Issue 
A&A
Volume 569, September 2014



Article Number  A106  
Number of page(s)  21  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201424352  
Published online  30 September 2014 
Pulsating lowmass white dwarfs in the frame of new evolutionary sequences
I. Adiabatic properties
^{1} Grupo de Evolución Estelar y Pulsaciones. Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina
^{2} IALP – CONICET, Argentina
email: acorsico@fcaglp.unlp.edu.ar, althaus@fcaglp.unlp.edu.ar
Received: 6 June 2014
Accepted: 31 July 2014
Context. Many lowmass white dwarfs with masses M_{∗}/M_{⊙} ≲ 0.45, including the socalled extremely lowmass white dwarfs (M_{∗}/M_{⊙} ≲ 0.20 − 0.25), have recently been discovered in the field of our Galaxy through dedicated photometric surveys. The subsequent discovery of pulsations in some of them has opened the unprecedented opportunity of probing the internal structure of these ancient stars.
Aims. We present a detailed adiabatic pulsational study of these stars based on full evolutionary sequences derived from binary star evolution computations. The main aim of this study is to provide a detailed theoretical basis of reference for interpreting present and future observations of variable lowmass white dwarfs.
Methods. Our pulsational analysis is based on a new set of Hecore whitedwarf models with masses ranging from 0.1554 to 0.4352 M_{⊙} derived by computing the nonconservative evolution of a binary system consisting of an initially 1 M_{⊙} ZAMS star and a 1.4 M_{⊙} neutron star. We computed adiabatic radial (ℓ = 0) and nonradial (ℓ = 1,2) p and g modes to assess the dependence of the pulsational properties of these objects on stellar parameters such as the stellar mass and the effective temperature, as well as the effects of element diffusion.
Results. We found that for white dwarf models with masses below ~ 0.18 M_{⊙}, g modes mainly probe the core regions and p modes the envelope, therefore pulsations offer the opportunity of constraining both the core and envelope chemical structure of these stars via asteroseismology. For models with M_{∗} ≳ 0.18 M_{⊙}, on the other hand, g modes are very sensitive to the He/H compositional gradient and therefore can be used as a diagnostic tool for constraining the H envelope thickness. Because both types of objects have not only very distinct evolutionary histories (according to whether the progenitor stars have experienced CNOflashes or not), but also have strongly different pulsation properties, we propose to define white dwarfs with masses below ~ 0.18 M_{⊙} as ELM (extremely lowmass) white dwarfs, and white dwarfs with M_{∗} ≳ 0.18 M_{⊙} as LM (lowmass) white dwarfs.
Key words: asteroseismology / stars: oscillations / white dwarfs / stars: evolution / stars: interiors / stars: variables: general
© ESO, 2014
1. Introduction
According to the current theory of stellar evolution, most low and intermediatemass (M_{∗} ≲ 11 M_{⊙}; Siess 2007) stars that populate our Galaxy will end their lives as white dwarf (WD) stars. These old and compact stellar remnants have encrypted inside them a precious record of the evolutionary history of the progenitor stars, providing a wealth of information about the evolution of stars, star formation, and the age of a variety of stellar populations, such as our Galaxy and open and globular clusters (GarcíaBerro et al. 2010; Althaus et al. 2010). Almost 80% of the WDs exhibit Hrich atmospheres; they define the spectral class of DA WDs. The mass distribution of DA WDs peaks at ~ 0.59 M_{⊙}, and also exhibits high and lowmass components (Kepler et al. 2007; Tremblay et al. 2011; Kleinman et al. 2013). The population of lowmass WDs has masses lower than 0.45 M_{⊙} and peaks at ~ 0.39 M_{⊙}. In recent years, many lowmass WDs with M_{∗} ≲ 0.20−0.25 M_{⊙} have been detected through the ELM survey and the SPY and WASP surveys (see Koester et al. 2009; Brown et al. 2010, 2012; Maxted et al. 2011; Kilic et al. 2011, 2012); they are commonly referred to as extremely lowmass (ELM) WDs.
Lowmass WDs are probably produced by strong massloss episodes at the red giant branch (RGB) phase before the Heflash onset. Since the ignition of He is avoided, these WDs are expected to harbour He cores, in contrast to average mass WDs, which all probably contain C/O cores. For solar metallicity progenitors, massloss episodes must occur in binary systems through masstransfer, since singlestar evolution is not able to predict the formation of these stars in a Hubble time. This evolutionary scenario is confirmed by the fact that most of lowmass WDs are found in binary systems (e.g., Marsh et al. 1995), and usually as companions to millisecond pulsars (van Kerkwijk et al. 2005). In particular, binary evolution is the most likely origin for ELM WDs (Marsh et al. 1995). The evolution of lowmass WDs is strongly dependent on their stellar mass and the occurrence of element diffusion processes. Althaus et al. (2001) and Panei et al. (2007) have found that element diffusion leads to a dichotomy in the thickness of the H envelope, which translates into a dichotomy in the age of lowmass Hecore WDs. Specifically, for stars with M_{∗} ≳ 0.18 − 0.20 M_{⊙}, the WD progenitor experiences multiple diffusioninduced CNO thermonuclear flashes that engulf most of the H content of the envelope, and as a result, the remnant enters the final cooling track with a very thin H envelope. As a result, the star is unable to sustain stable nuclear burning while it cools, and the evolutionary timescale is rather short (~ 10^{7} yr)^{1}. On the other hand, if M_{∗} ≲ 0.18−0.20 M_{⊙}, the WD progenitor does not experience H flashes at all, and the remnant enters its terminal cooling branch with a thick H envelope. This is thick enough for residual H nuclear burning to become the main energy source, which ultimately slows down the evolution, in which case the cooling timescale is of about ~ 10^{9} yr. The age dichotomy has also been suggested by observations of the lowmass Hecore WDs that are companions to millisecond pulsars (Bassa et al. 2003; Bassa 2006).
While some information such as surface chemical composition, temperature, and gravity of WDs can be inferred from spectroscopy, the internal structure of these compact stars can be unveiled only by means of asteroseismology, an approach based on the comparison between the observed pulsation periods of variable stars and appropriate theoretical models (Winget & Kepler 2008; Fontaine & Brassard 2008; Althaus et al. 2010). The first variable WD, HL Tau 76, was serendipitously discovered by Landolt (1968). Since then, many pulsating WDs have been detected. At present, there are several families of pulsating WDs known, which span a wide range in effective temperature and gravity (Fig. 1). Among them, the variables ZZ Ceti or DAVs (almost pure H atmospheres, 12 500 ≳ T_{eff} ≳ 11 000 K) are the most numerous ones. The other classes comprise the DQVs (atmospheres rich in He and C, T_{eff} ~ 20 000 K), the variables V777 Her or DBVs (atmospheres rich in He, 29 000 ≳ T_{eff} ≳ 22 000 K), and the variables GW Vir (atmospheres dominated by C, O, and He) that include the DOVs and PNNVs objects (180 000 ≳ T_{eff} ≳ 65 000 K). WD asteroseismology allows us to place constraints not only on global quantities such as gravity, effective temperature, or stellar mass, they provide, in addition, information about the thickness of the compositional layers, the core chemical composition, the internal rotation profile, the presence and strength of magnetic fields, the properties of the outer convective regions, and several other interesting properties (for a recent example in the context of ZZ Ceti stars, see Romero et al. 2012, 2013, and references therein).
Fig. 1 Location of the several classes of pulsating WD stars in the log T_{eff} − log g plane, marked with dots of different colours. In parenthesis we include the number of known members of each class. Two postVLTP (Very Late Thermal Pulse) evolutionary tracks are plotted for reference. We also show the theoretical blue edge of the instability strip for the GW Vir stars, V777 Her stars, the DQV stars (Córsico et al. 2006, 2009a,b, respectively), the ZZ Ceti stars (Fontaine & Brassard 2008), and the pulsating lowmass WDs. For this last class, we show the blue edge according to Hermes et al. (2013a; dotted red line), Córsico et al. (2012; solid red line), and Steinfadt et al. (2010; dashed red line). For reference, we also include two evolutionary tracks of lowmass Hecore white dwarfs from Althaus et al. (2013). Small black dots correspond to lowmass WDs that are nonvariable or have not been observed yet to assess variability. 

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The possible variability of lowmass WDs was first discussed by Steinfadt et al. (2010), who predicted that nonradial pulsation g modes with long periods should be excited in WDs with masses below ~ 0.2 M_{⊙}. The various attempts to find pulsating objects of this type were unsuccessful, however (Steinfadt et al. 2012). This situation drastically changed with the exciting discovery of the first pulsating lowmass WD, SDSS J184037.78+642312.3, by Hermes et al. (2012). Subsequent searches resulted in the detection of two additional pulsating objects, SDSS J111215.82+111745.0 and SDSS J151826.68+065813.2 (Hermes et al. 2013b), and finally another two, SDSS J161431.28+191219.4 and SDSS J222859.93+362359.6 (Hermes et al. 2013a), all of them belonging to the DA spectral class and with masses below 0.24 M_{⊙}. At present, this small group of five stars makes up a new, separate class of pulsating WDs. The effective temperatures of these stars are found to be between 10 000 K and 7800 K, which means that they are the coolest pulsating WDs known to date (see Fig. 1). Another distinctive feature of these stars is the length of their pulsation periods, Π ≳ 1180 s, longer than the periods found in ZZ Ceti stars (100 ≲ Π ≲ 1200 s). The period at Π = 6235 s detected in the power spectrum of SDSS J222859.93+362359.6 is the longest period ever measured in a pulsating WD star. On the other hand, SDSS J111215.82+111745.0 exhibits two short periods, at Π ~ 108 s and Π ~ 134 s, which might be caused by nonradial p modes or radial modes (ℓ = 0). If so, this would be the first detection of p or radial modes in a pulsating WD star. Finally, the existence of many nonvariable stars in the region where pulsating objects are found (Fig. 1) suggests that the instability strip for lowmass WDs probably is not pure (Hermes et al. 2013a). This might be a hint that lowmass WDs that populate the same region in the log T_{eff} − log g plane have substantially different internal structures and consequently quite different evolutionary origins.
Selected properties of our Hecore WD sequences (final cooling branch) at T_{eff} ≈ 10 000 K: the stellar mass, the mass of H in the outer envelope, the time it takes the star to cool from T_{eff} ≈ 10 000 K to ≈ 8000 K, and the occurrence (or not) of CNO flashes on the early WD cooling branch.
The discovery of pulsating lowmass WDs constitutes a unique opportunity to probe the interiors of these stars and eventually to test their formation channels by employing the tools of asteroseismology. A first step in the theoretical study of these variable stars has been given by Córsico et al. (2012), who have thoroughly explored the gmode adiabatic pulsation properties of the lowmass Hecore WD models with masses in the range 0.17 − 0.46M_{∗} coming from highmetallicity progenitors (Z = 0.03) and singlestar evolution computations. These authors also performed nonadiabatic pulsation computations and found many unstable g and p modes approximately at the correct effective temperatures and the correct range of the periods observed in pulsating lowmass WDs. This result has later been confirmed by Van Grootel et al. (2013).
To fully exploit the asteroseismological potential of this type of stars, accurate and realistic stellar models of lowmass WDs are crucial. To assess the correct thermomechanical structure of the WD and the thickness of the H envelope left by the preWD stage, the complete evolutionary history of the progenitor stars must be fully accounted for. A relevant physical ingredient that must be considered during the WD cooling phase is element diffusion to consistently account for the evolving shape of the internal chemical profiles (and, in particular, the chemical transition regions). Last but not least, stable H burning, which is particularly relevant in the case of ELM WDs (and might play a role in driving the pulsations), must be taken into account as well.
Motivated by the asteroseismological potential of pulsating lowmass WDs, and stimulated by the discovery of the first five variable objects of this type, we report a further step in the theoretical study of these stars by exploring the adiabatic pulsation properties of a new set of lowmass, Hecore WD models with masses ranging from 0.1554 to 0.4352 M_{⊙} (including the mass range for ELM WDs, M_{∗}/M_{⊙} ≲ 0.20−0.25, see Table 1) presented by Althaus et al. (2013), which were derived by computing the nonconservative evolution of a binary system consisting of an initially 1 M_{⊙} ZAMS star and a 1.4 M_{⊙} neutron star for various initial orbital periods. We here extend the study of Córsico et al. (2012) in two ways. First, we explore not only the nonradial gmode pulsation spectrum of lowmas WD models, but we also study the nonradial p modes and the radial (ℓ = 0) pulsation spectrum. Second, thanks to the availability of five WD model sequences with progenitors that have not suffered from CNOflashes (see Table 1), we are now able to explore the pulsation properties of ELM WDs in detail, which are characterised by very thick H envelopes. This is at variance with the work of Córsico et al. (2012), in which only one WD sequence (with mass M_{∗} = 0.17 M_{⊙}) corresponded to a progenitor that did not experience H flashes. In this paper, we examine the possible differences in the pulsation properties of models with stellar masses near ~ 0.18−0.2 M_{⊙}, which could be used as a seismic tool to distinguish stars that have undergone CNO flashes in their earlycooling phase from those that have not. We additionally discuss how our models match the observed properties of the known five pulsating lowmass WD stars. In particular, we determine whether our models are able to account for the short periods exhibited by SDSS J111215.82+111745.0 and evaluate the possibility that these modes might be p modes and/or radial modes. We also examine the hypothesis that one of these stars, SDSS J222859.93+362359.6, is not a genuine ELM WD, but instead a He preWD star undergoing a CNO flash episode. We defer to a second paper of this series a thorough nonadiabatic exploration of our complete set of Hecore WD models.
The paper is organised as follows: In Sect. 2 we briefly describe our numerical tools and the main ingredients of the evolutionary sequences we use to assess the pulsation properties of lowmass Hecore WDs. In Sect. 3 we present our pulsation results in detail. Section 4 is devoted to assessing how our models match the observations. Finally, in Sect. 5 we summarise our main findings.
2. Computational tools and evolutionary sequences
2.1. Evolutionary code and input physics
The evolutionary WD models employed in our pulsational analysis were generated with the evolutionary code LPCODE, which produces complete and detailed WD models that incorporate updated physical ingredients. While detailed information about LPCODE can be found in Althaus et al. (2005, 2009, 2013) and references therein, we list below only the ingredients employed that are relevant for our analysis of lowmass, Hecore WD stars.

