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arXiv:1308.6416v1 [astro-ph.EP] 29 Aug 2013

ISSN 0038-0946, Solar System Research, 2013, Vol. 47, No. 5, pp. 386-402. c Pleiades

Publishing, Inc., 2013. Original Russian Text c E.V. Pitjeva, 2013, published in Astronomicheskii

Vestnik, 2013, Vol. 47, No. 5, pp. 419-435

UDK 521.172:523.2

Updated IAA RAS Planetary Ephemerides-EPM2011

and Their Use in Scientific Research

c ?2013 г. E. V. Pitjeva

Institute of Applied Astronomy, Russian Academy of Sciences,

nab. Kutuzova 10, St. Petersburg, 191187 Russia

Received December 20, 2012

Abstract -The EPM (Ephemerides of Planets and the Moon) numerical ephemerides

were first created in the 1970s in support of Russian space flight missions and since then

have been constantly improved at IAA RAS. In the following work, the latest version of

the planetary part of the EPM2011 numerical ephemerides is presented. The EPM2011

ephemerides are computed using an updated dynamical model, new values of the parameters,

and an extended observation database that contains about 680 000 positional measurements

of various types obtained from 1913 to 2011. The dynamical model takes into account mutual

perturbations of the major planets, the Sun, the Moon, 301 massive asteroids, and 21 of the

largest trans-Neptunian objects (TNOs), as well as perturbations from the other main-belt

asteroids and other TNOs. The EPM ephemerides are computed by numerical integration

of the equations of motion of celestial bodies in the parameterized post-Newtonian n-body

metric in the BCRS coordinate system for the TDB time scale over a 400-year interval. The

ephemerides were oriented to the ICRF system using 213 VLBI observations (taken from

1989 to 2010) of spacecraft near planets with background quasars, the coordinates of which

are given in the ICRF system. The accuracy of the constructed ephemerides was verified by

comparison with observations and JPL independent ephemerides DE424.

The EPM ephemerides are used in astronavigation (they form the basis of the

Astronomical Yearbook and are planned to be utilized in GLONASS and LUNA-RESURS

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programs) and various research, including the estimation of the solar oblateness, the

parameters of the rotation of Mars, and the total mass of the asteroid main belt and TNOs,

as well as the verification of general relativity, the secular variations of the Sun’s mass and

the gravitational constant, and the limits on the dark matter density in the Solar System.

The EPM ephemerides, together with the corresponding time differences TT - TDB and

the coordinates of seven additional objects (Ceres, Pallas, Vesta, Eris, Haumea, Makemake,

and Sedna), are available at ftp://quasar.ipa.nw.ru/incoming/EPM.

DOI: 10.1134/S0038094613040059

HISTORICAL INTRODUCTION

Until the coming of the space age in the 1960s, the classic analytical theories of planetary

motion developed by Le Verrier, Hill, Newcomb, and Clemens, which were fully consistent

with optical observations in terms of accuracy, were being constantly refined in accordance

with the development of astronomical practice.

However, the launch of the first satellites exposed the demand for a more accurate

calculation of the coordinates and the speeds of planets. Deep-space experiments and

the introduction of new observational techniques (lunar and planetary ranging, trajectory

measurements, etc.) required the development of planetary ephemerides that would be far

more accurate than the classical ones. On the other hand, it was the new observational

facilities that made it possible to develop ephemerides of the new generation.

The errors of the current best ranging observations do not exceed several meters, which

makes it necessary to compute the ranging correctly up to the 12th significant digit. An

appropriate model of the motion of celestial bodies is required to achieve such high precision.

The construction of a proper model that would take into account all the significant factors

is a serious problem, and the current most feasible way to solve it is to perform numerical

integration of the equations of motion of the planets and the Moon on a computer.

In the late 1960s several research groups in the United States and Russia developed

numerical theories to support space flights. American groups worked at the California

Institute of Technology and the Massachusetts Institute of Technology. Russian high-

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precision numerical ephemerides of planets (Akim et al., 1986) were created as a result of

the research carried out at the Institute of Applied Mathematics, the Institute of Radio

Engineering and Electronics and the Space Flight Control Center, and the Institute of

Theoretical Astronomy, where N. I. Glebova, G. I. Eroshkin, and a group led by G. A.

Krasinsky developed theories independently. This work was continued at the Institute of

Applied Astronomy (IAA), where a series of EPM (Ephemerides of Planets and the Moon)

ephemerides was produced. In order to provide technological support for such research, a

large group of developers working at the IAA under the direction of G. A. Krasinsky created

a unique software system called ERA (Ephemeris Research in Astronomy) that uses a high-

level language targeted at astronomical and geodynamical applications. This ensures the

flexibility of the system, which is being constantly upgraded, and considerably simplifies the

development of various applications. The two dynamical models of planetary motion that are

being developed in the series of DE (Development Ephemeris, JPL) (Standish, 1998; 2004;

Folkner, 2010; Konopliv et al., 2011) and EPM (Krasinsky et al., 1993; Pitjeva, 2001; 2005a;

2012) ephemerides are currently the most complete, have the same precision, and are faithful

to modern radio observations. For the reasons of technological independence, researchers at

the Institut de Mecanique Celeste et de Calcul des Ephemerides (IMCCE) have started

constructing their own numerical planetary ephemerides INPOP (Fienga et al., 2008; 2011)

in 2006. The history of the creation of planetary ephemerides, the EPM2004 ephemeris and

the differences between the DE and EPM ephemerides are discussed in greater detail in

a paper by Pitjeva (2005a). In the present work the planetary part of the latest, updated

version of the EPM ephemerides (EPM2011) and its use in various scientific investigations

are discussed.

EPM DYNAMICAL MODEL OF PLANETARY MOTION

Construction of high-precision planetary ephemerides that are needed for space

experiments, and would guarantee the meter-level accuracy of modern observations, requires

creating a proper mathematical and dynamical model of the motion of planets, which takes

into account all the significant perturbing factors on the basis of general relativity (GR).

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The motion of the barycenter of the Earth-Moon system is appreciably perturbed by

the Moon itself. The Moon’s orbit is subject to perturbations from the asphericity of the

gravitational potentials of the Earth and the Moon, which makes it necessary to characterize

the positions of the equators of the Earth and the Moon with respect to an inertial coordinate

system (i.e., take into account the impact of precession, nutation, and physical libration) with

sufficient accuracy. The resonant behavior of the coupling between orbital and rotational

motions of the Moon makes it essential to reconcile various theories in a unified dynamical

model. As a consequence, modern numerical theories are built by simultaneous numerical

integration of the equations of motion of all planets and the Moon’s physical libration, while

also taking into account the perturbations on the figure of the Earth due to the Moon

and the Sun and the perturbations on the figure of the Moon due to the Earth and the Sun.

