# Signature of high-amplitude pulsations in seven $\delta$ Sct stars via TESS
observations

Fatemeh Vasigh Department of Physics, Faculty of Science, University of
Zanjan, University Blvd., Zanjan, 45371-38791, Zanjan, Iran Elham Ziaali,
Hossein Safari<EMAIL_ADDRESS><EMAIL_ADDRESS>Elham Ziaali Department of
Physics, Faculty of Science, University of Zanjan, University Blvd., Zanjan,
45371-38791, Zanjan, Iran  Instituto de Astrofísica de Andalucía – CSIC,
E-18008 Granada, Spain  Hossein Safari Department of Physics, Faculty of
Science, University of Zanjan, University Blvd., Zanjan, 45371-38791, Zanjan,
Iran  Observatory, Faculty of Science, University of Zanjan, University Blvd.,
Zanjan, 45371-38791, Zanjan, Iran

###### Abstract

The regular behavior of the pulsations of high-amplitude $\delta$ Sct (HADS)
stars gives a greater chance to investigate the interiors of stars. We
analyzed seven HADS stars showing peak-to-peak amplitudes of more than 0.3 mag
that were newly observed by TESS. We obtained that TIC 374753270, TIC 710783,
and TIC 187386415 pulsate in fundamental radial mode; also, TIC 130474019 and
TIC 160120432 show double radial modes. On the other hand, TIC 148357344 and
TIC 278119167 demonstrate triple-mode behavior. Our analysis shows that these
seven stars are close to the red edge of the (inside) instability strip in the
Hertzsprung–Russell diagram. The fundamental mode of these seven targets
follows the period-luminosity (PL) relation for $\delta$ Sct stars. However,
TIC 278119167 deviates slightly from the fundamental PL relation. The double-
mode and triple-mode HADS stars (TIC 130474019, TIC 160120432, TIC 148357344,
and TIC 278119167) are in agreement with the period ratio ranges (fundamental
to first and second overtones). Using the information of 176 HADS stars
(Netzel and Smolec), we find a scaling relation
(${\rm[Fe/H]}\propto\log(M^{7.95\pm 0.15}L^{-1.83\pm 0.11}{\rm P0^{0.79\pm
0.14}}T_{\rm eff}^{0.047\pm 0.02}$)) between the metallicity ([Fe/H]) and mass
($M$), luminosity ($L$), effective temperature ($T_{\rm eff}$), and the
fundamental period ($\rm P0$). We estimate the metallicity of the seven newly
identified HADS stars ranging from -0.62 to 0.37 dex.

Asteroseismology (73); Delta Scuti variable stars (370); Stellar oscillations
(1617)

,

##  1 Introduction

Asteroseismology is an approach to studying the internal characteristics of
pulsating stars through their oscillation modes (Aerts et al., 2010; Antoci et
al., 2019; Bedding et al., 2020; Holdsworth et al., 2024). The
asteroseismology approach has more details explanations for the interior
structure of different stars by increasing the resolution (temporal, spectral,
and spatial) of ground and space observations of the pulsating stars. While
overall instrument resolution (temporal and spatial) helps measure pulsation
frequencies more precisely, the spectral resolution provides extra detail of
chemical abundances (metallicity). Studies of Kepler (Gilliland et al., 2010)
data have had significant impacts on asteroseismology. Recently, the
Transiting Exoplanet Survey Satellite (TESS; Ricker et al. (2015)) provided
unprecedented space observations of numerous star systems. Antoci et al.
(2019) reported the information of 117 $\delta$ Sct stars from TESS short-
cadence observations for sectors 1 and 2. Very recently, Read et al. (2024)
identified 848 $\delta$ Sct stars having a narrow color range of 0.29
$<G_{BP}-G_{RP}<$ 0.31 (at the center of instability strip) as being nearer
than 500 pc by using sectors 27-55 of TESS observations and Gaia Data Release
3 (DR3) information. Investigating the pulsation property of $\delta$ Sct
stars could help to understand their internal structure, such as rotation,
chemical composition, density, temperature, and gravity.

In the Hertzsprung–Russell (H-R) diagram, $\delta$ Sct stars are pulsating
variables of spectral types A0-F5 that mainly lie at the overlap of the main
sequence and the instability strip (Breger, 1979; Murphy et al., 2020; Bowman
& Michielsen, 2021). The $\delta$ Sct stars have the intermediate mass located
at the transition region from low-mass to high-mass stars. Stars with lower
than 1 and higher than 2 $M_{\odot}$ are called low-mass and high-mass,
respectively. The effective temperature range of the $\delta$ Sct stars lies
between $6500$ and $9500$ K (Petersen & Christensen-Dalsgaard, 1996; Breger,
2000a; McNamara, 2011; Bowman et al., 2016; Pietrukowicz et al., 2020; Murphy
et al., 2019; Jayasinghe et al., 2020; Yang et al., 2021b). The $\delta$ Sct
stars pulsation frequencies are in the range of 5-60 $\rm day^{-1}$. They show
various modes of pulsations such as radial, nonradial pressure modes, and
mixed modes of the low radial order in their complex light curves (Houdek et
al., 1999; Breger, 2000b; Handler, 2009; Hasanzadeh et al., 2021; Soszyński et
al., 2021)

Their pulsations are primarily driven by the classical $\kappa$ mechanism in
the He II ionization zone. Some studies also suggested that intrinsically
stable, stochastically driven (solar-like) $p$-modes may be excited
simultaneously in such $\delta$ Sct star (Baker & Kippenhahn, 1962; Balmforth
& Gough, 1990; Houdek et al., 1999; Samadi et al., 2002; Antoci et al., 2011;
Antoci, 2013).

High-amplitude $\delta$ Sct (HADS) stars pulsate in $V$-band amplitudes over
0.3 mag. However, there are few reports on HADS stars showing peak-to-peak
value slightly smaller than 0.3 mag (Breger et al., 2011; Lv et al., 2023).
HADS stars are characterized by their high amplitude, short periods (less than
one day), and also $v\sin i\leq\ 30\ \rm kms^{-1}$. HADS stars oscillate as
single-periodic with a radial fundamental mode or double-periodic or triple-
periodic pulsators (Bono et al., 1997; Wils et al., 2008; McNamara, 2011; Yang
et al., 2021a; Netzel & Smolec, 2022; Lv et al., 2023; Xue et al., 2023).
However, the number of HADS stars having four independent radial frequencies
as quadruple-mode HADS stars has been increasing in recent years (Mow et al.,
2016; Lv et al., 2022).

Also, nonradial modes may exist in the HADS frequency spectrum (Uytterhoeven
et al., 2011). HADS stars are important because they can be used to study the
structure and evolution of stars. These stars can be used to investigate the
evolution of stellar populations, as they are found in old stellar
populations, while low-amplitude $\delta$ Sct (LADS) stars are young and
intermediate-age stars (Chang et al., 2013; Netzel & Smolec, 2022; Soszyński
et al., 2021). Recently, there has been a growing interest in HADS stars as
potential targets for exoplanet searches. High-precision observations of HADS
stars’ pulsations and signatures of orbiting planets could detect slight
variations in their light curves. Several exoplanets have already been
discovered around HADS stars (Hey et al., 2021; Guzik, 2021). Lv et al. (2023)
studied seven new HADS stars through photometric data of TESSand reported that
two stars may be RR Lyrae based on their light curves and the difference in
their placement with the HADS star in the Peterson and period-luminosity (PL)
diagrams. In these HADS stars, the metallicity can have a greater effect than
the rotation of the star on the period ratios (the ratio of overtones to the
fundamental mode). The period ratios decrease with increasing metallicity.

