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

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5. Discussion

We should summarize what we know about the 10 pulsational frequencies of [FORMULA] Tucanae. For better understanding we reproduce here (Fig. 7) the schematic frequency spectrum of [FORMULA] Tucanae published in Paper I. The extremely high regularity of frequency spacing is obvious. The spectrum is dominated by groups of closely spaced frequencies. The groups seem to be equally spaced and are divided by single frequencies.

[FIGURE] Fig. 7. Schematic frequency spectrum of [FORMULA] Tucanae

According to [FORMULA] we distinguish three groups in the present investigation. The middle group on Fig. 7 corresponds to Group I. The (right side) higher frequency group and the single frequency correspond to Group II. Group III consists of the (left side) lower frequency group and the single frequency.

Modes in Group III have speciality in u colour and in [FORMULA] amplitude ratio. The modes have lower amplitude in u colour and higher [FORMULA] amplitude ratio than the modes in other groups. The [FORMULA] amplitude ratio is the highest for Group III, although Group I and II have increasing [FORMULA] amplitude ratio with increasing frequency. Since Group III does not show any special behaviour in [FORMULA], the modes have to have smaller amplitude in y or larger amplitude in b than the other groups. Group I and II are slightly separated according to [FORMULA].

Concerning the special behaviour of modes in Group III it would be a logical conclusion that these modes are excited in the secondary component of higher temperature in the binary system. Unfortunately, no spectral type has been obtained for the secondary component. However, the low mass [FORMULA] obtained from spectroscopy (Sterken et al. 1997) seems to exclude [FORMULA] Scuti type pulsation in the secondary. Furthermore, the similar regularity in frequency spacing to the other modes would be a hard job to explain if we do not involve a very severe tidal synchronization in the oscillation.

If these modes belong to [FORMULA] Tuc, serious questions can be raised. Can we find a region of excitation with higher temperature and/or nonadiabacity (larger [FORMULA]) what we need to explain the higher [FORMULA] amplitude ratio of modes in Group III? According to the present theoretical point of view the region of excitation is indifferent in respect of the observable behaviour of modes. A new investigation of the location of theoretical modes on the traditional comparison plane based on the actual model atmospheres has been recently published by Garrido (2000). For l = 3 the amplitude ratio [FORMULA] is very different (much higher) from the lower l - values. Should we identify the modes in Group III with l = 3 value? Such a straightforward conclusion does not seem to be well-established since the present paper gives the arrangement of modes according to two parameters simultaneously not only according to l .

In fact, the behaviour of the 10 pulsational modes on the levels seems to be unified. The modes in Group III join the same levels as the other modes. Not all modes in a group belong to the same level. This is a normal behaviour if we see groups connected to the radial overtones. According to the modelling of [FORMULA] Tucanae by Templeton et al. (2000) the frequency distances between the consecutive low radial orders of the radial modes are [FORMULA] 2.9 c/d for most of the models. However, the distances between the non-radial consecutive radial orders show variety of values from 2.9-1.6 cycles/day for Models 1-3 and 2.85-0.55 cycles/day for the Model 4. In the observation the dominant frequency spacing is 2.2 c/d.

In our view the sign of [FORMULA] is opposite to the sign of [FORMULA]. As a first approximation, the first level with [FORMULA] mean value is regarded as the location of radial modes l = 0 , while the l = 1 modes are situated on the second level. The third level corresponds to l = 2 . The groups would be the consecutive radial orders.

However, such an explanation creates some problem. There are pairs of modes with the same radial quantum number situated on the same level. It is obvious, especially for the closely spaced frequencies, that such a simple interpretation does not work. These modes may not be consecutive radial orders with the same l value. There are two ways of avoiding the duplicating of modes.

[FORMULA] In the first explanation the duplicating of modes with the same radial and horizontal quantum numbers are caused by rotational splitting. In this case the member of pairs have different azimuthal quantum number (m ). However, the values of splitting are remarkably different (1.12, 0.169, 0.77 and 0.52 c/d) for the different pairs and only two modes are seen instead of tripletts. More serious problem that the rotational splitting explanation does not work for the radial modes on the first level. Two of the modes (17.06 and 19.02 c/d) behaving unusually according to [FORMULA] on Fig. 3 are involved in pairs on the first and second levels.

The pair of 20.11 and 20.28 c/d should be separately mentioned since it is connected to a unique effect. On the upper panel of Fig. 8 in the paper by De Mey et al. (1998) a node seems to exist for the mode at 20.28 c/d . However, in the model calculations the outermost mode is located in much deeper layers even for high overtones. The layers (temperature is above 2x[FORMULA] oK) cannot be reachable by spectroscopic observations. A preliminary mode identification of [FORMULA] Tucanae matching the observed and rotationally splitted theoretical modes has been recently published by Templeton et al. (2000). This mode is supposed to be a radial mode and the best fit is the fourth overtone. It is worth mentioning that this is the dominant mode of [FORMULA] Tuc with the largest amplitude.

