THE NEWLY DISCOVERED PULSATING LOW-MASS WHITE DWARFS: AN EXTENSION OF THE ZZ CETI INSTABILITY STRIP

White dwarfs with masses less than about 0.45 M_☉ are expected to harbor He cores. They are understood to be the products of stars experiencing strong mass loss on the red giant branch before the onset of He burning in their cores. The mass loss can be related to strong stellar winds in high-metallicity environments (D'Cruz et al. 1996) or mass transfer via binary evolution, leading, in particular, to double-degenerate systems (Ergma et al. 1998; Sarna et al. 2000; Bassa et al. 2003). And indeed, many low-mass white dwarfs are found in binary systems, and in particular are the usual companions of millisecond pulsars (Marsh et al. 1995; van Kerkwijk et al. 2005).

In this context, the search for pulsations in low-mass, He-core DA white dwarfs was spearheaded by Steinfadt et al. (2010), who made a strong case for the importance and significance of finding such objects. These authors built several representative sequences of He-core DA stars with masses ranging from 0.15 M_☉ to 0.20 M_☉ and analyzed their pulsation characteristics, particularly the question of mode propagation. They provided a first view of the basic adiabatic pulsation properties of such stars.

In terms of mode stability, in the absence of full nonadiabatic calculations, Steinfadt et al. (2010) estimated the location of the blue edge of a possible instability strip for low-mass DA white dwarfs using the semi-analytic criterion discussed by Brickhill (1991b) as applied to their stellar models. This criterion proposes a comparison of a "representative" period P—chosen as the one corresponding to the g-mode with ℓ = 1 and k = 1 in the models considered by Steinfadt et al.—with the thermal timescale τ_th at the base of the H convection zone of a given model according to the formula P ≃ 8πτ_th. This is based on the so-called convective driving mechanism originally proposed by Brickhill (1983, 1990, 1991a, 1991b) in his series of seminal papers to account for g-mode instabilities in ZZ Ceti white dwarfs. Steinfadt et al. (2010) thus obtained a blue edge representing an extension of the ZZ Ceti instability strip into the low-mass regime in the log g–T_eff plane (the circles in their Figure 3). Although the authors did not specify what convective efficiency they actually used in their calculations, it is likely that they employed the standard version of the mixing-length theory (denoted as ML1/α = 1.0 in white dwarf jargon) because they imported MESA subroutines (Paxton et al. 2011), where this is the default option.

In a follow-up effort on the observational side, Steinfadt et al. (2012) searched for luminosity variations in a handful of low-mass He-core DA white dwarfs, concentrating in particular on a few targets lying just to the right of their estimated blue edge in the spectroscopic Hertzsprung–Russell (HR) diagram. According to the theoretical estimates of Steinfadt et al. (2010), it was hoped that some of these objects would show g-mode pulsational instabilities. Unfortunately, none were detected, but given the rarity of confirmed low-mass white dwarfs, the uncertainties on the atmospheric parameters of the known ones, the limitations on the available photometric observations, and the lack of full nonadiabatic calculations, Steinfadt et al. (2012) still concluded that it would be worthwhile to pursue the search, especially in view of new samples of He-core white dwarfs becoming available.

The most significant effort on this front in the last few years has been the so-called ELM (as in "extremely low mass") white dwarf survey which has led to the first results published in a series of papers by Brown et al. (2010, 2012) and Kilic et al. (2011, 2012). Table 4 of Kilic et al. (2012), for example, provides basic data for 40 low-mass white dwarfs that have been observed so far for radial velocity variations within the ELM project. This sample has provided a growing list of potential candidates for variability, and an observing program was set up at the McDonald Observatory to monitor the most promising targets. This new program has led to the recent discovery of the first pulsating extremely low-mass white dwarf, SDSS J1840+6423, as reported by Hermes et al. (2012). It is a DA star with T_eff ≃ 9140 ± 170 K and log g ≃ 6.16 ± 0.06, according to Brown et al. (2012). It is part of a compact binary system with an invisible companion orbiting about their barycenter with a period of 4.59 hr (Kilic et al. 2012). Due to its most probable formation channel, common envelope evolution, SDSS J1840+6423 most likely possesses an He core. It shows multiperiodic modulations with a dominant period of about 4698 s, quite large by ZZ Ceti standards, but not unexpected given its very low surface gravity.

