# 4.3.3 Astrometric source model

Author(s): Lennart Lindegren

The astrometric model is a recipe for calculating the proper direction $\boldsymbol{u}_{i}(t)$ to a source ($i$) at an arbitrary time of observation ($t$) in terms of its astrometric parameters $\boldsymbol{s}_{i}$ and various auxiliary data, assumed to be known with sufficient accuracy. The auxiliary data include an accurate barycentric ephemeris of the Gaia satellite, $\boldsymbol{b}_{\text{G}}(t)$, including its coordinate velocity $\text{d}\boldsymbol{b}_{\text{G}}/\text{d}t$, and ephemerides of all other relevant solar-system bodies. The details of the model have been outlined in Section 3.2 of Lindegren et al. (2012) or Section 4.1.4 and only a short introduction is given here.

As explained in Section 4.1.3, the astrometric parameters refer to the ICRS and the time coordinate used is TCB. The reference epoch $t_{\text{ep}}$ is preferably chosen to be near the mid-time of the mission in order to minimize statistical correlations between the position and proper motion parameters.

The transformation between the kinematic and the astrometric parameters is non-trivial (Klioner 2003a), mainly as a consequence of the practical need to neglect most of the light-propagation time $t-t_{*}$ between the emission of the light at the source ($t_{*}$) and its reception at Gaia ($t$). This interval is typically many years and its value, and rate of change (which depends on the radial velocity of the source), will in general not be known with sufficient accuracy to allow modelling of the motion of the source directly in terms of its kinematic parameters according to Equation 4.1. The proper motion components $\mu_{\alpha*i}$, $\mu_{\delta i}$ and radial velocity $v_{ri}$ correspond to the ‘apparent’ quantities discussed by in Sect. 8 of Klioner (2003a).

The coordinate direction to the source at time $t$ is calculated with the same ‘standard model’ as was used for the reduction of the Hipparcos observations (ESA (1997), Vol. 1,  Sect. 1.2.8), namely

 $\boldsymbol{\bar{u}}_{i}(t)=\bigl{\langle}\boldsymbol{r}_{i}+(t_{\text{B}}-t_{% \text{ep}})(\boldsymbol{p}_{i}\mu_{\alpha*i}+\boldsymbol{q}_{i}\mu_{\delta i}+% \boldsymbol{r}_{i}\mu_{ri})-\varpi_{i}\boldsymbol{b}_{\text{G}}(t)/A_{\text{u}% }\bigr{\rangle}$ (4.100)

where the angular brackets signify vector length normalization, and $[\boldsymbol{p}_{i}~{}\boldsymbol{q}_{i}~{}\boldsymbol{r}_{i}]$ is the ‘normal triad’ of the source with respect to the ICRS (Murray 1983). In this triad, $\boldsymbol{r}_{i}$ is the barycentric coordinate direction to the source at time $t_{\text{ep}}$, $\boldsymbol{p}_{i}=\langle\boldsymbol{Z}\times\boldsymbol{r}_{i}\rangle$, and $\boldsymbol{q}_{i}=\boldsymbol{r}_{i}\times\boldsymbol{p}_{i}$. The components of these unit vectors in the ICRS are given by the columns of the matrix

 $\mathsf{C}^{\prime}[\boldsymbol{p}_{i}~{}\boldsymbol{q}_{i}~{}\boldsymbol{r}_{% i}]=\begin{bmatrix}-\sin\alpha_{i}&~{}-\sin\delta_{i}\cos\alpha_{i}&~{}\cos% \delta_{i}\cos\alpha_{i}\\ \phantom{-}\cos\alpha_{i}&~{}-\sin\delta_{i}\sin\alpha_{i}&~{}\cos\delta_{i}% \sin\alpha_{i}\\ 0&~{}\cos\delta_{i}&~{}\sin\delta_{i}\end{bmatrix}.$ (4.101)

$\boldsymbol{b}_{\text{G}}(t)$ is the barycentric position of Gaia at the time of observation, and $A_{\text{u}}$ the astronomical unit. $t_{\text{B}}$ is the barycentric time obtained by correcting the time of observation for the Römer delay; to sufficient accuracy it is given by

 $t_{\text{B}}=t+\boldsymbol{r}_{i}^{\prime}\boldsymbol{b}_{\text{G}}(t)/c\,,$ (4.102)

where $c$ is the speed of light. See Section 3.2 of Lindegren et al. (2012) for further details.

The iterative updating of the sources is described in Section 5.1 of Lindegren et al. (2012). There the method of identifying potential outliers and estimating the excess source noise $\epsilon_{i}$ is also outlined together with the calculation of the partial derivatives of the coordinate direction with respect to the source parameters.

The perspective acceleration modelled by $\mu_{r}=v_{r}\varpi/A_{\text{u}}$ in Equation 4.100 is totally negligible for the vast majority of sources, and $\mu_{r}=0$ (at the reference epoch) is therefore normally assumed, equivalent $v_{r}=0$. For Gaia EDR3 we used the radial velocity from the RVS instrument (as given in Gaia DR2), complemented by values from the literature for the 38 sources listed in Table 4.3. The accumulated effect over a time interval $T$ is $\Delta=|v_{r}|\,\mu\varpi T^{2}/A_{\text{u}}$, where $\mu=(\mu_{\alpha*}^{2}+\mu_{\delta}^{2})^{1/2}$ is the total proper motion and $A_{\text{u}}=977792221.680789$ mas yr km s${}^{-1}$ is the astronomical unit. The largest effect, $\Delta\simeq 4.9$ mas (for $T=2.76$ yr), is obtained for Barnard’s star (Gaia EDR3 4472832130942575872).
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