EPM2015 ephemerides contain coordinates and velocities of the Sun, the Moon, nine major planets, three largest asteroids (Ceres, Pallas, Vesta) and 4 TNO (Eris, Haumea, Makemake, Sedna) (in au, au/day) as well as lunar libration (in radians) and TT-TDB (in seconds).

Dynamical model

The numerical integration of the equations of motion of the celestial bodies has been performed in the Parameterized Post-Newtonian N-body metric for General Relativity in the TDB time scale. EPM2015 ephemerides are computed in the barycentric coordinate system—BCRS, over more than the 400-year interval (1787-2214) as in EPM2011, but using the other version of program package ERA (Ephemeris Research in Astronomy), ERA-8 [1], which is a complete rework of the original program package ERA-7 [2]. ERA comprises a domain-specific language SLON tailored for astronomical tasks [3]. ERA-8 is based on the Racket programming platform and has SQLite as the database engine. Most of the numerical algorithms of ERA-8 are implemented in C.

Ephemerides EPM2015 have been constructed in accordance with the B2 resolution of 28 GA IAU which fixed the value of the astronomical units of length (au) equal 149597870700 m and proposed the determination of GM_Sun in SI units.

For constructing planetary ephemerides using the best modern observations, it is necessary to take into account all influencing factors. The dynamical model of the planetary part of the EPM ephemerides includes:

  • mutual perturbations from major planets and Pluto, the Sun, and the Moon;
  • perturbations from 301 the most massive asteroids, and the 30 largest trans-neptunean objects (TNO);
  • perturbations from a modeled massive two-dimensional asteroid annulus (for the rest of small asteroids) with a uniform mass distribution and radii R1=2.06 au, R2= 3.27 and perturbations from a one-dimensional massive ring of TNO's in the ecliptic plane with a radius of 43 au (see [6]);
  • relativistic perturbations;
  • perturbations due to the solar oblateness.

Inclusion of the 30 largest and the very far TNO (Eris which surpasses Pluto being one of them) into the simultaneous integration causes a significant change to the barycenter of the solar system, so that barycentric positions of the Sun and all other objects change, but relative coordinates (heliocentric or geocentric) remain the same. Thus, only the comparison of relative coordinates shows real differences between EPM and DE or INPOP ephemerides. Moreover, as the TT-TDB differences depend on coordinates and masses of all objects, included in corresponding ephemerides, the TT-TDB differences for EPM2015 differ from DE430 TT-TDB by the linear trend about 15 ns/cy basically due to the 30 largest TNO.

Thereby, as compared to the previous available EPM2011, the updated dynamic model of the planetary part of EPM2015 includes [6-7]:

  • the massive two-dimensional asteroid annulus for modeling the total perturbation from the remaining small asteroids with estimated mass, instead of a one-dimensional asteroid ring of the EPM2011 model;
  • 30 individual TNO have been included into the simultaneous integration process instead of 21 TNO for the EPM2011 model;
  • new values of celestial bodies masses and other parameters;
  • expanded database of observations (1913-2014).

For the TT-TDB conversion, the differential equation from the paper by Klioner [12] has been used, and TT-TDB was obtained by numerically integrating the EPM2015.

For EPM2015 ephemerides, the parameters of the lunar and planetary parts of ephemerides are in agreement with each other.

In the past, George Krasinsky constructed a dynamic model of the lunar motion, which takes into account the tidal perturbation in the lunar rotational motion [14]. This model was implemented by Krasinsky's group as the lunar part of the EPM ephemerides [15-16]. In the course of the work on the EPM ephemeris another model of lunar orbital and rotational motion was implemented, based on the equations used in JPL DE430 ephemeris [17] with combination of up-to-date astronomical, geodynamical, and geo- and selenophysical models.

The Moon is considered an elastic body having a rotating liquid core. The following equations are included in the model:

  • perturbations of the orbit of the Moon in the gravitational potential of the Earth;
  • torque due to the gravitational potential of the Moon;
  • perturbations of the orbit of the Moon due to lunar and solar tides on the Earth;
  • distortion of the Moon's figure as a result of its rotation and Earth's gravity;
  • torque due to the interaction between the lunar crust and the liquid core.

EGM2008 was taken for the gravitational model of the Earth, while GL660B was used for the Moon.

For improvement of the planetary part of EPM2015, about 270 parameters are determined:

  • orbital elements of the planets and the 18 satellites of the outer planets;
  • the value of the solar mass parameter;
  • the ratio of the Earth and Moon masses;
  • three angles of orientation with respect to the ICRF2 frame;
  • parameters of the rotation of Mars and topography of the inner planets;
  • masses of 31 asteroids and the mean densities of three taxonomic classes (C, S, and M) of asteroids, masses of the asteroid belt and TNO's ring;
  • the time delay from the solar corona (the parameters of its model were determined from observations for different solar conjunctions);
  • phase effects for outer planets; for Pluto it is the difference between the dynamical barycenter and light barycenter of the Pluto-Charon system.

