We review the current state of Cepheid modeling and discuss its dominant deficiency, namely the use of time dependent mixing length. Notwithstanding, Cepheid modeling has achieved some excellent successes, and we mention some of the most recent ones. Discrepancies between observations and modeling appear not so much in the gross properties of single mode Cepheids, but rather when more subtle nonlinear effects are important, such as in double mode or even triple mode pulsations. Finally we discuss what we consider the most important challenges for the next decade. These are, first, the multidimensional modeling of convection in a pulsating Cepheid environment, and second, the nonlinear modeling of the nonradial pulsations, and, possibly relatedly, of the Blazhko like phenomenon, both of which recently been observed in Cepheids.

First Compton scattering of photons in a hot partially degenerate plasma is discussed, correcting an error in the literature and avoiding two previously made approximations. An analytical fitting formula for the Rosseland mean opacity is given. Next the Compton scattering functions for neutrini are worked out which are needed in the study of neutrino transport in a supernova model.

Context. A number of microlensing dark-matter surveys have produced tens of millions of light curves of individual background stars. These data provide an unprecedented opportunity for systematic studies of whole classes of variable stars and their host galaxies.Aims. We aim to use the EROS-2 survey of the Magellanic Clouds to detect and study the population of beat Cepheids (BCs) in both Clouds. BCs pulsating simultaneously in the first overtone and fundamental modes (FO/F) or in the second and first overtone modes (SO/FO) are of particular interest.Methods. Using special software designed to search for periodic variables, we have scanned the EROS-2 data base for variables in the typical period range of Cepheids. Metallicities of FO/F objects were then calculated from linear nonadiabatic convective stellar models.Results. We identify 74 FO/F BCs in the LMC and 41 in the SMC, and 173 and 129 SO/FO pulsators in the LMC and SMC, respectively; 185 of these stars are new discoveries. For nearly all the FO/F objects we determine minimum, mean, and maximum values of the metallicity.Conclusions. The EROS data have expanded the samples of known BCs in the LMC by 31%, in the SMC by 110%. The FO/F objects provide independent measures of metallicities in these galaxies. The mean value of metallicity is 0.0045 in the LMC and 0.0018 in the SMC.

A search for variable stars with ultra-low amplitudes (ULA), in the millimag range, has been made in the combined MACHO and OGLE data bases in the broad vicinity of the Cepheid instability strip in the HR diagram. A total of 25 singly periodic and 4 multiply periodic ULA objects has been uncovered. Our analysis does not allow us to distinguish between pulsational and ellipsoidal (binary) variability, nor between LMC and foreground objects. However, the objects are strongly clustered and appear to be associated with the pulsational instability strips of LMC Pop. I and II variables. When combined with the ULA variables of Buchler et al (2005) a total of 20 objects fall close to the classical Cepheid instability strip. However, they appear to fall on parallel period-magnitude relations that are shifted to slightly higher magnitude which would confer them a different evolutionary status. Low amplitude RV Tauri and Pop. II Cepheids have been uncovered that do not appear in the MACHO or OGLE catalogs. Interestingly, a set of binaries seem to lie on a PM relation that is essentially parallel to that of the RV Tauri/Pop. II 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.

The phase properties of hot, dense bulk nuclear matter with variable proton to neutron ratios are discussed. Such matter has the behaviour typical of a binary mixture and exhibits a line of critical points, a line of equal concentrations and a line of maximum temperatures.

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, Grevesse & Sauval (2005) solar mix is compared to that for for the older, "standard" solar mix of Grevesse & Noels (1993).

As shown by Mennickent, et al(2003), a subset of the blue variable stars in the Large Magellanic Cloud exhibit brightness variability of small amplitude in the period range 2.4 to 16 days as well as larger amplitude variability with periods of 140 to 600 days, with a remarkably tight relation between the long and the short periods. Our re-examination of these objects has led to the discovery of additional variability. The Fourier spectra of 11 of their 30 objects have 3 or 4 peaks above the noise level and a linear relation of the form f_a = 2(f_b - f_L) among three of the frequencies. An explanation of this relation requires an interplay between the binary motion and that of a third object. The two frequency relations together with the Fourier amplitude ratios pose a challenging modeling problem.

It is well known that the mere location of a beat Cepheid model in a period ratio versus period diagram (Petersen diagram) puts constraints on its metallicity Z. But these bounds are sensitive to the mixture of elements that are lumped into the parameter Z. In this short paper we update the previous results that were based on the Grevesse-Noels solar mixture to the recent, revised Asplund, Grevesse,& Sauval (AGS) solar mixture. We also examine the effect of the envelope depth on the accuracy of the computed pulsation periods. We find that for low-period Cepheids with high Z the customary approximation of envelope pulsation breaks down. It is necessary to compute stellar models that extend to the center and to include burning and composition inhomogeneities in the modeling. Fortunately, however, most beat Cepheids that have been observed so far seem to avoid that regime.

