A new instability domain of CNO-flashing low-mass He-core stars on their early white-dwarf cooling branches
Leila M. Calcaferro, Alejandro H. Córsico, Leandro G. Althaus, Keaton J. Bell
aa r X i v : . [ a s t r o - ph . S R ] J a n Astronomy & Astrophysicsmanuscript no. paper-pul-vii © ESO 2021January 11, 2021
A new instability domain of CNO-flashing low-mass He-core starson their early white-dwarf cooling branches
Leila M. Calcaferro , , Alejandro H. Córsico , , Leandro G. Althaus , , and Keaton J. Bell , 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 Instituto de Astrofísica La Plata, CONICET-UNLP, Paseo del Bosque s / n, 1900, La Plata, Argentina DIRAC Institute, Department of Astronomy, University of Washington, Seattle, WA-98195, USA NSF Astronomy and Astrophysics Postdoctoral Fellow and DIRAC Fellowe-mail: [email protected]
Received ;
ABSTRACT
Context.
Before reaching their quiescent terminal white-dwarf cooling branch, some low-mass helium-core white dwarf stellar modelsexperience a number of nuclear flashes which greatly reduce their hydrogen envelopes. Just before the occurrence of each flash, stablehydrogen burning may be able to drive global pulsations that could be relevant to shed some light on the internal structure of thesestars through asteroseismology, similar to what happens with other classes of pulsating white dwarfs.
Aims.
We present a pulsational stability analysis applied to low-mass helium-core stars on their early white-dwarf cooling branchesgoing through CNO flashes in order to study the possibility that the ε mechanism is able to excite gravity-mode pulsations. We assessthe ranges of unstable periods and the corresponding instability domain in the log g − T e ff plane. Methods.
We carried out a nonadiabatic pulsation analysis for low-mass helium-core white-dwarf models with stellar masses between0 . . M ⊙ going through CNO flashes during their early cooling phases. Results.
We found that the ε mechanism due to stable hydrogen burning can excite low-order ( ℓ = ,
2) gravity modes with periodsbetween ∼
80 and 500 s, for stars with 0 . . M ⋆ / M ⊙ . . g − T e ff diagram withe ff ective temperature and surface gravity in the ranges 15 000 . T e ff .
38 000 K and 5 . . log g . .
1, respectively. For the sequencesthat experience multiple CNO flashes, we found that with every consecutive flash, the region of instability becomes wider, and themodes, more strongly excited. The magnitudes of the rate of period change for these modes are in the range ∼ − - 10 − [s / s]. Conclusions.
Since the timescales required for these modes to reach amplitudes large enough to be observable are shorter thantheir corresponding evolutionary timescales, the detection of pulsations in these stars is feasible. Given the current problems indistinguishing some stars that are populating the same region of the log g − T e ff plane, the eventual detection of short-period pulsationsmay help in the classification of such stars. Furthermore, if a low-mass white dwarf star were found to pulsate with low-order gravitymodes in this region of instability, it would confirm our result that such pulsations can be driven by the ε mechanism. In addition,confirming a rapid rate of period change in these pulsations would support that these stars actually experience CNO flashes, aspredicted by evolutionary calculations. Key words. asteroseismology — stars: oscillations — white dwarfs — stars: evolution — stars: interiors
1. Introduction
White dwarf (WD) stars represent the end stage in the life of themajority of all stars, including our Sun (Winget & Kepler 2008;Fontaine & Brassard 2008; Althaus et al. 2010; Córsico et al.2019). Most WDs ( ∼ ∼ . M ⊙ (Kepler et al. 2019).According to the stellar evolution theory, they probably harborcarbon-oxygen (CO) cores, although the most massive ones mayhave cores made of O and neon (Ne). At variance with averageDA WDs, there is a population of WDs with low mass ( M ⋆ . . M ⊙ ), that probably harbor helium (He) cores. The forma-tion context for such low-mass WDs is thought to consist of alow-mass red giant-branch (RGB) star that experiences strongmass loss, mostly as a result of binary interaction taking placebefore the onset of the He flash (Althaus et al. 2013; Istrate et al.2016b), which is avoided, and the core of these stars is com-posed of He. The current