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Astron. Astrophys. 349, 151-168 (1999)

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2. Correlations with fundamental stellar parameters

In this section we study the correlations between slopes of SPh behaviours obtained in MZH with the following stellar parameters: [FORMULA], [FORMULA] (critical equatorial linear velocity of rigid rotation), T[FORMULA] and [FORMULA] which is used as an aspect angle indicator (see Sect. 2.3.1). In what follows, the SPh slopes are noted as:

[EQUATION]

Units of slopes [FORMULA] and [FORMULA] are mag dex-1, for [FORMULA] [FORMULA] and for [FORMULA] [FORMULA].

2.1. Data used

For each studied star the slopes [FORMULA], [FORMULA], [FORMULA], [FORMULA] and their respective statistical uncertainty are given in Table 1. When Be stars have two slopes corresponding to the same kind of SPh phase, they are both considered separately. The same was also done for a given star when it showed two different SPh phases.


[TABLE]

Table 1. Fundamental parameters and SPh slopes of the studied Be stars



[TABLE]

Table 1. (continued).
Note:
"E" and "A" in 2nd column stand respectively for SPh-E and SPh-A phases. Units of slopes [FORMULA] and [FORMULA] and their respective dispersions are mag/dex, for [FORMULA] and [FORMULA] µm/dex and for [FORMULA] and [FORMULA] [FORMULA]


Effective temperatures, radii and masses of our program Be stars were determined from the "photospheric" BD ([FORMULA]) and, when available, the BCD [FORMULA] parameter. For stars without a measured [FORMULA] parameter, we adopted the mean [FORMULA] corresponding to its MK luminosity class. Calibrations of ([FORMULA] in [FORMULA], [FORMULA] are from Divan & Zorec (1982), Zorec (1986). With [FORMULA] and [FORMULA] we determine both the stellar radius [FORMULA] and the stellar mass using Maeder & Meynet's (1988) evolutionary tracks. The critical velocity [FORMULA] is then derived assuming [FORMULA] (Sackmann 1970, Bodenheimer 1971, Zorec et al. 1988). The [FORMULA] parameters are mostly from Slettebak (1982) except for HD 37202 (Yang et al. 1990). The stellar parameters used in this work are given in Table 1.

2.2. Results

Instead of considering each SPh slope individually, for they are marred to some extent by uncertainties, we preferred to study mean values of slopes per interval of a given stellar parameter. Mean values are obtained independently for SPh-E and SPh-A phases. In this way, trends indicating possible underlying parametric dependencies can be made conspicuous more clearly. For some stars there are two slopes in a given SPh phase. As in each of them points correspond to different observation epochs, they were considered independently. In the calculation of mean slopes we did not take into account those for which the total detected variation of D was [FORMULA]. The sampling intervals used are: 50 km s-1 for velocities, 0.125 for the velocity ratio [FORMULA] and 1250 K for the effective temperature. The resulting means, with their respective bars representing 1 standard deviation, are shown in Figs. 2 to 5. In these figures, filled squares are for SPh-E phases and open squares for SPh-A phases.

[FIGURE] Fig. 2. Averaged SPh slopes a[FORMULA], a[FORMULA], a[FORMULA] and a[FORMULA] against [FORMULA]. "Filled squares" for SPh-E phase; "open squares" for SPh-A phase. Regression lines shown refer to SPh-E phases only.

[FIGURE] Fig. 3. Idem Fig. 2 but against [FORMULA]

[FIGURE] Fig. 4. Idem Fig. 2 but against [FORMULA]

[FIGURE] Fig. 5. Idem Fig. 2 but against T[FORMULA]

Results shown in Figs. 2 to 5 can be summed up as follows:

(i) We can immediately see that there is a clear distinction between the SPh-E and SPh-A slopes. Excepting mean values of a[FORMULA], all remaining averaged slopes corresponding to SPh-A behaviours are [FORMULA] [FORMULA], [FORMULA] [FORMULA] [FORMULA] [FORMULA]. In a number of stars slopes have high intrinsic errors. When [FORMULA] [FORMULA] we can consider that [FORMULA] [FORMULA]. Exceptions like HD 184279 definitely have [FORMULA]. Other stars in SPh-A phases have [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA].

(ii) T[FORMULA]-dependence is detected for a[FORMULA] and a[FORMULA].

(iii) Correlation with V[FORMULA] is detected for mean a[FORMULA] and a[FORMULA] slopes.

(iv) Correlations of slopes against [FORMULA] and [FORMULA] can be considered equivalent. Mean values of a[FORMULA] and a[FORMULA] slopes correlate with [FORMULA] and [FORMULA].

(v) No clear correlations with any of studied stellar parameters seem to be outstanding for the mean a[FORMULA] slopes.

For easy comparison of the present results with previous results dealing with SPh behaviours, we reproduce the regression lines of mean slopes [FORMULA], [FORMULA], [FORMULA] which can be used simply as indications for possible mean dependencies on the respective stellar parameters. Regression lines for SPh-E phases are:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The significance level of SPh-A mean slope regressions are too low to suggest any reliable correlation. The values of mean SPh-A slopes scatter near zero and are characterized by the following averages and [FORMULA] dispersions:

[EQUATION]

Comparing our results with those obtained in previous works, we observe that:

(a) Slopes [FORMULA] in SPh-E phases, which suggests that, as a mean, there is reddening of visible energy distribution as emission in the second BD increases. Nevertheless, two exceptions (HD 56014 and HD 91465) have been found that present some blueing as emission increases in the second BD;

