# Imprints of the nuclear symmetry energy slope in gravitational wave signals
emanating from neutron stars

Luiz L. Lopes<EMAIL_ADDRESS>Centro Federal de Educação Tecnológica de
Minas Gerais Campus VIII, Varginha/MG, CEP 37.022-560, Brazil  Victor B. T.
Alves  Departamento de Física, CCET - Universidade Federal do Maranhão, Campus
Universitário do Bacanga; CEP 65080-805, São Luís, MA, Brasil.  César O. V.
Flores<EMAIL_ADDRESS>Centro de Ciências Exatas, Naturais e
Tecnológicas, CCENT - Universidade Estadual da Região Tocantina do Maranhão;
C.P. 1300,
CEP 65901-480, Imperatriz, MA, Brasil.  Departamento de Física, CCET -
Universidade Federal do Maranhão, Campus Universitário do Bacanga; CEP
65080-805, São Luís, MA, Brasil.  Germán Lugones<EMAIL_ADDRESS>Universidade Federal do ABC, Centro de Ciências Naturais e Humanas, Avenida
dos Estados 5001 - Bangú, CEP 09210-580, Santo André, SP, Brazil.

###### Abstract

We investigate possible traces of the nuclear symmetry energy slope ($L$) in
the gravitational wave emission of neutron stars. For fixed stellar mass
values, we examine how the slope influences the stellar radius, compactness,
the tidal deformability, the frequency of the quadrupole fundamental fluid
mode, and the damping time of the mode due to the gravitational wave emission.
We demonstrate that all these physical quantities are sensitive to the slope
and could potentially impose significant constraints on it.

††preprint:

##  I Introduction

Our knowledge of nuclear physics has taken a great leap forward in the last
decade. However, as of today, there is a giant fog blurring our vision of a
crucial nuclear property: the slope of the symmetry energy at the saturation
point ($L$). In the early 2010s, most studies pointed to a relatively low
value of $L$. For instance, references [1, 2, 3] presented upper limits of
54.6, 61.9, and 66 $\mathrm{MeV}$, respectively. However, in the last couple
of years, much higher upper limits have appeared in the literature. For
example, a study of the spectra of pions in intermediate energy collisions
pointed to an upper limit of 117.5 $\mathrm{MeV}$[4], while one of the PREX2
analyses [5] suggested even larger values. On the other hand, in a recent
paper [6] a potential conflict among these results was highlighted: the CREX
group reported a slope in the range $0<L(\mathrm{MeV})<51$, while the PREX2
results pointed to $76<L(\mathrm{MeV})<165$; both at the 68% confidence level.
Although there is overlapping between them at the 90% confidence level [7],
the PREX results are in tension with CREX, as well as with predictions of
chiral effective field theory [5].

Due to the potentially conflicting results obtained in terrestrial
laboratories, we turn our attention to studying neutron stars, which may
provide valuable insights. Just like in nuclear physics, our knowledge of
neutron stars has dramatically increased in the last decade. We can highlight
the discovery of two-solar masses pulsars, such as PSR J0348+0432 [8], as well
as results from the NICER X-ray telescope [9, 10] and the LIGO/VIRGO
gravitational wave observatories [11, 12, 13].

In this work, we study how the symmetry energy slope $L$ affects some neutron
star properties. To accomplish this task, we use quantum hadrodynamics (QHD)
with the traditional $\sigma-\omega-\rho$ mesons [14]. Moreover, to keep the
symmetry energy fixed while varying the slope, we use two extensions of QHD:
to reduce the slope, we add the non-linear $\omega-\rho$ coupling as presented
in the IUFSU model [15, 16, 17], while to increase the slope, we add the
scalar-isovector $\delta$ meson [18, 19, 20]. Using two different interactions
allows us not only to understand the phenomenological point of view about the
influence of changes in $L$, but also the field theory point of view about the
influence of different fields and couplings.

We start by studying how the slope, $L$, affects the classical, macroscopic
mass-radius relation [16, 21]. Although we demonstrate how the slope affects
the entire neutron star family, we also focus explicitly on two specific
values: the 1.4 $M_{\odot}$ star, which is called the canonical star, and the
2.01 $M_{\odot}$ star, which is not only the most probable mass value of PSR
J0348+0432 [8], but also the lower limit of PSR J0740+6620, whose
gravitational mass is 2.08 $\pm$ 0.07 $M_{\odot}$ [22, 23]. Therefore, any
equation of state (EOS) unable to reach at least 2.01 $M_{\odot}$ must be
ruled out.

We then analyze the influence of the slope on the dimensionless tidal
parameter $\Lambda$ [21]. The tidal deformability of a compact object is
related to how easily the object is deformed when subjected to an external
tidal field. A larger tidal deformability indicates that the object is easily
deformable. Conversely, a compact object with a small tidal deformability
parameter is more compact and more difficult to deform [24]. Binary neutron
star mergers, such as GW170817 detected by the LIGO/VIRGO observatories [12,
11, 13], provide us with additional information about the neutron star’s EOS.
In these types of events, the neutron star components begin to react to their
mutual tidal fields before the merging, and this effect can be detected in the
phase modification of the gravitational wave impinging on the detector. This
tidal response strongly depends on the neutron star EOS, and therefore
important information can be obtained about it [25, 26, 27].

Finally, neutron star oscillations can provide valuable information about the
microphysical properties of dense matter. A wide variety of pulsation modes
can be excited in newly born compact objects, such as those associated with
the violent dynamics of core-collapse supernovae [28], starquakes and glitches
[29], accretion in a binary system, or the rearrangement of the star following
the conversion of a hadronic star into a quark star [30, 31]. Furthermore, in
the post-merger phase of a binary neutron star system, the violent dynamics of
the merging process can leave behind a massive neutron star that strongly
oscillates [32, 33]. Specifically, the fundamental quadrupolar fluid mode of
the remnant is strongly excited and dominates the post-merger gravitational
wave (GW) signal.

Neutron star oscillations can be studied using the quasi-normal mode formalism
[34, 35, 36]. Research conducted over the past four decades has demonstrated
that the $f$-mode of non-radial oscillations of neutron stars can yield
crucial insights into the internal structure of these objects. Specifically,
the frequency of this mode is highly dependent on the EOS, thereby providing a
valuable means of probing the dense matter in the core of neutron stars.
Additionally, the $f$-mode is particularly significant as it can be more
easily excited than other higher frequency modes and can be detected by
current gravitational wave detectors. In this work we will focus on the
influence of the slope, $L$, on the $f$-mode of neutron stars.

##  II Nuclear model

###  II.1 Quantum hadrodynamics

To describe the nuclear interaction, we use here an extended version of the
QHD [14], which includes the $\omega-\rho$ non-linear coupling [15, 16, 17],
as well the scalar-isovector $\delta$ meson [18, 19, 20]. In this study we
focus on purely nucleonic matter. The Lagrangian density in natural units
reads:

$\displaystyle\mathcal{L}_{QHD}=$
$\displaystyle\bar{\psi}_{N}\big{[}\gamma^{\mu}(\mbox{i}\partial_{\mu}-g_{\omega}\omega_{\mu}-g_{\rho}\tfrac{1}{2}\vec{\tau}\cdot\vec{\rho}_{\mu})$
(1) $\displaystyle-(M_{N}-g_{\sigma}\sigma-
g_{\delta}\vec{\tau}\cdot\vec{\delta})\big{]}\psi_{N}-U(\sigma)$
$\displaystyle+\tfrac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{s}^{2}\sigma^{2})$
$\displaystyle+\tfrac{1}{2}(\partial_{\mu}\vec{\delta}\cdot\partial^{\mu}\vec{\delta}-m_{\delta}^{2}\delta^{2})$
$\displaystyle-\tfrac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu}+\tfrac{1}{2}m_{v}^{2}\omega_{\mu}\omega^{\mu}$
$\displaystyle+\Lambda_{\omega\rho}(g_{\rho}^{2}\vec{\rho^{\mu}}\cdot\vec{\rho_{\mu}})(g_{\omega}^{2}\omega^{\mu}\omega_{\mu})$
$\displaystyle+\tfrac{1}{2}m_{\rho}^{2}\vec{\rho}_{\mu}\cdot\vec{\rho}^{\;\mu}-\tfrac{1}{4}\bf{P}^{\mu\nu}\cdot\bf{P}_{\mu\nu}.$

Here, $\psi_{N}$ is the Dirac field of the nucleons, and $\sigma$,
$\omega_{\mu}$, $\vec{\delta}$, and $\vec{\rho}_{\mu}$ are the mesonic fields.
The $g$’s are the Yukawa coupling constants that simulate the strong
interaction, $M_{N}$ is the nucleon mass, and $m_{s}$, $m_{v}$, $m_{\delta}$,
and $m_{\rho}$ are the masses of the $\sigma$, $\omega$, $\delta$, and $\rho$
mesons, respectively. The self-interaction term $U(\sigma)$, introduced in
Ref. [37] to fix the incompressibility, is given by:

$U(\sigma)=\frac{\kappa
M_{N}(g_{\sigma}\sigma)^{3}}{3}+\frac{\lambda(g_{\sigma}\sigma)^{4}}{4}.$ (2)

The Pauli matrices are denoted by $\vec{\tau}$, the antisymmetric mesonic
field strength tensors are given by their usual expressions:
$\Omega_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$, and
$P_{\mu\nu}=\partial_{\mu}\vec{\rho}_{\nu}-\partial_{\nu}\vec{\rho}_{\mu}-g_{\rho}(\vec{\rho}_{\mu}\times\vec{\rho}_{\nu})$.
The $\gamma^{\mu}$ are the Dirac matrices and $\Lambda_{\omega\rho}$ is a non-
linear isoscalar-isovector mixing coupling that provides a simple and
efficient method of softening the symmetry energy without compromising the
success of the model in reproducing well-determined ground-state observables
[15]. The detailed calculation of the EOS in the mean field approximation can
be found in [14, 15, 20, 38, 39] and the references therein.