We adopted the standard mixing length theory (MLT) for convection (see, e.g., Kippenhahn et al. 2013) with the free parameter α = 1.6. With this value, the present luminosity and effective temperature of the Sun, log T_{eff} = 3.7641 and L_{⊙} = 3.842 × 10^{33} erg s^{1}, at an age of 4570 Myr, are reproduced by LPCODE when Z = 0.0164 and X = 0.714 are adopted, in agreement with the Z/X value of Grevesse & Sauval (1998). A different convective efficiency that could change the temperature profiles of our models would have no direct impact on their adiabatic pulsation properties.

We assumed the metallicity of the progenitor stars to be Z = 0.01.

Radiative opacities for arbitrary metallicity in the range from 0 to 0.1 were taken from the OPAL project (Iglesias & Rogers 1996). At low temperatures, we used the updated molecular opacities with varying C/O ratios computed at Wichita State University (Ferguson et al. 2005) that were presented by Weiss & Ferguson (2009).

The conductive opacities were taken from Cassisi et al. (2007).

The equation of state during the mainsequence evolution is that of OPAL for a H and Herich composition.

Neutrino emission rates for pair, photo, and bremsstrahlung processes were taken from Itoh et al. (1996), and for plasma processes we included the treatment of Haft et al. (1994).

For the WD regime we employed an updated version of the Magni & Mazzitelli (1979) equation of state.

The nuclear network takes into account 16 elements and 34 thermonuclear reaction rates for ppchains, CNO bicycle, He burning, and C ignition.

Timedependent diffusion caused by gravitational settling and chemical and thermal diffusion of nuclear species was taken into account following the multicomponent gas treatment of Burgers (1969).

Abundance changes were computed according to element diffusion, nuclear reactions, and convective mixing. This detailed treatment of abundance changes by different processes during the WD regime constitutes a key aspect in evaluating the importance of residual nuclear burning for the cooling of lowmass WDs.

For the WD regime and for effective temperatures lower than 10 000 K, outer boundary conditions for the evolving models were derived from nongrey model atmospheres (Rohrmann et al. 2012).
2.2. Pulsation codes
We carried out a detailed adiabatic radial (ℓ = 0) and nonradial p and gmode (ℓ = 1,2) pulsation analysis. The pulsation computations were performed with the adiabatic version of the pulsation code LPPUL that is described in detail in Córsico & Althaus (2006), which is coupled to the LPCODE evolutionary code. The pulsation code is based on a general NewtonRaphson technique that solves the full fourthorder set of real equations and boundary conditions that govern linear, adiabatic, radial, and nonradial stellar pulsations following the dimensionless formulation of Dziembowski (1971) (see Unno et al. 1989). The prescription we followed to assess the run of the BruntVäisälä frequency (N) for a degenerate environment typical of the deep interior of a WD is the socalled Ledoux modified treatment (Tassoul et al. 1990). For our exploratory stability analysis described in Sect. 4.2, we employed the nonadiabatic version of the code LPPUL described in Córsico et al. (2006). The code solves the full sixthorder complex system of linearised equations and boundary conditions as given by Unno et al. (1989). The caveat is that our nonadiabatic computations rely on the frozenconvection approximation, in which the perturbation of the convective flux is neglected. While this approximation is known to give unrealistic locations of the gmode red edge of instability, it leads to satisfactory predictions for the location of the blue edge of the ZZ Ceti (DAV) instability strip (see, e.g., Brassard & Fontaine 1999; Van Grootel et al. 2012) and also for the V777 Her (DBV) instability strip (see, for instance, Beauchamp et al. 1999; Córsico et al. 2009a).
2.3. Evolutionary sequences
To derive realistic configurations for the lowmas Hecore WDs, Althaus et al. (2013) mimicked the binary evolution of progenitor stars. Since Hshell burning is the main source of star luminosity during most of the evolution of ELM WDs, computating realistic initial WD structures is a fundamental requirement, in particular for correctly assessing the Henvelope mass left by progenitor evolution (see Sarna et al. 2000).
Binary evolution was assumed to be fully nonconservative, and the loss of angular momentum through mass loss, gravitationalwave radiation, and magnetic braking was considered. All of the Hecore WD initial models were derived from evolutionary calculations for binary systems consisting of an evolving lowmass component of initially 1 M_{⊙} and a 1.4 M_{⊙} neutron star as the other component. Metallicity was assumed to be Z = 0.01. A total of 14 initial Hecore WD models with stellar masses between 0.155 and 0.435 M_{⊙} were computed for initial orbital periods at the beginning of the Rochelobe phase in the range 0.9 to 300 d. While full details about the procedure to obtain the initial models are provided in Althaus et al. (2013), we repeat here the prescriptions used by these authors to obtain the Hecore WD models employed in this work. The formalism of Sarna et al. (2000) is followed. If M_{1} is the mass of the secondary (masslosing) star, M_{2} the mass of the neutron star (primary), and Ṁ_{1} the massloss rate of the secondary, the change of the total orbital angular momentum (J) of the binary system can be written as (1)where , , and are the angular momentum loss from the system through mass loss, gravitationalwave radiation, and magnetic braking (which is relevant when the secondary has an outer convection zone). To compute these quantities we follow Sarna et al. (2000) (see also Muslimov & Sarna 1993): where a is the semiaxis of the orbit and R_{1} the radius of the secondary. All quantities are given in solar units. The massloss rate from the secondary is calculated as in Chen & Han (2002). Mass loss is considered as long as the secondary fills its Roche lobe r_{L}, given by (5)where q = M_{1}/M_{2} is the mass ratio. The semiaxis of the orbit is found by integrating the equation for the rate of change of a. If mass lost by the secondary is completely lost from the system, meaning that nothing of the mass lost by the secondary is accreted by the primary, a is given by (see Muslimov & Sarna 1993) (6)Mass loss is continued until the secondary star shrinks within its Roche lobe.
Fig. 2 log T_{eff} − log g diagrams for the Hecore WD sequences computed in Althaus et al. (2013). Sequences with masses in the range 0.18 ≲ M_{∗} ≲ 0.4 undergo CNO flashes during the earlycooling phase, which leads to the complex loops in the diagram. Green squares and magenta triangles correspond to the observed postRGB lowmass stars from Silvotti et al. (2012) and Brown et al. (2013), and filled blue circles correspond to the five pulsating lowmass WDs detected so far (Hermes et al. 2013a). Numbers in the left upper corner of each panel correspond to the stellar mass at the WD stage. 

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In Table 1, we provide some main characteristics of the whole set of Hecore WD models. The evolution of these models was computed down to the range of luminosities of cool WDs, including the stages of multiple thermonuclear CNO flashes during the beginning of the cooling branch. The unstable H burning in CNO flashes occurs at the innermost tail of the Habundance distribution. As the WD evolves along the cooling branch, chemical diffusion carries some H inwards to hotter and Crich layers, where it burns unstably. In this region, the C abundance distribution is not modified by gravitational settling (Althaus et al. 2004). Column 1 of Table 1 shows the resulting final stellar masses (M_{∗}/M_{⊙}). The second column corresponds to the total amount of H contained in the envelope (M_{H}/M_{∗}) at T_{eff} ≈ 10 000 K (at the final cooling branch), and Col. 3 displays the time spent by the models to cool from T_{eff} ≈ 10 000 K to ≈ 8000 K. Finally, Col. 4 indicates the occurrence (or not) of CNO flashes on the early WD cooling branch. We note that, in good agreement with previous studies (Sarna et al. 2000; Althaus et al. 2001; Panei et al. 2007) there exists a threshold in the stellar mass value (at ~ 0.18 M_{⊙}), below which CNO flashes on the early WD cooling branch are not expected to occur. Sequences with M_{∗} ≲ 0.18 M_{⊙} have thicker H envelopes and much longer cooling timescales than sequences with stellar masses above that threshold in mass. In numbers, this means that the H content is about four times higher and τ (the time to cool from T_{eff} ≈ 10 000 K to T_{eff} ≈ 8000 K) is about 22 times longer for the sequence with M_{∗} = 0.1762 M_{⊙} than for the sequence with M_{∗} = 0.1806 M_{⊙} (see Table 1). Note that in this example, we compare the properties of two sequences with virtually the same stellar mass (ΔM_{∗} ≈ 4 × 10^{3} M_{⊙}). The sequences without flashing evolve this slowly because the residual Hshell burning is the main source of surface luminosity, even at very advanced stages of evolution. It is clear that for these WDs, an appropriate treatment of progenitor evolution is required to correctly assess the evolutionary timescales. In contrast, an entirely different behaviour is expected for sequences that experience unstable Hshell burning on their early cooling branch. During the final cooling branch, evolution proceeds on a much shorter timescale than that characterising the sequences with M_{∗} ≲ 0.18 M_{⊙}. This is because CNO flashes markedly reduce the H content of the star, with the result that residual nuclear burning is much less relevant when the remnant reaches the final cooling branch. The log T_{eff} − log g diagrams for all the Hecore WD sequences are shown in Fig. 2, which is an update of Fig. 2 of Althaus et al. (2013). Sequences that undergo CNO flashes during the earlycooling phase exhibit complex loops in the diagram. In contrast, sequences without flashes show very simple cooling tracks.
The mass limit below which lowmass WDs are classified as ELM WDs is not yet very clear in the literature. For instance, Brown et al. (2010, 2012, 2013) and Hermes et al. (2013b,a) defined lowmass WDs with M_{∗} ≲ 0.25 M_{⊙} as ELM WDs, while Kilic et al. (2011, 2012) classified objects with M_{∗} ≲ 0.20 M_{⊙}as ELM WDs. Here, we propose to designate as ELM WDs those lowmass WDs that do not experience CNO flashes in their earlycooling branch. According to this classification, ELM WDs are lowmass WDs with masses below ~ 0.18 M_{⊙} in the frame of our computations^{2}. This criterium is by no means arbitrary because both types of objects not only differ in their evolutionary history, but also show markedly different pulsation properties.
We conclude this section by showing in Fig. 3 the log T_{eff} − log g plane in the region where pulsating lowmass WDs are found, along with the Hecore lowmass WD evolutionary tracks of Althaus et al. (2013). Note that there is a gap between the two sets of tracks, corresponding to a stellar mass of ~ 0.18 M_{⊙}. Interestingly, there are two pulsating stars, SDSS J184037.78+642312.3 and SDSS J111215.82+111745.0, which are located precisely in that transition region between ELM WDs and lowmass WDs, according to our definition. They are, therefore, very interesting targets for asteroseismology.
Fig. 3 T_{eff} − log g plane showing the lowmass Hecore WD evolutionary tracks of Althaus et al. (2013; thin black lines). Sequences with H flashes during the earlycooling phase are depicted with dashed lines, sequences without H flashes are displayed with solid lines. Numbers correspond to the stellar mass of each sequence. The locations of the five known pulsating lowmass WDs (Hermes et al. 2013a) are marked with a small circle (red). Stars not observed to vary are depicted with green triangles. Black circles and squares on the evolutionary tracks of M_{∗} = 0.1554 M_{⊙} and M_{∗} = 0.2389 M_{⊙} indicate the location of the template models analysed in Sect. 3.2. 

