# Study of Compact Stars in $\mathcal{R}+\alpha\mathcal{A}$ Gravity

M. Farasat Shamir<EMAIL_ADDRESS>National University of Computer
and Emerging Sciences,
Lahore Campus, Pakistan.  Eesha Meer<EMAIL_ADDRESS>National
University of Computer and Emerging Sciences,
Lahore Campus, Pakistan.

###### Abstract

The main goal of this work is to provide a comprehensive study of relativistic
structures in the context of recently proposed $\mathcal{R}+\alpha\mathcal{A}$
gravity, where $\mathcal{R}$ is the Ricci scalar, and $\mathcal{A}$ is the
anti-curvature scalar. For this purpose, we examine a new classification of
embedded class-I solutions of compact stars. To accomplish this goal, we
consider an anisotropic matter distribution for
$\mathcal{R}+\alpha\mathcal{A}$ gravity model with static spherically
symmetric spacetime distribution. Due to highly non-linear nature of field
equations, we use the Karmarkar condition to link the $g_{rr}$ and $g_{tt}$
components of the metric. Further, we compute the values of constant
parameters using the observational data of different compact stars. It is
worthy to mention here that we chose a set of twelve important compact stars
from the recent literature namely $4U~{}1538-52$, $SAX~{}J1808.4-3658$,
$Her~{}X-1$, $LMC~{}X-4$, $SMC~{}X-4$, $4U~{}1820-30$, $Cen~{}X-3$,
$4U~{}1608-52$, $PSR~{}J1903+327$, $PSR~{}J1614-2230$, $Vela~{}X-1$,
$EXO~{}1785-248$. To evaluate the feasibility of
$\mathcal{R}+\alpha\mathcal{A}$ gravity model, we conduct several physical
checks, such as evolution of energy density and pressure components, stability
and equilibrium conditions, energy bounds, behavior of mass function and
adiabatic index. It is concluded that $\mathcal{R}+\alpha\mathcal{A}$ gravity
supports the existence of compact objects which follow observable patterns.

Keywords: Compact stars, Karmarker, metric potentials,
$\mathcal{R}+\alpha\mathcal{A}$ gravity.

##  I Introduction

Investigation of relativistic stellar structures has now become an interesting
research pursuit in astrophysics. The fundamental features of compact stars
have encouraged several researchers not only in the prospect of usual general
relativity ($\mathcal{GR}$) but also in broadly emerging gravitational
alternative theories during the last few decades Skm ; Bhr ; Akp ; SMR ; Vum ;
Bmk ; Hin ; Nag ; Hat ; Add .

Compact stars are originated as the result of gravitational collapse that
happens due to the pressure of the relativistic objects. The gravitational
collapse occurs at the center of the massive star which is highly exothermic
and this happens when the internal pressure of the stars fails to maintain the
pressure against the external force. As the result of gravitational collapse,
either a compact star will be born, like neutron stars, white dwarfs, or it
will be extended as a black hole. $\mathcal{GR}$ provides most suitable
results in the study of stellar objects and develop basic understanding about
gravitational theories. However, $\mathcal{GR}$ alone does not provide
satisfactory outcomes in order to determine the mysterious nature of dark
energy. Due to the limitation of $\mathcal{GR}$, the modified theories have
gained the attention of researchers. As an alternative to the theory of
$\mathcal{GR}$, modified theories of gravity have played an important role to
somehow explain the accelerating expansion of the universe. Various modified
gravitational theories are presented in literature, some of mostly debated are
$f(\mathcal{R})$, $f(\mathcal{R},\mathcal{T})$, $f(\mathcal{G})$,
$f(\mathcal{G},\mathcal{T})$, $f(\mathcal{R},\mathcal{G})$ Hbl ; MFS ; Stn ;
Sma ; Ssz . A recently proposed theory namely $f(\mathcal{R},\mathcal{A})$
gravity has also gained some popularity due to the involvement of bivariate
function of Ricci and anti-curvature scalars Luca .

We require a precise solution to the Einstein field equations
($\mathcal{EFE}$) to examine celestial compact structure models. Schwarzschild
Sch  was the first who discovered the $\mathcal{EFE}$ solution for the
internal structure of compact stars in 1916. Tolman Tol  and Oppenheimer Opp
explored some feasible models of stellar objects to represent the relationship
between internal pressure and the gravitational force. Bowers and Liang Bal
introduced the idea of non-zero anisotropy in a stellar arrangement. Ruderman
Rud  was the first to discover that the nuclear density becomes anisotropic at
the center of compact objects. Errehymy et al. Err  investigated the solutions
of dense stars in the domain of $\mathcal{GR}$. A new class of solution
describing an anisotropic star which is satisfying Karmarkerﬂs condition was
reported by Maurya et al. Sbs . The discussion of compact structures in
modified gravity also seems interesting. Nojiri and Odintsov Nao  discussed
some viable and stable results of different models in the framework of
$f(\mathcal{R})$ gravity. Starobinsky Star  also presented the outcomes of
various models of $f(\mathcal{R})$ gravity. Hu and Sawicki Has  suggested
several $f(\mathcal{R})$ gravity models. Cognola et al. Cog  examined the
physical behavior of the compact stars in the exponential type models of
$f(\mathcal{R})$ gravity. Schlai Sli  identified the embedding problem on
geometrically significant spacetimes. Shamir et al. Mir  explored the
solutions of dense stars using Karmarkar condition in the background of
$f(\mathcal{R})$ theory of gravity. Naz et al. Naz  investigated the existence
of solutions of anisotropic compact objects through embedding approach in the
context of $f(\mathcal{R})$ gravity by applying Karmarkar condition.

Motivated from the interesting properties of stellar structures in modified
theories of gravity, we aim to investigate anisotropic compact structures in
the background of Ricci inverse ($\mathcal{RI}$) gravity. In particular, we
provide a detailed investigation of stellar structure for twelve important
compact stars from the recent literature namely $4U~{}1538-52$,
$SAX~{}J1808.4-3658$, $Her~{}X-1$, $LMC~{}X-4$, $SMC~{}X-4$, $4U~{}1820-30$,
$Cen~{}X-3$, $4U~{}1608-52$, $PSR~{}J1903+327$, $PSR~{}J1614-2230$,
$Vela~{}X-1$, $EXO~{}1785-248$. Our work is managed as follows: Section II
presents the fundamental concepts, structure, and the modified field equations
of $\mathcal{RI}$ gravity. In Section III, the matching constraints for the
chosen linear $\mathcal{RI}$ gravity model ($\mathcal{R}+\alpha\mathcal{A}$)
are developed using Schwarzschild’s geometry. Section IV is devoted to
calculate and analyze physical properties such as energy density, pressure
profiles etc. The last section contains final remarks.

##  II $f(\mathcal{R},\mathcal{A})$ Gravity and Modified Field Equations

The inverse of the Ricci tensor known as the anticurvature tensor helps to
form an alternative theory namely $f(\mathcal{R},\mathcal{A})$ gravity Luca ,
where $\mathcal{R}$ and $\mathcal{A}$ are the traces of curvature and
anticurvature respectively. This theory can be explored to verify the
widespread presence of spectral instabilities and to understand the effects of
perturbations such as the propagation of gravitational waves, perturbation
growth, or the Newtonian bound. Such theory is confirmed as a suitable choice
to be regarded as a cosmological model after deriving the general field
equations of the Lagrangian $f(\mathcal{R},\mathcal{A})$. According to a no-go
theorem, every Lagrangian action $f(\mathcal{R},\mathcal{A})$ containing terms
with any positive or negative exponent of anticurvature $\mathcal{A}$ can be
used to justify dark energy Luca . Thus, it would be exceptionally intriguing
to analyze this theory further. In this work, we focus ourselves to study
compact structures in this theory. The starting point to derive the field
equations is to define a tensor $\mathcal{A}^{\eta\xi}$ as inverse of
$\mathcal{R}_{\eta\xi}$, such that

$\mathcal{A}^{\eta\alpha}\mathcal{R}_{\alpha\xi}=\delta^{\eta}_{\xi}.$ (1)

In order to develop the modified field equations, we need the following
definition of the $f(\mathcal{R},\mathcal{A})$ gravitational action:

$S=\int d^{4}x\sqrt{-g}[f(\mathcal{R},\mathcal{A})+\mathcal{L}_{m}],$ (2)

where, the arbitrary function $f(\mathcal{R},\mathcal{A})$ depends on the
Ricci scalar $\mathcal{R}$ and the anticurvature scalar $\mathcal{A}$, which
are the traces of the Ricci tensor and the anticurvature tensor respectively.
Here, $\mathcal{L}_{m}$ is matter Lagrangian and $g$ represents the
determinant of the metric. We get the following $f(\mathcal{R},\mathcal{A})$
gravity field equation by varying the action mentioned in Eq. (2) with respect
to the metric tensor

