11institutetext: 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 11email<EMAIL_ADDRESS>22institutetext: IALP - CONICET, La Plata, Argentina 33institutetext:
Instituto de Física y Astronomía, Universidad de Valparaíso, Gran Bretaña
1111, Playa Ancha, Valparaíso 2360102, Chile 44institutetext: European
Southern Observatory, Alonso de Cordova 3107, Santiago, Chile 55institutetext:
Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-900,
Porto-Alegre, RS, Brazil 66institutetext: INAF-Osservatorio Astrofisico di
Torino, strada dell’Osservatorio 20, 10025 Pino Torinese, Italy
77institutetext: XCP-6, MS F-699 Los Alamos National Laboratory, Los Alamos,
NM 87545, USA 88institutetext: Uniwersytet Pedagogiczny, Obserwatorium na
Suhorze, ul. Podchora̧żych 2, 30-084 Kraków, Polska 99institutetext: Institut
für Theoretische Physik und Astrophysik, Universität Kiel, 24098 Kiel, Germany
1010institutetext: DIRAC Institute, Department of Astronomy, University of
Washington, Seattle, WA-98195, USA 1111institutetext: NSF Astronomy and
Astrophysics Postdoctoral Fellow 1212institutetext: Department of Astronomy &
Institute for Astrophysical Research, Boston University, 725 Commonwealth
Ave., Boston, MA 02215, USA 1313institutetext: European Southern Observatory,
Karl-Schwarzschild-Straße 2, Garching 85748, Germany

# Pulsating hydrogen-deficient white dwarfs and pre-white dwarfs observed with
TESS

V. Discovery of two new DBV pulsators, WDJ152738.4$-$450207.4 and WD
1708$-$871, and asteroseismology of the already known DBV stars PG 1351+489,
EC 20058$-$5234, and EC 04207$-$4748

Alejandro H. Córsico 1122 Murat Uzundag 3344 S. O. Kepler 55 Leandro G.
Althaus 1122 Roberto Silvotti 66 Paul A. Bradley 77 Andrzej S. Baran 88 Detlev
Koester 99 Keaton J. Bell 10101111 Alejandra D. Romero 55 J. J. Hermes 1212
Nicola P. Gentile Fusillo 1313

(Received ; accepted )

###### Abstract

Context.  The TESS space mission has recently demonstrated its great potential
to discover new pulsating white dwarf and pre-white dwarf stars, and to detect
periodicities with high precision in already known white-dwarf pulsators.

Aims. We report the discovery of two new pulsating He-rich atmosphere white
dwarfs (DBVs) and present a detailed asteroseismological analysis of three
already known DBV stars employing observations collected by the TESS mission
along with ground-based data.

Methods. We processed and analyzed TESS observations of the three already
known DBV stars PG 1351+489 (TIC 471015205), EC 20058$-$5234 (TIC 101622737),
and EC 04207$-$4748 (TIC 153708460), and the two new DBV pulsators
WDJ152738.4$-$50207.4 (TIC 150808542) and WD 1708$-$871 (TIC 451533898), whose
variability is reported for the first time in this paper. We also carried out
a detailed asteroseismological analysis using fully evolutionary DB white-
dwarf models built considering the complete evolution of the progenitor stars.
We constrained the stellar mass of three of these target stars by means of the
observed period spacing, and derived a representative asteroseismological
model using the individual periods, when possible.

Results. We extracted frequencies from the TESS light curves of these DBV
stars using a standard pre-whitening procedure to derive the potential
pulsation frequencies. All the oscillation frequencies that we found are
associated with $g$-mode pulsations with periods spanning from $\sim 190$ s to
$\sim 936$ s. We find hints of rotation from frequency triplets in some of the
targets, including the two new DBVs. For three targets, we find constant
period spacings, which allowed us to infer their stellar masses and constrain
the harmonic degree $\ell$ of the modes. We also performed period-to-period
fit analyses and found an asteroseismological model for three targets, with
stellar masses generally compatible with the spectroscopic masses. Obtaining
seismological models allowed us to estimate the seismological distances and
compare them with the precise astrometric distances measured with Gaia. We
find a good agreement between the seismic and the astrometric distances for
three stars (PG 1351+489, EC 20058$-$5234, and EC 04207$-$4748), although for
the other two stars (WD J152738.4$-$50207 and WD 1708$-$871), the
discrepancies are substantial.

Conclusions. The high-quality data from the TESS mission continue to provide
important clues to determine the internal structure of pulsating pre-white
dwarf and white dwarf stars through the tools of asteroseismology.

###### Key Words.:

stars — pulsations — stars: interiors — stars: evolution — stars: white dwarfs

##  1 Introduction

Pulsating white dwarfs (WD) and pre-WDs constitute one of the most thoroughly
studied classes in the zoo of pulsating stars (see, e.g, Catelan & Smith,
2015; Aerts, 2021; Kurtz, 2022). Their multiperiodic brightness fluctuations,
with periods in the range $100-7\,000$ s, and amplitudes up to $0.4$ mag, are
due to low-degree ($\ell\leq 3$) nonradial gravity ($g$) mode pulsations.
These modes are excited by a driving mechanism related to the partial
ionization of the dominant chemical species in the zone of driving, which is
located at the outer regions of these stars (Winget & Kepler, 2008; Fontaine &
Brassard, 2008; Althaus et al., 2010). The most populous class of pulsating
WDs is that of the ZZ Ceti or DAV stars, which are DA-type —hydrogen(H)-rich
atmospheres— at $T_{\rm eff}\sim 10\,500-13\,000$ K, of which $\sim 500$
objects are known so far (Córsico et al., 2019a; Vincent et al., 2020; Guidry
et al., 2021; Romero et al., 2022). At higher temperatures ($T_{\rm eff}\sim
21\,000-30\,000$ K), and with helium (He)-dominated atmospheres, we find the
pulsating WDs called V777 Her or DBVs. Although not many DBVs are known at
present (47 objects; see Córsico et al., 2019a; Duan et al., 2021; Vanderbosch
et al., 2022), these compact pulsating stars are very interesting because the
origin of DB WDs is not entirely clear. They are probably descendants of the
PG 1159 stars (oxygen (O)-, carbon (C)-, He-rich atmospheres) after going
through the DO WD (very hot He-rich atmospheres) stage (Dehner & Kawaler,
1995; Althaus et al., 2005; Battich et al., 2020; Bédard et al., 2022). A
large fraction of PG 1159 stars is believed to be the result of a born-again
episode, i.e., a very late thermal pulse (VLTP) experienced by a WD during its
early cooling phase (Fujimoto, 1977; Schoenberner, 1979; Iben et al., 1983;
Althaus et al., 2005). During the VLTP, most of the H content of the remnant
is violently burned (Herwig et al., 1999; Miller Bertolami & Althaus, 2006).
As a result, the remnant is forced to evolve rapidly back to the asymptotic
giant branch (AGB) and finally into the central star of a planetary nebula as
a hot H-deficient object. After that, gravitational settling acting during the
early stages of WD evolution causes He to float and heavier elements (C, O) to
sink, giving rise to an He-dominated surface, and turning the PG 1159 star
into a DO WD (Unglaub & Bues, 2000), and later into a DB WD (Althaus et al.,
2005). Alternatively, some DB WDs could be descendant of DO WDs resulting from
evolutionary channels that not involve only the PG 1159 stars. For instance,
they could be the result of post-merger evolution involving the giant,
H-deficient RCrB stars (Rauch et al., 2006; Althaus et al., 2009). Also, DB
WDs could be the result of the merger of two WDs (Webbink, 1984; Lauer et al.,
2019).

DBV stars are found to pulsate with periods between 120 and 1080 s (Winget &
Kepler, 2008). Their existence was anticipated theoretically (Winget et al.,
1982b) before being discovered shortly after (Winget et al., 1982a). $g$ modes
in DBVs are thought to be excited by a combination of the $\kappa$ mechanism
acting in the He partial ionization zone — and thus setting the blue edge of
the DBV instability strip (Winget et al., 1983), and the “convective driving”
mechanism (Brickhill, 1991) which is possibly dominant once the outer
convection zone has deepened enough (Van Grootel et al., 2017).

At the beginning of the exploration of pulsating WDs, the discovery of
variable objects with single-site observations (e.g. Landolt, 1968; Winget et
al., 1982a) and even with multi-site campaigns like those of the Whole Earth
Telescope (WET; Nather et al., 1990; Winget et al., 1994), was challenging and
slow. A very interesting account of those early days can be found in the
article of Bradley (2021). Later on, discoveries of pulsating WDs increased
enormously thanks to the identification of candidates from the Sloan Digital
Sky Survey (SDSS, York et al., 2000; Kleinman et al., 2013; Kepler et al.,
2015, 2016, 2019a), and in recent years, with space missions. In particular,
the Kepler/K2 mission (Borucki et al., 2010; Howell et al., 2014) resulted in
the study of 32 ZZ Ceti stars and three DBV stars (Østensen et al., 2011; Bell
et al., 2015; Hermes et al., 2017a, b; Bell et al., 2017; Córsico, 2020; Duan
et al., 2021). That mission ended in 2018. The Transiting Exoplanet Survey
Satellite (TESS mission, Ricker et al., 2015), on the other hand, is currently
in its fourth year of operation and has completed its initial 2-year mission
in 2019 and 2020 to observe $\sim 85$% of the whole sky searching for small
planets orbiting nearby stars. The mission is planned to observe 69 sectors of
the sky, covering it completely. Currently (April 2022), the mission is
observing sector 51. TESS has allowed the discovery of pulsating stars, and in
particular, pulsating WDs and pre-WDs with magnitude G$\lesssim$16–17.

The first pulsating WD observed extensively with TESS was the DBV pulsator EC
0158$-$160 (TIC 257459955), which was studied and modeled in detail by Bell et
al. (2019). Recently, a thorough asteroseismological analysis of the DBV star
GD 358 —the archetype of the class— based on the observations gathered by the
TESS mission combined with data taken from the Earth has been carried out by
Córsico et al. (2022). The authors detected 26 periodicities from $\sim 422$ s
to $\sim 1087$ s, along with eight combination frequencies with periods
between $\sim 543$ s and $\sim 295$ s. These data, combined with a huge amount
of ground-based observations (Bischoff-Kim et al., 2019), allowed Córsico et
al. (2022) to find a constant period spacing of $39.25$ s, compatible with a
stellar mass of $M_{\star}=0.581\pm 0.031M_{\odot}$. Also, they found an
asteroseismological model for GD 358 with a stellar mass
$M_{\star}=0.584^{+0.025}_{-0.019}M_{\odot}$, compatible with the stellar mass
derived from the period spacing, and in agreement with the spectroscopic mass
($M_{\star}=0.560\pm 0.028M_{\odot}$). The seismological distance of GD 358,
$d_{\rm seis}=42.85\pm 0.73$ pc, is in good agreement with the precise
astrometric distance measured by Gaia (EDR3), $d_{\rm Gaia}=43.02\pm 0.04$ pc.

In this work, we present new TESS observations of the already known DBV stars
PG 1351+489, EC 20058$-$5234, and EC 04207$-$4748\. In addition, we report for
the first time the variability of the DB WDs, WDJ 152738.4$-$450207.4, and WD
1708$-$871\. With the discovery of these two new DBVs, the number of known
stars of this class increases to 49. We perform a detailed asteroseismological
analysis of these stars on the basis of the fully evolutionary models of DB
WDs computed by Althaus et al. (2009). The present study is the fifth in our
series of papers devoted to the study of pulsating H-deficient WDs observed
with TESS. The first article is focused on six already known GW Vir stars
(Córsico et al., 2021), the second one is devoted to the discovery of two new
GW Vir stars, specifically DOVs (Uzundag et al., 2021), the third paper is
dedicated to the DBV star GD 358 (Córsico et al., 2022), and the fourth one is
focused on the discovery of two additional GW Vir stars (Uzundag et al.,
2022).

The paper is organized as follows. In Section 2 we provide a brief account of
the main characteristics of the studied DB stars. In Sect. 3, we describe the
methods we apply to obtain the pulsation periods of each target star. A brief
summary of the stellar models of DB WD stars employed for the
asteroseismological analysis of these stars is provided in Sect. 4. Sect. 5 is
devoted to the asteroseismological modelling of the target stars, including
the search for a constant period spacing in the sets of periods of each star
by applying statistical tests, the assessing of the stellar mass of each
object through the use of the period spacing when possible, and by performing
period-to-period fits with the aim of finding an asteroseismological model for
each DBV star. Finally, in Sect. 6, we summarize our main results and make
some concluding remarks.

Figure 1: Location of DB WDs in the $T_{\rm eff}-\log g$ diagram (Kepler et
al., 2019b), marked with small black circles. Also depicted is the location of
the published DBV stars (gray circles), according to the compilation by
Córsico et al. (2019a) and including also the newly discovered DBV stars by
Duan et al. (2021) and Vanderbosch et al. (2022). The target stars of the
present paper are highlighted with red circles corresponding to three already
known DBVs, and with blue circles associated to the two new DBVs observed with
TESS and reported here for the first time. The DB WD evolutionary tracks of
Althaus et al. (2009) are displayed with different colors according to the
stellar-mass values (in solar units). The blue-dashed line represents the
theoretical dipole ($\ell=1$) blue edge of the DBV instability strip,
according to Córsico et al. (2009).

##  2 The target stars

Table 1: The list of six DBV (V777 Her) stars studied in this work. Columns 1,
2, 3, 4, 5, and 6 correspond to the TESS input catalog number, name of the
object, effective temperature, surface gravity, parallax, and distance,
respectively. For details, see the text.

TIC | Name | $T_{\rm eff}$ | $\log g$ | $\pi$ | $d$
---|---|---|---|---|---
|  | [K] | [cgs] | [mas] | [pc]
471015205 | PG 1351+489 | $26\,010\pm 1536^{(a)}$ | $7.91\pm 0.07^{(a)}$ | $5.69_{-0.05}^{+0.05}$ | $175.73^{+1.53}_{-1.58}$
101622737 | EC 20058$-$5234 | $25\,500\pm 500^{(b)}$ | $8.01\pm 0.05^{(b)}$ | $8.48_{-0.05}^{+0.05}$ | $117.95^{+0.68}_{-0.75}$
153708460 | EC 04207$-$4748 | $27\,288\pm 545^{(c)}$ | $7.808\pm 0.058^{(c)}$ | $10.93_{-0.03}^{+0.03}$ | $91.48^{+0.22}_{-0.23}$
150808542 | WDJ 152738.4$-$450207.4 | $25\,228\pm 630^{(d)}$ | $8.124\pm 0.010^{(d)}$ | $10.57_{-0.04}^{+0.04}$ | $94.62^{+0.34}_{-0.37}$
451533898 | L 7$-$44=WD 1708-891 | $23\,980\pm 1686^{(e)}$ | $8.05\pm 0.03^{(e)}$ | $14.47_{-0.03}^{+0.02}$ | $69.09^{+0.14}_{-0.12}$

References: (a) Bergeron et al. (2011); (b) Koester et al. (2014); (c) Voss et
al. (2007); (d) This work; (e) Rolland et al. (2018).