evolutionary models predict that, once the mass-loss stage is finished, these low-mass stars would ex-perience a number of CNO nuclear flashes which greatly reducetheir H content before reaching their quiescent terminal WDcooling branch. Such binary-star evolution scenario is confirmedby observations, since most low-mass WDs are found in binarysystems (Marsh et al. 1995). Theoretical computations (see, forinstance, Althaus et al. 2013; Istrate et al. 2016b) predict thatlow-mass WDs with masses lower than ∼ . − . M ⊙ ,called extremely low-mass (ELM) WDs, do not experience CNOflashes in their past evolution. The absence of flashes wouldconsequently suggest that ELM WDs harbor thick H envelopes,and then, they would be characterized by very long coolingtimescales, and have pulsational properties di ff erent in compari-son with systems that had experienced flashes (see Althaus et al.2013; Córsico & Althaus 2014a). This threshold depends on the metallicity of the WD progenitors (e.g.Serenelli et al. 2002; Istrate et al. 2016b). Some authors adopt ∼ . M ⊙ as the upper mass limit for ELM WDs (Brown et al. 2016).Article number, page 1 of 9 & Aproofs: manuscript no. paper-pul-vii
In the last years, numerous low-mass and ELM WDs havebeen detected in the context of relevant surveys such as theSDSS, ELM, SPY and WASP (see, for instance, Koester et al.2009; Brown et al. 2010, 2016, 2020; Kilic et al. 2011, 2012;Gianninas et al. 2015; Kosakowski et al. 2020). The discoveryof their probable precursors, the so-called low-mass pre-WDs,has triggered the interest in this type of objects because ofthe possibility of studying the evolution of the progenitorsthat lead to the WD phase. Even more interesting, the detec-tion of multi-periodic brightness variations in low-mass WDs(Hermes et al. 2012, 2013b,a; Kilic et al. 2015, 2018; Bell et al.2017, 2018; Pelisoli et al. 2018), and low-mass pre-WDs(Maxted et al. 2013, 2014; Gianninas et al. 2016; Wang et al.2020) has brought about new classes of pulsating stars knownas ELMVs and pre-ELMVs, respectively. It has allowed thestudy of their stellar interiors with the tools of asteroseismol-ogy, similar to the case of other pulsating WDs such as ZZCeti stars or DAVs —pulsating WDs with H-rich atmospheres—and V777 Her or DBVs —pulsating WDs with He-rich at-mospheres (Winget & Kepler 2008; Fontaine & Brassard 2008;Althaus et al. 2010; Córsico et al. 2019). The pulsations ob-served in ELMVs are compatible with global gravity ( g )-modepulsations. In the case of pulsating ELM WDs, the pulsationshave large amplitudes mainly at the core regions (Steinfadt et al.2010; Córsico et al. 2012; Córsico & Althaus 2014a), allowingthe study of their core chemical structure. According to nona-diabatic computations (Córsico et al. 2012; Van Grootel et al.2013; Córsico & Althaus 2016) these modes are probably ex-cited by the κ − γ (Unno et al. 1989) mechanism, acting atthe H-ionization zone. In the case of pre-ELMVs, the nona-diabatic stability computations for radial (Je ff ery & Saio 2013)and nonradial p - and g -mode pulsations (Córsico et al. 2016;Gianninas et al. 2016; Istrate et al. 2016a) revealed that the ex-citation is probably due to the κ − γ mechanism acting mainlyat the zone of the second partial ionization of He, with a weakercontribution from the region of the first partial ionization of Heand the partial ionization of H. The presence of He in the driv-ing zone is crucial in order to have the modes destabilized by the κ − γ mechanism (Córsico & Althaus 2016; Istrate et al. 2016a).Additionally, Córsico & Althaus (2014b) showed that the ε mechanism due to stable H burning may contribute to destabi-lize some short-period g modes at the basis of the H envelope,in particular, for low-mass WD sequences with stellar masseslower than . . M ⊙ and e ff ective temperatures below ∼
