11institutetext: Instituto de Física y Astronomía, Universidad de Valparaíso,
Gran Bretaña 1111, Playa Ancha, Valparaíso 2360102, Chile
11email<EMAIL_ADDRESS>22institutetext: European Southern
Observatory, Alonso de Cordova 3107, Santiago, Chile 33institutetext:
Astronomical Institute of the Czech Academy of Sciences, CZ-251 65, Ondřejov,
Czech Republic 44institutetext: Astroserver.org, Fő tér 1, 8533 Malomsok,
Hungary 55institutetext: Instituto de Astrofísica de La Plata, UNLP-CONICET,
La Plata, Paseo del Bosque s/n, B1900FWA, Argentina 66institutetext: INAF-
Osservatorio Astrofisico di Torino, strada dell’Osservatorio 20, 10025 Pino
Torinese, Italy 77institutetext: ARDASTELLA Research Group, Institute of
Physics, Pedagogical University of Cracow, ul. Podchora̧żych 2, 30-084 Kraków,
Poland 88institutetext: Embry-Riddle Aeronautical University, Department of
Physical Science, Daytona Beach, FL 32114, USA 99institutetext: Nordic Optical
Telescope, Rambla José Ana Fernández P’erez 7, 38711 Breña Baja, Spain
1010institutetext: Department of Physics and Astronomy, Aarhus University, Ny
Munkegade 120, DK-8000 Aarhus C, Denmark 1111institutetext: Department of
Physics, Astronomy and Materials Science, Missouri State University, 901 S.
National, Springfield, MO 65897, USA 1212institutetext: Nicolaus Copernicus
Astronomical Centre of the Polish Academy of Sciences, ul. Bartycka 18, 00-716
Warsaw, Poland

# Asteroseismic analysis of variable hot subdwarf stars observed with _TESS_

I. The mean g-mode period spacings in hot subdwarf B stars††thanks: Based on
observations at the La Silla Observatory of the European Southern Observatory
for programs number 0103.D-0511 and 0104.D-0514. Based on observations
obtained at Las Campanas Observatory under the run code 0KJ21U8U.

Murat Uzundag 1122 Maja Vučković 11 Péter Németh 3344 M. Miller Bertolami 55
Roberto Silvotti 66 Andrzej S. Baran 77881111 John H. Telting 991010 Mike Reed
1111 K. A. Shoaf 1111 Roy H. Østensen 1111 and Sumanta K. Sahoo 771212

###### Abstract

Context. We present photometric and spectroscopic analyses of gravity (g-mode)
long-period pulsating hot subdwarf B (sdB) stars, also called V1093 Her stars,
observed by the TESS space telescope in both 120 s short-cadence and 20 s
ultra-short-cadence mode during the survey observation and the extended
mission of the southern ecliptic hemisphere.

Aims. We perform a detailed asteroseismic and spectroscopic analysis of five
pulsating sdB stars observed with TESS aiming at the global comparison of the
observations with the model predictions based on our stellar evolution
computations coupled with the adiabatic pulsation computations.

Methods. We process and analyze TESS observations of long-period pulsating hot
subdwarf B stars. We perform standard pre-whitening techniques on the datasets
to extract the pulsation periods from the TESS light curves. We apply standard
seismic tools for mode identification, including asymptotic period spacings
and rotational frequency multiplets. Based on the values obtained from
Kolmogorov-Smirnov and Inverse Variance tests, we search for a constant period
spacing for dipole ($l=1$) and quadrupole ($l=2$) modes. We calculate the mean
period spacing for $l=1$ and $l=2$ modes and estimate the errors by means of a
statistical resampling analysis. For all stars, atmospheric parameters were
derived by fitting synthetic spectra to the newly obtained low-resolution
spectra. We have computed stellar evolution models using LPCODE stellar
evolution code, and computed $l=1$ g-mode frequencies with the adiabatic non-
radial pulsation code LP-PUL. Derived observational mean period spacings are
then compared to the mean period spacings from detailed stellar evolution
computations coupled with the adiabatic pulsation computations of g-modes.

Results. We detect 73 frequencies, most of which are identified as dipole and
quadrupole g-modes with periods spanning from $\sim 3\,000$ s to $\sim
14\,500$ s. The derived mean period spacing of dipole modes is concentrated in
a narrow region ranging from 251 s to 256 s, while the mean period spacing for
quadrupole modes spans from 145 s to 154 s. The atmospheric parameters derived
from spectroscopic data are typical of long-period pulsating sdB stars with
the effective temperature ranging from 23 700 K to 27 600 K and surface
gravity spanning from 5.3 dex to 5.5 dex. In agreement with the expectations
from theoretical arguments and previous asteroseismological works, we find
that the mean period spacings obtained for models with small convective cores,
as predicted by a pure Schwarzschild criterion, are incompatible with the
observations. We find that models with a standard/modest convective boundary
mixing at the boundary of the convective core are in better agreement with the
observed mean period spacings and are therefore more realistic.

Conclusions. Using high-quality space-based photometry collected by the TESS
mission coupled with low-resolution spectroscopy from the ground, we have
provided a global comparison of the observations with model predictions by
means of a robust indicator such as the mean period spacing. All five objects
that we analyze in this work show a remarkable homogeneity in both seismic and
spectroscopic properties.

###### Key Words.:

asteroseismology — stars: oscillations (including pulsations) — stars:
interiors — stars: evolution — stars: horizontal-branch — stars: subdwarfs

##  1 Introduction

Hot subdwarf stars (sdB) are core-helium burning stars with a very thin
hydrogen (H) envelope ($M_{\rm env}$ $<$ 0.01 $M_{\odot}$), and a mass close
to the core-helium (He)-flash mass $\sim$0.47 $M_{\odot}$. The sdB stars are
evolved compact ($\log{g}$ = 5.0 - 6.2 dex) and hot ($T_{\rm eff}$ = 20 000 -
40 000 K) objects with radii between 0.15 $R_{\odot}$ and 0.35 $R_{\odot}$,
located on the so-called extreme horizontal branch (EHB; see Heber 2016, for a
review). They have experienced extreme mass-loss mostly due to binary
interactions at the end of the red giant phase, where they lost almost the
entire H-rich envelope, leaving a He burning core with an envelope too thin to
sustain H-shell burning. Hot subdwarf B stars will spend about $10^{8}$ yr
burning He in their cores. Once the He has been exhausted in their core, they
will start burning He in a shell surrounding a carbon/oxygen (C/O) core as
subdwarf O (sdO) stars, and eventually they will end their lives as white
dwarfs (Heber 2016).

One of the major progresses in our understanding of sdB stars was initiated by
Kilkenny et al. (1997), who discovered rapid pulsations in hot sdBs known as
V361 Hya stars (often referred to as short-period sdBV stars). The V361 Hya
stars show multiperiodic pulsations with periods spanning from 60 s to 800 s.
In this region, the pulsational modes correspond to low-degree, low-order
pressure p-modes with photometric amplitudes up to a few per cent of their
mean brightness (Reed et al. 2007; Østensen et al. 2010; Green et al. 2011).
These modes are excited by a classical $\kappa$-mechanism due to the
accumulation of the iron group elements (mostly iron itself), in the $Z$-bump
region (Charpinet et al. 1996, 1997). The authors also showed that radiative
levitation is a key physical process to enhance the abundances of iron group
elements in order to be able to excite the pulsational modes. The p-mode sdB
pulsators are found in a temperature range between 28 000 K and 35 000 K and
with the surface gravity $\log{g}$ in the interval 5-6 dex. Later on, the
long-period sdB pulsators known as V1093 Her stars were discovered by Green et
al. (2003), which show brightness variations with periods of up to a few hours
and have amplitudes smaller than 0.1 per cent of their mean brightness (Reed
et al. 2011). The oscillation frequencies are associated with low degree ($l<$
3) medium- to high-order (10 $<$ $n$ $<$ 60) g-modes, which are driven by the
same mechanism (Fontaine et al. 2003; Charpinet et al. 2011). The g-mode sdB
pulsators are somewhat cooler with temperatures ranging from 22 000 K to 30
000 K and $\log{g}$ from 5 dex to 5.5 dex. Between the two described families
of pulsating sdB stars, some ”hybrid” sdB pulsators, which simultaneosuly show
g- and p-modes, have been found. These hybrid sdB pulsators are located in the
middle of the region of the HR diagram between p- and g-mode sdB pulsators
(Green et al. 2011, Figure 5). These objects are of particular importance as
they enable us to study both the core structure and the outer layers of the
sdBVs via asteroseismology. A few examples were found from the ground (Oreiro
et al. 2005; Baran et al. 2005; Schuh et al. 2006; Lutz et al. 2008).

During the nominal Kepler mission (Borucki et al. 2010), 18 pulsating subdwarf
B stars were monitored in 1 min short-cadence mode. The majority of the stars
(16) were found to be long-period g-mode pulsators, while just two of them
were short period p-mode pulsators (Østensen et al. 2010, 2011; Baran et al.
2011; Reed et al. 2011; Kawaler et al. 2010). Additionally, 3 known sdBs stars
in the old open cluster NGC 6791 were found to pulsate (Pablo et al. 2011;
Reed et al. 2012). The temperature of these sdBs range from 21 500 K to 37 000
K with a median of 27 400 K and surface gravities between $\log{g}$ of 4.67
dex and 5.82 dex, with a median of 5.42 dex. In 2013, the Kepler mission was
re-initiated after the second reaction-wheel failure and it continued as the
K2 mission observing along the ecliptic (Haas et al. 2014). During the K2
mission more than 25 sdBs have been found to pulsate and the analyses are
still ongoing. Thus far 18 of these sdBVs have been published together with
their atmospheric parameters (Reed et al. 2016; Ketzer et al. 2017; Bachulski
et al. 2016; Reed et al. 2019; Baran et al. 2019; Silvotti et al. 2019; Reed
et al. 2020b; Østensen et al. 2020). These stars have $T_{\rm eff}$ in between
22 300 K and 37 000 K and $\log{g}$ is from 5.2 dex to 5.7 dex (Reed et al.
2018, for a review).

For many sdBVs observed with Kepler the asymptotic period sequences for g-mode
pulsations have been successfully applied, especially for dipole ($l=1$) and
quadrupole ($l=2$) modes, as more than 60% of the periodicities are associated
with these modes (Reed et al. 2018). The asymptotic approximation can be
perfectly applied for homogeneous stars. The period separation of g-modes
becomes approximately constant for high radial orders. This is called the
asymptotic regime of pulsations, and it is masterfully documented in Tassoul
(1980). It is important to note that the asymptotic g-mode theory is strictly
valid for completely radiative and chemically homogeneous stars. However, sdB
stars are stratified and diffusion processes (gravitational settling and
radiative levitation) contribute significantly to compositional
discontinuities, which disturb the pulsational modes and could break the
sequences of periods with constant spacing. This effect has been found in
several g-mode dominated sdBV stars. Furthermore, when the compositional
discontinuities become stronger in transition zones, some modes are trapped,
which was also detected for a few sdBV stars observed with Kepler (Østensen et
al. 2014; Uzundag et al. 2017; Baran et al. 2017; Kern et al. 2018). Mode
trapping is characterized by strong departures from a constant period spacing.
Trapped modes can be quite useful to provide a test of stellar evolution
models, and offer a unique opportunity to determine mixing processes due to
convective overshooting beyond the boundary of the helium burning core
(Ghasemi et al. 2017).

Another important asteroseismic tool, rotational multiplets, became available
for sdB stars thanks to the long baseline of Kepler data (Baran 2012). During
the nominal mission of Kepler, the rotation periods of sdBs have been found to
range from 10 d to 100 d (Reed et al. 2014). For the short-period sdB systems
with WD companions (P$\rm orb$ $<$ 15 d) the rotational periods of the sdBs
were found to be in the range from 28 d to 50 d (Baran et al. 2019, references
therein). For the short-period sdB binary systems with M-dwarf companions
(P$\rm orb$ $<$ 0.8 d) rotational periods in the range between 7 and 40 days
have been identified (Baran et al. 2019). Moreover, detecting rotational
multiplets in both g- and p-modes in hybrid sdB pulsators is of special
importance as it provides a way to determine the rotation of both the core and
the envelope of these stars. Also, rotational splittings allow us to assign a
harmonic degree to a pulsation mode.These remarkable stars have been found to
be either solid-body rotators (Baran 2012; Ketzer et al. 2017) or radially
differential rotators (Foster et al. 2015; Baran et al. 2017). The rotation
properties of evolved stars are further discussed in a recent review of
Charpinet et al. (2018).

