The determination of Baade-Wesselink radii and luminosities for pulsating stars are long-standing and highly desired goals since they provide the promise of being standard candles. In a modest contribution towards these goals, we have undertaken a programme to determine the radii and luminosities of the large-amplitude δ Scuti stars DY Herculis, EH Librae and DY Pegasi from optical and infrared photometry and cross-correlated radial velocity data. We use Fourier representations for V, I and J light curves and for the radial velocity curves in Baade-Wesselink analyses to derive minimum radii over the pulsation cycles. These radii and their errors and the mean bolometric luminosities and absolute magnitudes will be discussed here and in papers to follow. As a check, we also apply our method to the data and results of other groups.

Compact stars that result from extreme mass loss on the asymptotic giant branch and planetary nebula formation are observed to pulsate in a very large surface effective temperature range as they cool to become the classical white dwarfs. The hottest and most luminous of these display periods in excess of 1000 seconds because they are large, but when the stars arrive on the cooling line on the Hertzsprung-Russell diagram, their periods become generally less than 1000 seconds. Then the stars have masses near 0.6 M⊙ and radii near 109 cm. Their luminosity depends then almost entirely on the surface effective temperature as the entire star with its legacy of complicated internal luminosity peaks cools to the classical simple electron degenerate structure. Very thin surface layers of hydrogen and helium cover the bulk of the carbon- and oxygen-rich mass that results from hydrogen and helium burning in earlier intermediate mass stellar evolution. The cause of the nonradial pulsations of low angular degree, but rather high radial order, for the most luminous of these stars is the cyclical ionization of carbon and oxygen in layers not too deep that their effectiveness is limited by a long luminosity time scale. Thus the surface hydrogen and helium must be thin, probably thinner than the current period spacings interpretation for PG 1159-035 suggests. For the classical DBV and DAV pulsators, it appears that neither the hydrogen ionization K and γ effects or convection blocking at the bottom of a hydrogen convection zone can destabilize the observed pulsations when the overriding short time scale effects of time-dependent convection are included. It appears, however, that a thin CO convection shell can produce pulsations by its time-dependent effects, but again only very thin H and He surface layers are allowed. This new pulsation mechanism can alleviate the serious problem that DAV variables are observed hotter than the hydrogen K and γ effects and convection blocking can predict. The appearance of non-pulsators in the DAV and DBV instability strips can be explained by a too-thick hydrogen and helium surface layer that interferes with (poisons) the CO ionization convection zone. Finally time-dependent convection predicts that only a few of the many possible modes exist due to their internal amplitude structure that can result in both strong driving and strong damping. Thus actually observed pulsating modes can assist in mapping individual internal white dwarf composition structures, not only by their periods but also the fact that they pulsate.

We present the results of a parametric survey of evolutionary models of compositionally stratified white dwarfs with helium surface layers (DB white dwarfs). Because white dwarfs are the most common final end state of stellar evolution, determining their internal structure will offer us many clues about stellar evolution, the physics of matter under extreme conditions, plus the history of star formation and age of the local Galactic disk. As a first step towards determining the internal structure of DB white dwarf stars, we provide a comprehensive set of theoretical g-mode pulsation periods for comparison to observations.

Because DB white dwarfs have a layered structure consisting of a helium layer overlying the carbon/oxygen core, some modes will have the same wavelength as the thickness of the helium layer, allowing a resonance to form. This resonance is called mode trapping (see Brassard et al. 1992 and references therein) and has directly observable consequences, because modes at or near the resonance have eigenfunctions and pulsation periods that are similar to each other. This results in much smaller period spacings between consecutive overtone modes of the same spherical harmonic index than the uniform period spacings seen between non-trapped modes. We demonstrate with an example how one can use the distribution of pulsation periods to determine the total stellar mass, the mass of the helium surface layer, and the extent of the helium/carbon and carbon/oxygen transition zones. With these tools, we have the prospect of being able to determine the structure of the observed DBV white dwarfs, once the requisite observations become available.

We are grateful to C.J. Hansen, S.D. Kawaler, R.E. Nather, and D.E. Winget for their encouragement and many discussions. This research was supported by the National Science Foundation under grants 85-52457 and 90-14655 through the University of Texas and McDonald Observatory.

