4 Polarization and projected rotational velocities

4.1 Young stars

Figure 13: $\log p/v\sin{i}$-diagram for HAEBE+Vega-type stars

Figure 14: $\log p/v\sin{i}$-diagram for classical Be stars

The main critical comment to the first paper of Yudin ([1988]) was that in the case of disk-like distribution of scatterers around the stars (that is at present a general model for the observed behaviour of young stars and many direct evidence of such geometry now exists) no linear correlations between $\log p$ and IR excesses should be observed. The justification for this suggestion is that the polarization is proportional (for optically thin disks) to the value of $\sin^{2}{i}$ where i is an inclination angle of the disks to the line of sight. The referees have commented on the existence of a "triangle diagram" rather than a linear correlation. However as follows from Fig. 4, even the significantly larger statistics demonstrate again the pronounced relationship between $\log p$ and E(V-L) for young stars. Note however that at present we don't know which objects have exact pole-on orientation of CS disks for the majority of young stars (maybe with a few exceptions). Even though the $v\sin{i}$ value is small it is not uniquely determined that the CS disk has a pole-on orientation while some stars may have low rotational velocity themselves. All available data on the rotational velocities for young stars (about 250 objects; see Appendices 1-5) allow us to conclude that no correlation exists between $p_{\rm obs}$ and $v\sin{i}$(see Fig. 13).

Moreover, there is reason to believe that the compression of CS environment takes place along the lines-of force of the interstellar magnetic field (see for example Grinin et al. [1991] and Tamura & Sato [1989]). For the most part the young stars are located in the plane of the Galaxy but the orientation of interstellar magnetic field is mainly co-linear with the direction of the axes of spiral arms. In the paper of Andriasyan & Makarov ([1989]) the directions of the Galactic coordinates for which the orientation of interstellar magnetic field is perpendicular to the line of sight were separated out. The correlation of these coordinates with those for the stars in our sample shows that about 90% of the objects lie in those regions. Thus we can reject the suggestion of pole-on orientation of CS disks for most of the selected stars. In addition note that, even for the pole-on orientation of the disks polarization of radiation may take place due to the presence of the rotating dust inhomogeneities in CS shells, or if the disks have no circular symmetry (for example for an ellipse-like envelope and location of a star in one of the focus of this ellipse). In any case even a small departure of i from $0\hbox{$^\circ$ }$ gives an ellipse for the projection of the disk on the sky-plane which results in the detectable polarization of radiation (see Discussion in Yudin et al. [1999]). Nevertheless, some influence of the disk's inclination on the observed polarization does occur for individual stars (see for example Yudin et al. [1999]). One can suppose that the inclination effect may be pronounced for homogeneous and optically thin disks when the main mechanism of polarization is the scattering in these envelopes and no other sources of polarization exists (such as that of scattering on dust inhomogeneities or condensations revolving around a star).

We may also compare the values of intrinsic polarization and $v\sin{i}$for the group of HAEBE stars with Algol-like minima (most of them have the similar spectral types around A5). During the photometric minima due to eclipse of nonpolarized stellar light we can detect scattered radiation from their nonspherical dust envelopes. As can be seen in Fig. 12 there is no evidence that stars with larger values of $v\sin{i}$ show larger values of polarization. However, at present, the list of this type of HAEBE stars is not large enough to draw definite conclusions. It is interesting that large Algol-like minima of brightness are observed for some stars which have respectively small values of $v\sin{i}$ (60-90 km s^-1). Discussion on this behaviour will be given in Sect. 7.