The mere location of a Beat Cepheid model in a Period Ratio vs. Period diagram (Petersen diagram) puts very tight constraints on its metallicity Z. The Beat Cepheid Peterson diagrams are revisited with linear nonadiabatic turbulent convective models, and their accuracy as a probe for stellar metallicity is evaluated. They are shown to be largely independent of the helium content Y, and they are also only weakly dependent on the mass-luminosity relation that is used in their construction. However, they are found to show sensitivity to the relative abundances of the elements that are lumped into the metallicity parameter Z. Rotation is estimated to have but a small effect on the 'pulsation metallicities'. A composite Petersen diagram is presented that allows one to read off upper and lower limits on the metallicity Z from the measured period P0 and period ratio P1/P0.

We compute the phase lags between the radial velocity curves and the light curves Delta Phi(1)=Phi(Vr)(1)-Phi(mag)(1) for classical Cepheid model sequences both in the linear and the nonlinear regimes. The nonlinear phase lags generally fall below the linear ones except for high-period models, in which they lie above, and of course for low pulsation amplitudes, in which the two merge. The calculated phase lags show good agreement with the available observational data of normal amplitude Galactic Cepheids. The metallicity has but a moderate effect on the phase lag, while the mass-luminosity relation and the parameters of the turbulent convective model (time-dependent mixing length) mainly influence the modal selection and the period, which is then reflected in the period-Delta Phi(1) diagram. We discuss the potential application of this observable as a discriminant for pulsation modes and as a test for ultralow-amplitude (ULA) pulsation.

Our analysis of the deep CFHT M33 variability survey database has uncovered five beat Cepheids that are pulsating in the fundamental and first-overtone modes. With only the help of stellar pulsation theory and of massluminosity relations, derived from evolutionary tracks, we can accurately determine the metallicities Z of these stars. The [O/H] metallicity gradient of -0.16 dex kpc(-1) that is inferred from the M33 galactocentric distances of these Cepheids and from their "pulsation" metallicities is in excellent agreement with the standard spectroscopic metallicity gradients that are determined from H ii regions, early-B supergiant stars, and planetary nebulae. Beat Cepheids can thus provide an additional, independent probe of galactic metallicity distributions.

The MACHO variables of LMC field 77 that lie in the vicinity of the Cepheid instability strip are reexamined. Among the 144 variables that we identify as Cepheids, we find 14 that have Fourier amplitudes < 0.05 mag in the MACHO red band, of which seven have an amplitude < 0.006 mag: we dub the latter group of stars ultralow-amplitude (ULA) Cepheids. The variability of these objects is verified by a comparison of the MACHO red with the MACHO blue light curves and with those of the corresponding OGLE LMC stars. The occurrence of ULA Cepheids is in agreement with theory. We have also discovered two low-amplitude variables whose periods are about a factor of 5-6 smaller than those of F Cepheids of equal apparent magnitude. We suggest that these objects are Cepheids undergoing pulsations in a surface mode and that they belong to a novel class of Strange Cepheids ( or surface mode Cepheids) whose existence was predicted by Buchler et al. ( 1997).

The pulsational properties of the Cepheid models along the evolutionary tracks from the Padova group (Girardi et al.), as calculated with our turbulent convective pulsation code, are in good agreement with the resonance constraints imposed by the observational OGLE-2 data of the Small and Large Magellanic Clouds. Our study suggests that the P4/P1 = 1/2 resonance for the overtone Cepheids occurs for periods clustering around 4.2 d, in disagreement with the suggestion of Antonello & Poretti based on the observations of light curves, but in agreement with Kienzle et al. and Feuchtinger et al. For the fundamen- tal Cepheids the lowest order Fourier decomposition coefficients from the light curves, viz. R21 and φ21 can be used to locate the resonance region, but not so for the first overtone Cepheids. Here, the radial velocity curves can be used to locate the overtone resonance region, or in their absence, one needs to resort to numerical hydrodynamic modelling.