10 000K. The ε mechanism is thought to be a potential mechanism forexciting pulsations in several type of stars. Since nuclear burn-ing has a strong dependence on the temperature, it can lead toa pulsational instability induced by thermonuclear reactions. In-deed, this mechanism can be responsible for the the excitation of g modes not only in ELM WDs (Córsico & Althaus 2014b), butalso in other types of WDs and pre-WDs as well. For instance,Córsico et al. (2009) showed that this mechanism could driveshort-period g -mode pulsations in GW Virginis stars. In addi-tion, Maeda & Shibahashi (2014) found that this mechanism canexcite low-degree g modes in very hot DA WDs and pre-WDscoming from solar-metallicity progenitors. Later, Althaus et al.(2015) and Camisassa et al. (2016) showed that DA WDs com-ing from low-metallicity progenitors can sustain stable H burn-ing even at low luminosities, motivating a subsequent nona-diabatic exploration that demonstrates the excitation of low-order g modes in hot DA WDs and pre-WDs (Calcaferro et al.2017c), as well as in DA WDs with e ff ective temperatures typ-ical of ZZ Ceti stars (Camisassa et al. 2016) coming from sub-solar metallicity progenitors. Finally, attempts were done to ex- plain the pulsations observed in other kind of compact stars,the pulsating He-rich hot subdwarf stars, as triggered by nuclearburning through the ε mechanism (Miller Bertolami et al. 2011;Battich et al. 2018).Another relevant aspect for this type of stars is the secularchange of periods with time ( ˙ Π ), which reflects the evolution-ary timescale of these stellar remnants. In particular, the theo-retical computations of the rates of period change carried outby Calcaferro et al. (2017a) for the low-mass WD, pre-WD andpre-CNO flash stages, indicate that the magnitudes of ˙ Π of g modes for models evolving in stages prior to the CNO flashesare up to 10 − - 10 − [s / s], and more importantly, larger thanthe maximum magnitudes of ˙ Π predicted for the other two stagesanalyzed in that work (i.e., the WD and pre-WD stages).In this work, we present a nonadiabatic stability analysisconsidering the e ff ects of the ε mechanism in destabilizing g -mode periods for sequences of low-mass WDs evolving throughCNO flashes. The low-mass WD sequences expected to experi-ence CNO flashes are in the mass range of 0 . . M ⋆ / M ⊙ . . . − . M ⊙ ,it is possible to detect a star evolving prior to the flash whereevolution clearly slows down (see Fig. 4 of Althaus et al. 2013).A fundamental motivation in this paper is the possibility offinding g -mode pulsations in stars that go through the regionin the log g − T e ff diagram where the flashes take place. Thiswill allow to study the structure of such objetcs, and conse-quently complement the information that can be extracted byanalyzing the pulsations observed in ELMV and pre-ELMVstars (e.g., Córsico & Althaus 2014a, 2016; Córsico et al. 2016;Istrate et al. 2016b; Calcaferro et al. 2017a,b, 2018).We report the existence of a new instability strip for low-mass He-core stars evolving through CNO flashes. We study thedestabilization e ff ects produced by the ε mechanism due to sta-ble H burning. The paper is organized as follows. A brief sum-mary of the numerical codes and the stellar models employed isprovided in Sect. 2. In Sect. 3, we show the results of the stabil-ity analysis performed on the sequences of low-mass WDs understudy. Finally, in Sect. 4 we summarize the main findings of thiswork.
2. Numerical codes
We employed the evolutionary models of low-mass He-core WDs generated with the
LPCODE stellar evolution code(Althaus et al. 2013).
LPCODE evolutionary code computes thecomplete evolutionary stages that lead to the formation of theWD, thus allowing the study of the WD evolution consistentlywith the predictions of the evolutionary history of the progen-itors. Initial configurations for low-mass He-core WD modelswere computed by Althaus et al. (2013) by mimicking the bi-nary evolution of an initially 1 . M ⊙ solar metallicity donor starand a 1 . M ⊙ neutron star companion. Binary evolution was as-sumed to be fully nonconservative, and the losses of angularmomentum due to mass loss, gravitational wave radiation, andmagnetic braking were considered. Initial He-core WD mod-els with stellar masses between 0 . . M ⊙ char-acterized by thick H envelopes were derived from stable massloss via Roche-lobe overflow, see Althaus et al. (2013) for de-tails. During WD regime, time-dependent element di ff usion dueto gravitational settling and chemical and thermal di ff usion ofnuclear species was considered, following the multicomponentgas treatment of Burgers (1969). For this work, we analyzedthe sequences with stellar masses of 0 . . . Article number, page 2 of 9alcaferro et al.: A new instability domain for CNO-flashing low-mass WDs
Log ( g ) Teff[K]
Fig. 1.