The symmetry energy for symmetric nuclear matter is defined as [18]:

$\displaystyle S(n)=$
$\displaystyle\frac{n}{8}\left(\frac{g_{\rho}}{m_{\rho}^{*}}\right)^{2}+\frac{k_{f}^{2}}{6\,(M_{N}^{*2}+k_{f}^{2})^{1/2}}$
(3)
$\displaystyle-\left(\frac{g_{\delta}}{m_{\delta}}\right)^{2}\frac{M_{N}^{*2}~{}n/2}{(M_{N}^{*2}+k_{f}^{2})[1+(g_{\delta}/m_{\delta})^{2}A(k_{f})]},$

where $n$ is the baryon number density, $k_{f}$ is the Fermi momentum,
$m_{\rho}^{*}=(m_{\rho}^{2}+2\Lambda_{\omega\rho}g_{\rho}^{2}g_{\omega}^{2}\omega_{0}^{2})^{1/2}$
is the effective mass of the meson $\rho$,
$M_{N}^{*}=M_{N}-g_{\sigma}\sigma_{0}-g_{\delta}\tau_{3}\delta_{0}$ is the
nucleon effective mass (it is worth to emphasize that for symmetric nuclear
matter, the $\delta$ field is zero), and $A(k_{f})$ is given by:

$A(k_{f})=\frac{4}{(2\pi)^{3}}\int_{0}^{k_{f}}d^{3}k\frac{k^{2}}{(M_{N}^{*2}+k^{2})^{3/2}}.$
(4)

Finally, the slope of the symmetry energy at the saturation point, $L$, is
defined as:

$L=3n\bigg{(}\frac{\partial S}{\partial n}\bigg{)}\bigg{|}_{n=n_{0}}.$ (5)

We can also define the slope for an arbitrary density, $L(n)$, as in Ref.
[40]:

$L(n)=3n\bigg{(}\frac{\partial S}{\partial n}\bigg{)}.$ (6)

###  II.2 Parametrization and symmetric nuclear matter

In this work, we use a slightly modified version of the L3$\omega\rho$
parametrization proposed in Ref. [41]. The model parameters and their
predictions for symmetric nuclear matter are presented in Table 1. The nuclear
constraints at saturation density, taken from two extensive review articles
[42, 43], are also included in Table 1. We select the parameters
$(g_{\rho}/m_{\rho})^{2}$, $(g_{\delta}/m_{\delta})^{2}$, and
$\Lambda_{\omega\rho}$ in order to fix the symmetry energy at the saturation
point $S_{0}$ = 31.7 MeV (which is slightly higher than the original value,
$S_{0}$ = 31.2 MeV [41]), while varying the slope $L$. The values of these
parameters are reported in Table 2.

| Parameters |  | Constraints | Our model
---|---|---|---|---
$\left(g_{\sigma}/{m_{s}}\right)^{2}$ | $12.108\,\mathrm{fm}^{2}$ | $n_{0}(\mathrm{fm}^{-3})$ | 0.148 - 0.170 | 0.156
$\left(g_{\omega}/{m_{v}}\right)^{2}$ | $7.132\,\mathrm{fm}^{2}$ | $M^{*}/M$ | 0.6 - 0.8 | 0.69
$\kappa$ | 0.004138 | $K\mathrm{(MeV)}$ | 220 - 260 | 256
$\lambda$ | -0.00390 | $S_{0}\mathrm{(MeV)}$ | 28.6 - 34.4 | 31.7
- | - | $B/A\mathrm{(MeV)}$ | 15.8 - 16.5 | 16.2
Table 1: Model parameters used in this study and their predictions for symmetric nuclear matter at saturation density. The phenomenological constraints were taken from Refs. [42, 43]. $L$ (MeV) |  $(g_{\rho}/m_{\rho})^{2}$ ($\mathrm{fm}^{2}$) |  $(g_{\delta}/m_{\delta})^{2}$ ($\mathrm{fm}^{2}$) | $\Lambda_{\omega\rho}$
---|---|---|---
44 | 8.40 | 0 | 0.0515
60 | 6.16 | 0 | 0.0344
76 | 4.90 | 0 | 0.0171
92 | 4.06 | 0 | 0
100 | 7.23 | 0.92 | 0
108 | 10.41 | 1.85 | 0
116 | 13.48 | 2.76 | 0
Table 2: Model parameters selected to set the symmetry energy at $S_{0}$ =
31.7 MeV.
---
Figure 1: Symmetry energy $S(n)$ and its slope $L(n)$ as a function of nucleon
density up to twice the saturation density.

Fig. 1 shows the symmetry energy $S(n)$ and its slope $L(n)$ for densities up
to twice the saturation density. We observe that the symmetry energy behaves
oppositely for $n/n_{0}<1$ and $n/n_{0}>1$. Parametrizations with lower slopes
have higher values of $S(n)$ for $n<n_{0}$ and lower values of $S(n)$ for
$n>n_{0}$. This behavior is explained as follows. For parametrizations with
nonlinear $\omega-\rho$ coupling, Table 2 indicates that those with lower
slopes have higher values of $(g_{\rho}/m_{\rho})^{2}$ and
$\Lambda_{\omega\rho}$. The first term in Eq. (3) is more relevant at lower
densities than at higher ones, owing to the fact that at lower densities,
$m_{\rho}^{*}~{}\simeq~{}m_{\rho}$. As the density rises, $m_{\rho}^{*}$ also
increases (due to its dependence on the $\omega$ field, which in turn
increases with the number density), consequently causing a decrease in
$(g_{\rho}/m^{*}_{\rho})^{2}$. This decline diminishes the impact of the first
term in Eq. (3).

When we include the scalar-isovector $\delta$ meson, we observe that the
contribution of the terms with the parameters $(g_{\rho}/m_{\rho})^{2}$ and
$(g_{\delta}/m_{\delta})^{2}$ increase as the slope of the symmetry energy
increases. In this case, we have a competition between the repulsive vector
$\rho$ meson and the attractive scalar $\delta$ meson. Since the vector field
increases with the cube of the Fermi momentum, while the scalar field grows
linearly [14], the attractive field dominates at lower densities, while the
repulsive field dominates at higher densities. The symmetry energy at twice
the saturation density ranges from $45~{}\mathrm{MeV}$ for a slope of
$L=44~{}\mathrm{MeV}$ to $85~{}\mathrm{MeV}$ for a slope of
$L=116~{}\mathrm{MeV}$.

For the slope at an arbitrary density, $L(n)$, we obtain analogous results
i.e., the values of lower $L_{0}$ present a larger value of $L(n)$ up to
densities around $0.5n_{0}$, and then the results are reversed. We find that
the slope at $n=0.5n_{0}$ is approximately the same, $L(0.5n_{0})\approx
44~{}\mathrm{MeV}$. However, at twice the saturation density, the slope varies
from $L(2n_{0})=83~{}\mathrm{MeV}$ for $L=44~{}\mathrm{MeV}$ to
$L(2n_{0})=412~{}\mathrm{MeV}$ for $L=116~{}\mathrm{MeV}$. These behaviors can
be explained by the nature of the couplings, analogous to the behavior of the
symmetry energy.

##  III Stellar structure and tidal deformability

To investigate the effects of the nuclear symmetry energy slope on neutron
star properties, we obtain the EOS of locally charge-neutral nuclear matter by
including leptons (electrons and muons) in the system. Additionally, we assume
that nuclear matter is in equilibrium under weak interactions and that the
system is at zero temperature. To describe the outer crust and inner crust of
the neutron star, we utilize the Baym-Pethick-Sutherland (BPS) EOS [44] and
the Baym-Bethe-Pethick (BBP) EOS [45], respectively. We use the BPS+BBP EoS up
to the density of 0.0089 fm-3 for all values of $L$, and from this point on,
we use the QHD EoS, as suggested in Ref. [46]. In Ref. [47], the authors
compare the BPS+BBP crust EoS with a unified EoS. They show that for the
canonical star, there is a variation in the radius of
$60\mathrm{m}<~{}R_{1.4}~{}<150\mathrm{m}$. For a radius of 13 km, this
implies an uncertainty around 1$\%$. Once we have the crust+core EoS, the
stellar structure is calculated by means of the Tolman-Oppenheimer-Volkoff
(TOV) equations [48]:

$\displaystyle\frac{dp}{dr}$ $\displaystyle=$
$\displaystyle-\frac{m\epsilon}{r^{2}}\bigg{[}1+\frac{p}{\epsilon}\bigg{]}\bigg{[}1+\frac{4\pi
pr^{3}}{m}\bigg{]}\bigg{[}1-\frac{2m}{r}\bigg{]}^{-1},$ (7)
$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}\epsilon.$ (8)

Here, $\epsilon$, $p$, and $m$ represent the energy density, pressure, and
mass at the radial position $r$, respectively. We adopt units such that
$G=c=1$.

---
Figure 2: The figure depicts the EOS (upper panel) and mass-radius
relationship (lower panel) for different $L$ values, along with observational
constraints mentioned in the text. In this figure, we take $M_{0}$ to
represent the solar mass.. The EOS curves converge at approximately at
$0.5n_{0}$, where all models have the same $L$ (see Fig. 1). Beyond this
point, a larger $L$ results in a stiffer EOS.

Fig. 2 displays the EOS and the TOV solutions for different values of $L$.
Despite the EOS curves are beta-stable, while the slope $L(n)$ curves are
relative to symmetric matter, both present similar behavior; i.e., the EOS is
stiffer for low values of $L$ up to $n~{}\simeq$ 0.5 $n_{0}$. From this point
on, this feature is reversed, and low values of $L$ present softer EOS. This
can be understood by expanding the EoS of asymmetric matter in terms of its
symmetric components. Various theoretical studies (see, for instance Ref.
[49]) have shown that the energy per nucleon, $E$, of asymmetric nuclear
matter can be well approximated by:

$E(n,\alpha)=E(n,\alpha=0)+S(n)\alpha^{2}+O(\alpha^{4})$ (9)

in terms of the baryon number density $n=n_{n}+n_{p}$, the isospin asymmetry
$\alpha=(n_{n}-n_{p})/(n_{n}+n_{p})$, the energy per nucleon in symmetric
nuclear matter $E(n,\alpha=0)$, and the bulk nuclear symmetry energy $S(n)$.
The pressure is given by $p=n^{2}\frac{\partial(\epsilon/n)}{\partial n}$, and
the energy per nucleon is related to the energy density as $E=\epsilon/n$.
Therefore:

$p(n,\alpha)=p(n,\alpha=0)+\frac{n}{3}\alpha^{2}L(n)+n^{2}S(n)\frac{\partial(\alpha^{2})}{\partial
n}+\cdots$ (10)

where $p(n,\alpha=0)$ is the pressure of symmetric matter. Neglecting the
variation of $\alpha^{2}$ with $n$, we obtain that the lowest order correction
to $p(n,\alpha=0)$ is directly related to $L(n)$:

$p(n,\alpha)\approx p(n,\alpha=0)+\frac{n}{3}\alpha^{2}L(n).$ (11)

Given that the first term is the same for all choices of $L$, the latter
equation shows clearly why the slope $L(n)$ curves relative to symmetric
matter present similar behavior as the pressure of beta stable matter around
$n=0.5n_{0}$ However, as the inner-outer core transition, around $2n_{0}$, is
approached, the curves start to converge and exhibit similar levels of
stiffness since the nuclear interaction at high density is dominated by the
$\omega$ meson. The correlation between $L$ and the radii of NSs (i.e., that
models with larger $L$ have larger radii) has been well established in the
literature for the $\omega-\rho$ coupling [16, 21]. In our study, we show that
this correlation can also be extended to cases where the $\delta$ meson is
included. Regarding the maximum mass, it varies from $2.30M_{\odot}$ up to
$2.39M_{\odot}$. Additionally, we observe that parametrizations with smaller
values of $L$ \- and therefore with the $\omega-\rho$ coupling - exhibit
slightly lower maximum masses compared to those with higher slope values - and
therefore with the scalar-isovector $\delta$ coupling.

We present a comparison of our results with recent constraints on the
properties of neutron stars in Fig. 2. One of the most significant constraints
is the existence of two solar masses NSs, which has been well-established by
observations of pulsars such as PSR J1614-2230 [50], PSR J0348+0432 [8], and
PSR J0740+6620 [51]. Recent NICER observations have also constrained the
radius of PSR J0740+6620 to be between $11.41-13.69~{}\mathrm{km}$ [23].
Another, less stringent, constraint is the radius of the canonical
$1.4M_{\odot}$ star. While Ref. [52] suggests an upper limit of only
$11.1~{}\mathrm{km}$, Ref. [53] raises the limit to $11.9~{}\mathrm{km}$.
Meanwhile, two NICER teams have pointed to a limit of $13.85~{}\mathrm{km}$
[9] and $14.26~{}\mathrm{km}$ [10] ($68\%$ credibility). We adopt the larger
of these values as a conservative limit on the radius of the canonical star.
These constraints are represented by the hatched areas in Fig. 2. Notice that
the constraint on the mass and radius of PSR J0740+6620 is satisfied for all
values of $L$, although for $L=116~{}\mathrm{MeV}$, this constraint is close
to the boundary of the 68% credibility region. Concerning the radius of the
canonical star, we observe that even though PREX2 implies that the slope could
be as high as $143~{}\mathrm{MeV}$, a value of $L=116~{}\mathrm{MeV}$ already
leads to $R_{(1.4)}>14.26~{}\mathrm{km}$.

