We computed a large number of time series of line profiles on
two-dimensional grids. For each grid we varied 2 out of 7
relevant pulsational and stellar parameters: 
, m, 
, 
, i, 
, and W. From these
calculations, we selected a few grids to serve as illustrations in
this paper (Figs. 
). The domains of the parameters
in these grids are chosen to bring out the most illustrative examples,
while covering the physically relevant cases. The results of each
grid calculation are displayed in two parts. The left part displays
the time series of line profiles along with the corresponding behavior
of the first and the second velocity moment. The right part shows the
corresponding residual spectra, and the distributions of variational
amplitudes and phases across the line profile. See Fig. 2 (click here)
for an example.
Note that in Figs. 
we present time series of line
profiles during 3 pulsation cycles, without specifying the time scale
of the variability; the velocity axes are scaled to 
. This means that in some of the Figs. 
, the
parameters that change along the axes of the grid imply stars with
different mass, radius, rotation period and/or pulsation frequency.
As shown in Sect. 4.1 (click here), line-profile variability is in general
not only found at the input frequency of the variations but also at
its multiples (i.e. its harmonics), even for an assumed sinusoidal
behavior of the surface velocity field (e.g. for k = 0 and
= 0) (see also Fig. 1 (click here)). This
effect is due to the Doppler-mapping of the three-dimensional stellar
surface velocity field onto one-dimensional velocity (or wavelength)
space, which has only a one-to-one correspondence
in the limit of zero pulsation amplitude.
Sharply peaked moving bumps give rise to strongly
non-sinusoidal line-profile time-variability, which results in large
harmonic amplitudes.
The shape of the amplitude distribution across the line profile depends on all parameters, in many cases in a very complicated manner. Only the behavior of the amplitude distribution of sectoral modes with different values of k are easy to understand (e.g. see Sects. 6.2 (click here) and 6.3 (click here)). In general, the shape of the amplitude distribution of the variations at the harmonic frequency resembles that at the input frequency.
The relative magnitude of the variations at the different frequencies
is another aspect of the variability. The ratio of the variational
amplitude at the first harmonic frequency to that at the input
frequency, 
, is an indicator of the non-sinusoidal
nature of the line-profile variations, and depends on the position in
the line profile. We write the ratio of the maxima of the individual
distributions (that may occur at different velocity positions) as
. We find that for all investigated cases
> 1 and 
> 1. For
most but not all cases we find 
> 1.
We find that 
depends mostly on W and on
. This is illustrated in Fig. 13 (click here). The
relative harmonic amplitudes 
and 
increase for increasing pulsation amplitude 
.
The width of the intrinsic line profile causes a blurring of the
moving bumps. Therefore, the contrast (sharpness) of the bumps
will decrease with increasing intrinsic width, and consequently also
.
We find weaker dependences of 
and
on k and i. A small decrease of
is found towards the high k-values and low
inclination angles.
In Fig. 7 (click here) we illustrate the peculiar line-profile behavior
that we found for modes with 
- m an odd number and with an
inclination angle very close to 
. For these modes,
we find that the apparent number of moving bumps in the line profile
is doubled. This behavior leads to a variability that is found at
even multiples of the input frequency, but not
at the input frequency itself (Fig. 7 (click here), see Reid & Aerts
1993). Towards an inclination of 
the variation at the
input frequency disappears because of cancellation effects, whereas the
variability at the first harmonic is unaffected. The cancellation at
the input frequency is caused by the symmetric behavior of the
pulsation with respect to the stellar equator, and will already
disappear at inclinations that differ only a few degrees from
(see Paper III).
and |m| from the phase
differences across the line profile
If the power at the harmonics relative to that at the input frequency
is small, the blue-to-red phase difference 
is directly
related to the number of traveling bumps that migrate through the
line profile. In such cases of sinusoidal line-profile variations, the
phase difference across the line profile can be measured by counting the
number of traveling bumps. We find that the phase diagrams themselves
provide a more objective measure of the number of cycles across the
line profile, regardless of the importance of the harmonics.
For multiple modes, the number of traveling bumps in the line profile
is determined by the combined effect of all modes, in which case the
blue-to-red phase differences can only be successfully determined if the
apparent frequencies of each of the modes can be resolved (see Paper
II).
Many authors (e.g. Smith 1986; Gies & Kullavanijaya 1988; Kambe
& Osaki 1988; Yang et al. 1988; Kambe et al. 1990) have used
the number of visible bumps or, equivalently, the blue-to-red phase
difference 
to identify |m| according to
.
However, from our calculations we find that the line-profile behavior for both
tesseral and sectoral modes displays a blue-to-red phase difference
proportional to 
rather than to |m|.
