# Bardeen Compact Stars in Modified $f(R)$ Gravity with Conformal Motion

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

###### Abstract

The main emphasis of this paper is to find the viable solutions of Einstein
Maxwell fields equations of compact star in context of modified $f(R)$ theory
of gravity. Two different models of modified $f(R)$ gravity are considered. In
particular, we choose isotropic matter distribution and Bardeen’s model for
compact star to find the boundary conditions as an exterior space-time
geometry. We use the conformal Killing geometry to compute the metric
potentials. We discuss the behavior of energy density and pressure
distribution for both models. Moreover, we analyze different physical
properties such as behavior of energy density and pressure, equilibrium
conditions, equation of state parameters, causality conditions and adiabatic
index. It is noticed that both $f(R)$ gravity models are suitable and provides
viable results with Bardeen geometry.

Keywords: Bardeen Compact Stars, Conformal Motion, Modified $f(R)$ Gravity.
PACS: 04.50.Kd, 98.80.-k, 98.80.E.

##  I Introduction

The exploration of compact star is the most striking area for the researchers
in the last few years. The usual endpoint of stellar evolution is the
formation of a compact star when outward radiation pressure from the nuclear
fusion in its interior can no longer resist the gravitational forces. The
quarks star, white dwarfs, brown dwarfs, neutrons and black holes are the
class of compact star. Some regular black holes exits which are singularity
free. In 1968 Bardeen provided Bardeen black hole Bar  which is the resolution
of black hole by using spherically symmetry space time. Frenando and Corrrea
Fer  have analyzed quasi-normal modes of Bardeen star by using scalar
perturbations. Which provided the desirable results in different directions.
Sharif and Javed shj  have presented quantum corrections of Bardeen Black
hole. A study on electromagnetic and gravitational stability has been
discussed by Moreno and Sarbach msr . Eiroa and Sendra eir  have studied the
gravitational lensing of consistent black hole. In presence of charge the
values of mass, radius and redshifts may be affected. Shamir and Mustafa sha3
have developed the new family of charged anisotropic compact star using
Bardeen’s analysis as an exterior space time. Kumar et al kum  have analyzed
the exact thermodynamics quantities of Bardeen black hole. Gedala et al ged
discovered two new exact and analytic solution of Einstein Maxwell field
equation in the context of Bardeen black hole also satisfying embedding
conditions. Mustafa et al mus  presented new interior solution using Kermarker
conditions in light of Bardeen stellar structure.

Modified theories of gravity prolong the form of general relativity through
different methods using different field equation and cosmological effects.
They play vital role in modern cosmology, providing base for the current
understating of physical phenomena of the universe. A number of theories have
been introduced by different researchers. For example, theories like $f(R)$
sha2 ; carr ; cap ; sha5 ; nor ; bert ; odi , $f(R,G)$ ata ; nor2 , $f(R,T)$
har ; das  and $f(R,\phi,X)$ sha4  have been developed by combining curvature
scalars and their derivatives. Einstein’s general relativity has been
generalized by a kind of gravitational theory which depends on random function
$f$ of Ricci scalar ($R$). As a result of introducing random function, there
may be liberty to explain the speedy expansion and establishment of the
universe structure without accumulation unknown forms of dark energy or black
objects. Bunchdahl bun  presented $f(R)$ modified theory of gravity in 1970.
Nojiri and Odintsov noj  discussed general properties of $f(R)$ gravity along
with popular models and their applications for the combined explanation of
inflation with dark energy. Cognola et al cog  has presented a class of
modified $f(R)$ exponential models along with stability of presented models.
The discussion of compact stars in modified theories of gravity can be
interesting task

In non-perturbation $f(R)$ gravity asta22 , quark star structures with
equation of state (EoS) are explored. The mass-radius relationship for the
$f(R)$ model is found along with scalar-tensor gravity equivalent description.
However, its contribution to gravitational mass is small for distant
observers.The maximum limit of neutron star mass increases for particular EoS,
and hence such EoS can characterize realistic stellar structures asta44 ;
capo222 . The maximum mass of neutron stars (NS) and the minimum mass of black
holes are essential concerns that have been addressed by the authors in asta11
by estimating the maximum baryonic mass of NSs in the framework of $f(R)$
gravity using multiple EoS models. They have shown that the minimum limit for
the mass of astrophysical black holes is $M\sim 2.5$ to $3M_{\odot}$ in
context of $GW190814$ gravitational wave asta33 .

