Pulsating Stars

Modern long-baseline interferometers are capable of resolving stellar diameters in the range of one to several milli-arcseconds with measurement precision approaching a few percent. This level of precision allows astronomers to directly resolve diameter changes associated with pulsation for various classesc of stars, including Cepheids and Miras. For several nearby Cepheids itis relatively straightforward to apply a Baade-Wesselink analysis and hence directly measure the distance to the Cepheid with a high degree of precision. This field is still quite new; I expect that in the near term several new interferometers will provide direct Cepheid distances to perhaps a few tens of Galactic Cepheids. However, equally important will be direct measurements of the atmospheric properties (in particular limb darkening) of these stars. Such measurements should help reduce the considerable systematic uncertainties that remain. This revised version was published online in July 2006 with corrections to the Cover Date.     

Measurement of the limb darkening coefficient of stars

This paper introduces a new method of describing the limb darkening phenomenon of stellar atmospheres in the 1st and 2nd order approximations. The limb darkening coefficients are given by direct measurements of relevant physical quantities or measurements of flux and a supplementary quantity, namely, the star's surface temperature. The comparison either with solar observations or with the method of numerical simulation of the atmosphere shows that the new method can very accurately describe the star's limb darkening and determine its relevant coefficients. The accuracy of the new direct method is assessed using 176 observational data points of the Sun, and the mean fractional differences is found to be 0.38% for the 1st-, and 0.26% for the 2nd-order approximations. The mean differences of the indirect method are close to, and slightly larger than these values. When compared with numerical models of the atmosphere, the differences are in the range from under 1% to under 3%.     

Spotted star light curve numerical modeling technique and its application to HII 1883 surface imaging

We developed a code for imaging the surfaces of spotted stars by a set of circular spots with a uniform temperature distribution. The flux from the spotted surface is computed by partitioning the spots into elementary areas. The code takes into account the passing of spots behind the visible stellar limb, limb darkening, and overlapping of spots. Modeling of light curves includes the use of recent results of the theory of stellar atmospheres needed to take into account the temperature dependence of flux intensity and limb darkening coefficients. The search for spot parameters is based on the analysis of several light curves obtained in different photometric bands. We test our technique by applying it to HII 1883.     

Limb-darkening corrections for interferometric uniform disc stellar angular diameters

Stellar angular diameters determined interferometrically are generally established by fitting the observed visibility data with a curve appropriate for a uniformly illuminated disc. The resulting uniform-disc diameters must be corrected for the effects of limb darkening in order to determine the true angular diameters of the stars. An extensive grid of limb-darkening corrections, based directly on the centre-to-limb intensity variations for Kurucz model stellar atmospheres, has been computed without the intermediate step of a parametrized representation of the centre-to-limb variation. The limitations of this method of correction are discussed.     

The standard systems of stellar rotation measurements (II)

The limb darkening effect on the measurement of stellar rotation is discussed in this paper. It is shown that this effect plays an important role in the measurement of V_e sin i. In the extreme case with the limb darkening coefficient of 1.0, it may cause a difference up to 17%. As a sequel to paper [1], this work presents the following new explanation for the systematic differences between Slettebak's new and old systems. The main causes of the systematic differences are: (1) The old system made an inadequate twice repeated correction for the limb darkening. (2) Owing to the historical reason, the old system used too large limb darkening coefficients.     

