T. S. Metcalfe1 - R. E. Nather2 - T. K. Watson3 - S.-L. Kim4 - B.-G. Park4 - G. Handler5
1 - High Altitude Observatory, National Center for Atmospheric
Research, PO Box 3000, Boulder, CO 80307-3000, USA
2 - Department of Astronomy, University of Texas, Austin, TX 78712, USA
3 - Southwestern University, 1001 E. University Avenue, Georgetown,
TX 78626, USA
4 - Korea Astronomy Observatory, Daejeon 305-348, Korea
5 - Institut für Astronomie, Universität Wien,
Türkenschanzstraße 17, 1180 Wien, Austria
Received 23 December 2004 / Accepted 6 February 2005
Abstract
The helium-atmosphere (DB) white dwarfs are commonly thought to
be the descendants of the hotter PG 1159 stars, which initially have
uniform He/C/O atmospheres. In this evolutionary scenario, diffusion
builds a pure He surface layer which gradually thickens as the star cools.
In the temperature range of the pulsating DB white dwarfs (
K) this transformation is still taking place, allowing
asteroseismic tests of the theory. We have obtained dual-site observations
of the pulsating DB star CBS 114, to complement existing observations of
the slightly cooler star GD 358. We recover the 7 independent pulsation
modes that were previously known, and we discover 4 new ones to provide
additional constraints on the models. We perform objective global fitting
of our updated double-layered envelope models to both sets of
observations, leading to determinations of the envelope masses and pure He
surface layers that qualitatively agree with the expectations of diffusion
theory. These results provide new asteroseismic evidence supporting one of
the central assumptions of spectral evolution theory, linking the DB white
dwarfs to PG 1159 stars.
Key words: stars: evolution - stars: individual: CBS 114 - stars: individual: GD 358 - stars: interiors - stars: oscillations - white dwarfs
The origin and evolution of the helium-atmosphere (DB) white dwarf stars has long been a subject of debate. They comprise roughly 20% of the population of field white dwarfs, with most of the remaining 80% consisting of their hydrogen-atmosphere (DA) cousins. The majority of white dwarfs in both of these spectral classes are thought to arise from the evolution of isolated main-sequence stars with masses that are insufficient to ignite carbon fusion, which ultimately leave their hot carbon/oxygen cores to descend the white dwarf cooling track. The bifurcation into two spectral classes is thought to occur during post-asymptotic-giant-branch (post-AGB) evolution when, in some cases, a very late thermal pulse burns off the residual hydrogen in the envelope, producing a nearly pure helium atmosphere (Iben et al. 1983). Such objects are then supposed to return to the white dwarf cooling track as PG 1159 stars, which are widely believed to be the precursors of most DB white dwarfs.
The primary difficulty with this scenario is the paucity of DB white
dwarfs with temperatures between 45 000 and 30 000 K - a phenomenon
known as the "DB gap'' (Liebert et al. 1986), which persists even in the latest
results from the Sloan Digital Sky Survey (Kleinman et al. 2004). If the PG 1159
stars and DB white dwarfs are linked, then why don't we see the long-lived
intermediate phase? The spectral evolution theory advanced by
Fontaine & Wesemael (1987,1997) postulates that traces of hydrogen (
)
in the envelopes of hot PG 1159 stars float to the surface
and disguise them as DA stars within this temperature interval. A growing
helium convection zone eventually dilutes the hydrogen in the photosphere
until it is no longer detectable at 30 000 K, and the star continues its
evolution as a DB. A relative overabundance of DA stars inside the DB gap
supports this hypothesis (see Kleinman et al. 2004, their Fig. 11). But lower
than predicted surface hydrogen abundances, measured for several DB stars
just below the gap, remain a problem for spectral evolution theory
(Provencal et al. 2000).
If we assume that there is an evolutionary connection between the PG 1159 stars and the cooler DB white dwarfs, we can look to several independent groups who have used time-dependent diffusion calculations to follow the changes in the interior structure of these objects as they cool (Fontaine & Brassard 2002; Dehner & Kawaler 1995; Althaus & Córsico 2004). The hot PG 1159 stars, having recently emerged from the born-again phase, contain envelopes with a nearly uniform mixture of helium (He), carbon (C), and oxygen (O) out to the photosphere (Herwig et al. 1999; Dreizler & Heber 1998). As they cool, the He diffuses upward and gradually accumulates to form a chemically pure surface layer. This leads to a double-layered structure, with the pure He surface layer overlying the remainder of the uniform He/C/O envelope, all above the degenerate C/O core.
