# Studying the Impact of Latent Representations in Implicit Neural Networks
for Scientific Continuous Field Reconstruction

Wei Xu1, Derek Freeman DeSantis2, Xihaier Luo1, Avish Parmar1,
Klaus Tan3, Balu Nadiga2, Yihui Ren1, Shinjae Yoo1

###### Abstract

Learning a continuous and reliable representation of physical fields from
sparse sampling is challenging and it affects diverse scientific disciplines.
In a recent work, we present a novel model called MMGN (Multiplicative and
Modulated Gabor Network) with implicit neural networks. In this work, we
design additional studies leveraging explainability methods to complement the
previous experiments and further enhance the understanding of latent
representations generated by the model. The adopted methods are general enough
to be leveraged for any latent space inspection. Preliminary results
demonstrate the contextual information incorporated in the latent
representations and their impact on the model performance. As a work in
progress, we will continue to verify our findings and develop novel
explainability approaches.

## Introduction

Field reconstruction is the process involving the recreation of a physical
field denoted as $\boldsymbol{u}$, depicting the spatial and temporal
distribution of a specific parameter (e.g., temperature, velocity, or
displacement). Its significance spans diverse scientific disciplines such as
geophysics, climate science, and fluid mechanics, due to several key
functions. First, it facilitates the extrapolation of values at unmeasured
locations by leveraging available data (Kutz et al. 2016). It also contributes
to the identification of patterns, trends, and variations within the
investigated parameters (Shu, Li, and Farimani 2023; Wang et al. 2022).
Lastly, field reconstruction plays a crucial role in optimizing sensor
placement and refining data collection strategies, thereby enhancing the
effectiveness of experimental designs (Fukami et al. 2021; Krause, Singh, and
Guestrin 2008).

Recent progress in machine learning has sparked a growing interest in
addressing field reconstruction challenges through implicit neural
representations (Xie et al. 2022; Mildenhall et al. 2021). These
representations harness multilayer perceptrons (MLPs) to represent images,
videos, and 3D objects. Specifically, coordinate-based MLPs operate by taking
low-dimensional coordinates, such as spatial position $(\boldsymbol{x})$ or
time index $(t)$, as input and predicting the value of a learned signal
$(\boldsymbol{u})$ for each coordinate point. The coordinate-based MLP
effectively learns an implicit, continuous, and differentiable function,
mapping input coordinates to the target signal
$\boldsymbol{u}=F_{\theta}(\boldsymbol{x},t)$.

Driven by the quest for an enhanced representation method, we introduce a
context-aware indexing mechanism. Specifically, we integrate existing
contextual information into the temporal coordinate $(t)$. Given the dynamic
changes in the quantity and locations of available measurements over time $t$,
we propose a framework where an encoder extracts a latent representation from
real-time measurements. This latent representation is then utilized to steer
the model towards the desired time instance. Paired with an implicit neural
representation-based decoder, our proposed method excels in the seamless
reconstruction of continuous fields (Luo et al. 2024).

Meanwhile, as neural networks become more complicated, it is a grand challenge
to increase the overall interpretability and trust especially when adopted by
scientific users. For our field reconstruction model, the learned latent
representation is an essential component and incorporates information about
the measurements beyond temporal indexing. In this work, we present the
application of several dimension reduction approaches and use them as
explainability methods for latent representation understanding. Concretely, we
leverage explainability tools to demonstrate that the latent representation
has sufficiently encoded the real-time measurements. We take the use case of
continuous field reconstruction using implicit neural networks for climate
science and report the findings as a work in progress.

## Related Work

Explainable Artificial Intelligence (XAI) technique has flourished in the past
decade. For the latent space understanding, since the latent vectors are
difficult to interpret by humans directly, many methods are proposed to tackle
the dimensionality challenges. Embedding techniques e.g., t-Distributed
Stochastic Neighbor Embedding (t-SNE) (van der Maaten and Hinton 2008) are
used to project high-dimensional data into 2D space and study whether the
internal correlations of the data is preserved in the latent space. Clustering
is another common method to apply to the latent vectors and demonstrate the
relationship. Moreover, both linear and nonlinear methods are used to estimate
the similarity between latent representations (Mehrer et al. 2020). Developing
visual analytics frameworks is another branch of research to facilitate user-
directed analysis exploring the latent space interactively (Shen, Wang, and
Shen 2020). We find that existing works for dimensionality and decomposition
studies can be beneficiary for general-purpose latent understanding, which are
less commonly adopted beyond t-SNE visualization and ablation studies.

## Methodology

### The MMGN model

Traditional implicit neural networks incorporate the time index along with
spatial coordinates. Here, we present a neural network reconstruction method
called MMGN (Multiplicative and Modulated Gabor Network) (Luo et al. 2024),
characterized by an encoder-decoder architecture. For the encoder, we propose
to utilize measurements, represented as $u^{1}_{t}$, $u^{2}_{t}$,
$u^{3}_{t}$,… at time $t$, for implicit model guidance instead of relying on
$t$ for explicit pointing. Therefore, we construct an encoder $E(\cdot)$ to
convert the observed measurements into a latent code $\boldsymbol{z}_{t}$.
Subsequently, we employ a decoder $D(\cdot)$ that utilizes both the encoded
information and spatial coordinates $\boldsymbol{x}$ to reconstruct the
underlying physical field.

$\displaystyle\text{Baseline}:$ $\displaystyle
u(\boldsymbol{x},t)=F(\boldsymbol{x},t)$ (1) $\displaystyle\text{Proposed}:$
$\displaystyle
u(\boldsymbol{x},t)=D(\boldsymbol{x},\boldsymbol{z}_{t})=D(\boldsymbol{x},E(u^{1}_{t},u^{2}_{t},\dots))$

• Encoder: Autoencoders (AE) and their probabilistic variant, variational
autoencoders (VAE) (Kingma and Welling 2014), are frequently employed for
representation learning, leveraging their inherent latent variable structures.
However, conventional AE and VAE encounter challenges when dealing with the
variability of sensor locations and their fluctuations over time. While their
graph counterparts can manage spatial variability, and modified versions
address dynamic graphs, they become computationally demanding for extensive
graphs and encounter difficulties with long-range dependencies. In contrast,
auto-decoder, as exemplified in Park et al. (2019), demonstrates reduced
underfitting and heightened flexibility. It accommodates observation grids of
varying forms, including irregular or those existing on a manifold, without
necessitating a specialized encoder architecture, provided the decoder shares
the same property. Therefore, the auto-decoder is chosen as the backbone for
the encoder due to its ability to accommodate free-formed observations,
considering that the number and positions of available observations vary over
time.

• Decoder: The decoder inputs comprise two components: spatial coordinates
($\boldsymbol{x}$) and latent codes ($\boldsymbol{z}$). Initially, processing
$\boldsymbol{x}$ through fully-connected feed-forward layers results in a
coordinate-based MLP. Although such an MLP based on coordinates can provide a
continuous representation, it encounters challenges in learning high-frequency
signals, a phenomenon referred to as spectral bias. Recent studies suggest
that addressing this issue involves incorporating positional encoding with
Fourier features (Tancik et al. 2020) or introducing periodic non-linearities
in the first hidden layer (Sitzmann et al. 2020). In our approach, we employ
Gabor filters $g(\cdot)$ over Fourier bases to transform the coordinates.
Following the transformation of coordinates $\boldsymbol{x}$, we introduce a
modulation step where the transformed coordinates $g(\boldsymbol{x})$ undergo
modulation through a multiplicative layer, facilitating the integration of
$g(\boldsymbol{x})$ and $\boldsymbol{z}$. Specifically, the decoder
incorporates a multiplicative filter network (MFN) (Fathony et al. 2020) as
its backbone network. This choice is motivated by the recursive mechanisms
embedded in MFN, enabling improved fusion of spatial coordinates
($\boldsymbol{x}$) and latent codes ($\boldsymbol{z}$).

