3. Nonadiabatic observables
We follow the terminology of Cugier et al. (1994) used in Cephei models. They adopted the term nonadiabatic observables to denote amplitude ratios and phase difference for any pair of oscillating parameters such as light in a selected filter, colour or radial velocity. They believe that in addition to precise frequency measurements, the nonadiabatic observables should be regarded as important data for asteroseismology.
A multifrequency analysis of Tucanae photometry was performed with the MUFRAN program (Kolláth 1990) and described in detail in Paper I. Ten pulsational frequencies were found in the range of 15.86 to 20.28 cycles/day. Two frequencies (0.282 and 0.142 cycles/day) were found to be responsible for the mean light level variation of Tucanae. A third frequency at 0.99350 cycles/day having 1.9 mmag amplitude near the level of significance was also found in the low frequency region. It could be a sign of a not-perfect homogenization, although the coincidence of the nightly mean value of Tucanae obtained at different sites with a periodic curve suggests no problem involved in the homogenization. Regarding the 1.00 and 0.957 cycles/day frequency differences between three pulsational frequencies, the frequency at 0.99350 cycles/day may be due to a linear combination of and which is not properly resolved. In a simulation noiseless synthetic data were generated for the three sinusoides mentioned above with data points distributed in time according to the observational data. A single peak at 0.99904 cycles/day revealed in the low frequency domain confirming our guess for the two unresolved linear combinations.
Nevertheless, in the present paper the 13 frequencies, worked out in Paper I were accepted and used to get the nonadiabatic observables for Tucanae. In Table 1 amplitudes and phases of the 10 pulsational frequencies for each Strömgren bands and colour indices are given, respectively, in mmag and degrees. The amplitudes and phases were obtained by least squares solution. With this solution a synthetic light curve was calculated. Noise was generated 300 times by a random number generator and added to the synthetic light curve. The parameters were determined in each case. Errors in amplitudes and phases are given as standard deviation around the mean value.
Table 1. Amplitudes and phases of 10 excited pulsational modes of Tucanae in different colours and colour indices.
We would like to call the attention that such kind of error calculation gives only the error bar of finding the frequencies because of observational errors. As the stability investigation for Tuc (Paparó 2000) shows, larger uncertainties are involved in the solutions because of the length and distribution of data. The SAAO and ESO data were separately treated. Amplitudes and phases for both data sets can be found in Tables 1-4 of that paper.
Determination of amplitudes and phases for the 13 frequencies were given in Table 2 in Paper II using only the ESO colour data. The amplitudes in v and u are the same as here because the same data set was used. In y and b the increased amount of data resulted in different values for the y and b amplitudes.
Table 2. List of modes in Groups. Values are given in cycle/day
The phases differ in Paper II and the present paper because a different epoch was used. In the present calculation only the last four digits of HJD integer were applied and the HJD integer of the first observation, 9249, was used as an epoch. Since the phase differences between colours and colour indices are independent of the epoch, the accepted value does not play a special role.
The colour dependence of amplitudes and phases based on ESO colour data was discussed in Paper II and shown in Fig. 3. A common trend was pointed out namely that the amplitudes of the frequencies increase towards shorter wavelengths.
As a more sophisticated check based on the error bars of the present investigation shows three modes ( = 17.06289, = 15.86246 and = 15.94618) having unusual behaviour in u colour compared to the common trend for other modes. These modes have lower amplitude in u than in v. For two modes ( and ) not only the amplitude values are lower but even the error bars are separated. Model calculations for the variation of amplitude ratio in different passbands (Balona & Evers 1999 and Garrido 2000) also show a break down in the short wavelength.
Table 1 is used to calculate the value of the nonadiabatic observables, amplitude ratios and phase differences. A non-correlated error calculation was carried out for the error bars of nonadiabatic observables.
For amplitude ratio , for example,
and for phase difference -
equations were applied where are the amplitude values, , the corresponding errors from Table 1a. and are the errors of the phases in the given colours from Table 1b. For the most important nonadiabatic observables served as useful criteria the values and error bars are given later in Table 5. These calculations and the sophisticated results give a strong base for searching observational guidelines for mode identification for Tucanae based on multicolour photometry.
© European Southern Observatory (ESO) 2000
Online publication: October 30, 19100