---
Figure 3: The compactness (top), the second Love number (middle), and the dimensionless tidal parameter (bottom) as a function of the neutron star mass for different values of $L$. $L(\mathrm{MeV})$ | $M_{\mathrm{max}}(M_{\odot})$ |  $R_{(1.4)}$ (km) | $C_{(1.4)}$ | $k_{2(1.4)}$ | $\Lambda_{(1.4)}$ |  $R_{(2.01)}$ (km) | $C_{(2.01)}$ | $k_{2(2.01)}$ | $\Lambda_{(2.01)}$
---|---|---|---|---|---|---|---|---|---
44 | 2.31 | 12.58 | 0.165 | 0.0947 | 515 | 12.40 | 0.240 | 0.0500 | 42
60 | 2.30 | 12.74 | 0.163 | 0.0885 | 513 | 12.47 | 0.239 | 0.0473 | 40
76 | 2.30 | 12.99 | 0.160 | 0.0841 | 535 | 12.59 | 0.236 | 0.0452 | 40
92 | 2.34 | 13.48 | 0.155 | 0.0839 | 639 | 13.04 | 0.228 | 0.0469 | 51
100 | 2.37 | 13.73 | 0.151 | 0.0813 | 685 | 13.34 | 0.223 | 0.0476 | 58
108 | 2.38 | 13.99 | 0.149 | 0.0799 | 728 | 13.59 | 0.219 | 0.0478 | 63
116 | 2.39 | 14.30 | 0.146 | 0.0796 | 812 | 13.82 | 0.215 | 0.0482 | 70
Table 3: For each value of the slope $L$, we give the maximum mass and the
properties of the canonical $1.4M_{\odot}$ star and the $2.01M_{\odot}$ star.

In recent years, the theory of relativistic tidal effects in binary systems
has been the focus of intense research. Below, we provide a summary of the
procedure for computing the dimensionless tidal parameter $\Lambda$, which
measures how easily an object is deformed by an external tidal field. It is
defined as follows:

$\Lambda=\frac{2k_{2}}{3C^{5}},$ (12)

where $M$ is the mass of the compact object and $C=GM/R$ is its compactness.
The parameter $k_{2}$ is known as the second-order Love number and is given by
(see Ref. [54]):

$\displaystyle k_{2}=$
$\displaystyle\frac{8C^{5}}{5}(1-2C)^{2}\left[2-y_{R}+2C\left(y_{R}-1\right)\right]$
(13)
$\displaystyle\times\left\\{2C\left[6-3y_{R}+3C\left(5y_{R}-8\right)\right]\right.$
$\displaystyle+4C^{3}\left[13-11y_{R}+C\left(3y_{R}-2\right)+2C^{2}\left(1+y_{R}\right)\right]$
$\displaystyle\left.+3(1-2C)^{2}\left[2-y_{R}+2C\left(y_{R}-1\right)\right]\ln(1-2C)\right\\}^{-1}$

where $y_{R}=y(r=R)$ and $y(r)$ is obtained by solving:

$r\frac{dy}{dr}+y^{2}+yF(r)+r^{2}Q(r)=0.$ (14)

The coefficients $F(r)$ and $Q(r)$ are given by:

$\displaystyle F(r)$ $\displaystyle=$ $\displaystyle{\left[1-4\pi
r^{2}(\epsilon-p)\right]\left[1-\frac{2m}{r}\right]^{-1}},$ (15)
$\displaystyle Q(r)$ $\displaystyle=$ $\displaystyle
4\pi\left[5\epsilon+9p+\frac{\epsilon+p}{c_{s}^{2}}-\frac{6}{4\pi
r^{2}}\right]\left[1-\frac{2m}{r}\right]^{-1}$ (16)
$\displaystyle-\frac{4m^{2}}{r^{4}}\left[1+\frac{4\pi
r^{3}p}{m}\right]^{2}\left[1-\frac{2m}{r}\right]^{-2},$

where $c_{s}^{2}\equiv dp/d\epsilon$ is the squared speed of sound. The
boundary condition for Eq. (14) at $r=0$ is given by $y(0)=2$. To obtain the
tidal Love number, we use the EOS of Sec. II and integrate the TOV equations
along with Eq. (14).

In Fig. 3 we show $C$, $k_{2}$ and $\Lambda$ as a function of the neutron star
mass. As expected, the lower the slope, the higher the star’s compactness.
However, the Love number $k_{2}$, also increases when we reduce the slope,
which creates a competition between $C$ and $k_{2}$ in the determination of
$\Lambda$, at least for stars with $M<1.8M_{\odot}$. Nonetheless, it can be
seen that $\Lambda$, which depends on both $C$ and $k_{2}$, mostly decreases
with decreasing slope. This trend is valid for both models, i.e., the one with
the $\omega-\rho$ coupling and the one with the scalar-isovector $\delta$
meson coupling.

The tidal parameter of the canonical star, $\Lambda_{(1.4)}$, is also a useful
tool to constrain $L$, but different restrictions are presented in the
literature, and they are mutually exclusive. For instance, the analysis of the
GW170817 event in Ref. [13] constrained the dimensionless tidal parameter to
be between $70$ and $580$, which suggests a rather soft EoS. Meanwhile, in
Ref. [5] the PREX2 experiment, combined with results from the NICER X-ray
telescope, constrained the tidal parameter to be between $642$ and $955$,
which suggests a rather stiff EoS. (Nevertheless, it is worth mentioning that
PREX2 and CREX results are still being debated in the community [6]). As
observed, the parametrizations that incorporate the $\omega-\rho$ coupling,
and hence have lower values of $L$, align with the results from Ref. [13].
Conversely, the parametrizations that include the coupling with the scalar-
isovector $\delta$ meson and therefore have higher values of $L$ align with
Ref. [5]. For $L=92$ MeV, the value of $\Lambda_{(1.4)}=639$ falls into an
ambiguous range.

We will now specifically focus on the role of $L$ in two cases: the canonical
$1.4M_{\odot}$ star and the $2.01M_{\odot}$ star. As previously emphasized,
the radius of the canonical star increases with the slope $L$, ranging from
12.58 km to 14.30 km. The tidal parameter of the canonical star also increases
with $L$, although the models with the lowest $L$ values (44 MeV and 60 MeV)
are almost indistinguishable. When the parameter $L$ is larger, the EOS
becomes stiffer. A stiffer EOS means that matter is more resistant to
compression, leading to a larger radius for a given NS mass. With the mass of
the NS held constant, the tidal deformability scales with the fifth power of
the NS radius. Consequently, a stiffer EOS leads to a larger tidal
deformability for a given mass value. This accounts for the positioning of the
$\Lambda$ curves in the lower panel of Fig. 3; those corresponding to higher
$L$ values are situated above those associated with lower $L$ values. It is
worth noting that, except for $L=116$ MeV, all other parametrizations result
in $R_{(1.4)}<$ 14.26 km, which agrees with [10, 5], and
$\Lambda_{(1.4)}<800$, in agreement with Ref. [11]. Regarding the 2.01
$M_{\odot}$ neutron stars, we observe that their radii also increase with $L$
and fall in the range of 12.40 km to 13.82 km. However, the Love number
$k_{2(2.01)}$ exhibits a more erratic behavior, in contrast to $k_{2(1.4)}$
which exhibits a monotonically decreasing trend with $L$. Finally, we have
$40<\Lambda_{(2.01)}<70$. In Table 3, we provide a summary of some of the main
properties of neutron stars for various values of the slope $L$.

##  IV Neutron star oscillations

In the near future, gravitational wave observatories are expected to detect
pulsation modes of compact stars that are excited in binary mergers and newly
born compact objects associated with core-collapse supernovae. Among the vast
family of oscillation modes of compact stars, the quadrupolar ($\ell=2$)
fundamental ($f$) mode is crucial because it is highly probable that it will
be significantly excited, thereby enabling its detection by the upcoming
generation of instruments. Observations of $f$-mode oscillations can provide
valuable information about the internal structure and composition of neutron
stars. For example, the frequency and damping rate of the $f$-mode oscillation
can be used to constrain the EOS of the dense matter in the star’s core (see
[55, 56, 57, 58] and references therein). Here, we will explore the
possibility of constraining the nuclear symmetry energy slope from $f$-mode
observations.

Non-radial oscillations of neutron stars are analyzed using first-order
perturbation theory within the framework of general relativity. Inside the
star, the static Schwarschild like metric is perturbed and it is given by [59,
60]:

$\displaystyle ds^{2}=$
$\displaystyle-e^{\nu}(1+r^{\ell}H_{0}Y^{\ell}_{m}e^{i\omega t})dt^{2}$ (17)
$\displaystyle-2i\omega r^{\ell+1}H_{1}Y^{\ell}_{m}e^{i\omega t}dtdr+$
$\displaystyle+e^{\lambda}(1-r^{\ell}H_{0}Y^{\ell}_{m}e^{i\omega t})dr^{2}$
$\displaystyle+r^{2}(1-r^{\ell}KY^{\ell}_{m}e^{i\omega
t})(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$

where $Y_{lm}(\theta,\phi)$ are the spherical harmonics and the functions
$H_{0}$, $H_{1}$, and $K$ depend only on $r$. The Lagrangian 3-vector fluid
displacement $\xi^{j}$ describes the small amplitude motion of the perturbed
configuration and can be represented in terms of perturbation functions $W(r)$
and $V(r)$ as:

$\displaystyle\xi^{r}$ $\displaystyle=$ $\displaystyle
r^{\ell-1}e^{-\lambda/2}WY^{\ell}_{m}e^{i\omega t},$ (18)
$\displaystyle\xi^{\theta}$ $\displaystyle=$
$\displaystyle-r^{\ell-2}V\partial_{\theta}Y^{\ell}_{m}e^{i\omega t},$ (19)
$\displaystyle\xi^{\phi}$ $\displaystyle=$
$\displaystyle-r^{\ell}(r\sin\theta)^{-2}V\partial_{\phi}Y^{\ell}_{m}e^{i\omega
t}.$ (20)

As shown in Refs. [34, 59, 60], perturbed Einstein’s equations inside the star
($0<r<R$), are reduced to the following first-order linear system of
differential equations:

$\displaystyle H_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{r}\biggl{[}\ell+1+\frac{2Me^{\lambda}}{r}+4\pi
r^{2}e^{\lambda}(p-\epsilon)\biggr{]}H_{1}$ (21)
$\displaystyle+\frac{e^{\lambda}}{r}\left[H_{0}+K-16\pi(\epsilon+p)V\right],$
$\displaystyle K^{\prime}$ $\displaystyle=$
$\displaystyle\frac{H_{0}}{r}+\frac{\ell(\ell+1)}{2r}H_{1}-\left[\frac{(\ell+1)}{r}-\frac{\nu^{\prime}}{2}\right]K$
(22) $\displaystyle-8\pi(\epsilon+p)e^{\lambda/2}r^{-1}W\>,$ $\displaystyle
W^{\prime}$ $\displaystyle=$
$\displaystyle-(\ell+1)r^{-1}W+re^{\lambda/2}\bigg{[}e^{-\nu/2}\gamma^{-1}p^{-1}X$
(23) $\displaystyle-\ell(\ell+1)r^{-2}V+\tfrac{1}{2}H_{0}+K\bigg{]}\>,$
$\displaystyle X^{\prime}$ $\displaystyle=$ $\displaystyle-\ell
r^{-1}X+\frac{(\epsilon+p)e^{\nu/2}}{2}\Biggl{[}\left(\frac{1}{r}+\frac{\nu^{\prime}}{2}\right)H_{0}$
(24)
$\displaystyle+\left(r\omega^{2}e^{-\nu}+\frac{\ell(\ell+1)}{2r}\right)H_{1}+\left(\tfrac{3}{2}\nu^{\prime}-\frac{1}{r}\right)K$
$\displaystyle-\ell(\ell+1)r^{-2}\nu^{\prime}V-2r^{-1}\Biggl{(}4\pi(\epsilon+p)e^{\lambda/2}$
$\displaystyle+\omega^{2}e^{\lambda/2-\nu}-\frac{r^{2}}{2}\biggl{(}e^{-\lambda/2}r^{-2}\nu^{\prime}\biggr{)}^{\prime}\Biggr{)}W\Biggr{]}\>,$

where the prime denotes a derivative with respect to $r$, $\gamma$ is the
adiabatic index

