Asteroseismological analysis of the ultra-massive ZZ Ceti stars BPM~37093, GD~518, and SDSS~J0840+5222
Alejandro H. Córsico, Francisco C. De Gerónimo, María E. Camisassa, Leandro G. Althaus
aa r X i v : . [ a s t r o - ph . S R ] O c t Astronomy & Astrophysicsmanuscript no. paper-one-astro c (cid:13)
ESO 2019October 17, 2019
Asteroseismological analysis of the ultra-massive ZZ Ceti starsBPM 37093, GD 518, and SDSS J0840+5222
Alejandro H. Córsico , , Francisco C. De Gerónimo , , María E. Camisassa , , Leandro G. Althaus , Grupo de Evolución Estelar y Pulsaciones. Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseodel Bosque s / n, (1900) La Plata, Argentina Instituto de Astrofísica La Plata, IALP (CCT La Plata), CONICET-UNLPe-mail: [email protected]
Received ; accepted
ABSTRACT
Context.
Ultra-massive ( & M ⊙ ) hydrogen-rich (DA) white dwarfs are expected to have a substantial portion of their cores in acrystalline state at the e ff ective temperatures characterizing the ZZ Ceti instability strip ( T e ff ∼
12 500 K), as a result of Coulombinteractions in very dense plasmas. Asteroseismological analyses of these white dwarfs can provide valuable information related tothe crystallization process, the core chemical composition and the evolutionary origin of these stars.
Aims.
We present a thorough asteroseismological analysis of the ultra-massive ZZ Ceti star BPM 37093, which exhibits a rich periodspectrum, on the basis of a complete set of fully evolutionary models that represent ultra-massive oxygen / neon(ONe)-core DA whitedwarf stars harbouring a range of hydrogen (H) envelope thicknesses. We also carry out preliminary asteroseismological inferenceson two other ultra-massive ZZ Ceti stars that exhibit fewer periods, GD 518, and SDSS J0840 + Methods.
We considered g -mode adiabatic pulsation periods for ultra-massive ONe-core DA white dwarf models with stellar massesin the range 1 . . M ⋆ / M ⊙ . .
29, e ff ective temperatures in the range 10 000 . T e ff .
15 000 K, and H envelope thicknesses inthe interval − . log( M H / M ⋆ ) . −
6. We explore the e ff ects of employing di ff erent H-envelope thicknesses on the mode-trappingproperties of our ultra-massive ONe-core DA white dwarf models, and perform period-to-period fits to ultra-massive ZZ Ceti starswith the aim of finding an asteroseismological model for each target star. Results.
We found that the trapping cycle and trapping amplitude are larger for thinner H envelopes, and that the asymptotic periodspacing is longer for thinner H envelopes. We found a mean period spacing of ∆Π ∼
17 s in the data of BPM 37093, which islikely to be associated to ℓ = ∆Π with M ⋆ , T e ff , and M H , an intrinsic property of DAV stars. We foundasteroseismological models for the three objects under analysis, two of them (BPM 37093 and GD 518) characterized by canonical(thick) H envelopes, and the third one (SDSS J0840 + ff ective temperature and stellar mass ofthese models are in agreement with the spectroscopic determinations. The percentage of crystallized mass of these asteroseismologicalmodels is 92 %, 97 %, and 81 % for BPM 37093, GD 518, and SDSS J0840 + Gaia for these stars.
Conclusions.
Asteroseismological analyses like the one presented in this paper could lead to a more complete knowledge of theprocesses occurring during crystallization inside white dwarfs. Also, they could make it possible to deduce the core chemical compo-sition of ultra-massive white dwarfs, and in this way, to infer their evolutionary origin, i.e., if the star has a ONe core and then is theresult of single-star evolution, or if it harbour a carbon / oxygen (CO) core and is the product of a merger of the two components of abinary system. However, to achieve these objectives it is necessary to find more pulsating ultra-massive WDs, and also to carry outadditional observations of the already known pulsating stars to detect more pulsation periods. Space missions such as TESS can givea great boost in this direction.
Key words. stars — pulsations — stars: interiors — stars: evolution — stars: white dwarfs
1. Introduction
ZZ Ceti (also called DAV stars) stars are the most numerous andbest studied class of pulsating white dwarf (WD) stars. They arenormal DA WDs with e ff ective temperatures between ∼
10 400K and ∼
12 400 K and logarithm of surface gravities in the range[7 . − . g (gravity) modes with low harmonic degree ( ℓ ≤
2) withperiods in the interval [70 − ff orts with observations of bright targets from the ground (Fontaine & Brassard 2008), then from the Sloan Digital SkySurvey (SDSS; York et al. 2000), and, in the last years, with the Kepler space telescope (Borucki 2016), and the
Kepler ’s secondmission K2 (Van Cleve et al. 2016). Currently, there are 260 ZZCeti stars known (Córsico et al. 2019), and it is expected that theTransiting Exoplanet Survey Satellite (TESS; Ricker et al. 2014)will increase this number substantially.Asteroseismology is a powerful technique that o ff ers the ex-citing prospect of deducing the internal structure of stars bystudying their natural frequencies. In the case of pulsating WDs,the first asteroseismic studies of ZZ Ceti stars that comparedthe observed periods with the theoretical periods computed ona large grid of realistic DA WD models (the so-called forward Article number, page 1 of 13 & Aproofs: manuscript no. paper-one-astro method ) were carried out by Bradley (1998, 2001). These pi-oneering works showed that it would be possible, in principle,to infer the internal chemical structure, the stellar mass, sur-face gravity, e ff ective temperature, luminosity, radius, seismo-logical distance, and rotation rate of ZZ Ceti stars on the basisof the observed pulsation periods. Since then, detailed astero-seismological studies of DAV stars have been carried out, ei-ther through the use of fully evolutionary models (Romero et al.2012, 2013, 2017; De Gerónimo et al. 2017, 2018), or by us-ing static / parametric models (Bischo ff -Kim et al. 2008; Fu et al.2013; Bognár et al. 2016; Giammichele et al. 2017a,b). Bothmethods have strengths and weaknesses, but eventually they arecomplementary to each other (see discussion in Córsico et al.2019).Of particular interest in this paper is the asteroseismologicalanalysis of the rare ultra-massive ZZ Ceti stars ( M ⋆ & M ⊙ ).At variance with average-mass (0 . . M ⋆ / M ⊙ . .
70) andmassive (0 . . M ⋆ / M ⊙ . .
