Hot vibrating white dwarf models of pulsating X-ray sources

A number of white dwarf models have been calculated which correspond to various radial and nonradial modes of vibration with eigenfrequencies in agreement with the observed pulsation frequencies of the X-ray sources Hercules X-1 and Centaurus X-3. Most of the white dwarf models have hot interiors, but for calculational purposes these were simplified so that the bulk of the interior was isothermal, and the surface layers were designed to produce an energy generation rate of 10³⁷ erg s^?1 and to transport this energy continuously to the surface by radiative transfer. Cold white dwarfs have a fairly large spread of masses corresponding to the different overtone modes with the given eigenfrequencies, but in the hot models this spread of masses is greatly reduced, for both radial and nonradial modes. It is concluded that if the pulsating X-ray sources are hot white dwarfs, the mass of Cen X-3 probably lies in the range 0.7–1.2M _⊙, and the mass of Her X-1 probably lies in the range 1.1–1.25M _⊙ (in accord with observation).     

Evolution of a 3-M⊙ star from the main sequence to the ZZ Ceti stage: the role played by element diffusion

The purpose of this paper is to present new full evolutionary calculations for DA white dwarf stars with the major aim of providing a physically sound reference frame for exploring the pulsation properties of the resulting models in future communications. Here, white dwarf evolution is followed in a self-consistent way with the predictions of time-dependent element diffusion and nuclear burning. In addition, full account is taken of the evolutionary stages prior to white dwarf formation. In particular, we follow the evolution of a 3-M_⊙ model from the zero-age main sequence (the adopted metallicity is   Z =0.02)  , all the way from the stages of hydrogen and helium burning in the core up to the thermally pulsing phase. After experiencing 11 thermal pulses, the model is forced to evolve towards its white dwarf configuration by invoking strong mass loss episodes. Further evolution is followed down to the domain of the ZZ Ceti stars on the white dwarf cooling branch.
Emphasis is placed on the evolution of the chemical abundance distribution caused by diffusion processes and the role played by hydrogen burning during the white dwarf evolution. We find that discontinuities in the abundance distribution at the start of the cooling branch are considerably smoothed out by diffusion processes by the time the ZZ Ceti domain is reached. Nuclear burning during the white dwarf stage does not represent a major source of energy, as expected for a progenitor star of initially high metallicity. We also find that thermal diffusion lessens even further the importance of nuclear burning.
Furthermore, the implications of our evolutionary models for the main quantities relevant for adiabatic pulsation analysis are discussed. Interestingly, the shape of the Ledoux term is markedly smoother compared with previous detailed studies of white dwarfs. This is translated into a different behaviour of the Brunt–Väisälä frequency.     

The YNEV stellar evolution and oscillation code

We have developed a new stellar evolution and oscillation code YNEV,which calculates the structures and evolutions of stars,taking into account hydrogen and helium burning.A nonlocal turbulent convection theory and an updated overshoot mixing model are optional in this code.The YNEV code can evolve low-and intermediate-mass stars from the pre-main sequence to a thermally pulsing asymptotic branch giant or white dwarf.The YNEV oscillation code calculates the eigenfrequencies and eigenfunctions of the adiabatic oscillations for a given stellar structure.The input physics and numerical scheme adopted in the code are introduced.Examples of solar models,stellar evolutionary tracks of low-and intermediate-mass stars with different convection theories(i.e.mixing-length theory and nonlocal turbulent convection theory),and stellar oscillations are shown.     

Diffusion in helium white dwarf stars

This paper is aimed at exploring the effects of diffusion on the structure and evolution of low-mass helium white dwarfs. To this end, we solve the multicomponent flow equations describing gravitational settling and chemical and thermal diffusion. The diffusion calculations are coupled to an evolutionary code in order to follow the cooling of low-mass, helium core white dwarf models having envelopes made up of a mixture of hydrogen and helium, as recently suggested by detailed evolutionary calculations for white dwarf progenitors in binary systems. We find that diffusion causes hydrogen to float and the other elements to sink over time-scales shorter than evolutionary time-scales. This produces a noticeable change in the structure of the outer layers, making the star inflate. Thus, in order to compute accurately the mass–radius relation for low-mass helium white dwarfs we need to account for the diffusion processes during (at least) the white dwarf stages of the evolution of these objects. This should be particularly important when studying the general characteristics of binary systems containing a helium white dwarf and a pulsar.
In addition, we present an analytic, approximate model for the outer layers of the white dwarf aimed at interpreting the physical reasons for the change in the surface gravity for low-mass white dwarfs induced by diffusion.     

Dynamic phases in the evolution of a 1.25M ⊙ C−O star

The evolution of a 1.25M _⊙ carbon and oxygen (equal fractions by mass) homogeneous star is followed by means of a computer code capable of dealing with dynamic evolutionary phases. After carbon ignition at the center, followed by successive shell flashes and the formation of aT-inversion, convection begins at the surface and the model evolves through a very short but strong dynamic phase (viz. a pulsation) after which it settles down to a white dwarf configuration.     

