arXiv subjects

Leila M. Calcaferro

Publications and source records attributed to Leila M. Calcaferro.

16 recordsLinked to original sources

arXiv2026

Unveiling the properties of pulsating low-mass helium-core white dwarfs through TESS asteroseismology I. First results

Recent space-based photometry, particularly from TESS, has opened new possibilities for probing the internal structure of low-mass (LM) helium (He)-core white dwarfs (WDs). We present a homogeneous asteroseismological analysis of six pulsating LM WD stars, based on new and updated TESS photometry. We processed short- and ultra-short-cadence TESS observations of TIC 290904838 (J1112), TIC 156064657, TIC 33717565, TIC 344130696, TIC 72637474, and TIC 188087204, and analyzed the resulting pulsation spectra. We then carried out a detailed asteroseismological analysis using fully evolutionary models of LM He-core WDs that allow for varying hydrogen (H)-envelope thicknesses. We also estimated spectroscopic/photometric stellar masses when atmospheric parameters are available. We report the first TESS-based frequencies for J1112 and provide revised or expanded frequency solutions for the remaining targets. The asteroseismological analysis yields relatively well-constrained solutions for three stars, a representative but more tentative solution for one target, and constrained ranges for the remaining two. The inferred solutions span a broad range of H-envelope thicknesses, although some of the asteroseismological inferences remain tentative because of the limited number of observed periods available for the analysis. For most objects, the derived spectroscopic/photometric stellar masses are broadly compatible with the asteroseismological values. This is the first homogeneous TESS-based asteroseismological study of a small sample of pulsating LM WDs. Our results suggest that LM WDs can harbor H envelopes with a range of thicknesses, from canonical (thick) to very thin, as in average-mass H-rich pulsating WDs. They also provide a useful reference point for future studies of larger samples, which will hopefully benefit from richer mode sets and improved mode identification.

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arXiv2026

Observing bright pulsating white dwarfs with PLATO: A new window into the late stages of stellar evolution

We present the scientific case for exploiting the capabilities of the PLATO mission to study bright pulsating white dwarfs across a wide spectral range, including hydrogen-deficient types (GW Vir and DBV stars) and hydrogen-rich classes (classical DAVs, pulsating extremely low-mass DA white dwarfs, and ultra-massive DA white dwarfs). PLATOs exceptional photometric precision, long-duration continuous monitoring, and extensive sky coverage promise transformative advances in white dwarf asteroseismology. Our key objectives include probing the internal structure and chemical stratification of white dwarfs, detecting secular changes in pulsation modes over extended timescales, and discovering rare or previously unknown classes of pulsators. To assess feasibility, we constructed a sample of 650 white dwarf candidates identified within PLATOs Southern LOPS2 field using the PLATO complementary science catalogue combined with Gaia DR3, and derived atmospheric parameters through photometric modeling. This sample comprises 118 DA white dwarfs (including 23 ZZ Ceti candidates), and 41 non-DAs (including 35 DBV candidates). Simulated observations using PlatoSim demonstrate that PLATO will be capable of detecting white dwarf pulsation modes with amplitudes as low as 0.1 mma depending on stellar magnitude, observation duration, pixel location, and the number of contributing cameras. We provide detailed detection limits and visibility forecasts for known pulsators across a representative range of these parameters. Furthermore, we emphasize strong synergies with Gaia astrometry, TESS photometry, and targeted spectroscopic campaigns, which together will enable robust mode identification and detailed stellar modeling. Collectively, these efforts will unlock unprecedented insights into white dwarf origins, evolution and internal physics, and the fate of their planetary systems.

