3. The triplet
Any attempt to understand the frequency spectrum of 12 (DD) Lac must begin with the equidistant triplet , , . We shall consider the following three possibilities of accounting for the triplet: (1) true rotational splitting of an or 2 mode, (2) an roAp-like rotational splitting, and (3) nonlinear phase lock.
In addition to the value of the central frequency, , the triplet can be characterized by the mean separation,
and the asymmetry,
Taking the values of the frequencies from Table 1, we get and d-1.
3.1. True rotational splitting
As we showed in Sect. 2.4, the spherical harmonic degree of the mode is either 1 or 2. In the case of rotational splitting, the and modes will both have also equal to 1 or 2. Since for a given there are spherical harmonic orders, m, there is only one possibility of accounting for the triplet if , and four possibilities if . If , the order of frequencies in the triplet implies (retrograde) for , 0 for , and -1 for . Note that this case is the only one consistent with the identification of from the amplitude ratios.
The four possible cases of accounting for the triplet when are listed in Table 2.
Table 2. The four cases of accounting for the , , triplet when
A number of modes of or 2 and , 0 or 1 with frequencies close to are unstable in models of 12 (DD) Lac in the range of effective temperature and mass allowed by the position of the star in the plane. An example is presented in Fig. 3. In this figure, computed frequencies of two modes are plotted together with (horizontal solid line). The designations and refer to the properties of the modes on ZAMS, where the p- and g-mode spectra are separated in frequency.
As can be seen from Fig. 3, can be fitted with the frequency of the mode in the whole range of . On the other hand, for the mode such fit is possible only at the lower limit of the temperature range and for the lowest-mass models which have still consistent with the observed one. In addition, an , , mode in models with and 13 (not shown in Fig. 3) also has its frequency close to .
Having identified the modes that could reproduce (those shown in Fig. 3 and many others), we varied the rotation frequency, , until computed frequencies of the outlying members of the triplet, and , were also fitted. The rule we adopted was that the observed mean separation of the triplet, S, should be reproduced exactly. The interaction between rotation and pulsation was treated as in Soufi et al. (1998), except that only terms up to quadratic in were kept in the perturbation treatment. The effect of near degeneracy of modes differing in by 2 was taken into account.
Out of a large number of models computed in this way, we selected those which (1) had the smallest asymmetry, , and (2) reproduced - at least approximately - the two frequencies outside the triplet, and . The parameters of these models are presented in Table 3. The first three columns identify the mode (the asterisk indicates case 4 of Table 2), is the equatorial velocity of rotation in km s-1 , t is the time in years elapsed since ZA, is the core hydrogen content; the remaining columns should be self-explanatory. The four cases discussed below are indicated in the last column (see also Fig. 2).
Table 3. Reproducing observed frequencies: parameters of the models
For the four cases, A, B, C, and D, the computed frequencies of all modes that are unstable in the frequency range from 3.9 to 6.1 d-1 are compared with the observed ones in Fig. 4. Frequencies of the prograde modes () are plotted with minus signs, those of the retrograde ones, with plus signs.
As can be seen from Fig. 4, the number of unstable modes is much greater than the number of the observed ones. This problem is common to all linear pulsation analyses (see, for example, Dziembowski 1997).
In Table 3, the computed asymmetry of the triplet is always greater than the observed one. When is identified with an mode, computed is about two orders of magnitude greater than observed. Case A, in which is identified with the , , mode, is an example. In case B, with identified as the , , mode, d-1. Although quite small, this value is still an order of magnitude too large. Case C represents the best fit to all five frequencies. Now, however, d-1. Finally, there is case D, with the smallest in Table 3. In its vicinity we found a model with . But in this model, the two remaining frequencies, and , were not well fitted. It is possible, however, that the situation could be improved by changing the heavy elements content.
