3.3.1 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)):

min𝒔,𝒏𝒇obs-𝒇calc(𝒔,𝒏). (3.89)

Here 𝒔 is the vector of unknowns (parameters) describing the barycentric motions of the ensemble of sources used in the astrometric solution, and 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 𝒇obs which could for example contain the measured detector coordinates of all the stellar images at specific times. 𝒇calc(𝒔,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 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 Wl=wl/(σl2+ϵl2) of individual observations l is composed of a contribution from the formal standard uncertainty of the observation σl and the excess noise ϵl 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 3.3.3), or that the attitude (Section 3.3.5) sometimes cannot be represented by the model used to sufficient accuracy.

The excess noise term ϵl 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 wl 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,

ϵl2=ϵi2+ϵa2(tl), (3.90)

where ϵi is the excess noise associated with source i (if li, that is, l is an observation of source i), and ϵa(t) is the excess attitude noise, being a function of time. For a ‘good’ primary source, we should have ϵi=0, and for a ‘good’ stretch of attitude data we may have ϵa(t)=0. Calibration modelling errors are not explicitly introduced in Equation 3.90, but could be regarded as a more or less constant part of the excess attitude noise. The estimation of ϵi is described in Section 3.3.3, and the estimation of ϵa(t) in Section 3.3.5.