On the second-overtone stability among Small Magellanic Cloud Cepheids

We present a new set of Cepheid, full amplitude, non-linear, convective models which are pulsationally unstable in the second overtone (SO). Hydrodynamical models were constructed by adopting a chemical composition typical of Cepheids in the Small Magellanic Cloud (SMC) and for stellar masses ranging from 3.25 to 4 M_⊙. Predicted φ ₂₁ Fourier parameters agree, within current uncertainties, with empirical data for pure first- and second-overtone variables as well as for first-/second-overtone (FO/SO) double-mode Cepheids collected by Udalski et al. in the SMC. On the other hand, predicted I -band amplitudes are systematically larger than the observed ones in the short-period range, but attain values that are closer to the empirical ones for log  P _SO≥−0.12 and log  P _FO≥0.1 . We also find, in agreement with empirical evidence, that the region within which both second and first overtones attain a stable limit cycle widens when moving towards lower luminosities. Moreover, predicted P _SO/ P _FO and P _FO/ P _F period ratios agree quite well with empirical period ratios for FO/SO and fundamental/FO double-mode SMC Cepheids.
Interestingly enough, current models support the evidence that pure SO Cepheids and SO components in FO/SO Cepheids are good distance indicators. In fact, we find that the fit of the predicted period–luminosity–colour ( V , V – I ) relation to empirical SMC data supplies a distance modulus ( DM ) of 19.11±0.08 mag . The same outcome applies to pure FO Cepheids and FO components in FO/SO Cepheids, and indeed we find DM =19.16±0.19 mag . Current distance estimates do not account for, within current uncertainties on photometry and reddening, the so-called short distance scale.     

Bifurcation in hydrodynamic models of stellar pulsation

Phenomena of bifurcation in hydrodynamic stellar models of radial pulsation are reviewed. By changing control parameters of models, we can see qualitatively different pulsation behaviors in hydrodynamic models with transitions due to various types of bifurcation.In weakly dissipative models (classical Cepheids), the bifurcation is induced by modal resonances. Two types of the modal resonances found in models are discussed: The higherharmonic resonances of the second overtone mode in the fundamental mode pulsator and of the fourth overtone mode in the first overtone pulsator are relevant to observations. The subharmonic resonance between the fundamental and first overtone modes is confirmed in classical Cepheid models.In strongly dissipative models (less-massive supergiant stars), the bifurcation of nonlinear pulsation is induced by the hydrodynamics of ionization zones as well as modal resonances. The sequence of the bifurcation sometimes leads to chaotic behaviors in nonlinear pulsation. The transition routes from regular to the chaotic pulsations found in models are discussed with respect to the theory of chaos in simple dynamical systems: The cascade of period-doubling bifurcation is confirmed to cause chaotic pulsation in W Virginis models. For models of higher luminosity, the tangent bifurcation is found to lead intermittent chaos.Finally, hydrodynamic models for chaotic pulsation with small amplitudes observed in the post-AGB stars are briefly discussed.     

恒星临边昏暗系数的测定

介绍一种新的一级挖和二级近似描述恒星大气临边昏境现象方法,其临边昏境系数可通过有关物理量的直接测量或测量流量和辅助量恒星表面温度的方法间接给出。与太阳观测比较和数值模型大气方法比较,均显示新方法能够以很精确的方式描述恒星临边昏境关系,或确定其相应系数 方法对太阳观测的１７６个数据比较了统计方差,其中直接法的一级挖和二级挖分别为０．３８％和０．２６％;间接法的方差与此相近而略大;与数值模型大气比较其     

Chemically Consistent Evolutionary Models With Dust

As a tool for interpreting nearby and high-redshift galaxy data from the optical to K-band we present our chemically consistent spectrophotometric evolutionary synthesis models. These models take into account the increasing initial metallicity of successive stellar generations using recently published metallicity-dependent stellar evolutionary tracks, stellar yields and model atmosphere spectra. The influence of the metallicity is analysed. Dust absorption is included on the basis of gas content and abundance as it varies with time and galaxy type. We compare our models with IUE template spectra and are able to predict UV fluxes for different spectral types. Combining our models with a cosmological model we obtain evolutionary and k corrections for various galaxy types and show the differences from models using only solar metallicity input physics as a function of redshift, wavelength band and galaxy type. This revised version was published online in July 2006 with corrections to the Cover Date.     

Formation of P Cygni profiles with limb darkening of the stellar core

In the framework of the Sobolev approximation, we investigate the effects of limb darkening of the stellar core onto the formation of line profiles in rapidly expanding envelopes.For the case of outward-accelerating envelopes, it is shown that the P Cygni profiles, calculated with and without limb darkening, do appear almost identical as long as the size of the envelope remains large with respect to the radius of the star. As the extent of the envelope decreases, the resulting profiles are found to differ appreciably close to the line center.When considering outward-decelerating (or equivalently inward-accelerating) envelopes, calculations with the limb darkening effects reveal that the P Cygni profiles are more strongly affected than in the previous case. These effects are specially important in the blue wing and near the center of the resulting line profiles, irrespective of the size of the envelope.Chargé de Recherches au Fonds National de la Recherche Scientifique, Belgium.     