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Fig. 4 Dipole (ℓ = 1) asymptotic period spacing of g modes (left panel) and the asymptotic frequency spacing of p modes (right panel) in terms of the effective temperature for all of our lowmass Hecore evolutionary sequences. Dashed lines correspond to sequences with CNO flashes during the earlycooling phase, solid lines sequences without H flashes. 

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3. Pulsation results
In this section we present the results of our detailed adiabatic survey of radial modes (ℓ = 0) and nonradial p and g modes (ℓ = 1,2) for the complete set of lowmass, Hecore WD sequences of Althaus et al. (2013). We cover a wide range of effective temperatures (12 000 ≳ T_{eff} ≳ 7000 K) and stellar masses (0.155 ≲ M_{∗}/M_{⊙} ≲ 0.435). The set of the computed modes covers a very wide range of periods, embracing all the periodicities detected in pulsating lowmass WDs up to now. Specifically, the lower limit of the computed periods is between Π_{min} ~ 0.5 s (corresponding to M_{∗} = 0.4352 M_{⊙}) and Π_{min} ~ 10 s (corresponding to M_{∗} = 0.1554 M_{⊙}). They are associated with highorder radial modes and p modes, and were computed to account for the shortest periods detected in SDSS J111215.82+111745.0 (Π ~ 108 − 134 s). On the other hand, the upper limit of the computed periods (associated with highorder g modes) is Π_{max} ~ 7000 s, to account for the longest periods detected in SDSS J222859.93+362359.6 (Π ~ 6235 s).
In the next section we discuss our pulsational results by showing the predictions of the asymptotic theory of stellar pulsations.
3.1. Asymptotic period spacing (ΔΠ^{a}) and asymptotic frequency spacing (Δν^{a})
For g modes with high radial order k (long periods), the separation of consecutive periods ( Δk  = 1) becomes nearly constant at a value given by the asymptotic theory of nonradial stellar pulsations. Specifically, the asymptotic period spacing (Tassoul et al. 1990) is given by (7)where (8)The squared BruntVäisälä frequency (N, one of the critical frequencies of nonradial stellar pulsations) is computed as (9)where the compressibilities are defined as (10)The Ledoux term B is computed as (Tassoul et al. 1990) (11)where (12)The expression in Eq. (7) is rigorously valid for chemically homogeneous stars. In this equation (see also Eq. (8)), the dependence of on the BruntVäisälä frequency is such that the asymptotic period spacing is larger when the mass and/or effective temperature of the model is lower. This trend is clearly visible in the left panel of Fig. 4, in which we depict the evolution of the asymptotic period spacing of g modes for all the sequences considered in this work. The higher values of for lower M_{∗} comes from the dependence N ∝ g, where g is the local gravity (). On the other hand, the higher values of for lower T_{eff} result from the dependence , with χ_{T} → 0 for increasing electronic degeneracy (T → 0). The abrupt change in the slope of the curves representing at certain effective temperatures (that decrease for decreasing stellar mass) is due to the appearance of surface convection in the models, which induces a lower value of the integral in Eq. (8) and the consequent increase of .
Fig. 5 Internal chemical profiles of He and H (upper panel) and the propagation diagram – the run of the logarithm of the squared critical frequencies (N, L_{ℓ}) – (lower panel) corresponding to the ELM WD template model of M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 9600 K. Plus symbols (in red) correspond to the spatial location of the nodes of the radial eigenfunction of dipole (ℓ = 1) g modes, x symbols (in blue) represent the location of the nodes of dipole p modes. 

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The strong dependence of the period spacing on M_{∗} shown in Fig. 4 might be used, in principle, to infer the stellar mass of pulsating lowmass WDs, provided that enough consecutive pulsation periods of g modes were detected. However, this prospect is severely complicated because the period spacing of pulsating WDs also depends on the thickness of the outer H envelope (Tassoul et al. 1990), being larger for thinner envelopes. This is particularly true in the context of lowmass Hecore WDs, in which ELM WDs models, because of the absence of CNO flashes, harbour H envelopes that are several times thicker than more massive models. For the value of , and for a fixed T_{eff}, therefore, a model with a low mass and a thick H envelope can readily mimic a more massive model with a thinner H envelope. A clear demonstration of this ambiguity can be found in Fig. 4, which shows a notorious degeneracy of the asymptotic period spacing. This figure shows for instance that for T_{eff} ≳ 8500 K, ELM WD models with masses M_{∗} = 0.1762 M_{⊙} and M_{∗} = 0.1706 M_{⊙} have lower values of than the more massive models with M_{∗} = 0.1805 M_{⊙}, which is caused by the much thicker H envelopes of the former models. If for a given pulsating star a rich spectrum of observed periods were available, this degeneracy might be broken by including additional information of the modetrapping properties, which yield clues about the thickness of the H envelope.
The asymptotic frequency spacing of p modes (k ≫ 1), on the other hand, is given by (Unno et al. 1989) (13)where c_{s} is the local adiabatic sound speed, defined as . The asymptotic frequency spacing is related to the Lamb frequency (the other critical frequency of nonradial stellar pulsations) through c_{s} by means of (14)In the righthand panel of Fig. 4 we display the evolution of the asymptotic frequency spacing (in units of mHz ≡ 10^{3} Hz) for all the sequences considered in this work. As can be seen, Δν^{a} is larger for higher stellar masses and lower effective temperatures. This behaviour can be understood by realising that for higher M_{∗} and lower T_{eff}, the values of the sound speed of the models globally increases, as a result of which the value of the integral in Eq. (13) is lower, and consequently, Δν^{a} increases.
In contrast to what occurs for the for g modes, the asymptotic frequency spacing for p modes is not degenerate for sequences with CNO flashes (M_{∗} ≳ 0.18 M_{⊙}) and those without (M_{∗} ≲ 0.18 M_{⊙}, ELM WDs). This is because is rather insensitive to the thickness of the H envelope. While an indepth discussion of p modes in lowmass WDs is only of academic interest as yet, we can envisage that if these modes were confirmed in future observations, the eventual measurement of the mean frequency spacing for a real star might help in constraining its stellar mass.
3.2. Template models at the edges of the instability strip
To illustrate the adiabatic pulsation properties of our huge set of lowmass Hecore WD models, which comprises more than 10 000 stellar structures, we focused on some selected, representative models. Since the instability domain of the known pulsating lowmass WDs in the log T_{eff} − log g diagram is comprised between T_{eff} ~ 10 000 K (the empirical blue edge) and T_{eff} ~ 7800 K (the empirical red edge), we considered template models at both boundaries.
We first analyse two template models at T_{eff} ~ 9600 K with stellar mass M_{∗} = 0.1554 M_{⊙} (ELM template model) and M_{∗} = 0.2389 M_{⊙} (LM template model). The ELM template model is representative of structures without CNO flashes in the past evolution (M_{∗} ≲ 0.18 M_{⊙}), while the LM template model characterises the objects with H flashes (M_{∗} ≳ 0.18 M_{⊙}). The location of these template models in the log T_{eff} − log g diagram is indicated in Fig. 3 with black circles. Our models have a He core surrounded by a H outer envelope. In between, there is a smooth transition region shaped by the action of microscopic diffusion. In the upper panels of Figs. 5 and 6 we display the internal chemical profiles for He and H corresponding to the two template models. These template models are different in two important ways. To begin with, the ELM model has a H envelope that is about seven times thicker than the LM model. As mentioned, this is the result of the very different evolutionary history of the progenitor stars. Second, the He/H transition region is markedly wider for the ELM model than for the LM one. In particular, the H profile for the ELM model is characterised by a diffusionshaped doublelayered chemical structure, which consists of a pure H envelope on top of an intermediate remnant shell rich in H and He. We return to this in Sect. 3.4.
Fig. 6 Same as Fig. 5, but for the LM WD template model of M_{∗} = 0.2389 M_{⊙}. 

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The chemical transition regions leave notorious signatures in the run of the squared critical frequencies, in particular in N. This is clearly displayed in the propagation diagrams(Cox 1980; Unno et al. 1989) of the lower panels in Figs. 5 and 6 that correspond to the template models. g modes propagate in the regions where , and p modes in the regions where . Here, σ is the oscillation frequency. Note the very different shape of N^{2} for both models. In particular, one of the strongest differences is the location of the bump at the He/H transition region. Indeed, this bump is located at much deeper regions for the ELM model (r/R_{∗} ~ 0.2) than for the LM model (r/R_{∗} ~ 0.7). From the local maximum at r/R_{∗} ~ 0.22 (which coincides with the He/H transition region) outwards, N^{2} also decreases for the ELM model until it reaches a minimum at r/R_{∗} ~ 0.8. After that minimum, N^{2} reaches higher values in the surface layers. This is in contrast with the LM template model, in which the squared BruntVäisälä frequency exhibits higher values at the outer layers than at the core, which resembles the situation in the C/O core DAV WD models (see Fig. 3 of Romero et al. 2012). This notoriously different shape of the run of N^{2} has strong consequences for the propagation properties of eigenmodes. Specifically, and because of the particular shape of N^{2}, which is larger in the core than in the envelope (see also Fig. 3 of Córsico et al. 2012), the resonant cavity of g modes for the ELM model is circumscribed to the core regions (r/R_{∗} ≲ 0.25) and the opposite holds for p modes, whereas for the LM model the propagation region for both g and p modes extends roughly along the whole model. To summarise, g modes in ELM WD models (M_{∗} ≲ 0.18 M_{⊙}) mainly probe the core regions and p modes the envelope. This means that we have the opportunity to constrain both the core and envelope chemical structure of these stars via asteroseismology. Steinfadt et al. (2010) were the first to notice this important characteristic.
Fig. 7 Run of the density of kinetic energy dE_{ekin}/dr (normalised to 1) for radial modes (dashed blue) and dipole g (red) and p modes (solid blue curves) with k = 1 (upper panel), k = 10 (middle panel), and k = 60 (lower panel), corresponding to the ELM template model with M_{∗} = 0.1554 M_{⊙}. The vertical dashed line marks the location of the He/H chemical transition region. 

