11institutetext: Departamento de Física Teórica y del Cosmos, Universidad de
Granada, Campus de Fuentenueva s/n, 18071, Granada, Spain
11email<EMAIL_ADDRESS>
# Dating young open clusters using $\delta$ Scuti stars

Results for Trumpler 10 and Praesepe

D. Pamos Ortega  G. M. Mirouh  A. García Hernández  J. C. Suárez Yanes  S.
Barceló Forteza

###### Abstract

Aims. The main goal of this work is to date young open clusters using $\delta$
Sct stars. Seismic indices such as the large separation and the frequency at
maximum power can help to constrain the models to better characterise the
stars. We propose a reliable method to identify some radial modes, which gives
us greater confidence in the constrained models.

Methods. We extract the frequency content of a sample of $\delta$ Sct stars
belonging to the same open cluster. We estimate the low-order large separation
by means of different techniques and the frequency at maximum power for each
member of the sample. We use a grid of models built with the typical
parameters of $\delta$ Sct stars, including mass, metallicity and rotation as
independent variables, and determine the oscillation modes. We select the
observed frequencies whose ratios match those of the models. Once we find a
range of radial modes matching the observed frequencies, mainly the
fundamental mode, we add it to the other seismic parameters to derive the
stellar age. Assuming star groups have similar chemistry and age, we estimate
their mean age by computing a weighted probability density function fit to the
age distribution of the seismically constrained models.

Results. We estimate the age of Trumpler 10 to be $30_{-20}^{+30}$
$\mathrm{Myr}$, and that of Praesepe to be $580\pm 230$ $\mathrm{Myr}$. In
this latter case, we find two apparent populations of $\delta$ Sct stars in
the same cluster, one at $510\pm 140$ $\mathrm{Myr}$ and another at $890\pm
140$ $\mathrm{Myr}$. This may be due to two different formation events,
different rotational velocities of the members in our sample of stars (as
rapid rotation may modify the observed large separation), or to membership of
unresolved binary systems.

###### Key Words.:

Physical data and processes – asteroseismology – Stars: variables: delta Scuti
– The Galaxy: open clusters and associations: general

##  1 Introduction

Determining the age of a star is essential to know its internal physics.
Regarding the dating of a star cluster, the importance lies in understanding
the structure and evolution of the galaxy. However, age is not a direct
observable and inferring it accurately is not an easy task. In addition,
ambiguity arises since we cannot be sure that all the stars in the cluster
formed at the same epoch. Recent works have shown that different populations,
or generations, of stars may coexist within the same cluster (e.g. Bastian &
Lardo, 2018). For instance, Costa et al. (2019) find two distinct populations
of stars aged 176 $\mathrm{Myr}$ and 288 $\mathrm{Myr}$ in NGC 1866, by
combining an analysis of its best-studied Cepheids with that of a very
accurate colour-magnitude diagram obtained with the Hubble Space Telescope
photometry. Other works, such as Krause et al. (2020), assume that most open
clusters feature a single population, as they remain essentially clear of gas
and winds after one stellar formation event. We have assumed this hypothesis
in order to determine a mean age for each of thr clusters we analyse.

Traditionally, isochrone fitting on the Hertzsprung-Russell diagram (HRD) has
been used to date clusters. This method works when dealing with old globular
clusters, where we can find a large sample of stars leaving the main sequence
and evolving past the turn-off point. However, the ambiguity of this method is
greater with young clusters, in which a majority of stars still on the main
sequence (MS). The method based on spectroscopic observations of lithium
(Basri & Martín, 1999; Stauffer et al., 1999) also generates large ambiguities
because of unresolved binary stars (Martín et al., 2001). The relation between
the rotation rate and the age of late F to M stars, called gyrochronology
(Barnes, 2003; Angus et al., 2022; Messina et al., 2022), seems to provide a
method to reduce the uncertainty on the age of evolved clusters. Other methods
based on chemical clocks (da Silva et al., 2012; Spina et al., 2017; Moya et
al., 2022) seem to help reduce the uncertainties, making use of machine
learning techniques. The drawback is that these algorithms are trained with
models of highly evolved stars, for which it has been possible to obtain
reliable spectroscopic observations. Therefore, despite the progress achieved
with these new techniques, we still do not have a reliable method to date
young open clusters.

In Pamos Ortega et al. (2022) (Paper I from now on), we proposed the use of
seismic parameters to date a group of four $\delta$ Sct stars belonging to the
young open cluster $\alpha$ Per. One of these seismic indices is the large
separation, defined as the difference between acoustic modes of the same
degree and consecutive radial orders, related to the mean density and the
surface gravity of the star. This regularity in the frequency pattern is also
present in the low-order regime ($n=[2,8]$) (Suárez et al., 2014; García
Hernández et al., 2015, 2017; Mirouh et al., 2019), where $\delta$ Sct stars
show their oscillation modes. Another parameter is the frequency at maximum
power, directly related to the effective temperature, used in solar-type stars
and found in $\delta$ Sct stars as well (Barceló Forteza et al., 2018, 2020;
Bowman & Kurtz, 2018; Hasanzadeh et al., 2021).

In this work, we date the young open clusters Trumpler 10 and Praesepe, using
a corresponding sample of $\delta$ Sct stars. We use only seismic parameters,
such as the low-order large separation and the frequency at maximum power. We
also include the identification of the fundamental mode of each star,
estimated by comparing the frequency ratios of the observed frequencies with
that of the models. These clusters are of very different ages, which allows us
to establish the possibilities of the method.

The structure of the paper is as follows: in Sect. 2, we provide the estimated
ages of the clusters Trumpler 10 and Praesepe from previous works. In Sect. 3,
we introduce the sample of $\delta$ Sct stars used in this research. In Sect.
4, we present how their seismic parameters have been computed. In Sect. 5, we
describe the details of the grids of models built to characterise our $\delta$
Sct stars. In Sect. 6, we explain the method to estimate the mean age of the
cluster. In Sect. 7, we present the ages we estimate and discuss their
reliability by comparing them with the observed parameters and the literature.
And finally, in Sect. 8, we lay out the main conclusions of our research and
leads to improve our method.

##  2 The ages of Trumpler 10 and Praesepe in the literature

Trumpler 10 and Praesepe have been dated through many different techniques.
Table 1 summarises the ages and metallicities derived in the last twenty
years.