Symmetries has been used to find the relation between the geometry and matter
using Einstein field equations. Killing vectors are the examples of such
symmetries. Herrera’s herr  contributions are a staple in the adoption of
categories for a one group of conformal motion. Conformal symmetries are
useful in evolving mathematical related issues for some solutions in compact
star. Herrera et al herr ; herr1 ; herr2  have investigated the solution
avowing one parameter group of conformal motion for anisotropic matter. He
also showed that for the group of conformal motion, the equation of state is
Inimitably define through the Einstein equations. Herrera et al herr2  found
some analytical solutions of Einstein-Maxwell filed equations for perfect
fluid. They matched all solutions to RN metric which possess positive energy
density larger than stresses within the sphere. Shamir Sha  has discussed the
solution of Einstein-Maxwell equations by choosing Bardeen model admitting
conformal motion. He investigated the equilibrium parameters by using TOV
equation and stability of stellar structure in presence of charged. Esculpi
and Aloma esc  developed two new families of solution of charged anisotropic
compact star maintaining one parameter group of conformal motion. They
considered tangential pressure to be proportional to radial pressure. Mak and
Harko mak  found the solution with conformal symmetries of static, spherically
gravitational field equation in the vacuum in the brane. They have taken
conformal factor as parameter. Shamir and Malik sha2  developed the solution
of charged compact star using modified $f(R)$ gravity. They used the matching
condition of spherically symmetric space-time with Bardeen geometry to find
the exterior solution. Shamir and Mustafa sha3 , in this paper they found new
family of charged anisotropic compact star solution using conformal killing
symmetries.

This paragraph includes all the framework of current research paper. Section
II includes the field equation of modified theory of gravity $f(R)$, which
have been developed along with the formulation of effective energy density and
effective pressure. In section III, the conformal killing vectors are defined
and the metric coefficients have been developed by using the definition of
conformal killing vectors. In section IV, details of both presented models of
modified $f(R)$ gravity have been discussed. In section V, the values of
constants have been defined by using matching condition of Bardeen’s exterior
solution. In section VI, we have presented the physical analysis of our models
along with graphical representation in detail. In the last section conclusion
and comparison have been discussed for both models.

##  II EINSTEIN-MAXWELL Field EQUATION

The Hilbert action in presence of charge is as follows

$S=\int d^{4}x\sqrt{-g}\large[\dfrac{1}{16\pi G}f(R)+L_{e}+L_{m}\large]$ (1)

In above action $f$ is the function of Ricci scalar $R$, $L_{m}$ represents
the Lagrangian, $g$ is the determinant of line element $g_{\mu\nu}$ and
$L_{e}$ is the lagrangian for electromagnetic field is given by

$L_{e}=\dfrac{1}{16\pi}F_{\alpha\beta}F_{\xi\psi}g^{\alpha\xi}g^{\beta\psi},$
(2)

an electromagnetic field tensor $F_{\alpha\beta}$ with electromagnetic four
potential vector $A_{\alpha}$ is given as

$F^{\alpha\beta}=A_{\alpha,\beta}-A_{\beta,\alpha},$ (3)

Einstein Maxwell field equations are as follows

$F^{\alpha\beta}_{;\beta}=-4\pi j^{\alpha}.$ (4)

In above Einstein$-$Maxwell field equation $j^{\alpha}=\sigma\nu^{\alpha}$ is
the electromagnetic four current vector with $\sigma$ the charge density. In
this research we have taken spherically symmetric space-time of the form

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

With occurrence of charge the stress energy momentum tensor is given by

$T_{\chi\xi}=(\rho+p)u_{\chi}u_{\xi}-pg_{\chi\xi}-\dfrac{1}{4\pi}\large(g_{\xi\beta}F^{\alpha\beta}F_{\xi\alpha}-\dfrac{1}{4}g_{\chi\xi}F_{\alpha\psi}F^{\alpha\psi})\large.$
(6)

The energy density, radial pressure and tangential pressure are represented by
$\rho$, $p_{r}$ and $p_{t}$ respectively. The velocity four vectors are
satisfying the following conditions

$u^{\chi}u_{\chi}=1,\nu^{\chi}\nu_{\chi}=-1$ (7)

There is only one component of Einstein-Maxwell equation that exists is
$F^{01}$. By working on Eq.(3)and Eq.(4) we get the following

$F^{01}=\dfrac{q}{r^{2}}e^{-\frac{\chi+\lambda}{2}},$ (8)

where $q$ is the total charge which is a function of $r$.