$\gamma=\frac{(\epsilon+p)}{p}\frac{dp}{d\epsilon},$ (25)

$X$ is given by

$\displaystyle X=$
$\displaystyle~{}\omega^{2}(\epsilon+p)e^{-\nu/2}V-\frac{p^{\prime}}{r}e^{(\nu-\lambda)/2}W$
(26) $\displaystyle+\frac{1}{2}(\epsilon+p)e^{\nu/2}H_{0},$

and $H_{0}$ fulfills the algebraic expression

$\displaystyle\left[3M+\tfrac{(\ell+2)(\ell-1)}{2}r+4\pi
r^{3}p\right]H_{0}=8\pi r^{3}e^{-\nu/2}X$ (27)
$\displaystyle-\left[\tfrac{1}{2}\ell(\ell+1)(M+4\pi
r^{3}p)-\omega^{2}r^{3}e^{-(\lambda+\nu)}\right]H_{1}$
$\displaystyle+\big{[}\tfrac{1}{2}(\ell+2)(\ell-1)r-\omega^{2}r^{3}e^{-\nu}$
$\displaystyle-r^{-1}e^{\lambda}(M+4\pi r^{3}p)(3M-r+4\pi r^{3}p)\big{]}K.$

The fluid quantities outside the neutron star vanish, and as a result the
perturbation equations of the vacuum Schwarschild metric reduce to the Zerilli
equation:

$\frac{d^{2}Z}{dr^{*2}}=[V_{Z}(r^{*})-\omega^{2}]Z,$ (28)

where the Zerilli function $Z(r^{*})$ and its derivative $dZ(r^{*})/dr^{*}$
can be expressed as functions of the metric perturbations $H_{0}(r)$ and
$K(r)$ using the transformation provided in Eqs. (A27)$-$(A34) of Ref. [59]
(notice that there is a typographical error in Eq. (A29) of [59], see e.g.
[61]). In the equations above, $Z(r^{*})$ depends on the “tortoise”
coordinate, which is given by $r^{*}=r+2M\ln(r/(2M)-1)$ and the effective
potential $V_{Z}(r^{*})$ is:

$\displaystyle
V_{Z}(r^{*})=\frac{(1-2M/r)}{r^{3}(nr+3M)^{2}}[2n^{2}(n+1)r^{3}$
$\displaystyle+6n^{2}Mr^{2}+18nM^{2}r+18M^{3}],$ (29)

where $n=(\ell-1)(\ell+2)/2$.

The physical solution of the oscillation equations must satisfy the following
boundary conditions: (a) The functions describing the oscillatory behavior of
the star must be regular at $r=0$. (b) The Lagrangian perturbation in pressure
must be zero at the surface of the star, which implies that $X$ must vanish at
$r=R$. For given values for $\ell$ and $\omega$, there exists a unique
solution that satisfies the above boundary conditions inside the star. (c)
Finally, at $r=\infty$, the physical solution of the Zerilli equation must
describe purely outgoing gravitational radiation.

---
Figure 4: Frequency (top) and damping time (bottom) as a function of the
neutron star mass for different values of $L$.

The $f$-mode frequency and damping time are shown in Fig. 4 as a function of
the stellar mass for different values of the nuclear symmetry energy slope
$L$. The top panel shows that an increase in $L$ causes a systematic decrease
in the gravitational wave frequencies. This behavior can be explained by the
stiffening of the EOS as the value of $L$ increases. If a star has a stiffer
EOS, its radius tends to be larger for a given mass, which in turn decreases
its average density. Consequently, since the frequency of the $f$-mode scales
with $\sqrt{M/R^{3}}$, the stiffer the EOS, the smaller the oscillation
frequency, as demonstrated in Fig. 2 of Ref. [56]. A complementary effect was
noted in Ref. [62]. In this work, the authors also found that a stiffer EOS
results in a lower $f$-mode frequency by varying the effective nucleon mass.
In our work, this effect is noticeable for all the parameters examined.
However, for masses greater than $1.8M_{\odot}$ and for $L\leq
76~{}\mathrm{MeV}$, this effect is relatively small. The bottom panel shows
that an increase in the value of $L$ results in an increase in the damping
time. This effect is more pronounced for small masses and becomes less
noticeable for masses above $2M_{\odot}$. In Table 4, we present the $f$-mode
frequencies and damping times for stars with masses of 1.4$M_{\odot}$ and
2.01$M_{\odot}$. For both mass values, an increase in $L$ results in a
decrease in the $f$-mode frequency and an increase in its damping time.
However, as already discussed in Fig. 4, the effect of $L$ is less pronounced
for more massive stars compared to less massive ones.

$L(\mathrm{MeV})$ | $f_{(1.4)}(\mathrm{kHz})$ | $\tau_{(1.4)}(\mathrm{ms})$ | $f_{(2.01)}(\mathrm{kHz})$ | $\tau_{(2.01)}(\mathrm{ms})$
---|---|---|---|---
44 | 1.72 | 245 | 1.95 | 154
60 | 1.70 | 251 | 1.95 | 154
76 | 1.66 | 264 | 1.94 | 156
92 | 1.57 | 297 | 1.85 | 165
100 | 1.52 | 316 | 1.80 | 173
108 | 1.48 | 335 | 1.76 | 178
116 | 1.44 | 354 | 1.72 | 184
Table 4: Frequency and damping time for the $1.4M_{\odot}$ and $2.01M_{\odot}$
stars.

##  V Summary and Conclusions

In this work we investigated the impact of the symmetry energy slope $L$ on
various properties of neutron stars employing QHD with
$\sigma-\omega-\rho-\delta$ mesons. We maintained a fixed value of symmetry
energy at $n=n_{0}$ while varying the slope $L$ at $n_{0}$. Specifically, we
decreased $L$ by introducing the non-linear $\omega-\rho$ coupling and
increased it by including the scalar-isovector $\delta$ meson. A stronger
$\omega-\rho$ coupling led to a lower slope, whereas a stronger $\delta$ meson
coupling resulted in a higher slope. Parametrizations with higher slopes had
lower values of the symmetry energy at $n~{}<n_{0}$ and higher values at
$n>~{}n_{0}$. For $n\gtrsim 0.5n_{0}$, the slope $L(n)$ follows a trend
consistent with the slope $L$ defined at $n_{0}$. Specifically, as $L$
increases, so too does $L(n)$, as seen in Fig. 1. A similar behavior is
observed in the pressure of the beta-stable matter (Fig. 2).

The maximum mass of a neutron star tends to increase slightly (less than 4%)
with an increase in the slope $L$. Meanwhile, the stellar radius is more
sensitive to variations in $L$, increasing as $L$ increases. For instance,
Table 3 shows that when $L$ rises from 44 MeV to 116 MeV, the radius
$R_{(1.4)}$ of a canonical $1.4M_{\odot}$ star experiences an increase of
approximately 15%. Our $M$-$R$ curves for low values of $L$ (i.e. in the range
expected from neutron matter and nuclear binding energies, and also from
measurements excluding PREX) are in excellent agreement with the 68%
credibility contour for the mass and radius of PSR J0030+0451 [10] and PSR
J0740+ 6620 [22] and the 90% credible level of the coalescing objects in the
GW170817 event [13]. However, as the value of $L$ increases, the $M$-$R$
curves shift towards larger radii. At around $L\sim 116\,\mathrm{MeV}$, the
radius of the $1.4M_{\odot}$ star is outside the upper boundary of the NICER
constraint on the radius of PSR J0030+0451, which places an upper limit of
$14.26\,\mathrm{km}$ at the 68% level [10]. When even larger values of $L$ are
considered, specifically those falling within the upper segment of the range
provided by PREX2, we find that the resulting $M$-$R$ curves exhibit some
tension with the aforementioned astrophysical observations, specially with
GW170817.

Our study also involved calculating the dimensionless tidal deformability
$\Lambda$ across a range of values for $L$. Our results, depicted in Fig. 3,
show that $\Lambda$ generally increases as the slope $L$ increases. It is
worth noting that $\Lambda$ exhibits a significant sensitivity to variations
in $L$. For instance, Table 3 shows that $\Lambda_{(1.4)}$ varies between
$515-812\,\mathrm{MeV}$ and $\Lambda_{(2.01)}$ varies between
$40-70\,\mathrm{MeV}$ as $L$ increases from 44 to 116 $\mathrm{MeV}$,
corresponding to a variation in $\Lambda$ of approximately $60-70\%$. These
results are especially promising in light of the upcoming LIGO-Virgo-KAGRA
observing period O4 scheduled for 2023, which is expected to enable resolution
of the dimensionless effective tidal deformability with seven times greater
accuracy than was achieved during O2 [63]. If such precise determinations
become available, they could potentially impose significant constraints on the
parameter $L$.

Finally, we focused on the impact of $L$ on the frequency and damping time of
the $f$-mode. We have demonstrated that augmenting the value of $L$ results in
a consistent reduction in the frequency, which can be ascribed to the
increased stiffness of the core’s EOS. In fact, a higher stiffness results in
a greater radius for a given mass, thereby leading to a decrease in the star’s
average density. Hence, given that the frequency of the $f$-mode scales with
$\sqrt{M/R^{3}}$, the larger the $L$ (stiffer EOS), the lower the oscillation
frequency. Although the frequency exhibits a significant sensitivity to
variations in $L$ for $L\gtrsim 70~{}\mathrm{MeV}$, we note that for masses
beyond $1.8M_{\odot}$ and $L\leq 76~{}\mathrm{MeV}$, this effect is relatively
minor. Furthermore, our analysis demonstrated that an increase in $L$ resulted
in an increase in the $f$-mode damping time, with this effect being more
conspicuous for lower masses and higher $L$ values. Although challenging, the
detection of $f$-mode signals is expected to become a reality in the near
future, due to the continued improvements in the sensitivity of GW
observatories. The ability to detect $f$-mode frequencies and damping times
would provide an additional independent tool to limit the range of possible
values for the slope $L$.

## Acknowledgements

Luiz. L. Lopes would like to thank Professor Debora Peres Menezes for the help
with some Fortran codes. GL acknowledges the financial support of the
Brazilian agencies CNPq (grant 316844/2021-7) and FAPESP (grants 2022/02341-9
and 2013/10559-5). César O. V. Flores acknowledges the financial support of
the Brazilian agency CNPq (grant 304569/2022-4). L. L. L. was partially
supported by CNPq (Universal Grant No. 409029/2021-1).