In this study, we announce the discovery of two new DBVs, WD
J152738.4$-$450207.4, and WD 1708$-$871 (L 7$-$44), and also report new TESS
observations of the already known DBV stars PG 1351+489, EC 20058$-$5234, and
EC 04207$-$4748\. The location of the five stars in the $\log T_{\rm eff}$
versus $\log g$ diagram is displayed in Fig. 1. These stars were classified as
white dwarf candidates with a a probability of being a white dwarf $P_{\rm
WD}\geq 0.99$ by Gentile Fusillo et al. (2019) based on their colors and Gaia
DR2 parallaxes. The Gaia EDR3 parallaxes and corresponding distances are given
in Table 1 from Bailer-Jones et al. (2021). We also describe the basic
characteristics of these stars below and summarize their stellar properties in
Table 1.

PG 1351+489 (hereafter PG 1351; aka TIC 471015205, V*EM UMa) is an known DBV
star discovered by Winget et al. (1987). Since the beginning, it was realized
that it could be a candidate for the first measurement of a rate of period
change in a DBV star, because its power spectrum is dominated by a single
high-amplitude pulsation mode with a period at $\sim 489$ s. This star was
reanalyzed by Redaelli et al. (2011), providing more precise periods and also
an additional low-amplitude mode. More importantly, it was possible to obtain,
for the first time for a DBV star, an estimate of the rate of period change
for the period at $\sim 489$ s of $\dot{\Pi}=(2.0\pm 0.9)\times 10^{-13}$ s/s
(Redaelli et al., 2011). Regarding the effective temperature and gravity of PG
1351, there are several spectroscopic determinations for this star. The first
detailed analysis Beauchamp et al. (1999) provided $T_{\rm eff}=22\,600\pm
700$ K, $\log g=7.90\pm 0.03$ using pure He atmospheres, and $T_{\rm
eff}=26\,100\pm 700$ K, $\log g=7.89\pm 0.03$ employing atmospheres with
impurities of H. The analysis of Bergeron et al. (2011) indicates that $T_{\rm
eff}=26\,010\pm 1536$ K, $\log g=7.91\pm 0.07$ (see Fig. 1), obtained by
assuming H contamination ($\log{\rm H}/{\rm He}=-4.37\pm 0.82$). Koester &
Kepler (2015) estimate $T_{\mathrm{eff}}=28434\pm 124$, $\log g=7.89\pm 0.02$,
$\log{\rm H}/{\rm He}<-3.1$, and $\log{\rm Ca}/{\rm He}<-4.0$. PG 1351 has
been asteroseismologically modeled by Córsico et al. (2014), and were able to
place constraints on the neutrino magnetic dipole moment. The star has also
been used to infer bounds to the axion mass by Battich et al. (2016).

EC 20058$-$5234, also known as QU Tel (henceforth EC 20058; aka TIC 101622737)
is an already known DBV star whose variability was reported by Koen et al.
(1995), who found that the pulsation spectrum appeared to be stable. This star
was long scrutinized by Sullivan (2003, 2005), Sullivan et al. (2007, 2008),
and Sullivan (2017). In particular, Sullivan et al. (2008) increased the
number of detected frequencies and performed an asteroseismic global-model
analysis. Interestingly, these authors found that EC 20058 had very stable
pulsation periods, which meant that, in principle, it could be possible to
measure the rate of period change to place constraints on the plasma neutrino
emission rate (Winget et al., 2004; Dalessio et al., 2010). However, Dalessio
et al. (2013) found that the pulsation frequencies of this star undergoes
secular changes that are inconsistent with simple neutrino plus photon-cooling
models. A detailed asteroseismological analysis of EC 20058 was carried out by
Bischoff-Kim & Metcalfe (2011) to place constraints on helium diffusion in WD
envelopes. The effective temperature and gravity of EC 20058 are $T_{\rm
eff}=25\,500\pm 500$ K and $\log g=8.01\pm 0.05$ (Koester et al., 2014) with H
and C impurities.

EC 04207$-$4748 (hereafter EC 04207; aka TIC 153708460) is an already known
DBV WD whose variability was discovered by Kilkenny et al. (2009), who found a
pulsation spectrum dominated by a period of $\sim 447$ s. The star was re-
observed by Chote et al. (2013) who found at least four independent eigenmodes
in the star, with the dominant mode having a period of $\sim 447$ s. The
light-curve exhibits distinct non-sinusoidal shapes, which results in
significant harmonics of the dominant frequency appearing in the Fourier
transforms. The effective temperature and gravity of EC 04207 are $T_{\rm
eff}=27\,288\pm 545$ K and $\log g=7.808\pm 0.058$, respectively (Voss et al.,
2007). However, (Koester et al., 2014) quote $T_{\rm eff}=25\,970$ K and $\log
g=7.79$, with log H/He of $-5.0$.

WD J152738.4$-$450207.4 (hereafter WD J1527; aka TIC 150808542) is a new DB WD
star. We observed WD J1527 with the Southern Astrophysical Research (SOAR)
Telescope, a 4.1-meter aperture optical and near-infrared telescope (Clemens
et al., 2004), situated at Cerro Pachón, Chile. The Goodman spectrograph with
a setup of 400 l/mm grating with the blaze wavelength 5500 $\AA$ (M1:
3000-7050 Å) with a slit of 1 arcsec was used. The signal-to-noise ratio of
the final spectrum is $\sim$150 at 4250 $\AA$. The atmospheric parameters for
WD J1527 are derived by fitting synthetic spectra to the newly obtained low-
resolution spectra as described by Koester (2010). The reduced spectrum is
shown in Fig. 2 with blue line and the model fits is shown with red line. WD
J1527 has $T_{\mathrm{eff}}=25\,228\pm 630$ K and $\log g=8.124\pm 0.010$,
with minor H contamination. At that effective temperature, the star is well
within the DBV instability strip (Fig. 1). Pulsations with periods between 229
and 1170 s are confirmed by observation in TESS sectors 12 and 38.

Figure 2: Optical spectra from SOAR (blue line) of the new DBV WD J1527.
Overplotted is the best-fit model atmosphere solution (red line).

WD 1708$-$871 (hereafter L 7$-$44; aka TIC 451533898) is a DB WD star
characterized by $T_{\rm eff}=23\,980\pm 1686$ K and $\log g=8.05\pm 0.03$
(Rolland et al., 2018) and only an upper limit for possible H contamination.
Observations in Sectors 12, 13, and 39 of TESS show luminosity variations with
periods in the range $466-937$ s. The location of this star in the $T_{\rm
eff}-\log g$ diagram is displayed in Fig. 1.

Table 2: The list of the five DBV (V777 Her) stars reported in this paper from TESS observations, including the name of the targets, the TESS magnitudes, the observed sectors, and the date and length of the runs (columns 1, 2, 3, 4, 5 and 6). After Fourier transform, three different set of parameters (resolution, average noise level of amplitude spectra, and detection threshold, which is defined as $0.1\%$ false-alarm probability (FAP), are presented in columns 7, 8 and 9, respectively. TIC | Name | $T_{\rm mag}$ | Obs. | Start Time | Run Length | Resolution | Average Noise |  $0.1\%$ FAP
---|---|---|---|---|---|---|---|---
|  |  | Sector | (BJD-2 457 000) | [d] |  $\mu$Hz | Level [ppt] | [ppt]
471015205 | PG 1351+489 | 17 | 16, 22, 23, 49, 50 | 1738.6492 | 128.89 | 0.1 | 1.32 | 6.3
101622737 | EC 20058$-$5234 | 16 | 13, 27 | 1653.9251 | 52.74 | 0.22 | 0.66 | 2.54
153708460 | EC 04207$-$4748 | 15 | 4, 5, 30, 31, 32 | 1410.9047 | 137.48 | 0.08 | 0.42 | 1.79
150808542 | WDJ152738.4-450207.4 | 16 | 12 ,38 | 1627.6494 | 25.24 | 0.45 | 4.90 | 20.7
451533898 | L 7$-$44 | 15 | 12, 13, 39 | 1624.9589 | 84.40 | 0.14 | 0.28 | 1.39

##  3 TESS data: observations, data reduction and analysis

Our analysis is based on 2-min short-cadence (SC) TESS data for all targets,
or 20-sec ultra-short-cadence when available. We used the pre-search data
conditioned simple aperture photometry (PDCSAP) data (Jenkins et al., 2016)
downloaded from the ”Barbara A. Mikulski Archive for Space Telescopes”
(MAST)111http://archive.stsci.edu. We extracted times in Barycentric corrected
Julian days (”BJD - 245700”), and fluxes (“PDCSAP FLUX”) by using the Python
package $\tt{lightkurve}$ (Lightkurve Collaboration et al., 2018). Afterwards,
we removed outliers by applying a running $5\sigma$ clipping mask. The fluxes
were normalized and transformed to amplitudes in parts-per-thousand (ppt)
units. We also detrend the light curves applying a Savitzky–Golay filter with
a three-day window length to remove any additional low-frequency systematic.
The target pixel files (TPFs) are examined to estimate the amount of
contamination from nearby unresolved targets. We have checked the
contamination by looking at the $\tt{CROWDSAP}$ parameter, which is the ratio
between target flux and total flux in the aperture. If the $\tt{CROWDSAP}$ is
significant ($<$ 0.5), then we used Gaia EDR3 parallaxes to look at the
contaminant targets in the field of view. Severe contamination affects the
average noise level in Fourier space and thus affects also the false-alarm
probability (FAP) thresholds as described in Bognár et al. (2020). After
creating the final light curves, we calculated their Fourier transforms (FTs)
to search for periodic signals. The detection thresholds were defined as
$0.1\%$ FAP significance level so that there is a 0.1% chance that the peaks
are caused by the noise fluctuations. $0.1\%$ FAP thresholds were calculated
by randomization of the data 1000 times, as described in Kepler (1993) (see
Baran & Koen, 2021). We identified frequency, amplitude and phase of each
pulsation mode using a nonlinear least square (NLLS) method. Also, we computed
sliding FTs to inspect the temporal evolution of the detected frequencies. To
calculate the sliding FTs, we use a 6-day sliding window with a 2-day step
size. We then compute the Fourier transform for each subset and trail them in
time. In Table LABEL:DBVlist we list the six DBV stars studied in this paper
from TESS observations, including observed sectors, TESS magnitude along with
name of the targets, the date and length of the runs.

###  3.1 PG 1351+489

PG 1351+489 ($T_{\rm mag}$ = 16.7) was observed in short cadence (SC, 120s
exposures) mode during sector 16 between September 11 and October 07, 2019,
for 23.04 days. PG 1351 was also observed during Sector 22 and 23 (2020-Feb-18
to 2020-Apr-16) in SC, providing another 54.74 days of observations. It was
also observed in fast cadence (20 s exposures) in Sectors 49 and 50
(2022-Feb-26 to 2022-Apr-22), 51.11 days of observations. We examined the
field of view (FOV) of PG 1351 and saw that there is a bright star that is
1.44 arcsec away from PG 1351 in sector 16. The problem is that the nearby
star is about 2.4 magnitudes brighter than PG 1351, dominating the pixels. The
$\tt{CROWDSAP}$ parameter is around 0.3, meaning that only 30 percent of the
light is coming from PG 1351, affecting the average noise level of the FT, as
shown in Table 2. Including all sectors, PG 1351 was observed for 128.89 days
in total. From these observations of PG 1351, we extracted only two
periodicities from the light curve. The main pulsation frequency at 2043.9
$\mu$Hz is present in all sectors with S/N $\geq$ 19\. In Fig. 3, we present
the temporal FT (upper panel) and sFT (bottom panel) for 2043.9 $\mu$Hz. The
other peak at 4087.8 $\mu$Hz, which has a lower S/N of about 5.75, is a
combination frequency (often called the first harmonic, 2 $\times$ 2043.9
$\mu$Hz). These two identified frequencies are listed in Table 3.

Table 3: Identified frequencies and combination frequencies, periods, amplitudes (and their uncertainties) and the signal-to-noise ratio in the data of PG 1351. Peak | $\nu$ | $\Pi$ | $A$ | S/N
---|---|---|---|---
| ($\mu$Hz) | (s) | (ppt) |
f1 | 2043.901 (20) | 489.260(22) | 25.78(1.05) | 19.8
2f1 | 4087.802(87) | 244.630(22) | 6.57(1.05) | 5.1

Figure 3: Top: Main pulsational mode in PG 1351 presented in Table 3. Fourier
transform of data taken in sector 16, 22 and 23. The red dashed line denotes
the 0.1% FAP. Bottom: Sliding Fourier transform sector 22 and 23 of PG
1351+489 concentrating the main pulsational mode. The color-scale illustrates
amplitude in ppt units.

###  3.2 EC 20058$-$5234

EC 20058$-$5234 was observed in sector 13 (2019-Jun-19 to 2019-Jul-18) in
camera 2. It was also observed in sector 27 (2020-Jul-04 to 2020-Jul-30) in
camera 2. Sector 13 contains 19582 measurements that span 28.4 days, while
sector 27 includes 16745 measurements that span 24.3 days. There are two stars
within 5 arcsec with brightnesses of 15.7 mag and 16.4 mag close to EC 20058.
These two stars have only a modest effect on the noise level of the FT. The FT
average noise level of sector 27 is 0.61 ppt, while it is 0.66 ppt in sector
13. We first examined the Fourier transform from each sector. Sector 13 shows
two frequencies at 3558 and 3893 $\mu$Hz, which are above the $0.1\%$ FAP of
2.54 ppt. These two peaks are also present in sector 27 above the $0.1\%$ FAP
of 2.3 ppt. We also examined the FT of both sectors together, as shown in Fig.
4. These two detected frequencies are given in Table 4. This target is known
to have significant phase and amplitude variations. The sliding Fourier
Transform of two reported frequencies was calculated. However, the signal-to-
noise ratio is insufficient to show and assess these substantial variations.

Table 4: Independent frequencies, periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio in the data of EC 20058. Peak | $\nu$ | $\Pi$ | $A$ | S/N
---|---|---|---|---
| ($\mu$Hz) | (s) | (ppt) |
f1 | 3558.935(27) | 280.983(20) | 4.31(53) | 6.5
f2 | 3893.289(28) | 256.852(20) | 4.25(53) | 6.4
Figure 4: Main pulsational modes (f1 and f2) in EC 20058$-$5234 presented in
Table 4 based on sector 13. The red dashed line denotes the 0.1% FAP.

