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The application of the parallax of pulsation (PoP) technique to determinedistances of pulsating stars implies the use of a scaling parameter, theprojection factor (p-factor), required to transform disc-integrated radialvelocities (RVs) into photospheric expansion velocities. The value of thep-factor is poorly known and debated. Most PoP applications assume a constantp-factor. However, it may actually depend on the physical parameters of eachstar. We aim to calibrate p-factors for RR Lyrae stars (RRLs) and compare themwith classical Cepheids (CCs). Due to their higher surface gravity, RRLs havemore compact atmospheres, and provide a valuable comparison with theirsupergiant siblings. We determined the p-factor of 17 RRLs using the SPIPScode, constrained by Gaia DR3 parallaxes, photometry, and new RVs from theOHP/SOPHIE spectrograph. We carefully examine the different steps of the PoPtechnique, particularly the method to determine RV from spectra using theclassical cross-correlation function (CCF) approach. The method employed for RVextraction from the CCF has a strong impact on the p-factor, of up to 10%.However, this choice of method results in a global scaling of the p-factor,marginally affecting the scatter within the sample for a given method. Over ourRRL sample, we find a mean value of $p = 1.248 \pm 0.022$ for RVs derived usinga Gaussian fit of the CCF. There is no evidence for a different value of thep-factor of RRLs, although its distribution for RRLs appears significantly lessscattered than that for CCs. The p-factor does not appear to depend in a simpleway on fundamental stellar parameters. We argue that large-amplitude dynamicalphenomena occurring in the atmospheres of RRLs and CCs during their pulsationaffect the relative velocity of the spectral line-forming regions compared tothe velocity of the photosphere.

The Baade-Wesselink projection factor of RR Lyrae stars -- Calibration from OHP/SOPHIE spectroscopy and Gaia DR3 parallaxes