A key prediction of the diffusion models is that, for a given stellar mass, the pure He surface layer will steadily grow thicker as the DB star cools. The only available observational tests of this prediction come from asteroseismology - the study of the internal structure of stars through the interpretation of their pulsation periods. Fortunately, non-radial oscillations are naturally excited in otherwise normal DB white dwarf stars as they cool through a narrow range of temperatures near 25 000 K. However, there are only 9 such pulsating DB stars (DBVs) presently known, with effective temperatures between 22 400 and 28 400 K (Beauchamp et al. 1999).
The most thoroughly studied DBV star is GD 358, which has been the target
of three coordinated observing campaigns of the Whole Earth Telescope
(WET; Winget et al. 1994; Kepler et al. 2003; Nather et al. 1990; Vuille et al. 2000). These observations revealed
dozens of independent pulsation periods, including a sequence of 11 dipole
(
;
Kotak et al. 2003) modes of consecutive radial overtone
(k; Bradley & Winget 1994), each providing a complementary probe of the stellar
interior. These data formed the basis for the only asteroseismic tests of
the diffusion models to date (Fontaine & Brassard 2002; Dehner & Kawaler 1995). In the most extensive of
these tests, Fontaine & Brassard (2002) produced a targeted grid of double-layered
evolutionary models to try to fit the periods of GD 358 in detail. They
found evidence of two composition transition zones near outer mass
fractions of 10-5.8 and
10-3.0 M*, which they interpreted as
the base of the pure He surface layer and the still-uniform He/C/O
envelope, respectively. However, Montgomery et al. (2003) showed that there is a
potential ambiguity in the inferred locations of these internal
structures, due to an inherent core-envelope symmetry for models of DBV
stars. As a consequence, real structure at specific locations in the core
could be mis-interpreted as structure at certain symmetric points in the
envelope, and vice versa.
The best way of circumventing this potential ambiguity is to fit additional stars using the same models, to determine whether the independent observations are consistent with the same underlying physical processes. Such model-fitting attempts for DBV stars other than GD 358 have been hampered by the lack of suitable observational data. Handler et al. (2002, hereafter HMW) found a total of 7 independent pulsation periods in the slightly hotter DBV star CBS 114, and Metcalfe et al. (2003) attempted to fit these observations using double-layered envelope models that were also applied to GD 358. They found qualitative agreement with the predictions of diffusion theory (thinner surface He layers for hotter white dwarfs of comparable mass) when using models that also included an adjustable C/O profile in the core. However, the large number of free parameters in their model, relative to the number of observed periods in CBS 114, made these results somewhat uncertain.
Motivated by the need for additional pulsation periods to constrain the
model-fitting of CBS 114, we organized a dual-site campaign using 2 m
class telescopes in February 2004. Our primary goals were to confirm the
periods of the 7 pulsation modes found by HMW, and to discover additional
modes below the 5 milli-mag noise limit of the previous
observations (obtained over 3 weeks from the 0.75 m telescope at SAAO).
This campaign was awarded 7 nights (2004 Feb. 19-25) on the 1.8 m
telescope at Bohyunsan Optical Astronomy Observatory (BOAO) in Korea, and
7 nights (2004 Feb. 21-27) on the 2.1 m telescope at McDonald Observatory
(McD) in Texas.
At BOAO we used a thinned SITe 2K CCD camera with no filter. To minimize
the readout time, we binned the pixels
for an effective
resolution of 0.68 arcsec pixel-1, and we used only the central
portion of the detector for a field of view
4 arcmin on a side.
On the first night we used an exposure time of 8 s for a cycle time
of 20 s, but on subsequent nights we increased the exposure time to
20.5 s for a cycle time of 30 s. This system has a gain of
1.8
and a readout noise of 7
rms.
The observations at McD were obtained using the Argos time-series CCD
photometer (Nather & Mukadam 2004) with no filter. This instrument uses a
back-illuminated
frame-transfer CCD mounted at the f/3.9
prime focus, for a field of view 2.8 arcmin on a side and a resolution
of 0.33 arcsec pixel-1. We used an exposure time of 10 s,
which was also the cycle time - the transfer of each image to the masked
on-chip buffer requires only 310
s, and the 280 ms readout takes
place during the next exposure. This system has a gain of 2
and a readout noise of 8
rms. The observations collected
for this dual-site campaign are summarized in Table 1.
Table 1: Journal of observations.