Trumpler 10 (C 0846-423) is an open cluster located in the Vela constellation.
According to the Milky Way Star Clusters catalogue (MWSC, Kharchenko et al.,
2013), it is at a distance of about 417 pc from the Sun and has an age of
$\log({\rm age})=7.380$ ($\approx 34$ $\mathrm{Myr}$). According to Netopil et
al. (2016), it has an age of $40\pm 10$ $\mathrm{Myr}$, with a metallicity of
$[{\rm Fe}/{\rm H}]$ = $-0.12\pm 0.06$, obtained from various photometric
systems and calibrations. The work of Dias et al. (2021) computes the age for
this cluster in $\log({\rm age})=7.753\pm 0.026$ ($\approx 57$
$\mathrm{Myr}$), with a metallicity of $[{\rm Fe}/{\rm H}]=0.043\pm 0.050$.
For these estimates, they used Gaia DR2 photometry and a grid of Padova
isochrones. According to all these references, the age of Trumpler 10 lies
between 34 $\mathrm{Myr}$ and 57 $\mathrm{Myr}$.

Praesepe (M44, NGC2632) is an open cluster located in the Cancer
constellation. Being one of the closest clusters to the Sun, it is also one of
the most studied (see for example Suárez et al., 2002; Meibom & Mathieu, 2005;
Fossati et al., 2008; Brandt & Huang, 2015; Choi et al., 2016; Cummings et
al., 2017; Gaia Collaboration et al., 2018, and references therein). Also
taking as reference the MWSC survey, it is at a distance of about 187 pc and
has an age of $\log({\rm age})=8.920$ ($\approx 729$ $\mathrm{Myr}$).
According to Netopil et al. (2016), it has an age of $730\pm 190$
$\mathrm{Myr}$ and a metallicity of $[{\rm Fe}/{\rm H}]=0.13\pm 0.03$, also
obtained from different photometric systems, as for Trumpler 10. The work of
Dias et al. (2021) yields $\log({\rm age})=8.882\pm 0.035$ ($\approx 762$
$\mathrm{Myr}$), with a metallicity of $[{\rm Fe}/{\rm H}]=0.196\pm 0.039$.
Zhong et al. (2020) estimate the metallicity at $[{\rm Fe}/{\rm H}]=0.22\pm
0.08$, using LAMOST spectroscopy. Meibom & Mathieu (2005) provides an age
estimate for Praesepe of about 630 $\mathrm{Myr}$. They used a completely
different technique, by looking at the circularization of binary systems of
solar-like stars: this circularization happens over time, so systems become
circular at higher and higher separations, so that the measurement of the
period of circular systems yields an age value. Douglas et al. (2019) estimate
an age of $670\pm 67$ $\mathrm{Myr}$. Actually, they computed the age of the
open cluster Hyades, using a gyrochronology model tuned with slow rotators in
Praesepe and the Sun, assuming that the two clusters are coeval, based on the
similarity of their colour-magnitude diagrams, activity, rotation and lithium
abundance. In short, all these references of the last twenty years provide
ages for Praesepe between 590 and 790 $\mathrm{Myr}$.

Table 1: The ages and metallicities for Trumpler 10 and Praesepe, in some of
the main references of the last twenty years, used in this work.

Reference | Age Trumpler 10 ($\mathrm{Myr}$) | Age Praesepe ($\mathrm{Myr}$) | Z Trumpler 10 | Z Praesepe
---|---|---|---|---
Meibom & Mathieu (2005) | - | 630 | - | -
Fossati et al. (2008) | - | $590^{+150}_{-120}$ | - | $0.024\pm 0.002$
Kharchenko et al. (2013) | 34 | 729 | - | $0.025\pm 0.002$
Netopil et al. (2016) | $40\pm 10$ | $730\pm 190$ | $0.014\pm 0.002$ | $0.024\pm 0.002$
Douglas et al. (2019) | - | $670\pm 67$ | - | -
Zhong et al. (2020) | - | - | - | $0.031\pm 0.005$
Dias et al. (2021) | 57 | 762 | $0.018\pm 0.004$ | $0.028\pm 0.002$

##  3 The data

Firstly, we cross-match the VizieR Online Data Catalogue Gaia DR2 of open
cluster members (Cantat-Gaudin et al., 2018) and the TESS Input Catalogue
(TIC, Stassun, 2019), searching possible $\delta$ Sct stars belonging to the
same open cluster.

According to the definition of a pure $\delta$ Sct from Grigahcène et al.
(2010) and Uytterhoeven et al. (2011), we find five candidates in the field of
Trumpler 10 (Table 2) and six in the field of Praesepe (Table 3). For our
Praesepe stars, we find values for the projected rotational velocity $v\sin
i$, the metallicity and the spectral type, consulting the available references
in the Simbad Astronomy Database111https://simbad.unistra.fr/simbad/. For the
Trumpler 10 stars, only data about the spectral type is available. These
parameters are useful in our discussion of the results in Sect. 7.

We perform a frequency analysis using data from sector 35 of the TESS mission
(Ricker et al., 2014) for the Trumpler 10 sample, with approximately 13800
points; and from sector 45 for Praesepe, with approximately 15500 points. In
both cases, the cadence is about 2 minutes and the Rayleigh resolution is
approximately 0.041 $d^{-1}$. We use the Pre-Search Data Conditioned (PDC)
light curves, corrected for instrumental effects, that are publicly available
through the TESS Asteroseismic Science Consortium222https://tasoc.dk (TASC).

Using MultiModes333https://github.com/davidpamos/MultiModes, we extract the
frequency content of each star in our sample. It is a Python code that
calculates the Fast Lomb-Scargle periodogram (Press & Rybicki, 1989) of a
light curve. It extracts, one by one, a limited number of significant signals,
and uses their corresponding parameters (amplitudes, frequencies) to fit the
total signal to a multisine function with a non-linear least squares
minimisation. We adopt a signal-to-noise ratio S/N ¿ 4.0 as a stop criterion
(Breger et al., 1993), to avoid spurious frequencies, and we filter possible
frequency combinations. The code is presented in detail in Paper I and in the
public repository.

Fig. 1 and Fig. 2 show, respectively, the extracted frequency spectrum of each
$\delta$ Sct candidate in Trumpler 10 and Praesepe. The values of the ten
highest amplitudes frequencies for each star in both clusters are shown in
Table A.1 (Appendix A). The table of all extracted frequencies is available
on-line.

Table 2: Stellar parameters of the selected targets in Trumpler 10. From left
to right: TESS magnitude, luminosity, mass, radii, density, surface gravity,
effective temperature, cluster member probability and spectral type.
References: (1)Stassun (2019), (2)Cantat-Gaudin et al. (2018), (3)Skiff
(2014).

