The pulsating ZZ Ceti stars cover a narrow range of effective temperatures along the cooling sequence of DA white dwarfs (see, eg., Winget and Fontaine 1982). Fast-photometric searches for pulsating stars in that class have provided strong evidence that the ZZ Ceti phase is an evolutionary phase through which all cooling DA stars will eventually go through (Fontaine et al. 1982). Recent investigations, based on optical or ultraviolet photometry and spectrophotometry, have set the boundaries of the instability strip at temperatures near 10,000-11,000 K and 12,000-13,000 K, respectively (McGraw 1979; Greenstein 1982; Weidemann and Koester 1984; Fontaine et al. 1985; Wesemael, Lamontagne, and Fontaine 1986; Lamontagne, Wesemael, and Fontaine 1987, 1988).

The spectral evolution of white dwarfs is governed by diffusion processes which enter into competition with mechanisms such as mass loss, convective mixing, and accretion from the interstellar medium in various phases of the evolution. Until recently, our theoretical understanding of the chemical evolution of these stars has been limited by the very severe numerical difficulties which plague a time-dependent description of the problem. Indeed, diffusion problems in white dwarf interiors and envelopes are particularly demanding from a computational standpoint: they involve relative chemical abundances spanning many orders of magnitude, time integration length of a few billion years, and many physical processes operating with greatly different time constants. We have already introduced in the field a robust numerical technique based on an implicit finite difference scheme designed for nonlinear two-point boundary value problems (Pelletier 1986). This method has been used to investigate a number of problems related to the spectral evolution of white dwarfs (Pelletier 1986; Pelletier et al. 1986; Dupuis et al. 1987). As requirements for further progress in the field become more exacting and in the interest of improving the efficiency, we have sought to develop even more powerful numerical techniques. We briefly introduce here an efficient computational approach to diffusion problems in white dwarfs based on a Galerkin finite element method to solve the convective-diffusion equation in an evolving white dwarf model. As an illustrative example, we discuss some sample results of a detailed investigation of the problem of chemical sedimentation (H, He, and C) in the envelopes of hot white dwarfs and the formation of DA stars.

The surface composition of a white dwarf evolves as a result of the interaction of several mechanisms, the most important of which being gravitational settling. In the early phases of the evolution, theory shows that selective radiative levitation can occasionally defeat settling and, thus, prevent the formation of a pristine pure hydrogen (helium) atmospheric layer in a hot DA (non-DA) white dwarf (Fontaine and Michaud 1979; Vauclair, Vauclair, and Greenstein 1979). The exciting discovery of sharp metallic features in the ultraviolet spectra of several hot DA and non-DA stars alike resulting from the work of several investigators has provided the essential motivation for further theoretical investigations of radiative levitation in the atmospheres of white dwarfs. Additionnal impetus comes from the continuing investigations of hot DA white dwarfs carried out by Bruhweiler and Kondo which have already revealed a most interesting observational pattern of heavy elements in these stars (Bruhweiler 1985). Moreover the recent availability of theoretical equivalent widths of selected astrophysically important ultraviolet metal lines in hot DA white dwarfs (Henry, Shipman, and Wesemael 1985) makes a comparison between theory and observations -in at least this type of stars- a timely and useful exercise.

The ZZ Ceti star L 19-2 is a stable pulsator whose light curve has now been deciphered with the help of over 300 hours of white light, high-speed photometry (O’Donoghue and Vamer 1982, 1987, hereafter ODV). The analysis indeed reveals the presence in the light curve of five coherent oscillations, with periods ranging from 113s to 350s. Among those, the 192s oscillation possesses three components, almost equally separated in frequency. Most importantly, the slight, but statistically significant, inequality in the frequency spacing of the triplet has been interpreted by these authors as second-order splitting of rotationally-perturbed g-mode oscillations. And indeed, the measured splitting appears consistent with the theoretical predictions of Chlebowski (1978), which are based on somewhat archaic white dwarf models. As pointed out by ODV, it is clearly of great interest to investigate 1) to what extent theoretical predictions based on more realistic, current-generation white dwarf models agree with ODVs identification, and 2) to what extent such second order effects can, eventually, be used to identify individual pulsation modes or constrain the structural parameters of variable white dwarf stars. Motivated by these questions, we have initiated a study of second-order effects due to rotation in ZZ Ceti stars, and we report here the first results of this program.

