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).
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 , which is a complete rework of the original program package ERA-7 . ERA comprises a domain-specific language SLON tailored for astronomical tasks . 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 );
- 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  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 . 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  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  and .
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 , and a new analysis of photographic plates taken at Lowell Observatory from 1931 to 1951 .
Ten-year ranging data (2004-2014) of Cassini  and MESSENGER  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 0.2 mas (3σ) 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 , the detailed description of the model has been purposed to .
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 , .
Constants and determined 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)|
|9.7e-2 day||lunar tide delay|
|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|
|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 , 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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