# A New Analysis of the Two Classical ZZ Ceti White Dwarfs GD 165 and Ross 548. II. Seismic Modeling

N. Giammichele, G. Fontaine, P. Brassard, S. Charpinet

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N. Giammichele, G. Fontaine, P. Brassard, S. Charpinet. A New Analysis of the Two Classical ZZ Ceti White Dwarfs GD 165 and Ross 548. II. Seismic Modeling. The Astrophysical Journal Supplement Series, 2016, 223, 10.3847/0067-0049/223/1/10. insu-03670538

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# A NEW ANALYSIS OF THE TWO CLASSICAL ZZ CETI WHITE DWARFS GD 165 AND ROSS 548.
## II. SEISMIC MODELING

N. Giammichele¹, G. Fontaine¹, P. Brassard¹, and S. Charpinet²,³

¹Département de Physique, Université de Montréal, Montréal, QC H3C 3J7, Canada

²Université de Toulouse, UPS-OMP, IRAP, Toulouse F-31400, France

³CNRS, IRAP, 14 avenue Edouard Belin, F-31400 Toulouse, France

Received 2015 December 25; accepted 2016 January 28; published 2016 March 25

## ABSTRACT

We present the second of a two-part seismic analysis of the bright, hot ZZ Ceti stars GD 165 and Ross 548. In this second part, we report the results of detailed searches in parameter space for identifying an optimal model for each star that can account well for the observed periods, while being consistent with the spectroscopic constraints derived in our first paper. We find optimal models for each target that reproduce the six observed periods well within ~0.3% on the average. We also find that there is a sensitivity on the core composition for Ross 548, while there is practically none for GD 165. Our optimal model of Ross 548, with its thin envelope, indeed shows weight functions for some confined modes that extend relatively deep into the interior, thus explaining the sensitivity of the period spectrum on the core composition in that star. In contrast, our optimal seismic model of its spectroscopic sibling, GD 165 with its thick envelope, does not trap/confine modes very efficiently, and we find weight functions for all six observed modes that do not extend into the deep core, hence accounting for the lack of sensitivity in that case. Furthermore, we exploit after the fact the observed multiplet structure that we ascribe to rotation. We are able to map the rotation profile in GD 165 (Ross 548) over the outermost ~20% (~5%) of its radius, and we find that the profile is consistent with solid-body rotation.

Key words: stars: interiors – stars: oscillations – white dwarfs

## 1. INTRODUCTION

In the first part of this short series of two papers dedicated to a detailed seismic analysis of the two bright, hot ZZ Ceti stars GD 165 and Ross 548, Giammichele et al. (2015a, hereafter Paper I) presented the results of frequency extraction exercises based on time-series data sets of exceptional quality. A total of up to 13 pulsation modes, regrouped into six main frequency multiplets, were uncovered in GD 165. Likewise, up to 11 modes, also regrouped into six multiplets, were isolated in Ross 548. The most likely cause of the multiplet structure is rotational splitting, and we will explore that avenue in some detail below. For the purpose of comparing the observed pulsation periods with those of spherically symmetric (non-rotating) stellar models, each multiplet must be considered as a single-degenerate mode, so there are, in fact, six “main” modes useful for that purpose for each star. We note that Paper I has revealed the existence of two more of these main modes than previously known for GD 165, and an additional one for Ross 548, thus providing welcome and significant additional constraints in the seismic modeling of these two pulsators. Further independent constraints were also provided in Paper I with the release of updated estimates of the time-averaged atmospheric parameters of each star in light of recent developments on the front of atmospheric modeling for DA white dwarfs. Given the proven and widely acknowledged reliability of the spectroscopic method developed by our colleague Pierre Bergeron at Université de Montréal (Bergeron et al. 1992), we view these as fundamental additional constraints. Indeed, any seismic model leading to estimates of the effective temperature and the surface gravity that would be in serious conflict with the values obtained independently by high-quality spectroscopic modeling should be disqualified.

We selected GD 165 and Ross 548 on the basis of the simplicity of their pulsation properties, a general characteristic

of stars—such as those two—located near the blue edge of the ZZ Ceti instability strip in the spectroscopic H-R diagram. Our selection finds further justification in the fact that these are known to be near spectroscopic twins, yet they show significantly different dominant pulsation periods. This presents an interesting challenge from a seismic point of view. In our quest to revisit the problem of the seismic modeling of ZZ Ceti stars using the most recent techniques, we thought it appropriate to focus our efforts first on those two “simple” pulsators and solve this puzzle if possible. The current work is also to be considered as a benchmark for future studies of more complicated ZZ Ceti stars, notably GD 1212, an object near the red edge that has been observed in short-cadence mode with Kepler-2. Hence, we seek credible seismic models for both GD 165 and Ross 548, able to account for the six observed periods<fnref n="4" /> at a level of accuracy comparable to the best currently achieved in the field of asteroseismology. To our knowledge, the best quantitative results on that front have been obtained from hot subdwarf seismology, leading to typical average dispersions in the range 0.15%–0.30% in fitted periods (see, e.g., Charpinet et al. 2008, 2011; Van Grootel et al. 2010a, 2010b).

The credibility of a seismic model rests not only on its ability to reproduce simultaneously all of the observed periods at the highest possible accuracy but also on the statistical significance of the solution. Typically, in seismic searches for an optimal model in parameter space, a merit function that measures the quality of the fit between the observed and theoretical periods for a given model is optimized. To establish the statistical validity of an optimal solution, it is then essential that the vicinity of that solution in parameter space be thoroughly explored. Furthermore, a credible seismic model should be compatible with the expectations of pulsation theory.

<footnote n="4">⁴ Those are labeled $f_1$ through $f_6$ in Table 3 of Paper I for GD 165 and in Table 4 for Ross 548.</footnote>
In particular, for pulsating ZZ Ceti white dwarfs (see, e.g., Figure 7(c) of Van Grootel et al. 2012), linear nonadiabatic theory predicts that a range of modes with consecutive values of the radial order k should be excited (for a given value of the degree l). In other words, the mode identification inferred from a seismic model should be in tune with such expectations, at least for stars located near the blue edge of the instability strip as is the case here. In addition, a credible seismic model must be compatible with external independent constraints such as estimates of the atmospheric parameters derived from time-averaged spectroscopy (as mentioned above), estimates of distances provided by parallax measurements or spectroscopy, and previous mode identification of the degree index l obtained through multicolor measurements or time-dependent spectroscopy, if available. In the following, we apply these principles in our search for optimal seismic models for GD 165 and Ross 548.

## 2. METHOD

### 2.1. Search for Optimal Seismic Models in Parameter Space

Our method is based on the forward modeling approach, which consists in finding the best possible match between a set of oscillation periods detected in a given star and the periods computed from stellar models. Our original strategy along this avenue has been presented by Brassard et al. (2001) in the context of pulsating hot subdwarf stars. This strategy has since been considerably improved as discussed, in particular, by Charpinet et al. (2005) and Van Grootel et al. (2013a). The most recent discussion of our continuing quest for an objective approach to quantitative asteroseismology, with application to white dwarf stars in particular, has been presented by Charpinet et al. (2015).

Our technique relies on a double-optimization scheme that, first, best fits the six—in the case of both GD 165 and Ross 548—observed periods (denoted $P_{obs}$ in what follows) with six periods (denoted $P_{th}$) belonging to the spectrum of a given model (note that the match might be quite poor if the model has properties quite different from those of the real star) and, second, searches for the best-matching model in parameter space. No a priori mode identification is imposed on the observed periods, except that they have to belong to degree index $l=1$ or $l=2$ in keeping with what is known about identified modes in white dwarfs (see, in this regard, Section 7 of Fontaine & Brassard 2008). For a given model, theoretical periods are computed in a window that encompasses the range of observed periods for modes with $l=1$ and 2. For the present purposes, sufficiently accurate periods may be computed in the adiabatic approximation, so this is adopted as a major time-saving measure. The quality of the match between the observed and computed periods is measured quantitatively with a merit function defined by

$$S^2 = \sum_{i=1}^{N_{\text{obs}}} (P_{\text{obs}}^i - P_{\text{th}}^i)^2, \quad (1)$$

where $N_{obs} = 6$ in the present case. The goal of the exercise is to find, if possible, the minimum of $S^2$ in parameter space and, hopefully, a minimum that identifies a good and credible optimal model. To objectively and automatically carry out this search for the optimal model in parameter space requires considerable computing resources. Currently, a typical exercise

of this kind for white dwarfs necessitates the computations of tens of thousand models and of their period spectra. In the present application, we have used the optimization package LUCY developed by one of us (S.C.) in Toulouse in order to handle this demanding numerical problem. This code is a massively parallel hybrid genetic algorithm capable of multi-modal optimization and offers several advantages. It does not rely on model grids, as parameter space is explored essentially continuously, while making the economy of millions of “useless” models. This leads to much higher computational efficiency and, in addition, the approach avoids grid resolution limitations. The code is also particularly robust at finding the best-fit solution, as well as all potentially interesting secondary optima. Some illustrations of the capabilities of the code, as well as additional details, can be found in Charpinet et al. (2015).

### 2.2. Need for Parameterized Models

When some of us started investigating the application of the forward asteroseismological method to pulsating hot subdwarf and white dwarf stars more than a decade ago (see, e.g., Brassard et al. 2001; Fontaine & Brassard 2002), we realized at the outset that the most practical way to do that would be through the use of parameterized static models. It has been our assertion since that parameterized models, as compared to full evolutionary models, provide the best, if not the only, way to thoroughly search in parameter space for an optimal seismic model. In principle, state-of-the-art evolutionary models provide the best physical description of stars. However, they suffer from two important drawbacks in the context of searching for the best seismic model.

The first of these shortcomings is their lack of flexibility. It is indeed nearly impossible, even with large computer clusters, to cover finely all of the relevant domains of parameter space with evolutionary sequences; it would take too much computing time. The practical consequence is that evolutionary sequences, necessarily limited in number, may actually miss the “correct” region of parameter space where the best seismic model resides. In that context, it is virtually impossible to ensure that a potentially acceptable seismic model is the best one available from the traditional evolutionary approach.

The second drawback is that, by construction, it is implicitly assumed that the constitutive physics used in the construction of state-of-the-art evolutionary models is “perfect.” Of course, this cannot be the case, especially in domains of the phase diagram corresponding to extreme physical conditions, such as those, for example, encountered in white dwarf stars. It may very well be the case in these circumstances that the input constitutive physics is not realistic enough and, consequently, that the observed pulsation periods cannot be accounted for at an acceptable level of accuracy on the basis of such evolutionary models.

In contrast, parameterized static models, even though they are cruder structures than evolutionary ones, offer maximum flexibility and speed for searching in parameter space. They use the exact same constitutive physics, but can compensate partly for uncertainties and are particularly useful at that level for identifying the part of the input physics that needs improvement. They thus are most useful for providing feedback on the constitutive physics itself. Ultimately, they must of course be validated by improved evolutionary models. Successful
examples of this type of feedback from parameterized seismology have been discussed recently by Charpinet et al. (2013). We further note that the method has proven itself in a rather remarkable way in the case of the pulsating hot B subdwarf component of the close eclipsing binary system PG 1336–018, which has turned out to be a true “Rosetta Stone” of hot subdwarf seismology (Charpinet et al. 2008; Van Grootel et al. 2013a).

In the case of white dwarf stars, extra care should be taken when using evolutionary models in a seismological context, particularly the more detailed ones that begin on the zero-age main sequence (ZAMS). This is because they pile up numerous uncertainties and numerical noise by the time the modeled star becomes a white dwarf. These defects, often observed as very tiny bumps and discontinuities in the profile of the Brunt-Väisälä frequency in the final white dwarf model, have no importance for the evolution itself, but may play havoc with the period spectrum and the merit function. This is especially true for those pulsation modes with weight functions extending into the deep C–O core where diffusion has not enough time to erase these defects. Hence, relying on such models for accurate seismic analyses of individual white dwarfs can be misleading. It has been our experience that extreme smoothness in all the variables of interest is a sine qua non condition for successful quantitative asteroseismology. We thus prefer the smoothness and flexibility provided by parameterized static models. Even though they constitute more approximate structures, they do not suffer from accumulated numerical noise, which tends to destroy the shape of the $S^2$ hypersurface in parameter space.

## 2.3. Parameterized Models

To define a full (but static) ZZ Ceti white dwarf model, it is necessary to specify the surface gravity (or the mass via the mass–radius relation),<fnref n="5" /> the effective temperature, the envelope layering, the core composition, and the convective efficiency via a flavor or another of the mixing-length theory. Quite realistic static models of white dwarfs can be computed (as compared to full evolutionary models) since in these cooling bodies the approximate relationship between the local luminosity and integrated mass, $L(r) \propto M(r)$, becomes quite accurate and, thus, may be used to provide an excellent estimate of the luminosity profile. Section 8 of the review paper of Fontaine & Brassard (2008) provides a detailed discussion of how the periods of pulsating white dwarfs depend on these structural parameters.

In particular, with respect to the convective efficiency to be used in the model-building phase, Figure 42 of Fontaine & Brassard (2008) shows that the periods of low-order g-modes in ZZ Ceti star models practically do not depend on the choice of the assumed convective efficiency, so it is justified here to adopt one version of the mixing-length theory and not worry further about it. This is especially justified in stars like GD 165 and Ross 548, which show rather short periods in the range from ~114 to ~334 s, because such short periods have to be associated with low-order modes. In the present asteroseismological exercise, the so-called ML2/$\alpha$ = 1.0 version of the mixing-length theory was therefore adopted following the calibration derived in Van Grootel et al. (2013b).

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Comparison of the profile of the square of the Brunt-Väisälä frequency vs. fractional mass depth in the core of three representative ZZ Ceti star models that differ only in their core composition. One model has a pure carbon core, the second a mixed, homogeneous C and O core in equal proportions by mass, and the third a pure O core. The other defining characteristics of these models are $T_{\text{eff}}$ = 12,000 K, $\log g$ = 8.0, $\log M(\text{He})/M_*$ = -2.0, and $\log M(\text{H})/M_*$ = -4.0. The different composition is expected to perturb the periods of g-modes, at least for those modes that have weight function amplitudes that are non-negligible in the core.</figcaption></figure>

Concerning the core composition, it is well known that the exact chemical profile in the C–O core of a typical white dwarf is uncertain (see, e.g., the brief discussion in Giammichele et al. 2015b). Furthermore, characterizing such a detailed profile would require several free parameters. Given the limited number of modes available in the present context, we decided instead to opt for homogeneous cores with the C/O ratio as the single free parameter. The assumption of core homogeneity is admittedly a rather crude approximation, but, in this way, at least the *bulk core composition* can potentially be probed (see, e.g., Giammichele et al. 2013). Figure 1 illustrates the small but significant differences that exist in the profile of the Brunt-Väisälä frequency in the core of three ZZ Ceti models that are otherwise the same, except for their homogeneous core composition (pure C, mixture of C and O in equal proportions by mass, pure O). One can expect a priori that these differences would affect the periods of gravity modes, at least for those modes with weight functions reaching down into the core with significant amplitudes.

The question of envelope layering also deserves some comments. It is specified not only by the total amount of mass in the helium mantle, $M(\text{He})$, and the total amount of mass in the hydrogen outer envelope, $M(\text{H})$, but also by the actual composition profiles in the transition zones themselves. This is because mode trapping/confinement, which has a very significant effect on the g-mode period spectrum in a white dwarf, is very sensitive to the conditions encountered in the composition transition zones (see, e.g., Section 4 in Brassard et al. 1992b). One standard assumption, based on physics as opposed to using some arbitrary profiles at the composition interfaces, has been to invoque diffusive equilibrium in order to compute the chemical distributions in the transition zones. The experiments reported by Brassard & Fontaine (2006) have been quite enlightening in this respect, in that they have shown that the assumption of diffusive equilibrium does not hold in GD 165. At the depths corresponding to the base of a rather thick

<footnote n="5">⁵ We prefer to use the surface gravity rather than the mass in our approach because this allows a direct comparison with atmospheric parameters derived from spectroscopy.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Left panel: map of the 5D merit function $S^2$ projected onto the log $g-T_{\text{eff}}$ plane for models of GD 165. The merit function is shown on a logarithmic scale (base 10). The location of the optimal model in this plane is indicated by a white cross. The white dotted curves delimit the regions where the merit function has values within the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels relative to the best-fit solution. By construction, the search domain in this plane is centered on the spectroscopic box (solid lines) and covers 4 times the spectroscopic uncertainties in both directions. The dashed lines define another spectroscopic box based on an earlier determination by Bergeron et al. (1995). Right panel: similar to the left panel, but for models of Ross 548. In contrast to the GD 165 case showing an optimal solution at the bottom of a relatively shallow $S^2$ valley, there are two well-defined minima here corresponding to a best-fit model with a thin envelope (upper minimum) and one with a thick envelope (lower minimum).</figcaption></figure>

envelope, diffusion may not have had the time to reach equilibrium (the diffusion timescale increases rapidly with increasing depth in white dwarfs). This proposition was verified explicitly by Brassard & Fontaine (2006), who carried out detailed evolutionary calculations including diffusion at the composition interfaces. They were able to conclude that diffusive equilibrium is indeed not reached at the composition interfaces in a ZZ Ceti star model with thick hydrogen and helium layers. More importantly for the present purpose, they were able to “calibrate” the steepness of the composition profiles in a model of GD 165, and it is this calibration that is used here. To keep the number of free parameters to a minimum and to ensure that the exact same search procedure is used for Ross 548 as well, we assume, without further justification, that this calibration is also valid for models of that star.

