4.3.1 Overview

Author(s): Lennart Lindegren

The astrometric principles for Gaia were outlined already in the Hipparcos Catalogue (ESA 1997, Vol. 3, Ch. 23) where, based on the accumulated experience of the Hipparcos mission and the general principle of a global astrometric data analysis was succinctly formulated as the minimization problems (see Lindegren et al. (2012)):

 $\min_{\boldsymbol{s},\,\boldsymbol{n}}~{}\lVert\boldsymbol{f}^{\text{obs}}-% \boldsymbol{f}^{\text{calc}}(\boldsymbol{s},\boldsymbol{n})\rVert_{\,\mathcal{% M}}\,.$ (4.90)

Here $\boldsymbol{s}$ is the vector of unknowns (parameters) describing the barycentric motions of the ensemble of sources used in the astrometric solution, and $\boldsymbol{n}$ is a vector of ‘nuisance parameters’ describing the instrument and other incidental factors which are not of direct interest for the astronomical problem but are nevertheless required for realistic modelling of the data. The observations are represented by the vector $\boldsymbol{f}^{\text{obs}}$ which could for example contain the measured detector coordinates of all the stellar images at specific times. $\boldsymbol{f}^{\text{calc}}(\boldsymbol{s},\boldsymbol{n})$ is the observation model, e.g., the expected detector coordinates calculated as functions of the astrometric and nuisance parameters. The norm is calculated in a metric $\mathcal{M}$ defined by the statistics of the data; in practise the minimization will correspond to a weighted least-squares solution with due consideration of robustness issues. The statistical weight $W_{l}=w_{l}/(\sigma_{l}^{2}+\epsilon_{l}^{2})$ of individual observations $l$ is composed of a contribution from the formal standard uncertainty of the observation $\sigma_{l}$ and the excess noise $\epsilon_{l}$ represents modelling errors and should ideally be zero. However, it is unavoidable that some sources do not behave exactly according to the adopted astrometric model (Section 4.3.3), or that the attitude (Section 4.3.5) sometimes cannot be represented by the model used to sufficient accuracy.

The excess noise term $\epsilon_{l}$ is introduced to allow these cases to be handled in a reasonable way, i.e., by effectively reducing the statistical weight of the observations affected. It should be noted that these modelling errors are assumed to affect all the observations of a particular star, or all the observations in a given time interval. (By contrast, the down-weighting factor $w_{l}$ is intended to take care of isolated outliers, for example due to a cosmic-ray hit in one of the CCD samples.) This is reflected in the way the excess noise is modelled as the sum of two components,

 $\epsilon_{l}^{2}=\epsilon_{i}^{2}+\epsilon^{2}_{a}(t_{l})\,,$ (4.91)

where $\epsilon_{i}$ is the excess noise associated with source $i$ (if $l\in i$, that is, $l$ is an observation of source $i$), and $\epsilon_{a}(t)$ is the excess attitude noise, being a function of time. For a ‘good’ primary source, we should have $\epsilon_{i}=0$, and for a ‘good’ stretch of attitude data we may have $\epsilon_{a}(t)=0$. Calibration modelling errors are not explicitly introduced in Equation 4.91, but could be regarded as a more or less constant part of the excess attitude noise. The estimation of $\epsilon_{i}$ is described in Section 4.3.3, and the estimation of $\epsilon_{a}(t)$ in Section 4.3.5.