###  3.3 EC 04207$-$4748

EC 04207 was first observed in SC during cycle 1 in sector 4 and 5
(2018-Oct-18 to 2018-Dec-11). Since in sector 4 the scatter is much higher
between 1422.61 d and 1423.51 d, and between 1436.60 d and 1436.82 d, we
removed these regions. Then, we added sector 5 and calculated the Fourier
transform of 53 days of data detecting five independent frequencies above
$0.1\%$ FAP confidence level, which corresponds to 1.79 ppt (Fig. 5 top). The
FT median noise level is 0.39 ppt. The S/N of the detected frequencies ranges
from 5 to 69. EC 04207 was also observed during three consecutive sectors (30,
31 and 32) of the extended mission in ultra-short-cadence (USC) mode. These
observations started on 4 July 2020 and ended on 26 August 2020. From this
84-day data set, we calculated the FT of the USC data up to the Nyquist
frequency of 25 000 $\mu$Hz. In this way, we identified 3 additional
combination frequencies that are not detectable in the 120 s cadence data
(Fig. 5 bottom). These combination frequencies are given in Table 5. In Fig.
5, we compare the two data sets showing the FT from 120 s-resolution (upper
panel) and 20 s-resolution data (lower panel). From the USC data, we extracted
three independent periodicities at 2236 $\mu$Hz (f1), 2361 $\mu$Hz (f3) and
2972 $\mu$Hz (f4). 2359 $\mu$Hz (f2) and 3861 $\mu$Hz (f5) are not detected in
USC data. The frequencies at 2359.142 and 2361.678 $\mu$Hz can be considered
as two components of a rotationally split dipole mode. These two peaks are
shown in Fig. 6 with frequency splitting of 2.536 $\mu$Hz. If we assume that
the central azimuthal component ($m=0$) is missing, then the rotation period
of EC 04207 would be 2.28 d. On the other hand, if the missing frequency was
one of the side components ($m=+1$ or $-1$), then the rotation period of EC
04207 would be 1.14 d. This solution is consistent with what has been
discovered for other types of pulsating WDs, such as GW Vir pulsating stars,
which range from 5 hours to a few days (Córsico et al., 2019a, 2021; Uzundag
et al., 2022) and DAV pulsating stars, which range from 1 hour to a 4.2 days
(Kawaler, 2015; Hermes et al., 2017a).

Combining the results from sector 4, 5, 30, 31 and 32, we detect five
independent frequencies which are concentrated between 2236 $\mu$Hz and 2972
$\mu$Hz. We also detect three combination frequencies beyond 4400 $\mu$Hz. The
extracted frequencies are listed in Table 5 with their associated errors and
S/N.

Table 5: Identified frequencies (combination frequencies), periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio in the data of EC 04207. The frequencies that are detected only in sector 4 and 5 are unmarked. Frequency and amplitude from sector 30, 31 and 32 are marked with $\dagger$. Frequency and amplitude that are detected in all sectors marked with $\dagger\dagger$.  Peak | $\nu$ | $\Pi$ | $A$ | S/N
---|---|---|---|---
| ($\mu$Hz) | (s) | (ppt) |
f${}_{\rm 1}^{\dagger\dagger}$ | 2236.165(00) | 447.194(02) | 22.87(26) | 69.3
f2 | 2359.142(18) | 423.882(34) | 2.14(33) | 5.3
f${}_{\rm 3}^{\dagger\dagger}$ | 2361.678(03) | 423.427(06) | 5.66(26) | 17.1
f${}_{\rm 4}^{\dagger\dagger}$ | 2972.672(04) | 336.397(05) | 4.44(26) | 13.4
2f${}_{\rm 1}^{\dagger}$ | 4472.332(03) | 223.597(02) | 5.18(26) | 15.7
(f1+f${}_{\rm 3})^{\dagger}$ | 4597.837(10) | 217.493(05) | 1.93(26) | 5.8
(f1+f${}_{\rm 4})^{\dagger}$ | 5208.833(10) | 191.981(04) | 1.89(26) | 5.7
Figure 5: Top: Fourier transform of data taken in sector 4 and 5 of EC 04207.
Bottom: Fourier transform of data taken in sector 30, 31 and 32 of EC 04207.
The horizontal red line indicates the 0.1% FAP level. Figure 6: A close up of
the amplitude spectrum of EC 04207 focusing on the potential rotational
multiplet with components at 2369 and 2361 $\mu$Hz.

###  3.4 WDJ152738.4-450207.4

WDJ152738.4-450207.4 is a new pulsating DBV star that was observed in sector
12 (2019-May-21 to 2019-Jun-19, cycle 1) and in sector 38 (2021-Apr-28 to
2021-May-26) with 20-sec cadence. Within 21 arcsec, the field of view (FOV)
includes 21 targets. All these targets are fainter than $G_{mag}=18$, which is
beyond the detection limit of TESS. The crowded field causes enough light
contamination to significantly raise the FT noise level, as reported in Table
LABEL:DBVlist. From the short-cadence observations, we detect two significant
frequencies near 1341 and 1539 $\mu$Hz (f1 and f5 in Table 6) above the
0.1%FAP level of 20.7 ppt.

The data from sector 38 have a time span of 26.33 d (97.5% duty cycle)
including 109476 data points. The frequency resolution is 0.44 $\mu$Hz. We
detect two independent frequencies near 1422 and 1425 $\mu$Hz. Moreover, we
detect a combination frequency at 2845.25 $\mu$Hz ($\simeq$1422
$\times\textsterling$2). All the detected frequencies of WD J1527 are reported
in Table 6. In Fig. 7 we display the Fourier transform of data taken in sector
12 (upper panel) and sector 38 (lower panel) of WD J1527. The horizontal red
line indicates the 0.1% FAP level. In sector 38, a clear triplet pattern can
be seen from the sliding FT of the ultra-short-cadence data of WD J1527. We
detect a candidate for rotational triplet ($\ell=1$) at 1419.97 (f2), 1422.69
(f3) and 1425.00 $\mu$Hz (f4) with a rotational splitting of $\sim$ 2.51
$\mu$Hz. The lower-frequency component at $\sim$1420 $\mu$Hz is not present in
overall FT, while it is clearly seen in the sFT as shown in Fig. 8. Therefore,
even though this side component is below the detection threshold, we include
it in our final frequency list with a S/N of 3.5. Multiplets in the frequency
spectrum of a pulsating WD are useful both to identify the harmonic degree of
the pulsation modes and to measure the rotation period of the star. We show in
Fig. 8 the rotational triplet detected in WD J1527. The average of the
splitting is $\sim$ 2.51 $\mu$Hz, which translates into a rotation period
$P_{\rm rot}\sim 2.3$ days for WD J1527. Since the frequency spectrum of WD
J1527 is admittedly scarce, so that we cannot search for a dipole or
quadrupole period spacing using only these three periods extracted from the
TESS data.

Table 6: Identified frequencies (combination frequencies), periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio in the data of WD J1527. The frequencies that are detected only in sector 12 are unmarked. Frequency and amplitude from sector 38 are marked with $\dagger$.  Peak | $\nu$ | $\Pi$ | $A$ | S/N
---|---|---|---|---
| ($\mu$Hz) | (s) | (ppt) |
f1 | 1340.703(28) | 745.877(15) | 30.74(3.89) | 7.1
f${}_{\rm 2^{\dagger}}$ | 1419.967(55) | 704.241(27) | 6.91(1.61) | 3.5
f${}_{\rm 3^{\dagger}}$ | 1422.687(02) | 702.895(01) | 17.28(1.61) | 8.5
f${}_{\rm 4^{\dagger}}$ | 1425.000(25) | 701.754(12) | 15.24(1.61) | 7.5
f5 | 1539.889(25) | 649.397(11) | 37.41(3.89) | 7.9
2f3 | 2845.254(42) | 351.462(05) | 8.94(0.31) | 4.6
Figure 7: Top: Fourier transform of data taken in sector 12 of WD J1527. The
horizontal red line indicates the 0.1% FAP level. Bottom: Fourier transform of
data taken in sector 38. The horizontal red line indicates the 0.1% FAP level.

Figure 8: Detected rotational triplet in WDJ152738.4-450207.4. The top panel
shows the Fourier transform of data taken in sector 38. The horizontal red
line presents the 0.1% FAP level. We indicated the azimuthal orders (vertical
blue lines) of dipole modes and the rotational splitting. The bottom panel
depicts the Sliding Fourier Transform of the same sector covering the same
frequency region. The color-scale shows amplitude in ppt units.

Finally, we performed time series photometry of WD J1527 with the IxON camera
on the 1.6-m Perkin Elmer Telescope at the Pico Dos Dias Observatory during
two nights, one in April and one in June 2021. We observed a total of $\sim$6
h, with integration times from 6 s to 12 s, using a red blocking filter BG40.
We reduced the data with the software IRAF, using the task DAOFOT and obtain a
light curve using differential photometry. Finally, we obtained the Fourier
Transform using the software ${\tt PERIOD04}$ (Lenz & Breger, 2005). The list
of frequencies, periods and variation amplitudes are listed in table 7. In
Fig. 9 we depict the light curve (top) and the Fourier transform (bottom) for
WD J1527 obtained from these ground-based observations. The plotted data is
one 4 h run, so there are no gaps to produce aliases. The same peaks are
detected when we add the 2 h data from June 21. We note that the ground-based
data can detect periods not seen with TESS. This is because the 1.6 m
telescope used has an area more than 100 times larger than TESS and therefore
can reach fainter magnitudes.

Figure 9: Upper panel: the light curve of the ground-based observations of WD J1527 carried out with Pekin Elmer Telescope. The plotted data is one 4h run. Lower panel: the Fourier transform, with a false-alarm pobability FAP(1/1000)= 3.9 mma. Table 7: Detected periods for WDJ152738.4$-$450207.4 from ground-based observations performed at the Pico do Dias observatory. Column 2 corresponds to the frequencies, column 3 shows the periods, and column 4 the amplitudes. Finally, column 5 gives the S/N level for each mode. Peak | $\nu$ | $\Pi$ | $A$ | S/N
---|---|---|---|---
| ($\mu$Hz) | (s) | (ppt) |
f1 | $1425.7(67)$ | $701.89(33)$ | $24.16$ | 18.6
f2 | $2290.6(67)$ | $436.98(13)$ | $5.71$ | 4.4
f3 | $2831.2(67)$ | $352.93(8)$ | $5.60$ | 4.3
f4 | $4283.9(67)$ | $233.26(4)$ | $4.69$ | 3.6

###  3.5 L 7$-$44

L 7$-$44 is also a new pulsating DBV star, which was observed by TESS in short
cadence mode during sector 12 and 13 between 21 May 2019 and 18 July 2019,
covering about 57 days. This provides a frequency resolution of 0.14 $\mu$Hz.
The FT average noise level is 0.28 ppt. The $0.1\%$ FAP confidence level is
1.39 ppt. We detect four frequencies at 981.22 (f1), 1067.52 (f2), 1094.50
(f5) and 2141.81 $\mu$Hz (f6).

L 7$-$44 was also observed in sector 39 in 20-sec cadence during 27 d. The
frequency resolution of this dataset is 0.41 $\mu$Hz. We identified four
frequencies above $0.1\%$ FAP confidence level of 1.84 ppt, which are listed
in Table LABEL:table:L7-44. We found a complicated pattern around 1092
$\mu$Hz, that is depicted in Fig. 10. This unresolved structure is most likely
due to frequency and/or phase variations over the length of the data. Also, it
could be caused by mode incoherence, that is, since this is the lowest
frequency pulsation mode in this sample, it could be incoherent because it is
reflecting off the moving base of the outer convection zone, as described for
DAVs in Montgomery et al. (2020). We extracted three frequencies (f2, f3 and
f4) from this region. However, only f3 will be used for the asteroseismic
modeling of this star since it is the highest FT peak.

Figure 10: Top: Fourier transform of sector 39 data of L 7$-$44\. The horizontal red line indicates the 0.1% FAP level. The magenta line is the residual after extraction of the signals. Bottom: Sliding Fourier transform of sector 39 data of L 7$-$44\. The color-scale illustrates amplitude in ppt units. Table 8: Independent frequencies, periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio in the data of L 7$-$44\. Frequency and amplitude from both short and ultra-short cadence data are marked with $\dagger\dagger$. Frequency and amplitude from sector ultra-short cadence data are marked with $\dagger$. The frequencies that are detected only in sector short-cadence data are unmarked. Peak | $\nu$ | $\Pi$ | $A$ | S/N
---|---|---|---|---
| ($\mu$Hz) | (s) | (ppt) |
f1 | 981.221(21) | 1019.138(21) | 1.64(31) | 4.6
f2 | 1067.521(19) | 936.750(17) | 1.80(31) | 4.7
f${}_{\rm 3^{\dagger}}$ | 1091.211(28) | 916.413(23) | 2.49(30) | 7.1
f${}_{\rm 4^{\dagger}}$ | 1093.003(27) | 914.910(23) | 2.53(30) | 7.2
f${}_{\rm 5^{\dagger\dagger}}$ | 1094.528(35) | 913.636(29) | 1.96(30) | 5.6
f${}_{\rm 6^{\dagger\dagger}}$ | 2141.813(16) | 466.894(4) | 2.09(31) | 5.5

##  4 Evolutionary models and spectroscopic masses

We employ a set of state-of-the-art DB WD stellar models that take into
account the complete evolution of the progenitor stars, assuming that DB WDs
are descendants of the PG 1159 stars after going through the DO WD stage. The
models were extracted from the evolutionary calculations of Althaus et al.
(2009) produced with the LPCODE evolutionary code (see that paper for
details). These DB WD models have been employed in the asteroseismic analyses
of the DBV stars KIC 8626021 (Córsico et al., 2012), KUV 05134+2605 (Bognár et
al., 2014), PG 1351+489 (Córsico et al., 2014), WD 0158$-$160 (Bell et al.,
2019), and GD 358 (Córsico et al., 2022). The sequences of DB WD models have
been obtained taking into account a complete treatment of the evolutionary
history of progenitors stars, starting from the zero-age main sequence (ZAMS),
through the thermally pulsing asymptotic giant branch (TP-AGB) and born-again
(VLTP; very late thermal pulse) phases to the domain of the PG 1159 stars, and
finally the DB WD stage. As such, they are characterized by evolving chemical
profiles consistent with the prior evolution. We varied the stellar mass and
the effective temperature in our model calculations, while the He content, the
chemical structure at the CO core, and the thickness of the chemical
interfaces were fixed by the evolutionary history of progenitor objects. The
models employ the ML2 prescription of convection with the mixing length
parameter, $\alpha$, fixed to 1 (Bohm & Cassinelli, 1971; Tassoul et al.,
1990). Specifically, we considered nine DB WD sequences with stellar masses
of: $0.515,0.530,0.542,0.565,0.584,0.609,0.664,0.741$, and $0.870M_{\odot}$.
These DB WD sequences are characterized by the maximum He-rich envelope that
can be left by prior evolution if we assume that they are the result of a
born-again episode. The value of envelope mass ranges from $M_{\rm
He}/M_{*}\sim 2\times 10^{-2}$ ($M_{*}=0.515M_{\odot}$) to $M_{\rm
He}/M_{*}\sim 1\times 10^{-3}$ ($M_{*}=0.870M_{\odot}$). In Figure 1 we show
the complete set of DB WD evolutionary tracks (with different colors according
to the value of the stellar mass) along with the location of all the DBVs
known to date (Córsico et al., 2019a; Duan et al., 2021; Vanderbosch et al.,
2022) and the two new DBVs reported in this paper for the first time.