TIC | $T_{mag}^{(1)}$ | $\log(L/L_{\odot})^{(1)}$ | $M(M_{\odot})^{(1)}$ | $R(R_{\odot})^{(1)}$ | $\bar{\rho}(\bar{\rho}_{\odot})$ | $\log g^{(1)}$ | $T_{\mathrm{eff}}(K)^{(1)}$ | $P_{member}^{(2)}$ | Spectral type(3)
---|---|---|---|---|---|---|---|---|---
28943819 | $10.545\pm 0.007$ | $1.11\pm 0.04$ | $2.2\pm 0.3$ | $1.60\pm 0.06$ | $0.53\pm 0.14$ | $4.37\pm 0.08$ | $8646\pm 161$ | 1.0 | -
30307085 | $10.294\pm 0.006$ | $1.28\pm 0.04$ | $2.5\pm 0.3$ | $1.47\pm 0.04$ | $0.80\pm 0.17$ | $4.51\pm 0.06$ | $9931\pm 202$ | 0.8 | A0
28944596 | $10.354\pm 0.006$ | $1.12\pm 0.03$ | $2.1\pm 0.3$ | $1.72\pm 0.06$ | $0.41\pm 0.10$ | $4.28\pm 0.07$ | $8383\pm 149$ | 1.0 | A2
271061334 | $10.347\pm 0.006$ | $1.12\pm 0.03$ | $2.2\pm 0.3$ | $1.58\pm 0.05$ | $0.56\pm 0.13$ | $4.38\pm 0.07$ | $8773\pm 170$ | 0.9 | -
271062192 | $10.238\pm 0.006$ | $1.16\pm 0.03$ | $2.2\pm 0.3$ | $1.68\pm 0.05$ | $0.46\pm 0.11$ | $4.33\pm 0.07$ | $8689\pm 158$ | 1.0 | A3

Table 3: Stellar parameters of the selected targets in Praesepe. From left to
right: TESS magnitude, luminosity, mass, radii, density, surface gravity,
effective temperature, cluster member probability, projected rotational
velocity, metallicity and spectral type. References: (1)Stassun (2019),
(2)Cantat-Gaudin et al. (2018), (3)Cummings et al. (2018), (4)Fossati et al.
(2008), (5)Bochanski et al. (2018).

TIC | $T_{mag}^{(1)}$ | $\log(L/L_{\odot})^{(1)}$ | $M(M_{\odot})^{(1)}$ | $R(R_{\odot})^{(1)}$ | $\bar{\rho}(\bar{\rho}_{\odot})$ | $\log g^{(1)}$ | $T_{\mathrm{eff}}(K)^{(1)}$ | $P_{member}^{(2)}$ | v sini (km/s) | [Fe/H](4) | [Fe/H](5) | Spectral type(4)
---|---|---|---|---|---|---|---|---|---|---|---|---
175194881 | $7.966\pm 0.006$ | $1.18\pm 0.02$ | $1.9\pm 0.3$ | $2.10\pm 0.07$ | $0.20\pm 0.05$ | $4.07\pm 0.08$ | $7873\pm 125$ | 1.0 | $85^{(4)}$ | 0.26 | - | A7V
175264376 | $8.233\pm 0.008$ | $1.09\pm 0.01$ | $1.7\pm 0.3$ | $2.13\pm 0.09$ | $0.17\pm 0.05$ | $4.01\pm 0.08$ | $7416\pm 141$ | 1.0 | $200^{(4)}$ | 0.12 | 0.08 | F0Vn
175265807 | $8.134\pm 0.006$ | $1.10\pm 0.02$ | $1.9\pm 0.3$ | $1.92\pm 0.06$ | $0.26\pm 0.07$ | $4.14\pm 0.08$ | $7826\pm 126$ | 1.0 | $135^{(3)}$ | - | 0.01 | -
175291778 | $7.771\pm 0.006$ | $1.26\pm 0.01$ | $1.9\pm 0.3$ | $2.31\pm 0.06$ | $0.15\pm 0.04$ | $3.98\pm 0.08$ | $7865\pm 126$ | 1.0 | $150^{(4)}$ | 0.09 | -0.08 | A7V
184914505 | $8.255\pm 0.007$ | $1.00\pm 0.02$ | $1.7\pm 0.3$ | $1.94\pm 0.06$ | $0.23\pm 0.06$ | $4.09\pm 0.08$ | $7369\pm 108$ | 1.0 | $90^{(4)}$ | 0.31 | - | A5
184917633 | $8.155\pm 0.006$ | $1.08\pm 0.02$ | $1.7\pm 0.3$ | $2.08\pm 0.07$ | $0.19\pm 0.05$ | $4.03\pm 0.08$ | $7443\pm 122$ | 1.0 | $155^{(4)}$ | -0.02 | -0.02 | A5

Figure 1: Frequency spectrum of the sample of $\delta$ Sct stars candidates in
the field of Trumpler 10. The red dotted line is the significance threshold,
S/N=4. Figure 2: Same as Fig. 1 for the Praesepe sample.

##  4 Seismic parameters

Sometimes it is possible to find regularities in the complex frequency pattern
of a $\delta$ Sct star (García Hernández et al., 2009; Paparó et al., 2016;
Bedding et al., 2020). Following the same techniques as in García Hernández et
al. (2009, 2013); Ramón-Ballesta et al. (2021) and used in Paper I, we
estimate the large separation in the low-order regime, $\Delta\nu_{low}$. Fig.
3 shows an example of this, where we use the discrete Fourier transform (DFT),
the autocorrelation diagram (AC) applied on the frequencies, the frequency
difference histogram (FDH) and the échelle diagram (ED), in order to find
regularities in the frequency content of TIC 28943819 (see Appendix B for the
rest of our sample). The theoretical works of García Hernández et al. (2009);
Reese et al. (2017) use the AC and the FT to search for the low-order large
separation. They point out that we expect to see the large separation and its
submultiples in the DFT, and its multiples in the AC and the FDH (except in
the case where the $l=1$ modes lie halfway between the $l=0$ modes, as in
solar-type stars, where we also find half the large separation). No method, by
itself, is objective enough to yield a reliable measurement of
$\Delta\nu_{low}$, except in very few cases. Our criterion requires finding
the same peaks in at least two of the methods used for measuring
$\Delta\nu_{\mathrm{low}}$ or half $\Delta\nu_{\mathrm{low}}$. We estimate the
uncertainties using the width of the peaks in the DFT or the AC, depending on
the case. Of all the stars analysed this way, TIC 28944596 and TIC 271062192
are the most difficult cases. TIC 28944596 (see Fig. B.2) only has a few
frequencies. Its DFT shows a peak a bit above 20 $\mu\mbox{Hz}$ and another
around 40 $\mu\mbox{Hz}$. The AC also shows two peaks very close to 80
$\mu\mbox{Hz}$, and the FDH shows three peaks between 70 and 80
$\mu\mbox{Hz}$. For all these reasons, we estimate the large separation to be
80 $\mu\mbox{Hz}$ for this star. TIC 271062192 (Fig. B.2) is an even more
complicated case, because the AC, the FDH and the ED do not show regularities
in the frequency content. The only evidence here is the DFT, that shows three
congruent peaks around 19, 38 and 76 $\mu\mbox{Hz}$. This is why we have
retained a large separation estimate of 76 $\mu\mbox{Hz}$.