The Transiting Exoplanet Survey Satellite (TESS) was launched successfully on
18 April 2018. The primary goal of this mission is to discover exoplanets
around nearby and bright stars by means of the transit method (Ricker et al.
2014). The spacecraft has four identical 100 mm aperture cameras and is in a
highly eccentric lunar-resonance Earth orbit. The orbit allows the telescope
to observe the targets during $\sim$27 days continuously covering a huge area
in the sky ($24^{o}$ x $96^{o}$). During the first year, TESS surveyed 13
sectors in the southern hemisphere with both 2-min short cadence (SC) and
30-min long cadence (LC). Results from LC observations of the first year have
been reported by Sahoo et al. (2020b).

During the first year, TESS observed 1702 compact objects including hot
subdwarfs, pre-white dwarfs and white dwarfs with 2-min cadence. The first
results regarding asteroseismic analysis of hot sdB pulsating stars observed
by TESS have been reported in three papers (Charpinet et al. 2019; Reed et al.
2020a; Sahoo et al. 2020a). In this paper, we analyze 5 pulsating sdB stars,
which were observed in a single sector in SC mode by TESS during the survey
phase of the southern ecliptic hemisphere. For each of these TESS targets we
have obtained low-resolution spectra and fitted model atmospheres in order to
derive their fundamental atmospheric parameters. We present the details of
spectroscopic and photometric observations as well as the main characteristics
of the studied sdBVs in Sect. 2. We discuss the analysis of the TESS data in
Sect. 3 and give details on the frequency analysis along with detailed seismic
mode identification. In Sect. 4 and 5, we analyzed the spectroscopic data and
derive atmospheric parameters for each star by fitting synthetic spectra to
the newly obtained low-resolution spectra. We calculate asteroseismic models
in Sect. 6 and compare them with the observations. Finally, we summarize our
findings in Sect. 7.

##  2 Observations

###  2.1 Photometric observations — TESS

The TESS mission Cycle 1, covering most of the southern hemisphere, started on
25 July 2018 and ended on 18 July 2019. During this time a total of 13 sectors
was observed, where each sector covered $\sim$27 days of continuous
observations with a 2-min short-cadence (SC) mode. During Cycle 1, TESS
observed 806 subdwarf (sd) candidates including sdB stars, sdO stars and their
He-rich counterparts. Among them, we have found several rich oscillators
including short- and long-period sdB pulsators. By ”rich”, we mean that there
are sufficient pulsation frequencies present for asteroseismic methods (e.g.
rotational multiplets and/or asymptotic period spacing) to be applicable. Thus
far, 5 rich long-period sdB pulsators have been reported. Four stars have been
analyzed by Sahoo et al. (2020a) and Reed et al. (2020a) by applying the
asymptotic period spacing. For one target, Charpinet et al. (2019) have
produced a detailed model by best-matching all the observed frequencies with
those computed from models. In this paper, we have concentrated on 5 other
long-period pulsating sdB stars, TIC 260795163, TIC 080290366, TIC 020448010,
TIC 138707823 and TIC 415339307 that were observed by TESS. During Cycle 2,
the second year of the primary mission, TESS observed the northern hemisphere,
sectors 14-26, after which it re-observed the southern hemisphere in what is
referred to as extended mission. During the extended TESS mission, three of
the selected stars (TIC 260795163, TIC 080290366 and TIC 138707823) have been
observed with 20-sec ultra-short-cadence (USC) mode. While TIC 080290366 and
TIC 138707823 were observed in USC mode during only one sector (29), TIC
260795163 was observed in USC mode during two consecutive sectors (27 and 28).

Among the stars analyzed in this paper, the only star for which a photometric
variability was discovered before TESS is TIC 080290366 (Koen & Green 2010).
The remaining four V1093 Her stars are new discoveries. The details of the
photometric TESS observations are summarized in Table LABEL:table1, where we
also give the literature name of the targets, which are taken from SIMBAD
111http://simbad.u-strasbg.fr/simbad/, the TESS Input Catalog (TIC) number,
right ascension, declination and $T$-magnitude along with their corresponding
observed sectors. Using available magnitude values from the literature, we
have calculated TESS magnitude of all targets as described by Stassun et al.
(2018) using the tool of ticgen222https://github.com/tessgi/ticgen.

###  2.2 Spectroscopic observations

The spectroscopic follow-up observations of the sdB pulsators analyzed in this
paper were obtained with two instruments, the Boller and Chivens (B&C)
spectrograph mounted at the 2.5-meter (100-inch) Iréne du Pont telescope at
Las Campanas Observatory in Chile333For a description of instrumentation, see:
http://www.lco.cl/?epkb_post_type_1=boller-and-chivens-specs, and the European
Southern Observatory (ESO) Faint Object Spectrograph and Camera (v.2) (EFOSC2)
(Buzzoni et al. 1984) mounted at the Nasmyth B focus of the New Technology
Telescope (NTT) at La Silla Observatory in Chile.

We obtained low-resolution spectra in order to calculate the atmospheric
parameters, such as effective temperature $T_{\rm eff}$, surface gravity
$\log{g}$ and He abundance Even though the atmospheric parameters for TIC
260795163, TIC 080290366, TIC 020448010, TIC 138707823 and TIC 415339307 are
available in the literature (Heber et al. 1984; Heber 1986; Kilkenny et al.
1995, 1988a; Németh et al. 2012; Lei et al. 2018), we reobserved them in order
to ensure homogeneity in our analysis. The B&C spectra were obtained using the
600 lines/mm grating corresponding to the central wavelength of 5 000 Å,
covering a wavelength range from 3 427 to 6 573 Å. We used a 1 arcsec slit,
which provided a resolution of 3.1 Å. Depending on the brightness of the
targets, the exposure times were between 300 s and 480 s, which was enough to
obtain an optimal signal-to-noise ratio (S/N) to measure $T_{\rm eff}$ and
$\log{g}$ with 5% precision. For the EFOSC2 setup, we used grism #7 and a 1
arcsec slit and the exposure times were between 200 s and 300 s. This setup
provided a wavelength coverage from 3270 to 5240 Åwith a S/N of about 150. TIC
260795163, TIC 080290366 and TIC 138707823 were observed with 2x2 binning mode
at a resolution of 5.4 Å, while TIC 020448010 was observed using 1x2 binning,
such that the spectral resolution slightly improved to 5.2 Å. The details of
the spectroscopic observations are given in Table LABEL:tablespec1 including,
instrument, date, exposure time, resolution and S/N ratio at 4 200 Å.

Table 1: Five sdB stars studied in this work, including the TESS input catalog number, the name of the star from the Simbad database, right ascension, declination, TESS magnitude and TESS observed sectors (including ultra-short cadence (USC) observations) and distances from $Gaia$, respectively. TIC | Name | RA(J2000) | Dec(J2000) | $T_{\rm mag}$ | Observed Sectors (USC) | Distance (pc)
---|---|---|---|---|---|---
260795163 | EC23073-6905 | 23:10:35.5 | -68:49:30.2 | 11.73 | 1 (27-28) | 499.6 $\pm$ 14
080290366 | JL194 | 00:31:41.6 | -47:25:20.1 | 11.85 | 2 (29) | 502.1 $\pm$ 13
020448010 | GALEXJ11143-2421 | 11:14:22.0 | -24:21:29.0 | 12.18 | 9 | 509.1 $\pm$ 14
138707823 | FB1 | 00:03:22.1 | -23:38:58.0 | 12.70 | 2 (29) | 695.9 $\pm$ 28
415339307 | HS0352+1019 | 03:55:14.3 | +10:28:12.6 | 14.24 | 5 | 771.3 $\pm$ 33
Table 2: Observing log of the spectroscopic data obtained for the 5 sdB stars studied in this work. Columns correspond to the TESS input catalog number, the spectrograph that was used for observations (with corresponding number of spectra obtained), the observing date, the exposure times, the spectroscopic data resolution and signal-to-noise level at 4200 Å, respectively. TIC | Spectrograph | Date | $t_{exp}(s)$ | Resolution ($\Delta\lambda$ (Å)) | S/N (@4200 Å)
---|---|---|---|---|---
260795163 | B&C (2) - EFOSC2 (1) | 23 Aug, 30 Oct 2019, 11 Jan 2020 | 420 - 300 - 200 | 3.1 - 5.4 | 120 - 80
080290366 | B&C (2) - EFOSC2 (1) | 23 Aug, 30 Oct 11, Jan 2020 | 450 - 240 - 200 | 3.1 - 5.4 | 100 - 150
020448010 | EFOSC2 (2) | 11 Jan 2020 | 240 | 5.2 | 150
138707823 | B&C (1) - EFOSC2 (1) | 22, 24 Jan 2020 | 480 - 300 | 3.1 - 5.4 | 100 - 150
415339307 | B&C (1) | 20 Aug 2020 | 420 | 3.1 | 70

###  2.3 The targets

  * •

TIC 260795163 (EC 23073-6905) was discovered during Edinburgh-Cape survey-II
(Kilkenny et al. 1995) and was classified as an sdB star with low-dispersion
spectrograms and UBV photometry. Kilkenny et al. (1995) derived $T_{\rm
eff}\sim 27000$ K and $\log{g}\sim 5$ dex, respectively. They reported a
radial velocity variation of about $\pm$ 26 km/s. Afterwards, Magee et al.
(1998) and Copperwheat et al. (2011) measured the radial velocity of the star
and they did not find a significant variation. The Gaia DR2 parallax and
corresponding distance for this object are $\pi=2.002\pm 0.058$ mas and
$d=499.6\pm 14.4$ pc.

  * •

TIC 080290366 (alias JL 194, EC 00292-4741, CD-48 106) is a well-known
relatively bright hot subdwarf star with $T$-band magnitude of 11.85. The star
was observed several times and can be found in many surveys, including Hill &
Hill (1966); Jaidee & Lyngå (1969); Kilkenny & Hill (1975); Wegner (1980);
Kilkenny et al. (1988b, 2016). The atmospheric parameters of TIC 080290366
have been derived by Heber et al. (1984) and also given by Kilkenny et al.
(1988a). The authors showed that the effective temperature and surface gravity
of TIC 080290366 are $T_{\rm eff}=25\,200$ K and $\log g=5.20$ dex. The
evolutionary status of TIC 080290366 was discussed by (Newell 1973). The
presence of a potential weak magnetic field was investigated by Mathys et al.
(2012), however, the detection limit was not enough to be conclusive. The
parallax and corresponding distance for this star extracted from Gaia DR2 are
$\pi=1.992\pm 0.05$ mas and $d=502.1\pm 12.6$ pc.

  * •

TIC 020448010 (EC 11119-2405) was discovered during the Edinburgh-Cape Blue
Object Survey as a hot sdB star with $V$-band magnitude of 12.72 (Kilkenny et
al. 1997). The atmospheric parameters were obtained by Németh et al. (2012).
The authors found $T_{\rm eff}=23\,430\pm 900$ K, $\log{g}=5.29\pm 0.15$ dex
and a low surface He abundance of $\log({\rm He/H})=-2.52\pm 0.25$. Kawka et
al. (2015) included the target in their survey for hot subdwarf binaries, but
did not detect significant velocity variations. From Gaia DR2, the parallax
and distance of this object are $\pi=1.964\pm 0.055$ mas and $d=509.1\pm 14.2$
pc, respectively.

  * •

The discovery of TIC 138707823 (alias EC 00008-2355, FB 1, Ton S 135, PHL
2580, MCT 0000-2355) was led by Haro & Luyten (1962) who searched for faint
blue stars in the region near the south galactic pole. TIC 138707823 was
confirmed by Lamontagne et al. (2000) as an sdB star. Estimation of
atmospheric parameters of TIC 138707823 have been given in several papers
(Greenstein & Sargent 1974; Kilkenny et al. 1977, 1988a) and ranged from
effective temperature of 23 000 to 27 000 K and surface gravity of 5.4 to 5.6
dex. Heber (1986) measured $T_{\rm eff}$ and $\log{g}$ as 25 600 $\pm$ 1 250 K
and 5.60 $\pm$ 0.20 dex, respectively. Edelmann et al. (2005) found that TIC
138707823 is a binary system comprising of an sdB and a main sequence (MS) or
a white dwarf (WD) companion with an orbital period of Porb = 4.122 $\pm$
0.008 d. Geier & Heber (2012) calculated the atmospheric parameters of TIC
138707823 from UVES spectroscopy and found $T_{\rm eff}$ = 27 600 $\pm$ 500 K
and $\log{g}$ = 5.43 $\pm$ 0.05 dex. From Gaia DR2, the parallax and distance
of TIC 138707823 are $\pi=1.437\pm 0.057$ mas and $d=695.9\pm 27.7$ pc,
respectively.