Even more recently, the McDonald Observatory program has revealed the existence of two additional pulsating extremely low-mass DA white dwarfs (Hermes et al. 2013), thus opening up a new and exciting avenue of research in the field of asteroseismology. SDSS J1112+1117 is characterized by T_eff ≃ 9400 ± 490 K and log g ≃ 5.99 ± 0.12 (Brown et al. 2012), and shows at least five independent periodicities in its light curve, ranging from 1793 s up to 2856 s. For its part, SDSS J1518+0658 is characterized as having T_eff ≃ 9810 ± 320 K and log g ≃ 6.66 ± 0.06 (Brown et al. 2012), and its light curve shows 6 independent periodicities ranging from 1339 s to 3680 s. As pointed out by Hermes et al. (2012) and J. J. Hermes et al. (2012, in preparation), there is no doubt that the detected light modulations are caused by g-mode pulsational instabilities.

As is always the case when a new type of pulsating star is discovered, a few obvious questions arise: (1) What is the driving mechanism in these objects? (2) Can the observed periods be accounted for? (3) Are these variables part of an instability domain in the spectroscopic HR diagram and, if so, can that domain be mapped properly? A detailed stability survey must be carried out to answer these questions, and this is the precise aim of this paper. We are particularly well equipped to address this problem, since we have recently reinvestigated the question of the ZZ Ceti instability strip with the help of improved evolutionary models that take into account the feedback effect of convection on the atmospheric structure (as opposed to standard gray atmospheres). Our group used the nonradial, nonadiabatic pulsation code MAD, which is the only code that has practically implemented a full time-dependent treatment of convection for both p- and g-mode pulsations (Dupret 2002; Grigahcène et al. 2005). This is in contrast with the standard frozen convection (FC) approximation approach used in most nonadiabatic codes employed in white dwarf studies. The first results of this project have been published in Van Grootel et al. (2012), and we refer to that paper for technical details that are of importance in the present work.

It should be pointed out at the outset that in order to map the ZZ Ceti instability strip and its possible extension into the low-mass regime (assuming for the time being that the observed instabilities are indeed due to the same partial ionization mechanism at work in ZZ Ceti stars), only a detailed treatment of the H envelope is required, since this is where the excitation physics resides. Of course, the nature of the core composition as well as the precise envelope layering leaves distinct signatures on the distribution of excited periods (this is the realm of adiabatic seismology), but they do not affect the location of an unstable model in the log g–T_eff plane. In fact, a perfectly suitable mapping of the theoretical ZZ Ceti strip could be derived with the help of simple pure H envelope models.

This simplifying circumstance has allowed us to attack the problem with the help of simple white dwarf cooling sequences, with no a priori knowledge of their past history. We thus computed a first series of white dwarf sequences to connect with the work of Van Grootel et al. (2012) on the "normal" ZZ Ceti stars. These models are quite similar in the sense that they are made of a pure C core, surrounded by an He mantle containing a mass fraction log q(He) = −2.0, which is itself surrounded by a H envelope of mass fraction log q(H) = −4.0. Likewise, all of these sequences have been calculated under the assumption that convection in the H partial ionization zone could be modeled in terms of the so-called ML2/α = 1.0 version of the mixing-length theory, which is defined by a more potent efficiency than the one characterizing the standard ML1/α = 1.0 version (see Arcoragi & Fontaine 1980 for details). In addition, and of prime importance in the present mapping exercise, all of these sequences incorporate a treatment of the atmosphere that mimics the use of a non-gray atmosphere on the superficial H convection zone, namely, the feedback reaction of convection on the atmospheric structure, an effect neglected when using standard gray atmospheres in evolutionary calculations (see Van Grootel et al. 2012 for a recent discussion of this point, their Figure 5 in particular). The first series of C-core evolutionary models thus contains structures that differ only in their masses: 0.45, 0.40, 0.35, 0.30, 0.25, 0.20, 0.15, and 0.125 M_☉. We are fully aware that C-core white dwarfs with masses equal to or less than about 0.45 M_☉ are unlikely to exist in nature, but, again, this has no consequence on our mapping exercise, and these low-mass models allow us to connect with the more massive models (0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, and 1.2 M_☉) of Van Grootel et al. (2012).