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Fig. 8 Same as Fig. 7, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙}. 

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Fig. 9 ℓ = 1 forward periodspacing of g modes versus periods (upper lefthand panel), and the forward frequencyspacing of radial modes (hollow small circles) and p modes (filled small circles) versus frequencies (upper righthand panel) and the associated oscillation kinetic energy distributions (lower panels) for the ELM WD template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ~ 9600 K. The red horizontal lines in the upper panels correspond to the asymptotic periodspacing (left), computed with Eq. (7), and the asymptotic frequencyspacing (right), computed with Eq. (13). 

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The property described above is vividly illustrated in Fig. 7, in which we display the density of the oscillation kinetic energy dE_{kin}/dr, (see Appendix A of Córsico & Althaus 2006, for its definition) for radial modes, p and g modes with radial order k = 1,10, and 60 for the same ELM template model as was analysed in Fig. 5. It is apparent that most of the kinetic energy of g modes is confined to the regions below the He/H interface, meaning that most of the spatial oscillations are located in the region with r/R_{∗} ≲ 0.25. In contrast, the kinetic energy of p modes and radial modes is spread throughout the star, but concentrated more towards the surface regions and almost absent from the core.
In Fig. 8 we show the situation for the LM WD template model. At variance with the ELM model, there is no such a clear distinction in the behaviour of g modes with respect to p and radial modes. Indeed, the kinetic energy of g modes is spread throughout the model, although it is particularly concentrated in the region of the He/H interface and also at the surface. In this sense, lowmass Hecore WDs with M_{∗} ≳ 0.18 M_{⊙} behave in a qualitative similar way as their massive cousins, the C/Ocore DAV WD stars. p and radial modes are insensitive to the presence of the chemical interface because their kinetic energy is enhanced at the stellar surface. In summary, it is apparent that for LM WD models, g modes are very sensitive to the He/H compositional gradient and therfore can be a diagnostic tool to constrain the H envelope thickness of lowmass WD models with M_{∗} ≳ 0.18 M_{⊙}.
We now examine the modetrapping properties of our template models. Mode trapping of g modes is a wellstudied mechanical resonance that acts in WD stars through chemical composition gradients. This means that one or more narrow regions in which the abundances of nuclear species (and consequently, the average molecular weight μ) vary spatially modify the character of the resonant cavity in which modes should propagate as standing waves. Specifically, chemical transition regions act like reflecting walls that partially trap certain modes, forcing them to oscillate with larger amplitudes in specific regions and with smaller amplitudes outside of those regions (see, for details, Brassard et al. 1992; Bradley et al. 1993; Córsico et al. 2002a). Mode trapping translates into local maxima and minima in E_{kin}, which are usually associated with modes that are partially confined to the core regions and modes that are partially trapped in the envelope. Unfortunately, the kinetic oscillation energy is hard to estimate from observations alone. A more important signature, which in principle can be employed as an observational diagnostic of mode trapping – provided that a series of periods with the same ℓ and consecutive radial order k is detected – is the strong departure from uniform period spacing, ΔΠ (≡Π_{k + 1} − Π_{k}), when plotted in terms of the pulsation period Π_{k}. For stellar models characterised by a single chemical interface, like those we consider here, local minima in ΔΠ_{k} usually correspond to modes trapped in the H envelope, whereas local maxima in ΔΠ_{k} are associated with modes trapped in the core region.
Fig. 10 Same as Fig. 9, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙} and T_{eff} ~ 9600 K. 

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In the upper lefthand panel of Fig. 9 we show the dipole forward periodspacing of g modes versus periods for the ELM template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ~ 9600 K. The upper righthand panel corresponds to the forward frequencyspacing, Δν (≡ν_{k + 1} − ν_{k}), of radial modes and p modes in terms of the frequency (in mHz) for the same model. Finally, the lower panels depict the kinetic energy distributions. For periods longer than about 1000 s, the periodspacing distribution of g modes shows a regular pattern of mode trapping with a very short trapping cycle (period interval between two trapped modes) of ~ 100 s and amplitude, superimposed on a highamplitude variation characterised by a long trapping cycle (~ 1800 s). The simultaneous presence of these two patterns is the result of the doublelayered chemical structure of the H profile that characterises the ELM WD template model at that effective temperature. This doublelayered structure becomes a singlelayered one by the action of element diffusion (see Sect. 3.4). The period spacing does not reach the asymptotic value in the period interval shown in the figure. The E_{kin} distribution only shows the longtrapping cycle pattern. We examined the radial eigenfunctions of two modes, one of them with k = 29 and associated to a local minimum of E_{kin}, and the other one with k = 33, corresponding to a maximum E_{kin} value. We realised that the eigenfunction of the k = 33 mode has larger amplitudes at the core regions than the k = 29 mode. It is apparent that as a general rule, all the g modes in the ELM WD template model are confined to the core regions, but some of them have eigenfunctions with larger amplitudes than the remaining ones. They correspond to local maxima in the E_{kin} distribution. On the other hand, the frequency spacing of radial modes and p modes are close to the asymptotic value for radial order k ≤ 9 and k ≤ 16, respectively. In this case, no modetrapping signatures are visible in the kinetic energy distribution of the modes.
Fig. 11 Chemical profiles of He and H (upper panel) and the propagation diagram (lower panel) for the ELM WD template model of M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 7800 K. 

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Fig. 12 Same as Fig. 11, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙}. 

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The results for the LM WD template model with M_{∗} = 0.2389 M_{⊙} and T_{eff} ~ 9600 K are depicted in Fig. 10. For g modes, modetrapping signatures are quite evident, but the trapping cycle (~ 200 s) of the present pattern is much shorter than for the highamplitude variation of the ELM WD template model. Modetrapping features are absent from the E_{kin} distribution, which instead is very smooth. Note that the ΔΠ values approach to the asymptotic prediction for periods longer than ~ 4000 s. The frequency separations of radial modes and p modes, on the other hand, show some signatures of mode trapping for loworder modes, and do not reach the asymptotic value for the range of frequencies considered in the plot.
We now describe the results for template models located at an effective temperature close to the T_{eff} of the coolest known pulsating lowmass WD (SDSS J222859.93+362359.6, T_{eff} = 7870 ± 120 K), which defines the empirical red edge of the instability strip of this new kind of variable stars. Specifically, we analyse two template models at T_{eff} ~ 7800 K, one of them with a stellar mass M_{∗} = 0.1554 M_{⊙}, and the other one with M_{∗} = 0.2389 M_{⊙}. The location of these template models in the log T_{eff} − log g plane is shown in Fig. 3 with black squares. In Figs. 11 and 12 we show their chemical profiles (upper panels) and propagation diagrams (lower panels). For the ELM template model, the changes in the chemical profiles caused by element diffusion while the star cools from T_{eff} ~ 9600 K to T_{eff} ~ 7800 are very noticeable. The shape of the He/H interface has changed from a doublelayered to a singlelayered structure (see Sect. 3.4). At this lower T_{eff} the He/H transition is located at r/R_{∗} ~ 0.45 and results in a more external bump of the BruntVäisälä frequency than in the model at T_{eff} ~ 9600 K (compare with Fig. 5). For the LM WD template model, the changes in the He/H chemical transition region and in the BruntVäisälä frequency due to element diffusion are less noticeable. When we compare the upper panels of Figs. 12 and 6, we barely note a slight variation in the thickness of the He/H transition region, which is somewhat narrow for the cooler model. This results in a slightly more narrow bump in the profile of N^{2}.
Fig. 13 Run of the density of kinetic energy dE_{ekin}/dr (normalised to 1) for dipole g (red) and p modes (solid blue curves) with k = 1 (upper panel), k = 10 (middle panel), and k = 60 (lower panel), for the ELM template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 7800 K (analysed in Fig.11). 

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Fig. 14 Same as Fig. 13, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙} analysed in Fig. 12. 

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In Figs. 13 and 14 we show dE_{kin}/dr for p and g modes^{3} with k = 1,10, and 60 for the ELM and the LM template models at T_{eff} ~ 7800 K. For the ELM model there is a clear distinction between p and g modes for the part of the star that they probe: g modes are mostly confined to the core, p modes to the envelope. The situation is therefore the same as for the ELM WD template model at T_{eff} ~ 9600 K (see Fig. 7). A similar situation is found for the LM template model. At T_{eff} ~ 7800 the g modes are very sensitive to the presence of the He/H transition region, whereas p modes (and radial modes, not shown) are concentrated towards the stellar surface and are unaffected by the presence of that chemical interface.
Fig. 15 ℓ = 1 forward periodspacing of g modes versus periods (upper lefthand panel), and the forward frequencyspacing of radial modes (hollow small circles) and p modes (filled small circles) versus frequencies (upper righthand panel) and the corresponding kinetic energy distributions (lower panels) for the ELM WD template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ~ 7800 K. 

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Fig. 16 Same as Fig.15, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙}. 

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Finally, in Figs. 15 and 16 we show the distributions of ΔΠ (g modes), Δν (p modes), and log (E_{kin}) for the ELM and the LM WD template models at T_{eff} ≈ 7800 K. It is evident that the periodspacing pattern of g modes of the ELM template model at this low effective temperature is completely different from that for the ELM model at high T_{eff}. Although the asymptotic periodspacing (and consequently, the average periodspacing) has not changed much, the ΔΠ distribution exhibits only a single signal of mode trapping, characterised by a short trapping cycle. Moreover, at variance with the model at T_{eff} ≈ 9600 K, ΔΠ reaches the asymptotic value for Π ≳ 4000 s. The very distinct pattern of ΔΠ in comparison with the hot ELM template model is due to the strong differences in the chemical profiles, and ultimately in the run of the BruntVäisälä frequency (compare Figs. 5 and 11). The frequencyspacing pattern for T_{eff} ~ 7800 K of the p modes (and radial modes, not shown) is similar to that found for T_{eff} ~ 9600 K, but there is a strong difference in the asymptotic frequencyspacing (and thus, in the average frequencyspacing). In fact, Δν^{a} = 4.16 mHz for the cool model is about three times larger than for the hot model (Δν^{a} = 1.51 mHz). This marked difference originates in the fact that for the model at T_{eff} ~ 7800 K the Lamb frequency adopts much higher values than for the model at T_{eff} ~ 9600 K, and in turn, the integral in Eq. (13) is lower, and consequently, Δν is higher.
We did not find substantial differences in the patterns of ΔΠ and Δν (Fig. 16) in the LM WD template model compared with the hot template model (Fig. 10). For the cool template model, both ΔΠ^{a} and Δν^{a}, and in turn the average ΔΠ and Δν, are larger than for the hot template model, however. This is a direct consequence of the decrease in the BruntVäisälä frequency and the increase in the Lamb frequency for decreasing effective temperatures. This behaviour was anticipated in Sect. 4 where we discussed the dependence of ΔΠ^{a} and Δν^{a} on the effective temperature.
3.3. Effects of the total mass and the effective temperature
The period spacing and the periods themselves of g modes vary as the inverse of the BruntVäisälä frequency (see Sect. 3.1). The BruntVäisälä frequency, in turn, increases with higher stellar masses and with higher effective temperatures. As a result, the pulsation periods of g modes are longer for smaller mass (lower gravity) and lower effective temperature (increasing degeneracy). For pmodes and radial modes the sound speed decreases for lower M_{∗} and higher T_{eff}, and consequently, the periods increase. We study the effects of M_{∗} and T_{eff} separately below.
The effects of the stellar mass on the pulsation periods is shown in Fig. 17, where we plot Π for nonradial ℓ = 1g modes and p modes, and also for radial modes (ℓ = 0) in terms of M_{∗} for a fixed value of T_{eff} = 9500 K. The periods of the three types of eigenmodes increase with decreasing stellar mass, as expected. The period gap that separates the families of p modes/radial modes and g modes notoriously shrinks for ELM WD models. The striking step in the run of the periods in terms of the mass is associated with the limit stellar mass value, at ~ 0.18 M_{⊙}. Note the strong increase of the p and radialmode periods for ELM WDs (M_{∗} ≲ 0.18 M_{⊙}). As can be seen, the period of the fundamental radial mode (r_{0}) is substantially longer than the period corresponding to the k = 1p mode (p_{1}) for LM WDs (M_{∗} ≳ 0.18 M_{⊙}), but both periods adopt virtually the same values (as do other loworder pairs of modes, as well: r_{1}–p_{2}, r_{2}–p_{3}, etc.) for ELM WD models. Figure 18 displays the results for g, f and p modes with ℓ = 2. In this case, the g mode periods are substantially shorter than ℓ = 1, and the branches of g and p modes are clearly split by the period of the f mode.
In summary, the deep structural differences between LM and ELM WD models are clearly illustrated by the very different behaviour of the periods (in particular of p modes and radial modes), as documented in Figs. 17 and 18.
In Fig. 19 we show the evolution of the pulsation periods of ℓ = 1g and p modes, and also radial modes with T_{eff} for models with M_{∗} = 0.1917 M_{⊙}. The lengthening of gmode periods with decreasing effective temperature is evident, although the effect is much weaker than the decrease with the stellar mass (compare with Fig. 17). The p and radialmode periods, on the other hand, decrease with decreasing T_{eff}.
Fig. 17 Pulsation periods of ℓ = 1g and p modes and also radial modes (ℓ = 0) in terms of the stellar mass for T_{eff} = 9500 K. Periods increase with decreasing M_{∗}. 