• Trained models: The MMGN model used in this work is trained with 5% sampling
rate of the original dataset generated by the CESM2 climate model (Danabasoglu
et al. 2020) that simulates Earth’s climate states. We take monthly averaged
global surface temperature data for model testing. That said, the network can
be trained using a sampling rate at 5% of the training data to recover a
continuous represenation of it without substantial errors. Moreover, by
altering the hyperparameter $k$ defining the size of latent codes
$\boldsymbol{z}_{k}$, we can train different MMGN models. In our previous
study, we train $10$ models with latent sizes, ranging from $1$ to $512$ by
doubling the latent size at a time, and collect the corresponding learned
latent codes for all $T$ time steps. We found that increasing the latent size
can reduce the reconstruction error and the improvement becomes more subtle
when the latent size is moderate.

### Explainability approaches

We adopt several explainability methods to conduct additional studies that
further understand the impact of latent codes on the model performance and
illustrate the contextual information latent codes integrate.

• Embedding and Clustering Latent representations are spatiotemporal vectors
in high-dimensional space. To compare the representations in different latent
spaces, we first adopt embedding techniques to project data into a 2D plane.
For the demonstration, we use t-SNE to generate the distribution of $T$ latent
vectors of each latent space and use K-means clustering to partition them into
groups. Then, we visualize the distributions and quantify the statistics of
clusters for all the latent spaces so that the variations can be observed.
This helps to explain the impact of latent size on the acquired contextual
information by the encoder $E(\cdot)$.

• Correlation Analysis Latent codes can also be represented by a $T\times k$
matrix where $k$-dimensional latent vectors are stacked in raster order.
Comparing the latent spaces becomes the comparison of the corresponding
matrices so that the rows represent samples and the columns represent
features. In that sense, we can use correlation analysis to generate patterns
of different latent spaces and compare their correlations. We leverage both
Principal Component Analysis (PCA) (Hotelling 1936) and Canonical Correlation
Analysis (CCA) (Hä 2007) in this study.

|  |
---|---|---
Figure 1: The t-SNE distribution changes in three latent spaces: (left) 4-D,
(middle) 512-D, and (right) the original data; the color represents the
temporal indexing.

PCA identifies the direction i.e., the principal component, formed by a linear
combination of original features that captures the maximal variance of the
data. The explained variance ratio (i.e., normalized eigenvalue) is a
measurement used to explain the proportion of variance of the matrix explained
by the corresponding principal component. Thus, we generate $k$ principal
components with their corresponding explained variance ratios for the matrix
with $k$-dimensional latents and compare the trends. Meanwhile, we also apply
PCA to the original data and generate $512$ explained variance ratios. By
visualizing the trends of ratios, we could evaluate the coherence of the rank
of information preserved in latent spaces and the original data.

CCA provides a more direct way to identify and measure the associations
between two sets of features, here two matrices. It determines pairs of
canonical variates $u$ from the first set and $v$ from the second set so that
they are maximally correlated with each other. The difference between PCA and
CCA roots in that while PCA finds the linear combination of features that
maximizes variance in the data, CCA looks for a linear combination of features
in both data that have the strongest correlation between each other.

• Tensor Factorizations Tensor factorizations provide a method for effectively
explaining surrogate model complexity. Tensor factorizations are techniques
for decomposing multidimensional arrays (tensors) into simpler, constituent
components. Among various tensor factorization techniques, Tucker
Decomposition plays a pivotal role in analyzing and interpreting multiway
data. This method decomposes a tensor into a core tensor multiplied by factor
matrices along each dimension of the tensor. The factor matrices describe the
dominant modes along each variable, while the core tensor describes their
interactions or mixing. Both the data used, and output derived from field
reconstruction, intrinsically arises as a tensor. The Tucker decomposition is
a natural tool to reveal the dominant modes and interactions within these data
sets. See (Kolda and Bader 2009) for more details.

We leverage the Tucker Decomposition to assess how well MMGN represents the
dominant modes (i.e., $\boldsymbol{x}$ and $t$) and their mixing. By taking
the latent size at 256 for this study, the Tucker decomposition of both the
training set and the model output are computed. The factor matrices and the
mixing tensor are then compared to see how well MMGN does at reproducing the
modes and mixing complexity of the Earth system data.

To compare the modes, we compute the Pearson correlation coefficient between
the modes for each variable. A high matching across modes in a variable
indicates that MMGN has learned the data generating process for that
particular variable. To compare the cores, we compute the Tucker decomposition
for cores shaped $[r,r,r]$ for each $r=1,2,\dots 200$. The values of the core
are then normalized, and the entropy of the core is computed. The entropy
measures the complexity of the mixing at these different multiranks. One
should expect that the entropy of the MMGN model to be less than the entropy
of the training data for each $r$, with exact matching indicating it has
captured the mixing process between the variables.

• Ablation Study Finally, an ablation study is designed to complement the
previous effort inspecting the contribution of each dimension of the latent
vector. By ablating one dimension at a time during the inference, we observed
a reduced performance but the mean squared error (MSE) keeps mitigating when a
bigger latent size is adopted. In this study, we want to decompose the overall
error into the temporal and spatial counterparts and check if any resonance
exists.

|  |
---|---|---
Figure 2: Left: The standard deviations of clusters diminish across latent
spaces and they are compared with the ones of the original data (the rightmost
boxplot). Middle: The trendings of explained variance ratios of the principal
components for all latent spaces (illustrating only the top 5 sizes) and
compare them with the trending of the original data. Right: Ablation result
indicating spatial linkage of temporal indexing.

## Results

Figure 3: Top: Correlation plots for the three variables. Bottom: Complexity
of Tucker core for different multi-ranks.

• Embedding and Clustering The t-SNE visualizations are shown in Figure 1 to
illustrate how the distributions of latents in different latent spaces vary.
Only the 4-D, 512-D, and the original spaces are shown for parsimony. Our goal
is to understand the impact of the latent size on the learned representations.
We observe that the lower dimensional space has more diverse but compact
clusters and it gets more evenly spreading out when the dimension increases.
We also find more string-shaped local clusters having the same color in the
higher dimensional space. Since we use the temporal indexing $t$ as colors,
these clusters are formed by the temporally adjacent latent vectors. The same
pattern and distribution are observed in the original data. We conclude that
the higher latent space better captures the global distribution of the
original data, and to some extent, also maintains local coherence (string-
shaped small clusters). Note here the use of t-SNE projections of 2-D latents
instead of their original coordinates is meant for a fair comparison with
other latent spaces. Then, the clustering results are generated and we collect
the standard deviations $\sigma_{1},...,\sigma_{n}$ of the vectors in the
corresponding $n$ clusters for all the latent spaces. As a preliminary result,
the increased $\sigma$ pattern also confirms that the distribution shifts from
compact and diverse groups to a wider space spanning as shown in Figure
2(left).

• Correlation Analysis We aim to use correlation analysis to understand the
impact of latent size in terms of rank complexity and potential information
loss. First, the PCA results are shown in Figure 2(middle) where the explained
variance ratios of higher latent spaces are compared with the one of the
original data. The slope of the curve suggests the rank of data complexity.
For example, a steep slope means with the first few principal components the
most variances can be preserved. Compared to the slope of the original data,
we observe a similar slope starting from a latent size at 64 and beyond.
Furthermore, we also generate the CCA canonical variate pairs between two
cases: 1) two latent spaces and 2) a latent space and the original data. Then
we use Pearson correlation coefficient to gauge their relations. We find that
all the paired canonical variates are fully correlated for both cases. We
conclude that any latent code is a reduced representation presenting
consistent information about the original data.

• Tensor Factorizations Similar to the purpose of adopting correlation
analysis, the results of the tensor based analysis are shown in Figure 3. For
the factor matrices, we see that the MMGN model has strong agreement with the
true factors for the first $20$ temporal and longitudinal components, and
first $40$ latitudinal components. These top modes are responsible for the
majority of the reconstruction of the tensor (roughly $\sim 4\%$ relative
error), so agreement here indicates MMGN has learned the dominant spatial
temporal patterns of the dataset. The divergence of higher order modes
suggests that MMGN fails to capture the higher frequency spatial components of
the data (see Figure 4). The fact that there is stronger agreement in the
latitude modes indicates the model had an easier time capturing this
component.

