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Astron. Astrophys. 335, 605-621 (1998)

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8. Summary and conclusions

We have studied the periods and the period changes of the microvariations of 6 LBVs observed with the LTPV project in Strömgren photometry. We adopted two period search methods: the Fourier method and the Phase Dispersion Minimization Method (PDM). The Fourier method assumes that the pulsations are sinusoidal, whereas the PDM method makes no apriori assumption about the shape of the lightcurves. Both methods give very similar results. The variations are approximately sinusoidal but do not show a strict periodicity, even within one interval. It seems that the stars pulsate with one period only a few cycles before changing the period. This hampers an accurate study of the variability, because it would require long continuous series of observing times.

We have determined the periods and amplitudes in selected intervals ranging from 50 to 800 days, depending on the number of photometric observations available. The six LBVs show microvariability with half-amplitudes of about 0.01 to 0.1 magnitude in V and periods between 11 and 195 days. Whenever the variations were studied during more than one epoch, the periods and the amplitudes change. The changes in period are up to about a factor 3 or 4. The changes in amplitude are up to a factor 6.

The structure of the LBVs changes slowly due to the moderate variations, on a timescale of years to decades. In three LBVs (R 71, HR Car and R 127) the epochs of the microvariations occur when the star changes [FORMULA] by more than 0.3 mag. due to the slow moderate variations. These changes in [FORMULA] correspond to changes in the stellar radius of about 40 percent for R 71 and HR Car, and a factor two for R 127. This enables us to compare the pulsation periods when the star changes its radius. The data show a trend of increasing period with increasing optical brightness, i.e. with increasing radius (Fig. 3). If the pulsational parameter Q was constant and if [FORMULA] was constant, we would expect log P to vary as [FORMULA]. The observed trend is steeper. This steep trend indicates that Q is not constant when the star changes its radius, but that Q increases as the radius increases. This is qualitatively in agreement with the steepening of the density profile as the star expands, because only less than one percent of the mass of the star takes part in the expansion.

For each LBV the amplitudes of the microvariations increase as the period increases (Fig. 4). The slope of the period-amplitude relation depends on [FORMULA], in the sense that the relation is steeper for the stars with lower luminosity than for stars with higher luminosity. The star AG Car may be an exception to this rule, but for this star the slope of its period-amplitude relation is not well defined.

The most common microvariations of LBVs have [FORMULA] days. However the stars go through phases of slower pulsations when Q can increase by as much as a factor four. This is not related to a particular phase in the lightcurve. The largest Q values are found for S Dor ([FORMULA] and 0.34 days) and for one epoch of AG Car ([FORMULA] days). The high Q-values of S Dor are probably only temporary, because S Dor does not differ significantly in its characteristics (luminosity, radius, mass, circumstellar nebula) from the other LBVs. Telting (private communication) has suggested that the slow phases might be due to a beat of two frequencies. Unfortunately the photometric data of LBVs are too sparse and the observing runs are too short (one observing season is about five periods of microvariations) to detect multiple periodicities.

The minimum value of the pulsational constant of the LBVs, [FORMULA] days, is about 1.5 to 2 times as large as that for normal supergiants (Burki 1978). This is possibly due to the fact that LBVs may have a higher [FORMULA]-ratio than normal supergiants, because they have lost more mass.

Kiriakidis et al.(1993) argued that the variations of LBVs might be due to strange-mode instabilities. To test this suggestion, we compared the observed periods of the LBVs with those predicted for the strange-mode instability in Sect. 6.1. The predicted periods for stars with parameters close to those of the LBVs are in the range of only a fraction of a day to a few days. The observed periods are much longer and on the order of tens to hundreds of days. So the microvariations of LBVs are probably not due to strange-mode instabilities. Another argument against the strange-mode interpretation of the microvariations is the fact that the strange-modes are expected to occur in a restricted region of the HR diagram (Fig. 5), whereas the microvariations of LBVs and the very similar [FORMULA] Cygni variations of normal supergiants occur in a much wider region of the HRD than predicted.

A more promising explanation of the microvariations is suggested by the comparison of LBVs with the [FORMULA] Cephei stars and the Slowly Pulsating B-stars (SPBs). Waelkens et al. (1998) have shown that the variations of the normal supergiants are a logical extension of those found in SPBs and [FORMULA] Cepheids. This is supported by calculations by Pamyatnykh (1997). Since the SPBs and [FORMULA] Cepheids are known to pulsate in g-modes, the microvariations of the LBVs are probably also due to g-mode pulsation.

We have attempted to identify the pulsation modes of the LBVs by studying the relation between amplitude and wavelength, in a similar way as has been applied to [FORMULA] Cepheids by e.g. Heynderickx et al. (1994). This implies the calculation of the variations of the energy distributions over the stellar surface due to modes of different [FORMULA]. This is done using Kurucz model atmospheres. The results, shown in Figs. 6 and 7., suggest that the stars are pulsating in g-modes of low order, typically [FORMULA]. We did not find evidence for radial ([FORMULA]) modes.

During the period of observations used in this paper, the LBVs did not show the reversal of the colours-magnitude variation of the microvariations that was observed for visual maximum by van Genderen et al. (1997a), (see Sect. 1.). Therefore we have no information on this possible change of the nature of the pulsations during the maximum of the light curve.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998