In order to demonstrate explicitly that the instability region in the log g–T_eff plane is insensitive to the details of the interior structure of the equilibrium models used in the exercise, we computed a second family of evolutionary sequences for low-mass white dwarfs, but, this time, assuming a pure He core. These models all have an assumed pure H "thick" envelope containing log q(H) = −2.0 sitting on top of the He core, the rest being the same as before. They have masses given by 0.45, 0.40, 0.35, 0.30, 0.25, 0.20, 0.19, 0.18, 0.17, 0.16, and 0.15 M_☉. In addition, and for comparison purposes, we also calculated yet another similar sequence of mass 0.17 M_☉, but with a "thin" H envelope, with log q(H) = −3.7. We are aware that detailed modeling of low-mass He-core DA white dwarfs leads to a rather complicated situation as described by Driebe et al. (1998), Althaus et al. (2001), and, more recently, by Panei et al. (2007). Roughly speaking, He-core white dwarfs with masses less than about 0.2 M_☉ are expected to have a thick H envelope with residual stable burning at its base. Such H burning contributes significantly to the luminosity, even at relatively low effective temperatures, and leads to very long cooling timescales. In contrast, more massive He-core white dwarfs are expected to have reduced their H layer to the point where residual H burning is no longer a factor, and these objects are then expected to cool much faster than their less massive counterparts. We did not take into account residual H burning in our family of simple cooling models with a thick H envelope, so our cooling rates are certainly inadequate. However, as mentioned above, this does not affect our stability survey. We must point out, in this context, that residual H burning could lead, in principle, to pulsational instabilities through the -mechanism (as already noted by Steinfadt et al. 2010 and Hermes et al. 2012). However, this mechanism cannot explain the periods observed in the newly discovered low-mass pulsating DA white dwarfs. Indeed, only periods comparable to the local thermal timescale in the driving region can be generally excited in stars. In the present case, the potential driving region for the -mechanism, the base of the H envelope, is located quite deep in the star, where the local thermal timescale is very large (orders of magnitude larger) compared to the observed periods.

3.1. The Blue and Red Edge of the Theoretical ZZ Ceti Instability Strip

3.2. Universality of the Instability Domain

In the course of our survey, we first explicitly verified our assertion that the structural properties do not affect the question of mode stability, given DA models with the same values of log g and T_eff. This is shown in Figure 1, where our evolutionary tracks are plotted in the relevant domain of the spectroscopic HR diagram. For each track, we pulsated a number of models in order to find a blue edge (and a red edge) of a possible instability strip with modes excited by the convective driving mechanism associated with the H partial ionization zone. This was done, as a first step, with the nonadiabatic code MAD in its FC option. Only dipole modes were considered in this search.

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A blue edge corresponding to the straight extension of the blue side of the known theoretical ZZ Ceti instability strip into the low-mass regime was found. This is illustrated by the blue features in Figure 1. The dotted blue curve on the left and superposed on the evolutionary tracks connects the blue edges of each of the C-core tracks (dotted black curves). The locations of the latter were found by interpolation in log g and T_eff between the last stable model on a sequence and the first unstable model.

