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Astronomy& Astrophysicsmanuscript no. epsilon_revised-printer c©ESO 2016June 15, 2016

Pulsations powered by hydrogen shell burning in white dwarfsM. E. Camisassa1, 2, A. H. Córsico1, 2, L. G. Althaus1, 2, and H. Shibahashi3

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 e-mail:camisassa@fcaglp.unlp.edu.ar

2 Instituto de Astrofísica de La Plata, Centro Científico Tecnológico La Plata, Consejo Nacional de Investigaciones Científicas yTécnicas, Paseo del Bosque s/n, 1900 La Plata, Argentina

3 Department of Astronomy, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

Received ; accepted

ABSTRACT

Context. In the absence of a third dredge-up episode during the asymptotic giant branch phase, white dwarf models evolved fromlow-metallicity progenitors have a thick hydrogen envelope, which makes hydrogen shell burning be the most important energysource.Aims. We investigate the pulsational stability of white dwarf models with thick envelopes to see whether nonradialg-mode pulsationsare triggered by hydrogen burning, with the aim of placing constraints on hydrogen shell burning in cool white dwarfs andon a thirddredge-up during the asymptotic giant branch evolution of their progenitor stars.Methods. We construct white-dwarf sequences from low-metallicity progenitors by means of full evolutionary calculations thattakeinto account the entire history of progenitor stars, including the thermally pulsing and the post-asymptotic giant branch phases, andanalyze their pulsation stability by solving the linear, nonadiabatic, nonradial pulsation equations for the models in the range ofeffective temperaturesTeff ∼ 15 000− 8 000 K.Results. We demonstrate that, for white dwarf models with massesM⋆ . 0.71 M⊙ and effective temperatures 8 500. Teff . 11 600K that evolved from low-metallicity progenitors (Z = 0.0001, 0.0005, and 0.001), the dipole (ℓ = 1) and quadrupole (ℓ = 2) g1 modesare excited mostly due to the hydrogen-burning shell through theε-mechanism, in addition to othergmodes driven by either theκ− γor the convective driving mechanism. However, theε mechanism is insufficient to drive these modes in white dwarfs evolved fromsolar-metallicity progenitors.Conclusions. We suggest that efforts should be made to observe the dipoleg1 mode in white dwarfs associated with low-metallicityenvironments, such as globular clusters and/or the galactic halo, to place constraints on hydrogen shellburning in cool white dwarfsand the third dredge-up episode during the preceding asymptotic giant branch phase.

Key words. stars: evolution — stars: interiors — stars: oscillations —white dwarfs

1. Introduction

White dwarf stars are the most common end product of stellarevolution, with 97% of all stars becoming white dwarfs. As aresult, observations of white dwarfs provide us rich informationabout the stellar evolution of their progenitors and the star for-mation history in our galaxy. Furthermore, some white dwarfsare pulsating stars, hence asteroseismology of these starsmakesfurther information available about the internal structure and theevolutionary status of white dwarfs. Several types of pulsatingwhite dwarfs have been observed: GW Vir (pulsating PG1159)stars, hot DA variable (HDAV) stars, V777 Her (DBV) stars, hotDQ variable (HDQV) stars, ZZ Ceti (DAV) stars, and ExtremelyLow Mass variable (ELMV) stars (Winget & Kepler 2008;Fontaine & Brassard 2008; Althaus et al. 2010; Fontaine et al.2013). They exhibit luminosity variations with periods rang-ing from ∼ 100 s to∼ 6000 s, which are attributed to nonradialg modes excited by either theκ − γ mechanism that works inthe ionization zones of the abundant chemical elements or con-vection in cool white dwarfs (Brickhill 1983; Goldreich & Wu1999).