With the convective efficiency chosen and two transition zone parameters calibrated (H/He and He/C–O interfaces), the search for the optimal model in parameter space boils down to a 5D exercise in terms of $X(O)$ (the mass fraction of oxygen in the C–O core), $T_{\text{eff}}$, log $g$, log $M(\text{He})/M_*$, and log $M(\text{H})/M_*$, still a formidable numerical challenge. In order to include at the outset the independent constraints derived from spectroscopy, the search was confined to a range of $T_{\text{eff}}$ and a range of log $g$ corresponding to the $4\sigma$ “spectroscopic box” defined by the published values of these atmospheric parameters and their uncertainties. From Paper I, this corresponds to $T_{\text{eff}} = 12,165 \pm 760$ K and log $g = 8.082 \pm 0.192$ for GD 165 and to $T_{\text{eff}} = 12,204 \pm 760$ K and log $g = 8.012 \pm 0.192$ for Ross 548. Given the confidence we have in the reliability of these spectroscopic determinations, we choose to accept an optimal seismic model only if it falls within this $4\sigma$ box in the log $g-T_{\text{eff}}$ plane. For its part, the mass of the helium mantle was allowed to vary in the range $-4.0 \lesssim \log M(\text{He})/M_* \lesssim -1.5$,

while the mass of the hydrogen envelope was allowed to vary in the range $-9.0 \lesssim \log M(\text{H})/M_* \lesssim -4.0$, with the condition $M(\text{He})/M_* > M(\text{H})/M_*$. The composition of the (homogeneous) core was searched in the full range of $X(O)$, from 0.0 to 1.0.

### 3. RESULTS

#### 3.1. Optimal Seismic Models in Parameter Space

The results of our searches in parameter space turned out to be most encouraging as we found best-fit solutions with values of the surface gravity and the effective temperature well within the $4\sigma$ spectroscopic box for both stars of interest. In this context, we point out that there is never any guarantee at the outset that a minimum of the merit function $S^2$ will be found within the spectroscopic box. This is the kind of consistency that gives credibility to a seismic solution. Figure 2 shows a map of the merit function $S^2$ projected onto the surface gravity–effective temperature plane. Note the logarithmic scale used in the color coding. For each pair (log $g$, $T_{\text{eff}}$), the value given is the minimum of log $S^2$ that can be found among the values obtained by varying the other parameters, i.e., log $M(\text{H})/M_*$, log $M(\text{He})/M_*$, and $X(O)$.

The left panel of Figure 2, which refers to GD 165, shows a small white cross that indicates the best-fit solution that we found for that star. It is located about $0.9\sigma$ away from the formal spectroscopic values at the center of the plot. The white dotted contours show regions where the period fits have $S^2$ values within the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels relative to that best-fit solution. Judging from their shapes, our optimal seismic model for GD 165 belongs to a family of solutions defining a rather long valley along the effective temperature axis, meaning that the observed pulsation modes in GD 165 do not constrain very tightly that parameter.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Left panel: map of the 5D merit function $S^2$ projected onto the *D*(He)–*D*(H) plane for models of GD 165. The merit function is shown on a logarithmic scale. The location of the optimal model in this plane is indicated by a white cross. The white dotted curves delimit the regions where the merit function has values within the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels relative to the best-fit solution. Contrary to Figure 2(a), the optimal model corresponds to a relatively well defined minimum in this plane. The parameters *D*(H) and *D*(He) used in the model-building code are directly related to the mass fractions of hydrogen and helium through $\log M(H)/M_* \simeq D(H) + 0.8$ and $\log M(He)/M_* \simeq D(He) + 0.7$. The optimal model for GD 165 has rather thick H and He layers by white dwarf standards. Right panel: similar to the left panel, but for models of Ross 548. There are two very well defined minima corresponding to a best-fit model with a relatively thick He and H envelope (lower right) and one with a thin envelope (lower left). The mode identification for the thick-envelope model is the same as for the thin-envelope model, except for the important fact that the radial index *k* is shifted to *k* − 1 for all six modes from the latter to the former.</figcaption></figure>

In contrast, the right panel of Figure 2, referring to Ross 548, indicates two rather well-defined minima corresponding to two possible solutions with comparable values of the merit function $S^2$. Those two solutions differ in the thickness of their H–He envelope, the upper one having a relatively thin envelope and the lower one having a thick one. This difference is associated with a shift of the radial index from *k* to *k* − 1 for all six fitted modes from the thin-envelope to the thick-envelope model. Even though the thin-envelope solution is located farther away from the formal spectroscopic values at the center of the diagram than the thick-envelope solution (2.3*σ* vs. 1.6*σ*), we argue below that it should be the preferred model, and consequently, we identified it with a small white cross in the figure.

Figure 3 is similar in format to the previous plot, but it refers to the *D*(He)–*D*(H) plane. These quantities are directly related to the mass contained in the He mantle and the mass contained in the H outer envelope, respectively (see figure caption). For GD 165 (left panel), the optimal seismic model is rather well localized in this plane. The derived values of *D*(He) and *D*(H) imply thick envelope layers for GD 165. In comparison, the two solutions found for Ross 548 clearly stand out in the right panel and are even more localized than the GD 165 solution. Note as well that the thin-envelope solution is more sharply defined in this diagram.

Figure 4 refers to the fifth free parameter in our approach, the core composition—assumed homogeneous in this work—as measured by the oxygen mass fraction $X(O)$. The plot illustrates the behavior of $S^2$ versus $X(O)$. In the case of GD 165, the curve is rather flat, indicating a lack of sensitivity of the period fits on the core composition. The smallest value of $S^2$ obtained in that case is for $X(O) = 0.0$, which corresponds to the core composition of the optimal model discussed above.

This finding, i.e., the flatness of the curve, implies that the modes detected in GD 165 are not very sensitive to the core composition and, consequently, cannot be used to constrain it accurately. In sharp contrast, there is a clear sensitivity of the period fits on the core composition for Ross 548, with a relatively well defined minimum in $S^2$ at $X(O) \simeq 0.7$. Given that the two stars are near spectroscopic twins, this is a puzzling and interesting difference. Somehow, the modes detected in Ross 548 must be more sensitive to the core composition than are those found in GD 165. We come back to that problem below.

Table 1 lists the values of the five defining parameters needed in the construction of the optimal models obtained from our detailed searches in parameter space. It is to be understood that the inferred value of $X(O)$ can only be considered as an estimate of the bulk core composition in regions actually probed by the modes available. The last two columns give the value of the merit function $S^2$ for these optimal models and of the normalized merit function $s^2$. The latter is simply defined as the merit function divided by the number of fitted periods (six for both stars in the present case), and it will be used below for comparison purposes with the results of other investigators.

## 3.2. Mode Identification and Other Properties

Our search method yields automatically the mode identification (in terms of the indices *l* and *k* for spherical models) consistent with the best $S^2$ value as output.<fnref n="6" /> Table 2 summarizes the period match obtained and the mode identification inferred for the optimal model found for GD

<footnote n="6">⁶ Note that we adopt here the convention that negative values of the radial order index *k* refer to *g*-modes.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Behavior of the merit function $S^2$ in terms of the core composition—measured by the oxygen mass fraction $X(O)$—for the family of optimal models associated with each target star. This is obtained by fixing the core composition and searching for the best-fit solution in 4D space. The process is repeated for 11 values of $X(O)$ from 0.0 to 1.0, in steps of 0.1. While $S^2$ shows a well-defined minimum for $X(O) \simeq 0.7$ in the case of Ross 548, there is practically no sensitivity on the core composition for GD 165. This implies that, contrary to Ross 548, the modes detected in GD 165 do not probe the deep core.</figcaption></figure>

165. It gives the results in terms of both the period *P* and the frequency *v* = 1/*P*. In addition, the last three columns of the table give, respectively, the kinetic energy for each model mode listed (useful for recognizing trapped and/or confined modes; see below), the observed Fourier amplitude of each detected mode (actually, the amplitude of the largest component within a multiplet, if applicable), and an identification giving the ranking in terms of observed amplitude, from $f_1$ through $f_6$, as also given in Paper I. Table 3 uses the same format for presenting the equivalent data for the optimal model of Ross 548 that we retained, the one with a thin envelope. We point out that the mean relative dispersions of the global fits in period or frequency, $\Delta X/X = 0.26\%$ for GD 165 and $\Delta X/X = 0.22\%$ for Ross 548, are comparable to the best results obtained in recent years in the fields of white dwarf and hot subdwarf seismology. From this point of view, each fit is quite satisfactory. In the worst case for GD 165 (the 146.32 s mode), there is a difference of +0.71% between the observed period and the theoretical period of the assigned *l* = 2, *k* = −3 mode in the optimal model. The counterpart for Ross 548 is the 217.83 s pulsation, identified with an *l* = 2, *k* = −4 mode, for which a difference of −0.50% is found. Figure 5 provides a graphic representation of these excellent overall fits. Note that we use the reduced period, $P[l(l + 1)]^{0.5}$, as the abcissa in order to have a tighter diagram.

It may be significant that the search exercise has assigned the three largest-amplitude oscillations in GD 165 to *l* = 1 modes and the three smallest-amplitude ones to *l* = 2 modes. But for one mode ($f_3$), this behavior is also observed in Ross 548. This is indeed the expected hierarchy in view of geometrical cancellation effects, but this argument should be used with some caution as there are known cases of pulsating

stars (including white dwarfs) that show *l* = 2 modes with larger amplitudes than *l* = 1 modes.<fnref n="7" /> Nevertheless, we do point out that the light curve of each star is dominated by two relatively large oscillations ($f_1$ and $f_2$), both of which show a clear triplet structure when viewed at very high resolution (see Figures 5–8 in Paper I). This certainly reinforces their assignment in Tables 2 and 3 to dipole modes. Furthermore, for both GD 165 and Ross 548, the largest of the two oscillations is identified with the radial order $k = -1$, while the other one is identified with the value of $k = -2$, so there is a common pattern here. It is also of interest to point out that the assigned modes in Table 2 have consecutive values of the radial order: from $k = -1$ to $k = -3$ for the dipole modes, and from $k = -2$ to $k = -4$ for the quadrupole modes. Likewise, in the case of Ross 548, as can be seen in Table 3, the four oscillations assigned to dipole modes have consecutive values of the radial order, from $k = -1$ to $k = -4$. The two other pulsations, identified with quadrupole modes, also have consecutive values of the radial order with $k = -3$ and $k = -4$. This is again circumstantial evidence that adds to the credibility of our optimal models because nonadiabatic theory (see below) usually predicts that pulsation modes should be excited in a band of $\Delta k$ modes in radial order, for a given value of *l*.

We have verified explicitly that the mode identification proposed for GD 165 in Table 2 remains the same whatever the assumed core composition. That is, it does not change along the flat curve illustrated in Figure 4 above. In retrospect, this is not at all surprising since the modes detected in GD 165 seem to hardly probe the core at all and, consequently, their periods are not very sensitive to a change in core composition. The mode identification in such a case must rest mostly with the response of the pulsation periods upon varying the other parameters. Interestingly, even though the pulsation periods do depend on the core composition in models of Ross 548, we find that the mode assignment in Table 3 is rather robust, as it does not change in a rather wide interval $0.2 \lesssim X(O) \lesssim 1.0$ along the dotted curve in Figure 4. For values of $X(O)$ smaller than $\simeq 0.2$, the mode identification loses its coherence as the assigned modes no longer show consecutive values of *k* for given *l*.

We also point out that the second optimal model found for Ross 548, the one with a thick envelope, differs in its mode identification from the model we retained in Table 3 in that all six of the modes of interest have values of their radial order decreased by 1 unit. Hence, the dominant observed pulsation at 212.94 s is now assigned to a mode characterized by *l* = 1 and *k* = −2, the second-largest oscillation at 274.52 s is assigned to a mode with *l* = 1 and *k* = −3, and so on. This leaves an *l* = 1 and *k* = −1 mode (with a period of 131.11 s) in that optimal model that has no observed counterpart. In our view, this undermines somewhat the credibility of that model. Nevertheless, on the sole basis of the value of $S^2$, both optimal models for Ross 548 are equally acceptable. A comparison with the spectroscopic values of log *g* and $T_{\text{eff}}$ even suggests that the thick-layer solution is somewhat better, but the improvement over the thin-layer solution is only marginal in that respect. We do prefer the thin-layer optimal model for two principal reasons.

<footnote n="7">⁷ The case of the crystallized ZZ Ceti star BPM 37093 is interesting in that respect in that only one mode has been assigned a value of *l* = 1, while the seven other detected pulsations have been identified with *l* = 2 modes (Metcalfe et al. 2004; Brassard & Fontaine 2005). It may be that the solid core in that object has an incidence on the preferred excitation of quadrupole modes in that particular case.</footnote>
<table><caption>**Table 1**
Defining Parameters for the Seismic Models Found for GD 165 and Ross 548</caption><thead><tr><th>Model</th><th><i>T</i><sub>eff</sub><br>(K)</th><th>log <i>g</i><br>(cm s<sup>-2</sup>)</th><th><i>D</i>(H)</th><th><i>D</i>(He)</th><th><i>X</i>(O)</th><th><i>S</i><sup>2</sup><br>(s<sup>2</sup>)</th><th><i>s</i><sup>2</sup><br>(s<sup>2</sup>)</th></tr></thead><tbody><tr><td>GD 165</td><td>12,087.27</td><td>8.0432</td><td>-4.9490</td><td>-2.3311</td><td>0.000</td><td>1.918</td><td>0.320</td></tr><tr><td>Ross 548 (thin)</td><td>12,209.31</td><td>8.1205</td><td>-8.3113</td><td>-3.5719</td><td>0.673</td><td>2.112</td><td>0.352</td></tr><tr><td>Ross 548 (thick)</td><td>12,179.57</td><td>8.0885</td><td>-5.4300</td><td>-3.0967</td><td>0.448</td><td>2.086</td><td>0.348</td></tr></tbody></table>

First, linear nonadiabatic theory shows that the 131.11 s $l=1, k=-1$ mode in the thick-layer optimal model is as strongly driven as the 212.73 s $l=1, k=-2$ mode in that model. We recall here that the predictions of nonadiabatic pulsation theory are most reliable for stars near the blue edge of the instability strip as in the case for Ross 548. The growth rates and the kinetic energies of the two modes are indeed quite comparable, meaning that both modes have the same chances of growing an observable amplitude. In other words, both modes should be detectable if one is observed. As this is not the case, we propose that the 213 s mode is more likely an $l=1, k=-1$ mode, in which case it would have the largest amplitude, similarly to the 120 s dominant pulsation in GD 165 identified above as an $l=1, k=-1$ mode. And indeed, as already mentioned previously, the 213 s pulsation has the largest observed amplitude in the light curve of Ross 548. If our proposed mode identification is correct ($l=1, k=-1$), then the thin-layer solution must apply as there exists a correlation between the thickness of the envelope in a ZZ Ceti star model and the period of the lowest-order dipole mode in the sense that the thinner the envelope, the larger the period. Hence, if the 213.73 s mode is indeed an $l=1, k=-1$ mode, the envelope must be thin (and see Figure 1 of Fontaine et al. 1992 for more details on this behavior). Note that we explicitly checked that different calibrations for the transition profile did not impact the mode identification in this specific case of the thin-envelope solution.

Second, in the light of what is known about rotation in the pulsating pre-white dwarfs of the GW Vir type (Charpinet et al. 2009; Fontaine et al. 2013), it is very likely that isolated white dwarfs, as a group, rotate extremely slowly and as solid bodies. If this is true, then the first-order rotation coefficient $C_{kl}$ (see Appendix) can be used to verify the mode identification in the presence of rotational-splitting data. We find in this regard that the thick-layer solution is characterized by values of $C_{kl}$ for the two dominant triplets (213 and 274 s) that are in conflict with the observed frequency spacings in the multiplet data. In particular, that solution leads to the prediction that the frequency spacing should be smaller in the 213 s multiplet than in the 274 s complex, in direct opposition with the observations. In contrast, the thin-layer model predicts frequency spacings that go in the same sense as the observations. For those two reasons, we propose the thin-layer optimal model for Ross 548 as the more appropriate one. In the rest of this paper, we concentrate on the latter model for Ross 548.