Construction of the theory of the Moon’s orbital and rotational motions and its improvement

using lunar laser ranging (LLR) observations are the most difficult tasks in creating modern

ephemerides of planets and the Moon. This work was carried out at the IAA under the

direction of G. A. Krasinsky and is described in a series of papers (Aleshkina et al., 1997;

Krasinsky, 2002; Yagudina et al., 2012). The lunar theory takes into account the effects

associated with elasticity, tidal dissipation of energy, and the frictional interaction between

the Moon’s liquid core and its mantle, and cites selenodynamical parameters obtained

through the analysis of LLR observations made from 1970 to 2010.

The influence of solar oblateness on planetary motion was established theoretically a

long time ago, and some researchers even tried to attribute to it the anomalous motion of

Mercury’s perihelion which was discovered by Le Verrier in the late 19th century. The solar

oblateness causes secular variations of the orbital elements of planets, with the exception

of semimajor axes and eccentricities, and has to be taken into account when constructing

the model of planetary motion. The problem lies in the fact that the solar oblateness is

determined indirectly from some complex astrophysical measurements that are subject to

various systematic errors caused by equipment imperfection and the solar atmosphere and

activity. The use of modern equipment made it possible to give a more reliable estimate

J2= 2 · 10−7. This value is used for the construction of ephemerides starting with DE 405

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(Standish, 1998) and EPM2000 (Pitjeva, 2001). Recently, it became possible to determine

the dynamical solar oblateness while processing of high-precision radar observations when

constructing planetary ephemerides (see Pitjeva, 2005b).

A serious problem arises in the construction of modern high-precision planetary

ephemerides due to the necessity of taking into account the perturbations caused by asteroids.

The DE200 and EPM87 ephemerides considered the perturbations only from the 3-5 largest

asteroids; the experiments revealed that this was impossible to attane a proper representation

of high-precision observations of the Viking 1 and Viking 2 landers, i.e., a representation

which would match the a priori errors (6-12 meters) of these observations. Amplitudes of the

perturbations from asteroids were determined analytically by Williams (1984) considering

commensurability between the orbital periods of the asteroids and Mars. The perturbations

from 300 asteroids that were selected by Williams due to the significant perturbations of

the orbit of Mars caused by them (Williams, 1989) are taken into account starting with

the DE 403 (Standish et al., 1995) and EPM98 (Pitjeva, 1998) ephemerides. However, the

masses of the majority of these asteroids are either unknown or known with insufficient

accuracy, and Standish and Fienga (2002) showed that the accuracy of planetary ephemerides

deteriorated substantially with time due to this factor. Direct dynamical estimates of the

masses of asteroids may be obtained by analyzing their perturbations to other celestial

bodies caused by them. This technique may be applied when examining spacecraft near

asteroids, binary asteroids or asteroids with satellites, perturbations on the Mars and the

Earth caused by asteroids and revealed through the processing of radar observations of

Martian spacecraft and landers, and close encounters of asteroids. Applying the latter

(classical) method requires great caution, since optical observations may produce large errors

(Krasinsky et al., 2002). These techniques were used to measure the masses of several dozen

asteroids, but the construction of high-precision planetary ephemerides demands taking

into account the perturbations from about 300 large asteroids. If the estimates of the

diameters and densities of these asteroids are available, one may also estimate their masses.

The diameters of hundreds of asteroids were determined by processing the infrared data

from the Infrared Astronomical Satellite (IRAS) and Midcourse Space Experiment (MSX)

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satellites. When constructing the DE and EPM ephemerides, these asteroids were divided

into the C (Carbonic), S (Sillicum), and M (Metallic) taxonomic types according to their

spectral classes, and the estimates of their densities were derived from radar observations

while improving the ephemerides. Apart from the sufficiently large asteroids, thousands of

small asteroids, many of which are too small to be ever discovered from the Earth, produce

a substantial cumulative effect on the orbits of the inner planets. The majority of these

bodies travel within the main asteroid belt, and the distribution of their instantaneous

positions in the main belt may be considered uniform. Thus, the perturbations from the

small asteroids that were not considered individually in the integration may be modeled by

additional perturbations from a massive ring in the plane of the ecliptic with a uniform mass

distribution. Starting with EPM2004 (Pitjeva, 2005a), the two parameters characterizing the

ring (its mass Mrand radius Rr) are included in the set of parameters that are improved

from observations.

Hundreds of trans-Neptunian objects (TNOs) that were discovered lately also exert

influence on the motion of planets, especially the outer planets. The updated dynamical

model of the EPM ephemerides includes Eris (a dwarf planet discovered in 2003, which is

more massive than Pluto) and 20 of the largest TNOs into simultaneous integration. The

perturbations from the other TNOs were modeled by a homogeneous TNO ring lying in the

plane of the ecliptic and having a radius of 43 AU and an estimated mass (Pitjeva, 2010a).

Thus, the dynamical model created at the IAA RAS, takes into account (besides

the mutual perturbations of large planets and the Moon) a number of relatively weak

gravitational effects that contribute appreciably while processing modern high-precision

observations:

• perturbations from 301 of the most massive asteroids;

• perturbations from other minor planets in the main asteroid belt, modeled by a

homogeneous ring;

• perturbations from the 21 largest TNOs;

• perturbations from the other trans-Neptunian planets, modeled by a homogeneous ring

at a mean distance of 43 AU;

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• perturbations from the solar oblateness (2 · 10−7);

• relativistic perturbations from the Sun, the Moon, planets (including Pluto), and five

largest asteroids.

When constructing the EPM ephemerides, the equations of motion of n bodies with

masses m1,...mn in a non-rotating barycentric coordinate system were used. These

equations take the form of

¨ ri= A + B + C + D + E,

where A stands for the Newtonian gravitational accelerations:

A =

?

j?=i

µj(rj− ri)

r3

ij

;

B stands for the relativistic terms:

B =

?

?vj

µj

r3

ij

j?=i

µj(rj− ri)

r3

ij

?

−2(β + γ)

c2

?

3

2c2

k?=i

µk

rik

−2β − 1

c2

?

