# Anisotropic Spheres Via Embedding Approach in $f(R,\phi,X)$ Gravity

Adnan Malik<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematical Sciences, Zhejiang Normal
University,
Jinhua, Zhejiang, China.  Department of Mathematics, University of Management
and Technology,
Sialkot Campus, Pakistan.  Yonghui Xia<EMAIL_ADDRESS>School of Mathematical
Sciences, Zhejiang Normal University,
Jinhua, Zhejiang, China.  Ayesha Almas<EMAIL_ADDRESS>Department of
Mathematics, University of Management and Technology,
Sialkot Campus, Pakistan.  M. Farasat Shamir<EMAIL_ADDRESS>National University of Computer and Emerging Sciences,
Lahore Campus, Pakistan.

###### Abstract

Abstract

In this manuscript, we investigate the behavior of stellar structure through
embedding approach in $f(R,\phi,X)$ modified theory of gravity, where $R$
denotes the Ricci scalar, $\phi$ represents the scalar potential and $X$
indicates the kinetic potential. For this purpose, we consider the spherically
symmetric space-time with anisotropic fluid. We further choose three different
stars i.e. LMC X-4, Cen X-3, and EXO 1785-248 to demonstrate the behavior of
stellar structures. We further compare the Schwarzschild space-time as
exterior geometry with spherically symmetric space-time to calculate the
values of unknown parameters. In this regard, we investigate the graphical
features of stellar spheres such es energy density, pressure components,
anisotropic component, equation of state parameters, stability analysis and
energy conditions. Furthermore, we investigate some extra conditions such as
mass function, compactness factor and surface redshift respectively.
Conclusively, all the compact stars under observations are realistic, stable,
and are free from any physical or geometrical singularities. We find that the
embedding class one solution for anisotropic compact stars is viable and
stable.

Keywords: Anisotropic Spheres, Compact Stars, Embedding Approach,
$f(R,\phi,X)$ Gravity.

##  I Introduction

Recent explanations based on astrophysical facts have revealed an amazing
picture of the expansion of the universe 1a ; 2a ; 3a . Cosmologist have
brought out new ideas to introduce the critical innovations for this
accelerated expanding universe. There are two types of serious issues in
cosmological models known as dark energy and dark matter. The dark matter is
considered to occupy a major portion of this accelerating universe and it is
the main cause of acceleration of universe. It is expected that the
modification in the theory of general relativity may explain the accelerating
expansion of universe. However, this approach has some limitations because
dark energy has never been directly detected or observed. The concept of dark
energy, which is never detected experimentally, is also a one way for solving
this problem. According to Sloan Digital Sky Surveys (SDSS) 4a , the BICEP2
experiment 5a ; 6a ; 7a , Planck satellite 8a ; 9a ; 10a and the Wilkinson
Microwave anisotropy probe (WMAP) 11a ; 12a , it happen that 27% of universe
is composed of the dark matter, 68% is of dark energy and the rest is ordinary
matter. The modification in Einstein theory of general relativity seems to be
a good approach to justify the idea of dark energy. Through this approach,
many models are introduced for explaining the universe expansion and observing
some interesting aspects of nature. It has led to a search for modified or
extended gravitational theories capable of addressing such challenges. General
relativity as a physical theory has been a great success in the previous
century but there are still some issues which could not be addressed properly
like dark energy, dark matter, initial singularity, late-time cosmic
acceleration and flatness problems. Some alternative models of gravity are
proposed which are believed to be a real cause of this accelerating expansion
of the universe. Severalmodified theories have suggested as alternatives to
general relativity in recent years. Some of these gravitational theories
include $f(R)$ 4.01 ; 4.02 ; 4.1 ; 4.2 ; 4.3 , $f(R,G)$ 4.4 , $f(G)$ 4.5 ; 4.6
; 4.7 ; 4.8 , $f(R,T)$ 4.09 ; 4.9 ; 4.9a ; 4.9b , $f(R,\phi)$ 4.10 ; 4.11 ;
4.11a ; 4.11b and $f(R,\phi,X)$ 4.12 ; 4.13 modified theories of gravity.
These modified theories of gravity explain the phenomena of expansion of
universe. These theories explain the weak field regimes but some modifications
are still required to address the strong field for universe expansion.

The research for precise static spherically solutions for relativistic
structures is a difficult problem to solve because of the close association of
non-linear elements in the modified Einstein-Maxwell equations. To solve this
problem, we may use the embedding class I technique with the Eishenhart
condition to identify new physically acceptable solutions for the sphere’s
compact geometry. New anisotropic results may be generated from a perfectly
distributed fluid in a direct, ordered, and simple manner. Regarding the
creation of novel spherical solutions, the embedding class I technique
exhibits a vast array of noteworthy components. Nazar and Abbas a14
investigated a class of an exact analytical solutions of shear-free and
spherically symmetric gravitational collapse of Karmarkar star in the
minimally coupled $f(R)$ theory of gravity by incorporating the properties of
anisotropic radiating matter. Malik and his collaborators a16 used an embedded
class-I technique to show the evolution of anisotropic stellar formations
against the framework of $f(R)$ modification of gravity. Maurya et al., a17
presented a hierarchical solution-generating technique employing the minimum
gravitational decoupling method and the generalized concept of Complexity as
applied to Class I spacetime for bounded compact objects in classical general
relativity. Sharif and Naseer a18 studied several specific anisotropic stellar
models in $f(R,T,Q)$ modified gravity by employing spherically symmetric
configuration to create solutions of modified field equations corresponding to
distinct matter Lagrangian options using the embedding class-one method.
Gudekli and colleagues a19 investigated the spherically symmetric solutions of
embedding class-one in the $f(T,\tau)$ theory of gravity, providing an
extended compact star model. Sarkar et al., a20 suggested an entirely new
model for spherically symmetric anisotropic astrophysical objects with class I
solutions in the framework of $f(R,T)$ gravity. Errehymy et al., a21 examined
the presence of compact objects characterizing anisotropic matter
distributions in the context of $f(R,T)$ gravity and utilize the embedding
class-I approach to generate a comprehensive space-time interpretation on the
inside of the stellar structure.

Ditta and Xia a22 investigated stellar formations using the Karmarkar
condition and Tolman-Kuchowiz metric components with an anisotropic fluid
allocation within the context of Rastall teleparallel gravity. Sharif and
Naseer a23 examined the charged stellar models linked with an anisotropic
source of fluid allocation in $f(R,T,Q)$ gravity by analyzing a self-
gravitational spherical configuration in the involvement of an electromagnetic
field and generating solutions to the field equations by employing the
Karmarkar condition and the equation of state (MIT bag model). Sharif and
Hassan a25 addressed the methodology of a complexity factor for a dynamical
anisotropic stellar structure in the context of $f(G,T)$ modified gravity and
analyzed the formation scalars by orthogonal division of the Riemann tensor to
calculate a complexity factor that includes all the basic principles of the
structure. Zubair et al., a26 investigated the generalized symmetric, static
compact objects under anisotropic fluid in the background of Karmarkar
embedding class-1 condition by considering the gravitational Lagrangian as a
linear function of the Ricci scalar and the trace of the stressenergy tensor.
Using Pant’s inner solution, Pant et al., a27 investigated a novel embedding
of an anisotropic charged form of a solution to modified field equations in
the 4-dimentional configuration using the Karmarkar conditions and the
gravitational decoupling through the minimum geometric decoupling approach.
Usman and Shamir a28 examined the effects of gravitational breakdown by
looking at heat flux anisotropic sources of fluid in the context of $f(R)$
modified gravity and employing the non-static spherically symmetric spacetime
to describe the essence of the internal spacetime and compare it with the
Vaidya external configuration.