Besides those mechanisms, theε mechanism has been pre-dicted to lead to destabilization of some short-periodg modesin white-dwarf and pre-white-dwarf stars. This mechanism isinduced by thermonuclear reactions, which are highly depen-

dent on temperature. During a compression phase, the temper-ature and thus the nuclear energy production rates are higherthan at equilibrium, hence, in the layers where nuclear reac-tions take place, more thermal energy is produced. The oppositeoccurs during the following expansion phase, and as a conse-quence, the contribution of the perturbation of nuclear energygeneration to the work integral is always a destabilizationef-fect (Unno et al. 1989). The first study of theε mechanism inpre-white-dwarf stars was the seminal work by Kawaler et al.(1986), who found that someg modes with periods in the range70 − 200 s are indeed excited through this mechanism by he-lium shell burning in hydrogen-deficient pre-white-dwarf stars.Later on, Córsico et al. (2009) conducted a more thorough stabil-ity analysis, and showed the existence of an instability strip forGW Vir stars in the H-R diagram, for which theε mechanismby helium shell burning is responsible. As for hydrogen burning,Maeda & Shibahashi (2014) recently predicted that some low-order g modes are excited by theε mechanism due to vigor-ous hydrogen shell burning in very hot and luminous pre-whitedwarfs with hydrogen-rich envelopes. Also, Córsico & Althaus(2014) have shown that some low-ordergmodes could be drivenin ELMVs by theε mechanism due to hydrogen shell burning,although most of the observed pulsation modes, which are char-

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acterized by high- and intermediate- radial orders, are likely tobe excited by theκ − γ mechanism.

Recent detailed studies by Miller Bertolami et al. (2013) andAlthaus et al. (2015) have shown that white dwarf stars evolvedfrom low-metallicity progenitors (0.00003. Z . 0.001), in theabsence of carbon enrichment due to third dredge-up episodesduring the asymptotic giant branch (AGB) phase, have a thickenough hydrogen envelope, which makes stable hydrogen shellburning by pp-chain the most important energy source even atlow luminosities (logL/L⊙ . −3). Nuclear burning becomesweaker with an increase of metallicity. The importance of hy-drogen shell burning is also strongly dependent on the mass ofthe white dwarf, and no significant burning is expected to occurin white dwarfs with masses greater than∼ 0.6 M⊙. This extrasource of energy delays the white dwarf cooling. Even at lowluminosities, logL/L⊙ ∼ −4, residual hydrogen burning leads toan increase in the cooling times of the low- and intermediate-mass white dwarfs (M . 0.6 M⊙) by ∼ 20− 40%. These resultsare not affected by the mass loss rate in the preceding stages. Onthe other hand, the occurrence of overshooting during the AGBphase strongly favors third dredge-up episodes. This leadsto theformation of white dwarfs with thinner hydrogen envelopes,withthe consequence that no appreciable residual hydrogen burn-ing during the white dwarf regime is expected. The delay ofthe cooling times of white dwarfs evolved from low-metallicityprogenitors impacts on our understanding of stellar populations.In particular, Torres et al. (2015) employed the observed whitedwarf luminosity function of the low-metallicity globularclusterNGC 6397 to constrain the hydrogen nuclear burning in whitedwarfs from low-metallicity progenitors. They conclude that thelow-mass white dwarf population in NGC 6397 should be char-acterized by important residual burning and hence, the progeni-tors of these stars should not have experienced third dredge-up inthe AGB phase. Although Cojocaru et al. (2015) tried the samewith the isolated white dwarfs in the galactic halo, they couldnot reach a definitive conclusion, because the number of detectedwhite dwarfs in the galactic halo remains too small.

