# Relativistic Polytropic Models of Charged Anisotropic Compact Object

H. Nazara , M. Azam b, G. Abbasa , Riaz Ahmedc and R. Naeemb
aDepartment of Mathematics The Islamia University
of Bahawalpur, Bahawalpur, Pakistan
b Department of Mathematics,
Division of Science and Technology,
University of Education, Lahore, Pakistan
c Department of Mathematics,
Maharishi International University, Fairfield, IA USA
hammadnazar350<EMAIL_ADDRESS>azammath@gmail.comghulamabbas@iub.edu.pkrahmed<EMAIL_ADDRESS>
###### Abstract

In this paper, we have introduced new viable solutions of Einstein-Maxwell
field equations by incorporating the features of anisotropic matter
distribution in the realm of General theory of Relativity ($GR$). For this
procurement, we have employed a Finch-Skea spacetime along with a generalized
polytropic equation of state ($EoS$). We have constructed various models of
generalized polytropes by assuming the different choices of the polytropic
index i.e.,$\eta=\frac{1}{2},\frac{2}{3},1$ and $2$. The numerous physical
characteristics of these considered models have been studied via graphical
analysis, which obey all the essential conditions of the astrophysical compact
objects. Furthermore, such outcomes of charged anisotropic compact star models
can be regained to the various cases such as linear, quadratic and polytropic
$EoS$.

Keywords: General Relativity, Exact solutions, Charged polytropic models,
Anisotropy of pressure;

##  1 Introduction

By using the basic formulation of $GR$, the investigation of stellar
astrophysical compact objects ($COs$) is still a growing and interesting field
of research. It is essential for the modeling of these gravitational $COs$ to
be static or non-static and spherically symmetric regarding to observe their
stable configuration. To develop a realistic and potentially stable model of
astrophysical $CO$ together with the distribution of an electric charge is
ever being a solid choice among the theorists and astrophysicists. It is
necessary to tackle the system of Einstein-Maxwell field equations to reduce
into a less complex system when someone adopts a better mathematical tool.
However, numerous researchers have chosen distinctive ansatz to comprehend
these equations. In this view point, many authors such as Patel and Kopper
[1], Patel et al. [2], Tikekar and Singh [3], Sharma et al. [4] and Komathiraj
and Maharaj [5] have used a variety of numerical strategies to get the
definite solutions for the relativistic $COs$. These investigations reflect
that the system of field equations is necessarily depicted as astrophysical
$COs$.

In $GR$, the exact solutions are physically viable and robustly utilized in
numerous applications of astrophysical $COs$. To attain the analytical
solutions of the Einstein-Maxwell equations is one of the basic problems in
the $GR$ as well as among the modified versions of this theory of gravitation.
It is crucial to guarantee that the physical nature of these solutions should
be valid. In this perspective, the physical effectiveness of exact solutions
has been manifested by Delgaty and Lake [6]. They have fixed the geometrical
standards for the gravitational potential, which imperatively fulfill all the
physical traits.

The presence of electric charge within the matter content creates an
extraordinary theoretical interest for the electromagnetic effects on the
stellar astrophysical $COs$. Gupta and Maurya [7, 8] have developed the super-
dense star models in the presence of electric intensity. They have analyzed
that the obtained solutions are well-behaved and regular within the entire
domain of the star. Maurya and Gupta [9] have proposed some vital features of
charged objects by assigning the value of electric parameter. Pant et al. [10]
have explored various solutions of charge matter balls influence with finite
pressure and matter density that satisfied the casuality condition
$(\frac{dP_{r}}{d\rho}\leq 1)$ and also possessed the positive nature at the
center. Some well-behaved charged $COs$ have been investigated by Maurya and
Gupta [11, 12] with a metric potential $g_{44}=B(1+Cr^{2})^{n}$. Beside of
these exceptional work, several researchers [11]-[14] have contributed a
variety of novel solutions regarding the modeling of charged $COs$.

During the recent decades, numerous authors have given the exact solutions of
Einstein-Maxwell field equations with isotropic fluid. Some global and
physically realistic solutions of charged isotropic and Reissner-
Nordstr$\ddot{o}$m metric were studied by Ivanov [15]. In order to obtain the
exact solution, one needs to integrate the Einstein-Maxwell field equations.
Therefore, distinctive confinement has been set by researchers on the
spacetime geometry as well as on the matter distribution. Under the
supposition of constant hypersurface, Komathiraj and Maharaj [16] have
introduced the charged spherical models. By setting the constraint on
gravitational potential, Thirukkanesh and Maharaj [17] and Komathiraj and
Maharaj [5] have investigated the exact solutions of the Einstein-Maxwell
field equations in the presence of isotropic matter. Mak and Harko [18] have
proposed the different techniques to obtain the various physical aspects of
the isotropic spheres. In contrast to these inspirational consequences, many
researchers have shown that the pressure anisotropy is an underlying
ingredient in the modeling of $COs$. The modeling of anisotropic $COs$ is
still an extensive and large dynamical field among the research groups during
the last several decades. In this sense, Ruderman [19] provided the basic
foundation for the configuration of stellar anisotropic $COs$ and concluded
that the extreme dense regions of $COs$ contain anisotropic fluid. Further,
Bowers and Liang [20] have studied the possible causes of anisotropic stress.
Subsequently, the impact of anisotropic pressure on the system of general
relativistic sphere and its physical characteristics have been investigated by
Dev and Gleiser [21, 22]. Herrera and his co-researchers [23]-[28] have
considered a class of exact solutions to the set of field equations regarding
the physical emergence of anisotropic pressure in the self-gravitating
structures. Moreover, Fuloria and Pant [29] and Maurya and Maharaj [30] have
suggested the various types of analytical solutions to demonstrate the
physical implications of anisotropic compact spheres in the regime of
Karmarkar approach. Apart of these notable consequences, some class of exact
relativistic solutions to the Einstein field equations for anisotropic
spherically symmetric and static dense compact star models inspired by the
Buchdahl and Korkina-Orlyanskii ansatzes have been proposed in [31]-[34].
Recently, many researcher have made phenomenological considerations in the
assessment of physical influences of anisotropic stress in the realm of $GR$
as well as alternative gravitational theories [35]-[53].

No doubt, it is a basic issue to choose a suitable $EoS$ for the demonstration
of astrophysical $COs$. For instance, Varela et al. [54] have discussed the
physical reliability of $EoS$ in the demonstration of $COs$. Sharma and
Maharaj [55], Thirukkanesh and Maharaj [56] and Takisa and Maharaj [57] have
employed the linear $EoS$ to develop the models of dense $COs$. For
spherically symmetric systems, Hansraj and Maharaj [58] have studied the
isotropic matter distribution that satisfied the barotropic $EoS$, i.e.,
$p_{r}=p_{r}(\rho)$. In context of quadratic $EoS$, Feroze and Siddique [59]
and Maharaj and Takisa [60] have explored the physical implications of charged
anisotropic relativistic star models. In the composition of astronomical
compact star models, the polytropic $EoS$ plays a vital role to explain the
various features of compact stars. Chandrasekhar [61] have discussed the
outcomes of Newtonian polytropic theory in the context of thermodynamical
laws. Herrera and Barreto [62] have organized a detail study for the Newtonian
polytropic models when the fluid is considered as anisotropic. Thirukkanesh
and Ragel [63, 64] have depicted the realistic features of uncharged compact
star models in the regime of polytropic $EoS$. Furthermore, Takisa and Maharaj
[65] have reviewed the physical consequences of polytropic $EoS$ on charged
anisotropic compact star models. Nilsson and Ugla [66] have adopted the
standard procedure to obtain a set of equations by admitting the polytropic
$EoS$. This $EoS$ can be assessed numerically to finite radius corresponding
to the various values of polytropic index $\eta$ defined from $0$ to 3.339 and
for $\eta\geq 5$ to infinite radius.

Despite of the above chosen polytropic $EoS$, the combination of linear and
polytropic $EoS$ gives the generalized form of polytropic $EoS$ i.e.,
$P_{r}=\alpha\rho+\beta\rho^{1+1/\eta}$. Several authors have employed the
generalized polytropic $EoS$ to study the definite stellar features of $COs$
like as, Chavanis [67] constructed the models of the early and the late
Universe with a generalized polytropic $EoS$ and concluded that the models can
be seem in fine symmetric tune for an early and late epoch of inflation. Azam
et al. [68] have formulated the general framework to explore the behavior and
reliability of Newtonian and relativistic polytropes in the presence of
generalized polytropic $EoS$. Nasim and Azam [69] have proposed the physically
viable charged anisotropic models by considering the generalized polytropic
$EoS$. To model the ultra relativistic $CO$, Finch and Skea [70] have
developed the analytical study in the background of Duorah and Ray ansatz.
Considering the Finch-Skea ansatz, some viable implications for charged
anisotropic compact star models have been proposed by Hansraj and Maharaj [58]
by employing the barotropic $EoS$. Sharma and Ratanpal [71] have reviewed the
various fruitful outcomes of static and spherically symmetric objects in the
framework of Finch-Skea ansatz. Pandya et al. [72] have shown the generalized
results of Finch-Skea geometry for the modeling of strange quark stars by
considering the Chaplygin gas $EoS$. Recently, several notable characteristics
of astrophysical $COs$ have been obtained by assuming the Finch-skea
geometries in various gravitational theories [73]-[77].

This manuscript is outlined as follows: In Sec. 2, we have investigated an
equivalent set of Einstein-Maxwell field equations for models of anisotropic
astrophysical $COs$ by implementing a generalized polytropic $EoS$ with the
transformation introduced by Durgapal and Bannerji [78]. The smooth matching
conditions have also been investigated between the interior metric and
exterior Reissner-Nordstr$\ddot{\ddot{o}}$m solution at the hypersurface. In
Sec. 3, we have assumed a gravitational potential $Z$ as Finch-Skea ansatz,
which leads us to a restricted system of field equations in comprehensive
manner. We have obtained the various feasible class of charged polytropic
models corresponding to the particular choices of polytropic index $\eta$ in
Sec. 4. In Sec. 5, deals with physical adequacy of presented models to study
the various attributes such as, material quantities, effective mass function,
anisotropy, electric charge and causality condition. The last section includes
the summary of our findings.

