeding the corrections to the central mass themselves in several times or even by several orders of magnitude. It should be remembered that the integral estimation of the dark matter mass falling into a spherically symmetric (relative to the Sun) volume depends on the accuracy of knowledge of all body masses into this volume. Basically, it is the inaccurate knowledge of the masses of asteroids.
More accurate results were obtained for estimates of the perihelion precessions (Table 4) allowing us to estimate the local density of dark matter at the mean orbital distance of a planet. Here, the uncertainties of determination of the corrections are comparable with the values themselves. Therefore, the estimates from Table 4 were actually used.
Planets . | |$\dot{\pi }$| . | ||${\sigma _{\dot{\pi }} / \dot{\pi\, }}$|| . |
---|---|---|
. | (mas yr−1) . | . |
Mercury | −0.020 ± 0.030 | 1.5 |
Venus | 0.026 ± 0.016 | 0.62 |
Earth | 0.0019 ± 0.0019 | 1.0 |
Mars | −0.000 20 ± 0.000 37 | 1.9 |
Jupiter | 0.587 ± 0.283 | 0.48 |
Saturn | −0.0032 ± 0.0047 | 1.5 |
Planets . | |$\dot{\pi }$| . | ||${\sigma _{\dot{\pi }} / \dot{\pi\, }}$|| . |
---|---|---|
. | (mas yr−1) . | . |
Mercury | −0.020 ± 0.030 | 1.5 |
Venus | 0.026 ± 0.016 | 0.62 |
Earth | 0.0019 ± 0.0019 | 1.0 |
Mars | −0.000 20 ± 0.000 37 | 1.9 |
Jupiter | 0.587 ± 0.283 | 0.48 |
Saturn | −0.0032 ± 0.0047 | 1.5 |
Planets . | |$\dot{\pi }$| . | ||${\sigma _{\dot{\pi }} / \dot{\pi\, }}$|| . |
---|---|---|
. | (mas yr−1) . | . |
Mercury | −0.020 ± 0.030 | 1.5 |
Venus | 0.026 ± 0.016 | 0.62 |
Earth | 0.0019 ± 0.0019 | 1.0 |
Mars | −0.000 20 ± 0.000 37 | 1.9 |
Jupiter | 0.587 ± 0.283 | 0.48 |
Saturn | −0.0032 ± 0.0047 | 1.5 |
Planets . | |$\dot{\pi }$| . | ||${\sigma _{\dot{\pi }} / \dot{\pi\, }}$|| . |
---|---|---|
. | (mas yr−1) . | . |
Mercury | −0.020 ± 0.030 | 1.5 |
Venus | 0.026 ± 0.016 | 0.62 |
Earth | 0.0019 ± 0.0019 | 1.0 |
Mars | −0.000 20 ± 0.000 37 | 1.9 |
Jupiter | 0.587 ± 0.283 | 0.48 |
Saturn | −0.0032 ± 0.0047 | 1.5 |
The investigation of the additional perihelion precession of the planets was carried out taking into account all other known effects affecting the perihelion drift. Indeed, if there is an additional gravitating medium, then a negative drift of the perihelion and aphelion occurs from revolution to revolution in accordance with the formula (11). Since the growth of the perihelion drift is accumulated, this criterion can be sensitive enough for verifying the presence of additional matter.
All the uncertainties of Table 4 are comparable or larger than the absolute values obtained for perihelion precessions. These uncertainties |$\sigma _{\dot{\pi }}$| may be treated as the upper limits for the possible additional drifts of the secular motion of the perihelia, and can give the upper limit for the density of the distributed matter by using (11). The resulting estimates ρdm are shown in Table 5.
