The early-type variables

We review the observational status of several different kinds of intrinsic variables among the early-type stars and attempt to interpret the variations in terms of our current understanding of stellar pulsation. Four distinct types of intrinsic variable can be defined: the Cep, 53 Per, Oph and Eri stars. A simple observational classification scheme, which is readily interpreted in terms of pulsation properties, is proposed. The limits of the instability strip and pulsation constants for the Cep and 53 Per stars is discussed. Problems with the interpretation of Eri stars in terms of pulsation are pointed out. The observations are consistent with rotational modulation. A problem with mode identification in Eri stars is discussed.     

Present Observational Status of High Mass Pulsating Stars

In this short review we present the current observationalstatus of high mass pulsating stars.We give an overview of the results of the first asteroseismicstudies performed for some of the best known Cepheiand slowly pulsating B stars and discuss the asteroseismicpotential of these pulsators for future ground-and space-based data-sets.     

Structural Properties of Pulsating Star Light Curves Through Fuzzy Divisive Hierarchical Clustering

The aim of this paper is to explore the possibility of using a fuzzy divisive clustering technique to the classification of pulsating stars according to the shape of their light curves. The considered sample contains Cepheides, RR Lyrae stars, and high amplitude Scuti and SX Phoenicis stars. Our investigations proved the ability of this method to identify morphological groups in a given class of pulsating stars. The method is also useful for establishing the membership of an unknown variable star in a given class.     

FBS 0137+400: A Mass-Losing AGB Carbon Mira?

The absolute K-band magnitude determined for the object FBS 0137+400 is typical for Asymptotic Giant Branch (AGB) carbon stars. The large J - K color index as well as large amplitude variability (K > 0.4) and the presence of H in the emission indicate also that this star is a mass-losing long-period Carbon Mira variable with thick circumstellar shell. The IRAS LRS spectrum indicates the presence of a dust shell surrounding this object. Monitoring of the K-band magnitude is necessary for the determination of the pulsation period of FBS 0137+400.     

Population I pulsating stars

Luminosities of Population I pulsating stars (Delta Scuti variables and classical cepheids) are investigated. From data for 80 Delta Scuti stars, semi-empirical period-luminosity-colour (P-L-C) relations and period-luminosity (P-L) relations are obtained for the four lowest modes of radial pulsations. The improvement of the accuracy of the stellar luminosity is determined when a P-L-C relation is used instead of the corresponding P-L relation. From data for 155 classical cepheids, empirical P-L relations are derived for short-period stars (logP1.1), long-period stars (logP>1.1), and s-cepheids. The comparison of the P-L relations for the two types of variable stars shows good agreement, but between them there is a gap with a dim nature.     

Contributing factors to flux changes in nonradial stellar pulsations

The linearized analytic representation of observed flux changes due to nonradial pulsations is examined under conditions applicable to a variety of pulsating star types. Histograms are presented as a function ofl, documenting the relative importance of local temperature, geometry, pressure, and limb-darkening variations, which are all contributing factors to the predicted flux changes. The most favourable situation for light and colour data to assist in mode discrimination for lowl requires, amongst other things, that the geometry-related factor be significant. This is noted to occur only if theQ value of the pulsation is not grossly different from theQ value of a radial fundamental pulsation. Published light and colour data for Cephei, 53 Persei, Scuti, rapidly oscillating Ap, Cepheid, and ZZ Ceti variables are compared with model predictions on an amplitude ratio versus phase difference plane . With the notable exception of the rapidly oscillating Ap stars, these comparisons suggest consistency of the nonradial flux change model with other known constraints on the different variable star groups.     

Do red giant stars pulsate in high overtones?

About 30 photometrically variable red giant stars have periods less than 10 d, as determined by the compilers of the Hipparcos Catalogue from Hipparcos photometric measurements. These periods, when combined with estimates of the radii and masses of these stars, and with pulsation theory, imply that these stars are pulsating in very high overtones. We present several pieces of evidence which suggest that the periods may be spurious, as a result of the particular aliasing properties of the Hipparcos photometry. We conclude that the evidence for high-overtone pulsation in red giant stars is equivocal.     