Frequency analysis of Cepheids in the Large Magellanic Cloud: new types of classical Cepheid pulsators

We have performed a detailed systematic search for multiperiodicity in the Population I Cepheids of the Large Magellanic Cloud. In this process, we have identified for the first time several new types of Cepheid pulsational behaviour. We have found two triple-mode Cepheids pulsating simultaneously in the first three radial overtones. In 9 per cent of the first overtone (FO) Cepheids, we have detected weak but well-resolved secondary periodicities. They appear either very close to the primary pulsation frequency or at a much higher frequency with a characteristic period ratio of 0.60–0.64. In either case, the secondary periodicities must correspond to non-radial modes of oscillation. This result presents a major challenge to the theory of stellar pulsations, which predicts that such modes should not be excited in Cepheid variables. Non-radial modes have also been found in three of the fundamental first overtone (FU/FO) double-mode Cepheids, but no such oscillations have been detected in single-mode Cepheids pulsating in the FU mode.
In 19 per cent of double-mode Cepheids pulsating in the first two radial overtones (FO/SO type), we have detected a Blazhko-type periodic modulation of amplitudes and phases. Both modes are modulated with a common period, which is always longer than 700 d. Variations of the two amplitudes are anticorrelated, and maximum of one amplitude always coincides with minimum of the other. We have compared observations of modulated FO/SO Cepheids with predictions of theoretical models of the Blazhko effect, showing that the currently most popular models cannot account for properties of these stars. We propose that the Blazhko effect in FO/SO Cepheids can be explained by a non-stationary resonant interaction of one of the radial modes with another, perhaps non-radial, mode of oscillations.     

Period changes of Cepheid variables

Secular period changes of one hundred northern Cepheids are investigated with the help of O–C diagrams. With the classical Cepheids the rate of observed period changes is in good agreement with that determined from stellar evolution theory. The period noise cannot mask the evolutionary period changes especially in longer period Cepheids for which the occurrence of parabolic O–C graphs is unusually frequent.     

Microlensing of close binary stars

The gravity due to a multiple-mass system has a remarkable gravitational effect: the extreme magnification of background light sources along extended so-called caustic lines. This property has been the channel for some remarkable astrophysical discoveries over the past decade, including the detection and characterization of extrasolar planets, the routine analysis of limb darkening, and, in one case, limits set on the apparent shape of a star several kiloparsec distant. In this paper, we investigate the properties of the microlensing of close binary star systems. We show that in some cases it is possible to detect flux from the Roche lobes of close binary stars. Such observations could constrain models of close binary stellar systems.     

Physics and Gas Dynamics of Wind-Blown Bubbles

We review the recent hydrodynamic modelling of wind-blown bubbles (WBB) which are result of interaction of a stellar wind with the circumstellar matter or the wind(s) emitted during the previous stages of the central star evolution. The much faster computers becoming available in the last decade allow a more complete picture of the physics of these objects to be built. Recent hydrodynamic models are capable of treating in detail different mechanisms as radiative plasma cooling, electron thermal conduction and the effects of magnetic fields. We discuss the various mechanisms proposed for shaping these objects and we emphasize on the problems related to the development of various instabilities and the X-ray emission from WBB. This revised version was published online in July 2006 with corrections to the Cover Date.     

Non-adiabatic effects on the pulsation periods

Weight functions for the non-adiabatic radial pulsations are introduced. It is shown from behavior of these functions that the pulsation periods in classical Cepheids are determined essentially in the adiabatic region of stellar envelopes and, on the other hand, those of low surface-gravity models are strongly affected in the region where the acoustic waves are strongly coupled with the radiation fields. The fact is important for understanding basic difference of the pulsation properties between classical Cepheids and low surfacegravity models.The non-adiabatic weight functions deviate from adiabatic ones in two ways in the stellar envelope layers. In the region where the acoustic waves are tightly coupled with radiation fields, the non-adiabatic weight functions have larger values than the corresponding adiabatic ones. On the contrary, the functions are smaller in the outer isothermal region.These results are discussed from the viewpoint on the propagation of the acoustic waves in radiation nelds.     

