# Effect of Inner Crust EoS on Neutron star properties.11footnotemark: 1

Ishfaq A. Rather<EMAIL_ADDRESS>A. A. Usmani
<EMAIL_ADDRESS>S. K. Patra<EMAIL_ADDRESS>Department of Physics,
Aligarh Muslim University, Aligarh 202002, India  Institute of Physics,
Bhubaneswar 751005, India  Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

The neutron star maximum mass and the radius are investigated within the
framework of the relativistic mean-field (RMF) model. The variation in the
radius at the canonical mass, $R_{1.4}$, using different inner crust equation
of state (EoS) with different symmetry energy slope parameter is studied. It
is found that although the NS maximum mass and the corresponding radius do not
vary much with different inner crust EoSs, the radius and the tidal
deformability at 1.4$M_{\odot}$ vary with the different choice of crust EoS
and variation of about 1-2 km is seen in the radius at the canonical mass. For
non-unified EoSs, the crust with a low symmetry energy slope parameter
produces a low NS radius at the canonical mass. The properties of maximally
rotating neutron stars are also studied. The variation in the radius of
rotating star at the canonical mass 1.4$M_{\odot}$ is also seen with the slope
parameter. Similar to the static neutron star, the radius at 1.4$M_{\odot}$ of
rotating neutron star is affected by slope parameter of the inner crust. Other
important quantities like moment of inertia, frequency, rotational kinetic
energy to gravitational energy ratio are also calculated. The variation in
these quantities with the crust slope parameter is found to be more
proportional to the mass and the radius of NS.

###### keywords:

Unified EoS , Inner crust, Neutron star, Symmetry energy slope parameter

††journal: Journal of LaTeX Templates

##  1 Introduction

The structure and the properties of Neutron Stars (NS) have been studied
effectively from experimental as well as theoretical models. Such studies
reveal the inner structure of NS and the presence of exotic phases. The
results obtained from the astrophysical observations require several
theoretical inputs for the interpretation. A coextensive effort from theory
and experiments have improved and provided new insights into the field. The
recently observed Gravitational Waves (GW) GW170817 [1, 2] and GW190425 [3]
have constrained the NS maximum mass and radius. The presence of exotic phases
like quarks inside NS has also been observed recently [4]. After the discovery
of GW170817, more theoretical work has been done to understand the relation
between EoS and NS properties through various aspects like a phase transition,
symmetry energy [5, 6, 7, 8, 9, 10, 11, 12, 13]. However, there are still
numerous fundamental problems related to the matter under extreme conditions
that are yet to be answered. The most important one is the unified model which
can describe the overall structure of a NS, from the outer crust to the inner
core.

The structure and the properties of a NS are determined by the Equation of
State (EoS) which describes the state of matter under given physical
conditions. The discovery of massive NS’s provide constrain on NS maximum mass
and radius [14, 15, 16]. To determine the pressure and the energy density of a
NS matter, the isospin asymmetry and the binding energy act as the key inputs.
The symmetry energy $S(\rho)$ and its derivatives ($L_{sym}$,$K_{sym}$,
$Q_{sym}$) affect the EoS and have been constrained in the narrow limits
through the finite nuclei properties [17, 18, 19]. For the density-dependent
symmetry energy, a strong correlation is found between the pressure at
saturation density and the radius of a NS [20]. It has been shown [21, 22]
that the slope parameter $L_{sym}$ is strongly correlated to the neutron skin
thickness of heavy nuclei like ${}^{208}Pb$. A higher $L_{sym}$ value
describes neutron matter with higher pressure and hence thicker neutron skin
[23, 24, 25]. The curvature parameter $K_{sym}$ determines the crust-core
transition density and the gravitational binding energy of the NS [26]. The
skewness parameter $Q_{sym}$ has a large saturation in its value due to
different values from various models. The skewness parameter is related to the
nuclear incompressibility of the system [27]. All these quantities affect the
EoS directly or indirectly.

The unified EoS describes the NS from its outer crust to the inner core.
However, a unified EoS is generally not available. Hence the complete EoS is
divided into three different phases: the outer crust phase, the inner crust,
and the core phase. It has been shown [28, 29] that the NS properties like
mass and radius do not depend upon the outer crust EoS, but a particular
choice of inner crust EoS and the matching of inner crust to the core EoS is
critical and the variations larger than 0.5 km have been obtained for a 1.4
M⊙. For the outer crust which lies in the density range 104-1011 g/cm3, the
Baym-Pethick-Sutherland (BPS) EoS [30], the Haensel-Pichon (HP) EoS [31] are
widely used in the literature. Both these EoSs do not affect the mass and
radius of a NS. For the matter beyond the neutron drip density (4$\times$1011
g/cm3) ranging from density 1011-1014 g/cm3, the inner crust EoS follows. The
Baym-Bethe-Pethick (BBP) EoS [32] is usually used. However to avoid the large
uncertainties in mass and radius of a NS, studies have shown that for the
complete unified EoS, the inner crust EoS should be either from the same model
as core EoS or the symmetry energy slope parameter should match [29, 33].

The observation of the mass and the radius of static as well as rotating NSs
may provide some useful constraints on the EoS. The NSs, due to their
compactness, can rotate very fast as compared to the other astrophysical
observables [34]. Hence the measurements of rotating neutron star properties
like mass, radius, frequency, moment of inertia etc. close to the mass-
shielding limit may lead to more constraints on the composition of the nuclear
matter at higher densities and the EoS. Rotating NSs proivde more information
about the structure and the composition of the star through the measurement of
more quantities than the non-rotating ones. Quantities like moment of inertia,
eccentricity provide the information about how fast an object can spin with a
given angular momentum and the deformation of the mass while spinning. The
universal relations between these bulk properties of NSs may help to impose
constraints on the radius of a NS. The choice of inner crust EoS on the static
and rns will affect the mass and radius and consequently the other star matter
properties.

This paper is organized as follows. In section 2, we explain the relativistic-
mean field formalism to the study of the EoS. The Thomas-Fermi approximation
within RMF framework to describe the inner crust part and the transition from
crust to the core is also discussed. Section 3 discusses the nuclear matter
(NM) properties for parameter sets used. The unified EoS by combining the
outer crust, inner crust, and the core EoS will be discussed in section 3. The
NS properties for static as well as rotating NS’s with different EoSs are also
discussed. Finally, the summary and conclusions are outlined in section 4.

##  2 Theory and Formalism

In this section, we summarize the formulations of our EoS for NM. The
Relativistic Mean Field (RMF) theory in which the nucleons interact through
the meson exchange is adopted. The simplest relativistic Lagrangian contains
the contribution from $\sigma$, $\omega$, and $\rho$ mesons without any cross-
coupling between them [35]. The prediction of large NM incompressibility [36]
by this model was reduced to an acceptable range by the addition of self-
coupling terms [37]. The addition of other self- and cross-couplings improved
the NM properties. The Effective field theory motivated RMF (E-RMF) is an
extension to the basic RMF theory in which all possible self- and cross-
couplings between the mesons are included [38, 39, 40]. The E-RMF Lagrangian
used in the present work, which contains the contribution from $\sigma$-,
$\omega$\- mesons upto 4th order expansion, and $\rho$\- and $\delta$\- mesons
upto 2nd order is given by [41]

$\displaystyle\mathcal{L}=\sum_{\alpha=n,p}\bar{\psi}_{\alpha}\Biggl{\\{}\gamma_{\mu}(i\partial^{\mu}-g_{\omega}\omega^{\mu}-\frac{1}{2}g_{\rho}\bm{\tau}_{\alpha}\cdot\bm{\rho}^{\mu})-(M-g_{\sigma}\sigma)-g_{\delta}\bm{\tau}_{\alpha}\cdot\bm{\delta}\Biggr{\\}}\psi_{\alpha}$
$\displaystyle+\frac{1}{2}\partial^{\mu}\sigma\partial_{\mu}\sigma-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{\zeta_{0}}{4!}g_{\omega}^{2}(\omega^{\mu}\omega_{\mu})^{2}-g_{\sigma}\frac{m_{\sigma}^{2}}{M}\Bigg{(}\frac{k_{3}}{3!}+\frac{k_{4}}{4!}\frac{g_{\sigma}}{M}\sigma\Bigg{)}\sigma^{3}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}-\frac{1}{4}{\textbf{F}}^{\mu\nu}\cdot{\textbf{F}}_{\mu\nu}+\frac{1}{2}\frac{g_{\sigma}\sigma}{M}\Bigg{(}\eta_{1}+\frac{\eta_{2}}{2}\frac{g_{\sigma}}{M}\sigma\Bigg{)}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}$
$\displaystyle+\frac{1}{2}\eta_{\rho}\frac{m_{\rho}^{2}}{M}g_{\sigma}\sigma(\bm{\rho}^{\mu}\cdot\bm{\rho}_{\mu})+\frac{1}{2}m_{\rho}^{2}\rho^{\mu}\cdot\rho_{\mu}-\frac{1}{4}{\textbf{R}}^{\mu\nu}\cdot{\textbf{R}}_{\mu\nu}$
$\displaystyle-\Lambda_{\omega}g_{\omega}^{2}g_{\rho}^{2}(\omega^{\mu}\omega_{\mu})(\bm{\rho}^{\mu}\cdot\bm{\rho}_{\mu})+\frac{1}{2}\partial^{\mu}\bm{\delta}\partial_{\mu}\bm{\delta}-\frac{1}{2}m_{\delta}^{2}\bm{\delta}^{2},$
(1)

where $\psi$ is the nucleonic field and $M$ is the nucleonic mass.
$m_{\sigma}$, $m_{\omega}$,$m_{\rho}$, and $m_{\delta}$ are the masses and
$g_{\sigma}$, $g_{\omega}$,$g_{\rho}$, and $g_{\delta}$ are the coupling
constants of $\sigma$, $\omega$, $\rho$, and $\delta$ mesons respectively. The
Euler-Lagrangian equations of motion for the meson fields are obtained using
the relativistic mean field approximation [8].

