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Astron. Astrophys. 341, 151-162 (1999)

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4. Pulsation models for FG Vir

Since the initial discovery of multiperiodicity of FG Vir, several studies attempted to fit the observed and theoretical frequency spectra of the star, viz. Breger et al. (1995), Guzik, Templeton & Bradley (1998), and Viskum et al. (1998). We will now calculate new models utilizing the newly discovered pulsation frequencies and mode identifications.

4.1. Method of computation

To compute models of FG Vir we used a standard stellar evolution code which was developed in its main parts by B. Paczyski, M. Kozowski and R. Sienkiewicz (private communication). The same code was used in our recent studies of period changes in [FORMULA] Scuti stars (Breger & Pamyatnykh 1998) and in a seismological study of XX Pyx (Pamyatnykh et al. 1998). These two papers include detailed descriptions of the model computations, so that the present decription can be brief. For the opacities, we used the latest version of the OPAL or the OP tables (Iglesias & Rogers 1996 and Seaton 1996, respectively) supplemented with the low-temperature data of Alexander & Ferguson (1994). In all computations the OPAL equation of state was used (Rogers et al. 1996).

The computations were performed starting with chemically uniform models on the ZAMS, assuming typical Population I values of hydrogen abundance, X, and heavy element abundance, Z. The initial heavy element mixture of Grevesse & Noels (1993) was adopted.

In some models, a possibility of overshooting from the convective core was taken into account. The overshooting distance, [FORMULA], was chosen to be [FORMULA], where [FORMULA] is the local pressure scale height at the edge of the convective core. Examples of evolutionary tracks for [FORMULA] Scuti models computed with and without overshooting are given in Breger & Pamyatnykh (1998).

In the stellar envelope the standard mixing-length theory of convection with the mixing-length parameter [FORMULA] = 1.0 or 2.0 was used. As we will see below, the choice of the mixing-length parameter [FORMULA] has only a small effect on our models, because they are too hot to have an effective energy transfer by convection in the stellar envelope.

In all computations we assumed uniform (solid-body) stellar rotation and conservation of global angular momentum during evolution from the ZAMS. These assumptions were chosen due to their simplicity. The influence of rotation on the evolutionary tracks of [FORMULA] Scuti models was demonstrated by Breger & Pamyatnykh (1998). We studied models of FG Vir with equatorial rotational velocities from, approximately, 30 to 90 km/s (on the ZAMS, the values are 5-10 km/s higher). This range is consistent with the values of [FORMULA]km/s and [FORMULA] found by Mantegazza et al. (1994) and an equatorial velocity of [FORMULA]km/s obtained by Viskum et al. (1998). At such low rotational velocities, the evolutionary tracks are located very close to those for non-rotating stellar models. The main effect of rotation to be considered is the splitting of multiplets in the oscillation frequency spectra. This splitting is non-symmetric even for slowly rotating stars, if second-order effects are included.

The linear nonadiabatic analysis of low-degree oscillations ([FORMULA] [FORMULA] 4) was performed using the code developed by Dziembowski (1977). In the modern version of the code, effects of slow stellar rotation on oscillation frequencies are taken into account up to second order in the rotational velocity (Dziembowski & Goode 1992, Soufi et al. 1998).

4.2. Model constraints using oscillation data

The models for FG Vir were constructed with the observed mode [FORMULA] (12.154 c/d) being identified with the radial fundamental mode (=F ) (see Sect. 3). Note that this determines the mean density of all possible models of FG Vir: with the pulsation constant of about 0.032 - 0.034 days, which is typical for [FORMULA] Scuti variables, we obtain [FORMULA] 0.15-0.17. A considerably more accurate value of the density will be obtained later in this section.

