Mode Identification from Combination Frequency Amplitudes in ZZ Ceti Stars

The lightcurves of variable DA stars are usually multi-periodic and non-sinusoidal, so that their Fourier transforms show peaks at eigenfrequencies of the pulsation modes and at sums and differences of these frequencies. These combination frequencies provide extra information about the pulsations, both physical and geometrical, that is lost unless they are analyzed. Several theories provide a context for this analysis by predicting combination frequency amplitudes. In these theories, the combination frequencies arise from nonlinear mixing of oscillation modes in the outer layers of the white dwarf, so their analysis cannot yield direct information on the global structure of the star as eigenmodes provide. However, their sensitivity to mode geometry does make them a useful tool for identifying the spherical degree of the modes that mix to produce them. In this paper, we analyze data from eight hot, low-amplitude DAV white dwarfs and measure the amplitudes of combination frequencies present. By comparing these amplitudes to the predictions of the theory of Goldreich & Wu, we have verified that the theory is crudely consistent with the measurements. We have also investigated to what extent the combination frequencies can be used to measure the spherical degree (ell) of the modes that produce them. We find that modes with ell > 2 are easily identifiable as high ell based on their combination frequencies alone. Distinguishing between ell=1 and 2 is also possible using harmonics. These results will be useful for conducting seismological analysis of large ensembles of ZZ Ceti stars, such as those being discovered using the Sloan Digital Sky Survey. Because this method relies only on photometry at optical wavelengths, it can be applied to faint stars using 4 m class telescopes.Comment: 73 pages, 22 figures, accepted in the Ap

High Resolution Spectroscopy of the Pulsating White Dwarf G29-38

We present the analysis of time-resolved, high resolution spectra of the cool white dwarf pulsator, G29-38. From measuring the Doppler shifts of the H-alpha core, we detect velocity changes as large as 16.5 km/s and conclude that they are due to the horizontal motions associated with the g-mode pulsations on the star. We detect seven pulsation modes from the velocity time-series and identify the same modes in the flux variations. We discuss the properties of these modes and use the advantage of having both velocity and flux measurements of the pulsations to test the convective driving theory proposed for DAV stars. Our data show limited agreement with the expected relationships between the amplitude and phases of the velocity and flux modes. Unexpectedly, the velocity curve shows evidence for harmonic distortion, in the form of a peak in the Fourier transform whose frequency is the exact sum of the two largest frequencies. Combination frequencies are a characteristic feature of the Fourier transforms of light curves of G29-38, but before now have not been detected in the velocities, nor does published theory predict that they should exist. We compare our velocity combination frequency to combination frequencies found in the analysis of light curves of G29-38, and discuss what might account for the existence of velocity combinations with the properties we observe. We also use our high-resolution spectra to determine if either rotation or pulsation can explain the truncated shape observed for the DAV star's line core. We are able to eliminate both mechanisms: the average spectrum does not fit the rotationally broadened model and the time-series of spectra provides proof that the pulsations do not significantly truncate the line.Comment: 24 pages, 9 figures, Accepted for publication in ApJ (June

Gravity-Modes in ZZ Ceti Stars: I.Quasiadiabatic Analysis of Overstability

We analyze the stability of g-modes in variable white dwarfs with hydrogen envelopes. In these stars, the radiative layer contributes to mode damping because its opacity decreases upon compression and the amplitude of the Lagrangian pressure perturbation increases outward. The overlying convective envelope is the seat of mode excitation because it acts as an insulating blanket with respect to the perturbed flux that enters it from below. A crucial point is that the convective motions respond to the instantaneous pulsational state. Driving exceeds damping by as much as a factor of two provided $\omega\tau_c\geq 1$, where $\omega$ is the radian frequency of the mode and $\tau_c\approx 4\tau_{th}$ with $\tau_{th}$ being the thermal time constant evaluated at the base of the convective envelope. As a white dwarf cools, its convection zone deepens, and modes of lower frequency become overstable. However, the deeper convection zone impedes the passage of flux perturbations from the base of the convection zone to the photosphere. Thus the photometric variation of a mode with constant velocity amplitude decreases. These factors account for the observed trend that longer period modes are found in cooler DAVs. The linear growth time, ranging from hours for the longest period observed modes ($P\approx 20$ minutes) to thousands of years for those of shortest period ($P\approx 2$ minutes), probably sets the time-scale for variations of mode amplitude and phase. This is consistent with observations showing that longer period modes are more variable than shorter period ones. Our investigation confirms many results obtained by Brickhill in his pioneering studies of ZZ Cetis.Comment: 26 pages, including 5 figures, uses aaspp4.sty, submitted to Ap

