Understanding the dynamical structure of pulsating stars: The center-of-mass velocity and the Baade-Wesselink projection factor of the

Context. High-resolution spectroscopy of pulsating stars is a powerful tool to study the dynamical structure of their atmosphere. Lines asymmetry is used to derive the center-of-mass velocity of the star, while a direct measurement of the atmospheric velocity gradient helps determine the projection factor used in the Baade-Wesselink method of distance determination. Aims. We aim at deriving the center-of-mass velocity and the projection factor of the β Cephei star α Lup. Methods. We present HARPS high spectral resolution observations of α Lup. We calculate the first-moment radial velocities and fit the spectral line profiles by a bi-Gaussian to derive line asymmetries. Correlations between the γ-velocity and the γ-asymmetry (defined as the average values of the radial velocity and line asymmetry curves respectively) are used to derive the center-of-mass velocity of the star. By combining our spectroscopic determination of the atmospheric velocity gradient with a hydrodynamical model of the photosphere of the star, we derive a semi-theoretical projection factor for α Lup. Results. We find a center-of-mass velocity of Vγ = 7.9 ± 0.6 km s-1 and that the velocity gradient in the atmosphere of α Lup is null. We apply to α Lup the usual decomposition of the projection factor into three parts, p = p0fgradfog (originally developed for Cepheids), and derive a projection factor of p = 1.43 ± 0.01. By comparing our results with previous HARPS observations of classical Cepheids, we also point out a linear relation between the atmospheric velocity gradient and the amplitude of the radial velocity curve. Moreover, we observe a phase shift (Van Hoof effect), whereas α Lup has no velocity gradient. New HARPS data of a short-period β Cephei star, τ1 Lup, are also presented in this paper. Conclusions. By comparing Cepheids and β Cephei stars, these results bring insight into the dynamical structure of pulsating star atmospheres, which helps to better understand the k-term problem and the Baade-Wesselink p-factor for Cepheids