The most fundamental aspect of white dwarf seismology is the determination of the gravity-mode (g-mode) period structures of models of isolated pulsating white dwarfs. These stars show multiperiodic luminosity variations which result from the superposition of excited pulsation modes. Among the many oscillation modes available in the very rich nonradial g-mode spectra of white dwarfs, the observed modes are selectively chosen by a filtering mechanism. Although the period evolution is strongly tied to the core temperature evolution in a white dwarf, the period structure remains largely specified by the mechanical properties of the star. The most basic structural feature of a white dwarf is its highly degenerate interior, which leads to nearly isothermal and nearly isentropic stratifications in the core region containing more than 99% of the mass of the star. In particular, because the density gradient is almost adiabatic throughout the interior of a white dwarf, the Brünt-Väisälä frequency (see below) is very small there and low-order g-modes cannot propagate. As a result, g-modes are essentially envelope modes in white dwarfs, with large amplitudes occuring only in the non-degenerate outer layers. One can thus expect that g-modes in white dwarfs are extremely sensitive to envelope properties such as compositional stratification and partial ionization mechanisms. Compositional stratification is, in fact, the second structural feature of a white dwarf model which has strong effects on the period structure. Indeed, trapped modes result when a resonance or near-resonance occurs between the local g-mode radial wavelength and the thickness of one of the composition layers. This results in a period structure which strongly bears the signature of compositional stratification in the outer layers. Thus, it has been widely accepted that white dwarf pulsations probe primarily the outer layers of these stars. This point of view has been borne out by detailed pulsation calculations carried out by several independant groups (see Winget 1987 and references therein).

Non-equilibrium heating of white dwarfs during two-step neutronization and formation of small but finite core of the new phase is a main source of energy in these stars when the temperature is sufficiently small and Coulomb crystal becomes degenerate.

Existence of Chandrasekhar mass limit of white dwarfs Mch = 5.83 /μ2z, μz = A/z, is connected with the prevalence of the relativistic degenerate electrons in pressure. When neutronization is taken into account, the maximal value of mass is smaller and is reached at finite density (Schatzman, 1958). The central density of the white dwarf with limiting mass is larger than threshold density of neutronization. Such star has a small, but finite core of new phase (Seidov, 1967). The neutronization threshold for the iron is equal to ρ = 1.15 109 g/cm3 The neutronization goes through two steps

Evolutionary sequences of a binary system consisting of a helium star in the stage of core helium burning and a white dwarf are presented. Different ways of He burning in the envelope of a dwarf are discussed. Incomplete helium burning is discussed with relation to occurence of possible observational features.

For the last two decades, plasma physics developments have led to a better understanding of physical conditions in white dwarfs interiors. Following the pioneering work of Mestel (1952), the problem of white dwarf cooling has been a subject of continuous interest until the present time. In the early sixties, Kirzhnits (1960), Abrikosov (1960), and Salpeter (1961) recognised the importance of Coulomb interactions in the dense plasma which forms the white dwarf interior. A first-order transition from liquid to solid phase was predicted and the resultant release of latent heat was shown to somewhat affect the cooling rate (Mestel and Ruderman, 1967). Subsequently, improved theoretical luminosity functions (number of white dwarfs per pc9 and per magnitude interval as a function of luminosity) taking into account not only Coulomb interactions but also neutrino losses, and using detailed atmosphere models (Van Horn, 1968; Koester, 1972; Lamb and Van Horn, 1975; Shaviv and Kovets, 1976; Sweeney, 1976). Recently, Iben and Tutukov (1984) have discussed the evolution of a 0.6 M⊙ carbon-oxygen white dwarf from its nuclear burning stages to complete crystallization. Their luminosity function agrees reasonably well with observations in the range −4 ≤ log(L/L⊙) ≤ 4 but it predicts an excess of white dwarfs at low luminosities. Indeed, the luminosity function derived from observations grows monotonically until log(L/L⊙) ≃ −4.5 (Mv ≤ 16) and then makes an abrupt shortfall (Liebert, Dahn and Monet, 1988). The agreement between theory and observations is so good in the aforementioned range luminosity that we can wonder as to whether it is possible not only to test the theory of white dwarf cooling but also to obtain information on the galactic structure and evolution. One example of that is the use of the cutoff in the white distribution to determine the age of the galactic disk (Schmidt, 1959). Using this method, Winget et al. (1987) have found that the galactic disk age could be of the order of 9 Gyr old, in agreement with some predictions from nucleocosmochronology (Fowler et al. 1987).