From Figure 3 we see that the core of the MMGN model has complexity that
matches the true data up to multi-rank $[75,75,75]$. At this high of a rank,
the tensor decomposition for both the true data and the MMGN output have a
relative reconstruction error of $\sim 1\%$. The agreement of model complexity
up to this large multi-rank indicates that MMGN accurately captures the mixing
processes of the underlying physical phenomenon.

Figure 4: Latitude modes $1,2,3$ in blue versus latitude modes $101,102,103$
in red. Higher order modes capture higher frequency information compared to
lower order modes.

• Ablation Study We collect the sets of temporal and spatial errors for the
model of a latent size at 64. A preliminary result for the spatial linkage of
the latent dimension is shown in Figure 2(right). We use colors to indicate
the latent dimension index (ranging between 0 and 63) so that once ablated it
causes the most performance reduction at a spatial location. It is interesting
to observe that individual latent dimensions contribute to adjacent regions
for low-fidelity areas. This plot also shows the MMGN model can capture long-
range dependencies. We will further verify this finding in future work.

## Conclusion

The present studies leveraged dimensionality reduction approaches to explain
the effect of the encoder latent space dimension has on the representation.
These tools have improved understanding of the learned latent representations.
As a work in progress, we will keep verifying the current findings and work on
potential extensions, developing feature importance methods attributing the
input features for the prediction.

## Acknowledgments

This paper is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research and Office
of Biological and Environmental Research, Scientific Discovery through
Advanced Computing (SciDAC) program under Award Number 9233218CNA000001. This
work is also partly sponsored by Science Undergraduate Laboratory Internship
(SULI) at Brookhaven National Laboratory.

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# Determining the seismic age of the young open cluster $\alpha$ Per using
$\delta$ Scuti stars

David Pamos Ortega1, Antonio García Hernández1, Juan Carlos Suárez1, Javier
Pascual Granado2, Sebastià Barceló Forteza1, José Ramón Rodón2
1Departamento de Física Teórica y del Cosmos, Universidad de Granada, Campus
de Fuentenueva s/n, 18071, Granada, Spain
2Instituto de Astrofísica de Andalucía (CSIC). Glorieta de la Astronomía s/n.
18008, Granada, Spain

(Accepted 2022 March 24. Received 2022 March 24; in original form 2021
December 17)

###### Abstract

In this work we aim at constraining the age of the young open cluster Melotte
20, known as $\alpha$ Per, using seismic indices. The method consists of the
following steps: 1) Extract the frequency content of a sample of stars in the
field of an open cluster. 2) Search for possible regularities in the frequency
spectra of $\delta$ Sct stars candidates, using different techniques, such as
the Fourier transform, the autocorrelation function, the histogram of
frequency differences and the échelle diagram. 3) Constrain the age of the
selected stars by both the physical parameters and seismic indices by
comparing them with a grid of asteroseismic models representative of $\delta$
Sct stars. 4) Find possible common ages between these stars to determine the
age of the cluster. We performed the pulsation analysis with MultiModes, a
rapid, accurate and powerful open-source code, which is presented in this
paper. The result is that the age of $\alpha$ Per could be between 96 and 100
Myr. This is an improvement over different techniques in the past. We
therefore show that space astroseismology is capable of taking important steps
in the dating of young open clusters.

###### keywords:

asteroseismology – delta Scuti stars

††pubyear: 2022††pagerange: Determining the seismic age of the young open
cluster $\alpha$ Per using $\delta$ Scuti stars–Determining the seismic age of
the young open cluster $\alpha$ Per using $\delta$ Scuti stars

##  1 Introduction

Open clusters are laboratories of stellar and galactic astrophysics. Having at
our disposal a group of tens of stars of similar chemistry and ages, within a
narrow field of observation, allows us to better constrain models with which
we can compare our observations. We can better characterize the stars in this
way, thereby determining their evolutionary stages and, by extension, the
origin and evolution of the galaxy to which they belong.

There are many sources that can contribute to ambiguity when determining the
age of a cluster by isochrone fitting on the Hertzsprung-Russell (HR) diagram:
uncertain distance modulus, cluster membership, binarity of individual stars,
reddening and extinction, metallicity, different treatments of the physics of
the models used to derive the theoretical isochrones, and the uncertainties
from isochrone fitting methods. Ambiguity is even greater in young clusters,
with a sparse population of stars leaving the main sequence (turn off). The
tip of the red giant branch is used to date globular clusters but cannot be
used in young open clusters.

Pulsating stars can allow a better characterization of the cluster, because
some of their seismic indices are directly related to internal characteristics
of the stars that we want to date. Some types of classical pulsating stars,
such as $\delta$ Sct stars, have a rich and varied frequency spectra. They
correspond to A and F stars of intermediate mass, very common in the main
sequence (MS) for young open clusters. Space missions such as CoRoT (Baglin et
al., 2006), Kepler (Koch et al., 2010) and TESS (Ricker et al., 2014) have
lowered the detection threshold for their signals. Being grouped around the
MS, it is not easy to estimate the age by isochrone fitting. In addition, it
has been very difficult, until recently, to find regularities in the frequency
spectra of these stars, as in the case of solar-type. There are studies
showing that it is possible to find regularities in the frequency pattern of
$\delta$ Sct stars, even out of the asymptotic regime (see for example, García
Hernández et al., 2009; Paparó et al., 2016; Barceló Forteza et al., 2017;
Bedding et al., 2020, and references therein). This is the named large
separation, like for solar-type stars, but in the low-order regime (from n=2
to n=8, Suárez et al., 2014). It is defined as the frequency difference
between modes of the same degree and consecutive orders:

$\Delta\nu_{\mathrm{low}}=\nu_{n,\ell}-\nu_{n,\ell-1}$ (1)

We used here the subscript $low$ to differentiate it from that of the
asymptotic regime. However, as for solar-type stars, this
$\Delta\nu_{\mathrm{low}}$ is also related to the stellar mean density (Suárez
et al., 2014; García Hernández et al., 2015, 2017).

Other seismic indices are the rotational splittings of modes of the same order
and degree, related to the angular rotation, and the frequency at maximum
power, used in solar-type stars, directly related to the effective
temperature, that has also been found in $\delta$ Sct stars (Barceló Forteza
et al., 2018, 2020; Bowman & Kurtz, 2018; Hasanzadeh et al., 2021, BF2018,
BF2020, BK2018 and H2021 from now on).

Here we use the asteroseismic indices to study the age of the open cluster
Melotte 20 (also known as $\alpha$ Per). The structure of the paper is as
follows: in Sec. 2 we provide the main characteristics of the cluster and
former estimates of its age. In section Sec. 3 we describe the sample of stars
used in this research. In Sec. 4 we introduce the code MultiModes, a new tool
for the analysis of pulsating stars. In order to show its reliability, we have
compared it to one of the most reliable and most used in the field, SigSpec
(Reegen, 2007). We explain the details of the code in this section, and also
the comparative results obtained between them, using synthetic and real light
curves, in terms of accuracy and computing time. In this section we also
present the list of $\delta$ Sct stars candidates identified in our sample
with the frequency analysis. In Sec. 5 we show how, in four of them, it has
been possible to measure the low-order large separation. The angular rotation
and the frequency at maximum power of two of these stars have been measured.
In Sec. 6 we explain the details of the grid of pulsation models calculated
with mesa (Paxton, 2019) and filou (Suárez & Goupil, 2008), in order to
constrain the models that better fit to the seismic observations. In Sec. 7 we
compare our asteroseismic age with previous works and discuss the reliability
of the proposed method to date open clusters. Finally, in Sec. 8, the main
conclusions are exposed.

