# 6.3.4 Wavelength calibration

The most important calibration necessary for estimating $V_{\rm rad}$ is the wavelength calibration. It is implemented in the pipeline which calculates the associated calibration parameters and their spatial and temporal dependence. The basic principle of the wavelength calibration is that the wavelength associated to a sample can be expressed as a function of the FoV coordinates $(\eta,\zeta)$ of the source at the time when the sample crosses the CCD fiducial line (i.e. AL pixel 2253.5).

A self-calibration approach is adopted and some of the RVS spectra are used as calibrators (Figure 6.4 shows an example of these). The pipeline detects the spectral lines in the calibrator spectra, identifies them and assigns a rest wavelength by comparison with a synthetic spectrum template. The dispersion function used is:

 $\lambda_{\rm observed}=\lambda_{\rm rest}\left(1+\frac{V(s)}{c}\right)=C_{00}+% C_{10}\eta+C_{20}\eta^{2}+C_{01}\zeta+C_{11}\eta\zeta$ (6.4)

where:

• $\lambda_{\rm observed}$ is the wavelength associated to the centre of the sample;

• $C_{mn}$ are the unknown calibration coefficients;

• $V(s)$ is the unknown Gaia-centric radial velocity of the calibration star;

• $\eta$ and $\zeta$ are the field angles FoV coordinates of the calibration stars at the fiducial time of the CCD.

To avoid the degeneracy due to the fact that a shift in the dispersion law can be compensated by a shift in the radial velocity $V(s)$ of the calibration stars, the auxiliary standard stars (Section 6.2.3) are used to fix the zero-point. More details of how Equation 6.4 is implemented are in Sartoretti et al. (2018).

The wavelength calibration process is repeated for all the calibration units for all CCDs and FoV of a trending epoch. Once the coefficients for the entire trending epoch are obtained, a trending module fits a curve to describe the emerging trend for each coefficient. The trending functions are used to calibrate the spectra acquired at any time within the trending epoch, and at any position (see Figure 6.7 for examples of trending functions).