PULSE: a finite element code for solving adiabatic nonradial pulsation equations

We briefly present the nonradial adiabatic pulsation code PULSE first developped for white dwarf asteroseismology and now used to compute adiabatic oscillation properties for various types of stellar objects. Numerical tests show that the code is able to provide the accuracy (for a given stellar model) required to deal with the precision in frequency expected from the COROT long runs. While the ultimate objective is to compare the output of various pulsation codes (see these proceedings), we already emphasize problems that need to be addressed concerning, in particular, the mesh resolution of the input stellar models and its impact on the accuracy at which frequencies can be computed.     

Analysis of the oscillations in HST observations of the quiescent SU UMa type dwarf nova WZ Sagittae

An analysis of the UV oscillations in WZ Sge is presented, in which we obtain the oscillation amplitude spectra. We find a strong 27.9-s oscillation in our Hubble Space Telescope ( HST ) UV and zeroth-order light curves as well as weaker oscillations at 28.4 s in the UV and 29.1 s in the zeroth order. We find that the main oscillation amplitude spectrum can be fitted with static white dwarf spectra of about 17 000 K, an accretion hotspot of only a few 100 K hotter than the underlying white dwarf temperature or a variety of cool (<14 500 K) white dwarf pulsation amplitude spectra. A pulsating white dwarf can also explain the very blue colour of oscillations of different periods previously found in the optical. Comparing our results with those of Welsh et al., we see that the amplitude spectra of the main oscillations in WZ Sge measured with different periods in data sets from different epochs are similar to each other. Our results raise questions about using the magnetically accreting rotating white dwarf model to explain the oscillations. We suggest that the pulsating white dwarf model is still a viable explanation for the oscillations in WZ Sge.     

Mode identification from time-resolved spectroscopy of the pulsating white dwarf G29‐38

We have used time-resolved spectroscopy to measure the colour dependence of pulsation amplitudes in the DAV white dwarf G29-38. Model atmospheres predict that mode amplitudes should change with wavelength in a manner that depends on the spherical harmonic degree ℓ of the mode. This dependence arises from the convolution of mode geometry with wavelength-dependent limb darkening. Our analysis of the six largest normal modes detected in Keck observations of G29-38 reveals one mode with a colour dependence different from the other five, permitting us to identify the ℓ-value of all six modes and to test the model predictions. The Keck observations also show pulsation amplitudes that are unexpectedly asymmetric within absorption lines. We show that these asymmetries arise from surface motions associated with the non-radial pulsations (which are discussed in detail in a companion paper). By incorporating surface velocity fields into line profile calculations, we are able to produce models that more closely resemble the observations.     

The pulsation of Delta-Scuti stars

Evolution, linear pulsation, and nonlinear pulsation codes were combined to produce nonlinear models of Delta-Scuti stars in an instability region extending over 3.8T _e<3.95 and 0.6L/L <2.0. The linear analysis upheld the consensus that they are normal Population I stars of about 2M , in stages of evolution corresponding to central hydrogen burning and shell hydrogen burning. The growth rates were very slow; driving was due to an opacity mechanism in the second helium ionization region; periods and period ratios of the lowest modes of the models were in the same range as those observed. A wide range of nonlinear models was investigated. When eigenfunctions from the linear analysis were used as initial velocity profiles, it was found that the dominant peak in the periodogram of the light curve corresponded to the mode initiated. For a small subset of models, limiting amplitudes were identified, and were found to be in close agreement with observed light amplitudes.     

Deciphering the pulsations of G 29-38 with optical time series spectroscopy

We present optical time series spectroscopy of the pulsating white dwarf star G 29-38 taken at the Very Large Telescope (VLT). By measuring the variations in brightness, Doppler shift and line shape of each spectrum, we explore the physics of pulsation and measure the spherical degree (ℓ) of each stellar pulsation mode. We measure the physical motion of the g modes correlated with the brightness variations for three of the eight pulsation modes in this data set. The varying line shape reveals the spherical degree of the pulsations, an important quantity for properly modelling the interior of the star with asteroseismology. Performing fits to the Hβ, Hγ and Hδ lines, we quantify the changing shape of the line and compare them to models and previous time series spectroscopy of G 29-38. These VLT data confirm several ℓ identifications and add four new values, including an additional ℓ= 2 and a possible ℓ= 4. In total, from both sets of spectroscopy of G 29-38, eleven modes now have known spherical degrees.     

