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

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2. Photometry and program stars

2.1. The photometry

The photometric data used in this study are from Spoon et al. (1994). He collected Strömgren and Walraven photometry of the program stars, mainly from the Long-Term Photometry of Variables project organized by Sterken (1983) and published by Manfroid et al. (1991, 1994) and Sterken et al. (1993, 1995). The data are summarized in Table 1. The photometry other than from the LTPV project is indicated separately. Spoon et al. homogeneized the data and calculated the brightness of the stars in the Johnson V magnitude. We study the variations in the V magnitudes, which have a typical uncertainty of [FORMULA], to search for periodicities. The colour variations of the microvariability will be used for the mode identification.


[TABLE]

Table 1. The number of observations for the program stars.
Notes: a: Manfroid et al. (1991) and Sterken et al. (1993a,b); b: van Genderen (1979; 1982) and by van Genderen et al. (1985; 1988; 1990); c: Kilkenny et al. (1985); d: Sterken (1977; 1982).


2.2. The luminosities of the program stars

The distance, bolometric magnitude and [FORMULA] of the program stars are listed in Table 2. We do not list the spectral type, temperature or radius because they vary, whereas the luminosity remains approximately constant during the moderate photometric variations upon which the microvariations are superimposed. We discuss the parameters listed in Table 2:

  1. R 71 : We show two sets of parameters for R 71. The first is from Wolf et al. (1981). They derive a luminosity of log [FORMULA]=5.3 by integrating the spectrum during minimum state. They adopt an LMC mean value of [FORMULA]. No correction for internal extinction within the LMC was made. The second set is from model calculations by Leitherer et al. (1989). They find that the model reproduces the observed energy distribution if [FORMULA] during maximum state and [FORMULA]. We use both sets throughout this paper. Note that a difference in [FORMULA] of 0.1 results in a difference in [FORMULA]  of 0.3 if [FORMULA].

  2. 164 G Sco : The bolometric magnitude is from Humphreys & Davidson (1994) and was estimated from parameters derived by Sterken (1977).

  3. HR Car : We calculated the bolometric magnitude with the parameters given by van Genderen et al. (1991) and the more accurate distance from Hutsemékers & van Drom (1991). This gives [FORMULA]=-9.05.

  4. S Dor : From the maximum of [FORMULA] de Koter (1993) derived [FORMULA]. Leitherer et al. (1985) derived a luminosity during maximum of 0.5 magnitude lower than the one derived by de Koter. Both assume an LMC mean value of [FORMULA]. De Koter finds that his value of [FORMULA] is in good agreement with the observed and predicted photometry at visual minimum. Therefore we adopt the value of [FORMULA].

  5. R 127 : We show three sets of parameters of R 127. The first set of parameters is derived by assuming the LMC mean value [FORMULA], which gives [FORMULA] = -10.14. The second set is from de Koter (1993) who compared the observed and predicted energy distributions and concluded that at minimum state the observations fit his models if [FORMULA], which leads to [FORMULA] = -10.35. The third set is from Stahl et al. (1983) who derived [FORMULA]=-10.6 during maximum state with [FORMULA]. The extinction is obtained during minimum state by a colour comparison with an O9 supergiant (Schmidt-Kaler, 1982).

  6. AG Car : The parameters have been determined by Humphreys et al. (1989) from UV and visual observations in maximum and minimum, based on a new distance determination of 6.2 kpc. The extinction is determined by comparison of the energy distribution with a set of standard Ia supergiants. This results in a bolometric magnitude of [FORMULA] = -10.8 [FORMULA] 0.4.


[TABLE]

Table 2. Parameters of the program stars.
Notes: a: Panagia et al. 1991; b: Wolf et al. 1981; c: Leitherer et al. 1989; d: Sterken 1977; e: Humphreys & Davidson 1994; f: Hutsemékers & van Drom 1991; g: van Genderen et al. 1990; h: de Koter 1993; i: Stahl & Wolf 1982; j: Stahl et al. 1983; k: Humphreys et al. 1989.