Given that our adopted seismic model for Ross 548 is characterized by a much thinner H-He envelope than our seismic model for GD 165, we find a natural explanation for the fact that the dominant pulsation periods are significantly longer in Ross 548 (213 and 274 s) than in GD 165 (120 and 192 s), even though, once again, those two stars are near spectroscopic twins. Indeed, all other things being the same, the period of a g-mode—specified by the pair of indices $l$ and $k$—

increases with decreasing envelope mass. This is very well explained in the detailed study of the adiabatic properties of models of ZZ Ceti stars presented by Brassard et al. (1992a; see, in particular, their Tables 3–5 and related text). Likewise, we find a natural explanation for the fact that our optimal model of GD 165 shows only a weak sensitivity on the core composition, while that of Ross 548 shows a much stronger dependence. Indeed, an examination of the weight functions of each of the assigned modes as shown in Figure 6 reveals that all six modes in the GD 165 model (left panel) have weight functions that do not extend into the deep core and, consequently, their periods are not sensitive to a variation of the core composition in that region. At best, these modes probe the outermost region of the core, which is expected to be C dominated, and it may therefore be significant that our seismic model suggests a value of $X(O) = 0.0$ in that case. In comparison, three of the six modes in the Ross 548 model are partly confined below the thin envelope and, therefore and a contrario, bear a strong sensitivity on the core composition. It has been shown, in this context, that thinner envelopes in models of ZZ Ceti stars have generally better capacities for trapping and confining g-modes (Brassard et al. 1992a).<fnref n="8" /> We submit that this remarkable internal consistency bodes well in favor of the credibility of our seismic models for these spectroscopic look-alikes.

We finally draw the attention of the reader to the correlation that exists between the behavior of the weight function of a pulsation mode and its global kinetic energy. A mode that extends deep into a model has a relatively large kinetic energy ($E_{kin} = 0.5 \int_0^V \rho v^2 dV$) because the density increases with depth. Conversely, a mode that would be trapped in the outermost layers of a model would show a much reduced value of its kinetic energy. For example, Tables 2 and 3, in conjunction with Figure 6, show that the $f_2$ mode in Ross 548 is completely trapped above the H/He composition transition layer and has a low value of $\log E_{kin} = 43.92$, while the $f_2$ mode in GD 165, with $\log E_{kin} = 47.51$, is largely confined to the He mantle in that model. Another example is the $f_6$ mode in Ross 548, which extends deep into the core and, consequently, shows a maximum in kinetic energy (log $E_{kin} = 45.75$) compared to its adjacent $l=2$ neighbors. This mode is largely confined below the thin H-He envelope.

### 3.3. Multiplet Structure and Rotation

On the basis of our optimal seismic models, we next analyze the multiplet structures detected in the light curves of GD 165 and Ross 548, which we ascribe to rotation. In the case of GD 165 (Ross 548), five (three) of the six main modes show

<footnote n="8">⁸ In the nomenclature suggested by Brassard et al. (1992a), a mode is said to be trapped above a composition transition layer if its amplitude and weight function are negligible below that layer. Conversely, a mode is said to be confined below a composition transition layer if its amplitude and weight function are negligible above that layer.</footnote>
<table><caption>**Table 2**
Mode Identification and Details of the Period Fit Obtained for the Optimal Solution ($S^2 = 1.918$, $s^2 = 0.320$) for GD 165</caption><thead><tr><th>l</th><th>k</th><th>ν<sub>obs</sub><br>(μHz)</th><th>ν<sub>th</sub><br>(μHz)</th><th>P<sub>obs</sub><br>(s)</th><th>P<sub>th</sub><br>(s)</th><th>ΔX/X<br>(%)</th><th>ΔP<br>(s)</th><th>Δν<br>(μHz)</th><th>log E<sub>kin</sub><br>(erg)</th><th>A<sub>obs</sub><br>(%)</th><th>ID</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>8308.691</td><td>8324.524</td><td>120.36</td><td>120.13</td><td>+0.19</td><td>+0.23</td><td>-15.833</td><td>46.55</td><td>0.507</td><td>f<sub>1</sub></td></tr><tr><td>1</td><td>-2</td><td>5189.950</td><td>5187.720</td><td>192.68</td><td>192.76</td><td>-0.04</td><td>-0.08</td><td>+2.230</td><td>47.51</td><td>0.231</td><td>f<sub>2</sub></td></tr><tr><td>1</td><td>-3</td><td>3997.459</td><td>3983.975</td><td>250.16</td><td>251.01</td><td>-0.34</td><td>-0.85</td><td>+13.484</td><td>45.31</td><td>0.057</td><td>f<sub>3</sub></td></tr><tr><td>1</td><td>-4</td><td>...</td><td>3456.083</td><td>...</td><td>289.34</td><td>...</td><td>...</td><td>...</td><td>45.09</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-1</td><td>...</td><td>14409.838</td><td>...</td><td>69.40</td><td>...</td><td>...</td><td>...</td><td>46.55</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-2</td><td>8753.931</td><td>8768.477</td><td>114.23</td><td>114.04</td><td>+0.17</td><td>+0.19</td><td>-14.546</td><td>47.41</td><td>0.043</td><td>f<sub>4</sub></td></tr><tr><td>2</td><td>-3</td><td>6834.527</td><td>6883.292</td><td>146.32</td><td>145.28</td><td>+0.71</td><td>+1.04</td><td>-48.764</td><td>45.30</td><td>0.044</td><td>f<sub>5</sub></td></tr><tr><td>2</td><td>-4</td><td>5945.614</td><td>5951.902</td><td>168.19</td><td>168.01</td><td>+0.11</td><td>+0.18</td><td>-6.288</td><td>45.07</td><td>0.027</td><td>f<sub>6</sub></td></tr><tr><td>2</td><td>-5</td><td>...</td><td>4858.771</td><td>...</td><td>205.81</td><td>...</td><td>...</td><td>...</td><td>44.44</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-6</td><td>...</td><td>4449.892</td><td>...</td><td>224.72</td><td>...</td><td>...</td><td>...</td><td>44.34</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-7</td><td>...</td><td>4170.575</td><td>...</td><td>239.78</td><td>...</td><td>...</td><td>...</td><td>44.24</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-8</td><td>...</td><td>3716.483</td><td>...</td><td>269.07</td><td>...</td><td>...</td><td>...</td><td>43.62</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-9</td><td>...</td><td>3449.181</td><td>...</td><td>289.92</td><td>...</td><td>...</td><td>...</td><td>43.52</td><td>...</td><td>...</td></tr></tbody></table>

Note. The mean relative dispersion of the fit is $\overline{\Delta X/X} = 0.26\%$ (or $\overline{\Delta P} = 0.43$ s and $\overline{\Delta \nu} = 16.858$ μHz).

multiplet structure. This is summarized in Table 4 (Table 5), where we provide the detailed mode identification in terms of the set of indices $(k, l, m)$ for each of the detected multiplet components (12 for GD 165 and 8 for Ross 548). Using data from Paper I, we also provide the observed Fourier amplitude $A_{\text{obs}}$, the observed period $P_{\text{obs}}$, and the observed frequency $\nu_{\text{obs}}$ ($=1/P_{\text{obs}}$) for each of these independent pulsation modes, as well as the frequency spacings $\delta\nu_{\text{obs}}$ between multiplet components.

Under the assumption of slow, solid-body rotation, one can use each multiplet structure in conjunction with a seismic model in order to obtain an independent estimate of the rotation period of a star. From the details provided in the Appendix, the solid-body rotation period may be obtained from the following equation (based on Equation (12)) while remembering that $\sigma = 2\pi\nu$ and $\Omega_{\text{rot}} = 2\pi/P_{\text{rot}}$:

$$P_{\text{rot}} = (1 - C_{kl})/\delta\nu_{\text{obs}}, \qquad (2)$$

where $C_{kl}$ is the first-order solid-body rotation coefficient. This quantity depends only on the unperturbed adiabatic eigenfunctions $\xi_r$ and $\xi_h$ associated with our purely spherically symmetric seismic models. Its value is provided for each multiplet structure listed in Tables 4 and 5.<fnref n="9" />

In the case of GD 165, each of the five detected multiplet structures provides a very similar estimate of the rotation period as given in the last column of Table 4. These estimates are quite coherent with the a priori assumption of slow, solid-body rotation, at least in those regions actually probed by the available modes. This internal coherency provides us with a strong test of the mode identification achieved previously in our objective and automatic search for an optimal seismic model in parameter space. Indeed, we note that the value of $C_{kl}$ coming from our seismic model of GD 165 varies significantly from mode to mode. This is because we are dealing with very low order g-modes, well away from the asymptotic regime of high-order modes for which the value of the first-order solid-

body rotation coefficient reduces to $1/(l(l+1))$, independent of the index $k$ (see the Appendix). Quite on the contrary for low-order modes, slow and rigid rotation leads to $k$-specific frequency spacings for a given value of the degree $l$. Hence, the observed frequency spacings $\delta\nu_{\text{obs}}$ bear the signature of both the $l$ and $k$ indices. We find that the results presented in Table 4 for GD 165 confirm remarkably well the mode identification obtained in Section 3.2 for the five modes showing rotational splitting.

For Ross 548, we find that the three independent estimates of the rotation period listed in the last column of Table 5 are formally inconsistent with the hypothesis of rigid rotation. Taken at face value, these different estimates imply some mild differential rotation between the different regions of the star actually probed by the three multiplets of interest. If true, we cannot use the observed frequency spacings to test the mode identification because the (differential) rotation law is unknown in that case. Before adopting that conclusion, however, it is worthwhile to take a critical look at each of these three multiplets. While the existence of the two triplets at 213 and 274 s cannot be questioned, we recall that the multiplet centered on 318 s may not be a real detection. Indeed, as discussed in Paper I, the low-amplitude component of that doublet, the one with a period of 318.7657 s, is only a possible detection with a value for signal-to-noise ratio (S/N) of 3.1 and a false alarm probability value of 0.34, formally smaller than the “passing value” of 0.5, but not by very much. Removing the 318 s doublet from Table 5 goes a long way toward improving the consistency of the two remaining estimates of $P_{\text{rot}}$, within the hypothesis of rigid rotation, since these estimates overlap within $2.7\sigma$ and are based on the implicit assumption in Equation (2) that our values of $C_{kl}$ have no uncertainties. Given that our optimal model of Ross 548 is certainly perfectible, we submit that the hypothesis of rigid rotation is still viable for Ross 548. As briefly alluded to above, slow and rigid rotation is generally expected in a cool, isolated white dwarf such as Ross 548. If we adopt that point of view, we find that the observed frequency spacings in the 213 and 274 s triplets are, in fact, consistent with their respective identification as $k = -1, l = 1$ and $k = -2, l = 1$ multiplets. In particular, and contrary to GD 165, the frequency spacings are predicted (and observed) to be larger in the $k = -1, l = 1$ triplet than in the $k = -2, l = 1$ triplet. In the preceding subsection,

<footnote n="9">⁹ As can be seen in these tables, there exists a strong correlation between the value of the first-order solid-body rotation coefficient and the kinetic energy of a g-mode, in the sense that a mode trapped in the outermost H layer tends to have a relatively large value of $C_{kl}$ and a small value of $\log E_{\text{kin}}$, and conversely for a confined mode. The interested reader will find a detailed discussion of that behavior in Brassard et al. (1992b); see their Figure 18, in particular.</footnote>
<table><caption>**Table 3**
Mode Identification and Details of the Period Fit Obtained for the Retained Optimal Solution ($S^2 = 2.112$, $s^2 = 0.352$) for Ross 548</caption><thead><tr><th>l</th><th>k</th><th>ν<sub>obs</sub><br>(μHz)</th><th>ν<sub>th</sub><br>(μHz)</th><th>P<sub>obs</sub><br>(s)</th><th>P<sub>th</sub><br>(s)</th><th>ΔX/X<br>(%)</th><th>ΔP<br>(s)</th><th>Δν<br>(μHz)</th><th>log E<sub>kin</sub><br>(erg)</th><th>A<sub>obs</sub><br>(%)</th><th>ID</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>4696.019</td><td>4690.631</td><td>212.94</td><td>213.19</td><td>-0.11</td><td>-0.24</td><td>+5.387</td><td>46.51</td><td>0.658</td><td>f<sub>1</sub></td></tr><tr><td>1</td><td>-2</td><td>3642.709</td><td>3637.563</td><td>274.52</td><td>274.90</td><td>-0.14</td><td>-0.39</td><td>+5.146</td><td>43.92</td><td>0.430</td><td>f<sub>2</sub></td></tr><tr><td>1</td><td>-3</td><td>3140.496</td><td>3143.119</td><td>318.42</td><td>318.15</td><td>+0.08</td><td>+0.26</td><td>-2.622</td><td>45.00</td><td>0.072</td><td>f<sub>4</sub></td></tr><tr><td>1</td><td>-4</td><td>2997.200</td><td>2996.045</td><td>333.64</td><td>333.77</td><td>-0.03</td><td>-0.12</td><td>+1.154</td><td>45.04</td><td>0.062</td><td>f<sub>5</sub></td></tr><tr><td>1</td><td>-5</td><td>...</td><td>2327.315</td><td>...</td><td>429.67</td><td>...</td><td>...</td><td>...</td><td>44.58</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-1</td><td>...</td><td>8096.161</td><td>...</td><td>123.51</td><td>...</td><td>...</td><td>...</td><td>46.48</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-2</td><td>...</td><td>6299.070</td><td>...</td><td>158.75</td><td>...</td><td>...</td><td>...</td><td>43.92</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-3</td><td>5351.199</td><td>5329.143</td><td>186.87</td><td>187.64</td><td>-0.41</td><td>-0.77</td><td>+22.055</td><td>44.76</td><td>0.088</td><td>f<sub>3</sub></td></tr><tr><td>2</td><td>-4</td><td>4590.660</td><td>4567.538</td><td>217.83</td><td>218.93</td><td>-0.50</td><td>-1.10</td><td>+23.121</td><td>45.75</td><td>0.032</td><td>f<sub>6</sub></td></tr><tr><td>2</td><td>-5</td><td>...</td><td>3921.390</td><td>...</td><td>255.01</td><td>...</td><td>...</td><td>...</td><td>44.26</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-6</td><td>...</td><td>3504.982</td><td>...</td><td>285.30</td><td>...</td><td>...</td><td>...</td><td>43.79</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-7</td><td>...</td><td>3266.365</td><td>...</td><td>306.15</td><td>...</td><td>...</td><td>...</td><td>42.61</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-8</td><td>...</td><td>3035.180</td><td>...</td><td>329.46</td><td>...</td><td>...</td><td>...</td><td>42.84</td><td>...</td><td>...</td></tr><tr><td>2</td><td>-9</td><td>...</td><td>2789.663</td><td>...</td><td>358.46</td><td>...</td><td>...</td><td>...</td><td>43.16</td><td>...</td><td>...</td></tr></tbody></table>

Note. The mean relative dispersion of the fit is $\bar{\Delta X}/\bar{X} = 0.22\%$ (or $\bar{\Delta P} = 0.48$ s and $\bar{\Delta \nu} = 9.915$ μHz).

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Left panel: comparison of the six observed periods in GD 165 (red) with the theoretical g-mode spectrum of the optimal model (blue). The dipole (quadrupole) modes are illustrated in the lower (upper) half of the plot, and the radial order of each mode is indicated. The “reduced” period is used as the abscissa in order to have comparable values for both dipole and quadrupole modes. Right panel: similar to the left panel, but for Ross 548.</figcaption></figure>

we used the fact that this coherency is not found in the thick-envelope model for Ross 548 as an argument to discredit it.

It is highly instructive to locate the region of a seismic model that is actually probed for rotation, given a specific multiplet structure. In the regime of slow rotation, that information is contained in the so-called first-order rotation kernel, $K_{kl}$, discussed in the Appendix. That depth-dependent function is computed from the eigenfunctions $\xi_r$ and $\xi_h$ associated with the degenerate mode $(k, l)$ identified, in the (nonrotating) seismic model, with the multiplet structure. From Equation (9), $K_{kl}$ clearly plays a role similar to the weight functions encountered above, as it indicates, in the present case, the layers contributing most to the integral giving the frequency spacing between multiplet components.