$q=4\pi\int 4\sigma\omega^{2}e^{-\frac{\lambda}{2}}d\omega.$ (9)

We obtain the electric field intensity $E$ as

$E=-F^{01}F_{01}=\dfrac{q}{r^{2}}.$ (10)

The modified field equation of $f(R)$ gravity is as follows

${G^{eff}}_{\chi\xi}=\kappa{T^{eff}}_{\chi\xi}$ (11)

where

$T^{eff}_{\chi\xi}=\dfrac{1}{f_{R}}(\kappa
T_{\chi\xi}+E_{\chi\xi}+\dfrac{1}{2}g_{\chi\xi}f(R)+\triangledown_{\chi}\triangledown_{\xi}f_{R}-\dfrac{1}{2}Rf_{R}g_{\chi\xi}-g_{\chi\xi}\square
f_{R})$ (12)

All above information have been used to determine the field equations of the
following form

$\rho^{eff}+\dfrac{E^{2}}{f_{R}}=\dfrac{1}{f_{R}}\Big{(}\kappa\rho+\dfrac{f-Rf_{R}}{2}+\dfrac{1}{e^{\lambda}}\Big{[}f^{{}^{\prime\prime}}_{R}+(\frac{\lambda^{{}^{\prime}}}{2}+\frac{2}{r})f^{{}^{\prime}}_{R}\Big{]}\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}$
(13) $p^{eff}-\dfrac{E^{2}}{f_{R}}=\dfrac{1}{f_{R}}\Big{(}\kappa
p+\dfrac{f-Rf_{R}}{2}-\dfrac{f^{{}^{\prime}}_{R}}{e^{\lambda}}(\frac{\chi^{{}^{\prime}}}{2}+\lambda^{{}^{\prime}}+\frac{2}{r})\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(14)

$\sigma=\dfrac{e^{-\frac{\lambda}{2}}}{4\pi r^{2}}(r^{2}E)^{{}^{\prime}}.$
(15)

##  III Killing Vectors and Conformal Motion

Conformal killing vector can provide the best results as it gives a deeper
understanding of space geometry. The work which have been done previously on
these equation (discussed in section I) shows that conformal symmetry in the
manifold is useful technique to establish the exact solution of Einstein
Maxwell equation. The conformal killing vector are given by the following
equation

$\pounds_{\xi}g_{ab}=\phi g_{ab}$ (16)

where $a$, $b=0,1,2,3$ showing four dimensional space-time and
$\pounds_{\xi}g_{ab}$ is the Lie derivative of a metric tensor with respect to
a vector field $\xi$. Now using Eq.(16) along with the space time (5) we
acquire the following conformal killing equations

$\xi^{0}=L,~{}~{}~{}~{}~{}~{}\xi^{1}\chi^{1}=\phi,~{}~{}~{}~{}~{}\xi^{1}=\dfrac{\phi
r}{2},~{}~{}~{}~{}~{}~{}\xi^{1}\lambda^{{}^{\prime}}+2\xi^{1}_{,1}=\phi.$ (17)

By solving Eqs.(17) simultaneously we get the following metric potentials

$e^{\chi}=X^{2}r^{2},~{}~{}~{}~{}~{}e^{\lambda}=(\dfrac{Y}{\phi})^{2},~{}~{}~{}~{}~{}\xi^{i}=W\delta^{i}_{4}+(\frac{r\phi}{2})\delta^{i}_{1},$
(18)

where $X$, $Y$ and $W$ are arbitrary constants. we suppose $\phi$ as a non-
zero linear function of $r$ i.e. $\phi=X+Zr$, where $Z$ is a random constant.
We also assume the electric charge of the form given as

$E^{2}=\dfrac{gkr}{gr+1}$ (19)

where $g$ and $k$ are the arbitrary constants.

##  IV Realistic Models of $f(R)$ theory gravity

In the current paper the analysis has been done by using two different
realistic and viable models of $f(R)$ gravity. The details are given in the
following sections.