2.3. The masses of the program stars

The masses of LBVs are not well known, because they cannot be derived from spectroscopic analysis in a reliable way. For instance, Pauldrach & Puls (1990) derived a spectroscopic mass for P Cyg of only 23 [FORMULA]. However, the mass cannot be smaller than about 30 [FORMULA], otherwise the star would not have any hydrogen left at its surface. Therefore we derived the mass of the program stars from evolutionary tracks from Schaller et al. (1992) for Galactic stars (Z=0.02), and from Schaerer et al. (1993) for S Dor, R 71 and R 127 in the LMC (Z=0.008). We assume that a star reaches the LBV phase when its surface composition has reached a He/H ratio of 0.40 by number. This is the composition derived from spectroscopic studies of several LBVs (Najarro et al., 1997; Crowther, 1997). The ratio He/H=0.4 corresponds to the phase where X=0.377, Y=0.603 and Z=0.020 for Galactic stars and X=0.382, Y=0.610 and Z=0.008 for LMC stars.

We adopt conservative upper and lower limits of the mass. The LBV phase occurs after the main sequence phase and before the Wolf Rayet phase. So, the maximum mass of a star in the LBV phase is the mass at the end of the H-core burning phase. The minimum mass of an LBV is the mass at the beginning of the N rich WNL phase of Wolf Rayet stars. From the models of Schaller et al. (1992) and Schaerer et al. (1993) we derived the luminosities and masses at the three phases where (a) He/H=0.40 and (b) at the beginning of the WNL phase and (c) after the core contraction at the end of the main sequence phase, for stars with initial masses of 85, 60, 40 and 25 [FORMULA]. These [FORMULA]-[FORMULA] relations were interpolated logarithmically to derive the mass and its upper and lower limits for each LBV of a given luminosity. The estimated masses with their upper and lower limit are listed in Table 2. The relation between mass and luminosity of the individual stars can be fitted by the relation

[EQUATION]

2.4. Estimates of [FORMULA] and log  g

For the purpose of this paper we need an estimate of [FORMULA] and log  g of the stars during the various phases of their variability. These quantities are not well known because they would require a detailed study of the energy distribution or the spectrum of each star during the various phases. This has not been done. Therefore we estimate [FORMULA] and log  g in a simpler, approximate way.

The variations in V are largely due to variations in the Bolometric Corrections (BC) of the stars, because LBVs vary at approximately constant bolometric magnitude (Wolf et al. (1981) for R71; Leitherer et al. (1985) for S Dor; Stahl & Wolf (1982) for R127; Lamers et al. (1989) for AG Car). This implies that the variations in V of the typical LBV variations (not of the micro variations!) can be related to variations in BC, which can be used to estimate the variations in [FORMULA] and log  g. We adopted the empirical relation between BC and [FORMULA] for supergiants from Schmidt-Kaler (1982). In the temperature range of 12000 to 35000 K, the BC can be approximated quite accurately, i.e. within about [FORMULA], as

[EQUATION]

with [FORMULA]. Using this expression we can estimate the values of [FORMULA] during the various phases of the LBVs, by deriving BC from the difference between [FORMULA] and [FORMULA]. With [FORMULA] and [FORMULA] we can derive [FORMULA] in the usual way. The value of log  g then follows from [FORMULA] and [FORMULA]. It is easy to show that for a constant value of [FORMULA] and Eq. (2) the gravity varies as [FORMULA]. So if the star gets fainter in V by half a magnitude the gravity increases by a factor 1.6, if [FORMULA] is constant.

For AG Car, which we study at the epoch of visual minimum [FORMULA], we adopt a value of [FORMULA] K. This value was derived from a detailed comparison between its energy distribution (visual and UV) at a phase when [FORMULA] and those of other supergiants by Lamers et al. (1989) (see also Humphreys et al. 1989).

There is some doubt about the constancy of the [FORMULA] of LBVs during their variations, because the IUE data used for these studies have an accuracy of only about 25 percent. A study by Vennix (see Lamers, 1995) of the changes in the energy distribution of S Dor and a comparison with extended model atmospheres has suggested that [FORMULA] of this star can vary by as much as [FORMULA] in [FORMULA] during a complete cycle of the typical LBV variations with [FORMULA] being fainter during visual maximum. If this trend is confirmed for other stars as well, it implies that the changes in BC and the resulting changes in [FORMULA] are larger than for constant [FORMULA]. However, the change in [FORMULA] then counteracts this effect on the determination of log  g. So, given the maximum changes in V of about [FORMULA] in the time intervals studied here, we conclude that the assumption of constant [FORMULA] results in sufficiently accurate estimates of the variations in log  g for the purpose of this paper.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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