Figure 7 (left panel) shows the relevant quantities for the five modes from our seismic model associated with multiplets in the light curve of GD 165. In the case of Ross 548 (Figure 7, right panel), based on the discussion just above, we retained only the two secure triplets at 213 and 274 s. In the upper part of each panel, we plotted the first-order rotation kernel for each mode of interest. To compare properly the kernel distribution associated with one mode with that associated with another mode, we plotted the relative rotation kernels, after having normalized arbitrarily the largest amplitude of all kernels involved to 1.0. We illustrate the outermost 20% of the normalized radius for both seismic models because, as it turns out, the rotation kernels have non-negligible amplitudes only in those outer layers.
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Left panel: normalized weight function vs. normalized radius for the six modes of the retained seismic model for GD 165 identified with the six pulsations detected in that star. Each individual weight function is normalized to a maximum value of 1.0. The weight function of a mode indicates the layers contributing most to the integral giving the frequency of the mode according to a well-known variational principle in linear pulsation theory (see, e.g., Unno et al. 1989). The middle subpanel illustrates the chemical stratification of the model. All of the weight functions have negligible amplitudes below a normalized radius of 0.6 (containing some 74% of the total mass of the star) and, consequently, cannot probe the chemical composition in that region. Right panel: similar to the left panel, but for the retained best-fit seismic model for Ross 548. In contrast to the case of GD 165, three of the six identified modes have non-negligible values of their weight functions below a normalized radius of 0.6 (containing some 72% of the total mass of the star in this case). The periods of these modes are therefore sensitive to the chemical composition in the inner core.</figcaption></figure>

In order to best identify the region over which rotation can be probed, we computed the weighted sum of kernels, which is shown by the dashed curve in the lower subpanel of each plot. The weighted sum takes into account the number of observed splittings observed in a given multiplet. Hence, the rotation kernels for the 120 and 192 s modes in GD 165 were multiplied by a weight of 2 in the sum because two frequency spacings were detected in their case (see Table 4). In comparison, a weight of 1 was assigned to the 250, 114, and 146 s doublets in that same sum. We thus find that rotation can be probed in GD 165 over approximately the outer 20% of its radius. In comparison, from the weighted sum of the kernels of the 213 and 274 s triplets in Ross 548, rotation can be probed in a more restricted region covering, approximately, the outer 5% of the radius in that case. In both instances, the coverage corresponds to the H-He envelope.

It is unfortunate, in this context, that the three confined modes sensitive to the core composition in Ross 548 (see Figure 6, right panel) have rather small amplitudes with no detected (218 and 333 s) or an unreliable (318 s) multiplet structure. Indeed, the coverage of the weight function of a mode and that of its first-order rotation kernel are comparable, meaning that the internal dynamics of the core of Ross 548 could have been tested, had those modes revealed rotational splitting. In comparison, even though five of the six main modes observed in GD 165 have secure multiplet structures, none of them have weight functions or rotation kernels extending down into the core according to Figures 6 and 7. The low-amplitude singlet at 168 s is also restricted to the H-

He envelope according to Figure 6 (left panel). This implies that the potential for sounding the rotation in the core of GD 165 does not exist on the basis of the available data and of our seismic model.

### 3.4. Formal Analysis of the Multiplet Data

In this subsection, we use and develop further the formal approach that was introduced by Charpinet et al. (2008, 2009) for exploiting quantitatively the fine structure caused by rotational splitting in seismic data. The method consists in analyzing simultaneously the available rotational data under the assumptions that (1) the star rotates slowly and (2) it rotates as a solid body. The inferred uniform rotation period, $P_{\text{rot}}$, can be used after the fact to verify whether the star indeed rotates slowly, in which case $P_{\text{rot}}$ should be much larger than the periods of the pulsation modes of interest and first-order perturbation theory should apply. The hypothesis that the star rotates as a rigid body is next verified through stringent tests based on the theory summarized in the Appendix. To ensure success, reliable multiplet data must be available, the modes must have been identified correctly, and a credible seismic model with appropriate eigenfunctions (rotation kernels) must be on hand.

It should be clear, at the outset, that testing for rigid rotation requires the availability of at least two multiplets, and preferably many more, as each main mode probes a specific region of the star as seen in Figure 7 above, for example. In this regard, the lowest-order modes contain much more information
<table><caption>**Table 4**
Mode Identification, Observed Multiplet Structures in GD 165, and Inferred Rotation Period under the Assumption of Solid-body Rotation</caption><thead><tr><th>l</th><th>k</th><th>m</th><th>C<sub>kl</sub></th><th>log E<sub>kin</sub><br>(erg)</th><th>A<sub>obs</sub><br>(%)</th><th>P<sub>obs</sub><br>(s)</th><th>ν<sub>obs</sub><br>(μHz)</th><th>δν<sub>obs</sub><br>(μHz)</th><th>P<sub>rot</sub><br>(hr)</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td></td><td></td><td>0.1131 ± 0.0042</td><td>120.3201 ± 0.0006</td><td>8311.16 ± 0.04</td><td>2.47 ± 0.05</td><td></td></tr><tr><td>1</td><td>-1</td><td>0</td><td>0.491</td><td>46.55</td><td>0.5070 ± 0.0042</td><td>120.3559 ± 0.0001</td><td>8308.69 ± 0.01</td><td>(2.46 ± 0.03)</td><td>57.43 ± 0.70</td></tr><tr><td>1</td><td>-1</td><td>+1</td><td></td><td></td><td>0.1592 ± 0.0042</td><td>120.3916 ± 0.0005</td><td>8306.23 ± 0.03</td><td>2.46 ± 0.03</td><td></td></tr><tr><td>1</td><td>-2</td><td>-1</td><td></td><td></td><td>0.2311 ± 0.0065</td><td>192.5717 ± 0.0006</td><td>5192.87 ± 0.03</td><td>2.92 ± 0.05</td><td></td></tr><tr><td>1</td><td>-2</td><td>0</td><td>0.399</td><td>47.51</td><td>0.2255 ± 0.0065</td><td>192.6801 ± 0.0001</td><td>5189.95 ± 0.03</td><td>(2.92 ± 0.05)</td><td>57.20 ± 0.98</td></tr><tr><td>1</td><td>-2</td><td>+1</td><td></td><td></td><td>0.0631 ± 0.0065</td><td>192.7886 ± 0.0046</td><td>5187.03 ± 0.12</td><td>2.92 ± 0.13</td><td></td></tr><tr><td>1</td><td>-3</td><td>0</td><td>0.486</td><td>45.32</td><td>0.0566 ± 0.0056</td><td>250.1589 ± 0.0074</td><td>3997.46 ± 0.12</td><td>2.49 ± 0.30</td><td>57.30 ± 6.90</td></tr><tr><td>1</td><td>-3</td><td>+1</td><td></td><td></td><td>0.0243 ± 0.0056</td><td>250.3150 ± 0.0173</td><td>3994.97 ± 0.28</td><td></td><td></td></tr><tr><td>2</td><td>-2</td><td>0</td><td>0.084</td><td>47.41</td><td>0.0428 ± 0.0041</td><td>114.2344 ± 0.0015</td><td>8753.93 ± 0.12</td><td>8.95 ± 0.32</td><td>56.88 ± 2.04</td></tr><tr><td>2</td><td>-2</td><td>+2</td><td></td><td></td><td>0.0167 ± 0.0041</td><td>114.3513 ± 0.0039</td><td>8744.98 ± 0.30</td><td></td><td></td></tr><tr><td>2</td><td>-3</td><td>0</td><td>0.155</td><td>45.30</td><td>0.0439 ± 0.0051</td><td>146.3159 ± 0.0030</td><td>6834.53 ± 0.14</td><td>3.90 ± 0.35</td><td>60.16 ± 5.40</td></tr><tr><td>2</td><td>-3</td><td>+1</td><td></td><td></td><td>0.0191 ± 0.0051</td><td>146.3995 ± 0.0068</td><td>6830.63 ± 0.32</td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 5**
Mode Identification, Observed Multiplet Structures in Ross 548, and Inferred Rotation Period under the Assumption of Solid-body Rotation</caption><thead><tr><th>l</th><th>k</th><th>m</th><th>C<sub>kl</sub></th><th>log E<sub>kin</sub><br>(erg)</th><th>A<sub>obs</sub><br>(%)</th><th>P<sub>obs</sub><br>(s)</th><th>ν<sub>obs</sub><br>(μHz)</th><th>δν<sub>obs</sub><br>(μHz)</th><th>P<sub>rot</sub><br>(hr)</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>-1</td><td></td><td></td><td>0.4105 ± 0.0066</td><td>212.7684 ± 0.0011</td><td>4699.95 ± 0.02</td><td>3.93 ± 0.08</td><td></td></tr><tr><td>1</td><td>-1</td><td>0</td><td>0.468</td><td>46.51</td><td>0.1361 ± 0.0066</td><td>212.9463 ± 0.0033</td><td>4696.02 ± 0.07</td><td>(4.02 ± 0.05)</td><td>36.73 ± 0.46</td></tr><tr><td>1</td><td>-1</td><td>+1</td><td></td><td></td><td>0.6578 ± 0.0066</td><td>213.1326 ± 0.0007</td><td>4691.92 ± 0.02</td><td>4.11 ± 0.08</td><td></td></tr><tr><td>1</td><td>-2</td><td>-1</td><td></td><td></td><td>0.4299 ± 0.0072</td><td>274.2508 ± 0.0019</td><td>3646.30 ± 0.03</td><td>3.59 ± 0.10</td><td></td></tr><tr><td>1</td><td>-2</td><td>0</td><td>0.498</td><td>43.92</td><td>0.1155 ± 0.0072</td><td>274.5209 ± 0.0071</td><td>3642.71 ± 0.10</td><td>(3.48 ± 0.07)</td><td>40.09 ± 0.81</td></tr><tr><td>1</td><td>-2</td><td>+1</td><td></td><td></td><td>0.3107 ± 0.0072</td><td>274.7745 ± 0.0027</td><td>3639.35 ± 0.04</td><td>3.36 ± 0.10</td><td></td></tr><tr><td>1</td><td>-3</td><td>-1</td><td>0.399</td><td>45.00</td><td>0.0722 ± 0.0077</td><td>318.0763 ± 0.0163</td><td>3143.90 ± 0.16</td><td>6.80 ± 0.42</td><td>49.11 ± 3.03</td></tr><tr><td>1</td><td>-3</td><td>+1</td><td></td><td></td><td>0.0303 ± 0.0077</td><td>318.7657 ± 0.0391</td><td>3137.10 ± 0.39</td><td></td><td></td></tr></tbody></table>

than high-order modes, which tend to show very similar rotation kernels. The sensitivity of the tests for rigid rotation proposed below rests heavily on the diversity of the applicable rotation kernels. We consider first the case of GD 165, which is outstanding in that seven well-measured frequency spacings belonging to five well-identified low-order modes are available (see Table 4), along with a credible seismic model that accounts remarkably well for the seismic and spectroscopic data on hand.

Under the a priori assumption that GD 165 spins slowly and rigidly, the rotational data are best explained globally with a value of the rotation period $P_{\text{rot}} = 57.294 \pm 0.338$ hr. This is obtained by minimizing the merit function given in Equation (14) (with $N_{\text{obs}} = 7$) in conjunction with Equations (12) and (13) for solid-body rotation. In this particular case, the optimization problem is a simple 1D exercise featuring the behavior of the merit function $S^2$ (not to be confused with the merit function above related to period matching) in terms of the uniform rotation period $P_{\text{rot}}$. The result of the procedure is shown in the upper left panel of Figure 8. Not surprisingly, the derived value is perfectly consistent with the individual estimates listed in the last column of Table 4 and indicates that GD 165 indeed rotates very slowly. For instance, in comparison to 57.294 hr, the detected pulsation periods have very low values, between 114 and 250 s.

The second hypothesis, that the star rotates as a solid body (at least in the region where the weighted sum of the rotation kernels has a non-negligible amplitude), is also nicely verified here through the results of three different tests as shown in the other panels of Figure 8. Test #1 relies on the assumption that the rotation law, $\Omega_{\text{rot}}(r)$, is linear with the radial coordinate and is defined by two free parameters: $P_c$, the central rotation period, and $P_s$, the superficial rotation period. The merit function $S^2$ given by Equation (14), in conjunction this time with Equations (9) and (10), is now a quantity that must be optimized in 2D, as a function of $P_c$ and $P_s$. Hence, if GD 165 rotates as a rigid body, the merit function must show a minimum when $P_c = P_s = P_{\text{rot}}$. This is precisely what the 2D map shown in the upper right panel of Figure 8 indicates. Note here that no use has been made of the fact that the rotation kernels are concentrated in amplitude in the outer layers; the computations took into account the full kernel distributions, from the center to the surface. The very small amplitudes of the rotation kernels in the core explain, a posteriori, why the central rotation period is much less constrained than the superficial rotation period is in that test. Nevertheless, the results are consistent with the original hypothesis of rigid rotation.

Test #2 uses a fixed step rotation law, i.e., one supposes that the star is divided into two zones: an inner one assumed to rotate rigidly with a period $P_i$, and an outer one assumed to rotate rigidly and independently with a period $P_o$. The location
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Left panel: relative rotation kernels vs. normalized radius for the five modes of the optimal model of GD 165 identified with the five multiplet structures observed in the light curve of that star (upper subpanel), chemical stratification of the model (middle subpanel), and weighted sum of the kernels vs. normalized radius (lower subpanel). The region over which the rotation profile can be probed corresponds to those layers where the weighted sum has a non-negligible amplitude, i.e., approximately over the outer 20% of the radius of the model in the present case. The vertical dotted line in the lower subpanel defines the layer where half of the running integral of the weighted sum is below and the other half above. Right panel: similar to the left panel, but pertaining to the retained optimal seismic model of Ross 548 and the two triplet structures observed in the light curve of that star. In this other case, the rotation profile can be probed over only the outermost ~5% of the radius.</figcaption></figure>

of the transition between the two zones must be chosen at some suitable fixed depth. A sensible choice is given by the location of the vertical dotted line in the lower plot of Figure 7 (left panel), which divides the integral of the weighted sum of kernels into two equal parts. In this way, the information that the overall kernel distribution is concentrated in the outer region of the star is built into test #2. The 2D merit function $S^2$ must now be optimized as a function of $P_i$ and $P_o$. Hence, if GD 165 rotates as a rigid body, the merit function must show a minimum when $P_i = P_o = P_{\text{rot}}$. The lower left panel of Figure 8 illustrates superbly that the hypothesis of solid-body rotation is verified (in the region where the kernels have sensible amplitudes).

Finally, test #3 is designed after a variable step rotation law, i.e., one supposes again that the star is divided into two zones, an inner one assumed to rotate rigidly with a period $P_i$ that may be varied, and an outer one assumed to rotate rigidly with the fixed uniform period $P_{\text{rot}}$, but, this time, the location of the transition between the two zones is allowed to freely vary from the center to the surface on a normalized radius scale, $r/R$. The 2D merit function $S^2$ must now be optimized as a function of $P_i$ and $r/R$. Hence, if GD 165 rotates as a rigid body, the merit function must show a minimum when $P_i = P_{\text{rot}}$, irrespective of the depth $r/R$. This is exactly what the lower right panel of Figure 8 illustrates, at least down to depths beyond which the method loses its sensitivity owing to the fact that the amplitudes of the rotation kernels plunge to very small values (below the outermost ~20% of the radius in the present case). Test #3 is probably the most useful one in terms of presentation, since it indicates explicitly and visually the

region over which the hypothesis of rigid rotation is effectively tested.

In the case of Ross 548, we dispose of only two secure triplets ($N_{\text{obs}} = 4$), the minimum required for testing for rigid rotation. Following an approach similar to that used for GD 165, we find the results summarized in Figure 9. Thus, under the assumption that Ross 548 spins slowly and rigidly, the available rotational data are best explained if the uniform rotation period takes the value $P_{\text{rot}} = 37.836 \pm 1.991$ hr. Quite noticeably in the upper right panel of Figure 9, we find that test #1 fails, in the sense that the central rotation period is left unconstrained. In retrospect, this is not at all surprising given the concentration of kernel amplitude in the outermost 5% of the radius as already observed in the right panel of Figure 7 above. On the other hand, the results of test #2 and test #3 are perfectly consistent with the proposition of solid-body rotation over that outermost region in Ross 548. Again, it is unfortunate that the three modes detected in that star, which extend into the deep core (those with periods of 318, 333, and 218 s), have amplitudes too small to reveal their multiplet structure. Those hold the potential for probing the rotation rate in very deep layers in Ross 548.