Semi-regular and alternating pulsations in hydrodynamical models for low-mass giants

On the basis of hydrodynamical simulations the semi-regular and alternating pulsations of the W Vir and RV Tau stars are studied including non-steady radiative transfer effects in gray sphericalsymmetric atmospheres. It is shown that the transition from regular to semi-regular oscillations occurs at periodsP13–15 days due to pulsation energy imbalance provoqued by strong radiating shocks. The differences between the periods and amplitudes of W Vir and RV Tau variables are explained as a result of this imbalance. It is suggested that the semi-regular alternating pulsations of the RV Tau stars appear following a period-doubling phenomenon, which can be also understood in terms of the pulsation energy variations between odd and even fundamental cycles. The pulsational characteristics of the period-doubled models are in general accordance with those observed in the RV Tau stars. On the basis of numberical results, a theoretical estimation of the upper limit for the luminosity of the RV Tau stars is derived asL10³ L .     

Optical monitoring of the central region of the globular cluster M 15 = NGC 7078: New variable stars

We obtained a series of more than two hundred R-band CCD images for the crowded central (115″×77″) region of the metal-poor globular cluster M 15 with an angular resolution of \(0\mathop .\limits^{''} 5 - 0\mathop .\limits^{''} 9\) in most images. Optimal image subtraction was used to identify variable stars. Brightness variations were found in 83 stars, 55 of which were identified with known cluster variables and the remaining 28 are candidates for new variables. Two of them are most likely SX Phe variables. The variability type of two more stars is uncertain. The remaining stars were tentatively classified as RR Lyrae variables. A preliminary analysis of published data and our results shows that the characteristics of RR Lyrae variables in the densest part (r<35″) of the cluster probably change. More specifically, the maximum of the period distribution of first-and second-overtone (RR1, RR2) pulsating stars shifts toward shorter periods; i.e., there is an increase in the fraction of stars pulsating with periods \( < 0\mathop .\limits^d 3\) and a deficiency of stars with \(0\mathop .\limits^d 35 - 0\mathop .\limits^d 40\). The ratio of the number of these short-period RR Lyrae variables to the number of fundamental-tone (RR0) pulsating variables changes appreciably. We found and corrected the error of transforming the coordinates of variables V128–155 in M 15 into the coordinate system used in the catalog of variable stars in globular clusters.     

Vibrational stability of stars in thermal imbalance

We discuss, in general, the problem of deriving a vibrational stability coefficient of stars in thermal imbalance on the basis of an energy principle.The damped harmonic oscillator with variable spring constant provides a typical illustration of the problem. In fact, for systems with variable parameters (as evolving stars), the vibrational stability coefficients, defined respectively in terms of the behaviour of the small displacement and of the total pulsation energy, are basically different. Davey and Cox's so-called modified energy method, proposed in order to recover the small displacement stability coefficient, is also shown to be unsatisfactory.The use of results obtained in the first two papers by asymptotic methods yields a simple compact general expression for the vibrational stability coefficient relative to the pulsation energy of arbitrary evolving stars. This allows the easy recovery of results obtained by more complicated approaches due to Kato and Unno, Simon and Sastri and Axel and Perkins, valid only in particular cases.     

Effects of rotation and tidal distortions on the shapes of radial velocity curves of polytropic models of pulsating variable stars

Anharmonic oscillations of rotating stars have been studied by various authors in literature to explain the observed features of certain variable stars. However, there is no study available in literature that has discussed the combined effect of rotation and tidal distortions on the anharmonic oscillations of stars. In this paper, we have created a model to determine the effect of rotation and tidal distortions on the anharmonic radial oscillations associated with various polytropic models of pulsating variable stars. For this study we have used the theory of Rosseland to obtain the anharmonic pulsation equation for rotationally and tidally distorted polytropic models of pulsating variable stars. The main objective of this study is to investigate the effect of rotation and tidal distortions on the shapes of the radial velocity curves for rotationally and tidally distorted polytropic models of pulsating variable stars. The results of the present study show that the rotational effects cause more deviations in the shapes of radial velocity curves of pulsating variable stars as compared to tidal effects.     