We computed $\ell=1,2$ $g$-mode pulsation periods in the range $80-1500$ s
with the adiabatic and nonadiabatic versions of the pulsation code LP-PUL
(Córsico & Althaus, 2006; Córsico et al., 2006) and the same methods we
employed in the previous works of La Plata Stellar Evolution and Pulsation
Research Group222http://fcaglp.fcaglp.unlp.edu.ar/evolgroup/. Employing the
evolutionary tracks presented in Fig. 1 and the values of the spectroscopic
surface gravity and temperature shown in Table 1, we derive a value of the
spectroscopic mass of each of the five analyzed stars by interpolation. It is
worth noting that we use the same set of DB WD models both for estimate the
spectroscopic mass and also to assess the stellar mass from the period spacing
(next Section). For PG 1351 we obtain $M_{\star}=0.558\pm 0.027\ M_{\sun}$. In
the case of EC 20058 and EC 04207, we derive $M_{\star}=0.614\pm 0.030\
M_{\sun}$ and $M_{\star}=0.515\pm 0.023\ M_{\sun}$, respectively. Finally, we
obtain $M_{\star}=0.675\pm 0.022\ M_{\sun}$ for WD J1527, and
$M_{\star}=0.630\pm 0.016\ M_{\sun}$ for L 7$-$44\. The uncertainties in the
stellar mass are estimated from the uncertainties in $T_{\rm eff}$ and $\log
g$ by adopting the extreme values of each parameter when interpolating between
the evolutionary tracks of Fig. 1.

##  5 Asteroseismological analysis

The asteroseismological methods we use in this paper to extract information of
the stellar mass and the internal structure of the DBV stars PG 1351, EC
20058, EC 04207, WD J1527, and L 7$-$44 are the same used in, for instance,
Córsico et al. (2022) for the DBV star GD 358 observed with TESS.

First, we attempt to infer the stellar mass, when possible, by comparing the
observed mean period spacing, $\Delta\Pi$, with the average of the computed
period spacings ($\overline{\Delta\Pi_{k}}$). The mean period spacing is
assessed through the Kolmogorov-Smirnov (K-S; Kawaler, 1988), and the inverse
variance (I-V; O’Donoghue, 1994) significance tests. On the other hand, the
average of the computed period spacings is calculated as
$\overline{\Delta\Pi_{k}}=(N-1)^{-1}\sum_{k}\Delta\Pi_{k}$, where the
”forward” period spacing ($\Delta\Pi_{k}$) is defined as
$\Delta\Pi_{k}=\Pi_{k+1}-\Pi_{k}$ ($k$ being the radial order) and $N$ is the
number of computed periods laying in the range of the observed periods. The
present method relies on the spectroscopic effective temperature, so the
results are unavoidably affected by its associated uncertainty. This method
takes full advantage of the fact that the period spacing of DBV stars
primarily depends on the stellar mass and the effective temperature, and very
weakly on the thickness of the He envelope (see, e.g., Tassoul et al., 1990).

Another asteroseismological tool to disentangle the internal structure of DBV
stars is to seek theoretical models that best match the individual pulsation
periods of the target star. To measure the goodness of the match between the
theoretical pulsation periods ($\Pi_{\ell,k}$) and the observed individual
periods ($\Pi_{i}^{\rm o}$), we compute the quality function
$\chi^{2}(M_{\star},T_{\rm eff})=\frac{1}{N}\sum_{i=1}^{N}{\rm
min}[(\Pi_{\ell,k}-\Pi_{i}^{\rm o})^{2}]$, where $N$ is the number of observed
periods. In order to find the stellar model that best fits the observed
periods exhibited by each target star — the “asteroseismological” model —, we
evaluate the function $\chi^{2}$ for stellar masses $0.515\leq
M_{\star}/M_{\odot}\leq 0.870$. For the effective temperature, we employ a
much finer grid ($\Delta T_{\rm eff}=10-30$ K). For each target star, the DB
WD model that shows the lowest value of $\chi^{2}$ is adopted as the best-fit
asteroseismological model. To quantitatively assess the quality of our period
fit, we compute the average of the absolute period differences,
$\overline{\delta\Pi_{i}}=\left(\sum_{i=1}^{N}|\delta\Pi_{i}|\right)/N$, where
$\delta\Pi_{i}=(\Pi_{\ell,k}-\Pi_{i}^{\rm o})$, and the root-mean-square
residual,
$\sigma=\sqrt{(\sum_{i=1}^{N}|\delta\Pi_{i}|^{2})/N}=\sqrt{\chi^{2}}$. To have
a global indicator of the goodness of the period fit that takes into account
the number of free parameters, the number of fitted periods, and the proximity
between the theoretical and observed periods, we compute the Bayes Information
Criterion (BIC; Koen & Laney, 2000), defined as ${\rm BIC}=n_{\rm
p}\left(\frac{\log N}{N}\right)+\log\sigma^{2}$, where $n_{\rm p}$ is the
number of free parameters of the stellar models, and $N$ is the number of
observed periods, that is, the number of periods that we want to match. The
smaller the value of BIC, the better the quality of the fit. In our case,
$n_{\rm p}=2$ (stellar mass and effective temperature). Note that this
criterion penalizes for an excess of parameters ($n_{\rm p}$). Below, we
employ the tools described above to extract information from the DBV stars
considered in this work.

###  5.1 PG 1351+489

Table 9: List of periods of PG 1351 available for the asteroseismological analysis. Column 1 corresponds to four $\ell=1$ periods measured by Redaelli et al. (2011) (REA11), and column 2 corresponds to the single period detected by TESS (Table 3). The third, fourth and fifth columns give the theoretical periods of the asteroseismological model of PG 1351, the harmonic degree and the radial order, respectively. Columns 6 and 7 correspond to the period difference ($\delta\Pi_{k}=\Pi_{i}^{\rm O}-\Pi_{k}$) and the rate of period change, respectively. Finally, the last column provides information about the pulsational stability or instability nature of the modes. $\Pi_{i}^{\rm O}$ | $\Pi_{i}^{\rm O}$ | $\Pi_{\ell,k}$ | $\ell$ | $k$ | $\delta\Pi_{\ell,k}$ | $\dot{\Pi}_{\ell,k}$ | Unstable
---|---|---|---|---|---|---|---
(s) | (s) | (s) |  |  | (s) |  $(10^{-13})$ s/s |
REA11 | TESS |  |  |  |  |  |
335.26 |  | 336.81 | 2 | 13 | $-1.55$ | 0.60 | yes
489.33 | 489.26 | 489.47 | 1 | 11 | $-0.21$ | 0.81 | yes
584.68 |  | 586.99 | 2 | 25 | $-2.31$ | 1.02 | yes
639.63 |  | 639.37 | 1 | 15 | $0.26$ | 1.19 | yes

For this star, TESS detects a single period of $489.26$ s which is likely
associated to an eigenmode of the star (see Table 3). The other period present
($\sim 244.630$ s) corresponds to twice the frequency of that single period.
The period of $\sim 489$ s is present also in the data acquired through the
ground-based monitoring by Redaelli et al. (2011), who identified four
independent periods (see their Table 2). In Fig. 11 we schematically show the
period spectrum detected with TESS (upper panel), and the periods detected by
Redaelli et al. (2011) (lower panel), with amplitudes set to unity to
facilitate visualization. Unfortunately, the TESS data does not broaden the
period spectrum of PG 1351 in relation to the periods already known with
observations from the ground. The list of periods of the star is shown in
Table 9. With this very limited list of periods, we have tried to find a
possible mean period spacing to infer some hint of the stellar mass of PG
1351.

Figure 11: Schematic distribution of the pulsation periods of PG 1351
according to TESS (6 periods, black lines, upper panel), and according to
Redaelli et al. (2011) (4 periods, blue lines, lower panel). The amplitudes
have been arbitrarily set to one for clarity. Figure 12: I-V (upper panel) and
K-S (lower panel) significance tests to search for a constant period spacing
in PG 1351. The tests are applied to the set of 4 pulsation periods of Table
9, that includes the TESS and Redaelli et al. (2011)’s periods. Several
possible constant period spacings are evident, the most notorious being that
of $\sim 50$ s. See text for details. Figure 13: Dipole (upper panel) and
quadrupole (lower panel) average of the computed period spacings,
$\overline{\Delta\Pi_{k}}$, assessed in a range of periods that includes the
periods observed in PG 1351, shown as curves of different colors according to
the different stellar masses. We consider the effective temperature $T_{\rm
eff}=26\,010\pm 1536$ K (Bergeron et al., 2011), and the possible period
spacing values derived from the four periods of PG 1351 according to Table 9.
We include the error bars associated to the uncertainties in
$\overline{\Delta\Pi_{k}}$ and $T_{\rm eff}$.

In Fig. 12 we show the results of applying the I-V and K-S statistical tests
to the set of four periods of Table 9. We are aware that with only four
periods, it is feasible to find different period spacing values that fit the
distribution of those periods. Indeed, the two tests show hints of several
possible constant period spacings of $\sim 11$ s, $\sim 14$ s, $\sim 19$ s,
$\sim 25$ s, $\sim 31$ s, and another at $\sim 50$ s. Since the period spacing
in DB WDs depends mainly on the stellar mass and the effective temperature, we
can explore which of these possible period spacings are compatible with the
range of stellar masses of DB WDs. We have computed the average of the
theoretical period spacings for $\ell=1$ and $\ell=2$,
$\overline{\Delta\Pi_{k}}$, in terms of the effective temperature for all the
masses considered and a period interval of $300-650$ s. In Fig. 13 we show
$\overline{\Delta\Pi_{k}}$ with curves of different colors to distinguish the
different stellar masses, corresponding to $\ell=1$ (upper panel) and $\ell=2$
(lower panel) modes. We note that the only period spacings compatible with
reasonable stellar masses are $\sim 31$ s for $\ell=1$ and $\sim 19$ s for
$\ell=2$. Both possible period spacings point to a stellar mass somewhat high
for DB WD standards, but still possible, in the range
$0.74-0.87M_{\odot}$333Note that there is also a possible period spacing of
$\sim 25$ s. If this were a $\ell=1$ period spacing, the mass of PG 1351
should be very high ($\sim 1M_{\odot}$). Alternatively, if this were a
$\ell=2$ period spacing, then the mass of PG 1351 should be too low ($\sim
0.40M_{\odot}$). Both cases have to be discarded because they are in serious
conflict with the spectroscopic determination of $\log g$, that indicates a
stellar mass in the range $0.53\lesssim M_{\star}/M_{\odot}\lesssim 0.59$ (see
Fig. 1).. However, we emphasize that, since the number of available periods is
very low, the derivation of the period spacing is not robust, and the stellar
mass by this mean is not significantly constrained.

Figure 14: Chemical profiles (upper panel) and the squared Brunt-Väïsälä and
Lamb frequencies for $\ell=1$ and $\ell=2$ (lower panel) corresponding to our
asteroseismological DB WD model of PG 1351 with a stellar mass
$M_{\star}=0.664M_{\odot}$ and an effective temperature $T_{\rm eff}=25\,775$
K. Table 10: The main characteristics of the DBV star PG 1351.

Quantity | Spectroscopy | Asteroseismology
---|---|---
| Astrometry |
$T_{\rm eff}$ [K] | $26\,010\pm 1536$ | $25\,775\pm 150$
$M_{\star}$ [$M_{\odot}$] | $0.558\pm 0.027$ | $0.664\pm 0.013$
$\log g$ [cm/s2] | $7.91\pm 0.07$ | $8.103\pm 0.020$
$\log(L_{\star}/L_{\odot})$ | $\ldots$ | $-1.244\pm 0.03$
$\log(R_{\star}/R_{\odot})$ | $\ldots$ | $1.912\pm 0.015$
$(X_{\rm C},X_{\rm O})_{\rm c}$ | $\ldots$ | $0.32,0.65$
$M_{\rm He}/M_{\star}$ | $\ldots$ | $5.42\times 10^{-3}$
$d$ [pc] | $175.73^{+1.53(a)}_{-1.58}$ | $167.05^{+2.31}_{-2.26}$
$\pi$ [mas] | $5.69\pm 0.05^{(a)}$ | $5.99^{+0.08}_{-0.09}$

References: (a) Gaia EDR3.

A detailed asteroseismological analysis of PG 1351 employing the same DB WD
models considered in this paper was already carried out by Córsico et al.
(2014). The only difference is that in that work, the authors used a period of
489.33 s, instead of 489.26 s as in the present paper (see Table 3). Since
this difference is completely negligible to the limit of our investigation,
the asteroseismological model that we obtain in this work is the same as in
Córsico et al. (2014) and we will not repeat that analysis here. We show in
Table 9 the theoretical periods and the harmonic degree and radial order of
this model, along with the period differences and the rates of period change.
We obtain $\overline{\delta\Pi_{i}}=1.08$ s, $\sigma=1.40$ s, and BIC= 0.59
for the asteroseismological model. We note that Córsico et al. (2022) obtain
${\rm BIC}=1.13$ for the asteroseismological model of GD 358 and Bischoff-Kim
et al. (2019) obtain ${\rm BIC}=1.2$ for their best period fit to the same
star. So, the low value of BIC obtained in this work for PG 1351 indicates
that our period fit is very good. We show the characteristics of this model
($M_{\star}=0.664M_{\sun},T_{\rm eff}=25\,775$ K) in Table 10444Here, $M_{\rm
He}/M_{\star}$ is the total mass content of He in units of the stellar mass.,
and we plot its internal chemical profiles (upper panel) and the logarithm of
the squared Brunt-Väisälä and Lamb frequencies (lower panel) in Fig. 14.