We started with the construction of evolutionary tracks of 1.75 - 1.95 [FORMULA] models for initial abundances [FORMULA] and [FORMULA] and using OPAL opacities. No overshooting from the convective core was allowed. The initial equatorial rotational velocity on the ZAMS was chosen to be 50 km/s. With our assumption of conservation of global angular momentum, the equatorial rotational velocity is decreasing during the MS-evolution from 50 km/s at the ZAMS to about 40-41 km/s at the TAMS (Terminal-Age-Main-Sequence). The evolutionary tracks are shown in Fig. 4 together with the range in effective temperature and gravity of FG Vir (see Introduction) derived from photometric calibrations ([FORMULA] K and [FORMULA]). This range requires MS models and constrains the mass of the models to 1.75-1.95 [FORMULA]. The position of the models, whose F -mode has a value of 12.154 c/d is also shown. This further constrains the mass to 1.815-1.875 [FORMULA]. We stress that this strong seismological mass constraint depends on an accurate effective temperature determination.

[FIGURE] Fig. 4. Evolutionary tracks of 1.75-1.95 [FORMULA] standard models. The equatorial rotational velocity on the ZAMS was chosen to be 50 km/s. On the TAMS (at the turn-off points to the left), equatorial velocities are of about 40-41 km/s. Dashed lines show effective temperature and [FORMULA] ranges of FG Vir from photometric calibrations. The thin solid line connects models whose radial fundamental mode frequency is 12.154 c/d. The rotational velocity of the models along this line is about 45 km/s

The identification of [FORMULA] F agrees well with the gravity estimate for FG Vir. This provides no additional constraints on the stellar mass since the lines of constant frequencies are approximately parallel to those of constant gravity, as shown in the lower panel of Fig. 4.

For a given family of stellar models, the radial fundamental pulsation constant, Q, is constant with a quite high accuracy due to the homologous structure of models of different masses. For M = 1.80 - 1.90 [FORMULA] in the range [FORMULA] = 3.869 - 3.881 ([FORMULA] = 7400 - 7600 K), the pulsation constant Q = 0.0326 with relative accuracy of about 0.2%. The accuracy will be still higher by approximately one order of magnitude if we consider only models based on [FORMULA] F . For these models we determine a mean density of [FORMULA]. However, such an extremely high accuracy is based on a fixed choice of input physics: stellar opacity, initial chemical composition, rotational velocity and parameters of convection. We will see in the next subsection that changing these parameters results in significantly larger spread of mean density for FG Vir models.

The strict constraint on mass is demonstrated in Fig. 5, where the changes of radial and dipole frequencies during MS evolution are shown. This agreement is an independent qualitative argument in favour of the proposed models for FG Vir. Another important result is the good agreement of the predicted frequency range of unstable modes with the observed frequency range of 9-34 c/d. An additional test shows that should we identify [FORMULA] with the first radial overtone instead of the F -mode, we cannot achieve agreement between the theoretical and observed frequency ranges: in the corresponding models of [FORMULA] the instability occurs in the frequency range of 8-30 c/d. The tendency in models of higher mass to shift the instability range to lower frequencies can be also seen in Fig. 5. There is an even stronger argument against these higher-mass models: their luminosities are too high ([FORMULA]) to be consistent with both the photometric calibrations and the Hipparcos parallax.

[FIGURE] Fig. 5. Main-sequence evolution of low-order frequency spectra of radial and dipole oscillations of stellar models with masses 1.80, 1.85 and 1.90 [FORMULA]. In each panel, lefmost and rightmost points correspond to the ZAMS and to the TAMS models, respectively. Large filled circles denote unstable modes. For simplicity, for [FORMULA] only axisymmetric ([FORMULA]) components of the dipole multiplets are shown. Rectangular boxes mark the observational frequency and effective temperature range of FG Vir. The vertical line in each panel denotes a model whose radial fundamental frequency (F -mode) fits the observational frequency [FORMULA] = 12.154 c/d. Only models with masses 1.815-1.875 [FORMULA] fit the allowed temperature range