  * •

TIC 415339307 (HS 0352+1019) was included in the KISO Survey and Hamburg
Quasar Survey (Wegner & Boley 1993; Edelmann et al. 2003). The atmospheric
parameters were determined by Edelmann et al. (2003), who found $T_{\rm eff}$
= 24 900 $\pm$ 600 K, $\log{g}$ = 5.34 $\pm$ 0.1 dex and $\log{(n_{\rm
He}/n_{\rm H})}$ = -2.7 $\pm$ 0.2. Recently, Lei et al. (2018) has derived
$26\,340\pm 150$ K, $5.33\pm 0.01$ dex and $\log{(n_{\rm He}/n_{\rm
H})}=-2.68\pm 0.06$. From Gaia DR2, the parallax and distance of TIC 415339307
are $\pi=1.296\pm 0.056$ mas and $d=771.36\pm 33.04$ pc, respectively.

##  3 Analysis of TESS data

We analyzed TESS observations using the SC mode, which samples every 2-minutes
allowing us to analyze the frequency range up to the Nyquist frequency at
about 4 167 $\mu$Hz. Given that the TESS USC data, with 20-sec sampling time,
became available recently for 3 of the analyzed stars, we have included the
analysis of the USC data when available. The Nyquist frequency of the USC data
is at about 25 000 $\mu$Hz which permits to analyse the short period range of
the pulsation spectra. The light curves were processed by the Science
Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016), which is
based on the Kepler Mission science pipeline and made available by the NASA
Ames SPOC center and at the MAST archive444http://archive.stsci.edu.

We first downloaded the target pixel file (TPF) of interest from the MAST
archive, which is maintained by the Lightkurve Collaboration; Lightkurve
Collaboration et al. (2018). The TPFs include an 11x11 postage stamp of pixels
from one of the four CCDs per camera that the target is located on. The TPFs
are examined to determine the amount of crowding and other potential bright
sources near the target. We used TPFs to optimize the aperture if it was
needed. In most of the cases we have used the pipeline aperture as it gave the
most optimal result with respect to signal-to-noise ratio. Given that the
pixel size of TESS is huge, 21 arcsec, we need to pay special attention to the
possible contamination. The contamination factor is indicated with the keyword
$\tt{CROWDSAP}$, which gives the ratio of the target flux to the total flux in
the TESS aperture. For the each target, we have checked the contamination by
looking at $\tt{CROWDSAP}$ parameter which is listed in Table LABEL:Table_FT.
For the 3 stars (TIC 260795163, TIC 080290366 and TIC 138707823) that were
also observed during the extended mission we give the relevant parameters in
parenthesis in Table LABEL:Table_FT. For 4 targets, TIC 080290366, TIC
020448010, TIC 138707823 and TIC 415339307, the $\tt{CROWDSAP}$ value is
higher than 0.9, which implies that less than 10% of the total flux originally
measured in the TESS aperture comes from other unresolved sources. For TIC
260795163, the $\tt{CROWDSAP}$ is smaller than 0.7 (for SC mode observations)
and almost 40% of the flux comes from the other background sources. Since the
difference in magnitude between the target and nearby object is more than 4,
we can safely conclude that the flux variations indeed come from the sdB star.
For the extended mission, the $\tt{CROWDSAP}$ is much better for TIC 260795163
with 0.81 and for TIC 080290366 and TIC 138707823 it is the same as for the SC
data, i.e., bigger than 0.9.

We have generated the light curves by integrating the fluxes (’PDCSAP
FLUX’555The pre-search data conditioning (PDC) flux, which corrects the simple
aperture photometry (SAP) to remove instrumental trends.) within the aperture
mask as a function of time in barycentric corrected Julian days (’BJD -
2457000’). After that, we removed the outliers that vary significantly from
the local standard deviation ($\sigma$) by applying a running $5\sigma$
clipping mask. Then, we detrended the light curves to remove any additional
low-frequency systematics that may be present in the data. To do this, we
applied Savitzky–Golay filter with a three-day window length computed with the
Python package lightkurve. Detrending with this method suppressed any spurious
frequencies below 1 $\mu$Hz, which is the typical region where spurious
frequencies are seen in TESS data. We have also examined the light curves
before applying the low-frequency fitting in order to search for any potential
binary signals that might be affected by detrending. However, we did not find
binary signatures in the FT of any of the 5 stars. The fluxes were then
normalized and transformed to amplitudes in parts-per-thousand (ppt) unit
(($\Delta I$/$I-1)\times 1000$). For TIC 260795163, which was observed during
two consecutive sectors in USC mode, we have combined the light curves.

###  3.1 Frequency analysis

The Fourier transforms (FT) of the light curves were computed to examine the
periodicities present in the data, aiming at identifying the frequency of all
pulsation modes, along with their amplitude and phase.

We adopt a rather conservative detection threshold of $0.1$ per cent false
alarm probability (FAP), which means that if the amplitude reaches this limit,
there is a $0.1$ percent chance that it is just result of noise fluctuations.
We calculated the $0.1\%$ FAP threshold following the method described in
Kepler (1993).

The temporal resolution of the data is around 0.6 $\mu$Hz (1.5/T, where T is
the data length which is between 24 and 27 d). In Table LABEL:Table_FT, we
listed all relevant information regarding the Fourier transform including
number of data points and $0.1\%$ FAP level of each dataset.

For all the peaks that are above the accepted threshold and up to the
frequency resolution of the particular dataset, we have performed a nonlinear
least square (NLLS) fit in the form of $A_{i}\sin(\omega_{i}\ t+\phi_{i})$,
with $\omega=2\pi/P$, where $P$ is the period. In this way, we determined the
values of frequency (period), phase and amplitude corresponding to each
periodicity. Using the parameters of NLLS fit, we have prewhitened the light
curves until no signal above the $0.1\%$ FAP level was left in the FT of each
star unless there were unresolved peaks. For all frequencies that still had
some signal left above the threshold after prewhitening we carefully checked
if there was a close-by frequency within the frequency resolution and in such
case only the highest amplitude frequency was fitted and prewhitened, as shown
in Fig. 3 and 5. All prewhitened frequencies for each of the 5 stars are given
in Table LABEL:260795163, LABEL:080290366, LABEL:020448010, LABEL:138707823
and LABEL:415339307, showing frequencies (periods) and amplitudes with their
corresponding errors and the S/N ratio. The Fourier transforms of the
prewhitened light curves of all 5 analysed stars are shown in Figures 13, 14,
15, 16 and 17.

For all 5 stars analyzed in this paper, a total of 73 frequencies were
extracted from their light curves. The detected frequencies are distributed in
a narrow region between 68 $\mu$Hz and 315 $\mu$Hz. This corresponds to the
g-mode region seen in V1093 Her type sdB pulsators (e.g. Reed et al. 2018).
The amplitude spectra of all 5 stars are shown in Fig. 1, where we also give
atmospheric parameters derived in Section 4 for each star. It has been
recently reported by Reed et al. (2020a) that there is a correlation between
effective temperature and the frequency of the highest amplitude of g-modes
detected in V1093 Her type sdB pulsators observed by Kepler and K2. As can be
seen in Fig. 1, the five g-mode sdB pulsators analyzed in this paper do not
deviate from this finding.

Table 3: The parameters of Fourier Transforms of g-mode sdB pulsators from this work. Columns 1 to 7 correspond to the TESS input catalog number, $Gaia$ magnitudes, duration of each dataset, the $\tt{CROWDSAP}$ keyword, the number of data points, data resolution and the significance level, respectively. In parenthesis, the details from the extended mission observations are given. For more details, see the text. TIC | $G_{mag}$ | Observations [d] | CROWDSAP | N.data | Resolution [$\mu$Hz] |  $0.1\%$ FAP
---|---|---|---|---|---|---
260795163 | 12.56 | 27.88 (49.39) | 0.61 (0.81) | 18099 (192385) | 0.623 (0.352) | 0.467 (0.278)
080290366 | 12.38 | 27.40 (24.25) | 0.93 (0.94) | 18312 (88731) | 0.634 (0.716) | 0.36 (0.325)
020448010 | 12.77 | 24.20 | 0.96 | 15946 | 0.717 | 0.429
138707823 | 13.27 | 27.40 (23.85) | 0.99 (0.99) | 18317 (85563) | 0.634 (0.728) | 0.563 (0.66)
415339307 | 14.15 | 25.99 | 0.95 | 17660 | 0.668 | 1.270
Figure 1: Fourier transform of all 5 sdB stars observed in single SC sectors,
concentrating on on the g-mode region of the frequency spectrum. The panels
are sorted with decreasing effective temperature from top to bottom. In each
panel we give the TESS input catalog number, effective temperature (in Kelvin)
and surface gravity (in dex), respectively. The horizontal red lines
correspond to $0.1\%$ FAP confidence level.

TIC 260795163 ($T_{mag}=11.73$) was observed in SC during sector 1 between
July 25 and August 22 2018. The observations yielded 18 099 data points with
the temporal resolution of 0.62 $\mu$Hz (1.5/T, where T is 27.87 d). From
sector 1 data, we detected 12 frequencies above $0.1\%$ FAP confidence level,
which corresponds to 0.467 ppt. The median of noise level including the entire
FT is equal to 0.09 ppt. The signal-to-noise level of detected frequencies
ranges from 4.75 to 30.87.

TIC 260795163 was also observed during two consecutive sectors of the extended
mission. These observations started on 04 July 2020 and ended on 26 August
2020. From this 49 day long dataset, we calculate the FT of the USC data up to
Nyquist frequency of 25 000 $\mu$Hz. The frequency resolution of USC data is
0.35 $\mu$Hz. The average noise level of the entire FT of the USC dataset is
0.055 ppt and the $0.1\%$ FAP threshold level is 0.26 ppt. All frequencies
with amplitudes above this threshold are concentrated in a narrow region
between 100 and 320 $\mu$Hz, which is almost identical to what we detected
from the SC observations. Beyond 320 $\mu$Hz, there is no peak detected above
the 0.1 % FAP threshold up to the Nyquist frequency. We find only one peak
reaching the 4.5 $\sigma$ level at 23 975.8 $\mu$Hz. However, this frequency
seems too high to be excited in an sdB pulsator.

Concerning the g-mode region, we extracted 23 significant frequencies from
both sector 1 and the extended mission (sector 27 and 28) dataset. The 13
frequencies that are found in both sectors (sector 1 and sector 27+28) are
tabulated in Table LABEL:260795163 and marked with $\dagger\dagger$. We found
7 frequencies in the extended mission dataset, which were not detected in
sector 1. These frequencies are also tabulated in Table LABEL:260795163 and
marked with $\dagger$. As can be seen in the Fig. 2, there are several
frequencies above the threshold level in the extended mission (sector 27 and
28) which are not detected in sector 1 (see Table LABEL:260795163 frequencies
tagged with $\dagger$).

The amplitude spectrum of this object is dominated by a number of frequencies
between 100 and 315 $\mu$Hz as it is shown in Fig.2. The top panel presents
the amplitude spectrum from sector 1, while the bottom panel shows the
amplitude spectrum from sectors 27 and 28. In the lower frequency region, the
four peaks (f1, f2, f3 and f8) were detected in sectors 27 and 28. The
frequency f5 is within 4.5 $\sigma$ level in 120-sec cadence data and in the
USC data it is becoming a significant peak above 0.1% FAP level. However,
there is a frequency, f4, that is detected only in sector 1, with a S/N of
4.75, while it is not detected in sectors 27 and 28. We recognized that there
are residuals in the FT of the USC data after prewhitening. These residuals
are shown in Fig. 3. The strong residuals at f13 and f23 could be either due
to unresolved close-by frequencies or due to amplitude/frequency/phase
variations over the length of the data. For these residuals, we did not
prewhiten further. Overall, combining the nominal and extended mission
dataset, we detect 23 g-modes spanning from 127 to 315 $\mu$Hz.

Figure 2: Top: Fourier transform of sector 1 of TIC 260795163. The horizontal
red line indicates the 0.1% FAP level. The orange line is the residual after
extraction of the signals. Bottom: Fourier transform of sector 27 and 28 of
TIC 260795163. The horizontal red line indicates the 0.1% FAP level. The
orange line is the FT of the prewhitened light curve.  Figure 3: Fourier
transform of USC data of TIC 260795163. The first raw shows the window
function calculated from USC data. The second and third panels from the top
display two different frequency regions in which the signals show strong
residual after the extraction. These frequencies are given in Table
LABEL:260795163. The horizontal red line indicates the 0.1% FAP level. The
orange line is the FT of the prewhitened light curve.