We describe a methodology that allows us to follow the pulsational behavior of an RR Lyrae model consistently and automatically along its evolutionary track throughout the whole instability strip. It is based on the powerful amplitude equation formalism, and resorts to a judicious combination of numerical hydrodynamical simulations, the analytical signal time-series analysis, and amplitude equations. A large-scale survey of the nonlinear pulsations in RR Lyr instability strip is then presented, and the mode selection mechanism is delineated throughout the relevant regions of parameter space. We obtain and examine two regions with hysteresis, where the pulsational state depends on the direction of the evolutionary tracks, namely a region with either fundamental (RRab) or first overtone (RRc) pulsations and a region with either fundamental (RRab) or double-mode (RRd) pulsations. The regions where stable double-mode (DM, or RRd) pulsations occur are very narrow and hard to find in astrophysical parameter (L, M, T_eff, X, Z) space with hydrodynamic simulations, but our systematic and efficient methodology allows us to investigate them with unprecedented detail. It is shown that by simultaneously considering the effects of mode selection and of horizontal branch evolution we can naturally solve one of the extant puzzles involving the topologies of the theoretical and observed instability strips, namely the slope of the fundamental blue edge. The importance of the interplay between mode selection and stellar evolutionary effects is also demonstrated for the properties of double-mode RR Lyr. Finally, the Petersen diagram of double-mode RR Lyr models is discussed for the first time.

We present an analysis of the photometric observations of the light curves of the five large-amplitude, irregularly pulsating stars R UMi, RS Cyg, V CVn, UX Dra, and SX Her. First, multiperiodicity is eliminated for these pulsations; i.e., they are not caused by the excitation of a small number of pulsation modes with constant amplitudes. Next, on the basis of energetics we also eliminate stochasticity as a cause, leaving low-dimensional chaos as the only alternative. We then use a global flow reconstruction technique in an attempt to extract quantitative information from the light curves and to uncover common physical features in this class of irregular variable stars that straddle the RV Tau to the Mira variables. Evidence is presented that the pulsational behavior of R UMi, RS Cyg, V CVn, and UX Dra takes place in a four-dimensional dynamical phase space, suggesting that two vibrational modes are involved in the pulsation. A linear stability analysis of the fixed points of the maps further indicates the existence of a two-mode resonance, similar to the one we had uncovered earlier in R Sct. The irregular pulsations are the result of a continual energy exchange between two strongly nonadiabatic modes, a lower frequency pulsation mode and an overtone that are in a close 2: 1 resonance. The evidence is particularly convincing for R UMi, RS Cyg, and V CVn, but much weaker for UX Dra. In contrast, the pulsations of SX Her appear to be more complex and may require a six-dimensional space.

We describe a methodology that allows us to follow the pulsational behavior of an RR Lyrae model consistently and automatically along its evolutionary track throughout the whole instability strip. It is based on the powerful amplitude equation formalism, and resorts to a judicious combination of numerical hydrodynamical simulations, the analytical signal time-series analysis, and amplitude equations.A large-scale survey of the nonlinear pulsations in RR Lyr instability strip is then presented, and the mode selection mechanism is delineated throughout the relevant regions of parameter space. We obtain and examine two regions with hysteresis, where the pulsational state depends on the direction of the evolutionary tracks, namely a region with either fundamental (RRab) or first overtone (RRc) pulsations and a region with either fundamental (RRab) or double-mode (RRd) pulsations.The regions where stable double-mode (DM, or RRd) pulsations occur are very narrow and hard to find in astrophysical parameter (L, M, T-eff, X, Z) space with hydrodynamic simulations, but our systematic and efficient methodology allows us to investigate them with unprecedented detail.It is shown that by simultaneously considering the effects of mode selection and of horizontal branch evolution we can naturally solve one of the extant puzzles involving the topologies of the theoretical and observed instability strips, namely the slope of the fundamental blue edge.The importance of the interplay between mode selection and stellar evolutionary effects is also demonstrated for the properties of double-mode RR Lyr. Finally, the Petersen diagram of double-mode RR Lyr models is discussed for the first time.

We demonstrate how, with a purely empirical analysis of the irregular lightcurve data, one can extract a great deal of information about the stellar pulsation mechanism. An application to R Sct thus shows that the irregular lightcurve is the result of the nonlinear interaction of two highly nonadiabatic pulsation modes, namely a linearly unstable, low frequency mode, and the second mode that, although linearly stable, gets entrained through a 2:1 resonance. In the parlance of nonlinear dynamics the pulsation is the result of a 4 dimensional chaotic dynamics.

The numerical hydrodynamic modelling of beat Cepheid behavior has been a long-standing quest in which purely radiative models have failed consistently. We find that beat pulsations occur quite naturally when turbulent convection is accounted for in our hydrodynamics code. The developments of a relaxation code and of a Floquet stability analysis greatly facilitate the search for and the analysis of beat Cepheid models. The conditions for the occurrence of beat behavior can be understood easily and at a fundamental level with the help of amplitude equations.