4.2 Classical Be stars

Figure 15: Histogram of $v\sin{i}$ values for HAEBE+Vega-type stars

Figure 16: Histogram of $v\sin{i}$ values for classical Be stars

It is interesting to compare polarimetric and $v\sin{i}$ data discussed here for young stars with those for classical Be stars. The correlation between p and $v\sin{i}$ values for classical Be stars has been investigated by many authors. McLean & Brown ([1978]) plotted the values of pversus $v\sin{i}$ for a sample of 67 classical Be stars and found a "triangle" distribution (see their Fig. 2). In Fig. 14 we plot the values of polarization versus $v\sin{i}$ for the larger group of classical Be stars (285 objects). With six times better statistics (compared with previous studies) we can conclude that the relation shows again a "triangle" distribution but not in the same sense as has been discussed by McLean & Brown ([1978]). The differences are that the polarization degree has reached maximum values for stars with intermediate values of rotational projected velocities 200 km s^-1 $< v\sin{i} < 300$ km s^-1 and there are a small portion of stars (mainly Be stars with observed polarization) with large polarization among the stars with relatively low ( $v\sin{i} < 150$ km s^-1, and very high ( $v\sin{i} >$ 350 km s^-1, projected rotational velocities. The detailed dicsussion of this behaviour and the results of a statistical study of different observational characteristics of classical Be stars may be found in Yudin ([2000]).

The investigation of $v\sin{i}$ distributions was not the initial aim of the present study. However large statistics make it possible to compare the rotational velocity distributions for the groups of stars which are at different stages of evolution, which may be interesting in the context of the present work.

Note that Davis et al. ([1983]) have noted the difference in the frequency distribution of $v\sin{i}$ between Herbig Be stars and classical Be stars (but for a small sample). B $\rm\ddot{o}$hm & Catala ([1995]) recently compared $v\sin{i}$ values for low, intermediate and high-mass HAEBE stars and have estimated the mean projected rotational velocities (MRPV) for 27 HAEBE stars: $v\sin{i}\approx 105\pm 35$ km s^-1. The histograms of $v\sin{i}$distributions for HAEBE+Vega-type stars from our sample and classical Be stars are presented in Figs. 15 and 16 respectively. Despite the fact that both distributions are broad it is reasonably safe to suggest that these distributions diverge considerably. Simple comparison of $v\sin{i}$ values for stars of different groups leads to the following results: the MPRV for HAEBE+Vega-type stars (164 objects) and classical Be stars are $\approx$105 km s^-1 and $\approx$220 km s^-1 (with the standard deviation 80 km s^-1and 90 km s^-1) respectively.

One can however note that comparison of the $v\sin{i}$distributions is inapplicable for stars having a wide range of spectral classes. However even for 83 Herbig Be+B Vega-type stars the MPRV is $\approx$130 km s^-1 or excluding a few stars which were early classified as classical Be-105 km s^-1. Various statistical tests indicate immediately that the difference between mean values of $v\sin{i}$for Herbig Be+B Vega-type stars and classical Be stars is statistically significant even at 99% confidence level. In spite of small statistics the average value of $v\sin{i}$ for young solar-type stars is smaller than for TT stars (see Appendices 4-5).

As is evident from the theory of stellar evolution, within similar spectral classes the rotational velocities should be smaller for more evolved stars due to the loss of AM (see for example Soderblom et al. [1993] or Dudorov & Pudritz [1994]). However classical Be stars are much more rapid rotators in comparison with young Herbig Be and B Vega-type stars in spite of the former being much more evolved. On the one hand it can be supposed that the increasing of rotational velocities in Be stars takes place due to further contraction and the associated conservation of AM, while a spin down for HAEBE stars is due to net AM loss (by stellar wind) as was supposed by Finkenzeller ([1985]). But on the other hand the young stars may lose AM by interaction with dust circumstellar shells because classical Be stars have no dust in their environment and this is the primary difference between them. However this leads us to conclude possible differences in initial conditions of formation of Be and young Herbig Be stars because if classical Be stars were surrounded by dust shells in earlier evolutionary phases they would also have low $v\sin{i}$ values at present (see also discussion in Zorec & Briot [1997]). A detailed discussion of various mechanisms for the redistribution of AM can be found in Brown & Verschueren ([1997]).

An important by-product of this study is that the average values of $v\sin{i}$ differ strongly for classical Be and young HAEBE + Vega-type stars.

Figure 17: $\log p/E(V-L)$ diagram for HAEBE stars, Vega-type stars and classical Be stars. The group of classical Be stars (1) contains the objects with observed polarization whereas the group (2) - the objects with calculated intrinsic polarization. The box indicates the position of supergiants from Serkowski et al. ([1975])