Instability domain corresponding to the ε mechanism on thelog g vs T e ff diagram for the low-mass WD sequence with 0 . M ⊙ .The stages of pulsational instability are emphasized with thick blacklines along the evolutionary tracks. Low-order g modes are driven be-fore each flash. The track begins after the end of Roche-lobe overflow(upper right branch of the curve) and proceeds downward, toward highervalues of T e ff , until the first CNO flash takes place. Numbers denote ev-ery consecutive flash. Red arrows along the curve indicate the courseof the evolution. The cyan diamond before the ninth flash indicates thelocation of the template model analyzed in Fig. 5. . M ⊙ . These sequences were evolved through the stages ofmultiple thermonuclear CNO flashes that take place during theirearly cooling branch.We carried out a pulsation stability analysis of nonradialdipole ( ℓ =
1) and quadrupole ( ℓ = g modes employingthe nonadiabatic version of the LP-PUL pulsation code (see,Córsico et al. 2006, 2009, for details). This pulsation code isbased on a general Newton-Raphson technique that solves thesixth-order complex system of linearized equations and bound-ary conditions (see Unno et al. 1989). The Brunt-Väisälä fre-quency ( N ) was computed following the so-called “LedouxModified” treatment (Tassoul et al. 1990; Brassard et al. 1991).Our nonadiabatic computations are based on the frozen-convection approximation, that neglects the perturbation of theconvective flux. The set of pulsation modes considered in thiswork covers a very wide range of periods (up to ∼ ff ects of nuclear energy release on the nonadiabatic pulsa-tions, for which we set ε = ε ρ = ε T =
0, being ε the nuclearenergy production rate, and ε ρ and ε T the corresponding loga-rithmic derivatives ε ρ = ( ∂ ln ε/∂ ln ρ ) T and ε T = ( ∂ ln ε/∂ ln T ) ρ .This prevents the ε mechanism from operating, however nuclearburning was still taken into account in the evolutionary calcula-tions.
3. Stability analysis
To illustrate the results of our nonadiabatic study, we displayin Fig. 1 the evolutionary track of the low-mass He-core se-quence with 0 . M ⊙ on the log g vs T e ff plane. As shownby Althaus et al. (2013), this sequence experiences nine CNOflashes before entering the final cooling track, that reduce con-siderably the thickness of the H envelope. Before and along eachloop described by this evolutionary sequence, our nonadiabatic exploration shows that the ε mechanism is able to destabilizelow-order ℓ = g modes. We highlight the corresponding regionof instability of each part of the track with thick black lines. Itis clear that the extension of the region of instability in the log g vs T e ff plane grows with every consecutive flash. The periods ofunstable ℓ = g modes in terms of the e ff ective temperaturefor each one of the loops described by this sequence are shownin Fig. 2. In the panels, the color coding indicates the logarithmof the e -folding time (in years) of the unstable modes, whichrepresents a measure of the time taken for the perturbation thatcauses the oscillation to reach an observable amplitude. Its def-inition is given by the expression τ e = / |ℑ ( σ ) | , where ℑ ( σ )is the imaginary part of the complex eigenfrequency σ . For thepanel representing the first CNO flash (top left panel), markedas "1", there are only a few unstable ℓ = g modes with lowradial orders ( k =
2, 3 and 4) corresponding to periods between ∼
260 and 450 s, in a very limited range of e ff ective temperature( ∼
16 000 −
18 000 K), and with relatively large values for the e -folding time (being its minimum value ∼ . × yr). TheFigure shows that with every consecutive flash, the instabilityregion becomes wider, and with more modes being destabilized.For instance, in the third flash (top right panel), one additionalmode is excited, corresponding to k =
1, and in general, thevalues of the e -folding time shorten as indicated by the colorcoding. It is apparent that for the last three panels ("7", "8" and"9") the instability domains are considerably extended and the e -folding times significantly shorten, that is, the excitation be-comes stronger, with the implication that these modes may havea larger chance to reach observable amplitudes. For the ninthpanel (bottom right panel), it is evident the large region in thediagram where we can find ℓ = g -mode periods destabilizedby the ε mechanism. In this case, the periods are characterizedby k =
1, 2, 3 and 4 corresponding to periods in the range of ∼
200 and 480 s, e ff ective temperatures between ∼
15 000 and31 000 K, and with e -folding times that, in its lowest, reach downto ∼ . × yr. We note that, in general, there is a trend for theperiods to lengthen as the evolution proceeds to lower e ff ectivetemperatures, before entering the loop, and to shorten afterward.In order to estimate if it would be possible to observe astar pulsating by the ε mechanism while evolving through oneof these loops, we consider the example of the models evolv-ing through the ninth flash for the 0 . M ⊙ sequence. Giventhat the time spent by this sequence in the region of unstable g -mode periods for the ninth loop is ∼ . × yr, and thevalues of the e -folding times for many modes are significantlylower, ∼ . × yr, then these modes would have enough timeto reach observable amplitudes. Since the duration of the wholepre-WD stage for this sequence, throughout the flashes and un-til the sequence gets to the maximum e ff ective temperature, is5 . × yr, when we compare it to the time spent by the se-quence during this ninth stage of instability, we see that it mightbe possible to detect a star with 0 . M ⊙ pulsating in low-order g modes via ε mechanism while evolving during this stage.Such possibility considerably increases if we take into accountall the nine stages of instability experienced by this sequence.This can be visualized by showing how these pulsations varywith time. In Fig. 3 we display the periods of unstable ℓ = g modes as in Fig. 2, but in terms of the elapsed time (in Myr)since the appearance of the first unstable mode (correspondingto the first flash) for this sequence. It is clear that the flashesoccur sooner each time, leading to less (and even almost negli-gible) temporal gaps with every consecutive flash. In addition,in Fig. 4 we show the temporal evolution of the surface lumi-nosity, L (green line), the luminosity given by the pp chain, L pp Article number, page 3 of 9 & Aproofs: manuscript no. paper-pul-vii P [ s ] ( un s t ab l e ) l og t e [ y r ] T eff [K] Fig. 2.
