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Astron. Astrophys. 333, 565-570 (1998)

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5. Discussion

In Fig. 7 we show the amplitudes resulting from a simultaneous fit of [FORMULA] and [FORMULA] to the linearly-corrrected uvby data. The fitted amplitudes for [FORMULA] show an increase towards the ultraviolet part of the spectrum (but this could be the result of combining the different data sets). [FORMULA], as expected, displays a decreasing amplitude at shorter wavelengths. The colour index behaviour associated with [FORMULA] very distinctly shows that the light maxima in the 1300 d cycle are associated with a reddening, implying that the low-frequency oscillations superimposed on the steady increase in light may be identified with normal SD phases, while the quasi-linear trend may very well be the rising branch of a VLT-SD phase as seen by van Genderen et al. (1997a) in S Dor and AG Car.

[FIGURE] Fig. 7. Amplitudes for the two principal frequencies in function of wavelength of the Strömgren filters. Each group of bars represents, from left to right, the frequencies [FORMULA] and [FORMULA]

At this point we wish to draw the attention to both approaches of de-trending as described in Sects.  4.1and 4.2. The correction for the rising trend through subtraction of a curve enveloping the magnitude values at light minimum effectively removes all possible low-frequency variation, and assigns the highest spectral power to a cycle length of the order of 100 d (or to one of the alias frequencies). The formal representation of the rising branch by a linear function allows a more objective de-trending, but the forcing of the linear model may, of course, introduce a spurious signal since the rising branch may not be simply linear. However, we find corroboration in long-term light curves of LBVs, viz. AG Car: see Fig. 1 of van Genderen et al. (1997a) where a long sequence of SD phases is superimposed on rather straight stretches of the VLT-SD cycle in the descending branch (carrying maxima 27-30), the quiescent section (supporting maxima 30-35) and the rising part (maxima 35-38). The SD cycle of AG Car is short (371:d4) and the occurring VLT-SD cycle is slightly longer than 20 y; in the case of R 40, the SD cycle is is three to four times longer (1300 d), with a VLT-SD cycle of the same length as in AG Car, i.e. [FORMULA] 20 y. R 40 thus resembles S Dor (SD cycle of 6.8 y and VLT-SD cycle of the order of 40 y, discovered as a result of the very long base line of photometric data).

It is clear that a formal Fourier expression of the light variations with intermediate frequencies does not account for the full variability in all bands, even if a second high-frequency oscillation is taken into account, and this fact is also illustrated in the phase diagrams shown in Fig. 5, where one sees some branches that deviate markedly from the more general light curve. This reflects the well-known fact that the microvariations of LBVs are semi-periodic. The SD and VLT-SD cycles, too, are of somewhat variable length; see the case of AG Car as described by Sterken et al. 1996 and van Genderen et al. 1997a). It is indeed very difficult to distinguish between semi-regular (multi-)cyclic variations and the additional (stochastic) variability that also characterises all luminous stars of these types. What is important is the fact that the microvariations are not irregular, that they are visible during almost the complete duration of the rising VLT-SD branch, and that they have a more or less constant cycle length.

Using [FORMULA], Szeifert et al. (1993) derive the following stellar parameters for R 40: [FORMULA] = 8700, [FORMULA] at the time of the S Dor maximum in 1991, and [FORMULA] = 10000, [FORMULA] at the time of the pre-maximum in 1987. In both cases [FORMULA] ; it is seen that [FORMULA] does not vary through the S Dor variations. The 1991 values for [FORMULA] and L applied to Fig. 14 of van Genderen et al. (1992) yield [FORMULA], in good agreement with our [FORMULA]. Considering the uncertainties on the stellar parameters used, the application of the [FORMULA] relation derived by Burki (1978) also yields an acceptable agreement, [FORMULA].

Though pulsation models are not available for stars that are not purely periodic, we did calculate the pulsation constant [FORMULA] using the above parameters, and obtained [FORMULA] for [FORMULA] with stellar parameters for the SD maximum, and [FORMULA] in the time interval preceding the phase of SD maximum.

Lovy et al. (1984) calculated the (radial) pulsation properties of stars with [FORMULA] for the radial fundamental mode ([FORMULA]) and the first and second overtones, but none of their models exactly fits the stellar parameters of R 40. Model 302 ([FORMULA], and [FORMULA]) corresponds to R 40 in its bright and cool phase, and yields [FORMULA], very close to [FORMULA] with [FORMULA], twice as large as the radial-pulsator Q values derived by these authors. The star, at its hottest phase (1986-87), reached [FORMULA], so that from Eq. 2 of Lovy et al. (1984) it follows that [FORMULA] d is the shortest radial-pulsator period to be expected. We did not find oscillations of such short cycle length, which adds evidence to the conjecture that R 40 might not be a radial pulsator. Nor does the 1986-89 observed dominant oscillation (cycle length [FORMULA] d) lead to a Q value that is acceptable for radial pulsation.

Our multicolour photometry, in fact, provides observables that could be useful to discriminate between radial and non-radial modes and may even yield unambiguous l -values provided suitable models for the gravity and temperature range of R 40 would exist. Watson (1988) derived diagrams of relative colour-to-visual amplitude versus colour-minus-visual phase differences ([FORMULA] and [FORMULA]) as a function of l. Apart from l, the points in such a diagram, naturally, depend on the equilibrium parameters of the star. Our data provide for [FORMULA]: [FORMULA] and [FORMULA]. A straightforward application of our observables (even if transformed to the UBV system) to Watson's diagrams is not permitted because the parameter space used in these diagrams is incompatible with ranges occupied by LBVs. Moreover, these observables are sensitive to the metal content parameter (Z), another obstacle when dealing with stars in the SMC. Our data should allow more precise assessment of the pulsation mode(s) of R 40 as soon as proper phase-amplitude diagrams become available.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998