0) ZZ Ceti stars which likelyhave C / O cores , ultra-massive ZZ Ceti stars are supposed toharbour cores made mostly of O and Ne if they are the re-sult of single-star evolution. However, it cannot be ruled outthat ultra-massive WDs could have CO cores if they are the re-sult of the merger of two WDs (García-Berro et al. 2012). Byvirtue of their very high masses, these stars are expected tohave a large fraction of their cores crystallized at the e ff ectivetemperatures characterizing the ZZ Ceti instability strip. Thecrystallization process is due to Coulomb interactions in verydense plasmas. It was theoretically predicted to take place inthe cores of WDs six decades ago (Kirzhnits 1960; Abrikosov1961; Salpeter 1961; van Horn 1968), but it was not until re-cent times that the existence of crystallized WDs was inferredfrom the study of WD luminosity function of stellar clusters(Winget et al. 2009; García-Berro et al. 2010), and the galacticfield (Tremblay et al. 2019). The e ff ects of crystallization on thepulsational properties of ZZ Ceti star models have been stud-ied by Montgomery & Winget (1999); Metcalfe et al. (2004);Córsico et al. (2004, 2005); Brassard & Fontaine (2005).In the specific case of ultra-massive WDs with ONe cores,the first attempt of studying their pulsational properties from atheoretical perspective was made by Córsico et al. (2004), whoshowed that the forward and mean period spacing of ONe-coreWDs are markedly di ff erent from those of CO-core WDs. Re-cently, De Gerónimo et al. (2019) revisited the topic by assess-ing the adiabatic pulsation properties of ultra-massive DA WDswith ONe cores on the basis of a new set of fully evolution-ary models generated by Camisassa et al. (2019). These mod-els incorporate the most updated physical ingredients for mod-elling the progenitor and WD evolution. Specifically, the chemi-cal profiles of the WD models of Camisassa et al. (2019), whichwere adopted from Siess (2010), are consistent with the predic-tions of the progenitor evolution with stellar masses in the range9 . < M ZAMS / M ⊙ < . O and Ne plasma of Medin & Cumming(2010).In this paper, we perform for the first time a detailed astero-seismological analysis of the ultra-massive ZZ Ceti stars known There are also the pulsating Extremely Low-Mass (ELM) and Low-Mass (LM) WDs, also called ELMVs ( M ⋆ . . M ⊙ ), which showH-rich atmospheres and are thought to have cores composed by He. eff [K]8,899,29,4 l og g [ c m / s ] o o o o GD 518SDSS J084021
BPM 37093WD J212402
Fig. 1.
Evolutionary tracks (red solid lines) of the ultra-massive DAWD models computed by Camisassa et al. (2019) in the T e ff − log g plane. Blue dashed lines indicate 0 , , , , , , , , , , up to date on the basis of the new grid of ONe-core WD mod-els presented in Camisassa et al. (2019). At present, there arefour objects of this class known: BPM 37093 ( M ⋆ = . M ⊙ ;Kanaan et al. 1992), GD 518 ( M ⋆ = . M ⊙ ; Hermes et al.2013), SDSS J084021 ( M ⋆ = . M ⊙ ; Curd et al. 2017), andWD J212402 ( M ⋆ = . M ⊙ ; Rowan et al. 2019). The locationof these stars in the spectroscopic Hertzsprung-Russell (HR) di-agram is shown in Fig. 1, along with the evolutionary tracks ofCamisassa et al. (2019). The observed pulsation periods of thesestars are shown in Tables 1, 3, 5, and 7. The star with the rich-est pulsation spectrum, BPM 37093 (Table 1), allows for a de-tailed asteroseismological analysis. This star is the main targetin the present paper. The remaining stars exhibit just three peri-ods (GD 518 and SDSS J084021, Tables 3 and 5, respectively)and only one period (WD J212402; Table 7). In view of this,for GD 518 and SDSS J084021 only a preliminary seismolog-ical analysis is possible, while for WD J212402 it is not pos-sible at present to carry out any asteroseismological inference.The stellar models on which we base our study consider time-dependent element di ff usion and crystallization with chemicalrehomogeneization due to phase separation. In order to have aset of models suitable for a detailed asteroseismological anal-ysis, we have expanded our set of models by generating newsequences of WD models characterized by H envelopes thinnerthan the (thick) canonical envelopes. In this way, we extend theparameter space to be explored in our asteroseismological anal-ysis.The paper is organized as follows. A brief description of thenumerical codes and the evolutionary models employed is pro-vided in Sect. 2. In Sect. 3 we present a brief description of thepulsation properties of our models. In Section 4 we perform a de-tailed asteroseismological analysis of the ultra-massive ZZ Cetistar BPM 37093, and in Section 5 we carry out period-to-periodfits to the stars GD 518 and SDSS J084021. Finally, in Sect. 6we summarize the main findings of this work. Article number, page 2 of 13órsico et al.: Asteroseismology of ultra-massive ZZ Ceti stars
2. Numerical codes and evolutionary models
The ultra-massive DA WD evolutionary models employedin this work were computed with the
LPCODE evolutionarycode (see Althaus et al. 2005b, 2010a; Renedo et al. 2010;Miller Bertolami 2016, for detailed physical description). Thisnumerical tool has been employed to study multiple as-pects of the evolution of low-mass stars (Wachlin et al.2011; Althaus et al. 2013, 2015), the formation of horizon-tal branch stars (Miller Bertolami et al. 2008), extremely low-mass WDs (Althaus et al. 2013), AGB and post-AGB evo-lution (Miller Bertolami 2016), the evolution of DA WDs(Camisassa et al. 2016) and H-de ffi cient WDs (Camisassa et al.2017), among others. More recently, the code has been em-ployed to assess the impact of the uncertainties in progenitorevolution on the pulsation properties and asteroseismologicalmodels of ZZ Ceti stars (De Gerónimo et al. 2017, 2018). Theinput physics of the version of the LPCODE evolutionary codeemployed in this work is described in Camisassa et al. (2019).We refer the interested reader to that paper for details. Of par-ticular importance in this study, is the treatment of crystalliza-tion. Theoretical models predict that cool WD stars must crys-tallize due to the strong Coulomb interactions in their very denseinteriors (van Horn 1968). The two additional energy sourcesinduced by crystallization, namely, the release of latent heat,and gravitational energy associated to changes in the chemi-cal profiles induced by crystallization, are consistently takeninto account. The chemical redistribution due to phase sepa-ration and the associated release of energy have been consid-ered following Althaus et al. (2010c), appropriately modified byCamisassa et al. (2019) for ONe plasmas. To assess the enhance-ment of Ne in the crystallized core, we used the azeotropic-type phase diagram of Medin & Cumming (2010).The pulsation code used to compute the nonradial g -modepulsations of our complete set of models is the adiabatic versionof the LP-PUL pulsation code described in Córsico & Althaus(2006). We did not consider torsional modes, since thesemodes are characterized by very short periods (up to 20 s;see Montgomery & Winget 1999) which have never been ob-served in ZZ Ceti stars. To account for the e ff ects of crys-tallization on the pulsation spectrum of g modes, we adoptedthe “hard-sphere” boundary conditions (Montgomery & Winget1999; Córsico et al. 2005), which assume that the amplitude ofthe radial displacement of g modes is drastically reduced belowthe solid / liquid boundary layer because of the non-shear modu-lus of the solid, as compared with the amplitude in the fluid re-gion (Montgomery & Winget 1999). The squared Brunt-Väisäläfrequency ( N ) for the fluid part of the models is computed asin Tassoul et al. (1990). The Ledoux term B , that explicitly con-tains the contributions of the chemical interfaces to the Brunt-Väisälä frequency, has been appropriately generalized in orderto include the presence of transition regions in which multiplenuclear species vary in abundance. The asteroseismological analysis presented in this work is basedon a set of four evolutionary sequences of ultra-massive WDmodels with stellar masses M ⋆ = . , . , .