$\displaystyle
m_{\sigma}^{2}\sigma=g_{\sigma}\rho_{s}(r)-\frac{m_{\sigma}^{2}g_{\sigma}}{M}\sigma^{2}\Bigg{(}\frac{k_{3}}{2}+\frac{k_{4}}{6}\frac{g_{\sigma}\sigma}{M}\Bigg{)}$
$\displaystyle+\frac{g_{\sigma}}{2M}\Bigg{(}\eta_{1}+\eta_{2}\frac{g_{\sigma}\sigma}{M}\Bigg{)}m_{\omega}^{2}\omega^{2}+\frac{\eta_{\rho}}{2M}g_{\sigma}m_{\omega}^{2}\rho^{2},$
(2)

$\displaystyle
m_{\omega}^{2}\omega=g_{\omega}\rho(r)-\frac{g_{\sigma}}{M}\Bigg{(}\eta_{1}+\eta_{2}\frac{g_{\sigma}\sigma}{M}\Bigg{)}m_{\omega}^{2}\omega-\frac{1}{6}\zeta_{0}g_{\omega}^{2}\omega^{3}$
$\displaystyle-2\Lambda_{\omega}(g_{\omega}g_{\rho}\rho)^{2}\omega,$ (3)

$\displaystyle
m_{\rho}^{2}\rho=\frac{1}{2}g_{\rho}\rho_{3}(r)-\eta_{\rho}\frac{g_{\sigma}\sigma}{M}m_{\rho}^{2}\rho-2\Lambda_{\omega}(g_{\omega}g_{\rho}\omega)^{2}\rho,$
(4)

$m_{\delta}^{2}\delta=g_{\delta}\rho_{s3}(r),$ (5)

where,

$\displaystyle\rho_{s}(r)=\sum_{\alpha=n,p}<\bar{\psi}_{\alpha}\gamma_{0}\psi_{\alpha}>=\rho_{sn}+\rho_{sp}$
$\displaystyle=\frac{2}{(2\pi)^{3}}\sum_{\alpha}\Bigg{(}\int\frac{M^{*}_{\alpha}}{(k_{\alpha}^{2}+M_{\alpha}^{*2})^{1/2}}d^{3}k\Bigg{)},$
(6)

$\displaystyle\rho(r)=\sum_{\alpha}<\bar{\psi}_{\alpha}\psi_{\alpha}>=\rho_{n}+\rho_{p}$
$\displaystyle=\frac{2}{(2\pi)^{3}}\sum_{\alpha}\int d^{3}k,$ (7)

$\displaystyle\rho_{3}(r)=\sum_{\alpha}<\bar{\psi}_{\alpha}\tau_{3}\psi_{\alpha}>=\rho_{p}-\rho_{n},$
(8)

and

$\displaystyle\rho_{s3}(r)=\sum_{\alpha}<\bar{\psi}_{\alpha}\tau_{3}\gamma_{0}\psi_{\alpha}>=\rho_{sp}-\rho_{sn}.$
(9)

are the scalar, baryon, and isovector densities respectively.
$M_{\alpha}^{*}(\alpha=n,p)$ is the effective mass of nucleons given by the
relation

$\displaystyle M_{n}^{*}=M-g_{\sigma}\sigma+g_{\delta}\delta,$ (10)
$\displaystyle M_{p}^{*}=M-g_{\sigma}\sigma-g_{\delta}\delta.$ (11)

The expression for the energy density and pressure are obtained from the given
Lagrangian using energy momentum tensor relation given by [42]

$T^{\mu\nu}=\sum_{i}\partial_{\nu}\phi_{i}\frac{\partial\mathcal{L}}{\partial(\partial^{\mu}\phi_{i})}-g_{\mu\nu}\mathcal{L}.$
(12)

The energy density follows from the zeroth component and the pressure from the
third component of the energy momentum tensor. Their expressions are given as
[41]

$\displaystyle\mathcal{E}=\frac{2}{(2\pi)^{3}}\sum_{\alpha=n,p}\int_{0}^{k_{\alpha}}d^{3}kE_{i}^{*}(k)+\rho
g_{\omega}\omega+m_{\sigma}^{2}\sigma^{2}\Bigg{(}\frac{1}{2}+\frac{k_{3}}{3!}\frac{g_{\sigma}\sigma}{M}+\frac{k_{4}}{4!}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}$
$\displaystyle-\frac{1}{4!}\zeta_{0}g_{\omega}^{2}\omega^{4}-\frac{1}{2}m_{\omega}^{2}\omega^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma}\sigma}{M}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}+\frac{1}{2}\rho_{3}g_{\rho}\rho$
$\displaystyle-\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma}\sigma}{M}\Bigg{)}m_{\rho}^{2}\rho^{2}-\Lambda_{\omega}g_{\rho}^{2}g_{\omega}^{2}\rho^{2}\omega^{2}+\frac{1}{2}m_{\delta}^{2}\delta^{2},$
(13)

and

$\displaystyle
P=\frac{2}{3(2\pi)^{3}}\sum_{\alpha=n,p}\int_{0}^{k_{\alpha}}d^{3}k\frac{k^{2}}{E_{i}^{*}(k)}-m_{\sigma}^{2}\sigma^{2}\Bigg{(}\frac{1}{2}+\frac{k_{3}}{3!}\frac{g_{\sigma}\sigma}{M}+\frac{k_{4}}{4!}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}+\frac{1}{4!}\zeta_{0}g_{\omega}^{2}\omega^{4}$
$\displaystyle+\frac{1}{2}m_{\omega}^{2}\omega^{2}\Bigg{(}1+\eta_{1}\frac{g_{\sigma}\sigma}{M}+\frac{\eta_{2}}{2}\frac{g_{\sigma}^{2}\sigma^{2}}{M^{2}}\Bigg{)}+\frac{1}{2}\rho_{3}g_{\rho}\rho$
$\displaystyle+\frac{1}{2}\Bigg{(}1+\frac{\eta_{\rho}g_{\sigma}\sigma}{M}\Bigg{)}m_{\rho}^{2}\rho^{2}+\Lambda_{\omega}g_{\rho}^{2}g_{\omega}^{2}\rho^{2}\omega^{2}-\frac{1}{2}m_{\delta}^{2}\delta^{2}.$
(14)

The symmetry energy usually defined as the difference in the energy per
nucleon of pure neutron matter (PNM) and symmetric NM (SNM) plays a major role
in the nuclear EoS [43]. The symmetry energy and its derivatives have a major
impact on NS properties like mass and radius. The symmetry energy is defined
as

$S(\rho)=\frac{1}{2}\Bigg{(}\frac{\partial^{2}\mathcal{E}(\rho,\beta)}{\partial\beta^{2}}\Bigg{)}_{\beta=0}$
(15)

where, $\beta$ is the asymmetric parameter which is 0 for SNM and 1 for PNM.

$\beta=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}$ (16)

The derivatives of symmetry energy, slope parameter $L_{sym}$, symmetry energy
curvature $K_{sym}$, and the skewness parameter $Q_{sym}$ are defined at the
saturation density $\rho_{0}$ as

$L_{sym}=3\rho\frac{\partial S(\rho)}{\partial\rho}\Bigg{|}_{\rho=\rho_{0}},$
(17)
$K_{sym}=9\rho^{2}\frac{\partial^{2}S(\rho)}{\partial\rho^{2}}\Bigg{|}_{\rho=\rho_{0}},$
(18)

and

$Q_{sym}=27\rho^{3}\frac{\partial^{3}S(\rho)}{\partial\rho^{3}}\Bigg{|}_{\rho=\rho_{0}},$
(19)

respectively.

The inner crust which contains the inhomogenous NM is defined by applying the
Thomas-Fermi(TF) approximation [44, 45] within the same RMF model [46, 47, 48,
49]. The Skyrme type interactions like Lattimer and Swesty EoS [50] and the
compressible liquid drop model have been widely used in the literature by
several authors to describe the nonuniform matter [51, 52, 53].

The outer crust of a NS consists of nuclei distributed in a solid body-
centered-cubic (bcc) lattice filled by free electron gas. Moving from outer
crust to the inner crust part, the increasing density leads to the complete
ionization of all the atoms and beyond a certain density, leads to the
formation of a quasiuniform gas. As we move to the greater density regions,
more and more electrons are captured by the nuclei, which thus become neutron-
rich. At the density around 4.2$\times$1011 g cm-3, neutron drip sets in. With
the help of nuclear models, the neutron drip density determines the boundary
between the outer crust and the inner crust. The inner crust part contains the
free electrons and the neutron gases, forming different types of pasta
structures. Both the inner and the outer crust densities are a fraction of the
normal nuclear density.

