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

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7. A first attempt for mode identification from the colour variations

If we accept that the pulsational behaviour of the LBVs is similar to the one of SPBs, we can attempt to identify the modes from multicolour photometry in the same way as it is achieved for these well-known non-radial g-mode pulsators. From a comparison of the photometric amplitudes in different filters of a photometric system, it is in principle possible to derive information on the degree of the pulsation mode in the context of linear pulsation theory. This method of mode identification is referred to as "the photometric amplitudes method" and was introduced by Dziembowski (1977), refined by Watson (1988), and by Heynderickx et al. (1994).

To achieve identification, a prediction for the photometric amplitude of a non-rotating pulsating star is derived as a function of the wavelength and of the degree [FORMULA] of the pulsation mode (see Heynderickx et al. 1994). The method is, up to now, mostly used for identification in [FORMULA]Cep stars and also in some SPBs. In the derivation of the relation between colour and amplitude, one assumes a phase difference between the temperature variation and the radial displacement of [FORMULA], while non-adiabatic effects are allowed for by means of a free parameter [FORMULA] ([FORMULA]: adiabatic pulsation). These approximations are good for [FORMULA]Cep stars and SPBs. In view of the fact that LBVs seem to be an extension of the instability domains of these two groups of non-radial pulsators towards massive stars (Waelkens et al., 1998), the use of the same approximations seems justified as a first attempt to identify modes in LBVs.

By calculating the relation between the photometric amplitude and wavelength for different values of the degree of the pulsation mode [FORMULA] ([FORMULA] for a radial pulsation) and S, and by comparing the results with the observed photometric amplitudes as a function of [FORMULA], one determines the value of [FORMULA] that best fits the observations. This is achieved by considering ratios of amplitudes, such that wavelength independent constants are eliminated. We have implemented Heynderickx et al.'s (1994) method for the Strömgren photometric system.

We considered only those epochs for which an accurate period was determined and for which at the same time sufficient multicolour data were available in the Strömgren system. The amplitudes of the variations were determined by subtracting the slower variation and fitting a sine to the data with the periods and the trend described in Table 4. The results are listed in Table 7, together with the fraction of the variability that is explained by the model. It can be seen that the fits are rather poor in the case of R 71. We therefore did not attempt a mode identification for this star.


[TABLE]

Table 7. Amplitudes [FORMULA] (in mag) of the multicolour data we considered. The Nr. listed corresponds to the epoch given in Tables 3 and 4. We fitted a sine function as well as a linear correction to the data according to the periods and the trend listed in Table 4. The fraction of the variance explained by the fit with the considered period in each of the filters is indicated as well


We show the variations in the four filters of HR Car and 164 G Sco in the upper panels of Fig. 6, and of R 127 and S Dor in the upper panels of Fig. 7. The amplitudes were then normalised with respect to the u-filter. For R 127 and 164 G Sco the relations between amplitude and wavelength change from one epoch to another (see the lower panels of Figs. 6 and 7). This points towards a switching of modes of different character. In the case of S Dor and HR Car this is not clear.

[FIGURE] Fig. 6. Top four panels: multicolour data for the four passbands of the Strömgren system of HR Car (two left panels) for the period 2 and 3 (see Table 4) and for 164 G Sco (two right panels) for the period 1 and 2. The dots are the observations while the full lines are fits with the pulsation periods listed in Table 4 and given on top of each panel. Middle panels: the amplitude differences (in magnitudes) as a function of the non-adiabaticity parameter (see text for an explanation) for the most likely degrees of the pulsations. Different symbols denote different degrees: full line: [FORMULA], dashed line: [FORMULA], dotted line: [FORMULA], dashed-dotted line: [FORMULA], dashed-dot-dot-dot line: [FORMULA], [FORMULA], [FORMULA]. Lower panels: the observed amplitude ratios [FORMULA] for [FORMULA] (crosses) are compared with those of the best theoretical model (connected by the full line) for the four filters of the Strömgren system