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arXiv2025

Post-common Envelope Evolution of Helium-core White Dwarfs

Helium-core white dwarfs (He WDs) formed through common envelope (CE) evolution offer valuable insight into binary interaction channels and compact remnant formation. Their cooling rates critically impact both detectability and age estimates in close binaries. Compared to He WDs formed via stable Roche-lobe overflow (SRLOF), those from the CE channel undergo markedly different mass-loss histories, resulting in distinct post-CE evolutionary behavior. We explore how the mass of the residual hydrogen envelope (Mh) shapes the cooling evolution of CE He WDs, focusing on the role of the bifurcation point in setting Mh and enabling residual hydrogen burning. Using the LPCODE stellar evolution code, we computed models of He WDs with masses from 0.20 to 0.42 solar masses, evolving from post-CE conditions to the white dwarf cooling track. Two evolutionary branches emerge: (i) non-flashing sequences, which cool rapidly with negligible hydrogen burning, and (ii) flashing sequences, where hydrogen shell flashes alter the envelope structure prior to cooling. Minimal-envelope models cool within 5-130 million years for effective temperatures between 12,000 and 27,000 K, and reach in approx. 300 million years at lower temperatures, remaining much younger than SRLOF counterparts. In contrast, models with more hydrogen retain active nuclear burning, delaying cooling and yielding ages of several billion years. Flashing sequences prolong the pre-white dwarf phase, though still shorter than in SRLOF evolution. The value of Mh also affects WD mass and surface gravity estimates, introducing systematic shifts with respect to SRLOF WDs. Our results show that CE He WDs follow distinct evolutionary paths, with important implications for interpreting the nature and fate of compact binaries hosting He WDs.

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arXiv2024

An analysis of spectroscopic, seismological, astrometric, and photometric masses of pulsating white dwarf stars

A central challenge in the field of stellar astrophysics lies in accurately determining the mass of isolated stars. However, for pulsating white dwarf (WD) stars, the task becomes more tractable due to the availability of multiple approaches such as spectroscopy, asteroseismology, astrometry, and photometry. The objective of this work is to compare the asteroseismological and spectroscopic mass values of WDs in detail and, in turn, to compare them with the masses derived using astrometric parallaxes/distances and photometry. Our analysis encompasses a selection of pulsating WDs with different surface chemical abundances that define the main classes of variable WDs. We calculated their spectroscopic masses, compiled seismological masses, and determined astrometric masses. We also derived photometric masses, when possible. Subsequently, we compared all the sets of stellar masses obtained through these different methods. To ensure consistency and robustness in our comparisons, we used identical WD models and evolutionary tracks across all four methods. The analysis suggests a general consensus among the these methods regarding the masses of pulsating WD with H-rich atmospheres, known as DAV or ZZ Ceti stars, especially for objects with masses below approximately $0.75 M_{\sun}$, although notable disparities emerge for certain massive stars. For pulsating WD stars with He-rich atmospheres, called DBV or V777 Her stars, we find that astrometric masses generally exceed seismological, spectroscopic, and photometric masses. Finally, while there is agreement among the sets of stellar masses for pulsating WDs with C-, O-, and He-rich atmospheres (designated as GW Vir stars), outliers exist where mass determinations by various methods show significant discrepancies.

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arXiv2024

Pulsating hydrogen-deficient white dwarfs and pre-white dwarfs observed with TESS VI. Asteroseismology of the GW Vir-type central star of the Planetary Nebula NGC 246

Significant advances have been achieved through the latest improvements in the photometric observations accomplished by the recent space missions, substantially boosting the study of pulsating stars via asteroseismology. The TESS mission has already proven to be of relevance for pulsating white dwarf and pre-white dwarf stars. We report a detailed asteroseismic analysis of the pulsating PG 1159 star NGC 246 (TIC3905338), the central star of the planetary nebula NGC 246, based on high-precision photometric data gathered by the TESS space mission. We reduced TESS observations of NGC 246 and performed a detailed asteroseismic analysis using fully evolutionary PG 1159 models computed accounting for the complete prior evolution of their progenitors. We constrained the mass of this star by comparing the measured mean period spacing with the average of the computed period spacings of the models and also employed the observed individual periods to search for a seismic stellar model. We extracted 17 periodicities from the TESS light curves from the two sectors where NGC246 was observed. All the oscillation frequencies are associated with g-mode pulsations, with periods spanning from ~1460 to ~1823s. We found a constant period spacing of $ΔΠ= 12.9$s, allowing us to deduce that the stellar mass is larger than ~0.87 Mo if the period spacing is assumed to be associated with l= 1 modes, and ~ 0.568 Mo if it is associated with l= 2 modes. The less massive models are more consistent with the distance constraint from Gaia parallax. Although we were not able to find a unique asteroseismic model for this star, the period-to-period fit analyses suggest a high-stellar mass ($\gtrsim$0.74 Mo) when the observed periods are associated with modes with l= 1 only, and both a high ($\gtrsim$ 0.74 Mo) and intermediate (~0.57 Mo) stellar mass when the observed periods are associated with modes with l= 1 and 2.