In this work, we intend to consider a generalized modified gravity theory,
$f(R,\phi,X)$ gravity, where $R$ is the Ricci scalar, $\phi$ a scalar field
and $X$ a kinetic term. This theory contains a wide range of known dark energy
and modified gravity models, for instance $f(R)$ gravity models or Galileons.
In particular, several cosmological solutions are studied within the framework
of these theories, specifically solutions that can provide cosmic acceleration
at late times, and even the exact $\Lambda$CDM evolution. Reconstruction
techniques are implemented in order to obtain the $f(R,\phi,X)$ of the action
given a particular Hubble parameter. This provides a way to efficiently check
the viability of any gravitational action by just considering a particular
cosmological evolution and then analyzing the gravitational action. An action
of the form $f(R,\phi,X)$ is in fact quite natural, as it removes any
assumptions on the underlying theory of gravity with the exception of being
second order. We can think of the field $\phi$ as the effective field
controlling the strength of the gravitational force. Bahamonde et al. ad1
explored a generalized $f(R,\phi,X)$ modified gravity theory, which contains a
wide range of dark energy and modified gravity models. They also considered
specific models and applications to the late-time cosmic acceleration.
Bahamonde along with his collaborators ad2 investigated new exact spherically
symmetric solutions in $f(R,\phi,X)$ theory of gravity by Noethers symmetry
approach, and some of these solutions can represent new black holes solutions
in this extended theory of gravity. Malik, et al. ad3 investigated the
behavior of anisotropic compact stars in generalized modified gravity, namely
$f(R,\phi,X)$ by considering the spherically symmetric spacetime to analyze
the feasible exposure of compact stars. Malik along with his collaborators ad4
investigated some cylindrically symmetric solutions in a very well known
modified theory named as $f(R,\phi,X)$ by taking the cylindrically symmetric
space-time to discuss the cylindrical solutions in some realistic regions.
Shamir, et al. ad5 considered the spherically symmetric static spacetime with
an anisotropic fluid source to discuss the wormhole solutions by using two
well-known distributions like Gaussian and Lorentzian non-commutative geometry
in $f(R,\phi,X)$ theory of gravity. The same authors ad5 considered a
particular equation of state parameter to study the behavior of traceless
fluid and examined the physical behavior of wormhole solutions in the
background of $f(R,\phi,X)$ theory of gravity. Recently, Malik et al., ad7
investigated the concept of cracking and overturning to analyze the impact of
local density perturbations on the stability of self-gravitating compact
objects in the framework of $f(R,\phi,X)$ theory of gravity. Recently, Malik
along with his collaborators ad8 provided a new model of anisotropic strange
star corresponding to the exterior Schwarzschild metric and the Einstein field
equations have been solved by utilizing the Krori-Barua ansatz in
$f(R,\phi,X)$ theory of gravity. Malik et al., ad9 investigated and analyzed
the behavior of charged compact stars in the modified $f(R,\phi,X)$ theory of
gravity by assuming the Krori Barua space-time.

To the best of our knowledge, no attempt has been made so far to discuss the
spherically symmetric solutions of embedding class I technique in
$f(R,\phi,X)$ theory of gravity. In this paper, we are inspired to investigate
the nature of stellar structure in the $f(R,\phi,X)$ theory of gravity
utilizing the Karmarkar condition. The arrangement of this paper is as
follows: Section II deals with the field equations of $f(R,\phi,X)$ gravity in
the presence of embedding class I. In Section III, we investigate the matching
conditions for finding the unknown parameters. All the graphical
representation of the stellar configuration has been discussed in Section IV.
The last section deals with the concluding remarks.

##  II Basic Formulism of $f(R,\phi,X)$ gravity

The action for the modified $f(R,\phi,X)$ theory of gravity ad10 ; ad11 is
given as

$S=\int\sqrt{-g}\bigg{(}\textit{L}_{m}+\frac{1}{2\kappa^{2}}f(R,\phi,X)\bigg{)}d^{4}x,$
(1)

where, $g$ represents the determinant of $g_{\eta\xi}$, $\textit{L}_{m}$ is
lagrangian matter, and $X$ is kinetic term, which is defined as

$X=\frac{-\varepsilon}{2}\partial^{\nu}\phi\partial_{\nu}\phi.$ (2)

Here, $\varepsilon$ is a parameter. By applying variation to Eq. (1) with
respect to $g_{\eta\xi}$, we get the field equation as

$f_{R}G_{\eta\xi}-\frac{1}{2}(f-Rf_{R})g_{\eta\xi}-\nabla_{\eta}\nabla_{\xi}f_{R}+g_{\eta\xi}\nabla_{\nu}\nabla^{\nu}f_{R}-\frac{1}{2}f_{X}(\nabla_{\eta}\phi)(\nabla_{\xi}\phi)=\kappa^{2}T_{\eta\xi}.$
(3)

For our convenience, we take $f\equiv f(R,\phi,X)$ and $f_{R}$ is a partial
derivative of $f$ w.r.t $R$. Whereas, $\nabla_{\nu}$ is a covariant derivative
and $T_{\eta\xi}$ is the energy-momentum tensor (EMT) ad12 yields the
following expression,

$T_{\eta\xi}=(\rho+p_{t})\upsilon_{\eta}\upsilon_{\xi}-p_{t}g_{\eta\xi}+(p_{r}-p_{t})\vartheta_{\eta}\vartheta_{\xi},$
(4)

where, $\upsilon_{\eta}$ and $\vartheta_{\xi}$ are four vector velocity
components, i.e., $\upsilon_{\eta}=e^{\frac{\lambda}{2}}\delta^{0}_{\eta}$ and
$\vartheta_{\xi}=e^{\frac{\zeta}{2}}\delta^{1}_{\xi}$. Whereas, $\rho$,
$p_{r}$, and $p_{t}$ represent the energy density, radial pressure and
tangential pressure respectively. This anisotropy feature influences the
physical properties, such as gravitational redshift, energy density, and total
mass ad13 . Moreover, theoretical studies indicate that the pressure within
compact stars with extreme internal density and strong gravity may be
anisotropic ad13a . We consider the spherically symmetric spacetime as

$ds^{2}=e^{\lambda(r)}dt^{2}-e^{\zeta(r)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (5)

where, $\lambda$ and $\zeta$ are the function of radial coordinate $r$. For
further analysis, we choose the following model of $f(R,\phi,X)$ gravity ad7
as

$f(R,\phi,X)=R+\gamma R^{2}+X-V(\phi).$ (6)

The model defined in Eq. (6) often referred to particular well-known
Starobinsky $R+\gamma R^{2}$ model, has been proposed as an alternative to
explain various astrophysical phenomena including compact object. We choose
this model because it shows exponential growth for early-time cosmic
expansion. By adding term like $\gamma R^{2}$ introduce modification to the
standard general relativity that can potentially account for gravitational
effect not explained by general relativity ad14 . Moreover, this term
interpolate a curvature dependence that affect gravitational dynamics also it
allows the possibility of a non-linear relation between the spacetime
curvature and the field due to gravity. It is likely to mention here that if
the value of parameter $\gamma$ chosen to be non-positive then beyond a
maximum mass limit is reached. However, this give rise to a problem,
particularly, the Ricci scalar $R$ exhibits damped oscillation. Conversely, as
we move towards infinity with non-negative values of $\gamma$, the R gradually
decreases to zero, leading to a maximum mass for the star that is lower then
$2M/M_{\odot}$. Furthermore, the term $V(\phi)=w_{0}\phi^{n}$ including
$w_{0}$ and $n$ are arbitrary non-zero constant, which illustrates a potential
energy function, which can be related to concept such as dark energy or
inflation. As far as the viability of the model in terms of solar system tests
is concerned, it is a debateable issue and require some extensive analysis.
However, it has been shown that the $f(R)$ modified theories may pass the
solar system observational constraints even if the scalar field is added ad1 .
This shows that our consider $f(R,\phi,X)$ gravity model may pass solar system
tests. Now using Eq. (5) along with Eq. (4) and (6) in Eq. (3), we get the
following equations:

$\begin{split}\rho=&\frac{1}{8r^{4}}e^{-2\zeta(r)}(8(-1+e^{\zeta(r)})(e^{\zeta(r)}(r^{2}-2\gamma)+10\gamma)-2e^{\zeta(r)}r^{2+2\alpha}\alpha^{2}+4e^{2\zeta(r)}r^{4}(r^{\alpha})^{n}w_{0}+r(r^{3}\gamma\lambda^{{}^{\prime}}(r)^{4}+8e^{\zeta(r)}(r^{2}-8\gamma)\\\
\\\
&\zeta^{{}^{\prime}}(r)-2r^{2}\gamma\lambda^{{}^{\prime}}(r)^{3}(-4+r\zeta^{{}^{\prime}}(r))+r\gamma\lambda^{{}^{\prime}}(r)^{2}(16+r(8\zeta^{{}^{\prime}}(r)-3r\zeta^{{}^{\prime}}(r)^{2}+4r(\lambda^{{}^{\prime\prime}}(r)+2\zeta^{{}^{\prime\prime}}(r))))+4r\gamma\lambda^{{}^{\prime}}(r)(-8r\zeta^{{}^{\prime}}(r)^{2}\\\
\\\
&+r^{2}\zeta^{{}^{\prime}}(r)^{3}+\zeta^{{}^{\prime}}(r)(4+5r^{2}(\lambda^{{}^{\prime\prime}}(r)-\zeta^{{}^{\prime\prime}}(r)))+2r(-2\lambda^{{}^{\prime\prime}}(r)+6\zeta^{{}^{\prime\prime}}(r)-2r\lambda^{{}^{\prime\prime\prime}}(r)+r\zeta^{{}^{\prime\prime\prime}}(r)))-4r\gamma(-4r\zeta^{{}^{\prime}}(r)^{3}+\zeta^{{}^{\prime}}(r)^{2}\\\
\\\
&(12+5r^{2}\lambda^{{}^{\prime\prime}}(r))-8\zeta^{{}^{\prime\prime}}(r)-4r\zeta^{{}^{\prime}}(r)(6\lambda^{{}^{\prime\prime}}(r)-5\zeta^{{}^{\prime\prime}}(r)+2r\lambda^{{}^{\prime\prime\prime}}(r))+r(3r\lambda^{{}^{\prime\prime}}(r)^{2}-8r\lambda^{{}^{\prime\prime}}(r)\zeta^{{}^{\prime\prime}}(r)+16\lambda^{{}^{\prime\prime\prime}}(r)-8\zeta^{{}^{\prime\prime\prime}}(r)\\\
\\\ &+4r\lambda^{{}^{\prime\prime\prime\prime}}(r))))),\end{split}$ (7)
$\begin{split}p_{r}=&\frac{1}{8r^{4}}e^{-2\zeta(r)}(-8(-1+e^{\zeta(r)})(e^{\zeta(r)}(r^{2}-2\gamma)-14\gamma)-2e^{\zeta(r)}r^{2+2\alpha}\alpha^{2}-r(4e^{2\zeta(r)}r^{3}(r^{\alpha})^{n}w_{0}+r^{3}\gamma\lambda^{{}^{\prime}}(r)^{4}+2r^{3}\gamma\lambda^{{}^{\prime}}(r)^{3}\\\
\\\
&\zeta^{{}^{\prime}}(r)+r^{2}\gamma\lambda^{{}^{\prime}}(r)^{2}(24\zeta^{{}^{\prime}}(r)+5r\zeta^{{}^{\prime}}(r)^{2}+4r(-\lambda^{{}^{\prime\prime}}(r)+\zeta^{{}^{\prime\prime}}(r)))-4\gamma(8r^{2}\zeta^{{}^{\prime}}(r)^{3}-6r\zeta^{{}^{\prime}}(r)^{2}(-2+r^{2}\lambda^{{}^{\prime\prime}}(r))+r(16\lambda^{{}^{\prime\prime}}(r)\\\
\\\
&-r^{2}\lambda^{{}^{\prime\prime}}(r)^{2}-16\zeta^{{}^{\prime\prime}}(r)+8r\lambda^{{}^{\prime\prime\prime}}(r))+4\zeta^{{}^{\prime}}(r)(-4+4e^{\zeta(r)}-2r^{2}\zeta^{{}^{\prime\prime}}(r)+r^{3}\lambda^{{}^{\prime\prime\prime}}(r)))+8\lambda^{{}^{\prime}}(r)(-e^{\zeta(r)}r^{2}+\gamma(8+r(r\zeta^{{}^{\prime}2}\\\
\\\
&-r^{2}\zeta^{{}^{\prime}}(r)^{3}+\zeta^{{}^{\prime}}(r)(12+r^{2}(-\lambda^{{}^{\prime\prime}}(r)+\zeta^{{}^{\prime\prime}}(r)))-r(6\lambda^{{}^{\prime\prime}}(r)-4\zeta^{{}^{\prime\prime}}(r)+r\lambda^{{}^{\prime\prime\prime}}(r))))))),\end{split}$
(8)
$\begin{split}p_{t}=&\frac{1}{8r^{4}}e^{-2\zeta(r)}(-16(-1+e^{\zeta(r)})(7+e^{\zeta(r)})\gamma+2e^{\zeta(r)}r^{2+2\alpha}\alpha^{2}-4e^{2\zeta(r)}r^{4}(r^{\alpha})^{n}w_{0}+r(r^{3}\gamma\lambda^{{}^{\prime}}(r)^{4}+2r^{2}\gamma\lambda^{{}^{\prime}}(r)^{3}(2-3r\zeta^{{}^{\prime}})\\\
\\\
&+4e^{\zeta(r)}(-((r^{2}-4\gamma)\zeta^{{}^{\prime}}(r))+r^{3}\lambda^{{}^{\prime\prime}}(r))+r\lambda^{{}^{\prime}}(r)^{2}(2e^{\zeta(r)}r^{2}+\gamma(-16+r(-32\zeta^{{}^{\prime}}(r)+9r\zeta^{{}^{\prime}}(r)^{2}+12r(\lambda^{{}^{\prime\prime}}(r)-\zeta^{{}^{\prime\prime}}))))\\\
\\\
&+2\lambda^{{}^{\prime}}(r)(e^{\zeta(r)}(2r^{2}+24\gamma-r^{3}\zeta^{{}^{\prime}}(r))-2\gamma(4+r(-11r\zeta^{{}^{\prime}}(r)^{2}+r^{2}\zeta^{{}^{\prime}}(r)^{3}+\zeta^{{}^{\prime}}(r)(-4+5r^{2}(2\lambda^{{}^{\prime\prime}}(r)-\zeta^{{}^{\prime\prime}}(r)))+2r(-\\\
\\\
&5\lambda^{{}^{\prime\prime}}(r)+7\zeta^{{}^{\prime\prime}}(r)-3r\lambda^{{}^{\prime\prime\prime}}(r)+r\zeta^{{}^{\prime\prime\prime}}(r)))))+4\gamma(-4r^{2}\zeta^{{}^{\prime}}(r)^{3}+5r^{3}\zeta^{{}^{\prime}}(r)^{2}\lambda^{{}^{\prime\prime}}(r)+4\zeta^{{}^{\prime}}(r)(3+r^{2}(-5\lambda^{{}^{\prime\prime}}(r)+5\zeta^{{}^{\prime\prime}}(r)-\\\
\\\
&2r\lambda^{{}^{\prime\prime\prime}}(r)))+r(5r^{2}\lambda^{{}^{\prime\prime}}(r)^{2}-8\lambda^{{}^{\prime\prime}}(r)(1+r^{2}\zeta^{{}^{\prime\prime}}(r))+4r(3\lambda^{{}^{\prime\prime\prime}}(r)-2\zeta^{{}^{\prime\prime\prime}}(r)+r\lambda^{{}^{\prime\prime\prime\prime}}(r)))))).\end{split}$
(9)