Periods of unstable ℓ = g modes in terms of the e ff ective temperature for the nine flashes experimented by our template low-masssequence with 0 . M ⊙ . Color coding indicates the logarithm of the e -folding time ( τ e ) of each unstable mode (right scale). Blue numbers at thebottom right corner of each panel indicate the number of the flash, as in Fig. 1. The cyan diamond on the bottom right panel indicates the locationof the template model analyzed in Fig. 5 (as in Fig. 1). (light-blue dashed line), and the luminosity due to the CNO bicy-cle, L CNO (red dotted line), along with a thick black line empha-sizing, particularly, the location of the models within the ninthinstability region (as in Fig. 1). In the Figure, two very narrowgrey vertical strips represent the evolutionary stages where con-vection (either internal or external) is present. The Figure showsthat the region of instability starts when L and L CNO consider-ably drop after the occurrence of the eighth CNO flash and endsbefore the beginning of the ninth CNO flash. It is clear that dur- ing the flashes (and very shortly before, although not noticeableby its narrowness), an internal induced-flash convective zone de-velops, that quickly moves toward the stellar surface to rapidlyvanish. In summary, at the stages of pulsation instability drivenby the ε mechanism, there is no convection inside our models.We conclude that convection does not a ff ect any of the instabilityregions presented in this work.Additionally, our calculations show that the ε mechanism isalso able to destabilize low-order ℓ = g modes in all the se- Article number, page 4 of 9alcaferro et al.: A new instability domain for CNO-flashing low-mass WDs
0 25 50 75 100 125 150 175 200 225 250 275 300 D t [Myr]
150 200 250 300 350 400 450 500 P [ s ] ( un s t ab l e ) l og t e [ y r ] Fig. 3.
Same as Fig.2 for the nine flashes of the sequence with0 . M ⊙ , but in terms of the elapsed time since the appearance ofthe first unstable mode for the first flash. −3−2−1 0 1 2 3 4 5 9740 9760 9780 9800 9820 98409 Log ( L / L o ) Age [Myr]
Fig. 4. log( L / L ⊙ ) vs age (in Myr) for the sequence with 0 . M ⊙ ,corresponding to the last instability region. The temporal evolution ofthe surface luminosity (green line), the luminosity given by the pp chain(light-blue dashed line) and the luminosity due to the CNO bicycle (reddotted line) are shown, along with the ninth instability region empha-sized by a thick black line. Grey vertical strips mark the narrow regionswhere convection is present (either internal or external). A blue numberindicates the number of the flash (as in Fig. 1), while the cyan diamondindicates the location of the template model analyzed in Fig. 5 (and 1). quences analyzed. Although the ranges of e ff ective temperatureof the models in which these modes are destabilized are approx-imately the same as in the ℓ = ε mechanism destabilizes a higher number of ℓ = . M ⊙ we found in the ℓ = k from 1 to 4 are excited, in the ℓ = k = e -folding time is, in general, lower for the ℓ = ℓ = g modes is shifted towardshorter values when compared to the ones found for the ℓ = ℓ = g modes spans from ∼
80 to ∼
250 s. For brevity, in whatfollows we will focus on the ℓ = ε mechanism as a destabiliz-ing agent in CNO-flashing low-mass WDs on their early-coolingbranches, we pick out a representative unstable pulsation modecorresponding to a template model evolving at the stage beforethe ninth flash for the sequence with 0 . M ⊙ , indicated inFigs. 1 and 2 with a cyan diamond. In the left panel of Fig. 5, wedisplay the Lagrangian perturbation of the temperature, ( δ T / T ),in terms of the mass fraction coordinate [ − log(1 − M r / M ⋆ )],for the unstable ℓ = k = g mode ( Π = . ε ), and the frac-tional abundances of H and He ( X H and X He ). The eigenfunc-tion δ T / T has its maximum value at − log(1 − M r / M ⋆ ) ∼ . ε mechanism yieldssubstantial driving to those g modes that have their maximum of δ T / T at the narrow region of the burning shell (Kawaler et al.1986). It is illustrated for the ℓ = k = ff erential work function ( dW / dr ) in terms of − log(1 − M r / M ⋆ ), and also the scaled running work integral( W ) with dashed black curves. It is evident that there is con-siderable driving ( dW / dr >