## References

  * Paar  _et al._ [2014] N. Paar, C. C. Moustakidis, T. Marketin, D. Vretenar, and G. A. Lalazissis, Neutron star structure and collective excitations of finite nuclei, Phys. Rev. C 90, 011304 (2014).
  * Lattimer and Lim [2013] J. M. Lattimer and Y. Lim, Constraining the Symmetry Parameters of the Nuclear Interaction, Astrophys. J. 771, 51 (2013).
  * Lattimer and Steiner [2014] J. M. Lattimer and A. W. Steiner, Constraints on the symmetry energy using the mass-radius relation of neutron stars, Eur. Phys. J. A 50, 40 (2014).
  * Estee  _et al._ [2021] J. Estee _et al._ (SpiRIT), Probing the Symmetry Energy with the Spectral Pion Ratio, Phys. Rev. Lett. 126, 162701 (2021).
  * Reed  _et al._ [2021] B. T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, Implications of PREX-2 on the Equation of State of Neutron-Rich Matter, Phys. Rev. Lett. 126, 172503 (2021).
  * Tagami  _et al._ [2022] S. Tagami, T. Wakasa, and M. Yahiro, Slope parameters determined from CREX and PREX2, Results in Physics 43, 106037 (2022).
  * Zhang and Chen [2022] Z. Zhang and L.-W. Chen, Bayesian Inference of the Symmetry Energy and the Neutron Skin in 48Ca and 208Pb from CREX and PREX-2, arXiv (2022), arXiv:2207.03328 [nucl-th] .
  * Antoniadis  _et al._ [2013] J. Antoniadis, P. C. C. Freire, N. Wex, _et al._ , A massive pulsar in a compact relativistic binary, Science 340, 1233232 (2013).
  * Riley  _et al._ [2019] T. E. Riley _et al._ , A NICER View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation, Astrophys. J. Lett. 887, L21 (2019).
  * Miller  _et al._ [2019] M. Miller _et al._ , PSR J0030+0451 Mass and Radius from NICER Data and Implications for the Properties of Neutron Star Matter, Astrophys. J. Lett. 887, L24 (2019).
  * Abbott  _et al._ [2017] B. P. Abbott _et al._ (LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  * Abbott  _et al._ [2019] B. Abbott _et al._ , Properties of the binary neutron star merger gw170817, Phys. Rev. X 9, 011001 (2019).
  * Abbott  _et al._ [2018] B. P. Abbott, R. Abbott, T. D. Abbott, _et al._ (The LIGO Scientific Collaboration and the Virgo Collaboration), GW170817: Measurements of Neutron Star Radii and Equation of State, Phys. Rev. Lett. 121, 161101 (2018).
  * Serot [1992] B. D. Serot, Quantum hadrodynamics, Rep. Progr. Phys. 55, 1855 (1992).
  * Fattoyev  _et al._ [2010] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, and G. Shen, Relativistic effective interaction for nuclei, giant resonances, and neutron stars, Phys. Rev. C 82, 055803 (2010).
  * Cavagnoli  _et al._ [2011] R. Cavagnoli, D. P. Menezes, and C. Providencia, Neutron star properties and the symmetry energy, Phys. Rev. C 84, 065810 (2011).
  * Dexheimer  _et al._ [2019] V. Dexheimer, R. de Oliveira Gomes, S. Schramm, and H. Pais, What do we learn about vector interactions from GW170817?, J. Phys. G 46, 034002 (2019).
  * Kubis and Kutschera [1997] S. Kubis and M. Kutschera, Nuclear matter in relativistic mean field theory with isovector scalar meson, Phys. Lett. B 399, 191 (1997).
  * Liu  _et al._ [2002] B. Liu, V. Greco, V. Baran, M. Colonna, and M. Di Toro, Asymmetric nuclear matter: The role of the isovector scalar channel, Phys. Rev. C 65, 045201 (2002).
  * Lopes and Menezes [2014] L. Lopes and D. Menezes, Effects of the symmetry energy and its slope on neutron star properties, Braz. J. Phys. 44, 774 (2014).
  * Li and Magno [2020] B.-A. Li and M. Magno, Curvature-slope correlation of nuclear symmetry energy and its imprints on the crust-core transition, radius, and tidal deformability of canonical neutron stars, Phys. Rev. C 102, 045807 (2020).
  * Miller  _et al._ [2021] M. Miller _et al._ , The Radius of PSR J0740+6620 from NICER and XMM-Newton Data, Astrophys. J. Lett. 918, L28 (2021).
  * Riley  _et al._ [2021] T. Riley _et al._ , A NICER View of the Massive Pulsar PSR J0740+6620 Informed by Radio Timing and XMM-Newton Spectroscopy, Astrophys. J. Lett. 918, L27 (2021).
  * Chatziioannou [2020] K. Chatziioannou, Neutron star tidal deformability and equation of state constraints, Gen. Rel. Grav. 52, 109 (2020).
  * Flanagan and Hinderer [2008] E. E. Flanagan and T. Hinderer, Constraining neutron-star tidal love numbers with gravitational-wave detectors, Phys. Rev. D 77, 021502 (2008).
  * Read  _et al._ [2009] J. S. Read, C. Markakis, M. Shibata, _et al._ , Measuring the neutron star equation of state with gravitational wave observations, Phys. Rev. D 79, 124033 (2009).
  * Flores  _et al._ [2020] C. Flores, L. Lopes, L. Benito, and D. Menezes, Gravitational wave signatures of highly magnetized neutron stars, Eur. Phys. J. C 80, 1142 (2020).
  * Camelio  _et al._ [2017] G. Camelio, A. Lovato, L. Gualtieri, O. Benhar, J. A. Pons, and V. Ferrari, Evolution of a proto-neutron star with a nuclear many-body equation of state: Neutrino luminosity and gravitational wave frequencies, Phys. Rev. D 96, 043015 (2017).
  * Warszawski and Melatos [2012] L. Warszawski and A. Melatos, Gravitational-wave bursts and stochastic background from superfluid vortex avalanches during pulsar glitches, Mon. Not. Roy. Astron. Soc. 423, 2058 (2012).
  * Lugones  _et al._ [2002] G. Lugones, C. R. Ghezzi, E. M. de Gouveia Dal Pino, and J. E. Horvath, Asymmetric combustion in neutron stars and a potential mechanism for gamma-ray bursts, Astrophys. J. Lett. 581, L101 (2002).
  * Abdikamalov  _et al._ [2009] E. B. Abdikamalov, H. Dimmelmeier, L. Rezzolla, and J. C. Miller, Relativistic simulations of the phase-transition-induced collapse of neutron stars, Mon. Not. Roy. Astron. Soc. 394, 52 (2009).
  * Andersson  _et al._ [2011] N. Andersson _et al._ , Gravitational waves from neutron stars: Promises and challenges, Gen. Rel. Grav. 43, 409 (2011).
  * Bauswein  _et al._ [2016] A. Bauswein, N. Stergioulas, and H.-T. Janka, Exploring properties of high-density matter through remnants of neutron-star mergers, Eur. Phys. J. A 52, 56 (2016).
  * Thorne and Campolattaro [1967] K. S. Thorne and A. Campolattaro, Non-Radial Pulsation of General-Relativistic Stellar Models. I. Analytic Analysis for L $\geq$ 2, Astrophys. J. 149, 591 (1967).
  * Lindblom and Detweiler [1983] L. Lindblom and S. L. Detweiler, The quadrupole oscillations of neutron stars, Astrophys. J. Suppl. 53, 73 (1983).
  * Detweiler and Lindblom [1985] S. L. Detweiler and L. Lindblom, On the nonradial pulsations of general relativistic stellar models, Astrophys. J. 292, 12 (1985).
  * Boguta and Bodmer [1977] J. Boguta and A. R. Bodmer, Relativistic Calculation of Nuclear Matter and the Nuclear Surface, Nucl. Phys. A 292, 413 (1977).
  * Lopes and Menezes [2022] L. L. Lopes and D. P. Menezes, On the Nature of the Mass-gap Object in the GW190814 Event, Astrophys. J. 936, 41 (2022).
  * Choi  _et al._ [2021] S. Choi, T. Miyatsu, M.-K. Cheoun, and K. Saito, Constraints on Nuclear Saturation Properties from Terrestrial Experiments and Astrophysical Observations of Neutron Stars, Astrophys. J. 909, 156 (2021).
  * Roca-Maza  _et al._ [2011] X. Roca-Maza, X. Viñas, M. Centelles, P. Ring, and P. Schuck, Relativistic mean-field interaction with density-dependent meson-nucleon vertices based on microscopical calculations, Phys. Rev. C 84, 054309 (2011).
  * Lopes [2022] L. L. Lopes, Hyperonic neutron stars: reconciliation between nuclear properties and nicer and ligo/virgo results, Commun. Theor. Phys. 74, 015302 (2022).
  * Dutra  _et al._ [2014] M. Dutra, O. Lourenço, S. S. Avancini, _et al._ , Relativistic mean-field hadronic models under nuclear matter constraints, Phys. Rev. C 90, 055203 (2014).
  * Oertel  _et al._ [2017] M. Oertel, M. Hempel, T. Klähn, and S. Typel, Equations of state for supernovae and compact stars, Rev. Mod. Phys. 89, 015007 (2017).
  * Baym  _et al._ [1971a] G. Baym, C. Pethick, and P. Sutherland, The ground state of matter at high densities, Astrophys. J. 170, 299 (1971a).
  * Baym  _et al._ [1971b] G. Baym, H. A. Bethe, and C. J. Pethick, Neutron star matter, Nucl.Phys. A 175, 225 (1971b).
  * Glendenning [2000] N. K. Glendenning,  _Compact stars:_ (2 ed. Edition, Springer New York, 2000).
  * Fortin  _et al._ [2016] M. Fortin _et al._ , Neutron star radii and crusts: Uncertainties and unified equations of state, Phys. Rev. C 94, 035804 (2016).
  * Oppenheimer and Volkoff [1939] J. R. Oppenheimer and G. M. Volkoff, On massive neutron cores, Phys. Rev. 55, 374 (1939).
  * Li  _et al._ [2008] B.-A. Li, L.-W. Chen, and C. M. Ko, Recent progress and new challenges in isospin physics with heavy-ion reactions, Phys. Rep. 464, 113 (2008).
  * Demorest  _et al._ [2010] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Shapiro Delay Measurement of A Two Solar Mass Neutron Star, Nature 467, 1081 (2010).
  * Cromartie  _et al._ [2019] H. Cromartie _et al._ (NANOGrav), Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar, Nature Astron. 4, 72 (2019).
  * Özel  _et al._ [2016] F. Özel, D. Psaltis, T. Güver, G. Baym, C. Heinke, and S. Guillot, The Dense Matter Equation of State from Neutron Star Radius and Mass Measurements, Astrophys. J. 820, 28 (2016).
  * Capano  _et al._ [2020] C. D. Capano, I. Tews, S. M. Brown, _et al._ , Stringent constraints on neutron-star radii from multimessenger observations and nuclear theory, Nature Astronomy 4, 625 (2020).
  * Hinderer [2008] T. Hinderer, Tidal love numbers of neutron stars, Astrophys. J. 677, 1216 (2008).
  * Andersson and Kokkotas [1998] N. Andersson and K. D. Kokkotas, Towards gravitational wave asteroseismology, Mon. Not. Roy. Astron. Soc. 299, 1059 (1998).
  * Vásquez Flores and Lugones [2018] C. Vásquez Flores and G. Lugones, Gravitational wave asteroseismology limits from low density nuclear matter and perturbative QCD, JCAP 08, 046.
  * Flores and Lugones [2017] C. V. Flores and G. Lugones, Constraining color flavor locked strange stars in the gravitational wave era, Phys. Rev. C 95, 025808 (2017).
  * Flores and Lugones [2014] C. V. Flores and G. Lugones, Discriminating hadronic and quark stars through gravitational waves of fluid pulsation modes, Class. Quant. Grav. 31, 155002 (2014).
  * Lindblom and Detweiler [1983] L. Lindblom and S. L. Detweiler, The quadrupole oscillations of neutron stars., Astrophys. J. Suppl. 53, 73 (1983).
  * Detweiler and Lindblom [1985] S. Detweiler and L. Lindblom, On the nonradial pulsations of general relativistic stellar models, Astrophys. J. 292, 12 (1985).
  * Tonetto and Lugones [2020] L. Tonetto and G. Lugones, Discontinuity gravity modes in hybrid stars: assessing the role of rapid and slow phase conversions, Phys. Rev. D 101, 123029 (2020), arXiv:2003.01259 [astro-ph.HE] .
  * Pradhan and Chatterjee [2021] B. K. Pradhan and D. Chatterjee, Effect of hyperons on $f$-mode oscillations in neutron stars, Phys. Rev. C 103, 035810 (2021).
  * Coupechoux  _et al._ [2023] J. F. Coupechoux, R. Chierici, H. Hansen, J. Margueron, R. Somasundaram, and V. Sordini, Impact of O4 future detection on the determination of the dense matter equations of state, to appear in Phys. Rev. D (2023), arXiv:2302.04147 [astro-ph.HE] .