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arXiv2023

Exploring the internal rotation of the extremely low-mass He-core white dwarf GD 278 with TESS asteroseismology

(Abridged) We present an exploration of the internal rotation of GD 278, the first pulsating extremely low-mass white dwarf that shows rotational splittings within its periodogram. We assess the theoretical frequency splittings expected for different rotation profiles and compare them to the observed frequency splittings of GD 278. To this aim, we employ an asteroseismological model representative of the pulsations of this star, obtained by using the LPCODE stellar evolution code. We also derive a rotation profile that results from detailed evolutionary calculations carried out with the MESA stellar evolution code and use it to infer the expected theoretical frequency splittings. We found that the best-fitting solution when assuming linear profiles for the rotation of GD 278 leads to values of the angular velocity at the surface and the center that are only slightly differential, and still compatible with rigid rotation. The values of the angular velocity at the surface and the center for the simple linear rotation profiles and for the rotation profile derived from evolutionary calculations are in very good agreement. Also, the resulting theoretical frequency splittings are compatible with the observed frequency splittings, in general, for both cases. The results obtained from the different approaches followed in this work to derive the internal rotation of GD 278 agree. The fact that they were obtained employing two independent stellar evolution codes gives robustness to our results. Our results suggest only a marginally differential behavior for the internal rotation in GD 278, and considering the uncertainties involved, very compatible with the rigid case, as has been observed previously for white dwarfs and pre-white dwarfs. The rotation periods derived for this star are also in line with the values determined asteroseismologically for white dwarfs and pre-white dwarfs in general.

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arXiv2021

Discovery, TESS Characterization, and Modeling of Pulsations in the Extremely Low Mass White Dwarf GD 278

We report the discovery of pulsations in the extremely low mass (ELM), likely helium-core white dwarf GD 278 via ground- and space-based photometry. GD 278 was observed by the Transiting Exoplanet Survey Satellite (TESS) in Sector 18 at a 2-min cadence for roughly 24 d. The TESS data reveal at least 19 significant periodicities between 2447-6729 s, one of which is the longest pulsation period ever detected in a white dwarf. Previous spectroscopy found that this white dwarf is in a 4.61 hr orbit with an unseen >0.4 solar-mass companion and has Teff = 9230 +/- 100 K and log(g) = 6.627 +/- 0.056, which corresponds to a mass of 0.191 +/- 0.013 solar mass. Patterns in the TESS pulsation frequencies from rotational splittings appear to reveal a stellar rotation period of roughly 10 hr, making GD 278 the first ELM white dwarf with a measured rotation rate. The patterns inform our mode identification for asteroseismic fits, which unfortunately do not reveal a global best-fit solution. Asteroseismology reveals two main solutions roughly consistent with the spectroscopic parameters of this ELM white dwarf, but with vastly different hydrogen-layer masses; future seismic fits could be further improved by using the stellar parallax. GD 278 is now the tenth known pulsating ELM white dwarf; it is only the fifth known to be in a short-period binary, but is the first with extended, space-based photometry.

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arXiv2021

A new instability domain of CNO-flashing low-mass He-core stars on their early white-dwarf cooling branches

Before reaching their quiescent terminal white-dwarf cooling branch, some low-mass helium-core white dwarf stellar models experience a number of nuclear flashes which greatly reduce their hydrogen envelopes. Just before the occurrence of each flash, stable hydrogen burning may be able to drive global pulsations that could be relevant to shed some light on the internal structure of these stars through asteroseismology. We present a pulsational stability analysis applied to low-mass helium-core stars on their early white-dwarf cooling branches going through CNO flashes in order to study the possibility that the $\varepsilon$ mechanism is able to excite gravity-mode pulsations. We carried out a nonadiabatic pulsation analysis for low-mass helium-core white-dwarf models going through CNO flashes during their early cooling phases. We found that the $\varepsilon$ mechanism due to stable hydrogen burning can excite low-order ($\ell= 1, 2$) gravity modes with periods between $\sim 80$ and $500\ $s, for stars with $0.2025 \lesssim M_{\star}/M_{\odot} \lesssim 0.3630$ located in an extended region of the $\log g - T_{\rm eff}$ diagram with effective temperature and surface gravity in the ranges $15\,000 \lesssim T_{\rm eff} \lesssim 38\,000\ $K and $5.8 \lesssim \log g \lesssim 7.1$, respectively. Since the timescales required for these modes to reach amplitudes large enough to be observable are shorter than their corresponding evolutionary timescales, the detection of pulsations in these stars is feasible. If a low-mass white dwarf star were found to pulsate with low-order gravity modes in this region of instability, it would confirm our result that such pulsations can be driven by the $\varepsilon$ mechanism. In addition, confirming a rapid rate of period change in these pulsations would support that these stars actually experience CNO flashes, as predicted by evolutionary calculations.