Metallicity is often expressed as a ratio of the abundance of iron (Fe) to
hydrogen (H) compared to the solar ratio, written as [Fe/H] (Kotoneva et al.,
2002; Chruslinska & Nelemans, 2019). A metallicity value greater than -0.5 dex
indicates metal-rich $\delta$ Sct stars, while a value less than -1.5 dex
shows a metal-poor star (McNamara, 2011). Determining a star’s metallicity
involves sophisticated astronomical techniques based on spectroscopic
investigations (Lianou et al., 2011; Liu et al., 2020), photometric methods
(Li et al., 2023; Dékány & Grebel, 2022) and astroseismology approach (Netzel
& Smolec, 2022).

Here, we analyzed the light curves of seven newly identified HADS stars from
TESS observations. We selected these HADS from the TESS catalog of several
thousand of $\delta$ Sct stars applying three criteria: (1) the peak-to-peak
amplitude is greater than 0.3 mag, (2) the related extinction is less than 0.6
mag, and (3) the relative parallax error is less than 0.05. Therefore, we
applied more analyses to seven HADS having short-cadence TESS observation that
satisfied the above criteria for more asteroseismic analyses. To do this, we
extracted the frequencies and amplitudes of these seven stars using the
periodogram analysis.

We studied the PL behavior and H-R diagram. To determine the metallicity of
targets, we obtained a scaling relation between the metallicity of 176 HADS
stars (Netzel & Smolec, 2022) and their physical parameters (mass, luminosity,
and effective temperature) and the fundamental periods. Section 2 gives the
HADS star information used in this paper. Sections 3 and 4 explain the methods
and results, respectively. Finally, section 5 summarizes the essential
findings of the present work.

##  2 Data

TESS is a space mission for probing exoplanets (Ricker et al. (2015)). This
telescope divides the sky into sectors and observes each sector for 27 days.
The TESS space telescope was designed to record objects in a red passband of
approximately 600–1000 nm (Aguirre et al., 2015; Barclay et al., 2018). TESS
data are available in two forms, target pixel file (TPF) and light curve,
which can be searched applying the Lightkurve tool (Lightkurve Collaboration
et al., 2018). Figure 1 illustrates TPF and aperture (red squares) for TIC
374753270 (left, first row), TIC 130474019 (right, first row), TIC 148357344
(left, second row), TIC 160120432 (right, second row), TIC 278119167 (left,
third row), TIC 710783 (right, third row), and TIC 187386415 (fourth row). As
shown in the figure, each aperture covers the primary pixels of each target
without considerable contamination of another object in the field of view. In
this paper, we used photometric data processed by SPOC (Jenkins et al., 2016),
which are short-cadence (2 minutes). We transformed the flux to magnitude and
subtracted from its average value for each light curve to get the corrected
time series.

Table 1: TIC number, variable name, and TESS short-cadence sector(s) for seven HADS stars. TIC | Star Name | Sector
---|---|---
374753270 | - | 36,37
130474019 | - | 7, 34
148357344 | - | 7
160120432 | YZ UMi | 14, 19, 20, 40, 47, 52, 53
278119167 | - | 40, 53
710783 | - | 5
187386415 | - | 51

http://dx.doi.org/10.17909/4a25-pg87

##  3 Method

###  3.1 Frequency analysis

We selected seven $\delta$ Sct light curves (related to TPFs of Figure 1) with
peak-to-peak amplitudes greater than 0.3 mag as an essential characteristic
for identifying HADS stars. We checked the TPFs for each target to ensure the
SPOC light curve’s validity due to the SPOC aperture’s size, which would not
be contaminated by the neighbor objects in the field. Our investigation showed
that the SPOC light curves were perfect for our targets. Then, their SPOC-
prepared light curves were analyzed to extract pulsation frequencies. We
removed outliers from each star’s light curve. We rejected the outlier data
points from light curves (data points were far from the average). We first
calculated the average and standard deviation for each light curve. The
outlier data points have amplitudes more significant than 4 standard
deviations (4$\sigma$). Also, the mean of each single light curve was
subtracted to get the corrected systematic error flux. To study our samples’
pulsating behavior, the PERIOD04 (Lenz & Breger, 2005) software was applied. A
prewhitening procedure carried out the frequency extraction. The Nyquist
frequency is half of the sampling rate, $f_{N}$ = 360.048 $\rm day^{-1}$ for 2
minute cadence observations. This value is well above the limit range (5 $<f<$
60 $\rm day^{-1}$) expected for a typical $\delta$ Sct star pulsation. We used
a resolution frequency $f_{res}$ = 1.5/$\Delta$$T$ to identify nearby
frequencies in the amplitude spectrum, where $\Delta$$T$ represents the length
of the light curve (Loumos & Deeming, 1978). Two frequencies are resolved if
the frequency difference is more than the resolution frequency. The light
curves are modeled by

$m=m_{0}+\Sigma_{i=1}^{N}\ A_{i}\sin(2\pi(f_{i}t+\phi_{i})),$ (1)

where $m_{0}$ is the zero point, $A_{i}$ is the amplitude, $f_{i}$ is the
frequency, and $\phi_{i}$ is the phase.

The PERIOD04 algorithm gives the frequency, amplitude, and phase of each
significant mode by applying the multifrequency least-square fit (Equation 1)
for the stellar light curve. We considered the modes with a signal-to-noise
ratio (S/N) more significant than 5.2 (Baran et al., 2015). The residual is
determined by subtracting the modeled light curve from the original light
curve that is expected to show the lack of significant modes (S/N$>5.2$). The
uncertainty of the peaks is determined using the approach by Montgomery &
O’Donoghue (1999).

Figure 1: Target Pixel File (TPF) for TIC 374753270 (left-first row), TIC
130474019 (right-first row), TIC 148357344 (left-second row), TIC 160120432
(right-second row), TIC 278119167 (left-third row), TIC 710783 (right-third
row), and TIC 187386415 (fourth-row). The aperture for each target star is
indicated by the red squares.

###  3.2 Asteroseismic analysis

The stellar oscillations are helpful to infer information about the stars’
interior. Each mode can be described by three quantum numbers: $n$ (radial
number), $l$ (degree), and $m$ (azimuthal number), which are related to the
characteristics of oscillations. In most pulsating stars, we expect the
highest amplitude mode to be the fundamental radial ($n=1$, $l=0$) oscillation
(Aerts et al., 2010; Murphy et al., 2020).

The $\delta$ Sct stars’ power spectrum, the frequency corresponding to the
highest peak amplitude, is called $f_{A_{max}}$. The $f_{A_{max}}$ might
change over time due to modulation of amplitude and other factors such as
noise (Bowman, 2017). Radial modes can be identified based on the ratio of
their periods. Frequency (period) ratios for radial modes (fundamental (F0),
first (F1), second (F2) and third (F3) overtones) were first presented by
Stellingwerf (1979) based on theoretical models as $\rm F0/F1=0.756-0.787$,
$\rm F0/F2=0.611-0.632$, and $\rm F0/F3=0.50-0.525$.