0) at the region of the H-burningshell for the mode analyzed. This mode is globally unstable,as indicated by the positive value of W at the stellar surface[ − log(1 − M r / M ⋆ ) ∼ ε mechanism by forcing ε = ε ρ = ε T = dW / dr are shown in the rightpanel of Fig. 5 with solid violet curves. Note that, in this case,a strong damping ( dW / dr <
0) takes place in the burning-shellregion, resulting in a value W < k = g mode is unstable due to thedestabilizing e ff ect of the H-burning shell via the ε mechanism.In Fig. 6 we show the evolutionary tracks of the low-massHe-core WDs with 0 . . . . M ⊙ onthe log g vs T e ff plane. As in Fig. 1, thick black lines su-perimposed on every evolutionary track represent the regionswhere the ε mechanism is able to excite low-order g modes.We have included a sample of ELM WD stars (shown withred asterisks; Brown et al. 2010, 2013, 2016, 2020; Vennes et al.2011; Gianninas et al. 2015; Kawka et al. 2015; Pelisoli & Vos2019). Also, we have included the location of the knownELMVs (Hermes et al. 2012, 2013b,a; Kilic et al. 2015, 2018;Bell et al. 2017, 2018; Pelisoli et al. 2018) marked with light-blue squares with dots, and pre-ELMVs (Maxted et al. 2013,2014; Gianninas et al. 2016; Wang et al. 2020), indicated withblack circles with dots (both regions have been emphasizedwith light-blue and light-red shaded areas, respectively). Inaddition, we have included the location of some sdBV stars(Green et al. 2011), BLAPs (Pietrukowicz et al. 2017), as well asHigh-Gravity (HG)-BLAPs (Kupfer et al. 2019) indicated withgreen circles, and orange and pink triangles, respectively (theirregions emphasized with green, orange and pink shaded ar-eas, respectively). The Figure shows that every evolutionary se-quence considered in this work has an extended zone of pulsa-tion instability which, altogether, results in a wide region in thelog g vs T e ff plane (grey shaded area) where low-order ℓ = g modes can be destabilized by the ε mechanism. This region cov- Article number, page 5 of 9 & Aproofs: manuscript no. paper-pul-vii −0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9 dT / T −Log (1−M r /M ★ ) e /2000X H X He −30−20−10 0 10 20 30 40 50 60 70 80 2 4 6 8 10 1250 x W(r)50 x W(r) d W / d r , W (r) −Log (1−M r /M ★ )( e , e T , e r = 0 ) e /50 Fig. 5.
Left panel:
Lagrangian perturbation of the temperature ( δ T / T ), along with the scaled nuclear generation rate ( ε ), and the fractionalabundances of H and He ( X H and X He ) in terms of the mass fraction coordinate ( − log(1 − M r / M ⋆ )), for a representative unstable pulsation modewith ℓ = k = Π = . T e ff = . M ⊙ sequence before the occurrence ofthe ninth CNO flash (see Figs. 1 and 2). Right panel: di ff erential work function ( dW ( r ) / dr ) in terms of − log(1 − M r / M ⋆ ) for the case in whichthe ε mechanism is allowed to operate (solid black curve) and when it is suppressed (solid violet curve). Also shown are the corresponding scaledrunning work integrals, W (dashed curves). ers the approximate ranges in T e ff and log g of [15 000 −
38 000]K and [5 . − . T e ff .
10 000 K and similarvalues of log g , nor with the one of the pre-ELMVs, that corre-spond to T e ff .
12 000 K and log g .