##  2 Maxwell field equations

We choose a static and spherically symmetric relativistic stellar interior
model in the standard coordinates as
$x^{a}=(x^{1},x^{2},x^{3},x^{4})=(t,r,\theta,\phi)$, which has the form

$ds^{2}=-e^{2\lambda(r)}dt^{2}+e^{2\psi(r)}dr^{2}+r^{2}d\Omega^{2},$ (1)

where $\lambda(r)$ and $\psi(r)$ are metric potentials for a static
gravitational field, and $d\Omega^{2}=d\theta^{2}+\sin\theta^{2}d\phi^{2}$. We
assume that the interior region of the stellar model is filled by the charged
anisotropic matter, whose total energy-momentum tensor is given by

$T_{a}^{b}=diag\Big{[}-\rho-\frac{\mathcal{E}^{2}}{2},P_{r}-\frac{\mathcal{E}^{2}}{2},P_{t}+\frac{\mathcal{E}^{2}}{2},P_{t}+\frac{\mathcal{E}^{2}}{2}\Big{]}.$
(2)

Here $\rho$ illustrates the energy density, $P_{r}$ and $P_{t}$ are radial and
tangential pressure, respectively, whereas $\mathcal{E}$ is the electric field
intensity. For static interior compact sphere (1) and anisotropic energy-
momentum tensor (2), the Einstein-Maxwell field equations get the following
form

$\displaystyle 8\pi\ \rho+\mathcal{E}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\ \Big{[}r(1-e^{-2\psi})\Big{]}^{\prime},$ (3)
$\displaystyle 8\pi\ P_{r}-\mathcal{E}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}}\
\Big{(}1-e^{-2\psi}\Big{)}+\frac{2\lambda^{\prime}}{r}e^{-2\psi},$ (4)
$\displaystyle 8\pi\ P_{t}+\mathcal{E}^{2}$ $\displaystyle=$ $\displaystyle
e^{-2\psi}\ \Big{(}\lambda^{\prime\prime}+\lambda^{\prime
2}+\frac{\lambda^{\prime}}{r}-\lambda^{\prime}\psi^{\prime}-\frac{\psi^{\prime}}{r}\Big{)},$
(5) $\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\frac{1}{r^{2}}\
e^{-\psi}\ \Big{(}r^{2}\ \mathcal{E}\Big{)}^{\prime}.$ (6)

Here, we use the geometric units as $G=c=1$, whereas ′ and ′′ signs are first
and second derivatives with respect to $r$. The underlying equations
describing the fundamental stellar interior model for charged anisotropic
fluid sphere are provided by the system of Eqs. (3)-(6). These Einstein-
Maxwell field equations are a system of four independent equations in six
independent unknowns ($\lambda$, $\psi$, $\rho$, $P_{r}$, $P_{t}$,
$\mathcal{E}$ or $\sigma$). Remember, if we assume a specific form of the
electric field intensity $\mathcal{E}$ then the system of Eqs. (3)-(6) becomes
a system of three independent equations in five independent unknowns. A
charged solution can be generated by specifying the forms for two unknown
functions or each combination of unknown functions and equations of state
associating the matter quantities. In order to evolve a simplest system, we
consider an important relation between the energy density and radial pressure,
which is described by the generalized polytropic $EoS$.

$\displaystyle P_{r}=\mathcal{A}\ \rho+\mathcal{B}\ \rho^{\Gamma},$ (7)

where $\Gamma=1+\frac{1}{\eta}$ in which $\eta$ refers to the polytropic
index, while $\mathcal{A}$ and $\mathcal{B}$ are arbitrary constants. In
deciding criterions for a certain $EoS$ to construct the stellar relativistic
astrophysical $CO$ rather than the self-gravitating compact model the radial
pressure $P_{r}$ should be non-negative and regular (finite) at every point
inside the stars and it should be vanished at the surface $r=r_{b}$ of the
sphere [79]. Therefore, when pressure is zero at the boundary, implies that
the density also vanishes on the surface of the star. To obtain a concise
system of the field equations, we imply the transformation priory proposed by
Durgapal and Banerji [78] given in the following way

$x=r^{2},\ \ \ \mathcal{Z}(x)=e^{-2\psi(r)},\ \ \ y^{2}(x)=e^{2\lambda(r)},$
(8)

Using the above transformation, Eqs(3)-(6) along with the generalized
polytropic $EoS$ can be rewritten as

$\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle\frac{1}{x}(1-\mathcal{Z})-2\dot{\mathcal{Z}}-\mathcal{E}^{2},$
(9) $\displaystyle P_{t}$ $\displaystyle=$ $\displaystyle P_{r}+\Delta,$ (10)
$\displaystyle P_{r}$ $\displaystyle=$ $\displaystyle\mathcal{A}\
\rho+\mathcal{B}\ \rho^{1+\frac{1}{\eta}},$ (11) $\displaystyle 8\pi P_{r}$
$\displaystyle=$ $\displaystyle
4\mathcal{Z}\frac{\dot{y}}{y}+\frac{1}{x}\Big{(}\mathcal{Z}-1\Big{)}+\mathcal{E}^{2},$
(12) $\displaystyle 8\pi P_{t}$ $\displaystyle=$ $\displaystyle
4x\mathcal{Z}\frac{\ddot{y}}{y}+\Big{(}4\mathcal{Z}+2x\dot{\mathcal{Z}}\Big{)}\frac{\dot{y}}{y}+\dot{\mathcal{Z}}-\mathcal{E}^{2},$
(13) $\displaystyle 8\pi\Delta$ $\displaystyle=$ $\displaystyle
4x\mathcal{Z}\frac{\ddot{y}}{y}+2x\frac{\dot{y}}{y}\dot{\mathcal{Z}}+\dot{\mathcal{Z}}+\frac{1}{x}\Big{(}1-\mathcal{Z}\Big{)}-2\mathcal{E}^{2},$
(14) $\displaystyle\sigma^{2}$ $\displaystyle=$
$\displaystyle\frac{4\mathcal{Z}}{x}\Big{(}\mathcal{E}+\dot{\mathcal{E}}x\Big{)}^{2}.$
(15)

From Eqs.(7-(9), we get

$\displaystyle\frac{\dot{y}}{y}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}}{4\mathcal{Z}}\Big{(}\frac{1-\mathcal{Z}}{x}-2\dot{\mathcal{Z}}-\mathcal{E}^{2}\Big{)}+\frac{\mathcal{B}(8\pi)^{-\frac{1}{\eta}}}{4\mathcal{Z}}\Big{(}\frac{1-\mathcal{Z}}{x}-2\dot{\mathcal{Z}}-\mathcal{E}^{2}\Big{)}^{1+\frac{1}{\eta}}+\frac{1-\mathcal{Z}}{4\mathcal{Z}x}-\frac{\mathcal{E}^{2}}{4\mathcal{Z}},$
(16)

where dot represents differentiation with respect to $x$, and anisotropy
factor is $\Delta$, which reflect the difference between the radial and the
tangential pressure. In this description, Thirukkanesh and Maharaj [56] have
assumed the linear $EoS$ to develop an analogous equatorial system for the
modeling of anisotropic dense compact sphere. Feroze and Siddique [59] and
Maharaj and Takisa [60] have employed the quadratic $EoS$ to obtain the
physically realistic compact star models. Furthermore, the polytropic $EoS$ is
assumed by Takisa and Maharaj [57] in order to construct a charged anisotropic
compact star model. The gravitating mass within the sphere of radius $r$ is
expressed as

$M(r)=4\pi\int^{r}_{0}\omega^{2}\rho(\omega)d\omega.$ (17)

The mass function after implementing the transformation takes the following
form

$M(x)=2\pi\int^{x}_{0}\sqrt{\omega}\rho(\omega)d\omega.$ (18)

###  2.1 Matching conditions

The matching conditions are generally employed to join interior and exterior
geometries of the star at the boundary surface $r=r_{b}$. The choice of the
exterior region entirely depends on the interior matter distribution of the
star. For spheroidal structure, if the fluid in the interior of an object is
electrically charged and anisotropic, we assume the Reissner-
Nordstr$\ddot{\ddot{o}}$m metric as the exterior region,

$ds^{2}=-\Big{(}1-\frac{2\tilde{M}}{r_{b}}+\frac{Q^{2}}{r_{b}^{2}}\Big{)}dt^{2}+\Big{(}1-\frac{2\tilde{M}}{r_{b}}+\frac{Q^{2}}{r_{b}^{2}}\Big{)}^{-1}dr^{2}+r_{b}^{2}\Big{(}d\theta^{2}+\sin^{2}\theta
d\phi^{2}\Big{)},$ (19)

where $\tilde{M}$ is the total mass of the stellar interior and $Q$ indicates
the total electric charge of the fluid. Matching conditions play a vital role
in the study of the compact star. These conditions tell whether the junction
of two geometries give a realistic solution or not when a boundary surface
separates the region into inner and outer regions. The smooth matching of
inner and outer metrics via the continuity of the first and second fundamental
form over the boundary surface [80]-[82] determines the following results:

$\displaystyle e^{2\lambda(r)}$ $\displaystyle=$ $\displaystyle
e^{-2\psi(r)}=\Big{(}1-\frac{2\tilde{M}}{r_{b}}+\frac{Q^{2}}{r_{b}^{2}}\Big{)},~{}~{}~{}~{}~{}\textit{M}=\tilde{M},~{}~{}~{}~{}~{}q=Q,$
$\displaystyle P_{r}$ $\displaystyle=$ $\displaystyle 0.$ (20)

The mass function of the stellar interior $CO$ defined by Misner and Sharp
[79] and Nielsen and Yeom [83], is given by

M $\displaystyle=$
$\displaystyle\frac{r}{2}\Big{(}1-\frac{1}{e^{2\psi}}+\frac{q^{2}}{r^{2}}\Big{)},$
(21)

where $q=2\pi\int_{0}^{x}\sigma e^{\psi(x)}\sqrt{x}dx$ is the total charge
enclosed by a sphere of radius $r$. In spheroidal system as a bounded object
the mass of the star can be computed as a measure of the total energy within a
sphere of a radius $r$. The concept of polytropes is based on the presumption
of hydrostatic stability and polytropic $EoS$. In this analysis, we study the
polytropic models in the scenario of a generalized polytropic $EoS$, which is
a combination of linear and polytropic $EoS$.