As a comparison, the blue dots on the left define the corresponding blue edges for each of the He-core tracks with log q(He) = −2.0. Likewise, the open blue circle on the left defines the blue edge of the He-core sequence with a thin H envelope (the appropriate track is plotted as a dashed curve). Clearly, in view of the finite resolution of the evolutionary sequences, the blue edge is indeed independent of total mass, core composition, or envelope layering for models with the same values of log g and T_eff. That is not to say that the spectra of excited periods are the same. On the contrary, these spectra bear the specific signature of the overall structure of the various models.

To the right of the blue edge depicted in Figure 1, all models are unstable according to our nonadiabatic calculations, including those belonging to the 0.15 M_☉ He-core sequence in the right upper corner of the diagram. As described in the previous subsection, this is a well-known shortcoming of nonadiabatic calculations on convective stars in general: the red edge is predicted to be much cooler than the empirical value. In our present approach with FC convection (and this is also true in TDC), the nonadiabatic red edge along these various sequences is found to be well below 7000 K, the limiting value considered in Figure 1.

However, if we are correct in our suggestion that the red edge is really due to energy leakage through the atmosphere, then the application of the criterion τ_th ≃ P_crit readily leads to the red features in Figure 1 for an assumed value of ℓ = 1. We thus find a red edge that is only a few to several hundred degrees cooler than the blue edge in this part of the HR diagram. Furthermore, and of immediate interest, Figure 1 also demonstrates that, as in the case of the blue edge, the location of the proposed red edge in the log g–T_eff plane is insensitive to mass, core composition, or envelope layering.

3.3. Main Results

The main results of our mapping exercise are presented in Figure 2. We find that the "normal" ZZ Ceti instability strip (associated with convective driving) extends well into the low-mass regime, covering some four orders of magnitude in surface gravity. Our best estimate of the blue edge of the strip is provided by the blue solid curve obtained in TDC. It is systematically hotter than the prediction based on the FC approximation (the blue dashed curve). Likewise, our best estimate of the red edge (based on the criterion τ_th ≃ P_crit) is illustrated by the red solid curve on the right. We thus find a relatively narrow theoretical instability strip, which gets narrower with decreasing surface gravity as the predicted red edge shows a larger slope than the blue edge in this log g–T_eff diagram. The illustrated theoretical strip specifically refers to the ML2/α = 1.0 convection and to g-modes with ℓ = 1. A slight shift of the boundaries of the strip to higher effective temperatures is found for ℓ = 2 modes (ΔT_eff ≃ 110 K at the blue edge and ΔT_eff ≃ 90 K at the red edge for M = 0.6 M_☉, log g ≃ 8.0 models).

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A comparison of our predicted instability strip with the empirical data is particularly encouraging, as can be seen in Figure 2. On the first account, we plotted the positions of 48 ZZ Ceti stars (the filled circles) as well as those of 84 DA white dwarfs that were observed not to vary (the open circles). These objects belong to the bright spectroscopic sample of DA white dwarfs that have been analyzed over the years at the Université de Montréal with the rigorous and homogeneous method developed by Bergeron et al. (1995), and followed subsequently by Fontaine et al. (2001, 2003), Bergeron et al. (2004), and Gianninas et al. (2005, 2006, 2007). This sample provides the best available view of the standard empirical ZZ Ceti instability strip.

On the second account, we also plotted the positions of the three ELM pulsators (the filled circles with error crosses) reported by Hermes et al. (2012) and J. J. Hermes et al. (2012, in preparation), as well as those of 12 low-mass DA white dwarfs observed not to vary (the solid crosses). In the latter case, nine of these objects come from the McDonald Observatory survey as reported in Hermes et al. (2012) and J. J. Hermes et al. (2012, in preparation), four come from Steinfadt et al. (2012), with one object in common. Those are stars with detection limits less than 10 mmag in white light. In all cases, the atmospheric parameters have been inferred in a homogeneous way within the ELM project (see Brown et al. 2012 and references therein), using model atmospheres provided by D. Koester, equivalent in principle to those of P. Bergeron. In addition, the Koester models are based on the same parameterization of the mixing-length theory as that calibrated initially by Bergeron et al. (1995), the ML2/α = 0.6 version. This implies that the relative positions of the circles (solid and open) and of the crosses in Figure 2 should be secure, unless there are some systematic effects between the two sets of models or in the reduction and fitting procedure between the two groups of modelers. We are not aware of such effects, and we consider the actual locations of the stars plotted in Figure 2 at face value.