For $\delta$ Sct stars with the first and the second overtones, the
characteristic frequency (period) ratios are around $\rm F1/F2=0.8$ (Soszyński
et al., 2021). The combination of different frequencies or the linear sum of
the independent frequencies, $n\nu_{i}\pm m\nu_{j}$, was identified by the
Loumos & Deeming (1978) criteria. The modes with no agreement with combination
or harmonic frequencies might be identified as nonradial modes. Whereas the
frequency (period) of pulsations is linked to stars’ interior characteristics,
the frequency is related to the radius changes. The critical factor is the
density of stars that is accurately related to the period. The period-density
relation is defines as

$Q=P\sqrt{\overline{\rho}/\overline{\rho}_{\odot}},$ (2)

where $\rm\overline{\rho}$ describes the mean densities of the target star and
$\rm\overline{\rho}_{\odot}$ is the mean density for Sun. $P$ is the pulsation
period. The other forms of Equation (2) were given by Breger (1990) that
included the gravity acceleration and bolometric magnitudes. Here, we used
Equation (2) to validate the periods of seven targets. The range of pulsations
constant ($Q$) for $\delta$ Sct stars’ fundamental, first, second, and third
overtone frequencies are $0.027-0.04$, $0.021-0.027$, $0.018-0.021$, and
$0.016-0.017$, respectively (Breger & Bregman, 1975; Milligan & Carson, 1992).

####  3.2.1 Period–Luminosity relation

The PL relation is a dependency between the star’s luminosity and density.
Consequently, this relation occurs between the luminosity and the pulsation
period of a group of pulsating stars that have a relatively narrow range of
effective temperature. The first PL was given by Leavitt & Pickering (1912)
for Cepheid stars. We computed the absolute $V$-band magnitudes ($M_{\rm V}$)
for our seven targets using the following equation:

$M_{\rm V}=m_{\rm v}+5\log\pi+5-A_{\rm V},$ (3)

where $m_{\rm v}$ is the apparent magnitude in the $V$ band and $\pi$ denotes
the star’s parallax (in arcseconds) collected from Gaia DR3. Since
interstellar dust can impact the flux of stars (luminosity), the $M_{\rm V}$
values were corrected for extinction ($A_{\rm V}$). Following Green et al.
(2018), we determined the extinction for our seven targets to correct the
$V$-band magnitudes. All seven samples have absolute relative parallax error
(the ratio of parallax uncertainty to the parallax) in the range of
0.010-0.020 and extinctions less than 0.6 mag. By applying the Tycho $V_{T}$
and $B_{T}$ magnitudes, we calculated the Johnson $V$ apparent magnitudes as
(Høg et al., 2000; 1997ESA, 1997),

$V=V_{T}-0.090(B_{T}-V_{T}).$ (4)

We investigated the validity of the obtained fundamental frequencies by using
the PL relation for our targets. Using Gaia Data Release 2 parallaxes, Ziaali
et al. (2019) gave the PL relation for $\delta$ Sct stars observed by Kepler,
as follows:

$M_{\rm V}=(-2.94\pm 0.06)\log(P)+(-1.34\pm 0.06),$ (5)

While Barac et al. (2022) reported a similar relation for stars observed by
TESS using the Gaia DR3 parallaxes,

$M_{\rm V}=(-3.01\pm 0.07)\log(P)+(-1.40\pm 0.07).$ (6)

Also, using multicolour photometric ground-based observations, Poro et al.
(2021) gave

$M_{\rm V}=(-3.200\pm 0.010)\log(P)-(1.599\pm 0.010).$ (7)

By using Equation (3) the absolute magnitude for each single star was
calculated, and the PL diagram is examined (Section 4.8). These PL relations,
Equations 5-7, are very close together in a narrow ridge for $\delta$ Sct
stars. So, the magnitudes and fundamental periods should be consistent with
the PLs within the observational uncertainties.

##  4 Results

We applied three criteria to identify the HADS stars, including the peak-to-
peak amplitude more significant than 0.3 mag (from the TESS catalog), relative
parallax error less than 0.05 (from Gaia DR3 catalog), and its measured
extinction less than 0.6 mag. Applying the frequency analysis for the TESS
light curves, we investigated the properties of the amplitude spectrum. We
considered the frequencies with S/N $>$ 5.2 to ensure the validity of the
identified frequencies. We obtained the period ratio and $Q$ value. As
essential criteria, if the targets (Table 1) have the $\delta$ Sct star
characteristic, we expect their absolute magnitudes and fundamental
frequencies to satisfy the PL relations, Equations 5-7. Please note that these
PL relations are in good agreement with the slight differences that are
derived from different observations.

Figure 2: H-R diagram including our seven HADS stars indicated with different-
colored star markers. The observational red and blue edge of the instability
strip for $\delta$ Sct stars along with the evolutionary tracks with $X=0.71$,
$Z=0.014$, and $\alpha_{MLT}$=1.8 were obtained by Murphy et al. (2019). The
temperatures are from Stassun et al. (2018) and the $V$ band luminosities were
calculated by using the apparent magnitudes, parallaxes, and the interstellar
extinctions (Eq. 3). The error bars indicate the errors for luminosities.

Figure 2 displays the position of the seven HADS stars (colored star markers)
between the instability strip edges in the H-R diagram given by Murphy et al.
(2019). As displayed in the figure, these seven targets tend toward the
instability strip’s red edge with lower temperatures. Murphy et al. (2019)
investigated the instability strip boundaries based on the position of 15000
$\delta$ Sct stars in the Kepler field of view using time-dependent convection
and mixing length ($\alpha_{MLT}$=1.8) reported by Dupret et al. (2005). In
the remainder of this paper, we examined the HADS star criteria for these
seven targets.

###  4.1 TIC 374753270

Figure 3 represents the light curve (top panel), amplitude spectrum (middle
panel), and amplitude spectrum of the residual (bottom panel) for TIC
374753270 (TYC 8943-3384-1). Barac et al. (2022) introduced TIC 374753270
($R.A.=10^{h}03^{m}20^{s}.62$,
$DEC=-61^{\circ}51^{\prime}15.63^{\prime\prime}$) as a $\delta$ Sct star. We
obtained the significant frequencies for combined sectors 36 and 37 using the
frequency analysis approach. By frequency analysis of the combined light
curve, we detected 14 significant frequencies with S/N $>5.2$ and frequency
resolution greater than 0.029 $\rm day^{-1}$. The frequencies are listed in
the Table 2, including one radial frequency (monomode HADS) of $\rm
F0=7.246052$ $\rm day^{-1}$ ($\rm P0$ = $0.138$ days). Other frequencies might
be called harmonics ($f_{2}$, $f_{3}$, $f_{4}$, $f_{5}$, $f_{7}$, $f_{8}$,
$f_{9}$) and nonradial modes ($f_{6}$, $f_{10}$, …, $f_{14}$). These
frequencies are in the reported range for $\delta$ Sct stars (Breger, 2000a;
Liakos & Niarchos, 2017). Removing these 14 identified modes, we obtained the
amplitude spectrum of the residual that indicated the absence of significant
modes with S/N greater than 5.2. The $Q$ value of the radial fundamental mode
is 0.032, which is in the reported range of fundamental mode for $\delta$ Sct
stars. The rectified light curve shows a peak-to-peak amplitude of about $\sim
0.43$ mag, which is slightly more than 0.3 mag (expected for HADS stars).