Nonlinear effects in time-resolved spectra of DAVs

Numerical simulations of light curves of variable DA white dwarfs (ZZ Ceti stars) predict flux amplitudes with surface distributions different from the spherical harmonics of the pulsation mode in deeper layers. In contrast to the results of the perturbation analysis by Goldreich and Wu this is also true for the fundamental period of the flux variation. As a consequence normalized amplitude spectra depend not only on the mode number l but also on pulsation amplitude and inclination. Another new result is that with increasing amplitude of the pressure variation below the convection zone the flux variation at the surface goes through a maximum and then decreases again

Time-resolved optical spectroscopy of the pulsating DA white dwarf HS 0507+0434B: New constraints on mode identification and pulsation properties

We present a detailed analysis of time-resolved optical spectra of the ZZ Ceti white dwarf, HS 0507+0434B. Using the wavelength dependence of observed mode amplitudes, we deduce the spherical degree, l, of the modes, most of which have l=1. The presence of a large number of combination frequencies (linear sums or differences of the real modes) enabled us not only to test theoretical predictions but also to indirectly infer spherical and azimuthal degrees of real modes that had no observed splittings. In addition to the above, we measure line-of-sight velocities from our spectra. We find only marginal evidence for periodic modulation associated with the pulsation modes: at the frequency of the strongest mode in the lightcurve, we measure an amplitude of 2.6+/-1.0 km/s, which has a probability of 2% of being due to chance; for the other modes, we find lower values. Our velocity amplitudes and upper limits are smaller by a factor of two compared to the amplitudes found in ZZ Psc. We find that this is consistent with expectations based on the position of HS 0507+0434B in the instability strip. Combining all the available information from data such as ours is a first step towards constraining atmospheric properties in a convectionally unstable environment from an observational perspective.Comment: 16 pages, 12 figs.; accepted for publication in A&

A new look at the pulsating DB white dwarf GD 358:Line-of-sight velocity measurements and constraints on model atmospheres

We report on our findings of the bright, pulsating, helium atmosphere white dwarf GD 358, based on time-resolved optical spectrophotometry. We identify 5 real pulsation modes and at least 6 combination modes at frequencies consistent with those found in previous observations. The measured Doppler shifts from our spectra show variations with amplitudes of up to 5.5 km/s at the frequencies inferred from the flux variations. We conclude that these are variations in the line-of-sight velocities associated with the pulsational motion. We use the observed flux and velocity amplitudes and phases to test theoretical predictions within the convective driving framework, and compare these with similar observations of the hydrogen atmosphere white dwarf pulsators (DAVs). The wavelength dependence of the fractional pulsation amplitudes (chromatic amplitudes) allows us to conclude that all five real modes share the same spherical degree, most likely, l=1. This is consistent with previous identifications based solely on photometry. We find that a high signal-to-noise mean spectrum on its own is not enough to determine the atmospheric parameters and that there are small but significant discrepancies between the observations and model atmospheres. The source of these remains to be identified. While we infer T_eff=24kK and log g~8.0 from the mean spectrum, the chromatic amplitudes, which are a measure of the derivative of the flux with respect to the temperature, unambiguously favour a higher effective temperature, 27kK, which is more in line with independent determinations from ultra-violet spectra.Comment: 14 pages, 11 figures; accepted for publication in A&

Prospects for Measuring Differential Rotation in White Dwarfs Through Asteroseismology