A numerical study of the rotational behavior of the white dwarf in DQ Herculis, II: the turn-over scenario

The optical pulsations in DQ Her are universally believed to be due to emission from the magnetic poles of the white dwarf. However, there is no way for a pulsation to be seen if the magnetic axis and the spin axis are aligned; whether the optical pulsation is seen directly from the magnetic poles or as a result of re-processing this beam from sides in an accretion disc, the magnetic and spin axes must be offset. This fact explains why the `oblique rotator' model has been adopted as a standard model for the DQ Her primary. In a recent paper, we have computed several axisymmetric models simulating the DQ Her white dwarf before its `turn-over' (where the term `turn-over' describes the process by which the magnetic axis gets inclining relative to the spin axis at a gradually increasing angle, the so-called `turn-over angle'). For such models, we have found that the moment of inertia along the rotation axis, I ₃₃, is less than the moment(s) of inertia along the two other principal axes, I ₁₁=I ₂₂. The situation I ₁₁>I ₃₃ is known as `dynamical asymmetry', and can cause a spontaneous turn-over of the magnetic axis with respect to the rotation axis . Consequently, the DQ Her white dwarf is either an oblique rotator undergoing its turn-over phase, or it is already qalmostequal a `perpendicular rotator', i.e., its turn-over angle is almost equal to 90^°. Assuming the first case, we study numerically the so-called `turn-over scenario', that is, a scenario on rotational evolution in which the turn-over phase is taken into account. We give emphasis on computations concerning the spin-down time rate of the DQ Her white dwarf due to turn-over (not to be confused with its spin-uptime rate due to accretion) for several possible values of the magnetic field. This revised version was published online in July 2006 with corrections to the Cover Date.     

Eigenfrequencies of rotationally and tidally distorted white dwarf models of stars

In the present paper we have studied the eigenfrequencies of small adiabatic barotropic pseudo-radial and nonradial modes of oscillations of the white dwarf models of rotating stars in binary systems. In this work the methodology of Mohan and Saxena (in Astrophys. Space Sci. 113:155, 1985) has been used that utilizes the averaging technique of Kippenhahn and Thomas (in Proc. IAU Colloq., vol. 4, p. 20, 1970) and certain results on Roche equipotential as that given by Kopal (in Advances in Astronomy and Astrophysics, Academic Press, 1972). The objective of this study is to investigate the effects of rotation and/or tidal distortion on the periods of oscillations of rotationally and/or tidally distorted white dwarf models of stars assuming it to be the primary component of the binary system and rotating uniformly. The results of present study show that the eigenfrequencies (both radial and nonradial modes) of the rotationally distorted and rotationally and tidally distorted white dwarf model of stars in binary systems tend to decrease under the influence of rotational distortions and rotational and tidal distortions, respectively. However, results are contrary for tidally distorted white dwarf model of stars.     

A new, efficient stellar evolution code for calculating complete evolutionary tracks

We present a new stellar evolution code and a set of results, demonstrating its capability at calculating full evolutionary tracks for a wide range of masses and metallicities. The code is fast and efficient, and is capable of following through all evolutionary phases, without interruption or human intervention. It is meant to be used also in the context of modelling the evolution of dense stellar systems, for performing live calculations for both normal star models and merger products.
The code is based on a fully implicit, adaptive-grid numerical scheme that solves simultaneously for structure, mesh and chemical composition. Full details are given for the treatment of convection, equation of state, opacity, nuclear reactions and mass loss.
Results of evolutionary calculations are shown for a solar model that matches the characteristics of the present sun to an accuracy of better than 1 per cent; a  1 M_⊙  model for a wide range of metallicities; a series of models of stellar Populations I and II, for the mass range 0.25 to  64 M_⊙  , followed from pre-main-sequence to a cool white dwarf or core collapse. An initial–final mass relationship is derived and compared with previous studies. Finally, we briefly address the evolution of non-canonical configurations, merger products of low-mass main-sequence parents.     

The impact of element diffusion on the formation and evolution of helium white dwarf stars

The aim of this work is to investigate the effect of element diffusion on the evolution of helium white dwarfs. To this end, we couple the multicomponent flow equations that describe gravitational settling, chemical and thermal diffusion to an evolutionary code. We compute the evolution of a set of helium white dwarf models with masses ranging from 0.169 to 0.406 M_⊙. In particular, several low-mass white dwarfs have been found in binary systems as companion to millisecond pulsars. In these systems, pulsar emission is activated by mass transfer episodes so that, if we place the zero-age point at the end of such mass transfer, then the pulsar and the white dwarf ages should be equal. Interestingly enough, available models of helium white dwarfs neglect element diffusion. Using such models, good agreement has been found between the ages of the components of the PSR J1012+5307 system. However, recent observations of the PSR B1855+09 system cast doubts on the correctness of such models, which predict a white dwarf age twice as long as the spin-down age of the pulsar. In this work, we find that element diffusion induces thermonuclear hydrogen shell flashes for models in the mass interval 0.18≲ M /M_⊙ ≲ 0.41 . We show, in particular, that the occurrence of these diffusion-induced flashes eventually leads to white dwarf models with hydrogen envelope masses too small to support any further nuclear burning, thus implying much shorter cooling ages than in the case when diffusion is neglected. In particular, excellent agreement is found between the ages of PSR B1855+09 system components, solving the age discrepancy from first principles.     