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arXiv2019

On the existence of warm H-rich pulsating white dwarfs

The possible existence of warm ($T_{\rm eff}\sim19\,000$ K) pulsating DA white dwarf (WD) stars, hotter than ZZ Ceti stars, was predicted in theoretical studies more than 30 yr ago. However, to date, no pulsating warm DA WD has been discovered. We re-examine the pulsational predictions for such WDs on the basis of new full evolutionary sequences. We analyze all the warm DAs observed by TESS satellite up to Sector 9 in order to search for the possible pulsational signal. We compute WD evolutionary sequences with H content in the range $-14.5 \lesssim \log(M_{\rm H}/M_{\star}) \lesssim -10$, appropriate for the study of warm DA WDs. We use a new full-implicit treatment of time-dependent element diffusion. Non-adiabatic pulsations were computed in the effective temperature range of $30\,000-10\,000$ K, focusing on $\ell= 1$ $g$ modes with periods in the range $50-1500$ s. We find that extended He/H transition zones inhibit the excitation of $g$ modes due to partial ionization of He below the H envelope, and only in the case that the H/He transition is assumed much more abrupt, models do exhibit pulsational instability. In this case, instabilities are found only in WD models with H envelopes in the range of $-14.5 \lesssim \log(M_{\rm H}/M_{\star}) \lesssim -10$ and at effective temperatures higher than those typical of ZZ Ceti stars, in agreement with previous studies. None of the 36 warm DAs observed so far by TESS satellite are found to pulsate. Our study suggests that the non-detection of pulsating warm DAs, if WDs with very thin H envelopes do exist, could be attributed to the presence of a smooth and extended H/He transition zone. This could be considered as an indirect proof that element diffusion indeed operates in the interior of WDs.

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arXiv2019

TESS first look at evolved compact pulsators: asteroseismology of the pulsating helium-atmosphere white dwarf TIC 257459955

Pulsation frequencies reveal the interior structures of white dwarf stars, shedding light on the properties of these compact objects that represent the final evolutionary stage of most stars. Two-minute cadence photometry from TESS will record pulsation signatures from bright white dwarfs over the entire sky. We aim to demonstrate the sensitivity of TESS data to measuring pulsations of helium-atmosphere white dwarfs in the DBV instability strip, and what asteroseismic analysis of these measurements can constrain about their stellar structures. We present a case study of the pulsating DBV WD 0158$-$160 that was observed as TIC 257459955 with the 2-minute cadence for 20.3 days in TESS Sector 3. We measure the frequencies of variability of TIC 257459955 with an iterative periodogram and prewhitening procedure. The measured frequencies are compared to calculations from two sets of white dwarf models to constrain the stellar parameters: the fully evolutionary models from LPCODE, and the structural models from WDEC. We detect and measure the frequencies of nine pulsation modes and eleven combination frequencies of WD 0158$-$160 to $\sim0.01 μ$Hz precision. Most, if not all, of the observed pulsations belong to an incomplete sequence of dipole ($\ell=1$) modes with a mean period spacing of $38.1\pm1.0$ s. The global best-fit seismic models from both codes have effective temperatures that are $\gtrsim3000$ K hotter than archival spectroscopic values of $24{,}100-25{,}500$ K; however, cooler secondary solutions are found that are consistent with both the spectroscopic effective temperature and distance constraints from Gaia astrometry.

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arXiv2018

Pulsating low-mass white dwarfs in the frame of new evolutionary sequences VI. Thin H-envelope sequences and asteroseismology of ELMV stars revisited

