3. Pulsation mode identifications from photometric phase differences
The relative phase difference between the temperature and radius variations of a pulsating star leads to an observable phase difference between the light curves at different wavelengths. The sizes of these phases differences depend not only on the properties of the star, but also on the type of pulsation mode. The observed phase difference can then be used for mode typing. This was already pointed out by Watson (1988). Garrido et al. (1990) presented detailed calculations and predictions for Scuti stars. They find that measurements through different filters of the Strömgren uvby system provide discrimation between radial and low-order nonradial pulsation, i.e. help determine the value 1.
We have chosen the v and y filters to provide a relatively large baseline in wavelength. The u filter was not used by us because of the very large potential for systematic observational errors. These details of the measurements can be found in Breger et al. (1998). The phase differences were determined in the following manner: The values of the 24 known and well-determined frequencies were optimized by making a common solution of the available y data from 1992 - 1996, while allowing for the amplitude variability of . As discussed in Breger et al., all CCD measurements were given a weight of 0.19. With these optimized frequencies, for the year 1995 the best amplitudes and phases were calculated from the available 412 hours of y and 292 hours of v data. Separate trial solutions indicate that the resulting phase differences are relatively insensitive to the weights adopted. For the year 1996, an additional 82 hours of uvby photometry are available (Viskum et al. 1998). The data were combined with the larger data set from 1995 while allowing for variable amplitudes of .We note that the calculated uncertainties of the phase differences are not reduced by including the additional data: the reason is that the 1995 data have smaller deviations, e.g. 4 vs. 6 mmag in v.
The resulting phase differences are shown in Table 2. The phase errors (in degrees) were estimated from the formula , where a is the amplitude and is the uncertainty of each data point (average deviation per point from the fit).
Table 2. Phase differences and mode identifications of FG Vir
We can now compare the observed phase differences with theoretical modelling in order to determine the values. The ATLAS9 models of Kurucz (1993) were used to construct a model atmosphere for FG Vir. Garrido et al. (1990) presented calculations using values of ranging from to . For FG Vir, Viskum (1997) determined a smaller range, viz. . This allowed us to refine the calculations, although the results are very similar. Another required constant, the deviation from adiabaticity, R, has been changed slightly from the value used by Garrido et al. (1990). Values of 0.20 (instead of 0.25) to 1.00 were used. This change was indicated by measurements of high-amplitude Scuti stars. The theoretical predictions are shown in Fig. 2 together with the observations. The importance of considering the dependence on the pulsation constant, Q, can be seen for Q = 0.02, where one can even find negative values of the phase difference for radial modes, although the separation between radial and nonradial modes is always maintained.
Our best mode identifications based on Strömgren photometry are shown in Table 2. We obtained five unambiguous values, while for three further modes we cannot distinguish between two adjacent values. The frequency , shown in the middle panel, is situated to the right of the = 0 by 1.9 . We note that the deviation is caused by only one subset of data (the CCD measurements from Siding Spring Observatory, see Stankov et al. 1998), without which a phase shift of is found. Irrespective of which of the two values for is accepted, an identification with radial pulsation is consistent within the statistical uncertainties.
We can now compare the results from the photometric method with those derived from a promising new technique of examining the equivalent width variations of selected lines. Bedding et al. (1996) have shown that for low degree pulsation, the -values of pulsation modes can be inferred from simultaneous observations of several selected absorption lines combined with simultaneous photometric observations. Viskum et al. (1998) have applied this method to the star FG Vir. In particular, the equivalent-width changes of the H and H lines turned out to be good discriminators. In their paper, identifications have been presented for the eight dominant modes.
We note that on the observational side the photometric and spectroscopic methods are independent. However, both methods rely on similar model-atmosphere calculations, so that they cannot be considered to be completely independent of each other.
The agreement between the photometric and spectroscopic mode determinations is remarkable. It appears prudent to examine the comparison of the results of the two methods in more detail, especially with consideration of the (unavoidable) observational uncertainties. In order to compare independent parameters with each other, we pick the amplitude ratio of A(H)/A(FeI) given by Viskum et al. (1998). The comparison is shown in Fig. 3, where the numbers next to the points refer to the frequency numbering in Table 1. The figure shows that some of excellent agreement may be accidental once the observational uncertainties are considered. Nevertheless, the viability of both methods to determine values has been demonstrated and for at least six modes the values have been observationally determined. These determinations now need to be used as input for pulsation models.
© European Southern Observatory (ESO) 1999
Online publication: November 26, 1998