Figure 3: The short-cadence light curve (sectors 36 and 37) of TIC 374753270 during 1.5 days (top panel), amplitude spectrum (middle panel), and amplitude spectrum of residual with S/N less than 5.2 (bottom panel). Table 2: A complete list of the 14 identified frequencies for TIC 374753270 (denoted by $f_{i}$).  $f_{i}$ | Frequency | Amplitude | phase | S/N | ID | period ratio |  $Q$ value
---|---|---|---|---|---|---|---
| ($\rm day^{-1}$) | (mmag) | (radians/2$\pi$) |  |  |  |
1 | 7.246052±0.000002 | 159.91±0.03 | 0.50029±0.00003 | 1430.94 | F0 | - | 0.032±0.004
2 | 14.491993±0.000004 | 74.91±0.03 | 0.31175±0.00006 | 1196.53 | 2F0 | - | -
3 | 21.737972±0.000008 | 39.41±0.03 | 0.9745±0.0001 | 825.07 | 3F0 | - | -
4 | 28.98394±0.00002 | 14.57±0.03 | 0.6471±0.0003 | 394.95 | 4F0 | - | -
5 | 36.22991±0.00004 | 7.83±0.03 | 0.3280±0.0006 | 236.33 | 5F0 | - | -
6 | 6.01456±0.00005 | 5.81±0.03 | 0.6383±0.0008 | 142.08 | nonradial | - | -
7 | 43.47588±0.00007 | 4.46±0.03 | 0.961±0.001 | 140.41 | 6F0 | - | -
8 | 50.7219±0.0001 | 3.14±0.03 | 0.264±0.001 | 101.92 | 7F0 | - | -
9 | 57.9679±0.0001 | 2.45±0.03 | 0.023±0.002 | 40.28 | 8F0 | - | -
10 | 12.0289±0.0002 | 1.34±0.03 | 0.939±0.003 | 18.88 | nonradial | - | -
11 | 18.0437±0.0006 | 0.51±0.03 | 0.367±0.01 | 17.45 | nonradial | - | -
12 | 13.2607±0.0007 | 0.42±0.03 | 0.059±0.01 | 15.06 | nonradial | - | -
13 | 8.477±0.001 | 0.21±0.03 | 0.324±0.02 | 6.42 | nonradial | - | -
14 | 20.506±0.001 | 0.18±0.03 | 0.902±0.02 | 5.33 | nonradial | - | -
Note. The frequency resolution is about $f_{res}=0.029\rm day^{-1}$. |  |  |  |  |  |

###  4.2 TIC 130474019

Figure 4 shows the light curve (top panel), amplitude spectrum (middle panel),
and amplitude spectrum of the residual (bottom panel) for TIC 130474019 (TYC
7636-1969-1). Barceló Forteza et al. (2020) determined the physical parameters
(temperature, gravity, and frequency scaling relation) of TIC 130474019
($R.A.=+6^{h}59^{m}54^{s}.82$,
$DEC=-42^{\circ}01^{\prime}15.12^{\prime\prime}$) using TESS observation,
while Barac et al. (2022) introduced this star as a $\delta$ Sct sample. This
star’s light curves recorded in sectors 7 and 34 include 16339 and 16850 data
points, respectively. Table 3 shows the 16 frequencies, which include two
radial frequencies of $\rm F0=12.28231$ $\rm day^{-1}$ (fundamental, $\rm P0$
= $0.0814$ days) and $\rm F1=15.8872$ $\rm day^{-1}$ (first overtone) with $Q$
values as 0.04 and 0.031 respectively. The frequency ratio ($\rm F0/\rm F1$)
is obtained to be 0.78, which indicates the independent property and validity
of the fundamental and first overtone. As tabulated in Table3, we observed the
harmonics frequencies ($f_{2}$, $f_{3}$, $f_{4}$), combination frequencies
($f_{6}$, $f_{10}$, $f_{11}$, $f_{13}$), and nonradial modes ($f_{7}$,..,
$f_{9}$, $f_{12}$, $f_{14}$,…, $f_{16}$). The peak-to-peak amplitude is about
$\sim 0.3$ mag, and the range of the frequencies might suggest TIC 130474019
is a double-mode HADS star.

Figure 4: The short-cadence light curve (sector 7) of TIC 130474019 during 1.5 days (top panel), amplitude spectrum (middle panel), and amplitude spectrum of residual with S/N less than 5.2 (bottom panel). Table 3: A complete list of the 16 identified frequencies for TIC 130474019 (denoted by $f_{i}$).  $f_{i}$ | Frequency | Amplitude | phase | S/N | ID | period ratio |  $Q$ value
---|---|---|---|---|---|---|---
| ($\rm day^{-1}$) | (mmag) | (radians/2$\pi$) |  |  |  |
1 | 12.28231±0.00001 | 121.25±0.06 | 0.95194±0.00008 | 1372.4 | F0 | - | 0.04±0.004
2 | 24.56457±0.00003 | 37.73±0.06 | 0.9304±0.0002 | 639.01 | 2F0 | - | -
3 | 36.8468±0.0001 | 11.96±0.06 | 0.7588±0.0008 | 279.6 | 3F0 | - | -
4 | 49.1293±0.0002 | 4.92±0.06 | 0.375±0.002 | 106.2 | 4F0 | - | -
5 | 15.8872±0.0007 | 1.94±0.06 | 0.499±0.005 | 18.10 | F1 | 0.78 | 0.027±0.003
6 | 28.173±0.001 | 1.17±0.06 | 0.904±0.008 | 12.57 | F0+F1 | - | -
7 | 16.577±0.002 | 0.69±0.06 | 0.70±0.03 | 8.31 | nonradial | - | -
8 | 16.015±0.002 | 0.60±0.06 | 0.44±0.01 | 7.96 | nonradial | - | -
9 | 15.416±0.002 | 0.63±0.06 | 0.23±0.01 | 8.29 | nonradial | - | -
10 | 8.672±0.002 | 0.47±0.06 | 0.02±0.02 | 7.18 | 2F0-F1 | - | -
11 | 40.452±0.002 | 0.46±0.06 | 0.71±0.02 | 9.34 | 2F0+F1 | - | -
12 | 28.299±0.003 | 0.41±0.06 | 0.99±0.02 | 8.24 | nonradial | - | -
13 | 52.737±0.003 | 0.34±0.06 | 0.09±0.02 | 8.31 | 3F0+F1 | - | -
14 | 15.973±0.003 | 0.25±0.06 | 0.90±0.03 | 5.67 | nonradial | - | -
15 | 20.611±0.003 | 0.23±0.06 | 0.62±0.03 | 5.44 | nonradial | - | -
16 | 27.699±0.003 | 0.22±0.06 | 0.43±0.03 | 5.23 | nonradial | - | -
Note. The frequency resolution is about $f_{res}=0.061\rm day^{-1}$. |  |  |  |  |  |