The observed falloff in the white dwarf luminosity function at log (L/L⊙) ≈ −4.5 (Liebert, Dahn, and Monet 1988) is most easily explained as a result of the finite age of the Galactic disk. Using a simple perturbation approach as our method and a white dwarf evolution code as our tool, we have mapped the sensitivity of the ages of the model sequences in this low-luminosity regime to the uncertainties in the input physics and model parameters. We present here a preliminary overview of what we’ve learned.

We have updated the White Dwarf Evolution Code of Lamb and Van Horn (1975) to include both carbon and oxygen in the core. We interpolate to the mixture composition using our pure-C and pure-O tables and the additive-volume technique (Fontaine, Graboske, and Van Horn 1977). The envelope subroutines calculate stratified H/He/C envelopes of essentially arbitrary layer masses within the range 0 to ∼ 10−2M*, and treat the composition transition zones as discontinuities. Because the equation of state tables referenced by the envelope routines do not include crystallization, a given sequence ends when the crystallization front reaches the core/envelope boundary.

Currently, there are disagreements on the theoretical pulsation properties of DA white dwarf models. Winget and his group (cf Winget 1981, Winget and Fontaine 1982, Winget et al. 1981, 1982) use 0.6M⊙ evolutionary white dwarf models, to find that (Teff)blue is sensitive to the mass of the hydrogen layer and very sensitive to the treatment of convective efficiency. However, (Teff)blue is relatively insensitive to the helium layer mass. Winget and collaborators also find that the hydrogen layer must be restricted to 10−8 > MH/M* > 10−12 for (Teff)blue to occur at ∼ 10,500K in models with ML1 convection (see below). More efficient (ML2) convection pushes the blue edge up to ∼ 12,200K, which is still short of the observed blue edge of ∼ 13,000K.

Though widely spread across the H-R diagram the compact pulsators have much in common. All are multi-periodic, and most have extremely complex light curves. All appear to be pulsating in nonradial g-modes, with temperature variations responsible for the bulk of the light modulations. The g-modes are global in nature and, typically, many are excited in each pulsator, so they are a rich source of seismological information about the interior regions of the white dwarf stars.

The pulsating DB white dwarf stars (DBV’s) form one of the three distinct classes of pulsating compact objects currently known. We find these classes nearly uniformly distributed in log Te spanning virtually the whole range of the white dwarf cooling sequence, from the hot DOV stars at log Te ~ 5 to the DAV (ZZ Ceti) stars at log Te ~ 4; the DBV stars, with logTe ~ 4.5, fall in the middle.

V471 Tau is a spectroscopic and eclipsing binary system located in the Hyades cluster. The binary consists of a K2V and a hot DA white dwarf star (Nelson and Young 1970). Soft x-ray observations reveal strong modulation at periods of 554.7 ± 0.3s and 277.5 ± 0.1s (Jensen 1985, Jensnen et al. 1986).

Robinson et al. (1988) reported the detection of the 555 s period in the optical. This period was about a factor of 20 reduced in mean amplitude in their data relative to the soft x-ray amplitude. They also found that it varied in amplitude from run to run by more than a factor of 2.5, dropping below detectability on several runs. They found some evidence for the 277.5 s period in several runs, but never at high enough amplitude to measure reliably. In addition, they noted that several of their runs had statistically significant power at other frequencies, but noted that in their six runs none of the additional frequencies repeated themselves. Robinson et al. also used observations near the eclipse of the white dwarf to demonstrate that most of the pulsed light is coming from the white dwarf.

White dwarf stars provide important boundary conditions for the understanding of stellar evolution. An adequate understanding of even these simple stars is impossible without detailed knowledge of their interiors. PG1346+082, an interacting binary white dwarf system, provides a unique opportunity to view the interior of one degenerate as it is brought to light in the accretion disk of the second star as the primary strips material from its less massive companion (see Wood et at. 1987).

