Author(s): Francois Mignard
The processing of the Gaia observations requires not only the knowledge of the position and velocity vectors of the spacecraft with respect to the barycentre of the solar-system, which is computed bu DPAC from the geocentric data provided regularly by ESOC from Gaia tracking. There are also several other important usages of the ephemeris for example to compute the relativistic light deflection by the major planets to the utmost accuracy for each observation. Likewise the identification and the processing of the observations of the minor planets requires that a computed state vector (position and velocity vectors) should be available for all known asteroids. This is derived by a numerical integration using initial conditions and the gravitational interaction with the Sun, the eight major planets and the Moon. Therefore access to solar-system ephemeris was a requirement of the DPAC system, with accuracy constraints and efficient access. To achieve full consistency thorough the processing it was also important that a unique source of ephemeris should be used by the different DPAC groups, in particular for the astrometric processing (single stars, multiple systems, solar-system objects) where the accuracy requirements are the most stringent. For specific analysis, relativistic tests and transits predictions, It was also desirable to have a quick access to the ephemeris of the natural satellites bright enough to be seen by Gaia.
The DPAC decided at an early stage, actually even before DPAC was formally formed, that planetary ephemeris will be obtained from IMCCE in Paris, and later on, INPOP (Intégration Numérique Planétaire de l’Observatoire de Paris) with position and velocity vectors given in the BCRS with TCB as independent time variable was chosen. The access is uniform for all the DPAC Coordination Units through the CU1-provided GaiaTools library. A TDB-compatible version is used at ESOC for the Gaia orbit tracking and modelling, so that there is a no risk of systematic differences that may have arisen from the use of a different ephemeris in the data processing and in the orbit reconstruction.
The accuracy requirements for the Gaia ephemeris are very constraining for the Earth as any error in the Earth ephemeris will propagate into a similar error in the reconstructed orbit of Gaia. Quantitatively the requirements for Gaia were:
Velocity in the BCRS to 2.5 mm s${}^{-1}$ for each component (1–$\sigma $ error) and no systematic error over the mission length larger than 1 mm s${}^{-1}$,
Position in the BCRS to 0.15 km over each component(1–$\sigma $ error).
This implied that the BCRS ephemeris of the Earth is better that these requirements. With the exclusion of the Earth, the velocity of the planets is not critical and could be taken in the 0.1 – 1 m s${}^{-1}$ range. A good ephemeris of the Moon is also required for dynamical modelling. It was known that for outer planets this accuracy could not be guaranteed and DPAC expected to receive the state-of-the-art ephemeris in these cases with an estimation of the external accuracy.
Symbol | Meaning | Value | Unit |
EMRAT | Earth to Moon mass ratio | 8.1300569999999990E+01 | – |
AU | Astronomical Unit | 1.4959787070000000E+08 | km |
CLIGHT | Velocity of light in vacuum | 2.9979245800000000E+05 | km s${}^{-1}$ |
GM–Sun | Sun $GM$ | 1.3271244210789468E+11 | km${}^{3}$ s${}^{-2}$ |
GM–Mer | Mercury $GM$ | 2.2032080834196266E+04 | km${}^{3}$ s${}^{-2}$ |
GM–Ven | Venus $GM$ | 3.2485859679756965E+05 | km${}^{3}$ s${}^{-2}$ |
GM–EMB | Earth–Moon $GM$ | 4.0350325101102696E+05 | km${}^{3}$ s${}^{-2}$ |
GM–Mar | Mars $GM$ | 4.2828375886337897E+04 | km${}^{3}$ s${}^{-2}$ |
GM–Jup | Jupiter $GM$ | 1.2671276453465731E+08 | km${}^{3}$ s${}^{-2}$ |
GM–Sat | Saturn $GM$ | 3.7940585442640103E+07 | km${}^{3}$ s${}^{-2}$ |
GM–Ura | Uranus $GM$ | 5.7945490985393422E+06 | km${}^{3}$ s${}^{-2}$ |
GM–Nep | Neptune $GM$ | 6.8365271283644792E+06 | km${}^{3}$ s${}^{-2}$ |
R–Sun | Solar radius | 6.9600001079161780E+05 | km |
R–Earth | Earth radius | 6.3781366988942700E+03 | km |
R–Moon | Moon radius | 1.7380000269480340E+03 | km |
J2–Sun | Solar J2 | 1.8000000000000000E-07 | – |
J2–Earth | Earth J2 | 1.0826222418469980E-03 | – |
