EPM2017 and EPM2017H

Download EPM2017

While IAA binary and ASCII formats, originally used for EPM, are still supported, users are encouraged to use the SPK and PCK formats, also known as the SPICE formats [12, 13].

Download EPM2017H

Long version of EPM2017 (10107 BC — AD 3036).

EPM User Manual

List of software to access the EPM ephemeris in SPICE and other formats.

Online Ephemeris Service

Interactive web tool for obtaining ephemeris tables according to various EPM versions, including EPM2017.

Overview

EPM2017 ephemerides contain coordinates and velocities of the Sun, the Moon, eight major planets, Pluto, three largest asteroids (Ceres, Pallas, Vesta) and 4 TNO (Eris, Haumea, Makemake, Sedna), as well as lunar libration and TT-TDB. EPM2017 covers a time span of more than 400 years (1787–2214).

EPM2017H ephemerides are the longer version of EPM2017, covering a time span of more than 13100 years (10107 BC — AD 3036, H stands for “Holocene”). Within the EPM2017 time span, EPM2017 and EPM2017H are practically equal with the only exception of the Moon.

Dynamical model

The dynamical model of EPM is based on the Parameterized Post-Newtonian N-body metric for General Relativity in the barycentric coordinate system (BCRS) and the TDB time scale.

The motion of the Sun, the planets (including Pluto), and the Moon (as point-masses) obeys the Einstein–Infeld–Hoffmann relativistic equations, with additional perturbations from: solar oblateness, 301 largest asteroids and 30 largest trans-Neptunian objects (TNO), as well as two discrete annuli: the first for the asteroid belt, and the second for the Kuiper belt.

For improvement of the planetary part of EPM2017, about 270 parameters were determined:

• orbital elements of the planets and the 18 satellites of the outer planets;
• the value of the solar mass parameter (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 \(GM_\mathrm{Sun}\) in SI units);
• the ratio of the Earth and Moon masses;
• the Sun's quadrupole moment (J2) (has been defined from the MESSENGER data);
• the three angles of orientation with respect to the ICRF2 frame;
• the parameters of the rotation of Mars and topography of the inner planets;
• the masses of 30 asteroids and the mean densities of three taxonomic classes (C, S, and M) of asteroids, the masses of the asteroid and TNO rings;
• the total masses of the main asteroid belt and Kuiper belt;
• 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 model of orbital and rotation motion of the Moon in EPM [1] is based on the equations used in the JPL DE430 ephemeris [2] 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.

Earth Gravitation Model 2008 (EGM2008 [3]) was taken for the gravitational model of the Earth, while GL660B [4] was used for the Moon. 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 1970–2016. Observations from the following stations were processed: Haleakala, McDonald/MLRS1/MLRS2, OCA, Apache, and Matera.

For the TT−TDB conversion, the differential equation from [5] has been used, and TT−TDB was obtained by numerical integration.

The parameters of the lunar and planetary parts of EPM2017 are in agreement with each other.

EPM2017 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 [6].

Changes in the “Holocene” version

The dynamical model used during the production of EPM2017H differs in two pieces. First, EPM2017H uses the model of the precession of the Earth valid over thousands of years [7]. Second, the lunar model of EPM2017H does not have friction between crust and core, to avoid exponential growing of the angular velocity of the lunar core in the past. That decision is similar to what has been done for the DE431 ephemeris model as compared to DE430. The lunar parameters of the solution were then re-fit to the EPM2017H model.

Changes since EPM2015

The most important changes in EPM2017 since the previous version of EPM are:

• inclusion of the Lense–Thirring acceleration into model;
• improved relativistic barycentre definition;
• modelling the acceleration of the Sun as a regular body;
• the new models of the asteroid belt and the Kuiper belt;
• recent estimates of masses and orbits of asteroids;
• usage of recent observations of spacecraft and LLR normal points that appeared after EPM2015 was released;
• the “Holocene” version of ephemeris, EPM2017H.

Lense–Thirring acceleration

where:

• \(\mathbf{r}_i\) is the barycentric position of the \(i\)-th body

• \(\mathbf{r}_{iS}\) is the heliocentric position of the \(i\)-th body

• \(r_{ij} = |\mathbf{r_j} - \mathbf{r_i}|\)

• \(\mathbf{z} = (0, 0, 1)\)

• \(R_\mathrm{Sun}\) — rotation matrix from \(\mathbf{z}\) to the Sun's pole. The following value is used: \(R_\mathrm{Sun} = \mathrm{R}_z(16^{\circ}.13)\,\mathrm{R}_x(26^{\circ}.13)\)

• \(G\) is the gravitational constant

• \(S_\mathrm{Sun}\) is the angular momentum of the Sun. Following the decision from [8], we take \(S_\mathrm{Sun} = 190 \cdot 10^{39}\ \mathrm{kg\ m}^2/\mathrm{s}\)

Relativistic barycentre definition

The relativistic barycentre of point-masses \({\mu_i}\), where \(\mu_{i} = GM_i\) is the \(i\)-th body's gravitational parameter, is defined as

where \(\mathbf{r}_i\) is the position of the \(i\)-th body, while \(\mu_i^{*}\) is the body's relativistic gravitational parameter:

Initial positions and velocities of the bodies are equally adjusted so that the position of the center of mass and the momentum are zero:

In previous versions of EPM, the \(\dot{\mu_i^{*}}\) was neglected.