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Figure 1:
Reduced light curves for CBS 114 during our dual-site
campaign in 2004, showing fractional intensity variations of
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The raw CCD frames were analyzed on-the-fly during the observing campaign to monitor the data quality, but final reductions were done uniformly using the IRAF high-speed photometry (HSP) package developed by Antonio Kanaan (personal communication). After correcting each image for bias level and flat field effects, we extracted aperture photometry for CBS 114 and three nearby comparison stars. The aperture was typically set to have a radius between 2 and 4 arcsec with a surrounding annulus to determine the local sky level, but the apertures were adjusted for each data set to maximize the S/N of the resulting light curves. The time-series measurements were then imported into XQED (Riddle 2003), the photometry analysis software for the WET. For each light curve, we subtracted the local sky contribution and performed a point-by-point division of the target star by the average of the three comparison stars. We then corrected for any low-frequency variations (e.g., caused by differential color extinction) by fitting and removing a second-order polynomial. Finally, we subtracted the mean light level and applied barycentric timing corrections to the exposure mid-times. The resulting light curves for CBS 114 are shown in Fig. 1.
The Fourier Transforms (FTs) of these light curves, for each observatory
and for the combined data set, are shown in Fig. 2. The
corresponding window functions - the FT of a single sinusoid sampled in
the same way as the data - are shown to the same scale in each panel. The
distributed sampling of the BOAO data yields better effective frequency
resolution, while the slightly larger aperture and higher cadence of the
McD data leads to a higher S/N. The full data set combines all of these
strengths, and reduces the amplitude of the 1 d aliases (separated from
the main frequency by
Hz).
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Figure 2: Fourier Transforms (FTs) of the light curves shown in Fig. 1 over the range of independent frequencies exhibited by CBS 114. The three panels show the FTs for the data from BOAO ( top), McD ( middle), and for the combined data set ( bottom), along with the corresponding window functions (inset) and significance cuts (dotted line). |
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To determine the significance level of peaks in each of these FTs, we
followed the analysis of Kepler (1993). Since power is the square of the
amplitude, the average power out to the Nyquist frequency is a measure of
the mean-square deviation from zero amplitude, which is dominated by the
noise since the signal is confined to a relatively narrow range of
frequencies. The square-root of this value is the root-mean-square
amplitude of the noise, which is analogous to the standard deviation
(
). By adopting a significance cut at
,
we ensure that only
0.3% of the peaks with this amplitude
will not be real signals (assuming the noise is Gaussian). Peaks with
amplitudes much above the significance cut, and those detected in multiple
independent data sets, have an even lower chance of being due to noise.
The
significance level is indicated by the dotted
line in each panel of Fig. 2, which is at 3.52, 2.33, and
2.26 mma for the BOAO, McD, and combined data sets respectively. For the
range of frequencies shown, we expect only
4 noise peaks to appear
above these cuts.
We performed a frequency analysis of the combined data set using the
program PERIOD98 (Sperl 1998). We started by iteratively fitting and
subtracting the frequencies with the largest amplitudes, refining our
estimates of the period, amplitude, and phase for each signal using
simultaneous multiple sine-wave fitting. This allowed us to recover the 7 independent pulsation frequencies found by HMW, with a few important
differences. We resolved both of the two largest amplitude modes near 1520
and 1610 Hz into triplets, though the splitting is close to the
1 d aliases (perhaps explaining why the earlier single-site data failed
to detect them, despite a longer time baseline). We also detected an
additional component in the 1520
Hz multiplet at 1514.1
Hz,
which could be caused by amplitude and/or frequency modulation during the
observations or might be caused by higher-order combination frequencies,
comparable to the "
'' mode seen in GD 358 in 1994
(Vuille et al. 2000). Our frequencies for the modes near 1830 and 1960
Hz
are both systematically lower than in HMW, but by less than would be
expected from 1 d alias problems. This might be the result of an
underlying multiplet structure with intrinsic amplitude modulation among
the various components (e.g., see Robinson et al. 1976, their Fig. 2). We also
found that our mode near 2300
Hz was the +1 d alias of the
frequency found by HMW - a possibility that they could not rule out.
Finally, we resolved the mode near 2510
Hz, as well as a newly
discovered mode near 2090
Hz, into doublets with a splitting
comparable to the frequency shifts seen in the 1830 and 1960
Hz
modes.