###  4.3 TIC 148357344

Figure 5 displays the light curve (top panel), amplitude spectrum (middle
panel), and amplitude spectrum of the residual (bottom panel) for TIC
148357344 (TYC 5968-1693-1). TIC 148357344 ($R.A.=+07^{h}05^{m}28^{s}.9$,
$DEC=-17^{\circ}05^{\prime}16.75^{\prime\prime}$) was first identified as a
variable star by Heinze et al. (2018), then it was introduced as $\delta$ Sct
by Barac et al. (2022). TIC 148357344 was observed only in sector 7 (short-
cadence) with a duration of 27 days and consisting of 16056 data points. We
identified 12 frequencies for this star (Table 4), which includes three radial
frequencies $\rm F0=8.43$ $\rm day^{-1}$ ($\rm P0$ = $0.119$ days), $\rm
F1=11.09951$ $\rm day^{-1}$ and $\rm F2=13.835$ $\rm day^{-1}$, as
fundamental, first, and second overtones, respectively. The frequency ratios
for the first ($\rm F0/F1$) and the second ($\rm F0/F2$) overtones were
obtained as 0.76 and 0.61, respectively, determining the validity of our
analysis along with the $Q$ values as 0.028, 0.023, and 0.018 for fundamental,
first, and second overtones, respectively. Other frequencies may be known as
harmonics ($f_{2}$, $f_{3}$,…, $f_{5}$) and combination frequencies ($f_{7}$,
$f_{8}$, $f_{10}$, $f_{11}$, $f_{12}$). It is noticeable that the first
overtone’s amplitude is more significant than that of the fundamental mode,
probably owing to amplitude modulation mechanisms that are well known in
$\delta$ Sct stars (Bowman et al., 2016; Lv et al., 2021; Sun et al., 2021).
Consequently, further studies are necessary to illuminate the driving
mechanisms, including the mode-selecting processes in $\delta$ Scts. The light
curve’s peak-to-peak value of about $\sim 0.4$ mag and the range of
frequencies propose that TIC 148357344 may be a triple-mode HADS.

Figure 5: The short-cadence light curve of TIC 148357344 during 1.5 days (top panel), amplitude spectrum (middle panel), and amplitude spectrum of residual with S/N less than 5.2 (bottom panel). Table 4: A complete list of the 12 identified frequencies for TIC 148357344 (denoted by $f_{i}$). $f_{i}$ | Frequency | Amplitude | phase | S/N | ID | period ratio |  $Q$ value
---|---|---|---|---|---|---|---
| ($\rm day^{-1}$) | (mmag) | (radians/2$\pi$) |  |  |  |
1 | 11.09951±0.00001 | 151.003±0.08 | 0.96259 ±0.00009 | 1948.012 | F1 | 0.76 | 0.023±0.003
2 | 22.19904±0.00003 | 55.44±0.08 | 0.3230±0.0002 | 942.301 | 2F1 | - | -
3 | 33.29853±0.00009 | 19.09±0.08 | 0.6087±0.0002 | 334.345 | 3F1 | - | -
4 | 44.3980±0.0001 | 10.92±0.08 | 0.8494±0.0007 | 198.396 | 4F1 | - | -
5 | 55.4977±0.0006 | 6.01±0.08 | 0.483±0.001 | 134.900 | 5F1 | - | -
6 | 13.835±0.001 | 2.93±0.08 | 0.592±0.004 | 55.0831 | F2 | 0.61 | 0.018±0.002
7 | 24.935±0.001 | 1.85±0.08 | 0.426±0.007 | 34.443 | F1+F2 | - | -
8 | 47.132±0.003 | 0.56±0.08 | 0.67±0.01 | 11.070 | 3F1+F2 | - | -
9 | 8.43±0.003 | 0.51±0.08 | 0.98±0.02 | 12.999 | F0 | - | 0.028±0.004
10 | 58.233±0.003 | 0.41±0.08 | 0.22±0.02 | 11.216 | 4F1+F2 | - | -
11 | 30.559±0.004 | 0.39±0.08 | 0.610±0.003 | 9.018 | 4F1-F2 | - | -
12 | 19.463±0.004 | 0.38±0.08 | 0.717±0.003 | 6.32 | 3F1-F2 | - | -
Note. The frequency resolution is about $f_{res}=0.061\rm day^{-1}$. |  |  |  |  |  |

###  4.4 TIC 160120432

Figure 6 shows the light curve (top panel), amplitude spectrum (middle panel),
and amplitude spectrum of the residual (bottom panel) for TIC 160120432 (TYC
4638-455-1). Using NSVS and ASAS-3 databases, Otero (2007) suggested TIC
160120432 ($R.A.=+15^{h}35^{m}30^{s}.40$,
$DEC=+85^{\circ}37^{\prime}38.94^{\prime\prime}$) as a HADS star. We applied
frequency analysis for combined sectors (19 and 20), individual sectors (14,
40, and 47), and combined sectors (52 and 53). We pick up the significant
frequencies (S/N$>5.2$ and frequency resolution greater than 0.028 $\rm
day^{-1}$ and 0.061 $\rm day^{-1}$ for combined and individual sectors,
respectively). In Table 5, we found 25 frequencies for TIC 160120432, which
include two radial frequencies $\rm F0=10.350856$ $\rm day^{-1}$ ($\rm P0$ =
$0.0966$ days) and $\rm F1=13.612$ $\rm day^{-1}$, with $Q$ values as 0.031
and 0.023, respectively. As shown in Table5, we specified the harmonic
frequencies ($f_{2}$, $f_{3}$,…, $f_{5}$), combination frequencies ($f_{7}$,
$f_{11}$, $f_{12}$, $f_{14}$), and nonradial modes ($f_{6}$, $f_{8}$, $f_{9}$,
$f_{13}$, $f_{15}$…, $f_{25}$). The light curve’s peak-to-peak value is about
$\sim 0.33$ mag. The frequencies of TIC 160120432 are ranging from 9 to 51
$\rm day^{-1}$. The frequency ratio ($\rm F0/F1$) is 0.76, and the $Q$ values
were calculated as 0.031 and 0.024 for the fundamental and first overtone,
respectively. These four findings strongly verified the HADS property for TIC
160120432 as a new double-mode HADS star.