22, and 1 . M ⊙ resulting from the complete evolution of the progenitor starsthrough the S-AGB phase (Camisassa et al. 2019). The core andinter-shell chemical profiles of our models at the start of the X i H He C O Ne Na Mg X i X i X i r /M * )00,20,40,60,81 X i log(M H /M * )= -6log(M H /M * )= -7log(M H /M * )= -8log(M H /M * )= -9log(M H /M * )= -10 M * = 1.29 M o T eff = 12000 K Fig. 2.
Abundances by mass of H, He, C, O, Ne, Na, and Mg as a function of the fractional mass, corresponding to ONe-coreWD models with M ⋆ . M ⊙ , T e ff ∼
12 000 K and log( M H / M ⋆ ) = − , − , − , −
9, and −
10 (from top to bottom). The models were com-puted taking into account time-dependent element di ff usion, and latentheat release and chemical redistribution caused by phase separation dur-ing crystallization. The solid part of the models is emphasized with agray tone. The crystallized mass fraction (in percentage) is 99 . WD cooling phase were derived from Siess (2010). The coresare composed mostly of O and Ne and smaller amounts of C, Na, and Mg (see Figs. 2 and 3 of Camisassa et al.2019). Since element di ff usion and gravitational settling oper-ate throughout the WD evolution, our models develop pure Henvelopes. The He content of our WD sequences is given by theevolutionary history of progenitor star, but instead, the H contentof our canonical (thick) envelopes [log( M H / M ⋆ ) ∼ −
6] has beenset by imposing that the further evolution does not lead to H ther-monuclear flashes on the WD cooling track. We have expandedour grid of models by artificially generating new sequences har-bouring thinner H envelopes [log( M H / M ⋆ ) = − , − , − , − ff ects of this procedure become irrelevantmuch before the models reach the ZZ Ceti regime. Details aboutthe method to compute the chemical rehomogeneization at thecore regions during crystallization are given in Camisassa et al.(2019) and De Gerónimo et al. (2019). The temporal changes ofthe chemical abundances due to element di ff usion are assessedby using a new full-implicit treatment for time-dependent ele- Article number, page 3 of 13 & Aproofs: manuscript no. paper-one-astro r /M * )-3-2-1012345 l og ( N ) [ / s ] log(M H /M * )= -6-7-8-9-10M * =1.29 M o T eff = 12000 K Fig. 3.
Logarithm of the squared Brunt-Väisälä frequency, correspond-ing to the same ONe-core WD models with M ⋆ = . M ⊙ , T e ff ∼
12 000 K and log( M H / M ⋆ ) = − , − , − , −
9, and −
10 shown in Fig.2. They gray zone corresponds to the crystallized part of the models. ment di ff usion described in detail in Althaus et al. (2019, sub-mitted).In Fig. 2 we show the H, He, C, O, Ne, Na,and Mg chemical profiles in terms of the fractional mass for1 . M ⊙ ONe-core WD models at T e ff ∼
12 000 K and H enve-lope thicknesses log( M H / M ⋆ ) = − , − , − , −
9, and −
10. Notethat a pure He bu ff er develops as we consider thinner H en-velopes (from the top to the bottom panel).At this e ff ective temperature, the chemical rehomogeneiza-tion due to crystallization has already finished, giving rise to acore where the abundance of O ( Ne) increases (decreases)outward. In Fig. 3 we show the logarithm of the squared Brunt-Väisälä frequency corresponding to the same models shown inFig. 2. The step at the triple chemical transition between C, O, and Ne seen in Fig. 2 [ − log(1 − M r / M ⋆ ) ∼ .
4] is withinthe solid part of the core, thus, it is irrelevant for the mode-trapping properties of these models. This is because, accordingto the hard-sphere boundary conditions adopted for the pulsa-tions, the eigenfunctions do not penetrate the solid region (grayzone). In view of this, the mode-trapping properties of the mod-els illustrated in Fig. 2 and 3 are entirely determined by the pres-ence of the He / H transition and the associated bump in the Brunt-Väisälä frequency, which is located in more external regions forthinner H envelopes (see the next Section).
3. Pulsation calculations
We computed adiabatic pulsation periods of ℓ = , g modesin a range of periods covering the period spectrum that is typ-ically observed in ZZ Ceti stars (70 s . Π . g modes in WDs is a well-studied me-chanical resonance for the mode propagation, that acts due tothe presence of density gradients induced by chemical transi-tion regions. Specifically, chemical transition regions, which in-volve non-negligible jumps in density, act like reflecting wallsthat partially trap certain modes, forcing them to oscillate withlarger amplitudes in specific regions and with smaller amplitudesoutside those regions (see, for details, Brassard et al. 1992a,b;Bradley et al. 1993; Córsico et al. 2002). From an observationalpoint of view, a possible signature of mode trapping in a WDstar is the departure from uniform period spacing. According tothe asymptotic theory of stellar pulsations, in absence of chemi-cal gradients , the pulsation periods of g modes with high radial order k (long periods) are expected to be uniformly spaced witha constant period separation given by (Tassoul et al. 1990): ∆Π a ℓ = Π / p ℓ ( ℓ + , (1)where Π = π "Z fluid Nr dr − . (2)Actually, the period separation in chemically stratified WDmodels like the ones considered in this work is not constant, ex-cept for very-high radial-order modes. We define the forwardperiod spacing as ∆Π k = Π k + − Π k . The left panels of Fig. 4show Π k − ∆Π k diagrams for the same WD models depicted inFigs. 2 and 3. These models are characterized by M ⋆ = . M ⊙ at T e ff ∼
12 500 K, and di ff erent thicknesses of the H enve-lope. In each panel, the horizontal dashed line corresponds tothe asymptotic period spacing, computed with Eqs. (1) and (2).Models with decreasing H envelope thicknesses are displayedfrom top to bottom, starting with the case of the canonical en-velope. By examining the plots, several aspects are worth men-tioning. Firstly, the asymptotic period spacing increases for de-creasing H envelope thickness. This is because the integral inEq. (2) for the quantity Π is smaller for thinner H envelopes,by virtue that the bump in the Brunt-Väisälä frequency inducedby the He / H chemical interface becomes progressively narrowin the radial coordinate r as this interface is located at moreexternal layers. Since Π is larger for thinner H envelopes, theasymptotic period spacing increases (Eq. 1). ∆Π a ℓ experiences anincrease between 37% and 60% when we go from the canon-ical envelope [log( M H / M ⋆ ) = −
6] to the thinnest envelope[log( M H / M ⋆ ) = −
10] for this sequence. Other outstanding fea-ture to be noted from the left panels of Fig. 4 is connected withthe changes in the mode-trapping properties when we considerH envelopes progressively thinner. Indeed, we note that for thickenvelopes, including the canonical one, the period-spacing distri-bution of g modes shows a regular pattern of mode trapping witha very short trapping cycle —the k interval between two trappedmodes. When we consider thinner H envelopes, the trappingcycle and the trapping amplitude increase. A common featurefor all the values of log( M H / M ⋆ ) considered is that the mode-trapping signatures exhibited by ∆Π k vanish for very large ra-dial orders (very long periods), in which case ∆Π k approaches to ∆Π a ℓ , as predicted by the asymptotic theory.Mode-trapping e ff ects also translate into local maxima andminima in the kinetic energy of oscillation, E kin , which are usu-ally associated to modes that are partially confined to the core re-gions and modes that are partially trapped in the envelope. Thiscan be appreciated in the right panels of Fig. 4. The behaviourdescribed above for ∆Π k is also found in the case of E kin , that is,the mode-trapping cycle and amplitude increase with decreasingH envelope thickness.
