Robert Buchler's carrier as seen by his long term friend and colleague Jean Perdang (Note of the Eds.).

One of the hardest questions of stellar pulsation theory is the modal selection in multi-mode oscillations. Linear stability analyses predicts a dense spectrum of excited modes; however, the amplitude of these modes cannot be determined by these calculations. Only nonlinear hydrodynamic models can answer the above question; however, codes exist only for radial pulsations. Amplitude-equation formalism works well for predicting and understanding the results of radial hydrocodes. We extend our results of double mode pulsations for the interaction of dozens of modes and check the possible scenarios in multi-mode oscillations.

Klein-Gordon operators with complex potentials are formulated as model systems to examine the possible impact of non-normality on the stellar pulsation eigenvalue problem. Computations of spectra, ε-pseudospectra and resolvent norms for these operators show that extended regions of complex potential, corresponding in stars to non-adiabatic regions, leads to strong non-normality. Introduction of an advective term further increases non-normal properties. Prospective model improvements and implications for resonances and for nonlinear pulsation are considered.

We present a new numerical scheme describing the convective velocity field using two parallel radial columns in order to represent up- and downstream flows. The two columns are coupled by horizontal exchange via fluid flow and radiation over their common interface. For this geometrical setup, the equations of radiation hydrodynamics are solved time dependently using an implicit scheme which has the advantage of allowing very large time steps. Finally we demonstrate our approach for the example of a Cepheids convection zone.

We develop a new version of a Cellular Automaton (CA) for the simulation of the dynamics of a stellar atmosphere sitting on top of an inert core, and specified by the following physical input parameters: mass, radius and luminosity of core, and mass of atmosphere. The CA incorporates various parametrised simulation schemes of the instability mechanism (essentially ionisation). The initial state in all of our numerical experiments is a radially symmetric atmosphere of exponential density run and uniform temperature (input parameters: density scale– height and temperature of atmosphere). The initial atmosphere is not in hydrostatic and thermal equilibrium. After a transient stage, the system relaxes, for certain ranges of the parameters of the instability mechanism, towards a state of nontrivial dynamical behaviour: Local heat–driven circulations are set up which may range from nearly stationary and spatially symmetric cellular patterns to temporally and spatially irregularly fluctuating velocity fields. The traditional radial symmetry of the density pattern is broken, so that the star acquires a globally non–spherical shape. The residual non–stationary component, when integrated over the star to produce the counterpart of an observational velocity curve of a variable star, shows an irregular cyclic behaviour which does not have the signature of low–dimensional deterministic chaos.

Behaviour of classical Cepheids is summarized from the observational point of view. The paper concentrates on trends in recent observational studies of, and important empirical results obtained on Cepheids.

We give a brief overview of Cepheids and of their modeling, with particular emphasis on F/O1 Beat Cepheids. Then we revisit the use of Period Ratio vs. Period diagram (Petersen diagram) for fundamental/first overtone Beat Cepheids, because they allow one to put very tight constraints on their metallicity Z. The Petersen diagram is shown to be largely independent of the helium content Y, of the mass-luminosity relation that is used in their construction, and of stellar rotation rates. However, it shows sensitivity to the chemical makeup of the elements that are lumped into the metallicity parameter Z. The Petersen diagram for the new Asplund, M., Grevesse, N., & Sauval, A.J. (2005, ASP Conf. Ser., 336, 25) solar mix is compared to that for for the older, “standard” solar mix of Grevesse, N., & Noels, A., (1993).

Systematic survey for multiperiodicity in the LMC Cepheids (Moskalik et al. 2004, IAU Coll. 193, ASP Conf. Ser., 310, 498 and Mem. Soc. Astron. Ital., 77, 563) has led to discovery of several new forms of pulsational behaviour. One of them is periodic amplitude and phase modulation observed in many first/second overtone (FO/SO) double mode Cepheids. In the current paper we present detailed discussion of this newly discovered phenomenon, based on a combined OGLE+ MACHO sample of double mode pulsators.

With the aid of stellar atmosphere models, we derive iron abundances [Fe/H] from the OGLE B, V, I photometry on the Magellanic Cloud Cepheids. We show that in both clouds the average metallicities of the first overtone variables are lower than those of the fundamental ones (by ~0.2 and ~0.3 dex in the LMC and SMC, respectively). Consequently, there is a correlation between the overall [Fe/H] and luminosity; the lower luminosity stars tend to be also of lower metallicity. These metallicity dependencies are concordant with the ones derived for the two types of double-mode Cepheids from pulsation theory. Yet another support of this dichotomy comes from the evolution theories that require lower metallicities for blue-looping low-luminosity stars than for high-luminosity ones. We also comment on the possibility of using period-luminosity-color relations to derive more accurate metallicities.

It is shown that – in addition to the long-known effect of metallicity on luminosity – the abundance of heavy-elements has an influence on both the light-curve stability of anomalous Cepheids and the period ratio of double-mode Cepheids. Moreover, metallicity is an important controlling factor of the value of the pulsation amplitude of Cepheids.

