# 3.3.4 Pseudo-colours in source update

Under certain conditions it is possible to derive an approximate estimate of the colour of a source from its astrometric solution (source update), utilizing the chromatic displacements of the image centroids, known as chromaticity. Effectively, the astrometric instrument acts as a spectrometer with extremely low resolution. Astrometric colour information obtained in this way are called pseudo-colours. The necessary conditions for the determination of pseudo-colours are

• that the astrometric instrument has some significant chromaticity – this is in practice always the case for Gaia;

• that the chromaticity was uncorrected in the pre-processing – this is the case for Gaia DR2, where the PSF/LSF calibration was still chromatic;

• that the chromaticity is adequately calibrated in the AGIS primary solution by means of sources with known colours, i.e. $G_{\text{BP}}-G_{\text{RP}}$ from the photometric processing.

As described in Section 3.3.6 the last condition was also satisfied for Gaia DR2: the along-scan geometric instrument model includes terms of the form

 $\eta_{\text{cal}}=\cdots+(\nu_{\text{eff}}-\nu_{\text{eff}}^{\text{ref}})\chi\,,$ (3.102)

where $\chi$ is the chromaticity parameter and $\nu_{\text{eff}}$ the effective wavenumber (in $\mu$m${}^{-1}$) of the source. The latter was computed from the colour index $C=G_{\text{BP}}-G_{\text{RP}}$ using the analytical approximation

 $\nu_{\text{eff}}=2.0-\frac{1.8}{\pi}\arctan\bigl{(}0.331+0.572C-0.014C^{2}+0.0% 45C^{3}\bigr{)}\,,$ (3.103)

derived from pre-launch calibrations of the photometric bands and standard stellar flux libraries. The arctan transformation constrains $\nu_{\text{eff}}$ roughly to the passband of $G$, or $\simeq\,$340–910 nm, even for extreme (spurious) values of $C$. A reference value $\nu_{\text{eff}}^{\text{ref}}=1.6$ close to the mean value for solar-type stars was adopted. Once $\chi$ has been calibrated as a function of time, CCD, etc. in a primary solution for sources with known colours, it can be used to estimate the pseudo-colour $\Psi$ of an arbitrary sources by including the corresponding term in the source update model:

 $\eta_{\text{obs}}=\cdots+(\Psi-\nu_{\text{eff}}^{\text{ref}})\chi\,.$ (3.104)

In this equation $\chi$ is regarded as known (from the geometric instrument calibration) and $\Psi$ is an additional unknown per source, solved together with the usual five astrometric parameters. This solution also provides an estimate of standard uncertainty of $\Psi$. From Equation 3.102 and Equation 3.104 it is clear that $\Psi\simeq\nu_{\text{eff}}$ is expected.

Figure 3.11 shows the relation between $\nu_{\text{eff}}$ and $\Psi$ for a random sample of the sources with colours from the photometric processing. The agreement is reasonable at least when the formal uncertainty of $\Psi$ is not too large. The plot includes a small number of points to the right, at unrealistically large $\nu_{\text{eff}}$, indicating that for these sources the colour index is too blue and $\Psi$ may provide a less biased colour estimate.