A number of gravity modes must be excited in FG Vir, if the assumption of [FORMULA] F is true, because the two lowest frequencies are more than 25% lower than the F -mode. Moreover, during the MS-evolution the frequencies of low-order g-modes increase and approach consecutively the frequencies of low-order p-modes resulting in mode interactions and avoided crossings (see Unno et al. 1989 and references therein). The frequency spectrum is much more complicated than in the case of pure p- or g-modes, as shown in Fig. 5 for dipole modes. The avoided crossing phenomenon takes place approximately in the middle of the observed frequency interval. Therefore, most of the excited modes at these and at lower frequencies are of mixed character: they behave like p-modes in the envelope and like g-modes in the interior. In the 1.85 [FORMULA] model with [FORMULA], modes [FORMULA], [FORMULA] and [FORMULA] are of mixed character. The frequencies of modes at avoided crossing are sensitive to the structure of the deep stellar interior. Consequently, the detection of these modes is important for testing convective overshooting theories (Dziewbowski & Pamyatnykh 1991).

Avoided crossings for quadrupole modes in the models of FG Vir occur close to the upper border of the observed frequency interval and also close to the F -mode. This means that most of [FORMULA] p-modes in the interval already interacted with gravity modes and are of mixed character.

4.3. Effects of different input parameters on the FG Vir models

The 1.85 [FORMULA] model for FG Vir, which was discussed in the previous subsection, will be referred to as the standard or reference model with the input parameters: [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] km/s and OPAL opacities. To examine the effects of varying input parameters on the predicted frequency spectrum, all these and the stellar mass were varied, under the condition that [FORMULA].

The changes introduced by using different opacities or non-standard chemical composition were mainly compensated by changes in mass, in order to fulfill the only identification we use.

The main characteristics of twelve models of that series are given in Table 3. Model 2 differs from model 1 (our reference model) in mass; models 3, 4, 5 - in rotational velocity; model 6 versus 1 will demonstrate effect of changing the mixing-length parameter [FORMULA]; model 7 versus 1 will show effect of the overshooting; models 8 and 9 have non-standard chemical composition; finally, models 10, 11 and 12 differ from model 1 in opacity (additionally, overshooting is taken into account in the model 12).


Table 3. Parameters of FG Vir models with [FORMULA]. The symbols have their usual meaning (see text). For the opacity, [FORMULA], the OPAL, OP or arficially modified OPAL data were used. [FORMULA] is the ratio of frequency of radial fundamental mode, [FORMULA], to that of third overtone, [FORMULA]

Note the significantly larger spread in stellar mass between different models (1.72-2.00 [FORMULA]) than for the mass interval of 1.815-1.875 [FORMULA] in the case of the standard choice of input parameters as was discussed in the previous subsection. The same is true for the mean density range: in Table 3 it varies between [FORMULA] = 0.1542 and 0.1597 (or between 0.1558 and 0.1584 when using only OPAL opacities). This spread is at least one order of magnitude larger than for the standard input data. Nevertheless, this seismic estimate of the mean density, which is based both on the well determined effective temperature and the one identification we are using, provides a strong constraint on possible FG Vir models  2.

We note that besides quite different stellar masses of 1.7 - 2.0 [FORMULA] (see Table 3) the evolutionary tracks for all 12 models in their MS-part lie well inside the region of 1.80 - 1.90 [FORMULA] of the standard set. Including a luminosity estimation [FORMULA] from trigonometric parallax determined by Hipparcos, all MS model tracks pass the error box, as well as the error box in the [FORMULA]-[FORMULA]-diagram. On the contrary, none of the post-MS models fits such a combination of parameters.

4.4. The problem of the radial frequency ratio

Viskum et al. (1998) identified [FORMULA] as a radial mode. We note here that the phase-difference method presented earlier in this paper allows both [FORMULA] = 0 and 1 for [FORMULA], i. e. radial as well as nonradial pulsation. We will now examine the radial hypothesis. In the last column of Table 3 the ratio of frequencies of the radial fundamental mode, [FORMULA], and of the third overtone, [FORMULA], is given ([FORMULA]). For models 1-10 these values are close to, but not equal to the observed ratio of [FORMULA], independent of which parameter was changed. This can also be seen in Fig. 6, where the ratio [FORMULA], is plotted against [FORMULA] for a wide range of parameters of [FORMULA] Scuti star models. There are well-defined monotonic variations of this ratio with changing mass, effective temperature or chemical composition, but the observed ratio disagrees with all these results.