TIC 080290366 was found to be a pulsating star by Koen & Green (2010), who
detected 5 oscillation frequencies ranging from 127 $\mu$Hz to 233 $\mu$Hz.
TIC 080290366 ($T_{mag}=11.85$) was observed in SC mode during sector 2
(2018-Aug-22 to 2018-Sep-20) for 27.4 d, with a frequency resolution of 0.63
$\mu$Hz. Other parameters such as the length of the observations,
contamination, number of data points and $0.1\%$ FAP confidence level are
given in Table LABEL:Table_FT. From sector 2, we detected 16 frequencies
between 81 and 257 $\mu$Hz.

TIC 080290366 was also observed during the extended mission in sector 29
(2020-Aug-26 to 2020-Sep-22). The data length of sector 29 is 24.3 d, implying
a lower frequency resolution of 0.72 $\mu$Hz. The FT average noise level is
0.067 ppt. The $0.1\%$ FAP confidence level is 0.316 ppt. We did not detect
any high-frequency p-mode from the USC observations, while we detected 17
g-mode frequencies, most of them being present already in sector 2. All the
frequencies that were detected in both the nominal and extended mission are
listed in Table LABEL:080290366. In total we have detected 18 frequencies
spanning from 81.6 $\mu$Hz ($\sim$12 200 s) to 257.4 $\mu$Hz ($\sim$3 900 s)
with amplitudes between 0.3 and 2.5 ppt. Fifteen frequencies are detected in
both datasets, the frequency near 141.7 $\mu$Hz was detected only in sector 2,
while the three frequencies near 90.9, 114.7 and 168.0 $\mu$Hz were found only
in sector 29.

The FTs of sector 2 (upper panel) and sector 29 (lower panel) are shown in
Fig. 4, where several frequencies clearly show differences. In Fig. 5, we show
the amplitude spectrum of sector 29 in more detail around the regions in which
an excess of power is left after prewhitening, compared with the window
function.

Figure 4: Top: Fourier transform of data taken in sector 2 of TIC 080290366.
The horizontal red line indicates the 0.1% FAP level. The orange line is the
residual after extraction of the signals. Bottom: Fourier transform of data
taken in sector 29 of TIC 080290366. The horizontal red line indicates the
0.1% FAP level. The orange line is the FT of the prewhitened light curve.
Figure 5: The Fourier Transform of sector 29 of TIC 080290366, compared with
the window function (top panel). The lower panels show an excess of power
after prewhitening (orange line), see Section 3.3 for more details. The
horizontal red line indicates the $0.1\%$ FAP level.

TIC 020448010 ($T_{mag}=12.18$) was observed in SC mode during sector 9 for a
period of 24.2 d, which provides a frequency resolution of 0.717 $\mu$Hz. The
FT average noise level is 0.088 ppt. The rest of the parameters, including
length of the observations, contamination, number of data points and $0.1\%$
FAP confidence level, are given in Table LABEL:Table_FT. The star was
discovered by TESS as a g-mode sdB pulsator with 15 frequencies concentrated
in a narrow region ranging from 68 $\mu$Hz to 187 $\mu$Hz abd with amplitudes
between 0.36 ppt and 1.26 ppt. The extracted frequencies are listed in Table
LABEL:020448010 with their associated errors and S/N ratio. In Fig. 15, we
show all detected frequencies (light grey) and residuals (orange) after
prewhitening.

TIC 138707823 ($T_{mag}$=12.7) was observed in SC mode during sector 2 between
August 22 and September 20, 2018 for 27.4 days. From these SC observations of
TIC 138707823, we have extracted 7 periodicities from the light curve. All
frequencies above $0.1\%$ FAP significance level of 0.563 ppt are listed in
Table LABEL:138707823. The frequencies are located in a narrow range from 87
to 237 $\mu$Hz.

TIC 138707823 was also observed during Sector 29 (2020-Aug-26 to 2020-Sep-22)
with USC mode. The length of these observations (23.85 d) is almost 4 days
shorter than the sector 2 dataset, resulting in somewhat worse frequency
resolution (0.728 $\mu$Hz) than the one obtained for the SC dataset (0.634
$\mu$Hz). The average noise level of the FT is 0.13 ppt. We extracted 3
significant frequencies above the threshold of 0.62 ppt. These 3 frequencies,
which were also detected in sector 2, are marked with $\dagger\dagger$ in
Table LABEL:138707823. The four frequencies at 87, 132, 181 and 191 $\mu$Hz,
which were not detected during the extended mission, are given without symbol
in Table LABEL:138707823.

The amplitude spectrum of the sector 2 data of TIC 138707823 is relatively
poor in comparison with the four stars presented in this work, displaying only
7 frequencies above the threshold level. These seven frequencies can be seen
in the upper panel of Fig. 16. The bottom panel of the same figure shows only
3 significant frequencies from sector 29. Combining the results from sector 2
and 29, we detect 7 frequencies which are concentrated between 87 $\mu$Hz and
237 $\mu$Hz.

Figure 6: Top: Fourier transform of sector 2 data of TIC 138707823. The
horizontal red line indicates the 0.1% FAP level. The orange line is the
residual after extraction of the signals. Bottom: Fourier transform of sector
29 data of TIC 138707823. The horizontal red line indicates the 0.1% FAP
level. The orange line is the FT of the prewhitened light curve.

TIC 415339307 ($T_{mag}=14.2$) was observed by TESS in SC mode during sector 5
between 15 November 2018 and 11 December 2018, covering about 26 days. The
amplitude spectrum of this object, shown in Fig. 17, contains 9 frequencies
above the $0.1\%$ FAP significance level of 1.27 ppt. The average noise level
of the FT is 0.26 ppt. We note that there is a frequency at 184.122 $\mu$Hz
(f3) just below the $0.1\%$ FAP confidence level, albeit at 4.5 $\sigma$,
which we keep as a candidate frequency and discuss it’s nature in section 3.3.

The photometric and FT parameters, including average noise level,
contamination factor, number of data points and $0.1\%$ FAP confidence level,
are given in Table LABEL:Table_FT. In Table LABEL:415339307, we list all (10)
frequencies (periods) and their amplitudes with corresponding errors and we
show all detected frequencies in Fig.17.

###  3.2 Rotational multiplets

The main goal of our analysis is to identify modes of detected pulsations in
order to constrain theoretical models of pulsating sdB stars. For rotating
stars, the existence of non-radial oscillations allows identification of the
pulsation modes via rotational multiplets (Aerts et al. 2010). The non-radial
pulsations are described by three quantized numbers, $n,l$ and $m$, where $n$
is the number of radial nodes between center and surface, $l$ is the number of
nodal lines on the surface, and $m$ is the azimuthal order, which denotes the
number of nodal great circles that connect the pulsation poles of the star.

In rotating stars, the pulsation frequencies are split into 2$l+$1 azimuthal
components due to rotation, revealing an equally spaced cluster of 2$l+$1
components. This 2$l+$1 configuration can be resolved with high-precision
photometry if the star has no strong magnetic field and the rotational period
is not longer than the duration of the observation.

Detection of rotational splitting is important as it is one of the two methods
used to identify the pulsational modes of a star and at the same time it gives
the information of the rotation period of the star. In particular for the
g-mode pulsators, it unveils the rotation of the deep part of the radiative
envelope close to the convective core.

In several sdBVs rotational multiplets have been detected (Charpinet et al.
2018, references therein). Typical rotation periods detected in sdB stars are
of the order of 40 d (Baran et al. 2019), unless the stars are in close binary
systems.

Even though TESS allows us to obtain uninterrupted time series, especially for
stars observed in multiple sectors, it is not ideally suited for the detection
of rotationally split multiplets in sdBVs. For the stars that have been
observed in just 1 sector ($\sim$27 d), which translates in a frequency
resolution of $\sim$ 0.6 $\mu$Hz, we are limited to the detection of
rotational periods shorter than about 13 days. For each star, we have searched
for a coherent frequency splitting, $\Delta\nu$ in the g-mode region and we
did not find any consistent solution. Therefore, we conclude that for neither
of the sdBV stars analysed in this paper, it is possible to perform mode
identification and determine rotational period based on rotational splitting.
Given that 4 of the analyzed stars are not members of close binaries, we do
not expect them to have short rotation periods such that they could be
detected in a single sector TESS data. For TIC 138707823 however, which is a
short period binary system with an orbital period of about 4 d (Edelmann et
al. 2005), we do not detect any significant signal that might be attributed to
this orbital period.

###  3.3 Frequency and amplitude variations

From the continuous light curves produced by the space missions such as Kepler
and K2 and now TESS, it has been observed that oscillation frequencies in
compact stars, including pulsating sdB stars, DBV and DOV pulsating white
dwarfs, may not be stable (Silvotti et al. 2019; Zong et al. 2016b; Córsico et
al. 2020). It is known that the frequency and amplitude variations mostly
occur due to beatings of unresolved peaks or unresolved multiplets. Recently,
the complex patterns that have been observed were interpreted as evidence of
frequency/amplitude/phase modulations due to weak non-linear mode
interactions, as discussed in Zong et al. (2016a). Furthermore, the
variability may occur due to the photon-count noise caused by contamination of
the background light in the aperture.

The continuous photometric measurements of 5 stars allow us to construct
sliding FTs (sFTs) to examine the temporal evolution of the detected
frequencies over the course of the TESS observations. Therefore, we have
computed and examined the sFT of each target. Since three targets were
observed in more than one sector, for these stars we selected the sectors in
which we see the largest number of pulsation modes, i.e. sectors 27 and 28 for
TIC 260795163, sector 29 for TIC 080290366, sector 2 for TIC 138707823.

The sFTs are computed in a similar way as described in Silvotti et al. (2019):
we run a 5-day sliding window with a step size of 0.2 days. The amplitudes are
shown as color-scale in ppt units. We set a lower limit on the amplitudes by
running the 3 times average noise level of 5-d chunk. Afterwards, we calculate
the Fourier transform of each subset and trail them in time.

In Fig. 7, 8, 9, 10 and 11 we show the sFTs for the g-mode region of each
star. In most of the cases, the high amplitude frequencies (S/N $\geq$ 10) are
stable in both frequency and amplitude over the length of the data for all
stars. However, some pulsational frequencies are not stable throughout the
TESS run and here we discuss each case.

In the case of TIC 260795163, the highest amplitude frequencies (at 250.447,
268.360, 288.186 and 315.537 $\mu$Hz) are stable in frequency, although the
one at 250 $\mu$Hz shows a small wobble. A few frequencies, mostly lower than
250 $\mu$Hz, are not stable, at least in amplitude. However, they have a low
S/N and therefore are below the detection threshold throughout part of the
run. In the case of the frequency at 293.996 $\mu$Hz (f22), it is quite stable
up to day $\sim$2050 and then it becomes weaker in amplitude. Indeed, we can
see this effect in the bottom panel of Fig. 3, in which we show the strong
residual after prewhitening, implying that either the amplitude, phase or
frequency could be variable over the length of the data.

In the case of TIC 080290366 (see Fig. 8), the amplitudes of the frequencies
at 196.144 (f14) and 228.658 $\mu$Hz (f17) are variable and it is exactly
these two frequencies that show residuals after prewhitening in Fig. 5.

The highest amplitude frequency at 105.313 $\mu$Hz (f5) of TIC 020448010 is
quite stable over the run, along with the frequency at 133.516 $\mu$Hz (f10)
(Fig. 9). The low-amplitude frequencies are mostly unstable in amplitude, e.g.
the frequency at 83.096 $\mu$Hz is absent between days 1551 and 1555.

In the case of TIC 138707823 (Fig. 10), we safely extracted 7 peaks from the
SC observations and all the frequencies are visible in the sFT except for the
frequency at 237.895 $\mu$Hz, whose S/N is low. The highest amplitude
frequency at 151.497 $\mu$Hz is stable over the length of the data, while the
rest of the frequencies (87.82, 132.67, 173.09, 181.38 and 191.02 $\mu$Hz) are
not stable in amplitude, while they seem to be stable in frequency.

In the case of TIC 415339307 (Fig. 11), the frequency at 175.929 $\mu$Hz shows
a strong variability in amplitude, being at very low S/N in the first half of
the run while in the second half it has the highest amplitude of about 4 ppt.
In Fig. 17, this effect is seen as a significant residual at 5684 s.
Frequencies at 203.126 and 214.359 $\mu$Hz are not stable in either frequency
nor amplitude. In the case of the frequency at 184.122 $\mu$Hz (f3) (S/N =
4.45 in the entire FT), it is definitely present in the sFT, however it is not
stable in either frequency nor amplitude during the first half of the run,
while it appears more stable in frequency during the second half of the run.
The frequency at 255.863 $\mu$Hz is stable up to almost 1448 d, while it is
absent beyond 1448 d. Since this frequency is above the $0.1\%$ FAP level in
the FT, we include it in our analysis as well.