We examine the potential of asteroseismology for exploring the internal rotation of white dwarf stars. Data from global observing campaigns have revealed a wealth of frequencies, some of which show the signature of rotational splitting. Tools developed for helioseismology to use many solar p-mode frequencies for inversion of the rotation rate with depth are adapted to the case of more limited numbers of modes of low degree. We find that the small number of available modes in white dwarfs, coupled with the similarity between the rotational-splitting kernels of the modes, renders direct inversion unstable. Accordingly, we adopt what we consider to be plausible functional forms for the differential rotation profile; this is sufficiently restrictive to enable us to carry out a useful calibration. We show examples of this technique for PG 1159 stars and pulsating DB white dwarfs. Published frequency splittings for white dwarfs are currently not accurate enough for meaningful inversions; reanalysis of existing data can provide splittings of sufficient accuracy when the frequencies of individual peaks are extracted via least-squares fitting or multipeak decompositions. We find that when mode trapping is evident in the period spacing of g modes, the measured splittings can constrain dOmega/dr.Comment: 26 pages, 20 postscript figures. Accepted for publication in The Astrophysical Journa

The Peculiar Pulsations of PY Vul

The pulsating white dwarf star PY Vul (G~185-32) exhibits pulsation modes with peculiar properties that set it apart from other variable stars in the ZZ Ceti (DAV) class. These peculiarities include a low total pulsation amplitude, a mode with bizarre amplitudes in the ultraviolet, and a mode harmonic that exceeds the amplitude of its fundamental. Here, we present optical, time series spectroscopy of PY Vul acquired with the Keck II LRIS spectrograph. Our analysis has revealed that the mode with unusual UV amplitudes also has distinguishing characteristics in the optical. Comparison of its line profile variations to models suggests that this mode has a spherical degree of four. We show that all the other peculiarities in this star are accounted for by a dominant pulsation mode of l=4, and propose this hypothesis as a solution to the mysteries of PY Vul.Comment: 30 pages, 14 figures, Accepted for publication in Ap

Re-defining the Empirical ZZ Ceti Instability Strip

We use the new ZZ Ceti stars (hydrogen atmosphere white dwarf variables; DAVs) discovered within the Sloan Digital Sky Survey (Mukadam et al. 2004) to re-define the empirical ZZ Ceti instability strip. This is the first time since the discovery of white dwarf variables in 1968 that we have a homogeneous set of spectra acquired using the same instrument on the same telescope, and with consistent data reductions, for a statistically significant sample of ZZ Ceti stars. The homogeneity of the spectra reduces the scatter in the spectroscopic temperatures and we find a narrow instability strip of width ~950K, from 10850--11800K. We question the purity of the DAV instability strip as we find several non-variables within. We present our best fit for the red edge and our constraint for the blue edge of the instability strip, determined using a statistical approach.Comment: 14 pages, 5 pages, ApJ paper, accepte

The pulsating DA white dwarf star EC 14012-1446: results from four epochs of time-resolved photometry

The pulsating DA white dwarfs are the coolest degenerate stars that undergo self-driven oscillations. Understanding their interior structure will help to understand the previous evolution of the star. To this end, we report the analysis of more than 200 h of time-resolved CCD photometry of the pulsating DA white dwarf star EC 14012-1446 acquired during four observing epochs in three different years, including a coordinated three-site campaign. A total of 19 independent frequencies in the star's light variations together with 148 combination signals up to fifth order could be detected. We are unable to obtain the period spacing of the normal modes and therefore a mass estimate of the star, but we infer a fairly short rotation period of 0.61 +/- 0.03 d, assuming the rotationally split modes are l=1. The pulsation modes of the star undergo amplitude and frequency variations, in the sense that modes with higher radial overtone show more pronounced variability and that amplitude changes are always accompanied by frequency variations. Most of the second-order combination frequencies detected have amplitudes that are a function of their parent mode amplitudes, but we found a few cases of possible resonantly excited modes. We point out the complications in the analysis and interpretation of data sets of pulsating white dwarfs that are affected by combination frequencies of the form f_A+f_B-f_C intruding into the frequency range of the independent modes.Comment: 14 pages, 6 figures, 6 tables. MNRAS, in pres