According to Miller Bertolami et al. (2013) andAlthaus et al. (2015), the contribution of nuclear energygeneration to the luminosity of the white dwarfs evolved fromlow-metallicity progenitors is the largest, being about 80% ofthe luminosity of the star, when logTeff ∼ 4. It is in this effectivetemperature range that many DAVs are found to pulsate ingmodes driven by either theκ − γ mechanism or the convectivedriving mechanism. Motivated by these results, in this paperwe assess the possibility that DA white dwarfs belonging tolow-metallicity stellar populations (like globular clusters and thegalactic halo) can develop short-period variabilities, analogousto the short-periodg modes exhibited by ZZ Ceti stars. Asdescribed above, the low-mass and low-metallicity progenitorsof this kind of DA white dwarfs should avoid the third dredge-upepisodes in the AGB phase, such that residual hydrogen burningcontinues at a significant level, and overstability ofg modes isinduced by theε mechanism. Along this study, we aim to usewhite dwarf pulsations to verify the presence of hydrogen shellburning in some white dwarfs, and eventually, to constrain theefficiency of third dredge-up episodes in the AGB phase.

The paper is organized as follows: In Sect. 2 we briefly de-scribe our stellar evolution and stellar pulsation tools and themain ingredients of the evolutionary sequences. In Sect. 3 wepresent our pulsation results in detail. Finally, in Sect. 4we sum-marize our main findings.

2. Evolutionary models and numerical tools

We have constructed equilibrium models of low-mass whitedwarfs evolved from low-metallicity progenitors, using theLPCODE stellar evolutionary code (Althaus et al. 2003, 2005,2012, 2015; Camisassa et al. 2016), which were initially evolvedfrom the zero age main sequence (ZAMS), all the way to thewhite-dwarf stage. Additionally, we computed an evolutionarysequence of solar metallicity (Z = 0.018) in order to see if theεmechanism could also excite oscillations in white dwarfs com-ing from solar-metallicity progenitors. The physical inputs of ourmodels were described in detail by Althaus et al. (2015). Below,we list the main ingredients which are relevant for this work:

– Extra mixing due to diffusive convective overshooting hasbeen considered during the core hydrogen and helium burn-ing phases. Following Althaus et al. (2015), extra mixingwas not considered during the thermally-pulsing AGB. TheOvershooting parameter was set tof = 0.015 at each con-vective boundary.

– Mass loss during the red giant branch phase was takenfrom Schröder & Cuntz (2005). For the AGB phase, we useagain that of Schröder & Cuntz (2005) for pulsation peri-ods shorter than 50 days. For longer periods, mass loss istaken as the maximum of the rates of Schröder & Cuntz(2005) and Groenewegen et al. (2009) for oxygen rich stars,or the maximum of the rates of Schröder & Cuntz (2005) andGroenewegen et al. (1998) for carbon-rich stars. In all of ourcalculations, mass loss was suppressed after the post-AGBremnants reach logTeff = 4.

– Radiative and conductive opacities are taken from OPAL(Iglesias & Rogers 1996) and from Cassisi et al. (2007),respectively. For the low-temperature regime we usedmolecular opacities with varying carbon to oxygen ratios(Ferguson et al. 2005; Weiss & Ferguson 2009). These opac-ities are necessary for a realistic treatment of progenitorevo-lution during the thermally-pulsing AGB phase.

– For the white-dwarf stage we employed an updated versionof the equation of state of Magni & Mazzitelli (1979) for thelow-density regime, whereas for the high-density regime weconsidered the equation of state of Segretain et al. (1994).

– Changes in the chemical abundances have been computedconsistently with the predictions of nuclear burning and con-vective mixing all the way from the ZAMS to the whitedwarf phase. In addition, abundance changes resulting fromatomic element diffusion were considered during the whitedwarf regime.

– For the white-dwarf stage and for effective temperatureslower than 10,000K, outer boundary conditions for theevolving models are derived from non-gray model atmo-spheres (Rohrmann et al. 2012).