We have not found strong evidence of rotational splitting in any of the
targets analysed in this paper, and hence all of them must have rotation
periods considerably longer than the length of their TESS dataset.
Consequently, any peak we have observed must be considered an unresolved
multiplet, consisting of a summation of 3, for $l=1$, or more sinusoids with
independent phase and amplitude, and with each sinusoid having slightly
different unresolved frequency. This produces beating effect on time scales
longer than the analysed TESS dataset, which then may appear as any form of
frequency and/or amplitude variation. Beating of unresolved multiplets is
therefore the default cause of any such variations observed in datasets that
are too short to reveal rotational splitting.

Figure 7: The sliding Fourier Transform (sFT) of TIC 260795163 from USC data.
The color-scale amplitude is given in ppt units. See text for more details on
sFT computations.  Figure 8: The sFT of TIC 080290366. The sFT is calculated
from USC data. Figure 9: The same figure as Fig. 8 but for TIC020448010. The
sFT is calculated from SC data. Figure 10: The same figure as 8 but for TIC
138707823. The sFT is calculated from SC data. Figure 11: The same figure as 8
but for TIC 415339307. The sFT is calculated from SC data.

###  3.4 Asymptotic g-mode period spacing

Fontaine et al. (2003) showed that the oscillation modes detected in long-
period pulsating subdwarf B stars are associated with high-order g-modes. In
the asymptotic limit, when n >> l the consecutive radial overtones of high-
order g-modes are evenly spaced in period such that the consecutive g-modes
follow the equation:

$P_{l,n}=\frac{\Delta P_{0}}{\sqrt{l(l+1)}}n+\epsilon_{l}$ (1)

where $\Delta P_{0}$ is just the asymptotic period spacing for g-modes, which
is defined as $\Delta P_{0}\propto[\int_{r_{1}}^{r_{2}}\frac{|N|}{r}dr]^{-1}$,
$N$ being the Brunt–Väisälä frequency, the critical frequency of non-radial
g-modes (Tassoul 1980) and $\epsilon_{l}$ is a constant (Unno et al. 1979).

The existence of a nearly constant period spacing of g-modes in the asymptotic
regime means that we can search for the patterns of modes with a given
harmonic degree in the observed period spectra of pulsating stars. This
method, so called the asymptotic period spacing method, has been used to
identify the degree of pulsational modes in many g-mode pulsators such as
$\gamma$-Dor stars, slowly pulsating B stars (SPBs) and white dwarfs (Aerts
2019, and references therein).

However, only with the onset of space based light curves such as $Kepler$,
this method became plausible for identifying the degrees of the pulsational
modes in sdB stars (Reed et al. 2011, e.g.). However, with this method it is
not possible to determine the absolute radial order n of a given mode from the
equation (1) without detail modelling. What we can do is choose a relative
number for a radial order ($n_{l}$) corresponding to a modal degree $l$ such
that all other consecutive radial orders for this modal degree will have an
offset with respect to $n_{l}$. The ratio between consecutive overtones is
then derived from the equation (1), such that the ratio for dipole (l = 1) and
quadrupole (l = 2) modes is $\sqrt{3}$. Based on the theoretical models by
Charpinet et al. (2000), the period spacing of l = 1 modes is around 250 s for
sdB pulsators. The Kepler and K2 observations of g-mode pulsating sdB stars
find that the average period spacing for $l=1$ modes is ranging from 227 s to
276 s (Reed et al. 2018). Recent results from the TESS mission on five g-mode
sdB pulsators find similar results, with the average period spacing ranging
from 232 s to 268 s (Charpinet et al. 2019; Sahoo et al. 2020a; Reed et al.
2020a). Here, we search for constant period spacing in our 5 target stars
observed by TESS using the Kolmogorov-Smirnov (K-S; Kawaler 1988) and the
inverse variance (I-V; O’Donoghue 1994) significance tests.

In the K-S test, $Q$ is the quantity that defines the probability distribution
of observed modes. If the data consist of non-random distribution in the
period spectrum, then the distribution will have a peak at minimum value of
$Q$. In the I-V test, on the other hand, a maximum of the inverse variance
will indicate a consistent period spacing.

In Figure 12, we show the K-S (top panel) and I-V (bottom panel) results
obtained for 5 stars. For both panels, we applied a vertical, arbitrary
offset, for a visualization purpose. The same color coding is applied for both
panels. For all targets, the statistical tests display a clear indication of
the mean period spacing at around 250 s. For the case of TIC 260795163 and TIC
080290366, the I-V test is not as conclusive as for the other 3 stars.
However, the K-S test does show an indication for a possible mean period
spacing of $l=1$ at 250 s. Moreover, TIC 260795163 and TIC 080290366 show
peaks at around 150 s referring to the possible period spacing of quadrupole
modes.

Based on the potential period spacings obtained from the K-S and I-V tests, we
search for the sequences of dipole and quadrupole modes in the entire FT of
each star. First, we search for the $l=1$ and $l=2$ sequences examining
consecutive modes in period domain of the FT of each star. Given that the
higher degree modes are more sensitive to geometrical suppression due to mode
cancellation effect (Aerts et al. 2010), we assume that the highest amplitude
frequencies are corresponding to low-degree modes ($l=1$ and $l=2$). This
assumption is valid only if all the modes have the same intrinsic amplitude.
For sdB pulsators, the majority of detected frequencies have been identified
with low-degree modes ($l\leq$ 2). However, a few exceptional examples have
been reported. For instance, in two sdB pulsators observed during the nominal
mission of Kepler, the high-degree g-modes were assigned up to $l=6$ (Kern et
al. 2018) or $l=8$ (Telting et al. 2014) using the method of rotational
multiplets. Silvotti et al. (2019) also reported several high-degree modes up
to $l=12$ using solely asymptotic period spacing in the brightest ($V=10.2$)
sdB pulsator HD 4539 (EPIC 220641886) observed during the K2 mission. We
proceed in the following way: we assign arbitrary radial orders ($n$) for each
identified $l$ degree and calculate the mean period spacing using a linear
regression fit. In this way we have calculated the mean period spacing
($\Delta P$) for all stars and found that the mean period spacing for the
$l=1$ mode is ranging from 251 s and 255 s. In order to assess the errors of
the mean period spacing obtained in our analysis, we perform a bootstrap
resampling analysis as described by Efron (1979); Simpson & Mayer-Hasselwander
(1986). We used this method because many possible modes are not detected in
the amplitude spectra. Furthermore, in some cases the individual pulsational
period has no unique modal degree solution, and in some specific cases the
modes could be altered due to mode trapping.

In order to make a realistic error assignment, we simulated $10^{4}$ datasets
from the determined $l$ modes. For each target, we created sets of randomly
chosen observed periods that are already identified as $l=1$ or $l=2$ modes,
in order to obtain the mean period spacing from each different sub-sequences.
The same data point can occur multiple times and ordering is not important,
such that for $N$ data points the total number of possible different bootstrap
samples is $\frac{(2N-1)!}{N!(N-1)!}$ (Andrae 2010). For instance, for a given
dipole sequence, which consists of 10 modes, the total number of potential
different bootstrapping includes about $10^{5}$ subsets. For 5 stars analyzed
in this paper, we detected the dipole sequences that range from 7 to 13 modes
depending on a star. For that reason, we restricted ourselves with $10^{4}$
subsets. For each such subset which includes a series of dipole or quadrupole
modes, we derive the mean period spacing with the linear regression fit. The
most probable solution is obtained as a mean period spacing (which corresponds
to the 50 percentile of the $\Delta P_{l=1}$ distribution). The errors are
then estimated as 1 $\sigma$ and given in Table LABEL:Table_Seismic.

The right panels of Figure 13, 14, 15, 16 and 17 show the residuals between
the observed periods and the periods derived from the mean period spacing for
the $\Delta P_{l=1}$ in where we can see the deviation of the modes. The
scatter of the residuals for all stars is up to 50 s and for the stars that we
have more $l=1$ modes detected, we notice the oscillatory pattern which is a
characteristic feature that was found in several V1093 Hya type sdB pulsators
(Telting et al. 2012; Baran 2012, e.g.). Detecting all modes of $l=1$ sequence
with an expected period spacing of 250 s (Charpinet et al. 2002) is unlikely
as the sdB stars are chemically stratified, which causes the observed modes to
be scattered around this value. Such small deviations from the mean period
spacing are to be expected in those stars where diffusion processes have had
enough time to smooth out the H-He transition zone (Miller Bertolami et al.
2012). On the other hand, the efficiency of trapping diminishes with
increasing the radial order as discussed in Charpinet et al. (2014). This is
because the local wavelength of the modes decreases with increasing the radial
order and therefore, the higher order g-modes become less affected by the H-He
transition zone. In this way the higher order g-modes may present the nearly
constant period spacing even without having the H-He transition zone smoothed.

Figure 12: Kolmogorov-Smirnov (top panel) and Inverse-Variance (lower panel)
tests to search for a constant period spacing in 5 stars. The offset on
$log\,Q$ for the Kolmogorov-Smirnov is 6 and the offset on y-axis for the
Inverse-Variance test is 80. All stars present a significant probability for a
constant period spacing.

####  3.4.1 TIC 260795163

The K-S test for TIC 260795163 shows an indication for both dipole and
quadrupole modes, see the upper panel of Fig. 12. Assuming that the highest
amplitude frequency at 288.186 $\mu$Hz (f21 in Table LABEL:260795163) is an
$l=1$ mode, then the sequence of $l=1$ mode is fulfilled with the following
frequencies: f10, f14, f17 and f19. Beyond 6000 s, 6 additional frequencies
(f1 to f6) were found to fit the dipole sequence.The following six
frequencies: f8, f9, f11, f15, f16 and f20 showed consistent period spacing
for quadrupole modes with a unique solution. For 4 frequencies we could not
find a unique solution, namely f4, f6, f10 and f21, and they could be
interpreted either as $l=1$ or $l=2$ modes.

Out of the total of 23 periodicities detected in TESS data of TIC 260795163,
there were 5 frequencies that did not fit neither $l=1$ nor $l=2$ sequences.
These periodicities could be either higher-degree modes or $l=1$ and/or $l=2$
modes which are severely affected by mode trapping and do not fit the $l=1$
and $l=2$ patterns. However, without detecting rotational multiplets, it is
impossible to test any of the two possibilities. The amplitude spectra of TIC
260795163 with our mode identification are presented in Fig. 13. Based on this
mode identification, we calculate the mean period spacing of dipole modes,
$\Delta P_{l=1}$ = $254.83^{+2.14}_{-2.28}$ s and quadrupole modes, $\Delta
P_{l=2}$ = $150.51^{+3.27}_{-2.28}$ s.

