Context. One way to understand the still mysterious Be phenomenon is to study the time variations of particular Be stars with a long observational history. ζ Tau is one obvious candidate. Aims. Using our rich series of spectral and photometric observations and a critical compilation of available radial velocities, spectrophotometry of Hα, and UBV photometry, we characterize the pattern of time variations of ζ Tau over about a century. Our goal is to find the true timescales of its variability and confront them with the existing models related to various aspects of the Be phenomenon. Methods. Spectral reductions were carried out using the IRAF and SPEFO programs. The HEC22 program was used for both photometric reductions and transformations to UBV. Orbital solutions were derived with the latest publicly available version of the program FOTEL, period analyses employed both the PDM and Fourier techniques -programs HEC27 and PERIOD04.

Context. Many scientists studying Be stars currently adopt the model of one-armed oscillations as the correct explanation of the cyclic longterm V/R variations observed for a number of Be stars. We test the ability of this model to be used for the predictions of V/R variations in real observed Be stars. Aims. The behavior of the one-armed oscillations can be described as a solution of linearized hydrodynamical equations with the presence of "distorted" gravitational potential and a radiation force. Methods. We developed a new computer program to model one-armed oscillations in Be star disks, which includes both the pressure force and the quadrupole term in the gravitational potential, related to the obliquity of a rapidly rotating star inside the disk. Moreover, we slightly improved the model in an effort to decrease the number of input parameters with the help of NLTE stellar atmosphere models. Results. We carried out detailed tests of the dependence of V/R "periods" predicted by the model on all individual input parameters. We arrived at the following results: (1) the model has great potential to explain not only the cause of the cyclic long-term V/R changes but also some of the observed statistical properties of the phenomenon. (2) The model in its present linear form cannot be considered as proven. Its ability to predict the duration of V/R cycles for individual well observed Be stars is insufficient. Changing some of the input parameters of the model, which are still loosely constrained by observations and/on current understanding of the disks, like the radial density distribution in the disk, one can easily arrive at any desired cycle length from, say, 1 to 20 years. Conclusions. Clearly, a much more sophisticated non-linear and self-consistent model of disk structure and its oscillations will be needed before a truly quantitative test of a one-armed model vs. observations will be possible.

We prove the existence and stability of flat steady states of the Vlasov-Poisson system, which in astrophysics are used as models of disk-like galaxies. We follow the variational approach developed by Guo and Rein [5,6,7] for this type of problems and extend previous results of Rein [11]. In particular, we employ a reduction procedure which relates the stability problem for the Vlasov-Poisson system to the analogous question for the Euler-Poisson system.