$\displaystyle
f_{\mathcal{R}}\mathcal{R}^{\eta\xi}-f_{\mathcal{A}}\mathcal{A}^{\eta\xi}-\frac{1}{2}fg^{\eta\xi}$
$\displaystyle+g^{\mu\eta}\nabla_{\beta}\nabla_{\mu}(\
f_{\mathcal{A}}\mathcal{A}_{\sigma}^{\beta}\mathcal{A}^{\xi\sigma})\
-\frac{1}{2}\nabla^{\kappa}\nabla_{\kappa}(\
f_{\mathcal{A}}\mathcal{A}_{\sigma}^{\eta}\mathcal{A}^{\xi\sigma})\ $
$\displaystyle-\frac{1}{2}g^{\eta\xi}\nabla_{\beta}\nabla_{\mu}(\
f_{\mathcal{A}}\mathcal{A}_{\sigma}^{\beta}A^{\mu\sigma})\
-\nabla^{\eta}\nabla^{\xi}f_{\mathcal{R}}+g^{\eta\xi}\nabla^{\kappa}\nabla_{\kappa}f_{\mathcal{R}}=\mathcal{T}^{\eta\xi},$
(3)

where, $f=f(\mathcal{R},\mathcal{A})$, $f_{\mathcal{R}}=\frac{\partial
f}{\partial\mathcal{R}}$, $f_{\mathcal{A}}=\frac{\partial
f}{\partial\mathcal{A}}$, $\mathcal{A}^{\eta\xi}$ is the anticurvature tensor
which is the inverse of the Ricci tensor $\mathcal{R}_{\eta\xi}$. The stress-
energy momentum tensor $\mathcal{T}_{\eta\xi}$ is chosen as

$\mathcal{T}_{\eta\xi}=(\rho+p_{t})u_{\eta}u_{\xi}-p_{t}g_{\eta\xi}+(p_{r}-p_{t})\mathcal{\chi}_{\eta}\mathcal{\chi}_{\xi}.$
(4)

Here, $\rho,$ $p_{r}$ and $p_{t}$ are energy density, radial and transverse
pressure components respectively, while $u_{\eta}$ and $\mathcal{\chi}_{\eta}$
are four velocity vectors, which satisfy the relations
$u^{\eta}u_{\eta}=-\mathcal{\chi}^{\eta}\mathcal{\chi}_{\eta}=1$. Furthermore,
we consider a static spherically symmetric line element which is represented
as

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

where, $\nu$ and $\lambda$ are the functions of radial coordinate. Our stress-
energy tensor is assumed to be anisotropic, with relativistic geometrized
units $8\pi G=1$. Moreover, the embedded class-I family is represented by the
metric tensor (5), if it fulfills the Karmarkar condition, i.e.

$\mathcal{R}_{1414}\mathcal{R}_{2323}=\mathcal{R}_{1212}\mathcal{R}_{3434}+\mathcal{R}_{1224}\mathcal{R}_{1334},$
(6)

with $\mathcal{R}_{2323}\neq 0$. The following differential equation for the
given metric (5) is developed by using the well-known Karmarkar condition as

$\lambda^{\prime}\nu^{\prime}-2\nu^{\prime\prime}-(\nu^{\prime})^{2}=\frac{\lambda^{\prime}\nu^{\prime}}{1-e^{\lambda}},$
(7)

where $e^{\lambda}\neq 1$. Integrating Eq. (7), we obtain

$e^{\nu}=\left[(A+B\int\sqrt{e^{\lambda}-1}dr)^{2}\right].$ (8)

In this case, $A$ and $B$ are integration constants. Further, due to the
highly non-linear nature of modified field equations, we have no other choice
but to choose the component $g_{rr}=e^{\lambda(r)}$ in the following shape Bhr
; Spt

$e^{\lambda}=1+a^{2}r^{2}(1+br^{2})^{x},$ (9)

where one can choose any real value of $x$ other than zero. It is really
important to mention here that in order to establish embedded class-I
spacetime, the parameters $a$ and $b$ should not be zero. In the domain of
$\mathcal{GR}$, Singh and Pant Spt  investigated the properties of stellar
structures by considering some positive values of $x$ for embedding class-I
solutions of different models of compact stars. Later, Bhar et al. Bhr  also
used this metric potential by taking the value of $x=-4$, and the
corresponding results are found stable in the context of $\mathcal{GR}$.
Inspired by the work of Bhar et al. Bhr , we aim to expand the study in the
framework of $\mathcal{RI}$ gravity by choosing $x=-4$. Interestingly, we
tried some other possibilities for the choice of $x$ (both positive and
negative), however, we could not obtain physically acceptable results in those
cases. Manipulating Eqs. (8) and (9), the metric potential expression becomes

$e^{\nu}=\left[\left(A-\frac{aB}{2b(1+br^{2})}\right)^{2}\right].$ (10)

|  |
---|---|---
Figure 1: Graphical Behavior of $g_{tt}$ and $g_{rr}$ of $S_{1}$ (
$\bigstar$), $S_{2}$ ( $\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ (
$\bigstar$), $S_{5}$ ( $\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ (
$\bigstar$), $S_{8}$ ( $\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ (
$\bigstar$), $S_{11}$ ( $\bigstar$), $S_{12}$ ( $\bigstar$).

Further, to analyze the properties of compact stars, we probe the necessary
conditions for the metric potentials of static spherically symmetric
spacetime, namely $e^{\lambda(0)}=1$ and
$((e^{\lambda(r)})^{\prime})_{r=0}=0$. The graphical response of $g_{rr}$
should not contain any singularities, and the curvature must be regular. From
Fig. 1, it is evident that all above properties are satisfied and also the
plot of $e^{\lambda}$ exhibits monotonically increasing behavior, reaching its
maximum value near the boundary.

##  III Viable $\mathcal{RI}$ gravity model and matching conditions

In this section, we provide some discussions about linear
$\mathcal{R}+\alpha\mathcal{A}$ gravity model and the matching conditions. By
substituting this model in Eq.(II), the field equations are simplified as

$\mathcal{R}^{\eta\xi}-\frac{1}{2}\mathcal{R}g^{\eta\xi}-\alpha\mathcal{A}^{\eta\xi}-\frac{1}{2}\alpha\mathcal{A}g^{\eta\xi}+\frac{\alpha}{2}(2g^{\mu\eta}\nabla_{\beta}\nabla_{\mu}\mathcal{A}_{\sigma}^{\beta}\mathcal{A}^{\xi\sigma}-\nabla^{\kappa}\nabla_{\kappa}\mathcal{A}_{\sigma}^{\eta}\mathcal{A}^{\xi\sigma}-g^{\eta\xi}\nabla_{\beta}\nabla_{\mu}\mathcal{A}_{\sigma}^{\beta}\mathcal{A}^{\mu\sigma})=\mathcal{T}^{\eta\xi}$
(11)

Using Eq. (11) and spacetime (5), modified field equations turn out to be

$\displaystyle\rho$
$\displaystyle=e^{-\nu}\Gamma_{1}-\frac{1}{2}\Gamma_{2}-\alpha
e^{\nu}\Gamma_{3}-\frac{1}{2}\alpha\Gamma_{4}+\frac{\alpha}{2}(e^{\nu-\lambda}\Gamma_{5}^{\prime\prime}+e^{\nu-\lambda}\nu^{\prime\prime}\Gamma_{5}+2e^{\nu-\lambda}\nu^{\prime}\Gamma_{5}^{\prime}+e^{\nu-\lambda}(\nu^{\prime})^{2}\Gamma_{5}-\frac{\lambda^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}^{\prime}$
$\displaystyle-\frac{\nu^{\prime}\lambda^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}+\frac{2}{r}e^{\nu-\lambda}\Gamma_{5}^{\prime}+\frac{2\nu^{\prime}}{r}e^{\nu-\lambda}\Gamma_{5}-\Gamma_{6}^{\prime\prime}-\lambda^{\prime\prime}\Gamma_{6}-\frac{3}{2}\lambda^{\prime}\Gamma_{6}^{\prime}-\frac{(\lambda^{\prime})^{2}}{2}\Gamma_{6}-2re^{-\lambda}\Gamma_{7}^{\prime}-\frac{4}{r^{2}}\Gamma_{6}),$
(12)

