Telluric Doppler shifts obtained for one spectral order (22730−22870 Å), where the telluric lines dominate and nearly no stellar lines are present. Left: Telluric Doppler shifts as a function of time (same colour coding as in the previous plots). The rms of the 134 measurements is 11.7 m/s. Right: Telluric Doppler shift obtained from the time series observations of GJ 588 from June 2021. The rms of the 25 measurements is 6 m/s over a time range of around 3.75 hours.