Planets . | |$\sigma _{\dot{\pi }}$| . | ρdm . |
---|---|---|
. | (arcsec yr−1) . | (g cm−3) . |
Mercury | 0.000 030 | <9.3 × 10−18 |
Venus | 0.000 016 | <1.9 × 10−18 |
Earth | 0.000 0019 | <1.4 × 10−19 |
Mars | 0.000 000 37 | <1.4 × 10−20 |
Jupiter | 0.000 283 | <1.7 × 10−18 |
Saturn | 0.000 0047 | <1.1 × 10−20 |
Planets . | |$\sigma _{\dot{\pi }}$| . | ρdm . |
---|---|---|
. | (arcsec yr−1) . | (g cm−3) . |
Mercury | 0.000 030 | <9.3 × 10−18 |
Venus | 0.000 016 | <1.9 × 10−18 |
Earth | 0.000 0019 | <1.4 × 10−19 |
Mars | 0.000 000 37 | <1.4 × 10−20 |
Jupiter | 0.000 283 | <1.7 × 10−18 |
Saturn | 0.000 0047 | <1.1 × 10−20 |
Planets . | |$\sigma _{\dot{\pi }}$| . | ρdm . |
---|---|---|
. | (arcsec yr−1) . | (g cm−3) . |
Mercury | 0.000 030 | <9.3 × 10−18 |
Venus | 0.000 016 | <1.9 × 10−18 |
Earth | 0.000 0019 | <1.4 × 10−19 |
Mars | 0.000 000 37 | <1.4 × 10−20 |
Jupiter | 0.000 283 | <1.7 × 10−18 |
Saturn | 0.000 0047 | <1.1 × 10−20 |
Planets . | |$\sigma _{\dot{\pi }}$| . | ρdm . |
---|---|---|
. | (arcsec yr−1) . | (g cm−3) . |
Mercury | 0.000 030 | <9.3 × 10−18 |
Venus | 0.000 016 | <1.9 × 10−18 |
Earth | 0.000 0019 | <1.4 × 10−19 |
Mars | 0.000 000 37 | <1.4 × 10−20 |
Jupiter | 0.000 283 | <1.7 × 10−18 |
Saturn | 0.000 0047 | <1.1 × 10−20 |
The values in Table 5 may be considered as the limits of the density ρdm at various distances. In a relatively narrow interval of the radial distances caused by the eccentricity of the planetary orbit, the density of dark matter can be considered to be approximately constant. The potential existence of the dark matter Mdm distributed between the Sun and the orbit of a planet gives very small contribution (the tenths or elevenths fraction of the magnitude) to the total attractive central mass determined by the solar mass. Therefore, one can use the formula (11) and obtain the local restrictive estimations for ρdm in the neighbourhood of the planet orbit (Table 5).
With the assumption of the concentration to the centre, 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 of Table 5 for Saturn (ρdm < 1.1 × 10− 20 g cm−3), Mars (ρdm < 1.4 × 10− 20 g cm−3) and Earth (ρdm < 1.4 × 10− 19 g cm−3) were used. Based on the data for Saturn and Mars a very flat trend of the density curve (13) between Mars and Saturn was obtained with ρ0 = 1.47 × 10−20 g cm−3 and c = 0.0299 au−1. From these parameters, the mass in the space between the orbits of Mars and Saturn is Mdm < 7.33 × 10− 11 M⊙. The obtained trend of the density curve (13) 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 cm−3 and c = 4.42 au−1. For these parameters, the mass (14) between the Sun and the orbit of Mars is Mdm < 0.55 × 10− 11 M⊙.
Summing masses for both intervals, the upper limit for the total mass of dark matter was estimated as Mdm < 7.88 × 10− 11 M⊙ between the Sun and the orbit of Saturn, taking into account its possible tendency to concentrate in the centre. This value is less than the uncertainty ±1.13 × 10−10 M⊙ (3σ) of the total mass of the asteroid belt. The value Mdm does not change perceptibly compared to the hypothesis of a uniform density (12), although the trend of the density curve in the second case provides the significant (by three orders of magnitude) increase to the centre.
CONCLUSION
The estimations of the gravitational PPN parameters, the solar oblateness, the secular change of the heliocentric gravitation constant GM⊙ and the gravitation constant G, as well as the possible gravitational influence of dark matter on the motion of the planets in the Solar system have been made on the basis of the EPM2011 planetary ephemerides of IAA RAS using about 678 000 positional observations of planets and spacecraft, mostly ranging ones.
The PPN parameters turned out to be β − 1 =−0.000 02 ± 0.000 03, γ − 1 = +0.000 04 ± 0.000 06 (σ). Our estimation for the change of the heliocentric gravitational constant is |$\dot{G{\rm M}}_{\odot }/G\,\mathrm{M}_{{\odot }} = (-6.3 \pm 4.3)\times 10^{-14}\,{yr^{-1}}\,(2\sigma )$|. It was found also that the limits for the time variation of the gravitational constant G are |$-7.0\times 10^{-14} < \dot{G}/G < +7.8\times 10^{-14}\,(2\sigma )$| yr−1.
The mass and the level of dark matter density in the Solar system, if any, was obtained to be substantially lower than the modern uncertainties of these parameters. The density of dark matter was found to be lower than ρdm < 1.1 × 10− 20 g cm−3 at the distance of the Saturn orbit, and the mass of dark matter in the area inside the orbit of Saturn is less than 7.9 × 10−11 M⊙, even taking into account its possible tendency to concentrate in the centre.
We would like to thank Professor V.A. Brumberg for support, invaluable advice and improving the text of this paper.