Some low-mass white-dwarf (LMWD) stars with H atmospheres show long-period g-mode pulsations, comprising the class of pulsating WDs called extremely low-mass variable (ELMV) stars. It is generally believed that these stars have thick H envelopes. However, from stellar evolution considerations, the existence of LMWDs with thin H envelopes is also possible. We present a thorough asteroseismological analysis of ELMV stars based on a complete set of fully evolutionary models that represents low-mass He-core WD stars harboring a range of H envelope thicknesses. Although there are currently nine ELMVs, here we only focus on those that exhibit more than three periods and whose periods do not show significant uncertainties. We considered g-mode adiabatic pulsation periods for low-mass He-core WD models with $M_*$ in the range [0.1554-0.4352]$M_{\odot}$, $T_{\rm eff}$ in the range [6000-10000]K, and H envelope thicknesses in the range -5.8<log($M_{\rm H}/M_*$)<-1.7. We explore the effects of employing different H-envelope thicknesses on the adiabatic pulsation properties of low-mass He-core WD models, and perform period-to-period fits to ELMVs to search for a representative model. We found that the mode-trapping effects of g modes depend sensitively on $M_{\rm H}$, with the trapping cycle and trapping amplitude larger for thinner H envelopes. Also, the asymptotic period spacing is longer for thinner H envelopes. Finally, we found asteroseismological models (when possible) characterized by canonical (thick) and by thin H envelope, with $T_{\rm eff}$ and $M_*$ in agreement with the spectroscopic determinations. The fact that we have found asteroseismological solutions with H envelopes thinner than canonical gives a clue of the possible scenario of formation of these stars. Indeed, in the light of our results, some of these stars could have been formed by binary evolution through unstable mass loss.

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arXiv2018

A Refined Search for Pulsations in White Dwarf Companions to Millisecond Pulsars

We present optical high-speed photometry of three millisecond pulsars with low-mass ($< 0.3 M_{\odot}$) white dwarf companions, bringing the total number of such systems with follow-up time-series photometry to five. We confirm the detection of pulsations in one system, the white dwarf companion to PSR J1738+0333, and show that the pulsation frequencies and amplitudes are variable over many months. A full asteroseismic analysis for this star is under-constrained, but the mode periods we observe are consistent with expectations for a $M_{\star} = 0.16 - 0.19 M_{\odot}$ white dwarf, as suggested from spectroscopy. We also present the empirical boundaries of the instability strip for low-mass white dwarfs based on the full sample of white dwarfs, and discuss the distinction between pulsating low-mass white dwarfs and subdwarf A/F stars.

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arXiv2018

The coolest extremely low-mass white dwarfs

Extremely low-mass white dwarf (ELM WD; $M_{\star}$ $\lesssim$ $0.18-0.20\ $ $M_{\odot}$) stars are thought to be formed in binary systems via stable or unstable mass transfer. Although stable mass transfer predicts the formation of ELM WDs with thick hydrogen (H) envelopes, and hence characterized by dominant residual nuclear burning along the cooling branch, the formation of ELM WDs with thinner H envelopes from unstable mass loss cannot be discarded. We compute new evolutionary sequences for helium (He) core WD stars with thin H envelope with the main aim of assessing the lowest $T_{\rm eff}$ that could be reached by this type of stars. We generate a new grid of evolutionary sequences of He core WD stars with thin H envelope in the mass range from $0.1554$ to $0.2025 M_{\odot}$, and assess the changes in both the cooling times and surface gravity induced by a reduction of the H envelope. We also determine, taking into account the predictions of progenitor evolution, the lowest $T_{\rm eff}$ reached by the resulting ELM WDs. We find that a slight reduction in the H envelope yields a significant increase in the cooling rate of ELM WDs. Because of this, ELM WDs with thin H envelope could cool down to $\sim 2500\ $K, in contrast with their canonical counterparts that cool down to $\sim 7000\ $K. In addition, we find that a reduction of the thickness of the H envelope increases markedly the surface gravity ($g$) of these stars. If ELM WDs are formed with thin H envelopes, they could be detected at very low $T_{\rm eff}$. The detection of such cool ELM WDs would be indicative that they were formed with thin H envelopes, thus opening the possibility of placing constraints to the possible mechanisms of formation of this type of stars. Last but not least, the increase in $g$ due to the reduction of the H envelope leads to consequences in the spectroscopic determinations of these stars.

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arXiv2017

Pulsating low-mass white dwarfs in the frame of new evolutionary sequences: V. Asteroseismology of ELMV white dwarf stars

