Robert Buchler's carrier as seen by his long term friend and colleague Jean Perdang (Noteof 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 systemsto examine the possible impact of non-normality on the stellar pulsation eigenvalueproblem. Computations of spectra, ε-pseudospectra and resolvent normsfor these operatorsshow that extended regions of complex potential, corresponding in stars to non-adiabaticregions, leads to strong non-normality. Introduction of an advective term furtherincreases non-normal properties. Prospective model improvements andimplications 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 particularemphasis on F/O1 Beat Cepheids.Then we revisit the use of Period Ratio vs. Period diagram(Petersen diagram) for fundamental/first overtone Beat Cepheids, becausethey 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 isused in their construction, and of stellar rotation rates.However, it shows sensitivity to thechemical makeup of the elements that are lumped into the metallicityparameter Z. The Petersen diagram for the new Asplund, M., Grevesse, N., & Sauval, A.J. (2005, ASP Conf. Ser., 336, 25) solar mix iscompared 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 pulsationalbehaviour. One of them is periodic amplitude and phase modulationobserved in many first/second overtone (FO/SO) double modeCepheids. In the current paper we present detailed discussion ofthis newly discovered phenomenon, based on a combined OGLE+ MACHOsample 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 theFlorida-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 amethod for mode identification that is complementary to that of Fourierdecomposition 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 andCepheids in particular. We have obtained more than 30 epochs ofg'r'i' data with the CFHT and theone-square-degree camera MegaCam. We present first results from thissurvey, including the search for variable objects and a basiccharacterization of the various groups of variable stars.

The link between the dynamical structure of Cepheid atmosphere andthe distance scale calibration in the universe is now clearlyestablished through a period-projection factor relation (Pp).However, to support future observations, we currently need fullyconsistent hydrodynamical models, including pulsating andevolutionary theories, convective energy transport, adaptivenumerical meshes, and a refined calculation of the radiativetransfer within the pulsating atmosphere and the expectedcircumstellar envelope (hereafter CSE). Confronting such models withobservations (spectral line profiles, spatial- and spectral-visibility curves), will permit to resolve and/or strengthen subtlequestions concerning (1) the limb-darkening, (2) the dynamicalstructure of Cepheids' atmosphere, (3) the expected interactionbetween the atmosphere and the CSE, and (4) it will bring newinsights in determining the fundamental parameters of Cepheids. Allthese physical quantities are supposed furthermore to be linked tothe pulsation period of Cepheids. From these studies, it will bepossible to paint a glowing picture of all Cepheids within theinstability strip, allowing an unprecedent calibration of theperiod-luminosity relation, leading to new insights in the fields ofextragalactic 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-periodicCepheid 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 tothe radial Cepheid pulsation, while the Hα and some otherstrong lines reveal very strange behaviour which is not consistent withusual Cepheid pulsation. We conclude that l Car must have anextended 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, forour model WN8 with M = 60 M๏, L = 8×105 L๏ and Teff = 25 000 Kwith 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 of1.28×10-4 M๏/yr. Further implications of such theoreticalmodels 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 amagnetic Taylor-Couette (mTC) flow in a thin-gap gives rise,for very small magnetic Prandtl numbers $({\calP}_{\rm m})$, to a real Ginzburg-Landau equation for thedisturbance amplitude. The saturation amplitude As is found to scalein 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 withnumerical calculations performed by using a spectral code. They suggest thatthe transport due to the nonlinearly developed MRI may bevanishingly 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.