$\gamma $ | PPN $\gamma $ | 1.0000000000000000E+00 | – |
$\beta $ | PPN $\beta $ | 1.0000000000000000E+00 | – |
INPOP is a consistent numerical integration of the equations of motion in the solar-system, based in a dynamical model and a set of physical constants and initial conditions, eventually adjusted on observations. The dynamical model used to build INPOP follows the recommendations of the International Astronomical Union regarding the definition of the reference frame, the relativistic metric and the associates relativistic equations of motion, the compatibility between the time scales TT and TDB. The basic scale factor comes from a fixed value of the astronomical unit instead of the traditional use of a value for $G{M}_{\odot}$. The latter is fitted with the other free parameters, the most important of which are listed in Table 3.1. The integration is adjusted to a large set of observations ranging from classical optical astrometry to range or VLBI tracking of planetary spacecrafts or Lunar Laser ranging on the Moon. There is no single way to estimate the accuracy of an ephemeris, which remains a numerical model of a complex system, but comparisons between independent ephemeris is a first approach and was used for INPOP10e. Typically the maximum difference between this version of INPOP and the JPL DE423 in the period is sub-km for barycentric position of the inner planets, of a few km for Jupiter and Saturn and 1000 km for Uranus and Neptune. However this provides the floor error since the observational data and the dynamical models are very similar in both ephemeris and the external error is very likely larger.
The link between INPOP ephemerides and the ICRF is realised by the use of VLBI differential observations of spacecraft relative to ICRF sources. This method gives milliarcsecond (mas) positions of a spacecraft orbiting a planet directly in the ICRF. When combined with spacecraft navigation, positions of planets can be deduced relatively to the ICRF sources. It is considered that the link between the INPOP10e reference frame and the ICRF has an accuracy of about 1 mas, which may mean about 1 km for the position of Gaia relative to the BCRS and 0.1 mm s${}^{-1}$ for the velocity. Actually only Gaia with its accurate observations of QSOs and solar-system objects will be in position to tell of the difference between ICRF and the INPOP frame.
During the development phase of the data processing, a first version of the INPOP ephemeris was delivered to DPAC by IMCCE in 2007, referred to as INPOP06b, and used to generate simulations, validate the DPAC implementation and its access through the GaiaTools. The operational delivery based on INPOP10e (Fienga et al. (2011)) was provided in 2012. It includes the barycentric ephemeris (position and velocity) for the Sun, the planets from Mercury to Neptune, the Earth–Moon barycentre and the Earth–Moon vector. The delivery included also an ephemeris for TT–TDB derived with a numerical integration consistent with the planetary motions.
Overall consistency with specifications was checked independently by F. Mignard and S. Klioner with a last digit reproducibility of the test data before the files were sent to ESAC for integration in the DPAC framework. Similar tests were successfully conducted as well on the DPAC implemented version in Java against two independent implementations in Fortran and Java.
The ephemeris is represented in the form of coefficients of a Chebyshev expansion, with order and granule sizes adjusted to meet Gaia requirement in terms of precision and efficiency. Given the extensive number of accesses to the solar-system ephemeris during the iterative astrometric solution, DPAC has favoured computational efficiency with polynomials of low degree (between 4 and 8, according to planets) at the expense of a larger storage volume and smaller time granule. For inner planets the granule size is of one day and increases to 2 or 4 days for Jupiter, Uranus and Neptune. The time coverage was designed to extend beyond any foreseeable mission delay and extension up to 2032, with the plan that no ephemeris change will be done during the data processing, if not justified by new accuracy requirements.