##  2 $\boldsymbol{\alpha\ P{er}}$

The open cluster $\alpha$ Per is located in the constellation of Perseus, at a
distance of $174.89\pm 0.16$ pc, obtained from Gaia DR2 parallaxes (Gaia
Collaboration et al., 2018). In Lodieu et al. (2019) a total of 517
astrometric member candidates had been identified in the tidal radius of
$\alpha$ Per, using a kinematic method combined with the statistical treatment
of parallaxes and proper motions. Netopil & Paunzen (2013) estimated solar
metallicity for all members of the cluster and an extinction along the line of
sight of around $A_{V}=0.3$ (Prosser, 1992).

Recent age estimates by isochrone fitting range from 20 to 90 Myr. Makarov
(2006) estimates the age in around 52 Myr, fitting isochrones in the $M_{V}$
vs. (B-V) diagram, with Z = 0.02, E(B-V) = 0.055, and overshooting computed
from the models by Pietrinferni et al. (2004). According to Silaj & Landstreet
(2014), the age of $\alpha$ Per is $60\pm 7$ Myr, determined with isochrones
by Girardi et al. (2000), with Z = 0.02, fitted onto an HR diagram,
considering that the uncertainties are less ambiguous than using a colour-
magnitude diagram. Isochrones are most constrained around the one that passes
throw the brightest main sequence star, $\psi$ Per (HD 22192), which lies
close to the terminal age main sequence (TAMS). In both works, the estimated
age of the cluster relies on the parameters of a single star, with a
difference of around 8 Myr in their results. Such an approach must be taken
with caution, since errors in the estimate of luminosities and temperatures
for stars in different evolutionary stages can be quite large. These errors
may come from, e.g. the type of photometry to use, bolometric corrections and
the correct modulus distance, or even the lack of corrections for rotation
effects (see e.g. Suárez et al., 2002, for more details).

Other works use spectroscopic observations of lithium in low-mass stars to
estimate the age of young clusters (see for example Basri & Martín (1999),
Stauffer et al. (1999)). The idea is that for stars near the substellar mass
limit, the age is very sensitive to the lithium depletion boundary, when they
start to burn lithium in their cores. As these stars are fully convective in
their range of masses, the abundance of lithium is visible in their
atmospheres through the 6708 Å Li I doublet. The problem with this method is
that unresolved binaries makes the cluster appear younger, as claimed by
Martín et al. (2001). For this reason, they used the faintest members to
derive the age of the cluster, for which lithium has not been detected.
Another difficulty with this method is its dependency on good models that take
into account the characteristics of the stellar atmosphere. In Stauffer et al.
(1999) the estimated age with this method is around $90\pm 10Myr$, noticeably
older than Makarov and Silaj & Landstreet.

Lodieu et al. assumes an age of 90 Myr to compile the census of stars that are
members of the cluster, because it is the most common age in the literature.

##  3 Our sample

Firstly, we have done a cross match between VizieR Online Data Catalogue Gaia
DR1 open cluster members (Gaia Collaboration et al., 2017) and TESS Input
Catalogue (TIC) (Stassun et al., 2019), searching possible targets belonging
to the same open cluster, from which we could obtain light curves from the
TESS mission. We have found a list of 112 stars in the field of $\alpha$ Per,
with measured values for parallax, mean G magnitude, E($B-V$) reddening,
bolometric corrections estimated by Andrae et al. (2018), and effective
temperature. They are all contained in the census prepared by Lodieu et al.,
obtained from Gaia Collaboration et al. (2018). We have estimated the G
magnitude with the equation:

$M_{G}=G-5\log r+5-A_{G},$ (2)

where the distance $r$ is taken as the inverse of the Gaia parallax. The
extinction $A_{G}$ is calculated from reddening values as in Stassun et al.:

$A_{G}=2.72\,\mathrm{E}(B-V).$ (3)

The G magnitude is converted to luminosity using the bolometric correction
BC(Teff) with the equation:

$-2.5\log
L=M\textsubscript{G}+\mathrm{BC}\textsubscript{G}(T\textsubscript{eff})-M_{\mathrm{bol}\odot}$
(4)

Figure 1 depicts the HR diagram of our sample, including
MIST111http://waps.cfa.harvard.edu/MIST/ isochrones (Dotter, 2016; Choi et
al., 2016; Paxton et al., 2011; Paxton et al., 2013, 2015), with solar
metallicity and different ages. They all have been computed from the pre-main
sequence phase (PMS) to the end of hydrogen burning. In rotating models,
solid-body rotation with $\Omega/\Omega_{crit}=0.4$ is initialized at zero age
mean sequence (ZAMS). Rotation makes the star hotter or cooler depending on
the efficiency of rotational mixing in the envelope. If rotational mixing
introduces a sufficient amount of helium into the envelope, increasing the
mean molecular weight, the star becomes more compact and hotter. On the
contrary, if rotational mixing is not efficient, the centrifugal effect
dominates, making the star cooler and more extended. The ages of the
isochrones are taken to cover the wide range of values found in the literature
(see previous section). We can see that all the calculated isochrones between
20 and 200 Mrs mostly overlap in the MS zone. We also have located the
position of $\psi$ Per on the HR diagram, which is compatible with an age of
200 Myr, quite far from the age estimated by Silaj & Landstreet.

This is where asteroseismology plays an important role, since it provides
independent seismic indices that provide an accurate determination of global
stellar parameters, including the age of the star.

We analyzed a set of 32 stars using data from sector 18 of the TESS mission,
with approximately 14700 points, a Rayleigh resolution of approximately 0.045
$d^{-1}$ and a cadence of two minutes. We used the Pre-Search Data Conditioned
(PDC) light curves, corrected for instrumental effects, that are publicly
available through the MAST222https://archive.stsci.edu/. These light curves
have a 2.5 days gap caused by the satellite downlink, when the spacecraft
passed through the shadow of the Earth at the start of orbit 43 (Fig. 2).
Pascual-Granado et al. (2018) showed that the analysis of light curves with
gaps can produce an undesirable amount of spurious peaks. Following Pascual-
Granado et al. (2015) we interpolated our data using the MIARMA code, which
uses autoregressive and moving average models. This method is aimed to
preserve the original frequency content of the light curve, making the
frequency extraction more reliable than leaving the gaps or using other gap-
filling methods.

Figure 1: HR for the sample of 112 stars from the field of $\alpha$ Per, taken
from TESS Input Catalogue. Isochrones between 20 and 200 Myr with solar
metallicity are taken from MIST. (bluehttp://waps.cfa.harvard.edu/MIST/). The
borders of the instability strip are also drawn, calculated according to
Murphy et al. (2019). The position of $\psi$ Per is also indicated, one of the
most luminous stars in the cluster Figure 2: Top: Light curve of TIC
252851046. Bottom: Interpolated light curve of TIC 25851046 with MIARMA
algorithm

##  4 Frequency analysis

###  4.1 MultiModes: a new tool for analysis of pulsating stars

Many of the light curves provided by space missions are uniformly sampled, but
show some gaps, due to lack of observations, instrumental issues or
environmental effects. The Lomb Scargle Periodogram (from now on LS) is very
powerful for analyzing non-uniformly sampled time series (Scargle, 1982). For
uniform sampling, LS provides the classical periodogram. The algorithm
calculates, at each frequency of interest, a time phase shift, and which in
turn is used to evaluate the power spectra at that frequency. This time phase
shift makes the LS to be independent of shifting all the points by any
constant. In this way, the calculation of LS is equivalent to performing a
least squares fit of data to a sinusoidal function for each evaluated
frequency. So there is a deep connection between the Fourier Transform and
least squares analysis in the LS.

Computing LS can be very slow because it requires a number of calculations of
the order of $N^{2}$, $N$ being the number of points in the sample. This
problem is partially solved by the implementation of the so-called Fast Lomb
Scargle Evaluation (Press & Rybicki, 1989), based on
extirpolation333Extirpolation consists of the substitution of actual points of
the series by others constructed by means of an evenly sampled mesh. On this
basis, the order of the calculations for the evaluation of the periodogram is
reduced to $N\log N$ from unevenly sampled data.