# KIC 12602250 : A low-amplitude Double-mode $\delta$ Scuti star with
Amplitude Modulation

Chenglong Lv  Xinjiang Astronomical Observatory, Chinese Academy of Sciences,
Urumqi, Xinjiang 830011, People’s Republic of China  School of Astronomy and
Space Science, University of Chinese Academy of Sciences, Beijing 100049,
People’s Republic of China  Ali Esamdin<EMAIL_ADDRESS>Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, People’s
Republic of China  Xiangyun Zeng<EMAIL_ADDRESS>Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, People’s
Republic of China  School of Astronomy and Space Science, University of
Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
J.Pascual-Granado  Instituto de Astrofísica de Andalucía - CSIC, 18008
Granada, Spain  Taozhi Yang  School of Physics, Xi’an Jiaotong University,
Xi’an 710049, People’s Republic of China  Junhui Liu  Department of Astronomy,
Xiamen University, Xiamen, Fujian 361005, China

###### Abstract

We propose for the first time that KIC 12602250 is a low-amplitude radial
double-mode $\delta$ Scuti star with amplitude modulation. The detailed
frequency analysis is given for the light curve of KIC 12602250 which is
delivered from the Kepler mission. The Fourier analysis of the long cadence
data (i.e. Q0 - Q17, spanning 1471 days) reveals that the variations of the
light curve are dominated by the strongest mode with frequency F0 = 11.6141
$\rm{d^{-1}}$, suggesting that KIC 12602250 is a $\delta$ Scuti star. The
other independent mode F1 = 14.9741 $\rm{d^{-1}}$ is newly detected. The
amplitude of the light variations of KIC 12602250 is $\sim$ 0.06 mag, which
indicates that this is a low-amplitude $\delta$ Scuti star, but the ratio of
F0/F1 is estimated as 0.7756 which is typical of HADS, and a slow amplitude
growth is detected in F1 and $f_{3}$, which could be due to stellar evolution,
suggesting that KIC 12602250 could be a post main sequence $\delta$ Scuti
which is crossing the instability strip for the first time.

stars: oscillations; stars: variable: $\delta$ Scuti

##  1 Introduction

The high-precision photometric data provided by Kepler provides an
unprecedented opportunity to explore stellar interiors by using the natural
oscillation mode of stars, thus greatly expanding the research field of
asteroseismology (Chaplin et al., 2010). The ultra-high precision photometric
observations at $\mu$ mag level have greatly improved our understanding of
many types of pulsating variables (Balona et al., 2012). Bedding et al. (2011)
proposed that the observed period spacings of gravity modes could be applied
to distinguish the hydrogen and helium burning stars in red giants, while
Giammichele et al. (2018) suggested that there may be a suppressed oxygen-
dominated core in pulsating white dwarfs. As a group of conventional variable
stars, the intrinsic homogenous pulsating modes of $\delta$ Scuti stars make
them excellent targets in the study of asteroseismology (Yang et al., 2018;
Breger et al., 2011; Balona & Dziembowski, 2011). The fundamental mode, first,
second, even the third and fourth radial pulsation mode could be indicators to
the interiors burning mechanisms of $\delta$ Scuti stars (Breger, 2000a).

The $\delta$ Scuti type pulsating stars lie inside the classical Cepheid
instability strip main sequence on the Hertzsprung Russell (HR) diagram.
$\delta$ Scuti stars typically range from A2 to F2 in spectral type with
luminosity classes from III to V (e.g., Breger, 2000a; Lopez de Coca et al.,
1990; Rodríguez & Breger, 2001), and within the effective temperature range of
6300K $\leq$ Teff $\leq$ 8600 K (Uytterhoeven et al., 2011). The pulsating
amplitudes are in the range of 0.003-0.9 mag in the V band, with periods
usually between 0.02 and 0.3 d (Breger, 1979). Some stars show amplitude
modulation of pulsation modes caused by different reasons, e.g., beating, mode
coupling, rotation, and Blazhko effect (Bowman & Kurtz, 2014; Bowman et al.,
2016; Yang et al., 2018b; Yang & Esamdin, 2019). These targets are excellent
samples for asteroseismic study, as they could improve our knowledge of the
stellar structure and evolution of stars.

As a subclass of $\delta$ Scuti stars, the HADS stars usually pulsate with a
light amplitude larger than 0.3 mag and generally rotate slowly with $v$ sin
$i$ $\leq$ 30 kms-1 (Breger, 2000a). Compared with the low-amplitude $\delta$
Scuti stars, the HADS stars possess a more restrictive instability strip with
a width in temperature of about 300 K and tend to shift to a lower temperature
with evolution (McNamara, 2000). Lee et al. (2008) reveal that only $\sim$
0.24 percent of the stars suited in the $\delta$ Scuti region belong to HADS
stars. The majority of HADS stars are typically young and metal-rich
Population I stars; some have been confirmed to be SX Phe variables, and are
Population II metal-deficient stars (Breger, 2000a; Balona & Nemec, 2012). In
general, the HADS stars pulsate with only one or two modes (e.g., AE UMa, Niu
et al., 2017; YZ Boo, Yang et al., 2018; etc.), and most of their pulsations
belong to radial modes.

KIC 12602250 is classified as a $\delta$ Scuti star with a pulsation period of
2.07 hrs by Debosscher et al. (2011). According to the frequency analysis, we
propose for the first time that KIC 12602250 is a low-amplitude radial double-
mode $\delta$ Scuti star with no non-radial modes. The frequency spectra of
KIC 12602250 is shown in Figure 2 while the basic parameters are tabulated in
Table 1.

In this paper, the observations of KIC 12602250 are introduced in Section 2.
Section 3 presents the frequency analysis while Section 4 discusses the radial
triple modes and amplitude modulation of this star. We summarize in Section 5.

Table 1: KIC 12602250 observational (photometry) data characteristics

Parameters | Value in Catalog |
---|---|---
Kepler ID | 12602250 |
2MASS ID | J19213193+5140320 |
Gaia ID | 2139231542654343552 |
RA, Dec | +19h:21m:31.9s, +51$\arcdeg$:40$\arcmin$:32.1$\arcsec$ |
BJD0 | 2454953.5388 |
Rayleigh$~{}f_{\mathrm{res}}$ | 0.000679 | d-1
Period | 2.07 | hr
Kmag | 13.277 |
Contamination | 0.024 |
$T_{\mathrm{eff}}$ | 6879 | K
$\log{g}$ | 4.131 | cgs
$\frac{R}{R_{\sun}}$ | 1.597 |
$\frac{Fe}{H}$ | -0.138 | cgs

Note. — These parameters are available in the KASOC:
https://kasoc.phys.au.dk/.

##  2 OBSERVATIONS AND DATA REDUCTION

KIC 12602250 was observed from BJD 2454953.538 to 2456424.001 by $Kepler$
space telescope, including eighteen quarters (i.e. Q0-Q17). There are only LC
photometric observations of KIC 12602250 available through the Kepler
Asteroseismic Science Operations Center (KASOC) database4 (Kjeldsen et al.,
2010) with two types: one is the raw flux, which is reduced by the NASA Kepler
Science pipeline, and the other is corrected flux, which is provided by KASOC
Working Group 4 (WG # 4: $\delta$ Scuti targets). The second type has been
corrected for systematic errors such as the cooling down, cooling up,
outliers, and jumps. We use the corrected flux and convert it to magnitudes.
Then the mean value of each quarter is subtracted to obtain the rectified time
series. The final rectified light curve was obtained with 65264 data points,
spanning over about 1471 days. Due to the gaps in time series cause spurious
frequencies in the power spectra, in light curves of pulsating stars, this
hampers identifying the theoretical oscillation modes. We used a forward-
backward predictor based on autoregressive moving-average modeling (ARMA) in
the time domain. The algorithm MIARMA is particularly suitable for replacing
invalid data (Pascual-Granado et al., 2015, 2018). We have used this algorithm
to fill the gaps in the light curve of KIC 12602250. Figure 1 shows a portion
of the rectified light curve of KIC 12602250 covering 7 days. From this
figure, the peak-to-peak amplitude of KIC 12602250 obtained from the rectified
light curve is $\sim$0.06 mag, suggesting that this star belongs to the low-
amplitude class of $\delta$ Scuti stars.

Figure 1: A portion of the long cadence light curve of KIC 12602250 . The
amplitude of the light curve is about 0.06 mag.

##  3 FREQUENCY ANALYSIS

The software PERIOD04 (Lenz & Breger, 2005) is employed to analyze the
pulsating behavior of KIC 12602250 . The rectified light curve was fitted with
the following formula:

$m=m_{0}+\sum_{\mbox{\scriptsize\
$i$=1}}^{N}\mathnormal{A}_{i}sin(2\pi(\mathnormal{f}_{i}\mathnormal{t}+\phi_{i})),$
(1)

where $m_{0}$, $A_{i}$, $f_{i}$, and $\phi_{i}$ are zero-point constant,
amplitude, frequency and the corresponding phase, respectively.

Nyquist frequency sets the minimum sampling frequency that needs to be defined
to prevent alias frequencies, while some potential significant frequencies
could beyond Nyquist frequency (Yang et al., 2018; Southworth et al., 2011;
Bowman et al., 2016). The Nyquist frequency of LC observations is $f_{N}$ =
24.469 d-1 (Murphy et al., 2013; Holdsworth et al., 2014). In order to detect
all potential significant frequencies, we choose the frequency range is 0 $<$
$\nu$ $<$ 50 d-1, which is somewhat wider than the typical pulsation frequency
of $\delta$ Scuti stars (Breger, 1979, 2000a; Uytterhoeven et al., 2011;
Balona & Dziembowski, 2011). As LC observations apply longer integration time,
the data signal-to-noise of LC observations is much higher than that of SC
observations. So that the noise in the LC spectrum is lower than that in the
SC observations. (Gilliland et al., 2010; Yang et al., 2018). Owing to the
combined effects of a high signal-to-noise and a non-regular sampling in LC
light curves, we might detect signal beyond the Nyquist limit and discriminate
it from aliases.