4. Asteroseismological analysis of BPM 37093
Kanaan et al. (1992) discovered the first ultra-massive ZZ Cetistar, BPM 37093. This star is characterized by T e ff =
11 370K and log g = .
843 (Nitta et al. 2016). Detailed theoreticalcomputations carried out by Winget et al. (1997), Montgomery(1998), and Montgomery & Winget (1999), suggested thatBPM 37093 should have a crystallized core. This star has beenthe target of two multisite observing campaigns of the Whole
Article number, page 4 of 13órsico et al.: Asteroseismology of ultra-massive ZZ Ceti stars ∆ Π k [ s ] l og E k i n ∆ Π k [ s ] l og E k i n ∆ Π k [ s ] l og E k i n ∆ Π k [ s ] l og E k i n ∆Π k [s]20 ∆ Π k [ s ] ∆Π k [s]44 l og EK i n M * = 1.29 M o , T eff = 12000 K, log(M H /M * )= -6log(M H /M * )= -7log(M H /M * )= -8log(M H /M * )= -9log(M H /M * )= -10 ∆Π a = 21.3 s ∆Π a = 25.1 s ∆Π a = 29.3 s ∆Π a = 32.3 s ∆Π a = 33.9 s Fig. 4.
Left panels: the forward period spacing, ∆Π k , in terms of the pulsation periods, Π k , for WD models with M ⋆ = . M ⊙ , T e ff ∼
12 000 Kand di ff erent thicknesses of the H envelope. The thin horizontal dashed lines correspond to the value of the asymptotic period spacing, ∆Π a . Rightpanels: the oscillation kinetic energy versus the periods for the same WD models shown in the left panel. The normalization ( δ r / r ) r = R ⋆ = δ r being the radial displacement), has been assumed to compute the kinetic energy values. Earth Telescope (WET; Nather et al. 1990). Preliminary resultsfrom these campaigns were published by Kanaan et al. (2000).The 1998 observations (XCov 16) revealed a set of regularlyspaced pulsation frequencies in the range 1500-2000 µ Hz. The1999 observations (XCov 17) revealed a total of four indepen-dent modes, including two new modes and two that had beenseen in the previous campaign. By comparing pulsation ampli-tudes in the UV to the optical spectra, Nitta (2000) identifiedthe harmonic degree of the BPM 37093 pulsation modes, con-cluding that they can not be ℓ = ℓ =
2. Metcalfe et al. (2004) obtained new single-siteobservations of BPM 37093 from the Magellan 6.5 m telescopeon three nights in 2003 February. These data showed evidenceof five independent modes, all of which had been detected inthe two previous multisite campaigns. Kanaan et al. (2005) re-ported on WET observations of BPM 37093 obtained in 1998and 1999 and, on the basis of a simple analysis of the averageperiod spacing, they concluded that a large fraction of the totalstellar mass of the star should be crystallized. On the basis of as-teroseismological techniques, Metcalfe et al. (2004) reported tohave "measured" the crystallized mass fraction in BPM 37093and determined a value ∼
90 %. However, employing similarasteroseismological methods, Brassard & Fontaine (2005) ques-tioned those conclusions, suggesting instead that the percentageof crystallized mass of BPM 37093 probably should be between32 % and 82 %. In the next sections, we carry out a detailed as-troseismological analysis that involves the assessment of a meanperiod spacing and its comparison with the theoretical values, I V l og Q ∆Π [s]05e-071e-061,5e-062e-06 | A | Inverse-Variance TestKolmogorov-Smirnov TestFourier-Transform TestBPM37093
Fig. 5.
I-V (upper panel), K-S (middle panel), and F-T (bottom panel)significance tests applied to the period spectrum of BPM 37093 tosearch for a constant period spacing. The periods used here are the 8periods shown in Table 1. and also period-to-period fits with the intention of finding an as-teroseismological model.
Article number, page 5 of 13 & Aproofs: manuscript no. paper-one-astro
Table 1.
The independent frequencies and periods in the data ofBPM 37093 from Metcalfe et al. (2004), along with the theoretical peri-ods, harmonic degrees, radial orders, and period di ff erences of the best-fit model described in Sect. 4.3. Π O ν Π T ℓ k δ i [sec] [ µ Hz] [sec] [sec]511.7 1954.1 512.4 2 29 − . − . . − . − . . − . . For the asteroseismological analysis of this star, we adopt the setof eight modes considered by Metcalfe et al. (2004) (see Table1). This list of periods is based on the set of periods detected byNitta (2000). We searched for a constant period spacing in thedata of BPM 37093 by using the Kolmogorov-Smirnov (K-S;see Kawaler 1988), the inverse variance (I-V; see O’Donoghue1994) and the Fourier Transform (F-T; see Handler et al. 1997)significance tests. In the K-S test, any uniform or at least sys-tematically non-random period spacing in the period spectrumof the star will appear as a minimum in Q . In the I-V test, amaximum of the inverse variance will indicate a constant periodspacing. Finally, in the F-T test, we calculate the Fourier trans-form of a Dirac comb function (created from the set of observedperiods), and then we plot the square of the amplitude of the re-sulting function in terms of the inverse of the frequency. Andonce again, a maximum in the square of the amplitude will in-dicate a constant period spacing. In Fig. 5 we show the resultsof applying the tests to the set of periods of Table 1. The threetests indicate the existence of a mean period spacing of about 17s. According to our set of models, the asymptotic period spacing(Eq. 1) for ultra-massive DA WDs with masses between 1 .