?2

k?=j

µk

rjk

+ γ

?vi

c

?2

+

+(1 + γ)

c

?2

{[ri− rj] · [(2 + 2γ)˙ ri− (1 + 2γ)˙ rj]}(˙ ri− ˙ rj) +3 + 4γ

−2(1 + γ)

c2

˙ ri· ˙ rj−

?(ri− rj) · ˙ rj

rij

+

1

2c2(rj− ri) ·¨ rj

?

+

+1

c2

?

j?=i

2c2

?

j?=i

µj¨ rj

rij

;

C stands for the terms caused by the solar oblateness (the solar quadrupole moment):

C = 3J2µSR2

r4

iS

??

5

2

?(ri− rS)

riS

· p

?2

−1

2

?

(ri− rS)

riS

−

?(ri− rS)

riS

· p

?

p

?

;

D stands for the terms caused by the asteroid and TNO rings to the inner planets:

D =1

2

Mr

R3

r

F

?

1.5,1.5,2,ri2

Rr2

?

ri;

and E stands for the terms caused by the asteroid ring to the outer planets:

E = −Mr

ri3

?

F

?

0.5,0.5,1,Rr2

ri2

?

+1

2

Rr2

ri2F

?

1.5,1.5,2,Rr2

ri2

??

ri.

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Here the following designations were introduced: ri, ˙ ri, ¨ ri (barycentric vectors) are the

coordinate, velocity, and acceleration vectors of the ith body; µj = Gmj, where G is the

gravitational constant and mj is the mass of the jth body; rij = |rj− ri|; β,γ are the

parameters of the PPN (parameterized post-Newtonian) formalism; vi= |˙ ri|; c is the speed

of light; J2is the second zonal harmonic of the Sun; R is the equatorial solar radius; p is the

unit vector pointing to the Sun’s north pole; Mr= Gmr, mr, Rrare the masses and radii of

the rings; and F is the hypergeometric function.

The summation in the equation that pertains to the Newtonian gravitational accelerations

(A) includes (besides planets, the Sun, and the Moon) 301 asteroids and 21 TNOs. The

five main asteroids (Ceres, Pallas, Vesta, Iris, and Bamberga) are entered not only in A,

but also in the equations B (the relativistic terms) and C (the terms caused by the solar

oblateness). Thus, the equations of motion for the 16 main objects incorporate all the mutual

perturbations, including relativistic ones and the perturbations due to the solar oblateness.

The variable ¨ rj, that appears in two terms in the right side of the equations stands for the

barycentric acceleration of the jth body due to the Newtonian acceleration of other bodies.

It should be noted that only the equations of motion of planets, asteroids, TNOs, and

the Moon are actually integrated. The barycentric coordinates and velocities of the Sun are

derived from the following equation:

?

i

µ∗

iri= 0,

where

µ∗

i= µi

?

1 +

1

2c2v2

i−

1

2c2

?

j?=i

µj

rij

?

.

All modern high-precision ephemerides are based on relativistic time scales and

relativistic equations of motion of celestial bodies and radio and light rays. The main common

feature of the DE, EPM, and INPOP series of ephemerides is the simultaneous numerical

integration of the equations of motion of nine major planets, the Sun, the Moon, and the

lunar physical libration carried out in the post-Newtonian approximation for GR (β = γ = 1)

in a harmonic coordinate system (α = 0).

Thus, the terms A, B, and C are identical in all those major planetary ephemerides.

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Various versions of ephemerides differ in modeling the lunar libration, reference frames in

which the ephemerides are computed, adopted values of the solar oblateness and other

parameters, modeling of perturbations from asteroids, and used sets of observations and

estimated parameters. The main distinction of the latest EPM ephemerides (starting with

EPM2008, as described in Pitjeva, 2009) from the DE and INPOP ephemerides is the

inclusion of the perturbations from TNOs that are actually present in the Solar System.

The inclusion of any additional objects into the simultaneous integration leads to the shift

of the barycenter of the Solar System. Since TNOs are located beyond the orbit of Neptune,

and there are many large objects (for example, Eris) among them, the said shift becomes

significant. In the process of calculations, the barycenter remains in its place, while the

coordinates of all objects involved in the integration change. Therefore, comparing the EPM

ephemerides with the DE and INPOP ephemerides requires using relative (heliocentric,

geocentric, etc.) coordinates of objects, but not barycentric ones. Such a comparison was

carried out for DE421, EPM2008, and INPOP08 by Hilton and Hohenkerk (2011). Since any

observations are relative (are usually made from the Earth), the shift of the barycenter does

not influence the representation of observations.

In recent years, a large number of high-precision radiometric observations of spacecraft,

revolving around or passing close to planets, and optical observations of the satellites of

planets carried out by both terrestrial observatories and the Hubble Space Telescope became

available. This enabled the researchers to derive new masses of planets and other bodies of

the Solar System. These values were adopted as the current best values of the constants

of dynamical astronomy by XXVII IAU GA in 2009 (Luzum et al., 2011) and are used in

updated versions of the EPM ephemerides (starting with EPM2008).

The integration in the barycentric coordinate system at the J2000.0 epoch was done using

Everhart’s method over a 400-year interval (from 1800 to 2200) by a lunar and planetary

integrator of the ERA-7 software system.

OBSERVATIONAL DATA, THEIR REDUCTION, AND TT - TDB

The observations that were used to improve the accuracy of the EPM2011 ephemerides

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included 677670 positional measurements of various types (from classical meridian

observations to modern radio observations of planets and spacecraft) obtained from 1913

to 2011. Optical observations dating from 1913, when an improved micrometer was installed

at the United States Naval Observatory and the measurements became more accurate

(∼0.′′5), and all the available radio observations (up to the year 2011) were used. It should

be noted that the accuracies of modern CCD observations approach a few hundredths of

an arcsecond. A real revolution in dynamical astronomy started in 1961 when the first

successful radiolocation of Venus was carried out simultaneously in the United States (at the

California Institute of Technology and the Massachusetts Institute of Technology), the USSR

(at the Institute of Radio Engineering and Electronics), and England (at the Jodrell Bank

Observatory). The significance of astronomical radar observations stems from two factors.

