We presented a description of the oscillatory displacement field at the surface of a non-radially pulsating rotating star. This description accounts for the effects of the Coriolis force on the pulsation, and is relatively simple to implement in models.
We modeled time series of line profiles of rotating stars that exhibit non-radial pulsations, and obtained amplitude and phase diagrams by means of a Fourier decomposition of the intensity variations in the line profile. We found a number of interesting aspects of the line profiles and their temporal behavior, which are either new or have not been mentioned elsewhere.
The precise shape of the intrinsic profile function only affects the
characteristics of the line-profile variability if the width of the
intrinsic line profile becomes of the order of the projected
equatorial rotation velocity 
.
Limb darkening does not fundamentally change the line-profile behavior. The phase diagrams are hardly affected by limb darkening. However, the amplitudes of intensity variations in the normalized line profiles do depend somewhat on the limb-darkening coefficient.
For a sufficiently inclined star, tesseral modes can produce large line-profile variations; the variations can become even larger than those of a sectoral mode.
Figure 4: a)
Time series of line profiles and their first and second velocity
moments, calculated for a grid of different combinations of 
and
m. The remaining fixed parameters are: i = 
,
= 0.0, W = 
,
= 
, 
= 0.20.
The first velocity moments are all plotted at the same scale, as well
as the variations of the second moment. The scales at which the
moments are plotted are therefore determined by the maximum moment
variations that are found in the grid. See Fig. 2 (click here) for a
detailed description of the figures. Here we show that the variations
of the velocity moments are negligible for high-degree modes
(
). Variations of the first two velocity moments of
high-degree modes (say 
) will be very hard to detect with
present observational techniques. Due to the nearly edge-on
perspective, the line-profile variations of the tesseral modes are
systematically low; they increase for smaller inclination angles (see
Figs. 6 (click here))
Figure 5: b)
Residual spectra, amplitude distributions 
and 
,
and phase distributions 
and 
of the time
series of line profiles displayed in Fig. 4a.
For a detailed description of the figures, see Fig. 2 (click here).
Note that in the gray-scale images we show three pulsation cycles of
the line-profiles without specifying the time-scale of the
variability. In the panels with amplitude and phase diagrams we give
the maxima of the amplitude distributions 
, 
,
in units of 
, and the total blue-to-red phase
differences 
, 
, and 
in radians. We only show amplitude and phase diagrams for the input
frequency (
, 
) and its first harmonic
(
, 
). The grid on this page gives a typical
example of the blue-to-red phase difference of the variations at the
input frequency, 
, being an indicator of 
. This
finding rejects earlier suggestions that the number of bumps in the
line profiles is proportional to |m|. The blue-to-red harmonic
phase difference of the variations, 
, does show
to be an indicator of |m|
For large pulsational velocities, harmonics of the observed pulsation
frequency are needed to describe the line-profile variability. The
larger the line-profile variations caused by the velocity variations,
the larger is the contribution of harmonics to the variations. The
harmonic variability increases also for a decreased intrinsic line
profile width. We found a steep increase of the relative harmonic
contribution, i.e. an apparent doubling of the number of bumps in the
line profile, for modes with odd 
- m with an inclination
close to 
.
The apparent number of bumps and troughs in the line profile is a measure of
only if the harmonics of the line profile variations are
relatively unimportant, and if
the star is pulsating in a single mode.
However, one can circumvent these restrictions with a Fourier analysis
of the time behavior of line profiles: the degree 
and the
azimuthal order m are related to the blue-to-red phase differences
at the apparent frequency and its first harmonic respectively. It
should therefore be possible to derive 
and to put constraints
on |m| from IPS phase diagrams.
The ability to detect high-k characteristics in the line-profile
variations depends on the width of the intrinsic line profile, W:
detection is only possible for cases with W 
0.4
. For tesseral modes, we find no characteristics that
distinguish between low and high k-values; a large subset of the
tesseral modes with a high value of k still gives variability in the
line center. In the general case, without presuming that the observed
line-profile variations are due to a sectoral mode, we find no
characteristics of the variations that make it possible to derive a
conclusive value of k. Only the cases with a double-peaked
amplitude distribution give unambiguous information by explicitly
implying a high k-value, provided that the corresponding phase
difference 
exceeds 
.
The effects of slow rotation on the line-profile behavior of
sectoral modes, are only important for modes of low-degree.
Line-profile variations of tesseral modes can be heavily
affected by the effects of rotation. The rotationally induced
toroidal movements of tesseral modes with even 
- |m| and
with large k-values give rise to a line-profile behavior which is
similar to that of sectoral modes with low k-values. The
line-profile variability of a radial mode in a rotating star can
easily be confused with that of a low-degree non-radial mode, if the
Coriolis force is important.
Our study of line-profile behavior has led to an atlas containing line profiles and their characteristics, for various values of the pulsation and relevant stellar parameters. This atlas can serve as a useful guide for those who plan to perform an analysis of observed line-profile variations in many types of rotating pulsating stars.
Figure 6: a)
Same as Fig. 4a
but for inclination i and azimuthal
order m. Fixed parameters are: 
= 6,
= 0.0, W = 0.10 
,
= 0.15 
, 
= 0.20.
Note that the velocity moments, which are very sensitive to the
inclination angle, have a far more detailed dependence on i than
what can be seen from this 
grid
Figure 7: b)
Residual spectra, amplitude distribution and phase distribution of
the time series of line profiles shown in Fig.