The comparison between our theoretical instability strip and the empirical data shown in Figure 2 is most noteworthy. Our results are perfectly consistent with the idea that the ZZ Ceti instability strip is pure, including its extension into the low-mass regime (see, e.g., Section 4 of Fontaine & Brassard 2008 for a further discussion of this issue). All of the pulsators considered in the figure, including the three newly found ELM stars, fall within the predicted strip or close to one of its edges, while all of the objects found not to vary fall outside the strip.

We also point out that our TDC blue edge is hotter than our FC blue edge by about 200 K on average. This is a significant difference, given that the width of the predicted instability strip ranges from about 1000 K at log g ∼ 8 down to about 650 K at log g ∼ 6.5. The TDC blue edge obtained in this paper provides a qualitatively superior fit to the observational data than the FC blue edge does, particularly in the domain of normal ZZ Ceti stars. Still, the differences between the predictions of TDC and those of FC are not dramatically different (see Van Grootel et al. 2012 for further details on this). Given that the FC approximation is unjustified in white dwarfs, this result may seem surprising. Saio (2012) has recently provided an explanation for this state of affairs.

In the list of 40 DA white dwarfs in the ELM sample (Table 1 of Brown et al. 2012), we have searched and identified eight objects that fall close to one or the other of the boundaries of our theoretical instability strip. These stars are identified by dotted crosses in Figure 2, and their atmospheric parameters are listed in Table 1. To our knowledge, they have yet to be monitored for variability, and we suggest that they are worthwhile candidates for future searches.

Table 1. Candidates for Variability

Object	g	log g	Teff	Reference
	(mag)	(cm s−2)	(K)
SDSS J0016−0043	19.70	7.40 ± 0.08	11190 ± 260	Brown et al. (2012)
SDSS J0117+2513	19.30	7.69 ± 0.08	10710 ± 190	Brown et al. (2012)
SDSS J0421+8302	19.79	7.94 ± 0.07	11870 ± 200	Brown et al. (2012)
SDSS J0702+1110	16.06	6.00 ± 0.12	8800 ± 600	Brown et al. (2012)
SDSS J0918+3753	18.61	7.91 ± 0.08	12060 ± 250	Brown et al. (2012)
SDSS J1235+4750	19.29	7.67 ± 0.08	10600 ± 200	Brown et al. (2012)
SDSS J1614+1912	16.19	5.64 ± 0.12	8590 ± 540	Brown et al. (2012)
SDSS J2337+1533	19.68	7.63 ± 0.07	11700 ± 180	Brown et al. (2012)

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3.4. Discussion

We found it quite instructive to examine the convection and pulsation properties of models along evolutionary sequences. As a representative case, we focus here on the He-core sequence with M = 0.2 M_☉ and a thick H envelope with log q(H) = −2.0. In particular, we show in Figure 3 the spectrum of excited dipole g-modes as it develops with cooling along that sequence. These results have been obtained with the nonadiabatic code MAD in TDC. The general pattern is qualitatively similar to that obtained in more massive ZZ Ceti star models (see, e.g., Figure 7 of Van Grootel et al. 2012). It is characterized by a relatively important broadening of the band of unstable modes immediately below the blue edge, followed by a subsequent reduction in width as the effective temperature decreases. The predicted pulsational instabilities persist well into the regime of very low effective temperatures compared to the empirical evidence. This is the typical "failure" of detailed nonadiabatic calculations in the presence of convection as discussed above. It is possible that the situation might improve on this front by introducing additional physics into our calculations, such as turbulent pressure, for example. On the other hand, if we are correct about the question of energy leakage, then the red edge would be located in effective temperature close to the value indicated by the dotted vertical line. We also note that the lowest order excited mode first appears in the model with T_eff = 8461 K and has a value of k = 4. We also point out that low-order p-modes are excited in our models, exactly as in the case of normal mass ZZ Ceti stars (e.g., Saio et al. 1983), but such modes have never been observed. Since the g-modes with the lowest possible values of k (1, 2, ...) are not excited in our low-mass DA models, there is a gap of stability between the excited p-modes and the band of excited g-modes.