Table 4: Frequency solution from the _TESS_ light curve of TIC 260795163 including frequencies, periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio. Errors are given in parenthesis to 2 significant digits. Identified modal degree and relative radial orders are listed column 5 and 6, respectively. The frequencies detected in both sector 1 and the extended mission of sector 27 and 28 are tagged with $\dagger\dagger$. The frequencies detected only in sector 27 and 28 are tagged with $\dagger$. The frequencies that are detected only in sector 1 are given without tag. See text for more details on mode identification. ID | Frequency | Period | Amplitude | S/N | $l$ | $n$ |
---|---|---|---|---|---|---|---
|  $\mu$Hz | [sec] | [ppt] |  |  |  |
f${}_{\rm 1^{\dagger}}$ | 127.915 (20) | 7817.63 (1.27) | 0.269(44) | 4.84 | 1 | 30 |
f${}_{\rm 2^{\dagger}}$ | 132.358 (14) | 7555.25 (81) | 0.394(44) | 7.09 | 1 | 29 |
f${}_{\rm 3^{\dagger}}$ | 137.178 (16) | 7289.77 (85) | 0.349(44) | 6.28 | 1 | 28 |
f4 | 147.752 (36) | 6768.08 (1.66) | 0.475 (76) | 4.75 | 1/2 | 26/44 |
f5 | 159.298 (10) | 6277.541 (35) | 0.26 (43) | 4.74 | 1 | 24 |
f${}_{\rm 6^{\dagger\dagger}}$ | 165.001 (10) | 6060.55 (37) | 0.554(44) | 9.98 | 1/2 | 23/39 |
f${}_{\rm 7^{\dagger}}$ | 166.194 (14) | 6017.04 (53) | 0.394(45) | 7.09 |  |  |
f8 | 207.634 (10) | 4816.172 (20) | 0.28 (43) | 5.17 | 2 | 31 |
f${}_{\rm 9^{\dagger\dagger}}$ | 213.093 (08) | 4692.77 (18) | 0.657(44) | 11.83 | 2 | 30 |
f${}_{\rm 10^{\dagger}}$ | 221.251 (14) | 4519.75 (29) | 0.392(44) | 7.06 | 1/2 | 17/29 |
f${}_{\rm 11^{\dagger}}$ | 228.75 (12) | 4371.58 (23) | 0.465(44) | 8.36 | 2 | 28 |
f${}_{\rm 12^{\dagger\dagger}}$ | 230.251 (15) | 4343.08 (29) | 0.356(44) | 6.41 |  |  |
f${}_{\rm 13^{\dagger\dagger}}$ | 231.428 (13) | 4320.98 (24) | 0.428(44) | 7.71 |  |  |
f${}_{\rm 14^{\dagger\dagger}}$ | 235.444 (08) | 4247.28 (14) | 0.683(44) | 12.28 | 1 | 16 |
f${}_{\rm 15^{\dagger\dagger}}$ | 237.532 (10) | 4209.94 (18) | 0.536(44) | 9.64 | 2 | 27 |
f${}_{\rm 16^{\dagger\dagger}}$ | 245.960 (08) | 4065.69 (14) | 0.654(44) | 11.77 | 2 | 26 |
f${}_{\rm 17^{\dagger\dagger}}$ | 250.447 (04) | 3992.84 (06) | 1.349(44) | 24.27 | 1 | 15 |
f${}_{\rm 18^{\dagger}}$ | 254.972 (18) | 3921.98 (28) | 0.308(44) | 5.54 |  |  |
f${}_{\rm 19^{\dagger\dagger}}$ | 268.360 (03) | 3726.32 (04) | 1.624(44) | 29.20 | 1 | 14 |
f${}_{\rm 20^{\dagger\dagger}}$ | 275.588 (16) | 3628.59 (22) | 0.335(44) | 6.03 | 2 | 23 |
f${}_{\rm 21^{\dagger\dagger}}$ | 288.186 (03) | 3469.97 (03) | 1.716(44) | 30.87 | 1/2 | 13/22 |
f${}_{\rm 22^{\dagger\dagger}}$ | 293.996 (12) | 3401.44 (11) | 1.469 (76) | 14.70 |  |  |
f${}_{\rm 23^{\dagger\dagger}}$ | 315.537 (05) | 3169.19 (05) | 1.043(44) | 18.77 | 2 | 20 |
Figure 13: The FT of TIC 260795163 (presented here are the sectors 1, 28 and
29 together). The residuals after prewhitening are shown with orange color.
The red dots are showing the dipole modes and the blue ones are displaying the
quadrupole modes. If there is no unique identification, the period is shown
with both colors. The horizontal red line correspond to $0.1\%$ FAP confidence
level. The right panel shows the residuals between the observed and the fitted
periods. See text for more details on mode identification and the mean period
spacing computations.

####  3.4.2 TIC 080290366

Based on the results from the K-S and I-V tests in which there is a peak at
around 255 s, indicating a possible $\Delta P_{l=1}$ and also a peak at 155 s
indicating a possible $\Delta P_{l=2}$ (see upper and lower panel of Fig. 12)
we searched for dipole and quadrupole sequences in the FT of TIC 080290366.
Starting from the highest amplitude frequency at 149.133 $\mu$Hz (f7 in Table
LABEL:080290366) as an $l=1$ mode, we find two frequencies that fit the dipole
mode sequence, f8 and f9. Beyond 7 500 s, there are 5 frequencies which are
also following dipole mode sequences: f1, f2, f3, f4 and f5. We found only 3
frequencies that uniquely fit the $l=2$ sequence. For 5 frequencies (f5, f8,
f9, f14 and f17) the solution is degenerate as the frequencies fit both dipole
and quadrupole mode sequences. Out of 18 detected frequencies there are 4
frequencies that did not fit dipole nor quadrupole mode sequences. These
frequencies could be high-order degree modes or trapped modes. The amplitude
spectra of TIC 080290366 with our mode identification are presented in Fig.
14. Based on the above explained mode identification, we calculated both
$\Delta P_{l=1}$ = $253.32^{+0.78}_{-0.84}$ s and $\Delta P_{l=2}$ =
$154.16^{+2.47}_{-7.97}$ s. The final seismic result for TIC 080290366 is
given in Table LABEL:Table_Seismic.

Table 5: Frequency solution from the _TESS_ light curve of TIC 080290366 including frequencies, periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio. Errors are given in parenthesis to 2 significant digits. Identified modal degree and relative radial orders are listed in column 5 and 6, respectively. The frequencies detected in both sector (2 and 29) are tagged with $\dagger\dagger$. The frequencies detected only in sector 29 are tagged with $\dagger$. The frequencies that are detected only in sector 2 are given without tag. See text for more details on mode identification.  ID | Frequency | Period | Amplitude | S/N | $l$ | $n$ |
---|---|---|---|---|---|---|---
|  $\mu$Hz | [sec] | [ppt] |  |  |  |
f${}_{\rm 1^{\dagger\dagger}}$ | 81.649 (31) | 12247.525 (4.62) | 0.412 (53) | 6.87 | 1 | 47 |
f${}_{\rm 2^{\dagger}}$ | 90.912 (41) | 10999.552 (4.86) | 0.313 (53) | 5.3 | 1 | 42 |
f${}_{\rm 3^{\dagger}}$ | 114.712 (08) | 8717.476 (63) | 1.560 (53) | 26.00 | 1 | 33 |
f${}_{\rm 4^{\dagger\dagger}}$ | 118.217 (28) | 8459.017 (2.06) | 0.452 (53) | 7.54 | 1 | 32 |
f${}_{\rm 5^{\dagger\dagger}}$ | 129.967 (13) | 7694.241 (78) | 0.982 (53) | 16.37 | 1/2 | 29/48 |
f6 | 141.123 (30) | 7086.02 (1.45) | 0.390 (52) | 6.11 | 2 | 44 |
f${}_{\rm 7^{\dagger\dagger}}$ | 149.133 (5) | 6705.43 (22) | 2.452 (52) | 37.02 | 1 | 25 |
f${}_{\rm 8^{\dagger\dagger}}$ | 154.754 (24) | 6461.854 (1.02) | 0.532 (53) | 8.87 | 1/2 | 24/40 |
f${}_{\rm 9^{\dagger\dagger}}$ | 162.579 (10) | 6150.830 (38) | 1.266 (53) | 21.11 | 1/2 | 23/38 |
f${}_{\rm 10^{\dagger\dagger}}$ | 165.129 (10) | 6055.88 (62) | 1.142 (52) | 17.41 |  |  |
f${}_{\rm 11^{\dagger}}$ | 168.027 (02) | 5951.40 (43) | 0.371 (53) | 5.55 |  |  |
f${}_{\rm 12^{\dagger\dagger}}$ | 195.836 (34) | 5106.29 (88) | 1.51 (51) | 22.98 |  |  |
f${}_{\rm 13^{\dagger\dagger}}$ | 196.144 (11) | 5098.29 (30) | 1.03 (52) | 15.63 | 2 | 31 |
f${}_{\rm 14^{\dagger\dagger}}$ | 202.916 (16) | 4928.14 (39) | 0.737 (52) | 11.11 | 1/2 | 18/30 |
f${}_{\rm 15^{\dagger\dagger}}$ | 224.994 (07) | 4444.544 (14) | 1.784 (53) | 29.73 | 2 | 27 |
f${}_{\rm 16^{\dagger\dagger}}$ | 228.658 (09) | 4373.344 (18) | 1.392 (54) | 23.20 |  |  |
f${}_{\rm 17^{\dagger\dagger}}$ | 240.884 (8) | 4151.37 (13) | 1.518 (52) | 22.93 | 1/2 | 15/25 |
f${}_{\rm 18^{\dagger\dagger}}$ | 257.402 (18) | 3884.97 (27) | 0.635 (51) | 9.76 | 1 | 14 |
Figure 14: Same as Fig. 13 but for TIC 080290366 (presented here is sector
29).

####  3.4.3 TIC 020448010

The K-S and IV tests display a clear indication of a period spacing at 253 s
as it is shown in Fig. 12. Starting with the highest amplitude peak at 9 495.5
s (f5 in Table LABEL:020448010), we search for a dipole sequence in the FT.
Out of 15 detected frequencies, 12 frequencies uniquely fit the $l=1$
sequence, while only 2 frequencies can be evaluated as $l=2$ mode as the
difference is corresponding to 2 x $\Delta P_{l=2}$. One periodicity (at
8778.2 s) however, does not have a unique solution, it could be either $l=1$
or $l=2$ mode.

The amplitude spectra of TIC 020448010 together with the mode identification
are presented in Fig. 15. Based on our mode identification we calculate the
average period spacing for dipole modes, $\Delta P_{l=1}$ =
$251.70^{+0.87}_{-0.96}$ s. In Table LABEL:Table_Seismic, we also gave $\Delta
P_{l=2}$ using the ratio between consecutive overtones derived from the
equation (1).

Table 6: Frequency solution from the _TESS_ light curve of TIC 020448010 including frequencies, periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio. Identified modal degree and relative radial orders are listed column 5 and 6, respectively. ID | Frequency | Period | Amplitude | S/N | $l$ | $n$ |
---|---|---|---|---|---|---|---
|  $\mu$Hz | [sec] | [ppt] |  |  |  |
f1 | 68.845(24) | 14525.3(5.1) | 0.731(7) | 8.5 | 1 | 56 |
f2 | 70.203(32) | 14244.3(6.5) | 0.561(7) | 6.5 | 1 | 55 |
f3 | 83.096(46) | 12034.3(6.7) | 0.393(7) | 4.4 | 1 | 46 |
f4 | 95.327(25) | 10490.2(2.7) | 0.742(7) | 8.2 | 1 | 40 |
f5 | 105.313(10) | 9495.5(9) | 1.752(7) | 20.4 | 1 | 36 |
f6 | 113.918(49) | 8778.2(3.7) | 0.358(7) | 4.2 | 1/2 | 33/49 |
f7 | 118.227(28) | 8458.3(2.0) | 0.615(7) | 7.3 | 2 | 48 |
f8 | 121.244(39) | 8247.8(2.7) | 0.442(7) | 5.2 | 1 | 31 |
f9 | 129.282(27) | 7735.0(1.6) | 0.677(7) | 7.7 | 1 | 29 |
f10 | 133.516(14) | 7489.7(8) | 1.260(7) | 14.6 | 1 | 28 |
f11 | 148.722(38) | 6724.0(1.7) | 0.449(7) | 5.3 | 1 | 25 |
f12 | 154.898(21) | 6455.9(9) | 0.865(7) | 9.8 | 1 | 24 |
f13 | 167.668(34) | 5964.2(1.2) | 0.508(7) | 6.0 | 1 | 22 |
f14 | 183.706(22) | 5443.5(6) | 0.798(7) | 9.4 | 1 | 20 |
f15 | 187.244(16) | 5340.64(46) | 1.122(7) | 12.6 |  |  |
Figure 15: Same as Fig. 13 but for TIC 020448010 (presented here is sector 9).

####  3.4.4 TIC 138707823

The K-S and IV tests show a clear indication of a possible $\Delta P_{l=1}$ at
255 s as can be seen in Fig. 12. Assuming the highest peak at 6 600.78 s (f3)
is an $l=1$ mode, we find that the dipole sequence can be completed with f4,
f5 and f6. In the longer period region beyond 6 600 s, there are two more
frequencies with the large gap in between that fit the $l=1$ sequence. We
interpreted these two frequencies as $l=1$ modes and added them to the fitting
procedure in order to find the mean period spacing of $\Delta P_{l=1}$. The
same case is valid for the shorter period region and between f6 and f7 there
is a big gap, of about 1 000 s (which is 4 x $\Delta P_{l=1}$). We noted that
the periodicity (at 6600.78 s) is deviated from the mean period spacing
significantly. This mode can be interpreted as a candidate of trapped mode,
however, without knowing where the quadrupole modes, it is not allowed to be
conclusive. The two frequencies f2 and f4 could be also interpreted as $l=2$
modes since three times $\Delta P_{l=1}$ equals to five times $\Delta
P_{l=2}$. If we exclude these two degenerate modes and calculate the average
period spacing of $l=1$, we find $\Delta P_{l=1}$ = 256.38 $\pm$ 1.43 s. If we
include these two modes and calculate the average period spacing, we found
$\Delta P_{l=1}$ = $256.09^{+6.97}_{-1.21}$ s. We list all modes identified in
Table LABEL:138707823 and show them in Fig. 16.