The final “product” of our formal approach designed to exploit quantitatively rotational splitting data is presented in Figure 10, which compares the observed (in red) and calculated (in blue) rotational splittings in GD 165 (left panel) and Ross 548 (right panel). The calculated values are based on the optimal seismic model retained for each star, in conjunction with the assumption of slow, rigid rotation with a period of $57.294 \pm 0.338$ hr for GD 165 and $37.836 \pm 1.991$ hr for
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Upper left panel: result of the optimization procedure under the hypothesis that GD 165 rotates slowly and rigidly. This shows the behavior of the merit function $S^2$ in terms of the assumed rotation period. The merit function exhibits a very well defined minimum, corresponding to a rotation period of 57.294 ± 0.338 hr. The dotted horizontal lines correspond, from bottom to top, to the $1\sigma$, $2\sigma$, and $3\sigma$ limits. Upper right panel: result of test #1 for solid-body rotation using a linear rotation law. This is a contour plot of the 2D merit function associated with that particular test. Note the logarithmic scale (base 10) used for that function, indicating a relatively well defined optimal solution. The latter is depicted by a small white cross in that plane, well within the $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels illustrated by the dotted white curves. Lower left panel: result of test #2 for solid-body rotation using a fixed step rotation law. This is similar in format to the panel associated with test #1, but note the more localized and better-defined minimum obtained with that different approach. This provides a more stringent test of the hypothesis of rigid rotation. Lower right panel: result of test #3 for solid-body rotation using a variable step rotation law. This is a contour map indicating the behavior of the 2D merit function $S^2$ in terms of depth (expressed in fractional radius) and in terms of the rotation period of the inner region of our two-zone approach. Again, the scale used for $S^2$ indicates a very well defined solution. That solution is illustrated by the nearly vertical white curve and the dotted white curves depicting its associated $1\sigma$, $2\sigma$, and $3\sigma$ contours. The fact that these contours diverge out at the larger depths considered here indicates that the rotationally split gravity modes available in the present case lose their capacity to measure the local rotation rate there. The horizontal dot-dashed line indicates the layer above which there is 1% of the mass of the star.</figcaption></figure>

Ross 548. The uncertainties on the calculated values of the frequency of the components are based on the uncertainty of 0.338 hr (1.991 hr) on the rotation period, while the central component (not affected by rotation in the context of our approach) is assumed to be known to perfect accuracy. The width of each of the blue boxes corresponds to the uncertainty on the calculated frequency of the multiplet components, expressed in units of microhertz. In comparison, the uncertainties on each of the detected (red) frequency components are the actual measured values coming out of the analysis presented in Paper I.

Not all of the multiplet components have been detected owing to the finite sensitivity of the available observations. This is a rather common situation encountered in pulsating white dwarfs and reflects the fact that the amplitudes of the

components vary significantly from mode to mode. There are five multiplets of interest in GD 165 (three dipole and two quadrupole modes). A total of seven individual spacings has been found in the available data. In comparison, two triplet structures (both of which are dipole modes) have been exploited for Ross 548. It is striking that the two dominant periodicities in both stars (120 and 192 s in GD 165; 213 and 274 s in Ross 548) have opposite behavior in the sense that the ($l=1, k=-1$) multiplet shows smaller frequency spacings than the ($l=1, k=-2$) multiplet for GD 165, while the opposite is true for Ross 548. As discussed above, this underlines the strong potential of frequency splitting for testing the mode identification. Figure 10 argues strongly in favor of the proposition that slow, solid-body rotation is indeed at work
<figure><img src="image_11.png" /><figcaption>Figure 9. Similar to Figure 8, but for Ross 548 and based on only two multiplet structures, compared to five in the case of GD 165. With an estimate of the rotation period of 37.836 ± 1.991 hr, the hypothesis of slow, rigid rotation is generally verified, although the test with the linear rotation law (upper right panel) is inconclusive. In addition, the available rotation data can be used to sample only the outer ~5% of the radius.</figcaption></figure>

in both stars, at least in the regions probed by the rotation kernels.

### 3.5. Statistical Analysis and Inferred Properties

An essential component of our approach to quantitative seismology is the requirement that the neighborhood of an optimal model in parameter space be thoroughly explored in order to assess the statistical significance of the solution. That information is contained in the shape of the merit function $S^2$ in parameter space.<fnref n="10" /> We thus seek to statistically estimate the value of each parameter of interest, as well as its uncertainty, given a best-fit seismic model. This is best done using a new procedure introduced in Van Grootel et al. (2013a), where details can be found. In a nutshell, however, we calculate the likelihood function in 5D parameter space (in the present case),

$$ \mathcal{L}(a_1, a_2, a_3, a_4, a_5) \propto e^{-\frac{1}{2}S^2}, \quad (3) $$

from the merit function, $S^2(a_1, a_2, a_3, a_4, a_5)$, which has been sampled by the optimization code during the search for the best-fit models, along with additional grid calculations covering

the regions of interest in parameter space. For each parameter of interest, say, $a_1$, this function is integrated over the full parameter range covered by the other free parameters, thus defining a density of probability function for the chosen parameter,

$$ \mathcal{P}(a_1) da_1 \propto da_1 \int \mathcal{L}(a_1, a_2, a_3, a_4, a_5) da_2 da_3 da_4 da_5. \quad (4) $$

This density of probability function is then normalized assuming that the probability that the value of $a_1$ is in the range specified for the search of a solution is equal to 1. In other words, the normalization factor is such that

$$ \int \mathcal{P}(a_1) da_1 = 1, \quad (5) $$

over the allowed parameter range. This method permits the construction of histograms for the probability distribution of each primary model parameter. Secondary model parameters, like the total mass $M_*$, the radius $R_*$, the luminosity $L_*$, and others, can also be evaluated in a similar way. For instance, let $b_1$ be one of these secondary parameters. The density

<footnote n="10">¹⁰ We refer here to the merit function defined by Equation (1).</footnote>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Left panel: observed and calculated rotational splittings in GD 165. This shows the comparison of the observed multiplet structures (in red) with those predicted (in blue) on the basis of the optimal seismic model assuming solid-body rotation with a period of 57.294 ± 0.338 hr. There are five multiplets of interest in GD 165 (three dipole and two quadrupole modes). Right panel: similar to the left panel, but pertaining to the two triplet structures (both of which are dipole modes) detected in Ross 548. The calculated rotational splittings are based on the assumption of slow, rigid rotation with a period of 37.836 ± 1.991 hr.</figcaption></figure>

probability function is then

$$ \mathcal{P}(b_1)db_1 \propto db_1 \int_{b \in [b_1, b_1+db_1]} \mathcal{L}(a_1, a_2, a_3, a_4, a_5; b) \times da_1da_2da_3da_4da_5, \quad (6) $$

where the integration is done with the additional constraint that for a given set of primary parameters, the corresponding value $b$ of the secondary parameter considered must be within $b_1$ and $b_1 + db_1$. Again, the density of probability for the secondary parameters is normalized such that

$$ \int \mathcal{P}(b_1)db_1 = 1. \quad (7) $$

The series of Figures 11(a) through (f) summarizes the results of the statistical analysis carried out in parameter space about our optimal seismic model for GD 165. The derived density of probability for each retained model parameter is illustrated as a histogram. The red-hatched region between the two red vertical solid lines defines the statistically acceptable range containing 68.3% of the distribution. For each parameter, it can be verified that the optimal model value, indicated in the figure by the blue vertical dashed line, belongs to that population as it ideally should. We adopt for simplicity and as our best final estimates of that quantity the central value of that interval as indicated in red in each panel.

In the case of GD 165, four of the five primary parameters are illustrated: (1) log $g$ (Figure 11(a)), (2) $T_{\text{eff}}$ (Figure 11(b)), (3) log $M(H)/M_*$ (Figure 11(c)), and (4) log $M(\text{He})/M_*$ (Figure 11(d)). Since we found no sensitivity of the pulsation periods on the core composition for our seismic model of GD 165, we did not compute the probability distribution for $X(O)$, the fifth primary quantity. The latter would have been quite flat and uninteresting. We also note that the effective temperature of GD 165 is relatively less constrained than the other parameters, as can be appreciated from Figure 11(b) in

comparison to the others. This was noticed above and should come as no surprise. Indeed, looking back at Figure 2 (left panel), one observes an elongated “valley” along the $T_{\text{eff}}$ axis in that $S^2$ map, so, clearly, the available seismic data (the six periods of Table 2) do not constrain that quantity well. To depict the behavior of some secondary parameters in this approach, we also provide in Figures 11(e) and (f) the histograms related to the total mass and total radius, respectively.

The results of our statistical analysis for Ross 548 are presented in a series of similar figures (Figures 12(a) through (f)), to which we have added Figure 12(g) pertaining to the probability distribution associated with the fifth parameter, $X(O)$. As indicated above, contrary to the case of GD 165, the pulsation periods in our seismic model for Ross 548 are sensitive to a variation of the core composition. The statistical exploration of parameter space in that direction leads to our best estimates of the composition of the C–O core (supposed homogeneous) for that star, i.e., $X(O) = 0.70 \pm 0.06$.

We have assembled in Table 6 (Table 7) the values of the parameters and properties that define GD 165 (Ross 548) as inferred from our asteroseismological exercises. The first five entries correspond to the values of our standard primary parameters as derived from the statistical approach described just above. To those, we added the related values of three interesting secondary parameters, i.e., total mass, radius, and luminosity, all in solar units. Using next the results of our multiplet analysis, we also listed the derived uniform rotation period $P_{\text{rot}}$, which, combined with the radius $R_*$, leads to an estimate of the equatorial velocity $V_{\text{eq}}$.

Another interesting quantity may be derived if we make the important assumption that both stars rotate rigidly throughout their interior. We recall that this is an inference that could not be made, given that the available multiplet data limited the coverage of the internal rotation profile to the outer 20% of the
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Results of the statistical analysis carried out in parameter space about the optimal seismic model for GD 165. Each histogram shows the derived probability density function for a given model parameter. The red hatched region between the two vertical solid red lines defines the ±1σ range, containing 68.3% of the distribution. The blue vertical dashed line indicates the value of the parameter of the optimal model solution. The mean, median, and mode values are also indicated. The final adopted estimate of the parameter of interest is listed in red and corresponds to the central value of the ±1σ interval. The various panels correspond to the following model parameters: (a) surface gravity, (b) effective temperature, (c) mass fraction in the hydrogen envelope, (d) mass fraction in the helium mantle, (e) total mass, and (f) total radius.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Similar to Figure 11, but for the optimal seismic model for Ross 548, with the addition of panel (g) referring to the oxygen mass fraction in the core.</figcaption></figure>

radius of GD 165 and only to the outer 5% in the case of Ross 548. Nevertheless, if the hypothesis of rigid rotation is adopted, an estimate of the total angular momentum *J* of each pulsator is obtained by combining the value of the uniform period *P*<sub>rot</sub> with a numerical integration over the structure of the seismic model. This leads to the value of *J* reported in each table, as well as the particularly interesting ratio, *J*/*J*<sub>☉</sub>, i.e., the value of the global angular momentum with respect to that of the Sun. These very small values of *J*/*J*<sub>☉</sub> suggest very strongly that these isolated ZZ Ceti stars have very likely shed essentially all of their initial angular momentum. This is in line with what has been found previously by Charpinet et al. (2009) and Fontaine et al. (2013) in the very hot pulsating white dwarfs of the GW Vir type.

The DA model atmospheres of P. Bergeron (2016, private communication) may also be used to estimate the value of the

absolute magnitude in the V band, given the seismic values of the surface gravity, the effective temperature, and the mass (or radius). We thus find $M_V = 11.73 \pm 0.12$ for GD 165 and $M_V = 11.78 \pm 0.06$ for Ross 548. The larger uncertainty in $M_V$ for the former object is due mostly to the less accurate value of the effective temperature derived in that case. The combination of $M_V$ with the observed visual magnitude $V$ ($V = 14.32 \pm 0.01$ for GD 165; $V = 14.16 \pm 0.01$ for Ross 548) then leads to the “seismic” distance *d*, listed as an additional entry in both Tables 6 and 7. No reddening was considered in the derivation of the distance, an approximation that is fully justified over the very short distances involved.

Finally, our seismic models may also be used to provide estimates of rates of period change of various pulsation modes due to cooling, which, at least in principle, are observables. To do this requires the computations of suitable evolutionary
<figure><img src="image_15.png" /><figcaption>Figure 12. (Continued.)</figcaption></figure>

sequences, and we describe such efforts in the next subsection. At this point, we simply report in the last two lines of Table 6 (Table 7) our best estimates of the rates of period change for the two largest-amplitude modes present in the light curve of GD 165 (Ross 548), i.e., the 120 and 192 s (213 and 274 s) main periodicities. The rates should be easier to determine for the larger-amplitude pulsations owing to their higher S/N. The range of values reported in the tables simply refers to the variation associated with a change of core composition, from pure C to pure O, and not to a detailed error analysis involving the uncertainties of all the other parameters that may have an impact on these values. For our present needs, the tabulated estimates are amply sufficient. We recall that the rate of period change is mode specific, i.e., it bears the signature of the *k* and *l* indices (see Section 8.3 of Fontaine & Brassard 2008 for more details). In this context, it is interesting to point out here that $\dot{P}_{120} < \dot{P}_{192}$ in GD 165, while $\dot{P}_{213} > \dot{P}_{274}$ in Ross 548, as an

illustration of the effects of different chemical layering and trapping/confinement properties in the two seismic models.

The results reported in Tables 6 and 7 represent the main outcome of this paper. In that context, it is quite fortunate that the values of some of the listed parameters have been estimated by means totally independent of asteroseismology, as we can use them to test further the credibility and validity of our seismic models. In particular, independent estimates of the surface gravity and of the effective temperature have been derived from the spectroscopic study reported in Paper I. Likewise, estimates of the absolute visual magnitude and of the distance for both stars have been presented in the very detailed spectroscopic analysis of DA white dwarfs presented by Gianninas et al. (2011). Also, a very reliable estimate of the distance to GD 165 through a parallax measurement has been reported by Kirkpatrick (2005) in his review of the properties of brown dwarfs (including GD 165B orbiting around the
<table><caption>**Table 6**
Inferred Properties of GD 165 on the Basis of Asteroseismology</caption><thead><tr><th>Quantity</th><th>Estimated Value</th></tr></thead><tbody><tr><td>log g (cm s<sup>-2</sup>)</td><td>8.052 ± 0.035</td></tr><tr><td>T<sub>eff</sub> (K)</td><td>12.130 ± 450</td></tr><tr><td>log M(H)/M<sub>*</sub></td><td>-4.23 ± 0.15</td></tr><tr><td>log M(He)/M<sub>*</sub></td><td>-1.70 ± 0.13</td></tr><tr><td>X(O)</td><td>0.0 (outer core)</td></tr><tr><td>M<sub>*</sub>/M<sub>☉</sub></td><td>0.64 ± 0.02</td></tr><tr><td>R<sub>*</sub>/R<sub>☉</sub></td><td>0.0124 ± 0.0003</td></tr><tr><td>L<sub>*</sub>/L<sub>☉</sub></td><td>0.0030 ± 0.0006</td></tr><tr><td>P<sub>rot</sub> (hr)</td><td>57.29 ± 0.34</td></tr><tr><td>V<sub>eq</sub> (km s<sup>-1</sup>)</td><td>0.263 ± 0.008</td></tr><tr><td>J (kg m<sup>2</sup> s<sup>-1</sup>)<sup>a</sup></td><td>(4.88 ± 0.03) × 10<sup>38</sup></td></tr><tr><td>J/J<sub>☉</sub></td><td>1/393</td></tr><tr><td>M<sub>V</sub><sup>b</sup></td><td>11.73 ± 0.12</td></tr><tr><td>d (pc)</td><td>33.0 ± 2.0</td></tr><tr><td>P<sub>120</sub> (s/s)</td><td>(6.98–7.96) × 10<sup>-16</sup></td></tr><tr><td>P<sub>192</sub> (s/s)</td><td>(2.47–2.71) × 10<sup>-15</sup></td></tr></tbody></table>

**Notes.**

a Assuming solid-body rotation throughout the star.

b Based on a model atmosphere with the seismic values of log g, T<sub>eff</sub>, and M<sub>*</sub>.