Figure 6: The short-cadence light curve (sector 14) of TIC 160120432 during 1.5 days (top panel), amplitude spectrum (middle panel), and amplitude spectrum of residual with S/N less than 5.2 (bottom panel). Table 5: A complete list of the 25 identified frequencies for TIC 160120432 (denoted by $f_{i}$). $f_{i}$ | Frequency | Amplitude | phase | S/N | ID | period ratio |  $Q$ value
---|---|---|---|---|---|---|---
| ($\rm day^{-1}$) | (mmag) | (radians/2$\pi$) |  |  |  |
1 | 10.350856±0.000003 | 126.46±0.05 | 0.213±0.00006 | 899.46 | F0 | - | 0.031±0.00323
2 | 20.701511±0.000007 | 50.991±0.05 | 0.400±0.0001 | 424.95 | 2F0 | - | -
3 | 31.05230±0.00001 | 21.50±0.05 | 0.394±0.0003 | 245.05 | 3F0 | - | -
4 | 41.40302±0.00003 | 10.378±0.05 | 0.710±0.0008 | 168.428 | 4F0 | - | -
5 | 51.75381±0.00008 | 4.86±0.05 | 0.051±0.001 | 102.31 | 5F0 | - | -
6 | 9.6819±0.0006 | 1.167±0.05 | 0.503±0.007 | 10.71 | nonradial | - | -
7 | 19.5715±0.0007 | 0.956±0.05 | 0.833±0.008 | 11.01 | 6F1-6F0 | - | -
8 | 16.1607±0.0009 | 1.069±0.05 | 0.641±0.007 | 15.85 | nonradial | - | -
9 | 18.8482±0.0001 | 0.912±0.05 | 0.754±0.002 | 9.24 | nonradial | - | -
10 | 13.612±0.001 | 0.219±0.05 | 0.062±0.03 | 8.880 | F1 | 0.76 | 0.023±0.003
11 | 29.9194±0.0008 | 0.625±0.05 | 0.249±0.01 | 9.96 | 6F1-5F0 | - | -
12 | 9.219±0.001 | 0.546±0.05 | 0.626±0.01 | 6.89 | 6F1-7F0 | - | -
13 | 26.510±0.001 | 0.524±0.05 | 0.745±0.01 | 9.74 | nonradial | - | -
14 | 40.269±0.001 | 0.487±0.05 | 0.836±0.01 | 7.45 | 6F1-4F0 | - | -
15 | 19.6427±0.0009 | 0.555±0.05 | 0.416±0.01 | 8.24 | nonradial | - | -
16 | 40.0626±0.0009 | 0.511±0.05 | 0.167±0.01 | 9.80 | nonradial | - | -
17 | 19.4962±0.0005 | 0.539±0.05 | 0.876±0.01 | 7.772 | nonradial | - | -
18 | 30.380±0.001 | 0.458±0.05 | 0.680±0.01 | 7.127 | nonradial | - | -
20 | 18.8807±0.0002 | 0.531±0.05 | 0.280±0.02 | 8.329 | nonradial | - | -
21 | 29.8476±0.0006 | 0.57±0.05 | 0.002±0.02 | 9.66 | nonradial | - | -
22 | 22.0570±0.0008 | 0.48±0.05 | 0.22±0.02 | 9.64 | nonradial | - | -
23 | 40.336±0.001 | 0.35±0.05 | 0.74±0.02 | 7.47 | nonradial | - | -
24 | 15.520±0.001 | 0.34±0.05 | 0.25±0.02 | 7.51 | nonradial | - | -
25 | 9.737±0.001 | 0.36±0.05 | 0.31±0.02 | 6.56 | nonradial | - | -

###  4.5 TIC 278119167

Figure 7 represents the light curve (top panel), amplitude spectrum (middle
panel), and amplitude spectrum of the residual (bottom panel) for TIC
278119167 (TYC 1578-1754-1). Heinze et al. (2018) categorized TIC 278119167
($R.A.=18^{h}38^{m}50^{s}.36$,
$DEC=+19^{\circ}07^{\prime}44.61^{\prime\prime}$) as a pulsating variable
component of a contact/overcontact binary system using a probabilistic
machine-learning algorithm. Our investigation indicates that TIC 278119167 is
triple-mode HADS. We analyzed the light curves for two individual sectors, 40
and 53, with a duration of 28.1 and 24.98 days, respectively. Interestingly,
the set of significant frequencies for the light curve of both sections is
approximately the same. We identified nine frequencies constructing the star’s
light curve (Table 6), which includes three radial frequencies $\rm F0=9.579$
$\rm day^{-1}$ ($\rm P0$ = $0.105$ days), $\rm F1=12.58960$ $\rm day^{-1}$ and
$\rm F2=15.671$ $\rm day^{-1}$, with $Q$ values as 0.031, 0.023, and 0.018
respectively. Other frequencies may be represented as harmonics ($f_{2}$,
$f_{3}$, $f_{4}$, $f_{9}$) and combination frequencies ($f_{6}$, $f_{7}$).
Again, in this star, the amplitude modulation phenomenon is probable due to
the first overtone’s amplitude being more significant than the fundamental
frequency’s amplitude (Bowman et al., 2016; Lv et al., 2021; Sun et al.,
2021). Consequently, further studies are necessary to illuminate the driving
mechanisms and mode-selecting processes, which are still not fully understood
in $\delta$ Sct stars. The rectified light curve shows a peak-to-peak
amplitude of about $\sim 0.43$ mag, which may be suggested as a new triple-
mode HADS.

Figure 7: The short-cadence light curve (sector 40) of TIC 278119167 during 1.5 days (top panel), amplitude spectrum (middle panel), and amplitude spectrum of residual with S/N less than 5.2 (bottom panel). Table 6: A complete list of the Nine identified frequencies for TIC 278119167 (denoted by $f_{i}$).  $f_{i}$ | Frequency | Amplitude | phase | S/N | ID | period ratio |  $Q$ value
---|---|---|---|---|---|---|---
| ($\rm day^{-1}$) | (mmag) | (radians/2$\pi$) |  |  |  |
1 | 12.58960±0.00002 | 176.2±0.1 | 0.5448±0.0002 | 797.74 | F1 | 0.76 | 0.023±0.002
2 | 25.17914±0.00007 | 49.6±0.1 | 0.1273±0.0006 | 565.91 | 2F1 | - | -
3 | 37.7685±0.0003 | 9.7±0.1 | 0.051±0.003 | 142.05 | 3F1 | - | -
4 | 50.3583±0.0006 | 5.3±0.1 | 0.745±0.005 | 95.015 | 4F1 | - | -
5 | 15.671±0.002 | 1.5±0.1 | 0.46±0.01 | 15.85 | F2 | 0.61 | 0.018±0.002
6 | 18.746±0.002 | 1.4±0.1 | 0.19±0.02 | 18.098 | 2F2-F1 | - | -
7 | 28.260±0.003 | 1.0±0.1 | 0.48±0.02 | 16.202 | F1+F2 | - | -
8 | 9.579±0.005 | 0.2±0.1 | 0.47±0.04 | 5.31 | F0 | - | 0.031±0.003
9 | 31.323±0.005 | 0.7±0.1 | 0.64±0.08 | 11.72 | 2F2 | - | -
Note. The frequency resolution is about $f_{res}=0.053\rm day^{-1}$. |  |  |  |  |  |

###  4.6 TIC 710783

Figure 8 represents the light curve (top panel), amplitude spectrum (middle
panel), and amplitude spectrum of the residual (bottom panel) for TIC 710783
(ATO J074.1485-27.6801). The TIC 710783 ($R.A.=04^{h}56^{m}35^{s}.63$,
$DEC=-27^{\circ}40^{\prime}48.56^{\prime\prime}$) was first recognized as a
variable star by Heinze et al. (2018), and then its physical parameters
(temperature, gravity, and frequency scaling relation) were measured by
Barceló Forteza et al. (2020) as a $\delta$ Sct star. We suggest that TIC
710783 shows signs of a single-mode HADS. The TESS space telescope recorded
TIC 710783 during sector 5 for 25.43 days and consisting 17,023 data points.
We identified 11 frequencies constructing the star’s light curve (Table 7),
which includes a radial frequency $\rm F0=12.74451$ $\rm day^{-1}$ ($\rm P0$ =
$0.078$ days), with $Q$ value as 0.039. We observed the harmonics frequencies
($f_{2}$, $f_{3}$, $f_{4}$) and nonradial modes ($f_{5}$,$f_{6}$,…,$f_{11}$).
The light curve’s peak-to-peak value of about $\sim 0.47$ mag and the range of
frequencies propose that TIC 710783 may be a new monomode HADS.

