Context. The projection factor is a key quantity for the interferometric Baade-Wesselink (hereafter IBW) and surface-brightness (hereafter SB) methods of determining the distance of Cepheids. Indeed, it allows a consistent combination of angular and linear diameters of the star. Aims. We aim to determine consistent projection factors that include the dynamical structure of the Cepheids' atmosphere.Methods. Hydrodynamical models of δ Cep and Car have been used to validate a spectroscopic method of determining the projection factor. This method, based on the amplitude of the radial velocity curve, is applied to eight stars observed with the HARPS spectrometer. The projection factor is divided into three sub-concepts : (1) a geometrical effect, (2) the velocity gradient within the atmosphere, and (3) the relative motion of the "optical" pulsating photosphere compared to the corresponding mass elements (hereafter f o−g ). Both, (1) and (3) are deduced from geometrical and hydrodynamical models, respectively, while (2) is derived directly from observations.Results. The Fe i 4896.439 Å line is found to be the best one to use in the context of IBW and SB methods. A coherent and consistent period-projection factor relation (hereafter Pp relation) is derived for this specific spectral line: p = [−0.064 ± 0.020] log P + [1.376 ± 0.023]. This procedure is then extended to derive dynamic projection factors for any spectral line of any Cepheid. Conclusions. This Pp relation is an important tool for removing bias in the calibration of the period-luminosity relation of Cepheids. Moreover, it reveals a new physical quantity f o−g to investigate in the near future.

Abstract. The distance of galactic Cepheids can be derived through the interferometric Baade-Wesselink method. The interferometric measurements lead to angular diameter estimations over the whole pulsation period, while the stellar radius variations can be deduced from the integration of the pulsation velocity. The latter is linked to the observational velocity deduced from line profiles by the so-called projection factor p. The knowledge of p is currently an important limiting factor for this method of distance determination. A self-consistent and time-dependent model of the star δ Cep is computed in order to study the dynamical structure of its atmosphere together with the induced line profile. Different kinds of radial and pulsation velocities are then derived. In particular, we compile a suitable average value for the projection factor related to different observational techniques, such as spectrometry, and spectral-line or wide-band interferometry. We show that the impact on the average projection factor and consequently on the final distance deduced from this method is of the order of 6%. We also study the impact of a constant or variable p-factor on the Cepheid distance determination. We conclude on this last point that if the average value of the projection factor is correct, then the influence of the time dependence is not significant as the error in the final distance is of the order of 0.2%.

Abstract. This paper presents statistics of the line-doubling phenomenon in a sample of 81 long-period variable (LPV) stars of various periods, spectral types and brightness ranges. The set of observations consists of 315 highresolution optical spectra collected with the spectrograph ELODIE at the Haute-Provence Observatory, during 27 observing nights at one-month intervals and spanning two years. When correlated with a mask mimicking a K0III spectrum, 54% of the sample stars clearly showed a double-peaked cross-correlation profile around maximum light, reflecting double absorption lines. Several pieces of evidence are presented that point towards the double absorption lines as being caused by the propagation of a shock wave through the photosphere. The observation of the Balmer lines appearing in emission around maximum light in these stars corroborates the presence of a shock wave. The observed velocity discontinuities, ranging between 10 and 25 km s −1 , are not correlated with the brightness ranges. A comparison with the center-of-mass (COM) velocity obtained from submm CO lines originating in the circumstellar envelope reveals that the median velocity between the red and blue peaks is blueshifted with respect to the COM velocity, as expected if the shock moves upwards. The LPVs clearly exhibiting line-doubling around maximum light with the K0III mask appear to be the most compact ones, the stellar radius being estimated from their effective temperatures (via the spectral type) and luminosities (via the period-luminosity relationship). It is not entirely clear whether or not this segregation between compact and extended LPVs is an artefact of the use of the K0III mask. Warmer masks (F0V and G2V) applied to the most extended and coolest LPVs yield asymmetric cross-correlation functions which suggest that line doubling is occurring in those stars as well. Although a firm conclusion on this point is hampered by the large correlation noise present in the CCFs of cool LPVs obtained with warm masks, the occurrence of line doubling in those stars is confirmed by the double CO ∆v = 3 lines observed around 1.6 µm by Hinkle et al. (1984, ApJS, 56, 1). Moreover, the Hδ line in emission, which is another signature of the presence of shocks, is observed as well in the most extended stars, although with a somewhat narrower profile. This is an indication that the shock is weaker in extended than in compact LPVs, which may also contribute to the difficulty of detecting line doubling in cool, extended LPVs.

Context. The projection factor p is the key quantity used in the Baade-Wesselink (BW) method for distance determination; it converts radial velocities into pulsation velocities. Several methods are used to determine p, such as geometrical and hydrodynamical models or the inverse BW approach when the distance is known. Aims. We analyze new HARPS-N spectra of δ Cep to measure its cycle-averaged atmospheric velocity gradient in order to better constrain the projection factor. Methods. We first apply the inverse BW method to derive p directly from observations. The projection factor can be divided into three subconcepts: (1) a geometrical effect (p0), (2) the velocity gradient within the atmosphere (f grad ), and (3) the relative motion of the optical pulsating photosphere with respect to the corresponding mass elements (fo−g). We then measure the f grad value of δ Cep for the first time.Results. When the HARPS-N mean cross-correlated line-profiles are fitted with a Gaussian profile, the projection factor is pcc−g = 1.239 ± 0.034(stat.) ± 0.023(syst.). When we consider the different amplitudes of the radial velocity curves that are associated with 17 selected spectral lines, we measure projection factors ranging from 1.273 to 1.329. We find a relation between f grad and the line depth measured when the Cepheid is at minimum radius. This relation is consistent with that obtained from our best hydrodynamical model of δ Cep and with our projection factor decomposition. Using the observational values of p and f grad found for the 17 spectral lines, we derive a semi-theoretical value of fo−g. We alternatively obtain fo−g = 0.975 ± 0.002 or 1.006 ± 0.002 assuming models using radiative transfer in plane-parallel or spherically symmetric geometries, respectively. Conclusions. The new HARPS-N observations of δ Cep are consistent with our decomposition of the projection factor. The next step will be to measure p0 directly from the next generation of visible interferometers. With these values in hand, it will be possible to derive fo−g directly from observations.