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Astron. Astrophys. 362, 245-254 (2000)

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4. Observational guidelines for mode identification

Each mode excited in a pulsating star is characterized by three quantum numbers. The process of determining l, m and n is known as mode identification. The azimuthal quantum number (m ) is mostly obtained from spectroscopic observations. The horizontal quantum number (l ) and the radial order (n ) can be found from photometric observations.

Many authors (Watson 1988; Stamford & Watson 1981; Cugier et al. 1994; Garrido et al. 1990; Breger et al. 1999; Balona & Evers 1999 and Garrido 2000) discussed which are the best choices of filter bands for getting nonadiabatic observables. In principle, a relatively large baseline in wavelength is suggested. The light of different wavelengths comes from different radial layers of the star. The limb-darkening has a different effect for light of different wavelengths. The nonadiabatic observables based on a larger baseline in wavelength contain more information about the star. However, because of practical observational reasons Johnson B, V and Strömgren b, y colours are most commonly used for mode identification.

Although we have uvby colours of [FORMULA] Tucanae, the most important part of our investigation is based on only b, y colours and b-y colour index. A conclusion of the stability test by Paparó (2000) suggests that our almost 1500 u, v data give criteria for mode identification only in a limited way. It seems that neither the SAAO nor the ESO data are long and properly distributed enough to get a stable solution for the nonadiabatic observables. As soon as we put the two data sets together we step over the critical length and distribution of the data in respect to the given frequency spectrum of [FORMULA] Tucanae and from the stable solution the correlations between the nonadiabatic observables reveal. The phrase stable solution is used in the present paper if the solutions of Fourier parameters (frequencies, amplitudes and phases) are the same within the error bars for two distinct data sets or for a shorter and longer data sets.

The first result that we should like to emphasize is that how important the critical length and distribution of data are in a given mode identification. In the past the mode identification of stars with complex frequency spectrum perhaps failed because of the unstable solution of nonadiabatic observables, not because of the small baseline in wavelength. The statistical method of Balona & Evers (1999) can be regarded as a bridge over the unstable solutions. However, combining the different colours in a statistical method we lose part of the information coming from different layers of the star.

In the present investigation we check the planes of nonadiabatic observables based on small baseline in wavelength but the solutions are stable for the whole data set of the multisite campaign.

A speciality of the present investigation is that not only the traditionally suggested (Watson 1988 and Garrido 2000) planes but each combination of the nonadiabatic observables including phase difference vs. phase difference planes were checked for observational guidelines.

A search for any kind of correlations between the nonadiabatic observables was carried out. Any definite, regular structure of the planes was recognized but only the higher level structures are presented. We do not mention the planes where no definite structures were noticed.

4.1. Straight lines

Distribution of points along a straight line used to be regarded as a definite correlation between parameters investigated. The straight line structure for different nonadiabatic observables is presented in Fig. 1. Symbols are interpreted in the next paragraph. Mostly the same kind of nonadiabatic observables, amplitude ratio vs. amplitude ratio or phase difference vs. phase difference for different colours, distribute along a straight line in our investigation. The modes along the straight lines do not show any systematic arrangements in the different panels except the first one. We should call, however, attention to two facts. Each graph presented in Fig. 1. includes parameters in u or v colours which are the least precisely obtained observables. Furthermore, graphs are presented in the paper by Paparó (2000) where distribution of modes along straight lines are caused by the unstable solution of nonadiabatic observables as a consequence of inproper data sets. We present these straight lines as examples which are not suggested to be used for any kind of conclusion in mode identification.

[FIGURE] Fig. 1. Unstable solutions of amplitude ratios and phase differences are distributed along straight lines

4.2. Groups

The second definite structure that we noticed is the group of modes. We call the closely spaced modes a group if the error bars of the modes are practically distinct from the error bars of modes in another group. On the other hand the mean value of the closely spaced modes in a certain group differs from the mean value of the other group by more than 10% of the lower mean value. Of course the grouping is more definite if the scatter of modes around the mean value is smaller.

Both definite and looser groupings are presented in Fig. 2Three definite groups of modes exist according to [FORMULA] shown in the first panel . The grouping of modes is so distinct, the scatter around the mean value is so small that the modes in a certain group are marked by the same symbol in each figure. It is supposed that the modes in a group have the same value for one of the quantum numbers and may have similar behaviour according to other nonadiabatic observables. The frequency values which belong to modes in a certain group and the symbols are listed in Table 2. These groups contain closely spaced modes not only according to [FORMULA] amplitude ratio but according to the frequency values. These modes are referred to in the rest of the paper as Group I, II and III.

[FIGURE] Fig. 2. Grouping according to [FORMULA] and unusual behaviour of modes of Group III in u

The mean value of amplitude ratios involved in our graphs are given in Table 3 for each group. Error bars are not given here and in Table 4 since the error bar of the individual modes presented on each figure gives more severe constraint for the groups. Nevertheless, the numerical values confirm how definite is the grouping according to [FORMULA] amplitude ratio.


Table 3. Mean value of amplitude ratio for the Groups


Table 4. Mean value of the phase differences for the levels

A more loose grouping of the same modes along a straight line according to [FORMULA] can be seen in the second panel of Fig. 2. The scatter of modes is larger around the mean value in a certain group but the mean values in Table 3 definitely show the grouping of modes.