The final seismic result for TIC 138707823 and the rest of the stars analyzed
in this paper are given in Table LABEL:Table_Seismic.

Table 7: Frequency solution from the _TESS_ light curve of TIC 138707823 including frequencies, periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio. Errors are given in parenthesis to 2 significant digits. Identified modal degree and relative radial orders are listed column 5 and 6, respectively. The frequencies detected in both sector (2 and 29) tagged with $\dagger\dagger$. The frequencies detected only in sector 29 are tagged with $\dagger$. The frequencies that are detected only in sector 2 data given without tag. See text for more details on mode identification. ID | Frequency | Period | Amplitude | S/N | $l$ | $n$
---|---|---|---|---|---|---
|  $\mu$Hz | [sec] | [ppt] |  |  |
f1 | 87.829 (26) | 11385.70 (3.33) | 0.858 (97) | 6.98 | 1 | 44
f2 | 132.679 (27) | 7537.01 (1.51) | 0.833 (97) | 6.77 | 1/2 | 29/45
f${}_{\rm 3^{\dagger\dagger}}$ | 151.497 (17) | 6600.78 (74) | 1.297 (97) | 10.55 | 1 | 25
f${}_{\rm 4^{\dagger\dagger}}$ | 173.099 (29) | 5777.02 (97) | 0.890 (10) | 6.84 | 1/2 | 22/33
f5 | 181.380 (39) | 5513.27 (1.17) | 0.576 (98) | 4.68 | 1 | 21
f6 | 191.027 (25) | 5234.87 (69) | 0.874 (98) | 7.11 | 1 | 20
f${}_{\rm 7^{\dagger\dagger}}$ | 237.895 (32) | 4203.52 (57) | 0.794 (10) | 6.10 | 1 | 16
Figure 16: Same as Fig. 13 but for TIC 138707823 (presented here is sector 2).

####  3.4.5 TIC 415339307

The FT of TIC 415339307 shows that all the frequencies are concentrated in a
narrow region between 156 $\mu$Hz (3 371 s) and 296 $\mu$Hz (6 399 s) as can
be seen in Fig. 17. In Fig. 12, K-S and I-V tests show a sign of a potential
average period spacing at 250 s. We identified 5 dipole modes and 2 quadrupole
modes, while 3 modes could fit both solutions. Between 4 500 s and 6 000 s,
there are four frequencies that can only be fitted by dipole mode sequence. We
find that the frequencies f1, f6, f8 and f10 also fit the dipole sequence. The
frequencies f7 and f9 can only be quadrupole modes as the difference between
the modes is around 300 s. The frequencies f1, f6 and f10 could be identified
as $l=2$ modes.

Including all these identified modes, we calculated $\Delta P_{l=1}$ =
$252.84^{+3.30}_{-2.86}$ s and $\Delta P_{l=2}$ = $151.50^{+6.69}_{-2.06}$ s.
The final seismic result for TIC 415339307 are given in Table
LABEL:Table_Seismic. We list all identified modes in Table LABEL:415339307 and
show them in Fig. 17.

Table 8: Frequency solution from the _TESS_ light curve of TIC 415339307 including frequencies, periods, and amplitudes (and their uncertainties) and the signal-to-noise ratio. Identified modal degree and relative radial orders are listed column 5 and 6, respectively. ID | Frequency | Period | Amplitude | S/N | $l$ | $n$
---|---|---|---|---|---|---
|  $\mu$Hz | [sec] | [ppt] |  |  |
f1 | 156.261 (20) | 6399.55 (79) | 2.661 (21) | 9.89 | 1/2 | 24/41
f2 | 175.929 (19) | 5684.11 (63) | 2.619 (21) | 9.74 | 1 | 21
f3 | 184.122 (44) | 5431.18 (1.26) | 1.198 (21) | 4.45 | 1 | 20
f4 | 203.126 (38) | 4923.05 (92) | 1.346 (21) | 5.00 | 1 | 18
f5 | 214.359 (37) | 4665.07 (82) | 1.362 (21) | 5.06 | 1 | 17
f6 | 240.309 (36) | 4161.31 (67) | 1.324 (21) | 4.92 | 1/2 | 15/26
f7 | 250.302 (33) | 3995.16 (54) | 1.514 (21) | 5.63 | 2 | 25
f8 | 255.863 (34) | 3908.34 (60) | 1.3 (21) | 4.83 | 1 | 14
f9 | 272.299 (23) | 3672.43 (56) | 2.188 (21) | 8.13 | 2 | 23
f10 | 296.648 (18) | 3371.00 (21) | 2.736 (21) | 10.17 | 1/2 | 12/21
Figure 17: Same as Fig. 13 but for TIC 415339307 (presented here is sector 5). Table 9: The seismic properties of 5 pulsating sdB stars studied in this work. Columns 1, 2, 3, 4, 5, 6 and 7 correspond to the TESS input catalog number, the number of g-modes extracted from the light curve, the number of identified dipole modes, the number of identified quadrupole modes, the range of radial orders, the average period spacing of dipole modes and the average period spacing of quadrupole modes, respectively. TIC | # g | $l=1$ | $l=2$ | $n$ | $\Delta P_{l=1}$ | $\Delta P_{l=2}$
---|---|---|---|---|---|---
260795163 | 23 | 11 | 7 | 14-44 | $254.83^{+2.14}_{-2.28}$ | $150.51^{+3.27}_{-2.28}$
080290366 | 18 | 11 | 3 | 14-48 | $253.32^{+0.78}_{-0.84}$ | $154.16^{+2.47}_{-7.97}$
020448010 | 15 | 14 | 1 | 20-56 | $251.70^{+0.87}_{-0.96}$ | 145.32
138707823 | 7 | 7 | - | 16-45 | $256.09^{+6.97}_{-1.21}$ | 147.83
415339307 | 10 | 8 | 2 | 12-41 | $252.84^{+3.30}_{-2.86}$ | $151.50^{+6.69}_{-2.06}$

##  4 Analysis of spectroscopic data

The data from the EFOSC2 spectrograph were reduced and analyzed using standard
PyRAF666http://www.stsci.edu/institute/software_hardware/pyraf (Science
Software Branch at STScI 2012) procedures. First, bias correction and flat-
field correction have been applied. Then, the pixel-to-pixel sensitivity
variations were removed by dividing each pixel with the response function.
After this we applied wavelength calibrations using the spectra obtained with
the internal He-Ar comparison lamp. In a last step, flux calibration was
applied using the standard stars EG 21 and EG 274. The SNR of the final
spectra is between 80 and 150 (see Table LABEL:tablespec1).

The data obtained with the B&C spectrograph were reduced in similar way using
PyRAF. All frames were bias subtracted, flat-field corrected and cosmic ray
events were removed. Afterwards, the wavelength calibration was performed with
calibration spectra taken immediately after target observations. All
wavelength-calibrated spectra were corrected for atmospheric extinction using
coefficients provided by PyRAF. Finally, all spectra were flux calibrated
using the spectrophotometric standard star EG 21. The final spectra have SNR,
that is ranging from 70 to 120.

##  5 Spectral analysis with XTgrid

All stars have been analyzed and atmospheric parameters were derived by
fitting synthetic spectra to the newly obtained low-resolution spectra.
Synthetic spectra have been calculated from Tlusty non-Local Thermodynamic
Equilibrium stellar atmosphere models (Hubeny & Lanz 2017) using H and He
composition. These models were utilized in the steepest-descent $\chi^{2}$
minimizing fitting procedure XTgrid (Németh et al. 2012) using the web-service
provided by Astroserver777https://xtgrid.astroserver.org. The iterative
procedure starts out from a starting model and by successive corrections
converge on the best-fit. We applied a convergence limit of 0.5% relative
change of all model parameters over three successive iterations. Error bars
were calculated by mapping the $\chi^{2}$ landscape around the best fit until
the 3$\sigma$ confidence limit for the given degree-of-freedom was reached.
The error calculations are performed in one dimension for the He abundance,
but include the correlations between surface temperature and gravity.

Figure 18 shows the new observations together with their best-fit
Tlusty/XTgrid models and Table LABEL:tablesresult lists the atmospheric
parameters. The sample is very homogeneous, all spectra in Figure 18 are
dominated by Balmer-lines and only weak He I lines are seen. This, together
with the Balmer-decrement suggest, that the stars must have very similar
atmospheric parameters, which lines up with earlier observations, that sdBVs
stars (g-mode pulsators) form a compact group on the EHB. However, the
spectral analysis revealed a systematic difference between the parameters
derived from EFOSC2 and B&C data. The poor blue coverage of the NTT/EFOSC2
wavelength calibration lamp results in serious flexure along the dispersion
axis and cause discrepancies in fitting. For completeness we include all
results in Table LABEL:tablesresult. Where B&C spectra are available, we
consider them superior in quality and the results from B&C data as final.

Figure 18: Best fit Tlusty/XTgrid models for the five hot subdwarf stars
analysed in this paper. The spectra are dominated by the H Balmer-series
marked at the bottom and few weak He I lines marked at the top of the figure.
The observed continua have been adjusted to the theoretical models.

Among the stars analyzed in this paper, there is a confirmed binary system TIC
138707823. The spectrum of TIC 138707823 in Figure 18 shows a clean sdB
spectrum without any significant optical contribution from a companion. This
confirms earlier results (Edelmann et al. 2005; Geier & Heber 2012), that the
companion is a compact object, most likely a WD.

Table 10: Spectroscopic results of five sdBV stars analyzed in this paper. The errors are 3$\sigma$ statistical errors. Systematic differences exist between parameters from different instruments. Where available, B&C observations are adopted. The final parameters are given with bold TIC numbers. TIC | Spectrograph | Sp. Type |  $T_{\rm eff}$ (K) |  $\log{g}$ (cm s-2) | $\log{(n_{\rm He}/n_{\rm H})}$
---|---|---|---|---|---
260795163 | B&C | sdB | 27600 ($\pm$460) | 5.37 ($\pm$0.08) | -2.76 ($\pm$0.02)
260795163 | EFOSC2 | sdB | 26260 ($\pm$550) | 5.12 ($\pm$0.09) | -2.80 ($\pm$0.09)
080290366 | B&C | sdB | 26310 ($\pm$410) | 5.30 ($\pm$0.05) | -2.61 ($\pm$0.08)
080290366 | EFOSC2 | sdB | 25770 ($\pm$380) | 5.21 ($\pm$0.06) | -2.69 ($\pm$0.06)
138707823 | B&C | sdB+WD | 24800 ($\pm$310) | 5.49 ($\pm$0.06) | -2.57 ($\pm$0.05)
138707823 | EFOSC2 | sdB+WD | 26310 ($\pm$330) | 5.37 ($\pm$0.08) | -2.53 ($\pm$0.07)
020448010 | EFOSC2 | sdB | 23720 ($\pm$480) | 5.45 ($\pm$0.08) | -2.57 ($\pm$0.52)
415339307 | B&C | sdB | 25920 ($\pm$530) | 5.44 ($\pm$0.10) | -3.00 ($\pm$0.03)

##  6 Asteroseismic models

A proper asteroseismic interpretation of the observed frequencies found in
g-mode sdB pulsators requires the computation of oscillations in stellar
models (Charpinet et al. 2000, 2002). To this end we have computed stellar
evolution models with LPCODE stellar evolution code (Althaus et al. 2005), and
computed $l=1$ g-mode frequencies with the adiabatic non-radial pulsation code
LP-PUL (Córsico & Althaus 2006). Opacities, nuclear reaction rates, thermal
neutrino emission and equation of state are adopted as in Miller Bertolami
(2016) and Moehler et al. (2019). In particular, it is worth noting that
atomic diffusion was not included in the present computations. Diffusion is
expected to reduce mode-trapping features in the latter stages of the He-core
burning phase (Miller Bertolami et al. 2012). In the present work we aim at
the global comparison of the model predictions with observations by means of a
robust indicator such as the mean period spacing ($\Delta P$), leaving
detailed period-to-period comparisons of individual stars for future works. As
such, we expect trapping features to play no important role in the
comparisons. SdB models were constructed from an initially $M_{i}=1M_{\odot}$
ZAMS model with $Z_{\rm ZAMS}=0.02$, and $Y=0.245+2\times Z$, as in Miller
Bertolami (2016). Mass loss was artificially enhanced prior to the He-core
flash in order to produce sdB models during the core He burning stage in the
$T_{\rm eff}$ range of g-mode sdB pulsators. The resulting models at the
beginning of the He-core burning stage have masses of $M_{\rm ZAHB}=0.46738$,
0.4675, 0.468, 0.469, 0.47, and 0.473 $M_{\odot}$. For the sake of comparison,
models during the He-core burning stage were computed under two different
assumptions regarding convective boundary mixing (CBM). One is the extreme
assumption of a strict Schwarzschild criterion (Schwarzschild 1906) at the
convective core, and the other corresponds to the inclusion of CBM at the
boundary of the convective core. In the latter case, CBM is adopted as an
exponentially decaying velocity field following Freytag et al. (1996) and
Herwig et al. (1997), with free parameters taken as in Miller Bertolami
(2016). This corresponds to the assumption of a moderate CBM (see section 3.1
in De Gerónimo et al. 2019). It is worth noting that although strong
theoretical arguments indicate that strict Schwarzschild criterion is not
physically sound, it serves as a useful estimation of the smallest possible
convective core size (Castellani et al. 1971, 1985; Gabriel et al. 2014). The
value of $\Delta P$ was computed from the periods of individual $l=1$ g-modes
in the range 2000 s—10000 s, which is a typical range for the periods observed
in V1093 Her stars. The computed value of $\Delta P$ shows oscillations due to
two completely different reasons. On one hand, as the models evolve structural
changes push different periods inside, or outside, the range in which we
computed $\Delta P$, leading to small fluctuations in the value of $\Delta P$.
For the sake of clarity in Fig. 19 we show its time-averaged value in a moving
window of 3 Myr. The variance around the mean value is shown by the confidence
bands in the bottom right panel of Fig. 19. On the other hand, larger
fluctuations occurring on longer timescales also appear in the sequences that
include moderate CBM due to the oscillating behaviour of the boundary of the
convective core.