The above Eqs. (7)-(9) are very complex and non-linear differential equation.
Now, we explore a major tool of the present study, which is the Karmarkar
condition ad14 , suggested by Karmarkar. By using this, we apply an embedding
technique to get all possible embedding class one spherically symmetric
solutions. Eishenhart ad15 proposes the following necessary and significant
conditions for second-order $\chi_{\eta\omega}$ tensor and Riemann tensor
$\textrm{{{R}}}_{\eta\xi\omega m}$:

$\textrm{{{R}}}_{\eta\xi\omega m}=\Sigma(\chi_{\eta\omega}\chi_{\xi
m}-\chi_{\eta
m}\chi_{\xi\omega}),~{}~{}~{}~{}~{}~{}~{}~{}\chi_{\eta\xi;\omega}-\chi_{\eta\omega;\xi}=0.$

Here, $\Sigma=\pm 1$. All the Riemann tensor for embedded class one given in
the above equation are as follows:

$\textrm{{{R}}}_{1414}=\frac{e^{\lambda(r)}(2\lambda^{{}^{\prime\prime}}(r)+\lambda^{{}^{\prime}}(r)^{2}-\lambda^{{}^{\prime}}(r)\zeta^{{}^{\prime}}(r))}{4},~{}~{}~{}\textrm{{{R}}}_{2323}=\frac{r^{2}\sin^{2}\theta(e^{\zeta(r)}-1)}{e^{\zeta(r)}},~{}~{}~{}\textrm{{{R}}}_{1334}=\textrm{{{R}}}_{1224}\sin^{2}\theta,$
$\textrm{{{R}}}_{1212}=\frac{r\zeta^{{}^{\prime}}(r)}{2},~{}~{}~{}~{}~{}\textrm{{{R}}}_{3434}=\frac{r\sin^{2}\theta\zeta^{{}^{\prime}}(r)e^{\lambda(r)-\zeta(r)}}{2},~{}~{}~{}~{}~{}\textrm{{{R}}}_{1224}=0.$

Now, Karmarkar’s condition is described as

$\textrm{{{R}}}_{1414}\textrm{{{R}}}_{2323}=\textrm{{{R}}}_{1224}\textrm{{{R}}}_{1334}+\textrm{{{R}}}_{1212}\textrm{{{R}}}_{3434},~{}~{}~{}\textrm{{{R}}}_{2323}\neq
0.$ (10)

Pandey and Sharma abc1 acquired the following symmetric spacetime

$ds^{2}=e^{\lambda}dt^{2}-dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$
(11)

which fulfills Karmarkar condition, still, is not embedded class one because
$\textrm{{{R}}}_{2323}=0$. They argued that in symmetric spacetime, Karmarkar
condition does not suffice to constitute a class one model. Therefore, the
exact solution of field equations in the case of metric (5) can be restricted
as class one model if it sufficient (10) along with $\textrm{{{R}}}_{2323}\neq
0$. Now, by using Eq. (10), we get a following differential equation as

$\frac{\lambda^{{}^{\prime}}\zeta^{{}^{\prime}}}{1-e^{\zeta}}-(\lambda^{{}^{\prime}}\zeta^{{}^{\prime}}+\lambda^{{}^{\prime
2}}-2(\lambda^{{}^{\prime\prime}}+\lambda^{{}^{\prime 2}}))=0,$ (12)

where, ${e^{\zeta}\neq 1}$. By integrating Eq. (10), we get link between two
metric potential components of the line element as

$e^{\zeta}=e^{\lambda}\lambda^{{}^{\prime 2}}+1+D,$ (13)

where $D$ is an integration constant. Now, $g_{tt}$ can be chosen as

$e^{\lambda}=\varrho(1+hr^{2})^{\tau}.$ (14)

where, $\tau$ is a positive integer, whereas $\varrho$ and $h$ are parameters.
It is mentioned that at $r\rightarrow 0$, $e^{\lambda(r)}=\varrho$, which
shows that the metric potential, we have selected is finite and regular at the
core. Similarly, Lake abc2 demonstrated that a metric potential $e^{\lambda}$
must be non-negative and increasing monotonically for any physically viable
stellar composition throughout the configuration. Hence, the metric potential
we assume by (14) is obeying all the mandatory requirement. Now, by
substituting Eq. (14) in (13), we get the $g_{rr}$ component given as

$e^{\zeta(r)}=1+ohr^{2}(1+hr^{2})^{-2+\tau},$ (15)

where ${o=4\tau^{2}h\varrho D}$. Now, by plugging Eq. (14) and Eq. (15) into
Eqs. (7)-(9), we obtain the following set of equations for the stellar
configuration as