White dwarfs are the final end state for the majority of stars, and hold clues to help solve many current pressing astrophysical problems. We can perform asteroseismology on the pulsating white dwarfs to better understand their internal structure and input physics, paving the way to a better understanding of astrophysics, stellar evolution, and the history of our Galaxy. I describe briefly the potential of asteroseismology by using it to infer the internal structure of PG1159-035.

Over the last several years, we have developed a comprehensive program aimed at better understanding the properties of pulsating DA white dwarfs (or ZZ Ceti stars). These stars are nonradial pulsators of the g-type, and their study can lead to inferences about their internal structure. For instance, the period spectrum of a white dwarf is most sensitive to its vertical chemical stratification, and one of the major goals of white dwarf seismology is to determine the thickness of the hydrogen layer that sits on top of a star. This can be done, in principle, by comparing in detail theoretical period spectra with the periods of the observed excited modes. Likewise, because the cooling rate of a white dwarf is very sensitive to the specific heat of its core material (and hence to its composition), it is possible to infer the core composition through measurements and interpretations of rates of period change in a pulsator.

A white dwarf model with cylindrical symmetry is used to investigate the effect of a uniform vertical magnetic field on the periods of g-modes trapped in the compositionally stratified surface layers. For those modes for which trapping is most effective, the periods wander by approximately 10% about their zero-field values as B0 increases. As the field strength increases further, the period may abruptly increase by some 25% to a new, more stable value; see Figure 1. The periods of less efficiently trapped modes do not show this sensitivity for field strengths up to 1 MG, the upper limit examined.

Since the discovery of the first rapidly oscillating Ap (roAp) star in 1978 by Kurtz, this class of magnetic chemically-peculiar pulsators has grown to over two dozen. The eigenfrequency spectra of roAp stars (with periods of ∼ 6 – 15 min) are consistent with nonradial p- modes of low degree and high overtone n, not unlike the Sun's five-minute oscillations seen in integrated light. However, unlike the Sun, the strong global dipole fields of roAp stars significantly affect the pulsations.

Although much of the effort in the last decade has been towards detecting new roAp candidates and refining the frequencies of known variables, initial “seismic” analyses have already yielded important results. Measurements of fundamental frequency spacings constrain the luminosities and radii of some roAp stars. In addition, mode splitting provides: (1) an independent determination of rotation period, even in the absence of longer-term light variations; (2) limits on the rotational inclination i and magnetic obliquity β; and (3) an indication of the relative internal field strengths of certain roAp stars. Very recently, the temperature - optical depth structure of the atmosphere of HR 3831 was inferred from optical and IR photometry of its oscillations.

Judging from current developments, the next decade promises exciting results on both observational and theoretical fronts. Several roAp stars have now been monitored for over a decade, allowing us to investigate long-term period changes due to evolution, binarity, etc. Eigenfrequency models for stars in the mass and radius range appropriate for Ap stars are becoming available, as well as explicit treatments of the perturbations due to magnetic fields. Armed with these, we may be able to place some roAp stars on a theoretical (or “asteroseismological H-R“) diagram to derive independently their masses and main-sequence ages.

Magnetic field appears to play a major role in the pulsations of rapidly oscillating Ap (roAp) stars. Understanding of the behaviour of these objects thus requires knowledge of their magnetic field. Such knowledge is in particular essential to interpret the modulation of the amplitude of the photometric variations (with a frequency very close to the rotation frequency of the star) and to understand the driving mechanism of the pulsation. Therefore, a systematic programme of study of the magnetic field of roAp stars has been started, of which preliminary (and still very partial) results are presented here.

Magnetic fields of Ap stars can be diagnosed from the Zeeman effect that they induced in spectral lines either from the observation of line-splitting in high-resolution unpolarized spectra (which only occurs in favourable circumstances) or from the observation of circular polarization of the lines in medium- to high-resolution spectra.