###  IV.1 Model I

First we take the starobinsky like model star2  given by

$f(R)=R+mR^{2},$ (20)

where we take $m$ as constant which shows exponential growth of cosmic
expansion.It is of great significance as all graph depends on parameter m.
By using Eqs.(13) to(15), Eq.(18) and Eq.(19) we attain the physical
quantities given as

$\displaystyle\rho^{eff}=\dfrac{1}{r^{2}(1+gr)Y^{2}(r^{2}Y^{2}+4m(-Y^{2}+3(X+rZ)(X+2rZ)))}[-2m(1+gr)(-3X^{4}-2X^{2}Y^{2}+Y^{4}+$
$\displaystyle
6rX(3X^{2}-Y^{2})Z+r^{2}(75X^{2}-4Y^{2})Z^{2}+90r^{3}XZ^{3}+36r^{4}Z^{4})+r^{4}Z^{4}(K\rho+gr(-k+k\rho))],$
(21) $\displaystyle
p^{eff}=\frac{1}{r^{2}(1+gr)Y^{2}(r^{2}Y^{2}+4m(-Y^{2}+3(X+rZ)(X+2rZ)))}[gkr^{2}Y^{5}+Kpr^{4}(1+gr)Y^{4}+2m(1+gr)$
$\displaystyle(45X^{4}-18X^{2}Y^{2}+Y^{4}+2rX(78X^{2}-17Y^{2})Z+r^{2}(201X^{2}-16Y^{2})Z^{2}+126r^{3}XZ^{3}+36r^{4}Z^{4})]$
(22)

###  IV.2 Model II

In present research we choose second model of $f(R)$ gravity for the analysis
of compact star with conformal motion is logarithmic corrected $R^{2}$ model
as discussed in sha5  and odi  given as

$f(R)=R+\alpha R^{2}+\beta R^{2}ln(\beta R)$ (23)

where $\alpha$ and $\beta$ are arbitrary constants. By using Eqs.(13) to(15),
Eq.(18) and Eq.(19) we get the physical quantities given as

$\displaystyle\rho^{eff}=\dfrac{1}{a_{1}r^{2}(1+gr)Y^{2}(r^{2}Y^{2}+a_{3}(2\alpha+\beta)+2a_{3}\beta\log(\frac{a_{3}\beta}{r^{2}Y^{2}}))}(-a_{1}gkr^{5}Y^{4}-2a_{1}^{3}gr\alpha+4a_{1}a_{2}X^{2}\alpha+4a_{1}a_{2}grX^{2}\alpha+$
$\displaystyle
12a_{1}a_{2}rXZ\alpha+12a_{1}a_{2}gr^{2}XZ\alpha-36a_{1}rX^{3}Z\alpha-36a_{1}gr^{2}X^{3}Z\alpha+8a_{1}a_{2}r^{2}Z^{2}\alpha+8a_{1}a_{2}r^{3}Z^{2}\alpha-72a_{1}r^{2}X^{2}Z^{2}\alpha-$
$\displaystyle
72a_{1}gr^{3}X^{2}Z^{2}\alpha-36a_{1}r^{3}XZ^{3}\alpha-36a_{1}gr^{4}XZ^{3}\alpha-2a_{1}^{3}\beta-2a_{1}^{3}gr\beta+6a_{1}a_{2}X^{2}\beta+4a_{2}^{2}X^{2}\beta+6a_{1}a_{2}grX^{2}\beta+$
$\displaystyle
4a_{2}^{2}grX^{2}\beta+18a_{1}a_{2}rXZ\beta+8a_{2}^{2}rXZ\beta+18a_{1}a_{2}gr^{2}XZ\beta+8a_{2}^{2}gr^{2}XZ\beta-54a_{1}rX^{3}Z\beta-54a_{1}gr^{2}X^{3}Z\beta$
$\displaystyle+12a_{1}a_{2}r^{2}Z^{2}\beta+4a_{2}^{2}r^{2}Z^{2}\beta+12a_{1}a_{2}gr^{3}Z^{2}\beta+4a_{2}^{2}gr^{3}Z^{2}\beta-108a_{1}r^{2}X^{2}Z^{2}\beta-108a_{1}gr^{3}X^{2}Z^{2}\beta-54a_{1}r^{3}XZ^{3}\beta$
$\displaystyle-54a_{1}gr^{4}XZ^{3}\beta+a_{1}\kappa
r^{4}Y^{4}\rho+a_{1}g\kappa r^{5}Y^{4}\rho-
a_{1}^{3}\beta\log(4)-2a_{1}^{3}(1+gr)\beta\log(\frac{a_{3}\beta}{r^{2}Y^{2}}))$
(24) $\displaystyle
p^{eff}=\dfrac{1}{r^{2}(1+gr)Y^{2}(r^{2}Y^{2}+a_{3}(2\alpha+\beta)+2a_{3}\beta\log(\frac{a_{3}\beta}{r^{2}Y^{2}}))}(\kappa
p_{r}r^{4}Y^{4}+gkr^{5}Y64+g\kappa
p_{r}r^{5}Y^{4}-2a_{1}^{2}\alpha+2a_{1}a_{3}\alpha-$ $\displaystyle
2a_{1}^{2}gr\alpha+2a_{1}a_{3}gr\alpha+12a_{2}X^{2}\alpha+12a_{2}grX^{2}\alpha+16a_{2}rXZ\alpha+16a_{2}gr^{2}XZ\alpha+4a_{2}r^{2}Z^{2}\alpha+4a_{2}gr^{3}Z^{2}\alpha+a_{1}a_{3}\beta$
$\displaystyle+a_{1}a-3gr\beta+18a_{2}X^{2}\beta+18a_{2}grX^{2}\beta+24a_{2}rXZ\beta+24a_{2}gr^{2}XZ\beta+6a_{2}r^{2}Z^{2}\beta+6a_{2}gr^{3}Z^{2}\beta-2a_{1}^{2}(1+gr)\beta\log(\frac{2a_{1}\beta}{r^{2}Y^{2}})$
$\displaystyle+2(1+gr)(a_{1}a_{3}+6a_{2}X^{2}+8a_{2}rXZ+2a_{2}r^{2}Z^{2})\beta\log(\frac{a_{3}\beta}{r^{2}Y^{2}})))$
(25)

