2.1. The effective temperature
The MK spectral type of 12 (DD) Lac is B 1.5 III (Lesh 1968). Five stars of similar MK type (that is, spectral type B 1 or B 2 and luminosity class from IV to II) are among the 32 stars for which Code et al. (1976) determined the empirical effective temperatures from the OAO-2 absolute fluxes and the intensity interferometer angular diameters. A mean for these five stars is equal to 4.369, and a mean of the standard deviations is 0.017.
Another value of can be obtained from the Strömgren and indices. Using the recent calibration of Napiwotzki et al. (1993) and the observed and of 12 (DD) Lac (Sterken and Jerzykiewicz 1993, Table 2A), we get .
The mean of the above two values, equal to 4.374, we shall adopt as of 12 (DD) Lac. Note, however, that since Napiwotzki et al. (1993) based their calibration on the empirical effective temperatures of Code et al. (1976), the two values are independent of each other only insofar as different observations of 12 (DD) Lac were used to derive them. Taking into account the mean standard deviation of the empirical values and uncertainties of the MK type and photometric indices, we estimate the standard deviation of the value we adopted to be equal to 0.020.
2.2. Surface gravity
Using solar-composition model atmospheres of Kurucz (1979) and procedures of Schmidt & Taylor (1979) and Schmidt (1979), Smalley & Dworetsky (1995) computed a grid of synthetic Strömgren indices as a function of and . In the Appendix we check this grid against empirical values of early-B components of several binary systems and then derive of 12 (DD) Lac from its observed index. The value we obtained is
In Fig. 2, 12 (DD) Lac is plotted using the above-mentioned values of and as coordinates. Also shown in the figure are evolutionary tracks computed by means of the new version of the Warsaw-New Jersey stellar evolution code with the updated OPAL opacities (Iglesias & Rogers 1996) for and . The code assumes uniform rotation and global angular momentum conservation and takes into account averaged effects of the centrifugal force. For the tracks shown in Fig. 2, and for all models that will be discussed below, the equatorial velocity of rotation on the ZAMS, , is assumed to be equal to 100 km s-1 .
2.3. The frequencies
The observed values of the frequencies we shall use for comparison with computed eigenfrequencies are given in the second column of Table 1. They are equal to , where are the mean periods derived by Pigulski (1994) from all available photometric and radial-velocity data. We omitted the combination frequency, .
Table 1. Observed pulsation frequencies of 12 (DD) Lac
Pigulski (1994) has also investigated the secular stability of the amplitudes and periods of the six terms. The only variation he detected in the amplitudes was a 30 percent increase of the light amplitude of the second term between 1961 and 1983. On the other hand, he found all six periods to vary on time scales comparable with the total span of the data, that is, about 75 years. The variations have very small amplitudes, however. The relative amplitudes, estimated from Pigulski's (1994) Fig. 1 and expressed in terms of frequency, are given in the third column of Table 1. They can be regarded as observational uncertainties of the frequencies.
2.4. The discrepant values
For the term, the spherical harmonic degree, , has been recently derived from the observed light, colour and radial-velocity amplitudes by Cugier et al. (1994). Using nonadiabatic eigenfunctions of Dziembowski & Pamyatnykh (1993), Cugier et al. (1994) examined diagnostic properties of several diagrams with different amplitude ratios as coordinates. They showed that modes of different are particularly well resolved when the colour to light amplitude ratio, @ is used as abscissa, and the radial-velocity to light amplitude ratio, , as ordinate. In this diagram, the term falls on the sequence.
For the term, the 1962-1976 UBV photometry of Sato (1973, 1977, 1979) and all available radial-velocities (Pigulski 1994) yield and km s-1 mag-1 . In the above-mentioned diagnostic diagram of Cugier et al. (1994), these coordinates indicate or 2, but certainly not .
From the same data, the amplitude ratios for the term are and km s-1 mag-1. Unfortunately, a point with these coordinates lies outside the area in the diagnostic diagram: while the first ratio falls within the range of abscissa corresponding to , the second one is more than five standard deviations below the smallest ordinate. However, if the first ratio were decreased by two standard deviations, the point would move into the sequence. Since a decrease by four standard deviations would be required to reach the sequence, we conclude that is the best identification for from the amplitude ratios now available.
For the remaining terms, and , the errors of the amplitude ratios become so large that discrimination between different values is no longer possible.
The above identifications can be compared with the results obtained from the line-profile observations. Unfortunately, the three modern line-profile studies of 12 (DD) Lac yield conflicting results although they were all based on observations of the Si III lines. Smith (1980) maintains that his data can be accounted for only if the triplet terms are identified with the states and the mode is assumed to be radial, while Mathias et al. (1994) conclude that the mode is either a sectoral one or a tesseral one with . More recently, the line-profile data of Mathias et al. (1994) have been re-analyzed by Aerts (1996). She finds for and for , but for . Thus, according to Aerts (1996), the triplet , , does not consist of three m states of the same . For , Aerts (1996) gets or 3 and . Note that both these possibilities contradict Smith's (1980) finding of for this mode. Another difference between these two investigations is that Smith (1980) determines the star's equatorial velocity of rotation, , to be equal to 75 km s-1, while Aerts (1996) finds = 30 km s-1.
The only constraint which follows from these discrepant results is that the mode is nonradial, with equal to either 1 or 2.
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998