The transition density at the crust-core interface is uncertain due to the
insufficient knowledge of the neutron-rich NM EoS. Furthermore, the
determination of the transition density is very complicated as the inner crust
part may contain some internal structures (pasta phases). A well defined
approach is to find the density at which the uniform liquid phase becomes
unstable against small density fluctuations, indicating the formation of
nuclear clusters. This boundary between the inhomogenous solid crust and the
liquid core is connected to the isospin dependence of nuclear models below the
saturation value by the widely used thermodynamic method [54, 55, 56, 57, 58].
In the present work, we use the thermodynamic method to determine the crust-
core transition boundary, which allows the stability of the NS’s to be
determined in terms of its bulk properties. The following conditions determine
the boundary between the crust and the core:

$-\Bigg{(}\frac{\partial P}{\partial v}\Bigg{)}_{\mu}>0,$ (20)

$-\Bigg{(}\frac{\partial\mu_{np}}{\partial q}\Bigg{)}_{v}>0.$ (21)

Here $P$ is the pressure of the $\beta$-stable matter, $\mu_{np}$ is the
difference between neutron and proton chemical potentials, $v=1/\rho$ is the
volume per baryon, and $q$ is the charge per baryon. The stability conditions
in equations (20) and (21) refer to the mechanical and chemical stabilities of
the system respectively. The total pressure $P$ in equation (20) contains the
contribution from baryonic part and the leptonic part. The leptonic
contribution to the pressure becomes a function of chemical potential $\mu$
only when the system is in beta-equilibrium and hence does not contribute to
the stability condition in equation (20), which now becomes

$-\Bigg{(}\frac{\partial P_{N}}{\partial v}\Bigg{)}_{\mu}>0.$ (22)

The charge fraction $q=Y_{p}-Y_{e}=Y_{p}-\rho_{e}/\rho$, where
$\rho_{e}=\mu_{e}^{3}/(3\pi^{2})$ follows that

$-\Bigg{(}\frac{\partial q}{\partial\mu}\Bigg{)}_{v}=-\Bigg{(}\frac{\partial
Y_{p}}{\partial\mu}\Bigg{)}_{v}+\frac{1}{\rho}\Bigg{(}\frac{\partial\rho_{e}}{\partial\mu}\Bigg{)}_{v}.$
(23)

With $v=1/\rho$ and $\mu=-\partial e(\rho,Y_{p})/\partial Y_{p}$, the
equations (22) and (23) reduce to

$\rho^{2}\Bigg{(}\frac{\partial
P_{N}}{\partial\rho}\Bigg{)}_{\mu}=2\rho^{3}\Bigg{(}\frac{\partial
e(\rho,Y_{p})}{\partial\rho}\Bigg{)}+\rho^{4}\Bigg{(}\frac{\partial^{2}e(\rho,Y_{p})}{\partial\rho^{2}}\Bigg{)}-\rho^{4}\frac{\Big{(}\frac{\partial^{2}e(\rho,Y_{p})}{\partial\rho\partial
Y_{p}}\Big{)}^{2}}{\Big{(}\frac{\partial^{2}e(\rho,Y_{p})}{\partial
Y_{p}^{2}}\Big{)}}>0,$ (24)

$-\Bigg{(}\frac{\partial
q}{\partial\mu}\Bigg{)}_{v}=\Bigg{(}\frac{\partial^{2}e(\rho,Y_{p})}{\partial
Y_{p}^{2}}\Bigg{)}^{-1}+\frac{\mu^{2}}{\pi^{2}\rho}>0.$ (25)

The first two terms on the rhs of eq. (24) refer to the nuclear pressure and
NM incompressibility which are positive. The third term comes from the
leptonic part and contributes negatively. The second term on the rhs of
eq.(25) can be recognized as the screening length, while the first term on rhs
is proportional to the nuclear symmetry energy $E_{sym}(\rho)$ under parabolic
approximation. The condition in eq.(25) is satisfied as the heavy-ion
collision studies favor an increasing behaviour of $E_{sym}(\rho)$. The
stability condition in eq.(24) thus reduces to

$V_{thermal}=2\rho\Bigg{(}\frac{\partial
e(\rho,Y_{p})}{\partial\rho}\Bigg{)}+\rho^{2}\Bigg{(}\frac{\partial^{2}e(\rho,Y_{p})}{\partial\rho^{2}}\Bigg{)}-\frac{\Big{(}\frac{\partial^{2}e(\rho,Y_{p})}{\partial\rho\partial
Y_{p}}\Big{)}^{2}}{\Big{(}\frac{\partial^{2}e(\rho,Y_{p})}{\partial
Y_{p}^{2}}\Big{)}}>0,$ (26)

where $V_{thermal}=\Big{(}\frac{\partial P_{N}}{\partial\rho}\Big{)}_{\mu}$
and $e(\rho,Y_{p})=H(\rho,Y_{p})/\rho$ is the energy per particle of
asymmetric nuclear matter (ANM) with $H(\rho,Y_{p})$ as the energy density.
From the above relation, the determination of the transition density for the
crust-core interface using the EoS of ANM is evident. For a given nucleonic
density $\rho$, once the proton fraction is ascertained from the charge
neutral and beta stability conditions, the nucleonic energy per particle can
be calculated and hence the expression (26) can be evaluated. The stability
condition of $V_{thermal}$ is a signature of the crust-core transition in the
beta-stable NS matter. A more detailed description of the crust-core
transition is given in ref’s [56, 55, 58].

##  3 Results and Discussion

To study the effect of crust EoS on NS properties, we have chosen several
parameter sets to construct the core EoS. Since the outer crust EoS does not
affect the NS maximum mass and radius, therefore the BPS EoS [30] has been
used for the outer crust part. For the inner crust part, the relativistic
mean-field model with constant couplings, non-linear terms [59], and density-
dependent couplings [60] have been used. These include NL3 [61] set with non-
linear $\sigma$ terms, TM1 [62] with non-linear $\sigma$ and $\omega$ terms,
NL3$\omega\rho$ [63, 25] which includes the non-linear $\omega\rho$ terms in
addition to the previous couplings, FSU [64] and IU-FSU [65], and the density-
dependent DD-ME2 [66] and DD-ME$\delta$ [67].

The NM properties at saturation density for the above considered crust EoSs
are shown in table 1. The symmetry energy slope parameter $L_{sym}$ of the
given sets lies in the range 47-118 MeV. We have considered these sets for the
inner crust part to determine the variation in the maximum mass and the
corresponding radius of a NS.

Table 1: NM properties for the inner crust EoS at saturation density ($\rho_{0}$), energy ($E_{0}$), symmetry energy ($S$), slope parameter ($L_{sym}$), incompressibility coefficient ($K$), and skewness parameter ($Q_{sym}$). All the values are in MeV except for the ($\rho_{0}$) which is in fm-3. Model | $\rho_{0}$ | $E_{0}$ | $S$ | $L_{sym}$ | $K$ | $Q_{sym}$
---|---|---|---|---|---|---
NL3 | 0.148 | -16.24 | 37.3 | 118.3 | 270.7 | 203
TM1 | 0.145 | -16.26 | 36.8 | 110.6 | 280.4 | -295
FSU | 0.148 | -16.30 | 32.6 | 60.5 | 230.0 | -523
IU-FSU | 0.155 | -16.40 | 31.3 | 47.2 | 213.2 | -288
NL3$\omega\rho$ | 0.148 | -16.30 | 31.7 | 55.2 | 272.0 | 203
DD-ME2 | 0.152 | -16.14 | 32.3 | 51.4 | 250.8 | 478
DD-ME$\delta$ | 0.152 | -16.12 | 32.4 | 52.9 | 219.1 | -741

The E-RMF formalism is used to construct the core EoS. We covered a wide range
of models containing only $\sigma$ self-coupling terms to the models with all
types of self- and cross-couplings along with the $\delta$ meson inclusion.
NL3 [61], TM1 [62], IU-FSU [68], IOPB-I [8], and G3 [40] parameter sets are
used to study the NS core. All the coupling constants and the meson masses for
the above parameter sets are given in [62, 65, 8]. The parameter sets for the
core EoS used in the present study cover a NS maximum mass range from
$\approx$2-2.8$M_{\odot}$. This allows us to determine the variation in the
properties of the NS for a range of maximum mass with the symmetry energy.

The complete EoS consisting of the outer crust, the inner crust, and the core
are constructed using the above defined parameter sets. The unified and non-
unified EoSs follow as: BPS (for the outer crust)+ BBP, NL3, TM1,
NL3$\omega\rho$, FSU, IU-FSU, DD-ME2, and DD-ME$\delta$ (for the inner crust)+
NL3, TM1, IU-FSU, IOPB-I, and G3 (for the core). The EoS without inner crust
is also constructed to see the impact of inner crust on NS properties. All the
inner crust EoSs used are taken from the reference [33]. The different unified
and non-unified EoSs produced are shown in figure 1.

Figure 1: Unified EoS with different inner crust and core EoS. The inset shows
the matching of outer crust with the inner crust. The green dot shows the
matching point of BPS EoS with the inner crust. The black dot represents the
point of crust-core transition.

The green dot in the inset (as well as in the main plot) of figure 1 shows the
matching of outer crust with inner crust EoS while the black dot in the main
plot represents the crust-core transition point. The NL3 parameter set
produces stiff core EoS as compared to other parameter sets. IU-FSU produces
soft EoS at low density. G3 set produces soft EoS as compared to NL3, TM1 and
IOPB-I at both low and high energy densities. Among the inner crust EoS, NL3
and TM1 set produce soft EoS at very low density and become stiff as the
density increases. IU-FSU crust initially produces stiff EoS but becomes soft
at higher energy density ($\mathcal{E}$ $\approx$ 45 MeV/fm3). For outer
crust, the BPS EoS is used for all different combinations of inner crust and
core EoSs as the outer crust EoS doesn’t affect the mass and radius of a NS.

To determine the maximum mass and the corresponding radius of a stationary and
spherical NS’s obtained using different EoSs, we use the Tolman-Oppenheimer-
Volkoff (TOV) equations [69, 70].

$\frac{dP(r)}{dr}=-\frac{[\mathcal{E}(r)+P(r)][M(r)+4\pi
r^{3}P(r)]}{r^{2}(1-2M(r)/r)},$ (27)

and

$\frac{dM(r)}{dr}=4\pi r^{2}\mathcal{E}(r).$ (28)

Here $M(r)$ is the gravitational mass. For the given boundary conditions
$P(0)=P_{c}$, $M(0)=0$, with $P_{c}$ being the central pressure, the equations
(27) and (28) are solved to obtain the NS properties. Apart from obtaining the
properties of a static NS, we also see the impact of inner crust on maximally
rotating NS (rns). The properties of rotating NS are obtained using the RNS
code [71, 72, 73, 74].