$\displaystyle p_{r}$
$\displaystyle=e^{-\lambda}\Gamma_{8}+\frac{1}{2}\Gamma_{2}-\alpha
e^{\lambda}\Gamma_{9}+\frac{1}{2}\alpha\Gamma_{4}+\frac{\alpha}{2}(-\frac{(\nu^{\prime})^{2}}{2}e^{\nu-\lambda}\Gamma_{5}+\frac{\nu^{\prime}\lambda^{\prime}}{2}\Gamma_{6}+\frac{\nu^{\prime}}{2}\Gamma_{6}^{\prime}+4e^{-\lambda}\Gamma_{7}+\frac{2\lambda^{\prime}}{r}\Gamma_{6}+\frac{2}{r}\Gamma_{6}^{\prime}$
$\displaystyle-\frac{\nu^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}^{\prime}+2re^{-\lambda}\Gamma_{7}^{\prime}),$
(13)

$\displaystyle p_{t}$
$\displaystyle=\frac{1}{r^{2}}\Gamma_{10}+\frac{1}{2}\Gamma_{2}-\alpha
r^{2}\Gamma_{11}+\frac{1}{2}\alpha\Gamma_{4}+\frac{\alpha}{2}(\frac{\nu^{\prime}}{2}r^{2}e^{-\lambda}\Gamma_{7}^{\prime}+re^{-\lambda}\nu^{\prime}\Gamma_{7}+r^{2}e^{-\lambda}\Gamma_{7}^{\prime\prime}+4e^{-\lambda}\Gamma_{7}+6re^{-\lambda}\Gamma_{7}^{\prime}$
$\displaystyle-\frac{\lambda^{\prime}}{2}r^{2}e^{-\lambda}\Gamma_{7}^{\prime}-re^{-\lambda}\lambda^{\prime}\Gamma_{7}-\frac{\nu^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}^{\prime}+\frac{(\nu^{\prime})^{2}}{2}\Gamma_{6}+\Gamma_{6}^{\prime\prime}+\lambda^{\prime\prime}\Gamma_{6}+\frac{3}{2}\lambda^{\prime}\Gamma_{6}^{\prime}+\frac{(\lambda^{\prime})^{2}}{2}\Gamma_{6}+\frac{2}{r^{2}}\Gamma_{6}),$
(14)

where, the values of $\Gamma_{i}$’s $(i=1...11)$ are mentioned in Appendix and
the prime represents the derivative with respect to $r$.

The star’s geometrical structure, whether viewed from the outside or the
inside, has no impact on the interior boundary metric. The metric components
must be continuous to the boundary regardless of the referential frame in this
emerging situation. While evaluating several matching conditions and exploring
compact objects in the context of $\mathcal{GR}$, the Schwarzschild solution
seems to be the right approach. According to the Jebsen-Birkhoff theorem,
given spherically symmetric spacetime, the solution of the Einstein field
equations $\mathcal{(EFE)}$ must be asymptotically flat and static. In
modified gravitational theories, the exterior geometry solution may vary from
the Schwarzschild solution when we consider modified $\mathcal{(TOV)}$
equations Tol ; Opp  with zero pressure and energy density. Furthermore, the
Schwarzschild solution can be achieved in $\mathcal{RI}$ gravity by choosing a
realistic and viable $\mathcal{R}+\alpha\mathcal{A}$ gravity model with non-
zero density and pressure components. The Birkhoff theorem contradicts
modified theories as a result Vfi . Numerous studies using Schwarzschild
solution in matching constraints have produced impressive findings Ava ; Acn ;
Agy ; Dmi . The radial pressure $p_{r}(r=R)=0$ is used to calculate the field
equations solution under the specified boundary conditions at $r=R$. Thus, we
compare the internal geometry solution to the Schwarzschild external geometry
provided by

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (15)

where the entire mass contained within the star is represented by $M$. The
following expressions are derived at $r=R$ by probing at the metric potentials
as follows:

$\displaystyle g_{tt}^{+}=g_{tt}^{-},\qquad
g_{rr}^{+}=g_{rr}^{-},\qquad\frac{\partial g_{tt}^{+}}{\partial
r}=\frac{\partial g_{tt}^{-}}{\partial r},$ (16)

where, (+) refers to the external geometry and (-) represents the internal
geometry. The values of the required unknown quantities can be obtained by
comparing the inner and outer matrices as shown below:

$\displaystyle a$
$\displaystyle=\frac{(1+bR^{2})^{2}}{R}\sqrt{\frac{\frac{2M}{R}}{1-\frac{2M}{R}}},$
$\displaystyle b$ $\displaystyle=\frac{4M-R}{R^{2}(9R-20M)},$ (17)
$\displaystyle A$
$\displaystyle=\frac{aB}{2b(1+bR^{2})}+\sqrt{1-\frac{2M}{R}},$ $\displaystyle
B$ $\displaystyle=\left(\frac{1}{2R}\right)\sqrt{\frac{2M}{R}}.$ (18)

We will use these matching conditions to analyze different compact star models
in the background of $\mathcal{R}+\alpha\mathcal{A}$ gravity. It is worthwhile
to mention here that we provide a detailed comparison by considering $12$
compact star models namely $4U~{}1538-52$, $SAX~{}J1808.4-3658$, $Her~{}X-1$,
$LMC~{}X-4$, $SMC~{}X-4$, $4U~{}1820-30$, $Cen~{}X-3$, $4U~{}1608-52$,
$PSR~{}J1903+327$, $PSR~{}J1614-2230$, $Vela~{}X-1$, $EXO~{}1785-248$. The
values of the input parameters radius and mass of considered compact stars are
given in Table I:

Star Model | $M(M_{\Theta})$ | $R(km)$
---|---|---
4U 1538-52 $(S_{1})$ | 0.87 $\pm$ 0.07 SiU  | 7.866 $\pm$ 0.21
SAX J1808.4-3658 $(S_{2})$ | 0.9 $\pm$ 0.3 Ele  | 7.951 $\pm$ 1.0
Her X-1 $(S_{3})$ | 0.85 $\pm$ 0.15 MKA  | 8.1 $\pm$ 0.41
LMC X-4 $(S_{4})$ | 1.04 $\pm$ 0.09 SiU  | 8.301 $\pm$ 0.2
SMC X-4 $(S_{5})$ | 1.29 $\pm$ 0.05 SiU  | 8.831 $\pm$ 0.09
4U 1820-30 $(S_{6})$ | 1.58 $\pm$ 0.06 GT  | 9.1 $\pm$ 0.4
Cen X-3 $(S_{7})$ | 1.49 $\pm$ 0.08 SiU  | 9.178 $\pm$ 0.13
4U 1608-52 $(S_{8})$ | 1.74 $\pm$ 0.01 Wro  | 9.3 $\pm$ 0.10
PSR J1903+327 $(S_{9})$ | 1.667 $\pm$ 0.021 PCC  | 9.48 $\pm$ 0.03
PSR J1614-2230 $(S_{10})$ | 1.97 $\pm$ 0.04 DEM  | 9.69 $\pm$ 0.2
Vela X-1 $(S_{11})$ | 1.77 $\pm$ 0.08 SiU  | 9.56 $\pm$ 0.08
EXO 1785-248 $(S_{12})$ | 1.30 $\pm$ 0.2 Oze  | 10.10 $\pm$ 0.44

Table 1: Approximated values of input parameters

By using the values of the input parameters from Table I and manipulating Eqs.
(17), the values of the output parameters $a$, $b$, $A$, and $B$ are shown in
the following Table:

Star Model | $a(km)$ | $b(km)$ | $A(km)$ | $B(km)$
---|---|---|---|---
4U 1538-52 $(S_{1})$ | 0.0784715 | -0.00097169 | -0.744456 | 0.0364083
SAX J1808.4-3658 $(S_{2})$ | 0.0793101 | -0.000920891 | -0.851084 | 0.0364385
Her X-1 $(S_{3})$ | 0.0726951 | -0.000977202 | -0.538845 | 0.0344393
LMC X-4 $(S_{4})$ | 0.0838586 | -0.000705186 | -1.50207 | 0.0367192
SMC X-4 $(S_{5})$ | 0.0934356 | -0.000366545 | -4.13713 | 0.0372694
4U 1820-30 $(S_{6})$ | 0.115014 | 0.0000941913 | 24.5842 | 0.039431
Cen X-3 $(S_{7})$ | 0.103164 | -0.000104682 | -18.0737 | 0.0378044
4U 1608-52 $(S_{8})$ | 0.127851 | 0.000368416 | 7.40205 | 0.0400518
PSR J1903+327 $(S_{9})$ | 0.112672 | 0.000126983 | 17.4001 | 0.0380914
PSR J1614-2230 $(S_{10})$ | 0.156003 | 0.0009043 | 3.91745 | 0.0413433
Vela X-1 $(S_{11})$ | 0.122018 | 0.000306813 | 8.16831 | 0.0387589
EXO 1785-248 $(S_{12})$ | 0.0708703 | -0.000447232 | -1.75321 | 0.0305887

