  The ratio of pulsation to radial velocity (the projection factor) is
currently limiting the accuracy of the interferometric Baade-Wesselink method.
This work aims at establishing a link between the line asymmetry evolution over
the Cepheids' pulsation cycles and their projection factor, with the final
objective to improve the accuracy of the Baade-Wesselink method for distance
determinations. We present HARPS high spectral resolution observations of nine
galactic Cepheids having a good period sampling. We fit spectral line profiles
by an asymmetric bi-Gaussian to derive radial velocity, Full-Width at
Half-Maximum in the line (FWHM) and line asymmetry for all stars. We then
extract correlations curves between radial velocity and asymmetry. A geometric
model providing synthetic spectral lines, including limb-darkening, a constant
FWHM (hereafter sigma_c) and the rotation velocity is used to interpret these
correlations curves. For all stars, comparison between observations and
modelling is satisfactory, and we were able to determine the projected rotation
velocities and sigma_c for all stars. We also find a correlation between the
rotation velocity (Vrot sin i) and the period of the star: Vrot sin i = (11.5
+- 0.9) log(P) + (19.8 +- 1.0) [km/s]. Moreover, we observe a systematic shift
in observational asymmetry curves (noted gamma_O), related to the period of the
star, which is not explained by our static model: gamma_O = (10.7+-0.1) log(P)
+ (9.7+-0.2) [in %] . For long-period Cepheids, in which velocity gradients,
compression or shock waves seem to be large compared to short- or medium period
Cepheids we observe indeed a greater systematic shift in asymmetry curves.
(abridged)