6 (click here)a.
One of the things shown here is the canceling of line-profile
variability for modes with 
an odd number, that are observed
at an inclination of nearly 
. Another feature seen in
this figure is the fact that line-profile variations are not
necessarily at largest for inclinations around 
(Sect. 6.1.1 (click here)). Opposite to sectoral modes, the amplitudes
of line-profile variations from tesseral modes generally tend to
increase for decreasing inclinations. This illustrates that tesseral
modes are equally well capable of producing large moving bumps, if the
star is sufficiently inclined. We find that this behavior is not
affected by any other parameter. Also note the wiggles in the
amplitude distribution, which are commonly, but not exclusively,
found for low inclination profiles
Figure 8: a)
Same as Fig. 4a
but for m against relative rotation
rate 
. Fixed parameters are: 
= 7,
i = 
, W = 
, 
= 
, 
= 5.0
Figure 9: b)
Residual spectra, amplitude distribution and phase distribution of the
time series of line profiles displayed in Fig. 8 (click here)a.
This figure shows that the toroidal term due to the Coriolis force
hardly affects the line profiles of high-degree sectoral modes. On
the contrary, tesseral modes are affected. Extensive
calculations, of tesseral modes at the highest rotation rate allowed
in our model, revealed that the cases with 
an even number
show the zero-rotation characteristics of a low 
-value. This
is best illustrated by the amplitude distributions in the column with
m = -5. Also illustrated here (rightmost 2 columns) is the
breaking of the symmetry between prograde and retrograde modes, when
effects of rotation become important. Furthermore, the right column
is an illustration of the 
phase jump in 
, that
can occur when another source of variability in the line center (in
this case toroidal movements) dominates the effect of radial motions
(see Sect 6.1.2 (click here) and Fig. 15 (click here))
Figure 10: a)
Same as Fig. 4 (click here)a but for intrinsic line-profile width
W (expressed in units of 
) against the ratio of
the horizontal to the vertical velocity amplitudes 
. Fixed
parameters are: 
= 8, m = -8, i = 
,
= 0.15, 
= 
. From the right column it can be seen that, for high values
of 
, the bumps that travel from blue to red through the line
profile tend to disappear at the blue side of the line center, and
reappear at the red side. This behavior of high 
, low W
profiles disappears for the larger values of the intrinsic profile
width W
Figure 11: b)
Residual spectra, amplitude distribution and phase distribution of the
time series of line profiles displayed in Fig. 7a.
The
figure shows the blurring of the line features for an increasing width
of the intrinsic profile. The change in contrast of the bumps with W
is most evident from the grey-scale residual plots. For large
intrinsic profile widths W and any value of k, the amplitude
distribution always has a shape that was previously considered as
characteristic for a lower k-value and more narrow intrinsic
profile. Also, the relation between the blue-to-red harmonic phase
difference and the value of |m|, is much less evident for the
higher values of W
Figure 12: a)
Same as Fig. 4a
but for velocity amplitude 
against intrinsic line-profile width W (both expressed in
units of 
). Fixed parameters are: 
= 6,
m = -6, 
= 0.5, i = 
,
= 0.10. This figure clearly illustrates the
increase of line-profile variability with increasing pulsation
velocity and with decreasing intrinsic width. Also illustrated here,
is that the variations of 
and 
are independent of W
Figure 13: b)
Residual spectra, amplitude distribution and phase distribution of the
time series of line profiles displayed in Fig. 8a.
This figure shows the dependence of the maximum ratio of amplitudes
on the surface velocity amplitude 
and the intrinsic profile width W (Sect. 6.1.1 (click here)). A
higher value of W or a lower value of 
leads to
more sinusoidal line-profile variability, i.e. a smaller value of
Figure 14: a)
Same as Fig. 4a
but for the ratio of the horizontal to
the vertical velocity amplitudes 
against the relative
rotation rate 
. Fixed parameters
are: 
= 5, m = -5, i = 
, W =
, 
= 
. The
empty fields are for combinations of 
and
that imply an equatorial rotation velocity of
more than 
of break-up (see Sect. 5.1 (click here))
Figure 15: b)
Residual spectra, amplitude distribution and phase distribution of the
time series of line profiles displayed in Fig. 9a.
It is clear that for this pulsation mode the effects of rotation on the
line-profile variability are very small. This figure also serves as
an example of the 
jump in 
(see
Sect 6.1.2 (click here)), which occurs for intermediate to high
k-values, if additional variability (in this case the toroidal
motion induced by the Coriolis force) is present at the line center.
An additional source of variability at the line center can reconnect a
bump at the blue side to a different one at the red side, leading to a
jump in the blue-to-red phase difference
Acknowledgements
We are grateful to Ilse De Boeck for the use of her computer code, which determines the k-values by means of numerical integration of the system of equations describing the pulsations up to first order in the rotation frequency, prior to publication. We thank Jan van Paradijs for his supportive remarks and his careful reading of the manuscript. CA is grateful to Dr. Alex Fullerton for initializing her into the field of Doppler Imaging and CLEAN. This research is supported by the Netherlands Foundation for Research in Astronomy (NFRA) with financial aid from the Netherlands Organization for Scientific Research (NWO), under project 781-71-043 (JHT).