Table 2: Approximated values of output parameters with $\alpha=1.3\times
10^{-11}$

Further, the requirements for a well-behaved compact structure are specified
as:

$\bullet$ The plots of $\rho$, $p_{r}$, and $p_{t}$ must always be positive,
finite, and maximum at the center.
$\bullet$ The gradient of $\rho$, $p_{r}$, and $p_{t}$ should be negative.
$\bullet$ All of the energy requirements must be achieved.
$\bullet$ The equation of state ($EoS$) should satisfy the mandatory
constraints, namely $0<\omega_{r}$, $\omega_{t}<1$.
$\bullet$ The equilibrium stability condition must be fulfilled by all of the
forces.
$\bullet$ The parameters of speed of sound must be between $[0,1]$, i.e.
$0<\upsilon_{r}^{2}$, and $\upsilon_{t}^{2}<1$.
$\bullet$ Adiabatic index must be greater than $\frac{4}{3}$ for an
anisotropic fluid sphere.

##  IV Physical characteristics of the compact star models

In this section, we discuss some important physical properties of the observed
compact stars in the context of considered $\mathcal{R}+\alpha\mathcal{A}$
gravity model. For this purpose, we provide the graphical representations of
energy density, pressure components, energy conditions, mass function,
adiabatic index, stability and equilibrium conditions. Moreover, for our given
model, we take the parameter $\alpha=1.3\times 10^{-11}$ for various compact
stars. The parameter $\alpha$ has a very limited range which provides us
physically accepted results. We have checked two other values of parameter
$\alpha=1.0\times 10^{-9}$, $\alpha=1.9\times 10^{-8}$ and found that there is
a very slight difference in the behavior of the graphs towards the boundary.
That difference is shown for one star $Vela~{}X-1$ as a magnified image in
Fig. 2.

###  IV.1 Energy Density and Pressure Profiles

We display the graphical behavior of $\rho$, $p_{r}$ and $p_{t}$ for
$\mathcal{R}+\alpha\mathcal{A}$ model in this subsection. The graphical action
of these aspects is presented in Fig. 2. In addition, we also show the
graphical response of gradients of $\rho$, $p_{r}$ and $p_{t}$ in the Fig 3.
It is clearly seen from the Fig. 2 that plots of energy density, radial and
transverse pressures are positive as well as finite. One can easily notice
that the demonstration of $\rho$, $p_{r}$ and $p_{t}$ components reach the
maximum values at the center of compact star and decline towards the stellar
surface boundary.

|  |  |
---|---|---|---
Figure 2: Behavior of $\rho$, $p_{r}$ and $p_{t}$ of $S_{1}$ ( $\bigstar$),
$S_{2}$ ( $\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ ( $\bigstar$), $S_{5}$ (
$\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ ( $\bigstar$), $S_{8}$ (
$\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ ( $\bigstar$), $S_{11}$ (
$\bigstar$), $S_{12}$ ( $\bigstar$). The magnified images in the figure show
the behavior of the star $Vela~{}X-1$ for two different values of parameter
$\alpha=1.0\times 10^{-9}$, $\alpha=1.9\times 10^{-8}$.

In Fig. 3, the gradients of $\rho$, $p_{r}$ and $p_{t}$ are plotted which are
finite but negative. These elements indicate the high compactness character of
the compact stars.

|  |  |
---|---|---|---
Figure 3: Variation of $\frac{d\rho}{dr}$, $\frac{dp_{r}}{dr}$ and
$\frac{dp_{t}}{dr}$ of $S_{1}$ ( $\bigstar$), $S_{2}$ ( $\bigstar$), $S_{3}$ (
$\bigstar$), $S_{4}$ ( $\bigstar$), $S_{5}$ ( $\bigstar$), $S_{6}$ (
$\bigstar$), $S_{7}$ ( $\bigstar$), $S_{8}$ ( $\bigstar$), $S_{9}$ (
$\bigstar$), $S_{10}$ ( $\bigstar$), $S_{11}$ ( $\bigstar$), $S_{12}$ (
$\bigstar$).

###  IV.2 Anisotropy

In this portion, we discuss the graphical response of anisotropy parameter Cgb
which is represented by $\Delta$ and defined as the difference between the
transverse and radial pressures i.e. $\Delta=p_{t}-p_{r}$. The nature of
$\Delta$ is attractive if $p_{t}<p_{r}$ and repulsive if $p_{t}>p_{r}$.
Therefore, the anisotropy must indicate repulsive nature Mkg  for the compact
stars to exist. From Fig. 4, it is obvious that anisotropy remains positive in
our case i.e. $\Delta>0$, which implies that anisotropic force is drawn
outwards.

|
---|---
Figure 4: Evolution of $\Delta$ of $S_{1}$ ( $\bigstar$), $S_{2}$ (
$\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ ( $\bigstar$), $S_{5}$ (
$\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ ( $\bigstar$), $S_{8}$ (
$\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ ( $\bigstar$), $S_{11}$ (
$\bigstar$), $S_{12}$ ( $\bigstar$).

###  IV.3 Energy Conditions

The significant role of energy conditions in describing the existence of
anisotropic compact stars is widely known in literature Zya . These energy
limitations are classified namely null energy, strong energy, weak energy and
dominant energy bounds represented by $\mathcal{NEC}$, $\mathcal{SEC}$,
$\mathcal{WEC}$ and $\mathcal{DEC}$, sequentially and expressed as

$\displaystyle\mathcal{NEC}:\rho+p_{r}\geq 0,\quad\rho+p_{t}\geq
0,\qquad\mathcal{WEC}:\rho\geq 0,\quad\rho+p_{r}\geq 0,\quad\rho+p_{t}\geq 0,$
$\displaystyle\mathcal{SEC}:\rho+p_{r}+2p_{t}\geq
0,\qquad\mathcal{DEC}:\rho\geq 0,\quad\rho-|p_{r}|\geq 0,\quad\rho-|p_{t}|\geq
0.$ (19)

From Fig. 5, it is clearly observed that all energy conditions are satisfied
for considered $\mathcal{R}+\alpha\mathcal{A}$ gravity model.

|  |  |
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Figure 5: Profiles of the energy conditions of $S_{1}$ ( $\bigstar$), $S_{2}$
( $\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ ( $\bigstar$), $S_{5}$ (
$\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ ( $\bigstar$), $S_{8}$ (
$\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ ( $\bigstar$), $S_{11}$ (
$\bigstar$), $S_{12}$ ( $\bigstar$).

###  IV.4 Equilibrium Conditions

Now, we investigate the equilibrium condition of our model under the existence
of hydrostatic force, gravitational force and anisotropic force. The
equilibrium condition among all these forces can be written as

$\frac{M_{G}(r)(\rho+p_{r})}{r}e^{\frac{\lambda-\upsilon}{2}}+\frac{dp_{r}}{dr}-\frac{2}{r}(p_{t}-p_{r})=0.$
(20)

The effective gravitational mass $M_{G}(r)$ is expressed as

$M_{G}(r)=\frac{1}{2}\upsilon^{\prime}e^{\frac{\upsilon-\lambda}{2}}.$ (21)

Inserting value of $M_{G}(r)$ in Eq. (21), we get

$\frac{\upsilon^{\prime}}{2}(\rho+p_{r})+\frac{dp_{r}}{dr}-\frac{2}{r}(p_{t}-p_{r})=0,$
(22)

where, $\mathcal{F}_{g}=-\frac{\upsilon^{\prime}}{2}(\rho+p_{r})$,
$\mathcal{F}_{h}=-\frac{dp_{r}}{dr}$ and $\mathcal{F}_{a}=\frac{2}{r}\Delta$
with $\mathcal{F}_{g}$, $\mathcal{F}_{h}$ and $\mathcal{F}_{a}$ identified as
gravitational force, hydrostatic force and anisotropic force. The sum of these
three forces should be equal to zero for the system to be balanced. Therefore,
Eq. (23) gives

$\mathcal{F}_{g}+\mathcal{F}_{h}+\mathcal{F}_{a}=0.$ (23)

It can be easily noticed from Fig. 6, that all these forces express the
required equilibrium condition.

|
---|---
Figure 6: Behavior of $\mathcal{F}_{h}$, $\mathcal{F}_{g}$ and
$\mathcal{F}_{a}$ of $S_{1}$ ( $\bigstar$), $S_{2}$ ( $\bigstar$), $S_{3}$ (
$\bigstar$), $S_{4}$ ( $\bigstar$), $S_{5}$ ( $\bigstar$), $S_{6}$ (
$\bigstar$), $S_{7}$ ( $\bigstar$), $S_{8}$ ( $\bigstar$), $S_{9}$ (
$\bigstar$), $S_{10}$ ( $\bigstar$), $S_{11}$ ( $\bigstar$), $S_{12}$ (
$\bigstar$).