10 and1 . M ⊙ and e ff ective temperatures within the ZZ Ceti instabilitystrip (13 500K −
10 500 K) varies between ∼
22 s and ∼
34 s for ℓ =
1, and between ∼
12 s and ∼
19 s for ℓ =
2. Clearly, theperiod spacing evidenced by the 3 tests for BPM 37093 corre-sponds to modes ℓ =
2. This indicates that the period spectrumof this star is dominated by quadrupole modes, being this in con-cordance with the finding of Nitta (2000). By averaging the pe-riod spacing derived from the three statistical tests, we found ∆Π ℓ = = . ± . ℓ = In principle, the stellar mass of pulsating WDs can be derivedby comparing the average of the period spacings (or the asymp-totic period spacing ) computed from a grid of models with dif-ferent masses, e ff ective temperatures, and envelope thicknesseswith the mean period spacing exhibited by the star, if present.This method takes full advantage of the fact that the period spac-ing of DBV (pulsating DB WDs) and GW Vir stars (pulsatingPG1159 stars) primarily depends on the stellar mass and the ef-fective temperature, and very weakly on the thickness of the Heenvelope in the case of DBVs (see, e.g., Tassoul et al. 1990) andthe thickness of the C / O / He envelope in the case of the GW Virstars (Kawaler & Bradley 1994). In the case of ZZ Ceti stars,however, the the average of the period spacings and the asymp-totic period spacing depend on the stellar mass, the e ff ective tem-perature, and the thickness of the H envelope with a comparablesensitivity. Consequently, the method is not –in principle– di-rectly applicable to ZZ Ceti stars due to the intrinsic degeneracyof the dependence of ∆Π with the three parameters M ⋆ , T e ff , and M H (Fontaine & Brassard 2008).In spite of this caveat, we tried to derive the stellar mass ofBPM 37093 from the measured quadrupole period spacing. Tothis end, we assessed the average quadrupole period spacingscomputed for our models as ∆Π ℓ = = ( n − − P nk ∆Π k , where ∆Π k is the forward period spacing for ℓ = n isthe number of theoretical periods considered from the model.For BPM 37093, the observed periods are in the range [511,635]s. In computing the averaged period spacings for the models,however, we have considered the range [500 , ℓ =
2) in terms of the e ff ective temperaturefor our ultra-massive DA WD evolutionary sequences for all thethicknesses of the H envelope, along with the observed periodspacing for BPM 37093. As can be appreciated from the figure,it is not possible in this instance to put very strong constraints onthe mass of BPM 37093, and the only thing that can be assuredis that the mass of the star could be M ⋆ = . M ⊙ with a thick(canonical) H envelope [log( M H / M ⋆ ) = − M ⋆ = . M ⊙ and with a H envelope 100 timesthinner [log( M H / M ⋆ ) = − Here, we search for a pulsation model that best matches the in-dividual pulsation periods of BPM 37093. The goodness of thematch between the theoretical pulsation periods ( Π T k ) and the ob-served individual periods ( Π O i ) is measured by means of a meritfunction defined as: χ ( M ⋆ , M H , T e ff ) = N N X i = min[( Π O i − Π T k ) ] , (3) Generally, the use of the asymptotic period spacing (computed ac-cording to Eq. 1), instead of the average of the computed period spac-ings, can lead to an overestimation of the stellar mass, except forstars that pulsate with very high radial orders, such as PNNV stars(Althaus et al. 2008).Article number, page 6 of 13órsico et al.: Asteroseismology of ultra-massive ZZ Ceti stars ∆ Π l = [ s ] log(M H /M * )= -6-7-8-9-10 eff [K]1015202530 ∆ Π l = [ s ] eff [K]1015202530 BPM 37093 M * = 1.16 M o M * = 1.22 M o M * = 1.29 M o M * = 1.10 M o BPM 37093 BPM 37093BPM 37093
Fig. 6.
Comparison between the quadrupole ( ℓ =
2) period spacing derived for BPM 37093 ( ∆Π = . ± . ℓ = ∆Π ℓ = , for all the considered stellar masses and di ff erent H-envelope thicknesses, in terms of the e ff ective temperature. where N is the number of observed periods. The WD model thatshows the lowest value of χ , if exists, is adopted as the “best-fit model”. We assess the function χ = χ ( M ⋆ , M H , T e ff ) forstellar masses of 1 .
10, 1 .
16, 1 .
22, and 1 . M ⊙ . For the e ff ec-tive temperature we cover a range of 15000 & T e ff & M H / M ⋆ ) = − , − , − , − , −
10. The quality of our periodfits is assessed by means of the average of the absolute perioddi ff erences, δ = (cid:16)P Ni = | δ i | (cid:17) / N , where δ i = Π O i − Π T k , and by theroot-mean-square residual, σ = q ( P Ni = | δ i | ) / N = p χ .We assumed two possibilities for the mode identification: (i) that all of the observed periods correspond to g modes associatedto ℓ =
1, and (ii) that the observed periods correspond to a mixof g modes associated to ℓ = ℓ =
2. We first considered the8 periods employed by Metcalfe et al. (2004) (see Table 1). Thecase (i) did not show clear solutions compatible with BPM 37093in relation to its spectroscopically-derived e ff ective temperature.Instead, the case (ii) , in which we allow the periods of the starto be associated to a combination of ℓ = ℓ = M ⋆ = . M ⊙ , T e ff =
11 650 K and log( M H / M ⋆ ) = −
6, as it canbe appreciated from Fig. 7. In Table 1 we show the periods of thebest-fit model along with the harmonic degree, the radial order,and the period di ff erences. For this model, we obtain δ = .
00 sand σ = .
28 s. In order to have an indicator of the quality of theperiod fit, we computed the Bayes Information Criterion (BIC; Koen & Laney 2000):BIC = N p log NN ! + log σ , (4)where N p is the number of free parameters of the models, and N is the number of observed periods. The smaller the value of BIC,the better the quality of the fit. In our case, N p = ff ective temperature, and thickness of the H envelope), N = σ = .
28 s. We obtain BIC = .
55, which means that our fitis very good. In Table 2, we list the main characteristics of thebest-fit model. The seismological stellar mass is in good agree-ment with the spectroscopic inference based on the evolutionarytracks of Camisassa et al. (2019). The quadrupole ( ℓ =
2) meanperiod spacing of our best fit model is ∆Π = .
63 s, in excellentagreement with the mean period spacing derived for BPM 37093( ∆Π = . ± . ∼
92% of its mass in crystalline state.We repeated the process of period fit considering the prelim-inary set of 13 periods observed by Nitta et al. (2016), but wedid not find a clear seismological solution neither when we con-sidered the case (i) nor when we adopted the case (ii) . We have assessed the uncertainties in the stellar mass ( σ M ⋆ ), thethickness of the H envelope ( σ M H ), and the e ff ective tempera- Article number, page 7 of 13 & Aproofs: manuscript no. paper-one-astro
Table 2.
The main characteristics of BPM 37093. The second col-umn corresponds to spectroscopic and astrometric results, whereas thethird column presents results from the asteroseismological model of thiswork.
Quantity Spectroscopy Asteroseismology T e ff [K] 11 370 ± (a)
11 650 ± M ⋆ / M ⊙ . ± . (b) . ± . g [cm / s ] 8 . ± . (a) . ± . L ⋆ / L ⊙ ) — − . ± . R ⋆ / R ⊙ ) — − . ± . M H / M ⋆ ) — − ± . M He / M ⋆ ) — − . M cr / M ⋆ . (b) . X O cent. — 0 . X Ne cent. — 0 . ∆Π ℓ = [s] — 29.70 ∆Π ℓ = [s] 17 . ± . Gaia ) Asteroseismology d [pc] 14 . ± .
01 11 . ± . π [mas] 67 . ± .
