Astron. Astrophys. 324, 121-132 (1997)

1. Introduction

The Double-Mode Cepheids (DMCs) play an important role in the study of stellar evolution. In recent years a substantial improvement was made to reconcile the pulsational mass (i.e. the mass predicted by the pulsation law ), the beat mass (i.e. the mass derived from the ratio between the observed periods) and the evolutionary mass (i.e. the mass predicted from evolutionary tracks and observed luminosity). The introduction of new opacities allowed theoretical studies to fill not only the large gap between the beat and pulsation masses, but also to match the evolutionary masses (Christensen-Dalsgaard & Petersen 1995).

In the same years, following the idea first expressed by Antonello et al. (1990), Mantegazza & Poretti (1992) and Poretti (1994) carefully studied the light curves of s -Cepheids using the Fourier decomposition technique; they redefined the s -Cepheids as the stars which do not follow the Hertzsprung progression (described by the Classical Cepheids) in the space of Fourier parameters. To explain this different behaviour it was suggested that the two classes are pulsating in two different modes, i.e. the fundamental radial (F) mode and the first overtone radial (1O) mode, respectively. The DMCs provide the obvious laboratory where this suggestion can be verified since it is a well established fact that in 13 cases out of 14 the two excited modes are indeed the fundamental and the first overtone mode; the data on V371 Per (Schmidt et al. 1995), the most promising candidate, are too scanty to establish its DMC nature. In the meantime, the large amount of data collected in the framework of the MACHO (Alcock et al. 1995) and the EROS (Beaulieu et al. 1995) projects yielded the first confirmation of the different pulsation modes since the Classical and s -Cepheids are separated in a plane exactly by the shift due to the 1 ratio. Moreover, new arguments were added to the debate owing to the large number of DMCs discovered in the LMC, against the only 14 cases observed in the Galaxy. To define in an accurate way the properties of the small number of galactic DMCs is mandatory to perform a significant comparison with the properties of the more numerous LMC DMCs.

The light curve of a DMC can be considered as the sum of the contributions of a number of frequencies, of which two only are independent ( and ). Since each of these two curves is not, as a rule, perfectly sine-shaped, we also have to observe the 2 , 3 , 4 , ..., 2 , 2 , 3 , 4 ... harmonics; moreover, the two modes are interacting and the cross coupling terms (i.e. their combination ; the two cases ,  are the most frequent) are expected to be observed. Even if systematic photoelectric surveys of DMCs were performed from 1947 onward (TU Cas; Oosterhoof 1959), no exhaustive study of their light curves was carried out; the most complete analysis was surely the one outlined by Stobie & Balona (1979). However, in that important paper also the light curve description was made on the basis of an a priori choice, i.e. the application of a -order fit to the collected points. This approach was also used by Faulkner (1977) to study the light variation of U TrA: he applied three different fits (, , order), but he did not investigate whether all the components were really present in the data, since the major result (i.e. the presence and the strength of the cross-coupling terms) is slightly affected by the completeness of the frequency content. Stobie & Balona were mainly interested in the phasing of the magnitude, colour and radial velocity observations and they showed, in the particular case of VX Pup, that the effect of additional high-order terms was to change only slightly the amplitude and the phases of the low-order terms, not affecting their main result. We can conclude that in previous works no attempt was done to detect how many harmonics of and are necessary to fit the observed light curves and which cross coupling terms are excited by their interaction. More recently, this incomplete approach was used by Matthews et al. (1992) in reexamining the TU Cas data: a -order fit was a priori applied to the data, thus obtaining incorrect values for the phase parameters and inconsistent amplitude ratios (see also Poretti 1994 and Subsect. 4.4).

Therefore, it seems crucial to submit all the available photometry on DMCs to a careful frequency analysis:

1. To detect the importance of the harmonics and of the cross coupling terms for each star and to evaluate the similarities. Frequency and amplitude variations can be investigated and the search for a third independent periodicity can be carried out;
2. To compare the values of the low-order Fourier parameters with those of the galactic single-mode Cepheids. This comparison will allow us to establish the similarities between the two classes and to give an independent identification of the pulsation mode observed in single-mode Cepheids;
3. To establish the properties of the Fourier parameters by determining boundary values in order to compare observed and theoretical light curves;
4. To search for the signature of resonances between modes in the Fourier parameter progression.

The first two items are discussed in this paper, the last two will be studied in a successive paper (Poretti & Pardo 1997; Paper II).

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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