###  IV.5 Equation of State

Further, we determine ($EoS$) parameters for radial and tangential pressures,
presented by

$\displaystyle\omega_{r}=\frac{p_{r}}{\rho},\qquad\omega_{t}=\frac{p_{t}}{\rho}.$
(24)

The graphical illustration of both parameters in Fig. 7, clearly shows that
$0<\omega_{r}$, $\omega_{t}<1$. Thus, it is concluded that the $EoS$ satisfy
the necessary conditions for compact stars under discussion.

|  |
---|---|---
Figure 7: Profiles of $\omega_{r}$ and $\omega_{t}$ of $S_{1}$ ( $\bigstar$),
$S_{2}$ ( $\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ ( $\bigstar$), $S_{5}$ (
$\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ ( $\bigstar$), $S_{8}$ (
$\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ ( $\bigstar$), $S_{11}$ (
$\bigstar$), $S_{12}$ ( $\bigstar$).

###  IV.6 Causality Condition

The causality condition is subjected to the sound speed of both pressure
components, i.e. radial and transverse pressures which are expressed as

$\displaystyle\upsilon_{r}^{2}=\frac{dp_{r}}{d\rho},\qquad\upsilon_{t}^{2}=\frac{dp_{t}}{d\rho}.$
(25)

Here, $\upsilon_{r}^{2}$ and $\upsilon_{t}^{2}$ stand for radial and
tangential sound speed. Using the concept of Herrera Lhr  i.e.,
$(0\leq\upsilon_{r}^{2},\upsilon_{t}^{2}\leq 1)$, both of the sound speeds of
pressure components must vary from 0 to 1 to stabilize our system. Further, we
examine the graphical response of our selected model for the Andreasson
condition Han , $|\upsilon_{r}^{2}-\upsilon_{t}^{2}|\leq 1$. The graphical
representations in Fig. 8, indicate the stability and balanced nature of
$\mathcal{R}+\alpha\mathcal{A}$ model satisfying the causality condition.

|  |
---|---|---

|  |
---|---|---

Figure 8: Behavior of speeds of sound parameters, and Abreu condition of
$S_{1}$ ( $\bigstar$), $S_{2}$ ( $\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ (
$\bigstar$), $S_{5}$ ( $\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ (
$\bigstar$), $S_{8}$ ( $\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ (
$\bigstar$), $S_{11}$ ( $\bigstar$), $S_{12}$ ( $\bigstar$).

###  IV.7 Mass Function, Compactness Factor and Redshift Analysis

The mass function Bhr  attained by making use of the metric potential
$g_{rr}^{-}=g_{rr}^{+}$ is given as

$\mathcal{M}(r)=\frac{a^{2}r^{3}}{2[(1+br^{2})^{4}+a^{2}r^{2}]}.$ (26)

Moreover, we compute the compactness parameter $\mathcal{U}(r)$ Mkm  and the
redshift function $\mathcal{Z}(r)$ Cgb , derived as the following expressions

$\mathcal{U}(r)=\frac{2\mathcal{M}(r)}{r}=\frac{a^{2}r^{2}}{[(1+br^{2})^{4}+a^{2}r^{2}]},$
(27) $\mathcal{Z}(r)=e^{-\frac{\nu}{2}}-1.$ (28)

Fig. 9 shows the graphical behavior of $\mathcal{M}(r)$, $\mathcal{U}(r)$ and
$\mathcal{Z}(r)$. The nature of mass function meets the Buchdahl Hab  and
Bondi Hbi  constraint i.e., $\frac{2M}{R}<\frac{8}{9}$. In addition, it can be
noticed that as we advance towards the boundary, mass function and compactness
parameter increase monotonically while the plot of redshift indicates
monotonically decreasing behavior. Therefore, we sum up that these graphs
illustrate the satisfactory behavior for our model.

|  |  |
---|---|---|---
Figure 9: Graphs of mass function (left panel), compactness factor (middle
panel), and redshift(right panel) of $S_{1}$ ( $\bigstar$), $S_{2}$ (
$\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ ( $\bigstar$), $S_{5}$ (
$\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ ( $\bigstar$), $S_{8}$ (
$\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ ( $\bigstar$), $S_{11}$ (
$\bigstar$), $S_{12}$ ( $\bigstar$).

###  IV.8 Adiabatic Index

The stiffness of ($EoS$) for a given energy density is described by adiabatic
index. It verifies the stability of stellar objects, both relativistically and
non relativistically. Chandrasekhar Scd  presented the theory of dynamical
stability for the infinitesimal, radial adiabatic oscillations of stellar
spheres. His idea has been tested by many authors for isotropic and
anisotropic stellar structures Hhn ; Whi ; Dho ; Ddd ; Hos ; Iba . The value
of adiabatic index should be greater than $\frac{4}{3}$, for compact star
models to be stable. Adiabatic index of pressure components, $p_{r}$ and
$p_{t}$ have the following expressions respectively

$\displaystyle\gamma_{r}=\frac{\rho+p_{r}}{p_{r}}(\frac{dp_{r}}{d\rho})=\frac{\rho+p_{r}}{p_{r}}v_{r}^{2},\qquad\gamma_{t}=\frac{\rho+p_{t}}{p_{t}}(\frac{dp_{t}}{d\rho})=\frac{\rho+p_{t}}{p_{t}}v_{t}^{2}.$
(29)

It is concluded from Fig. 10 that the considered
$\mathcal{R}+\alpha\mathcal{A}$ model is potentially balanced under the above
stability condition.

|  |
---|---|---
Figure 10: Variation of $\gamma_{r}$, and $\gamma_{t}$ of $S_{1}$ (
$\bigstar$), $S_{2}$ ( $\bigstar$), $S_{3}$ ( $\bigstar$), $S_{4}$ (
$\bigstar$), $S_{5}$ ( $\bigstar$), $S_{6}$ ( $\bigstar$), $S_{7}$ (
$\bigstar$), $S_{8}$ ( $\bigstar$), $S_{9}$ ( $\bigstar$), $S_{10}$ (
$\bigstar$), $S_{11}$ ( $\bigstar$), $S_{12}$ ( $\bigstar$).

##  V Summary and Concluding Remarks

We assume a realistic $\mathcal{R}+\alpha\mathcal{A}$ gravity model by using
the Karmarkar condition to investigate a novel family of embedded class-I
solutions in the anisotropic framework. Since the Karmarkar condition provides
a link between $g_{rr}$ and $g_{tt}$, it simplifies the solution generating
process of $\mathcal{EFE}$ to a single metric potential. Assuming the metric
potential $e^{\lambda}=1+\frac{a^{2}r^{2}}{(1+br^{2})^{4}}$ for this purpose,
we use the Karmarkar condition to determine the second metric potential,
$e^{\nu}=\left[\left(A-\frac{aB}{2b(1+br^{2})}\right)^{2}\right]$, where $a$,
$b$, $A$, and $B$ are arbitrary constants. We also determine the unknown
constants by applying matching constraints between the internal and
Schwarzschild external geometries. We examine the physical behavior of twelve
different compact stars to confirm that the discussed solutions are physically
feasible. The primary objective of our research is to study stellar structures
using a viable $\mathcal{R}+\alpha\mathcal{A}$ gravity model in the context of
an anisotropic matter source. The notable results are listed below.

  * •

Metric potentials are frequently used to characterize the properties of
spacetime. According to Fig. 1, both the metric potentials
$g_{rr}=e^{\lambda}$ and $g_{tt}=e^{\nu}$ exhibit graphical behavior that is
finite, positive, singularity-free, and satisfies the criteria, i.e.,
$e^{\lambda(r=0)}=1$ and $e^{\nu(r=0)}=\left[A-\frac{aB}{2b}\right]^{2}$.
Since these graphs expand monotonically and reach their maximum values at the
boundary, the progression of both potentials exhibits excellent outcomes,
supporting the $\mathcal{R}+\alpha\mathcal{A}$ gravity model.