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Fig. 18 Same as Fig. 17, but for pulsation periods of ℓ = 2g, f, and p modes. 

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3.4. Effects of element diffusion
Here, we describe the effects of the evolving chemical profiles on the pulsation properties of lowmass, Hecore WDs. This matter has been extensively explored in previous papers (see Córsico et al. 2002b, in the context of DAV stars). Timedependent element diffusion strongly modifies the shape of the He and H chemical profiles as the WD cools, causing H to float to the surface and He to sink down. In particular, diffusion not only modifies the chemical composition of the outer layers, but also the shape of the He/H chemical transition region itself. This is clearly documented in Fig. 20 for the 0.1554 M_{⊙} ELM sequence in the T_{eff} interval (9600−8000 K). For the model at T_{eff} = 9600 K, the H profile is characterised by a diffusionshaped doublelayered chemical structure, which consists of a pure H envelope on top of an intermediate remnant shell rich in H and He. This structure still remains, although to a much weaker extent, in the model at T_{eff} = 9000 K. Finally, from T_{eff} ~ 8040 K down, the H profile adopts a singlelayered chemical structure. This type of transitions of the shape of chemical profiles caused by element diffusion has previously been studied in detail in DB WDs (see, e.g., Althaus & Córsico 2004). Element diffusion processes affect all the sequences considered in this paper, although the transition from a doublelayered to a singlelayered structure occurs at different effective temperatures. The low surface gravity that characterises the model shown in Fig. 20 (log g ~ 5.5−6.2), which results in a weak impact of gravitational settling, and the very long timescale that characterises element diffusion processes at the depth where the He/H interface is located (r/R_{∗} ~ 0.2), eventually lead to a wider chemical transition than for more massive models. Because of this, the sequences with larger masses (M_{∗} ≳ 0.19 M_{⊙}) reach the singlelayered configuration at effective temperatures higher than ~ 10 000 K, beyond the domain in which pulsating objects are currently found.
Fig. 19 Pulsation periods of ℓ = 1g modes and p modes/radial modes in terms of the effective temperature for M_{∗} = 0.1917 M_{⊙}. gmode periods increase with decreasing T_{eff}; the opposite holds for p modes and radial modes. 

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Fig. 20 Internal chemical profile of H (upper panel), the Ledoux term B entering in the computation of the BruntVäisälä frequency (see Eq. (11)) (middle panel), and the logarithm of the squared BruntVäisälä frequency (lower panel), for an ELM WD model with M_{∗} = 0.1554 M_{⊙} at different effective temperatures, as indicated. 

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The evolution of the shape of the He/H chemical transition region are translated into noticeable changes in the run of the BruntVäisälä frequency (lower panel of Fig. 20), as a result of the changing Ledoux term B (middle panel of Fig. 20). In fact, at high effective temperatures, N^{2} is characterised by two bumps (the most prominent one located at r/R_{∗} ~ 0.25 and the smaller at r/R_{∗} ~ 0.62), which merge into a single bump (at r/R_{∗} ~ 0.5) when the chemical interface He/H adopts a singlelayered structure. As was first emphasised by Córsico et al. (2012), and in the light of our results, diffusive equilibrium is not a valid assumption in the He/H transition region for ELM WD stars. Exploratory computations without element diffusion result in changes of up to 5% in the value of the periods compared with computations with element diffusion.
The effects of element diffusion for a more massive model (an LM WD model with M_{∗} = 0.2389 M_{⊙}) at the same range of effective temperatures is less impressive, as is shown in Fig. 21. In this case, the H chemical profile does not change dramatically, and the same occurs with B and the BruntVäisälä frequency. Even so, this slight evolution in the shape of the chemical profiles is translated into nonnegligible changes in the pulsation periods. Again, for LM WD stars it is necessary to properly account for element diffusion processes to accurately compute the period spectrum.
3.5. Template models with M_{∗} near the threshold mass
It is interesting to examine the pulsation properties of lowmass WD sequences with stellar masses near the critical mass for the development of CNO flashes (M ~ 0.18 M_{⊙}). This is because at least three of the five ELM pulsating WDs reported by Hermes et al. (2013a) have stellar masses near this threshold value (see Fig. 3). Therefore it might be quite important to find an asteroseismic prospect to distinguish ELM from LM WDs (Althaus et al. 2013). Specifically, we focus on the most massive ELM WD sequence (M_{∗} = 0.1762 M_{⊙}), and the lowest mass LM WD sequence (M_{∗} = 0.1805 M_{⊙}). In Figs. 22 and 23 we display the chemical profiles (upper panels) and the propagation diagrams (central panels) for two models at T_{eff} ~ 10 000 K. Interestingly enough, the H content and the shape of the chemical interface of He/H are very different even though the difference in the stellar mass of these two models is almost negligible: ΔM_{∗} = 0.0043 M_{⊙}. The differences in the shape and location of the He/H interface (by virtue of the different thicknesses of the H envelope) for both models are translated into distinct features in the run of the squared critical frequencies, in particular in the BruntVäisälä frequency (middle panels). Note, in particular, the presence of two bumps in N^{2} in the LM model with M_{∗} = 0.1805 M_{⊙}, which result from a doublelayered chemical structure at the He/H interface. It differs from the single bump of N^{2} for the ELM model with M_{∗} = 0.1762 M_{⊙}, which results from the singlelayered shape of the He/H chemical transition region.
Fig. 21 Same as Fig. 20, but for an LM WD model with M_{∗} = 0.2389 M_{⊙}. 

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Fig. 22 Chemical profiles of He and H (upper panel), the propagation diagram (centre panel), and the kinetic energy density for dipole g (red) and p modes (solid blue curves) with k = 1,10, and 20 (lower panels), for an ELM WD model with M_{∗} = 0.1762 M_{⊙} and T_{eff} ≈ 10 000 K. 

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Fig. 23 Same as Fig. 22, but for an LM WD model with M_{∗} = 0.1806 M_{⊙}. 

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Fig. 24 Upper panels: ℓ = 1 forward periodspacing of g modes vs. periods. Lower panels: forward frequencyspacing of p modes vs. frequency. Lefthand panels: ELM WD model with M_{∗} = 0.1762 M_{⊙}. Righthand panels: LM WD model with M_{∗} = 0.1805 M_{⊙}, both at T_{eff} ~ 10 000 K. Red dashed lines correspond to the asymptotic predictions. 

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The consequences of these differences are clearly illustrated in the propagation characteristics of the pulsation modes. The lower panels of Figs. 22 and 23 display the kinetic energy density of pulsation for ℓ = 1g and p modes with k = 1,10, and 20. As we found in Sect. 3.2, for the ELM WD model g modes are confined to the He core, and p modes have most of their kinetic energy located at the surface regions (Fig. 22). For g modes in the LM WD model, the kinetic energy is distributed throughout the model, and they are sensitive to the presence of the He/H interface. p modes, on the other hand, are rather insensitive to the chemical gradient and have most of their kinetic energy placed at the stellar surface (Fig. 23).
In Fig. 24 we show the forward periodspacing in terms of the dipole gmode periods (upper panels) and the forward frequencyspacing in terms of the dipole p mode frequencies (lower panels) for the ELM WD template model with M_{∗} = 0.1762 M_{⊙} (left) and the LM WD template model with M_{∗} = 0.1805 M_{⊙} (right), the same models as considered in Figs. 22 and 23. Modetrapping features in the form of strong departures of uniform periodspacing of g modes can be appreciated in both models, in particular for periods shorter than about 2000 s for the ELM WD model, and Π ≲ 4000 s for the LM WD model. Longer periods approach the asymptotic periodspacings, although even with lowamplitude deviations because of mode trapping. For periods shorter than ~ 3000 s, the amplitudes of deviations from a constant period separation are markedly larger for the LM than for the ELM model. In principle, this difference might be considered as a practical tool for distinguishing stars with CNO flashes in their earlycooling phase from those without, provided that enough consecutive low and intermediateorder g modes with the same harmonic degree were observed in pulsating lowmass WDs (Althaus et al. 2013). This optimistic view is somewhat dampened because the most useful low radialorder g modes are most likely pulsationally stable, as shown from the nonadiabatic analysis of Córsico et al. (2012) and demonstrated by most of the observed pulsating lowmass WD stars, in which no periods shorter than about 1400 s are seen.
Finally, as for the pmodes, there are clear nonuniformities in the frequency spacing distributions, which are of similar amplitudes in both WD models. However, the asymptotic frequencyspacing for the LM WD model is about twice that for the ELM WD model, even though the difference in stellar mass is only ~ 0.004 M_{⊙}. This strong difference in the mean frequencyspacing could also be exploited to distinguish between the two types of objects. To be able to employ this property as a useful seismological tool, however, it will be necessary to detect many p mode consecutive periods in pulsating lowmass WDs. At present, only a few short periods (Π ~ 108 − 134 s) have been detected in only one lowmass pulsating WD (SDSS J111215.82+111745.0), and still it remains to be determined whether they are genuine p modes or not (see Sect. 4.1 ). Thus, this potential asteroseismological tool is for now only of academic interest.
4. Interpretation of observations
Until today, five pulsating lowmass WD stars have been detected: SDSS J184037.78+642312.3 (Hermes et al. 2012), SDSS J111215.82+111745.0, SDSS J151826.68+065813.2 (Hermes et al. 2013b), SDSS J161431.28+191219.4, and SDSS J222859.93+362359.6 (Hermes et al. 2013a). They have very long pulsation periods, from ~ 1180 s up to ~ 6240 s, although one object (SDSS J111215.82+111745.0) also has shortperiod pulsations (108−134 s). Table 4 of Hermes et al. (2013a) lists the main properties of the five known pulsating lowmass WDs. These authors discuss the common properties of this set of pulsators at some length. In Fig. 3 we display the location of the stars in the log T_{eff} − log g diagram, while in Figs. 25 and 26 we show the period spectrum of these pulsating stars in terms of the effective temperature and surface gravity. The longperiod pulsations are most likely the result of intermediate and highorder g modes (12 ≲ k ≲ 50 for ℓ = 1) excited by the κ − γmechanism that acts in the H partial ionisation zone, as indicated by stability computations (Córsico et al. 2012; Van Grootel et al. 2013). On the other hand, the short periods detected in SDSS J111215.82+111745.0 might be caused by p modes or even radial modes of low radial order, as was suggested by stability computations, although the precise nature of these periodicities still remains to be defined^{4}.
Fig. 25 Pulsation periods of the five known pulsating lowmass WD stars in terms of the effective temperature, according to Table 4 of Hermes et al. (2013a). 