In addition to recovering the 7 pulsation modes found by HMW, we also
discovered 4 new ones. The largest is the doublet near 2090 Hz, which
nicely fills the gap in the sequence of consecutive radial overtones
identified by HMW. A marginal peak near this frequency also appeared in
the discovery data of Winget & Claver (1989,1988), but it was below the detection
threshold of HMW. The three other new modes appear at frequencies near
1190, 1250, and 1380
Hz. The period spacing of these peaks suggests
that they are additional independent pulsation modes with higher radial
overtones than those previously seen. Two additional frequencies had
amplitudes near our significance threshold. These could tentatively be
identified as an even higher radial overtone near 1140
Hz, and a
possible rotationally split component of the mode near 1720
Hz. These
marginal detections prompted us to compute weighted FTs, following the
suggestion of Handler (2003). The results were nearly identical to the
unweighted case, and the marginal nature of these two frequencies was
unchanged.
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Figure 3: The Fourier Transform of our combined data set for CBS 114 before and after removing the 28 sinusoids listed in Table 2. The independent and linear combination frequencies are indicated by the solid and open triangles respectively. The frequencies of our optimal model matched to the observed modes are shown as solid squares, while open squares denote model frequencies with no detected counterpart. |
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Aside from the independent pulsation modes observed in CBS 114, we also detected 11 significant linear combination frequencies. Fundamentally, these frequencies arise in our analysis because the Fourier Transform decomposes the signal into sinusoidal components. A casual inspection of the light curves shown in Fig. 1 reveals that they are not perfectly sinusoidal. Instead, they tend to have sharp peaks and shallow troughs, which is commonly interpreted to be associated with a nonlinear response of the surface convection zone to the pulsations (e.g., see Ising & Koester 2001; Montgomery 2005; Wu 2001; Brickhill 1992). The earlier observations by HMW revealed two such frequencies, both of which are among those we detected. The combination frequencies identified in our data involve the six modes or multiplets with the largest amplitudes.
Our final simultaneous fit of 28 sinusoids to the data using PERIOD98
contains a total of 17 independent modes and 11 linear combination
frequencies, which are all listed in Table 2. In Fig. 3
we show the FT of our combined data set before and after the removal of
these 28 frequencies (indicated by solid and open triangles for the
independent and combination frequencies, respectively). The marginal
signals near 1140, 1514, and 1720 Hz are apparent in the residuals.
Table 2: The multifrequency solution for CBS 114 from PERIOD98. Also shown is the mode identification from our optimal model.
Table 3: Optimal model parameters for CBS 114 and GD 358.
To investigate whether these new observations can fit comfortably with the
predictions of diffusion theory, we repeated the global model-fitting
procedure described by Metcalfe et al. (2003). We used an updated version of the
code that incorporates the OPAL radiative opacities (Iglesias & Rogers 1996) rather
than the older LAO data (Huebner et al. 1977), which are known to produce
systematic errors in the derived temperatures (Fontaine & Brassard 1994). The fitting
procedure uses a parallel genetic algorithm (Metcalfe & Charbonneau 2003) to minimize the
root-mean-square residuals between the observed and calculated periods
(
)
for models with effective temperatures (
)
between 20 000 and 30 000 K, and stellar masses (M*) between 0.45 and
0.95
.
It allows the base of the uniform He/C envelope to be
located at an outer mass fraction
between -2.0and -4.0. The base of the pure He surface layer can assume values of
between -5.0 and -7.0. To ensure a smooth
transition between the self-consistent cores and the static envelopes in
our models, we fixed the core composition to be pure C. We cannot
presently include oxygen in the envelopes of our models.
We applied this fitting procedure to our new observations of CBS 114 and to the WET observations of GD 358 from 1990 (Winget et al. 1994) for comparison. For CBS 114 we assumed that each single mode had an azimuthal order m=0(see Metcalfe 2003a), and we used the central frequency for each of the observed triplets. For the two doublets, we used the average of the two observed frequencies. The resulting sets of optimal model parameters for these two stars are shown in Table 3 with internal errors for each parameter set by the resolution of our search. The theoretical pulsation frequencies for CBS 114 are shown in Fig. 3 as solid squares where they match an observed frequency, and the corresponding mode identifications are listed in Table 2. For reference, the open squares in Fig. 3 show additional model frequencies with no clearly detected counterpart.
The spectroscopic temperature determinations of Beauchamp et al. (1999) suggest that
CBS 114 could be as much as 1500 K hotter than GD 358. Our model-fitting
results suggest a larger temperature difference, placing CBS 114 at
K hotter than GD 358. On an absolute scale, the temperature
and mass of our fit to CBS 114 are consistent with the spectroscopic
values, assuming that any traces of H in the atmosphere are small. Our fit
to GD 358 is consistent with the spectroscopic mass, but our temperature
determination is slightly low. Our value is also low compared to a more
recent and precise determination using UV spectroscopy from the Hubble
Space Telescope (
K; Castanheira et al. 2005).