$\begin{split}\rho=&\frac{1}{4r^{2}t_{2}(r)^{5}}\bigg{(}4o^{5}h^{5}r^{8}t_{1}(r)^{5\tau}(r^{2}-2\gamma)+8\tau
h^{2}r^{2}t_{1}(r)^{6}t_{3}(r)\gamma+r^{2\alpha}t_{1}(r)^{10}\alpha^{2}+4o^{3}h^{3}r^{4}t_{1}(r)^{1+3\tau}(6(2+\tau
hr^{2})(r+hr^{3})^{2}\\\ \\\
&+2t_{4}(r)\gamma-r^{2+2\alpha}t_{1}(r)^{3}\alpha^{2})+4ohr^{2}t_{1}(r)^{4+\tau}(3-r^{2\alpha}\alpha^{2}+2h^{2}r^{2}t_{5}(r)+2ht_{6}(r)+2h^{3}r^{4}t_{7}(r)+h^{4}r^{6}t_{8}(r))o^{4}h^{4}r^{6}\\\
\\\
&t_{1}(r)^{1+4\tau}(-56\gamma+r^{2}(24+8h((1+\tau)r^{2}+(9-8\tau)\gamma)-r^{2\alpha}t_{1}(r)\alpha^{2}))+2o^{2}h^{2}r^{2}t_{1}(r)^{2+2\tau}(-276\gamma+r^{2}t_{9}(r))+2r^{2}\\\
\\\
&(r^{\alpha})^{n}(t_{1}(r)^{2}+ohr^{2}t_{1}(r)^{\tau})^{5}w_{0}\bigg{)},\end{split}$
(16)
$\begin{split}p_{r}=&\frac{1}{4r^{4}t_{2}(r)^{5}}\bigg{(}8t_{1}(r)^{4}t_{10}(r)t_{11}(r)\gamma
t_{2}(r)-128t_{1}(r)^{6}t_{10}(r)^{3}(2+(2+\tau)hr^{2})\gamma-8t_{1}(r)(-21+hr^{2}t_{12})\gamma
t_{2}(r)^{3}-32t_{1}^{2}\\\ \\\ &(30+hr^{2}t_{13}(r))\gamma
t_{2}(r)^{2}+t_{1}(r)(112\gamma+r^{2}t_{14}(r))t_{2}(r)^{4}-2r^{4}(r^{\alpha})^{n}w_{0}-4r^{2}+8\gamma\bigg{)},\end{split}$
(17)
$\begin{split}p_{t}=&\frac{1}{4r^{2}t_{2}(r)^{5}}\bigg{(}-8o^{5}h^{5}r^{8}t_{1}(r)^{5\tau}\gamma+8\tau^{4}h^{4}r^{6}t_{1}(r)^{2}(t_{1}(r)^{4}+16o^{2}h^{2}r^{4}t_{1}(r)^{2\tau}+23ohr^{2}t_{1}(r)^{2+\tau}\gamma)+32\tau^{3}h^{3}r^{4}t_{1}(r)^{2}\\\
\\\
&(-((-2+hr^{2})t_{1}(r)^{4})+120^{2}h^{2}h^{2}r^{4}t_{1}(r)^{2\tau}+9ohr^{2}(4+hr^{2})t_{1}(r)^{2+\tau})\gamma+4\tau^{2}h^{2}r^{2}t_{1}(r)^{2}t_{15}(r)+8\tau
hr^{2}t_{1}(r)t_{16}(r)\\\ \\\
&+r^{2\alpha}t_{1}(r)^{10}\alpha^{2}+4ohr^{2}t_{1}(r)^{5+\tau}(-1+hr^{2}(-1+hr^{2}+h^{2}r^{4})+32h(5+hr^{2}(-17+2hr^{2}))\gamma+r^{2\alpha}t_{1}(r)^{3}\alpha^{2})+2o^{2}\\\
\\\
&h^{2}r^{2}t_{1}(r)^{3+2\tau}(6t_{1}(r)(r+hr^{3})^{2}+4(37+hr^{2}(261+hr^{2}(89+79hr^{2})))\gamma+3r^{2+2\alpha}t_{1}(r)^{3}\alpha^{2})+o^{4}h^{4}r^{6}t_{1}(r)^{1+4\tau}(72\gamma\\\
\\\
&+r^{2}(4+4h(r^{2}+26\gamma)+r^{2\alpha}t_{1}(r)\alpha^{2}))+4o^{3}h^{3}r^{4}t_{1}(r)^{1+3\tau}t_{17}(r)-2r^{2}(r^{\alpha})^{n}(t_{1}(r)^{2}+ohr^{2}t_{1}(r)^{\tau})^{5}w_{0}\bigg{)}.\end{split}$
(18)

where $t_{\textit{i}}(r)$, {$\textit{i}=1,2,...,17$}, are given in the
Appendix. Whereas the expressions of the parameters $h,$ $o$, $\varrho$ will
be determine from the matching condition.

Table 1: Values of $M$, $a$, $c_{1}$, $\rho_{c}$ and $p_{c}$ for LMC X-4 Stars with $R=10~{}km$. $\tau$ | $o$ | $h$ | $\varrho$ | $\gamma$
---|---|---|---|---
3 |  4.7820 |  0.001273 |  0.4329 |  0.05584110
5 |  7.6640 |  0.000729 |  0.4363 |  0.03277414
10 |  14.900 |  0.000352 |  0.4387 |  0.01556560
20 |  29.392 |  0.000173 |  0.4399 |  0.00702774
50 |  72.882 |  0.000068 |  0.4406 |  0.00702774
100 |  145.373 |  0.000034 |  0.4408 |  0.00023905
500 |  725.285 |  $0.000007$ |  0.4410 |  0.00011337
Table 2: Numerical values of $o$, $\varrho$, $h$, and $\gamma$ for Cen X-3 star model 2. $\tau$ | $o$ | $h$ | $\varrho$ | $\gamma$
---|---|---|---|---
3 |  4.5683 |  0.001421 |  0.3765 |  0.05584110
5 |  7.2751 |  0.000805 |  0.3807 |  0.03277414
10 |  14.082 |  0.000386 |  0.3838 |  0.01556560
20 |  28.267 |  0.000186 |  0.3852 |  0.00702774
50 |  68.656 |  0.000074 |  0.3861 |  0.00702774
100 |  138.251 |  0.000036 |  0.3864 |  0.00023905
500 |  682.743 |  $7.447003\times 10^{-6}$ |  0.3866 |  0.00011337
Table 3: Numerical values of $o$, $\varrho$, $h$, and $\gamma$ for EXO 1785-248 star model 3. $\tau$ | $o$ | $h$ | $\varrho$ | $\gamma$
---|---|---|---|---
3 |  4.5698 |  0.001862 |  0.3768 |  0.07644824
5 |  7.2777 |  0.001055 |  0.3811 |  0.04422569
10 |  14.361 |  0.000496 |  0.3842 |  0.02079960
20 |  27.733 |  0.000248 |  0.3856 |  0.0093616181
50 |  68.685 |  0.000098 |  0.3864 |  0.00259253
100 |  136.944 |  0.000048 |  0.3867 |  0.00035201
500 |  683.029 |  $9.759977\times 10^{-6}$ |  0.3869 |  0.00014346

##  III Matching Conditions

In order to find the values of parameters $o$, $\varrho$, $h$, which depict
the model of relativistic anisotropic sphere, we compare the spherically
symmetric space-time with the Schwarzschild metric as exterior space-time and
develop some matching conditions for finding the solutions. The external
Schwarzschild spacetime is defined as

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2},$ (19)

where ‘$M$’ represents the total mass enclosed on the stellar surface. In
order to compare the internal solution with the external, we impose some
continuity condition for the metric potentials on the boundary, i.e., $r=R$ as
follows

${g_{tt}}^{+}={g_{tt}}^{-},~{}~{}~{}~{}{g_{rr}}^{+}={g_{rr}}^{-},~{}~{}~{}~{}\frac{\partial
g_{tt}}{\partial r}^{+}=\frac{\partial g_{tt}}{\partial r}^{-}.$ (20)

Now, by utilizing the above boundary condition, we obtain the unknown
constants as

$\displaystyle\varrho$ $\displaystyle=$
$\displaystyle\frac{(R-2M)\bigg{(}1-\frac{M}{2\tau M-\tau
R+M}\bigg{)}^{-\tau}}{R},$ (21) $\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{M}{R^{2}(\tau R-(2\tau+1)M)},$ (22) $\displaystyle o$
$\displaystyle=$ $\displaystyle 2\tau\bigg{(}1-\frac{M}{2\tau M-\tau
R+M}\bigg{)}^{1-\tau}.$ (23)

The above-mentioned Eqs. (21)-(23) are very important for further graphical
analysis and the physical properties of stellar structure. These equations
depend on the constants, $\tau$, $M$, and $R$. Here the numerical values of
$o$, $\varrho$, $h$, and $\gamma$ for the considered star models are given
below in Table 1-3.

---
Figure 1:  Graphical variation of $\rho$ against radial coordinate.