Such a relation was already suggested by Merryfield & Kennelly
(1993), who found for one tesseral mode that their apparent azimuthal
order 
corresponds more closely to 
than to |m|. In
addition, we find that the phase difference 
at the
first harmonic frequency 
is an indicator of |m|.
Figure 5 (click here) gives a typical example of this behavior of the phase
diagrams.
We tentatively conclude that it is possible to determine both the
degree 
and the azimuthal order |m| from the phase
information provided by a Fourier analysis of an observed time series
of high-resolution spectra with sufficient signal to noise ratio; in
Paper II we elaborate on this subject. An independent check of the
(
,|m|)-value can be performed by an application of the moment
method (for l 
4, and if effects of rotation and temperature
variation can be neglected).
Figure 3:
Line-profile variability for two different pulsation modes as a
function of 
. See Fig. 2 (click here) for a
description of the figures. For clarity we do not plot 
and
. Top row: radial pulsation (
= 0). Bottom row:
non-radial pulsation mode (
= 2, m = -2).
The values of the other parameters are (for both modes): 
= 0.15 
, W = 0.1 
,
= 0.8 (not relevant if 
= 0), and
i = 
. The effect of the rotation is large for these
low-degree modes. In both cases power is added to the line center,
leading to similar line-profile behavior. For the radial mode this is
caused by the rotationally induced toroidal term; for the non-radial
mode the effect is mainly caused by the first-order correction of k
In several of our diagnostic diagrams (see Figs. 9 (click here) and
15 (click here)) for modes with intermediate to high
k-values, we find a blue-to-red phase difference 
that differs by 
from the empirical relation with 
that is
shown by the majority of our calculations. Modes with low k-values
show line-profile variations mainly caused by radial motions, causing
a phase of variability that changes across the line profile along with
the change of phase of the traveling waves at the surface of the
star. For intermediate to high k-values the profile variations do
not follow the continuous phase change at the stellar surface. A
positive azimuthal motion (
) will cause a negative
Doppler-shift at the blue wing of the profile, but a positive shift at
the red wing. This results in an additional change of phase by 
around the line center. Other sources of variability at the line
center determine whether this change will be positive or negative,
which leads to differences of 
in 
for modes
with the same 
-value. The blue-to-red phase difference of modes with
intermediate to high k is therefore co-determined by other possible
sources of variability at the line center (e.g. toroidal terms; radial
and 
-movements; local brightness and EW changes). In this
work, differences in 
of 
are found by using
different parameters i and 
(see Figs. 9 (click here) and 15 (click here)).
A detailed analysis of the determination of
and |m| from the phase diagrams at the input frequency and
its harmonics is the subject of investigation in Paper II.
The effects of the Coriolis force on the line-profile variations are hard to classify, because they depend on the specific combination of parameter values. However, the effects of the Coriolis force increase for higher values of k.
For high-degree sectoral modes (say 
> 4), the effects of
rotation are negligible for values of 
up to 0.5.
This is because the toroidal amplitude of these sectoral modes is too
small, compared to the spheroidal amplitude, to have a large effect on
the line profiles. Under the same conditions the value of k is also
not modified by rotation (Eq. 22 (click here)). The effects of rotation
increase in importance for sectoral modes of lower degree. If 
is small enough (0, 1 or 2) the first-order terms can lead to
drastic changes in the line profiles and their variations.
Figure 3 (click here) shows the line-profile behavior of a prograde
sectoral 
= 2 mode and a radial (
= 0) mode. It shows
that radial modes in rotating stars can easily be confused with
= -m = 2 modes. In spite of the completely different
surface velocity fields of these modes, the presence of rotation
induces a very similar line-profile behavior. The reason for the
similarity is purely accidental; in case of the 
= 2 mode the
dominant rotational effect is a change in k-value, whereas the
radial mode is changed only by one toroidal term (see Eq. 12 (click here))
which induces a periodic increase/decrease of the rotational velocity
at the surface. Note the similarity with the line-profile variations
presented by Gies (1994).
Tesseral modes are much more affected by the Coriolis force than
sectoral modes since for tesseral modes the rotation gives rise to a
relatively large second toroidal correction term (see
Eq. 12 (click here)), which is zero for sectoral modes. We find
remarkable line-profile behavior for tesseral modes with a high
-value and no node line at the equator (
- m
even). In the slow-rotation model, these tesseral modes lead to
dominant variability at the line center, similar to a sectoral mode
with small 
-value (see Figs. 8 (click here)).
In his discussion of the so called k-problem, Smith (1986) mentioned
that for high 
-values, the toroidal term(s) caused by
rotation might be able to mimic the amplitude-distribution
characteristics of a low-
mode. For sectoral modes with
> 3 and values of 
0.5, we
find that the toroidal term induced by the Coriolis force is not
capable of generating enough variability near the line center.
