The velocity field of pulsation, described in the corotating frame, is
given by the time derivative of the Lagrangian displacement field
which we calculate on a spherical grid of typically more than 5000 equally sized visible surface elements. For each visible element, the corresponding rotational velocity of a uniformly rotating sphere is added. Neglecting all atmospheric effects due to the pulsation, we attribute to each surface element a Gaussian intrinsic line profile with width W
where the free parameter is a measure of the line depth and the central velocity position of the Doppler-shifted line profile of the regarded surface element. The line profile I(V) is then obtained by a weighted integration of the Doppler-shifted Gaussian profiles of all visible surface elements. Finally, the profile I(V) is normalized by means of a division by its continuum value.
We note that our conclusions will not depend on the actual shape of the intrinsic line profile. The difference between profiles calculated with either a Gaussian or, for example, a rectangular shape is only noticeable in the wings of the rotationally broadened line profile. If the rotational broadening is large enough (i.e. > W), only the effective width of the intrinsic line profile is important.
The weights in the integration are determined by the aspect angle of
each element, neglecting the rotational and pulsational deformation of
the stellar surface, and by a linear limb-darkening correction of the
in which is the angle between the surface normal of the sphere and the line of sight. We adopt a limb-darkening coefficient , which is appropriate for early B-type stars in the optical region (Wade & Rucinski 1985). For simplicity we apply the same limb-darkening correction to the continuum and to the wavelength region of the intrinsic line profile itself. This means that we keep the shape of the intrinsic profile constant over the stellar disk. For very strong lines this might not be a good approximation.
We checked whether or not the use of other values of the limb-darkening coefficient affects the validity of any of the conclusions in this paper. The phases of variability across the profiles do not change, but the amplitude distribution (see Sect. 4 (click here)) does. In our representation of the amplitudes of line-profile variability (see Sect. 4.1 (click here)), a high limb-darkening coefficient results in higher variational amplitudes at the line center, and lower amplitudes at the wings of the profile. A limb-darkening coefficient of 1.0 can lead to changes up to 20% of the variational amplitudes with respect to the case with no limb darkening. Nevertheless, our conclusions do not depend on our choice for the limb-darkening coefficient.
In this work we aim to investigate the observational characteristics of line-profile variations in the best situation one can consider, namely that of perfect data. Following this approach, we can establish which pulsation parameters one may ultimately derive from observations. For this reason, we generate data that perfectly sample the pulsation period, with sufficient spectral resolution, and we do not include noise in the synthesis of our spectra.
In the case of real data, the data sampling and data quality (e.g.\ signal to noise ratio, S/N, spectral resolution, and correct continuum rectification) govern the detectability of the pulsation characteristics. The interplay between the time sampling and the S/N is complicated and may hamper the detection of characteristics of certain pulsation modes in a complex fashion. Generally, we expect that from lengthy data sets the most conclusive results can be derived.