Figure 3: Estimated low-order large separation for TIC 28943819, using the
autocorrelation diagram (top left), the discrete Fourier transform (top
right), the frequency difference histogram (bottom left), and the échelle
diagram (Hey & Ball, 2020, bottom right).

Barceló Forteza et al. (2020) find a scaling relation between the frequency at
maximum power ($\nu_{\mathrm{max}}$) and the effective temperature in a sample
of more than 2000 $\delta$ Sct stars. They define $\nu_{\mathrm{max}}$ as the
average frequency weighted over the amplitudes, $A_{i}$, of the frequencies,
$\nu_{i}$, of the envelope,

$\nu_{\rm max}=\frac{\sum{(A_{i}\cdot\nu_{i})}}{\sum{A_{i}}}.$ (1)

Another independent study (Hasanzadeh et al., 2021) finds a similar
correlation with a sample of more than 400 $\delta$ Sct stars. They use the AC
method on several windows of the S/N periodogram, to obtain the range of
frequencies for the modes’ envelope. Then, a Gaussian curve is fitted to the
mean collapsed correlation with each window. The peak of this curve is their
$\nu_{\mathrm{max}}$. However, such correlation between $\nu_{\mathrm{max}}$
and the effective temperature is recently debated by Bedding et al. (2023),
who do not see it with a sample of 35 $\delta$ Sct stars in the Pleiades
cluster. The $\nu_{\mathrm{max}}$ definitions of Barceló Forteza et al. (2020)
and Hasanzadeh et al. (2021), both based on the envelope, are more stable than
choosing the frequency with the highest amplitude, as is done by Bedding et
al. (2023), since this frequency is more likely to vary with time in the
presence of non-linear effects than the envelope. Despite the large
uncertainty involved in using $\nu_{\mathrm{max}}$ as a seismic parameter, we
decide to trust it as a seismic indicator.

Following Barceló Forteza et al. (2020), we relate $\nu_{\rm max}$ and
$\tilde{T}_{\mathrm{eff}}$. The relation depends on the value of $\log g$, so
for Trumpler 10 ($\log g\approx 4.3)$ we use

$\tilde{T}_{\mathrm{eff}}=(3.5\pm 0.1)\nu_{\rm max}^{(\mu Hz)}+(6460\pm
40)^{(K)},$ (2)

while for Praesepe ($\log g\approx 4.0)$ we use

$\tilde{T}_{\mathrm{eff}}=(3.8\pm 0.2)\nu_{\rm max}^{(\mu Hz)}+(6750\pm
40)^{(K)}.$ (3)

The estimated values for $\Delta\nu_{\rm low}$, $\nu_{\mathrm{max}}$ and their
corresponding seismic temperatures, $\tilde{T}_{\mathrm{eff}}$, for our sample
of $\delta$ Sct stars in Trumpler 10 and Praesepe are shown in Table 4.

Comparing the values of TIC effective temperature and the seismic temperature,
we see that the agreement is good for the majority of the sample, taking into
account the large uncertainties of the seismic temperature. However, TIC
30307085 and TIC 271062192 are discrepant, with differences between the two
estimates above 1000K, which can be the signature of a binary companion or
gravity darkening induced by rapid rotation.

Table 4: Seismic indices of the selected targets from Trumpler 10 and
Praesepe: the low-order large separation, the frequency at maximum power and
its corresponding seismic and TIC effective temperature.

TIC |  $\Delta\nu_{\mathrm{low}}$($\mu\mbox{Hz}$) |  $\nu_{\mathrm{max}}$($\mu\mbox{Hz}$) |  $\tilde{T}_{\mathrm{eff}}$ (K) | TIC $T_{\mathrm{eff}}$ (K)
---|---|---|---|---
Trumpler 10 |
28943819 | $82\pm 2$ | $510\pm 30$ | $8250\pm 200$ | $8646\pm 161$
30307085 | $84\pm 1$ | $710\pm 60$ | $8950\pm 320$ | $9931\pm 202$
28944596 | $80\pm 2$ | $330\pm 80$ | $7620\pm 350$ | $8383\pm 149$
271061334 | $80\pm 2$ | $650\pm 50$ | $8740\pm 280$ | $8773\pm 170$
271062192 | $76\pm 2$ | $290\pm 90$ | $7480\pm 380$ | $8689\pm 158$
Praesepe |
175194881 | $58\pm 1$ | $350\pm 30$ | $8080\pm 220$ | $7873\pm 125$
175264376 | $52\pm 3$ | $210\pm 60$ | $7550\pm 310$ | $7416\pm 141$
175265807 | $57\pm 2$ | $360\pm 40$ | $8120\pm 260$ | $7826\pm 126$
175291778 | $52\pm 3$ | $200\pm 70$ | $7510\pm 350$ | $7865\pm 126$
184914505 | $56\pm 1$ | $320\pm 60$ | $7970\pm 330$ | $7369\pm 108$
184917633 | $56\pm 1$ | $270\pm 80$ | $7780\pm 400$ | $7443\pm 122$

##  5 The grids of models

As $\delta$ Sct are usually moderate to fast rotators ($v\sin i$ ¿ 100
$\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$), the models have to take
into account the stellar structure deformation that occurs at these speeds.
This centrifugal flattening reduces the value of the mean density, directly
related to one of the seismic indices that we are using here, the large
separation. For this reason, we compute our models with the mesa code, version
15140, (Paxton, 2019), and the related oscillations with the filou code
(Suárez & Goupil, 2008), taking rotation into account up to second order in
the perturbative theory for the adiabatic oscillation computation (including
near-degeneracy effects and stellar structure deformation).