Figure 8: The short-cadence light curve of TIC 710783 during 1.5 days (top panel), amplitude spectrum (middle panel), and amplitude spectrum of residual with S/N less than 5.2 (bottom panel). Table 7: A complete list of the 11 identified frequencies for TIC 710783 (denoted by $f_{i}$). $f_{i}$ | Frequency | Amplitude | phase | S/N | ID | period ratio |  $Q$ value
---|---|---|---|---|---|---|---
| ($\rm day^{-1}$) | (mmag) | (radians/2$\pi$) |  |  |  |
1 | 12.74451±0.00005 | 164.6±0.4 | 0.4635±0.0003 | 738.43 | F0 | - | 0.039±0.003
2 | 25.4890±0.0001 | 57.8±0.4 | 0.828±0.001 | 350.83 | 2F0 | - | -
3 | 38.2334±0.0004 | 19.7±0.4 | 0.29±0.03 | 126.32 | 3F0 | - | -
4 | 50.977±0.001 | 7.8±0.4 | 0.922±0.008 | 68.48 | 4F0 | - | -
5 | 7.255±0.003 | 2.8±0.4 | 0.89±0.02 | 10.83 | nonradial | - | -
6 | 36.277±0.003 | 2.7±0.4 | 0.18±0.02 | 22.87 | nonradial | - | -
7 | 5.489±0.003 | 2.4±0.4 | 0.73±0.02 | 7.88 | nonradial | - | -
8 | 32.745±0.003 | 2.3±0.4 | 0.37±0.02 | 18.55 | nonradial | - | -
9 | 54.508±0.006 | 1.4±0.4 | 0.63±0.04 | 11.05 | nonradial | - | -
10 | 18.231±0.008 | 1.0±0.4 | 0.74±0.06 | 8.35 | nonradial | - | -
11 | 11.838±0.009 | 0.9±0.4 | 0.76±0.07 | 5.44 | nonradial | - | -
Note. The frequency resolution is about $f_{res}=0.058\rm day^{-1}$. |  |  |  |  |  |

###  4.7 TIC 187386415

Figure 9 represents the light curve (top panel), amplitude spectrum (middle
panel), and amplitude spectrum of the residual (bottom panel) for TIC
187386415 (ATO J236.5561-00.4351). TIC 187386415
($R.A.=15^{h}46^{m}13^{s}.48$,
$DEC=-00^{\circ}26^{\prime}06.34^{\prime\prime}$) was classified by Hey et al.
(2021) as a variable star. This study shows that TIC 187386415 is a single-
mode HADS. We analyzed this star for sector 51 with a duration of 19.13 days
and consisting of 6775 data points. We identified four frequencies
constructing the star’s light curve (Table 8), which includes a radial
frequency $\rm F0=14.49465$ $\rm day^{-1}$ ($\rm P0$ = $0.069$ days), with $Q$
value as 0.033. Other frequencies may be introduced as harmonic frequencies
($f_{2}$, $f_{3}$, $f_{4}$). The peak-to-peak amplitude ($\sim 0.31$ mag) and
the frequencies’ range propose that TIC 187386415 may be a new monomode HADS
star.

Figure 9: The short-cadence light curve of TIC 187386415 during 1.5 days (top panel), amplitude spectrum (middle panel), and amplitude spectrum of residual with S/N less than 5.2 (bottom panel). Table 8: A complete list of the Four identified frequencies for TIC 187386415 (denoted by $f_{i}$). $f_{i}$ | Frequency | Amplitude | phase | S/N | ID | period ratio |  $Q$ value
---|---|---|---|---|---|---|---
| ($\rm day^{-1}$) | (mmag) | (radians/2$\pi$) |  |  |  |
1 | 14.49465±0.00003 | 113.8±0.1 | 0.5514±0.0001 | 473.8 | F0 | - | 0.033±0.0045
2 | 28.98936±0.00009 | 41.0±0.1 | 0.8682±0.0005 | 333.91 | 2F0 | - | -
3 | 43.4840±0.0002 | 15.6±0.1 | 0.223±0.001 | 196.24 | 3F0 | - | -
4 | 57.9786±0.0005 | 7.5±0.1 | 0.989±0.003 | 93.54 | 4F0 | - | -
Note. The frequency resolution is about $f_{res}=0.078\rm day^{-1}$. |  |  |  |  |  |

###  4.8 Asteroseismical parameters vs. physical quantities

Using Equation (3), we obtained the absolute magnitudes and their errors
(applying errors of apparent magnitudes and parallaxes) for seven HADS stars.
We calculated the period ($\rm P0$) of each star using the fundamental
frequency (Tables 2-8). Figure 10 (left panel) shows the PL relation for our
HADS stars (colored star markers), the PL relation of Equation (5) (solid
black), Equation (6) (red dashed line), and Equation (7) (cyan dashed–dotted
line). For stars TIC 374753270 and TIC 130474019, the fundamental period and
absolute magnitude agreed with Barac et al. (2022). As shown in Figure 10, all
seven cases satisfy the PL relation for $\delta$ Sct stars. that indicates the
validity of our frequency analysis.

Figure 10 (right panel) examines the first and second overtone periods of TIC
278119167 by PL relations for the first (dashed line) and second
(dashed–dotted line) overtones. The PL relation for the first and second
overtones was obtained by Poro et al. (2021). While the TIC 278119167 star’s
fundamental period (shown as the largest star marker) slightly deviates from
the fundamental PL relation, the first overtone period (the middle-sized star
marker) and the second overtone period (the smallest star marker) entirely
agree with the first and second overtone PL relations. This implies the
validity of the identified fundamental frequency for TIC 278119167.

Figure 10: (Left panel) The period-luminosity (PL) diagram of our HADS stars
(colored star markers). The solid black line shows the relation from Ziaali et
al. (2019) (see Equation 5). The diagonal dashed red line shows the relation
from Barac et al. (2022) (see Equation 6). The cyan dashed–dotted line shows
the relation from Poro et al. (2021) (see Equation 7). The error bars indicate
the errors for absolute magnitudes. (Right panel) The PL relation of TIC
278119167 for the first (cyan dashed line) and second (cyan dotted line)
overtones. While the fundamental mode (the largest star marker) slightly
deviates from fundamental lines, the first (the middle-sized star marker) and
second (the smallest star marker) overtones are entirely consistent with their
PL relations. Figure 11: The scaling metallicity $\rm[Fe/H]_{s}$ (Equation 8)
versus the metallicity [Fe/H] for 176 HADS stars (Netzel & Smolec, 2022).

Stellar metallicity is the abundance of elements heavier than hydrogen and
helium. However, in astrophysics, it is often written as the ratio of iron to
hydrogen relative to the solar value ([Fe/H]; Kotoneva et al. (2002);
Chruslinska & Nelemans (2019)). Netzel & Smolec (2022) obtained the [Fe/H] for
176 HADS stars by applying the MESA, Warsaw code, and optimization procedure.
These 176 HADS stars are mostly multimode ($\rm P0$, $\rm P1$, $\rm P2$,…)
cases. The metallicity distribution for these 176 HADS stars ranges from -0.2
to 0.5 dex, which may be slightly related to the dispersion of period ratio
($\rm P1/\rm P0$ and $\rm P2/\rm P0$). Lv et al. (2023) obtained a linear
relation for period ratios ($\rm P1$/$\rm P0$ and $\rm P2$/$\rm P0$) and
fundamental period $\rm P0$ (Figures 10 and 11 therein). Therefore, we may
consider the $\rm P1$ and $\rm P2$ as a function of $\rm P0$ that allows us to
investigate the scaling relation between metallicity, physical parameters, and
fundamental period ($\rm P0$).