Figure 2 shows the mass-radius relation for a static and maximally rotating NS
with NL3 core with different inner crust EoS. The shaded regions represent the
constraints on the maximum mass of a NS by pulsars PSR J1614-2230
(1.928$\pm$0.017)$M_{\odot}$ [16], PSR J0348+0432(2.01$\pm$0.04)$M_{\odot}$
[14], and PSR J0740+6620 (2.04${}^{+0.10}_{-0.09}$)$M_{\odot}$ [75], and
GW190814 (2.50-2.67 $M_{\odot}$) [76]. The NICER constraints on the stellar
radius obtained from the measurement of millisecond pulsar (MSP) PSR
J0030+0451 at the inferred mass $M=1.34_{-0.16}^{+0.15}M_{\odot}$ and
$M=1.44_{-0.14}^{+0.15}M_{\odot}$ given by $R=12.71_{-1.19}^{+1.14}$km and
$R=13.04_{-1.06}^{+1.24}$km are also shown [77, 78]. The upper limit on the
radius at the canonical mass of a NS is found to be $R_{1.4}\leq 13.76$ km
[9]. The constraints on the maximum mass show that the theoretical prediction
of a NS maximum mass should reach the limit $\approx$ 2.0 $M_{\odot}$. But the
recent observation of gravitational wave data GW190814 has a secondary
component with a maximum mass in the range 2.5-2.67 $M_{\odot}$. This
secondary object is considered to be either a light black-hole or a
supermassive NS.

Figure 2: Mass-Radius profile of SNS and RNS for NL3 core with different inner
crust EoS. The solid (dashed) lines represent the rotating NS’s (static NS’s)
The recent constraints on the mass [16, 14, 75, 76] and radius from NICER
measurements [77, 78] of NS are also shown.

From figure 2, it is clear that the NS maximum mass produced using different
inner crust EoS varies by small margins and lies in the range 2.764-2.787
$M_{\odot}$. The corresponding radius varies from 13.027-13.378 km. However,
the radius at canonical mass is much more affected than the radius at maximum
mass. For a NS without inner crust, the radius at the canonical mass is found
out to be $R_{1.4}$ =14.987 km. The large value for the radius of a NS at
1.4$M_{\odot}$ without inner crust is due to the direct transition from outer
crust to the core part of the star. The neutron drip density determines the
outer crust-inner crust boundary and the thermodynamic method provides the
transition between the inner crust and the core. With no inner crust
considered, the transition density obtained for the outer crust-core boundary
affects the core EoS which results in an unexpected large value of the radius.
With the addition of the inner crust, the proper measurement of the transition
density leads to a true value of the NS radius which varies from 14.496-13.853
km. The NS with a small radius at the canonical mass is produced by using NL3
as inner crust EoS which satisfies the constraints by GW170817. The NL3 set
has higher value of symmetry energy slope parameter $L_{sym}$ = 118.3 MeV,but
matches completely with the core EoS and hence forms a unified EoS. Thus, a
unified EoS produces a NS with smaller radius at the canonical mass. The other
inner crust EoSs have a smaller value of slope parameter than the NL3 set and
such low value slope parameter sets like IU-FSU produces a larger radius at
the canonical mass as seen in the figure. Also, the NL3 inner crust EoS
satisfies the radius constraints from the NICER measurements. Thus we see that
the $R_{1.4}$ has a significant relation with the slope parameter. This is
consistent with the work in references [9, 11].

Figure (2) also shows the MR profile for maximally rotating stars (RNS).
Similar to the Static NS (SNS), the RNS maximum mass and the corresponding
radius are not much affected by the inner crust EoS, but the radius at
1.4$M_{\odot}$ varies in a similar fashion as SNS in the range R1.4=19.5-21.4
km.

Figure 3: a) Same as figure 2

The Mass-Radius profile for TM1 and IU-FSU parameter sets are shown in figure
3. For TM1 core EoS with different inner crust EoS, the maximum mass and the
corresponding radius lie in the range 2.141-2.177 $M_{\odot}$ and
12.234-12.805 km, respectively as shown in figure. The radius at the canonical
mass varies from 13.572 km for TM1 crust EoS to 15.549 km produced without
using the inner crust. As seen in the figure, although every EoS for TM1 core
along with different inner crust satisfies the mass constraint from recently
observed GW data, only the unified EoS (TM1 inner crust + TM1 core) satisfies
the radius constraint at the canonical mass, $R_{1.4}\leq 13.76$ km [9] . The
radius of PSR J0030+0451, $R=12.71_{-1.19}^{+1.14}$ km by NICER observations
[79, 77] is also satisfied by the unified TM1 EoS. For IU-FSU core EoS, as
shown in the figure, the maximum mass and radius vary from 1.931-1.940
$M_{\odot}$ and 11.030-11.263 km, respectively. The radius at the canonical
mass varies from 12.295 km for IU-FSU crust EoS to 12.778 km without inner
crust, all satisfying the radius constraint from NICER measurements and
$R_{1.4}\leq 13.76$ km. The maximum mass and the radius for RNS with TM1 and
IU-FSU core are almost identical with mass in the range 2.57-2.63$M_{\odot}$
satisfying the GW190814 mass constraint and radius around 16 km. This also
shows that possibility of the secondary component of GW190814 to be a
maximally RNS. The unified EoS in TM1 (Lcrust=110.6 MeV + Lcore=110.6 MeV) and
IU-FSU (Lcrust=47.2 MeV + Lcore=47.2 MeV) produce a NS with smaller radius for
SNS as well as RNS.

Figure 3 shows the MR profile for IOPB-I and G3 core EoS. For IOPB-I set, the
maximum mass varies from 2.141-2.156 $M_{\odot}$ and the radius 11.872-12.029
km. $R_{1.4}$ varies from 13.118-13.508 km. Similar results follow for G3 EoS,
where the NS maximum mass and the corresponding radius vary slightly with
different inner crust EoS. However, as usual, the radius $R_{1.4}$ varies from
12.436-14.447 km. For RNS, the radius at the canonical mass varies from 18.65
to 19.18 km and 17.72 to 20.86 km for IOPB-I and G3 EoS, respectively. It is
to be mentioned here that both IOPB-I anf G3 sets do not form a unified EoS
i.e, inner crust and core EoS with same symmetry energy slope parameter.
However, we see that for IOPB-I core EoS, the FSU inner crust with a similar
value of slope parameter as IOPB-I, predicts smaller radius for NS at
1.4$M_{\odot}$ than any other crust EoS. Similar follows for G3 set ($L_{sym}$
= 49.3 MeV), the IU-FSU inner crust with a slope parameter value of 47.2 MeV
gives a smaller radius NS.

Figure 4: a) Same as figure 2

From the above MR relations, we see that the maximum mass and the radius of
both static and rotating NS’s do not change by a large amount using the inner
crust with different slope parameters. The unified EoSs (NL3, TM1, and IU-FSU)
produce NS’s with a small radius at the canonical mass. However, for non-
unified EoSs (IOPB-I and G3), the inner crust EoS from same or different model
with smaller symmetry energy slope parameter $L_{sym}$ predicts a smaller
radius at canonical mass of a NS.

The effect of inner crust on the static NS tidal deformability $\lambda$ is
also studied for all core EoSs. In addition to this, the variation in the RNS
properties like Moment of Inertia (MI) is also discussed. The tidal
deformability of a NS is defined as [80, 81]

$\lambda=-\frac{Q_{ij}}{\mathcal{E}_{ij}}=\frac{2}{3}k_{2}R^{5},$ (29)

where, $Q_{ij}$ is the induced quadrupole mass and $\mathcal{E}_{ij}$ is the
external field. The dimensionless tidal deformability then follows from the
$\lambda$ as

$\Lambda=\frac{\lambda}{M^{5}}=\frac{2k_{2}}{3C^{5}},$ (30)

where, $k_{2}$ is the second love number and $C=M/R$ is the compactness
parameter. The expression for the love number is given as [80]

$\displaystyle
k_{2}=\frac{8}{5}(1-2C)^{2}[2C(y-1)]\Bigl{\\{}2C(4(y+1)C^{4}+(6y-4)C^{3}+(26-22y)C^{2}+3(5y-8)C-3y+6)$
$\displaystyle-3(1-2C)^{2}(2C(y-1)-y+2)log\Big{(}\frac{1}{1-2C}\Big{)}\Bigr{\\}}^{-1}.$
(31)

The value of $y=y(R)$ can be computed by solving the following differential
equation [81, 82]

$r\frac{dy(r)}{dr}+y(r)^{2}+y(r)F(r)+r^{2}Q(r)=0,$ (32)

where,

$F(r)=\frac{r-4\pi r^{3}[\mathcal{E}(r)-P(r)]}{r-2M(r)},$ (33) $\displaystyle
Q(r)=\frac{4\pi
r\Big{(}5\mathcal{E}(r)+9P(r)+\frac{\mathcal{E}(r)+P(r)}{\partial
P(r)/\partial\mathcal{E}(r)}-\frac{6}{4\pi
r^{2}}\Big{)}}{r-2M(r)}-4\Bigg{[}\frac{M(r)+4\pi
r^{3}P(r)}{r^{2}(1-2M(r)/r)}\Bigg{]}^{2}.$ (34)

Figure 5: The relation between dimensionless tidal deformability and the mass
of a NS for NL3 core EoS with different inner crust EoS. The overlaid arrows
(green and orange) represents the combined constraints on the tidal
deformability at 1.4$M_{\odot}$ from PSR J0030+0451 and GW170817 and x ray
data [2, 83, 84] . The grey dashed line represents the upper limit on the
$\Lambda_{1.4}$ value [1] .

Figure (5) shows the variation of the dimensionless tidal deformability with
the NS mass for NL3 core EoS with different inner crust EoS. The constraints
on the $\Lambda$ from the recent GW data is also shown. The green overlaid
arrow shows the recent constraints on the $\Lambda_{1.4}$ from the combined
data of PSR J0030+0451 and GW170817, $\Lambda_{1.4}=370_{-130}^{+360}$ [83],
while the orange one shows the non-parametric constraints from PSRs + GWs + x
ray, $\Lambda_{1.4}=451_{-279}^{+241}$ [84].The grey dashed line represents
the upper limit on the dimensionless tidal deformability at the canonical mass
from GW170817 data, $\Lambda_{1.4}$=800 [1]. The NL3 unified EoS predicts the
lowest value of the dimensionless tidal deformability at 1.4$M_{\odot}$,
$\Lambda_{1.4}$=800, due to the small radius at the canonical mass. The other
non-unified NL3 EoSs (NL3 core + other crust EoSs except NL3) predict a value
in the range 800-1400 with $\Lambda_{1.4}$=1400 for NL3 without the inner
crust. This shows that the unified EoS is important in determining the NS
properties that support the contraints from recent GW data.