Following Paper I, we build two grids of representative models to characterise
$\delta$ Sct stars during their stay on the Pre-Main Sequence (PMS) and the
MS, one for Trumpler 10 and another for Praesepe. In Table 5 we introduce
their main parameters. We used the default nuclear reactions network,
basic.net, provided by mesa. At the zero-age main sequence (ZAMS), X being
constant, when Z increases, Y decreases by the same amount, so that X + Y + Z
= 1. After testing exponential overshooting values $f_{0}=0.002$ and
$f=0.022$, we find no significant impact and thus use no overshooting. We
include internal differential rotation. We computed models initiating rotation
at the ZAMS. The values for the initial angular velocity to critical velocity
ratio are between $0.1\Omega_{c}$ and $0.5\Omega_{c}$, avoiding higher values
that may lie beyond the limits of the perturbative theory. We compute $p$
modes between $n=1$ and the cut-off frequency, the limiting frequency up to
which acoustic modes can propagate without damping. We use the modes in the
low-order regime, between $n=2$ and $n=8$, and with degrees between $l=0$ and
$l=3$, to calculate the low-order large separation, as proposed by Suárez et
al. (2014).

Table 5: Parameters of the stellar model grids built with the mesa code. From
top to bottom: mass, metallicity (both for Trumpler 10 and Praesepe), the
initial angular velocity to critical velocity ratio and the mixing-length
parameter.

Parameter | Range | Step
---|---|---
$M({\rm M}_{\odot})$ | [1.60, 2.50] | 0.01 ${\rm M}_{\odot}$
Z (Trumpler 10) | [0.016, 0.020] | 0.002
Z (Praesepe) | [0.028, 0.032] | 0.002
$\Omega/\Omega_{c}$ | [0.1, 0.5] | 0.1
$\alpha$ | 2.0 | Fixed

##  6 The method for estimating ages

We estimate the seismic age of each cluster following these steps:

  1. 1.

For each star, we constrain the models using its estimated values of
$\Delta\nu_{\mathrm{low}}$ and $\tilde{T}_{\mathrm{eff}}$, taking into account
their corresponding uncertainties.

  2. 2.

We compute the ratios of the observed frequencies, in order to select the
frequencies with ratios that match those of the models (Table 6).

  3. 3.

Once we find a range of radial modes matching the observed frequencies, we use
them with the models selected at step 1, to better constrain the stellar ages.

  4. 4.

After applying steps 1-3 to all stars in the same group, we plot the age
distribution weighted histograms of all the constrained models.

  5. 5.

We compute the best weighted probability density function (WPDF) over the
histogram. For that, we assume a normal distribution, and we use, as initial
guesses, the maximum likelihood estimation: the weighted mean of all the ages
of the constrained models, and its corresponding weighted standard deviation.
We finally take, for the age of the cluster and its corresponding standard
deviation, the parameters of the fitted WPDF in a $\chi^{2}$ minimisation
process.

We compute the weight of each constrained model, $p$ given in Eq. (4), taking
into account the following assumptions:

  1. 1.

Each evolutionary track computed with mesa is oversampled at low ages. To
compensate for this effect, our WPDF is proportional to the time step, $\Delta
t$, divided by the total time, $t$, of the model in its evolutionary track.

  2. 2.

We also evaluate the probability of the models corresponding to each star in
the sample. Models corresponding to stars with better measured
$\Delta\nu_{\mathrm{low}}$ and $\tilde{T}_{\mathrm{eff}}$ contribute with a
greater probability in the estimated age of the cluster. Then the WPDF is
inversely proportional to the relative uncertainties of
$\frac{e_{\Delta\nu_{\rm low}}}{\Delta\nu_{\rm low}}$ and
$\frac{e_{\tilde{T}_{\mathrm{eff}}}}{\tilde{T}_{\mathrm{eff}}}$ of each star.

  3. 3.

Regarding the probability that a model is the age of the whole cluster, we
assign it a weight proportional to the number of stars that are this same age,
$n_{\rm stars}$ divided by the total number of stars in the sample, ${N_{\rm
stars}}$.

  4. 4.

The number of constrained models with the same age, $n$, divided by the total
number of models in the grid, $N$.

Combining everything, we obtain

$p=\frac{\Delta t}{t}\frac{\Delta\nu_{\rm low}}{e_{\Delta\nu_{\rm
low}}}\frac{\tilde{T}_{\mathrm{eff}}}{e_{\tilde{T}_{\mathrm{eff}}}}\frac{n_{\rm
stars}}{N_{\rm stars}}\frac{n}{N}.$ (4)

The formula for $\chi^{2}$ (Eq. 5) has been applied over the densities of the
histogram, his(age), in order to obtain the best possible fit to a normal
distribution, norm(age):

$\chi^{2}=\sum_{\rm bins}{\frac{\rm(his(age)-norm(age))^{2}}{\rm
norm(age)^{2}}}.$ (5)

Table 6: The fundamental mode to radial overtone ratios in our mesa/filou
grids of models, with their corresponding standard deviations.

Relationship | Value with 1$\sigma$ uncertainty
---|---
$f_{1}/f_{2}$ | $0.77\pm 0.01$
$f_{1}/f_{3}$ | $0.63\pm 0.02$
$f_{1}/f_{4}$ | $0.53\pm 0.02$
$f_{1}/f_{5}$ | $0.45\pm 0.02$
$f_{1}/f_{6}$ | $0.40\pm 0.02$
$f_{1}/f_{7}$ | $0.35\pm 0.01$
$f_{1}/f_{8}$ | $0.31\pm 0.01$

Fig. 4 show the positions and the ranges of the possible radial overtones in
the frequency spectrum of TIC 28943819 (see Appendix C for the rest of our
sample). These ranges are too wide in some cases because we have sampled the
whole grid for identification. Then, the inclusion of the fundamental mode has
a minimal impact on the constraints we derive on the models, but it helps
confirm what we obtain from the other seismic parameters. The identification
has failed estimating the ranges for the possible radial overtones only in the
cases with fewer than 30 extracted significant frequencies: TIC 175194881 in
Praesepe, and TIC 30307085 and TIC 271061334 in Trumpler 10. That’s why in
Appendix C there are seven figures, instead of 10, the total number of stars
in our sample, plus TIC 28943819.

Figure 4: Ranges for the possible radial overtones in the frequency spectrum
of TIC 28943819.

##  7 Results and discussion

###  7.1 Trumpler 10

The HRD of Fig. 5 (left panel) shows the ages of the seismically constrained
models of our sample of $\delta$ Sct stars in Trumpler 10. We can see that
they are very close to the ZAMS. Focussing on models between 1.60 and 2.00
${M_{\odot}}$ (Fig. 5, right panel), the constrained models show that TIC
28944596 and TIC 271062192, the least massive stars of the sample, seem to be
older than the rest of the sample. Maybe both stars are actually older, or
maybe these apparent older ages have to do with a gravity-darkening effect or
a possible binary companion. The observed larger radii, the lower densities
and higher luminosities (presented in Table 2) could explain the three
hypotheses.

