In the zero-rotation approximation, our model of the line-profile
variations has six parameters.
In the slow-rotation model, the relative rotation rate
is the seventh parameter. Of course, real line
profiles and their variations also depend on 
and
. These two parameters can be removed from the
model by proper scalings, which are described in Sects. 4.1 (click here)
and 5.5 (click here). A single representation of the line-profile
variability is then applicable to a wide range of rotational
velocities and line strengths.
Following AW, we use values for 
up to 0.5 for
the slow-rotation model. The parameters k and 
are closely related because of their joint dependence on
. Nevertheless, we have chosen to keep them as separate
parameters, and consequently, every combination corresponds to a star
with different mass, radius and/or rotational frequency. For a given
rotational velocity, a large value for 
corresponds to a large k-value. This implies that certain domains
in the 
-(
)-plane become irrelevant,
depending on the ratio, 
, of the centrifugal force to the
gravitational force at the equator of the star
In our work we only use combinations of 
and
that satisfy 
. This corresponds
to an equatorial rotation velocity that is at most 
of the
equatorial break-up velocity. Clement (1994) has found that the
spheroidal part of axisymmetric modes (zonal, m = 0), does
not maintain its basic zero-rotation spatial distribution. Therefore,
our model might be inaccurate for zonal modes with high
.
The frequencies that we use for our line-profile calculations are those as observed in the corotating frame of the star. In this way the retrograde modes lead to bumps that move from red to blue in the line profiles, while the bumps of prograde modes move from blue to red. This makes it easier to distinguish prograde and retrograde modes in our figures. In the zero-rotation approach, the line-profile variations associated with prograde (m < 0) and retrograde (m > 0) modes behave symmetrically in time (see Fig. 8 (click here), right two columns). This symmetry is broken by the effects of rotation on the mode, which necessitates a separate investigation of prograde and retrograde modes in the case of the slow-rotation model.
Symmetries in the expressions of the displacement field allow us to
restrict the inclination angle i to the interval
(Aerts 1993).
instead of 
A comparative study of line profiles with different values of 
,
m, 
, or 
, in which the model amplitude
is kept constant, is not convenient, since this
implies physical situations with substantially different velocities,
and correspondingly different amplitudes of the line-profile
variations. To facilitate a fair comparison between the line-profile
variations resulting from different parameter settings, we adjust the
displacement amplitude 
in each case such that the
calculated maximum pulsational surface velocity
is kept the same whereas the other parameters may vary.
as a scaling factor
Two profiles will have identical shapes if the combination of
and 
is the
same for both profiles.
Such profiles differ only in velocity (wavelength) and intensity scale.
We therefore use 
as a scaling factor for the
velocity amplitude, for the width of the intrinsic profile, and for
the velocity scale of the line profile. We adopt only one specific
value of 
and vary the pulsation amplitude and
intrinsic width to investigate the response in the line-profile
variations.
The amplitude of pulsation 
and the intrinsic profile
width W have a broad range of physically relevant values. The more
rapid rotators are best accounted for by relatively small values of
and 
, while
the slower rotators have values of 
and W of order
.
of the horizontal to the vertical
amplitudes
Theoretically, the value for k can range from approximately zero up
to infinity (Eq. 22 (click here)). If we keep 
the
same for each calculated mode, the profiles do not change much for k
values higher than a certain value 
. Above this value,
the magnitude of the velocity vector is almost proportional to k,
because the radial velocity component becomes negligible. For all
investigated modes we found 
1 , depending somewhat
on the considered 
value. In Figs. 10 (click here) and 15 (click here),
where 
is one of the two running parameters, we adjust the
range of 
according to the value of 
. The
line-profile characteristics for the highest values that we use for
can be considered as representative for all higher values as
well.
For the calculation of line profiles, we have chosen to use 
as model parameter and to calculate the corrected k from
Eq. (22 (click here)). Such an approach allows to separate the
effect of the rotation on k, and to study the line profiles by
comparing different rotation rates for the same 
-value
(Fig. 14 (click here)).