Figure 6: Same as figure (5), but for a) TM1 and b) IU-FSU core EoS. Figure 7:
Same as figure (5), but for a) IOPB-I and b) G3 core EoS.

Figure (6) shows the dimensionless tidal deformability for TM1 and IU-FSU core
EoS with different crust EoSs. The unified TM1 EoS (TM1 crust + TM1 core)
provides a low value for tidal deformability, $\Lambda_{1.4}$ =1060. This
value increases upto to $\Lambda_{1.4}$= 1587 for TM1 NS without inner crust.
For IU-FSU, the unified EoS gives $\Lambda_{1.4}$=368, and $\Lambda_{1.4}$=637
without inner crust EoS. For other non-unified EoSs, the variation in the
tidal deformability at 1.4$M_{\odot}$ is very small for both TM1 and IU-FSU
core EoSs.

Figure (7) shows the dimensionless tidal deformability for IOPB-I and G3 core
sets. For IOPB-I, the FSU crust EoS predicts the smallest value of
$\Lambda_{1.4}$=637, while the other crust EoSs determine the value in the
range 640-730. Similarly for G3 set, the IU-FSU gives a low value for tidal
deformability, $\Lambda_{1.4}$=349, while others provide a value in the range
393-450. The value increases with the increase in the value of symmetry energy
slope parameter. The NS without inner crust provides a value of
$\Lambda_{1.4}$=914 and 620 for IOPB-I and G3 sets respectively.

In the simplest form, the moment of inertia $I$ is defined as the ratio of the
angular momentum to the angular velocity of a NS, $I=J/\Omega$. In terms of
the angular frequency $\omega$, the moment of inertia is defined as [85, 86]

$I\approx\frac{8\pi}{3}\int_{0}^{R}(\mathcal{E}+P)e^{-\phi(r)}\Big{[}1-\frac{2m(r)}{r}\Big{]}^{-1}\frac{\bar{\omega}}{\Omega}r^{4}dr,$
(35)

where, $\bar{\omega}$ is the dragging angular velocity of a rotating star,
satisfying the boundary conditions

$\bar{\omega}(r=R)=1-\frac{2I}{R^{3}},\frac{d\bar{\omega}}{dr}|_{r=0}=0,$ (36)

The slow rotation approximation for the NS moment of inertia neglects the
deviation from the spherical symmetry and becomes independent of the angular
velocity $\Omega$. The MI for the slowly rotating NS computed throught the
above formalism can be approximated by the following empirical relation [87]

$I\approx(0.237\pm
0.008)MR^{2}\times\Bigg{[}1+4.2\frac{M}{M_{\odot}}\frac{km}{R}+90\Bigg{(}\frac{M}{M_{\odot}}\frac{km}{R}\Bigg{)}^{4}\Bigg{]}.$
(37)

In the present work, the NSs rotating at the kepler frequency are studied,
hence the numerical calculations for the moment of inertia are performed using
the RNS code. The MI of a NS has been calculated by various groups [71, 34,
88], but the variation in the value of MI with different inner crust EoS
hasn’t been calculated.

Figure 8: Variation of moment of inertia $I$ with the mass of a NS for NL3
core EoS with different crust EoS. The green arrow represents the constraints
on the moment of inertia from PSR J0737-3039A [89, 90]obtained from the GW
170817 data analysis [1, 2] .

The variation in the moment of inertia for RNS with NL3 core EoS and different
crust EoSs is shown in figure (8). As clear from the figure, the change in the
moment of inertia with different crust EoSs is very small. For unified NL3
EoS, the value of $I$ is found to be 1.53$\times$ 1045g cm2 well satisfying
the constraint from PSR J0737-3039A $I$=1.53${}_{-0.24}^{+0.38}\times$ 1045g
cm2. For the non-unified EoSs, the moment of inertia increases with the
symmetry energy slope parameter $L_{sym}$ as they predict a large radius.

For TM1, IU-FSU, IOPB-I, and G3 core EoSs, the moment of inertia variation
with the NS mass is shown in figure (9). For the TM1 and IU-FSU, the unified
EoS provides a small moment of inertia $I$=1.31 & 1.29$\times$ 1045g cm2,
respectively. With no unified EoS available for IOPB-I and G3 sets, the low
symmetry energy slope parameter crust EoS provides a lower value of the moment
of inertia, $I$=1.27 & 1.22$\times$ 1045g cm2, respectively. The moment of
inertia is approximated by the relation $I\propto MR^{2}$ which shows that it
increases almost linearly with the NS mass for all models. The NS radius
starts to decrease as soon as the maximum mass is achieved which allows the
moment of inertia to drop sharply. Since $I$ is proportional to the mass
linearly and square of the radius, it is more sensitive to the density
dependence of nuclear symmetry energy and its derivatives like slope
parameter, which influence the NS radius [86, 91]. By using different inner
crust equation of states for a given core EoS, the change in the radius at the
canonical mass is observed, which affects the moment of inertia. Therefore the
contribution to the moment of inertia due to the crust part of the star is
much less than the core part.

Figure 9: Same as figure (8), but for a) TM1, b) IU-FSU, c) IOPB-I , and d) G3
core EoSs.

To study the effect of different crust EoSs on the rns properties, we have
calculated the kepler frequency for NL3, TM1, and IU-FSU parameter sets only
as they contain both unified as well as non-unified EoSs.

Figure 10: Kepler frequency $\nu_{k}$ vs the gravitational mass of NS for NL3
parameter set using different crust EoSs. The grey band represents the
observational limits on the frequency from rapidly rotating pulsars PSR
J1748-2446ad ($\nu$=716 Hz) and XTE J1739-285 ($\nu$=1122 Hz).

Figure (10) shows the NS gravitational mass as a function of kepler frequency.
Since the maximum mass obtained for the NL3 core with different crust EoSs
does not vary too much, the frequency at the maximum mass remains almost same
for all equation of states. The maximum frequency lies around 1400 Hz. With
different crust EoS, the variation at the 1.4$M_{\odot}$ is observed. For the
EoS with NL3crust +NL3core, the frequency at the canonical mass is found to be
$\approx$ 800 Hz. For other crust EoSs, the frequency at 1.4$M_{\odot}$ is
less than 800 Hz with around 700 Hz for EoS without inner crust. This
variation in the frequency with different inner crust follows from the change
in the radius of NS, which is affected by the change in the symmetry energy
slope parameter of the crust EoS.

Figure 11: Same as fig. (10), but for a) TM1 and b) IU-FSU parameter sets.

Figure (11) shows the kepler frequency variation with the NS mass using
different crust EoS for TM1 and IU-FSU parameter sets. For TM1 and IU-FSU, all
the EoSs obtained rotate with a maximum frequency of $\approx$ 1400 Hz. The
variations at the canonical mass are also too small. The EoS obtained without
the inner crust part rotates with a frequency of around 820 Hz at
1.4$M_{\odot}$ for both TM1 and IU-FSU sets. Thus we see that the change in
the symmetry energy slope parameter of crust EoS doesn’t affect the rotational
frequency of the rns directly.

An important quantity that characterizes the rotation of a star is the T/W
ratio, which is defined as the ratio of the rotational kinetic energy T to the
gravitational potential energy W. For a given rotating star, if this ratio is
greater than a critical value, the star becomes dynamically unstable. The
critical value of T/W ratio is not fixed. Some measurements predict a value of
0.27 as the critical limit [92, 93] and some show it to be in the range
0.14-0.27 [94, 95, 96]. Figure (12) displays the variation in T/W ratio of a
rns with the gravitational mass. With the increase in the central density, the
angular velocity increases which in turn produces a star with higher value of
T/W ratio. The unified EoS for NL3, TM1, and IU-FSU parameter sets predict the
highest value of the T/W ratio as compared to other non-unified equation of
states. For NL3 set, the unified EoS (L${crust}$=Lcore= 118.3 MeV) has T/W
ratio of 0.15, while for TM1 and IU-FSU unified EoSs, the ratio is 0.12 and
0.13, respectively. The EoS without inner crust part predicts a value of 0.13,
0.11, and 0.12 for NL3, TM1, and IU-FSU sets respectively.

Figure 12: The variation of the Rotational kinetic energy to the gravitational
potential energy ratio with the NS mass for a) NL3, b) TM1 and c) IU-FSU
parameter sets with different crust EoSs.

From the measurements of the rns properties like the moment of inertia,
frequency, and T/W ratio for NL3, TM1, and IU-FSU unified as well as non-
unified equation of states, we see that such quantities vary with a change in
the symmetry energy slope parameter $L_{sym}$ of the inner crust part, but
such variation is more proportional to the NS mass and radius. A change in the
NS radius at the canonical mass alters the structure of the star which
influences the other properties of the star. A relation between the slope
parameter and the above defined rns properties for the inner crust part will
allow us to study these properties in great detail leading to a proper choice
of the crust EoS.