##  3 Comprehensive overview on Finch-Skea ansatz

In this perspective, our fundamental endeavor is to develop a fine tune model
for stable configuration of the stellar relativistic dense object, when no
expedient proofs concerning the evolution and nature of the particle
interactions are commodious. It might be formulated by determining the viable
solutions of equations of motion describing the static stellar interior core
of spherical $CO$. However, analytical solution of equations of motion is not
an easy task because of the highly non-linear differential equations. In this
view point, several appropriate approaches have been successfully imposed to
tackle the system of differential equations. Since, Einstein’s field equations
permits a joint affinement between the matter field and its geometry, we shall
affirm a structural approach to deal with such a constraint. To do so, an
authentic form of one of the metric potentials with clear attribution of an
analogous metric will be specified to point out the other. Such approach was
developed by Finch and Skea [70] for the composition of an interior spheroidal
geometry. Duorah and Ray [84] were the pioneer, who gave the idea of such
ansatz in which they did not provide an ideal form of such ansatz to meet the
equations of motion for the modeling of astrophysical $COs$. This type of
ansatz has procured lot of insights in the composition of astrophysical $COs$
and consequences are much suitable and satisfy all the constraints, which are
necessary for the acceptability of the models [6]. Various fruitful findings
regarding to the such Finch-Skea ansatz have been reviewed to identify an
extensive group of paradigms for the stellar astrophysical $COs$ by
incorporating electric charge, dissipative matter content, pressure
anisotropy, quadratic $EoS$, quintessence matter, etc. [58, 71, 72, 85, 86].

Inspiring from this motivational information, we reported here that in the
connection of such ansatz too many realistic features of cosmological and
stellar astrophysical $COs$ have been studied in $GR$ as well as in
alternative theories of gravitation. In the discipline of the stellar
astrophysical bodies, two parameters viable class of an exact solutions of
compact relativistic star model in the presence of strange quark source have
proposed by Tikekar and Jotania [87]. Bhar [88] has presented the implications
of spherical $CO$ supported by the Chaplygin $EoS$ and examined that the $CO$
is composed of a quark matter content. Banerjee et al. [89] have contemplated
the viable features of an astrophysical compact body with an exterior $BTZ$
metric under the framework of Finch-Skea ansatz. Bhar et al. [90] have
explored a viable class of analytical solutions of Einstein field equation by
using the $MIT$ bag $EoS$ in the presence of Finch-Skea geometry for the
formulation of stellar relativistic compact sphere. In additionally, various
theoretical physicists have done a variety of novel work by applying the
Finch-Skea ansatz as well as its generalization in the higher dimensional
gravity theories [91]-[95]. In spite of the astrophysical implications, a
quite fascinating and qualitative analysis has been disclosed in the regime of
Finch-Skea geometries regarding to observe the cosmological evidences by
adopting the Chaplygin gas model, barotropic, quadratic and MIT bag $EoS$.
These explicit models of $EoS$ have described the immense influence of matter
content in the forms of dark matter and dark energy, because they have
robustly affirmed our thoughts regarding the conceptions of unseen features of
inflationary Universe. Over the last few decades, some researchers have
reviewed the models of Chaplygin gas $EoS$ into the account of evolutionary
expansion of the Universe and composition of massive scale objects [96]-[97].
In some of recent investigations, Chanda et al. [76] have proposed quite
fascinating work within the $f(T)$ theory of gravitation for the modeling of
stellar anisotropic $COs$ by implementing the Finch-Skea geometry. Singh et
al. [77] have proposed a class of analytical solutions to formulate a model of
anisotropic compact sphere by using the Finch-Skea ansatz. Maurya and his
collaborators [98]-[101] have investigated various viable class of an exact
solutions for the modeling of charged and without charged anisotropic
relativistic $COs$ by proposing the MIT bag model $EoS$ and Finch-Skea ansatz
within the context of $GR$ and alternative theories of gravity. They explored
that under described conditions these solutions are well behaved and satisfy
all the necessary bounds for models of stellar anisotropic $CO$. After an
inspirational overview on such ansatz, now our basic purpose to develop a new
family of an exact solutions for the Einstein-Maxwell field equations. For
this purpose, we need to assume one of the metric potentials as $\mathcal{Z}$
given in Ref.[70]. Hence, we take

$e^{2\psi}=1+\frac{r^{2}}{R^{2}},$ (22)

where $R$ describes the curvature parameter. The geometric approach of such
solution has been successfully imposed to satisfy the necessary conditions for
the composition of stellar relativistic $COs$. By using Eq. (8) and (22), we
get

$\mathcal{Z}(x)=e^{-2\psi}=\frac{1}{1+\frac{r^{2}}{\textit{R}^{2}}}.$ (23)

We assume a real arbitrary constant $\xi$ in above form, which gives

$\mathcal{Z}(x)=({1+\xi x})^{-1},\ \ \ \ \xi\neq 0.$ (24)

The ansatz $\mathcal{Z}$ is physically admissible as it is non-singular and
continuous at the central core of star. Maharaj et al [102] have studied new
feasible class of analytical solutions in the Finch-Skea spacetime to develop
a model of anisotropic fluid sphere. Sharma et al. [103] have constructed
singularity free solutions for anisotropic compact body by setting a parameter
$\xi=1$ within the Finch-Skea spacetime. Consequently, the choice of ansatz
$\mathcal{Z}$ is trustworthy to obtain physically realistic charged sphere
models having anisotropic fluid configuration through generalized polytropic
$EoS$. Our consideration for electromagnetic charge $\mathcal{E}$ is

$\mathcal{E}^{2}=\frac{\alpha x}{(1+\xi x)^{2}}.$ (25)

Here $\alpha$ is a real arbitrary constant. For modeling of charged
relativistic $CO$, it is vital to make sure that two generic aspects are
contained in the model. First, the model must be physically acceptable, i.e.,
the metric potentials, electric charge and matter components should be regular
and free from physical singularities within the entire distribution of the
sphere, the inner region joins smoothly with the outer metric and causality is
not breached. Finally, we must retrieve neutral solution (uncharged) of the
equations of motion when the electric charge vanishes, an uncharged sphere
must be retrievable as a potentially stable final fate. Due to the closed
bounded interior configuration, the above choice of $\mathcal{E}$ is quite
physically reliable and sustainable to predict a potentially viable
astrophysical compact star model. For the appropriate choice of independent
variable $x$, the choice of $\mathcal{E}$ gives the constant term. Hansraj and
Maharaj [58] and Nasim and Azam [69] have applied analogous choice of an
electric charge to compose a potentially stable charged anisotropic compact
star models. Moreover, Maurya et al. [104] have investigated the singularity
free solution of charged anisotropic fluid sphere by assuming the specific
choice of electric intensity into an account of $GR$. Consequently, this
choice is most capable to construct these models with generalized polytropic
$EoS$ in spacetime geometry of Finch and Skea. Hence, Eqs.(9)-(16) become

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{3\xi+\xi^{2}x-\alpha
x}{8\pi(1+\xi x)^{2}},$ (26) $\displaystyle P_{r}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}}{8\pi}\Big{[}\frac{3\xi+\xi^{2}x-\alpha
x}{(1+\xi
x)^{2}}\Big{]}+\frac{\mathcal{B}}{(8\pi)^{1+\frac{1}{\eta}}}\Big{[}\frac{3\xi+\xi^{2}x-\alpha
x}{(1+\xi x)^{2}}\Big{]}^{1+\frac{1}{\eta}},$ (27) $\displaystyle\sigma^{2}$
$\displaystyle=$ $\displaystyle\frac{\alpha\ (3+\xi x)^{2}}{(1+\xi x)^{5}},$
(28) $\displaystyle P_{t}$ $\displaystyle=$ $\displaystyle P_{r}+\Delta,$ (29)

whereas, the pressure anisotropy for our charged interior compact sphere model
can be described as

$\displaystyle 8\pi\Delta$ $\displaystyle=$ $\displaystyle\frac{4x}{1+\xi
x}\Big{(}\frac{\dot{y}}{y}\Big{)}^{2}-\frac{2\xi x}{(1+\xi
x)^{2}}\Big{(}\frac{\dot{y}}{y}\Big{)}+\frac{4x}{1+\xi
x}\Big{[}\frac{d}{dx}\Big{(}\frac{\dot{y}}{y}\Big{)}\Big{]}-\frac{2\alpha
x}{(1+\xi x)^{2}}+\frac{\xi^{2}x}{(1+\xi x)^{2}},$ (30)

we have the following principal equation which helps to generate various
charged models of anisotropic compact relativistic object. It is crucial to
integrate Eq.(31), because of its non-linearity that could be reformulated for
different values of polytropic index $\eta$.

$\displaystyle\frac{\dot{y}}{y}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}}{4}\left[\frac{3\xi+\xi^{2}x-\alpha x}{1+\xi
x}\right]+\frac{\mathcal{B}(1+\xi
x)}{4(8\pi)^{\frac{1}{\eta}}}\left[\frac{3\xi+\xi^{2}x-\alpha x}{(1+\xi
x)^{2}}\right]^{1+\frac{1}{\eta}}-\frac{\alpha x}{4(1+\xi x)}+\frac{\xi}{4},$
(31)

and

$\displaystyle\frac{d}{dx}\left(\frac{\dot{y}}{y}\right)$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{A}(2\xi^{2}+\alpha)}{4(1+\xi
x)^{2}}-\left[\frac{3\xi+\xi^{2}x-\alpha x}{(1+\xi
x)^{2}}\right]^{\frac{1}{\eta}}\left[\frac{\xi^{3}x-\alpha\xi
x+\alpha(1+\eta)+\xi^{2}(5+2\eta)}{\eta(1+\xi
x)^{2}}\right]\frac{\mathcal{B}}{4(8\pi)^{\frac{1}{\eta}}}$ (32)
$\displaystyle-$ $\displaystyle\frac{\alpha}{4(1+\xi x)^{2}}.$

For our compact sphere model, the mass function is given by

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{[}\frac{\sqrt{x}\ (2\xi^{3}x-3\alpha-2\xi
x\alpha)}{\xi^{2}(1+\xi x)}+\frac{3\ \alpha\ arctan\sqrt{\xi
x}}{\xi^{\frac{5}{2}}}\Big{]}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(33)

The compactness of the star can be expressed by

$\displaystyle u=\frac{M}{\sqrt{x}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{[}\frac{(2\xi^{3}x-3\alpha-2\xi
x\alpha)}{\xi^{2}(1+\xi x)}+\frac{3\ \alpha\ arctan\sqrt{\xi
x}}{\xi^{\frac{5}{2}}\sqrt{x}}\Big{]}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(34)

Here, we compare the novelty of our class of exact analytical solutions of the
Finch-Skea model with previous variety of works correspond to the four
different polytropic index, i.e., $\eta=1,2,\frac{2}{3},\frac{1}{2}$. In the
literature, there have been numerous fruitful investigations available with
Finch-Skea geometries in the context of $GR$. Very recently, Malaver and Iyer
[105] have proposed new charged Finch-Skea model for configuration of
astrophysical $COs$ by admitting the linear dark energy $EoS$. Dey and Paul
[106] have studied various notable features of charged anisotropic
relativistic solutions for higher dimensional Finch-Skea spacetime by
considering the different viable compact star candidates. The impacts of
electromagnetic field on the spherically symmetric Finch-Skea star model in
the background of two specific choices of polytropic index $\eta=1$ and $2$
have determined by Ratanpal [107]. The consequences of static and spherically
symmetric anisotropic charged Bardeen spheres have reviewed by indulging the
Finch-Skea ansatz obeying the Karmarkar condition [108]. In some recent
insights [75, 109]-[111], several class of exact solutions of anisotropic
uncharged Finch-Skea models have been put forward into the account of
different mathematical approaches by proposing the observational data of
various well-known $COs$. In comparison to these concrete evidences, our
results are quite novel and specifically more generic in the background of
generalized polytropic $EoS$ rather than the previous finding [107].