To determine the mean age of the group, we first compute the age weighted
histograms corresponding to every star of the sample, from the seismically
constrained models (Fig. 6, left panel). We then estimate the mean age of the
whole group, as a single population, by calculating the best possible
distribution on the histograms, using a normal WPDF, as explained in Sec. 6
(Fig. 6). The result is a mean age of around $30_{-20}^{+30}$ $\mathrm{Myr}$,
very close to the ZAMS. This is a younger age estimate than those referenced
in Sect. 2, compatible with estimates of Kharchenko et al. (2013); Netopil et
al. (2016), of around 34 $\mathrm{Myr}$ and 40 $\mathrm{Myr}$, respectively.
Uncertainties probably emerge because seismic parameters, including the large
separation, evolve rapidly for stars on the PMS.

Recent theoretical works show that the PMS is a complex phase. For example,
Kunitomo et al. (2017) claim that the spread in luminosity during the PMS can
be explained through different efficiencies at which the accreted material is
converted into internal energy for each star. For these very young clusters,
it seems that we need other parameters, in addition to the seismic ones we
use, to date the stars with greater accuracy. This is confirmed by Steindl et
al. (2022), according to whom different PMS accretion scenarios cause
differences in the pulsation modes, thus leaving an imprint on the frequency
content of a $\delta$ Sct star. Seismology of PMS stars has a lot to say about
their interior structure.

Compared to the works of Murphy et al. (2021); Steindl et al. (2022), our
uncertainties are one order of magnitude larger with the stars of Trumpler 10.
Murphy et al. (2021) compute $\Delta\nu_{\mathrm{low}}$ $=6.83\pm 0.01\
d^{-1}$ for the PMS star HD 139614, a value one order of magnitude more
precise than our measured large separations. By scanning a variety of models
for mode identification, we sample the entire relevant parameter space, which
makes uncertainties larger but more realistic. We limit our mode
identification precision by avoiding over-reliance on our models.

Figure 5: Left: HRDs of the evolutionary tracks of our grid of representative
models for the sample of $\delta$ Sct stars in Trumpler 10, the ages of the
seismically constrained models have been colour coded. Right: Zoom between
1.60 $M_{\odot}$ and 2.00 $M_{\odot}$, distinguishing between models for each
of the stars in the sample by different colours. Figure 6: Age weighted
histograms of our sample of stars in Trumpler 10. Left: Weighted histogram of
each member of the sample. Right: Weighted histogram of the whole sample. The
red solid line shows the computed PDF with a mean age of $30_{-20}^{+30}$
$\mathrm{Myr}$. Table 7: Constrained parameters of the models corresponding to
our selected targets in Trumpler 10, with their corresponding standard
deviations.

TIC | $M(M_{\odot})$ | $R(R_{\odot})$ | $\bar{\rho}(\bar{\rho}_{\odot})$ | $\log g$ | $\tilde{T}^{\mathrm{eff}}(K)$ | $log(L/L_{\odot})$ |  $v_{rot}$($\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$) | Age ($\mathrm{Myr}$)
---|---|---|---|---|---|---|---|---
28943819 | $1.72\pm 0.04$ | $1.51\pm 0.02$ | $0.50\pm 0.02$ | $4.31\pm 0.01$ | $8250\pm 120$ | $0.97\pm 0.03$ | $70_{-40}^{+40}$ | $30_{-20}^{+30}$
30307085 | $1.87\pm 0.05$ | $1.53\pm 0.02$ | $0.52\pm 0.01$ | $4.34\pm 0.01$ | $8920\pm 180$ | $1.12\pm 0.04$ | $60_{-10}^{+20}$ | $20_{-10}^{+20}$
28944596 | $1.64\pm 0.03$ | $1.51\pm 0.02$ | $0.48\pm 0.02$ | $4.29\pm 0.01$ | $7780\pm 120$ | $0.87\pm 0.03$ | $80_{-40}^{+40}$ | $40_{-30}^{+40}$
271061334 | $1.87\pm 0.05$ | $1.58\pm 0.03$ | $0.48\pm 0.02$ | $4.31\pm 0.01$ | $8740\pm 160$ | $1.11\pm 0.04$ | $80_{-50}^{+50}$ | $20_{-15}^{+30}$
271062192 | $1.64\pm 0.03$ | $1.55\pm 0.02$ | $0.44\pm 0.02$ | $4.27\pm 0.01$ | $7670\pm 120$ | $0.87\pm 0.03$ | $110_{-50}^{+50}$ | $40_{-30}^{+40}$

Table 8: Same as Table 7 for the Praesepe sample.

TIC | $M(M_{\odot})$ | $R(R_{\odot})$ | $\bar{\rho}(\bar{\rho}_{\odot})$ | $\log g$ | $\tilde{T}^{\mathrm{eff}}(K)$ | $log(L/L_{\odot})$ |  $v_{rot}$($\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$) | Age (Myr)
---|---|---|---|---|---|---|---|---
175194881 | $1.83\pm 0.03$ | $1.92\pm 0.02$ | $0.26\pm 0.01$ | $4.14\pm 0.01$ | $7680\pm 100$ | $1.06\pm 0.03$ | $130\pm 60$ | $470\pm 120$
175264376 | $1.73\pm 0.05$ | $2.03\pm 0.05$ | $0.21\pm 0.01$ | $4.06\pm 0.02$ | $7190\pm 160$ | $0.99\pm 0.05$ | $110\pm 50$ | $880\pm 130$
175265807 | $1.85\pm 0.04$ | $1.94\pm 0.03$ | $0.25\pm 0.01$ | $4.13\pm 0.01$ | $7730\pm 130$ | $1.08\pm 0.04$ | $140\pm 60$ | $480\pm 130$
175291778 | $1.72\pm 0.06$ | $2.02\pm 0.06$ | $0.21\pm 0.02$ | $4.06\pm 0.02$ | $7150\pm 180$ | $0.98\pm 0.06$ | $120\pm 50$ | $890\pm 150$
184914505 | $1.82\pm 0.05$ | $1.96\pm 0.03$ | $0.24\pm 0.01$ | $4.11\pm 0.01$ | $7590\pm 160$ | $1.05\pm 0.05$ | $130\pm 60$ | $580\pm 130$
184917633 | $1.77\pm 0.06$ | $1.93\pm 0.03$ | $0.25\pm 0.01$ | $4.11\pm 0.01$ | $7430\pm 200$ | $1.01\pm 0.06$ | $140\pm 50$ | $600\pm 160$

###  7.2 Praesepe

Fig. 7 shows the HRD of the seismically constrained models for the $\delta$
Sct stars group in Praesepe. Two stars, TIC 175264376 and TIC 175291778,
clearly appear to be older than the rest of the sample (right panel). This is
more evident in Fig. 8 (top left panel), where we plot the age weighted
histogram for every star of the sample. If we consider the sample as a single
population, then the computed WPDF (bottom panel) give us a mean age of
$580\pm 230$ $\mathrm{Myr}$, younger if we compare it to the references cited
in Sect. 2, but in good agreement with them. It is significant that our
estimate is very near of the age used by Fossati et al. (2008),
$590_{-120}^{+150}$ $\mathrm{Myr}$, where they calculated the metallicity of
the cluster from an abundance analysis of A- and F-type stars. Five of them
have been used in the present work.