Table 2: Variation (in percent) in the maximum mass, corresponding radius, and the radius at 1.4$M_{\odot}$ of a NS without inner crust and the corresponding inner crust. Model |  | BBP | IU-FSU | DD-ME2 | DD-ME$\delta$ | NL3$\omega\rho$ | FSU | TM1 | NL3
---|---|---|---|---|---|---|---|---|---
|  $\Delta M_{max}$ | 0.25 | 0.14 | 0.11 | 0.07 | 0.11 | 0.11 | 0.32 | 0.58
NL3 |  $\Delta R_{max}$ | 1.90 | 2.62 | 0.87 | 1.22 | 0.93 | 0.79 | 0.73 | 1.19
|  $\Delta R_{1.4}$ | 3.28 | 3.12 | 3.22 | 3.30 | 3.54 | 4.74 | 6.51 | 7.57
|  $\Delta M_{max}$ | 0.60 | 1.65 | 1.51 | 1.70 | 1.38 | 1.56 | 0.64 | 1.65
TM1 |  $\Delta R_{max}$ | 4.51 | 2.12 | 1.85 | 1.92 | 4.68 | 2.98 | 4.46 | 3.29
|  $\Delta R_{1.4}$ | 9.69 | 10.22 | 10.19 | 10.16 | 10.13 | 10.11 | 12.73 | 11.28
|  $\Delta M_{max}$ | 0.10 | 0.00 | 0.05 | 0.26 | 0.10 | 0.05 | 0.15 | 0.21
IU-FSU |  $\Delta R_{max}$ | 1.35 | 2.07 | 1.41 | 2.23 | 1.25 | 1.26 | 1.71 | 1.79
|  $\Delta R_{1.4}$ | 2.91 | 3.78 | 3.55 | 3.42 | 3.30 | 3.19 | 2.78 | 2.53
|  $\Delta M_{max}$ | 0.04 | 0.02 | 0.06 | 0.04 | 0.04 | 0.07 | 0.04 | 0.04
IOPB-I |  $\Delta R_{max}$ | 1.11 | 1.17 | 0.66 | 0.61 | 0.73 | 0.54 | 0.92 | 1.30
|  $\Delta R_{1.4}$ | 1.84 | 2.57 | 2.34 | 2.29 | 2.26 | 2.89 | 1.85 | 1.79
|  $\Delta M_{max}$ | 0.05 | 0.00 | 0.15 | 0.20 | 0.15 | 0.30 | 0.04 | 0.50
G3 |  $\Delta R_{max}$ | 3.94 | 5.74 | 4.02 | 3.79 | 4.24 | 3.28 | 0.92 | 3.50
|  $\Delta R_{1.4}$ | 11.50 | 13.91 | 11.82 | 11.66 | 9.45 | 11.36 | 9.56 | 10.17

Table 2 shows the deviation in the properties of a static NS like maximum
mass, the corresponding radius, and the radius at the canonical mass for the
given parameter sets. The deviations are calculated by considering the NS with
different inner crust EoS with respect to the NS without inner crust. The NS
without inner crust predicts a very large radius at the canonical mass, while
the unified EoS (same crust and core EoS) for any parameter set gives the low
radius NS, which satisfies the radius constraints as explained before. The
deviation between a NS without inner crust and with crust is large, which
implies that the unified EoS gives a better estimate of the radius at the
canonical mass as compared to other non-unified EoSs. It is clear that the
variation in the maximum mass and the corresponding radius are very small for
EoSs, but the radius at 1.4 $M_{\odot}$ is highly impacted by the inner crust
EoS. For NL3 core EoS, the variation in the radius at $R_{1.4}$ is maximum for
NL3 inner crust $\approx$ 7.57$\%$ which is around 1.2 km. The maximum
variation in the radius, $R_{1.4}$, for IOPB-I and G3 core EoSs is with the
FSU and IU-FSU inner crust EoS, respectively. Such large deviations in the
radius, $R_{1.4}$ show that a proper choice of inner crust EoS is important to
calculate the mass and radius of a NS with small uncertainities in these
values. The unified EoSs for NL3, TM1, and IU-FSU predict small values of
radius, tidal deformability, at the canonical mass. For RNS, the radius at
1.4$M_{\odot}$ varies with different crust EoS. The moment of inertia doesn’t
vary largely. For the core EoSs without having a unified EoS, the crust with a
smaller symmetry energy slope parameter $L_{sym}$ predicts small values of
radius, the tidal deformability for SNS and radius, moment of inertia for RNS.
Thus the crust-core transition allows the construction of a stellar EoS and a
precise measurement of the NS properties for both static and rotating stars.

The constraints on the inner crust EoS of a NS and the proper matching of
inner crust with core EoS is helpful in considering the nuclear and
astrophysical applications of the RMF model. A core EoS with a smaller
symmetry energy slope parameter implies small symmetry energy at high
densities [97]. For a model with higher symmetry energy at sub-saturation
density, the inner crust properties of a NS are affected in addition to the
pasta phases as shown in refs. [98, 99]. Studies have shown that for a
complete unified EoS, the inner crust part should either be from the same
model or the symmetry energy slope parameter should match. However, as we have
seen in the above plots, the inner crust EoS with smaller slope parameter
$L_{sym}$ predicts a smaller radius at 1.4 $M_{\odot}$ as compared to the one
with large $L_{sym}$. The same trend is followed by all parameter sets with
unified EoS. For non-unified EoS, the crust from same or different model but
with smaller symmetry energy slope parameter gives a low radius NS. While the
inner crust does affect the radius of RNS, the moment of inertia varies only
by a small factor.

##  4 Summay and conclusion

The NS properties like mass and radius were investigated using the
relativistic mean-field (RMF) model. To study the effect of symmetry energy
and its slope parameter on a NS, the inner crust EoSs with different symmetry
energy slope parameters have been used. For the outer crust, the BPS EoS is
used for all sets as the outer crust part doesn’t affect the NS maximum mass
and radius. For the inner crust part, the NL3, TM1, FSU, NL3$\omega\rho$, DD-
ME$\delta$, DD-ME2, and IU-FSU parameter sets have been used whose slope
parameter varies from 118.3-47.2 MeV. For the core part, NL3, TM1, IU-FSU,
IOPB-I, and G3 parameter sets are used. The unified EoSs are constructed by
properly matching the inner crust EoS with outer crust and core EoS using the
thermodynamic method. The EoSs constructed for the spherical and symmetrical
NS under charge neutral and $\beta$-equilibrium conditions are taken as the
input into the TOV equation to obtain NS properties. It is seen that although
the NS maximum mass and the corresponding radius do not change by a large
amount, the radius at the canonical mass, $R_{1.4}$ are largely impacted by
using inner crust EoSs with different symmetry energy slope parameter. By
varying the slope parameter from low to high values, the radius $R_{1.4}$ also
increases. Parameter sets with different nuclear matter properties for the NS
core. The effect of $L_{sym}$ on the NS maximum mass, radius, and the radius
at 1.4$M_{\odot}$ are calculated and the variation of about 2 km is found in
the radius at the canonical mass. The properties like mass, radius, the moment
of inertia, T/W ratio of a maximally rotating stars are also calculated using
same combination of EoSs. It is seen that similar to SNS, the maximum mass and
the corresponding radius for RNS do not vary much, but the radius at the
canonical mass is affected by the slope parameter. The moment of inertia
doesn’t vary too much with change in the symmetry energy slope parameter
$L_{sym}$. The kinetic to potential energy ratio also varies with the change
in the symmetry energy slope parameter of the crust. Such rns properties are
more related to the mass and the radius of the star and the variation in the
radius at the canonical mass influences the other properties of the star.
There are several different aspects that need to be further studied in the
current work. A unified EoS for the parameter sets like IOPB-I and G3 with
both crust and core part described by the same model with different slope
parameter $L_{sym}$ will be a better investigation to see the behavior of
radius and other NS properties at canonical mass.

## Acknowledgement

We are thankful to the referee for enhancing our understanding on the subject
and the quality of the paper. A.A.U. acknowledges the Inter-University Centre
for Astronomy and Astrophysics, Pune, India for support via an associateship
and for hospitality.

## References

  * [1] B. P. Abbott, et al., Phys. Rev. Lett. 119 (2017) 161101.  doi:10.1103/PhysRevLett.119.161101.
  * [2] B. P. Abbott, et al., Phys. Rev. Lett. 121 (2018) 161101.  doi:10.1103/PhysRevLett.121.161101.
  * [3] B. P. Abbott, et al., The Astrophys. J. 892 (2020) L3.  doi:10.3847/2041-8213/ab75f5, [link].
URL https://doi.org/10.3847%2F2041-8213%2Fab75f5

  * [4] E. Annala, T. Gorda, et al., Nat. Phys. 16 (2020) 907–910.  doi:10.1038/s41567-020-0914-9, [link].
URL https://doi.org/10.1038/s41567-020-0914-9

  * [5] E. Annala, T. Gorda, A. Kurkela, A. Vuorinen, Phys. Rev. Lett. 120 (2018) 172703\.  doi:10.1103/PhysRevLett.120.172703, [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.120.172703

  * [6] V. Paschalidis, K. Yagi, D. Alvarez-Castillo, D. B. Blaschke, A. Sedrakian, Phys. Rev. D 97 (2018) 084038.  doi:10.1103/PhysRevD.97.084038, [link].
URL https://link.aps.org/doi/10.1103/PhysRevD.97.084038

  * [7] T. Malik, N. Alam, M. Fortin, C. Providência, B. K. Agrawal, T. K. Jha, B. Kumar, S. K. Patra, Phys. Rev. C 98 (2018) 035804.  doi:10.1103/PhysRevC.98.035804, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.98.035804

  * [8] B. Kumar, S. K. Patra, B. K. Agrawal, Phys. Rev. C 97 (2018) 045806.  doi:10.1103/PhysRevC.97.045806, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.97.045806

  * [9] F. J. Fattoyev, J. Piekarewicz, C. J. Horowitz, Phys. Rev. Lett. 120 (2018) 172702\.  doi:10.1103/PhysRevLett.120.172702, [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.120.172702

  * [10] E. R. Most, L. R. Weih, L. Rezzolla, J. Schaffner-Bielich, Phys. Rev. Lett. 120 (2018) 261103.  doi:10.1103/PhysRevLett.120.261103, [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.120.261103

  * [11] Y. Lim, J. W. Holt, Phys. Rev. Lett. 121 (2018) 062701.  doi:10.1103/PhysRevLett.121.062701, [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.121.062701