Both panels of Fig. 2 contain information for the unusual behaviour of modes in u for Group III. In the first panel the [FORMULA] amplitude ratio is less than 1.0 while for the other modes this value is larger than 1.0. In the second panel there is a break in the straight line because of the lower [FORMULA] value of modes in Group III. Both panels obviously confirm the conclusion what we previously mentioned based on Table 1 namely, that modes in Group III have lower amplitude in u than in v colour. This can be seen also from the appropriate columns of Table 3.

A specific model calculation for [FORMULA] Tucanae obtained by Luis Balona shows that [FORMULA] ratio is lower than 1.0 for modes with larger frequency. However, Group III consists of modes with the lowest frequency values.

In Fig. 3 a special arrangement of modes on the [FORMULA] vs. [FORMULA] plane can be seen. The grouping of modes according to [FORMULA] is similar to the grouping based on [FORMULA]. However, one mode from each group has a special behaviour according to [FORMULA] and they are situated on a crossing straight line or simply shifted according to [FORMULA]. The frequency values for the unusually behaving modes are given in the figure. Since the v colour data is shorter than b and y , the panel is worth showing because of the definite structure but no conclusion is given.

[FIGURE] Fig. 3. Special arrangement of modes in [FORMULA] versus [FORMULA] plane

One axis of each panel in Fig. 4 shows the special behaviour of modes in Group III according to [FORMULA]. A higher, distinct level of [FORMULA] exists in the traditional plane used for mode identification. Modes in Group III display 25% higher mean values in [FORMULA]. It can be seen in the appropriate coloumn of Table 3. Group I and II are separated not only according to [FORMULA] but a slight separation according to [FORMULA] also seems to exist in the third panel of Fig. 4. Of course, the error bars of amplitude ratio including the colour index amplitude are larger because of the smaller amplitude variation in colour indices.

[FIGURE] Fig. 4. Higher, distinct level of modes in Group III according to [FORMULA]

4.3. Levels

The third definite structures are the levels. The novelty of the present study is a check for correlation among the phase differences. Beside the traditionally used [FORMULA], the [FORMULA] phase difference proved to be a surprisingly good criterion for discrimination among the modes. We would like to emphasize at the first moment that both phase differences are obtained from a six-week long multisite campaign. The values plotted definitely belong to a stable solution.

In both panels of Fig. 5 the discriminative power of the [FORMULA] phase difference is shown. In the first panel the location of modes on the plane of the two phase differences is given. It seems obvious that the [FORMULA] phase difference is more discriminative for the modes than [FORMULA]. Along the [FORMULA] axis the modes are continuously distributed, while according to [FORMULA], two definite levels can be seen besides the third one. Since the third level contains only one mode (17.85 c/d) with the lowest amplitude among the frequencies, only the first two levels are mentioned as definite. It is worth mentioning that a certain level contains modes from different groups. The extention of error bar to the first level in the case of the first mode on the second level is acceptable since this mode (17.54 c/d) has one of the lowest amplitudes. Its amplitude is as low as for the single mode on the third level. Nevertheless, its location on the second level is convincing. The mean values of the levels for different colours are given in Table 4.

[FIGURE] Fig. 5. Distinct level of modes according to [FORMULA]

The [FORMULA] phase difference to [FORMULA] and [FORMULA], respectively, is also checked but the error bars are larger and the structure is not so clear. The lower levels are similar to the panel for [FORMULA] but the higher levels are not constant lines anymore but decreasing straight lines.

Keeping the traditionally used [FORMULA] amplitude ratio, the [FORMULA] phase difference is used on the horizontal axis of the second panel. Not only the higher distinct level of [FORMULA] for modes in Group III can be noticed but the two vertical levels of modes according to [FORMULA] is displayed. This new version of amplitude ratio vs. phase difference plane contains more information about the behaviour of modes to each other than the traditionally used plane shown in the first panel of Fig. 4.

No doubt that the phase difference [FORMULA] proves to be a new, useful criterion for mode identification , at least in [FORMULA] Tucanae.

In Fig. 6 the two strongest criteria, [FORMULA] versus [FORMULA] are plotted. The modes are obviously arranged, according to two parameters, on levels in groups. The values and error bars of the two strongest criteria are given in Table 5. This plane can serve as an observational guideline for mode identification. Although it does not give the exact value of quantum numbers but predicts which theoretical modes must share the same behaviour. As we have previously mentioned a finding of the horizontal quantum number (l ) and the radial order (n ) based on nonadiabatic observables has theoretical base. The two parameters in Fig. 6 may be the radial order (n ) and the spherical harmonic degree (l ). Following the theoretical predictions for [FORMULA] Scuti stars the phase difference gives the horizontal quantum number of modes. [FORMULA] amplitude ratio gives the amplitude ratio of eigenfunction in a different radial layer of the star. It is plausible to suppose that it gives information about the radial order of modes.

[FIGURE] Fig. 6. Groups and levels according to the two strongest observational criteria, [FORMULA] and [FORMULA]. Discriminative plane for mode identification


Table 5. Values and error bars of the most important nonadiabatic observables

As we have shown in this paragraph observational guidelines for mode identification seem to exist for [FORMULA] Scuti stars. These guidelines can help to reduce the number of suitable models to a unique solution. However, calibration of the pure observational guidelines by new, delicate theoretical calculation for planes of nonadiabatic observables concerning the location of theoretical modes is needed not only for l but with the radial order at the same time.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100