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Fig. 26 Pulsation periods of the five known pulsating lowmass WD stars in terms of the surface gravity, according to Table 4 of Hermes et al. (2013a). 

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Although there are very few objects currently available to establish any trend, Figs. 25 and 26 reveal a certain correlation of the long periods (those presumably associated with g modes) with g and T_{eff}. Specifically, periods are longer for lower gravities and effective temperatures. These trends are expected from theoretical grounds, as discussed by Hermes et al. (2013a). For instance, in the context of the κ − γmechanism of mode driving, the trend of the periods to be longer for cooler stars is related to the fact that the outer convection zone in a WD star deepens with cooling, with the consequent increase in the thermal timescale (τ_{th}) at its base. Since the periods of the excited modes are of the order of τ_{th}, modes with increasingly longer periods are gradually excited as the WD cools. This is an oftenstudied property of ZZ Ceti stars (Mukadam et al. 2006)^{5}. Similarly, the trend of periods with gravity (and thus, with M_{∗}) can be understood by realising that lower g imply lower mean densities (ρ), and that the pulsation periods roughly scale with the dynamical timescale for the whole star, Π ∝ ρ^{1/2} (Hermes et al. 2013a).
We emphasise that while these arguments qualitatively explain the trends observed in the period spectrum of the five known pulsating lowmass WDs, the final decision on this matter relies on detailed nonadiabatic pulsation calculations like those performed by Córsico et al. (2012) and Van Grootel et al. (2013). We defer a complete and detailed study of the nonadiabatic pulsation properties of our set of lowmass Hecore WD models to a forthcoming paper. In the next two sections, we interpret the short periods in SDSS J111215.82+111745.0 by considering adiabatic periods alone (Sect. 4.1), and the possibility that SDSS J222859.93+362359.6 is a preWD star instead of a genuine ELM WD by using adiabatic computations and some exploratory nonadiabatic results (Sect. 4.2).
4.1. ELM WD J111215.82+111745.0: the first p modes/radial modes detected in a pulsating WD star?
Notwithstanding many theoretical predictions (Ledoux & SauvenierGoffin 1950; Ostriker & Tassoul 1969; Cohen et al. 1969; Vauclair 1971a,b; Starrfield et al. 1983; Saio et al. 1983; Kawaler 1993) and several observational efforts (Robinson 1984; Kawaler et al. 1994; Silvotti et al. 2011; Chang et al. 2013; Kilkenny et al. 2014), no p modes or radial modes have ever been detected so far in a pulsating WD star of any kind. Thus, the discovery of shortperiod pulsations in an ELM WD by Hermes et al. (2013a) appears to be the first detection of these elusive types of pulsation modes in a WD star and need to be confirmed. Here, we examine whether our lowmass Hecore WD models are able to account for the shortperiod pulsations observed in J111215.82+111745.0 and explore the possibility that they might be genuine p modes and/or radial modes.
Fig. 27 Pulsation periods of our set of lowmass Hecore WD models in terms of the stellar mass, for radial (blue dashed lines), nonradial pmodes (blue solid lines) and g modes (red solid lines) at T_{eff}, roughly the effective temperature measured for J111215.82+111745.0. Horizontal black segments (framed in a grey rectangle) represent the periods exhibited by this pulsating star. 

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In Fig. 27 we plot the pulsation periods of our set of models in terms of M_{∗}, for radial modes (ℓ = 0) and nonradial p and g modes with ℓ = 1 at roughly the spectroscopic effective temperature derived for J111215.82+111745.0 (T_{eff} = 9590 ± 140 K). The mass assumed for this star, M_{∗} = (0.179 ± 0.0012) M_{⊙} is that derived by Althaus et al. (2013) according to the spectroscopic estimation of the surface gravity (log g = 6.36 ± 0.06) as quoted by Hermes et al. (2013a, their Table 4). The periods exhibited by this star (107.5600 s, 134.2750 s, 1792.905 s, 1884.599 s, 2258.528 s, 2539.695 s, 2855.728 s; Hermes et al. 2013b) are indicated by horizontal black segments. Clearly, the longperiod pulsations are well accounted for by high radialorder g modes (18 ≲ k ≲ 30). However, at the effective temperature and mass (gravity) of SDSS J111215.82+111745.0 as predicted by spectroscopy, our models are unable to explain the presence of the short periods. They lie in the forbidden region in between g and p modes. An alternative might be to consider modes with higher harmonic degree. In Fig. 28 we show the results for ℓ = 2. Even in this case, the short periods exhibited by SDSS J111215.82+111745.0 are not accounted for by our models, although now they are very close to the lowestorder gmode periods. We did not perform pulsation computations for higher harmonic degrees, but we expect that the periods at 107.5600 s, 134.2750 s can readily been accounted for by loworder g modes with ℓ = 3. In this case, however, it would be difficult to conceive the detection of ℓ = 3 modes due to geometric cancellation effects (Dziembowski 1977). If we relax the constraint of the stellar mass (gravity), these short periods might be attributed to loworder p modes and/or radial modes, if the stellar mass were somewhat lower (M_{∗} ~ 0.16 M_{⊙}). Alternatively, they might be associated with loworder g modes if the stellar mass were substantially larger (M_{∗} ~ 0.43 M_{⊙}). Finally, we might relax the constraint imposed by the effective temperature to determine whether we can accommodate the short periods observed with theoretical periods. The width of the gap between p and g modes decreases slightly for higher effective temperatures (see Fig. 19). However, since the sensitivity of the periods with the effective temperature is by far weaker than with the stellar mass (see Sect. 3.3), the change in the periods (for reasonable variations of T_{eff}) is insufficient for the theoretical periods of loworder (k ~ 1 − 3) g and p modes, and radial modes be similar to the observed ones.
Fig. 28 Same as Fig. 27, but for nonradial ℓ = 2p, f, and g modes. 

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In summary, if the temperature and mass (gravity) of SDSS J111215.82+111745.0 are correct, our models are unable to explain the short periods, and in particular, they cannot be attributed to p modes and/or radial modes.
4.2. SDSS J222859.93+362359.6: an ELM WD or a preWD star?
The pulsating ELM WD star SDSS J222859.93+362359.6 is the coolest pulsating WD known so far, with T_{eff} = 7870 ± 120 K. At this low effective temperature, it is remarkably cooler than the other pulsating objects of the class. That this star pulsates at all is somewhat surprising because there are many ELM WDs that do not pulsate in between of this star and SDSS J161431.28+191219.4, the secondcoldest pulsating object (T_{eff} = 8800 ± 170 K). Figure 2 reveals the very interesting fact that the star SDSS J222859.93+362359.6 has T_{eff} and log g values that are degenerate with those of more massive preWDs that are undergoing a CNO flash episode. This occurs for preWD tracks of the sequences with M_{∗} = 0.3624 M_{⊙}, M_{∗} = 0.3605 M_{⊙}, M_{∗} = 0.2707 M_{⊙}, M_{∗} = 0.2389 M_{⊙}, M_{∗} = 0.2019 M_{⊙}, and M_{∗} = 0.1805 M_{⊙}^{6}. This makes it natural to wonder whether the star is a genuine ELM WD star with a stellar mass of M_{∗} ~ 0.16 M_{⊙}, or is a more massive preWD star going through a CNO flash. The hypothesis that this star might be a preWD is interesting on its own and deserves to be explored, even taking into account that the evolution of the preWDs is much faster than that of the ELM WDs, and therefore there are far fewer opportunities of observing it. To find some clue, we computed the pulsation spectrum of g modes for a template model belonging to the sequence with M_{∗} = 0.2389 M_{⊙} when it is looping through one of its CNO flashes and is briefly located at T_{eff} = 7870 K, log g = 6.03. In Fig. 29 we depict the evolutionary track of the 0.2389 M_{⊙} sequence and the location of this template model on the T_{eff} − log g diagram. We also include in the analysis a template ELM WD model with similar T_{eff} and log g values and M_{∗} = 0.1554 M_{⊙} (see lefthand panel of Fig. 29) to compare its pulsation spectrum with that of the 0.2389 M_{⊙} preWD model.
Fig. 29 Location of the star SDSS J222859.93+362359.6 (red star symbol) in the log T_{eff} − log g diagram, along with the template ELM WD model on its evolutionary track (lefthand panel) and its twin template preWD model with virtually the same T_{eff}/ log g values on its evolutionary track (righthand panel). 

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In Fig. 30 we show the chemical profiles and propagation diagrams of the two template models, including the three frequencies detected in SDSS J222859.93+362359.6. The strongly different internal chemical structure of both models is evident, as is expected because they represent very different evolutionary stages, although they share the same effective temperature and surface gravity. In particular, the preWD model (which is going through a CNO flash episode) is characterised by a very thick internal convective zone from r/R_{∗} ~ 0.3 to r/R_{∗} ~ 0.9. This convection zone forces most g modes to be strongly confined to the He core (r/R_{∗} ≲ 0.3), but a few g modes are instead strongly trapped in the H envelope. Note also the very thin outer convection zone, which extends from r/R_{∗} ~ 0.988 up to r/R_{∗} = 1. Of course, g modes are evanescent in the convective regions.
Fig. 30 Internal chemical profiles of He and H (upper panels) and the propagation diagrams (lower panels) for the ELM WD template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 7860 K (left) and the preWD template model with M_{∗} = 0.2389 M_{⊙} and a similar T_{eff} (right). The grey areas in the 0.2389 M_{⊙} model indicate a thick inner convective zone at 0.3 ≲ r/R_{∗} ≲ 0.9 and another, very thin outer convection zone, at 0.988 ≲ r/R_{∗} ≤ 1. Dashed horizontal lines represent the three frequencies detected in SDSS J222859.93+362359.6. 

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The period spacing and kinetic energy of the g modes with ℓ = 1 in terms of periods for both template models are shown in the upper and middle panels of Fig. 31, in which we include, for illustrative purposes, the three periods observed in SDSS J222859.93+362359.6. In addition to a slight difference in the asymptotic periodspacing (ΔΠ^{a} = 106.5 s for the ELM WD model, and ΔΠ^{a} = 91.85 s for the preWD model), very pronnounced minima in ΔΠ appear for the preWD model, which are associated with modes with k = 2, k = 17, k = 33, k = 54, and k = 78, which exhibit very low kinetic energies (up to six orders of magnitude lower than the average). These modes are strongly trapped in the outer (radiative) H envelope. They should be easy to excite up to observable amplitudes by virtue of their very low kinetic oscillation energy (low inertia). If this were true, then the spectrum of the preWD model should be characterised by a few excited modes with clearly separated periods instead of a continuous spectrum of unstable periods  which should be found, instead, in the 0.1554 M_{⊙} ELM WD model, by extrapolating to the results of Córsico et al. (2012) for an ELM WD model with M_{∗} = 0.17 M_{⊙}. If all this were true, the period spectrum of SDSS J222859.93+362359.6, which is itself discrete and composed of only three periods, should be more nearly compatible with the preWD model than with the ELM WD model.
Fig. 31 ℓ = 1 period spacing (upper panels), the kinetic oscillation energy (middle panels) and the growth rates (lower panels) in terms of the periods for the template ELM WD model (left) and the template preWD model (right). η> 0 implies pulsationally unstable modes. Vertical dashed lines represent the periods observed in SDSS J222859.93+362359.6. 