The width of a peak does not depend on the number of sampled points or the
signal to noise ratio (SNR), it only depends on the Rayleigh resolution, which
is inversely proportional to the total sample size (VanderPlas, 2018). Two
peaks within the Rayleigh resolution, in theory, could not be perfectly
resolved by the periodogram. Furthermore, the stochastic nature of the series
can cause numerous spurious frequencies to emerge, many of them of low
amplitude but above the mean noise level. As detailed in Balona (2014), the
problem of using the pre-whitening technique lies in the large number of
artificial frequencies that are generated with a high SNR. When, at each step,
a signal very similar to the one to be extracted is added to the analyzed
light curve, if the fit has not enough accuracy, it can generate an
interference pattern around the added frequency.

These spurious frequencies can be avoided, to a great extent, by making a
simultaneous fit to all the extracted frequencies, through non-linear
optimization. We use least squares fit to a multisine function, taking as
parameters the frequencies, amplitudes and phases of each of the peaks to be
extracted.

The well-known SigSpec algorithm (Reegen, 2007, SS from now on) uses pre-
whitening to extract all those frequencies that are above a certain criterion
of significance, defined from the False Alarm Probability (FAP) as:

$\mathrm{sig}=-\log(\mathrm{FAP})$ (5)

taking into account the dependency of the FAP with the phases.

Widely used in asteroseismology, it is not open source, and no longer updated.

Inspired by the SigSpec methodology, we developed MultiModes (MM from now on),
which is a Python routine developed to extract the most significant peaks of a
sample of classical pulsating stars. It is available in a public Github
repository444https://github.com/davidpamos/MultiModes and it is customizable.
This routine has been designed using the Astropy package
555bluehttps://www.astropy.org to calculate the periodograms and the LMFIT
package 666bluehttps://lmfit.github.io/lmfit-py/ for non-linear optimization
of the extracted signals.

The algorithm (see Fig. 3) fits frequency, amplitude and phase through non-
linear optimization, using a multisine function. This function is redefined
with the new calculated parameters. It does a simultaneous fit of a number of
sinusoidal components (20 are usually enough).

Then they are subtracted from the original signal and the algorithm goes back
to the beginning of the loop with the residuals as input, repeating the same
process, until the stop criterion is reached. After that, the code can filter
suspicious spurious frequencies, those of low amplitude above the Rayleigh
resolution, and possible combinations of modes.

###  4.2 Accuracy test with simulated spectra

We tested the reliability of MM using synthetic light curves. Following Balona
(2014), we constructed artificial light curves with 50, 100, 200 and 400
frequencies, with uniformly distributed values between 0 and 30 d-1,
amplitudes ranging between 0 and 10 mmag, exponentially distributed towards
low values, phases uniformly distributed, and adding a Gaussian noise of about
0.5 mmag.

The frequency deviation of MM is similar to that of SS in the curves with 50
and 100 frequencies, around $10^{-4}\ d^{-1}$, when we used an unlimited
number of components for the simultaneous fit of the simulated signal (Fig.
4). With a higher number of frequencies, 200 and 400, SS is more accurate than
MM by one order of magnitude, because MM used a limited number of components
for the simultaneous fit, no more than 50. Still, results with MM have a high
enough accuracy for a precise frequency extraction. As the frequency density
increases, the problem becomes more unstable since more spurious frequencies
appear, with both MM and SS. Therefore, it makes sense to limit the number of
components of the simultaneous fit of the light curve when frequency density
is very high. In this sense, MM allows us to have greater control over the
frequency analysis, by limiting the number of components for the fit, and
working with frequency packs.

Figure 3: Workflow of the MultiModes algorithm Figure 4: Comparative analysis
of accuracy between MM and SS, using simulated light curves with different
number of frequencies, 50 (upper left panel), 100 (upper right panel), 200
(bottom left panel) and 400 (bottom right panel). Each plot represents the
frequency deviation of the extracted peaks with MM (blue filled circles) and
SS (red crosses). The dotted green line is the level of the Rayleigh
resolution.

###  4.3 Frequency content of the $\boldsymbol{\delta}$ Scuti sample

From the sample of 32 analysed stars analyzed with MM, we found that 11 of
them are $\delta$ Sct stars, or hybrids, a type of pulsating stars with
intermediate mass that show low-order acoustic oscillations (p-modes) and also
high order gravity modes (g-modes), more typical of $\gamma$ Dor stars
(Grigahcène et al., 2010; Uytterhoeven et al., 2011). Except for TIC 252829836
and TIC 347570557, 9 stars in our sample had not been identified as $\delta$
Sct stars so far.

We have analysed the frequency content of these 11 stars using MM and SS, for
greater reliability. The frequency comparison assumes that both codes identify
the same frequency when their difference is smaller than the Rayleigh
resolution. We established the degree or percentage of coincidence based on
whether each frequency extracted by MM is among those extracted by SS or not.
The degree of coincidence between MM and SS in the values of the extracted
frequencies, extracting the most significant frequencies, is above $90\%$ in
all the cases (Tab. 1). We have also done a comparative analysis on the
computing speed between MM and SS with our sample of 11 $\delta$ Sct stars
stars (see Ap. A).

Table 1: Comparative analysis between SS and MM extracted peaks, using ARMA-
interpolated light curves with MIARMA code, for the 11 $\delta$ Sct stars
sample from $\alpha$ Per. $N$SSar and $N$MMar are, respectively, the number of
extracted frequencies by SigSpec and MultiModes. SSarMMar is the degree of
coincidence between SigSpec and MultiModes

TIC |  $N$SSar |  $N$MMar | SSarMMar
---|---|---|---
104319359 | 249 | 238 | 97.9
116011834 | 233 | 222 | 97.7
252829836 | 82 | 76 | 97.4
252851046 | 213 | 222 | 93.7
285935852 | 45 | 40 | 97.5
347570557 | 189 | 188 | 94.7
354638295 | 97 | 89 | 98.9
354792288 | 55 | 47 | 100.0
401079326 | 39 | 35 | 100.0
410732825 | 69 | 55 | 100.0
428320122 | 71 | 64 | 100.0

##  5 Seismic indices of $\boldsymbol{\delta\ S{cuti}}$ stars

Among our sample of 11 $\delta$ Sct stars, in four of them, TIC 410732825, TIC
354792288, TIC 285935852 and TIC 252829836 (Fig. 5), we found a pattern that
we identify as a low-order large separation ($\Delta\nu_{\mathrm{low}}$),
following the techniques from García Hernández et al. (2009) and Ramón-
Ballesta et al. (2021). The autocorrelation function (AC), the Fourier
transform (FT), the histogram of frequency differences (HFD), and the échelle
diagram (ED) were applied to the 30 frequencies with highest amplitudes and
above 5 $\mbox{d}^{-1}$ to avoid g modes. These frequencies are selected by
amplitude but they are given equal amplitudes when computing the
transformations.

Fig. 3 and Fig. 7 show, respectively, the cases of TIC 354792288 and TIC
410732825 (see the other two stars in Ap. B). For TIC 354792288, the FT, AC
and HFD show a peak around $42\ \mu\mathrm{Hz}$. It may be half the value of
the low-order large separation. The AC and HFD also show a peak around $84\
\mu\mathrm{Hz}$. The ED shows the alignment of several frequencies when 86
$\mu\mbox{Hz}$ is chosen as the low-order large separation. Thus, we took 84
$\mu\mbox{Hz}$ as $\Delta\nu_{\mathrm{low}}$, being the most common value
obtained by the different diagnostic techniques.

