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 pseudocolours.
The necessary conditions for the determination of pseudocolours 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 preprocessing – 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 alongscan geometric instrument model includes terms of the form

$${\eta}_{\text{cal}}=\mathrm{\cdots}+({\nu}_{\text{eff}}{\nu}_{\text{eff}}^{\text{ref}})\chi ,$$ 

(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

$${\nu}_{\text{eff}}=2.0\frac{1.8}{\pi}\mathrm{arctan}\left(0.331+0.572C0.014{C}^{2}+0.045{C}^{3}\right),$$ 

(3.103) 
derived from prelaunch 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 solartype
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
pseudocolour $\mathrm{\Psi}$ of an arbitrary sources by including the corresponding term
in the source update model:

$${\eta}_{\text{obs}}=\mathrm{\cdots}+(\mathrm{\Psi}{\nu}_{\text{eff}}^{\text{ref}})\chi .$$ 

(3.104) 
In this equation $\chi $ is regarded as known (from the geometric instrument calibration)
and $\mathrm{\Psi}$ is an additional unknown per source, solved together with the usual five
astrometric parameters. This solution also provides an estimate of standard uncertainty
of $\mathrm{\Psi}$. From Equation 3.102 and
Equation 3.104 it is clear that $\mathrm{\Psi}\simeq {\nu}_{\text{eff}}$
is expected.
Figure 3.11 shows the relation between ${\nu}_{\text{eff}}$ and $\mathrm{\Psi}$
for a random sample of the sources with colours from the photometric processing. The agreement
is reasonable at least when the formal uncertainty of $\mathrm{\Psi}$ 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 $\mathrm{\Psi}$ may provide a less biased colour
estimate.