Seismology of an Ensemble of ZZ Ceti Stars
aa r X i v : . [ a s t r o - ph . S R ] N ov Seismology of an Ensemble of ZZ Ceti Stars
J. C. Clemens , P. C. O’Brien , Bart. H. Dunlap , J. J. Hermes , University of North Carolina at Chapel Hill, Chapel Hill, NC, USA; [email protected] Hubble Fellow
Abstract.
We combine all the reliably-measured eigenperiods for hot, short-periodZZ Ceti stars onto one diagram and show that it has the features expected from evo-lutionary and pulsation theory. To make a more detailed comparison with theory weconcentrate on a subset of 16 stars for which rotational splitting or other evidence givesclues to the spherical harmonic index ( ℓ ) of the modes. The suspected ℓ = k =
1. Introduction
Long after the recognition that ZZ Ceti pulsations might be used to understand theinternal structures of white dwarfs, these stars remain enigmatic. For most individualZZ Ceti stars with stable, resolved pulsation spectra, the number of periods is smalland the radial order ( k ) and spherical harmonic degree ( ℓ ) unknown. As a result, anycomparison between the eigenmodes of pulsational models and observed periods isunderconstrained. Thus, even when a few observed periods in a star are well-matchedby a few eigenperiods from a model, we cannot be certain that this match implies themodel has a structure resembling that of the actual star. This shortcoming in ZZ Cetiseismology has been addressed in a variety of ways: by using fully evolutionary modelswith di ff usion to reduce the number of free parameters that must be fitted (Romero et al.2012); by improving the spectroscopic determinations of mass and gravity (Gianninaset al. 2011; Tremblay et al. 2013); and by long-timebase or high-sensitivity observingcampaigns to detect more modes and resolve them into multiplets that can constrain ℓ if not k (Giammichele et al. 2015).Clemens (1993, 1994) noticed that the short period modes from hot ZZ Ceti starscompiled into one diagram defined a pattern with distinct period groupings separatedby gaps, and suggested that studying the structure in this ensemble diagram might helpto resolve the degeneracy in the fitting process. One reason for this is that geometriccancellation should result in the largest modes being mostly ℓ =
1, so uncertaintiesarising from individual mode identification are reduced in the ensemble diagram. An-1other is that the mean periods of the groups formed by the largest modes depend on theaverage mass and temperature of the ensemble, which are more robustly known fromspectroscopy than those for any star alone, reducing the number of free parameters.For stars with a narrow distribution in mass, as is known to be true for whitedwarfs, a pattern like that found by Clemens (1994) is expected from pulsation theory.The left hand panel of Figure 1, reproduced from Romero et al. (2012) illustrates thispoint. The figure shows low-order ℓ = M H ), which is treated as a free parameter, the periods of the low order modes willaccumulate near one of the solid yellow horizontal lines in the diagram. Althoughthese theoretical periods will be di ff erent for models with mass not equal to the value inthis plot (0.593 M ⊙ ), the narrowness of the observed field white dwarf mass distribution(Tremblay et al. 2016) allows the ℓ = Figure 1. The left-hand panel, reproduced from Romero et al. (2012), shows asequence of pulsation periods derived from their evolutionary models with M = . M ⊙ . Given random choices for M H , ℓ = M H will give periods nearthe dotted green lines. The histogram on the right shows the distribution of modeperiods for all 239 modes measured in 75 hot ZZ Ceti stars.