Period distributions in pulsating and binary stars

We analyze the hypothesis of quantization in bands for the angular momenta of binary systems and for the maount of actionA _c in stable and pulsating stars. This parameter isA _c=Mv _eff R _eff, where the effective velocity corresponds to the kinetic energy in the stellar interior and the effective radius corresponds to the potential energyGM ²/R _eff. Analogous parameters can be defined for a pulsating star withm=M where is the rate of the massm participating in the oscillation to the total massM andv _osc,R _osc the effective velocity and oscillation radius.From an elementary dimensional analysis one has thetA _c (energy x time) (period)^1/3 independently ifA _c corresponds to the angular momentum in a binary system, or to the oscillation in a pulsating star or the inner energy and its time-scaleP _eff in a stable star.From evolving stellar models one has that P _effP _eff(solar)1.22 hr a near-invariant for the Main Sequence and for the range of masses 0.6M <M<1.6M .With this one can give scalesn _k=kn ₁ withk integers andn ₁=(P/P ₁)^1/3 withP ₁=P _eff1.22 hr. In these scales proportional toA _c, one sees that the periods in binary and pulsating stars are clustered in discrete unitsn ₁,n ₂,n ₃, etc.This can be seen in pulsating Scuti, Cephei, RR Lyrae, W Virginis, Cephei, semi-regular variables, and Miras and in binary stars as cataclysmic binaries, W Ursa Majoris, Algols, and Lyrae with the corresponding subgroups in all these materials. Phase functions (n _k) in RR Lyrae and Cephei are also associated with discrete levelsn _k.the suggested scenario is that the potential energies and the amounts of actionE _p(t), A_c(t) are indeed time-dependent, but the stars remain more time in determinated most proble states. The Main Sequence itself is an example of this. These most probable states in binary systems, or pulsating or stable stars, must be associated with velocities sub-multiplesc/ _F , given by the velocity of light and the fine structure constant.Additional tests for such a hypothesis are suggested when the sufficient amount of observational data are available. They can made with oscillation velocities in pulsating stars and velocity differences of pairs of galaxies.     

Identification of temperature anomaly RR Lyrae stars in LAMOST survey,misclassification and binarities

RR Lyrae stars,a well-known type of pulsating variable stars,have been known about for more than a century.A large amount of photometric data on RR Lyrae stars has been accumulated by space-and ground-based sky surveys,but the spectral data are relatively poor.Fortunately,the LAMOST sky survey project provides an opportunity to view them from the point of view of spectra.We collect the atmospheric parameters of 1685 RR Lyrae stars provided by the LAMOST catalog,and carry out research by using the reliable T_(eff).We find that there is a clear correlation between their Teff and pulsation periods,which is consistent with the pulsation and evolution theories of RR Lyrae stars.In addition,we focus on those RR Lyrae stars with abnormal temperatures.After analyzing the data from several photometric surveys,we find that some of these temperature anomalies are misclassified variable stars(e.g.,eclipsing binaries,pulsating stars on main sequence),and some are RR Lyrae binary candidates.For the latter,the temperatures of potential companions should be lower and their luminosities should not be neglected(e.g.,red giant stars).We obtain that the ratio of temperature anomaly stars to all the sample stars is 4%,which means that the impact on the further analysis(e.g.,kinematics analysis) is low.We also present the catalogs of spectral anomaly RR Lyrae stars.     

δ Scuti Pulsations in Pre-main Sequence Stars

Intermediate mass pre-main sequence stars (1.5 to 5 M )cross the instability strip on their way to the main sequence.They are therefore expected to be pulsating in a similar way as the Scuti stars. In this review, I present the status of observational studies ofpulsations in these stars, and comment on prospects for futureinvestigations of these pulsations from the ground and from space.     

Long-term amplitude and period variations ofδ Scuti stars: A sign of chaos?