##  IV Physical features and visual investigation

In this section, we discuss the physical features and various stellar
configurations in the $f(R,\phi,X)$ modified gravity model. Here, we use three
different star candidates, i.e., LMC X-4, Cen X-3, and EXO 1785-248. Moreover,
to identify the stable configuration in the present study, we utilize the
observational data of three compact star candidates, i.e., LMC X-4, Cen X-3,
and EXO 1785-248 whose masses are measured by
$1.29M/M_{\odot},~{}1.49M/M_{\odot},~{}1.30M/M_{\odot}$ with their
corresponding radii $9.711~{}km,~{}10.136~{}km,8.849~{}km$. These compact
stars are X-ray binaries detected by using X-ray telescope, such as NASA’s
Chandra X-ray observatory or European space Agency’s XMM-Newton. Both LMC X-4
and Cen X-3 are categorized as High-Mass X-ray Binaries (HMXB’s) due to the
nature of their companion stars. In HMXB’s, the companion star is typically a
massive, young star, often of spectral type O or B. The compact star EXO
1785-248 is also an X-ray binary, but its nature is not well-studied. It is
classified as a Low-Mass X-ray Binary (LMXB), which typically consists of a
less massive companion star, such as a main-sequence or a compact object,
potentially a neutron star or a black hole. Several investigations abc4 ; abc5
have been carried out in order to examine the existence of these three compact
stars within the framework of general relativity and modified gravity
theories. This analysis includes the energy density, pressure components,
equation of state parameters, stability analysis, metric potentials, energy
condition, adiabatic index and anisotropy parameter. We also investigate some
extra features of stellar structuresi.e., mass function, compactness factor
and surface redshift. Further, we choose the constants $n=0.1$, $w_{0}=1\times
10^{-5}$, and $\alpha=1\times 10^{-6}$ to obtain the desired results.
Furthermore, the remaining parameters are already given in tabular form.

---
Figure 2:  Graphical variation of $p_{r}$ against radial coordinate.
---
Figure 3:  Graphical variation of $p_{t}$ against radial coordinate.

###  IV.1 Energy Density and Pressure Progression

The graphical nature of physical quantities including energy density and
pressure components are illustrated in Figs. 1-3. From Fig. 1, it can be seen
that the graphical analysis of energy density is positive, decreasing and
maximum at the core throughout the internal configuration. The graphical
analysis of radial pressure is maximum at the center of stellar structure and
ultimately vanishes at the boundary of the star as seen in Fig. 2.
Furthermore, the graphical representation of tangential pressure is positive
and decreasing, when we move towards the boundary as seen in Fig. 3. The
behavior of energy density and pressure components suggest the high
compactness of the center of the star, which represent that our model under
inspection is feasible for the exterior region of the core. Additionally, we
examine the nature of our model by using the first-order derivative test as

$\frac{d\rho}{dr}<0,~{}~{}~{}~{}~{}~{}\frac{dp_{r}}{dr}<0,~{}~{}~{}~{}~{}~{}\frac{dp_{t}}{dr}<0.$
(24)

The derivative of energy density, radial pressure and tangential pressure with
respect to radial component show negative behavior as seen in Figs. 4-6. These
gradients must be vanish at the core of the compact star, which is
demonstrated as

$\frac{d\rho}{dr}\mid_{r=0}=0,~{}~{}~{}~{}~{}~{}\frac{dp_{r}}{dr}\mid_{r=0}=0,~{}~{}~{}~{}~{}~{}\frac{dp_{t}}{dr}\mid_{r=0}=0.$
(25)

It can also be seen from the Figs. 4-6 that all these components are zero at
the center of the considered stellar structures. This is the first derivative
test, which demonstrates that all these functions are positive and attained
maximum output at the center of the star, which is again a valid condition for
the compact star.

---
Figure 4:  Graphical variation of gradient of $\rho$ against radial
coordinate.
---
Figure 5:  Graphical variation of gradient of $p_{r}$ against radial
coordinate.
---
Figure 6:  Graphical variation of gradient of $p_{t}$ against radial
coordinate.

###  IV.2 Anisotropy Evolution

Anisotropy is a well-known parameter that depicts the information regarding
anisotropic behavior of stellar configuration. The difference of tangential
pressure and radial component (i.e., $p_{t}-p_{r}$) is known as anisotropic
function, which is considered a very attractive feature to study the compact
star. Anisotropy is denoted by $\Delta$ and denoted by

$\Delta=p_{t}-p_{r}.$ (26)

There is an interesting fact that if $\Delta=0$, then there is isotropic
pressure in the matter distribution. If anisotropic measurement is positive
$\Delta>0$, then anisotropic force is outward. On the other hand, force is
inward if an anisotropic measurement is negative ($\Delta<0$) ad15 . In our
current manuscript, the anisotropy is positive, directed outward and zero at
the core of the considered stars as seen in Fig. 7.

---
Figure 7:  Graphical variation of anisotropy $(\Delta)$ against radial
coordinate.

###  IV.3 Equations of State Parameters

In literature, there are several equations of state (EoS) parameters but we
prefer radial EoS parameter $(EoS)_{r}$ and tangential EoS parameter
$(EoS)_{t}$ ad16 , which is defined as

$(EoS)_{r}=\frac{p_{r}}{\rho},~{}~{}~{}~{}~{}~{}~{}~{}~{}(EoS)_{t}=\frac{p_{t}}{\rho}.$
(27)

---
Figure 8:  Graphical variation of $(EoS)_{r}$ parameter against radial
coordinate.

In stellar configuration, the EoS parameters plays an important role in
determining the hydrostatic equilibrium which refers to the balance between
inward gravitational force and the outward pressure force within a star.
Furthermore, the mandatory and adequate condition for these two parameters is
that the values of $(EoS)_{r}$ and $(EoS)_{t}$ lies between $0$ and $1$. If
this parameter is too high (greater than 1), it can lead to an unstable
configuration and gravitational collapse. Similarly, if EoS parameter is too
low (less than 0), it can also lead to instability and expansion of the
stellar configuration. Moreover, the value of EoS parameter between 0 and 1 is
crucial for compact star stability because it implies the presence of
degeneracy pressure, which counteracts gravitational collapse, preventing the
star from further compression. The EoS value in this range signifies the
existence of stable white dwarfs, neutron stars, and other compact objects. It
can be observed from Figs. 8 and 9 that the behavior of radial and tangential
EoS parameters is maximum at core of star, monotonically decreasing towards
the boundary, and lying between 0 and 1. From these figures, we note that the
tangential pressure and radial pressure are less than the energy density
throughout the internal configuration.

---
Figure 9:  Graphical variation of $(EoS)_{t}$ parameter against radial
coordinate.

###  IV.4 Gravitational Metric Potential

In the study of stellar spheres, the presence of geometric singularity is
always a crucial feature. We investigate the compulsory condition for the
metric potential, which is $e^{\zeta(r=0)}=1$ and $e^{\lambda(r=0)}\neq 0$.
From Fig. 10, it can be observed that the behavior of metric potentials is
finite, positive at the center, and increasing towards the boundary, which
shows that the present stellar structures are free from singularity. Moreover,
the matching of the metric potentials at the boundary of the star is a
consequence of the requirement of continuity and smoothness in the
gravitational field. At the boundary of the star, the interior solution
describing the gravitational field must smoothly match with exterior solution,
which corresponds to the gravitational field outside the star.

---
Figure 10:  Graphical representation of metric potentials against radial
coordinate.

###  IV.5 Stability Analysis

To examine the stability of compact stars within the context of modified
gravity is an important part of stellar strucuture study Pretel et al., rv01
derived the hydrostatic equilibrium equation and the modified Chandrasekhars
pulsation equation with in the framework of $f(R,T)$ gravity. In another work,
Sarmah et al rv02 obtained the Chandrasekhar limiting masses as well as the
dynamical instability criteria for white dwarfs in $f(R)$ gravitational
theory. In this work, we discuss the stability using the sound speeds, i.e.,
(radial sound speed $\nu^{2}_{sr}$ and transverse sound speed $\nu^{2}_{st}$).
These sound speeds can be defined as

$\nu^{2}_{sr}=\frac{dp_{r}}{dr}\times\frac{dr}{d\rho}=\frac{dp_{r}}{d\rho},$
(28)
$\nu^{2}_{st}=\frac{dp_{t}}{dr}\times\frac{dr}{d\rho}=\frac{dp_{t}}{d\rho}.$
(29)

According to Herrera’s cracking condition ad17 , these velocities must lie
between the interval [0,1]. The graphical analysis of radial sound speed is
positive and decreasing in nature as seen in Fig. 11. Similarly, the
transverse sound speed has similar nature as radial sound speed as shown in
Fig. 12. It can also be noticed from these Figs. 11 and 12 that these sounds
satisfy the Herrera condition, i.e., $0\leq\nu^{2}_{sr}$ and $\nu^{2}_{st}\leq
1$. Abreu et al., ad18 discussed another condition to check either stellar
structure is stable or unstable. The Abreu’s condition is describe as
$-1\leq\nu^{2}_{t}-\nu^{2}_{r}\leq 0$ and $0\leq\nu^{2}_{r}-\nu^{2}_{t}\leq
1$. It can be observed from Fig. 13 that the Abreu condition has also been
satisfied.