However, as mentioned above, tesseral modes are modified by a second
toroidal term that can indeed contribute to the amplitude at the line
center, mimicking the characteristics of a low-
mode.
For the line-profile variations that are significantly affected by the
effects of the Coriolis force, we find differences between the
prograde and the retrograde modes. We find only modest differences
between the amplitude diagrams of both cases. As mentioned in
Sect. 6.1.2 (click here), the blue-to-red phase difference of modes
with intermediate to high k values is co-determined by the toroidal
motions caused by the Coriolis force. In some cases this leads to a
difference of 
between the prograde and the retrograde
case (see rightmost 2 columns in Fig. 9 (click here)).
The determination of the k-value from observed line-profiles has been discussed by several authors (e.g. Smith 1986; Kambe et al.\ 1990; Lee & Saio 1990). Many factors may play a role in the appearance of modes with high k-values, of which a few are discussed below.
For sectoral modes, the most significant effects of the parameter k
on the line-profile behavior can be described as follows (see
Fig. 11 (click here), top row). A convex amplitude distribution is found
for 
. For a somewhat higher value of k (depending on the
values of m and 
), the amplitude distribution takes a
rectangular shape, as a consequence of larger horizontal motions.
This trend persists towards the higher k-values so that above a
certain value of k all variability is concentrated in the wings of
the absorption profile, and the amplitude distribution has acquired a
double-peaked shape with almost no power in the line center.
Proceeding to still higher values of k, while keeping 
constant, has no further effect on the line profiles and their
amplitude distribution, since the horizontal motions already dominate.
The rectangularly shaped and double-peaked amplitude distributions of
sectoral modes with intermediate to high k-values are blurred at
high values of the intrinsic width 
(see
Fig. 11 (click here)). If 
is sufficiently large, a
change in k does not lead to any effect on the amplitude
distribution at all. A correct determination of k, by means of
amplitude distributions, should therefore include a reasonable
estimate for the intrinsic line-profile width 
.
This is especially relevant for line profiles of light ions in
slow/moderate rotators.
The k-characteristics of tesseral modes depend crucially on the
relative rotation rate 
. As we have shown in
Sect. 6.2 (click here) and Fig. 9 (click here), slow rotation changes the
amplitude distribution of many high-k tesseral modes in such a way
that the highest amplitude is found at the line center, whereas for
other high-k tesseral modes the maximum amplitudes are found in the
wings of the profile. Therefore, we cannot identify high-k
characteristics for tesseral modes in general.
Finally, it has been shown previously (Lee & Saio 1990; Lee et al.\ 1992; Townsend 1996) that if temperature effects (i.e. local surface brightness and EW variations) co-determine the line-profile formation, the k-characteristics may generally disappear. We intend to describe the influence of these temperature effects on the amplitude and phase diagrams in a separate paper (Paper IV).
We conclude that in general it is difficult to determine the k-value
of a pulsation mode from the amplitude distributions of observed time
series of line profiles alone. However, a double-peaked amplitude
distribution always reflects a high k-value, if the blue-to-red
phase difference 
exceeds 
radians.
A double-peaked amplitude distribution together with a blue-to-red
phase difference of less than 
, can also occur for a radial or
an 
mode.
It is well known that, if the star is seen equator-on, sectoral modes
are much more effective in producing line-profile variations than
tesseral modes. For smaller inclination angles, this effectiveness is
reduced. For tesseral modes, this efficiency behaves different with
respect to the inclination angle. Towards smaller inclination angles,
tesseral modes are increasingly efficient in producing line-profile
variability. In general one could state that at inclinations
around 
the tesseral modes are more efficient in
producing line-profile variations than sectoral modes, for equal
surface velocities. This aspect of tesseral modes is illustrated in
Figs. 6 (click here).
As expected, the line-profile amplitude characteristics of tesseral modes are much more sensitive to the inclination angle, than those of sectoral modes. Especially in the cases that the Coriolis force causes significant toroidal terms, the line-profile formation of tesseral modes is very complex (see Sect. 6.2 (click here)).
We have not found a clear indication that one can derive the
inclination from the IPS diagnostics. The presence of red-to-blue
moving bumps, due to the traveling waves at the far side of the star,
coexisting with blue-to-red moving bumps might be indicative for a low
i (see Baade 1984, 1987). The presence of coexisting bumps moving in
opposite directions, has been discussed by Kambe & Osaki (1988) for
the case of toroidal modes with small inclinations. Since for some
modes there is enough information in the line-profile variability to
derive i with the moment method, we conclude that it should be
possible to estimate the inclination by fitting generated time-series
of line profiles or IPS amplitude and phase diagrams to the observed
ones. However, such an attempt can only be successful if both 
and m are precisely known.