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The series of Figures 4(a)–(c) reveals some interesting properties of the structure of the outer H convection zone that develops with cooling in the reference sequence of interest. The focus is on the part of the sequence that is relevant for pulsations. The tiny dots in Figure 4(a) show isocontours of the opacity in units of log κ. It can be seen that recombination of H leads to a huge opacity peak culminating above a value of log κ = 5.5 in the coolest models depicted in the figure. As a consequence of increasing opacity with cooling, the material becomes convective, and the boundaries (top and bottom) of the convection zone are indicated in the plot by the small filled circles. It is important to realize that the top of the H convection zone always resides very near the photosphere, which moves inward with cooling on the fractional mass depth scale used here as the ordinate. At the same time, the base of the convection zone, which resides in the unobservable optically thick layers, sinks into the star with cooling, at least in the illustrated range of effective temperature. Our TDC blue edge for that sequence, as well as our estimate of the red edge based on the energy leakage argument, is indicated by the dashed vertical lines. All of the action in terms of pulsations is confined between these boundaries.

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Figure 4(b) shows the equivalent, but for the ratio of the convective to total flux, F_c/F. It can be seen that the profile of F_c/F at the blue edge rises progressively outward from the base of the convection zone, reaches a maximum value of ∼0.97 about halfway into the zone, and subsequently decreases toward the top of the zone. This convection flux profile is shallower than that found in a more massive ZZ Ceti star model (see Figure 3(a) of Van Grootel et al. 2012 showing that F_c/F ∼ 0.99 at the base of the convection zone for a 0.6 M_☉ model). This can be explained by the fact that our current 0.2 M_☉ model is a lot less degenerate than the former. This implies less efficient electron thermal conductivity in the core, a shallower temperature gradient between the core and the surface, and a more extended but less efficient convection zone in the envelope for the less massive, less degenerate model. However, the driving mechanism is fundamentally the same in the 0.2 and 0.6 M_☉ models: the blue edge appears when the convection zone is sufficiently deep and developed to efficiently carry the energy of the pulsation mode.

Figure 4(c), of the same general format as the two previous plots, brings together the various timescales of interest. In particular, the tiny dots indicate various values of the local convective turnover timescale. The figure demonstrates, among other things, that the convective turnover timescale at the base of the convection zone of the model at the blue edge is about 10 s, much smaller indeed than the periods excited in that model (see Figure 3). For its part, the solid curve represents the thermal timescale τ_th at the base of the convection zone. It grows rapidly with decreasing effective temperature and ultimately overtakes the critical period P_crit for ℓ = 1 g-modes, shown here by the dashed curve. According to the energy leakage argument, the red edge is defined by the crossover of these two curves.

Finally, in Figure 5, we show representative period spectra of excited dipole g-modes for models specifically computed for each of the three known pulsating ELM white dwarfs. These models are simply meant to verify if we can qualitatively account for the observed periods; they are not seismic fits. Hence, for each ELM pulsator, we calculated a static ML2/α = 1.0 model specified by the values of log g and T_eff tabulated in Brown et al. (2012), and an assumed thick H layer with log q(H) = −2.0 sitting on top of a pure He core. For SDSS J1518+0658, we also computed another model, similar on all accounts, except for the fact that it has a thin H layer with log q(H) = −4.0. This is to illustrate specifically our point made above that models with the same values of log g and T_eff are part of the same instability strip, but that their period spectra still bear the signature of their internal structure (which may be different).