A large model survey with the Florida-Budapest hydrodynamical code has revisited the Cepheid phase-lag problem. A good agreement between the models and the available observational data has been found for normal amplitude Cepheids. The phase-lag is a method for mode identification that is complementary to that of Fourier decomposition coefficients. In this contribution we exploit its potential in identifying the pulsational mode of Cepheids. The possible application of the phase lag as a test for ultra-low amplitude pulsation is discussed as well.

We analyse the presence of nonradial oscillations in Cepheids, a problem that has not been theoretically revised since the work of Dziembowsky (1977, Acta Astron., 27, 95) and Osaki (1977, Publ. Astron. Soc. Japan, 29, 235). Our analysis is motivated by a work of Moskalik et al. (2004, ASPC, 310, 498), which reports the detection of low-amplitude periodicities in a few Cepheids of the large Magellanic cloud. These newly discovered periodicities were interpreted as nonradial modes.
Based on linear nonadiabatic stability analysis, our goal is to reanalyse the presence and stability of nonradial modes, taking into account improvement in the main input physics required for the modelling of Cepheids.

We have started a survey of M 33 in order to find variable stars and Cepheids in particular. We have obtained more than 30 epochs of g'r'i' data with the CFHT and the one-square-degree camera MegaCam. We present first results from this survey, including the search for variable objects and a basic characterization of the various groups of variable stars.

The link between the dynamical structure of Cepheid atmosphere and the distance scale calibration in the universe is now clearly established through a period-projection factor relation (Pp). However, to support future observations, we currently need fully consistent hydrodynamical models, including pulsating and evolutionary theories, convective energy transport, adaptive numerical meshes, and a refined calculation of the radiative transfer within the pulsating atmosphere and the expected circumstellar envelope (hereafter CSE). Confronting such models with observations (spectral line profiles, spatial- and spectral- visibility curves), will permit to resolve and/or strengthen subtle questions concerning (1) the limb-darkening, (2) the dynamical structure of Cepheids' atmosphere, (3) the expected interaction between the atmosphere and the CSE, and (4) it will bring new insights in determining the fundamental parameters of Cepheids. All these physical quantities are supposed furthermore to be linked to the pulsation period of Cepheids. From these studies, it will be possible to paint a glowing picture of all Cepheids within the instability strip, allowing an unprecedent calibration of the period-luminosity relation, leading to new insights in the fields of extragalactic distance scales and cosmology.

Until recently, Cepheids were considered implicitely as "nak- ed" stars, i.e. devoid from circumstellar material. In 2005, VINCI/ VLTI and FLUOR/CHARA interferometric observations revealed circumstellar envelopes (hereafter CSEs) around Cepheids. Their presence may have an impact on the determination of distances, and could bear testimony of their mass-loss history. Although their observation is made difficult by the brightness of the Cepheids themselves, many observation techniques have the potential to improve our knowledge of their physical properties. We discuss in particular long-baseline interferometry, visible and infrared imaging, spectroscopy and radio observations.

A nonlinear radiative pulsating model for the long-periodic Cepheid l Carinae is presented. The LTE metallic lines and non-LTE hydroen lines are calculated and compared with the HARPS observations. We found that the weak and medium metallic lines well correspond to the radial Cepheid pulsation, while the Hα and some other strong lines reveal very strange behaviour which is not consistent with usual Cepheid pulsation. We conclude that l Car must have an extended circumstellar envelope, and discuss its possible parameters.

Based on 1-dimensional numerical simulations including the rotational acceleration in a quasi-spherical approximation we present pulsation-driven mass loss from luminous stars. In particular, for our model WN8 with M = 60 M๏, L = 8×105L๏ and Teff = 25 000 K with a rotation rate of v sin i = 160 km s-1 given at the stellar photosphere of Rp = 47.6 R๏ and a stellar pulsation of P = 4.144 days we compute a time-dependent mass loss with an averaged rate of 1.28×10-4M๏/yr. Further implications of such theoretical models are briefly discussed for observations of variable stars with large amplitudes.

We show, by means of a perturbative weakly nonlinear analysis, that the axisymmetric magneto-rotational instability (MRI) in a magnetic Taylor-Couette (mTC) flow in a thin-gap gives rise, for very small magnetic Prandtl numbers $({\cal P}_{\rm m})$, to a real Ginzburg-Landau equation for the disturbance amplitude. The saturation amplitude As is found to scale in this regime as ${\cal P}_{\rm m}^\delta$, with 1/2 < δ < 2/3 (depending on the boundary conditions adopted). The asymptotic results are shown to comply with numerical calculations performed by using a spectral code. They suggest that the transport due to the nonlinearly developed MRI may be vanishingly small for ${\cal P}_{\rm m} \ll$ 1.

As a by-product of β Cephei and δ Cephei radiative model surveys we have found interesting cases of multimode pulsation. In case of β Cephei stars, we found two multimode domains with two or three modes being involved. The origin of the multimode pulsation can be traced to one of the two different mechanisms: either to the non-resonant coupling of the two excited modes (double-mode pulsation) or to the 2ω1$\simeq$ω0 + ω2 parametric resonance (triple-mode pulsation). In case of δ Cephei models, a triple-mode pulsation domain, connected with the 2:1 resonance, 2ω0$\simeq$ω2, is found. These models are of theoretical interest only, since they do not obey a proper mass-luminosity relation.