Asteroseismology of White Dwarf Stars

G. Fontaine, P. Brassard

Département de Physique, Université de Montréal
Electronic-mail: fontaine, brassard@astro.umontreal.ca

Abstract:

We briefly describe some of our ongoing efforts at the CFHT to understand better the properties of the nonradially pulsating stars of the ZZ Ceti type. We emphasize the need for high sensitivity and the need for high temporal resolution in the game of white dwarf seismology.

Introduction

  For several years now, we have been pursuing actively an observational program at the CFHT aimed at understanding better the properties of a fascinating class of pulsating stars, the ZZ Ceti white dwarfs. Those are H-atmosphere white dwarfs found to vary in a narrow range of effective temperature, $12,500{\rm K}\lower 0.5ex\hbox{$\buildrel\gt\over\sim\ $}{\rm T_{eff}}\lower 0.5ex\hbox{$\buildrel\gt\over\sim\ $}11,200{\rm K}$, and which show multiperiodic luminosity variations with periods in the range 100-1200 s. Their broadband optical light curves show typical peak-to-peak variations ranging from about 5 millimag to upward of 0.1 mag in the largest amplitude variables. The luminosity variations are caused by nonradial pulsation instabilities of the gravity type (g-modes).

Given that the average apparent visual magnitude of the 26 ZZ Ceti stars currently known is $\langle V\rangle\sim15$, and considering the relatively low amplitudes and short periods of the observed pulsations, measurements of their variability have intrinsically low S/N ratios. Consequently, the vast majority of the observations of ZZ Ceti stars have been carried out through fast photometry (with typical sampling times of 5-10 s) in unfiltered or ``white'' light. The very best such light curves that currently exist in this field have been obtained through our CFHT program using our own specialized instrument, LAPOUNE, the Montréal 3-channel portable photometer. This rugged instrument has proved ideally suited for the task.

In the following, we emphasize the need for high sensitivity and high temporal resolution in the field of white dwarf seismology. Some of the specific contributions of the CFHT are highlighted.

The Need for High Sensitivity

The major difficulty in interpreting the light curves of ZZ Ceti stars in asteroseismological terms resides in the fact that these pulsators tend to show only a few excited modes out of the relatively rich period spectra available to them. It is believed that the layered chemical stratification of ZZ Ceti white dwarfs acts as a filtering mechanism, allowing only a certain number of modes to be excited. Clearly, if an asteroseismological analysis is to be carried out successfully, as many excited pulsation modes as possible must be uncovered, thus providing as many constraints on the models.

 

Figure 1: Comparison of the Fourier amplitude spectrum of the light curve of GD165 obtained in white light (1) at the CFHT during a 6-day run (upper curve) and (2) during the 10-day long 1990 WET campaign (lower curve, plotted upside down).

Figure 1 shows, in its upper half section, the total Fourier amplitude spectrum of the light curve of the low-amplitude ZZ Ceti star GD165 (V = 14.3) we have gathered at the CFHT in May 1995. The light curve was sampled during 27.8 h distributed over 6 consecutive nights, thus corresponding to a duty cycle of 21.9$\%$ and a temporal resolution of 2.2 $\mu$Hz. The Fourier spectrum is illustrated in the 0-10 mHz bandpass (corresponding to pulsation periods $\geq$ 100 s). At least 8 harmonic oscillations are detected in this bandpass.

In contrast, the results of the 1990 Whole Earth Telescope (WET) campaign on GD165 turned out to be quite disappointing, even though that campaign necessitated considerably more efforts than our single CFHT run. Indeed, the luminosity variations of GD165 were monitored during 233 h distributed over 10 consecutive nights by a network of 1 to 2 m telescopes located at 6 different sites in longitude. Despite the substantially larger duty cycle (35$\%$) and improved resolution (1.2 $\mu$Hz), only 3 harmonic oscillations were detected in the light curve of GD165. This can be seen in the lower half of the figure where, plotted upside down, is shown the total Fourier spectrum of the WET campaign. Only the peaks labelled f₁, f₂, and f₃ are clearly present above the noise in the WET Fourier spectrum. This is insufficient for mode identification in GD165. In sharp contrast, a complete, successful, and unambiguous asteroseismological analysis has been possible on the basis of our more sensitive CFHT data.

