6 Conclusions

We have examined how the weak-line force and the rotational deformation of the star affect the confinement of one-armed oscillations in near-Keplerian discs of Be stars. For simplicity, we assumed an isothermal disc with and , where and are the disc temperature and the effective temperature of the star, respectively, and the density gradient index of the disc. The advective motion in the unperturbed disc was neglected. We took into account the effect of rotation by including the quadrupole contribution to the potential around the rotationally-distorted central star. For the radiative force due to an ensemble of optically thin lines, we adopted the parametric form proposed by Chen & Marlborough (1994).

Based on these assumptions, we have numerically studied the linear, isothermal m=1 eigenmodes confined to the inner part of the disc. Examining the eigenvalue problem for a wide range of parameters characterizing the effects of rotation and radiation, we have derived the following conclusions:

1. The present study strongly suggests a scenario in which the mechanism that causes the confinement of one-armed oscillations in early-type Be stars is different from that in late-type Be stars. In late-type Be stars, the confinement of oscillations in the inner part of the disc occurs because of the deviation from the point-mass potential around the rotationally-deformed star. The radiative effect plays a minor role in these stars. On this point, we confirm the conclusions obtained by Papaloizou et al. (1992) and Savonije & Heemskerk (1993). In early-type Be stars, however, it is the weak-line force that mainly contributes to the confinement. For these stars, the rotational effect plays a minor role.

2. The period of a one-armed eigenmode confined to the inner part of the disc is sensitive to the value of the parameter characterizing the effect of rotation or radiation. This sensitiveness, together with the period range of observed V/R variations, enables us to place rather narrow constraints on the parameter values. In particular, the distribution of the V/R periods expected from the hybrid scenario agrees well with the distribution of the observed V/R periods. Furthermore, this scenario predicts that the periods of V/R variations of early-type Be stars have little correlation with the rotation parameter of the central star, whereas, in late-type Be stars, a star with smaller rotation parameter exhibits either a V/R variation with a longer period or no V/R variation if the star rotates too slowly. Observational tests of this prediction are desired.

3. The characteristics of the eigenfunction are very similar in early- and late-type Be stars, in spite of the fact that the mechanism that causes the confinement of the mode is different. They agree well with the periodicity and line-profile variability of the observed V/R variations.

In the formulation of the current problem, we adopted the parametric form of the radiative force due to an ensemble of optically thin lines, and assumed that the radiative force decreases, with increasing radius, less rapidly than the gravity. Some of the above conclusions have been derived on the basis of this assumption. If the radiative force decreases with radius more rapidly than the gravity, it never contributes to the confinement of the m=1 oscillations. We need, then, another unknown mechanism for the confinement of oscillations in discs of early-type Be stars. It is, therefore, highly desirable to derive a more detailed form of the radiative force due to an ensemble of optically thin lines.

acknowledgements: The author thanks H.F. Henrichs for careful reading of the manuscript and stimulating discussions based on which Sect. 5 was written. The author also thanks the Astronomical Institute 'Anton Pannekoek' for their warm hospitality. This work was partially supported by a grant from Hokkai-Gakuen Educational Foundation.

Appendix: List of stars showing long-term V/R variations

In their statistical study of long-term V/R variations, Hirata & Hubert-Delplace (1981) used a total of 28 sample stars collected from McLaughlin (1932, 1958), Copeland & Heard (1963), and Hubert-Delplace et al. (1982). Later, Mennickent & Vogt (1991) statistically analyzed short- and long-term V/R variations based on their plates for southern Be stars and data collected by a literature search. Among their sample stars, a total of 29 stars show long-term V/R variations. Since the sample stars of Hirata & Hubert-Delplace (1981) and Mennickent & Vogt (1991) overlap only partly, we combined the data for these two sets to make a larger data set [we have missed data for four stars in the sample of Hirata & Hubert-Delplace (1981)]. We added data for some additional stars from the literature. The results are shown in Table 1. Table 1 also includes information on shell episodes and binarity for individual stars, which were taken mainly from Hanuschik (1996) and Pols et al. (1991), respectively. Many stars studied by Copeland & Heard (1963) are fainter than , for which we could not find recent studies. We consider this table as not exhaustive but rather fairly representative, except for stars later than B4, for which there are not sufficient data available. We also note that about 1/3 of the Be stars have not shown long-term V/R variations (Copeland & Heard 1963; Mennickent & Vogt 1991).

: Periods of long-term V/R variations for 53 stars

: (Continued)