where

$\displaystyle a_{1}$
$\displaystyle=3X^{2}-Y^{2}+9rXZ+6X^{2}Z^{2},a_{2}=6X^{2}-2Y^{2}+9rXZ,a_{3}=2(-Y^{2}+3(X+rZ)(X+2rZ)).$

##  V Matching with Bardeen Model

In current section, for finding the solution we match spherically symmetric
space time with Bardeen exterior space time, which is defined as

$ds^{2}=j(r)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta d^{2}-j(r)dt^{2},$
(26)

where

$j(r)=1-\dfrac{2Mr^{2}}{(q^{2}+r^{2})^{\frac{3}{2}}},$ (27)

Where $M$ is the mass of usual astrophysical formation, the asymptotic
behavior of Bardeen star is given by

$j(r)=1-\dfrac{2M}{r}+\dfrac{3Mq^{2}}{r^{3}}+O(\dfrac{1}{r^{5}}).$ (28)

In above equation, the term having fraction of $r^{3}$ makes this model
different from RN solution. We have taken
$j(r)=1-\dfrac{2M}{r}+\dfrac{3Mq^{2}}{r^{3}}$ for current analysis. We are
analyzing the interior and exterior geometries for physical aspects.Wwe use
the following continuity conditions for the metric potentials on the boundary
$(r=R_{b})$ and the solution obtained from conformal killing vector Eq.(18).
We have the matching conditions,

$1-\dfrac{2M}{r}+\dfrac{3Mq^{2}}{r^{3}}=X^{2}r^{2}$ (29)
$\Big{(}1-\dfrac{2M}{r}+\dfrac{3Mq^{2}}{r^{3}}\Big{)}^{-1}=(\frac{Y^{2}}{(X+Zr)^{2}})$
(30) $\frac{\partial g^{-}_{tt}}{\partial r}=\frac{\partial
g^{+}_{tt}}{\partial r}$ (31)

Where $(+)$ and $(-)$ shows the exterior and interior solution respectively.
By using Eqs.(29)-(31) we get,

$X=-\sqrt{\dfrac{R_{b}(1+gR_{b})-2M(1+gR_{b})+3mgkR_{b}^{3}}{R_{b}^{3}+gR_{b}^{4}}},$
(32)

$Y=-\dfrac{\sqrt{-2M+R_{b}-2gMR_{b}+gR_{b}^{2}+3gkMR_{b}^{3}}-ZR_{b}^{\frac{5}{2}}\sqrt{1+gR_{b}}}{-R_{b}\sqrt{-2M+R_{b}-2gMR_{b}+gR_{b}^{2}+3gkMR_{b}^{3}}},$
(33)

$Z=-\dfrac{2\sqrt{1+gR_{b}}(R_{b}+gR_{b}^{2}-2M-2MgR_{b}+3MgkR_{b}^{3})^{\frac{3}{2}}}{R_{b}^{\frac{5}{2}}(2R_{b}(1+gR_{b})^{2}-6M-12MgR_{b}-6Mg^{2}R_{b}^{2}+3Mg^{2}kR_{b}^{3})}$
(34)

These parameters plays significant role for defining the behavior of charged
star. To match interior space-time we discuss the appropriate boundary
conditions.