<table><caption>**Table 7**
Inferred Properties of Ross 548 on the Basis of Asteroseismology</caption><thead><tr><th>Quantity</th><th>Estimated value</th></tr></thead><tbody><tr><td>log g (cm s<sup>-2</sup>)</td><td>8.108 ± 0.025</td></tr><tr><td>T<sub>eff</sub> (K)</td><td>12.281 ± 125</td></tr><tr><td>log M(H)/M<sub>*</sub></td><td>-7.45 ± 0.12</td></tr><tr><td>log M(He)/M<sub>*</sub></td><td>-2.92 ± 0.10</td></tr><tr><td>X(O)</td><td>0.70 ± 0.06</td></tr><tr><td>M<sub>*</sub>/M<sub>☉</sub></td><td>0.65 ± 0.02</td></tr><tr><td>R<sub>*</sub>/R<sub>☉</sub></td><td>0.0118 ± 0.0002</td></tr><tr><td>L<sub>*</sub>/L<sub>☉</sub></td><td>0.0029 ± 0.0002</td></tr><tr><td>P<sub>rot</sub> (hr)</td><td>37.84 ± 1.99</td></tr><tr><td>V<sub>eq</sub> (km s<sup>-1</sup>)</td><td>0.379 ± 0.020</td></tr><tr><td>J (kg m<sup>2</sup> s<sup>-1</sup>)<sup>a</sup></td><td>(7.18 ± 0.38) × 10<sup>38</sup></td></tr><tr><td>J/J<sub>☉</sub></td><td>1/267</td></tr><tr><td>M<sub>V</sub><sup>b</sup></td><td>11.78 ± 0.06</td></tr><tr><td>d (pc)</td><td>30.0 ± 0.9</td></tr><tr><td>P<sub>213</sub> (s/s)</td><td>(2.87–2.91) × 10<sup>-15</sup></td></tr><tr><td>P<sub>274</sub> (s/s)</td><td>(1.41–1.51) × 10<sup>-15</sup></td></tr></tbody></table>

**Notes.**

a Assuming solid-body rotation throughout the star.

b Based on a model atmosphere with the seismic values of log g, T<sub>eff</sub>, and M<sub>*</sub>.

pulsating ZZ Ceti GD 165A). A much less reliable estimate of the distance to Ross 548, based on an old ground-based parallax measurement, is also available from Bergeron et al. (1995). And last but not least, Mukadam et al. (2013) have summarized some 41 yr of efforts to derive a value of the rate of period change of the dominant 213 s pulsation mode (more precisely, of the largest-amplitude component in that triplet) in the light curve of Ross 548. They found $P_{213} = (3.3 \pm 1.1) \times 10^{-15}$ s/s through the O–C technique.

We compare these estimates in Table 8. First, we find remarkable agreement between the seismic and spectroscopic estimates of the surface gravity and of the effective temperature. These values overlap within their $1\sigma$ uncertainties, except for the surface gravity of Ross 548, which is still only

1.3σ away from its spectroscopic counterpart. We see this as a very important and significant test of the reliability of our seismic models. Second, the spectroscopic values of the absolute magnitude and of the distance obtained by Gianninas et al. (2011) are entirely consistent with our derived seismic values, and this is true for both stars. We note also the excellent agreement between our seismic distance to GD 165 and that based on the parallax value of Kirkpatrick (2005). In contrast, the only parallax measurement available for Ross 548 indicates a distance much larger than that obtained through the spectroscopic method ($d = 31.0$ pc) or the seismic method ($d = 30.0 \pm 0.09$ pc). As pointed out originally by Bergeron et al. (1995), there is very likely a problem with the old parallax value for Ross 548. The problem has also been commented on in an asteroseismological context by Bradley (1998). It is hoped that the issue will be settled with the release of appropriate data from Gaia. The last entry of Table 8 demonstrates rather nicely that our proposed mode identification and seismic model for Ross 548 pass another major test. Indeed, it shows that our computed rate of period change for the dominant 213 s mode is in perfect agreement with the observed value of Mukadam et al. (2013).

### 3.6. Evolutionary Calculations and Rates of Period Change

In Section 2.2 above, we discussed the need for and the use of parameterized static models for quantitative seismology. Having identified and characterized seismic models in this way for our target stars, it is essential at this stage to verify whether they can be reproduced by full evolutionary calculations. In other words, are our static models artificial structures that cannot be computed from evolutionary calculations, or, on the contrary, are they compatible with traditional evolutionary models? We show, in this subsection, that the latter alternative is the correct one, and that our static seismic models also pass this important additional test. We then take advantage of the evolutionary sequences computed in this context to obtain reliable estimates of the rates of period change for the two dominant modes in the light curves of both GD 165 and Ross 548, as reported above.

We used the Montréal evolutionary code to first compute a model sequence with defining parameters compatible with those inferred from our optimal seismic model of GD 165. We also computed a similar sequence for Ross 548. Those defining parameters are summarized in Table 9. Hence, for full evolutionary models, the total mass M<sub>*</sub>, the fractional mass in the H envelope log M(H)/M<sub>*</sub>, the fractional mass in the He mantle log M(He)/M<sub>*</sub>, and the oxygen mass fraction X(O) in the (assumed) homogeneous core must be specified. Likewise, for the purpose of comparing with our static models, the same version of the convective efficiency (ML2/α = 1.0) was employed, and we also used the same calibration of the composition transition zone profiles as discussed above.

The sequences we computed all started as purely contracting, very hot models of the former nuclei of red giant stars. No residual thermonuclear burning was allowed, but neutrino cooling, which is significant in the hot phases of the evolution, was included. No connection was made with previous evolutionary phases. Since the time stepping is controlled by an internal algorithm that checks the convergence of each model, we end up with values of the effective temperature that cannot be predicted in advance. Hence, in the case of GD 165, we ended up with three adjacent models with T<sub>eff</sub> values of
<table><caption>**Table 8**
Comparison with Parameters Obtained Independently</caption><thead><tr><th>Star</th><th>Quantity</th><th>Seismology</th><th>Spectroscopy</th><th>Parallax</th><th>O - C</th></tr></thead><tbody><tr><td rowspan="4">GD 165</td><td>log g (cm s<sup>-2</sup>)</td><td>8.052 ± 0.035</td><td>8.082 ± 0.048<sup>a</sup></td><td>...</td><td>...</td></tr><tr><td>T<sub>eff</sub> (K)</td><td>12,130 ± 450</td><td>12,165 ± 190<sup>a</sup></td><td>...</td><td>...</td></tr><tr><td>M<sub>V</sub></td><td>11.73 ± 0.12</td><td>11.80<sup>b</sup></td><td>...</td><td>...</td></tr><tr><td>d (pc)</td><td>33.0 ± 2.0</td><td>32.0<sup>b</sup></td><td>31.5 ± 2.5<sup>c</sup></td><td>...</td></tr><tr><td rowspan="4">Ross 548</td><td>log g (cm s<sup>-2</sup>)</td><td>8.108 ± 0.025</td><td>8.012 ± 0.048<sup>a</sup></td><td>...</td><td>...</td></tr><tr><td>T<sub>eff</sub> (K)</td><td>12,281 ± 125</td><td>12,204 ± 190<sup>a</sup></td><td>...</td><td>...</td></tr><tr><td>M<sub>V</sub></td><td>11.78 ± 0.06</td><td>11.68<sup>b</sup></td><td>...</td><td>...</td></tr><tr><td>d (pc)</td><td>30.0 ± 0.9</td><td>31.0<sup>b</sup></td><td>67.1 ± 9.0<sup>d</sup></td><td>...</td></tr><tr><td></td><td>P&#x0307;<sub>213</sub> (s/s)</td><td>(2.87–2.91) × 10<sup>-15</sup></td><td>...</td><td>...</td><td>(3.3 ± 1.1) × 10<sup>-15</sup> e</td></tr></tbody></table>

**Notes.**

a Paper I.

b Gianninas et al. (2011).

c Kirkpatrick (2005).

d Bergeron et al. (1995).

e Mukadam et al. (2013).

<table><caption>**Table 9**
Defining Parameters for the Evolutionary Models for GD 165 and Ross 548</caption><thead><tr><th>Model</th><th>M<sub>*</sub>/M<sub>☉</sub></th><th>log M(H)/M<sub>*</sub></th><th>log M(He)/M<sub>*</sub></th><th>X(O)</th></tr></thead><tbody><tr><td>GD 165</td><td>0.6315</td><td>-4.1402</td><td>-1.6368</td><td>0.0</td></tr><tr><td>Ross 548</td><td>0.6571</td><td>-7.5125</td><td>-2.9376</td><td>0.6732</td></tr></tbody></table>

12,218.55, 12,015.36, and 11,823.38 K, in the vicinity of that characterizing our seismic model for that star, that is, $T_{\text{eff}} = 12,087.27$ K (see Table 1). Note that we could have slowed down the evolutionary calculations on purpose to obtain a better resolution in effective temperature, but, in any case, this underlines the lack of flexibility that we pointed out above (Section 2.2) concerning full evolutionary calculations as opposed to static models.

We have explicitly compared the $S^2$ values associated with those three evolutionary models against the values obtained from dedicated static models computed with exactly the same effective temperatures and other relevant parameters. Comparing the values of the merit function indeed represents the most stringent test that we could come up with in this exercise. We find that the ratio $S_{\text{evol}}^2/S_{\text{static}}^2$ varies from about 1.25 to 1.75 over these three models, which is not perfect (the ideal ratio is 1), but nevertheless extremely gratifying. We also find, using parabolic interpolation in conjunction with the values of $S^2$ and $T_{\text{eff}}$ associated with the three evolutionary models, that the minimum of the merit function occurs for $T_{\text{eff}} = 12,094.47$ K, only 7.2 K away from our seismic model. Likewise, on the basis of a similar sequence for Ross 548, we isolated the three consecutive evolutionary models in the neighborhood of our seismic model in terms of effective temperature, i.e., $T_{\text{eff}} = 12,447.99$, 12,226.41, and 12,021.01 K, compared to the seismic value of 12,209.31 (Table 1). In that case, the ratio $S_{\text{evol}}^2/S_{\text{static}}^2$ varies from 1.02 to 1.49, and the minimum in $S^2$ is found at 12,193.23 K (only 16.08 K away from the seismic value of 12,209.31 K).

We thus find that our seismic models are extremely well reproduced from evolutionary calculations, but not absolutely perfectly. And there is a reason why perfection cannot be

<figure><img src="image_16.png" /><figcaption>**Figure 13.** Comparison of the run of the Brunt-Väisällä frequency in a representative evolutionary model of a ZZ Ceti star with that of its equivalent static model. The abscissa is the fractional mass depth. The upper panel refers to the full model, while the lower one provides a zoomed-in view of the inner regions. The well in the outer layers illustrated in the upper panel corresponds to that H convection zone.</figcaption></figure>
achieved here. Indeed, as we found out long ago, it is impossible to implement in *exactly* the same way the composition transition profiles in an evolutionary and a static model. To illustrate the consequences of this technicality, as well as to impress upon the reader the notion that the resulting effects are particularly small, we provide in Figure 13 a comparison of the profile of the Brunt-Väisälä frequency of an evolutionary model culled from the GD 165 sequence above (the one with $T_{eff} = 12,015.36$ K) with that of its static counterpart. We do point out, in this context, that the run of this quantity (as compared to others such as density, pressure, adiabatic exponents, etc.) bears the strongest signature on the g-mode period spectrum. Hence, a small difference between the two curves necessarily implies a small difference in the period spectrum. On the scale of the upper panel of Figure 13, the tiny differences that exist between the evolutionary and static model cannot be seen. In the zoomed-in view presented in the lower panel, however, one can notice very small differences in the upper part of the H/He transition zone (the right side of the upper bump) and even smaller differences again in the upper part of the He/C transition zone (the right side of the lower bump). As pointed out just above, a technical reason is the culprit for this state of affairs. In principle, these tiny differences could be compensated, at least partly, by moving so slightly around in parameter space. We submit that, in fact, such related evolutionary models could be found with values of the merit function getting very close to our above value of $S^2 = 1.918$ for GD 165. The price to pay, however, would be to compute a large (and still undefined) number of full sequences, a very time-consuming and totally impractical way to prove our point.

We end this subsection by mentioning that we took advantage of the two evolutionary sequences described here, as well as supplementary ones (a pure O core sequence for GD 165, and a pure C core sequence and another one with a pure O core for Ross 548) to derive estimates of the rates of period change associated with the two largest-amplitude modes of both stars. These values are reported at the bottom of Tables 6 and 7. They were obtained by simply taking the ratio of the period difference over the age difference for two adjacent models that would sandwich the seismic model in effective temperature. For ZZ Ceti pulsators, cooling leads to a systematic (and mode-specific) increase of the period of a g-mode, so the expected values of the rates of period change are all positive.

### 3.7. Nonadiabatic Considerations

It is instructive to analyze each of the optimal models we retained for GD 165 and Ross 548 with a nonadiabatic pulsation code to verify whether the modes identified with the observed ones are indeed predicted to be unstable. This provides a last consistency check and, if successful, adds further credibility to these seismic models. As stated previously, the periods of the low-order modes found in GD 165 and Ross 548 are quite insensitive to the choice of the convective efficiency that has been made in the model-building phase. However, the question of the stability of the modes does depend sensitively on the choice of the assumed convective efficiency as has been known for a long time (see, e.g., Tassoul et al. 1990). The models that were built in the present search exercise used the $ML2/\alpha = 1.0$ version of convection, which, not by accident, is the version that was most recently

<table><caption>**Table 10**
Nonadiabatic Properties of the Optimal Model for GD 165</caption><thead><tr><th>l</th><th>k</th><th>P<br>(s)</th><th>σ<sub>R</sub><br>(rad s<sup>-1</sup>)</th><th>σ<sub>I</sub><br>(rad s<sup>-1</sup>)</th><th>τ<sub>e</sub><br>(yr)</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>120.127</td><td>5.23 × 10<sup>-2</sup></td><td>-1.18 × 10<sup>-13</sup></td><td>2.69 × 10<sup>5</sup></td></tr><tr><td>1</td><td>-2</td><td>192.763</td><td>3.26 × 10<sup>-2</sup></td><td>-1.64 × 10<sup>-13</sup></td><td>1.93 × 10<sup>5</sup></td></tr><tr><td>1</td><td>-3</td><td>251.006</td><td>2.50 × 10<sup>-2</sup></td><td>-5.61 × 10<sup>-11</sup></td><td>5.65 × 10<sup>2</sup></td></tr><tr><td>1</td><td>-4</td><td>289.345</td><td>2.17 × 10<sup>-2</sup></td><td>-2.05 × 10<sup>-10</sup></td><td>1.55 × 10<sup>2</sup></td></tr><tr><td>2</td><td>-1</td><td>69.397</td><td>9.05 × 10<sup>-2</sup></td><td>-1.21 × 10<sup>-13</sup></td><td>2.62 × 10<sup>5</sup></td></tr><tr><td>2</td><td>-2</td><td>114.045</td><td>5.51 × 10<sup>-2</sup></td><td>-2.48 × 10<sup>-13</sup></td><td>1.28 × 10<sup>5</sup></td></tr><tr><td>2</td><td>-3</td><td>145.279</td><td>4.32 × 10<sup>-2</sup></td><td>-6.66 × 10<sup>-11</sup></td><td>4.76 × 10<sup>2</sup></td></tr><tr><td>2</td><td>-4</td><td>168.013</td><td>3.74 × 10<sup>-2</sup></td><td>-2.38 × 10<sup>-10</sup></td><td>1.33 × 10<sup>2</sup></td></tr><tr><td>2</td><td>-5</td><td>205.813</td><td>3.05 × 10<sup>-2</sup></td><td>-2.16 × 10<sup>-9</sup></td><td>1.47 × 10<sup>1</sup></td></tr><tr><td>2</td><td>-6</td><td>224.724</td><td>2.80 × 10<sup>-2</sup></td><td>-1.85 × 10<sup>-9</sup></td><td>1.71 × 10<sup>1</sup></td></tr><tr><td>2</td><td>-7</td><td>239.775</td><td>2.62 × 10<sup>-2</sup></td><td>-1.08 × 10<sup>-8</sup></td><td>2.93 × 10<sup>0</sup></td></tr><tr><td>2</td><td>-8</td><td>269.072</td><td>2.34 × 10<sup>-2</sup></td><td>-3.02 × 10<sup>-8</sup></td><td>1.05 × 10<sup>0</sup></td></tr><tr><td>2</td><td>-9</td><td>289.924</td><td>2.17 × 10<sup>-2</sup></td><td>-7.15 × 10<sup>-8</sup></td><td>4.43 × 10<sup>-1</sup></td></tr></tbody></table>