Firstly, they added two new types of measurements, namely, the measurement of the delay

time (ranging) that could be converted to distance using the known speed of light and the

measurement of the Doppler frequency shift that gives the relative radial velocity of the

reflecting surface. Secondly, radar observations are highly accurate. Nowadays the relative

accuracy that ranges from 10−11to 10−12has become ordinary for trajectory measurements

of spacecraft. These values are five orders of magnitude better than the accuracy of classical

optical measurements. However, only the terrestrial planets are fully provided by with radio

observations. Fewer observations of this type are made for Jupiter and Saturn, and there

exists only one three-dimensional normal point provided by Voyager 2 for Uranus (and

Neptune). Therefore, optical observations still retain their significance for the outer planets.

The main factors that limit the accuracy of photographic and CCD observations of planets

are the brightness of planets compared to reference stars (the equalization of brightness);

the distortion of photographic images due to meteorological, instrumental, and astronomical

(the phase effect) causes; and the difficulty of measuring an extended object of a non-

uniform density. This applies especially to bright planets (Jupiter and Saturn) with large

visible disks. Positional observations of planetary satellites are not prone to any of these

restrictions. Since the position of a satellites relative to the stars is determined both by the

planetary motion and the satellite’s own motion around the planet, the measurements of

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the positions of satellites may be used to define the planetary orbits more accurately. The

astrometric photographic observations of the satellites of Jupiter and Saturn were started

in the Nikolaev Observatory in 1962. In 1998, astronomers in Flagstaff began observing the

satellites of the outer planets (in addition to the observations of the outer planets themselves),

and all their measurements are referred to the ICRF system with the use of reference stars

from the AST and TYCHO2 catalogues. Observations of satellites are also carried out at a

number of other observatories. Theories of the motion of satellites are required to process such

observations. Analytical theories of the motion of the satellites of Jupiter (Lieske), Saturn

(Vienn and Duriez), and Uranus (Lascar and Jacobson) are incorporated in the ERA-7

software system. The drawback of these analytical theories lies in the fact that they do not

provide an opportunity to correctly introduce the parameters of the satellite’ motion when

improved from observations. Therefore, the researchers at the IAA RAS, construct their own

numerical theories of the motion of the satellites of Mars and the outer planets (Poroshina

et al., 2012). These theories are successfully used to improve the ephemerides of satellites

and planets alike. Lately, the previous observations (prior to 2005) were supplemented with

the new data from spacecraft, namely, measurements of ranging made using Odyssey, Mars

Reconnaissance Orbiter (MRO), Mars Express (MEX), and Venus Express (VEX); VLBI

observations of Odyssey and MRO; and three-dimensional normal point observations of

Cassini and Messenger. These measurements were complemented by CCD observations of the

outer planets and their satellites made at the Flagstaff and Table Mountain observatories.

The observations used are shown on the page 12 (1 mas = 0.′′001); the numbers in the

headings (57560, 58112, and 561998) indicate the number of observations.

The majority of these observations were taken from the Jet Propulsion Laboratory (JPL)

database (http://iau-comm4.jpl.nasa.gov/plan-eph-data/index.html) which was created by

E. M. Standish and is now maintained and expanded by W.M. Folkner. This data

set was supplemented by Russian radar observations of planets made from 1961 to

1995 (http://www.ipa.nw.ru/PAGE/DEPFUND/LEA/ENG/rrr.html) and data from Venus

Express and Mars Express obtained through the courtesy of A. Fienga.

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Astrometric observations of planets and spacecraft

Optical observations of the outer planets and their satellites made from 1913 to 2011 (57560)

Radar observations of Mercury, Venus, and Mars (58112)

Radio data provided by spacecraft from 1971 to 2010 (561998)

USNO

Pulkovo

Nikolaev

Tokyo

Bordeaux

LaPalma

Flagstaff

TMO

Observation type IntervalA priori accuracy

Transits 1913–19941′′→0.′′5

0.′′8 →0.′′25

1′′→0.′′2

0.′′2 →0.′′05

Photoelectric transits1963–1998

Photographic 1913–1998

CCD 1995–2011

Millstone

Haystack

Arecibo

Goldstone

Crimea

Observation typeInterval A priori accuracy

Ranging1961–1997100 km → 150 m

Mariner − 9

Pioneer − 10,−11 Jupiter

V oyager

Phobos

Ulysses

Magellan

Galileo

V iking − 1,−2

Pathfinder

MGS

Odyssey

MRO

Cassini

V EX

Messenger

MEX

Venus

Jupiter

Mars

Jupiter

Venus

Jupiter

Mars

Mars

Mars

Mars

Mars

Saturn

Venus

Mercury

Mars

Observation typeInterval A priori accuracy

Ranging 1971–2009 6 km → 1 m

Dif.range1976–19971.3 → 0.1 mm/s

Rad.velos. 1992–1994 0.1 → 0.002 mm/s

Flybys1973–2010 400 mas → 0.4 mas

∆VLBI1990–2010 12 mas → 0.2 mas

The processing of observational data was done using proven and reliable techniques with

due account for all the needed reductions (Pitjeva, 2005a). The following reductions were

applied to radar data:

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• reduction of time moments to a uniform scale;;

• relativistic corrections, namely, the delay of radio signals in the gravitational field of

the Sun, Jupiter, and Saturn (the Shapiro effect) and the transition from the coordinate

time (the argument of ephemerides) to the proper time of the observer;

• the delay of radio signals in the Earth’s troposphere;

• the delay of radio signals in the plasma of the solar corona;

• correction for topography of the surfaces of planets (Mercury, Venus, and Mars).

The following reductions were applied to optical data:

• reduction to the ICRF system: from reference catalogues to FK4, then to the FK5

catalog, and at last to the ICRF frame;

• correction for additional phase effect;

• correction for gravitationa deflection of light by the Sun.

The transition from the observing time (UTC = TAI + an integer number of seconds)

to the barycentric dynamic time (TDB) of the ephemerides requires knowing the differences

between the terrestrial time (TT = TAI +32.184 s) and TDB. Until recently, these differences

were computed by applying the analytical expansions for the DE405 ephemerides. However,

the differences TT - TDB depend on the coordinates of all bodies that are involved in the

integration of the corresponding ephemerides. Therefore, the construction of these differences

by numerical integration using the corresponding ephemerides is more correct.

The following differential equation taken from the paper by Klioner (2010) was used for

connection between TT and TDB:

d(TT − TDB)

dTDB

=LB− LG

1 − LB

+1 − LG

1 − LB

?1

c2α′+1

c4β′

?

,

where LB= 1.550519768 · 10−8, LG= 6.969290134 · 10−10, c is the speed of light,

α′= −1

2v2

E−

?

A?=E

GMA

rEA

,

β′= −1

8v4

E+1

2

??