2. Updating the Hot ZZ Ceti Period Ensemble
Clemens (1994) collected the observed periods for 11 hot ZZ Ceti stars and foundsuggestions of an emergent pattern, but small-number statistics and lack of certain ℓ identification limited the application of the ensemble technique. Now, 22 years later, wehave repeated this exercise for 75 known hot ZZ Cetis, most of them still without securemode identification. As the top panel of Figure 2 shows, the evidence for groupingis still apparent. The gap in periods between 220 and 250 s is particularly striking,and aligns with the region between predicted mode periods in the models of Figure1. The existence of this gap and other features argues against allowing the hydrogenlayer mass to vary freely, because ℓ = k = M H = − M ⋆ would populate the gap, conflicting with the observations. Likewise, theappearance of a grouping of periods in the 100-120 s range suggests that a significantfraction of the stars have the thickest possible M H allowed by nuclear burning, which isaround 10 − M ⋆ for this model. To illustrate these points we have converted the perioddistribution in the top panel of Figure 2 into a histogram and added it to the right ofthe Romero diagram in Figure 1. This ensemble of observed periods still su ff ers fromthe drawback that we do not know which modes are ℓ =
1, so the observed distributionlikely combines modes of di ff erent ℓ , while the theoretical diagram has only periodsfor ℓ = m of the modes,but for periods below 400 s and typical rotation periods of 1 d, the ambiguities from m identification are typically less than 1 s. Figure 2. The top panel shows all reliably-measured eigenperiods for the 75known short-period (hot) ZZ Cetis. The bottom panel are the likely ℓ = ℓ identifi-cation. These appear to form a sequence of continuous radial overtone, k, from 1 to4. All combination frequencies have been ignored, and multiplets are plotted once atthe central period of the multiplet (assumed m = Fortunately, it is now possible to improve this diagram by restricting ourselves toa subset of stars for which we have independent knowledge of the sperical degree ofthe eigenperiods observed. The number of stars for which this is possible has recentlyincreased owing to discovery and short cadence observations of new ZZ Ceti stars bythe
Kepler and K2 missions. These new ZZ Cetis will be the subject of a series offorthcoming papers by Hermes et al. (2016). Figure 3 shows the Fourier Transformof K2 data accumulated over 78.7 days at 60 s cadence for one such star. In addition,improved ground-based observations of GD 165 and R 548 (ZZ Ceti) (Giammichele etal. 2015), GD 66 and G238-53 (Mullally et al. 2008) , LP133-144 (Bognár et al. 2016),and other well-known stars have added to our knowledge of their multiplet structure.The subset of hot ZZ Ceti stars with ℓ identification we consider in this paper includeseight new stars from Kepler and eight previously known examples. The lower panel ofFigure 2 shows the periods of those modes in these stars which we identify as ℓ = Figure 3. Typical multiplet structure from
Kepler short cadence observations of ahot ZZ Ceti star. We identify the modes at 121.6, 194.7, and 253.6 s in this star as ℓ =
1, and discard 277.5 s because it cannot be securely identified.
3. Monte Carlo Simulations using the Romero et.al Period Grid
In order to compare these measured ℓ = k from 1 to 4. Tocompare these periods to the seismological prediction of evolutionary models, we usedthe published grid of ℓ = k foreach star that match the ones observed in its period spectrum.We have plotted these simulations for a number of assumptions in the four lowerpanels of Figure 4. The second and third panels show the simulated periods drawnfrom the Romero models under the assumption that all stars have the canonically thick M H . The temperatures and gravities used in the second panel are from the 1D mod-els of Gianninas et al. (2011) while the third panel shows the values corrected for thethree-dimensional dependence of convection (Tremblay et al. 2013). In their overallappearance, these two simulations most resemble the data. They have a group at 100-120 s, and clean gaps between the lower k modes. However, the periods of the k > k inthe models is systematically too small. We experimented with uniformly decreasingthe masses in the models to spread out the overtones, and found that a roughly 10% de-crease would be required, and this would still not improve the match to the k = M He . We have used quasi-evolutionarymodels to experiment with thinner layers and this approach shows promise. We willpursue this approach further in a forthcoming paper.Finally, we have also calculated simulations in which M H is allowed to vary fromthe canonical limit. These do not match the data as well inasmuch as they lack a cleargroup at 100-120 seconds, and the gaps between consecutive k are not clean. Thiscomports with the expectations we developed based on Figure 1; varying M H allowsstars to have periods in between the yellow accumulation lines on the diagram and doesnot select the correct fraction of stars with modes near 100 s. The fourth panel showsmodels in which M H was treated as a random variable, while the bottom panel uses thedistribution of M H found by (Romero et al. 2012) using individual seismological fits toan ensemble of stars with mostly unknown values of ℓ and k .
4. Conclusions
Examining the ensemble of hot ZZ Ceti pulsators with known ℓ shows a pattern ofconsecutive groups that we have identified with the lowest k non-radial g -mode pulsa-tions. Analyzing this pattern o ff ers a promising way to constrain seismological modelparameters like M H and M He . The 16 stars in our sample suggest that most hot ZZCetis have M H values at or near the canonical limit and that their helium layers arethinner than those calculated by evolutionary models. Detailed model fits to individualstars informed by these results should allow us to extract believable asteroseismologicalmeasurements of ZZ Ceti interior properties. Acknowledgments.
The authors acknowledge support from the National ScienceFoundation under award AST-1413001. Support for this work was also provided byNASA through Hubble Fellowship grant // home.gna.org / veusz / . References
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