The two sets of sdB models are shown in Fig. 19 together with the properties
observed in the V1093 Her stars studied in the present and previous works.
From Fig. 19 it becomes apparent that models with a small convective core,
close to those predicted by a strict Schwarzschild criterion, are too compact
(and consequently too dim) to fit the surface gravities observed in known
pulsators. This is reinforced by the comparison between the evolution of the
mean period spacing ($\overline{\Delta P_{\ell}}$) against $\log g$ and $\log
T_{\rm eff}$ in Fig. 19. Models with small convective cores give mean period
spacings too small to account for the observations. On the contrary, models
with a moderate CMB prescription are able to reach the range of mean period
spacings observed in V1093 Her stars (Fig. 19). The reason behind this is two
fold, and can be understood by looking at the expression of the asymptotic
mean period spacing $\Delta P_{\ell}^{a}$,

$\Delta
P_{\ell}^{a}=\frac{P_{o}}{\sqrt{\ell(\ell+1)}}=\frac{2\pi^{2}}{\sqrt{\ell(\ell+1)}}\left[\int_{r_{1}}^{r_{2}}\frac{N}{r}dr\right]^{-1}.$
(2)

The buoyancy (Brunt-Väisälä) frequency for an ideal monoatomic gas with
radiation can be written as

$N^{2}=g^{2}\frac{\mu(4-3\beta)}{\Re\,T}\left[\nabla_{\rm
ad}-\nabla+\frac{\beta\,\nabla_{\mu}}{4-3\beta}\right],$ (3)

where in the case of sdB stars radiation pressure is almost negligible and
$\beta\sim 1$. We see that period spacing, in general, scales as $\Delta
P\propto\,{\overline{g}}^{-1}R_{\star}/(R_{\star}-R_{\rm core})$. As models
that include CBM have cores that grow larger than those using a strict
Schwarzschild criterion the g-mode propagation cavity (of size $\sim
R_{\star}-R_{\rm core}$) is reduced and leads to larger values of the period
spacings. This is in fact the reason why models that include CBM show
oscillations in the value of $\Delta P$, as those models show oscillations in
the size of the convective core (see Fig. 19). In addition, models with larger
convective cores are able to increase the mean molecular weight of the star to
larger values than their small convective core counterparts. This leads to
larger luminosities in the sdB models that include CBM, and consequently to
smaller surface gravities for the same effective temperature. The decrease in
the mean value of the local surface gravity then also leads to an increase in
the period spacing of the models. Finally, close to the end of the He-core
burning stage, sdB models with CBM develop one of more breathing pulse
instabilities (Castellani et al. 1985). This creates the loops in the $\log
g-\log T_{\rm eff}$ diagram at relatively low gravities and leads to an
extension of the He-core burning lifetime.

From the previous discussion it is apparent that small convective cores close
to those predicted by a bare Schwarzschild criterion can be discarded on the
base of the observations. This is in agreement with theoretical arguments
(Castellani et al. 1985; Gabriel et al. 2014) as well as with independent
observational constraints (Charpinet et al. 2011; Bossini et al. 2015;
Constantino et al. 2015, 2016). Although models with a moderate CBM
prescription at the burning core cover the range of observed period spacings,
a closer inspection shows that observed mean period spacings are about 10 to
20 s larger ($\sim 5$ to 10%) than those predicted by the models. Such a shift
could be attained by a small increase in the size of the convective core. The
consequent small decrease in the surface gravity of the models would still be
in agreement with observations.

Figure 19: Properties of the two sets of stellar evolution models discussed in
the text as compared with those observed in pulsating sdB stars. Blue circles
indicate the stars studied in this work while green circles indicate those
from previous studies (Reed et al. 2018; Charpinet et al. 2019; Reed et al.
2020a; Sahoo et al. 2020a). Red tracks correspond to models computed with a
moderate CBM prescription while cyan tracks correspond to those computed under
the extreme assumption of a strict Schwarzschild criterion. Masses of the
models are $M_{\rm ZAHB}=0.467$, 0.4675, 0.468, and 0.47 $M_{\odot}$, darker
colors correspond to more massive models. Thick magenta lines show the locus
of model at the Zero Age Extreme Horizontal Branch (ZAEHB). Light coloured
bands around the curves in the bottom right panel indicate the typical
variance of $\Delta P$ around the mean value.

##  7 Summary and conclusions

We present here the analysis of data collected on 5 pulsating hot subdwarf B
stars TIC 260795163, TIC 080290366, TIC 020448010, TIC 138707823 and TIC
415339307 observed with the TESS mission. From the five analyzed stars four
are new detections of long-period pulsating sdB (V1093 Her) stars, namely TIC
260795163, TIC 020448010, TIC 138707823 and TIC 415339307. This high-duty
cycle space photometry delivered by the TESS mission provides data with
excellent quality to detect and identify the modes in long-period sdBV stars.

The pulsations detected in these 5 stars are concentrated in the short
frequency region from $\sim$70 to $\sim$300 $\mu$Hz, which is in line with
what has been discovered during the second half of the survey phase of Kepler
(Baran et al. 2011). We have detected 73 oscillation frequencies which we
associate with g-modes in sdBVs. We did not find any p-modes for any of the
targets. With the 120-sec observations, it is difficult to find p-modes since
we are limited to Nyquist frequency of about 4 200 $\mu$Hz. However, we did
not find any p-modes either in the 20-sec data of TIC 260795163, TIC 080290366
and TIC 138707823 even though the Nyquist frequency in that data set is about
25 000 $\mu$Hz. This might imply that the analyzed stars are most likely pure
g-mode sdB pulsators.

We have analyzed the data using the asteroseismic methods of rotational
multiplets and asymptotic period spacing in order to identify pulsational
modes. Although we detect many pulsation frequencies, we did not find evidence
for complete rotational multiplets in any of the analyzed stars. Relying
solely on asymptotic period spacing relationships, we identify the observed
periods as mainly dipole and quadrupole g-modes.

We constructed sFTs to examine the temporal evolution of the pulsation modes.
The pulsation spectra and sFTs of all analyzed stars do not show clear pattern
indicating rotational multiplets. Moreover, the highest signal-to-noise
frequencies seem to be stable over the course of these TESS observations (from
25 to 49 days). However, the low-amplitude frequencies do show complex pattern
in their sFTs. With a higher number of data points and with a longer baseline
such as for stars that have been observed in multiple sectors with TESS, a
better frequency resolution will be achieved and also noise level will be
lower, which will make detection of possible rotational multiples viable. In
particular, we believe that for the stars which lie in the continuing viewing
zone of TESS it would be possible to detect rotational multiplets and make the
mode identification more secure. We will concentrate on the stars which have
been observed in more than a single sector, and on a star which is in the
southern continuous viewing zone of TESS in forthcoming papers.

In the 5 V1093 Her stars analyzed in this paper TIC 260795163, TIC 080290366,
TIC,020448010 TIC 138707823 and TIC 415339307, we have identified 49
frequencies out of 73 detected ones as $l=1$ modes by using solely asymptotic
period spacing. In some cases we were not able to identify the modes with a
unique modal degree. We mention that for 15 periodicities, we did not find a
unique identification, therefore they can be interpreted as either $l=1$ or
$l=2$ modes.

We apply the method of bootstrapping with $10^{4}$ times randomization in
order to calculate the mean period spacing obtained in our analysis and also
to assess the errors. The mean period spacing for $l=1$ modes obtained in this
way for the 5 analyzed stars is ranging from 251 s to 255 s, while the average
period spacing for $l=2$ modes is spanning from 145 s to 155 s.

We have derived atmospheric parameters for all 5 stars analyzed in this paper
by fitting synthetic spectra to the newly obtained low-resolution Dupont/B&C
and NTT/EFOSC2 spectra. The effective temperature of the observed sdB stars is
spanning from 23,700 K to 27 600 K and their surface gravity ($\log{g}$) is in
the range from 5.3 to 5.5 dex, confirming that they are indeed occupying the
g-mode sdBV parameter space.

We have computed stellar evolution models with LPCODE stellar evolution code
(Althaus et al. 2005), and we have computed $l=1$ g-mode frequencies with the
adiabatic non-radial pulsation code LP-PUL (Córsico & Althaus 2006). We
compared the derived mean period spacings ($\Delta P$) of dipole g-modes
derived from TESS observations of 5 sdBV stars analysed in this paper and 33
found in the literature with the predictions of the adiabatic pulsation
computations performed on stellar evolutionary models. In agreement with the
expectations from theoretical arguments and previous asteroseismological works
(Castellani et al. 1985; Gabriel et al. 2014; Charpinet et al. 2011; Bossini
et al. 2015; Constantino et al. 2015; Schindler et al. 2015), and recently
(Ostrowski et al. 2020), we find that models relying on a simplistic
implementation of the Schwazschild criterion leads to small convective cores
and values of $\Delta P$ which are too low to match the observations. On the
contrary, models with standard treatment of convective boundary mixing at the
convective core are able to match the observed values of $\Delta P$, although
more intense convective boundary mixing cannot be discarded.

For any of the targets analysed in this work, we did not attempt to constrain
envelope properties because of the following reasons. First, we did not detect
any p-modes. Although lower order g-modes are sensitive to envelope
properties, one needs to attempt to do a model fit of the observed periods as
it has been done by (Charpinet et al. 2019), which is beyond the scope of this
paper. Lastly, we did not detect any trapped modes either, which can be used
to investigate the envelope by identifying the period spacing between the
trapped modes.

###### Acknowledgements.

We wish to acknowledge the suggestions and comments of an anonymous referee
that improved the original version of this work. We thank Alejandro H. Córsico
for kindly providing us with the codes for the K-S and I-V tests and for his
valuable comments on the earlier version of this manuscript. M.U. thanks Özgür
Bastürk and M.Ekrem Esmer for valuable discussions. M.U. acknowledges
financial support from CONICYT Doctorado Nacional in the form of grant number
No: 21190886 and ESO studentship program. P.N. acknowledges support from the
Grant Agency of the Czech Republic (GAČR 18-20083S). This research has used
the Sandbox services of www.Astroserver.org. Financial support from the Polish
National Science Centre under projects No. UMO-2017/26/E/ST9/00703 and
UMO-2017/25/B ST9/02218 is acknowledged. This paper includes data collected by
the TESS mission. Funding for the TESS mission is provided by the NASA
Explorer Program. Funding for the TESS Asteroseismic Science Operations Centre
is provided by the Danish National Research Foundation (Grant agreement no.:
DNRF106), ESA PRODEX (PEA 4000119301) and Stellar Astrophysics Centre (SAC) at
Aarhus University. We thank the TESS team and staff and TASC/TASOC for their
support of the present work. We thank Brad Barlow, who made possible TESS
Cycle 3 observations of variable hot subdwarf stars observations with the
proposal number G03221. And finally we thank the TASC WG8 team for supporting
this project and providing valuable feedback.

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