Córsico et al. (2014) show that there are other possible asteroseismological
solutions, as can be seen in their Figure 9 and Table 1. We note (as those
authors do) that the possible solution that assumes only $\ell=1$ modes has a
very low quality of the period fit, and can be discarded. The two solutions
with $M_{\star}=0.87M_{\odot}$ and assuming a mix of $\ell=1$ and $\ell=2$
modes can be discarded too, in one case due to poor quality of the period fit,
and in the other one because the $T_{\rm eff}$ is too low. As an additional
argument to discard these three possible solutions, we emphasize the fact that
they correspond to a stellar mass ($M_{\star}=0.87M_{\odot}$) that is too
large as compared with the spectroscopic one ($M_{\star}=0.56M_{\odot}$), and
in severe contradiction with the surface gravity derived for this star.
Finally, none of these three possible solutions fits the Gaia’s distance
better than the asteroseismological model of Table 10 does (see below).

We have examined the pulsational stability/instability nature of the modes
associated with the periods fitted to the observed ones. We adopted the
frozen-in convection approximation (Unno et al., 1989). In particular, we
examined the sign and magnitude of the linear nonadiabatic growth rates,
$\eta_{k}=-\Im(\sigma_{k})/\Re(\sigma_{k})$, where $\Re(\sigma_{k})$ and
$\Im(\sigma_{k})$ are the real and the imaginary parts, respectively, of the
complex eigenfrequency $\sigma_{k}$. We find that the periods of the
asteroseismological model of PG 1351 are associated with unstable modes, in
line with the observational evidence. A common fact in all nonadiabatic
calculations of pulsations in WDs, but also in any other class of pulsating
stars, is that in general the models predict many more unstable modes than are
observed. PG 1351 is no exception. There must be some (unknown) filter
mechanism that selects some unstable modes and gives them large, observable
amplitudes, while the other unstable modes never reach those amplitudes and
are not observed.

Finally, using the asteroseismological model of PG 1351, we can assess the
seismic distance of the star. We employed the effective temperature and
gravity of our best-fit model to infer the absolute $G$ magnitude ($M_{G}$) in
the Gaia photometry. This was done by computing a DB WD atmosphere model using
$T_{\rm eff}$ and $\log g$ of the asteroseismological model. We find
$M_{G}=10.564$ mag. With the observed Gaia magnitude, $m_{G}=16.678$ mag, a
distance modulus was calculated as $m_{G}-M_{G}$, and this was converted to a
seismological distance $d$ using $\log(d_{\rm s})=(m_{G}-M_{G}+5)/5$. We
obtained $d_{\rm s}=167.05^{+2.31}_{-2.26}$ pc, where the errors were
calculated from the errors of $T_{\rm eff}$, $\log g$, and $m_{G}$. The
atmosphere models used $\log({\rm H}/{\rm He})=-5$. A test calculation showed
that the difference to pure He models is negligible. The seismological
distance is in agreement with the value derived by Bergeron et al. (2011)
($\sim 166$ pc), and is just $\sim 5\%$ lower than the astrometric distance
measured by Gaia EDR3, of $175.73^{+1.53}_{-1.58}$ pc.

Because TESS has detected a single eigenfrequency in PG 1351 ($\nu\sim
2044\leavevmode\nobreak\ \mu$ Hz), we have not been able to confirm (nor
discard) the rotation period of 8.9 h derived by Redaelli et al. (2011).

###  5.2 EC 20058$-$5234

Similar to the case of PG 1351, the pulsation spectrum of EC 20058 detected by
TESS is not rich. Indeed, TESS detected only two large-amplitude periods,
compared to a total of 11 periods measured with observations from the ground
by Sullivan et al. (2008) (see Table 11). In Fig. 15 we schematically show the
two periods detected with TESS (upper panel), and the periods detected by
Sullivan et al. (2008) (lower panel). For our analysis, in the case of the two
periods that are common to both sets, we adopt those detected by TESS, as they
are more accurate.

Table 11: Enlarged list of periods of EC 20058. Column 1 corresponds to 11 $\ell=1$ periods derived by Sullivan et al. (2008) (SEA08; see also Table 1 of Bischoff-Kim & Metcalfe, 2011), and column 2 corresponds to two periods detected by TESS (Table 4). The periods with one (two) asterisk(s) are used in a linear least square fit for modes with $\ell=1$ ($\ell=2$). $\Pi_{i}^{\rm O}$ (s) |  $\Pi_{i}^{\rm O}$ (s) |  $\Pi_{\rm fit}^{\ell=1}$ (s) |  $\delta\Pi^{\ell=1}$ (s) |  $\Pi_{\rm fit}^{\ell=2}$ (s) |  $\delta\Pi^{\ell=2}$ (s) | $\ell^{\rm O}$
---|---|---|---|---|---|---
SEA08 | TESS |  |  |  |  |
195.0(*) |  | 193.662 | 1.338 |  |  | 1
204.6 |  |  |  |  |  | ?
207.6(**) |  |  |  | 208.572 | $-0.972$ | 2
256.9 | 256.852(**) |  |  | 256.162 | 0.69 | 2
274.7(*) |  | 276.485 | $-1.785$ |  |  | 1
281.0 | 280.983(**) |  |  | 279.957 | 1.026 | 2
286.6 |  |  |  |  |  | ?
333.5 |  |  |  |  |  | ?
350.6(**) |  |  |  | 351.342 | $-0.742$ | 2
525.4(*) |  | 524.954 | 0.446 |  |  | 1
539.8 |  |  |  |  |  | ?
Figure 15: Schematic distribution of the pulsation periods of EC 20058
according to TESS (2 periods, black lines, upper panel), and according to
Sullivan et al. (2008) (11 periods, blue lines, lower panel). The amplitudes
have been arbitrarily set to one for clarity.

In Fig. 16 we display the results of applying the statistical tests to the set
of 11 periods of the first column of Table 11. The tests show hints of several
possible constant period spacings,
$\Delta\Pi=12.78,15.84,23.90,26.36,41.11,47.67$ s (emphasized with vertical
arrows in the figure), suggesting that no simple pattern of period spacing is
apparent in the EC 20058 spectrum, in line with previous works (Sullivan et
al., 2008). Fig. 17 depicts $\overline{\Delta\Pi_{k}}$ (computed considering a
period interval of $200-600$ s) with curves of different colors to distinguish
the different stellar masses, corresponding to $\ell=1$ (upper panel) and
$\ell=2$ (lower panel) modes. Based on this figure, the possible extreme
values for constant period spacings, that is 12.78 s and 47.67 s, can be
discarded from the analysis because they imply a mass too high or extremely
low, respectively. The remaining four values could actually be constant period
spacings, and they are plotted in Fig. 17 with red circles. Concentrating on
the upper panel (dipole modes), we see that a possible spacing of periods of
$41.11$ s s would indicate a stellar mass of $\sim 0.55M_{\odot}$ if it
corresponds to modes with $\ell=1$. Regarding the lower panel ($\ell=2$), we
can discard the possible period spacing of $\sim 26$ s because it would
indicate a very low mass for EC 20058 ($\lesssim 0.4M_{\odot}$). The possible
solutions that remain indicate a stellar mass of $\sim 0.530M_{\odot}$ if
$\Delta\Pi=23.90$ s, or a stellar mass $\sim 0.95M_{\odot}$ if
$\Delta\Pi=15.84$ s. We note that a stellar mass as high as $\sim
0.95M_{\odot}$ is not compatible with the spectroscopic mass of this star,
$\sim 0.61M_{\odot}$. On the other hand, such massive DB WDs are very unusual.
For these reasons, we discard the possible solution of $\sim 16$ s for the
period spacing. Then, we are left with the possible period spacing of $23.90$
s and $\ell=2$.

Figure 16: I-V (upper panel) and K-S (lower panel) significance tests to
search for a constant period spacing in EC 20058. The tests are applied to the
set of 11 pulsation periods of Table 11, that includes the TESS plus Sullivan
et al. (2008)’s periods. Possible values of constant period spacing are marked
with arrows in the upper panel. Figure 17: Dipole ($\ell=1$, upper panel) and
quadrupole ($\ell=2$, lower panel) average of the computed period spacings,
$\overline{\Delta\Pi_{k}}$, assessed in a range of periods that includes the
periods observed in EC 20058, shown as curves of different colors according to
the different stellar masses. We consider the effective temperature $T_{\rm
eff}=25\,500\pm 500$ K (Koester et al., 2014), and the possible period spacing
values $\Delta\Pi=12.78,15.84,23.90,26.36,41.11,47.67$ s (see Fig. 16). For a
discussion, see the text.

In summary, the two most probable period spacings present in the star are
$\Delta\Pi_{\ell=1}=41.11$ s and $\Delta\Pi_{\ell=2}=23.90$s. We immediately
note that $41.11/23.90=1.72$, which is very close to the asymptotic
prediction, $\Delta\Pi_{\ell=1}/\Delta\Pi_{\ell=2}=\sqrt{3}\sim 1.732$
(Tassoul et al., 1990). While it is appealing to conclude that the pulsational
spectrum of EC 20058 is composed of two patterns of periods, one with $\ell=1$
and the other with $\ell=2$, it is not trivial to explain why both patterns
suggest different stellar masses ($\sim 0.53$ and $\sim 0.55M_{\odot}$).
However, this small discrepancy could be alleviated if more realistic
uncertainties in the effective temperature were adopted. Beyond this, the fact
that the ratio of the period spacings is so close to that predicted by the
asymptotic theory is quite encouraging, and prompts us to try to identify the
harmonic degree of each period of EC 20058. To determine the precise period
spacings, we performed linear least-squares fits (plotted in the upper panel
of Fig. 18) using three periods for the case of $\ell=1$, and four periods for
the case of $\ell=2$, marked with a single asterisk and two asterisks,
respectively, in Table 11. We obtain a dipole period spacing
$\Delta\Pi_{\ell=1}=41.41\pm 0.39$ s and a quadrupole period spacing
$\Delta\Pi_{\ell=2}=23.80\pm 0.28$ s. We have four remaining periods (204.6 s,
286.6 s, 333.5 s, and 539.8 s) that do not fit in any of the two period
spacing patterns found. They could have harmonic degree $\ell=1$ and/or
$\ell=2$ and the reason for their deviation from the spacings derived for both
$\ell=1$ and $\ell=2$ modes could be that they are affected by mode trapping
(see, e.g., Bradley et al., 1993).

We investigated how the results of the K-S test change with variations in the
input frequency list, that is, excluding different periods, one at a time, and
ran the test again. We have found that the $\sim 24$ sec period spacing is
very robust, since, in general, the corresponding minimum in the K-S test
becomes steeper or stays the same when one ignores each period. On the other
hand, the $\sim 41$ sec period spacing is much more sensitive to the specific
list of periods on which the test is computed.

A completely different assumption than the one described above is that the
period spectrum of EC 20058 has a combination of $\ell=1$ and $\ell=2$ modes,
but with the possible presence of rotational frequency multiplets. It has been
the hypothesis favored by Sullivan et al. (2008) (see their Fig. 8). In this
scenario, the frequency $4816.8\leavevmode\nobreak\ \mu$Hz (period 207.6 s)
could be one component of a rotational triplet centered at
$4887.8\leavevmode\nobreak\ \mu$Hz (204.6 s), the other component being
inhibited by an unknown mechanism, and the frequencies
$3640.1\leavevmode\nobreak\ \mu$Hz (274.7 s) and $3489.0\leavevmode\nobreak\
\mu$Hz (286.6 s) could be the two non-zero $m$ components of a rotational
triplet centered at $3558.935\leavevmode\nobreak\ \mu$Hz (280.983 s).
Implicitly, the periods 204.6 s and 280.983 are assumed here to be $\ell=1$.
Using a mean frequency splitting of $\sim 70\leavevmode\nobreak\ \mu$Hz and
the asymmetry observed in the frequency splitting of the triplet centered at
the frequency $3558.935\leavevmode\nobreak\ \mu$Hz, Sullivan et al. (2008)
derived a rotation period of $\sim 2$ hours and a magnetic field of $\sim 3$
kG for EC 20058. Under the present hypothesis, it is interesting to look for a
constant period spacing with the statistical tests, this time neglecting the
periods 207.6 s, 274.7 s, and 286.6 s. Our exercise does not indicate any
clear constant period spacing in this reduced set of periods. The non-
existence of a constant period spacing could be explained by the fact that
some of the periods exhibited by the star are not in the asymptotic regime.

Putting it all together, we face two possible scenarios for EC 20058.
According to one of them, we are seeing a combination of ($m=0$) $\ell=1$ and
$\ell=2$ excited modes, that conform two patterns of constant period spacing.
In this scenario, no frequency multiplets due to rotation are present. In this
case, the assignment of the $\ell$ value for some of the periods is that of
Table 11 (column 7). In addition, following Sullivan et al. (2008), we can
assume the existence of rotational triplets, in this way identifying the
periods 204.6 s and 280.983 s with $m=0$ $\ell=1$ modes. We cannot rule out or
accept conclusively either of these two scenarios. Thus, when performing the
period-to-period fits (see below), we have to consider both possibilities for
the identification of the harmonic degree of the modes.