2aA· rEA+1

A?=E

GMA

rEA

?2

+

?

A?=E

GMA

rEA

?

4vA· vE−

−3

2v2

E− 2v2

A+1

2

?vA· rEA

rEA

?2

+

?

B?=A

GMB

rAB

?

.

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Fig. 1. Differences in (TT–TDB) for the EPM2004 and EPM2008

ephemerides expressed in nanoseconds.

Figure 1 shows, as an example, differences in TT - TDB for the EPM2004 and EPM2008

ephemerides expressed in nanoseconds.

EPM2011 PARAMETERS AND REPRESENTATION OF OBSERVATIONS

About 270 parameters were determined in the process of improving the planetary part

of the EPM2011 ephemerides:

• the orbital elements of planets and 18 satellites of the outer planets;

• the value of the astronomical unit or GM⊙;

• the angles of orientation of the ephemerides with respect to the ICRF;

• parameters of the rotation of Mars and the coordinates of three Martian landers;

• the masses of 21 asteroids and the mean densities of three taxonomic classes (C, S,

and M) of asteroids;

• the mass and radius of the asteroid ring and the mass of the TNO ring;

• the ratio of the Earth and Moon masses;

• the Sun’s quadrupole moment and parameters of the solar corona for different

conjunctions of planets with the Sun;

• the coefficients of Mercury’s topography and corrections to the level surfaces of Venus

and Mars;

• the coefficients for additional phase effect of the outer planets;

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• the constant shifts for the series of observations of Venus in Goldstone (1964) and Venus

(1969) and Mercury (from 1986 to 1989) in Crimea, as well as the shifts (and, in certain

cases, their derivatives) for all spacecraft that were interpreted as the calibration errors;

• post model parameters, such as the PPN parameters (β, γ), ˙ πi,

˙

GM⊙/GM⊙, ˙ ai/ai.

Mean values and rms’s of the residuals of observations are shown in the tables 1, 2, where

“n. p.” stands for normal points (with the exception of Viking and Pathfinder for which the

total number of observations is given).

Table 1. Mean values and rms’s of the residuals of radio observations

Planet Observation typeInterval Number of n. p. < O − C >σ

Mercury

τ [м]1964–19977460.0610

КА τ [м]1974–200951.318.9

Venus

τ [м]1961–199513540.0594

Magellan dr [мм/с]1992–19941950.00.007

MGN,VEX VLBI [mas] 1990–2010470.02.7

Cassini τ [м]1998–19992-2.62.4

VEX τ [м]2006–201017210.02.8

Mars

τ [м]1965–19954030.0745

КА τ [м]1971–1989644-13.743.9

Viking τ [м]1976–198212580.09.5

Viking dτ [мм/с]1976–1978149780.00.89

Pathfinder τ [м]1997900.02.7

Pathfinder dτ [мм/с]199775740.00.09

MGS τ [м]1998–200673410.01.3

Odyssey τ [м]2002–200981870.01.1

MRO τ [м]2006–20099300.01.2

MEX τ [м]2009–20109700.01.5

КА VLBI [mas]1989–20101440.00.8

JupiterКА τ [м]1973–200070.012.4

КА VLBI [mas]1996–199724-1.011.4

SaturnКА τ [м]1979–2006340.02.8

Uranus

τ [м]198611.7105

NeptuneVoyager-2 τ [м]198910.014

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Table 2. Mean values and rms’s of the residuals of optical observations and data from

spacecraft near planets (marked by∗) obtained from 1913 to 2011 for α and δ expressed in

mas

PlanetNumber of observations < O − C >α σα < O − C >δ

σδ

Mercury∗

Venus∗

6 0.00.71.01.8

4 0.31.7 1.86.5

Jupiter

Jupiter∗

13364 12181-28194

16-1.0 2.2-4.97.9

Saturn

Saturn∗

15056-1.0 160 -1.0 157

920.10.3 0.00.8

Uranus

Uranus∗

11846 3.01710.4203

2 -458.0 -2712

Neptune

Neptune∗

11634 4.9 152 6.4195

2-12 3.5-134.0

Pluto56600.4 138 3.0140

The residuals of ranging for Odyssey, MRO, MEX, and VEX are shown in Fig. 2. In

ranging the increase of the dispersion O–C is evident during solar conjuctions when the

signal passes through the solar corona. The delay in the solar corona was taken into account

with the improvement of the coefficients of the corona model from observations, but getting

rid of the solar corona noise completely requires the two frequencies measurements. The rms

errors of the residuals amount to 1.1 m (Odyssey), 1.2 m (MRO), 1.5 m (MEX), and 2.8 m

(VEX).

Fig. 2. Residuals of ranging (expressed in meters) for observations made by Odyssey,

Mars Reconnaissance Orbiter (MRO), Mars Express (MEX), and Venus Express (VEX).

The residuals of observations of right ascensions (or, to be more precise, αcosδ) and

declinations for the outer planets and their satellites are presented in Fig. 3.

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Fig. 3. Residuals of observations of αcosδ and δ

(1913-2011) for the outer planets on a scale of ±5′′.

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Tables 1 and 2 show that the majority of observations that form the basis of the

ephemerides are classified as radio observations, mostly ranging obtained with the use of

spacecraft. These measurements allow us to obtain all the orbital elements of planets with

the exception of the three angles of the Earth’s orientation, which is equivalent to the

orientation of the whole system of the ephemerides (angles εx,εy, and εz). The earliest

numerical planetary ephemerides (DE118 and EPM87) were referred to the FK4 catalogue

system, while the DE200 ephemerides were referred to the system of the dynamical equator

and equinox. At present, planetary ephemerides are oriented with respect to the international

ICRF system through the use of VLBI observations of various spacecraft near planets with

background quasars, the coordinates of which are given in the ICRF system. The accuracy

of these observations has improved considerably from 2001 to 2010 and reached several

tenths of a milliarcsecond for Saturn and Mars (Jones et al., 2011). This made it possible to

significantly improve the orientation of the EPM2011 ephemerides. The angles of rotation

between the EPM ephemerides and the ICRF system and their errors obtained at present

and previously are presented in Table 3. Figure 4 shows the residuals of VLBI observations

of various spacecraft near Mars and of Cassini near Saturn.