5. Note that the low-gravityboundary of the instability domain reported in this paper slightlyoverlaps with the high-gravity limit of the instability domain ofthe sdBV stars.We summarize in Table 1 the main results of the stabilityanalysis carried out for all the evolutionary sequences consid-ered in this work. In the Table, the second column shows the to-tal evolutionary timescale, ∆ t evol , which represents the time spentby the sequence between the beginning of the pre-WD phase andthe maximum e ff ective temperature reached before entering thefinal cooling branch. The third column indicates the time interval( ∆ t exc ) in which stellar models exhibit pulsation instability dueto the ε mechanism, where we have considered the summationof the time taken by every instability phase for those sequencesthat experience multiple CNO flashes (that is, the total time thatthe models of a given sequence spend while evolving along theblack line marked on its evolutionary track as indicated in Fig.6). The rest of the columns represent the radial order, the rangeof e ff ective temperature of the instability, the average value ofthe periods, and the minimum value of the e -folding time of theunstable ℓ = g modes destabilized by the ε mechanism, re-spectively (where we have taken all the flashes into account forthe multiple-flashing sequences). Comparing the values of ∆ t exc to the e -folding times, we see that for all the sequences thereis plenty of time for the instabilities to reach observable ampli-tudes, being the unstable modes of the sequences with 0 . . M ⊙ the ones more likely to be observed due to thelarger di ff erences between τ e and ∆ t exc . When we compare the values of the total evolutionary timescale ∆ t evol to ∆ t exc , we findthat it would be possible to detect one of these low-mass WDstars while evolving along these stages of instability where the ε mechanism excites low-order g modes.It is interesting to estimate how many stars pulsating by the ε mechanism are expected to be found. According to Table 1,considering once again the total time spent by the models of ev-ery evolutionary sequence during their stages of instability withrespect to the evolutionary timescale, we can roughly estimatethat between ∼ −
80% of these low-mass WD stars may befound pulsating due the ε mechanism. Considering the samplesshown in Fig. 6, by virtue of their spectroscopic parameters, andtaking the minimum value for the estimated probability, it mightindicate that roughly 7 of the stars in the catalogues could befound pulsating in low-order g modes by the ε mechanism. Oneof the reasons why these short-period pulsations have not beendetected yet may be possibly attributed to the pulsation ampli-tudes being smaller than the detection limits.In addition, the ε mechanism is also able to destabilize awide range of g -mode periods during the stages of the evolu-tion between flashes, for all the sequences analyzed in this work.However, the e -folding times are larger than (or of the order of)the corresponding evolutionary timescales, and also, these evo-lutionary stages occur faster than the rest of the stages of theevolution (e.g., ∼ . × yr for the sequence with 0 . M ⊙ ;see also Althaus et al. 2013), so it would be unlikely to detect apulsating star while evolving between flashes, let alone, exhibit-ing detectable pulsations. Then, we have discarded such regionsas possible locations for stars pulsating by this mechanism.We close this section by noting that we have carried out addi-tional nonadiabatic calculations on model sequences with stellarmasses . . − . M ⊙ , which, as we already mentioned, donot experience CNO flashes and evolve very slowly. This was Article number, page 6 of 9alcaferro et al.: A new instability domain for CNO-flashing low-mass WDs o Log ( g ) T eff [K] ELMVssdBVs
BLAPs
HG−BLAPs
Fig. 6.
Instability domain (grey shaded area) of low-order g modes excited by the ε mechanism on the log( g ) vs T e ff for each sequence of low-massWD analyzed in this work. Stellar sequences are in units of solar mass. Red asterisks represent ELM (and pre-ELM) stars (Brown et al. 2010, 2013,2016, 2020; Vennes et al. 2011; Gianninas et al. 2015; Kawka et al. 2015; Pelisoli & Vos 2019). Light-blue squares with dots represent pulsatinglow-mass WDs (ELMVs) (Hermes et al. 2012, 2013b,a; Kilic et al. 2015, 2018; Bell et al. 2017, 2018; Pelisoli et al. 2018), while black circleswith dots correspond to pulsating low-mass pre-WDs (pre-ELMVs) (Maxted et al. 2013, 2014; Gianninas et al. 2016; Wang et al. 2020). Greencircles correspond to sdBVs (Green et al. 2011), orange triangles represent BLAPs (Pietrukowicz et al. 2017), and pink triangles correspond toHG (High Gravity)-BLAPs (Kupfer et al. 2019). The evolutionary track of the He-core WD sequence with 0 . M ⊙ is included as a reference. done in order to explore if in some part of the pre-WD evolu-tion, the ε mechanism is able to drive pulsations in these se-quences. In stages previous to the maximum e ff ective tempera-ture, H-shell burning via CNO bicycle is the dominant nuclearsource for these sequences. We have found that the ε mecha-nism is not capable of destabilizing g modes for sequences with M ⋆ . . − . M ⊙ .For the sake of completeness, we have also performed addi-tional calculations for sequences with M ⋆ & . − . M ⊙ , butthis time artificially disabling the action of the element di ff usionin the evolution of our stellar models, which has the e ff ect of sup-pressing (or diminishing) the occurrence of the CNO flashes. Asa consequence, for instance, the sequence with 0 . M ⊙ doesnot experience any CNO flashes, in agreement with the litera-ture (e.g., Driebe et al. 1998; Istrate et al. 2016b). We have foundthat, although the ε mechanism continues to destabilize some(but significantly less) g -mode period pulsations due to residualH burning, the corresponding values of the rate of period changeof g modes ( ∼ − - 10 − [s / s]) are orders of magnitude lowerthan for modes of stellar models that go through flashes (as men-tioned, ∼ − - 10 − [s / s]). Therefore, if one low-mass WDwas found to pulsate in this region of the log g - T e ff diagram, and the rate of period change could be measured, it would helpin discerning whether or not the star experiences CNO flashes.At the same time, since the values of the rate of period changeare much lower for the non-flashing sequences, it would be verydi ffi cult to detect those. However, if a rate of period change wasmeasured and resulted in a value lower than ∼ − [s / s] then,based on our results, it would be possible to rule out that such astar is going through a flashing cycle.