The the Sun's quadrupole moment (J2_S = 2.3E-7) has been taken from [11] and [13].

The following masses were estimated in EPM2015: the solar mass parameter (GM_S) from ranging data, masses of the Earth and the Moon from ranging and LLR data, masses of 31 asteroids and and the mean densities of three taxonomic classes (C=1.270 g/cm^3, S=1.714 g/cm^3, and M=3.383 g/cm^3) of asteroids. These mean densities used for estimation of the masses other 255 asteroids whose diameters were taken from IRAS and MSE surveys. The masses of 13 asteroids (from 301) having satellites, as well as Vesta and Eros exploring by spacecraft, are known well and have been fixed (marked by * further). The masses of TNOs have been taken from online sources.

The other planet masses (except the Earth and the Moon) correspond to the planet masses of DE432 obtained by different authors from data spacecraft near planets and optical observations of natural satellites of planets.


The EPM2015 ephemerides have been fitted to 120000+ observations and normal points of different types, spanning 1913-2014, from classical meridian observations to modern planetary and spacecraft ranging. New planet data were added to the database of EPM2011 including the observations obtained in 2010-2014 for Odyssey, Mars Reconnaissance Orbiter (MRO), Mars Express (MEX) and Venus Express (VEX) spacecraft, the VLBI data (2011-2014) for VEX, Odyssey, MRO, Cassini. These new data were obtained through the courtesy of William Folkner (JPL) and Agnes Fienga (IMCCE) via private communications. Moreover, new CCD observations were added obtained in 2012-2013 at Flagstaff and TMO observatories, and new data for Pluto obtained in 1950-2013 at Brazilian Pico dos Dias observatory [8], and a new analysis of photographic plates taken at Lowell Observatory from 1931 to 1951 [9].

Ten-year ranging data (2004-2014) of Cassini [10] and MESSENGER [11] observations was added most recently. Due to these data, EPM2015 ephemerides for Saturn and Mercury are significantly more accurate as compared to EPM2011.

The ephemerides of the inner planets are based fully on radio-technical observations (mostly, measurements of time delays). The accuracy of observations of ranging has improved from about 6 km to several meters for today's spacecraft data. The ephemerides of the outer planets are mainly based on optical measurements taken since 1913, when they become accurate enough (0."5). In addition to optical observations of these planets, for the construction of ephemerides, positional observations of the satellites of the outer planets are used, as these observations are more precise and practically free from the phase effect, which is difficult to take into account. The modern optical data are CCD observations, and their accuracy reaches 0."05.

Radar observations have been reduced using relativistic corrections - the time delay of the propagation of radio signals in the gravitational fields of the Sun, Jupiter, Saturn (the Shapiro effect), and the reduction of observations from the coordinate time of the ephemerides to the proper time of the observer, and for the extra delay of electromagnetic signals in the Earth's troposphere and in the solar corona. In addition, the radar observations of surface of Mercury, Venus, and Mars were corrected for their topography.

The main reductions of optical observations of planets include the correction for the additional phase effect, the corrections for referring the observations to the ICRF reference frame, and the relativistic correction for light bending.

EPM2015 has been oriented to the ICRF2 with an accuracy better than 0.1 mas by including into the total solution 266 ICRF2-based VLBI measurements of spacecraft taken from 1989-2014 near Venus, Mars, and Saturn.

Common models recommended by IERS Convensions 2010 were used for the rotation of the Earth, displacement of stations and troposheric signal delay. Refinement of parameters was done basing on the lunar laser ranging (LLR) observational data in the time span of 1970-2013. Observations from the following stations were processed: Haleakala, McDonald/MLRS1/MLRS2, CERGA, Apache, and Matera. The first result of this new model was published in [18], the detailed description of the model has been purposed to [19].

Comparison to other ephemerides

Our experiments have shown that including the asteroid annulus and the TNO ring in the dynamical model enlarges slightly the estimation of the solar mass parameter, so our EPM2015 value of GM_S is somewhat greater than the DE432 value.

The lunar and planetary ephemerides EPM2015 obtained as a result is comparable to similar ephemerides worldwide [20], [13].