Table 1 lists the initial masses of the sequences, the resultingwhite dwarf masses, the hydrogen mass fraction, log (MH/M⊙),and the surface carbon to oxygen ratio, C/O, at the begin-ning of the white dwarf phase. Our prediction for the initial-to-final mass relation is similar to that calculated by Romero etal.(2015). The C/O ratio indicates that, with the exception of theM⋆ = 0.738 M⊙ sequence withZ = 0.0001, none of thesemodel stars experienced carbon enrichment of the envelope dur-ing the AGB phase. The total amount of hydrogen at the begin-ning of the white dwarf stage (MH) is a key quantity for assess-ing the importance of nuclear burning, since the more massivethe hydrogen envelope is, the more intense the burning is. Inthiswork, we have not considered that some white dwarfs could have

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Fig. 1. Evolutionary tracks of white dwarf models withM⋆ ∼ 0.51 - 0.54 M⊙ on the logTeff - logg plane. Each panel corresponds to a differentmetallicity value (Z) of the white dwarf progenitors. For the case ofZ = 0.0001 we include two additional sequences with massesM⋆ = 0.666 M⊙andM⋆ = 0.738 M⊙. Red thick segments on the tracks at logTeff ∼ 4 correspond to DA white dwarf models having the dipoleg1 mode destabilizedmostly by theε mechanism acting at the hydrogen burning shell. The violet thick segment in the case of theM⋆ = 0.535 M⊙ andZ = 0.0001sequence corresponds to models with unstable low-orderg-modes (g1 to g4) destabilized solely by theε mechanism. The dashed green linedisplays the theoretical red edge of the instability domainof pre-white-dwarfs with hydrogen-rich envelopes evolvedfrom MZAMS ≥ 1.5 M⊙ withsolar metallicity found by Maeda & Shibahashi (2014). The almost vertical dotted blue line corresponds to the empiricalblue edge of the instabilitydomain of ZZ Ceti stars reported by Gianninas et al. (2011).

thin hydrogen envelopes (Castanheira & Kepler 2008, 2009;Romero et al. 2012, 2013)

The pulsation computations were performed with the lin-ear, nonradial, nonadiabatic version of theLP-PUL pulsationcode described by Córsico et al. (2006) (see also Córsico et al.2009). Although we have considered both dipole (ℓ = 1) andquadrupole (ℓ = 2) g modes, the results are qualitatively similarand therefore we will focus on results forℓ = 1. The pulsationcode solves the sixth-order complex system of linearized equa-tions and boundary conditions as given by Unno et al. (1989).The code considers the mode excitation by both theκ − γ mech-anism and theε mechanism. Our computations ignore the per-turbation of the convective flux, that is, we are adopting the

“frozen-in convection” approximation. While this approxima-tion is known to give unrealistic locations of the g-mode rededgeof the ZZ Ceti instability strip, it leads to satisfactory predic-tions for the location of the blue edge (Van Grootel et al. 2012;Saio 2013). The same is true for the blue edge of the instabilitydomain of the ELMV stars (Córsico & Althaus 2016). We havealso performed additional calculations disregarding the effects ofnuclear energy release on the nonadiabatic pulsations, forwhichwe fix ε = ερ = εT = 0, whereε is the nuclear energy productionrate, andερ andεT are their logarithmic derivatives, defined asερ = (∂ ln ε/∂ lnρ)T andεT = (∂ ln ε/∂ ln T )ρ, respectively. Inthis way, we prevent theε mechanism from operating, although

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Table 1.Basic model properties for sequences with Z=0.0001, 0.0005,0.0010 and 0.0180.

Z MZAMS(M⊙) MWD(M⊙) log(MH/M⊙) C/O0.0001 0.85 0.535 −3.304 0.3010.0005 0.80 0.509 −3.231 0.2410.0010 0.85 0.538 −3.434 0.3070.0180 1.00 0.521 −3.543 0.0050.0001 1.50 0.666 −3.851 0.2640.0001 2.00 0.738 −4.347 11.477

Notes.MZAMS: initial mass,MWD: white dwarf mass, log(MH/M⊙): log-arithm of the mass of the hydrogen envelope at the maximum effectivetemperature at the beginning of the cooling branch, C/O: surface carbonto oxygen mass ratio at the beginning of the cooling branch.