Figure 18: Upper panel: linear least-squares fits to the three dipole periods of EC 20058 marked with an asterisk and the four quadrupole periods marked with two asterisks in Table 11. The derived period spacings from these fits are $\Delta\Pi_{\ell=1}=41.4115\pm 0.3864$ s and $\Delta\Pi_{\ell=1}=23.795\pm 0.284$ s, respectively. Middle panel: residuals of the $\ell=1$ period distribution relative to the dipole mean period spacing. Lower panel: residuals of the $\ell=2$ period distribution relative to the quadrupole mean period spacing. Modes with consecutive radial order are connected with a thin black line. Table 12: Observed and theoretical periods of the asteroseismological solution for EC 20058 in the case 1, using 11 periods [$M_{\star}=0.664M_{\odot}$, $T_{\rm eff}=26\,380$ K]. Periods are in seconds and rates of period change (theoretical) are in units of $10^{-13}$ s/s. $\delta\Pi_{i}=\Pi^{\rm O}_{i}-\Pi_{k}$ represents the period differences, $\ell$ the harmonic degree, $k$ the radial order, $m$ the azimuthal index. The last column gives information about the pulsational stability/instability nature of the modes. $\Pi_{i}^{\rm O}$ | $\ell^{\rm O}$ | $\Pi_{k}$ | $\ell$ | $k$ | $m$ | $\delta\Pi_{k}$ | $\dot{\Pi}_{k}$ | Unstable
---|---|---|---|---|---|---|---|---
(s) |  | (s) |  |  |  | (s) | ($10^{-13}$ s/s) |
195.0 | 1 | 193.39 | 2 | 6 | 0 | 1.61 | 0.40 | yes
204.6 | ? | 204.19 | 1 | 3 | 0 | 0.41 | 0.44 | yes
207.6 | 2 | 206.45 | 2 | 7 | 0 | 1.15 | 0.47 | yes
256.852 | 2 | 248.42 | 2 | 9 | 0 | 8.43 | 0.60 | yes
274.7 | 1 | 275.87 | 1 | 5 | 0 | $-1.17$ | 0.53 | yes
280.983 | 2 | 275.87 | 1 | 5 | 0 | 5.11 | 0.53 | yes
286.6 | ? | 293.03 | 2 | 11 | 0 | $-6.43$ | 0.69 | yes
333.5 | ? | 332.28 | 2 | 13 | 0 | 1.22 | 0.750 | yes
350.6 | 2 | 352.48 | 2 | 14 | 0 | $-1.88$ | 0.62 | yes
525.4 | 1 | 524.05 | 1 | 12 | 0 | 1.35 | 1.11 | yes
539.8 | ? | 538.01 | 2 | 23 | 0 | 1.79 | 1.24 | yes
Table 13: Same as Table 12, but for case 2, in which eight periods are employed in the period fit. The possible asteroseismological solution for EC 20058 corresponds in this case to a DB WD model with $M_{\star}=0.664M_{\odot}$, $T_{\rm eff}=25\,467$ K. $\Pi_{i}^{\rm O}$ | $\ell^{\rm O}$ | $\Pi_{k}$ | $\ell$ | $k$ | $m$ | $\delta\Pi_{k}$ | $\dot{\Pi}_{k}$ | Unstable
---|---|---|---|---|---|---|---|---
(s) |  | (s) |  |  |  | (s) | ($10^{-13}$ s/s) |
195.0 | ? | 196.99 | 2 | 6 | 0 | $-1.99$ | 0.29 | yes
204.6 | 1 | 208.24 | 1 | 3 | 0 | $-3.64$ | 0.33 | yes
256.852 | ? | 254.22 | 2 | 9 | 0 | 2.63 | 0.48 | yes
280.983 | 1 | 280.94 | 1 | 5 | 0 | 0.04 | 0.42 | yes
333.5 | ? | 339.06 | 2 | 13 | 0 | $-5.56$ | 0.54 | yes
350.6 | ? | 349.46 | 1 | 7 | 0 | 1.14 | 0.43 | yes
525.4 | ? | 526.18 | 2 | 22 | 0 | $-0.78$ | 0.92 | yes
539.8 | ? | 534.71 | 1 | 12 | 0 | 5.09 | 0.88 | yes

In the second part of our analysis of EC 20058, we have performed period fits
considering two sets of periods and identifications. One of them corresponds
to that indicated in Table 12 (case 1), in which we follow the $\ell$
identification derived from the period spacing analysis (see Table 11), and
the other one to that shown in Table 13 (case 2), that corresponds to the
identification assumed by Sullivan et al. (2008). We show our results in Figs.
19 and 20, where we display the inverse of the quality function for case 1 and
case 2, respectively. Focusing on case 1, there is no good possible
seismological solution in the range of effective temperatures allowed by
spectroscopy, so we can adopt a relatively good fit model with $T_{\rm
eff}=26\,380$ K (slightly outside the allowed range of $T_{\rm eff}$) and
$M_{\star}=0.664M_{\odot}$, marked with a down arrow in Fig. 19. The non-
existence of a defined and clear seismological model in this case would
indicate that the identification of the modes established in Table 11 based on
possible period spacings with $\ell=1$ and $\ell=2$ could not be robust. In
case 2, however, it is possible to find a small family of possible
seismological solutions consisting of models with $M_{\star}=0.664M_{\odot}$
and $T_{\rm eff}$ in the range $25\,000-26\,000$ K. In particular, there is a
good asteroseismological solution for a model with $T_{\rm eff}=25\,467$ K
(marked with a down arrow in Fig. 20), well within the allowed range.
Curiously, the mass of this seismological model is the same as that of the
model with good period fit in case 1, although with a much lower effective
temperature and in excellent agreement with the spectroscopic $T_{\rm eff}$ of
EC 20058 ($T_{\rm eff}=25\,500\pm 500$ K). The theoretical periods of the
models selected for case 1 and case 2 are included in Tables 12 and 13, along
with the harmonic degree, the radial order, the difference with the observed
periods, the rates of period change, and the pulsational stability nature of
the modes. In particular, our identification of the harmonic degree and radial
order of the modes for the asteroseismological solution of case 2 is
remarkably similar to the Sullivan et al. (2008)’s identification (see their
Table 5), even though those authors used DB WD models completely different
than ours. Similar considerations hold when we compare the results of our case
2 with the results of Bischoff-Kim & Metcalfe (2011), even when they consider
higher effective temperatures for EC 20058. A similar overall agreement
between seismological results derived using the La Plata group
evolution/pulsation models (using the LPCODE and LP-PUL numerical codes) and
those from the Texas group (using the WDEC code) has been found in the case of
the DBV pulsator WD 0158$-$160 (TIC 257459955), the first DBV studied with
TESS (Bell et al., 2019). This agreement is encouraging, since the global
asteroseismological results are not severely dependent on the WD modeling and
period-fit methods used, which makes this type of analysis more robust.

We have also carried out period fits assuming that all the modes exhibited by
EC 20058 are $\ell=1$, but we have not found any possible seismological
solution. Indeed, in general it is not possible with our models to reproduce
the periods satisfactorily if they are considered as $\ell=1$ only. This
confirms that the pulsation spectrum of EC 20058 must be a mixture of dipole
and quadrupole modes. Finally, we have tried leaving the identification free
of all the modes (allowing them to be $\ell=1$ or $\ell=2$ from the outset).
In this case, we find the same asteroseismological model as in case 2,
although with a slightly poorer agreement between observed and theoretical
periods.

Figure 19: The inverse of the quality function of the period fit in terms of
the effective temperature, shown with different colors for the different
stellar masses, corresponding to case 1 (11 periods). The vertical black
dashed line corresponds to the spectroscopic $T_{\rm eff}$ of EC 20058 and the
vertical dotted lines its uncertainties ($T_{\rm eff}=25\,500\pm 500$ K;
Koester et al., 2014). A local maxima (marked with an arrow), corresponds to a
possible asteroseismological solution compatible with spectroscopy (see the
text). Figure 20: Same as in Fig. 19, but for case 2, which involves eight
periods. A clear maxima for a model with $T_{\rm eff}=26\,080$ K and
$M_{\star}=0.664M_{\sun}$, compatible with the spectroscopic effective
temperature of EC 20058, is marked with an arrow (see the text).

In view of this analysis, we adopt the DB WD model with
$M_{\star}=0.664M_{\odot}$ and $T_{\rm eff}=25\,467$ K, that corresponds to
the best fit of periods of the case 2 (see Table 13), as the
asteroseismological model for EC 20058. We obtain
$\overline{\delta\Pi_{i}}=2.61$ s, $\sigma=3.22$ s, and BIC= 1.24. We show the
characteristics of this model in Table 14. The internal chemical profiles and
the Brunt-Väisälä and Lamb frequencies of this model are quite similar to
those corresponding to the asteroseismological model of PG 1351 (Fig. 14).
Indeed, the only difference between both models is the effective temperature
($\Delta T_{\rm eff}\sim 300$ K), which has a negligible impact on the
chemical profiles and critical frequencies. Thus, Fig. 14 is also
representative of the seismological model of EC 20058. We mention that the
possible asteroseismological solution for case 1 is characterized by
$\overline{\delta\Pi_{i}}=2.77$ s, $\sigma=3.74$ s, and BIC= 1.34, which
reflects that the period fit is worse than that of the adopted
asteroseismological model (case 2). In addition, the model associated to the
possible solution for case 1 is outside the range of the effective temperature
allowed by spectroscopy.

Regarding the stability/instability of the modes of the asteroseismological
model, we find that all the periods of EC 20058 are predicted to be excited
according to our nonadiabatic computations, in line with the very existence of
these oscillation periods in the spectrum of EC 20058 (see last column of
Table 13). The same happens with regard to the possible seismological solution
in case 1 (see last column of Table 12).

Based on the asteroseismological model for EC 20058, we derive a seismological
distance, as we did for PG 1351. We find a magnitude absolute in the Gaia
photometry of $M_{G}=10.518$ mag using a DB WD model atmosphere. Using the
observed Gaia magnitude, $m_{G}=15.756$ mag, we obtained $d_{\rm
s}=111.60^{+1.53}_{-1.50}$ pc. The seismological distance is $\sim 5\%$ lower
than the astrometric distance measured by Gaia EDR3, of
$117.95^{+0.68}_{-0.75}$ pc, similar to we found for PG 1351.

Table 14: The main characteristics of the DBV star EC 20058 according to the
adopted asteroseismological model (case 2).

Quantity | Spectroscopy | Asteroseismology
---|---|---
| Astrometry |
$T_{\rm eff}$ [K] | $25\,500\pm 500$ | $25\,467\pm 130$
$M_{\star}$ [$M_{\odot}$] | $0.614\pm 0.030$ | $0.664\pm 0.013$
$\log g$ [cm/s2] | $8.01\pm 0.05$ | $8.062\pm 0.020$
$\log(L_{\star}/L_{\odot})$ | $\ldots$ | $-1.223\pm 0.03$
$\log(R_{\star}/R_{\odot})$ | $\ldots$ | $-1.901\pm 0.015$
$(X_{\rm C},X_{\rm O})_{\rm c}$ | $\ldots$ | $0.32,0.65$
$M_{\rm He}/M_{\star}$ | $\ldots$ | $5.42\times 10^{-3}$
$d$ [pc] | $117.95^{+0.68(a)}_{-0.75}$ | $111.60^{+1.53}_{-1.50}$
$\pi$ [mas] | $8.48\pm 0.05^{(a)}$ | $8.96\pm 0.12$

References: (a) Gaia DR3.

###  5.3 EC 04207$-$4748

Table 15: Enlarged list of periods of EC 04207. Column 1 corresponds to four periods derived by Chote et al. (2013), and column 2 corresponds to the periods detected by TESS (Table 5). In the case of the couple of periods of TESS of 423.882 s and 423.427 s, we adopt a single value of 423.655 s (see the text). $\Pi_{i}^{\rm O}$ (s) |  $\Pi_{i}^{\rm O}$ (s) |  $\Pi_{\rm fit}$ (s) |  $\delta\Pi$ (s) | $\ell^{\rm O}$
---|---|---|---|---
CEA13 | TESS |  |  |
336.4 | 336.397 | 335.035 | $1.362$ | 1
423.5 | 423.655 |  |  | ?
447.2 | 447.194 | 448.116 | $-0.922$ | 1
599.1 |  | 598.891 | $0.209$ | 1

In Fig. 21, we schematically show for the DBV star EC 04207 the periods
detected with TESS (upper panel), and the periods detected by Chote et al.
(2013) (lower panel). There are three common periods which are detected in
both data sets ($\sim 336$ s, $\sim 423$ s, and $\sim 447$ s), and there is a
period detected by Chote et al. (2013) ($\sim 599$ s) but not by TESS. In the
case of the three periods that are common to both sets of data, we adopt for
the asteroseismological analysis those detected by TESS as they are more
precise. In summary, we have four periods in total to carry out the
asteroseismological analysis. In the TESS data, the period of $\sim 423$ s is
actually a pair of periods very close to each other, of 423.882 s and 423.427
s. Assuming that they are the $m=-1$ and $m=+1$ component of a rotational
triplet, the period corresponding to the central component ($m=0$) is 423.655
s. We adopt this value of the period for this mode in the following
analysis555We have also considered the possibility that the periods of 423.882
s and 423.427 s are the $m=0$ and $m=\pm 1$ components of a rotational
triplet. The results we obtain in these cases are exactly the same as if we
assume that the periods correspond to the components $m=+1$ and $m=-1$..

Figure 21: Schematic distribution of the pulsation periods of EC 04207 according to TESS (3 periods, black lines, upper panel), and according to Chote et al. (2013) (4 periods, blue lines, lower panel). Actually, the period of $\sim 423$ s from TESS is a couple of very similar periods, of 423.882 s and 423.427 s. The amplitudes have been arbitrarily set to one for clarity. Figure 22: I-V (upper panel) and K-S (lower panel) significance tests to search for a constant period spacing in EC 04207. The tests are applied to the set of 4 pulsation periods of Table 15, that includes the TESS plus Chote et al. (2013)’s periods. A period spacing of $\sim 38$ s and its sub-harmonic of $\sim 19$ s are evident. Figure 23: Upper panel: dipole ($\ell=1$) average of the computed period spacings, $\overline{\Delta\Pi_{k}}$, assessed in a range of periods that includes the periods observed in EC 04207, shown as curves of different colors according to the different stellar masses. We consider the effective temperature $T_{\rm eff}=27\,288\pm 545$ K (Voss et al., 2007). Lower panel: same as in upper panel, but for quadrupole ($\ell=2$) modes. The red circles correspond to different possible period spacings as predicted by the statistical tests (see Fig. 22). Figure 24: The inverse of the quality function of the period fit with $\ell=2$ in terms of the effective temperature, shown with different colors for the different stellar masses, corresponding to the four periods of EC 04207 in Table 15. The vertical black dashed line corresponds to the spectroscopic $T_{\rm eff}$ of EC 04207 and the vertical dotted lines its uncertainties ($T_{\rm eff}=27\,288\pm 545$ K; Voss et al., 2007). A prominent maxima corresponds to the best-fit model selected for EC 04207, with $M_{\star}=0.542M_{\odot}$ and $T_{\rm eff}=27\,614$ K. Table 16: Observed and theoretical periods of the asteroseismological solution for EC 04207 [$M_{\star}=0.542M_{\odot}$, $T_{\rm eff}=27\,614$ K]. Periods are in seconds and rates of period change (theoretical) are in units of $10^{-13}$ s/s. $\delta\Pi_{i}=\Pi^{\rm O}_{i}-\Pi_{k}$ represents the period differences, $\ell$ the harmonic degree, $k$ the radial order, $m$ the azimuthal index. The last column gives information about the pulsational stability/instability nature of the modes. $\Pi_{i}^{\rm O}$ | $\ell^{\rm O}$ | $\Pi_{k}$ | $\ell$ | $k$ | $\delta\Pi_{k}$ | $\dot{\Pi}_{k}$ | Unstable
---|---|---|---|---|---|---|---
(s) |  | (s) |  |  | (s) | ($10^{-13}$ s/s) |
336.397 | 2 | 335.356 | 2 | 12 | 1.041 | 1.510 | yes
423.655 | 2 | 422.653 | 2 | 16 | 1.002 | 1.594 | yes
447.194 | 2 | 446.476 | 2 | 17 | 0.718 | 1.754 | yes
599.100 | 2 | 599.646 | 2 | 24 | $-0.546$ | 1.606 | yes