Many pulsating low-mass white-dwarf stars have been detected in the last years in the field of our Galaxy. Given that some of them exhibit multiperiodic variation of brightness, it is possible to probe their interiors through asteroseismology. We present a detailed asteroseismological study of all the known low-mass variable white dwarf stars based on a complete set of fully evolutionary models representative of low-mass He-core white dwarf stars. We employed adiabatic radial and nonradial pulsation periods for low-mass white dwarf models with stellar masses ranging from $0.1554$ to $0.4352\ M_{\odot}$, that were derived by simulating the nonconservative evolution of a binary system consisting of an initially $1 M_{\odot}$ ZAMS star and a $1.4 M_{\odot}$ neutron star companion. We estimated the mean period spacing for the stars under study and then we used the comparison between the observed period spacing with the average of the computed period spacings for our grid of models to constrain the stellar mass. We also employed the individual observed periods of every known pulsating low-mass white dwarf star, to search for a representative seismological model. We found that even though the stars under analysis exhibit few periods and the period fits show multiplicity of solutions, it is possible to find seismological models whose mass and effective temperature are in agreement with the values given by spectroscopy, for most of the cases. In the two cases where we could extract the period spacing from the set of observed periods, this method led to values of the stellar masses substantially larger than expected for this type of stars. The results presented in this work show in the one hand, the need for further photometric searches, and on the other hand, some improvements of the theoretical models, in order to place the asteroseismological results on a firmer ground.

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arXiv2017

Pulsating low-mass white dwarfs in the frame of new evolutionary sequences: IV. The secular rate of period change

We present a theoretical assessment of the expected temporal rates of change of periods ($\dotΠ$) for low-mass ($M_{\star}/M_{\sun} \lesssim 0.45$) and extremely low-mass (ELM, $M_{\star}/M_{\sun} \lesssim 0.18-0.20$) white-dwarf stars, based on fully evolutionary low-mass He-core white dwarf and pre-white dwarf models. Our analysis is based on a large set of adiabatic periods of radial and nonradial pulsation modes computed on a suite of low-mass He-core white dwarf and pre-white dwarf models with masses ranging from $0.1554$ to $0.4352 M_{\sun}$. We compute the secular rates of period change of radial ($\ell= 0$) and nonradial ($\ell= 1, 2$) $g$ and $p$ modes for stellar models representative of ELMV and pre-ELMV stars, as well as for stellar objects that are evolving just before the occurrence of CNO flashes at the early cooling branches. We found that the theoretically expected magnitude of $\dotΠ$ of $g$ modes for pre-ELMVs are by far larger than for ELMVs. In turn, $\dotΠ$ of $g$ modes for models evolving before the occurrence of CNO flashes are larger than the maximum values of the rates of period change predicted for pre-ELMV stars. Regarding $p$ and radial modes, we found that the larger absolute values of $\dotΠ$ correspond to pre-ELMV models. We conclude that any eventual measurement of a rate of period change for a given pulsating low-mass pre-white dwarf or white dwarf star could shed light about its evolutionary status. Also, in view of the systematic difficulties in the spectroscopic classification of stars of the ELM Survey, an eventual measurement of $\dotΠ$ could help to confirm that a given pulsating star is an authentic low-mass white dwarf and not a star from another stellar population.

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arXiv2016

Asteroseismology of the GW Virginis stars SDSS J0349-0059 and VV 47

We present an asteroseismological study of SDSS J0349-0059 and VV 47 aimed mainly at deriving their total mass on the basis of state-of-the-art PG 1159 evolutionary models. We compute adiabatic nonradial $g$-mode pulsation periods for PG 1159 evolutionary models with stellar masses ranging from $0.515$ to $0.741\ M_{\sun}$, that take into account the complete evolution of the progenitor stars. We first estimate a mean period spacing for both SDSS J0349-0059 and VV 47. By comparing the observed period spacing with the asymptotic period spacing we obtain $M_{\star}\sim 0.569\ M_{\sun}$ for SDSS J0349-0059 and $M_{\star}\sim 0.523\ M_{\sun}$ for VV 47. If we compare the observed period spacing with the average of the computed period spacings we found $M_{\star}\sim 0.535\ M_{\sun}$ for SDSS J0349-0059 and $M_{\star}\sim 0.528 M_{\sun}$ for VV 47. Searching for the best period fit we found, in the case of SDSS J0349-0059, an asteroseismological model with $M_{\star}= 0.542\ M_{\sun}$ and $T_{\rm eff}= 91\, 255\ $K. For VV 47, we could not find a unique and unambiguous asteroseismological model. Finally, for SDSS J0349-0059, we determined the rotation period as being $P_{\rm rot}= 1/Ω\sim 0.407$ days. The results presented in this work constitute a further step in the study of GW Vir stars through asteroseismology in the frame of fully evolutionary models of PG 1159 stars. In particular, once again it is shown the potential of asteroseismology to derive stellar masses of PG 1159 stars with an unprecedented precision.

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