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

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1. Introduction

A successful identification of modes excited in [FORMULA] Scuti stars could make them suitable for asteroseismological investigations since both radial and nonradial modes are excited. The problem of mode identification in [FORMULA] Scuti stars is definitely situated on the meeting point of observational and theoretical investigation. From theoretical point of view the problem is not so simple as in the case of the Sun and white dwarfs where the high-order modes obey the asymptotic theory which predicts the systematic arrangement of excited modes. The excited modes in [FORMULA] Scuti stars are of low-order and do not obey simple asymptotic relations. The present linear [FORMULA] Scuti models predict very dense frequency spectrum which does not explain the observational facts, i.e. the limited number of excited modes above the observable level of amplitudes. It is generally accepted that some mode selecting and/or amplitude limiting mechanisms caused by nonlinearity are missing from the present models.

As Pamyatnykh at al. (1998) wrote: "it is clear that improvement on the side of theory is needed before we will be able to produce a credible seismic model" of a [FORMULA] Scuti star and mode identification remains impossible unless we discover the clue to mode selection. They urged that observational determination of l values for some of the excited modes could significantly change the situation.

It has been discussed by many authors (Dziembowski 1977; Balona & Stobie 1979; Stamford & Watson 1981; Watson 1988; Garrido et al. 1990; Garrido 2000) that multicolour photometry contains information about l because local temperature, geometry, pressure and limb-darkening variations are important contributing factors to the predicted flux changes.

However, the location of theoretical modes on the comparison plane is based on assumed ranges of (R, [FORMULA]) which are very uncertain. R is a parameter which describes departure from adiabacity of the atmospheres of pulsating stars. [FORMULA] phase lag gives the angle between maximum temperature and minimum radius.

In the past few years new theoretical investigations concerning the effect of rotation (Soufi et al. 1998) and limb-darkening (Heynderickx et al. 1994) for mode indentification have been carried out. Cugier et al. (1994) found that nonadiabatic observables are useful not only to determine l but also the radial order of the observed modes. It is a fact that this statement is deduced for [FORMULA] Cep stars, where the mode identification has a different problem based mostly on amplitude ratios, not phase differences.

Since mode identification of stars cannot be done independently of calculations involving construction of equilibrium models and their oscillation properties, all of the theoretical investigations are very important to find a final solution for mode identification. However, we definitely need additional guidelines to reduce the number of suitable models to a unique solution.

In the past few years many attempts have been carried out to obtain reliable criteria for mode identification from observational point of view in [FORMULA] Scuti stars. As a most plausible solution the observable level of amplitude of the excited modes, based on remarkably longer data sets, were decreased to have more numerous excited modes above the observable level. While the problem is rather complex in matching the observed modes by theoretical models, considerable progress has been achieved as shown by Breger et al. (1995) and Guzik et al. (1998).

A promising tool is a search for regular frequency spacing. Of course, for low-overtone pulsators as [FORMULA] Scuti stars, one cannot expect to find the asymptotic frequency spacing. However, the deviations from a regular frequency spacing are small even for low radial overtones and so some regularity among the pulsation modes excited to visible amplitude (Handler et al. 1997, Breger et al. 1999) can be expected. Beside the traditionally accepted line profile analysis (both time series or moment method) as a pure spectroscopic method for mode identification, a new complex method (equivalent width method) based on similtaneous observation of selected absorption lines combined with simultaneous photometric observations, has been established (Kjeldsen et al. 1995, Bedding et al. 1996) and applied to FG Vir (Viskum et al. 1998).

A common, pure multiphotometric method of determining the mode for [FORMULA] Scuti stars is to plot amplitude ratio versus phase difference for two colours. According to Balona & Evers (1999) it is difficult to combine results from numerous two-colour diagrams. They improved a rigourous, statistically based, method of deducing the mode which does not require these plots and which gives the probability of correct identification for [FORMULA] Scuti stars. A similar method has been proposed by Fontaine et al. (1996) for white dwarfs where only the amplitude dependence on wavelength is sensitive to the mode.

In the present paper we modify the common multiphotometric method of determining the mode for [FORMULA] Scuti stars. On the basis of extended multisite, multiphotometric observations not only the generally used plane ([FORMULA] versus [FORMULA] - [FORMULA]) but each combination of the nonadiabatic observables including amplitude ratio vs. amplitude ratio and phase difference vs. phase difference planes are checked for observational guidelines to mode identification.

Additional information on data comparing to Paparó et al. 1996 (Paper I) and Sterken 1997 (Paper II) is discussed in Sect. 2. Nonadiabatic observables (amplitudes and phases) with their error bars are presented in Sect. 3. The most useful set of plots for observational guidelines to mode identifications are shown in Sect. 4. A discussion concerning the conclusion of plots is given in Sect. 5.

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Online publication: October 30, 19100