We also searched for a pattern connected to the rotational splitting in the
p-mode regime. At first order in the perturbative theory, rotation splits the
oscillation modes of the same order and spherical degree in the form:

$\omega_{nlm}=\omega_{nl}+m\Omega(1-C_{L}),$ (6)

where $\omega$ is the angular frequency, $\Omega$ is the angular rotation
frequency and $C_{L}$ is the Ledoux constant, to take into account the effects
of the Coriolis force (Aerts et al., 2010). According to Eq. 6, this would
allow us to find a pattern corresponding to the rotation frequency in the
periodogram, i.e., a regular structure related to $\Omega$. However, this
simple distribution of frequencies is only valid for slow rotators. Even at
moderate rotations (around 50 $\mathrm{km}~{}\mathrm{s}^{-1}$) the picture
changes: rotational splittings are not symmetric anymore and even the m = 0
mode can be displaced (see e.g. Suárez et al., 2006, for more details). These
effects may hamper the identification of a rotational splitting.

Nonetheless, some theoretical works computing 2D non-pertubative models and
rapid rotation (Reese et al., 2017) combined the AC and the FT (following
García Hernández et al., 2009) to search for the patterns of the low-order
large frequency separation and the rotational splitting. They pointed out that
$\Delta\nu_{\mathrm{low}}$ or its half value comes up usually clearer in the
FT, whereas twice the rotational splitting can be better found with AC
diagnostic.

From the observational side, Paparó et al. (2016) searched for regularities in
the frequency spectra of $\delta$ Sct stars observed by CoRoT using a visual
inspection and by means of a semi-automatic algorithm that looks for repeated
spacings between frequencies. They claimed that some of these spacings might
be not only the large separation but also the rotational splitting or a linear
combination of both. The work of Barceló Forteza et al. (2017) used the AC,
HFD, ED and DFT of the power spectrum, following the techniques from Régulo &
Roca Cortés (2002), to find rotational splittings in a sample of four $\delta$
Sct stars. The main idea was that structural characteristics of the star, such
as rotation, alter the power spectra, particularly in the flat plateau, where
the density of low amplitude frequencies increases with the rotation rate.
Ramón-Ballesta et al. (2021) related regularities found in the frequency
spectra of binary $\delta$ Sct stars to their rotational splitting. In their
work, they used the FT, the AC and the HFD as in the present research.

The HFD for two of our stars, TIC 410732825 (Fig. 7) and TIC 285935852 (Fig.
A.2) shows a prominent peak around 9-11 $\mu\mbox{Hz}$ that could be the
identified as the angular rotation. In fact, Fig. 7 also shows significant
peaks at both sides of the one identified as $\Delta\nu_{\mathrm{low}}$,
implying the presence of spacings corresponding to the large separation plus
and minus the rotational splitting.

In the case of TIC 285935852, Fig. 8 displays the periodogram with the most
visible modes, between 40 and 60 $\mu Hz$. It is indicated the low-order large
separation, with a value of around $\Delta\nu_{\mathrm{low}}=7d^{-1}$. Some
other periodicity can also be noticed that might correspond to twice the
rotational splitting of around 0.9-1.0 $d^{-1}$, or equivalently, 9-11
$\mu\mbox{Hz}$. This is consistent with the results by Reese et al. (2017).
The $m\neq 0$ modes are not symmetric with respect to the $m=0$ mode, that it
is displaced but a spacing corresponding to double the rotational splitting
still remains. To check this behaviour, the mode distribution is compared to a
compatible model and set of pulsation frequencies computed with the mesa and
filou codes (see Sec. 6), taking rotation into account up to second order in
the pertubative theory for the adiabatic oscillation computation (including
near-degeneracy effects and stellar structure deformation). The mode
distribution is similar in both the model and the observation although the
limitations of our calculations are clear. Some theoretical modes lie close to
the observed ones but others do not. There are missing or completely off modes
from the model. Fortunately, the spacings remain although our mode
identification might mismatch if we use non-rotating models. We emphasise the
importance of taking rotation into account when modelling $\delta$ Sct stars,
as also pointed out by Murphy et al. (2021), although they used non-rotating
models to carry out a mode identification.

Another seismic index, recently discovered for $\delta$ Sct stars, is the
frequency at maximum power, related to the effective temperature of the star,
as shown in the works of BF2018, BF2020, BK2018 and H2021. There are different
ways of defining the frequency at maximum power. BK2018 defines it with the
peak of maximum amplitude in the spectrum, $\nu$max,0. BF2018 and BF2020 do it
in two different ways, one of them with $\nu$max,0 and the other through the
mean of the most significant extracted peaks weighted by their amplitudes,
$\nu$max,w. H2021 do it through the maximum of the autocorrelation of the
power spectra, $\nu$max,2D. As is explained in Hasanzadeh et al. (2021), and
with more detail in Viani et al. (2019), the AC method is applied on several
windows of the SNR 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 $\nu$max,2D.

This relation is very dependent on the evolutionary stage of the star through
log g, and it is affected by gravity darkening, as a result of the rapid
rotation. This quite large dispersion is probably intrinsic to the
relationship itself, and also due to observational errors. The relation
probably does not have the same origin as for solar-like stars since the
excitation mechanism of the modes is very different in these two types of
pulsators.

Tab. 2 shows the values of all these seismic indices that will serve to
constrain the models to try to date these four stars accurately, and
therefore, the cluster. The uncertainty of $\nu$max,0 is the error of the
single frequency extracted with MM. The uncertainty in the location of
$\nu$max,w has been calculated from the standard deviation of the amplitude
weighted data. For $\nu$max,2D, we have taken the standard deviation of the
Gaussian fit as its corresponding error.

TIC 354792288 and TIC 285935852 show a set of very grouped frequencies between
40 and 60 $\mu\mbox{Hz}$ (Fig. 9), and don’t show significant modes below
$5d^{-1}$, which indicates that they can probably be pure $\delta$ Sct stars,
as it is defined in Grigahcène et al. (2010); Uytterhoeven et al. (2011). With
both stars, we have used $\nu$max,0 for BF2018’s and BK2018’s relations,
$\nu$max,w for BF2020’s, and $\nu$max,2D for H2021’s, in order to determine a
mean effective temperature, $\widetilde{T}$eff, (see Tab. 3), considering that
the gravity darkening effect due to rotation makes the star hotter on the
poles and cooler on the equator. Regarding the uncertainties of the effective
temperature, we have calculated them using the error propagation method, from
the corresponding errors of $\nu$max,0, $\nu$max,w and $\nu$max,2D and the
coefficients of their respective relations.

Figure 5: The four selected targets in the field of $\alpha$ Per for which we
have obtained regularities in the frequency spectra Figure 6: Measured
regularities in the frequency spectra of TIC 354792288. The top left panel
represents the autocorrelation diagram (AC) of periodicities calculated with
the first 30 frequencies extracted by MM, showing peaks around 42 $\mu$Hz
(black dotted line) and 84 $\mu$Hz. Top right, the Fourier transform (FT) for
the periodicities of those first 30 frequencies, showing a peak around 42
$\mu$Hz (black dotted line). Bottom left, the histogram of frequency
differences (HFD) with the first 30 frequencies, showing peaks around 42
$\mu$Hz (black dotted line) and 84 $\mu$Hz. Bottom right, the echelle diagram
(ED) showing two vertical ridges (orange circles) when
1/2$\Delta\nu_{\mathrm{low}}$ is chosen around 3.7 $d^{-1}$, equivalent to 43
$\mu$Hz. ED is done with Python package Echelle 1.5.1, developed by Hey & Ball
2020333Echelle 1.5.1 Figure 7: Measured regularities in the frequency spectra
of TIC 410732825. The autocorrelation diagram (AC) (top left), the Fourier
transform (FT) (top right), and the histogram of frequency differences (HFD)
(bottom left), calculated with the first 30 frequencies extracted by MM, show
a prominent peak around 63 $\mu$Hz (black dotted line). And also, the echelle
diagram (ED) (bottom right) show a very clear vertical ridge (orange circles)
when $\Delta\nu_{\mathrm{low}}$ is around 5.3 $d^{-1}$, equivalent to 61
$\mu$Hz