Table 3. Rotation angles for orientation of the EPM ephemerides into the ICRF system

ObservationNumber of

εx

εy

εz

intervalobservations mas mas mas

1989-1994 20

4.5 ± 0.8−0.8 ± 0.6−0.6 ± 0.4

1989-2003 62

1.9 ± 0.1−0.5 ± 0.2−1.5 ± 0.1

1989-2007 118

−1.528 ± 0.0621.025 ± 0.0601.271 ± 0.046

1989-2010213

−0.000 ± 0.042−0.025 ± 0.0480.004 ± 0.028

The improvement of the orientation of the EPM ephemerides made it possible to reach

the accuracy of the Earth’s heliocentric coordinates (X,Y,Z) over a 100-year interval (from

1950 to 2050) of at least 250 m and the accuracy of velocities (˙X,˙Y ,˙Z) of at least 0.05 mm/s

(see Fig. 5). The knowledge of the Earth’s accurate heliocentric coordinates is particularly

important when studying pulsars, variable stars, and exoplanets. However, the comparison of

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Fig. 4. Residuals of VLBI observations of various spacecraft near Mars

and of Cassini near Saturn expressed in mas.

the EPM2011 and JPL DE424 ephemerides showed that differences of heliocentric distances

of the Earth, determined by ranging, for these ephemerides over the same interval are much

smaller and do not exceed 6 m (see left side of Fig. 6 for the geocentric Sun).

Fig. 5. Differences EPM2011 – DE424 in heliocentric coordinates (X,Y,Z)

and velocities (˙X,˙Y ,˙Z) of the Earth from 1950 to 2050.

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Some parameters determined in the process of improving the DE and EPM ephemerides

(Pitjeva and Standish, 2009) and adopted as the current best values for ephemeris astronomy

by XXVII IAU GA in 2009 (Luzum et al., 2011) were taken as initial in EPM2011 and

were then improved from all observations. Among them are such parameters as the ratio

of masses of the Earth and the Moon MEarth/MMoon= 81.30056763 ± 0.00000005 and

the masses of largest asteroids (Ceres, Pallas, and Vesta) and 18 other asteroids. Table 4

gives the masses and the estimates of these masses with ones taken from papers by Konopliv

et al. (2011) and Fienga et al. (2011), where they were obtained in the same way using

the DE423 and INPOP10a ephemerides. All parameters obtained in the present work and

mentioned in this section are given with uncertainties that correspond to 3σ (formal standard

error of the least squares method). Experience shows that formal standard errors are overly

optimistic. Uncertainties given by Konopliv et al. (2011) are obtained with a special method

that is characterized by the fact that the uncertainties of the masses of asteroids that are

not estimated are taken into account while calculating all the adjusted parameters. The

uncertainties obtained in this way are probably close to the actual errors. Uncertainties

specified in a paper by Fienga et al. (2011) are larger than the ones obtained here due to the

large quantity (145) of the estimated masses of asteroids. The data presented in Table 4 point

to the fact that the estimates of masses of asteroids largely agree with each other within the

limits of their errors. The two exceptions are the masses of (52) Europa and (511) Davida for

the INPOP10a ephemerides. On August 16, 2011, the Dawn spacecraft approached Vesta,

one of the largest asteroids. The spacecraft studied the asteroid for a year and determined

its mass to be (130.2927 ± 0.0005) · 10−12GM⊙ (Russel et al., 2012). This value virtually

coincides with the estimate of the mass of Vesta obtained in the present work.

Special effort was given to producing an accurate estimate of the total influence of

asteroids on the motion of planets, the majority of which lie in the main asteroid belt. In

EPM the main belt is modeled by the motion of the 301 largest asteroids and a homogeneous

material ring that represents the influence of numerous other small asteroids. Parameters

Mringand Rringthat characterize the ring of small asteroids were determined through the

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processing of observations:

Mring= (1.06 ± 1.12) · 10−10M⊙,Rring= (3.57 ± 0.26)AU

Table 4. Estimates of the masses of asteroids obtained by using the observations of ranging

for the EPM 2011, DE423 (Konopliv et al., 2011), and INPOP10a (Fienga et al., 2011)

ephemerides and expressed in (GMi/GM⊙) × 10−12

AsteroidEPM2011DE423INPOP10a

(1) Ceres

472.17 ± 0.79467.90 ± 3.25475.8 ± 2.8

(2) Pallas

104.72 ± 0.92103.44 ± 2.55111.4 ± 2.8

(3) Juno

14.67 ± 0.2512.10 ± 0.9111.6 ± 1.3

(4) Vesta

129.70 ± 0.45130.97 ± 2.06133.1 ± 1.7

(6) Hebe

4.05 ± 0.466.73 ± 1.647.1 ± 1.2

(7) Iris

6.54 ± 0.305.53 ± 1.327.7 ± 1.1

(8) Flora

2.05 ± 0.182.01 ± 0.424.07 ± 0.63

(9) Metis

1.64 ± 0.253.28 ± 1.08

—

(10) Hygiea

41.61 ± 1.3444.97 ± 7.36

—

(14) Irene

3.61 ± 0.281.91 ± 0.81

—

(15) Eunomia

14.45 ± 0.5514.18 ± 1.4918.8 ± 1.6

(16) Psyche

12.75 ± 1.0312.41 ± 3.4411.2 ± 5.2

(19) Fortuna

4.36 ± 0.133.20 ± 0.53

—

(23) Thalia

1.24 ± 0.211.11 ± 0.71

—

(29) Amphitrite

5.39 ± 0.507.42 ± 1.49

—

(41) Daphne

4.17 ± 0.444.24 ± 1.779.2 ± 2.6

(52) Europa

9.06 ± 1.3211.17 ± 8.4042.3 ± 8.0

(324) Bamberga

5.10 ± 0.145.34 ± 0.994.67 ± 0.38

(511) Davida

6.11 ± 1.748.58 ± 5.9319.9 ± 4.1

(532) Herculina

7.07 ± 0.624.97 ± 2.812.89 ± 0.76

(704) Interamnia

12.22 ± 0.9619.97 ± 6.57

—

The total mass Mbeltof the main belt asteroids is expressed as the sum of the masses of

301 largest asteroids and the asteroid ring and is equal to Mbelt= (12.3 ± 2.1) · 10−10M⊙

(about 3 times the mass of Ceres). The gravitational attraction of trans-Neptunian objects

is modeled in much the same way by summing the influences of 21 known TNOs and an

additional homogeneous ring with a radius of 43 AU that represents numerous other smaller

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objects. The mass of the TNO ring MTNOringwas determined to be equal to MTNOring=

(501 ± 249) · 10−10M⊙while processing observations.