In Fig. 22 we show the statistical tests applied to the set of four periods in
Table 15. The tests strongly suggest a possible constant period spacings of
$\Delta\Pi\sim 21.9$ s. There is another possible period spacings at 17 s, 29
s, 37 s, and 43 s, that only are seen only in the K-S test, but not in the I-V
test. To interpret these possible spacings of periods and assume an harmonic
degree value for them, it is useful to first examine how they compare to the
theoretical period-spacing values of stellar models. In other words, we can
assign the values $\ell=1$ or $\ell=2$, or simply rule out any indication of
period spacing by plotting them together with the average of the theoretical
period spacings in terms of effective temperature for different stellar
masses. This is shown in Fig. 23, where we display the average of the computed
period spacing $\overline{\Delta\Pi_{k}}$ (computed considering a period
interval of $200-600$ s) with curves of different colors to distinguish the
different stellar masses, corresponding to $\ell=1$ (upper panel) and $\ell=2$
(lower panel) modes. We note that the strongest peak in the statistical tests,
that corresponds to $\Delta\Pi=21.85$ s (averaging the values that arise from
the K-S and I-V tests), must be associated to $\ell=2$ modes, indicating a
stellar mass of $M_{\star}\sim 0.565M_{\star}$. The corresponding period
spacing with $\ell=1$ should be close to what the asymptotic theory predicts,
that is $21.85\ {\rm s}\times\sqrt{3}=37.85$ s. Indeed, the peak at
$\Delta\Pi=36.91$ s in the K-S test is likely that dipole period spacing,
indicating, in this case, a stellar mass of $M_{\star}\sim 0.609M_{\odot}$ for
EC 04207. Alternatively, the presence of the possible period spacings at
$\Delta\Pi=17.45$ s ($\ell=2$) and $\Delta\Pi=29.05$ s ($\ell=1$) would
indicate a stellar mass of $M_{\star}\sim 0.82M_{\odot}$ and $M_{\star}\sim
0.85M_{\odot}$, respectively. Finally, the possible period spacing
$\Delta\Pi=43.62$ s, that should be associated to $\ell=1$, would indicate a
stellar mass too low ($M_{\star}\sim 0.50M_{\odot}$) and in contradiction with
the indications of its spectroscopic parameters. Although all these possible
period spacings are estimated on the basis of very few periods (four in total)
—and therefore we cannot conclude anything definite regarding the stellar mass
based on the period spacing— the strong signal of a possible period spacing of
21.85 s in both statistical tests would be indicating a stellar mass of
$M_{\star}\sim 0.565M_{\star}$. It remains to explain why the corresponding
dipole period spacing period, of $\sim 37$ s, appears not so noticeably and
only in the K-S test. We have recomputed the K-S test for the cases in which
we ignore the period of $\sim 424$ s and the period of $\sim 447$ s (one at a
time). We still get the same strong indication of a $\sim 22$ s period spacing
in both cases. As for the possible dipole period spacing of $\sim 37$ s, the
associated minimum appears enhanced when we ignore the $\sim 424$ s period,
and remains unchanged when we ignore the $\sim 447$ s period. We conclude that
the $\sim 22$ s period spacing is robust, and that the $\sim 37$ s potential
period spacing is sensitive to the presence or not of the $\sim 424$ s period.

We have performed a period-to-period fit employing the four observed periods
of EC 04207 from Table 15. We have fixed the harmonic degree $\ell=2$ for the
all the periods, since they do fit the pattern of quadrupole period spacing
derived above. The results of our period fit are shown in Fig. 24. The best
fit corresponds to a model with a mass of $M_{\star}=0.542M_{\odot}$ and an
effective temperature $T_{\rm eff}=27\,614$ K, compatible with the effective
temperature of EC 04207. There is also other possible solution for
$M_{\star}=0.565M_{\odot}$ and $T_{\rm eff}=27\,095$ K, although the quality
of the period match is poorer. We have repeated the analysis leaving free the
harmonic degree of the four periods of EC 04207 (allowing them to be $\ell=1$
or $\ell=2$), that is, disregarding the constraints imposed by the period
spacing derived above. In this case (not shown), we also find the same
solution of $0.542M_{\odot}$ and $T_{\rm eff}=27\,614$ K as before. We adopt
the $0.542M_{\odot}$ model as the asteroseismological model for EC 04207. The
average of the $\ell=2$ period spacing for this model is
$\overline{\Delta\Pi}_{\ell=2}=22.09$ s, close to the period spacing derived
from the statistical tests ($\Delta\Pi_{\ell=2}=21.85$ s). A DBV pulsator with
only $\ell=2$ modes seems to be very unusual. In the case of DAVs, there is
the ultra-massive ZZ Ceti star BPM 37094 that appears to be pulsating mostly
with $\ell=2$ modes (Metcalfe et al., 2004; Brassard & Fontaine, 2005; Córsico
et al., 2019b). As far as we are aware, there is no other DBV star with all
its periods associated to $\ell=2$ modes, apart from EC 04207 according to our
analysis. It must be taken into account, however, that to put this result on a
firmer basis, it would be necessary to detect more periods and, on the other
hand, to carry out other independent asteroseismological analyses of EC 04207.

In Table 16 we show a comparison of the observed and theoretical periods and
mode identifications, along with the theoretical rate of period change and
stability nature of each mode. We get, for this model,
$\overline{\delta\Pi_{i}}=0.83$ s, $\sigma=0.85$ s, and BIC= 0.16. We show the
characteristics of this model in Table 17. The chemical composition profiles
and the Brunt-Väisälä and Lamb frequencies of this model are displayed Fig.
25). We find that all the periods of EC 04207 are unstable according to our
nonadiabatic computations, in line with the fact that we observe these
oscillation periods in the pulsation spectrum of this star (see last column of
Table 16).

Based on the asteroseismological model for EC 04207, we can assess a
seismological distance, as we did for PG 1351 and EC 20058. We find a
magnitude absolute in the Gaia photometry of $M_{G}=10.126$ mag using a DB WD
model atmosphere. Using the observed Gaia magnitude, $m_{G}=15.229$ mag, we
obtained $d_{\rm s}=104.88^{+1.54}_{-1.94}$ pc. The seismological distance is
$\sim 12\%$ higher than the astrometric distance measured by Gaia EDR3, of
$91.48^{+0.22}_{-0.23}$ pc. This discrepancy is an indication of the existence
of uncertainties in the modeling of the interior of this DB WD star, and also
of the scarcity of periods exhibited by this target, which make that the
asteroseismological methods, in this case, lead to a seismological model that
is not entirely well constrained.

We close this section by noting that, if instead of using the $T_{\rm
eff}=27\,288\pm 545$ K of (Voss et al., 2007) we used the effective
temperature derived by (Koester et al., 2014), that is $T_{\rm eff}=25\,970\pm
500$ K, the stellar mass suggested by the derived quadrupole period spacing of
$\sim 19$ s would be slightly higher, of $\sim 0.585M_{\odot}$ (as compared
with $\sim 0.565M_{\odot}$). On the other hand, adopting the lower $T_{\rm
eff}$ assessed by (Koester et al., 2014) we could not find a seismological
model for EC 04207.

Table 17: The main characteristics of the DBV star EC 04207 according to the
adopted asteroseismological model.

Quantity | Spectroscopy | Asteroseismology
---|---|---
| Astrometry |
$T_{\rm eff}$ [K] | $27\,288\pm 545$ | $27\,614\pm 100$
$M_{\star}$ [$M_{\odot}$] | $0.515\pm 0.023$ | $0.542^{+0.017}_{-0.008}$
$\log g$ [cm/s2] | $7.808\pm 0.058$ | $7.872^{+0.027}_{-0.021}$
$\log(L_{\star}/L_{\odot})$ | $\ldots$ | $-0.980\pm 0.008$
$\log(R_{\star}/R_{\odot})$ | $\ldots$ | $-1.850\pm 0.007$
$(X_{\rm C},X_{\rm O})_{\rm c}$ | $\ldots$ | $0.15,0.83$
$M_{\rm He}/M_{\star}$ | $\ldots$ | $1.32\times 10^{-2}$
$d$ [pc] | $91.48^{+0.22(a)}_{-0.23}$ | $104.88^{+1.54}_{-1.94}$
$\pi$ [mas] | $10.93\pm 0.03^{(a)}$ | $9.53^{+0.18}_{-0.13}$

References: (a) Gaia DR3.

Figure 25: Chemical profiles (upper panel) and the squared Brunt-Väïsälä and
Lamb frequencies for $\ell=1$ and $\ell=2$ (lower panel) corresponding to our
asteroseismological DB WD model of EC 04207 with a stellar mass
$M_{\star}=0.542M_{\odot}$ and an effective temperature $T_{\rm eff}=27\,614$
K.

###  5.4 WDJ152738.4-450207.4

Figure 26: The inverse of the quality function of the period fit in terms of the effective temperature, shown with different colors for the different stellar masses, corresponding to three of the five periods of WD J1527 in Table 6. The vertical black dashed line corresponds to the spectroscopic $T_{\rm eff}$ of WD J1527 and the vertical dotted lines its uncertainties ($T_{\rm eff}=25\,228\pm 630$ K). A local maxima (marked with a green arrow), corresponds to a possible representative model for WD J1527 compatible with spectroscopy (see the text). Table 18: Observed and theoretical periods of the representative model of WD J1527 [$M_{\star}=0.542M_{\odot}$, $T_{\rm eff}=25\,780$ K]. Periods are in seconds and rates of period change (theoretical) are in units of $10^{-13}$ s/s. $\delta\Pi_{i}=\Pi^{\rm O}_{i}-\Pi_{k}$ represents the period differences, $\ell$ the harmonic degree, $k$ the radial order, $m$ the azimuthal index. The last column provides information about the pulsational stability/instability nature of the modes. $\Pi_{i}^{\rm O}$ | $\ell^{\rm O}$ | $\Pi_{k}$ | $\ell$ | $k$ | $\delta\Pi_{k}$ | $\dot{\Pi}_{k}$ | Unstable
---|---|---|---|---|---|---|---
(s) |  | (s) |  |  | (s) | ($10^{-13}$ s/s) |
233.26 | ? | 235.619 | 2 | 7 | $-2.189$ | 0.927 | yes
352.93 | ? | 351.457 | 2 | 12 | 1.023 | 1.340 | yes
436.98 | ? | 440.544 | 2 | 16 | $-3.484$ | 1.562 | yes
649.397 | ? | 646.283 | 2 | 25 | 3.114 | 2.282 | yes
702.895 | 1 | 701.448 | 1 | 15 | 1.244 | 2.344 | yes
745.877 | ? | 744.651 | 1 | 16 | 1.226 | 2.751 | yes

This is a new DBV star discovered with the TESS data. The star exhibits 5
periods between $\sim 649$ s and $\sim 746$ s (Table 6). Three periods
($701.754$ s, $702.895$ s, and $704.241$ s) appear to be the components of a
rotational triplet, suggesting a rotation period of $P_{\rm rot}\sim 2.3$ d.
This rotation-period value is within what is usual to find in WDs (from 1 h to
4.2 d; see Kawaler, 2015; Hermes et al., 2017a). The rotation period of $\sim
2.3$ d is assessed by considering the _average_ of the frequency splitting
observed during the sector 38 of _TESS_ , since clearly the frequency
splittings are time dependent (for some unknown reason). In summary, we have
only three independent pulsation periods (assumed to be $m=0$ modes) of WD
J1527 available from TESS for our asteroseismological analysis. If these
periods were associated with low radial-order modes, they would be useful to
establish strong asteroseismological constraints on the internal structure of
WD J1527 (see, for example, the cases of the ZZ Ceti stars G226$-$29,
G117$-$B15A, and R 548; Fontaine et al., 1992; Kepler et al., 1995; Bradley,
1998). However, these three periods correspond to intermediate or high radial
orders, lowering their asteroseismic potential. As we mentioned in Sect. 3.4,
we acquired additional ground based observations in the hopes of increasing
the number of detected periods. Observations with the 1.6 m OPD Telescope
allowed us to find additional periods of 233.26 s, 352.93 s, 436.98 s, and
701.89 s. They complement the periods detected by TESS in such a way as to
cover the short-period regime of the pulsation spectrum of the star. The
addition of the short periods of 233.26 s, 352.93 s, 436.98 s enables an
asteroseismological analysis. Since the period of 701.4 s detected from the
ground corresponds to the rotationally triplet centered at 702.895 s, we will
not take it into account in our period fit. First, we have searched for a
uniform period spacing in the pulsational spectrum of WD J1527 by employing
the statistical tests, but we did not find any conclusive result. Then, we
performed a period-to-period fit, assuming that the period 702.895 s
corresponds to a mode with $\ell=1$, $m=0$ (the central component of a
rotational triplet), and leaving free the value of $\ell$ for the other five
periods (233.43 s, 352.48 s, 437.06 s, 649.397 s and 745.877 s). Our results
for the inverse of the quality function versus the effective temperature for
different stellar masses are shown in Fig. 26. There are several models that
reproduce well the three periods of WD J1527, particularly a model with a
stellar mass $M_{\star}=0.609M_{\odot}$ at $T_{\rm eff}=23\,658$ K. However,
its effective temperature is not in good agreement with the spectroscopic
effective temperature. Restricting ourselves to models that provide good
agreement with the observed periods (local maxima of the inverse of
$\chi^{2}$) but at the same time satisfy the effective-temperature constraint,
the best-fit model has $M_{\star}=0.542M_{\odot}$ and $T_{\rm eff}=25\,780$ K
(green arrow in Fig. 26). We note that the mass of this model is substantially
smaller than the spectroscopic mass of $M_{\star}=0.675\pm 0.022M_{\sun}$. In
Table 18 we compare the observed pulsation periods of WD J1527 with the
theoretical periods of this representative model and their $\ell$ and $k$
values. The asymptotic period spacing of this model is $\Delta\Pi^{\rm
a}=42.107$ s. As we did for the other stars in this paper, we derive a
seismological distance for WD J1527 using the $\log g$ and $T_{\rm eff}$ of
the representative model. We find $d_{\rm s}=111.98$ pc, that is about $20\%$
larger than the Gaia astrometric distance, of 94.01 pc. Clearly, more
observations of the WD J1527 variability from space and/or from the ground
will be necessary to be able to analyze this DBV star in depth.