[FIGURE] Fig. 6. Frequency ratio of the radial fundamental mode to the third overtone for a wide range of parameters of [FORMULA] Scuti star models and of some FG Vir models from Table 3 (asterisks). The large filled circle corresponds to the observed frequency ratio, [FORMULA], of 0.5193. Only the models with artificially modified opacities (such as model 12 of Table 3) can fit the observed ratio

The only exception is the model 12 with artificial opacities, which was constructed in the following way. For FG Vir models, the frequency ratio is most sensitive to the choice of opacities as it can be seen from comparison of models 10 and 1 (OP versus OPAL opacities). Physically, OP data differ from those of OPAL by underestimation of collective effects in stellar plasma, therefore OP opacities are systematically lower than OPAL in deep stellar interiors. For FG Vir models, this difference in opacity is about 20% at temperatures above [FORMULA] K. In the envelope, at lower temperatures, OP opacity varies slightly more monotonously along radius than does OPAL opacity: some dips are slightly shallower and some bumps are more flat. The differences do not exceed 8%: for example, at a temperature of [FORMULA] K the OP opacity is 4% smaller and at a temperature of [FORMULA] K it is 7.5% higher than the OPAL opacity.

Using the fact that the difference in frequency ratios between model 1 (OPAL) and model 10 (OP) is comparable with the difference between model 1 and the observations (but these differences are of opposite sign, see Fig. 7), we performed a very simple numerical experiment: we artificially scaled OPAL opacities with a factor, which is the ratio of OPAL to OP data. More clearly, we used [FORMULA]. Models 11 and 12 were constructed using [FORMULA]. For model 12 we additionally set [FORMULA] to [FORMULA] and lowered the rotational velocity. This model fits the observed frequency ratio very nicely as demonstrated in Fig. 7. However, this agreement should not be construed as an indicator for a new revision of atomic physics data on opacity, since the mode identification from Viskum et al. (1998) may not be unique due to the size of the error bars. Moreover, we cannot exclude additional effects like nonlinear mode interaction or rotational mode coupling, which may influence the frequency spectrum. In the last section the problem of rotational mode coupling will be briefly discussed. The observed variability of the amplitude of mode [FORMULA] is another argument in favour of possible nonlinear mode interaction.

[FIGURE] Fig. 7. Frequency spectra of radial, dipole and quadrupole oscillations of various FG Vir models. Axisymmetric modes ([FORMULA]) are marked by enlarged circles. Model numbers (see Table 3 for parameters) together with some model indicators are given to the right of the panels. Vertical solid and dashed lines correspond to observed frequencies - statistically significant and probable, respectively. Numbers above some observed frequencies give identifications for the degree of the modes ([FORMULA]) based on multicolor photometry data and on the results by Viskum et al. (1998)

Viskum et al. (1998) were able to interpret the observed frequency ratio [FORMULA]/[FORMULA] as the radial frequency ratio [FORMULA]. They did not construct full evolutionary models but scaled a model of 2.2[FORMULA], which was selected to match the observed frequency ratio with the radial frequency ratio [FORMULA]. Using the homology argument, they estimated the mean density of the true FG Vir model by multiplying the mean density of the 2.2[FORMULA] model by the square of the ratio [FORMULA]/[FORMULA]. In such a way an agreement between observed frequencies [FORMULA], [FORMULA] and a pair of radial modes of the scaled model was achieved by definition. The estimated gravity, luminosity and distance of the scaled model were found to be in good agreement with the photometric and the spectroscopic data and with the Hipparcos parallax. The authors noted that the high precision of their asteroseismic density estimate ([FORMULA]) is based on a fixed (solar) metallicity for FG Vir. Indeed, with our standard choice of chemical abundances and opacity and assuming the fit [FORMULA] we estimated the mean density of the FG Vir model with even five times higher accuracy (see Sect. 4.2), but possible variations of the global parameters result in at least one order of magnitude worse precision of this estimate.