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We pulsated these individual models in TDC, and the results are shown in Figure 5. For SDSS J1840+6423 (first row in the figure), the band of excited periods spans a range from ∼860 s (k = 7) to ∼7833 s (k = 81).⁵ In the case of the SDSS J1518+0658 models, the band of excited periods spans a range from ∼406 s (k = 2) to ∼4383 s (k = 46) for the thin envelope structure (second row), and a range from ∼530 s (k = 5) to ∼4657 s (k = 62) for the thick envelope structure (third row). We point out that the average period spacing between adjacent modes is significantly less for the thick than for the thin H envelope model, in line with the detailed adiabatic investigation of Brassard et al. (1992) on standard ZZ Ceti stars. This clearly illustrates that the predicted period spectra do bear the signature of the structure of the models, in this case a different H envelope mass. For the SDSS J1112+1117 model (lower row), the band of excited periods spans a range from ∼1235 s (k = 10) to ∼5661 s (k = 53).

A comparison with the available period data in Figure 5 shows that our approach can account qualitatively for these relatively long periods. The observed periods are still very preliminary and come from discovery light curves where it is not easy to correctly identify the right aliases. This comparison with theoretical periods has to be seen as being only qualitative here. Furthermore, in all three cases, the observational data are quite sparse compared to the predicted period distributions, very much like the situation encountered from the ground for long period g-mode oscillations in cooler hot B subdwarf stars (see, e.g., Randall et al. 2006a, 2006b). This is not a cause for concern at this time, given that the pulsating ELM DA white dwarfs have not yet been observed over long campaigns. It is likely that higher sensitivity observations, beyond the discovery light curves, will reveal many more modes.

The detection of g-mode pulsational instabilities in three extremely low-mass, He-core DA white dwarfs by Hermes et al. (2012) and J. J. Hermes et al. (2012, in preparation) is a very significant discovery. As foreseen initially by Steinfadt et al. (2010), such a discovery opens the way for the eventual application of asteroseismology to a whole domain of the spectroscopic HR diagram that has remained out of reach so far. In particular, it is hoped that asteroseismology will one day reveal the detailed internal structure and, perhaps, even constrain the age of an ELM white dwarf pulsator.

We have presented in this paper the results of a detailed stability study aimed at (1) explaining the very existence of the pulsating ELM white dwarfs, (2) accounting for their observed periods (at the qualitative level), and (3) assigning them to a known (or new) instability strip in the spectroscopic HR diagram. For those purposes, we built on the experience of Van Grootel et al. (2012), and first computed several evolutionary sequences of low-mass DA white dwarfs which incorporate the feedback effect of convection on the atmospheric structure. This effect—usually neglected in evolutionary calculations of white dwarf stars—is important for pulsation studies, particularly for accurately mapping the boundaries of the instability strip. In addition, we took advantage of the nonradial nonadiabatic code MAD, which incorporates a detailed treatment of TDC (Dupret 2002; Grigahcène et al. 2005). This code has been adapted successfully to pulsate DB white dwarf models (Dupret et al. 2008; Quirion et al. 2008), and, more recently, to pulsate DA models as well (Van Grootel et al. 2012). It provides the best available physical description of g-mode pulsations in convective white dwarf models, particularly at or near the blue edge of the instability strip. However, in its current implementation, MAD fails (as all other nonadiabatic codes used so far) to provide a realistic red edge. In this connection, we have tested the idea of Hansen et al. (1985) that energy leakage might be the physical mechanism responsible for the red edge of the ZZ Ceti instability strip.