Constants and determined parameters

Selected parameters:

JD 2446000.5 (27.10.1984) date of the starting epoch of the integration
JD 2374000.5 (10.09.1787) left boundary date
JD 2530000.5 (22.10.2214) right boundary date
299792.458 km/s the speed of light
149597870700 m Astronomical Unit in meters
81.3005676344 the Earth-Moon mass ratio
2.0321572e-4 dynamical form factor of the Moon
6.310230e-4 lunar (C-A)/B (beta)
2.277332e-4 lunar (B-A)/C (gamma)
2.42e-2 lunar k2
9.7e-2 day lunar tide delay
2.3×10-7 solar oblateness
1.0 PPN parameter β
1.0 PPN parameter γ
0.0 the variation of the gravitational constant
R1 = 2.06 au, R2 = 3.27 au the radii of the asteroid annulus
43.00 au the radius of the TNO ring

Masses (in TDB scale) of barycenters of systems of planets with their satellites:

GM in 10-15 au3/day2 GM in km3/sec2
SUN 295912208310.4427 132712440053.08499
MOON 10931.8946 4902.80009
MERCURY 49124.8045 22031.78000
VENUS 724345.2333 324858.59200
EARTH 888769.2466 398600.43638
MARS 95495.4870 42828.37521
JUPITER 282534582.5972 126712764.13345
SATURN 84597060.7325 37940585.20000
URANUS 12920265.7963 5794556.46575
NEPTUNE 15243573.4789 6836527.10058
PLUTO 2175.0991 975.50118
Ceres-1 138.2384 61.99795
Pallas-2 30.5678 13.70925
Juno-3 3.8819 1.74099
*Vesta-4 38.547481 17.288245
Hebe-6 1.3134 0.58904
Iris-7 1.9416 0.87079
Flora-8 0.8986 0.40299
Metis-9 0.2554 0.11455
Hygiea-10 12.2371 5.48816
Parthenope-11 0.7420 0.33279
Egeria-13 1.3795 0.61867
Irene-14 0.8413 0.37732
Eunomia-15 5.0251 2.25369
Psyche-16 5.0874 2.28165
Fortune-19 1.2871 0.57725
Massalia-20 0.0920 0.04126
*Kalliope-22 1.1846 0.53127
Thalia-23 0.2537 0.11377
Amphitrite-29 1.8880 0.84672
Euphosyne-31 3.4477 1.54623
*Daphne-41 0.9390 0.42115
*Eugeria-45 0.8617 0.38644
Doris-48 2.6760 1.20016
Europa-52 1.1331 0.50817
Cybele-65 0.8908 0.39949
*Sylvia-87 2.1995 0.98646
Thisbe-88 1.7459 0.78302
*Antiope-90 0.1235 0.05540
*Minerva-93 0.5209 0.23360
Aegle-96 1.4588 0.65426
Artemis-105 0.2524 0.11322
*Camilla-107 1.6668 0.74752
*Hermione-121 0.7396 0.33171
*Elektra-130 0.9822 0.44050
Juewa-139 0.5138 0.23044
*Kleopatra-216 0.6905 0.30969
*Emma-283 0.2054 0.09211
Bamberga-324 1.6930 0.75930
Diotima-423 2.7845 1.24880
*Eros-433 0.00099490317 0.0004462
Patientia-451 0.3379 0.15153
Davida-511 4.7307 2.12167
Herculina-532 1.2655 0.56756
*Alauda-702 0.9014 0.40426
Interamnia-704 2.1583 0.96796
*Pulcova-762 0.2083 0.09344
Largest TNO
Eris-136199 2485.2618 1114.60477
Haumea-136108 596.1650 267.37157
Sedna-90372 148.8180 66.74277
Makemake-136472 446.4540 200.22830
Quaoar-50000 210.3840 94.35425
84522 223.2270 100.11415
Orcus-90482 94.3510 42.31509
Varuna-20000 55.0630 24.69498
Ixion-28978 44.6450 20.02265
307261 76.1670 34.15982
2006 QH_181 69.7520 31.28279
55565 61.0150 27.36436
208996 78.8740 35.37387
225088 415.8790 186.51585
Salacia-120347 64.8850 29.10000
Asteroid annulus (does not include 301 asteroids) 24.9431 11.18662
TNO ring (does not include 30 TNO) 11241.7569 5041.76897


EPM2015 They can be found on the FTP server of the IAA RAS: ftp://ftp.iaaras.ru/pub/epm/EPM2015. In addition to IAA binary and ASCII formats, a representation of ephemerides in SPK/PCK formats [4,5] has been added [1], which allows easy access to EPM2015 for users of CALCEPH and SPICE libraries.

Full list of provided software to access EPM ephemeris is given in the User manual.

An interactive web tool for obtaining ephemeris tables according to various versions of ephemerides (including EPM2015) is accessible at http://iaaras.ru/en/dept/ephemeris/online/.


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