(b) Similarly to the findings of other authors, slopes [FORMULA] in SPh-E phases, which corresponds to brightening in the V magnitude as the continuum Balmer emission is higher. Increased faintness in the V magnitude as emission in the second BD increases was however found in HD 32343, HD 68980 (during one of its SPh-E periods) and in HD 148184 during its period of strongest emission. For HD 148184, Dachs (1982) reports lowering of H[FORMULA] line emission from 1972 to 1979, when the star showed brightening in the V magnitude;

(c) Comparing our results with those of Hirata (1982) and Hirata & Hubert-Delplace (1981), we see that reddening of the visible energy distribution in SPh-E phases implied by a[FORMULA] in relations (5) or in Fig. 5 is smaller for lower effective temperatures, but that it never turns to a blueing effect for lower temperatures, as noted in Kogure & Hirata (1982). We note that blueings of photometric indices can sometimes be due to variable spectral lines produced in CE, so that they do not necessarily correspond to a temperature effect;

(d) Using the transformations from UBV photometric to SPh indices obtained in MZH [cf. (1), (2) and (3)], we readily derive the following relations between photometric and SPh slopes:

[EQUATION]

[EQUATION]

which allow us to compare the photometric results discussed by other authors with the SPh results used in the present work. Using the [FORMULA] and [FORMULA] slopes listed in Table 1 respectively in (7) and (8), we confirm the results previously obtained by Hirata (1982) and Hirata & Hubert-Delplace (1981), where it was noted that in SPh-E phases [FORMULA] and [FORMULA] have the same sign. However, we have found some exceptions to these rules which are observed in HD 58978, HD 63462, HD 173219 and HD 217543;

(e) Introducing relation (2) for [FORMULA] into (8), we see that [FORMULA] is steeper when [FORMULA] is higher as also claimed in previous works. Using now relation (5) for [FORMULA] in (8), we can still add that [FORMULA] is flatter when [FORMULA] is higher;

(f) Dachs et al. (1988) found that the ratio [FORMULA]. They also found that absolute values of this slope are twice as small as those predicted theoretically. Using the a[FORMULA] slopes given in Table 1, we find that in SPh-E phases this ratio not only has negative values as low as [FORMULA], but that it can also have positive values [FORMULA] as previously also noted by Hirata (1982).

2.3. Correlations of SPh parameters with stellar parameters

Noticeable correlations seem to exist for mean [FORMULA] and [FORMULA] SPh slopes with [FORMULA] (and/or [FORMULA]), [FORMULA] and [FORMULA]. A correlation between [FORMULA] and [FORMULA] is not expected in a sample where stars are of different luminosity classes. Nevertheless, the regression line obtained for stars in the SPh-E phase: [FORMULA] [FORMULA] [FORMULA] is obtained with a correlation coefficient [FORMULA]. On the other hand, [FORMULA] [FORMULA] [FORMULA] with [FORMULA] is derived by using the mean values of [FORMULA] and [FORMULA] corresponding to the sampling intervals of Figs. 4 and 5. This shows that mixing of luminosity classes in our sample is not high enough to prevent a rather clear relation between [FORMULA] and [FORMULA]. This also means that correlations of [FORMULA] and [FORMULA] against [FORMULA] and [FORMULA] can be considered synonymous.

2.3.1. Correlations with [FORMULA] and/or [FORMULA]

We have just seen that for our program stars there is a well defined relation between V[FORMULA] and T[FORMULA]. On the other, as we shall see, the "true" rotational velocity V of Be stars can be considered nearly proportional to [FORMULA]. So, this could imply that trends in Figs. 2 and 3 might be considered to be due to some temperature effect. However, the following argument supports the assumption of a dominant aspect angle effect.

Distributions of the angular velocity ratio [FORMULA] ([FORMULA] critical angular velocity for rigid rotation) of Be stars are strongly peaked, according to the spectral type and luminosity class, from [FORMULA] to [FORMULA] (Zorec 1986, Zorec & Briot 1997). In Fig. 6 is shown the distribution of the most numerous group of Be stars: B2+B2.5+B3 main sequence stars, obtained from their observed distribution of [FORMULA] (histogram) and corrected from errors in [FORMULA] (Smart 1958) and random distribution of the inclination i (Lucy 1974). It can be seen that 80% of these stars are rotating in the [FORMULA] interval. The distribution of true rotational velocities of Be stars of all subspectral types together (Porter 1996) also has the same appearance as the one for the B2+B2.5+B3 main sequence stars. We can then safely assume that the value [FORMULA] represents a reliable approach to the mean probable value of [FORMULA] at which the Be stars rotate. Slettebak et al. (1992) used [FORMULA]. Thus, to infer the probable viewing angle of Be stars we shall use the interval [FORMULA] (Sect. 3.1), so that [FORMULA] [FORMULA] [we used [FORMULA] [FORMULA] for the equatorial radius deformed by rigid rotation (Sackmann 1970, Bodenheimer 1971, Zorec et al. 1988)]. As the mean value of [FORMULA] in each sampling interval of [FORMULA] of Figs. 2 and 3 is amazingly the same: [FORMULA] km s-1, it means that trends shown in Figs. 2 and 3 are due to a genuine aspect angle effect. The high dispersion of points in Figs. 2 and 3 is not only due to measurement uncertainties of SPh slopes, but also to possible physical phenomena which are discussed in Sect. 3.3.

[FIGURE] Fig. 6. Distribution of B2+B2.5+B3 main sequence Be stars against rotational velocities [FORMULA], V and [FORMULA]

2.3.2. Correlations with [FORMULA]

The visible continuum emission is proportional to the emission measure. The latter is larger when the radius of the H-ionized region is more extended, so when [FORMULA] is higher. It is expected then that SPh variations will also be more noticeable if stars are hotter. Such a tendency is depicted by the trends in Fig. 5 of [FORMULA] and [FORMULA] SPh slopes against [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999
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