The total TNO mass MTNOthat includes the masses of Pluto, the 21 largest TNOs, and

the TNO ring is equal to MTNO= 790 · 10−10M⊙, (about 164 times the mass of Ceres or 2

times the mass of the Moon).

ACCURACY OF EPHEMERIDES AND COMPARISON BETWEEN

THE EPM2011 AND DE424 EPHEMERIDES

Firstly, the accuracy of the constructed ephemerides may be estimated from the

representation of observations, i.e., from comparison of the observal values (O) with the

computed values (C) of observations. Tables 1–3 and Figures 2–4 present the residuals,

their mean values, and their errors (σ) that do not exceed their a priori errors. Secondly,

the accuracy of ephemerides may be evaluated by comparing them with other ephemerides

constructed by independent research teams. Starting from the 1970s, the EPM ephemerides

computed at the IAA RAS, were regularly compared with the DE ephemerides created at

JPL. In a paper by Pitjeva (2005a), the differences in heliocentric distances of planets for the

EPM2004 and DE410 ephemerides over a 40-year interval (from 1970 to 2010) are presented.

In the present work, the differences in three coordinates over a 100-year interval are presented

in Figures 6 and 7. These coordinates – geocentric distances (D), right ascensions (α), and

declinations (δ) – fully characterize the accuracy of the ephemerides determined by comparing

the EPM2011 and DE424 ephemerides.

Since the coordinates of the inner planets were obtained through high-precision radio

observations, the differences calculated for them are much smaller for all the coordinates

(D, α, δ) than the differences for the outer planets (the geocentric position of the Sun

may be viewed as the heliocentric position of the Earth with an opposite sign). The fact

that the difference in Mercury distances is slightly larger than the one given in a paper by

Pitjeva (2005a) is explained by the use of new Messenger data, so far inaccessible to us, in

DE424. The differences in distances (over the interval considered in the 2005 paper) for all

the other planets have become less. In the case of Mars, the differences remain minor over

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Fig. 6. Differences EPM2011–DE424 in geocentric distances (D), right ascensions (α),

and declinations (δ) of Mercury, Venus, the Sun, and Mars over a 100-year interval

(from 1950 to 2050).

an interval which is somewhat wider than the one covered by observations. More precisely,

the differences in distance, α, and δ do not exceed 150 m, 0.7 mas, and 0.5 mas, respectively,

over a 58-year interval (from 1970 to 2028).

The availability of some radio observations of Jupiter and particularly Saturn (studied

by the Cassini spacecraft) allowed us to reconstruct their orbits with an accuracy greater

than that achievable for the other outer planets’ orbits defined virtually only by optical

observations. There exists only one three-dimensional point (D, α, δ) provided by Voyager -

2 for Uranus and Neptune. Besides that, not even one period of orbital rotation of Neptune

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and Pluto is covered with more or less accurate observations. The uncertainty of the Pluto’s

distance, which was specified by Folkner in his talk at XXVIII IAU GA, changes from 1100 to

3000 km over a 18-year interval (from 2000 to 2018). These values are roughly correspondent

to the uncertainty obtained in the present work (3300 km) by comparing the EPM2011 and

DE424 ephemerides (see left bottom part of Fig. 7).

Fig. 7. Differences EPM2011–DE424 in geocentric distances (D), right ascensions (α),

and declinations (δ) of the outer planets over a 100-year interval (from 1950 to 2050).

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In a paper by Fienga et al. (2011) the differences in geocentric distances, right ascensions,

and declinations of planets over a 120-year interval (from 1900 to 2020) are also shown

for the INPOP10a and DE421 ephemerides. The comparison of results on the common

interval (1950–2020) shows that all the differences in D, α, δ for the EPM2011 and DE424

ephemerides are lower than the corresponding differences for the INPOP10a and DE421

ephemerides. This may be attributed to the use of the new version of the EPM ephemerides.

Specifically, the INPOP10a ephemerides included the observations of the outer planets that

were carried out not later than 2008, whereas the EPM2011 ephemerides include data up

to the year 2011. The sole exception is the distance for Jupiter in the EPM2011 and DE424

ephemerides. The distances for Jupiter are determined for the most part by a few radar

observations carried out from 1974 to 2000. All 7 such observations, weighted according to

their accuracy, are used in the EPM2011 ephemerides, while the other ephemerides include

only the five most accurate ones.

It is interesting to look at the comparison of the same values given in a paper by Standish

(2004) for the DE200 and DE409 ephemerides over a 50-year interval (from 1970 to 2020).

It can be seen that all the differences are at least an order of magnitude larger. This leads

to the conclusion that modern ephemerides have made great progress in terms of accuracy

compared to DE200 (1982).

The comparison with modern observations and the DE ephemerides verifies that the

planetary part of the EPM ephemerides is sufficiently accurate.

THE USE OF THE EPM EPHEMERIDES IN SCIENTIFIC RESEARCH

The potential to construct and maintain fundamental ephemerides of the major planets,

the Sun, and the Moon may be viewed as one of the characteristics of a technologically mature

state. The reason for this lies in the fact that these ephemerides have various practical

applications. Specifically, they serve as an important element of terrestrial - , marine - ,

and space-based navigational systems. Nowadays the DE/LE series of ephemerides that are

developed in the United States and serve, first and foremost, to support the American space

research program are adopted as the international standard of fundamental ephemerides. The

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high accuracy of these ephemerides is preconditioned by the fact that enormous high-quality

sets of observational data obtained using terrestrial observatories and spacecraft are utilized

in creating the DE/LE ephemerides. However, the use of the American DE/LE ephemerides

may present some difficulties. Among them are the problems with licensing (the IAU did not

issue recommendations for the use of any DE ephemerides except DE200), openness (not all

the algorithms are described in detail), possible delays (the access to new versions of the

DE/LE ephemerides may remain restricted for a certain period of time), and reliability. Since

domestic ephemerides are not subject to these problems, the IAA RAS, developed its own

EPM ephemerides and uses them when preparing the Astronomical Yearbook (starting from

2006), the Nautical Astronomical Yearbook, and the Nautical Astronomical Almanac. Besides

that, it is planned to use these ephemerides in GLONASS and LUNA-RESURS programs.