###  5.5 L 7$-$44

Figure 27: The inverse of the quality function of the period fit in terms of the effective temperature, shown with different colors for the different stellar masses, corresponding to three of the five periods of L 7$-$44 in Table LABEL:table:L7-44. The vertical black dashed line corresponds to the spectroscopic $T_{\rm eff}$ of L 7$-$44 and the vertical dotted lines its uncertainties ($T_{\rm eff}=23\,980\pm 1686$ K; Rolland et al., 2018). Table 19: Observed and theoretical periods of a representative model of L 7$-$44 [$M_{\star}=0.565M_{\odot}$, $T_{\rm eff}=24\,690$ K]. Periods are in seconds and rates of period change (theoretical) are in units of $10^{-13}$ s/s. $\delta\Pi_{i}=\Pi^{\rm O}_{i}-\Pi_{k}$ represents the period differences, $\ell$ the harmonic degree, $k$ the radial order, $m$ the azimuthal index. The last column gives information about the pulsational stability/instability nature of the modes. $\Pi_{i}^{\rm O}$ | $\ell^{\rm O}$ | $\Pi_{k}$ | $\ell$ | $k$ | $\delta\Pi_{k}$ | $\dot{\Pi}_{k}$ | Unstable
---|---|---|---|---|---|---|---
(s) |  | (s) |  |  | (s) | ($10^{-13}$ s/s) |
466.894 | ? | 466.720 | 2 | 17 | 0.174 | 1.159 | yes
914.955 | 1 | 914.864 | 1 | 20 | 0.091 | 2.121 | yes
936.750 | ? | 939.282 | 2 | 37 | $-2.532$ | 2.358 | yes
1019.138 | ? | 1017.539 | 2 | 40 | 1.599 | 2.902 | yes

This is another new DBV star discovered with the TESS observations. Similarly
to WD J1527, the TESS light curve of L 7$-$44 shows six independent periods,
although three of them may belong to a rotational triplet, corresponding to a
rotation period $P_{\rm rot}\sim 3.5$ d. This rotation period is within the
expected range of WD rotation periods (Córsico et al., 2019a). The periods
detected fall in the range 467 s $-$ 1019 s. Since all but one of the periods
are quite long, and consequently they are in the asymptotic regime of
pulsations of DBV stars, this set of periods is not potentially useful for a
deep asteroseismological analysis. At variance with the case of WD J1527, for
this star we do not have additional periods detected with observations from
the ground available. With so few periods, for L 7$-$44 we limited ourselves
to perform an exploratory asteroseismological analysis.

We first tried to find evidence of uniform period spacings, but we find no
conclusive results and we ruled out the possibility of deriving a stellar mass
through this approach. We also tried a period-to-period fit for this star. We
considered the period of 914.955 s as corresponding to a mode with $\ell=1$
since it is likely the central component ($m=0$) of a rotational triplet,
being the other $m\neq 0$ components the periods 913.655 s and 916.436 s.
Thus, we have four periods available for the period fit: 466.894 s, 914.955 s,
936.750 s, and 1019.138 s. We show in Fig. 27 the inverse of the quality
function versus the effective temperature for different stellar masses, along
with the location of L 7$-$44 according to its effective temperature. There
are two models that reproduce very well the periods of L 7$-$44, but they have
very high effective temperatures. Among these two good fits, we have to
discard that of $M_{\star}=0.870M_{\odot}$ at $T_{\rm eff}=30\,634$ K, because
it corresponds to a model that is too massive and hot, in clear contradiction
with the spectroscopic mass and the effective temperature of the star,
$M_{\star}=0.630M_{\odot}$ at $T_{\rm eff}=23\,980\pm 1686$ K. The other peak
corresponds to a model with the stellar mass $M_{\star}=0.664M_{\odot}$ at
$T_{\rm eff}=28\,387$ K, which provides the best agreement with the observed
periods. In the range of effective temperatures allowed by spectroscopy, we
find several small peaks associated to period fits of low quality. Among them,
we consider that corresponding to a model with $M_{\star}=0.565M_{\odot}$ and
$T_{\rm eff}=24\,690$ K, since it represents the best-fit model in the allowed
interval of $T_{\rm eff}$. In Table 19 we show a comparison between the
detected periods of L 7$-$44 and the theoretical periods of the model with
$M_{\star}=0.565M_{\odot}$ at $T_{\rm eff}=24\,690$ K. The asymptotic period
spacing of this model is $\Delta\Pi^{\rm a}=42.07$ s. We derive a
seismological distance for L 7$-$44 using the $\log g$ and $T_{\rm eff}$ of
the representative model, and find $d_{\rm s}=75.26$ pc, that is $9\%$ larger
than the Gaia astrometric distance, of 69.24 pc. As in the case of WD J1527,
we emphasize that more short periods would be necessary before we can place
significant constraints on the structure of L 7$-$44 with asteroseismology.

Table 20: The name of the studied stars (column 1), the independent periods detected with TESS (column 2), and the periods measured from ground-based observations (column 3). Periods in boldface are the ones used in the period-to-period fits (Section 5). Name | Periods from TESS | Periods from ground-based observations
---|---|---
| [s] | [s]
PG 1351+489 | 489.260 |  335.26, 489.33, 584.68, 639.63
EC 20058 |  256.852, 280.983 |  195.0, 204.6, 207.6, 256.9,
|  | 274.7, 281.0, 286.6, 333.5,
|  |  350.6, 525.4, 539.8
EC 04207 |  336.397, 423.655, 447.194 | 336.4, 423.5, 447.2, 599.1
WD J1527 |  649.397, 701.754, 702.895, 704.241, 745.877 |  233.26, 352.93, 436.98, 701.89
L 4$-44$ |  466.894, 914.955, 936.750, 1019.138 | $\cdots$

##  6 Summary and conclusions

In this paper, we presented an analysis of five pulsating DBV stars observed
with the TESS mission. Three of the five analyzed targets are the already
known DBV stars PG 1351, EC 20058 and EC 04207. The other two objects are new
DBVs discovered with the TESS data: WD J1527 and L 7$-$44\. With the discovery
of these two new DBVs, the number of known stars of this class increases to
49. We examined the variability of these five DBV stars by analyzing their
short and ultra-short-cadence single-sector observations obtained with TESS
and found 18 significant independent oscillation frequencies.

We detected and measured just a single frequency and a combination frequency
(the first harmonic of the independent frequency) for PG 1351 (see Table 3).
In the case of EC 20058, we found two independent frequencies and no
combination frequencies (Table 4). These two stars have very simple FTs,
unlike the rest of the objects. Indeed, EC 04207, WD J1527 and L 7$-$44 have
relatively rich FTs. Specifically, for EC 04207 we detected four independent
frequencies along with three combination frequencies (Table 5), for WD J1527
we found five independent frequencies and one combination frequency (Table 6),
and finally, for L 7$-$44 we measured six independent frequencies and no
combination frequencies (Table LABEL:table:L7-44). In the case of WD J1527, we
measured four additional independent frequencies from ground-based
observations (Table 7). We also examined the running Fourier transform (sFT)
of each target to investigate the temporal evolution of the pulsation modes.
As in other compact pulsating stars, including sdBV, GW Vir, and DAV stars, we
find variable amplitudes in the sFTs of WD J1527 (Fig. 8) and L 7$-$44 (Fig.
10). For two targets, EC 04207 and WD J1527, we estimated the possible
rotation periods from $g$-mode frequency splittings. In the former case, the
rotation period of EC 04207 would be either 1.14 or 2.28 d depending on the
missing azimuthal order. We show in Fig. 6 the peaks corresponding to the
frequencies at $2359.142\leavevmode\nobreak\ \mu$Hz and
$2361.678\leavevmode\nobreak\ \mu$Hz that can be considered as two components
of a rotationally split dipole mode. In the latter case, the rotation period
of WD J1527 is constrained to 2.3 days. Fig. 8 shows a detected rotational
triplet with a splitting of $\sim 2.51\leavevmode\nobreak\ \mu$Hz for WD
J1527. These findings are in line with what has been discovered for other
types of pulsating WDs, such as GW Vir variables, the rotation periods of
which ranges from 5 hours to a few days (Córsico et al., 2019a, 2021; Uzundag
et al., 2022) and DAV pulsating stars, for which it ranges from 1 hour to a
4.2 days (Kawaler, 2015; Hermes et al., 2017a). According to the atmospheric
parameters derived for WD J1527 from spectroscopic data, we found that the
star is well within the DBV instability strip with an effective temperature 25
228 K and surface gravity 8.124 dex.

We also performed a detailed asteroseismological analysis of these stars on
the basis of the fully evolutionary models of DB WDs computed by Althaus et
al. (2009). In Table LABEL:summary-period-astero we show the periods employed
for the asteroseismological period fits of each target, including their
origins (ground-based and TESS data). We summarize below our
asteroseismological findings for each star:

Table 21: The name of the studied stars (column 1), the effective temperature and gravity of their asteroseismological or representative models (columns 2 and 3), the apparent and absolute Gaia magnitudes (columns 4 and 5), and the seismological and astrometric distances (columns 6 and 7). Column 8 gives the discrepancy in the distances. Name | $T_{\rm eff}$ | $\log g$ | $m_{G}$ | $M_{G}$ | $d_{\rm s}$ | $d_{\it Gaia}$ | Distance
---|---|---|---|---|---|---|---
| [K] | [cgs] | [mag] | [mag] | [pc] | [pc] | discrepancy
PG 1351+489 | 25 775 | 8.103 | 16.678 | 10.564 | $167.05^{+2.31}_{-2.26}$ | $175.73^{+1.53}_{-1.58}$ | 4.94 %
EC 20058$-$5234 | 25 467 | 8.062 | 15.756 | 10.518 | $111.60^{+1.53}_{-1.50}$ | $117.95^{+0.68}_{-0.75}$ | 5.38 %
EC 04207$-$4748 | 27 614 | 7.872 | 15.229 | 10.126 | $104.88^{+1.54}_{-1.94}$ | $91.48^{+0.22}_{-0.23}$ | 12.8 %
WD J1527$-$4502 | 25 780 | 7.877 | 15.482 | 10.236 | $111.98$ | $94.01^{+0.34}_{-0.37}$ | 16.0 %
L 7$-$44 | 24 690 | 7.921 | 14.737 | 10.354 | $75.26$ | $69.24^{+0.14}_{-0.12}$ | 8.0 %

  * •

PG 1351: TESS detected only one period for this star, which agrees with one of
the four periods detected from ground-based observations. We searched for a
constant period spacing and found hints of a period spacing of $\sim 31$ s for
$\ell=1$ modes and $\sim 19$ s for for $\ell=2$ modes. These possible period
spacings suggest a high mass for PG 1351, in the range $0.74-0.87M_{\odot}$.
The period-to-period fit carried out on this star indicates an
asteroseismological model with a stellar mass of $M_{\star}=0.664\pm
0.013M_{\odot}$, which is smaller than the mass indicated by the period
spacings, although larger than the spectroscopic mass ($M_{\star}=0.558\pm
0.027M_{\odot}$).

  * •

EC 20058: For this star, TESS detected only two periods, which are in perfect
agreement with two of the 11 periods found from ground-based observations. The
search for constant period spacings suggests the existence of a $\ell=1$
period spacing of $41.4$ s and a $\ell=2$ period spacing of $23.8$ s, which
indicate a stellar mass of $0.53M_{\odot}$ and $0.55M_{\odot}$, respectively.
According to this analysis, the 11 periods of EC 20058 are a mix of $\ell=1$
and $\ell=2$ periods, without rotational splitting multiplets present. An
alternative analysis, proposed by Sullivan et al. (2008), consists of a period
spectrum of both dipole and quadrupole periods, but with the presence of
rotational triplets. A period-to-period fit analysis considering the above two
possibilities results in the adoption of an asteroseismological model with a
stellar mass of $0.664M_{\odot}$, substantially larger than the mass range
indicated by the period spacings ($0.53M_{\odot}-0.55M_{\odot}$) and in
agreement with the spectroscopic mass, of $M_{\star}=0.614\pm 0.030M_{\odot}$.

  * •

EC 04207: For this star, TESS detected three periods, which are in agreement
with three of the four periods detected from the ground. TESS failed to detect
a period measured from ground-based telescopes ($\sim 600$ s). A search for a
constant period spacing strongly suggest a $\ell=2$ uniform period separation
of $\sim 21.9$ s, compatible with a stellar mass of
$M_{\star}=0.565M_{\odot}$. A period-to-period fit analysis for this star
allowed us to find an asteroseismological model with a stellar mass of
$0.542M_{\odot}$, somewhat higher than the spectroscopic mass ($0.515\pm
0.023M_{\odot}$), but lower than the mass derived from the period spacing.

  * •

WD J1527: This is one of the new DBV stars discovered with TESS. The star
exhibits five periods between $\sim 649$ s and $\sim 746$ s, but three periods
(701.754 s, 702.895 s, and 704.241 s) appear to be the components of a
rotational triplet. We have also considered three additional periods (233.43
s, 352.48 s, 437.0 s) detected with ground-based observations. We have not
found a robust period spacing, nor a conclusive asteroseismological model. We
have to wait for further observations of this star —either from space or from
the ground— in order to analyze it in depth.

  * •

L 7$-$44: This is the other new DBV star discovered with TESS. Similar to what
happened in the case of WD J1527, L 7$-$44 exhibits six periods, three of
which seems to be part of a rotational triplet. Thus, we have only four
independent periods. Clearly, this is insufficient for asteroseismological
modelling, so we are again limited to searching for a representative model
that replicates as closely as possible the observed periods. For a deeper
study, we will have to wait for new observations of the star.

We summarize in Table LABEL:distances the asteroseismological distances for
the five stars analyzed in this work, and their comparison with their
astrometric Gaia distances. We find a relatively good agreement ($\sim 5-8\%$)
between both sets of distance estimates for some stars (PG 1351, EC 20058, and
EC 04207), although for the cases of WD J1527 and L 7$-$44 the discrepancies
are substantial ($\sim 13-16\%$), meaning that the representative DB WD models
used to estimate the distances are not entirely reliable.

The results presented in this work demonstrate that TESS is having a great
impact in the asteroseismology of WDs and pre-WDs —as well as in other classes
of pulsating stars— as a worthy successor to the Kepler/K2 mission. We
envisage that, as this mission continues to provide uninterrupted high-quality
photometric data of pulsating WD and pre-WDs, it will be possible to advance
our understanding of the structure and evolution of these stars.

###### Acknowledgements.

We wish to acknowledge the detailed suggestions and comments of an anonymous
referee that strongly improved the original version of this work. This paper
includes data collected with the TESS mission, obtained from the MAST data
archive at the Space Telescope Science Institute (STScI). Funding for the TESS
mission is provided by the NASA Explorer Program. STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5–26555. M.U. acknowledges financial support from CONICYT
Doctorado Nacional in the form of grant number No: 21190886 and ESO
studentship program. Part of this work was supported by AGENCIA through the
Programa de Modernización Tecnológica BID 1728/OC-AR, and by the PIP
112-200801-00940 grant from CONICET. K.J.B. is supported by the National
Science Foundation under Award AST-1903828. J.J.H. acknowledges support
through TESS Guest Investigator Programs 80NSSC20K0592 and 80NSSC22K0737.
Financial support from the National Science Centre under project No.
UMO-2017/26/E/ST9/00703 is acknowledged. This research has made use of NASA’s
Astrophysics Data System.

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