Using the mass ($M$), luminosity ($L$), fundamental period ($\rm P0$), and the
effective temperature ($T_{\rm eff}$) of these 176 HADS stars, we determined
the following scaling relation for [Fe/H] as

${{\rm[Fe/H]}_{s}=\log_{10}\left(\left(\frac{M}{M_{\odot}}\right)^{7.95\pm
0.15}\left(\frac{L}{L_{\odot}}\right)^{-1.83\pm 0.11}\left(\frac{\rm P0(\rm
day)}{0.064}\right)^{0.79\pm 0.14}\left(\frac{T_{\rm eff}}{K}\right)^{0.047\pm
0.02}\right)},$ (8)

where $\rm[Fe/H]_{s}$ is the scaling metallicity obtained by an optimization
procedure for 176 HADS stars. The plus/minus values indicate the standard
error of the power indexes in the model that obtained by the optimization
algorithm. Figure 11 shows the comparison of [Fe/H] and scaling
$\rm[Fe/H]_{s}$ (Equation 8) for HADS stars. Using Equation (8), we obtained
the $\rm[Fe/H]_{s}$ of our seven newly identified HADS stars. We estimated the
metallicity ranging from -0.62 to 0.37 dex (Table 9) for these seven HADS
stars. Equation (8) is the scaling relation for the metallicity of multimode
HADS, while three of our HADS (TIC 374753270, TIC 710783, and TIC 187386415)
are the single-modes stars. Therefore, estimating the metallicity of three
targets using Equation (8) may contain some uncertainty.

The four HADS stars (TIC 374753270, TIC 148357344, TIC 160120432, and TIC
278119167) show lower iron abundance than the Sun, while the rest of the three
HADS (TIC 130474019, TIC 710783, and TIC 187386415) demonstrate positive
[Fe/H]. The metallicities of these seven HADS stars are more significant than
-0.5 dex, indicating metal-rich behavior (McNamara, 2011). The metal-rich
behavior is an essential characteristic of the evolved old stars.

Table 9: TIC number, mass ($M/M_{\odot}$), radius ($R/R_{\odot}$), surface
gravity (log g), effective temperature ($T_{\rm eff}$), absolute $V$-band
magnitude ($M_{\rm V}$), and metallicity ($\rm[Fe/H]_{s}$) for seven HADS
stars.

TIC number | $\ M/M_{\odot}$ | $R/R_{\odot}$ | log g | $T_{\rm eff}$ | $M_{\rm V}$ | $\rm[Fe/H]_{s}$
---|---|---|---|---|---|---
|  |  | (cm/$\rm s^{2}$) | (K) |  | (dex)
374753270 | $1.60\pm 20$ | $3.04\pm 0.22$ | $3.67\pm 0.11$ | $7166\pm 304.2$ | $1.61\pm 0.16$ | $-0.451\pm 0.161$
130474019 | $1.76\pm 0.21$ | $1.93\pm 0.09$ | $4.11\pm 0.10$ | $7573\pm 197.5$ | $2.20\pm 0.18$ | $0.243\pm 0.135$
148357344 | $1.61\pm 0.21$ | $2.99\pm 0.22$ | $3.69\pm 0.06$ | $7183\pm 270$ | $1.51\pm 0.23$ | $-0.465\pm 0.160$
160120432 | $1.59\pm 0.19$ | $2.51\pm 0.13$ | $3.84\pm 0.08$ | $7138\pm 153.8$ | $1.61\pm 0.16$ | $-0.266\pm 0.143$
278119167 | $1.68\pm 0.2$ | $2.68\pm 0.13$ | $3.80\pm 0.08$ | $7369\pm 185.1$ | $1.05\pm 0.16$ | $-0.271\pm 0.154$
710783 | $1.77\pm 0.23$ | $1.92\pm 0.09$ | $4.11\pm 0.07$ | $7604\pm 150.9$ | $2.04\pm 0.22$ | $0.237\pm 0.135$
187386415 | $1.67\pm 0.20$ | $1.93\pm 0.16$ | $4.08\pm 0.08$ | $7340\pm 97.2$ | $2.20\pm 0.08$ | $0.098\pm 130$

  * •

Note.These physical parameters (mass, radius, surface gravity, and effective
temperature) of seven targets are given by Stassun et al. (2018). Using
Equations (3, 8), the absolute $V$-band magnitude ($M_{\rm V}$) and
metallicity ($\rm[Fe/H]_{s}$) were calculated.

##  5 conclusion

To probe the new HADS stars from recent TESS observations of $\delta$ Sct
stars, we recognized seven new targets, namely TIC 374753270, TIC 13047401,
TIC 148357344, TIC 160120432, TIC 278119167, TIC 71083, and TIC 187386415,
which show the peak-to-peak amplitude slightly more significant than 0.3 mag
in their light curves. Using Equation (3) and the Gaia DR3 parallaxes, the
$V$-band absolute magnitude was calculated for these seven stars. Applying the
TESS effective temperatures from Stassun et al. (2018) and the obtained
absolute magnitudes, these seven HADS stars were located in the H-R diagram.
They lie in the $\delta$ Sct stars’ stability strip of the H-R diagram (Figure
2) close to the cooler boundary.

The HADS stars were expected to show a simple amplitude spectrum, including
one to a few large amplitudes. The period ratios, $Q$ values, and PL relation
between the fundamental radial frequency and luminosity were used to show the
validity of mode information for these seven stars. We applied the frequency
analysis to obtain information on the significant frequencies of these stars.
The significant frequencies followed an S/N greater than 5.2. Our analyses
show that the HADS stars were categorized into single modes (TIC 374753270,
TIC 710783, and TIC 187386415), double modes (TIC 130474019 and TIC
160120432), and triple modes (TIC 148357344 and TIC 278119167). We determined
the variety of harmonic and combination frequencies in the amplitude spectrum
of targets (Tables 2-8) as well as nonradial oscillation frequencies for most
of our targets. We derived a scaling relation using 176 HADS stars from Netzel
& Smolec (2022) between metallicity ([Fe/H]) and physical parameters (mass
($M$), luminosity ($L$), and the effective temperature ($T_{\rm eff}$)) and
fundamental period ($\rm P0$). Using Equation (8), we estimated the
metallicity of the seven HADS stars. The metallicity of all seven stars is
greater than -0.62 dex, as expected of the HADS stars.

##  6 Acknowledgments

This paper includes data collected by the TESS mission, which are publicly
available from the Mikulski Archive for Space Telescopes (MAST). 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
would like to thank the Gaia team for providing accurate data from the
European Space Agency (ESA) mission Gaia, processed by the Gaia Data
Processing and Analysis Consortium (DPAC). Funding for the DPAC has been
provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement. This work has been supported
by the Iran National Science Foundation (INSF) under grant No. 4002562. E.Z.
expresses gratitude for that. Also, E.Z. acknowledges financial support from
project PID2019-107061GB-C63 from the ‘Programas Estatales de Generación de
Conocimiento y Fortalecimiento Científico y Tecnológico del Sistema de I+D+i y
de I+D+i Orientada a los Retos de la Sociedad’ and from the grant
CEX2021-001131-S funded by MCIN/AEI/10.13039/501100011033.

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