The EPM ephemerides lie at the basis of much scientific research. For six years (1976–

1982) the Viking 1 and Viking 2 landers were observed from California, Madrid, and

Canberra, while the Pathfinder lander was observed for three months in 1997. These

unique observations made it possible to define more precisely the rotation of Mars when

constructing the EPM ephemerides. The determination of the more precise values of the

parameters of rotation of Mars is important for understanding its geophysics. Firstly,

the comparison of the observed and the calculated precessions of Mars coupled with the

oblateness coefficient of Mars makes it possible to calculate the normalized polar moment

of inertia that allows researchers to evaluate the density variations within the planet.

Secondly, the comparison of the determined amplitudes of short-period nutation terms with

the theoretical predictions enables exploration of the question of the distinctions between

Mars and a rigid body. The observations of Martian landers (on the basis of the EPM

ephemerides) made it possible to determine the coordinates of all the three landers and

define all parameters of rotation of Mars (precession, nutation, and seasonal rotation terms

governed by melting and condensation of carbonic acid at the polar caps) and the polar

moment of inertia, corresponding to the speed of precession of Mars (Pitjeva, 1999), more

precisely. The parameters of rotation of Mars and their accuracies were found to be close to

the corresponding values taken from a paper by Yoder and Standish (1997).

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Asteroids exert a significant influence on the motion of planets (especially Mars);

therefore, the masses of the largest asteroids (in the present work we examined the 21 largest

ones) and the total mass of the main asteroid belt may be estimated from radio observations.

Hundreds of trans-Neptunian objects, which also exert influence on the motions of planets,

were discovered recently; their total mass may also be estimated, as was already done by

Pitjeva (2010a). The knowledge of such characteristics is important not only for devising a

more precise description of the forces acting in the Solar System, but also for understanding

the general dynamics of the Solar System and the processes associated with its formation.

The passage of photons and motion of planets in the gravitational filed of the Sun allow us

to view the Solar System as a sufficiently convenient laboratory for testing gravity theories.

Modern radar observations of planets and spacecraft, that have meter-level accuracy, make

it possible to explore relativistic effects, estimate the value of the heliocentric gravitational

constant GM⊙(the Sun’s mass) and its possible variation, and estimate the solar oblateness.

The comparison of the results of determination of additional motion of the perihelia of

planets, which is not modeled by Newtonian interaction and GR, the PPN parameters (β, γ),

the quadrupole moment of the Sun, and GM⊙, that were cited in previous works (Krasinsky

et al., 1986; Pitjeva, 1993; 2005b; 2010b) and obtained in the present work:

β − 1 = −0.00002 ± 0.00003, γ − 1 = +0.00004 ± 0.00006, J2= (2.0 ± 0.2) · 10−7,

shows that, firstly, the uncertainties of these parameters did decrease significantly (at least

by an order of magnitude). This substantial progress may be attributed to the increase in

accuracy of the dynamical models of motion and the methods of reduction of observations

and to the improvement of observational data (i.e., boost in precision and widening of the

observational time interval). Secondly, the reduction of the uncertainties of these parameters

constrains the possible values of relativistic parameters and imposes increasingly tight

restrictions on the gravity theories that are competing with GR.

For the first time, the variation of the heliocentric gravitational constant

˙

GM⊙/GM⊙= (−5.0 ± 4.1) · 10−14

per year (3σ) has been deduced through the analysis of various types (mostly radio) of

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positional observations of planets and spacecraft. The value obtained, coupled with the

known upper limits on the possible variation of the Sun’s mass, allow us to place tighter

restrictions on the variation of the gravitational constant and infer that its annual value falls

in the interval

−4.2 · 10−14<˙G/G < +7.5 · 10−14

with a probability of 95%. The GM⊙variation is seemingly associated not with the variation

of G, but with the variation of the Sun’s mass. Therefore, the variation of M⊙, is reflective of

the balance between the mass lost through radiation and solar wind and the material falling

onto the Sun (Pitjeva and Pitjev, 2012).

Besides that, the search for and the estimation of a possible gravitational influence of

dark matter in the Solar System on the motion of planets has been carried out on the basis of

the EPM2011 planetary theory by studying the additional motion of the perihelia of planets

and the estimates of the heliocentric gravitational constant obtained through the analysis

of observations of certain planets. The estimates obtained of the density and mass of dark

matter at different distances from the Sun are, as a rule, exceeded by their errors (σ). This

points to the fact that the density of dark matter ρdm(if any) is very low and resides well

below the errors of determination of such parameters achievable nowadays. It was found

that ρdmat the distance of the orbits of Saturn, Mars, and the Earth should be lower than

1.1 · 10−20g/cm3, 1.4 · 10−20g/cm3, and 1.4 · 10−19g/cm3, respectively. The possibility of

dark matter concentrating at the center of the Solar System was also considered, and it was

found that the mass of dark matter located in the sphere inside the Saturn’s orbit would

still not exceed 1.7 · 10−10M⊙(Pitjev and Pitjeva, 2013).

CONCLUSIONS

The EPM series of high-precision ephemerides of planets and the Moon that is faithful

to modern observations and comparable in terms of accuracy with the latest versions of

the well-known DE ephemerides (JPL) was created at the IAA RAS. The use of a more

accurate dynamical model of planetary motion and a large number of additional high-

precision observations allows us to assert that the latest versions (EPM2004–EPM2011)

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of the EPM ephemerides are more accurate than the DE405 ephemerides, which are adopted

as an international standard. The EPM ephemerides have the following advantages over the

DE ones while using EPM for Russian astronavigation:

• They are constructed using independent and constantly updated software.

• They are promptly updated and improved according to incoming new data.

• The clients (GLONASS programs) may request additional needed data in any format.

Convenient access procedures (Bratseva et al., 2010) for external users were recently

devised at the IAA RAS. The users may access the EPM ephemerides of planets and the

Moon together with the corresponding differences TT−TDB, as well as the ephemerides,

computed simultaneously with the EPM ones, of seven additional objects (Ceres, Pallas,

Vesta, Eris, Haumea, Makemake, and Sedna) that are provisionally called dwarf planets.

The EPM ephemerides are available at ftp://quasar.ipa.nw.ru/incoming/EPM/.

The constructed EPM ephemerides used in practice form the basis of the Astronomical

Yearbook, and are needed to fulfill the GLONASS Federal Program and to carry out space

experiments in the Solar System. They also help us to solve some of the problems of

fundamental astrometry, including the determination of the dynamical structure of the Solar

System and a number of astronomical constants.

ACKNOWLEDGMENTS

This work was supported by a grant from the RAS Presidium Program 22 “Fundamental

Problems of Research and Exploration of the Solar System”.

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