The results of our mapping exercise in the HR diagram are illustrated in Figure 2. The comparison of the location, extent, and shape of our predicted instability strip with the empirical data is most encouraging. The expected blue edge (obtained from detailed nonadiabatic calculations in the TDC mode), as well as the expected red edge (obtained from the energy leakage argument), indeed accounts remarkably well for the data, including the positions of the three ELM pulsators. We did not expect this result at the outset, especially for the red edge. This suggests that we may have put our finger on the correct mechanism that is physically responsible for the observed red edge. That mechanism has remained elusive ever since the discovery of the very first pulsating DA white dwarf in 1968.

The good agreement between expectations and observations displayed in Figure 2 also provides a very credible calibration of the mixing-length theory in DA white dwarfs, since both edges are sensitive to the assumed convective efficiency. As explained above, our pulsation considerations can be used to measure the convective efficiency in layers near the base of the convection zone in these white dwarf models. On the other hand, spectroscopic and photometric constraints impose a different calibration in the atmospheric layers (Bergeron et al. 1995). The asteroseismic calibration does depend on the location of the stars in the HR diagram and, therefore, is related to the atmospheric calibration. In the present situation, the available empirical data are based on the ML2/α = 0.6 version. With that constraint, our asteroseismic approach leads rather to ML2/α = 1.0, which could be perceived negatively as an incoherence, but should be seen instead as the demonstration that the convective efficiency in a white dwarf increases with depth as predicted by hydrodynamic simulations (Ludwig et al. 1994). We note that for evolutionary calculations, the asteroseismic calibration of the mixing-length convective efficiency is the one to use. We note also that, should the empirical atmospheric parameter data be revised, this will imply a concomitant revision of the asteroseismic calibration.

We have thus established in this paper that the newly discovered ELM DA pulsators are genuine ZZ Ceti stars: they belong to the ZZ Ceti instability strip, and the driving mechanism responsible for exciting some of their g-modes is the same as the one at work in more massive pulsating DA white dwarfs. The relatively long periods observed in these stars are readily explained by the low values of the surface gravity. In this respect, Figure 5 provides a preview of what the spectrum of excited dipole modes might look like for each of the three ELM pulsators. Clearly, the predicted periods are consistent with the (sparse) observations and account for them at the qualitative level. We also recall that there exists a correlation between the excited period and the effective temperature or surface gravity for ZZ Ceti pulsators, as best seen in Figure 24 of Fontaine & Brassard (2008). Without pushing too much in the sense of overinterpretation, there could already be hints in the available data that the correlation is again at work. For instance, we find that the average observed period is the smallest in SDSS J1112+1117 and the largest in SDSS J1840+6423, consistent with their position near the blue edge and near the red edge, respectively. Likewise, SDSS J1518+0658 is rather nearer the red edge than the blue edge, but its surface gravity, being higher than in the two previous stars, compensates for this partly in the sense of producing shorter excited periods, also as expected.

Finally, we have provided in Table 1 a list of candidates for variability, which are ELM white dwarfs that have atmospheric parameters falling close to the boundaries of our theoretical instability strip.

The referee has kindly pointed out to us the existence of an independent paper dedicated to the seismic properties of low-mass, He-core white dwarfs (Corsico et al. 2012). We were not aware of that very recent work. That paper presents a pulsational study of ELM white dwarfs based on detailed evolutionary models, with a particular emphasis on adiabatic properties. To a large extent, our paper is complementary to theirs, as we focused exclusively on the nonadiabatic aspects of the problem. It should be pointed out that Corsico et al. (2012) also carried out nonadiabatic calculations which, we find, are in qualitative agreement with our findings. There are, however, significant differences in that (1) they used the FC approximation in their nonadiabatic computations, (2) they assumed a different convective efficiency (ML1/α = 1.6) in their models than we did in ours (ML2/α = 1.0), (3) they ignored the important effect of the feedback of convection on the atmospheric structure, and (4) they did not provide an estimate for the red edge. As a result, they could not map the ZZ Ceti instability strip and account for the empirical data (including the three newly found ELM pulsators) as we did in this paper.

This work was supported in part by the NSERC of Canada. G.F. also acknowledges the contribution of the Canada Research Chair Program.