<table><caption>**Table 11**
Nonadiabatic Properties of the Optimal Model for Ross 548</caption><thead><tr><th>l</th><th>k</th><th>P<br>(s)</th><th>σ<sub>R</sub><br>(rad s<sup>-1</sup>)</th><th>σ<sub>I</sub><br>(rad s<sup>-1</sup>)</th><th>τ<sub>e</sub><br>(yr)</th></tr></thead><tbody><tr><td>1</td><td>-1</td><td>213.191</td><td>2.95 × 10<sup>-2</sup></td><td>-1.81 × 10<sup>-11</sup></td><td>1.75 × 10<sup>3</sup></td></tr><tr><td>1</td><td>-2</td><td>274.909</td><td>2.29 × 10<sup>-2</sup></td><td>-3.52 × 10<sup>-9</sup></td><td>9.00 × 10<sup>0</sup></td></tr><tr><td>1</td><td>-3</td><td>318.155</td><td>1.97 × 10<sup>-2</sup></td><td>-3.53 × 10<sup>-9</sup></td><td>8.98 × 10<sup>0</sup></td></tr><tr><td>1</td><td>-4</td><td>333.773</td><td>1.88 × 10<sup>-2</sup></td><td>-9.91 × 10<sup>-10</sup></td><td>3.20 × 10<sup>1</sup></td></tr><tr><td>1</td><td>-5</td><td>429.679</td><td>1.46 × 10<sup>-2</sup></td><td>-3.09 × 10<sup>-10</sup></td><td>1.03 × 10<sup>2</sup></td></tr><tr><td>2</td><td>-1</td><td>123.515</td><td>5.09 × 10<sup>-2</sup></td><td>-2.04 × 10<sup>-12</sup></td><td>1.55 × 10<sup>4</sup></td></tr><tr><td>2</td><td>-2</td><td>158.753</td><td>3.96 × 10<sup>-2</sup></td><td>-4.07 × 10<sup>-9</sup></td><td>7.79 × 10<sup>0</sup></td></tr><tr><td>2</td><td>-3</td><td>187.647</td><td>3.35 × 10<sup>-2</sup></td><td>-1.07 × 10<sup>-9</sup></td><td>2.96 × 10<sup>1</sup></td></tr><tr><td>2</td><td>-4</td><td>218.936</td><td>2.87 × 10<sup>-2</sup></td><td>-2.45 × 10<sup>-10</sup></td><td>1.29 × 10<sup>2</sup></td></tr><tr><td>2</td><td>-5</td><td>255.011</td><td>2.46 × 10<sup>-2</sup></td><td>-4.29 × 10<sup>-9</sup></td><td>7.39 × 10<sup>1</sup></td></tr><tr><td>2</td><td>-6</td><td>285.308</td><td>2.20 × 10<sup>-2</sup></td><td>-2.36 × 10<sup>-8</sup></td><td>1.34 × 10<sup>0</sup></td></tr><tr><td>2</td><td>-7</td><td>306.150</td><td>2.05 × 10<sup>-2</sup></td><td>-2.49 × 10<sup>-7</sup></td><td>1.27 × 10<sup>-1</sup></td></tr><tr><td>2</td><td>-8</td><td>329.469</td><td>1.91 × 10<sup>-2</sup></td><td>-7.85 × 10<sup>-7</sup></td><td>4.03 × 10<sup>-2</sup></td></tr><tr><td>2</td><td>-9</td><td>358.466</td><td>1.75 × 10<sup>-2</sup></td><td>-2.77 × 10<sup>-7</sup></td><td>1.14 × 10<sup>-1</sup></td></tr></tbody></table>

“calibrated” by matching the theoretical with the empirical blue edge of the ZZ Ceti instability strip (Van Grootel et al. 2013). This calibration is based on the use of the Liège nonadiabatic code MAD (Dupret 2001), which takes properly into account the perturbations of the convective flux, unlike all other nonadiabatic codes employed in a white dwarf context (see Van Grootel et al. 2013b for details).

Table 10 (GD 165) and Table 11 (Ross 548) provide some of the results obtained with MAD in the present context. Along with the mode identification provided by the set of indices $(l, k)$, a mode is characterized in these tables by the value of its period $P$, the values of the real and imaginary parts of the complex frequency of the mode $\sigma_R (=2\pi/P)$ and $\sigma_I$, and the e-folding timescale $\tau_e (=1/\sigma_I)$. An excited (stable) mode has a negative (positive) value of $\sigma_I$. Hence, the tables immediately reveal that the modes identified with the observed ones in both GD 165 and Ross 548 are indeed excited in their respective optimal models. In fact, all dipole g-modes with radial order between $k=1$ and $k=13$ and all quadrupole modes with radial order between $k=1$ and $k=17$ are predicted to be driven in the optimal model of GD 165, as well as many low-order p-modes, including radial modes. A similar situation is encountered for Ross 548, which indicates that all dipole modes with radial order between $k=1$ and $k=9$ and all quadrupole modes with radial order between $k=1$ and $k=12$ are predicted to be
<figure><img src="image_17.png" /><figcaption>**Figure 14.** Comparison of the bands of detected periods in GD 165 and Ross 548 (thick lines) with the bands of excited periods (thin lines) in the seismic models of these stars as obtained from detailed nonadiabatic calculations. The reduced period is used as the abscissa in order to have comparable values for both dipole (in red) and quadrupole modes (in blue). The radial order *k* is indicated at both ends of each band.</figcaption></figure>

driven. This is a fairly common situation in nonadiabatic linear physics, which tends to predict wider bands of excited periods than those observed. But the important test here is that the modes in the optimal models assigned to the observed modes in GD 165 and Ross 548 are indeed predicted to be excited. Doubts could justifiably have been cast on the credibility of the seismic models if they had failed that test.

The tables also indicate that, for all modes considered, $|σ_R| \gg |σ_I|$, which is consistent with an earlier remark made above concerning the justification for using the adiabatic approximation to compute sufficiently accurate periods. Indeed, the imaginary part is so small compared to the real part that nonadiabatic effects have negligible impact on the values of the oscillation period in the stars of interest. In addition, the *e*-folding timescales listed in Tables 10 and 11 are all much smaller than the evolutionary timescales,<fnref n="11" /> which is a sine qua non condition for mode observability since the predicted unstable modes must have enough time to develop a detectable amplitude. This is clearly the case here. For instance, at its current cooling rate according to our evolutionary calculations, GD 165 needs to cool only by a further 2.9 K in order to cover $2.69 \times 10^5$ yr, the *e*-folding timescale associated with the dominant 120 s mode in that star. Likewise, a tiny drop of 0.016 K in the effective temperature of Ross 548 is sufficient to cover $1.75 \times 10^3$ yr, the *e*-folding time associated with the dominant 213 s mode in that star.

Figure 14 summarizes the situation by illustrating the bands of detected periods in relation to the bands of predicted unstable modes. It should be pointed out that linear

nonadiabatic pulsation theory is unable to predict which of the unstable modes will grow to observable amplitudes. This is rather in the realm of nonlinear theory, which is still being developed. Hence, it is not possible to offer an explanation for the fact that there seems to be a shift in radial order from the dipole to the quadrupole modes that are effectively detected in GD 165 and Ross 548. In particular, the lowest-order quadrupole modes (those with *k*=1 in GD 165 and *k*=1 and 2 in Ross 548) are not detected, and no credible explanation can be offered other than invoking unknown nonlinear processes. On the other hand, it is possible to narrow down the widths of the bands of predicted unstable periods by decreasing the convective efficiency used in the nonadiabatic modeling, but this needs to be justified in view of the results of Van Grootel et al. (2013b) showing that the blue edge of the ZZ Ceti instability strip is very well explained using a convective efficiency defined by $ML2/α = 1.0$. Given also that both GD 165 and Ross 548 are located near the blue edge, it is difficult to see how the convective efficiency could be significantly decreased in those two particular stars.

In this context, it is quite appropriate to mention the recent work of Tremblay et al. (2015), who provided a calibration of the mixing-length parameter to be used in 1D models of DA white dwarfs (such as those used here) on the basis of detailed 3D hydrodynamics calculations. These authors provided effective-temperature- and surface-gravity-dependent values of the parameter $α$ (the ratio of the mixing length to the local pressure scale height) to be used within the $ML2/α$ framework of the mixing-length theory employed in white dwarf models. An examination of their Table 2, taking into account the values of $T_{eff}$ and log *g* of both stars, indicates that the effective value of $α$ that should be used for modeling the flux is, in fact, rather close to 1.0 in both cases. This implies that the predictions of linear nonadiabatic pulsation theory presented in Figure 14 remain essentially unchanged.

## 4. COMPARISON WITH PREVIOUS STUDIES

There have been a few previous seismic analyses concerning the classical ZZ Ceti stars GD 165 and Ross 548, and it is of interest to compare the results of these independent studies with our own. In the case of GD 165, Bradley (2001) presented the first seismic model of that star using parameterized evolutionary models computed with the Rochester code WDEC. More recently, Romero et al. (2012) carried out a general seismic survey of some 44 ZZ Ceti stars, including GD 165 (and Ross 548). Their approach is characterized by the use of evolutionary models with a full history from the ZAMS. In the case of Ross 548, along with the Romero et al. (2012) paper, seismic models have been proposed by Bischoff-Kim et al. (2008a) and Castanheira & Kepler (2009), both of which use parameterized evolutionary models from WDEC.

Table 12 (Table 13) indicates the mode identification and normalized merit function inferred in the past available seismic studies of GD 165 (Ross 548) in comparison to the present work. We recall that the normalized merit function, $s^2$, is obtained by dividing our usual merit function (Equation (1)) by the number of fitted periods in order to put the different studies on the same footing. We also point out, in this context, that Paper I has revealed two more main modes than previously known for GD 165, as well as an additional one for Ross 548. These additions are quite significant, especially for low-order modes such as those involved here, because they contain extra

<footnote n="11">¹¹ It takes $(1-3) \times 10^8$ yr for a typical H-atmosphere white dwarf to cross the ZZ Ceti instability strip.</footnote>
<table><caption>**Table 12**
Observed Periods, Mode Identification, and Normalized Merit Function for Past Seismic Studies of GD 165 in Comparison to the Present Work</caption><thead><tr><th>Mode</th><th>120.36<br>(s)</th><th>192.68<br>(s)</th><th>250.16<br>(s)</th><th>114.23<br>(s)</th><th>146.32<br>(s)</th><th>168.19<br>(s)</th><th>s<sup>2</sup><br>(s<sup>2</sup>)</th><th>References</th></tr></thead><tbody><tr><td><i>l</i></td><td>1</td><td>1</td><td>1</td><td>2</td><td>2</td><td>2</td><td>0.32</td><td>This work</td></tr><tr><td>−<i>k</i></td><td>1</td><td>2</td><td>3</td><td>2</td><td>3</td><td>4</td><td>...</td><td>...</td></tr><tr><td><i>l</i></td><td>1</td><td>1</td><td>1</td><td>2</td><td>...</td><td>...</td><td>0.83–4.68</td><td>Bradley (2001)</td></tr><tr><td>−<i>k</i></td><td>1</td><td>2</td><td>4</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td><i>l</i></td><td>1</td><td>1</td><td>1</td><td>2</td><td>...</td><td>...</td><td>1.71</td><td>Romero et al. (2012)</td></tr><tr><td>−<i>k</i></td><td>1</td><td>2</td><td>3</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td></tr></tbody></table>

<table><caption>**Table 13**
Observed Periods, Mode Identification, and Normalized Merit Function for Past Seismic Studies of Ross 548 in Comparison to the Present Work</caption><thead><tr><th>Mode</th><th>212.94<br>(s)</th><th>274.52<br>(s)</th><th>318.42<br>(s)</th><th>333.64<br>(s)</th><th>186.87<br>(s)</th><th>217.83<br>(s)</th><th>s<sup>2</sup><br>(s<sup>2</sup>)</th><th>References</th></tr></thead><tbody><tr><td><i>l</i></td><td>1</td><td>1</td><td>1</td><td>1</td><td>2</td><td>2</td><td>0.35</td><td>This work</td></tr><tr><td>−<i>k</i></td><td>1</td><td>2</td><td>3</td><td>4</td><td>3</td><td>4</td><td>...</td><td>...</td></tr><tr><td><i>l</i></td><td>1<sup>a</sup></td><td>1<sup>a</sup></td><td>1</td><td>2</td><td>2</td><td>...</td><td>Not given<sup>b</sup></td><td>Bischoff-Kim et al. (2008a)</td></tr><tr><td>−<i>k</i></td><td>1</td><td>2</td><td>4</td><td>8</td><td>4</td><td>...</td><td>...</td><td>...</td></tr><tr><td><i>l</i></td><td>1</td><td>1</td><td>1</td><td>2</td><td>2</td><td>...</td><td>6.96</td><td>Castanheira & Kepler (2009)</td></tr><tr><td>−<i>k</i></td><td>1</td><td>2</td><td>3</td><td>9</td><td>3</td><td>...</td><td>...</td><td>...</td></tr><tr><td><i>l</i></td><td>1<sup>a</sup></td><td>1<sup>a</sup></td><td>2</td><td>2</td><td>1</td><td>...</td><td>11.72</td><td>Romero et al. (2012)</td></tr><tr><td>−<i>k</i></td><td>2<sup>a</sup></td><td>3</td><td>8</td><td>9</td><td>1</td><td>...</td><td>...</td><td>...</td></tr></tbody></table>

**Notes.**

<sup>a</sup> Fixed a priori.

<sup>b</sup> The authors use a different merit function than ours, $\Phi = \sum_{i=1}^{N_{\text{obs}}}|P_{\text{obs}}^i - P_{\text{th}}^i|/N_{\text{obs}}$, and state that they find a minimum value around $\lesssim 1.5$. Translated into their merit function, our results for the five periods in common give a value of $\Phi = 0.36$.

seismic information about the pulsating star. At the same time, the challenge of fitting simultaneously all of the observed periods increases greatly,<fnref n="12" /> which is, again, particularly true for low-order modes. Despite this, Tables 12 and 13 reveal that our approach leads to significantly better values of $s^2$ than in all previous studies, and sometimes by very large margins. This should not be overly surprising given that our use of parameterized static models allows us to explore parameter space much more finely than before.

With respect to mode identification, our automatic and blind procedure leads to results that are sometimes at odds with previous analyses, especially for Ross 548. For GD 165, the comparison is rather positive since our mode identification for the four fitted periods in common with Romero et al. (2012) gives the same results, while there is a difference with Bradley (2001) for the 250 s period. Things are not so good for Ross 548, since we differ from Bischoff-Kim et al. (2008a) for three modes out of five, from Castanheira & Kepler (2009) for two modes out of five, and from Romero et al. (2012) for all five modes. We point out that in some of the previous studies, the values of *k* and *l* for some modes were fixed a priori. We further note that our mode identification has the clear advantage of boosting consecutive values of *k* for a given value of *l* for both stars, which is the expected behavior from nonadiabatic theory as discussed above.

Table 14 (Table 15) lists the main inferred parameters for past seismic studies of GD 165 (Ross 548) in comparison to the

present effort. It should be pointed out here that the oxygen mass fraction in the core, X(O), corresponds to the value of the assumed homogeneous composition in our approach, while it corresponds to the central value in the other studies using more realistic composition profiles (except for Castanheira & Kepler [2009], who also assume homogeneous cores). Furthermore, we found no sensitivity of the fitted periods for GD 165 on the core composition, so this parameter is left unconstrained for that star in our study. This is not in conflict with the results of Bradley (2001) and Romero et al. (2012) because, in their cases, the core composition is not varied and is either fixed in advance (Bradley 2001) or comes from detailed calculations from the ZAMS (Romero et al. 2012). In contrast, because of the presence of three modes probing deep into the core in Ross 548, we found a definite sensitivity of the fitted periods on the core composition. In comparison, the core composition was fixed in advance in the studies of Bischoff-Kim et al. (2008a) and Castanheira & Kepler (2009), or resulted from evolutionary calculations from the ZAMS (Romero et al. 2012).

Overall, we find good consistency between the three seismic models of GD 165 in Table 14 and a very good agreement between our results for Ross 548 and the seismic model proposed by Bischoff-Kim et al. (2008a) as summarized in Table 15. This encouraging agreement is compounded by the fact that the rate of period change of the dominant 213 s pulsation in Ross 548 was found to be equal to $(2.91 \pm 0.29) \times 10^{-15}$ s/s by Bischoff-Kim et al. (2008b), which compares extremely well with our own value given in Table 8, $\dot{P}_{213} = 2.87-2.91 \times 10^{-15}$ s/s. It is interesting to point out that these two seismic models for Ross 548 agree very well, even though the mode identification is at odds for three modes out of five (see Table 13). This situation is not new, however, and has been pointed out before by Van Grootel et al. (2008) in the

<footnote n="12">¹² A clear indication of this is provided in Romero et al. (2012), from which we can deduce that their average merit function $\langle\Phi\rangle$ degrades quite significantly from a value of 0.57 for the objects in their sample with four or less fitted periods (about half of the sample) to 1.69 for the pulsators with more than four fitted periods. This different merit function is defined by $\Phi = \sum_{i=1}^{N_{\text{obs}}}|P_{\text{obs}}^i - P_{\text{th}}^i|/N_{\text{obs}}$.</footnote>
<table><caption>**Table 14**
Main Inferred Parameters for Past Seismic Studies of GD 165 in Comparison to the Present Work</caption><thead><tr><th>log g<br>(cm s<sup>-2</sup>)</th><th>T<sub>eff</sub><br>(K)</th><th>M<sub>*</sub>/M<sub>☉</sub></th><th>log M(H)/M<sub>*</sub></th><th>log M(He)/M<sub>*</sub></th><th>X(O)</th><th>References</th></tr></thead><tbody><tr><td>8.052 ± 0.035</td><td>12,130 ± 450</td><td>0.64 ± 0.02</td><td>-4.23 ± 0.15</td><td>-1.70 ± 0.13</td><td>Unrestricted</td><td>This work<sup>a</sup></td></tr><tr><td>~8.06</td><td>~12,190</td><td>0.65–0.68</td><td>-3.8, -4.0</td><td>-1.8, -2.0</td><td>~0.80</td><td>Bradley (2001)<sup>b</sup></td></tr><tr><td>8.05 ± 0.07</td><td>11,635 ± 330</td><td>0.632 ± 0.014</td><td>-4.17 ± 0.21</td><td>-1.76</td><td>0.76</td><td>Romero et al. (2012)<sup>c</sup></td></tr></tbody></table>

**Notes.**

a Parameterized static models.

b Parameterized evolutionary models.

c Evolutionary models from the ZAMS.