[FIGURE] Fig. 7. Top four panels: multicolour data for the four passbands of the Strömgren system of R 127 (two left panels) for the period 1 and 2 (see Table 4) and for S Dor (two right panels) for the period 1 and 2. The dots are the observations while the full lines are fits with the pulsation periods listed in Table 4 and given on top of each panel. Middle panels: The amplitude differences (in magnitudes) as a function of the non-adiabaticity parameter (see text for an explanation) for the most likely degrees of the pulsations. Different symbols denote different degrees: full line: [FORMULA], dashed line: [FORMULA], dotted line: [FORMULA], dashed-dotted line: [FORMULA], dashed-dot-dot-dot line: [FORMULA], [FORMULA], [FORMULA]. Lower panels: the observed amplitude ratios [FORMULA] for [FORMULA] (crosses) are compared with those of the best theoretical model (connected by the full line) for the four filters of the Strömgren system

The observed normalised amplitudes were subsequently compared with theoretical normalised amplitudes as a function of [FORMULA] and S. The logarithmic derivatives that appear in the theoretical expressions for the amplitude (Heynderickx et al., 1994) were determined by interpolation in the grid of atmosphere models by Kurucz (1979) with [FORMULA] and [FORMULA] as close as possible to the estimated values listed in Table 4. Accurate stellar models are not available for LBVs, and we had to consider [FORMULA] for all stars except AG Car.

For the comparison between the predicted and observed relative amplitude versus wavelength relations, we use the square root of the sum of squares of the differences between the observed and theoretical amplitudes divided by two as an indicator of the agreement. This quantity is defined as the "amplitude difference" and was determined for [FORMULA] in steps of 0.05 and for [FORMULA]. The most likely mode is the one with the smallest amplitude difference. The results of our analysis are shown in the middle panels of Fig. 6 for HR Car and 164 G Sco, and Fig. 7 for R 127, and S Dor. The mode identification of AG Car was considered as unreliable since the results were rather dependent on the gravity of the adopted Kurucz models. For the other four LBVs, we found the same results for [FORMULA] as for higher [FORMULA]-values. We now discuss the results for four LBVs.

HR Car (Fig. 6, left)
We find that the best fit occurs for an [FORMULA] mode for the first epoch (Nr 2 with P=41 days) while an [FORMULA] is obtained for the second epoch (Nr 3 with P=19 days). The large difference in the periods indicates that the pulsations have a different radial order. Our identification thus points towards a pulsation with the same degree [FORMULA], but another overtone at the two epochs. In the lower panel of Fig. 6 we compare the theoretical amplitudes with the observed ones and it can be seen that the observations can be accurately described by the pulsation model with the parameters given above, especially for the second epoch.

164 G Sco (Fig. 6, right)
For interval Nr 1 (P=55 days) we find an [FORMULA] solution, independent of the value of the non-adiabaticity parameter S. For interval Nr 2 (P=45 days) we find an [FORMULA] solution to give the best representation of the data. In both cases our simple linear non-radial pulsation model explains the observations very well. This star thus seems to switch between modes of different degree.

R 127 (Fig. 7, left)
For interval Nr 1 (P=35 days) we find a strange behaviour of the radial mode ([FORMULA]) below [FORMULA]. We do not know if this is a mathematical artefact of the model or if it is real. Although this radial solution is able to explain the observed amplitudes with high accuracy, we have to treat this identification with caution because of the large errors on the amplitudes. The latter are due to the fact that we had to correct for the long-term variations by means of a sine with a long period. For interval Nr 2 (P=111 days) we find that an [FORMULA] mode gives the best fit with the observations. Our preliminary suggestion then is that there is a switching of modes, similar to for 164 G Sco.

S Dor (Fig. 7, right)
The pulsational behaviour during interval Nr 1 (P=195 days) is difficult to describe with our model (see the lower panel of Fig. 7). The [FORMULA] mode gives the smallest amplitude difference. For interval Nr 2 (P=131 days), we find a good description by means of an [FORMULA] or [FORMULA] mode with intermediate S. It thus seems that we have a different radial order here, but not necessarily another type of mode.

We conclude that, besides the ambiguous result for the first epoch of R 127, the observed relations between amplitude and wavelength give no evidence for a radial pulsation in the considered LBVs. Instead we found evidence for g-modes of low [FORMULA]. S Dor (high Q) and HR Car (low Q) seem to switch from one overtone to another, while R 127 and 164 G Sco pulsate in modes of different degree. We found that the modes which fit the multicolour data of the LBVs have a degree between 1 and 4 in terms of linear pulsation theory.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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