| Abstract
| - Context. The application of the parallax-of-pulsation (PoP) technique to determine the distances of pulsating stars implies the use of a scaling parameter, namely the projection factor ( p-factor), which is required to transform disc-integrated radial velocities (RVs) into photospheric expansion velocities. The value of this parameter is poorly known and is still debated. Most present applications of the PoP technique assume a constant p-factor. However, it may actually depend on the physical parameters of each star, as past studies designed to calibrate the p-factor (predominantly for Cepheids) led to a broad range of individual values. Aims. We aim to calibrate the p-factors of a sample of RR Lyrae stars (RRLs) in order to compare them with classical Cepheids (CCs). Due to their higher surface gravity, RRLs have more compact atmospheres, and therefore provide a valuable comparison with their supergiant siblings. Methods. We determined the p-factor of 17 RRLs by modelling their pulsation using the SPIPS code. The models are constrained using Gaia DR3 parallaxes, photometry, and new RVs that we collected with the OHP/SOPHIE spectrograph. We carefully examine the different steps of the PoP technique, in particular the method used to determine the RV from spectra using the classical approach based on the cross-correlation function (CCF). Results. The method employed to extract the RV from the CCF has a strong impact on the p-factor, that is, of up to 10%. However, this choice of method results in a global scaling of the p-factor, and only marginally affects the scatter of p within the sample for a given method. Over our RRL sample, we find a mean value of p = 1.248 ± 0.022 for RVs derived using a Gaussian fit of the CCF. There is no evidence for a different value of the p-factor of RRLs, although its distribution for RRLs appears significantly less scattered ( σ ≈ 7%) than that for CCs ( σ ≈ 12%). Conclusions. The p-factor does not appear to depend in a simple way on fundamental stellar parameters (pulsation period, radius, metallicity, amplitude of the RV curve). We argue that large-amplitude dynamical phenomena occurring in the atmospheres of RRLs (and CCs) during their pulsation affect the relative velocity of the spectral line-forming regions compared to the velocity of the photosphere.
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