On short time-scales of under a year, the vast majority of Scuti stars studied in detail show completely regular multiperiodic pulsation. Nonradial pulsation is characterized by the excitation of a large number of modes with small amplitudes. Reports of short-term irregularity or nonperiodicity in the literature need to be examined carefully, since insufficient observational data can lead to an incorrect impression of irregularity. Some interesting cases of reported irregularities are examined.A few Scuti stars, such as 21 Mon, have shown stable variations with sudden mode switching to a new frequency spectrum. This situation might be an indication of deterministic chaos. However, the observational evidence for mode switching is still weak.One the other hand, the case for the existence of long-term amplitude and period changes is becoming quite convincing. Recently found examples of nonradial pulsators with long-term changes are 4 CVn, 44 Tau, Peg and HD 2724. (We note that other Scuti pulsators such as X Cae and ² Tau, have shown no evidence for amplitude variations over the years.) Neither the amplitude nor the period changes are periodic, although irregular cycles with time scales between a few and twenty years can be seen. While the amplitude changes can be very large, the period changes are quite small. This property is common in nonlinear systems which lead to chaotic behavior. There exists observational evidence for relatively sudden period jumps changing the period by about 10^–5 and/or slow period changes near dP/dt 10^–9. These period changes are an order of magnitude larger than those expected from stellar evolution.The nonperiodic long-term changes are interpreted in terms of resonances between different nonradial modes. It is shown that a large number of the nonradial acoustic modes can be in resonance with other modes once the mode interaction terms, different radial orders and rotationalm-mode splitting are considered. These resonances are illustrated numerically by the use of pulsation model. Observational evidence is presented that these interaction modes exist in the low-frequency domain.     

Evolution and Period Change in RR Lyr Variables of the Globular Cluster M 3

The grid of evolutionary tracks of population II stars with initial masses 0.81 M_⊙ ≤ M_ZAMS ≤ 0.85 M_⊙ and chemical composition of the globular cluster M3 is computed. Selected models of horizontal branch stars were used as initial conditions for solution of the equations of radiation hydrodynamics and time–dependent convection describing radial stellar oscillations. The boundaries of the instability strip on the Herztsprung–Russel diagram were determined using ≈100 hydrodynamic models of RR Lyr pulsating variables. For each evolutionary track crossing the instability strip the pulsation period was determined as a function of evolutinary time. The rate of period change of most variables is shown to range within ?0.02 ≤ \(\dot{\Pi}\) ≤ 0.05 day/10⁶ yr. Theoretical estimate of the mean period change rate obtained by the population synthesis method is 〈\(\dot{\Pi}\)〉 = 6.0 × 10^?3 day/10⁶ yr and agrees well with observations of RR Lyr variables of the globular cluster M3.     

On properties of the fourier harmonics in the limit-cycle models of classical cepheids

The Fourier coefficients of the hydrodynamic variables are calculated for the limit-cycle models of classical Cepheids having periods from 7.2 to 10.9 days. In adiabatically pulsating layers of the stellar envelope, each Fourier harmonic of orderk 8 is shown to be identified with a corresponding standing wave, so that the pulsation motions of the adiabatic layers may be represented as a superposition of standing waves. Each Fourier harmonic of orderk may also be identified with the eigenfunction of orderl of the linear adiabatic wave equation when the resonance condition _l /₀ =k is fulfilled. The spectra of the oscillatory moment of inertia and the spectra of kinetic energy obey the power law for the Fourier harmonics of orderk 15, the spectrum slope being steeper for shorter pulsation periods. In the helium and hydrogen ionizing regions all of the Fourier harmonics drive the pulsation instability, whereas in the radiative damping region the mechanical work done by each Fourier harmonic is negative. In classical Cepheids having periods shorter than 10 days the period dependence of the secondary bump is due to phase changes of the second order Fourier harmonic in the outer nonadiabatic layers of the stellar envelope. At a pulsation period of II 9.7 days the second order Fourier harmonic is identified with the second overtone. At periods II > 10 days the second order Fourier harmonic tends to be attracted by the fundamental mode in such a way that their phases coincide in the outer layers of the stellar envelope.     