##  4 Polytropic models

The polytropic $EoS$ has an adequate implications in the prediction of
astrophysical relativistic compact stars. For instance, Azam and Mardan [112]
and Mardan and Azam [113] have studied the charged anisotropic polytropes
within the framework of generalized polytropic $EoS$. Kippenhahn et al. [114]
have investigated the polytropes with the infinite radius for $\eta=5$. The
different properties of uncharged anisotropic stellar relativistic $COs$ with
polytropic $EoS$ and $MIT$ $Bag$ model have been proposed in [63]-[64]. Here,
we can generate new exact solutions for charged anisotropic compact star,
which are physically admissible corresponding to the different values of
polytropic index $\eta=\frac{1}{2},\frac{2}{3},1,2$.

###  4.1 Model-1: $\eta=1$

For $\eta=1$, the generalized polytropic $EoS$ converted into quadratic $EoS$,
i.e.,

$P_{r}=\mathcal{A}\rho+\mathcal{B}\rho^{2}.$ (35)

Integrating Eq.(31) and using above expression, we have

$\displaystyle\ln y$ $\displaystyle=$
$\displaystyle\left[\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}}+\frac{\mathcal{B}(\xi^{2}-\alpha)^{2}}{32\pi\xi^{3}}\right]\ln(1+\xi
x)-\frac{\mathcal{B}(2\xi^{2}+\alpha)}{64\pi\xi^{3}}\left[\frac{6\xi^{2}+4\xi^{3}x-3\alpha-4\xi
x\alpha}{(1+\xi x)^{2}}\right]$ (36) $\displaystyle+$
$\displaystyle\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}+\mathcal{C},$

where $\mathcal{C}$ is integration constant. In more simplest form this
solution can be written as

$y=\mathcal{C}\ (1+\xi x)^{i}\ \exp[L(x)].$ (37)

The form of function $L(x)$ and constant $i$ is given as

$\displaystyle L(x)$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}-\frac{\mathcal{B}(2\xi^{2}+\alpha)}{64\pi\xi^{3}}\Big{[}\frac{6\xi^{2}+4\xi^{3}x-3\alpha-4\xi
x\alpha}{(1+\xi x)^{2}}\Big{]},$ $\displaystyle i$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}}+\frac{\mathcal{B}(\xi^{2}-\alpha)^{2}}{32\pi\xi^{3}}.$
(38)

By assuming $\mathcal{D}$ as a square of integration constant, the line-
element gets the form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{2i}\ \exp[2\ L(r^{2})]\ dt^{2}+(1+\xi r^{2})\ dr^{2}+r^{2}\
d\Omega^{2}.$ (39)

Further, for $\alpha=0$, we have the following new uncharged anisotropic
compact sphere model

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{\mathcal{A}+\frac{\mathcal{B}\xi}{16\pi}}\ \exp\left[\frac{\xi
r^{2}}{2}(\mathcal{A}+1)-\frac{\mathcal{B}\xi}{16\pi}\left(\frac{6+4\xi
r^{2}}{1+\xi r^{2}}\right)\right]dt^{2}+(1+\xi r^{2})\ dr^{2}$ (40)
$\displaystyle+$ $\displaystyle r^{2}d\Omega^{2}.$

For $\eta=1$, the matter variable $P_{r}$ and $P_{t}$ as well as pressure
anisotropy $\Delta$ are obtained by using the value of $y$ in Eqs.(11)-(14),
the new values of matter variables are given in Appendix Eqs.(61)-(63),
respectively.

###  4.2 Model-2: $\eta=2$

For $\eta=2$, the generalized polytropic $EoS$ is written as

$P_{r}=\mathcal{A}\rho+\mathcal{B}\rho^{\frac{3}{2}}.$ (41)

From the solution of Eq.(31) along with the above choice, we have

$\displaystyle\ln y$ $\displaystyle=$
$\displaystyle\left[\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}}\right]\ln(1+\xi
x)+\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}$ (42) $\displaystyle+$
$\displaystyle\frac{3\mathcal{B}(\xi^{2}-\alpha)\sqrt{2\xi^{2}+\alpha}}{16\sqrt{2\pi}a^{\frac{5}{2}}}\ln\left[\frac{\sqrt{2\xi^{2}+\alpha}-\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)x}}{\sqrt{2\xi^{2}+\alpha}+\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)x}}\right]$
$\displaystyle-$
$\displaystyle\frac{\mathcal{B}}{8\sqrt{2\pi}}\left[\frac{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
[2\xi(\alpha-\xi^{2})x+3\alpha]}{\xi^{2}(1+\xi x)}\right]+\mathcal{C},$

where $\mathcal{C}$ indicates an integration constant. Using some algebraic
technique, we notice that

$\displaystyle y$ $\displaystyle=$ $\displaystyle\mathcal{C}\ (1+\xi
x)^{j}\left[\frac{\sqrt{2\xi^{2}+\alpha}-\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)x}}{\sqrt{2\xi^{2}+\alpha}+\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\beta)x}}\right]^{k}\exp[\mathcal{M}(x)].$
(43)

The expression of function $\mathcal{M}(x)$ and constants $j$, $k$ are
represented as

$\displaystyle\mathcal{M}(x)$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}-\frac{\mathcal{B}}{8\sqrt{2\pi}}\left[\frac{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
[2\xi(\alpha-\xi^{2})x+3\alpha]}{\xi^{2}(1+\xi x)}\right],$ $\displaystyle j$
$\displaystyle=$
$\displaystyle\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}},$
$\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{3\mathcal{B}(\xi^{2}-\alpha)\sqrt{(2\xi^{2}+\alpha)}}{16\sqrt{2\pi}\xi^{\frac{5}{2}}}.$
(44)

By taking $\mathcal{C}^{2}=\mathcal{D}$, the line-element becomes

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{2j}\times\left[\frac{\sqrt{2\xi^{2}+\alpha}-\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)r^{2}}}{\sqrt{2\xi^{2}+\alpha}+\sqrt{a}\sqrt{3\xi+(\xi^{2}-\alpha)r^{2}}}\right]^{2k}$
(45) $\displaystyle\times$ $\displaystyle\exp[2\mathcal{M}(r^{2})]\
dt^{2}+(1+\xi r^{2})\ dr^{2}+r^{2}d\Omega^{2}.$

Here, we can obtain new uncharged anisotropic compact star model for the
choice of $\eta=2$

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{\mathcal{A}}\left[\frac{\sqrt{2}-\sqrt{3+\xi
r^{2}}}{\sqrt{2}+\sqrt{3+\xi
r^{2}}}\right]^{\frac{3\mathcal{B}\sqrt{\xi}}{8\sqrt{\pi}}}\exp\left[\frac{\xi
r^{2}}{2}(\mathcal{A}+1)\right.$ (46)
$\displaystyle\left.+\frac{\mathcal{B}\xi^{\frac{3}{2}}\sqrt{2}\
r^{2}}{4\sqrt{\pi}}\ \left(\frac{\sqrt{3+\xi r^{2}}}{1+\xi^{2}}\right)\right]\
dt^{2}+(1+\xi r^{2})\ dr^{2}+r^{2}d\Omega^{2}.$

For $\eta=2$, the radial and tangential pressures as well as anisotropic
factor $\Delta$ can be determined by inserting the outcome of $y$ into Eqs.
(11)-(14), which have distinctively given in Appendix Eqs.(64)-(66),
respectively.

###  4.3 Model-3: for $\eta=2/3$

For the value of $\eta=2/3$, the generalized polytropic $EoS$ becomes

$P_{r}=\mathcal{A}\rho+\mathcal{B}\rho^{\frac{5}{2}}.$ (47)

The solution of Eq.(31) for the above consideration takes the form

$\displaystyle\ln y$ $\displaystyle=$
$\displaystyle\left[\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}}\right]\ln(1+\xi
x)+\frac{5\ \mathcal{B}\ (\xi^{2}-\alpha)^{3}}{1024\ \sqrt{2}\
\pi^{\frac{3}{2}}\ a^{\frac{7}{2}}\ \sqrt{2\xi^{2}+\alpha}}$ (48)
$\displaystyle\times$
$\displaystyle\ln\left[\frac{\sqrt{2\xi^{2}+\alpha}-\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)x}}{\sqrt{2\xi^{2}+\alpha}+\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)x}}\right]-\frac{\mathcal{B}\
\sqrt{3\xi+(\xi^{2}-\alpha)x}}{1536\ \sqrt{2}\ \pi^{\frac{3}{2}}\ \xi^{3}}$
$\displaystyle\times$ $\displaystyle\left[\frac{33(\xi^{2}-\alpha)^{2}}{(1+\xi
x)}+\frac{26(\xi^{2}-\alpha)(2\xi^{2}+\alpha)}{(1+\xi
x)^{2}}+\frac{8(2\xi^{2}+\alpha)^{2}}{(1+\xi x)^{3}}\right]$ $\displaystyle+$
$\displaystyle\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}+\mathcal{C}.$

Here, $\mathcal{C}$ is the constant of integration. The Eq.(48) can be
rewritten as

