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7. Conclusions

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 tex2html_wrap_inline3767.

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.

  figure1079
Figure 4: a) Time series of line profiles and their first and second velocity moments, calculated for a grid of different combinations of tex2html_wrap_inline3769 and m. The remaining fixed parameters are: i = tex2html_wrap_inline3777, tex2html_wrap_inline3779 = 0.0, W = tex2html_wrap_inline3789, tex2html_wrap_inline3791 = tex2html_wrap_inline3795, tex2html_wrap_inline3797 = 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 (tex2html_wrap_inline3803). Variations of the first two velocity moments of high-degree modes (say tex2html_wrap_inline3805) 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))

  figure1092
Figure 5: b)
Residual spectra, amplitude distributions tex2html_wrap_inline3807 and tex2html_wrap_inline3809, and phase distributions tex2html_wrap_inline3811 and tex2html_wrap_inline3813 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 tex2html_wrap_inline3815, tex2html_wrap_inline3817, tex2html_wrap_inline3819 in units of tex2html_wrap_inline3821, and the total blue-to-red phase differences tex2html_wrap_inline3823, tex2html_wrap_inline3825, and tex2html_wrap_inline3827 in radians. We only show amplitude and phase diagrams for the input frequency (tex2html_wrap_inline3829, tex2html_wrap_inline3831) and its first harmonic (tex2html_wrap_inline3833, tex2html_wrap_inline3835). The grid on this page gives a typical example of the blue-to-red phase difference of the variations at the input frequency, tex2html_wrap_inline3837, being an indicator of tex2html_wrap_inline3839. 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, tex2html_wrap_inline3843, 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 tex2html_wrap_inline3847 - m with an inclination close to tex2html_wrap_inline3853.

The apparent number of bumps and troughs in the line profile is a measure of tex2html_wrap_inline3855 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 tex2html_wrap_inline3857 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 tex2html_wrap_inline3861 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 tex2html_wrap_inline3871 0.4tex2html_wrap_inline3873. 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 tex2html_wrap_inline3883 exceeds tex2html_wrap_inline3885.

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 tex2html_wrap_inline3887 - |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.

  figure1124
Figure 6: a) Same as Fig. 4a but for inclination i and azimuthal order m. Fixed parameters are: tex2html_wrap_inline3901 = 6, tex2html_wrap_inline3907 = 0.0, W = 0.10 tex2html_wrap_inline3915, tex2html_wrap_inline3917 = 0.15 tex2html_wrap_inline3919, tex2html_wrap_inline3921 = 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 tex2html_wrap_inline3929 grid

  figure1135
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 tex2html_wrap_inline3931 an odd number, that are observed at an inclination of nearly tex2html_wrap_inline3933. Another feature seen in this figure is the fact that line-profile variations are not necessarily at largest for inclinations around tex2html_wrap_inline3935 (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

  figure1145
Figure 8: a) Same as Fig. 4a but for m against relative rotation rate tex2html_wrap_inline3939. Fixed parameters are: tex2html_wrap_inline3941 = 7, i = tex2html_wrap_inline3951, W = tex2html_wrap_inline3957, tex2html_wrap_inline3959 = tex2html_wrap_inline3963, tex2html_wrap_inline3965 = 5.0

  figure1157
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 tex2html_wrap_inline3971 an even number show the zero-rotation characteristics of a low tex2html_wrap_inline3973-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 tex2html_wrap_inline3981 phase jump in tex2html_wrap_inline3983, 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))

  figure1169
Figure 10: a) Same as Fig. 4 (click here)a but for intrinsic line-profile width W (expressed in units of tex2html_wrap_inline3987) against the ratio of the horizontal to the vertical velocity amplitudes tex2html_wrap_inline3989. Fixed parameters are: tex2html_wrap_inline3991 = 8, m = -8, i = tex2html_wrap_inline4007, tex2html_wrap_inline4009 = 0.15, tex2html_wrap_inline4015 = tex2html_wrap_inline4019. From the right column it can be seen that, for high values of tex2html_wrap_inline4021, 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 tex2html_wrap_inline4023, low W profiles disappears for the larger values of the intrinsic profile width W

  figure1184
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

  figure1190
Figure 12: a) Same as Fig. 4a but for velocity amplitude tex2html_wrap_inline4041 against intrinsic line-profile width W (both expressed in units of tex2html_wrap_inline4045). Fixed parameters are: tex2html_wrap_inline4047 = 6, m = -6, tex2html_wrap_inline4059 = 0.5, i = tex2html_wrap_inline4069, tex2html_wrap_inline4071 = 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 tex2html_wrap_inline4077 and tex2html_wrap_inline4079 are independent of W

  figure1201
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 tex2html_wrap_inline4083 on the surface velocity amplitude tex2html_wrap_inline4085 and the intrinsic profile width W (Sect. 6.1.1 (click here)). A higher value of W or a lower value of tex2html_wrap_inline4091 leads to more sinusoidal line-profile variability, i.e. a smaller value of tex2html_wrap_inline4093

  figure1212
Figure 14: a) Same as Fig. 4a but for the ratio of the horizontal to the vertical velocity amplitudes tex2html_wrap_inline4095 against the relative rotation rate tex2html_wrap_inline4097. Fixed parameters are: tex2html_wrap_inline4099 = 5, m = -5, i = tex2html_wrap_inline4115, W = tex2html_wrap_inline4121, tex2html_wrap_inline4123 = tex2html_wrap_inline4127. The empty fields are for combinations of tex2html_wrap_inline4129 and tex2html_wrap_inline4131 that imply an equatorial rotation velocity of more than tex2html_wrap_inline4133 of break-up (see Sect. 5.1 (click here))

  figure1227
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 tex2html_wrap_inline4135 jump in tex2html_wrap_inline4137 (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 tex2html_wrap_inline4141 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).


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