# 4.1.5 Relativistic model

Author(s): Sergei Klioner

Section 4.1.3 gives an overview of the set of relativistic reference systems used in Gaia data processing. The Barycentric Celestial Reference System (BCRS) is used to model the motion of celestial bodies both inside and outside the solar-system. From the relativistic point of view, the Gaia catalogue is the model of Universe expressed in the BCRS. All astrometric parameters — parallaxes (distances), proper motions, positions — are defined in the BCRS coordinates. The goal of the relativistic model — called Gaia Relativity Model (GREM) — is to compute (predict) the observed CoMRS direction towards a source given its parameters in BCRS. The details of the model can be found in Klioner (2003, 2004); Klioner and Peip (2003); Klioner and Zschocke (2010); Zschocke and Klioner (2011).

Using the standard model of stellar motion described in Section 4.1.4 the astrometric parameters of a source are used to compute the coordinate BCRS direction from the location of Gaia at the moment of observation to the source $\boldsymbol{\bar{u}}(t)$. This direction has to be transformed into the observed direction $\boldsymbol{u}$ with respect to CoMRS.

The transformation essentially consists of two steps. First, the light propagation from the source to the location of Gaia is modelled in the BCRS in full details required to reach the required numerical accuracy of about 0.1 $\mu$as. In this process, the influence of the gravitational field of the solar-system is taken into account. This includes the gravitational light-bending due to the Sun, the major planets and the Moon. More deflecting bodies are readily available and can be used for special purposes (e.g. special processing of the data close to Jupiter). Both post-Newtonian and post-post-Newtonian effects are calculated. In this process special care was given to the relation between the analytical order of smallness of the effects and their numerical magnitude (Klioner and Zschocke 2010). In particular, only the so-called enhanced post-post-Newtonian effects, which can exceed 1 $\mu$as in some special observational configurations, are taken into account.

For observations close to the giant planets the effects of their quadrupole gravitational fields are taken into account in the post-Newtonian approximation. The effective computation of the rather complicated quadrupole deflection of light represents a separate problem (Zschocke and Klioner 2011). To speed up the computations of the model, the post-Newtonian formula for the quadrupole deflection was simplified as much as possible to give the required numerical accuracy of at least 0.1 $\mu$as for the realistic observational configuration in Gaia. Besides that, a very efficient criterion was found allowing one to decide if the actual calculation of the quadrupole deflection is needed. The criterion allows one to estimate the quadrupole deflection using only three multiplications.

The non-stationarity of the gravitational field (in particular, due to translational motion of the solar-system bodies) is also properly taken into account (Klioner 2003, 2003; Klioner and Peip 2003, and references therein).

No attempt is made to account for effects of the gravitational field outside the solar-system. This plays a role only in cases when its influence is variable on time scales comparable with the duration of observations, e.g. in various gravitational lensing phenomena.

The second step is to compute the observed direction $\boldsymbol{u}$ in CoMRS from the computed BCRS direction of light propagation at the location of Gaia at the moment of observation (Klioner (2003, Section 5) and Klioner (2004, Section VI)). Technically, the transformation represents a closed-form Lorentz transformation with the velocity of Gaia as seen by an fictitious observer that is co-located with Gaia at the moment of observation, but having zero BCRS velocity. One can show that that ‘observed’ velocity $\boldsymbol{v}$ is the BCRS velocity of Gaia $\boldsymbol{v}_{\text{Gaia}}$ multiplied by a factor depending on the gravitational potential at the location of Gaia.

Besides astrometric parameters of the sources, GREM requires several kinds of auxiliary data:

• Gaia spatial ephemeris (BCRS position and velocity of Gaia for any moment of time covered by observations; Section 4.2.3);

• Gaia time ephemeris (the relation between the readings of the Gaia on-board clock and TCB; Section 4.1.6);

• Solar-system ephemeris; the INPOP10e ephemeris (Fienga et al. 2016) parametrized by TCB is used in the Gaia data processing (see Section 4.2.1);

• Various astronomical and physical constants; this includes the TCB-compatible constants used in INPOP10e (masses of all major bodies of the solar-system, etc.).