$y=\mathcal{D}\ (1+\xi
x)^{l}\left[\frac{\sqrt{2\xi^{2}+\alpha}-\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)x}}{\sqrt{2\xi^{2}+\alpha}+\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)x}}\right]^{m}\
\exp[N(x)],$ (49)

where the function $N(x)$ and constants $l$ and $m$ are

$\displaystyle N(x)$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}-\frac{\mathcal{B}\
\sqrt{3\xi+(\xi^{2}-\alpha)x}}{1536\ \sqrt{2}\ \pi^{\frac{3}{2}}\ \xi^{3}}$
$\displaystyle\times$
$\displaystyle\Big{[}\frac{33(\xi^{2}-\alpha)^{2}}{(1+\xi
x)}+\frac{26(\xi^{2}-\alpha)(2\xi^{2}+\alpha)}{(1+\xi
x)^{2}}+\frac{8(2\xi^{2}+\alpha)^{2}}{(1+\xi x)^{3}}\Big{]},$ $\displaystyle
l$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}},$
$\displaystyle m$ $\displaystyle=$ $\displaystyle\frac{5\ \mathcal{B}\
(\xi^{2}-\alpha)^{3}}{1024\ \sqrt{2}\ \pi^{\frac{3}{2}}\ a^{\frac{7}{2}}\
\sqrt{2\xi^{2}+\alpha}}.$ (50)

Assuming $\mathcal{C}^{2}=\mathcal{D}$, we can obtain the following line-
element

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{2l}\times\left[\frac{\sqrt{2\xi^{2}+\alpha}-\sqrt{a}\sqrt{3\xi+(\xi^{2}-\alpha)r^{2}}}{\sqrt{2\xi^{2}+\alpha}+\sqrt{\xi}\sqrt{3\xi+(\xi^{2}-\alpha)r^{2}}}\right]^{2t}\exp[2N(r^{2})]\
dt^{2}$ (51) $\displaystyle+(1+\xi r^{2})\ dr^{2}+r^{2}\ d\Omega^{2}.$

For $\eta=\frac{2}{3}$, the uncharged polytropic model is given by the
following line-element

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{\mathcal{A}}\times\left[\frac{\sqrt{2}-\sqrt{3+\xi
r^{2}}}{\sqrt{2}+\sqrt{3+\xi r^{2}}}\right]^{\frac{5\ \mathcal{B}\
\xi^{\frac{3}{2}}}{2^{11}\ \pi^{\frac{3}{2}}}}\exp\left[\frac{\xi
r^{2}}{2}(\mathcal{A}+1)\right.\left.+\frac{\mathcal{B}\xi^{\frac{3}{2}}\sqrt{(3+\xi
r^{2})}}{48(8\pi)^{\frac{3}{2}}(1+\xi r^{2})^{3}}\right.$ (52)
$\displaystyle\left.\times\left[117+118\xi r^{2}+33\xi^{2}r^{4}\right]\right]\
dt^{2}\ (1+\xi r^{2})\ dr^{2}+r^{2}\ (d\theta^{2}+\sin^{2}\theta\ d\phi^{2}).$

For $\eta=2/3$, the radial and tangential pressures and anisotropic pressure
$\Delta$ can be found by putting the value of $y$ into Eqs. (11)-(14), which
are given in Appendix Eqs.(67)-(69), respectively.

###  4.4 Model 4: for $\eta=1/2$

For $\eta=\frac{1}{2}$, the generalized polytropic $EoS$ can be obtained as

$P_{r}=\mathcal{A}\rho+\mathcal{B}\rho^{3}.$ (53)

We integrate Eq.(31) corresponding the value $\eta=\frac{1}{2}$, and get

$\displaystyle\ln y$ $\displaystyle=$
$\displaystyle\left[\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}}\right]\ln(1+\xi
x)-\frac{\mathcal{B}\xi^{2}}{256\pi^{2}}[2\xi(\alpha-\xi^{2})x+\alpha-4\xi^{2}]$
(54) $\displaystyle\times$
$\displaystyle\left[\frac{2\xi^{2}(\alpha-\xi^{2})^{2}x^{2}+2\xi(\alpha-4\xi^{2})(\alpha-\xi^{2})x\alpha^{2}-2\xi^{2}\alpha+10\xi^{4}}{4\xi^{4}(1+\xi
x)^{4}}\right]$ $\displaystyle+$
$\displaystyle\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}+\mathcal{C},$

where $\mathcal{C}$ is the constant of integration. We can rewrite the above
form in the following manner

$y=\mathcal{C}\ (1+\xi x)^{n}\ \exp[O(x)].$ (55)

The formulation of variable $O(x)$ and constant $n$ is described as

$\displaystyle O(x)$ $\displaystyle=$
$\displaystyle\frac{(\mathcal{A}+1)(\xi^{2}-\alpha)x}{4\xi}-\frac{\mathcal{B}\xi^{2}}{256\pi^{2}}[2\xi(\alpha-\xi^{2})x+\alpha-4\xi^{2}]$
$\displaystyle\times$
$\displaystyle\left[\frac{2\xi^{2}(\alpha-\xi^{2})^{2}x^{2}+2\xi(\alpha-4\xi^{2})(\alpha-\xi^{2})x\alpha^{2}-2\xi^{2}\alpha+10\xi^{4}}{4\xi^{4}(1+\xi
x)^{4}}\right]+\mathcal{C},$ $\displaystyle n$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}(2\xi^{2}+\alpha)+\alpha}{4\xi^{2}}.$ (56)

We assume $\mathcal{C}^{2}=\mathcal{D}$, then metric has the following form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{2n}\exp[2\ O(r^{2})]\ dt^{2}+(1+\xi r^{2})\ dr^{2}+r^{2}\
d\Omega^{2}.$ (57)

Further, we can get the following line-element for an uncharged polytropic
model by considering $\eta=\frac{1}{2}$,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\mathcal{D}\ (1+\xi
r^{2})^{\mathcal{A}}\ \exp\left[\frac{\xi
r^{2}}{2}(\mathcal{A}+1)+\frac{\mathcal{B}\xi^{2}}{128\pi^{2}}\right.$ (58)
$\displaystyle\left.\times\left(\frac{\xi^{3}r^{6}+6\xi^{2}r^{4}+13\xi
r^{2}+10}{(1+\xi r^{2})^{4}}\right)\right]\ dt^{2}+(1+\xi r^{2})\ dr^{2}+\
r^{2}\ d\Omega^{2}.$

For $\eta=1/2$, the radial and tangential pressure as well as anisotropic
factor $\Delta$ can be explored by putting the result of $y$ into Eqs.
(11)-(14), which are given in Appendix Eqs.(70)-(72), respectively. More
specifically, the results of charged anisotropic compact star models inspired
by a generalized polytropic $EoS$ can be retrieved into linear, quadratic and
polytropic $EoS$. When we set $\mathcal{B}=0$, $\eta=1$ and $\mathcal{A}=0$
the generalized polytropic $EoS$ recovered as linear, quadratic and polytropic
$EoS$, respectively. In the gravitational and astrophysical perspectives,
these $EoSs$ have observed variety of implications; for instance, the linear
$EoS$ has described the dust fluid ($\mathcal{A}=0$), radiation matter
($\mathcal{A}=\frac{1}{3}$) [68], and it has elaborated the quark made star
configuration [56, 57]. The quadratic $EoS$ has proposed the various stellar
astrophysical compact models like charged relativistic strange star and quark
star [59, 60]. Moreover, the polytropic $EoS$ has inspected the numerous
relativistic astrophysical $COs$ like white dwarfs, brown dwarfs, neutron
stars, whereas it has explained the early and late time Universe with positive
and negative values of the polytropic index [67, 115].

##  5 Physical attributes of polytropic models

We have observed the physical stability constraints for an under developed
polytropes models via generalized polytropic $EoS$ in the spacetime geometry
of Finch and Skea. It is predicted that the proposed ansatz is physically
reliable as it is continuous everywhere and potentially stable inside the
sphere. To choice the gravitational potential as seed ansatz is ever being a
challenged among the theoretical physicists and cosmologists for the
validation of maximum essential physical requirements of the system. In favor
of the purposed ansatz, there have been made wide range of substantial
explorations to check the sustainability and regularity of the cosmic objects.
It has admitted a key role in number of aspects by explaining the stellar
relativistic astrophysical as well as cosmological consequences. For instance,
this form has contemplated as seed ansatz to intimate that Universe is in a
phase of accelerated expansion by imposing the dark energy $EoS$ [105]. In the
usual four and higher dimensions the solution of Finch-Skea has disclosed to
understand some physical implications at extreme conditions of charged compact
stellar models [106]. This geometry has validated various stability tests with
physical variables of the astrophysical $COs$ under by the influence of
polytropic $EoS$ [107, 111]. In particular, the Finch-Skea model as an
interior spacetime has manifested several viable features with an exterior
solution of Bardeen black hole which corresponds to the magnetic monopole of
gravitationally collapsing remnants [108, 116, 117]. Moreover, it has
successfully employed to endorse the observational constraints of various type
of astrophysical compact stars under by the domination of various physical
conditions [75, 109]-[110]. Apart of these inspirational outcomes, in most of
the recent works the finch and skea geometry has positively revealed the
stellar relativistic astrophysical implications with contribution of the Adler
and buchdahl spacetimes [31, 100, 108, 116]-[118]. These spacetimes have
procured the solutions as like to Finch and skea model by obeying the various
astrophysical bounds of the compact stars. Therefore, it fulfills the
following conditions:

$\displaystyle e^{2\lambda(0)}$ $\displaystyle=$ $\displaystyle\rm{constant},$
$\displaystyle e^{2\psi(0)}$ $\displaystyle=$ $\displaystyle 1,$

Figure 1: Evolutionary behavior of matter density as a function of radial
coordinate $r$; for $\xi=3$ and $\alpha=1.15$.