04 87 . ± . References: (a) Nitta et al. (2016). (b) Camisassa et al. (2019) ture ( σ T e ff ), of the best-fit model by employing the expression(Zhang et al. 1986; Castanheira & Kepler 2008): σ i = d i ( S − S ) , (5)where S ≡ χ ( M ⋆ , M , T ff ) is the minimum of χ which isreached at ( M ⋆ , M , T ff ) corresponding to the best-fit model,and S is the value of χ when we change the parameter i (inthis case, M ⋆ , M H , or T e ff ) by an amount d i , keeping fixed theother parameters. The quantity d i can be evaluated as the mini-mum step in the grid of the parameter i . We obtain the follow-ing uncertainties: σ M ⋆ ∼ . M ⊙ , σ M H ∼ . × − M ⋆ , and σ T e ff ∼
40 K. The uncertainty in L ⋆ is derived from the width ofthe maximum in the function (1 /χ ) in terms of L ⋆ . We obtain σ L ⋆ ∼ . × − L ⊙ . The uncertainties in R ⋆ and g are derivedfrom the uncertainties in M ⋆ , T e ff , and L ⋆ .Table 2 includes the parameters of the best-fit model alongwith the uncertainties derived above. These are formal uncer-tainties related to the process of searching for the asteroseismo-logical model, and therefore they can be considered as internal uncertainties inherent to the asteroseismological process. We employ the e ff ective temperature and gravity of our best-fitmodel to infer the absolute G magnitude ( M G ) of BPM 37093 inthe Gaia photometry (D. Koester, private communication). Wefind M G = .
53 mag. On the other hand, we obtain the apparentmagnitude m G = . Gaia
Archive . Accordingto the well-known expression log d = ( m G − M G + /
5, we ob-tain d = . ± .
06 pc and a parallax π = . ± .
40 mas.These asteroseismological distance and parallax are somewhatdi ff erent as compared with those provided directly by Gaia , thatis, d = . ± .
01 pc and π = . ± .
04 mas. However,we note that the uncertainties in the asteroseismological distance ( https://gea.esac.esa.int/archive/ ). / χ -6-7-8-9-10 / χ / χ T eff [s] / χ M * = 1.10 M o M * = 1.16 M o M * = 1.22 M o M * = 1.29 M o l= 1, 2 modes BPM37093 Fig. 7.
The inverse of the quality function of the period fit in the casein which we allow the periods to be associated to ℓ = ℓ = ff ective temperature for the ultra-massive DAWD model sequences with di ff erent stellar masses ( M ⋆ ) and H envelopethicknesses [log( M H / M ⋆ )], as indicated. The vertical dashed line andthe gray strip correspond to the spectroscopic e ff ective temperature ofBPM 37093 and its uncertainties ( T e ff =
11 370 ±
500 K). Note thestrong maximum in ( χ ) − for M ⋆ = . M ⊙ and log( M H / M ⋆ ) = − T e ff ∼
11 650 K. This corresponds to our "best-fit" model (see text fordetails). and parallax come mainly from the uncertainties in the e ff ec-tive temperature and the logarithm of the gravity of the best-fitmodel ( ∼
40 K and ∼ . T e ff and log g , and thus in the errors in the asteroseismologicaldistance and parallax, the agreement with the astrometric valuescould substantially improve. If the stellar rotation is slow and rigid, the rotation frequency Ω of the WD is connected with the frequency splitting δν throughthe coe ffi cients C k ,ℓ —that depend on the details of the stellarstructure— and the values of m ( − ℓ, · · · , − , , + , · · · , + ℓ ), bymeans of the expression δν = m (1 − C k ,ℓ ) Ω (Unno et al. 1989).The period at 564.1 s in Table 1 is actually the average of twovery close observed periods which are assumed to be the com-ponents m = − m = + ℓ = m = − , , +
2) arenot visible for some unknown reason. Under this hypothesis, wederive a frequency splitting of δν = . µ Hz. Making the sameassumption for the pair of observed periods at 633.5 s and 636.7s (see Nitta et al. 2016), which, averaged, give the period 635.1
Article number, page 8 of 13órsico et al.: Asteroseismology of ultra-massive ZZ Ceti stars X i H He C O Ne Na Mg0 1 2 3 4 5 6 7 8 9 10-log(1-M r /M * )-3-2-10123 l og ( N ) , l og ( L l ) log(N )log(L )log(L )M * = 1.16 M o T eff = 11653 KM cr = 0.92 Fig. 8.
The internal chemical structure (upper panel), and the squaredBrunt-Vaïsälä and Lamb frequencies for ℓ = ℓ = M ∗ = . M ⊙ , an e ff ective tempera-ture T e ff =
11 653 K, a H envelope mass of log( M H / M ∗ ) ∼ −
6, and acrystallized mass fraction of M cr = . M ⋆ . s in Table 1), we have δν = . µ Hz. For our best-fit modelfor BPM 37093, we find that the 564 s and 635 s modes have C k ,ℓ = ∼ . C k ,ℓ , and the averaged fre-quency splitting, δν = . µ Hz, we obtain a rotation period of ∼
55 h. This rotation period is consistent with the rotation-periodvalues inferred from asteroseismology for WD stars (see Table10 of Córsico et al. 2019). We also can estimate what the rotationperiod would be if these periods were the components m = − m = + ℓ = ℓ = C k ,ℓ = ∼ .
498 from the best-fit model, and thenthe rotation period should be of ∼
33 h.