The Need For High Temporal Resolution

While observations at high sensitivity help uncover more pulsation modes (from which period spacings and ratios can be constructed and compared with theory), observations at high temporal resolution have also proved extremely useful at mode identification because they may reveal rotational splitting in the Fourier transforms of the light curves. Indeed, in a way entirely analogous to the Zeeman effect in atomic physics, slow rotation of a pulsating star lifts the frequency degeneracy associated with spherical symmetry, and a degenerate mode of index $\ell$ is split into its ($2\ell+1$) m components of slightly different frequencies (or periods) in the Fourier domain. Here, the $\ell$ and m indices refer to the indices of the spherical harmonic function $Y^m_\ell$which specifies the angular geometry of a given nonradial pulsation mode in linear pulsation theory. Rotational splitting produces a ``fine structure'' in the Fourier domain and, when detected, provides direct determinations of the pulsation indices $\ell$ and m. Obviously, to detect such splitting requires an observational window longer than the rotation timescale of the pulsator or, equivalently, a sufficient temporal resolution.

A nice example of rotational splitting is provided by the Fourier spectrum of the light curve of GD165 itself. If we expand the frequency scale and zoom on the dominant peak in the Fourier spectrum of our CFHT data (the peak labelled f₁ in Figure 1), we obtain the upper curve shown in Figure 2. This complex structure reflects the diurnal gaps in our light curve sampled over 6 consecutive nights and contains aliases of the true pulsation periods. It turns out, however, that this structure is an almost perfect triplet with components equally spaced in the frequency domain. This is shown in Figure 2, where the results of prewhitening our light curve in succession by 3 harmonic components with periods 120.356 s, 120.391 s, and 120.320 s are shown, from top to bottom. The lower curve in the figure represents the residuals of these operations; note, in particular, how remarkably low is the level of noise.

In an operation sometimes abusively named the computation of the ``window function'', we have constructed an artificial light curve by adding together the 3 harmonic components inferred in our prewhitening calculations (with their amplitudes, phases, and periods (frequencies) obtained by nonlinear least-square fitting to the actual light curve) and by sampling in exactly the same way as during our 6-night run. The Fourier spectrum of that artificial light curve is shown as a dotted curve superimposed on the upper curve in Figure 2. The fact that we can hardly see the dotted curve in the figure (it overlaps almost exactly with the solid curve) is the proof that we have ``reconstructed'' properly the complex f₁ peak. This peak is clearly a triplet with components equally separated by 2.46 $\mu$Hz. The simplest explanation is a $\ell = 1$ mode split by rotation into its $2\ell+1$ components (m = -1, 0, 1). This also leads to a rotation timescale for GD165 of $T = 1/(2.46 {\mu}Hz) = 4.70$ day. We note, for the record, that the WET analysis of the f₁ complex in that star led to the ambiguous result of a mode with either $\ell$ = 1 or 2.

 

Figure 2: Fine structure in the dominant peak of the Fourier transform of the CFHT light curve of GD165. The different curves correspond, from top to bottom, to the total Fourier spectrum, the spectrum of the residual light curve after subtracting the dominant harmonic component with a period of 120.356 s, the spectrum of the residual light curve after subtracting two harmonic components with periods 120.356 s and 120.391 s, and the spectrum of the residual light curve after subtracting three harmonic components with periods 120.356 s, 120.391 s, and 120.320 s. The spectra are arbitrarily shifted vertically for better visualization. The ``window function'' is also shown as the dotted curve that very nearly overlaps with the upper solid curve.