<table><caption>**Table 15**
Main Inferred Parameters for Past Seismic Studies of Ross 548 in Comparison to the Present Work</caption><thead><tr><th>log g<br>(cm s<sup>-2</sup>)</th><th>T<sub>eff</sub><br>(K)</th><th>M<sub>*</sub>/M<sub>☉</sub></th><th>log M(H)/M<sub>*</sub></th><th>log M(He)/M<sub>*</sub></th><th>X(O)</th><th>References</th></tr></thead><tbody><tr><td>8.108 ± 0.025</td><td>12,281 ± 125</td><td>0.65 ± 0.02</td><td>-7.45 ± 0.12</td><td>-2.92 ± 0.10</td><td>0.70 ± 0.06</td><td>This work<sup>a</sup></td></tr><tr><td>...</td><td>12,150 ± 550</td><td>0.63 ± 0.04</td><td>-7.46 ± 0.24</td><td>-2.40</td><td>0.76<sup>d</sup></td><td>Bischoff-Kim et al. (2008a)<sup>b</sup></td></tr><tr><td>...</td><td>12,100</td><td>0.635</td><td>-4.5</td><td>-2.0</td><td>0.50<sup>d</sup></td><td>Castanheira & Kepler (2009)<sup>b</sup></td></tr><tr><td>8.03 ± 0.05</td><td>11,627 ± 390</td><td>0.609 ± 0.012</td><td>-7.98 ± 0.16</td><td>-1.61</td><td>0.72</td><td>Romero et al. (2012)<sup>c</sup></td></tr></tbody></table>

**Notes.**

a Parameterized static models.

b Parameterized evolutionary models.

c Evolutionary models from the ZAMS.

d Fixed value.

context of pulsating hot B subdwarfs. The fact that an exact mode identification is not rigorously required to derive an acceptable seismic model is still considered a small “heresy” by many in the field of asteroseismology. Yet, we now have evidence that this can be the case in at least two different kinds of pulsating stars. In fact, this should not be so surprising for stars with dense period spectra.

The comparison of our seismic model for Ross 548 with those of Castanheira & Kepler (2009) and Romero et al. (2012) in Table 15 is less positive. There is conflict in terms of layering, in particular. Characterized by rather bad values of the merit function $s^2$ (see Table 13), those two models could be improved upon.

## 5. CONCLUSION

We have presented in this short series of two papers a new seismic analysis of the two classical, bright, hot ZZ Ceti stars GD 165 and Ross 548. Those were selected on the basis of the relative simplicity of their pulsation properties associated with their locations near the blue edge of the ZZ Ceti instability strip in the spectroscopic H-R diagram, where linear pulsation theory—in both its adiabatic and nonadiabatic flavors—is expected to work best. Given their near spectroscopic twin nature, those two stars also present some interesting challenges from a seismic point of view because the observed periods are significantly longer in Ross 548 than in GD 165. More fundamentally, as representative of the “simpler” ZZ Ceti pulsators, those two stars were picked to test, in a white dwarf context, the parameterized asteroseismic approach that has been quite successfully developed in recent years for pulsating hot subdwarf stars. This was deemed necessary before considering the exploitation of the exquisite observations gathered by *Kepler* and *Kepler-2* on other more complicated pulsating white dwarfs.

In Paper I, we took advantage of the availability of time-series data sets of exceptional quality (by ground-based standards) to uncover, in GD 165, a total of up to 13 pulsation modes regrouped into six main frequency multiplets. Likewise, up to 11 pulsation modes, also regrouped into six multiplets, were isolated in Ross 548. For the purpose of comparing with spherically symmetric (nonrotating) models of stars, Paper I revealed the existence of two more of these “main” modes than previously known in GD 165 ($f_5$ and $f_6$ in Table 2), and one more in Ross 548 ($f_6$ in Table 3). Given the high information content of such low-order modes, these constituted significant additional constraints and welcome additions to the seismic analyses to follow. Furthermore, Paper I provided updated estimates of the time-averaged atmospheric parameters of each target star in the light of recent developments on the front of atmospheric modeling for DA white dwarfs. These constitute fundamental external and independent constraints that, in our approach, seismic models must respect to be deemed acceptable.

In the present paper, we reported the results of detailed searches in parameter space for identifying an optimal model for each star that can account well and simultaneously for all the observed periods, while being consistent with the spectroscopic constraints derived in Paper I. This was done with the help of the dedicated code LUCY, which is a highly efficient optimization package that provides an objective and automatic way for identifying optimal seismic models while relying on a minimum of a priori assumptions. In the present exercises, we put no restrictions on the mode identification, except for constraining the modes to belong to the $l=1$ or $l=2$ families in order to exploit the fact that only dipole and quadrupole $g$-modes have been formally identified in ZZ Ceti stars. The equilibrium structures that we used are full, static, spherically symmetric parameterized models of DA white dwarfs. In the
spirit of keeping things as simple as possible in our seismic exercises, we varied only five basic parameters: (1) the composition of the C–O core (assumed homogeneous), (2) the surface gravity, (3) the effective temperature, (4) the mass contained in the He mantle surrounding the C–O core, and (5) the mass contained in the H layer sitting on top of that mantle. Two additional key parameters, the composition profile in the H/He transition zone and its equivalent in the He/C–O transition zone, were fixed to the shapes derived from detailed time-dependent diffusion calculations coupled to evolutionary computations carried out previously for GD 165. This “calibration” was also assumed to be valid for models of Ross 548, again for the sake of simplicity, along with the choice of a convective efficiency fixed to the ML2/α = 1.0 version.

We found optimal models for each target that reproduce the six observed periods well within ~0.3% on the average (see Tables 2 and 3), which is comparable to the best results achieved so far in asteroseismology. We also found that there is a sensitivity of the solution on the core composition for Ross 548, while there is practically none for GD 165. We submit that this somewhat surprising difference, along with the fact that the observed periods are significantly longer in Ross 548 than in its spectroscopic look-alike GD 165, may find a natural explanation in that the internal structures of the two stars differ in one significant aspect. Indeed, according to our seismic models, Ross 548 has a relatively thin outer envelope, while GD 165 has a rather thick one. It has been demonstrated in the past that a thinner envelope leads to a longer period for a given *g*-mode, all other things being the same. Furthermore, a thin envelope may allow for more efficient trapping/confinement properties. We explicitly verified that our optimal model of Ross 548 shows weight functions for three confined modes extending well into the deep core, thus explaining naturally the sensitivity of the period spectrum on the core composition in that star. In contrast, our optimal seismic model of its spectroscopic sibling, GD 165 with its thick envelope, does not trap/confine modes very efficiently, and we find weight functions for all six modes of interest that do not extend into the deep core, hence accounting for the lack of sensitivity in that case. This discovery that confined modes must be detected in order to probe the core composition of ZZ Ceti stars has implications for future work: it means that a certain amount of luck will be needed to fall onto a “cooperative” star.

The results reported in Tables 6 and 7 represent the ultimate outcome of this series of two papers. The reliability of the properties listed in those tables rests on the credibility of the proposed seismic models. In this connection, we have strived to verify that these models are consistent with all possible indicators, including external constraints. On the first account, the optimization method we used, based on static parameterized equilibrium models, allowed us to reproduce simultaneously all of the observed periods at the highest possible accuracy, given the (small) number of free parameters that were varied and the available constitutive physics. An intrinsic part of this approach included a detailed inspection of the shape of the 5D merit function in parameter space in order to put the derived seismic models on a firm statistical basis. We verified explicitly that our static seismic models can be reproduced quite satisfactorily from full evolutionary calculations. Likewise, we verified that the seismic models and the implied mode identifications are compatible with the expectations of linear pulsation theory, particularly in terms of amplitude hierarchy and nonadiabatic

considerations. We also fully exploited the observed multiplet structure to test the mode identification and to constrain the internal dynamics of the two target stars. In the case of GD 165 in particular, the mode identification was strengthened considerably through the (k,l) dependence of the observed frequency spacings within multiplets. Both GD 165 and Ross 548 were found to rotate extremely slowly by stellar standards, and very likely as solid bodies.

On the second account, as summarized in Table 8, we tested the compatibility of our seismic models with all of the external independent constraints that we could find in the literature. Notably, we found an excellent agreement for both stars between our seismic estimates of the surface gravity and the effective temperature and the values obtained independently through the spectroscopic method reported in Paper I. Also, again for both stars, our seismic estimates of the absolute V magnitude and of the distance turned out to be perfectly compatible with independent spectroscopic values of these quantities. Our seismic distance to GD 165 is also perfectly in line with a highly reliable parallax measurement for that star. Finally, we found a perfect consistency between the measured rate of period change for the dominant 213 s pulsation mode in Ross 548 and the value predicted on the basis of our seismic model. We conclude from these comparisons that our proposed seismic models for GD 165 and Ross 548 are credible and reliable, and that the method that was to be assessed passes the test with flying colors.

The next step in our continuing quest for improved seismic models of pulsating white dwarfs is defined by our discovery— in Ross 548—of the existence of deeply confined modes in white dwarfs as cool as ZZ Ceti stars. Obviously, such modes hold the potential for probing in detail the composition stratification in the core. This came as a small surprise to us since we expected that, generally, the increasing degeneracy in a cooling ZZ Ceti star would have pushed the significant part of the amplitudes of *g*-modes (and those of their associated weight functions) to the outer envelope, thus leaving very little sensitivity of the period spectrum on the structure of the core, very much like the case of GD 165. The existence of deeply confined modes in Ross 548 implies that our seismic model for that star, contrary to that of GD 165, could be perfected by considering a more realistic description of the core than the crude one-parameter homogeneous model used here.<fnref n="13" /> This is one important lesson that we retain for the future: since it is impossible to know in advance whether a given ZZ Ceti pulsator behaves more like Ross 548 or GD 165, it is wise to include from now on a better description of the composition profile in the core. We will therefore seek to parameterize the relatively complicated composition profile expected in the core of white dwarfs in a next effort and use this improved description in future seismic analyses.

This work was supported in part by the NSERC Canada through a doctoral fellowship awarded to N.G., and through a research grant awarded to G.F. The latter also acknowledges the contribution of the Canada Research Chair Program. S.C.

<footnote n="13">¹³ The inclusion of such a description, along with the relaxation of the assumption that the composition transition zone profiles calibrated for GD 165 also hold for Ross 548, would indeed likely improve our seismic model for the latter star. We suspect that solid-body rotation would be better constrained in such an improved model and, furthermore, that the derived mass in the He mantle, a bit on the low side by traditional evolutionary arguments, would be revised upward.</footnote>
acknowledges financial support from “Programme National de Physique Stellaire” (PNPS) of CNRS/INSU, France, and from the Centre National d'Études Spatiales (CNES, France). This work was granted access to the HPC resources of CALMIP under allocation number 2015-p0205. All of the authors are also grateful to Valérie Van Grootel, who kindly carried out the nonadiabatic calculations reported in Figure 14.

# APPENDIX
## SOME BACKGROUND ON PULSATIONS AND ROTATION

Within the framework of the linear theory of stellar pulsations, a mode is defined in terms of three “quantum numbers,” *k*, *l*, and *m*, the first one giving the number of nodes in the radial direction of the eigenfunctions associated with that mode, and the others (*l* and *m*) being the indices of the spherical harmonic function that specifies the angular geometry of the mode.

Nonrotating (spherical) stars have eigenfrequencies that are (2*l* + 1)-fold degenerate in *m*. However, a slowly rotating star has an eigenmode spectrum that is not degenerate in *m* as a result of the destruction of the spherical symmetry. To first order, with slow rotation considered as a perturbation, one can show that

$$ \sigma_{klm} \approx \sigma_{kl} - m(\delta\sigma_{kl}), \quad (8) $$

where $\sigma_{klm}$ is the angular frequency of mode (*k*, *l*, *m*), $\sigma_{kl}$ is the angular frequency of the degenerate mode (*k*, *l*) in the absence of rotation, and the frequency spacing is given by

$$ \delta\sigma_{kl} = \int_{0}^{R} \Omega_{\text{rot}}(r) K_{kl}(r) dr. \quad (9) $$

This leads to a set of equally spaced frequencies with a splitting between adjacent frequency components given by $\delta\sigma_{kl}$.

In the above expression, $\Omega_{\text{rot}}(r)$ is the (assumed) spherically symmetric rotation law (expressed in units of angular frequency), and $K_{kl}$ is the so-called first-order rotation kernel, which plays the role of a weight function. It is given by

$$ K_{kl}(r) = \frac{\xi_r^2 + \{l(l+1) - 1\}\xi_h^2 - 2\xi_r\xi_h}{\int_0^R \{\xi_r^2 + l(l+1)\}\xi_h^2 dr} \rho r^2, \quad (10) $$

where $\xi_r$ and $\xi_h$ are, respectively, the real parts of the radial and horizontal components of the displacement vector,

$$ \xi = \left[ \xi_r(r), \xi_h(r) \frac{\partial}{\partial \theta}, \xi_h(r) \frac{1}{\sin \theta} \frac{\partial}{\partial \phi} \right] Y_m^l(\theta, \phi) e^{i\sigma t}. \quad (11) $$

These eigenfunctions for mode (*k*,*l*) refer to the unperturbed configuration, i.e., to a purely spherical (nonrotating) stellar model.

Hence, to first order, slow spherically symmetric ($\Omega_{\text{rot}}(r)$) rotation produces a set of 2*l* + 1 equally spaced frequency components out of a degenerate mode with indices (*k*, *l*). The spacing between two adjacent components of this multiplet (modes that differ by $|\Delta m| = 1$) can be computed from the unperturbed eigenfunctions as can be appreciated from Equations (9) and (10). This fine structure within a multiplet is usually referred to as rotational splitting.

In the particular case of uniform rotation, $\Omega_{\text{rot}}$ does not depend on the radial coordinate *r*, and it is easy to derive the

following expression:

$$ \delta\sigma_{kl} = \Omega_{\text{rot}}(1 - C_{kl}), \quad (12) $$

where $C_{kl}$, the first-order solid-body rotation coefficient, is given by

$$ C_{kl} = \frac{\int_{0}^{R} \{\xi_{h}^{2} + 2\xi_{r}\xi_{h}\} \rho r^{2} dr}{\int_{0}^{R} \{\xi_{r}^{2} + l(l + 1)\xi_{h}^{2}\} \rho r^{2} dr}. \quad (13) $$

In the asymptotic limit of high radial order ($k \gg 1$) for g-modes, it is well known that the horizontal component of the displacement vector becomes much larger than the radial component ($\xi_h \gg \xi_r$). Hence, from Equation (13), the first-order solid-body rotation coefficient reduces to $C_{kl} = 1/(l(l+1))$ in that limit.

Given a credible seismic model for a slowly rotating pulsating star (this implies that each frequency multiplet has been properly identified in terms of its basic indices *k* and *l*, and that the corresponding unperturbed eigenfunctions $\xi_r$ and $\xi_h$ are available), and given a rotation law $\Omega_{\text{rot}}(r)$, it is possible to define a goodness-of-fit merit function inspired from a $\chi^2$ approach such that

$$ S^2 = \sum_{i=1}^{N_{\text{obs}}} \left( \frac{\delta\sigma_{\text{obs}}^{(i)} - \delta\sigma_{\text{th}}^{(i)}}{\Delta_{\text{obs}}^{(i)}} \right)^2, \quad (14) $$

where $\delta\sigma_{\text{obs}}^{(i)}$ is the ith observed frequency spacing between two rotationally split frequency components within a multiplet (modes with the same values of *k* and *l*, but different values of *m*), $\Delta_{\text{obs}}^{(i)}$ is its associated uncertainty, and $\delta\sigma_{\text{th}}^{(i)}$ is the computed spacing for the same two modes according to the previous recipe.

The merit function $S^2$ is minimized in parameter space to obtain the best possible fit. In the particular case of solid-body rotation, parameter space reduces to a single dimension, and $S^2$ is optimized as a function of the uniform rotation period $P_{\text{rot}}(=2\pi/\Omega_{\text{rot}})$.

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