3.2. Oblique magnetic pulsator
Equally split triplets are observed in rapidly oscillating Ap (roAp) stars such as HD 24712 = DO Eri (Kurtz et al. 1989). According to the oblique pulsator model, first proposed by Kurtz (1982), an equally split triplet is seen in the observer's frame if a single mode has its axis of symmetry (the pulsation axis) inclined to the rotation axis. In such a case is exactly equal to zero and , where is the rotation frequency of the pulsation axis. An equally spaced quintuplet, separated by , would be seen if an mode were excited instead of the mode. In general, the number of frequencies seen by the observer will be for each mode of degree .
In the context of the Cephei stars, the oblique pulsator model has been recently invoked by Shibahashi & Aerts (1998) in their attempt to account for the frequency spectrum of Cep itself. In order to find out whether this model could be also applied to the present case, we compared the effect of a magnetic field and that of rotation on the frequencies of low-order, low-degree modes. Sample results, computed for three modes in a 12 model with and , are presented in Table 4. Following Dziembowski & Goode's (1996) investigation of magnetic effects in roAp stars, we assumed a dipole magnetic field. In column 3 are given the magnetic frequency shifts, , due to a kG field for in the magnetic reference system. The magnetic shifts scale roughly as . The rotational shifts, listed in columns 4 and 5, were computed for km s-1 ; the linear shifts, , are for , the quadratic ones, , are for . The linear shifts scale as , while the quadratic ones, as , where . Therefore, even for the lowest in Table 3, remains an order of magnitude smaller than either or .
Table 4. The frequencies and the magnetic and rotational frequency shifts, in d-1, for low-order modes
This situation is typical for low-order modes. In roAp stars at similar field intensities the magnetic corrections are much larger. The reason is that the modes excited in these stars are acoustic modes of very high radial orders (). Such modes propagate into the outermost layers in which the magnetic perturbation is large because the perturbing effect scales as , where P is the gas pressure, and P grows much more rapidly inward than . In these magnetically perturbed layers the low degree modes are evanescent. In the model considered here the magnetic perturbation becomes comparable with the rotational one for .
Application of the oblique pulsator model to 12 (DD) Lac in order to account for the triplet would thus require postulating unrealistically strong magnetic field, much stronger than that observed in the helium star Ori C , which has the strongest known magnetic field in this part of the H-R diagram (Bohlender et al. 1987). In the case of weak magnetic perturbation, the departure from a single dependence in the rotational reference system for the individual eigenmodes is small. Thus, given the lack of observations of magnetic field in 12 (DD) Lac, we may contemplate only a weak amplitude modulation or, equivalently, only small amplitudes of the side peaks in the triplet. This is not what we see. In addition, there is the following difficulty with the model: since the and terms are both singlets, they must both be radial. For this, however, the observed frequency separation between them is too small, as can be seen from Fig. 4.
For all these reasons we regard the oblique magnetic pulsator explanation of the triplet as the least likely. However, we cannot reject it altogether because we made a specific assumption about the field geometry.
3.3. Nonlinear phase locking
Buchler et al. (1995) have investigated the nonlinear behavior of rotationally split mode. They found that - depending on the parameters of the model and the specific triplet considered - the predicted amplitudes are constant in time or exhibit long-term modulations. When all three component modes are excited and the amplitudes are constant then nonlinear phase locking causes the frequencies to appear exactly equidistant. For 12 (DD) Lac, this mechanism was recently considered by Goupil et al. (1998). The conclusion of their work is that in this case the phase lock is marginally possible.
The phase lock leading to exactly equidistant frequency separations may occur for any three modes whose linear frequencies obey the approximate relation , provided that the integral of , where 's denote spherical harmonics of the respective modes, is not equal to zero. This implies that the azimuthal orders of the three modes must satisfy the condition . Note that this condition is not fulfilled in the case of Aerts' (1996) and m identifications for the , , triplet (see above, Sect. 2.4). Consequently, her suggestion that "mode 4 might be excited through resonant coupling between modes 1 and 3" is invalid.
The nonlinear phase locking shifts the frequencies from their positions predicted by the linear theory. These shifts would have to be determined by means of nonlinear calculations and taken into account in fits such as those shown in Figs. 3 and 4. An order of magnitude example of this can be found in Goupil et al. (1998).
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998