Chaos theory has been applied to a variety of variable stars, but few convincing candidates for chaos have been identified. Here, well-established analysis methods have been applied to some very extensive data sets of rapidly oscillating Ap (roAp) stars and one white dwarf. It it shown that in spite of the amount of data, the signal-to-noise ratio makes positive detection of chaos extremely difficult, especially due to scintillation noise. A new form of dimension computation is presented and discussed. Simple models were constructed to show what noise levels can be tolerated before the detection of chaos is no longer possible and comparisons are drawn with data that could be obtained in the future from space. The lack of phase and amplitude stability in HD 134214 and mode switching in HD217522 and HD 137949 are pointed out as the possible results of chaos, making frequent monitoring of roAp stars desirable.

Recently, one of the rapidly oscillating Ap stars, HR 3831, has been found to have an equally split frequency septuplet, though its oscillation seems to be essentially an axisymmetric dipole mode with respect to the magnetic axis which is oblique to the rotation axis (Kurtz et al. 1992; Kurtz 1992). In order to explain this fine structure, we investigate oscillations of obliquely rotating magnetic stars by taking account of the perturbations due to the magnetic fields and the rotation. We suppose that the star is rigidly rotating and that the magnetic field is a dipole field and its axis is oblique to the rotation axis. We treat the effects of the rotation and of the magnetic field as perturbations. In doing so, we suppose that the rotation of the star is slow enough so that the effect of the rotation on oscillations is smaller than that of the magnetic field.

The review examines recent observational and theoretical results on the nonradial pulsation of δ Scuti stars. The dominant pulsation modes are rotationally split p modes with = 1 and 2, and radial orders 1 to 4. Line-profile analyses also reveal the existence of additional high-degree modes. Nonradial pulsation is usually accompanied by slow amplitude and period variability. The small-amplitude single-mode variables are shown to be typical nonradial pulsators.

The interesting cases of 1 Mon and θ2 Tau are examined in more detail. The present status of asteroseismolgy of the δ Scuti stars is reviewed.

The periods of the four evolved δ Scuti variables studied are found to be decreasing, although stellar evolution predicts period increases due to larger radii. New work on the multiple periods of AI Vel shows that, apart from stellar evolution, an additional mechanism must exist to cause temporary period changes. It is argued that for a larger sample of stars the average period change must nevertheless reflect evolutionary changes.

The amplitude and period of the first frequency increased, but the amplitude and period of the second frequency decreased. Two new frequencies (10.5011 and 2.0484 c/d) were found. The frequency (2.0484 c/d) might be an interaction term from which resonances can occur and then lead amplitude and period variations.

The identification of modes of oscillation is an important first step towards the seismology of stars. Low- and high-degree nonradial modes of oscillation may appear as variations in the line profiles of rapidly rotating δ Scuti stars. We present a technique whereby complex patterns in the line profiles are decomposed into Fourier components in both time and “Doppler space”. The technique is applied to the 7.3-hour time series of high-resolution data obtained from CFHT for the δ Scuti star τ Peg. In addition to the low-degree mode which has been identified in photometric studies (Breger 1991), we find evidence for at least three high-degree modes near 11 and 15. Correcting for the rotation of the star, most of these modes appear to oscillate with frequencies near 17 cycles day-1. Our results are found to be in good agreement with the theoretical limits imposed on the frequencies of oscillation by the models of Dziembowski (1990).

Numerous rapidly rotating δ Scuti stars exhibit variable line profiles containing traveling subfeatures [l]-[2], which may be signatures of nonradial pulsations having relatively high azimuthal order |m| We describe a procedure whereby a time series of spectral line profiles is Fourier analysed both in time and in a wavelength variable that is presumed to correspond to azimuthal position Φ on the star. What such an analysis can tell us is examined by analysing artificially-generated data. For an ideal example in which sin i = 1 and a single mode having is present, the two-dimensional Fourier transform yields a power spectrum in frequency and an apparent azimuthal order that provides a good indication of the actual and m. Such a straightforward interpretation is also possible when sin i < 1, and when multiple sectoral modes are present. For tesseral modes may correspond more closely to than to m.

We report some of the recent results of our studies on δ Scuti star pulsation which are based on observations made at Merate and La Silla Observatories, sometimes in two site campaigns. Our recent experiences on X Caeli and 44 Tauri have shown that quite complicate light curves can be resolved even with observations obtained from one site only, if compact and accurate datasets are collected on sufficiently long time baselines (Mantegazza and Poretti, 1992; Poretti et al., 1992). Here we report the preliminary results of the light curve analysis of three more stars surveyed for more than 120 hours each: BI CMi, HD 18878 and HD 224639.

