6 Discussion

There are two main obstacles in the investigation of the variablity patterns of EN Lac. The first is the need to separate safely the variations due to stellar oscillations from the orbital motion. Whereas the orbital period can be determined very accurately from the narrow eclipses, accurate orbital elements can only be derived from spectroscopy. To do that, a considerable contribution of the short-term RV changes has to be eliminated.

A proper analysis of the pattern of the rapid changes requires a solution of another problem, namely an appropriate modelling of the long-term amplitude modulation of the pulsational periods. Accurate RVs as well as observations covering several decades are needed to this task. The data set at our disposal fullfils these requirements. It consists of numerous recent observations of high quality and of numerous older observations covering a period of about 90 years. Analyzing this sample, we were able to derive

• a new, accurate orbital solution which confirms the presence of a small but not-zero eccentricity of about 0.04-0.05;
• accurate values of the triplet of fundamental periods of the rapid RV changes: $P_1=0\hbox{$.\!\!^{\rm d}$ }16916703$, $P_2=0\hbox{$.\!\!^{\rm d}$ }17085554$, and $P_3=0\hbox{$.\!\!^{\rm d}$ }18173256$;
• a period of the amplitude variation of the dominant period P₁ which amounts to about 74 years;
• probable periods of the amplitude changes of the P₂ and P₃ periods amounting to 331 d and 674 d, respectively - note that they differ by about a factor of two;
• measurable apsidal motion was not detected.

The accuracy of our final multiple-frequency model is limited by the accuracy of the older RVs obtained from photographic spectra. That is why we offer (in Tables 8 and 9) two possible solutions. The first one is based on all data and was obtained by successive prewhitenings. The second solution mainly rests on the recent RVs from electronic spectra which were given large weights. It includes one more period describing the amplitude modulation of P₁, and it also slightly modifies the period of the amplitude variation of P₂. Only the change in the orbital elements $\gamma$ and e (Table 10) is considerably larger than the errors calculated from the orbital fit itself. Note, however, that the smallest rms error per 1 observation was achieved for solution II where we allowed for local fits of rapid changes. This may indicate that either our description in terms of several sinusoidal terms is not complete or that the variations are not strictly periodic after all.

Pigulski & Jerzykiewicz (1988) derived an eclipse period of 12 $.\!\!^{\rm d}$09684 from photometric data obtained between 1952 and 1981, the epoch of minimum light is JD 2439054.568. The small differences between this period and our spectroscopic periods of orbital solutions II to IV give only rise to a negligible phase shift between the mean epochs of the photometric data and our RVs. So we can directly compare the epochs of minimum light (Tables 4, 9). The agreement is very good: the maximum deviation from the photometric value is of 0 $.\!\!^{\rm d}$048 which is only 13% of the total duration of the eclipse of about 0 $.\!\!^{\rm d}$36.

The pulsation periods derived for P₁ and P₃ are in good agreement with the periods known from the photometry of ENLac (Jerzykiewicz & Pigulski 1999, see also Table 1). In particular we observe an amplitude modulation period of the P₃ variation of the RVs of 674d corresponding to the same modulation period and splitted frequencies as observed by Jerzykiewicz & Pigulski (1999) for the light changes. Also the modulation time scale of P₁ of about 75yr is of the same order as observed photometrically. In the RV variation we did not find a modulation time scale of the order of several decades for P₂, however.

The 0 $.\!\!^{\rm d}$17085553 period obtained for P₂ agrees with the early values reported by Struve et al. (1952), Walker (1952), and Fitch (1969). This value had been revised in all later publications where the authors adopted the new value of 0.17078d and assumed the old one to be a one year alias of this value. In our extended data set the 0 $.\!\!^{\rm d}$170856 period is the dominating period during all reduction steps of successive prewhitening. The value of 0 $.\!\!^{\rm d}$17077 we found as a slight additional contribution in one of the last steps of this procedure. The doublet finally adopted for P₂ corresponds to a modulation period of 331d which is significantly different from the value of one year. A word of caution is appropriate, however. The window function (Fig. 1) has a complicated structure and there is also a peak corresponding to about 340 d, though with a smaller amplitude than the one-year alias. One should, therefore, still treat the 674/331-d modulation of the P₂ and P₃ oscillations with some reservation and perhaps to seek for other possible explanations. On the other hand, our tentative assumption that we deal with real modulation time scales is supported by the agreement of the value for P₁ with the value obtained from the light variation, and for P₂ by the fact that besides the P₂ doublet the exact one-year alias of P₂ can be observed in all periodograms.

We were not able to establish a self-consistent multiple frequency model for the line shape variations. Since we are limited to the high resolution data our time basis is too small to resolve frequencies small enough to give rise to modulation time scales of years or decades as it was observed for the RV variations. But we can put all periods into one scheme if we assume that the variations can be described by one fundamental frequency and its first harmonic. The three strongest contributions are near P₁, P₂, and P₃ (Table 11). All three contributions are of comparable total strengths. The fundamental period dominates in the P₂ variation, whereas the first harmonics dominate the P₁ and P₃ variations.

The observed line profile variations show a behaviour which is typical for low degree lmodes of nonradial pulsations: an anti-phase variation of line depth and line width preserving the equivalent width of the lines. There are different methods of mode identification based on line profile and RV analysis (Balona 1999). In general the identification of nrp-modes is much easier done by an analysis of line profile moments (Aerts et al. 1992; Aerts 1996) as by an investigation of the line variation in terms of line widths and depths. Results of this line moments analysis will be presented in a forthcomimg paper as well as the results of an analysis of the light variations of ENLac. By the light curve analysis of the eclipsing binary we will be able to set narrow limits for the physical parameters of ENLac and its secondary. By combining the results with the results of mode identification we should be able to give a physical explanation of most of the RV and line profile variations found and described here.

Acknowledgements

The authors are indebted to the referee, Guillermo Torres, for his comments which helped to improve the paper and for the careful proofreading of the manuscript. Research of P.Har. was partly supported by grants A3003805 of the Granting Agency of the Academy of Sciences of the Czech Republic and M402: Program KONTAKT of the Ministry of Education of the Czech Republic, and in final stages from the research plan J13/98: 113200004.