Discovery of magnetic fields in three He variable Bp stars with He and Si spots

It is essential for the understanding of stellar structure models of high mass stars to explain why constant stars, nonpulsating chemically peculiar hot Bp stars and pulsating stars co‐exist in the slowly pulsating B stars and β Cephei instability strips. We have conducted a search for magnetic fields in the four Bp stars HD55522, HD105382, HD131120, and HD138769 which previously have been wrongly identified as slowly pulsating B stars. A recent study of these stars using the Doppler Imaging technique revealed that the elements He and Si are inhomogeneously distributed on the stellar surface, causing the periodic variability. Using FORS 1 in spectropolarimetric mode at the VLT, we have acquired circular polarisation spectra to test the presence of a magnetic field in these stars. A variable magnetic field is clearly detected in HD55522 and HD105382, but no evidence for the existence of a magnetic field was found in HD131120. The presence of a magnetic field in HD138769 is suggested by one measurement at 3σ level. We discuss the occurrence of magnetic B stars among the confirmed pulsating B stars and find strong magnetic fields of order kG and oscillations to be mutually exclusive. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface trapping and leakage of low-frequency g modes in rotating early-type stars – II. Global analysis

A global analysis of the surface trapping of low-frequency non-radial g modes in rotating early-type stars is undertaken within the Cowling, adiabatic and traditional approximations. The dimensionless pulsation equations governing these modes are reviewed, and the boundary conditions necessary for solution of the equations are considered; in particular, an outer mechanical boundary condition, which does not enforce complete wave trapping at the stellar surface, is derived and discussed in detail. The pulsation equations are solved for a 7-M_⊙ model star over a range of rotation rates, using a numerical approach.
The results of the calculations confirm the findings of the preceding paper in the series: modes with eigenfrequencies below a cut-off cannot be fully trapped within the star, and exhibit leakage in the form of outwardly propagating waves at the surface. The damping rates resulting from leakage are calculated for such 'virtual' modes, and found to be appreciably larger than typical growth rates associated with opacity-driven pulsation. Furthermore, it is demonstrated that the surface perturbations generated by virtual modes are significantly changed from those caused by fully trapped modes; the latter result suggests differences in the line-profile variations exhibited by these two types of mode.
The findings are discussed in the context of the 53 Per, SPB and pulsating Be classes of variable star. Whilst wave leakage will probably not occur for overstable g modes in the 53 Per and slowly rotating SPB stars, the adoption of the new outer mechanical boundary condition may still affect the pulsational stability of these systems. Wave leakage for overstable modes remains a possibility in Be stars and the more rapidly rotating SPB stars.     

Population I pulsating stars

Evolutionary masses corresponding to various evolutionary phases of Population I pulsating stars (89 Delta Scuti variables and 155 classical cepheids) are interpolated in the systems of tracks of Iben (1967) and Paczyski (1970). The evolutionary masses are larger in the latter system than in the former one. The uncertainty of the evolutionary mass of a star is estimated, when various evolutionary phases are possible for this star (a smaller evolutionary mass corresponds to a later phase). Semi-empirical period-evolutionary mass-colour (P-M _e -C) and period-evolutionary mass (P-M _e ) relations are derived for various modes, groups of stars, colour indices (and effective temperature), and evolutionary phases. For Delta Scuti stars, the uncertainty of evolutionary masses calculated from theP-M _e relations for different modes, is estimated. The improvement of the evolutionary mass accuracy is estimated, when aP-M _e -C relation is used instead of the correspondingP-M _e relation. The theoretical and semi-empirical period ratios of radial pulsations derived from theP-M _e relations for Delta Scuti stars, are compared. There is a relatively good agreement between theP-M _e relations for the two types of Population I pulsating stars, but a gap exists between them. The evolutionary masses of these stars are closer in the two systems of tracks and are derived with a relatively higher accuracy in comparison with their ages.