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nuclear burning was still taken into account in the evolutionarycomputations.

While the temperature dependence of energy generation allthrough the pp-chain in equilibrium is governed by the slowestreaction between two protons andεT ≃ 4, we adopted the ef-fective temperature dependence of nuclear reaction through theperturbation, which is mainly governed by3He-3He reaction andεT ≃ 11 (Unno et al. 1989).

3. Results

Our stability analysis indicates the existence of many unstableg modes in all the white dwarf model sequences, most of themodes being destabilized by theκ − γ mechanism acting at thesurface hydrogen ionization zone. From observational grounds,we know that as the ZZ Ceti stars cool inside the instabilitystrip, modes with progressively higher radial orders are excited.This can be understood theoretically on the basis of the ther-mal timescale argument (Cox 1980; Winget & Kepler 2008). Asthe star cools, the base of the outer convection zone movesinwards, whereby the thermal timescale becomes increasinglyhigher there. Since the modes that can be excited have periodssimilar to the thermal timescale at the basis of the convectionzone, modes with progressively longer periods (i.e., with higher

radial orders) are excited as the star cools. When we shut downtheε mechanism, theg1 mode becomes stable for the completerange (or part) of effective temperatures analyzed. This indicatesthat these modes are excited to a great extent by theε mech-anism through the hydrogen-burning shell. Higher radial-ordermodes, on the other hand, are insensitive to the effects of nuclearburning, and remain unstable due only to theκ − γ mechanism.

Figure 1 shows the evolutionary tracks of all our white dwarfsequences in the logTeff - logg diagram. Red thick segments onthe evolutionary tracks indicate white dwarf models with thedipoleg1 mode excited mostly by theε mechanism. The figureshows that low-mass white dwarfs evolved from progenitor starscharacterized by metallicities in the range 0.00003. Z . 0.001are expected to undergog-mode pulsations powered by nuclearburning at the typical effective temperatures of ZZ Ceti stars.Nevertheless, in white dwarf models derived from solar metal-licity progenitors (right-hand lower panel), which are indeedrepresentative of ZZ Ceti stars (which are located in the solarneighborhood), nuclear burning does not play a role, and theε-mechanism is not operative. For the case of white dwarf mod-els evolved from progenitors withZ = 0.0001 (left-hand upperpanel), we have explored the efficiency of theε-mechanism bychanging stellar masses, and found that for models more massivethan∼ 0.71 M⊙, nog modes are excited by nuclear burning. Wehave included in Fig. 1 a dotted blue line, which correspondstothe empirical blue edge of the ZZ Ceti instability strip accordingto Gianninas et al. (2011). Clearly, models having the pulsation-ally unstableg1 mode excited by theε mechanism are locatedin (or near) the ZZ Ceti instability domain. We remark that theg1 mode found to be unstable in our computations is not merelyexcited by theε mechanism, because theκ − γ mechanism con-tributes somewhat to its destabilization.

Furthermore, we include in Fig. 1 the location of the theo-retical end of the instability domain of pre-white-dwarf modelswith hydrogen-rich envelope (dashed green line), as predictedby Maeda & Shibahashi (2014). Note that, at variance with theε-destabilizedg1 mode found in this paper in the ZZ Ceti insta-bility domain, the unstable modes in the pre-white-dwarf modelsconsidered by Maeda & Shibahashi (2014) are excited only bytheεmechanism. This is because at those high effective temper-atures and luminosities, hydrogen is completely ionized atthesurface layers, hence theκ − γ mechanism is inhibited to op-erate. We have performed additional stability computations toinvestigate whetherg modes are excited by theε mechanismat higher luminosities. For these exploratory computations, wehave considered a template white dwarf evolutionary sequencewith M⋆ = 0.535 M⊙ and progenitor metallicityZ = 0.0001,from Teff ∼ 100 000 K downwards (violet thick line). We foundthat dipole modesg1, g2, g3 andg4 are excited by theε mech-anism due to hydrogen burning down to an effective tempera-ture of∼ 36 000 K. These results are in line with those obtainedby Maeda & Shibahashi (2014), although the instability domainobtained here extends to lower effective temperatures, by virtuethat our white dwarf models are characterized by more vigoroushydrogen burning. A full exploration of the instability domain athigh luminosities by varying the stellar mass and the metallicityof the progenitors is, however, beyond the scope of the presentpaper, and will be presented in a future publication.