We consider two frequencies to be resolved if the difference between them is
larger than the resolution frequency, $f_{res}$ = 1 / $T$, which is 0.000679
$d^{-1}$ for the LC light curve. The highest peak is usually identified as a
potential significant frequency while extracting significant frequencies. Then
a multi-frequency least-square fit using Equation 1 applies to the light curve
with all significant frequencies detected and obtained the solutions for all
the frequencies. A theoretical light curve constructed using the above
solutions is subtracted from the rectified data while the residual is obtained
for the next search. The above steps are repeated until no significant peak is
detected in the frequency spectrum. The criterion of S/N $>$ 5.0 suggested by
Baran et al. (2015) is adopted to determine the significance of the detected
peaks. The uncertainties of frequencies were obtained following the method
proposed by Montgomery & O’donoghue (1999). Figure 2 shows the amplitude
spectra and the pre-whitening procedures of the light curve. The last panel
shows the residual amplitude spectra after pre-whiten eleven detected
frequencies. No significant peak could be detected in the residual spectrum,
which displays an overall distribution of typical noise.

Table 2: Extracted frequencies in LC data of KIC 12602250

$f_{i}$ | Frequency (d-1) | Amplitude (mmag) | S/N | Comment |
---|---|---|---|---|---
1 | 11.6141(1) | 13.695(6) | 498.0 | F0 |
2 | 14.9741(2) | 8.173(6) | 829.4 | F1 |
3 | 3.3600(3) | 4.874(6) | 62.6 | F1-F0 |
4 | 23.2282(4) | 2.697(6) | 38.3 | 2F0 |
5 | 26.5883(6) | 2.403(6) | 30.9 | F0+F1 |
6 | 38.1998(6) | 1.371(6) | 11.8 | F0+2F1 |
7 | 7.3734 (4) | 0.612(6) | 5.8 | 2$f_{N}$-F0-2F1 |
8 | 29.9483(7) | 0.473(6) | 5.6 | 2F1 |

Note. — Among these frequencies, three peaks are independent frequencies,
others are harmonic or combinations (denoted by $f_{i}$). $f_{7}$ is a alias
frequency.

A total of eight significant frequencies are detected in the spectra of KIC
12602250 and a full list is given in Table 2. Among these frequencies, two of
them are considered to be independent. It is reasonable that the strongest
peak $f_{1}$ was assumed to be the fundamental mode, since the light
variations were dominated by this frequency. Therefore, we marked $f_{1}$ with
’F0’ in the last column of Table 2, the other independent frequency $f_{2}$ is
labeled as ’F1’. In addition, harmonics (i.e., $f_{4}$, $f_{8}$) of ’F0’,’F1’
and three combination frequencies (i.e., $f_{3}$, $f_{5}$, $f_{6}$) of ’F0’,
’F1’ are also detected. The frequency $f_{7}$ consider as a alias frequency.

Figure 2: Fourier amplitude spectra and the prewhitening process for the light
curve of KIC 12602250 . The top panel shows two independent frequencies, two
harmonics and three combination frequencies. The independent frequencies F0,
F1 are marked. The bottom panel shows the residual after subtracting 8
significant frequencies and the S/N $<$ 5\. The dashed line refers to the
Nyquist frequency.

##  4 DISCUSSION

###  4.1 The Double-mode in KIC 12602250

From a large number of stars, it was demonstrated that $\delta$ Scuti stars
pulsate in low amplitude non-radial modes in addition to high-amplitude radial
modes (Bowman, 2016). However, low amplitude $\delta$ Scuti stars, with only
the radial pulsation mode are not common.

Stellingwerf (1979) statistically analyzed the pulsation modes of a number of
$\delta$ Scuti stars and presented the period ratios of the first four radial
modes as: $P_{1}$ / $P_{0}$ = (0.756 - 787), $P_{2}$ / $P_{0}$ = (0.611 - 632)
and $P_{3}$ / $P_{0}$ = (0.500 - 525), in which $P_{0}$, $P_{1}$, $P_{2}$ and
$P_{3}$ represent the fundamental mode, first overtone, second overtone and
third overtone, respectively. The ratio of $P_{1}$ / $P_{0}$ of KIC 12602250
is measured as 0.776. In general, the single or double radial mode is common
in HADS pulsators (Breger, 2000a) and a detailed diagram of double-mode HADS
stars and metallicities are given by Petersen & Christensen-Dalsgaard (1996).
But the light curve variation of peak-to-peak amplitude is lower than 0.3 mag,
indicating that KIC 12602250 is not a HADS star but a low-amplitude radial
double-mode $\delta$ Scuti star, this might indicate it is necessary to revise
the definition of a HADS which is, in addition, quite arbitrary.

Table 3: Comparison of the characteristics of triple-mode variables

| AC And | V823 Cas | V829 Aql | GSC 762-110 | GSC 03144-595 | KIC 10975348 | This paper |
---|---|---|---|---|---|---|---|---
Period F (d) | 0.7112 | 0.6690 | 0.2924 | 0.1945 | 0.2036 | 0.0977 | 0.0861 |
$F$ | 1.4059 | 1.4947 | 3.4150 | 5.1412 | 4.9099 | 10.231 | 11.614 |
$1O$ | 1.9043 | 1.9505 | 4.5290 | 6.7284 | 6.4319 | 13.498 | 14.974 |
$2O$ | 2.3749 | 2.4335 | 5.6640 | 8.3974 | 8.0351 | - | - |
$3O$ | - | - | - | - | - | 19.000 | - |
$F/1O$ | 0.7383 | 0.7663 | 0.7555 | 0.7641 | 0.7656 | 0.7579 | 0.7756 |
$F/2O$ | 0.5920 | 0.6142 | 0.6001 | 0.6120 | 0.6112 | - | - |
$F/3O$ | - | - | - | - | - | 0.5385 | - |
$1O/2O$ | 0.8018 | 0.8015 | 0.7997 | 0.8012 | 0.8001 | 0.7105 | 0.7885 |
$A_{F}$ (mmag) | 204 | 86 | 82 | 75 | 94 | 269 | 16.5 |
$A_{1O}/A_{F}$ | 0.84 | 1.51 | 1.04 | 1.00 | 0.97 | 0.0055 | 0.4864 |
$A_{2O}/A_{F}$ | 0.35 | 0.26 | 0.35 | 0.53 | 0.15 | - | - |
$A_{3O}/A_{F}$ | - | - | - | - | - | 0.0018 | - |

Note. — Reference (Fitch & Szeidl, 1976), (Jurcsik et al., 2006), (Handler et
al., 1998), (Wils et al., 2008), (Mow et al., 2016), (Yang et al., 2021).

Moreover, two frequencies of radial modes have been detected in KIC 12602250 ,
enriching the sample of multiple-mode variables. In the low amplitude $\delta$
Scuti stars, it is uncommon for all independent frequencies to be radial
pulsation modes. Therefore, the study of the double radial pulsation modes of
KIC 12602250 might shed light on the intrinsic evolution mode of $\delta$
Scuti stars. Since no low-amplitude $\delta$ Scuti stars with radial multi-
pulsation modes are similar to this star, we chose six HADS for comparison.
The basic pulsation parameters of KIC 12602250 and six triple-mode HADS from
literature are lists in Table 3. The detections of those weak frequencies in
KIC 12602250 are partly due to the high-precision observations of $Kepler$.
Compared with other comparison stars, KIC 12602250 has the smallest amplitude.
Breger & Bregman (1975) propose that the observed radial pulsation modes are
related to the temperature. Fundamental pulsators mainly lie on the cool side
of the instability strip, while the stars with overtones tend to be hot
pulsators. KIC 12602250 with a overtone would have a high temperature. The two
frequencies of radial modes detected in low-amplitude $\delta$ Scuti KIC
12602250 without non-radial modes is unusual, suggesting this star might be a
post main sequence $\delta$ Scuti crossing the instability strip for the first
time (Poleski et al., 2010). We suggest that detailed studies on the period
ratios and seismic modeling might help to understand the nature of the double
radial pulsation mode of KIC 12602250 . The observations of $\delta$ Scuti
stars from $TESS$ also provide well-sampled photometric data, which help shed
light on interiors pulsation modes of $\delta$ Scuti stars, and hence the
multiple radial modes.

###  4.2 Amplitude growth in KIC 12602250

Bowman et al. (2016) propose that the amplitude of KIC 12602250 might increase
somewhat with time. We then investigate the amplitude variation of the three
frequencies with the highest amplitude. Following the method described by
Murphy et al. (2012), we employ a non-linear least-squares fit of the
frequencies to the data of each quarter using the software PERIOD04 (Lenz &
Breger, 2005). The frequency, amplitude, and phase of the peak from the high-
resolution amplitude spectrum for all data are utilized as input parameters,
while the fixed frequency of each mode used for tracking amplitude and phase
of the whole data set can be obtained from the least square fitting. The fixed
frequency is then used to optimize the amplitude and phase of each time bin,
and the same method has been applied to track the amplitude and phase for the
other frequency.

Figure 3: The amplitude variations of F1 and F1-F0 as a function of time. The
red lines represent the linear fits of those frequencies using emcee (a Markov
chain Monte Carlo Ensemble sampler, Foreman-Mackey et al., 2013). Errors on
amplitude are of order of a few micromagnitudes, much small than the plot
symbols.

Figure 3 shows slow growth in the amplitude of F1 and $f_{3}$ (F1-F0), while a
relatively flat evolution in the amplitude of F0. According to Murphy et al.
(2012), two potential scenarios have been proposed for the apparent amplitude
growth: (1) the amplitude growth is in some way instrumental; and (2) the
amplitude growth is an intrinsic or extrinsic effect to the star.

The slight variation of F1 and $f_{3}$ might be somewhat a result of
systematic error from the instrument. So we check the contamination of KIC
12602250 from the Kepler Input Catalogue (KIC; (Brown et al., 2011)). The
contamination reported for KIC 12602250 is 0.024, significantly smaller than
KIC 3429637 (Murphy et al., 2012), suggesting that the amplitude increasing of
F1 is not an instrumental signal. Besides, Murphy et al. (2012) demonstrate
that it could not be an instrumental effect, as all modes would decrease or
increase with the same functional form if the modulation is instrumental, and
instrument trends generally affect low frequencies only.

The 4 CVn is the only one $\delta$ Scuti star with detailed studies of
amplitude modulation based on ground-based telescopes. (Breger, 2000b, 2009,
2016). Bowman et al. (2016) utilize observations of $\sim$1000 $\delta$ Scuti
stars from the Kepler Space Telescope (Borucki et al., 2010) to investigate
the amplitude modulation as well as its possible mechanism. The reason for
$\delta$ Scuti stars with variable pulsation amplitudes (and/or frequencies)
could roughly be classified as the intrinsic and extrinsic, i.e., physical,
interior variations to the star and external effects (Bowman et al., 2016).
The beating from pairs (or groups) of close-frequency (Breger & Bischof, 2002;
Breger & Pamyatnykh, 2006) and non-linearity or mode coupling (Bowman et al.,
2016) could be considered intrinsic effects.