The discrepant large separations and densities of TIC 175264376 and TIC
175291778 can be explained through rotation, a different age, binary systems
or less trustful estimated seismic parameters. First, a rapid rotation may
modify the value of the large separation, although not the scaling relation
between the large separation and the mean density (García Hernández et al.,
2015; Mirouh et al., 2019). Their relatively higher TIC radii and lower TIC
densities (Table 3), are in line with the lower low-order large separations
and frequencies at maximum power we estimate for both stars (Table 4). The
high value of the projected rotation velocity of TIC 175264376 (200
$\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$) is very significant in this
sense. Our 1D models cannot be reliably applied to such high rotational
velocities. We then need 2D models to characterise rapidly-rotating stars. In
a more evolved cluster as Praesepe, rotation and other internal mixing
phenomena can affect the stars differently over time.

Second, these bigger and lower-density stars with masses similar to the
others, can also simply be more evolved. In Table 3 we can see that these
stars, plus TIC 175194881, may be late A or earlier F stars, while the other
three may be middle A stars, according to Fossati et al. (2008). These could
point to two different populations of stars.

Third, the outlier stars may be in binary systems. Their luminosities would be
brighter than models of single stars would suggest, making them appear more
evolved.

Then, we revisit our one-population assumption, as there could be two
populations, in which the four younger stars would be ’Pop 1’ and TIC
175264376 and TIC 175291778 would be ’Pop 2’ (Fig. 8, top right panel). Once
the WPDF of both populations are computed, we obtain a mean age of $510\pm
140$ $\mathrm{Myr}$ for ‘Pop 1’, and a mean age of $890\pm 140$ $\mathrm{Myr}$
for ‘Pop 2’. These histograms are quite different from the histogram with a
single population, which brings us to a fourth explanation for this apparent
bimodality. Compared to ‘Pop 1’ stars, those in ‘Pop 2’ contribute less weight
to the WPDF of the totality of the constrained models, due to their larger
uncertainties in the measurements of the low-large separation and the seismic
effective temperature. Then, assuming one single population, the age of the
cluster is closer to the age of ‘Pop 1’ than the age of ‘Pop 2’.

As we can see in Tables 7 and 8, the models are not well constrained in terms
of rotational velocity. Estimating the rotation rate of some stars of the
group would yield further constraints on the models, especially given the
dependence of the seismic parameters such as the large separation or the
frequency ratios on rotation (Suárez et al., 2006).

Murphy et al. (2022) use a grid of non-rotating stars to model the three
slowest rotators from the Pleiades sample, in order to verify their mode
identification. However, they do not model the other two stars, rapid rotators
V1228 Tau (vsini = 200 $\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$) and
V650 Tau (vsini = 230 $\mathrm{km}\leavevmode\nobreak\ \mathrm{s}^{-1}$), for
which the échelle diagrams are more ambiguous. They also conclude that
rotating models are required for a more accurate inference of the
asteroseismic parameters, including the mode identification. A high rotation
mixes the modes in such a way that it is not easy to identify them within such
a complex spectrum. In this analysis, and that of Paper I, we have included
rotating models and defined a strategy that will serve as a stepping stone
towards a complete mode identification in rapid rotators. Then, to advance
this strategy, we need a method to help us confidently interpret the rotation
frequency in the dense frequency spectrum of $\delta$ Sct stars. It is also
crucial to obtain a more reliable mode identification, through longer exposure
observations that allow a higher resolution in the frequency spectrum. We hope
that future missions, such as the ESA projects
PLATO444https://sci.esa.int/web/plato (PLAnetary Transits and Oscillations of
stars, Rauer et al., 2014) and HAYDN555http://www.asterochronometry.eu/haydn/
(High-precision AsteroseismologY in DeNse stellar fields, Miglio et al.,
2021), will provide higher resolution photometry of stars belonging to
clusters, leading to accurate age estimates.

Figure 7: The same as Fig. 5 for Praesepe. Figure 8: Age weighted histograms
of the seismically constrained models of our sample of stars in Praesepe 10.
Top left: Weighted histograms for each member of the sample. Top right:
Weighted histograms for the apparent two $\delta$ Sct stars populations: Pop 1
groups the stars TIC 175194881, TIC 175265807, TIC 184914505 and TIC
184917633, and Pop 2 groups the stars TIC 175264376 and TIC 175291778. The red
solid line is the computed PDF of Pop 1, with a mean age of $510\pm 140$
$\mathrm{Myr}$. The blue solid line is the computed PDF of Pop 2, with a mean
age of $890\pm 140$ $\mathrm{Myr}$. Bottom: Weighted histogram of the whole
sample, considered as one single population. The green solid line is the
computed PDF, with a mean age of $580\pm 230$ $\mathrm{Myr}$.

##  8 Conclusions

The use of asteroseismology to date young open clusters provides promising
results, despite the limitations of statistical techniques applied to samples
with such small numbers of stars. We have tested a seismic method with three
open clusters of different ages. With $\alpha$ Per we obtained the first
results in Pamos Ortega et al. (2022). In this work we extend the research to
Trumpler 10 and Praesepe, and we estimate the ranges with the possible values
of some radial overtones and the fundamental mode.

Regarding Trumpler 10, we find five $\delta$ Sct star candidates never before
classified as such, with a mean age of $30_{-20}^{+30}$ $\mathrm{Myr}$. The
uncertainty is large due to how close they are to the PMS, where stars evolve
rapidly. Other parameters are needed to better constrain the models near the
ZAMS, and to more accurately date stars.