---
Figure 11:  Graphical representation of $\nu^{2}_{sr}$ against radial
coordinate.
---
Figure 12:  Graphical representation of $\nu^{2}_{st}$ against radial
coordinate.
---
Figure 13:  Graphical representation of $-1\leq\nu^{2}_{t}-\nu^{2}_{r}\leq 0$
and $0\leq\nu^{2}_{r}-\nu^{2}_{t}\leq 1$.

###  IV.6 Energy Conditions

Another essential condition for analyzing the stellar configuration is the
energy conditions. These energy conditions are crucial in classifying the
exotic and relativistic matter distributions in the stellar model. In the
context of extended gravity theories, Atazadeh and Darabi ad23 ; ad23a ; ad23b
; ad23c has been presented a formalism to check the validity of the energy
conditions. These conditions can be described as

  * •

Null energy condition${(NEC):\rho+p_{r}\geq 0,\rho+p_{t}\geq 0,}$

  * •

Week energy condition$(WEC):\rho\geq 0,\rho+p_{r}\geq 0,\rho+p_{t}\geq 0,$

  * •

Strong energy condition$(SEC):\rho+p_{r}\geq 0,\rho+p_{t}\geq
0,\rho+p_{r}+2p_{t}\geq 0,$

  * •

Dominant energy condition${(DEC):\rho\geq|p_{r}|,\rho\geq|p_{t}|.}$

The graphical trend of $\rho+p_{r}$ is initially maximum at the core of the
star and then becomes minimum as seen in Fig. 14. Similarly, $\rho+p_{t}$ has
similar trends like $\rho+p_{r}$ as shown in Fig. 14. It can be concluded that
NEC is satisfied due to positive nature of these two components i.e.,
$\rho+p_{r}$ and $\rho+p_{t}$. Furthermore, WEC is also satisfied because the
graphical representation of energy density and NEC is satisfied. It can also
be seen from Fig. 16 that the graphical analysis of $\rho+p_{r}+p_{t}$ is
positive and decreasing in nature, which means that SEC is satisfied. It can
also be noticed from Fig. 17 and Fig. 18 that DEC is also satisfied due to
positive behavior of $\rho-p_{r}$ and $\rho-p_{t}$. It can be concluded that
all the energy conditions are satisfied, which justifies that our star is
viable.

---
Figure 14:  Graphical representation of $\rho+p_{r}$ against radial
coordinate.
---
Figure 15:  Graphical representation of $\rho+p_{t}$ against radial
coordinate.
---
Figure 16:  Graphical representation of $\rho+p_{r}+2p_{t}$ against radial
coordinate.
---
Figure 17:  Graphical representation of $\rho-p_{r}$ against radial
coordinate.
---
Figure 18:  Graphical representation of $\rho-p_{t}$ against radial
coordinate.

###  IV.7 Mass, Compactness factor and Surface Redshift

The mass functions ad24 is defined as

$m(r)=4\pi\int_{0}^{R}\rho r^{2}dr.$ (30)

It can be noticed that $M(r)\longrightarrow 0$ as $r\longrightarrow 0$, which
represents that the mass function is regular at the center for the stellar
object under the background of embedding approach in $f(R,\phi,X)$ theory of
gravity as shown in Fig. 19. The compactness factor is denoted by $u(r)$ and
defined as

$u(r)=\frac{m(r)}{r}.$ (31)

The graphical analysis of compactness factor can be seen in Fig. 20, which is
positive and increasing in nature. One can also be noticed that compactness
factor has similar nature like mass function shown in Fig. 19. It is believed
that the Redshift function is also crucial component for the study of
astrophysical objects and these physical features are all linked with each
other. The surface redshift function ad25 is given as

$Z_{s}+1=[-2u(r)+1]^{\frac{-1}{2}}.$ (32)

The graphical plotting of surface redshift function is zero at the core and
then becomes increasing when we move towards the boundary as sen in Fig. 21.
Moreover, the graphical illustration redshift function is uniformly
increasing, which confirms the stable behavior of the proposed $f(R,\phi,X)$
gravity models.

---
Figure 19:  Graphical representation of mass functions against radial
coordinate.
---
Figure 20:  Graphical representation of compactness factor against radial
coordinate.
---
Figure 21:  Graphical representation of surface redshift against radial
coordinate.

##  V Conclusion

In this present study, we examined anisotropic stellar spheres in the
$f(R,\phi,X)$ theory of gravity with embedding class-one approach. For the
investigation of the stellar configuration, we use the compatible
$f(R,\phi,X)=R+\gamma R^{2}+X-V(\phi)$ gravity model along with the
spherically symmetric spacetime. These compact stars have different physical
characteristics which are graphically examined for the present model by using
different values of parameter $\tau$. Furthermore, we utilize the
Schwarzschild’s exterior geometry to find the unknown constants. All the
essential results that are satisfied in a stellar configuration are summarized
as follows:

  * •

The graphical analysis of energy density and tangential pressure is positive,
decreasing and maximum at the core throughout the internal configuration.
Similarly, the graphical analysis of radial pressure is maximum at the center
of stellar structure and ultimately vanishes at the boundary of the star. The
behavior of energy density and pressure components suggest the high
compactness of the center of the star, which represent that our model under
inspection is feasible for the exterior region of the core.

  * •

The derivative of energy density, radial pressure and tangential pressure with
respect to radial component show negative, which demonstrates that all these
functions are positive and attained maximum output at the center of the star,
which is again a valid condition for the compact star.

  * •

There is an interesting fact that if $\Delta=0$, then there is isotropic
pressure in the matter distribution. If anisotropic measurement is positive
$\Delta>0$, then anisotropic force is outward. On the other hand, force is
inward if an anisotropic measurement is negative ($\Delta<0$). In our current
manuscript, the anisotropy is positive, directed outward and zero at the core
of the considered stars as seen in Fig. 7.

  * •

The graphical behavior of radial and tangential EoS parameters is maximum at
core of star, monotonically decreasing towards the boundary, and lying between
0 and 1, which is again a crucial condition of stellar structure.

  * •

From Fig. 10, it can be observed that the behavior of metric potentials is
finite, positive at the center, and increasing towards the boundary, which
shows that the present stellar structures are free from singularity.

  * •

The graphical analysis of radial sound speed and transverse sound speed is
positive and decreasing in nature as seen in Fig. 11 and Fig. 12 and satisfy
the Herrera condition, i.e., $0\leq\nu^{2}_{sr}$ and $\nu^{2}_{st}\leq 1$. It
can be noticed that the Abreu condition i.e.,
$-1\leq\nu^{2}_{t}-\nu^{2}_{r}\leq 0$ and $0\leq\nu^{2}_{r}-\nu^{2}_{t}\leq 1$
is also satisfied for considered stars.

  * •

All the energy conditions namely NEC, WEC, SEC and DEC are satisfied, which
justifies that our star is viable.