The amplitude variation of KIC 12602250 may be inconsistent with beating
effects since no pairs of close-frequency have been detected. Breger et al.
(2012) propose that three frequencies within a family follow the frequency,
amplitude and phase relations described in equations $\nu_{1}$ $\simeq$
$\nu_{2}$ $\pm$ $\nu_{3}$, $A_{1}$ = $\mu$c ($A_{2}$ $A_{3}$), $\phi_{1}$ =
$\phi_{2}$ $\pm$ $\phi_{3}$, where $A_{i}$ and $\phi_{i}$ represent the
amplitude and phase of the child and parent modes, respectively, and $\mu_{\rm
c}$ is the coupling factor. The $\mu_{\rm c}$ could be small values for
combination frequencies from the non-linear distortion model. For KIC 12602250
, the $f_{3}$ is a combination frequency of F0 and F1 with a $\mu_{\rm c}$
estimated as 0.0025 which might consistent with the non-linear distortion
model. However, Dziembowski (1982) suggest that two linearly unstable low-
frequency parent modes can damp a high-frequency unstable child mode once they
reach the critical amplitude. We propose that the amplitude modulation of the
$f_{3}$ detected in KIC 12602250 could not originate from the non-linearity
for either of F0 nor F1 is low-frequency modes. Lares-Martiz et al. (2020)
utilize a different approach which is based on a Volterra expansion to
describe the nonlinearities. Murphy et al. (2012) propose that stellar
evolution is the primary reason for the amplitude variation of modes in KIC
3429637. The amplitude variation of KIC 12602250 could be result from stellar
evolution.

##  5 SUMMARY

We have analyzed the pulsating behavior of KIC 12602250 using high-precision
photometric observations from the $Kepler$ mission, and eight significant
frequencies are detected. While two of them are independent frequencies, i.e.
F0 = 11.6141 d-1, F1 = 14.9741 d-1. The ratio of P1/P0 of KIC 12602250 is
measured to be 0.776, suggesting that this target could be a new low amplitude
radial double-mode $\delta$ Scuti star. This may indicate the modification of
the definition of HADS is necessary. The low-amplitude, double-mode, and slow
amplitude growth of F1 and $f_{3}$ might suggest that KIC 12602250 could be a
post main sequence $\delta$ Scuti crossing the instability strip for the first
time (Poleski et al., 2010). The slow amplitude growth detected in F1 might an
indicator of stellar evolution. In order to confirm that the pulsation modes
detected using Kepler photometry are indeed overtones and to reveal the nature
of the amplitude variation found in KIC 12602250, a follow-up spectroscopic
campaign and a full seismic modelling are necessary.

We thank the anonymous referee for the suggestive comments, which improved the
manuscript. This research is supported by the National Natural Science
Foundation of China (grant No. U2031209 and 12003020). JPG acknowledge 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’, and from the State Agency for
Research through the ”Center of Excellence Severo Ochoa” award to the
Instituto de Astrofísica de Andalucía (SEV-2017-0709), all from the Spanish
Ministry of Science, Innovation and Universities (MCIU). We would like to
thank the $Kepler$ science team for providing such excellent data.

## References

  * (1)
  * Balona & Dziembowski (2011) Balona L. A., Dziembowski W. A., 2011, MNRAS, 417, 591-601
  * Balona et al. (2012) Balona L. A., Lenz P., Antoci V., et al., 2012, MNRAS, 419, 3028-3038
  * Balona & Nemec (2012) Balona, L. A. & Nemec, J. M. 2012, MNRAS, 426, 2413. doi:10.1111/j.1365-2966.2012.21957.x
  * Baran et al. (2015) Baran, A. S., Koen, C., & Pokrzywka, B. 2015, MNRAS, 448, L16. doi:10.1093/mnrasl/slu194
  * Bedding et al. (2011) Bedding T. R., Mosser B., Huber D., et al., 2011, Nature, 471, 608
  * Borucki et al. (2010) Borucki W. J., Koch D., Basri G., et al., 2010, Science, 327, 977
  * Bowman et al. (2016)
  * Breger et al. (2011) Breger, M., Balona, L., Lenz, P., et al. 2011, MNRAS, 414, 1721. doi:10.1111/j.1365-2966.2011.18508.x
  * Balona & Dziembowski (2011) Balona, L. A. & Dziembowski, W. A. 2011, MNRAS, 417, 591. doi:10.1111/j.1365-2966.2011.19301.x
  * Breger & Bregman (1975) Breger, M. & Bregman, J. N. 1975, ApJ, 200, 343. doi:10.1086/153794
  * Breger et al. (1993) Breger, M., Stich, J., Garrido, R., et al. 1993, A&A, 271, 482
  * Breger & Pamyatnykh (1998) Breger, M. & Pamyatnykh, A. A. 1998, A&A, 332, 958
  * Breger (2000a) Breger, M. 2000, Delta Scuti and Related Stars, 210, 3
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977. doi:10.1126/science.1185402
  * Balona et al. (2012) Balona, L. A., Lenz, P., Antoci, V., et al. 2012, MNRAS, 419, 3028. doi:10.1111/j.1365-2966.2011.19939.x
  * Breger (2000b) Breger, M. 2000, MNRAS, 313, 129. doi:10.1046/j.1365-8711.2000.03185.x
  * Breger (2009) Breger, M. 2009, Stellar Pulsation: Challenges for Theory and Observation, 1170, 410. doi:10.1063/1.3246530
  * Breger (2016) Breger, M. 2016, A&A, 592, A97. doi:10.1051/0004-6361/201628473
  * Brown et al. (2011) Brown, T. M., Latham, D. W., Everett, M. E., et al. 2011, AJ, 142, 112. doi:10.1088/0004-6256/142/4/112
  * Bowman et al. (2016) Bowman, D. M., Kurtz, D. W., Breger, M., et al. 2016, MNRAS, 460, 1970. doi:10.1093/mnras/stw1153
  * Breger & Bischof (2002) Breger, M. & Bischof, K. M. 2002, A&A, 385, 537. doi:10.1051/0004-6361:20020124
  * Breger & Pamyatnykh (2006) Breger, M. & Pamyatnykh, A. A. 2006, MNRAS, 368, 571. doi:10.1111/j.1365-2966.2006.10119.x
  * Bowman (2016) Bowman, D. M. 2016, Ph.D. Thesis
  * Breger et al. (2012) Breger, M., Fossati, L., Balona, L., et al. 2012, ApJ, 759, 62. doi:10.1088/0004-637X/759/1/62
  * Bowman & Kurtz (2014) Bowman, D. M. & Kurtz, D. W. 2014, MNRAS, 444, 1909. doi:10.1093/mnras/stu1583
  * Breger (1979) Breger, M. 1979, PASP, 91, 5. doi:10.1086/130433
  * Chaplin et al. (2010) Chaplin, W. J., Appourchaux, T., Elsworth, Y., et al. 2010, ApJ, 713, L169
  * Dziembowski (1982) Dziembowski, W. 1982, Acta Astron., 32, 147
  * Debosscher et al. (2011) Debosscher, J., Blomme, J., Aerts, C., et al. 2011, A&A, 529, A89. doi:10.1051/0004-6361/201015647
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., et al. 2013, PASP, 125, 306. doi:10.1086/670067
  * Fitch & Szeidl (1976) Fitch, W. S. & Szeidl, B. 1976, ApJ, 203, 616. doi:10.1086/154120
  * Giammichele et al. (2018) Giammichele N., Charpinet S., Fontaine G., et al. 2018, Nature, 554, 73
  * Gilliland et al. (2010) Gilliland, R. L., Jenkins, J. M., Borucki, W. J., et al. 2010, ApJ, 713, L160. doi:10.1088/2041-8205/713/2/L160
  * Handler et al. (1998) Handler, G., Pikall, H., & Diethelm, R. 1998, Information Bulletin on Variable Stars, 4549, 1
  * Holdsworth et al. (2014) Holdsworth, D. L., Smalley, B., Gillon, M., et al. 2014, MNRAS, 439, 2078. doi:10.1093/mnras/stu094
  * Jurcsik et al. (2006) Jurcsik, J., Szeidl, B., Váradi, M., et al. 2006, A&A, 445, 617. doi:10.1051/0004-6361:20052876
  * Kjeldsen et al. (2010) Kjeldsen H., Christensen-Dalsgaard J., Handberg R., et al., 2010, Astronomische Nachrichten, 331, 966
  * Lenz & Breger (2005) Lenz P., Breger M. 2005, CoAst, 146, 53
  * Lopez de Coca et al. (1990) Lopez de Coca, P., Rolland, A., Rodriguez, E., et al. 1990, A&AS, 83, 51
  * Lares-Martiz et al. (2020) Lares-Martiz, M., Garrido, R., & Pascual-Granado, J. 2020, MNRAS, 498, 1194. doi:10.1093/mnras/staa2256
  * Lee et al. (2008) Lee, Y.-H., Kim, S. S., Shin, J., et al. 2008, PASJ, 60, 551. doi:10.1093/pasj/60.3.551
  * McNamara (2000) McNamara, D. H. 2000, Delta Scuti and Related Stars, 210, 373
  * Murphy et al. (2012) Murphy, S. J., Grigahcène, A., Niemczura, E., et al. 2012, MNRAS, 427, 1418. doi:10.1111/j.1365-2966.2012.22062.x
  * Mow et al. (2016) Mow, B., Reinhart, E., Nhim, S., et al. 2016, AJ, 152, 17
  * Murphy et al. (2013) Murphy, S. J., Shibahashi, H., & Kurtz, D. W. 2013, MNRAS, 430, 2986. doi:10.1093/mnras/stt105
  * Montgomery & O’donoghue (1999) Montgomery M. H., & O’donoghue D., 1999, Delta Scuti Star Newsletter, 13, 28
  * Niu et al. (2017) Niu, J.-S., Fu, J.-N., Li, Y., et al. 2017, MNRAS, 467, 3122. doi:10.1093/mnras/stx125
  * Petersen & Christensen-Dalsgaard (1996) Petersen, J. O. & Christensen-Dalsgaard, J. 1996, A&A, 312, 463
  * Poleski et al. (2010) Poleski, R., Soszyński, I., Udalski, A., et al. 2010, Acta Astron., 60, 1
  * Pietrukowicz et al. (2013) Pietrukowicz, P., Dziembowski, W. A., Mróz, P., et al. 2013, Acta Astron., 63, 379
  * Pascual-Granado et al. (2015) Pascual-Granado, J., Garrido, R., & Suárez, J. C. 2015, A&A, 575, A78. doi:10.1051/0004-6361/201425056
  * Pascual-Granado et al. (2018) Pascual-Granado, J., Suárez, J. C., Garrido, R., et al. 2018, A&A, 614, A40. doi:10.1051/0004-6361/201732431
  * Rodríguez & Breger (2001) Rodríguez, E. & Breger, M. 2001, A&A, 366, 178. doi:10.1051/0004-6361:20000205
  * Stellingwerf (1979) Stellingwerf, R. F. 1979, ApJ, 227, 935
  * Southworth et al. (2011) Southworth, J., Zima, W., Aerts, C., et al. 2011, MNRAS, 414, 2413. doi:10.1111/j.1365-2966.2011.18559.x
  * Uytterhoeven et al. (2011) Uytterhoeven, K., Moya, A., Grigahcène, A., et al. 2011, A&A, 534, A125. doi:10.1051/0004-6361/201117368
  * Wils et al. (2008) Wils, P., Rozakis, I., Kleidis, S., et al. 2008, A&A, 478, 865
  * Yang et al. (2018) Yang, T.-Z., Esamdin, A., Fu, J.-N., et al. 2018, Research in Astronomy and Astrophysics, 18, 002
  * Yang et al. (2018b) Yang, T., Esamdin, A., Song, F., et al. 2018, ApJ, 863, 195
  * Yang & Esamdin (2019) Yang, T.-Z. & Esamdin, A. 2019, ApJ, 879, 59
  * Yang et al. (2021) Yang, T.-Z., Sun, X.-Y., Zuo, Z.-Y., et al. 2021, AJ, 161, 27. doi:10.3847/1538-3881/abcb8b
  * Zhou & Jiang (2011) Zhou, A.-Y. & Jiang, S.-Y. 2011, AJ, 142, 100