Figure 2: Physical evolution of radial and tangential pressures vs $r$; for
different choices of $\eta=1,2,\frac{1}{2},\frac{2}{3}$,
$\xi=3,\mathcal{A}=0.01,\mathcal{B}=0.7$ and $\alpha=1.15$.

and $(e^{2\psi(r)})^{\prime}=(e^{2\lambda(r)})^{\prime}=0$ when $r=0$. In this
analysis, we study the graphical impact of various polytropic models under
numerous physical properties like, matter density, pressure ($P_{r}$ and
$P_{t}$), pressure anisotropy, mass function, electric field, speed of sound,
compactness, electric charge and adiabatic index. We observe the physical
viability of matter density for the evolution of compact sphere model in Fig.
1. It can be seen from Fig. 1, that the behavior of matter density is non-
negative as well as non-singular throughout the whole interior region of the
star. It is also noticed here that the matter density is maximum near the
center as compared to the surface of the star. This characteristic suggests
that the presented model predict an ultra compact relativistic object. In Fig.
2, we depict the evolutionary nature of radial and tangential pressure for the
modeling of stellar interior compact stars. One can transparently see from
Fig. 2, that the radial and tangential pressures possess positive evolution as
they remain regular (finite) and non-singular at each point inside the compact
stellar body. It is also evident here that both material quantities are
increasing from center and are decreasing gently towards the boundary surface.

In formation of static and spherically symmetric anisotropic $COs$, we
described here some viable consequences of Einstein equations of motion with
their standard physical features. These consequences can be obtained in
absence of pressure, isotropic source and anisotropic fluid. Nevertheless,
there are many vigorous probes that steep excessive compact relativistic stars
are not formed of isotropic sources. In certain bounds, the structures with
unpredictable physical implications are formed, for the illustration of
pressure anisotropy. In Particular, we briefly review the anisotropic matter
field via energy-momentum tensor that characterizing the physical composition
of stellar interior $CO$. Indeed, anisotropic matter field is an exceptional
exotic choice for modeling of stellar $COs$ like, white dwarfs, quark made
stars, neutron stars, pulsars, etc. Keeping in view, Jeans [119] was the first
theorist who reviewed the effects of anisotropic pressure on the self-
gravitating relativistic dense star model. Moreover, it has been revealed that
the pressure anisotropy may appear within an interior configuration of the
compact stars due to various properties like, superluminal fluid [19, 20,
114], phase periods [119] and electromagnetic field [120]. Based on these
vital facts, we examine the profile of pressure anisotropy for our proposed
polytropic models in Fig. 3. It can be easily seen from Fig. 3, that the
evolution of pressure anisotropy is positive as well as regular (finite)
within the distribution of the star but at some limiting points it is slightly
deviated from the standard position because it might be due to exotic nature
of the fluid. The pressure anisotropy sharply proceeds from center towards the
boundary surface of the sphere where $r=1$. In addition, this trend indicates
that the pressure anisotropy have repulsive nature, i.e., $P_{r}<P_{t}$.
Actually, this characteristic identifies that an interior core of the star is
made up with high density, which is due to increase of pressure anisotropy. As
a result, we can reflect that our proposed models of spherically dense compact
stars are physically admissible as well as potentially stable.

Figure 3: Variational nature of pressure anisotropy versus $r$ for various
values of $\eta=1,2,\frac{1}{2},\frac{2}{3}$,
$\xi=3,\mathcal{A}=0.01,\mathcal{B}=0.7$ and $\alpha=1.15$.

Figure 4: Physical variation of effective mass function (left plot) and
electromagnetic field (right plot) w.r.t. radial distance $r$.

In Fig. 4, we analyze the physical evolution of effective mass function as
well as electromagnetic field for our compact star model. The behavior of
effective mass within an entire region of the star is non-negative as it is
continuous (regular) at every point. Also, it is rapidly progressive from
center towards the boundary of the sphere and attains maximum value at $r\sim
1.5$. From these features, we can predict that our proposed star model is able
to develop an ultra dense relativistic star as predicted in Ref. [88].
Moreover, one can notice from right plot (Fig. 4), that the electromagnetic
field is positive at every point inside the sphere as it is continuous
(regular) in everywhere. Also, it possess maximum evolutionary nature near the
core of the star and steadily decreases towards the boundary surface with the
increase of radial coordinate $r$.

Figure 5: Physical variation of sound speed $\frac{dP_{r}}{d\rho}$ (left plot)
and $\frac{dP_{t}}{d\rho}$ (right plot) versus $r$; for certain values of
$\eta=1,2,\frac{1}{2},\frac{2}{3}$, $\xi=3,\mathcal{A}=0.01,\mathcal{B}=0.7$
and $\alpha=1.15$.

We study the evolutionary nature of sound speed ($\frac{dP_{r}}{d\rho}$ and
$\frac{dP_{t}}{d\rho}$) for our polytropic models of compact stars analogous
to various choices of $\eta$ in Fig. 5. It is evident to observe from Fig. 5,
that the trend of all curves of sound speed (left plot) with respect to their
polytropic models suggest positive nature at any interior point of the star.
Moreover, it would be very interesting to mention here that for our proposed
polytropic models sound speed (left plot) remains within the certain limit and
does not cross the sufficient value, i.e.,
$0<\frac{d\mathrm{P_{r}}}{d\rho}\leq 1$. On the other hand, the behavior of
tangential speed of sound remains positive throughout the whole configuration
of the star for all polytropic models and it does not cross the sufficient
bound, i.e., $0<\frac{d\mathrm{P_{t}}}{d\rho}\leq 1$ (see Fig. 5 (right
plot)). Moreover, the nature of tangential sound speed regular (finite) at
every interior point and it is gradually decreases from center outward towards
the boundary surface of the star. These all features of sound speed once again
strongly recommend that our compact star models are stable as well as
compatible with the prior result of causality condition depicted in [6].

Figure 6: Physical variation of compactness (left plot) and total electric charge (right plot) versus radial distance $r$. Table 1: Numerically estimated value of total electric charge and compactness parameter corresponds to the arbitrary radius and mass of the compact star $r_{b}$ | $\textit{M}(\textit{M}_{\odot})$ | M | $Q$ | $\frac{Q}{r_{b}}$ | $\frac{Q^{2}}{r^{2}_{b}}$ | $\frac{\textit{M}}{r_{b}}$ | $\frac{2\textit{M}}{r_{b}}$
---|---|---|---|---|---|---|---
$(Km)$ |  | $(Km)$ | (Coulomb C) |  |  | $<\frac{4}{9}$ | $<\frac{8}{9}$
8 | 2.40661 | 3.54976 | 3.7880$\times 10^{20}$ | 0.406133 | 0.164944 | 0.443720 | 0.887441

A quite realistic relation between the compactness and an electric charge has
ever remained pivotal regarding the non-violation of its stable configuration
in the modeling of $COs$. For this assessment, the pioneering study of
compactness factor for static and spherically symmetric perfect fluid sphere
was proposed by Buchdahl [121]. According to Buchdahl, the maximum allowable
ratio of mass-radius for a $CO$ can be less than $\frac{4}{9}$ i.e.,
($\frac{2\textit{M}}{r_{b}}<\frac{8}{9}$). Mak et al. [122] and
Andr$\acute{e}$asson [123] have generalized such condition for an interior
charged compact star models. The term $\frac{\textit{M}}{r_{b}}$ is considered
as compactness factor which categorizes the astrophysical $COs$ as normal
star; $\frac{\textit{M}}{r_{b}}\sim 10^{-5}$, white dwarf;
$\frac{\textit{M}}{r_{b}}\sim 10^{-3}$, neutron star;
$0.1<\frac{\textit{M}}{r_{b}}<0.25$, ultra dense $CO$ ;
$0.25<\frac{\textit{M}}{r_{b}}<0.50$ and black hole;
$\frac{\textit{M}}{r_{b}}=0.50$ [124]. It would be very interesting to mention
here that the numerically estimated value of an electric charge and
compactness for our compact star model is shown in Table 1. One can easily
inspect from the given Table 1, that the numerically estimated ratio of charge
to the radius for the given compactness is in allowable limit and does not
violate the Buchdahl condition for charged $CO$. Despite of this allowable
limit, there is another upper limit that should necessarily be satisfied the
maximum allowed compactness and charge-radius ratio, i.e.,
($\frac{Q^{2}}{r^{2}_{b}}<\frac{2\textit{M}}{r_{b}}<\frac{8}{9}$). This whole
discussion vigorously confirmed that our compact star model fulfilled the
necessary Buchdahl condition for charged $CO$ as well as might be able to
predict a ultra dense $CO$. There is another interesting comparison presented
here with previous momentous works regarding the computed amount of total
electric charge in the coulomb unit given in Table 1. We observed that the
total charge on the surface in coulomb unit is 3.7880$\times 10^{20}$ [C] for
the proposed star whose arbitrary mass and radius are ${M}_{\odot}$=3.54976
[km] and $r_{b}=8$ [km]. As investigation of $f(R,T)$ gravity in the
background of two geometrical techniques analyzed by Maurya et al. [100], in
which they estimated the maximum amount of electric charge as 2.46693$\times
10^{20}$ [C] and 2.58885$\times 10^{20}$ [C] for two different charged compact
star models. In the realm of $GR$ the same author and his collaborators [98,
104] have shown the maximum amount of electric charge in coulomb unit for
anisotropic strange star candidates as 4.04378$\times 10^{19}$ [C] whose
radius is 8 [km], and 1.638631$\times 10^{20}$ [C] whose radius is 7.1 [km],
respectively, as well as they estimated 7.5317$\times 10^{19}$ [C] for PSR
J1664-2230 and 9.8951$\times 10^{19}$ [C] for PSR 1937 + 21. From this detail
analysis, our computed amount of electric charge on the surface in coulomb
unit is more better comparatively to these discussed results.

Moreover, the physical behavior of compactness and an electric charge for our
compact star model is shown in Fig. 6. It can be clearly observed from Fig. 6,
that the trend of compactness (left plot) and electric charge (right plot)
suggest positive evolution at every point of the stellar interior as they are
non-singular within the entire region of the star. The compactness and
electric charge become increasing function of radial coordinate $r$ and they
attain maximum value at the boundary sphere. Also, it is mentioned here that
the Buchdahl limit for charged $CO$ remains in good agreement and does not
violate the necessary condition. In spite of these facts, the mass-radius
relation yields $0.25<\frac{\textit{M}}{r_{b}}<0.50$, which correspond to
stable neutron star and ultra dense compact star.

Table 2: The numerical value of adiabatic index $\Gamma(0)$ along with critical value $\Gamma$(crit) for the proposed polytropic models are given as $\Gamma(0)$ | $\Gamma(0)$ | $\Gamma(0)$ | $\Gamma(0)$ |  $\Gamma$(crit)
---|---|---|---|---
(Model-1) | (Model-2) | (Model-3) | (Model-4) |
2.47297 | 2.12667 | 3.07882 | 2.79126 | 1.73479
Figure 7: Physical behavior of adiabatic index $\Gamma$ vs radial distance
$r$.