##  VI Physical Analysis

In current section, For physical validity of stellar structure, we illustrated
the graphical representation of energy density, pressure distribution, EoS
parameters, energy conditions, equilibrium conditions, mass$-$radius ratio and
adiabatic parameter. For model-I we fix the constant $k=0.8$ and the detail of
the rest of constants are given in table 1. For model-II we fix the constant
$\alpha$ and table 2 shows the details. The analysis have been done by
choosing massive stars with radius not less than $9$ and maximum masses must
be with in $M\sim 2.5$ to $3M_{\odot}$ as described in asta11 ; asta33 ; Sha .

Table 1: Values of unknown constants for $k=0.8$. $M(M_{\odot})$ |  $R_{b}$ |  $m$ |  $g$ |  $X$ |  $Y$ |  $Z$
---|---|---|---|---|---|---
$2.97979661$ |  $9.12904$ Sha  |  0.10 |  $4$ |  $-0.205752$ |  $-0.142473$ |  $-0.00873881$
$2.97644068$ |  $9.14507$ Sha  |  0.20 |  $4.2$ |  $-0.203263$ |  $-0.142542$ |  $-0.00864716$
$2.96298998$ |  $9.20902$ Sha  |  0.30 |  $3.9$ |  $-0.203996$ |  $-0.14254$ |  $-0.00867793$
$2.94349831$ |  $9.30119$ Sha  |  0.40 |  $4$ |  $-0.202295$ |  $-0.142611$ |  $-0.00861998$
$2.93093559$ |  $9.36021$ Sha  |  0.50 |  $4$ |  $-0.2033218$ |  $-0.142602$ |  $-0.00865751$
Table 2: Values of unknown constants for $g=0.3$ and $\alpha=0.2$. $M(M_{\odot})$ |  $R_{b}$ |  $k$ |  $\beta$ |  $X$ |  $Y$ |  $Z$
---|---|---|---|---|---|---
$2.97979661$ |  $9.12904$ |  0.6 |  $0.2$ |  $-0.770323$ |  $-0.138778$ |  $-0.0306343$
$2.97644068$ |  $9.14507$ |  0.7 |  $0.4$ |  $-0.83146$ |  $-0.138755$ |  $-0.0330464$
$2.96298998$ |  $9.20902$ |  0.8 |  $0.5$ |  $-0.883538$ |  $-0.137379$ |  $-0.0347578$
$2.94349831$ |  $9.30119$ |  0.9 |  $0.5$ |  $-0.943703$ |  $-0.141243$ |  $-0.0381605$
$2.93093559$ |  $9.36021$ |  0.9 |  $0.6$ |  $-0.939609$ |  $-0.140396$ |  $-0.0377677$

###  VI.1 Energy Density and Pressure

Here, the energy density ($\rho^{eff}$) and pressure ($p^{eff}$) behave
realistically for modified $f(R)$ gravity are show in Fig(1) for model-I and
Fig(2) for model-II, which are decreasing and positive. The pressure is
approaching to zero near the boundary of the star $r=R_{b}$. The concave up
graphs of $\rho^{eff}$ and $p^{eff}$ are because of conformal motion and due
to occurrence electric charge. The derivatives of energy density
$\frac{d\rho}{dr}$ and pressure $\frac{dp^{eff}}{dr}$ are shown in Fig(3) and
Fig(4) for both models. The negative behavior shows that they are physically
acceptable.

|  |
---|---|---
Figure 1: Behavior of energy density $\rho^{eff}$ and pressure $p^{eff}$ for model-I. |  |
---|---|---
Figure 2: Behavior of energy density $\rho^{eff}$ and pressure $p^{eff}$ for model-II. |  |
---|---|---
Figure 3: The gradient behavior of energy density $\rho^{eff}$ and pressure $p^{eff}$ for model-I. |  |
---|---|---
Figure 4: The gradient behavior of energy density $\rho^{eff}$ and pressure
$p^{eff}$ for model-II.