Modelling the acceleration of the Sun as a regular body

In earlier versions of the EPM, the Sun was explicitly placed to keep the barycenter at the origin. In EPM2017, the Sun obeys the Einstein–Infeld–Hoffmann equations of motion as all the other bodies. Similar approach has been taken in DE ephemeris starting from DE430 [2] and INPOP ephemeris starting from the first release INPOP06 [9].

The perturbations from solar oblateness and Lense–Thirring effect act not only on planets, but on Sun, too, following Newton's third law.

The mutual influence of the astroid belt and the Kuiper belt to the orbit of the Sun, neglected in previous versions of EPM, is properly taken into account in EPM2017.

Numerical round-off errors in the integration cause the barycenter, initially placed at the origin, to drift at a slow pace. In the timespan of 1900–2020, the displacement of the barycenter from the origin is less than 0.03 mm.

The new models of the asteroid belt and the Kuiper belt

The largest 301 asteroids and 31 TNO (including Pluto) are directly included in integration in EPM ephemeris. However, in order to take into account gravitational influence of other smaller asteroids and TNO, their perturbations were modeled as perturbations from uniform rings.

The first model [9, 10] consisted of a one-dimensional asteroid ring with estimated mass and radius, and a one-dimensional TNO ring of radius 43 au with estimated mass. Both rings lie in the ecliptic plane. Due to large correlation between radius and mass of the asteroid ring, later it was replaced with a “two-dimensional ring” (annulus) [5].

But these models did not quite reflected the mutual interactions between moving bodies of the belts and ordinary point-masses (Sun and planets). To remedy this shortcoming, a new discrete rotating models have been has been proposed.

In EPM2017, the asteroid and the Kuiper belt are modeled by a system of uniformly distributed point-masses lying in the plane of the ecliptic on three circular lines each. That also allows to model their influence to the orbit of the Sun (unlike previous versions of EPM) and planets.

The asteroid belt is approximated with 180 points with equal masses (60 points on each circle). The two border circles (inner and outer) lie on R1 = 2.06 au and R2 = 3.27 au that correspond to the average distances for orbital resonances 1:2 and 2:1 with the motion of Jupiter. The middle cicle lies on Rmid = 2.67 au.

The Kuiper belt is approximated with 160 points with equal masses. The two border circles, 40 points each, lie on R1=39.4 au and R2=48.7 au—the average distances of orbital resonances 3:2 and 2:1 with the motion of Neptune. The middle circle (80 points), approximating the dense core of the Kuiper belt, lies on Rmid = 44 au.

The belt points don't interact each other, only with other bodies of the Solar system. At the inital moment of the integration, the velocities of the points correspond to a circular motion.

The total masses of the belts were the fitted parameters that were determined from processing and fitting the spacecraft observational data.

Recent masses and orbits of some asteroids

Some asteroid data were updated. In particular, the new mass value was obtained during the approach of the Dawn spacecraft to Ceres [10]. The masses of 30 large asteroids (not including 13 double and multiple asteroids) were improved from observations of different spacecraft (their values are given below). Masses of other 255 asteroids were calculated from their diameters and densities.

Recent aditional observations

The planetary part of EPM2017 ephemerides has been fitted to about 800000 observations of different types, spanning 1913–2015, from classical meridian observations to modern planetary and spacecraft ranging.

However, spacecraft ranging near planets are correlated with each other on each tracking pass, so there is effectively only one independent range point for each pass. Based on this, Dr. Folkner formed normal points of JPL spacecraft ranging; the normal points for other spacecraft ranging were formed by authors. Moreover, the points near solar conjunction were deleted due to large solar plasma effects, as Dr. Folkner did with the MESSENGER data. The new radar data were obtained due to the courtesy of William Folkner (JPL) and Agnes Fienga (IMCCE) via private communication, a NASA JPL webpage “Observational Data for Planets and Planetary Satellites”, and a Geoazur “Astrometric Planetary Data” webpage.

The four-year MESSENGER (2011–2015) [8] observations were added in EPM2017. As a result, the Mercury ephemeris has become significantly more accurate than it was in EPM2015.