On the contrary of above discussed stability criterion, another most prominent
condition for ensuring the stability equilibrium of the model is adiabatic
index. The term adiabatic index is referred as the ratio of two specific heats
[125]. To check the consistency of the model, the relativistic adiabatic index
$\Gamma$ in the radial direction plays a prime role. Here, the radial
direction is vital, because, under the influence of anisotropy the system is
reformed from the spherically symmetric one in the radial direction only for
restraining gravitational collapse. In some former papers [126]-[127], the
detail analysis has been given in different mathematical mechanisms to compute
the relativistic adiabatic index along with critical value to ensure the
stability of the stellar compact models. The relativistic adiabatic index can
be expressed as

$\displaystyle\Gamma=\frac{\rho+P_{r}}{P_{r}}\frac{dP_{r}}{d\rho}.$ (59)

One can affirm that, for stability of the system becomes
$\Gamma\geq\Gamma_{crit}$. Whereas, the critical value of adiabatic
relativistic index can be evaluated by this expression

$\displaystyle\Gamma_{crit}=\frac{4}{3}+\frac{19u}{21}.$ (60)

In Fig. 7, we plot the physical behavior of relativistic adiabatic index
$\Gamma$ for different polytropic models. It can be eminently seen from Fig.
7, that all curves of the adiabatic relativistic index corresponding to
various polytropic models suggest positive evolution within the stellar object
as well as gradually decreases from center outward towards the boundary of the
sphere. Also, these polytropic models confirm the stability of our stellar
$CO$ by satisfying the relativistic condition as $\Gamma>\frac{4}{3}$ (see
Fig. 7). As a result, we can firmly say that the suggested models are
potentially well behaved and being able to model of $CO$. Moreover, we
computed the numerical values of $\Gamma$ along with critical value of
adiabatic relativistic index shown in Table 2. From these computed numerical
values, one can understand that the condition (60) is once again ensured the
stability equilibrium of our models.

##  6 Discussion

Inspirited from an under evolved polytropic models via generalized polytropic
$EoS$ in the Finch and Skea geometry, the formation of stellar relativistic
$CO$ might be quite conducive. It has been observed in Ref.[128] that a non-
linear $EoS$ has possessed the features of dark matter and dark energy which
is helpful to explain inflationary the phases of our Universe. In this work,
we have generated new exact analytical solutions for static and spherically
symmetric metric containing the properties of anisotropic matter. The obtained
solutions of generalized polytropic $EoS$ are constructed to model charged
anisotropic compact stars corresponding to Finch-Skea spacetime. We have
examined graphically that models are physically admissible as well as
potentially sustainable. It is found that the gravitational potential and
matter quantities are non-singular, as these are well-consistent inside the
celestial sphere. For $\eta=1$, we can simply concise our solution to the
particular case which recommended by Feroze and Siddique [59]. These solutions
may be fruitful in the modeling of relativistic $COs$ and consider particular
matter distribution due to such kind of generalized polytropic $EoS$. It might
be an interesting work to differentiate acquired solutions to manifest
astrophysical $COs$ such as $SAXJ1808.4$-$3658$ for charge and uncharge as
studied by Mafa Takisa an Maharaj [57] and Dey et al. [129]. It would be very
fine tune model to construct the charged relativistic astrophysical compact
sphere which is made up of high density. Also, we can revert our results into
new uncharged anisotropic solutions by vanishing electric charge. This work
reveals that Finch-Skea spacetime is physically suitable for the formation of
astronomical $COs$. Form this understanding, we provide the following thorough
authentication regarding our proposed solutions via the graphical analysis:

  * •

Our results of matter variables ($\rho$, $P_{r}$ and $P_{t}$) have confirmed
that they are regular and well behaved throughout the whole configuration of
stellar object as shown in Figs. 1 and 2. The non-negative and monotonically
decreasing nature of such variables can be evidently seen in the graphical
plots. Furthermore, we have noticed that near the central core of the star,
the nature of these variables is gradually increasing and then sharply
vanishing towards the boundary of the star where $r$ is maximum. These
implications ultimately suggest that our models is ultra dense relativistic
compact stars.

  * •

From Fig. 3, we have clearly observed that the anisotropy of pressure
possessed evolutionary nature and continuous at every point inside the stellar
body, also it is observed that the anisotropy of pressure have repulsive
trend, i.e., $P_{r}<P_{t}$. This behavior indicates that our models are well-
consistent and can predict an ultra dense $CO$.

  * •

The graphical behavior as observed in Fig. 4, shows that mass function (left
plot) is rapidly increasing from center to the boundary of the star, which
further endorse that the present model is viable and potentially stable. We
have also studied the physical evolution of electromagnetic field for our
compact star model in Fig. 4 (right plot). The electromagnetic field have non-
negative evolution at every point inside the sphere as it is regular
everywhere. It is a monotonically decreasing function of radial coordinate $r$
and vanishes at the boundary of the star.

  * •

Despite of these concrete evidences, we have investigated the physical impact
of causality condition ($\frac{d\mathrm{P_{r}}}{d\rho}$ and
$\frac{d\mathrm{P_{t}}}{d\rho}$) on our compact star as shown in Fig. 5. The
evolution of each curve of causality condition
($\frac{d\mathrm{P_{r}}}{d\rho}$) remains in the appropriate range and satisfy
$0<\frac{d\mathrm{P_{r}}}{d\rho}\leq 1$. Also, the transverse sound speed
behaves regular nature at each point inside the star as well as non-negative
throughout the entire region. Beside of this characteristics, the all trends
of the transverse sound of speed with respect to the various choices of $\eta$
show good agrement and doest not breach the sufficient limit as
$0<\frac{d\mathrm{P_{t}}}{d\rho}\leq 1$. The causality condition once again
strongly agreed that the presented models of compact star are stable.

  * •

The variation of compactness and electric charge are plotted against $r$ in
Fig. 6. One can transparently see from Fig. 6, that the nature of both plots
(left and right) is continuous at all points in the interior of the star. The
figure also indicates that the compactness and electric charge are
monotonically increasing function of $r$ and they attained maximum amount at
$r\sim 1.5$. Also, the Buchdahl condition for charged $CO$ remained in an
allowable range and satisfy the necessary bound.

  * •

We have depicted the physical behavior of adiabatic relativistic index
$\Gamma$ for the proposed polytropic models in Fig. 7. The all curves of
$\Gamma$ corresponding to various choices of $\eta$ propose good stable
agreement at every point within the compact sphere. It is also confirmed from
Table 2, the value of adiabatic relativistic index for all models remain
greater than critical value of adiabatic index throughout the celestial system
and it represents a stable equilibrium position to compose a viable $CO$.

Conclusively, the obtained features of the generalized polytropic $EoS$ along
with Finch-Skea ansatz are quite conducive and potentially stable in
hydrostatic equilibrium for describing the gravitational aspects of the
astrophysical $COs$. There have several physical conditions satisfied via
different criterions, viz. mass function, electric field, causality bounds,
Buchdahl condition with charge to radius ratio and adiabatic index along with
critical value. Particulary, our estimated value of compactness nearly close
to Buchdahl limit, and it has evidently justified with upper bound of charge-
radius ratio for charged $CO$ (see Table 1). It is another intriguing aspect
to note here that our observed amount of total electric charge at the boundary
in the coulomb unit as 3.7880$\times 10^{20}$ [C] correspond to the arbitrary
mass ${M}_{\odot}$=3.54976 [km] and radius $r_{b}=8$ [km], which is more
better than in comparison of the previous work proposed in [98, 100, 104]. We
have once again revitalized our results to ensure the stability of our models
by inspecting the relativistic adiabatic index along with critical value. Both
stability conditions are in good agreement and please the sufficient
relativistic physical bounds (see Table 2). These attributes have vigorously
confirmed the maximum gravitational stability constraints for composing the
viable ultra dense compact star model and neutron star. Such consequences in
presence of electric charge are exclusively novel and more generic in the
framework of generalized polytropic $EoS$ alternately to preceding works [107,
111], which were deduced only for the limited frameworks.

## Appendix

The outcomes of pressure ($P_{r}$ and $P_{t}$) and anisotropy of pressure
($\Delta$) analogous to different choices of polytropic index like,
$\eta=1,2,\frac{2}{3},\frac{1}{2}$ are given below

  * •

For $\eta=1$

$P_{r}=\frac{\mathcal{A}}{8\pi}\Big{(}\frac{3\xi+\xi^{2}x-\alpha x}{(1+\xi
x)^{2}}\Big{)}+\frac{\xi}{8\pi}\Big{(}\frac{3\xi+\xi^{2}x-\alpha x}{(1+\xi
x)^{2}}\Big{)}^{2},$ (61) $\displaystyle P_{t}$ $\displaystyle=$
$\displaystyle\mathcal{A}\ \Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\Big{]}+\xi\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\Big{]}^{2}+\frac{x}{2\pi(1+\xi x)}\Big{[}\frac{i(i-1)\xi^{2}}{(1+\xi
x)^{2}}+\frac{2\ i\ \xi\ \dot{L}(x)}{1+\xi x}$ (62) $\displaystyle+$
$\displaystyle\dot{L}(x)^{2}+\ddot{L}(x)\Big{]}-\frac{\xi x}{4\pi(1+\xi
x)^{2}}\Big{[}\frac{i\ \xi}{1+\xi
x}+\dot{L}(x)\Big{]}+\frac{\xi^{2}x}{8\pi(1+\xi x)^{2}}-\frac{\alpha
x}{4\pi(1+\xi x)},$ $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{x}{2\pi(1+\xi x)}\Big{[}\frac{i(i-1)\xi^{2}}{(1+\xi
x)^{2}}+\frac{2\ i\ \xi\ \dot{L}(x)}{1+\xi
x}+\dot{L}(x)^{2}+\ddot{L}(x)\Big{]}-\frac{\xi x}{4\pi(1+\xi
x)^{2}}\Big{[}\frac{i\ \xi}{1+\xi x}$ (63) $\displaystyle+$
$\displaystyle\dot{L}(x)\Big{]}+\frac{\xi^{2}x}{8\pi(1+\xi
x)^{2}}-\frac{\alpha x}{4\pi(1+\xi x)}.$

Here, the derivative w.r.t variable $x$ is represented with dot sign, whereas
the function $\dot{L}(x)$ and $\ddot{L}(x)$ are illustrated as