###  VI.2 Energy Conditions

In cosmology the energy conditions have foremost importance to show the
viability of the presented model. Many researchers have discussed the great
features of energy conditions. It is to satisfy the null energy conditions
(NEC),weak energy condition (WEC), strong energy condition (SEC) and dominant
energy condition (DEC). These energy conditions are given by

  * •

Null energy conditions

$\rho^{eff}+p^{eff}\geq 0,~{}~{}\rho^{eff}+\frac{q^{2}}{4\pi r^{4}}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (35)

  * •

Weak energy condition

$\rho+\frac{q^{2}}{8\pi r^{4}}\geq 0,~{}~{}\rho+p\geq
0,~{}~{}\rho+p+\frac{q^{2}}{4\pi r^{4}}\geq
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (36)

  * •

Strong energy conditions

$\rho^{eff}+p^{eff}\geq 0,~{}~{}\rho^{eff}+3p^{eff}+\frac{q^{2}}{4\pi
r^{4}}\geq 0,~{}~{}~{}~{}~{}~{}~{}$ (37)

Above mentioned all energy conditions are satisfied for the selected $f(R)$
model as shown in Fig(5) and Fig(6).

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

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

Figure 5: Behavior of energy conditions for model-I.

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

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

Figure 6: Behavior of energy conditions for model-II.

###  VI.3 Equilibrium Conditions

Here in this section, we discussed the equilibrium conditions in the scenario
of Bardeen geometry with conformal motions for the existing charged stellar
structure. The Tolman Oppenheimer Volkof $(TOV)$ equation is very useful to
examine the equilibrium conditions for stellar structure. Fig(7) shows that
all equilibrium conditions are satisfied for the current $f(R)$ models. The
TOV equations for the charged isotropic fluid is defined as

$-\dfrac{\chi^{{}^{\prime}}}{2}(\rho^{eff}+p^{eff})+\sigma(r)E(r)e^{\frac{\lambda(r)}{2}}-\dfrac{dp^{eff}}{dr}=0$
(38)

Above equation can be written as

$\mathcal{F}_{g}+\mathcal{F}_{e}+\mathcal{F}_{h}=0$ (39)

|  |
---|---|---
Figure 7: Evolution of $\mathcal{F}_{h}$,$\mathcal{F}_{g}$ and
$\mathcal{F}_{e}$ for both modeles

  * •

Gravitational Force

$\mathcal{F}_{g}=-\dfrac{\chi^{{}^{\prime}}}{2}(\rho^{eff}+p^{eff})$ (40)

  * •

Electric Force

$\mathcal{F}_{e}=\sigma(r)E(r)e^{\frac{\lambda(r)}{2}}$ (41)

  * •

Hydrostatic Force

$\mathcal{F}_{h}=-\dfrac{dp^{eff}}{dr}$ (42)

###  VI.4 Equation of State Parameter

|  |
---|---|---
Figure 8: Graph of equation of state parameter $w$ for model I and II.

In this section we presented the analysis of equation of state (EoS)
parameters. The equation is given by the following expression,

$w=\dfrac{p^{eff}}{\rho^{eff}}.$ (43)

The graphical behavior of is given in Fig(8), the graph of model-I shows that
for radius less than $8$, the effective pressure is increasing, this means
these equations of state is often referred to as finite strain theory to
compare their behavior from the predictions of elastic strain theory, in which
the magnitude of volumetric strains are assumed to be infinitesimal as
described in corm . Where as near outer boundary the equation of state is
within the range which shows that the star consists of the normal matter,
stable and indicates the high compactness at the outer boundary.

###  VI.5 Adiabatic Index

The adiabatic index has a key factor that reflects the stiffness of equation
of state. Chandrasekhar chnd  presented insatiability criteria depending upon
adiabatic index, which is a good separator between a large gravitational field
and a small abhorrent nuclear forces. The formulation of adiabatic index is
given by

$\Gamma=(1+\dfrac{\rho^{eff}}{p^{eff}}\dfrac{dp^{eff}}{d\rho^{eff}}).$ (44)

The adiabatic index must be greater than $4/3$, Fig(9) shows that the our
models are stable.

|  |
---|---|---
Figure 9: Evolution of adiabatic index.

###  VI.6 Mass-Radius Relation, Compactness Factor, Surface Red-Shift

The mass-radius relation for a compact star must be within the limits i.e.
$2M/R\leq 8/9$ buchd

$M(r)=4\pi\int_{0}^{\pi}\rho^{eff}r^{2}dr=\dfrac{R}{2}(1-e^{\lambda}+\frac{q^{2}}{r^{2}})$
(45)

For charged compact stars the mass function have to fulfill Andreasson’s limit
and  given as
$\sqrt{M}\leq\frac{\sqrt{R_{b}}}{3}+\sqrt{\frac{R_{b}}{9}+\frac{q^{2}}{3R_{b}}}$.
The details are given in table 3.