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To test this hypothesis, we computed nonadiabatic pulsations. Specifically, we performed an exploratory stability analysis of the ELM WD sequence with M_{∗} = 0.1554 M_{⊙} and the preWD sequence with M_{∗} = 0.2389 M_{⊙}. These computations were carried out with the help of the nonadiabatic version of the LPPUL pulsation code described in Sect. 2.2. We only computed ℓ = 1g modes with periods in the range of interest and effective temperatures in the range 9000 ≳ T_{eff} ≳ 7400 K, thus embracing the effective temperature of SDSS J222859.93+362359.6 (T_{eff} ~ 7900 K). The output of our code are the normalised nonadiabatic growth rates, η, which are defined as η = −ℑ(σ)/ℜ(σ), where ℜ(σ) and ℑ(σ) are the real and the imaginary part of the complex eigenfrequency σ (see Córsico et al. 2006, for details). Positive values of η mean unstable modes. The growth rates in terms of the pulsation periods are shown in the lower panels of Fig. 31. For the 0.2389 M_{⊙} preWD model, the few modes that are strongly trapped in the H envelope are unstable, as expected. In particular, modes with k = 54 and k = 78 are the only unstable modes for periods Π ≳ 3600 s. The differential work function (see Córsico et al. 2006, for a definition) shows that most of driving for these modes comes from the base of the outer convective zone. Below Π ~ 3600 s we found a continuous band of unstable modes with consecutive radial orders. The differential work functions for these unstable modes indicate that they are excited at the base of the inner convective zone, except for those with k = 2, k = 17, and k = 33, which are trapped in the outer H envelope and are driven at the base of the outer convection zone. On the other hand, the spectrum of unstable modes for the 0.1554 M_{⊙} ELM WD model consists of a continuous range of periods in the range 800 ≲ Π ≲ 7000 s with consecutive radial orders (8 ≤ k ≤ 64), much in line with the results of Córsico et al. (2012). The k = 1 and k = 2 modes are marginally unstable . We note that the growth rates η for the unstable modes of the 0.2389 M_{⊙} model are, on average, more than ten times larger than for the 0.1554 M_{⊙} model. This indicates that the unstable modes in the preWD model are more strongly excited than in the ELM WD model. Finally, in Fig. 32 a more general picture of the situation is shwon. In this plot we depict the unstable periods in terms of the effective temperature for the sequence of preWD models with M_{∗} = 0.1554 M_{⊙} and for the sequence of ELM WD models with M_{∗} = 0.2389 M_{⊙} . We included the three periods detected in SDSS J222859.93+362359.6. The discrete period spectrum of SDSS J222859.93+362359.6 is better represented qualitatively by the period spectrum of unstable modes of the 0.2389 M_{⊙} preWD model sequence than by those of the 0.1554 M_{⊙} ELM WD sequence.
Fig. 32 Instability domain on the T_{eff} − Π plane for ℓ = 1g modes for the set of ELM WD models with M_{∗} = 0.1554 M_{⊙} (left) and the sequence of preWD models with M_{∗} = 0.2389 M_{⊙} (right). For the M_{∗} = 0.1554 M_{⊙} sequence, the modes with k = 1 and k = 2 are marginally unstable . We also show the periodicities measured in SDSS J222859.93+362359.6, marked with red horizontal segments. 

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In summary, if we accept as valid the possibility that SDSS J222859.93+362359.6 might be a preWD star that we are only observing by chance at its brief incursion in the domain of the log T_{eff} − log g diagram where most ELM WDs are found, we would need to explain why we do not observe the continuous period spectrum as predicted by the nonadiabatic computation for the range of periods 200 ≲ Π ≲ 3600 s. Similarly, if we adopt the idea that SDSS J222859.93+362359.6 is a genuine ELM WD star, we need to explain why of the entire continuum spectrum of unstable periods, as predicted by theoretical models, we only observe three isolated periods. All in all, currently we can only state that this star might be a preWD instead of an ELM WD, but more observations aimed at detecting additional periods, together with extensive nonadiabatic calculations, are necessary to confirm or discard this statement.
5. Summary and conclusions
We have presented a comprehensive theoretical study of the seismic properties of lowmass, Hecore WDs with masses in the range 0.1554 − 0.4389 M_{⊙}. We employed stateoftheart evolutionary stellar structures representative of these stars, extracted from the sequences of lowmass Hecore WDs of Althaus et al. (2013). These models were derived by computing the nonconservative evolution of a binary system consisting of an initially 1 M_{⊙} ZAMS star and a 1.4 M_{⊙} neutron star for various initial orbital periods. The evolutionary computations were carried out by accounting for a timedependent treatment of the gravitational settling and chemical diffusion, as well as of residual nuclear burning. We explored the adiabatic pulsation properties of these models, including the asymptotic predictions, the expected range of periods, period spacings and frequency spacings, the propagation properties and mode trapping of pulsations, as well as the dependence on the effective temperature and stellar mass and the effects of element diffusion. In particular, we strongly emphasised the expected differences in the seismic properties of objects with M_{∗} ≳ 0.18 M_{⊙} with CNO flashes during the earlycooling phase, and the ELM WDs (M_{∗} ≲ 0.18 M_{⊙}) without H flashes.
The pulsation properties of lowmass Hecore WDs have been explored in detail recently by Córsico et al. (2012) on the basis of a set of evolutionary models derived by Althaus et al. (2009) considering progenitor stars with supersolar metallicities and singlestar evolution. In the present work, we recovered much of the results of Córsico et al. (2012) and extended that study in two ways. First, we explored not only the nonradial g mode pulsation spectrum of lowmass WD models, but also considered nonradial p modes (ℓ = 1,2) and radial (ℓ = 0) pulsation modes. Second, and thanks to the availability of five WD model sequences with progenitors without CNOflashes (see Table 1), here we were able to explore in detail the pulsation properties of ELM WDs, which are characterised by very thick H envelopes. This is at variance with the work of Córsico et al. (2012), in which only one WD sequence (that with mass M_{∗} = 0.17 M_{⊙}) belonged to a progenitor star without H flashes. Finally, our study relied on evolutionary models consistent with the expected binary evolution of progenitor stars.
We also discussed how our models match the observed properties of the known five pulsating lowmass WD stars. In particular, we tried to determine whether our models are able to account for the short periods observed in the star SDSS J111215.82+111745.0, and evaluated the possibility that these modes might be p modes and/ radial modes. In addition, we tested the hypothesis that one of these stars, SDSS J222859.93+362359.6, is not a genuine ELM WD, but instead a preWD star going through a CNO flash episode.
Although we included some exploratory nonadiabatic pulsation computations, most of our results rely on adiabatic pulsations. We defer a thorough nonadiabatic exploration of our complete set of Hecore WD models to a forthcoming paper.
We summarise our findings below.

Lowmass WDs have not been clearly classified as ELM WDs in the literature up to now. Here, we proposed to define as ELM WDs the lowmass WDs without CNO flashes in their earlycooling branch. According to this classification, and in the frame of our computations, ELM WDs are the lowmass WDs with masses below ~ 0.18 M_{⊙}. We call stars with M_{∗} ≳ 0.18 M_{⊙} LM (lowmass) WDs.

The asymptotic periodspacing of g modes in lowmass Hecore WDs is larger for lower mass and/or effective temperature. The strong dependence of the period spacing on M_{∗} might be used to infer the stellar mass of pulsating lowmass WDs, provided that enough consecutive pulsation periods of g modes were detected, although this prospect is complicated by the fact that the period spacing also depends on the thickness of the outer H envelope. In particular, there exist an ambiguity of for masses near the threshold mass, M_{∗} ~ 0.18 M_{⊙}.

The asymptotic frequencyspacing of p modes and radial modes is larger for higher stellar masses and lower effective temperatures. is insensitive to the thickness of the H envelope. If the detection of this type of modes were confirmed in future observations, the eventual measurement of the mean frequencyspacing for a real star might help to constrain its stellar mass.

g modes in ELM WDs (M_{∗} ≲ 0.18 M_{⊙}) mainly probe the core regions and p modes the envelope, which provides the opportunity of constraining both the core and envelope chemical structure of these stars via asteroseismology.

For LM WDs, g modes are very sensitive to the He/H compositional gradient and therefore they can be a diagnostic tool for constraining the Henvelope thickness of lowmass WD models with M_{∗} ≳ 0.18 M_{⊙}.

The gmode periods are longer for smaller mass and lower effective temperature, and in the case of p and radial modes, the periods increase with lower masses and higher effective temperatures. In both cases, the dependence on the effective temperature is much weaker than that on stellar mass.

Timedependent element diffusion strongly affects the gmode pulsation spectrum of lowmass WDs, in particular ELM WDs. Diffusion processes substantially alter the shape of the He/H chemical interface and in turn the resulting period spectrum. The effects are weaker but still nonnegligible for LM WDs. We claim that timedependent element diffusion must be taken into account in any pulsational analysis of lowmass WD stars.

The chemical structure, propagation diagrams, and consequently the pulsation properties of LM WD and ELM WD models with masses near the limit mass (M ~ 0.18 M_{⊙}) are markedly different. These differences are reflected in the period spacing of g modes and the frequency spacing of p modes. For period spacing of g modes, it might be possible to use these differences as a practical tool to distinguish stars with CNO flashes in their earlycooling phase from those without them, provided that enough consecutive low and intermediateorder g modes with the same harmonic degree were observed in pulsating lowmass WDs. This optimistic view is dampened, however, because the most useful low radialorder g modes are most likely pulsationally stable, as shown from the nonadiabatic analysis of Córsico et al. (2012), and also demonstrated by most of the observed pulsating lowmass WD stars, in which no periods shorter than about 1400 s are seen. Regarding the frequency spacing of p modes, Δν^{a} for the LM WD model is about twice that of the ELM WD model, even though the difference in stellar mass is only ~ 0.004 M_{⊙}. This property might constitute a useful seismological tool if many p mode consecutive periods in pulsating lowmass WDs were detected.

Although there are only five pulsating lowmass WDs known at present, and they are not enough in number to trace clear trends of their pulsation spectra, we can note that the observed periods are in general longer for lower gravities and effective temperatures. These trends are in line with theoretical considerations, as discussed by Hermes et al. (2013a).

The star SDSS J111215.82+111745.0 exhibits short period pulsations in the range ~ 107 − 140 s. For the temperature and mass (gravity) of this star according to spectroscopy, our models are unable to explain these short periods. In particular, if the temperature and mass (gravity) of the star are correct, the short periods cannot be attributed to p modes and/or radial modes. However, these periods might be caused by loworder p modes and/or radial modes if the stellar mass were lower (M_{∗} ≈ 0.16 M_{⊙}). Alternatively, they might be loworder g modes if the stellar mass were substantially larger (≈ 0.43 M_{⊙}). Another (unlikely) possibility is that at the spectroscopic mass (gravity) and T_{eff}, the observed periods might be caused by higherdegree (ℓ ≥ 3) loworder g modes.