Hydrogen-model atmospheres for white dwarf stars

We present a detailed calculation of model atmospheres for DA white dwarfs. Our atmosphere code solves the atmosphere structure in local thermodynamic equilibrium with a standard partial linearization technique, which takes into account the energy transfer by radiation and convection. This code incorporates recent improved and extended data base of collision-induced absorption by molecular hydrogen. We analyse the thermodynamic structure and emergent flux of atmospheres in the range 2500 T _eff60 000 K and 6.5log  g 9.0. Bolometric correction and colour indices are provided for a subsample of the model grid. Comparison of the colours is made with published observational material and results of other recent model calculations.
Motivated by the increasing interest in helium-core white dwarfs, we analyse the photometric characteristics of these stars during their cooling, using evolutionary models recently available. Effective temperatures, surface gravities, masses and ages have been determined for some helium-core white dwarf candidates, and their possible binary nature is briefly discussed.     

Multicolor Photometry for Mode Identification

The goal of asteroseismology is to discern the physical conditions of stars by comparing observed pulsations with models.To obtain this goal, the observed pulsation periods and the spherical harmonics (n, , and m) need to match the theoretical model.Typically the most difficult part in this process is the identification of the pulsation modes in the observations.Multicolour photometry is one method that has proven useful for identifying pulsation modes.By observing stars through various wavebands, and comparing the amplitudes and phases, it is possible to determine the spherical harmonics.This contribution will emphasize the work of Watson (1988), which has since been applied to many different types of variable stars including Scuti (Garrido et al., 1990), Doradus (Breger et al., 1997), Cepheid (Cugier et al., 1994), and EC 14026 (Koen, 1998) stars. I will also discuss the technique of summing spectra (especially UV) into various wavebands which are then used to identify modes as pioneered by Robinson, Kepler, and Nather (1982) and applied to white dwarf stars (Kepler et al., 2000).     

Evolutionary Timescale of the Pulsating White Dwarf G117-B15A: The Most Stable Optical Clock Known

We observe G117-B15A, the most precise optical clock known, to measure the rate of change of the main pulsation period of this blue-edge DAV white dwarf. Even though the obtained value is only within 1 sigma, P&d2;=&parl0;2.3+/-1.4&parr0;x10-15 s s-1, it is already constraining the evolutionary timescale of this cooling white dwarf star.     

The evolution of iron-core white dwarfs

Recent measurements by Hipparcos present observational evidence supporting the existence of some white dwarf (WD) stars with iron-rich core composition. In connection with this, the present paper is aimed at exploring the structure and evolution of iron-core WDs by means of a detailed and updated evolutionary code. In particular, we examined the evolution of the central conditions, neutrino luminosity, surface gravity, crystallization, internal luminosity profile and ages. We find that the evolution of iron-rich WDs is markedly different from that of their carbon–oxygen counterparts. In particular, cooling is strongly accelerated (up to a factor of 5 for models with pure iron composition) as compared with the standard case. Thus, if iron WDs were very numerous, some of them would have had time enough to evolve at lower luminosities than that corresponding to the fall-off in the observed WD luminosity function.     

The evolution of a rapidly accreting helium white dwarf to become a low-luminosity helium star

We have examined the evolution of merged low-mass double white dwarfs which become low-luminosity (or high-gravity) extreme helium stars. We have approximated the merging process by the rapid accretion of matter, consisting mostly of helium, on to a helium white dwarf. After a certain mass is accumulated, a helium shell flash occurs, the radius and luminosity increase and the star becomes a yellow giant. Mass accretion is stopped artificially when the total mass reaches a pre-determined value. As the helium-burning shell moves inwards with repeating shell flashes, the effective temperature gradually increases as the star evolves towards the helium main sequence. When the mass interior to the helium‐burning shell is approximately 0.25 M_⊙, the star enters a regime where it is pulsationally unstable. We have obtained radial pulsation periods for these models.
These models have properties very similar to those of the pulsating helium star V652 Her. We have compared the rate of period change of the theoretical models with that observed in V652 Her, as well as with its position on the Hertzsprung–Russell diagram. We conclude that the merger between two helium white dwarfs can produce a star with properties remarkably similar to those observed in at least one extreme helium star, and is a viable model for their evolutionary origin. Such helium stars will evolve to become hot subdwarfs close to the helium main sequence. We also discuss the number of low-luminosity helium stars in the Galaxy expected for our evolution scenario.