Recent time-dependent diffusion calculations by Althaus & Córsico (2004) suggest that
for a given stellar mass and envelope thickness, the pure He surface layer
should thicken by 0.15 dex between the spectroscopic temperatures of
CBS 114 and GD 358. We find a marginal thickening of only
dex. However, diffusion models with more massive envelopes
lead to thicker surface He layers more quickly as the star cools, since
there is a larger reservoir of He to draw from. Our fit to CBS 114 has a
slightly larger total envelope mass, which would tend to diminish the
expected difference between the thickness of the surface He layers for the
two stars to
0.12 dex. The limited grid of published diffusion
models currently makes more quantitative comparisons impossible.
Our new dual-site observations of the pulsating DB white dwarf CBS 114 have finally revealed enough independent modes to bring the model-fitting for this star to the same level of reliability as for GD 358. We used our updated double-layered envelope models to match the new observations of CBS 114, and the archival observations of GD 358 - fitting 11 pulsation modes in each star. The resulting pair of optimal models have pure He surface layers and total envelope masses that qualitatively agree with our expectations from diffusion theory. This provides new asteroseismic evidence supporting one of the central assumptions of spectral evolution theory, linking the DB white dwarfs to PG 1159 stars.
In addition to the new observational constraints for CBS 114, we improved our analysis by incorporating the OPAL radiative opacities into the models. Our results include higher temperatures and lower masses than the previous fits by Metcalfe et al. (2003), leading to better agreement with the spectroscopic estimates for both stars. The optimal locations for the base of the pure He surface layer and the uniform He/C envelope are systematically deeper for GD 358 relative to the previous fit. For CBS 114 the fit is now consistent with the envelope mass expected from simulations of carbon dredge-up in DQ stars (see Fontaine & Brassard 2002; Pelletier et al. 1986), and with the He layer mass expected from time-dependent diffusion calculations (Althaus & Córsico 2004).
These results have now met the challenge posed by Metcalfe (2003b), to fit the
observations of more than one DBV star and demonstrate that they are both
consistent with the same underlying physical process. In that paper, the
relevant process was the nuclear burning history during the red giant
phase leading to a C/O ratio in the core that could help to constrain the
rate of the
reaction. In the
present case, the relevant process is diffusion in the white dwarf
envelopes leading to a gradually thickening pure He surface layer as the
star cools. In effect, each of these studies has concentrated on an
isolated structure in the stellar interior, ignoring the contribution of
other possible features. Regardless of whether the asteroseismic signal
originates from the core or the envelope, it is now clear that the
relevant structure (C/O profile or He surface layer) is very nearly the
same in the two stars. The better absolute quality of the fits for
GD 358 and CBS 114 in Metcalfe (2003b) suggest that the C/O profile is the
more important structure from an asteroseismic standpoint. Indeed, this is
consistent with the finding by Montgomery et al. (2003) that the pure He surface layer
has a smaller effect on the pulsation modes than the uniform He/C
envelope, which is itself secondary in importance to the C/O profile in
the core.
Of course, there are sound physical reasons to expect complicated structures in both the core and the envelope of real white dwarfs. We have demonstrated that analyses of two completely independent sets of pulsation data with two physically distinct classes of models can both lead to reasonably self-consistent interpretations of the corresponding internal structures. Our challenge is now to synthesize hybrid models that can fit the core and envelope structure simultaneously. It is possible that such models will exhaust the information content of the 11 pulsation modes we have observed in these two stars. If so, our only recourse is to detect additional modes in GD 358 and CBS 114 and to discover additional DBV stars with rich pulsation spectra. Both options will probably require multi-site campaigns using 4 m class telescopes since the existing observations are already the result of campaigns using 1-2 m telescopes, and the newly discovered DBV stars from the Sloan Digital Sky Survey are significantly fainter (Nitta et al. 2005). Even so, the future continues to look bright for white dwarf asteroseismology.
Acknowledgements
We would like to thank Steve Kawaler and Mike Montgomery for their comments and suggestions. This work was partially supported by the Smithsonian Institution through a CfA Postdoctoral Fellowship, and by the National Science Foundation through an Astronomy & Astrophysics Postdoctoral Fellowship under award AST-0401441. Computational resources were provided by White Dwarf Research Corporation through grants from the Fund for Astrophysical Research and from the American Astronomical Society.