  * •

The trend of density and pressure components for
$\mathcal{R}+\alpha\mathcal{A}$ gravity model is shown in Fig. 2. For this
purpose, we consider the model parameter $\alpha=1.3\times 10^{-11}$. It is
important to mention here that the parameter $\alpha$ has a very limited range
which provides us physically accepted results. We have checked two other
values of parameter $\alpha=1.0\times 10^{-9}$, $\alpha=1.9\times 10^{-8}$ and
found that there is a very slight difference in the behavior of the graphs
towards the boundary. That difference is shown for one star only $Vela~{}X-1$
as a magnified image in Fig. 2. The variations of $\rho$, $p_{r}$, and $p_{t}$
with respect to $r$ are finite and regular at the center. It is also evident
that graphs of $\rho$, $p_{r}$, and $p_{t}$ reach their highest values at the
center and show a diminishing response toward the border, supporting the
stability of our model. The gradient of $\rho$, $p_{r}$, and $p_{t}$ has been
depicted in Fig. 3. These graphs exhibit negative behavior, demonstrating the
consistency of our suggested model.

  * •

It is clear from Fig. 4 that anisotropy is positive throughout the stellar
objects. This behavior demonstrates that anisotropy has a repulsive nature,
confirming its existence for compact objects.

  * •

The energy bounds for the suggested $\mathcal{R}+\alpha\mathcal{A}$ model are
shown in Fig. 5. We observed that our given model fulfills all energy bounds.

  * •

The equilibrium conditions of $\mathcal{F}_{g}$, $\mathcal{F}_{h}$, and
$\mathcal{F}_{a}$ of our physically acceptable model show balancing behavior
in Fig. 6.

  * •

The graphical response of $EoS$ must fulfill the requirements $0<\omega_{r}$,
$\omega_{t}<1$ for the consistency of the stellar objects. The corresponding
graphical representation in Fig. 7 depicts that both $EoS$ ratios have a
stable character.

  * •

The radial component sound velocity $\upsilon_{r}^{2}$ plot and the transverse
component sound velocity $\upsilon_{t}^{2}$ plot should fall within the range
[0, 1] for compact stars. According to Fig. 8, the stability constraints for
our model are stable.

  * •

From Fig. 9, it is easy to notice that the graphs of the mass function and the
compactness parameter demonstrate monotonically increasing responses. On the
other hand, the graphical illustration of redshift analysis reveals
monotonically decreasing behavior. The behavior shows that our given model
achieves the stability criteria.

  * •

In Fig. 10, the adiabatic index of $\gamma_{r}$ and $\gamma_{t}$ for our given
model is more than $\frac{4}{3}$, confirming the stability of our system.

As a final observation, we have used the Karmarkar condition to provide a
stellar system that is both well-stable and singularity-free. We conclude that
$\mathcal{R}+\alpha\mathcal{A}$ gravity model supports the existence of
compact objects which follow observable patterns. Furthermore, it is worth
mentioning that our suggested results are almost similar to the outcomes
investigated by Naz et al. Naz  in the context of $f(\mathcal{R})$ theory of
gravity. It may be noted that due to the involvement of inverse curvature
terms, we get more dense stars in our case.

## Acknowledgements

The authors appreciate important comments from anonymous reviewer which
improved the paper.

## References

## References

  * (1) S.K. Maurya, Y.K. Gupta, Saibal Ray, S.R. Chowdhury, Eur. Phys. J. C 75, 389 (2015).
  * (2) P. Bhar, K.N. Singh, T. Manna, Int. J. Mod. Phys. D 26, 1750090 (2017).
  * (3) A.K. Prasad, et al., Astrophys. Space Sci. 364, 66 (2019).
  * (4) S. K. Maurya, M. Govender, S. Kaur, R. Nag, Eur. Phys. J. C 82, 100 (2022).
  * (5) V.U.M. Rao, K.V.S. Sireesha, and D.Ch. Papa Rao, Eur. Phys. J. Plus 129, 17 (2014).
  * (6) B. Modak, K. Sarkar A. K Sanyal, Astrophys Space Sci. 353, 707–720 (2014).
  * (7) H. Azmat, M. Zubair, Eur. Phys. J. Plus 136, 112 (2021).
  * (8) R. Nagpal, S. K. J Pacif, Eur. Phys. J. Plus 136, 875 (2021).
  * (9) H. Azmat, M. Zubair, Z. Ahmad, Ann. Phys. 439, 168769 (2022).
  * (10) A. Addazi, S. Capozziello, S. Odintsovgh, Phys. Lett. B 816, 136257, (2021).
  * (11) H. A. Buchdahl, Mon. Not. R. Ast. Soc. 150, 1 (1970).
  * (12) M. F. Shamir, Eur. Phys. J. C 75, 354 (2015).
  * (13) M. F. Shamir, T. Naz, Phys. Lett. B 806, 135519 (2020).
  * (14) M. F. Shamir, M. Ahmad, Mod. Phys. Lett. A 32, 1750086 (2017).
  * (15) M. F. Shamir, S. Zia, Eur. Phys. J. C 77, 448 (2017).
  * (16) L. Amendola, L. Giani, G. Laverda, Phys. Let. B 811, 135923(2020).
  * (17) K. Schwarzschild, Sit. Dtsch. Akad. Wiss. Math. Phys. Ber. 24, 424 (1916).
  * (18) R.C. Tolman, Phys. Rev. 55, 364 (1939).
  * (19) J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55, 374 (1939).
  * (20) R.L. Bowers, E.P.T. Liang, Astrophys. J. 188, 657 (1974).
  * (21) R. Ruderman, Annu. Rev. Astron. Astrophys. 10, 427 (1972).
  * (22) A. Errehymy, Y. Khedif, M. Daoud, Eur. Phys. J. C 81, 266 (2021).
  * (23) S.K. Maurya, B.S. Ratanpal, M. Govender, Ann. Phys. 382, 36-49 (2017).
  * (24) S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59 (2011).
  * (25) A.A. Starobinsky, JETP Lett. 86, 157 (2007).
  * (26) W. Hu, I. Sawicki, Phys. Rev. D 76, 064004 (2007).
  * (27) G. Cognola, et al., Phys. Rev. D 77, 046009 (2008).
  * (28) L. Schlai, Ann. Mat. 5, 170 (1871).
  * (29) M. F. Shamir, A. Usman, T. Naz, Eur. Phys. J. Plus 136, 55 (2021).
  * (30) T. Naz, A. Usman, M. F. Shamir, Ann. Phys. 429, 168491(2021).
  * (31) K.N. Singh, N. Pant, Eur. Phys. C 76, 524 (2016).
  * (32) V. Faraoni, Phys. Rev. D 81, 044002 (2010).
  * (33) A.V. Astashenok, S. Capozziello, S.D. Odintsov, J. Cosmol. Astropart. Phys. 12, 040 (2013).
  * (34) A. Cooney, S.D. Deo, D. Psaltis, Phys. Rev. D 82, 064033 (2010).
  * (35) A. Ganguly, R. Gannouji, R. Goswami, S. Ray, Phys. Rev. D 89, 064019 (2014).
  * (36) D. Momeni, R. Myrzakulov, Int. J. Geom. Methods Mod. Phys. 12, 1550014 (2015).
  * (37) C. G. Bohmer, T. Harko, Cla. Qua. Grav. 23, 6479 (2006).
  * (38) M.K. Gokhroo, A.L. Mehra, Gen. Relativity Gravitation 26, 75 (1994).
  * (39) Z. Yousaf, Eur. Phys. J. Plus 132 (2017) 276.
  * (40) L. Herrera, Phys. Lett. A 165, 206 (1992).
  * (41) H. Andreasson, Comm. Math. Phys. 288, 715 (2009).
  * (42) M.K. Mak, T. Harko, Proc. R. Soc. Lond. A 459, 393 (2003).
  * (43) C. G. Bohmer, T. Harko, Cla. Qua. Grav. 23, 6479 (2006).
  * (44) H.A. Buchdahl, Phys. Rev. 116, 1027 (1959).
  * (45) H. Bondi, Proc. R. Soc. Lond. A 281, 39 (1964).
  * (46) S. Chandrasekhar, Astrophys. J. 140, 417 (1964).
  * (47) H. Heintzmann, W. Hillebrandt, Astron. Astrophys. 38, 51 (1975).
  * (48) W. Hillebrandt, K.O. Steinmetz, Astron. Astrophys. 53, 283 (1976).
  * (49) D. Horvat, S. Ilijic, A. Marunovic, Cla. Qua. Grav. 28, 025009 (2011).
  * (50) D.D. Doneva, S.S. Yazadjiev, Phys. Rev. D 85, 124023 (2012).
  * (51) H.O. Silva, C.F.B. Macedo, E. Berti, L.C.B. Crispino, Cla. Qua. Grav. 32, 145008 (2015).
  * (52) I. Bombaci, Astron. Astrophys. 305, 871 (1996).
  * (53) M.L. Rawls, et al., ApJ, 730, 25 (2011).
  * (54) P. Elebert, M. T. Reynolds, P. J. Callanan, et al., MNRAS, 395, 884 (2009).
  * (55) M. K. Abubekerov, E. A. Antokhina, A. M. Cherepashchuk, V. V. Shimanskii, Astronomy Reports, 52, 379 (2008).
  * (56) T. Guver, F. Ozel, A. Cabrera-Lavers, P. Wroblewski, ApJ, 712, 964 (2010).
  * (57) T. Guver, P. Wroblewski, L. Camarota, F. Ozel, ApJ, 719, 1807 (2010).
  * (58) P.C.C. Freire, et. al., MNRAS, 412, 2763 (2011)
  * (59) P.B. Demorest, T. Pennucci, S.M. Ransom, M.S.E. Roberts, J.W.T. Hessels, Nature, 467, 1081 (2010).
  * (60) F. Ozel, T. Guver, D. Psaltis, ApJ, 693, 1775 (2009).