4.5. Theoretical frequency spectra versus observations

Frequency spectra of radial, dipole and quadrupole modes for all 12 models from Table 3 are shown in Fig. 7. The effects of different choices of input parameters can be estimated by comparison of the results for different models. We discuss here both general properties and some peculiarities of these frequency spectra.

For nonradial oscillations, evolutionary overlapping of frequency intervals of g- and p-modes (see Fig. 5) results in avoided crossings, which disturb the approximately equidistant frequency spacing between acoustic multiplets. Gravity and mixed modes are very sensitive to the interior structure as can be seen for models of different chemical composition, different opacities and for models with and without overshooting. On the contrary, the change of [FORMULA] (model 6 versus model 1) has a negligible influence on the frequency spectrum due to ineffective convection in the relatively hot envelope of FG Vir.

Rotation splits nonradial multiplets and strongly complicates the frequency spectra. Except for models with slowest rotation, we observe a forest of quadrupole modes in the low-frequency part of the interval, with overlapping components of the different multiplets. The common property of the spectra is a large asymmetry of the rotational splitting, which is caused by the second-order effects of rotation (Dziembowski & Goode 1992). The asymmetry is higher the higher the order of the p-modes is.

It is not trivial to select a model reproducing the observed frequencies exactly. Simple attempts to minimize frequency differences (O-C) by a combined variation of input parameters of the reference model fail due to strong and non-linear sensitivity of gravity modes to interior structure. At the same time, this strong sensitivity may help to fit some chosen frequencies without changing the rest of the spectrum, (cf. models 1 and 7, for example). It is obvious from Fig. 7 that generally it is much easier to fit a low-frequency mode than a high-frequency mode, because the spectrum is more dense at lower frequencies. For some of the observed modes there is no satisfactory [FORMULA] solution: see, for example, the group [FORMULA], [FORMULA], [FORMULA] around 21 c/d or [FORMULA] at 33 c/d. Besides geometric cancellation, there are no objections to identify frequencies in the gaps with modes of degree [FORMULA]=3 and [FORMULA]=4. Note that even for [FORMULA] the number of unstable modes is a few times larger than the observed one: in the observational frequency interval there are 6-7 radial modes, 24 dipole modes and 50-55 quadrupole modes. Therefore, a presently unknown mode selection mechanism must exist.

Note that most of the models presented in Fig. 7 show a good fit of the dominant observed mode [FORMULA] (12.716 c/d) with [FORMULA] mode of [FORMULA] or 0. This is in agreement with the mode identification from photometric phase differences. On the contrary, it is quite difficult to achieve a similar fit with a dipole mode for the observed mode [FORMULA] (23.403 c/d). In Table 4 we present some results to quantify the fitting of 21 modes (corresponding to the observed frequencies [FORMULA] through [FORMULA], omitting [FORMULA]) for models 1 (reference model), 3 (low rotation), 8 ([FORMULA], [FORMULA]) and 12 (artificially modified opacity + overshooting). In the cases of close observed frequencies (for example, [FORMULA], [FORMULA], [FORMULA] around 24 c/d, or [FORMULA], [FORMULA], [FORMULA] around 21 c/d) we give a few possible identifications for each frequency. Moreover, if there is no [FORMULA] mode for a given observed frequency, we show the closest mode of [FORMULA] or 4: the frequency spectrum of these modes is so dense that practically everywhere in the observed interval a fitting within 0.1 c/d is possible.


Table 4. Best fits for some FG Vir models

Model 12 seems to be the best-fitting model in our series. Note the excellent agreements in both frequency and [FORMULA]-identifications for all ten dominant frequencies. The mean difference (O-C) is about 0.04 c/d for these modes. The fit for most of the remaining frequencies is also good, once [FORMULA] [FORMULA] modes are considered. Possible discrepances at low frequencies do not appear serious to us due to the strong sensitivity of g-modes to model parameters. Because of the effects of avoided crossing some of the quadrupole modes of higher order are also quite sensitive to small variations of parameters (cf. [FORMULA] modes for models 1 and 2 at approximately 32 c/d).