The pulsating ELM WD SDSS J222859.93+362359.6 is the coolest pulsating WD known so far. According to its location in the log T_{eff} − log g plane, this star might be a preWD star that is looping through one of its CNO flashes before it enters its final WD cooling track. According to our models, we can neither confirm nor discard this hypothesis at present, although some indications of its discrete period spectrum might promote the idea that the star is a preWD star.
Theoretical computations also predict the occurrence of Hshell flashes before the terminal cooling branch is reached even if element diffusion processes are excluded, but in this case, the H envelopes remain thick, with substantial H burning and long WD cooling ages (Driebe et al. 1998; Sarna et al. 2000).
Note that this theoretical mass threshold can vary according to the metallicity of the progenitor stars (Sarna et al. 2000). For instance, Panei et al. (2007), who assumed Z = 0.02 for the progenitor stars, obtained a value ~ 0.17 M_{⊙} for the mass threshold.
Note that the same generic trend of periods with T_{eff} is predicted in the frame of the convective driving mechanism, proposed by Brickhill (1990), since in this case the critical timescale is the convective response timescale τ_{C}, which itself is some multiple of τ_{th} (Montgomery et al. 2008).
Acknowledgments
We wish to thank our anonymous referee for the constructive comments and suggestions that greatly improved the original version of the paper. We also warmly thank J. J. Hermes for reading the paper and making enlightening comments and suggestions. Part of this work was supported by AGENCIA through the Programa de Modernización Tecnológica BID 1728/OCAR, and by the PIP 11220080100940 grant from CONICET. This research has made use of NASA Astrophysics Data System.
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All Tables
Selected properties of our Hecore WD sequences (final cooling branch) at T_{eff} ≈ 10 000 K: the stellar mass, the mass of H in the outer envelope, the time it takes the star to cool from T_{eff} ≈ 10 000 K to ≈ 8000 K, and the occurrence (or not) of CNO flashes on the early WD cooling branch.
All Figures
Fig. 1 Location of the several classes of pulsating WD stars in the log T_{eff} − log g plane, marked with dots of different colours. In parenthesis we include the number of known members of each class. Two postVLTP (Very Late Thermal Pulse) evolutionary tracks are plotted for reference. We also show the theoretical blue edge of the instability strip for the GW Vir stars, V777 Her stars, the DQV stars (Córsico et al. 2006, 2009a,b, respectively), the ZZ Ceti stars (Fontaine & Brassard 2008), and the pulsating lowmass WDs. For this last class, we show the blue edge according to Hermes et al. (2013a; dotted red line), Córsico et al. (2012; solid red line), and Steinfadt et al. (2010; dashed red line). For reference, we also include two evolutionary tracks of lowmass Hecore white dwarfs from Althaus et al. (2013). Small black dots correspond to lowmass WDs that are nonvariable or have not been observed yet to assess variability. 

Open with DEXTER  
In the text 
Fig. 2 log T_{eff} − log g diagrams for the Hecore WD sequences computed in Althaus et al. (2013). Sequences with masses in the range 0.18 ≲ M_{∗} ≲ 0.4 undergo CNO flashes during the earlycooling phase, which leads to the complex loops in the diagram. Green squares and magenta triangles correspond to the observed postRGB lowmass stars from Silvotti et al. (2012) and Brown et al. (2013), and filled blue circles correspond to the five pulsating lowmass WDs detected so far (Hermes et al. 2013a). Numbers in the left upper corner of each panel correspond to the stellar mass at the WD stage. 

Open with DEXTER  
In the text 
Fig. 3 T_{eff} − log g plane showing the lowmass Hecore WD evolutionary tracks of Althaus et al. (2013; thin black lines). Sequences with H flashes during the earlycooling phase are depicted with dashed lines, sequences without H flashes are displayed with solid lines. Numbers correspond to the stellar mass of each sequence. The locations of the five known pulsating lowmass WDs (Hermes et al. 2013a) are marked with a small circle (red). Stars not observed to vary are depicted with green triangles. Black circles and squares on the evolutionary tracks of M_{∗} = 0.1554 M_{⊙} and M_{∗} = 0.2389 M_{⊙} indicate the location of the template models analysed in Sect. 3.2. 

Open with DEXTER  
In the text 
Fig. 4 Dipole (ℓ = 1) asymptotic period spacing of g modes (left panel) and the asymptotic frequency spacing of p modes (right panel) in terms of the effective temperature for all of our lowmass Hecore evolutionary sequences. Dashed lines correspond to sequences with CNO flashes during the earlycooling phase, solid lines sequences without H flashes. 

Open with DEXTER  
In the text 
Fig. 5 Internal chemical profiles of He and H (upper panel) and the propagation diagram – the run of the logarithm of the squared critical frequencies (N, L_{ℓ}) – (lower panel) corresponding to the ELM WD template model of M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 9600 K. Plus symbols (in red) correspond to the spatial location of the nodes of the radial eigenfunction of dipole (ℓ = 1) g modes, x symbols (in blue) represent the location of the nodes of dipole p modes. 

Open with DEXTER  
In the text 
Fig. 6 Same as Fig. 5, but for the LM WD template model of M_{∗} = 0.2389 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 7 Run of the density of kinetic energy dE_{ekin}/dr (normalised to 1) for radial modes (dashed blue) and dipole g (red) and p modes (solid blue curves) with k = 1 (upper panel), k = 10 (middle panel), and k = 60 (lower panel), corresponding to the ELM template model with M_{∗} = 0.1554 M_{⊙}. The vertical dashed line marks the location of the He/H chemical transition region. 

Open with DEXTER  
In the text 
Fig. 8 Same as Fig. 7, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 9 ℓ = 1 forward periodspacing of g modes versus periods (upper lefthand panel), and the forward frequencyspacing of radial modes (hollow small circles) and p modes (filled small circles) versus frequencies (upper righthand panel) and the associated oscillation kinetic energy distributions (lower panels) for the ELM WD template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ~ 9600 K. The red horizontal lines in the upper panels correspond to the asymptotic periodspacing (left), computed with Eq. (7), and the asymptotic frequencyspacing (right), computed with Eq. (13). 

Open with DEXTER  
In the text 
Fig. 10 Same as Fig. 9, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙} and T_{eff} ~ 9600 K. 

Open with DEXTER  
In the text 
Fig. 11 Chemical profiles of He and H (upper panel) and the propagation diagram (lower panel) for the ELM WD template model of M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 7800 K. 

Open with DEXTER  
In the text 
Fig. 12 Same as Fig. 11, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 13 Run of the density of kinetic energy dE_{ekin}/dr (normalised to 1) for dipole g (red) and p modes (solid blue curves) with k = 1 (upper panel), k = 10 (middle panel), and k = 60 (lower panel), for the ELM template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 7800 K (analysed in Fig.11). 

Open with DEXTER  
In the text 
Fig. 14 Same as Fig. 13, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙} analysed in Fig. 12. 

Open with DEXTER  
In the text 
Fig. 15 ℓ = 1 forward periodspacing of g modes versus periods (upper lefthand panel), and the forward frequencyspacing of radial modes (hollow small circles) and p modes (filled small circles) versus frequencies (upper righthand panel) and the corresponding kinetic energy distributions (lower panels) for the ELM WD template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ~ 7800 K. 

Open with DEXTER  
In the text 
Fig. 16 Same as Fig.15, but for the LM WD template model with M_{∗} = 0.2389 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 17 Pulsation periods of ℓ = 1g and p modes and also radial modes (ℓ = 0) in terms of the stellar mass for T_{eff} = 9500 K. Periods increase with decreasing M_{∗}. 

Open with DEXTER  
In the text 
Fig. 18 Same as Fig. 17, but for pulsation periods of ℓ = 2g, f, and p modes. 

Open with DEXTER  
In the text 
Fig. 19 Pulsation periods of ℓ = 1g modes and p modes/radial modes in terms of the effective temperature for M_{∗} = 0.1917 M_{⊙}. gmode periods increase with decreasing T_{eff}; the opposite holds for p modes and radial modes. 

Open with DEXTER  
In the text 
Fig. 20 Internal chemical profile of H (upper panel), the Ledoux term B entering in the computation of the BruntVäisälä frequency (see Eq. (11)) (middle panel), and the logarithm of the squared BruntVäisälä frequency (lower panel), for an ELM WD model with M_{∗} = 0.1554 M_{⊙} at different effective temperatures, as indicated. 

Open with DEXTER  
In the text 
Fig. 21 Same as Fig. 20, but for an LM WD model with M_{∗} = 0.2389 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 22 Chemical profiles of He and H (upper panel), the propagation diagram (centre panel), and the kinetic energy density for dipole g (red) and p modes (solid blue curves) with k = 1,10, and 20 (lower panels), for an ELM WD model with M_{∗} = 0.1762 M_{⊙} and T_{eff} ≈ 10 000 K. 

Open with DEXTER  
In the text 
Fig. 23 Same as Fig. 22, but for an LM WD model with M_{∗} = 0.1806 M_{⊙}. 

Open with DEXTER  
In the text 
Fig. 24 Upper panels: ℓ = 1 forward periodspacing of g modes vs. periods. Lower panels: forward frequencyspacing of p modes vs. frequency. Lefthand panels: ELM WD model with M_{∗} = 0.1762 M_{⊙}. Righthand panels: LM WD model with M_{∗} = 0.1805 M_{⊙}, both at T_{eff} ~ 10 000 K. Red dashed lines correspond to the asymptotic predictions. 

Open with DEXTER  
In the text 
Fig. 25 Pulsation periods of the five known pulsating lowmass WD stars in terms of the effective temperature, according to Table 4 of Hermes et al. (2013a). 

Open with DEXTER  
In the text 
Fig. 26 Pulsation periods of the five known pulsating lowmass WD stars in terms of the surface gravity, according to Table 4 of Hermes et al. (2013a). 

Open with DEXTER  
In the text 
Fig. 27 Pulsation periods of our set of lowmass Hecore WD models in terms of the stellar mass, for radial (blue dashed lines), nonradial pmodes (blue solid lines) and g modes (red solid lines) at T_{eff}, roughly the effective temperature measured for J111215.82+111745.0. Horizontal black segments (framed in a grey rectangle) represent the periods exhibited by this pulsating star. 

Open with DEXTER  
In the text 
Fig. 28 Same as Fig. 27, but for nonradial ℓ = 2p, f, and g modes. 

Open with DEXTER  
In the text 
Fig. 29 Location of the star SDSS J222859.93+362359.6 (red star symbol) in the log T_{eff} − log g diagram, along with the template ELM WD model on its evolutionary track (lefthand panel) and its twin template preWD model with virtually the same T_{eff}/ log g values on its evolutionary track (righthand panel). 

Open with DEXTER  
In the text 
Fig. 30 Internal chemical profiles of He and H (upper panels) and the propagation diagrams (lower panels) for the ELM WD template model with M_{∗} = 0.1554 M_{⊙} and T_{eff} ≈ 7860 K (left) and the preWD template model with M_{∗} = 0.2389 M_{⊙} and a similar T_{eff} (right). The grey areas in the 0.2389 M_{⊙} model indicate a thick inner convective zone at 0.3 ≲ r/R_{∗} ≲ 0.9 and another, very thin outer convection zone, at 0.988 ≲ r/R_{∗} ≤ 1. Dashed horizontal lines represent the three frequencies detected in SDSS J222859.93+362359.6. 

Open with DEXTER  
In the text 
Fig. 31 ℓ = 1 period spacing (upper panels), the kinetic oscillation energy (middle panels) and the growth rates (lower panels) in terms of the periods for the template ELM WD model (left) and the template preWD model (right). η> 0 implies pulsationally unstable modes. Vertical dashed lines represent the periods observed in SDSS J222859.93+362359.6. 

Open with DEXTER  
In the text 
Fig. 32 Instability domain on the T_{eff} − Π plane for ℓ = 1g modes for the set of ELM WD models with M_{∗} = 0.1554 M_{⊙} (left) and the sequence of preWD models with M_{∗} = 0.2389 M_{⊙} (right). For the M_{∗} = 0.1554 M_{⊙} sequence, the modes with k = 1 and k = 2 are marginally unstable . We also show the periodicities measured in SDSS J222859.93+362359.6, marked with red horizontal segments. 

Open with DEXTER  
In the text 
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