  * [12] I. A. Rather, A. Kumar, H. C. Das, M. Imran, A. A. Usmani, S. K. Patra, Int. J. Mod. Phys. E 29 (07) (2020) 2050044.  doi:10.1142/S0218301320500445, [link].
URL https://doi.org/10.1142/S0218301320500445

  * [13] I. A. Rather, A. A. Usmani, S. K. Patra, J. Phys. G Nucl. Part. Phys. 47 (10) (2020) 105104.  doi:10.1088/1361-6471/aba116, [link].
URL https://doi.org/10.1088%2F1361-6471%2Faba116

  * [14] J. Antoniadis, et al., Science 340 (6131).  doi:10.1126/science.1233232, [link].
URL https://science.sciencemag.org/content/340/6131/1233232

  * [15] E. Fonseca, et al., Astrophys. J. 832 (2) (2016) 167.  doi:10.3847/0004-637x/832/2/167, [link].
URL https://doi.org/10.3847%2F0004-637x%2F832%2F2%2F167

  * [16] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, J. W. T. Hessels, Nature 467 (7319) (2010) 1081–1083.  doi:10.1038/nature09466.
  * [17] P. Morfouace, C. Tsang, Y. Zhang, W. Lynch, M. Tsang, D. Coupland, M. Youngs, Z. Chajecki, M. Famiano, T. Ghosh, G. Jhang, J. Lee, H. Liu, A. Sanetullaev, R. Showalter, J. Winkelbauer, Physics Letters B 799 (2019) 135045.  doi:https://doi.org/10.1016/j.physletb.2019.135045, [link].
URL http://www.sciencedirect.com/science/article/pii/S0370269319307671

  * [18] J. M. Lattimer, Y. Lim, Astrophys. Jour. 771 (1) (2013) 51.  doi:10.1088/0004-637x/771/1/51, [link].
URL https://doi.org/10.1088/0004-637x/771/1/51

  * [19] B. K. Agrawal, T. Malik, J. N. De, S. K. Samaddar (2020).  arXiv:2006.05758.
  * [20] J. M. Lattimer, M. Prakash, The Astrophys. J. 550 (1) (2001) 426–442.  [link].
URL https://ui.adsabs.harvard.edu/abs/2001ApJ...550..426L

  * [21] B. A. Brown, Phys. Rev. Lett. 85 (2000) 5296–5299.  doi:10.1103/PhysRevLett.85.5296, [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.85.5296

  * [22] S. Typel, B. A. Brown, Phys. Rev. C 64 (2001) 027302.  doi:10.1103/PhysRevC.64.027302, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.64.027302

  * [23] X. Roca-Maza, M. Centelles, X. Viñas, M. Warda, Phys. Rev. Lett. 106 (2011) 252501\.  [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.106.252501

  * [24] R. Furnstahl, Nucl. Phys. A 706 (1) (2002) 85 – 110.  doi:https://doi.org/10.1016/S0375-9474(02)00867-9, [link].
URL http://www.sciencedirect.com/science/article/pii/S0375947402008679

  * [25] C. J. Horowitz, J. Piekarewicz, Phys. Rev. Lett. 86 (2001) 5647–5650.  [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.86.5647

  * [26] M. Tsang, W. Lynch, P. Danielewicz, C. Tsang, Phys. Lett. B 795 (2019) 533 – 536\.  doi:https://doi.org/10.1016/j.physletb.2019.06.059, [link].
URL http://www.sciencedirect.com/science/article/pii/S0370269319304423

  * [27] S. Typel, Particles 1 (2018) 3.  doi:https://doi.org/10.3390/, [link].
URL https://www.mdpi.com/2571-712X/1/1/2

  * [28] M. Fortin, C. Providência, A. R. Raduta, F. Gulminelli, J. L. Zdunik, P. Haensel, M. Bejger, Phys. Rev. C 94 (2016) 035804.  doi:10.1103/PhysRevC.94.035804, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.94.035804

  * [29] H. Pais, C. Providência, Phys. Rev. C 94 (2016) 015808.  doi:10.1103/PhysRevC.94.015808, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.94.015808

  * [30] G. Baym, C. Pethick, P. Sutherland, Astrophys. J. 170 (1971) 299–317.  doi:10.1086/151216.
  * [31] P. Haensel, B. Pichon, Astron. Astrophys. 283 (1) (1994) 313–318.  [link].
URL https://ui.adsabs.harvard.edu/abs/1994A&A...283..313H

  * [32] G. Baym, H. A. Bethe, C. J. Pethick, Nucl. Phys. A 175 (2) (1971) 225 – 271.  doi:https://doi.org/10.1016/0375-9474(71)90281-8, [link].
URL http://www.sciencedirect.com/science/article/pii/0375947471902818

  * [33] F. Grill, H. Pais, C. m. c. Providência, I. Vidaña, S. S. Avancini, Phys. Rev. C 90 (2014) 045803.  doi:10.1103/PhysRevC.90.045803, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.90.045803

  * [34] J. L. Friedman, N. Stergioulas, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2013.  doi:10.1017/CBO9780511977596.
  * [35] C. Horowitz, B. D. Serot, Nucl. Phys. A 368 (3) (1981) 503 – 528.  doi:https://doi.org/10.1016/0375-9474(81)90770-3, [link].
URL http://www.sciencedirect.com/science/article/pii/0375947481907703

  * [36] J. Walecka, Ann. Phys. 83 (1974) 491–529.  doi:10.1016/0003-4916(74)90208-5, [link].
URL https://ui.adsabs.harvard.edu/abs/1974AnPhy..83..491W

  * [37] J. Boguta, A. Bodmer, Nucl. Phys. A 292 (3) (1977) 413–428.  doi:10.1016/0375-9474(77)90626-1, [link].
URL https://ui.adsabs.harvard.edu/abs/1977NuPhA.292..413B

  * [38] R. Furnstahl, B. D. Serot, H.-B. Tang, Nucl. Phys. A 598 (4) (1996) 539 – 582.  doi:https://doi.org/10.1016/0375-9474(95)00488-2, [link].
URL http://www.sciencedirect.com/science/article/pii/0375947495004882

  * [39] R. Furnstahl, B. D. Serot, H.-B. Tang, Nucl. Phys. A 615 (4) (1997) 441 – 482.  doi:https://doi.org/10.1016/S0375-9474(96)00472-1, [link].
URL http://www.sciencedirect.com/science/article/pii/S0375947496004721

  * [40] B. Kumar, S. Singh, B. Agrawal, S. Patra, Nucl. Phys. A 966 (2017) 197 – 207.  doi:https://doi.org/10.1016/j.nuclphysa.2017.07.001, [link].
URL http://www.sciencedirect.com/science/article/pii/S0375947417303196

  * [41] S. K. Singh, S. K. Biswal, M. Bhuyan, S. K. Patra, Phys. Rev. C 89 (2014) 044001\.  doi:10.1103/PhysRevC.89.044001, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.89.044001

  * [42] B. D. Serot, J. D. Walecka, Adv. Nucl. Phys. 16 (1986) 1–327.
  * [43] N. H. Tan, D. T. Loan, D. T. Khoa, J. Margueron, Phys. Rev. C 93 (2016) 035806.  doi:10.1103/PhysRevC.93.035806, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.93.035806

  * [44] H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, Progress of Theoretical Physics 100 (5) (1998) 1013–1031.  doi:10.1143/PTP.100.1013, [link].
URL https://doi.org/10.1143/PTP.100.1013

  * [45] H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, Nuclear Physics A 637 (3) (1998) 435 – 450.  doi:https://doi.org/10.1016/S0375-9474(98)00236-X, [link].
URL http://www.sciencedirect.com/science/article/pii/S037594749800236X

  * [46] T. Maruyama, T. Tatsumi, D. N. Voskresensky, T. Tanigawa, S. Chiba, Phys. Rev. C 72 (2005) 015802.  doi:10.1103/PhysRevC.72.015802, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.72.015802

  * [47] S. S. Avancini, D. P. Menezes, M. D. Alloy, J. R. Marinelli, M. M. W. Moraes, C. Providência, Phys. Rev. C 78 (2008) 015802.  doi:10.1103/PhysRevC.78.015802, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.78.015802

  * [48] S. S. Avancini, L. Brito, J. R. Marinelli, D. P. Menezes, M. M. W. de Moraes, C. Providência, A. M. Santos, Phys. Rev. C 79 (2009) 035804.  doi:10.1103/PhysRevC.79.035804, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.79.035804

  * [49] S. S. Avancini, S. Chiacchiera, D. P. Menezes, C. m. c. Providência, Phys. Rev. C 82 (2010) 055807.  doi:10.1103/PhysRevC.82.055807, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.82.055807

  * [50] J. M. Lattimer, F. Douglas Swesty, Nuclear Physics A 535 (2) (1991) 331 – 376\.  doi:https://doi.org/10.1016/0375-9474(91)90452-C, [link].
URL http://www.sciencedirect.com/science/article/pii/037594749190452C

  * [51] B. P. Abbott, R. A. et al, The Astrophys. Jour. 848 (2) (2017) L13.  doi:10.3847/2041-8213/aa920c, [link].
URL https://doi.org/10.3847%2F2041-8213%2Faa920c

  * [52] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 627 (4) (1997) 710 – 746.  doi:https://doi.org/10.1016/S0375-9474(97)00596-4, [link].
URL http://www.sciencedirect.com/science/article/pii/S0375947497005964

  * [53] N. Chamel, P. Haensel, Liv. Rev. Relat. 11 (4) (2008) 1433–8351.  doi:10.12942/lrr-2008-10, [link].
URL https://doi.org/10.12942/lrr-2008-10

  * [54] S. Kubis, Phys. Rev. C 70 (2004) 065804.  doi:10.1103/PhysRevC.70.065804, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.70.065804

  * [55] S. Kubis, Phys. Rev. C 76 (2007) 025801.  doi:10.1103/PhysRevC.76.025801, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.76.025801