In Fig. 8 we compare the range of observed frequencies and of excited modes explicitly for the reference model as well as three other models. FG Vir is located in the HR-diagram in the middle of the instability strip (see, for example, Breger & Pamyatnykh 1998). Consequently, the driving of oscillations is effective in a wide frequency region which extends over 7 radial overtones. The independence of the driving efficiency on the mode degree, [FORMULA], is typical for the oscillations excited by the opacity mechanism.

[FIGURE] Fig. 8. Normalized growth-rates, [FORMULA], of low-degree oscillation modes of some FG Vir models from Table 4. Only axisymmetric modes ([FORMULA]) are shown. Positive values correspond to unstable modes. Vertical lines mark observed frequencies from Table 1, probable frequencies [FORMULA] to [FORMULA] are shorter. The longest line at 12.716 c/d corresponds to the mode [FORMULA] with the highest amplitude

We also note that the observed frequency spectrum is divided clearly into two groups: at 9-13 c/d and at 19-25 c/d. However, the present models do not predict any instability gap in frequency, as can be seen in Fig. 8.

4.6. Rotational mode coupling

An additional factor which influences the frequencies of a rotating star is the coupling between modes with close frequencies whose azimuthal orders, m, are the same and whose degrees, [FORMULA], are also the same or differ by 2. The effect was described and discussed in detail by Dziembowski & Goode (1992) and by Soufi et al. (1998). The frequency distance between two near-degenerate modes increases when coupling is taken into account. The significance of this rotational frequency perturbation was demonstrated by Pamyatnykh et al. (1998) in application to XX Pyx. It was shown that at rotational velocity of about 90 km/s the frequency shifts of coupled radial and quadrupole (or dipole and octupole) overtones achieve 0.1-0.2 c/d. Therefore, in particular, a significant change of radial frequency ratios may be expected.

We estimated this effect in some of our FG Vir models and found it to be unimportant at rotational velocities of about and less than 45 km/s. For example, for the reference model 1, the frequencies of the radial fundamental mode, [FORMULA], and of the third overtone, [FORMULA], are changed due to coupling with closest axisymmetric quadrupole modes by -0.0035 c/d and 0.0091 c/d, respectively. The effect is much stronger for more rapidly rotating models: for model 5 with [FORMULA] = 90 km/s, the radial fundamental mode and third overtone are shifted by -0.0436 c/d and 0.1545 c/d, respectively, which results in the change of the frequency ratio from 0.5266 to 0.5212. As another example, we were able to reproduce the observed ratio [FORMULA] as the radial frequency ratio for a model with initial abundances [FORMULA], [FORMULA] and with [FORMULA] = 91-92 km/s. However, the rapid rotation seems to be rather improbable for FG Vir with [FORMULA]km/s, as it was discussed by Viskum et al. (1998).

Note that the coupling effect is higher for higher overtones: for model 5, the shift of the radial sixth overtone frequency (34.306 c/d) is 0.520 c/d due to interaction with the closest axisymmetric quadrupole mode of 34.219 c/d which is shifted by the same quantity 0.520 c/d in the opposite direction. Moreover, [FORMULA] modes are affected by coupling with [FORMULA] modes, and so on. We conclude that for rapidly rotating models it is necessary to take the rotational coupling into account in attempts to fit the observed frequency spectrum with the theoretical one.

The rotational coupling results also in a mutual contamination of amplitudes of spherical harmonic components of interacting modes (Soufi et al 1998, see examples in Table 4 of Pamyatnykh et al. 1998). This adversely affects the mode discrimination by means of multicolor photometry (see the footnote in Sect. 3) and should influence spectroscopic determinations as well. However, this effect was found to be important only in models with more rapid rotation than found for FG Vir. A more detailed discussion of the rotational mode coupling problem in connection with the interpretation of the observed multifrequency spectrum will be given by Dziembowski & Goupil (1998).

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© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998