Metcalfe et al. (2004) carried out a parametric asteroseismolog-ical analysis on BPM 37093 on the basis of the eight periodslisted in the first column of Table 1. These authors employed DAWD models characterized by chemical transition regions result-ing from the assumption of di ff usive equilibrium. The free pa-rameters of the analysis are the crystallized mass fraction (thatis, the location of the inner boundary conditions for the pulsa-tions, which coincides with the liquid / solid interface), the Heand H envelope thickness, and the e ff ective temperature. The au-thors consider pure C- and O-core WDs, and three fixed stellar-mass values. They obtain a family of asteroseismological mod-els characterized by di ff erent stellar parameters, but all of themwith 90 % of the mass crystallized. A second parametric as-teroseismological analysis of BPM 37093 was performed inde-pendently by Brassard & Fontaine (2005), who employed DAWD models with some improved aspects; for example, updated opacities, chemical transitions resulting from time-dependent el-ement di ff usion, and cores made of CO in addition to pure Cand O cores. In addition, the models of Brassard & Fontaine(2005) do not consider the crystallized mass fraction as a freeparameter, but instead, the value is fixed for each model and re-sults from the predictions of the EoS. The results of this anal-ysis largely di ff er from those of Metcalfe et al. (2004). Indeed,Brassard & Fontaine (2005) found a set of optimal asteroseis-mological models characterized by a percentage of crystallizedmass in the range 32-82 %. These authors emphasize that the in-formation contained in the eight periods employed in both anal-yses is not enough to unravel the core chemical structure norto derive the percentage of crystallized mass of this star, due tothe fact that the modes are characterized by high radial ordersand therefore, they are in the asymptotic regime of g -mode pul-sations. The strong di ff erences of the results of the works byMetcalfe et al. (2004) and Brassard & Fontaine (2005) could bedue to the fact that Metcalfe et al. (2004) only varied the crystal-lized mass fraction in increments of 10 % (i.e., 10%, 20%, 30%, · · · , 80%, 90%). Using a finer grid in the increments of the crys-tallized mass fraction could result in many other possible best-fitsolutions, potentially more in agreement with the larger set ofsolutions found by Brassard & Fontaine (2005).The DA WD models employed in the present paper aresubstantially di ff erent as compared with those employed byMetcalfe et al. (2004) and Brassard & Fontaine (2005), partic-ularly regarding the core chemical structure and composition. Infact, while those authors consider cores made of pure C, pure O,and mixtures of 50 % of C and 50 % of O, in the present analy-sis we consider cores made of O and Ne with evolving chemicalstructures as predicted by fully evolutionary computations. Inaddition, our asteroseismological approach, which is based onfully evolutionary models, largely di ff ers from that adopted inthe mentioned works, that is, the employment of structure mod-els with a number of adjustable free parameters to search for theoptimal asteroseismological models. For these reasons, a directcomparison of our results with those of Metcalfe et al. (2004)and Brassard & Fontaine (2005) is not possible. However, wecan emphasize that our analysis favours a WD model with a largefraction of mass in solid phase ( ∼ ℓ and the radial order k of the pulsation modes forthe asteroseismological solutions are similar. Indeed, our anal-ysis predicts that most of the modes are quadrupole modes, ex-cept the modes with periods at 531.1 s and 613.5 s, which aredipole modes. In the case of Metcalfe et al. (2004), most of thethe modes are ℓ =
2, except modes with periods 582.0 s and613.5 s which are ℓ = ℓ =
2, except the modewith periods 613.5 s, which is a ℓ = ≤ k ≤ ≤ k ≤
35. The surprising agreementof the identification of the radial order k of the modes accord-ing to Metcalfe et al. (2004) and Brassard & Fontaine (2005) ascompared with the current analysis (di ff ering only by 1) couldbe due to the fact that our best-fit model for BPM 37093 has alarge fraction of mass crystallized, so that g -mode pulsations areinsensitive to the ONe-core chemical features, and thus, the pul-sational properties of the model resemble those of a model witha similar mass but with a CO core. Article number, page 9 of 13 & Aproofs: manuscript no. paper-one-astro
Table 3.
The independent frequencies in the data of GD 518 fromHermes et al. (2013) along with the theoretical periods, harmonic de-grees, radial orders, and period di ff erences of the best-fit model. Π O [sec] ν [ µ Hz] Π T [sec] ℓ k δ i [sec]440 . ± . . ± . . ± . . ± . − . . ± . . ± . Table 4.
Same as Table 2, but for GD 518.
Quantity Spectroscopy Asteroseismology T e ff [K] 12 030 ± (a)
12 060 ± M ⋆ / M ⊙ . (b) . ± . g [cm / s ] 9 . ± . (a) . ± . L ⋆ / L ⊙ ) — − . ± . R ⋆ / R ⊙ ) — − . ± . M H / M ⋆ ) — − ± . M He / M ⋆ ) — − M cr / M ⋆ . (b) . X O cent. — 0 . X Ne cent. — 0 . Gaia ) Asteroseismology d [pc] 64 . ± . . ± . π [mas] 15 . ± .
08 20 . ± . References: (a) Hermes et al. (2013); (b) Camisassa et al. (2019)
5. Other ultra-massive ZZ Ceti stars
There are three other pulsating ultra-massive ZZ Ceti starsknown to date, apart from BPM 37093. They are GD 518,SDSS J084021.23 + − + − Pulsations in WD J165915.11 + T e ff ∼
12 030 K and log g ∼ .
08, which would correspond to amass of 1 . M ⊙ if the ONe-core WD models from Althaus et al.(2005a) are used, or 1 . M ⊙ if the CO-core WD models fromWood (1995) are employed. The value of the stellar mass of thestar is M ⋆ = . M ⊙ if the evolutionary tracks of ONe-coreWD models of Camisassa et al. (2019) are adopted. To date, noasteroseismological analysis has been performed to this star. Ourperiod-to-period fits for this star indicate that our best fit model—the one which minimizes the merit function from Eq. (3)— ischaracterized by a value of χ = . δ = .
74 s, σ = .
75 s, andBIC = .
22, and has a stellar mass of 1 . M ⊙ and T e ff =
12 060K (see Table 4 and Fig. 9). The stellar mass of the asteroseismo-logical model is consistent with the spectroscopic mass derivedfrom the evolutionary tracks of Camisassa et al. (2019). X i H He C O Ne Na Mg0 1 2 3 4 5 6 7 8 9 10-log(1-M r /M * )-3-2-10123 l og ( N ) , l og ( L l ) log(N )log(L )log(L )M * = 1.22 M o T eff = 12060 KM cr = 0.97 Fig. 9.
Same as in Fig. 8, but for the asteroseismological best-fit modelof GD 518.
The asteroseismological distance and parallax inferred forGD 518, derived in the same way than for BPM 37093, are d = . ± .
06 pc and π = . ± .
03 mas. These val-ues are somewhat di ff erent than those provided by Gaia , that is, d = . ± . π = . ± .
08 mas. The agreementbetween these sets of values could improve if we could employmore realistic values for the uncertainties in T e ff and log g of theasteroseismological model for GD 518, in a similar way than forBPM 37093 (see discussion at the end of Sect. 4.5). This ultra-massive ZZ Ceti star was discovered by Curd et al.(2017) from a sample of DA WD from the SDSS DR7 and DR10.Model-atmosphere fits indicate T e ff ∼
12 160 K, log g ∼ . M ⋆ ∼ . M ⊙ . These results are in good agreement withthe preliminary asteroseismological analysis performed by thesame authors, where their best-fit CO-core WD model has M ⋆ = . M ⊙ , M H = . × − M ⋆ , M He = . × − M ⋆ , 0 . ≤ M cr / M ⋆ ≤ .
70 and 11 850 ≤ T e ff ≤
12 350 K.Our best fit model is characterized by χ = . δ = . σ = .
38 s, and BIC = − .
37 with one period (797.4 s)identified as a ℓ = ℓ = T e ff =
12 550 K, M ⋆ = . M ⊙ , M H / M ⋆ = . × − , M He / M ⋆ = . × − , M cr / M ⋆ = .
81, with a central Neabundance of 0 .
52. The stellar mass derived from the astero-seismological model is somewhat smaller than the value de-rived spectroscopically on the basis of the evolutionary tracksof Camisassa et al. (2019). On the other hand, the disagreementregarding the mass of the crystallized part of the core as com-pared with the result found by Curd et al. (2017) is because herewe are employing ONe-core WD models, whereas those authorsconsider CO-core WD models. When searching for the best-fit
Article number, page 10 of 13órsico et al.: Asteroseismology of ultra-massive ZZ Ceti stars
Table 5.
The independent frequencies in the data ofSDSS J084021.23 + ff erences of the best-fit model. Π O [sec] ν [ µ Hz] Π T [sec] ℓ k δ i [sec]172 . ± . . . ± . . − . . ± . .
14 797.76 2 40 − . Table 6.
