INSPIRE: INvestigating Stellar Population In RElics III. Second data release (DR2): testing the systematics on the stellar velocity dispersion

This is the second data release (DR2) of the INvestigating Stellar Population In RElics (INSPIRE) project, comprising 21 new systems with observations completed before March 2022. For each system, we release four one-dimensional (1D) spectra to the ESO Science Archive, one spectrum for each arm of the X-Shooter spectrograph. In this paper, we focus on the line-of-sight velocity distribution, measuring integrated stellar velocity dispersions from the spectra, and assessing their robustness and the associated uncertainties. For each of the 21 new systems, we systematically investigated the effect of the parameters and set-ups of the full spectral fitting on the stellar velocity dispersion ($\sigma$) measurements. In particular, we tested how $\sigma$ changes when several parameters of the fit as well as the resolution and spectral coverage of the input spectra are varied. We found that the effect that causes the largest systematic uncertainties on $\sigma$ is the wavelength range used for the fit, especially for spectra with a lower signal-to-noise ratio (S/N $\leq$ 30). When using blue wavelengths (UVB arm) one generally underestimates the velocity dispersion (by $\sim$15 km/s). The values obtained from the near-IR (NIR) arm present a larger scatter because the quality of the spectra is lower. We finally compared our results with those in literature, finding a very good agreement overall. Joining results obtained in DR1 with those presented here, INSPIRE contains 40 ultra-compact massive galaxies, corresponding to 75% of the whole survey. By plotting these systems in a stellar mass-velocity dispersion diagram, we identify at least four highly reliable relic candidates among the new systems. Their velocity dispersion is larger than that of normal-sized galaxies of similar stellar mass.