  * •

It can be noticed that $M(r)\longrightarrow 0$ as $r\longrightarrow 0$, which
represents that the mass function is regular at the center for the stellar
object under the background of embedding approach in $f(R,\phi,X)$ theory of
gravity as shown in Fig. 19.

  * •

The graphical analysis of compactness factor can be seen in Fig. 20, which is
positive and increasing in nature. One can also be noticed that compactness
factor has similar nature like mass function shown in Fig. 19.

  * •

The graphical plotting of surface redshift function is zero at the core and
then becomes increasing when we move towards the boundary as sen in Fig. 21.
Moreover, the graphical illustration redshift function is uniformly
increasing, which confirms the stable behavior of the proposed $f(R,\phi,X)$
gravity models.

Modified $f(R,\phi,X)$ theory of gravity plays an alluring role in the study
of stellar structures. Hence, we conclude that our $f(R,\phi,X)$ model in the
frame of anisotropic fluid is stable and consistent, as all physical
characteristics of compact objects obey physically obtained patterns.
Moreover, it is mentioned here that our results are more identical to the past
relevant work in $f(R)$ gravity a16 .

## Data Availability Statement

No data was used for the research in this article. It is pure mathematics.

## Conflict of Interest

The authors declare that they have no conflict of interest.

## Contributions

We declare that all the authors have same contributions to this paper.

## Data Availability Statement

The authors declare that the data supporting the findings of this study are
available within the article.

## Acknowledgement

Adnan Malik acknowledges the Grant No. YS304023912 to support his Postdoctoral
Fellowship at Zhejiang Normal University, China. This paper was completed
during the postdoctoral fellowship of the first author under the supervision
of Professor Xia at Zhejiang Normal University, China.

## References

##  VI Appendix

$\displaystyle t_{1}(r)$ $\displaystyle=$ $\displaystyle(1+hr^{2}).$
$\displaystyle t_{2}(r)$ $\displaystyle=$
$\displaystyle(t_{1}(r)^{2}+ohr^{2}t_{1}(r)^{\tau}).$ $\displaystyle t_{3}(r)$
$\displaystyle=$ $\displaystyle
60-3\tau+2(-20+\tau(19+3\tau))hr^{2}+(1+\tau)(-4+\tau+\tau^{2})h^{2}r^{4}.$
$\displaystyle t_{4}(r)$ $\displaystyle=$ $\displaystyle
5+hr^{2}(55-20\tau+(47+4(-4+\tau)\tau)hr^{2}+(29-12\tau(1+\tau))h^{2}r^{4}).$
$\displaystyle t_{5}(r)$ $\displaystyle=$ $\displaystyle
3(1+\tau)r^{2}+(120+\tau(-312+241\tau))\gamma-3r^{2+2\alpha}\alpha^{2}.$
$\displaystyle t_{6}(r)$ $\displaystyle=$
$\displaystyle(4+\tau)r^{2}+12(-10+9\tau)\gamma-2r^{2+2\alpha}\alpha^{2}.$
$\displaystyle t_{7}(r)$ $\displaystyle=$ $\displaystyle 3\tau
r^{2}+216\gamma+2\tau(-70+\tau(-87+71\tau))\gamma-2r^{2+2\alpha}\alpha^{2}.$
$\displaystyle t_{8}(r)$ $\displaystyle=$
$\displaystyle(-1+2\tau)r^{2}+2(-24+\tau(56+\tau(-27+\tau(-30+17\tau))))\gamma-r^{2+2\alpha}\alpha^{2}.$
$\displaystyle t_{9}(r)$ $\displaystyle=$
$\displaystyle(20-3r^{2\alpha}\alpha^{2}+2h^{2}r^{2}16(4+3\tau)r^{2}+420\gamma-16\tau(3+14\tau)\alpha-9r^{2+2\alpha}\alpha^{2})+4h((14+3\tau)r^{2}+4(27-46\tau)\alpha-$
$\displaystyle=$ $\displaystyle
3r^{2+2\alpha}\alpha^{2})+4h^{3}r^{4}((2+9\tau)r^{2}+4(27+\tau(4(5-16\tau)\tau))\gamma-3r^{2+2\alpha}\alpha^{2})+h^{4}r^{6}(4(-1+3\tau)r^{2}+4(75+4\tau(-36$
$\displaystyle=$
$\displaystyle+\tau(21+2\tau-6\tau^{2})))\gamma-3r^{2+2\alpha}\alpha^{2}).$
$\displaystyle t_{10}(r)$ $\displaystyle=$ $\displaystyle(1+(-1+\tau)hr^{2}).$
$\displaystyle t_{11}(r)$ $\displaystyle=$ $\displaystyle
116+4(-3+56\tau)hr^{2}+(-84+\tau(16+155\tau))h^{2}r^{4}+(-1+\tau)(-44+\tau(68+39\tau))h^{3}r^{6}.$
$\displaystyle t_{12}(r)$ $\displaystyle=$ $\displaystyle
33-28\tau+(-15+4(20-3\tau)\tau)hr^{2}+(-5+4\tau(-1+9\tau))h^{2}r^{4}.$
$\displaystyle t_{13}(r)$ $\displaystyle=$ $\displaystyle
8+66\tau+(-12+\tau(16+59\tau))hr^{2}+(24+7\tau(6+\tau(-1+4\tau)))h^{2}r^{4}+2(-1+\tau)(1+\tau(11+\tau(4+3\tau)))h^{3}r^{6}.$
$\displaystyle t_{14}(r)$ $\displaystyle=$
$\displaystyle(4+4(1+2\tau)hr^{2}+16(-1+4\tau)h\gamma-r^{2\alpha}t_{1}(r)\alpha^{2}).$
$\displaystyle t_{15}(r)$ $\displaystyle=$ $\displaystyle
r^{2}t_{1}(r)^{6}+o^{3}h^{3}r^{8}t_{1}(r)^{3\tau}+2t_{1}(r)^{4}(11+3hr^{2}(-18+hr^{2}))\gamma+o^{2}h^{2}r^{4}t_{1}(r)^{2\tau}(3(r+hr^{3})^{2}+4(59+hr^{2}(4$
$\displaystyle=$
$\displaystyle-39hr^{2}))\gamma)+ohr^{2}t_{1}(r)^{2+\tau}(3(r+hr^{3})^{2}+h(-231+hr^{2}(194+45hr^{2}))\gamma).$
$\displaystyle t_{16}(r)$ $\displaystyle=$
$\displaystyle(t_{1}(r)^{7}+8o^{4}h^{4}r^{6}(t_{1}(r)^{4})^{\tau}\gamma+8ht_{1}(r)^{5}(-5+hr^{2}(8+hr^{2}))\gamma-
oh(1+hr^{2})^{3+\tau}(-3(r+hr^{3})^{2}+4(\tau+hr^{2}(-1$ $\displaystyle=$
$\displaystyle+hr^{2}(-52+17hr^{2})))\gamma)+o^{3}h^{3}r^{4}t_{1}(r)^{3\tau}(26\gamma+r^{2}t_{1}(r)+6h(6+7hr^{2})\gamma))+o^{3}h^{2}r^{2}t_{1}(r)^{2+2\tau}(226\gamma+r^{2}(3-$
$\displaystyle=$ $\displaystyle 80h\alpha+hr^{2}(3+158h\gamma))).$
$\displaystyle t_{17}(r)$ $\displaystyle=$
$\displaystyle-102\gamma+r^{2}(-3+3hr^{2}(-1+hr^{2}+h^{2}r^{4})-2h(129+hr^{2}(89+43hr^{2}))\gamma+r^{2\alpha}t_{1}(r)^{3}\alpha^{2}).$

##  VII References

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