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# EPM2017 and EPM2017H

## 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 the timespan of more than 400 years (1787–2214).

EPM2017H ephemerides are the longer version of EPM2017, covering the timespan of more than 13100 years (10107 BC – AD 3036, H stands for “Holocene”). Within the EPM2017 timespan, 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 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

$$\ddot{\mathbf{r}}_i^{\mathrm{LT}} = \frac{2}{c^2}GS_\mathrm{Sun}\frac{1}{r_{iS}^3}R_\mathrm{Sun}\left(\mathbf{\dot{r}}_{iS}\times\mathbf{z} + 3\frac{\mathbf{z}\cdot\mathbf{r}_{iS}}{r^2}\mathbf{r}_{iS}\times\mathbf{\dot{r}}_{iS}\right)$$

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 $$S_\mathrm{Sun} = 190\times 10^{39}\ \mathrm{kg\ m}^2/\mathrm{s}$$

### 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

$$\mathbf{b} = \frac{\sum_{i} \mu_i^{*} \mathbf{r}_i}{\sum_{i}\mu_i^{*}}$$

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

$$\mu_i^{*} = \mu_i\left(1+\frac{1}{2c^2}\dot{r_i}^2 - \frac{1}{2c^2}\sum_{j\neq i}\frac{\mu_j}{r_\mathrm{ij}}\right)$$

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

$$\begin{cases} \sum_{i} \mu_i^{*} \mathbf{r}_i = \mathbf{0} \\ \sum_{i} \dot{\mu_i^{*}} \mathbf{r}_i + \mu_i^{*} \mathbf{\dot{r}}_i = \mathbf{0} \end{cases}$$

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.

### 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.

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. Folker 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. Folker 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].

## Availability

EPM2017 can be found on the FTP server of the IAA RAS. 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]. The list of provided software to access the EPM ephemeris in SPICE and other formats is given in the User manual.

IAA RAS provides an interactive web tool for obtaining ephemeris tables according to various versions of ephemerides (including EPM2017).

## Constants and determined parameters

Selected constants:

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 the Astronomical Unit in meters
81.300568886 the Earth-Moon mass ratio
2.0321572×10-4 dynamical form factor of the Moon
6.310179×10-4 lunar (C-A)/B (beta)
2.277292×10-4 lunar (B-A)/C (gamma)
2.41×10-2 lunar k2
9.3×10-2 day lunar tide delay
2.366×10-7 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
R1 = 39.4 au, R2 = 48.7 au the radii of the TNO annulus

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:

GM in 10-15 au3/day2 GM in km3/sec2
Sun 295912208289.8587 132712440043.85333
Moon 10931.8945 4902.80008
Mercury 49124.8045 22031.78000
Venus 724345.2333 324858.59200
Earth 888769.2447 398600.43552
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
Asteroids
*Ceres-1 139.6418 62.62736
Pallas-2 30.3058 13.59173
Juno-3 4.1238 1.84947
*Vesta-4 38.547481 17.288245
Hebe-6 1.3284 0.59578
Iris-7 1.8658 0.83676
Flora-8 1.0298 0.46185
Metis-9 0.4020 0.18028
Hygiea-10 11.7734 5.28020
Parthenope-11 0.8442 0.37861
Egeria-13 1.8018 0.80807
Irene-14 1.4314 0.64198
Eunomia-15 4.9508 2.22034
Psyche-16 4.7941 2.15009
Fortune-19 1.3907 0.62372
Massalia-20 0.4277 0.19180
*Kalliope-22 1.1846 0.53127
Thalia-23 0.2537 0.11377
Amphitrite-29 0.9840 0.44129
Euphosyne-31 4.2720 1.91592
*Daphne-41 0.9390 0.42115
*Eugeria-45 0.8617 0.38644
Doris-48 2.4123 1.08189
Europa-52 2.7672 1.24104
Cybele-65 1.5472 0.69388
*Sylvia-87 2.1995 0.98646
Thisbe-88 2.6142 1.17245
*Antiope-90 0.1235 0.05540
*Minerva-93 0.5209 0.23360
Aegle-96 1.6885 0.75726
Artemis-105 0.3689 0.16543
*Camilla-107 1.6668 0.74752
*Hermione-121 0.7396 0.33171
*Elektra-130 0.9822 0.44050
Juewa-139 0.7809 0.35022
*Kleopatra-216 0.6905 0.30969
*Emma-283 0.2054 0.09211
Bamberga-324 1.6437 0.73718
Diotima-423 1.9738 0.88521
*Eros-433 0.00099490317 0.0004462
Patientia-451 2.4679 1.10681
Davida-511 4.6674 2.12167
Herculina-532 1.9482 0.87375
*Alauda-702 0.9014 0.40426
Interamnia-704 5.3123 2.38250
*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
Other
Asteroid annulus (does not include 301 asteroids) 15.0024 6.72837
TNO ring (does not include 30 TNO) 9846.5489 4416.03791

## 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)

Elena Pitjeva, e-mail evp@iaaras.ru,
Dmitry Pavlov, e-mail dpavlov@iaaras.ru,
Institute of Applied Astronomy RAS, St. Petersburg, Russia
7 November 2017.