Figure 2 shows the unstable periods of dipoleg modes interms of the effective temperature for our white dwarf model se-quence withM⋆ = 0.535 M⊙ andZ = 0.0001. The left panelshows the results for the standard nonadiabatic pulsation com-putations. A dense spectrum ofg modes is destabilized by theκ − γ mechanism for effective temperatures below∼ 11, 800 -

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Fig. 2.Periods (Π) of unstable dipolegmodes in terms of the effective temperature for our white dwarf model sequence withM⋆ = 0.535 M⊙ andZ = 0.0001. The left panel corresponds to the sequence in which theεmechanism was allowed to operate, whereas in the right panelwe show thesituation in which it was artificially suppressed. Color coding indicates thee-folding time (τe) of each unstable mode (right scale).

11, 400K. The right panel corresponds to the same sequence butfor the case in which theε mechanism is artificially inhibited,and consequently, only theκ − γ mechanism is allowed to act.The palette of colors (right-hand scale) indicates thee-foldingtime (in years) of each unstable mode, defined asτe = 1/|ℑ(σ)|,where ℑ(σ) is the imaginary part of the complex eigenfre-quencyσ. The e-folding time is a measure of the time takenfor the perturbation to reach an observable amplitude. For theM⋆ = 0.535 M⊙ andZ = 0.0001 sequence, the dipoleg1 mode isunstable in the standard computations (left panel), but themodebecomes stable when we neglect the effect of nuclear energy pro-duction on stability (right panel). Hence, we conclude thatthemode is unstable due to theε mechanism. Thee-folding time ofthis mode is shown in terms of the effective temperature in Fig-ure 3. When theε mechanism is allowed to operate (solid redline), thee-folding time is about 5× 106 years, whereas the timethe star needs to cross the instability domain is of about 1.2×109

years. Therefore, if the nuclear energy production is as intense asour models predict, the dipole g1 mode has enough time to reachobservable amplitudes while the star is still evolving in that ef-fective temperature range. A similar situation is found forthewhite dwarf models evolved from progenitors withZ = 0.0005andZ = 0.001 sequences, as shown in the upper right and thelower left panels of Fig. 1, respectively. In these cases, however,the efficiency of theε mechanism is lower (due to less vigor-ous nuclear burning), and as a result, the ranges inTeff in whichthe dipoleg1 mode is destabilized by this mechanism are nar-rower. In the extreme case of white dwarfs evolved from solar-metallicity progenitors, that mode is found to be no longer ex-cited by theε mechanism. This result is expected because, forthese white dwarfs, nuclear burning plays a minor role at ad-vanced stages of their evolution.

We also performed nonadiabatic computations for whitedwarf models withM⋆ = 0.666 M⊙ andM⋆ = 0.738 M⊙ comingfrom progenitors withZ = 0.0001. For the 0.666 M⊙ sequence,we still found that the dipoleg1 mode is unstable mostly due totheε mechanism, although for a narrow range of effective tem-peratures (see the upper left panel of Fig. 1). However, for the0.738 M⊙ sequence, we found that mode no longer excited by theε mechanism. Again, this is an expected result, since no appre-ciable nuclear burning occurs in these models (see Althaus et al.2015), and because the progenitor star of this sequence experi-

enced carbon enrichment in the AGB phase due to third dredge-up episodes, as revealed in the carbon to oxygen ratio listedinTable 1. We have estimated the expected threshold mass (MT)above which the epsilon mechanism is not operative anymore.Specifically, we artificially reduce the nuclear energy release inthe 0.666 M⊙ sequence down to a value below which the epsilonmechanism is not able to excite theg1 pulsation mode, and thenwe use the corresponding nuclear luminosity at which this occursto interpolate between our 0.666 M⊙ and 0.738 M⊙ sequences tofind MT. The stellar mass value turns out to be MT ≈ 0.71 M⊙.

