Astron. Astrophys. 335, 605-621 (1998) 5. The Q-values and the relations between the stellar parameters and the periods5.1. The Q-values of pulsating early-type starsThe periods of adiabatic pulsations are expected to depend on the mean density of the star via the well-known period mean-density relation The parameter Q depends on the adiabatic index , which determines the dynamical stability of the star, and on the density distribution within the star. The average Q-value for the radial fundamental mode of Cepheids is 0.040 days (e.g. Cox, 1980). The (non-)radial p-mode pulsations of Cephei stars have an average Q-value of 0.03, while the non-radial g-mode pulsations that appear in Slowly Pulsating B-stars have 0.68 as average Q-value (Heynderickx et al., 1994). The Q-values of BA-type supergiants range from 0.032 up to 0.172 (Burki 1978), while O-type supergiants reach values up to 0.6 (van Genderen, 1985). Lovy et al. (1984) have determined the period and the value of Q for radial modes (fundamental and first and second overtones) of supergiants, using the evolutionary tracks of Maeder (1981). They find that the observed periods in 40% of the supergiants are compatible with those predicted by their models. In the other cases, however, the periods are much longer than the ones predicted by their models and they conclude that these supergiants undergo non-radial g-mode pulsations or that they are in a post-red supergiant stage. 5.2. The Q-values of the microvariations of LBVsWe have determined the Q values for all the epochs in which periodicity was found. They are listed in Column 8 of Table 4. The Q-values are in the range of 0.058 to 0.909 days. We find that the Q-values differ significantly for different stars and moreover that they change considerably from one epoch to the other for some stars (e.g. AG Car and S Dor). In Table 5 we list the mean values for the stars. We also give an indication of typical LBV phase (visual minimum, visual maximum, or halfway) during the observations as shown by the lightcurves of Fig. 1. We give the data for the two values of for R 71 and for the three values of for R 127. Table 5. Empirical Q-values. The values of Q are sensitive to the adopted stellar masses, since . The conservative uncertainty of the masses are listed in Table 2. The resulting uncertainties in Q, expressed in terms of minimum and maximum corrections are also listed in Table 5. The data in this table show that the mean value of Q is between 0.07 and 0.18 days for most of the stars (R 71, HR Car, 164 G Sco, and R 127). The two exceptions are S Dor with and 0.67, and AG Car with , 0.19 and 0.11. The behaviour of AG Car shows that the period of the microvariations and Q can change by as much as a factor four, even when the visual magnitude of the star hardly changes. The data in Tables 4 and 5 suggest that the most common microvarations of LBVs have days. However the stars can go through phases of slower pulsations when Q increases by a factor up to four. This does not seem to be related to a specific phase in the lightcurve. The two high values of Q for S Dor possibly represent temporary phases of slow pulsation. The same behaviour might also be present in the microvariations of normal supergiants. Van Genderen (1985) lists three other supergiants that have abnormaly high Q-values. All other supergiants studied by him and by Burki (1978) have Q's that are at least a factor-of-two smaller than these three cases. Lovy et al. (1984) predicted Q values for the fundamental and the first overtone of the radial pulsations of a grid of supergiant models. For the early type supergiant models in the range of log , with masses in the range of , the predicted Q-values for the fundamental radial mode are . Our mean values of Q for the four stars R 71, HR Car, 164 G Sco, and R 127 are larger than this by a factor 1.5 or 2. This might be due to the higher - ratio of the LBVs compared to normal stars, because they may have lost more mass already. Post-main sequence stars which have lost mass will have a mass distribution that is more concentrated in the center than stars without mass loss, because the core is hardly affected by the mass loss, but the envelope mass has decreased. So we expect that LBVs, which have lost more mass than normal B supergiants, will have higher Q-values. 5.3. The period-M_{V} relationWhen the visual brightness of an LBV increases, its effective temperature decreases. At the same time the radius of the star increases. This change in structure of the star may also affect the period of the microvariations. In this section we search for relations between the visual magnitude of the stars and the period of the microvariations. The mean density can be expressed in terms of and while can be expressed in and or . This results in a predicted relation for adiabatic pulsations: The mass can be eliminated from this equation by means of the mass luminosity relation for LBVs, given in Eq. (1). The temperature can be expressed in terms of the bolometric correction, BC. The dependence of the bolometric correction on is taken from the empirical relation for normal supergiants, approximated by Eq. (2). This results in the following predictions for P: This equation is valid if Q and are both constant during the typical LBV variations. Equation (4) shows that the pulsational period of the LBVs with the same effective temperature are expected to vary with . Equation (5) shows that for an LBV that varies with constant luminosity or , the pulsational period is expected to vary as , if Q remains constant. Fig. 3 shows the relation between the observed values of and for the stars R 71, HR Car and R 127. We only show the relation for these three stars for which we either have several period determinations or where these determinations refer to significantly different values of .
We know beforehand that we cannot expect an excellent empirical correlation. This is simply due to the fact that the period of the microvariations changes irregularly after a few pulsation cycles, even if the visual magnitude of the star remains almost constant. This is most obvious in the data of the star HR Car (Table 4) for which we derived four reliable periods at almost constant visual magnitude between = -8.57 and -8.61. The period varies between 18.6 and 41 days, which is more than a factor 2. Fig. 3 also shows the slope of the expected relation (Eq. 5)
for constant Q and . The figure shows
that on the average the periods of the microvariations increase when
the star gets visually brighter. This is to be expected because the
radius increases. Although the data are scarce, the figure suggests
that the observed relation between P and
is steeper than predicted for constant Q and
. This could be due to at least two effects:
We conclude that the observations show the expected trend of increasing period with increasing visual brightness, but that the observed relation is steeper than predicted for constant Q and . The difference can be due to changes in Q due to changes in the density distribution as the outer envelope of the star expands. 5.4. The period-amplitude relationFig. 4 shows the amplitude of the variations versus the period. The stars are arranged in order of increasing luminosity. We find that larger amplitudes are seen for variations with longer periods. The lines in the six graphs are simple linear fits going through zero: . Except for AG Car (and possibly R 71) these linear fits agree with the data. The slope of the amplitude-period relations are listed in Table 6. The data in Fig. 4 and in Table 6 show that, apart from AG Car, there is a trend of decreasing slope with increasing luminosity. Van Genderen (1985) already noticed this trend. AG Car behaves differently in the sense that the linear fit through the origin has a much steeper slope than suggested by its high luminosity. However, if we would relax the condition that the relation should go through the origin, the slope would be very flat (dashed line).
Table 6. The Amplitude-Period relation. We conclude that the -relation of the microvariations of LBVs has a slope that decreases with increasing luminosity of the star. 5.5. Conclusion about the photometric "microvariations"
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