$\displaystyle\dot{L}(x)$ $\displaystyle=$
$\displaystyle\frac{(\xi^{2}-\alpha)}{4a}\ (\mathcal{A}+1)+\frac{\mathcal{B}\
(2\xi^{2}+\alpha)}{32\pi\xi^{2}}\left[\frac{4\xi^{2}+2\xi^{3}x-2\xi
x\alpha-\alpha}{(1+\xi x)^{3}}\right],$ $\displaystyle\ddot{L}(x)$
$\displaystyle=$ $\displaystyle-\frac{\mathcal{B}\
(2\xi^{2}+\alpha)}{32\pi\xi}\left[\frac{10\xi^{2}+4\xi^{3}x-4\xi
x\alpha-\alpha}{(1+\xi x)^{4}}\right].$

  * •

For $\eta=2$

$\displaystyle P_{r}$ $\displaystyle=$
$\displaystyle\mathcal{A}\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\Big{]}+\mathcal{B}\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\Big{]}^{\frac{3}{2}},$ (64) $\displaystyle P_{t}$ $\displaystyle=$
$\displaystyle\mathcal{A}\ \Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\Big{]}+\mathcal{B}\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\Big{]}^{\frac{3}{2}}+\frac{x}{2\pi(1+\xi
x)}\Big{[}\frac{d}{dx}\Big{[}\frac{\xi j}{1+\xi x}$ (65) $\displaystyle+$
$\displaystyle\frac{k\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{M}(x)\Big{]}+\frac{\dot{y}^{2}}{y^{2}}\Big{]}-\frac{\xi
x}{4\pi(1+\xi x)^{2}}\Big{[}\frac{\xi j}{1+\xi x}$ $\displaystyle+$
$\displaystyle\frac{k\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{M}(x)\Big{]}+\frac{\xi^{2}x}{8\pi(1+\xi x)^{2}}-\frac{\alpha
x}{4\pi(1+\xi x)},$ $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{x}{2\pi(1+\xi x)}\Big{[}\frac{d}{dx}\Big{[}\frac{\xi
j}{1+\xi x}+\frac{k\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{M}(x)\Big{]}+\frac{\dot{y}^{2}}{y^{2}}\Big{]}$ (66)
$\displaystyle-$ $\displaystyle\frac{\xi x}{4\pi(1+\xi x)^{2}}\Big{[}\frac{\xi
j}{1+\xi x}+\frac{k\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{M}(x)\Big{]}$ $\displaystyle+$
$\displaystyle\frac{\xi^{2}x}{8\pi(1+\xi x)^{2}}-\frac{\alpha x}{4\pi(1+\xi
x)}.$

The function $\dot{M}(x)$ and $\frac{\dot{y}}{y}$ are defined as

$\displaystyle\frac{\dot{y}}{y}$ $\displaystyle=$ $\displaystyle\frac{\xi
j}{1+\xi x}+\frac{k\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{M}(x),$ $\displaystyle\dot{M}(x)$ $\displaystyle=$
$\displaystyle\frac{(\xi^{2}-\alpha)}{4\xi}(\mathcal{A}+1)+\frac{\mathcal{B}\
[(\xi^{2}-\alpha)x+3\xi]^{-\frac{1}{2}}}{16\ \xi^{2}\ \sqrt{2\pi}\ (1+\xi
x)^{2}}\Big{[}\xi^{3}x(6\xi^{2}+2\xi^{3}x-9\alpha)+3\alpha^{2}(1$
$\displaystyle+$ $\displaystyle 3\xi
x)-4\xi^{4}(-3+x^{2}\alpha)+\xi^{2}\alpha(3+2x^{2}\alpha)\Big{]}.$

  * •

For $\eta=2/3$

$\displaystyle P_{r}$ $\displaystyle=$ $\displaystyle\mathcal{A}\
\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi x)^{2}}\Big{]}+\mathcal{B}\
\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi x)^{2}}\Big{]}^{\frac{5}{2}},$
(67) $\displaystyle P_{t}$ $\displaystyle=$ $\displaystyle\mathcal{A}\
\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi x)^{2}}\Big{]}+\mathcal{B}\
\Big{[}\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\Big{]}^{\frac{5}{2}}+\frac{x}{2\pi(1+\xi
x)}\Big{[}\frac{d}{dx}\Big{(}\frac{\xi l}{1+\xi x}$ (68) $\displaystyle+$
$\displaystyle\frac{m\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{N}(x)\Big{)}+\frac{\dot{y}^{2}}{y^{2}}\Big{]}-\frac{\xi
x}{4\pi(1+\xi x)^{2}}\Big{[}\frac{\xi l}{1+\xi x}$ $\displaystyle+$
$\displaystyle\frac{m\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{N}(x)\Big{]}+\frac{\xi^{2}x}{8\pi(1+\xi x)^{2}}-\frac{\alpha
x}{4\pi(1+\xi x)},$ $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{x}{2\pi(1+\xi x)}\Big{[}\frac{d}{dx}\Big{(}\frac{\xi
l}{1+\xi x}+\frac{m\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{N}(x)\Big{)}+\frac{\dot{y}^{2}}{y^{2}}\Big{]}$
$\displaystyle-$ $\displaystyle\frac{\xi x}{4\pi(1+\xi x)^{2}}\Big{[}\frac{\xi
l}{1+\xi x}+\frac{m\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\
(1+\xi x)}+\dot{N}(x)\Big{]}$ $\displaystyle+$
$\displaystyle\frac{\xi^{2}x}{8\pi(1+\xi x)^{2}}-\frac{\alpha x}{4\pi(1+\xi
x)}.$ (69)

The function $\dot{N}(x)$ and $\frac{\dot{y}}{y}$ are given as

$\displaystyle\frac{\dot{y}}{y}=\frac{\xi l}{1+\xi
x}+\frac{m\sqrt{\xi(2\xi^{2}+\alpha)}}{\sqrt{3\xi+(\xi^{2}-\alpha)x}\ (1+\xi
x)}+\dot{N}(x),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

$\displaystyle\dot{N}(x)$ $\displaystyle=$ $\displaystyle-\frac{\mathcal{B}\
(\xi^{2}-\alpha)\ \sqrt{(3\xi+(\xi^{2}-\alpha)x)}}{1536\sqrt{2}\ \xi^{2}\
\pi^{\frac{3}{2}}}\Big{[}\frac{66(\xi^{2}-\alpha)}{(1+\xi x)^{2}}+\frac{26\
(2\xi^{2}+\alpha)}{(1+\xi x)^{3}}\Big{]}$ $\displaystyle-$
$\displaystyle\frac{\mathcal{B}(\xi^{2}-\alpha)}{3072\sqrt{2}\
\xi^{3}\pi^{\frac{3}{2}}\sqrt{(3\xi+(\xi^{2}-\alpha)x)}}\Big{[}\frac{33(\xi^{2}-\alpha)^{2}}{(1+\xi
x)}+\frac{26(\xi^{2}-\alpha)(2\xi^{2}+\alpha)}{(1+\xi x)^{2}}$
$\displaystyle+$ $\displaystyle\frac{8(2\xi^{2}+\alpha)^{2}}{(1+\xi
x)^{3}}\Big{]}+\frac{\mathcal{B}\ \sqrt{(3\xi+(\xi^{2}-\alpha)x)}}{512\
\sqrt{2}\ \xi^{2}\
\pi^{\frac{3}{2}}}\times\Big{[}\frac{33(\xi^{2}-\alpha)^{2}}{(1+\xi
x)^{2}}+\frac{26(\xi^{2}-\alpha)}{(1+\xi x)^{3}}(2\xi^{2}+\alpha)$
$\displaystyle+$ $\displaystyle\frac{8(2\xi^{2}+\alpha)^{2}}{(1+\xi
x)^{4}}\Big{]}.$

  * •

For $\eta=1/2$

$\displaystyle P_{r}$ $\displaystyle=$ $\displaystyle\mathcal{A}\
\left[\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi x)^{2}}\right]+\mathcal{B}\
\left[\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi x)^{2}}\right]^{3},$ (70)
$\displaystyle P_{t}$ $\displaystyle=$ $\displaystyle\mathcal{A}\
\left[\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi x)^{2}}\right]+\mathcal{B}\
\left[\frac{3\xi+\xi^{2}x-\alpha x}{8\pi(1+\xi
x)^{2}}\right]^{3}+\frac{x}{2\pi(1+\xi x)}\Big{[}\frac{n(n-1)\xi^{2}}{(1+\xi
x)^{2}}$ (71) $\displaystyle+$ $\displaystyle\frac{2\ n\ \xi\
\dot{O}(x)}{(1+\xi x)}+\dot{O}(x)^{2}+\ddot{O}(x)\Big{]}-\frac{\xi
x}{4\pi(1+\xi x)^{2}}\left[\frac{n\ \xi}{(1+\xi
x)}+\dot{O}(x)\right]+\frac{\xi^{2}x}{8\pi(1+\xi x)^{2}}$ $\displaystyle-$
$\displaystyle\frac{\alpha x}{4\pi(1+\xi x)},$ $\displaystyle\Delta$
$\displaystyle=$ $\displaystyle\frac{x}{2\pi(1+\xi
x)}\left[\frac{n(n-1)\xi^{2}}{(1+\xi x)^{2}}+\frac{n\ \xi\ \dot{O}(x)}{(1+\xi
x)}+\dot{O}(x)^{2}+\ddot{O}(x)\right]-\frac{\xi x}{4\pi(1+\xi
x)^{2}}\Big{[}\frac{n\ \xi}{2(1+\xi x)}$ (72) $\displaystyle+$
$\displaystyle\dot{O}(x)\Big{]}+\frac{\xi^{2}x}{8\pi(1+\xi
x)^{2}}-\frac{\alpha x}{4\pi(1+\xi x)}.$

The function $\dot{O}(x)$ and $\ddot{O}(x)$ are described as

$\dot{O}(x)=\frac{(\xi^{2}-\alpha)}{4\xi}\
(\mathcal{A}+1)+\frac{\mathcal{B}}{256\pi^{2}}\left[\frac{(3\xi+\xi^{2}x-\alpha
x)^{3}}{(1+\xi x)^{5}}\right],$
$\ddot{O}(x)=-\frac{\mathcal{B}}{256\pi^{2}}\left[\frac{[2\xi^{2}(6+\xi
x)+\alpha(3-2\xi x)](3\xi+\xi^{2}x-\alpha x)^{2}}{(1+\xi x)^{6}}\right].$

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