BI CMi has been observed in a two-site campaign in January-February 1991, the observations being collected during 17 nights. The frequency analysis of the light curve allowed us to identify unambiguously 4 pulsation modes at 8.247, 8.863, 8.514 and 7.424 c/d with semi-amplitudes of 22, 19, 5 and 5 mmag respectively.

The large-amplitude δ Scuti star CY Aqr was observed from sites in the U.S.A., South Africa and Australia during August 1988. Coates et al. (1991) published 48 new times of maximum light derived from these observations and assembled, from the literature, previous times of maximum light. It is clear that the period of the star is changing with the balance of evidence favouring discrete changes in 1951 and 1966, rather than a continuous change.

It has been suggested by Fitch (1973) and Else (1972), from an analysis of the observations of Zissell (1968), that there is a secondary frequency present in CY Aqr. Coates et al. (1992) have analysed both the 1988 observations and those of Zissell. After subtracting the primary frequency and its harmonics, they find no stable secondary frequency above the noise level of two millimagnitudes.

The current theoretical status of understanding solar oscillations is reviewed. Interpretation of the thousands of well-determined frequencies refines our knowledge of the composition and convection structure of the Sun, since its mass, radius, luminosity, and age are better known from other sources. Recent issues that have been discussed are the solar center structure, bearing on the missing solar neutrino problem, the convection zone helium content, validating helium settling by diffusion, the variations of the oscillation frequencies over the solar cycle, indicating cyclical structure changes in the very outer magnetic layers, and the fine structure splittings of mode frequencies, revealing the internal rotation. Our ability to match observed frequencies to now within only a few microhertz has been enhanced by the recently improved MHD equation of state and the new Livermore OPAL opacities. Thus solar oscillations not only reveal solar structure data, but also they guide improvements for stellar astrophysics material properties. A new discussion of current investigations of the convection zone helium abundance and its depth is presented.

The periodicity of P0=160.0101 (±1) min discovered in the Sun's global oscillations, then in rapid variability of active galactic nuclei (AGNs) (Kotov & Lyuty 1990), might have a cosmological origin. Analysing the distribution of periods of ∼4000 close binaries we found that the most commensurate (with an odd-even parity taken into account) period of galactic binaries is equal to 160.0(±1) min (∼ 4σ). Accordingly, it is argued that the famous 2-3 hr gap of cataclysmic binaries might be closely associated with the “universal” P0- oscillation. A similar analysis applied to the ∼1200 RR Lyrae variables in galactic globular clusters (GCs) and 72 RR Lyraes in the LMC GCs (Graham 1985; Nemec et al. 1985) showed that for all those stars the P0- period appears to be the most “resonant” one (Fig. 1). This strongly supports the cosmological origin of the P0- oscillation. Since the P0- resonance for RR Lyraes appears to be most pronounced when the “odd-even” parity is taken into account, it is hypothesized that variability in the majority of GC RR Lyraes should be related to their binary nature and may be relevant to explanation of Blazhko effect.

The recent revision of metal opacities has opened up the possibility that the long-sought mechanism for driving pulsations in these stars has at last, been found. This hypothesis makes a testable prediction — that no β Cep variables should exist among metal-poor B-type stars. We report on the results of an intensive CCD monitoring campaign to test this prediction. The question of the pulsation mode of β Cep stars and the consequence of a revision of the absolute magnitudes to accommodate the fundamental radial mode among these stars is also discussed. Pulsational instability due to driving by ionization of metals has many other repercussions for the incidence of pulsation among the B-type stars. A classification scheme for other intrinsically variable B-type stars is suggested. It is shown that if, as generally supposed, pulsation is common among B-type stars then at least two different mechanisms must be in operation.

We review the latest results on searches for new β Cephei stars, and discuss the most recent findings from high-resolution spectrographic observations. The implications of analyses of archival data and new observations in the light of period changes, binarity, and aspects of the evolutionary state of β Cephei stars are evaluated.