Center-to-limb variation of the continuum intensity and linear polarization of stars with transiting exoplanets

The limb darkening and center-to-limb variation of the continuum polarization is calculated for a grid of one-dimensional stellar model atmospheres and for a wavelength range between 300 and 950 nm. Model parameters match those of the transiting stars taken from the NASA exoplanet archive. The limb darkening of the continuum radiation for these stars is shown to decrease with the rise in their effective temperature. For the λ = 370 nm wavelength, which corresponds to the maximum of the Johnson–Cousins UX filter, the limb darkening values of the planet transiting stars lie in a range between 0.03 and 0.3. The continuum linear polarization depends not only on the effective temperature of the star but also on its gravity and metallicity. Its value decreases for increasing values of these parameters. In the UX band, the maximum linear polarization of stars with transiting planets amounts to 4%, while the minimum value is approximately 0.3%. The continuum limb darkening and the linear polarization decrease rapidly with wavelength. At the R band maximum (λ = 700 nm), the linear polarization close to the limb is in fact two orders of magnitude smaller than in the UX band. The center- to-limb variation of the continuum intensity and the linear polarization of the stars with transiting planets can be approximated, respectively, by polynomials of the fourth and the sixth degree. The coefficients of the polynomials, as well as the IDL procedures for reading them, are available in electronic form. It is shown that there are two classes of stars with high linear polarization at the limb. The first one consists of cold dwarfs. Their typical representatives are HATS-6, Kepler-45, as well as all the stars with similar parameters. The second class of stars includes hotter giants and subgiants. Among them we have CoRoT-28, Kepler-91, and the group of stars with effective temperatures and gravities of approximately 5000 K and 3.5, respectively.     

Uncertainties of Synthetic Population Models

This paper investigates the uncertainties in the synthetic integrated colours of simple stellar populations, currently the most popular method of estimating the ages of unresolved stellar systems. Three types of uncertainties studied here originate from the stellar models, the population synthesis techniques, and from the stellar spectral libraries. Despite some skepticism, synthetic colours appear to be reliable age indicators as long as they are used for select age ranges. This revised version was published online in September 2006 with corrections to the Cover Date.     

Period–colour and amplitude–colour relations in classical Cepheid variables – V. The Small Magellanic Cloud Cepheid models

Period–colour (PC) and amplitude–colour (AC) relations at maximum, mean and minimum light are constructed from a large grid of full amplitude hydrodynamic models of Cepheids with a composition appropriate for the Small Magellanic Cloud (SMC). We compare these theoretical relations with those from observations. The theoretical relations are, in general, in good agreement with their observational counterparts, though there exist some discrepancy for short period  (log [ P ] < 1)  Cepheids. We outline a physical mechanism which can, in principle, be one factor to explain the observed PC/AC relations for the long and short period Cepheids in the Galaxy, Large Magellanic Cloud (LMC) and SMC. Our explanation relies on the hydrogen ionization front (HIF)–photosphere interaction and the way this interaction changes with pulsation period, pulsation phase and metallicity. Since the PC relation is connected with the period–luminosity (PL) relation, it is postulated that such a mechanism can also explain the observed properties of the PL relation in these three galaxies.     

Solar limb darkening

The coefficients of several polynomial representations of the limb darkening at 62 wavelengths in the UV and visible portions of the solar spectrum obtained at the McMath Solar Telescope are presented in tabular form. Full corrections for scattered light and seeing have been included in the reductions.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

An essay on stellar oscillations and evolution

It is cautioned that solar models adjusted in such a way as to achieve a match between theoretical solar oscillation characteristics and observed ones may produce neutrino fluxes inconsistent with the observations and that this is likely to be explicable as a deficiency in modeling that portion of the envelope which is most strongly affected by uncertainties in the treatment of convection. Then follows a summary of how the results of pulsation theory and of stellar evolution theory have been used together to learn about the structure and evolution of RR Lyrae stars, classical Cepheids, and high luminosity AGB stars.