Regarding Praesepe, we find a new possible $\delta$ Scuti star, TIC 184917633,
that we add to our sample of five previously known $\delta$ Sct stars. We
estimate the mean age of this star group to $580\pm 230$ $\mathrm{Myr}$, in
good agreement with the literature. Two of the six stars in the sample seem to
be older than the rest. The different values in their parameters, especially
the spectral type, support the thesis of two stellar populations: one with a
mean age of $510\pm 140$ $\mathrm{Myr}$ and another with a mean age of $890\pm
140$ $\mathrm{Myr}$. This apparent bimodality in the age distribution could
also be due to the effects of gravity-darkening in rapidly-rotating stars. The
lower values in the low-order large separation and the frequency at maximum
power, in addition to the measured large projected rotation velocity of both
stars, support this idea. The 1D models that we have used in this work are not
the most suitable to stars with such high rotation rates. Two-dimensional
models are needed in order to take more into account the deformation that
occurs in them, and that greatly impact the seismic parameters, such as the
large separation and a reliable determination of the rotation frequency.
Finally, we cannot rule out an unreliable estimate of the seismic parameters
of the outlier stars. A greater weight of the other four stars in the WPDF
corrects for this apparent bimodality.

###### Acknowledgements.

We appreciate the comments and questions from the anonymous referee, because
they have contributed to improving the paper. DPO and AGH acknowledge funding
support from ‘FEDER/junta de Andalucía-Consejería de Economía y Conocimiento’
under project E-FQM-041-UGR18 by Universidad de Granada. JCS, GMM and SBF
acknowledge funding support from the Spanish State Research Agency (AEI)
project PID2019-107061GB-064. This paper includes data collected with the TESS
mission, obtained from the TASC data archive. Funding for the TESS mission is
provided by the NASA Explorer Program. STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.

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##  Appendix A The ten modes with highest extracted amplitudes and associated
frequencies for each star in our $\delta$ Sct star sample.

Trumpler 10 | Praesepe
---|---
TIC | $f(d^{-1})$ | A (ppt) | TIC | $f(d^{-1})$ | A (ppt)
TIC 28943819 | 47.185 | 1.667 |  | 31.566 | 0.645
40.230 | 0.885 |  | 30.242 | 0.314
43.554 | 0.787 |  | 29.507 | 0.426
44.008 | 0.463 |  | 31.896 | 0.351
40.122 | 0.342 |  | 27.811 | 0.168
40.359 | 0.335 | TIC 175194881 | 26.556 | 0.117
37.046 | 0.318 |  | 36.742 | 0.079
19.721 | 0.282 |  | 21.457 | 0.081
27.977 | 0.277 |  | 23.489 | 0.051
36.830 | 0.251 |  | 38.623 | 0.047
TIC 30307085 | 66.752 | 2.630 |  | 14.845 | 1.471
62.280 | 2.009 |  | 12.021 | 0.914
55.489 | 1.809 |  | 19.260 | 1.055
53.359 | 0.810 |  | 22.514 | 0.971
59.387 | 0.703 |  | 27.996 | 0.919
60.692 | 0.548 | TIC 175264376 | 19.167 | 0.610
57.244 | 0.448 |  | 15.575 | 0.511
4.427 | 0.359 |  | 13.359 | 0.394
48.369 | 0.344 |  | 10.180 | 0.339
64.533 | 0.292 |  | 16.501 | 0.292
TIC 28944596 | 25.341 | 1.674 |  | 35.980 | 1.629
34.691 | 1.588 |  | 31.020 | 1.618
24.494 | 1.358 |  | 28.457 | 1.200
33.649 | 1.089 |  | 24.659 | 0.975
16.494 | 0.815 |  | 29.333 | 0.949
39.021 | 0.682 | TIC 175265807 | 33.417 | 0.829
17.145 | 0.535 |  | 33.544 | 0.812
35.790 | 0.513 |  | 26.086 | 0.708
22.821 | 0.398 |  | 33.794 | 0.300
32.201 | 0.344 |  | 28.901 | 0.262
TIC 271061334 | 60.529 | 1.163 |  | 23.855 | 2.336
48.627 | 0.684 |  | 22.667 | 1.093
58.877 | 0.474 |  | 7.908 | 0.790
50.485 | 0.455 |  | 20.512 | 0.702
53.414 | 0.201 |  | 7.332 | 0.640
55.330 | 0.192 | TIC 175291778 | 24.209 | 0.631
53.583 | 0.168 |  | 6.827 | 0.612
59.792 | 0.147 |  | 12.141 | 0.609
60.696 | 0.134 |  | 18.050 | 0.546
0.212 | 0.157 |  | 17.146 | 0.519
TIC 271062192 | 21.680 | 1.397 |  | 29.667 | 1.849
18.611 | 0.703 |  | 31.899 | 1.543
29.522 | 0.367 |  | 29.465 | 1.138
43.681 | 0.351 |  | 26.145 | 1.168
42.380 | 0.238 |  | 31.565 | 1.059
33.928 | 0.224 | TIC 184914505 | 29.257 | 0.869
3.527 | 0.209 |  | 25.231 | 0.782
7.028 | 0.178 |  | 12.482 | 0.591
59.146 | 0.157 |  | 12.233 | 0.417
5.777 | 0.154 |  | 27.690 | 0.323
|  |  |  | 13.309 | 1.359
|  |  | 30.241 | 1.019
|  |  | 27.305 | 0.613
|  |  | 16.610 | 0.581
|  |  | 26.031 | 0.531
|  | TIC 184917633 | 32.840 | 0.526
|  |  | 26.241 | 0.518
|  |  | 22.494 | 0.435
|  |  | 13.192 | 0.402
|  |  | 28.240 | 0.409

Table A.1: Ten highest-amplitude modes and their frequencies, for each star in
our samples of Trumpler 10 and Praesepe. The tables, corresponding to each
star in our sample, with all the extracted frequencies is available on-line at
CDS

##  Appendix B The estimated large separations of our $\delta$ Sct star sample

Figure B.1: The same as Fig. 3 for TIC 30307085.

Figure B.2: The same as Fig. 3 for TIC 28944596.

The same as Fig. 3 for TIC 271061334.

The same as Fig. 3 for TIC 271062192.

The same as Fig. 3 for TIC 175194881.

The same as Fig. 3 for TIC 175264376.

The same as Fig. 3 for TIC 175265807.

The same as Fig. 3 for TIC 175291778.

The same as Fig. 3 for TIC 184914505.

The same as Fig. 3 for TIC 184917633.

##  Appendix C The positions and ranges of the possible radial modes of our
$\delta$ Sct star sample

The same as Fig. 4 for TIC 28944596.

The same as Fig. 4 for TIC 271062192.

The same as Fig. 4 for TIC 175264376.

The same as Fig. 4 for TIC 175265807.

The same as Fig. 4 for TIC 175291778.

The same as Fig. 4 for TIC 184914505.

The same as Fig. 4 for TIC 184917633.