We now examine which regions of the models contribute tothe driving and damping of a given pulsation mode. Figure 4shows the Lagrangian perturbation of the temperature (δT/T )for the dipoleg1 mode (upper left-hand panel) and theg3 mode(lower left-hand panel), respectively, corresponding to aZ =0.0001,M⋆ = 0.535 M⊙ template model atTeff = 11 000K. Forthe g1 mode, the eigenfunctionδT/T has a local maximum atlog(1−Mr/M⋆) ∼ −3.5, where the hydrogen-burning shell is lo-cated. Theε mechanism acts as a “filter” that provides substan-tial driving to thoseg modes that have a maximum ofδT/T inthe region of the nuclear burning shell. For the dipoleg3 mode,on other hand,δT/T has negligible values at the burning shellregion, hence theεmechanism does not provide efficient drivingfor this mode.

It is useful to introduce the “running work integral”W(r),which represents the work done on the overlying layer by thesphere of radiusr (see Unno et al. 1989). Its surface value,W := W(R), gives the increase of the total energy over one pe-riod of oscillation. IfW > 0, the pulsation is gaining energy inone cycle, which means that the mode is unstable and grows.Otherwise, ifW < 0, the mode is losing energy and damps. Thederivative ofW(r) gives us information about the regions of driv-ing and damping of the oscillations. Regions in the star wheredW(r)/dr > 0 will contribute to driving, and zones in whichdW(r)/dr < 0 will provide damping.

We show dW(r)/dr for the dipoleg1 mode in the middle up-per panel of Fig. 4 with black dashed lines (red dashed lines)forthe case when theε mechanism is allowed (inhibited) to oper-ate. Theε mechanism provides significant driving at the hydro-gen burning shell region, but substantial damping occurs therewhen the mechanism is inhibited. As a result, the running workintegralW(r) increases at the burning shell region when theε

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ε [erg/s]Log(κ) [cm2/gr]

−15

−10

−5

0

5

10

15

20

25

30

2 4 6 8 10 12 14

W(r

)

−Log(1−mr/M★)

Π=261.3s

k=3, l=1

10 x W(r)10 x W(r) (ε=0, εT=0, ερ=0)

ε [erg/s]Log(κ) [cm2/g]

Fig. 4.Left panels: Lagrangian perturbation of temperature (δT/T ), nuclear generation rate (ε), and the mass fractions of hydrogen and helium (XH

andXHe, respectively) as functions of log(1−Mr/M⋆) for dipoleg1 mode (upper panel) andg3 mode (lower panel), respectively, for the model withZ = 0.0001,M⋆ = 0.535 M⊙ andTeff = 11, 000 K. The eigenfunction is normalized so that the relative displacement in the radial direction is unityat the stellar surface. Middle panels: same as left panels but for the differential work function (dW(r)/dr) for the case in which theεmechanism isallowed to operate (black dashed curves) and when it is suppressed (red dashed curves). Also, the scaled running work integrals (W(r)) are shownwith solid lines. Right panels: same as central panels, but the scaled running work integrals are shown for a wider range of the mass coordinate,along with the Rosseland mean opacityκ (radiative plus conductive).

mechanism operates (solid black line), while it substantially de-creases when the mechanism is switched off (solid red line). Ascan be seen in the right-hand upper panel of this figure, whenεmechanism operates,W(r) becomes positive at the surface of thestar, thanks to the driving contribution due to theκ − γ mecha-nism, and theg1 mode becomes globally unstable. In contrast,when theε mechanism is inhibited, the driving contribution bytheκ − γ mechanism cannot prevent thatW(r) becomes negativeat the surface, making theg1 mode stable in these circumstances.