Figure 8: Periodogram of TIC 285935852, showing a measured low-order large
separation of around $\Delta\nu_{\mathrm{low}}=7d^{-1}$, and a rotational
splitting of around red2$\omega_{r}=1.9d^{-1}$. The modes distribution is
compared to a suitable model calculated with mesa-filou(see Sec. 6). The blue
dotted lines show the positions of some of the theoretical modes, denoted as
$f_{nlm}$, where n is the corresponding radial order, l is the spherical
degree and m the azimutal order

. Some modes seem to follow the frequency distribution of the model. Others,
related to rotation, where the frequency splitting is visible, don’t do it.
Even the centroids are displaced from the model

Figure 9: Periodograms for TIC 354792288 and TIC 285935852, showing very
grouped frequencies between 40 and 60 $\mu Hz$ Table 2: Seismic indices of the
selected targets from $\alpha$ Per to constrain the models, in order to
determine its ages

TIC | $\Delta\nu_{\mathrm{low}}(\mu Hz)$ | $\omega_{r}(\mu Hz)$ | $\nu_{\text{max,0}}(\mu Hz)$ | $\nu_{\text{max,w}}(\mu Hz)$ | $\nu_{\text{max,2D}}(\mu Hz)$
---|---|---|---|---|---
410732825 | $63\pm 1$ | $10\pm 1$ | - | - | -
354792288 | $83\pm 1$ | - | $670.587\pm 0.001$ | $624\pm 15$ | $620\pm 70$
285935852 | $82\pm 1$ | $10\pm 1$ | $625.026\pm 0.001$ | $602\pm 40$ | $605\pm 70$
252829836 | $71\pm 1$ | - | - | - | -

Table 3: The different relations between the frequency at maximum power and
the mean effective temperature for TIC 354792288 and TIC 285935852.
References: 1Barceló Forteza et al. 2018, 2Bowman & Kurtz 2018, 3Barceló
Forteza et al. 2020, 4Hasanzadeh et al. 2021

Relation $\nu_{\mathrm{max}}-\widetilde{T}_{\mathrm{eff}}$ | $\sigma(\%)$ | TIC 354792288 $\widetilde{T}_{\mathrm{eff}}$ (K) | TIC 285935852 $\widetilde{T}_{\mathrm{eff}}$ (K)
---|---|---|---
${}^{1}(2.39\pm 0.20)\nu_{\mathrm{max,0}}$ ($\mu\mbox{Hz}$) + $(7110\pm 50)$ | 5.87 | [8530:8898] | [8429:8779]
${}^{2}(22.7\pm 4.0)\nu_{\mathrm{max,0}}(d^{-1})$ \+ $(6819\pm 21)$ | - | [7881:8387] | [7808:8282]
${}^{3}(3.5\pm 0.1)\nu_{\mathrm{max,w}}$ ($\mu\mbox{Hz}$) + $(6460\pm 40)$ | 3.36 | [8477:8773] | [8327:8807]
${}^{4}(1.14\pm 0.07)\nu_{\mathrm{max,2D}}+(1.22\pm 0.01)$ (solar units) | - | [8107:8697] | [8076:8662]

##  6 The grid of models

We have built a grid of 1D stellar models using the code mesa for the
evolution, and filou for calculating the oscillation modes, because it takes
into account the stellar distortion due to the centrifugal force in the
oscillation frequency computation, and rotation up to second order in the
perturbation approximation, including near-degeneracy effects. We have taken
orders $2\leq n\leq 8$ and degrees $0\leq l\leq 2$, for calculating the low-
order large separation of the models (Suárez et al., 2014; Rodríguez-Martín et
al., 2020).

Regarding the evolutionary models, we built a grid using the parameters of
Tab. 4, delimiting ages between 20 and 200 Myrs. According to Zbase values, we
selected the Type2 tables for opacities. For the mixing length parameter we
considered $\alpha=2.0$, and a diffusion coefficient for mixing of elements of
$D\\_mix=1/30$ (Heger et al., 2000). Overshooting was not considered. In
total, 24965 models were calculated.

Table 4: Selected values for the parameters of the grid of models built with
MESA

Parameter | Range | Step
---|---|---
Age | [20, 200] Myr | 1 Myr
M $(M_{\odot})$ | [1.6, 2.7] | 0.1 $M_{\odot}$
$Z_{0}$ | [0.014, 0.020] | 0.002
$\Omega/\Omega_{c}$ | [0.15, 0.25] | 0.05
$\alpha$ | 2.0 | Fixed

##  7 Seismic determination of the age of $\boldsymbol{\alpha\ P{er}}$

Fig. 10 shows the effective temperature vs age plots for all the calculated
models, using the different relations between the frequency at maximum power
and the effective temperature presented in Sec. 5. The constrained models for
TIC 410732825 are represented in green, for TIC 354792288 in blue, for TIC
285935852 in red and for TIC 252829836 in cyan.

We have compared the observed $\Delta\nu_{\mathrm{low}}$ value with the one
calculated directly from the models. If we use the $\nu$max,0-Teff relation
from BK2018 (top left panel) or $\nu$max,2D-Teff from H2021 (bottom left
panel), then the constrained models for our four stars show common ages
between 96 and 130 Myr. If we use the $\nu$max,0-Teff relation from BF2018
(top left panel) or the $\nu$max,w-Teff relation from BF2020 (bottom left
panel), the result is that the age of the cluster would be between 96 and 100
Myr, not too far from the most common age in the literature, around 90 Myr.

If we add as a constraint the rotation frequency obtained from the splittings
in TIC 410732825 and TIC 285935852, then the scenario changes. In Fig. 11 we
have plotted effective temperature vs age when using just the low-order large
separation (top left panel), when the rotation frequency of both stars are
added to constrain the models (top right panel), and also when using the
different relations between the frequency at maximum power and the effective
temperature (middle and bottom panels). When we have used BK2018 (middle right
panel) and H2021 (bottom right panel), two independent works, we obtained the
same common ages, between 96 and 130 Myr. However, in the cases of BF2018 and
BF2020, the result is that they don’t show common ages.

Tab. 5 shows the observed values of the main parameters of these four stars,
and Tab. 6 the values corresponding to the models when using as constraints
the low-order large separation and the relation BF2020 (see Fig. 10, bottom
left panel). The agreement between the different relationships is greater when
we do not constrain the models in rotation. Here we highlight the need to use
tighter parameters in models with rotation, such as overshooting or the mixing
length parameter, in order to improve the accuracy of the method. We have
chosen BF2020 as reference because it was obtained with a quite large sample
of pure $\delta$ Sct stars, as can be the case of the two stars with which we
have used the relation. BK2018 and H2021 also used hybrid stars, which can
impair the accuracy of the method.

Regarding TIC 410732825, the models tell us that it has a larger radius and a
lower density than the ones calculated from the observed parameters. The
density obtained from the large separation using the García Hernández et al.
(2017) relationship seems to corroborate this. The values of the projected
rotation of Kounkel et al. (2019) and the seismic rotation are very similar,
so the star seems to be equator on. This would explain why the possible values
of the observed luminosity are below those of the models. If the observed
parameters for TIC 252829836 were correct, then the star would have an age of
hundreds of millions of years in order to be compatible with the measured
large separation. This solution was therefore discarded, and points toward a
wrong estimate of its physical parameters.

Our four stars, which seem to be members of the cluster according to Lodieu et
al. (2019), and do not seem to be binary systems, according to Kounkel et al.
(2019), point to an age for the cluster of at least 96 Myr.