Same as Table 2, but for SDSS J084021.23 + Quantity Spectroscopy Asteroseismology T e ff [K] 12 160 ± (a)
12 550 ± . M ⋆ / M ⊙ . (b) . ± . g [cm / s ] 8 . ± . (a) . ± . L ⋆ / L ⊙ ) — − . ± . R ⋆ / R ⊙ ) — − . ± . M H / M ⋆ ) — − ± . M He / M ⋆ ) — − . M cr / M ⋆ . (b) . X O cent. — 0 . X Ne cent. — 0 . Gaia ) Asteroseismology d [pc] 138 . ± . . ± . π [mas] 7 . ± .
21 11 . ± . References: (a) Curd et al. (2017); (b) Camisassa et al. (2019)
Table 7.
The single frequency in the data of WD J212402.03 − Π O [sec] f [ µ Hz]357 2801model with all periods assumed to be associated to ℓ = χ = . d = . ± .
08 pc and π = . ± .
01 mas,which di ff er from the Gaia values, d = . ± . π = . ± .
21 mas. Again, a better estimate of the uncertainties ofthe e ff ective temperature and gravity of the asteroseismologicalmodel could contribute to bring the asteroseismological distanceand parallax values closer to those derived by Gaia . The variability of WD J212402 was discovered by Rowan et al.(2019) from time-series GALEX space-telescope observa-tions. This star has T e ff =
12 510 K and log g = . M ⋆ = . M ⊙ and the crystallized mass fraction should beof M cr / M ⋆ ∼ .
90 according to the evolutionary tracks ofCamisassa et al. (2019). Unfortunately, only a single period hasbeen detected (Table 7), preventing us from attempting an as-teroseismological analysis. It would be very important to haveadditional observations of this star to detect more pulsation peri-ods. X i H He C O Ne Na Mg0 1 2 3 4 5 6 7 8 9 10-log(1-M r /M * )-3-2-10123 l og ( N ) , l og ( L l ) log(N )log(L )log(L )M * = 1.10 M o T eff = 12550 KM cr = 0.81M * = 1.10 M o T eff = 12550 KM cr = 0.81 Fig. 10.
Same as in Fig. 8, but for the asteroseismological best-fit modelof SDSS J084021.23 +
6. Summary and conclusions
In this paper, we have conducted for the first time an astero-seismological study of the ultra-massive ZZ Ceti stars knownhitherto by employing an expanded set of grid of ONe-core WDmodels presented in Camisassa et al. (2019). The stellar mod-els on which this study is based consider crystallization withchemical rehomogeneization due to phase separation. We haveincluded ultra-massive WD models with di ff erent thicknesses ofthe H envelope, with the aim of expanding the parameter spacein our asteroseismological exploration.For the ultra-massive ZZ Ceti star BPM 37093, we have car-ried out a detailed asteroseismological analysis that includes thederivation of a mean period spacing of ∼
17 s, which is as-sociated to ℓ = g modes. We have not been able, however,to infer the stellar mass of the star by comparing the observedperiod spacing with the averaged theoretical period spacings.This is due to the intrinsic degeneracy of the dependence of ∆Π with the three parameters M ⋆ , T e ff and M H . On the other hand,we have derived a best-fit model for the star, by consideringtheir individual pulsation periods. This model is characterizedby T e ff =
11 650 K, M ⋆ = . M ⊙ , log( M H / M ⋆ ) = −
6, and M cr / M ⋆ = .
92 (see Table 2). In addition, we have derived anasteroseismological distance of 11.38 pc, which somewhat dif-fers from the astrometric distance measured by
Gaia , of 14.81pc. Finally, a rotation period of 55 h has been inferred, underthe assumption that the modes that exhibit frequency splittingsare associated to ℓ = M cr / M ⋆ = .
97 (GD 518) and M cr / M ⋆ = .
81 (SDSS J084021). The asteroseismological dis-tances inferred for these stars (50 pc and 90 pc, respectively) are
Article number, page 11 of 13 & Aproofs: manuscript no. paper-one-astro somewhat di ff erent to the distances measured by Gaia (65 pc and139 pc, respectively). Finally, for the ultra-massive ZZ ceti starWD J212402, which exhibits one single period, it is not possibleto do any kind of asteroseismological inference at this stage.Tables 2, 4, and 6 include the parameters of the best-fit mod-els for BPM 37093, GD 518, and SDSS J084021.23 + g -mode pulsations only sample the non-crystallized regions, and since these regions are dominated byO, C, He and H, it is not surprising that the best-fit seismologi-cal models are consistent with prior studies which assumed COcores, particularly in the case of BPM 37093.In Tables 2, 4, and 6 we have included the formal uncer-tainties related to the process of searching for the asteroseismo-logical model, and therefore they can be considered as internal uncertainties inherent to the asteroseismological process. An es-timation of more realistic uncertainties in the structural quan-tities that characterize the asteroseismological models of thesestars ( T e ff , M ⋆ , M H , M He , R ⋆ , etc) is very di ffi cult to obtain, sincethey depend on the uncertainties a ff ecting the physical processesof the progenitor evolution. An estimate of the impact of theuncertainties in the prior evolution on the structural parame-ters of the asteroseismological models has been carried out byDe Gerónimo et al. (2017, 2018) for ZZ Ceti stars of interme-diate masses harbouring CO cores. These authors derive typicaluncertainties of ∆ M ⋆ / M ⋆ . . ∆ T e ff .
300 K and a factorof two in the thickness of the H envelope. While we can not di-rectly extrapolate these results to our analysis of ultra-massiveDA WD models with ONe cores, we can adopt them as repre-sentative of the real uncertainties a ff ecting the parameters of ourasteroseismological models for BPM 37093, GD 518, and SDSSJ084021.23 + M ⋆ & M ⊙ ) come from single-star evolution and must haveONe cores. However, it cannot be discarded that these objects arethe result of mergers of two WDs (the so-called "double degen-erate scenario"; see, e.g., García-Berro et al. 2012; Schwab et al.2012) in a binary system, in which case it is expected that theyhave CO cores. The study of the evolutionary and pulsationalproperties of ultra-massive WDs resulting from WD + WD merg-ers is beyond the scope of the present paper and will be the focusof a future investigation.We close the article by emphasizing the need of new photo-metric observations from the ground or from space (e.g., TESS)in order to find more variable ultra-massive WDs, and also to re-observe the already known objects (for instance WD J212402) inorder to find more periods. This will result in reliable asteroseis-mological analyses that could yield valuable information aboutthe crystallization processes in WDs. Also, it could be possibleto derive the core chemical composition and, in turn, to infertheir evolutionary origin —that is, either single-star evolution orbinary-star evolution with the merger of two WDs.
Acknowledgements.
We wish to acknowledge the suggestions and comments ofan anonymous referee that strongly improved the original version of this work.We gratefully acknowledge Prof. Detlev Koester for providing us with a tabula-tion of the absolute magnitude of DA WD models in the
Gaia photometry. Part ofthis work was supported by AGENCIA through the Programa de ModernizaciónTecnológica BID 1728 / OC-AR, and by the PIP 112-200801-00940 grant fromCONICET. This research has made use of NASA’s Astrophysics Data System.
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