4. Summary and conclusions
In this paper, we performed a stability analysis focused on low-mass WD stars evolving through CNO flashes. We have shownthat the ε mechanism due to stable H burning is able to desta-bilize some low-order ℓ = , g modes in stellar models withmasses in the range of 0 . − . M ⊙ . As displayed inFigs. 1 and 2 for the template sequence with 0 . M ⊙ , thesequences have more modes destabilized in every consecutiveflash, and with shorter e -folding times. For several modes, the e -folding times are shorter than the corresponding evolution-ary timescales, and therefore, there would be enough time toexcite such pulsations to reach observable amplitudes. In gen- Article number, page 7 of 9 & Aproofs: manuscript no. paper-pul-vii
Table 1.
Stellar mass, total evolutionary timescale, time spent during the excitation phase, radial order, approximate range of e ff ective temperaturefor the instability, average period, minimum value of the e -folding time of unstable ℓ = g mode destabilized by the ε mechanism. M ⋆ [ M ⊙ ] ∆ t evol [10 yr] ∆ t exc [10 yr] k T e ff [kK] h Π i [s] τ e [10 yr]0 . . . . . . . . . . . . . . . . . . . . . T e ff ∼
15 000 −
38 000 K and log g ∼ . − . ℓ = g -mode periods desta-bilized by the ε mechanism spans from 150 to 500 s, with radialorder k between 1 and 4. For ℓ = g modes, the location in thelog g − T e ff diagram is similar, but less modes become excited inthis case.Up to our knowledge, no pulsating low-mass He-core WDon its early-cooling branch with M ⋆ & . − . M ⊙ has beendetected lying in the region of instability predicted in this work.However, there are some possible candidate stars, as illustratedby Fig. 6. The eventual detection of g mode pulsations in low-mass He-core stars populating this new instability domain wouldconfirm the theoretically predicted existence of the ε mecha-nism as an agent able to destabilize g -mode periods. Since themagnitudes of the rate of period change of g modes for mod-els evolving in stages prior to the CNO flashes are significantlylarge (particularly, in comparison to the WD and pre-WD stages,Calcaferro et al. 2017a), this quantity could be measured and, inthat case, support the predicted occurrence of the CNO flashes,providing a first proof of the existence of these flashes, and thusconfirming the predicted age dichotomy for low-mass He-coreWDs (Althaus et al. 2001, 2013). Last but not least, the detec-tion of these pulsations would also help in the classification ofseveral stars with uncertain nature.Although we are aware that detecting pulsations (and ratesof change of periods) in this type of objects is not an easy task,we consider that searches for low-amplitude variability are worthdoing. As already shown by the results from the Transiting Exo-planet Survey Satellite ( TESS , Ricker et al. 2015) in the case ofthe two new pre-ELMVs reported by Wang et al. (2020), and asthe future space missions like
Plato (Piotto 2018) and
Cheops (Moya et al. 2018) will probably show, the continuous improve-ment in the quality of the observations is likely to help in thisregard.
Acknowledgements.
We wish to thank our anonymous referee for the construc-tive comments and suggestions that greatly improved the original version of thepaper. Part of this work was supported by PICT-2017-0884 from ANPCyT, PIP112-200801-00940 grant from CONICET, grant G149 from University of LaPlata. K.J.B. is supported by the National Science Foundation under Award AST-1903828. This research has made use of NASA Astrophysics Data System.
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