6 Summary

In Paper I we proposed a model for the long-term V/R variations of Be stars. Examining the linear, one-armed, isothermal oscillations in isothermal equatorial disks, we found that this model well reproduces the periodicity of the observed long-term V/R variations for a range of disk sizes and density-gradient indices .

In this paper we have considered the profile variabilities of Balmer lines caused by the one-armed isothermal oscillations studied in Paper I. In order to examine the effects of one-armed oscillations on line profiles, we assumed nonlinear perturbation patterns similar to the linear m=1 eigenfunctions. The eigenfunctions were normalized so that the maximum value of the perturbed part of the angular velocity is 10% of the unperturbed part . In addition, we assumed that the source function is constant over the entire disk region. Based on these assumptions, we studied the profile variations by computing fluxes along a bundle of line-of-sights for various values of disk sizes , density gradient indices , characteristic optical depths , and inclination angles i. We took only the thermal broadening into account as the line-broadening mechanism. As a result of this study, the following conclusions have been derived:

1. For a line profile from a disk with a density distribution of , the peak separation is given by , and is independent of the line optical depth. This is because the largest contribution to the total emission for arises from the constant line-of-sight velocity region tangent to the disk outer radius. On the other hand, for a steeper () density distribution, the largest contribution arises from the constant line-of-sight velocity region tangent to the radius where the optical depths along the line-of-sights are about unity. Thus, for , the peak separation decreases towards along with an increase in the line optical depth.
2. For a wide range of disk parameters, the line profiles from disks with m=1 perturbation patterns exhibit remarkable V/R asymmetries. The profile as a whole shifts redward (blueward) when V/R > 1 (V/R < 1). These features are insensitive to the detailed disk structure adopted. We thus conclude that in general terms the one-armed oscillation scenario agrees well with the observed V/R variability. On the other hand, the amplitude of the profile shift associated with the V/R variation is sensitive to the perturbed disk structure. In the present model it is larger for disks with steep () density gradients than for disks with flat () density gradients. This is because the amplitude of the perturbation is larger for than for at the region from which the largest contribution to the total emission arises. In principle, this disk-structure dependent feature can be used to probe the structure of Be-star disks.

In this paper we assumed perturbation patterns similar to the linear eigenfunctions and the constant-source function to study the effects of the one-armed oscillations on line profiles. Consideration of the effect of the non-uniform source function is expected to provide only minor modifications to the above conclusions. Taking the nonlinear effects exactly into account, however, can modify the results obtained in this paper, since the amplitude of the V/R variation depends on the detailed forms of the eigenfunctions. In addition, the periods and eigenfunctions are sensitive to the disk models adopted. Therefore, it is highly desired to investigate the nonlinear one-armed modes in equatorial disks of Be stars.

It is a pleasure to thank R. Hirata, S. Kato, M. Mon, and T. Horaguchi for stimulating discussions. The hospitalities of the Department of Astronomy, Kyoto University and Astronomical Institute, University of Amsterdam are also greatly appreciated. The author wishes to acknowledge financial support of the Foundation for Inservice Trading and Welfare of the Private School Personnel. This work was partially supported by a grant from the Hokkai-Gakuen Educational Foundation.