## Appendix

Following are the values of $\Gamma_{i}$’s,

$\displaystyle\Gamma_{1}=-\frac{a(1+br^{2})(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})B(-2b(1+br^{2})A+aB)}{2b(a^{2}r^{2}+(1+br^{2})^{4})^{2}}$
$\displaystyle\Gamma_{2}=\frac{(2a(2a^{3}br^{2}(1+br^{2})A-2ab(1+br^{2})^{4}(-3+5br^{2})A-a^{4}r^{2}B+a^{2}(-3+br^{2})(1+br^{2})^{3}B+2b(-3+br^{2})(1+br^{2})^{6}B))}{((a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB))}$
$\displaystyle\Gamma_{3}=-\frac{2b(a^{2}r^{2}+(1+br^{2})^{4})^{2}}{a(1+br^{2})(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})B(-2b(1+br^{2})A+aB)}$
$\displaystyle\Gamma_{4}$
$\displaystyle=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(2b(1+br^{2})A-aB)}{2ab(1+br^{2})^{3}(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})B}$
$\displaystyle+\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB)}{2a(1+br^{2})^{3}(-1+3br^{2})(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)}$
$\displaystyle+\frac{2(a^{2}r^{2}+(1+br^{2})^{4})^{2}(2b(1+br^{2})A-aB)}{a(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)}$
$\displaystyle\Gamma_{5}=-\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{4}(2b(1+br^{2})A-aB)}{a^{2}(1+br^{2})^{4}(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})^{2}B^{2}(-2b(1+br^{2})A+aB)}$
$\displaystyle\Gamma_{5}^{\prime}=-\frac{8r(a^{2}r^{2}+(1+br^{2})^{4})^{3}(a^{4}r^{2}(-1+br^{2})+b(-5+br^{2})(1+br^{2})^{7}-a^{2}(1+br^{2})^{3}(2+br^{2}(-1+7br^{2})))}{a^{2}(1+br^{2})^{5}(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})^{3}B^{2}}$
$\displaystyle\Gamma_{5}^{\prime\prime}$
$\displaystyle=(8(a^{2}r^{2}+(1+br^{2})^{4})^{2}(2a^{8}r^{6}(3+br^{2}(-12+5br^{2}))+b(1+br^{2})^{14}(15+br^{2}(181+br^{2}(-67+7br^{2})))$
$\displaystyle-a^{6}r^{4}(1+br^{2})^{3}(-25+br^{2}(93+br^{2}(-151+35br^{2})))-2a^{2}(1+br^{2})^{10}(-3+br^{2}(-37+br^{2}(-71+br^{2}$
$\displaystyle(-211+34br^{2}))))+a^{4}r^{2}(1+br^{2})^{6}(31+3br^{2}(-23+br^{2}(125+br^{2}(-13+122br^{2}))))))/$
$\displaystyle(a^{2}(1+br^{2})^{6}(-2a^{2}r^{2}+(-3+br^{2})(1+br^{2})^{3})^{4}B^{2})$
$\displaystyle\Gamma_{6}=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{3}(2b(1+br^{2})A-aB)(-2b(1+br^{2})A+aB)}{4a^{2}(1+br^{2})^{2}(-1+3br^{2})^{2}(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)^{2}}$
$\displaystyle\Gamma_{6}^{\prime}$
$\displaystyle=(r(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-48b^{4}(-1+br^{2})(1+br^{2})^{9}A^{2}B-6a^{5}b(3+br^{2}+b^{2}r^{4}+3b^{3}r^{6})AB^{2}+a^{6}(3+br^{2}(-2+3br^{2}))B^{3}$
$\displaystyle+a^{4}b(1+br^{2})^{2}B(12b(3+br^{2}(-2+3br^{2}))A^{2}-(-1+br^{2})(1+br^{2})(19+3br^{2})B^{2})+4ab^{3}(1+br^{2})^{7}A(16b(-2+3br^{2})A^{2}$
$\displaystyle+(1+br^{2})(-11+9br^{2})B^{2})+4b^{2}(a+abr^{2})^{3}A(-2b(3+br^{2}(-2+3br^{2}))A^{2}+(1+br^{2})(-27+br^{2}(19+18br^{2}))B^{2})$
$\displaystyle+2a^{2}b^{2}(1+br^{2})^{5}B(-6b(-17+br^{2}(10+19br^{2}))A^{2}-(-5+3br^{2})(B+br^{2}B)^{2})))/(2a^{2}(1+br^{2})^{3}(-1+3br^{2})^{3}$
$\displaystyle(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)^{3})$
$\displaystyle\Gamma_{6}^{\prime\prime}$
$\displaystyle=-(((a^{2}r^{2}+(1+br^{2})^{4})(48b^{5}(1+br^{2})^{16}(1+br^{2}(23+br^{2}(-37+21br^{2})))A^{2}B^{2}-8a^{9}br^{2}(1+br^{2})(15+$
$\displaystyle
br^{2}(-8+br^{2}(94+3br^{2}(-16+9br^{2}))))AB^{3}+a^{10}r^{2}(15+br^{2}(-8+br^{2}(94+3br^{2}(-16+9br^{2}))))B^{4}$
$\displaystyle+4ab^{4}(1+br^{2})^{14}AB(-8b(7+25br^{2}(7+3br^{2}(-5+3br^{2})))A^{2}-(1+br^{2})(11+3br^{2}(79-93br^{2}+45b^{2}r^{4}))B^{2})$
$\displaystyle+4a^{3}b^{3}(1+br^{2})^{10}AB(-4b(35+br^{2}(1100+3br^{2}(-618+br^{2}(-28+561br^{2}))))A^{2}-(1+br^{2})(43+br^{2}(1042+$
$\displaystyle
br^{2}(-700-546br^{2}+513b^{2}r^{4})))B^{2})+4a^{7}b(1+br^{2})^{3}AB(-8b^{2}r^{2}(15+br^{2}(-8+br^{2}(94+$
$\displaystyle
3br^{2}(-16+9br^{2}))))A^{2}-(1+br^{2})(6+br^{2}(355+3br^{2}(4+br^{2}(98+39br^{2}(-2+3br^{2})))))B^{2})+4a^{5}b^{2}$
$\displaystyle(1+br^{2})^{6}AB(-4b(6+br^{2}(325+br^{2}(106+3br^{2}(-56+br^{2}(-8+105br^{2})))))A^{2}-(1+br^{2})(41+br^{2}(1376+$
$\displaystyle
br^{2}(-871+9br^{2}(-31+15br^{2}(6+5br^{2})))))B^{2})+a^{8}(B+br^{2}B)^{2}(24b^{2}r^{2}(15+br^{2}(-8+br^{2}(94+3br^{2}$
$\displaystyle(-16+9br^{2}))))A^{2}+(1+br^{2})(3+br^{2}(185+br^{2}(-24+br^{2}(280+3br^{2}(-49+45br^{2})))))B^{2})+2a^{2}b^{3}(1+br^{2})^{12}$
$\displaystyle(64b^{2}(2+br^{2}(59+3br^{2}(-56+45br^{2})))A^{4}+2b(85+br^{2}(2152+3br^{2}(-650-480br^{2}+603b^{2}r^{4})))$
$\displaystyle
A^{2}B^{2}+(1+br^{2})^{2}(5+br^{2}(101+3br^{2}(-23+9br^{2})))B^{4})+a^{4}b^{2}(1+br^{2})^{8}(16b^{2}(1+br^{2})(3+br^{2}$
$\displaystyle(149+3br^{2}(-77+45br^{2})))A^{4}+4b(114+br^{2}(3773+br^{2}(-2344+3br^{2}(-1310+9br^{2}(154+127br^{2})))))A^{2}B^{2}$