Table 5: Parameters of the selected targets. References: 1Stassun et al. 2019,
2Kounkel et al. 2019, 3García Hernández et al. 2017

TIC | $M(M_{\odot})^{1}$ | $R(R_{\odot})^{1}$ | $\bar{\rho}(\bar{\rho}_{\odot})$ | $logg^{1}$ | Teff$(K)^{1}$ | $log(L/L_{\odot})^{1}$ | $vsini(kms^{-1})^{2}$ | $\Delta\nu_{\mathrm{low}}(cd^{-1})$ | $\bar{\rho}_{\Delta\nu_{\mathrm{low}}}(\bar{\rho}_{\odot})^{3}$
---|---|---|---|---|---|---|---|---|---
410732825 | [1.983:2.651] | [1.592:1.686] | [0.41:0.66] | [4.30:4.45] | [8851:9453] | [1.18:1.28] | [71:107] | [62:64] | [0.31:0.33]
354792288 | [1.666:2.370] | [1.518:1.624] | [0.39:0.68] | [4.27:4.44] | [7849:8605] | [0.93:1.08] | [107:119] | [82:84] | [0.54:0.57]
285935852 | [1.465:2.063] | [1.545:1.645] | [0.33:0.56] | [4.20:4.36] | [7363:7815] | [0.84:0.92] | [68:73] | [81:83] | [0.53:0.56]
252829836 | [1.313:1.863] | [1.617:1.747] | [0.25:0.44] | [4.10:4.27] | [6994:7272] | [0.80:0.84] | [37:39] | [70:72] | [0.39:0.42]

Table 6: Constrained parameters of the models corresponding to our selected
targets, using the relation BF2020

TIC | $M(M_{\odot})$ | $R(R_{\odot})$ | $\bar{\rho}(\bar{\rho}_{\odot})$ | log g | Teff(K) | $log(L/L_{\odot})$ | $Z_{0}$ | $v(kms^{-1})$ | $\Delta\nu_{\mathrm{low}}(cd^{-1})$ | Age (Myr)
---|---|---|---|---|---|---|---|---|---|---
410732825 | [2.3:2.6] | [1.94:2.10] | [0.28:0.32] | [4.20:4.24] | [9731:11055] | [1.49:1.75] | [0.016:0.020] | [72:125] | [62:64] | [96:200]
354792288 | [1.75:1.85] | [1.51:1.54] | [0.49:0.52] | [4.32:4.34] | [8592:8666] | [1.05:1.06] | [0.016:0.020] | [71:116] | [82:84] | [20:100]
285935852 | [1.75:1.85] | [1.51:1.55] | [0.48:0.52] | [4.31:4.33] | [8579:8660] | [1.05:1.07] | [0.016:0.020] | [71:116] | [81:83] | [20:130]
252829836 | [1.9:2.6] | [1.68:1.94] | [0.36:0.40] | [4.24:4.30] | [8541:11248] | [1.13:1.71] | [0.016:0.020] | [69:126] | [70:72] | [20:200]

Figure 10: Constrained models for dating ages of the four stars, just using
the low regime large separation ($\Delta\nu$), and also adding the different
relationships between the frequency at maximum power and the effective
temperature: $\Delta\nu+BF2018$, $\Delta\nu+BK2018$, $\Delta\nu+BF2020$ and
$\Delta\nu+H2021$ Figure 11: Constrained models for dating ages of the four
stars, just using the low regime large separation ($\Delta\nu$), adding the
angular rotation for TIC 410732825 and TIC 285935852 ($\Delta\nu+\omega_{r}$)
and also adding the different relationships between the frequency at maximum
power and the effective temperature: $\Delta\nu+\omega_{r}+BF2018$,
$\Delta\nu+\omega_{r}+BK2018$, $\Delta\nu+\omega_{r}+BF2020$ and
$\Delta\nu+\omega_{r}+H2021$

##  8 Conclusions

We have tested an asteroseismic method in order to determine the age of the
young open cluster $\alpha$ Per. The technique is based on finding seismic
indices, such as the low-order large separation, the frequency at maximum
power and the rotation frequency in a sample of 11 $\delta$ Sct stars stars
belonging to the field of the cluster. With them we have used the FT, the AC,
the HFD and the ED. In four of them, TIC 410732825, TIC 354792288, TIC
285935852 and TIC 252829836 we have measured the low-order regime large
separation. With TIC 410732825 and TIC 285935852 we have found evidences of
the angular rotation frequency. With the necessary caution, considering that
they are near the ZAMS, we have used the relations between the frequency at
maximum power and the effective temperature to constrain the age of $\alpha$
Per between 96 and 100 Myr.

The grid we have used can be extended to take into account the variability of
parameters such as overshooting, the mixing length parameter or the diffusion
of angular momentum.

The estimated age for $\alpha$ Per must be considered with caution until new
studies with more $\delta$ Sct stars stars and/or a more detailed modeling
allow us to confirm it. In any case, the method based on seismic indices looks
like a promising tool for dating young stellar clusters. For the frequency
analysis, we have developed a new code, MultiModes, (blueMultiModes), that we
present in this paper. The code shows a good reliability after testing it with
synthetic light curves, and after being compared with one of the most reliable
codes in the field, SigSpec. Moreover, MultiModes is more versatile in terms
of frequency analysis control. It has allowed us to find nine previously
unknown $\delta$ Sct stars from the original sample of 32 analysed stars.

## Acknowledgements

This publication is part of the project "Contribution of the UGR to the
PLATO2.0 space mission. Phases C / D-1", funded by
MCNI/AEI/PID2019-107061GB-C64.

DPO acknowledges MNRAS for the opportunity to publish his first scientific
article, in particular to the assistant editor, Bella Lock, for her comments.
He also appreciates the comments of the anonymous reviewer of this article.
They have undoubtedly contributed to improving it. He also acknowledges all
the co-authors of this work, especially to JCS and AGH, his thesis project
directors, because without them, he would not have been able to publish this
work. And he also appreciates the patience of his wife, Pilar, and his son,
Fernando, for all the time they have allowed him to get to this point.

AGH acknowledges funding support from Spanish public funds for research under
project PID2019-107061GB-C64 by the Spanish Ministry of Science and Education,
and from ‘European Regional Development Fund/Junta de Andalucía-Consejería de
Economía y Conocimiento’ under project E-FQM-041-UGR18 by Universidad de
Granada.

JPG acknowledges funding support from Spanish public funds for research 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", as well as from the State
Agency for Research of the Spanish MCIU through the “Center of Excellence
Severo Ochoa” award to the Instituto de Astrofísica de Andalucía
(SEV-2017-0709).

SBF received financial support from the Spanish State Research Agency (AEI)
Projects No. PID2019-107061GB-C64. He also thanks the resources received from
the PLATO project collaboration with Centro de Astrobiología
(PID2019-107061GB-C61).

This paper includes data collected with the TESS mission, obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the TESS mission is provided by the NASA Explorer Program. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555.

## Data Availability

Tables of the most significant peaks, corresponding to each of the 11 $\delta$
Sct stars of our sample, extracted with MM, are available in blueVizieR
DataBase. In them we can find the values of the frequencies, amplitudes,
phases, corresponding errors and S/N of the extracted peaks.

## List of acronyms

  * •

AC: Autocorrelation Function

  * •

BF2018: Barceló Forteza et al. (2018)

  * •

BF2020: Barceló Forteza et al. (2020)

  * •

BK2018: Bowman & Kurtz (2018)

  * •

DFT: Discrete Fourier Transform

  * •

ED: Èchelle Diagram

  * •

HFD: Histogram of frequency differences

  * •

FT: Fourier Transform

  * •

H2021: Hasanzadeh et al. (2021)

  * •

MM: MultiModes code

  * •

SS: SigSpec code

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##  Appendix A Comparative analysis of efficiency between MM and SS

Fig. A.1 represents the executing time of both codes using the light curves of
11 $\delta$ Sct stars stars from our sample. MM is a factor two faster than SS
in the cases where the number of extracted peaks is above 200 approximately.

Figure A.1: Comparative analysis of efficiency between MM and SS, in terms of
computing time, done with the sample of 11 $\delta$ Sct stars stars

##  Appendix B Diagrams of regularities of TIC 285935852 and TIC 252829836

We include here the AC, FT, HFD and ED of TIC 285935852 (Fig. A.2) and TIC
252829836 (Fig. A.3), which together with TIC 410732825 (Fig. 7) and TIC
354792288 (Fig. 3), complete the four stars analyzed for which regularities
have been found in their corresponding frequency spectra.

Figure A.2: Measured regularities for TIC 285935852 Figure A.3: Measured
regularities for TIC 252829836