  * [56] C. C. Moustakidis, Phys. Rev. C 86 (2012) 015801.  doi:10.1103/PhysRevC.86.015801, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.86.015801

  * [57] W. M. Seif, D. N. Basu, Phys. Rev. C 89 (2014) 028801.  doi:10.1103/PhysRevC.89.028801, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.89.028801

  * [58] T. R. Routray, X. Viñas, D. N. Basu, S. P. Pattnaik, M. Centelles, L. B. Robledo, B. Behera, Jour. Phys. G: Nucl. and Part. Phys. 43 (10) (2016) 105101\.  doi:10.1088/0954-3899/43/10/105101, [link].
URL https://doi.org/10.1088/0954-3899/43/10/105101

  * [59] J. Boguta, A. R. Bodmer, Nucl. Phys. A 292 (3) (1977) 413–428.  doi:10.1016/0375-9474(77)90626-1.
  * [60] S. Typel, H. Wolter, Nucl. Phys. A 656 (1999) 331–364.  doi:10.1016/S0375-9474(99)00310-3.
  * [61] G. A. Lalazissis, J. König, P. Ring, Phys. Rev. C 55 (1997) 540–543.  doi:10.1103/PhysRevC.55.540, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.55.540

  * [62] Y. Sugahara, H. Toki, Nucl. Phys. A 579 (1994) 557–572.  doi:https://doi.org/10.1016/0375-9474(94)90923-7.
  * [63] C. J. Horowitz, J. Piekarewicz, Phys. Rev. C 64 (2001) 062802.  doi:10.1103/PhysRevC.64.062802, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.64.062802

  * [64] B. G. Todd-Rutel, J. Piekarewicz, Phys. Rev. Lett. 95 (2005) 122501.  doi:10.1103/PhysRevLett.95.122501, [link].
URL https://link.aps.org/doi/10.1103/PhysRevLett.95.122501

  * [65] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, G. Shen, Phys. Rev. C 82 (2010) 055803\.  doi:10.1103/PhysRevC.82.055803, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.82.055803

  * [66] G. A. Lalazissis, T. Nikšić, D. Vretenar, P. Ring, Phys. Rev. C 71 (2005) 024312.  doi:10.1103/PhysRevC.71.024312, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.71.024312

  * [67] X. Roca-Maza, X. Viñas, M. Centelles, P. Ring, P. Schuck, Phys. Rev. C 84 (2011) 054309.  doi:10.1103/PhysRevC.84.054309, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.84.054309

  * [68] A. Carbone, A. Schwenk, Phys. Rev. C 100 (2019) 025805.  doi:10.1103/PhysRevC.100.025805, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.100.025805

  * [69] J. R. Oppenheimer, G. M. Volkoff, Phys. Rev. 55 (1939) 374–381.  doi:10.1103/PhysRev.55.374, [link].
URL https://link.aps.org/doi/10.1103/PhysRev.55.374

  * [70] R. C. Tolman, Phys. Rev. 55 (1939) 364–373.  doi:10.1103/PhysRev.55.364, [link].
URL https://link.aps.org/doi/10.1103/PhysRev.55.364

  * [71] N. Stergioulas, Living Rev. Relativ. 6 (2003) 3.  doi:10.12942/lrr-2003-3, [link].
URL https://doi.org/10.12942/lrr-2003-3

  * [72] N. Stergioulas, J. L. Friedman, The Astrophys. Jour. 444 (1995) 306.  doi:10.1086/175605, [link].
URL http://dx.doi.org/10.1086/175605

  * [73] N. Stergioulas, Rapidly rotating neutron stars, RNS: Rapidly rotating neutron stars.
URL http://www.gravity.phys.uwm.edu/rns

  * [74] T. Nozawa, N. Stergioulas, E. Gourgoulhon, Y. Eriguchi, Astron. Astrophys. Suppl. Ser. 132 (3) (1998) 431–454.  doi:10.1051/aas:1998304, [link].
URL https://doi.org/10.1051/aas:1998304

  * [75] H. T. Cromartie, E. Fonseca et al., Nature Astronomy 4 (2019) 72–76.  doi:10.1038/s41550-019-0880-2, [link].
URL https://doi.org/10.1038/s41550-019-0880-2

  * [76] R. Abbott, T. D. A. et al., The Astrophys. Jour. 896 (2) (2020) L44.  doi:10.3847/2041-8213/ab960f, [link].
URL https://doi.org/10.3847%2F2041-8213%2Fab960f

  * [77] T. E. Riley et al., Astrophys. J. 887 (1) (2019) L21.  doi:10.3847/2041-8213/ab481c, [link].
URL https://doi.org/10.3847/2041-8213/ab481c

  * [78] M. C. Miller, et al., Astrophys. Jour. 887 (1) (2019) L24.  doi:10.3847/2041-8213/ab50c5, [link].
URL https://doi.org/10.3847/2041-8213/ab50c5

  * [79] M. C. Miller et al., Astrophys. J. 887 (1) (2019) L24.  doi:10.3847/2041-8213/ab50c5, [link].
URL https://doi.org/10.3847/2041-8213/ab50c5

  * [80] T. Hinderer, B. D. Lackey, R. N. Lang, J. S. Read, Tidal deformability of neutron stars with realistic equations of state and their gravitational wave signatures in binary inspiral, Phys. Rev. D 81 (2010) 123016\.  doi:10.1103/PhysRevD.81.123016.
URL https://link.aps.org/doi/10.1103/PhysRevD.81.123016

  * [81] B. Kumar, S. K. Biswal, S. K. Patra, Phys. Rev. C 95 (2017) 015801.  doi:10.1103/PhysRevC.95.015801, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.95.015801

  * [82] T. Hinderer, The Astrophys. J. 677 (2) (2008) 1216–1220.  doi:10.1086/533487, [link].
URL https://doi.org/10.1086%2F533487

  * [83] J.-L. Jiang, S.-P. Tang, Y.-Z. Wang, Y.-Z. Fan, D.-M. Wei, Astrophys. Jour. 892 (1) (2020) 55.  doi:10.3847/1538-4357/ab77cf, [link].
URL https://doi.org/10.3847/1538-4357/ab77cf

  * [84] P. Landry, R. Essick, K. Chatziioannou, Phys. Rev. D 101 (2020) 123007.  doi:10.1103/PhysRevD.101.123007, [link].
URL https://link.aps.org/doi/10.1103/PhysRevD.101.123007

  * [85] J. M. Lattimer, M. Prakash, Physics Reports 333-334 (2000) 121 – 146.  doi:https://doi.org/10.1016/S0370-1573(00)00019-3, [link].
URL http://www.sciencedirect.com/science/article/pii/S0370157300000193

  * [86] A. Worley, P. G. Krastev, B.-A. Li, The Astrophys. Jour. 685 (1) (2008) 390–399.  doi:10.1086/589823, [link].
URL https://doi.org/10.1086%2F589823

  * [87] J. M. Lattimer, B. F. Schutz, Constraining the equation of state with moment of inertia measurements, Astrophys. Jour. 629 (2) (2005) 979–984.  doi:10.1086/431543.
URL https://doi.org/10.1086/431543

  * [88] V. Paschalidis, N. Stergioulas, Living Rev. Relativ. 20 (1) (2017) 1433–8351.  doi:10.1007/s41114-017-0008-x, [link].
URL https://doi.org/10.1007/s41114-017-0008-x

  * [89] P. Landry, B. Kumar, The Astrophys. Jour. 868 (2) (2018) L22.  doi:10.3847/2041-8213/aaee76, [link].
URL https://doi.org/10.3847%2F2041-8213%2Faaee76

  * [90] B. Kumar, P. Landry, Phys. Rev. D 99 (2019) 123026.  doi:10.1103/PhysRevD.99.123026, [link].
URL https://link.aps.org/doi/10.1103/PhysRevD.99.123026

  * [91] B.-A. Li, P. G. Krastev, D.-H. Wen, N.-B. Zhang, Eur. Phys. Jour. A 55 (117) (2019) 390–399.  doi:10.1140/epja/i2019-12780-8, [link].
URL https://doi.org/10.1140/epja/i2019-12780-8

  * [92] D. McNall, Geophysical Journal International 21 (1) (1970) 103–104.  doi:10.1093/gji/21.1.103-a, [link].
URL https://doi.org/10.1093/gji/21.1.103-a

  * [93] J. E. Tohline, R. H. Durisen, M. McCollough, Astrophys. J.
  * [94] B. K. Pickett, R. H. Durisen, G. A. Davis, Astrophys. J. 458 (1996) 714.  doi:10.1086/176852, [link].
URL https://ui.adsabs.harvard.edu/abs/1996ApJ...458..714P

  * [95] J. N. Imamura, J. Toman, R. H. Durisen, B. K. Pickett, S. Yang, Astrophys. J. 444 (1995) 363.  doi:10.1086/175611, [link].
URL https://ui.adsabs.harvard.edu/abs/1995ApJ...444..363I

  * [96] J. M. Centrella, K. C. B. New, L. L. Lowe, J. D. Brown, Astrophys. J. Lett. 550 (2) (2001) L193–L196.  doi:10.1086/319634, [link].
URL https://ui.adsabs.harvard.edu/abs/2001ApJ...550L.193C

  * [97] N.-B. Zhang, B.-A. Li, Eur. Phys. J. A 55 (2019) 1434–601X.  doi:10.1140/epja/i2019-12700-0, [link].
URL https://doi.org/10.1140/epja/i2019-12700-0

  * [98] S. S. Bao, J. N. Hu, Z. W. Zhang, H. Shen, Phys. Rev. C 90 (2014) 045802.  doi:10.1103/PhysRevC.90.045802, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.90.045802

  * [99] S. S. Bao, H. Shen, Phys. Rev. C 89 (2014) 045807.  doi:10.1103/PhysRevC.89.045807, [link].
URL https://link.aps.org/doi/10.1103/PhysRevC.89.045807