$\displaystyle+(1+br^{2})^{2}(29+br^{2}(660+br^{2}(-242+3br^{2}(-100+63br^{2}))))B^{4})+a^{6}b(1+br^{2})^{4}(16b^{3}r^{2}(15+br^{2}$
$\displaystyle(-8+br^{2}(94+3br^{2}(-16+9br^{2}))))A^{4}+8b(1+br^{2})(9+br^{2}(510+br^{2}(95+3br^{2}(25+br^{2}(-68+177br^{2})))))$
$\displaystyle
A^{2}B^{2}+(1+br^{2})^{2}(22+br^{2}(745+3br^{2}(-174+br^{2}(212-60br^{2}+81b^{2}r^{4}))))B^{4})))/(2a^{2}(1-3br^{2})^{4}$
$\displaystyle(1+br^{2})^{4}(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)^{4}))$
$\displaystyle\Gamma_{7}=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{4}(2b(1+br^{2})A-aB)(-2b(1+br^{2})A+aB)}{a^{2}r^{2}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)^{2}}$
$\displaystyle\Gamma_{7}^{\prime}$
$\displaystyle=(2(a^{2}r^{2}+(1+br^{2})^{4})^{3}(-8b^{3}(1+br^{2})^{12}(-1+3br^{2})A^{2}B+6a^{7}br^{4}(1+br^{2})AB^{2}-a^{8}r^{4}B^{3}+2a^{2}b(1+br^{2})^{8}B(12b$
$\displaystyle(1+br^{2}(-12+7br^{2}))A^{2}-(-1+br^{2})(B+br^{2}B)^{2})+8ab^{2}(1+br^{2})^{9}A(-2b(1+br^{2}(-12+7br^{2}))A^{2}+(-1+2br^{2})(B+$
$\displaystyle
br^{2}B)^{2})+2(a+abr^{2})^{4}B(-2b^{2}r^{2}(9+br^{2}(8+63br^{2}))A^{2}+(-1+br^{2})(-1+11br^{2})(B+br^{2}B)^{2})+4a^{3}b(1+br^{2})^{5}A$
$\displaystyle(2b^{2}r^{2}(3+br^{2}(2+23br^{2}))A^{2}-(3+br^{2}(-36+25br^{2}))(B+br^{2}B)^{2})-a^{6}(r+br^{3})^{2}B(3B^{2}+br^{2}(12bA^{2}+(4+13b$
$\displaystyle
r^{2})B^{2}))+2a^{5}br^{2}(1+br^{2})^{3}A(9B^{2}+br^{2}(4bA^{2}+(10+53br^{2})B^{2}))))/(a^{2}r^{3}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})$
$\displaystyle(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)^{3})$
$\displaystyle\Gamma_{7}^{\prime\prime}$
$\displaystyle=(2(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-48b^{4}(1+br^{2})^{22}(1+br^{2}(-2+5br^{2}))A^{2}B^{2}+8a^{13}br^{8}(1+br^{2})AB^{3}-a^{14}r^{8}B^{4}+2a^{12}r^{6}(1+br^{2})$
$\displaystyle
B^{2}(-12b^{2}r^{2}(1+br^{2})A^{2}-(3+br^{2}(7+5br^{2}(7+11br^{2})))B^{2})+8a^{9}b(r+br^{3})^{4}AB(4b^{2}r^{2}(6+br^{2}(11+3br^{2}(38+67br^{2})))$
$\displaystyle
A^{2}+(1+br^{2})(17+br^{2}(16+br^{2}(564+br^{2}(270+257br^{2}))))B^{2})+4a^{8}br^{4}(1+br^{2})^{5}(-8b^{3}r^{2}(3+br^{2}(5+br^{2}(61$
$\displaystyle+107br^{2})))A^{4}-2b(1+br^{2})(51+br^{2}(36+br^{2}(1942+3br^{2}(348+533br^{2}))))A^{2}B^{2}+(1+br^{2})^{3}(-37+br^{2}(-48+br^{2}(-85$
$\displaystyle+358br^{2})))B^{4})+a^{10}r^{4}(1+br^{2})^{3}(-16b^{4}r^{4}(1+br^{2})A^{4}-16b^{2}r^{2}(9+br^{2}(18+7br^{2}(22+39br^{2})))A^{2}B^{2}+(1$
$\displaystyle+br^{2})(-17+br^{2}(-20+br^{2}(-462+br^{2}(-92+31br^{2}))))B^{4})+16ab^{3}(1+br^{2})^{19}AB(4b(3+br^{2}(-21+br^{2}(43-29br^{2})))$
$\displaystyle
A^{2}+3(1+br^{2}(-1+2br^{2}))(B+br^{2}B)^{2})+8b^{2}r^{4}(a+abr^{2})^{7}AB(4b(17+br^{2}(8+br^{2}(712+br^{2}(356+755br^{2}))))$
$\displaystyle
A^{2}-(-1+br^{2})(145+br^{2}(448+1367br^{2}))(B+br^{2}B)^{2})+16a^{3}b^{2}(1+br^{2})^{15}AB(12b(2+br^{2}(-23+br^{2}(186+br^{2}(-195$
$\displaystyle+58br^{2}))))A^{2}+(9+br^{2}(-51+br^{2}(-3+br^{2})))(B+br^{2}B)^{2})+16a^{5}b(1+br^{2})^{11}AB(2b^{3}r^{4}(139+br^{2}(455+br^{2}(1381$
$\displaystyle-983br^{2})))A^{2}+(6+br^{2}(-63+br^{2}(511+br^{2}(-853+231br^{2}))))(B+br^{2}B)^{2})-4a^{2}b^{2}(1+br^{2})^{16}(16b^{2}(3+br^{2}(-36$
$\displaystyle+br^{2}(290+br^{2}(-300+91br^{2}))))A^{4}-24b(1+br^{2})^{2}(-3+br^{2}(19+br^{2}(-21+13br^{2})))A^{2}B^{2}+(3+b^{2}r^{4})(B$
$\displaystyle+br^{2}B)^{4})+8a^{4}b(1+br^{2})^{12}(16b^{4}r^{4}(-17+br^{2}(-62+br^{2}(-173+112br^{2})))A^{4}-2b(1+br^{2})^{2}(18+br^{2}(-198+br^{2}$
$\displaystyle(1605+br^{2}(-1988+543br^{2}))))A^{2}B^{2}-(3+br^{2}(-15+br^{2}(-23+19br^{2})))(B+br^{2}B)^{4})+4a^{6}(1+br^{2})^{8}(-4b^{4}r^{4}(17+$
$\displaystyle
br^{2}(4+br^{2}(758+324br^{2}+897b^{2}r^{4})))A^{4}+4b^{3}r^{4}(1+br^{2})^{2}(-213+br^{2}(-586+3br^{2}(-623+584br^{2})))A^{2}B^{2}-(3$
$\displaystyle+br^{2}(-30+br^{2}(243+4br^{2}(-147+50br^{2}))))(B+br^{2}B)^{4})+8a^{11}br^{6}(1+br^{2})^{2}AB(6B^{2}+br^{2}(4b(1+br^{2})A^{2}+$
$\displaystyle(13+br^{2}(88+153br^{2}))B^{2}))))/(a^{2}r^{4}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B$
$\displaystyle+2b(1+br^{2})^{7}B)^{4})$
$\displaystyle\Gamma_{8}=\frac{2a(-1+3br^{2})(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)}{(1+br^{2})(a^{2}r^{2}+(1+br^{2})^{4})(2b(1+br^{2})A-aB)}$
$\displaystyle\Gamma_{9}=\frac{(1+br^{2})(a^{2}r^{2}+(1+br^{2})^{4})(2b(1+br^{2})A-aB)}{2a(-1+3br^{2})(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)}$
$\displaystyle\Gamma_{10}=\frac{ar^{2}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)}{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB)}$
$\displaystyle\Gamma_{11}=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB)}{ar^{2}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)}$