[FORMULA] In the second explanation we suppose that the levels give information only for the even (including zero) and odd consecutive l value of modes. The first level contains both l = 0 and l = 2 modes, while the second level contains modes with l = 1 and l = 3 . According to the asymptotic theory l = 0 and l = 2 further l = 1 and l = 3 modes, respectively, are closely spaced in frequency. However, what is the explanation for the third level in this scenario, l = 4 and 6 ? In the identification of Templeton et al. (2000) the two frequencies at 15.86 c/d and 15.94 c/d l = 2 n = 1 and l = 2 n = 2 values are given, respectively. In our figure these modes are situated on different levels. The identification of 15.94 c/d mode can be accepted but for 15.86 c/d l = 3 or 1 would agree with our discriminative plane.

The third level is connected to a hitherto not mentioned but interesting fact. The unusually behaving modes according to [FORMULA] on Fig. 3 (marked by asterisk in Table 6) are situated on three different levels and are different overtones. Two of them represent the single modes between groups in Fig. 7. The role of these modes is not clear. However, the frequency at 19.02 c/d exhibit strong g-mode type behaviour according to Templeton et al. (2000).


Table 6. Possible schematic and numerical identification of modes in [FORMULA] Tuc. The numerical values (marked as particular) are based on the second explanation if we accept the identification of 20.28 c/d given by Templeton et al. 2000 (Frequency f, [FORMULA] first explanation, [FORMULA] second explanation.

The following principles can help in the interpretation of the radial order. According to the asymptotic theory P(l, n) [FORMULA] P(l+2, n-1) relation is valid. If a pair exists in a group on a certain level the radial order has to be consecutive values. The radial orders in the same group but on different levels are shifted to each other by one consecutive radial order. The first level has the highest radial orders, the other levels have lower and lower values. The consecutive groups in frequency have consecutive radial orders decreasing to the direction of lower frequency. Fig. 8 gives the systematic arrangement of radial orders in this concept for [FORMULA] Tucanae. The groups are rearranged here according to increasing frequency but the location of modes to each other are kept as in the discriminative plane. Surprisingly simple arrangement of radial orders revealed. The rotational splitting is not involved in this explanation.

[FIGURE] Fig. 8. Systematics in radial orders for [FORMULA] Tucanae

According to the logical-theoretical calibration of Fig. 6 the possible schematic identification of modes are given in Table 6. [FORMULA], [FORMULA], [FORMULA] and [FORMULA] mean consecutive radial orders, [FORMULA] and [FORMULA] are different azimuthal order of rotationally splitted modes. The first column gives the frequencies. The 2 and 3 columns (marked as [FORMULA]) give the result of the first explanation, the 4 and 5 columns (marked as [FORMULA]) give the possible solution of the second explanation in general.

If we accept the l = 0, n = 4 identification of the frequency at 20.28 c/d given by Templeton et al. (2000) the resulting particular identification of modes based on the systematics are given in the last two columns of Table 6.

The exact calibration of the observational guidelines obtained in this investigation is definitely needed. How general the regularities are for other stars we do not know at this moment. How these regularities would be confirmed or modified by regularities in v and u we also do not know. How these regularities are really connected to the quantum numbers (l, n and m ) is a future task of detailed theoretical modelling. However, we can find some connection to investigations published in the literature.

Grouping in modes of FG Vir according to the [FORMULA] parameter has been published by Viskum et al. (1998). The modes in a group are interpreted as modes with the same l value comparing to other mode identification methods. However, one problem arises immediately. In the group l = 0 two modes are too closely spaced in frequencies for both to be radial modes. It is maybe the same duplicating effect what we have on the levels in our investigation. Fig. 6 of that paper can be an example for our interpretation concerning the stability of solution. The grouping of modes disappears, the modes are distributed along a straight line, as part of the data has been excluded. The stability of the solution is questionable for this particular data set.

Only a few published examples are mentioned but in the near future another [FORMULA] Scuti star (38 Eri) is going to be investigated by one of the authors (MP) following the concept of the present investigation.

The high level systematic behaviour of the observational facts in [FORMULA] Tucanae concerning both the regularities in frequency spacing, in grouping according to amplitude ratios and in leveling according to phase differences, give high probability to reduce the number of suitable models to a unique solution. The observational guidelines, hopefully, can serve as key to the mystic mode selecting and/or amplitude limiting mechanism in low amplitude [FORMULA] Scuti stars.

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Online publication: October 30, 19100