Table 3: Estimated values of $M$, $R_{b}$ and Andreasson’s limit for model-I and model-II. $M(M_{\odot})$ |  $R_{b}$ |  $\sqrt{M}$ |  Model-I |  Model-II
---|---|---|---|---
$2.97979661$ |  $9.12904$ |  1.665850658 |  1.92155 |  2.57812
$2.97644068$ |  $9.14507$ |  1.65850249 |  1.91807 |  2.59772
$2.96298998$ |  $9.20902$ |  1.65841051 |  1.91719 |  2.64241
$2.94349831$ |  $9.30119$ |  1.65806095 |  1.91338 |  2.66736
$2.93093559$ |  $9.36021$ |  1.65770708 |  1.91384 |  2.63862
|  |  |
---|---|---|---
Figure 10: Graphical behavior of mass function, compactness factor and red shifts for model-I. |  |  |
---|---|---|---
Figure 11: Graphical behavior of mass function, compactness factor and re
shifts for model-II.

Here the compactness parameter iva ; makh11  and surface red shift smh  are
denoted by $U(r)$ and $Z_{rs}$ defined as

$U(r)=\frac{M}{r}=\frac{1-e^{-\lambda}}{2}+\frac{q^{2}}{2r^{2}},$ (46)
$Z_{rs}=(1-U(r))^{-1/2}-1$ (47)

We have plotted the mass function, compactness parameter and surface red shift
in Fig(10) and Fig(11) for model-I and II respectively. The graphical behavior
of all these are monotonically increasing.

##  VII Conclusion And Comparison

In this paper, we select Bardeen black hole geometry with conformal motion and
applied matching condition on two different models of modified $f(R)$ gravity
i.e. $f(R)=R+mR^{2}$ and $f(R)=R+\alpha R^{2}+\beta R^{2}ln(\beta R)$ with
isotropic energy momentum tensor in presence of electric charge. The analysis
have been done for different values of $m$ which are $m=0.10$, $m=0.2$,
$m=0.3$, $m=0.4$, and $m=0.5$ for model I. For model II we fixed the $g$ and
$\alpha$ and varied the value of $\beta$ to get viable results. The analysis
of our work is enumerated below

  * •

The solutions found with conformal symmetries for the metric potentials are
finite, bounded and singularity free across the star and also at the boundary
from $r=0$ to $r=R_{b}$. The analysis of energy density, pressure are shown in
Fig(1). This study is done with $R_{b}=9.12904$, $R_{b}=9.14507$,
$R_{b}=9.20909$, $R_{b}=9.30119$ and $R_{b}=9,36021$. The derivatives
$d\rho/dr$ and $dp/dr$ are negative which illustrate that they are acceptable
for the present study.

  * •

We have presented the energy conditions in this paper which are WEC, NEC and
SEC. All the energy conditions are satisfied for both models under conformal
motion. The graphical illustration of all energy conditions are shown in
Fig(5) and Fig(6).

  * •

For the existing charged stellar structure, we presented the equilibrium
conditions by using TOV equations in the scenario of Bardeen geometry along
with conformal motions are shown in Fig(7), the positive graphs shows that all
equilibrium conditions are satisfied for the presented $f(R)$ models. The
balancing development of the graphs of forces shows that our models are viable
and stable in presence of electric charge through conformal motion with
Bardeen geometry.

  * •

The adiabatic index is the main element which shows the stiffness of equation
of state. Here in current scenario, the value of Adiabatic index is as per the
requirement i.e. $\Gamma$ ie greater $4/3$ as given in chnd .

  * •

It is observed that the model-I satisfied the equation of state parameters
near the outer boundary which implies that the effective pressure is
increasing and near outer boundary the equation of state is within the range
which shows that the star consists of the normal matter, stable and indicates
the high compactness at the outer boundary. The graph of model-II satisfied
the limits i.e. these parameters must lie between 0 and 1.

  * •

The mass function satisfied the Andreasson’s limit and  amd the details are
given in table 3. The compactness parameter and surface red-shift have
monotonically increasing behaviour which shows the stability of our results.

In the end, we have effectively conducted a stable and viable stellar
structure under the presence of charge by using two different models of
modified $f(R)$ gravity i.e. $f(R)=R+mR^{2}$ and $f(R)=R+\alpha R^{2}+\beta
R^{2}ln(\beta R)$ with Bardeen black hole geometry along with conformal
motion. Conclusively, the derived calculated solutions demonstrate authentic
evidences for better, realistic and viable stellar structure in presence of
charge with conformal motion with Bardeen geometry.

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