PG1346+082 is a photometric variable with a four magnitude variation over a four to five day quasi-period. A fast Fourier transform (FFT) of the light curve shows a complex, time-dependent structure of harmonics. PG1346+082 exhibits flickering – the signature of mass transfer. The optical spectra of the system contain weak emission features during minimum and broad absorption at all other times. This could be attributed to pressure broadening in the atmosphere of a compact object, or to a combination of pressure broadening and doppler broadening in a disk surrounding the compact accretor. No hydrogen lines are observed and the spectra are dominated by neutral helium. The spectra also display variable asymmetric line profiles.

Studies of the structure and evolution of low mass stars, brown dwarfs and giant planets require an equation of state (EOS) that includes a detailed, accurate model of the strongly non-ideal behavior of matter in these cool, compact objects [Fig. (1)]. Physical processes in the outer layers of white dwarfs, especially the analysis of the pulsation properties of ZZ Ceti stars, depend sometimes critically on the thermal properties of partially ionized hydrogen. In view of the substantial developments in the statistical physics of dense fluids and plasmas over the past decade, the computation of a new, independent EOS for astrophysical applications is justified. The statistical mechanical models which we present for hydrogen can be adapted to treat both pure helium and hydrogen-helium mixtures.

Just one year ago at the 2nd Conference on Faint Blue Stars, two review papers were presented summarizing the observational characteristics and attempts at spectral modeling of isolated magnetic white dwarfs (see Schmidt 1987 and Wickramasinghe 1987, respectively). Because much of that information is still relevant, this paper will concentrate on some of the progress which has occurred since the time of that meeting, and on highlighting current problems in the area of research. Using specific examples for illustration, the discussion concentrates on the single stars. Magnetic binary systems are considered from the point of view of the white dwarfs themselves.

There are now some 15 cataclysmic variables that have been recognised as being of the AM Her type. They are characterised by the presence of variable circular and linear polarisation in the optical and IR spectral regions, approaching 40% in some systems. The polarized radiation has been attributed to high harmonic cyclotron emission from hot (∼ 5 – 30 Kev) plasma located at accretion shocks near the poles of a highly magnetised (∼ 20 – 60 MG) white dwarf primary. There is hope that the study of the rich variety of phenomena exhibited by these systems will eventually enable us to probe the magnetic field structure of the underlying white dwarfs, perhaps even in more detail than has been possible with the isolated magnetic white dwarfs (Wickramasinghe (1987), Schmidt (1987)).

We present detailed theoretical models for polarisation and spectroscopic data from EXO 033319–2554.2 that suggest the presence of two cyclotron emission regions of field strengths B ∼ 5.6 × 107 G and B ∼ 2.8 × 107 G, and which are separated by about 90° on the surface of the white dwarf. The large difference in magnetic field strength and the proximity of the two regions imply a more complex field structure than that of a centred dipole field distribution if it is assumed that the regions are connected by closed field lines.

New photometric and polarimetric observations of the magnetic white dwarfs G 99–37 and G 99–47 are in accordance with a period of 4.117h and 0.97h respectively. Spectra taken at various phases could be correlated and analyzed by model atmosphere technique.

A new program for the calculation of theoretical spectra and wavelength dependend degrees of the linear and circular polarization for magnetic white dwarfs with hydrogen atmospheres has been developed. The main improvement compared to previous models is a more realistic treatment of the bound-free absorption and of the magnetooptical parameters arising from free electrons in the continuum. For the newly discovered magnetic white dwarf HS 1254+3430 a polar field strength of 18 MG was found. Calculations for the high field object Grw +70°8247 confirms the results of Wickramasinghe and Ferrario (320 MG) but gave a better fit to the spectrum at wavelengths larger than 6200 Å.

We show that the existence of the AM Her period spike implies (i) a unique white dwarf mass ≃ 0.6 − 0.7M⊙ for most magnetic CV’s (ii) nova explosions remove exactly the accreted mass from magnetic white dwarfs, and (iii) the maximum magnetic field for most CV’s is ≤ 4 × 107 G. The existence of the spike is very strong support for the idea that the period gap results from a drastic reduction of angular momentum losses when the secondary star becomes fully convective.