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6. Results

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: tex2html_wrap_inline3413, m, tex2html_wrap_inline3417, tex2html_wrap_inline3419, i, tex2html_wrap_inline3423, and W. From these calculations, we selected a few grids to serve as illustrations in this paper (Figs. tex2html_wrap3429). 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. tex2html_wrap3431 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 tex2html_wrap_inline3427. This means that in some of the Figs. tex2html_wrap3433, the parameters that change along the axes of the grid imply stars with different mass, radius, rotation period and/or pulsation frequency.

6.1. Line-profile variations with the pulsation frequency and its harmonics

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 tex2html_wrap_inline3437 = 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 correspondencegif 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.


6.1.1. The amplitude distribution at harmonic frequencies

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, tex2html_wrap_inline3441, 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 tex2html_wrap_inline3443. We find that for all investigated cases tex2html_wrap_inline3445 > 1 and tex2html_wrap_inline3451 > 1. For most but not all cases we find tex2html_wrap_inline3457 > 1.

We find that tex2html_wrap_inline3463 depends mostly on W and on tex2html_wrap_inline3467. This is illustrated in Fig. 13 (click here). The relative harmonic amplitudes tex2html_wrap_inline3469 and tex2html_wrap_inline3471 increase for increasing pulsation amplitude tex2html_wrap_inline3473. 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 tex2html_wrap_inline3475.

We find weaker dependences of tex2html_wrap_inline3477 and tex2html_wrap_inline3479 on k and i. A small decrease of tex2html_wrap_inline3485 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 tex2html_wrap_inline3489 - m an odd number and with an inclination angle very close to tex2html_wrap_inline3495. 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 tex2html_wrap_inline3497 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 tex2html_wrap_inline3499 (see Paper III).


6.1.2. Determination of tex2html_wrap_inline3501 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 tex2html_wrap_inline3505 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 tex2html_wrap_inline3507 to identify |m| according to tex2html_wrap_inline3511. 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 tex2html_wrap_inline3513 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 tex2html_wrap_inline3517 corresponds more closely to tex2html_wrap_inline3519 than to |m|. In addition, we find that the phase difference tex2html_wrap_inline3523 at the first harmonic frequency tex2html_wrap_inline3525 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 tex2html_wrap_inline3529 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 (tex2html_wrap_inline3533,|m|)-value can be performed by an application of the moment method (for l tex2html_wrap_inline3539 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 tex2html_wrap_inline3543. See Fig. 2 (click here) for a description of the figures. For clarity we do not plot tex2html_wrap_inline3545 and tex2html_wrap_inline3547. Top row: radial pulsation (tex2html_wrap_inline3549 = 0). Bottom row: non-radial pulsation mode (tex2html_wrap_inline3553 = 2, m = -2). The values of the other parameters are (for both modes): tex2html_wrap_inline3561 = 0.15 tex2html_wrap_inline3563, W = 0.1 tex2html_wrap_inline3567, tex2html_wrap_inline3569 = 0.8 (not relevant if tex2html_wrap_inline3573 = 0), and i = tex2html_wrap_inline3579. 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 tex2html_wrap_inline3585 that differs by tex2html_wrap_inline3587 from the empirical relation with tex2html_wrap_inline3589 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 (tex2html_wrap_inline3595) 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 tex2html_wrap_inline3597 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 tex2html_wrap_inline3599 in tex2html_wrap_inline3601 for modes with the same tex2html_wrap_inline3603-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 tex2html_wrap_inline3607-movements; local brightness and EW changes). In this work, differences in tex2html_wrap_inline3609 of tex2html_wrap_inline3611 are found by using different parameters i and tex2html_wrap_inline3615 (see Figs. 9 (click here) and 15 (click here)).

A detailed analysis of the determination of tex2html_wrap_inline3617 and |m| from the phase diagrams at the input frequency and its harmonics is the subject of investigation in Paper II.


6.2. The effects of rotation on the line-profile behavior

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 tex2html_wrap_inline3625 > 4), the effects of rotation are negligible for values of tex2html_wrap_inline3629 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 tex2html_wrap_inline3635 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 tex2html_wrap_inline3643 = 2 mode and a radial (tex2html_wrap_inline3647 = 0) mode. It shows that radial modes in rotating stars can easily be confused with tex2html_wrap_inline3651 = -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 tex2html_wrap_inline3657 = 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 tex2html_wrap_inline3663-value and no node line at the equator (tex2html_wrap_inline3665 - 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 tex2html_wrap_inline3671-value (see Figs. 8 (click here)).

In his discussion of the so called k-problem, Smith (1986) mentioned that for high tex2html_wrap_inline3675-values, the toroidal term(s) caused by rotation might be able to mimic the amplitude-distribution characteristics of a low-tex2html_wrap_inline3677 mode. For sectoral modes with tex2html_wrap_inline3679 > 3 and values of tex2html_wrap_inline3685 tex2html_wrap_inline3687 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-tex2html_wrap_inline3691 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 tex2html_wrap_inline3695 between the prograde and the retrograde case (see rightmost 2 columns in Fig. 9 (click here)).


6.3. The difficulty to derive k-values from amplitude diagrams

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 tex2html_wrap_inline3705. For a somewhat higher value of k (depending on the values of m and tex2html_wrap_inline3711), 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 tex2html_wrap_inline3719 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 tex2html_wrap_inline3723 (see Fig. 11 (click here)). If tex2html_wrap_inline3725 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 tex2html_wrap_inline3731. 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 tex2html_wrap_inline3735. 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 tex2html_wrap_inline3749 exceeds tex2html_wrap_inline3751 radians. A double-peaked amplitude distribution together with a blue-to-red phase difference of less than tex2html_wrap_inline3753, can also occur for a radial or an tex2html_wrap_inline3755 mode.

6.4. Amplitudes of line-profile variations of tesseral modes

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 tex2html_wrap_inline3757 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)).

6.5. Retrieval of the inclination

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 tex2html_wrap_inline3763 and m are precisely known.

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