New LLR data that appeared after EPM2015 release (till the end of 2016) was added into EPM2017 solution—most notably, the new high quality infrared data from the OCA observatory.

Software

Processing of observations, fitting of parameters, and numerical integration of EPM2017 were performed with a software package ERA-8 (Ephemeris Research in Astronomy) developed at IAA RAS [11].

Constants and determined parameters

Selected constants:

The following masses were estimated in EPM2017: the solar mass parameter (\(GM_\mathrm{Sun}\)) from ranging data, masses of the Earth and the Moon from ranging and LLR data, masses of 30 asteroids and the mean densities of three taxonomic classes (C = 1.270 g/cm³, S = 1.714 g/cm³, and M = 3.383 g/cm³) of asteroids. Those mean densities were taken to determine the masses of 255 asteroids whose diameters were taken from IRAS and MSE surveys. The masses of 13 asteroids (from 301) having satellites, as well as Vesta, Ceres, and Eros exploring by spacecraft, are known well and have been fixed (marked by * in the table below).

The masses of planets (except the Earth) and TNO in EPM2017 are equal to their EPM2015 counterparts.

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

References

1. Dmitry A. Pavlov, James G. Williams, Vladimir V. Suvorkin: Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models // Celestial Mechanics and Dynamical Astronomy 126(1), 61–88 (2016)

2. William M. Folkner, James G. Williams, Dale H. Boggs, Ryan S. Park, and Petr Kuchynka: The Planetary and Lunar Ephemeris DE430 and DE431 // JPL IPN Progress Report, 42-196 (2014)

3. Pavlis N. K., Holmes S. A., Kenyon S. C., Factor J. K.: The development and evaluation of the Earth Gravitational Model 2008 (EGM2008) // J. Geophys. Res.: Solid Earth, Vol. 117, Issue B4 (2012)

4. Konopliv A. S., Park R. S., Yuan D. N., Asmar S. W., Watkins M. M., Williams J. G., Fahnestock E., Kruizinga G., Paik M., Strekalov D., Harvey N., Smith D. E., Zuber M. T.: The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission // J. Geophys. Res: Planets 118(7), 1415–1434 (2013)

5. Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and Data Analysis / Edited by S.A. Klioner, P.K. Seidelmann, and M.H. Soffel // International Astronomical Union Symposium, Vol. 261. Cambridge, UK: Cambridge University Press, 2010.

6. E. V. Pitjeva, N. P. Pitjev: Development of planetary ephemerides EPM and their applications // Celestial Mechanics and Dynamical Astronomy 119(3), 237–256 (2014)

7. J. Vondrák, N. Capitaine, and P. Wallace: New precession expressions, valid for long time intervals // A&A 534, A22 (2011)

8. Park R. S., Folkner W. M., Konopliv A. S., Williams J. G., Smith D. E., Zuber M. T.: Precession of Mercury's Perihelion from Ranging to the MESSENGER Spacecraft // The Astronomical Journal, 2017, Volume 153, Issue 3, article id. 121, 7 pp.

9. A. Fienga, H. Manche, J. Laskar and M. Gastineau: INPOP06: a new numerical planetary ephemeris // A&A, Volume 477, Number 1, January I 2008.

10. R. S. Park, A. S. Konopliv, B. G. Bills, N. Rambaux, J. C. Castillo-Rogez, C. A. Raymond, A. T. Vaughan, A. I. Ermakov, M. T. Zuber, R. R. Fu, M. J. Toplis, C. T. Russell, A. Nathues, F. Preusker: A partially differentiated interior for (1) Ceres deduced from its gravity field and shape // Nature (Letter), 22 September 2016, Vol. 537, 515–428.

11. D. A. Pavlov, V. I. Skripcnichenko: Rework of the ERA software system: ERA-8 / Proceedings of the Journées 2014 “Systèmes de Référence Spatio-Temporels” // Z. Malkin and N. Capitaine (eds), Pulkovo observatory, 243–246 (2015)

12. Acton C., Bachman N., Folkner W., Hilton J.: SPICE as an IAU Recommendation for Planetary Ephemerides // IAU General Assembly, Meeting #29, id.2240327 (2015).

13. J. L. Hilton, C. Acton, J.-E. Arlot, S. A. Bell, N. Capitaine, A. Fienga, W. M. Folkner, M. Gastineau, D. Pavlov, E. V. Pitjeva, V. I. Skripnichenko, P. Wallace: Report of the IAU Commission Working Group on Standardizing Access to Ephemerides and File Format Specification: Update September 2014 // Proceedings of the Journées 2014 "Systèmes de Référence Spatio-Temporels" Z. Malkin and N. Capitaine (eds), Pulkovo observatory, 254–255 (2015)