On the other hand, for the dipoleg3 mode (lower panels ofFig. 4), no significant driving is provided at the hydrogen burn-ing shell region. However, since substantial driving occurs in thehydrogen-ionization zone due to theκ − γ mechanism,W(r) be-comes positive at the surface. Thus, theg3 mode is unstable, re-gardless of whether nuclear burning is taken into account ornotin the stability computations.

4. Summary and conclusions

Althaus et al. (2015) predicted that low mass white dwarfs —evolved from low metallicity progenitors that did not experiencecarbon enrichment due to third dredge-up episodes during theAGB phase— are born with hydrogen envelopes thick enoughto develop intense hydrogen shell burning at low luminosities(log(L/L⊙) . −3). Following these results, in this paper, wehave investigated whetherg modes in DA white dwarfs couldbe unstable due to theε mechanism powered by the hydrogenburning shell. For that purpose, we computed nonadiabatic pul-sation modes of white dwarf models withM⋆ ∼ 0.51 - 0.54 M⊙,evolved from progenitorial stars with four different metallici-ties: Z = 0.0001,Z = 0.0005,Z = 0.001, andZ = 0.018.The first three sequences correspond to low-metallicity stellarpopulations, while the latter one is representative of objectsin the solar neighborhood. We have found that for the whitedwarf sequences evolved from low-metallicity progenitors, both

dipole and quadrupoleg1 modes are unstable mostly due to theε mechanism acting at the hydrogen burning shell. We foundthat the ability of this mechanism to destabilize those pulsa-tion modes decreases with increasing metallicity. In the limit ofwhite dwarfs with solar metallicity progenitors (Z = 0.018), nopulsation is driven by theε mechanism, since nuclear burningdoes not play a role in these objects. We have also explored themass dependence of theε mechanism efficiency, by computingwhite dwarf sequences with higher masses (M⋆ = 0.666 M⊙ andM⋆ = 0.738 M⊙) evolved fromZ = 0.0001 progenitors. Ourcalculations showed that theε mechanism destabilizes both thedipole and quadrupoleg1 modes associated with the 0.666 M⊙sequence, but does not excite these modes for the sequence withM⋆ = 0.738 M⊙.

To summarize, hydrogen shell burning in DA white dwarfs ateffective temperatures typical of the instability strip of ZZ Cetistars triggers the dipole (ℓ = 1) and the quadrupole (ℓ = 2)g1 modes with periodΠ ∼ 70− 120 s, provided that the massof the white dwarf isM⋆ . 0.71 M⊙ and the metallicity of theprogenitor stars is in the range 0.0001 . Z . 0.001. Thee-folding times for these modes are much shorter than the evo-lutionary timescale, which means that they could have enoughtime to reach observable amplitudes. We hope that, in the future,these modes can be detected in white dwarfs in low-metallicityenvironments, such as globular clusters and/or the galactic halo,allowing us to place constraints on nuclear burning in the en-velopes of low-mass white dwarfs with low-metallicity. More-over, detection of these pulsation modes could eventually verifythat low-metallicity AGB stars do not experience carbon enrich-ment due to third dredge-up episodes. Once again, asteroseis-mology of white dwarfs can prove to be a powerful tool to soundthe interior of these ancient stars, and also to constrain the his-tory of their progenitors.

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Camisassa et al.: Pulsations powered by hydrogen shell burning in white dwarfs

Acknowledgements. We acknowledge the comments and suggestions of our ref-eree, S. O. Kepler, that improved the original version of this paper. Part of thiswork was supported by AGENCIA through the Programa de Modernización Tec-nológica BID 1728/OC-AR, by the PIP 112-200801-00940 grant from CON-ICET. This research has made use of NASA Astrophysics Data System.

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