# Physical Attributes of Bardeen Stellar Structures infR Gravity

Authors: M. Farasat Shamir; Ammara Usman; Tayyaba Naz
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

The main aim of our study is to explore some relativistic configurations of compact object solution in the background offR gravity, by adopting the Krori-Barua spacetime. In this regard, we establish the field equations for spherically symmetric spacetime along with charged anisotropic matter source by assuming the specific form of the metric potentials, i.e., νr=Br2+C and λr=Ar2. Further, to calculate the constant values, we consider the Bardeen model as an exterior spacetime at the surface boundary. To ensure the viability of the fR gravity model, the physical characteristics including energy density, pressure components, energy bonds, equilibrium condition, Herrera cracking concept, mass-radius relation, and adiabatic index are analyzed in detail. It is observed that all the outcomes by graphical exploration and tabular figures show that the Bardeen black hole model describes the physically realistic stellar structures.


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## Body

## 1. Introduction

In modern cosmology, the discussion of accelerated evolution of the universe is considered as the most significant topic. One can believe that the formation of the universe depends upon ordinary matter, dark matter, and dark energy [1–4]. Various astronomical observations suggested that the key intent of this mysterious expansion of our universe is dark energy, as it preserves huge negative pressure as compared to the pressure of dark matter. Although general relativity (GR) provides many realistic results in exploring the nature of the universe, in the case of dark energy, GR fails to provide satisfactory outcomes. Thus, to achieve the fruitful results, numerous modified theories have been proposed and gained much popularity in the preceding decades. These gravitational theories are formulated by modifying the Einstein-Hilbert action and provide the natural replacement of GR. The extensive range of these modified theories of gravity is fR,fR,T,fG,fR,G [5–12]. However, fR theory of gravity is one of the most famous and realistic alternatives to GR and gains the trust of researchers by providing the most interesting outcomes in the field of dark energy problems. Several forms of fR theory have already been presented by many researchers [13–15]. This theory has shown the significant outcomes in dealing with different cosmological constraints, such as galactic scale, the cosmic microwave background test, early time inflation, the late-time cosmic evolution, phantom fields, and most importantly in the exploration of the mystery of expansion of the universe [16–23]. It is worthwhile to mention here that several physical aspects like Newtonian limit [24–26], the solar system test [27–29], formulation of singularities [30], and gravitational stability [31–33] were comprehensively explored through fR gravity. The fR theory of gravity was first introduced by Buchdahl [34]. Later on, this modification was used to solve some cosmic acceleration and early inflation problems [35, 36]. Moreover, Nojiri and Odintsov [37] generalized the Einstein-Hilbert action by using the Ricci scalar function and observed some interesting results. Erickcek et al. [38] obtained the unique solution for the exterior spacetime of the stellar structure by using the matching constraints in the background of matter distribution. In this regard, Kainulainen et al. [39] discussed the interior spacetime of compact objects in the Palatini formulation of the fR theory of gravity. Some auxiliary features in the study of the massive compact objects have been added by the fR gravity modeling [40]. Moreover, some viable fR theory of gravity rainbow models was demonstrated by Hendi et al. [41]. A class of nonlinear electrodynamics ranging from three-dimensional spacetime is also explored by the same authors [42]. Starobinsky [43] presented a new family of fR theory models that represented the interesting outcomes for the solar system laboratory testing.In astrophysics, the gravitational collapse of stellar objects and the formulation of new compact stars is always considered as the most interesting topic. The gravitational collapse appears at the point where internal pressure of the stellar objects fails to keep up the pressure against the outer gravitational force. The degeneracy pressure produces stability against the collapsing of the star, and the outcomes of collapsing of these compact stars lead to the birth of white dwarfs, neutron stars, or black holes. These stars emerged with apex densities because these stars were considered to be massive objects but volumetrically smaller. However, the exact features of these compact stars are yet to be explored, one can believe that they are actually the massive ones with a very small radius. All these types of objects other than black holes are commonly categorized as degenerate stars. To investigate the vital existence of such stellar systems, the solutions of Einstein field equations (EFE) are mandatory. The very first solution of these EFE was determined by Schwarzschild [44]. In this regard, the realistic and complex nontraversable models of the celestial objects have been investigated by Tolman [45] and Oppenheimer and Volkoff [46] in the frame of observational data. Later, Oppenheimer and Snyder [47] demonstrated the effects of gravitational collapse with a homogeneity-based dust sphere. The realistic features of the compact objects show the connection between the interior pressure and the force of gravitation that ultimately led to an equilibrium state. This phenomenon has considerable importance in the study of interior structure of compact objects. Further, Baade and Zwicky [48] examined the compact stellar system and proposed that the supernova might convert into a smaller stellar structure once observing the strongly magnetized spinning neutrons. Folomeev and Singleton [49] analyzed the spherically symmetric polytropic matter distribution for nonminimal coupling and showed that the stellar system had regular, static, and asymptotically flat behavior. Dzhunushaliev et al. [50] explored the stability features of symmetric matter fluid by using scalar fields. Moreover, the idea that nuclear density of the compact object illustrates anisotropic nature at the center of the star was first presented by Ruderman [51].For modeling of static objects, the spherical symmetric geometry is considered as a very natural and effective source while there are several possibilities in the choice of matter distribution. In the beginning, it has been assumed that the formulation of a star’s core consists of perfect fluid which leads to the isotropic state. However, isotropy might be a suitable attribute, but it does not show a usual characteristic of a stellar compact, whereas, in case of pressure anisotropic and viscosity fluid, the anisotropy disturbs the stability of the structure relative to local isotropic cases. Anisotropy in fluid normally arises due to the existence of a combination of various kinds of fluids, rotation, magnetic field, viscosity, etc. Many models of anisotropic compact stars have been introduced in the literature [52–58]. The modeling of the mass-radius relation of the neutron star was studied by Egeland [59], and he claimed that density of the vacuum space is the main cause of cosmological constant. Mak and Harko [60] discussed the physical features of the strange stars and obtained an exact solution by using spherical symmetric matter distribution. Yet, the presence of electric charged anisotropy enhances the equilibrium state and stability of the celestial system [61–63]. Hossein et al. [64] described a couple of useful characteristics of anisotropic fluid with cosmological constants. Rahaman et al. [65] discussed the Chaplygin gas equation of state (EoS) and presented the extended version of the Krori-Barua [66] model. Further, Herrera and his collaborators [67–69] investigated the interesting physical aspects of anisotropic stars in different contexts. Lobo [70] studied the compact stars along with a barotropic EoS and provided the extension of Mazur-Mottola Gravatar models by employing the matching conditions between static source and Schwarzschild spacetime. Later, Sunzu et al. [71] investigated the theory of MIT bag EoS for quark stars and they concluded that an increase in anisotropic parameters will gradually decrease the energy density, due to which the EoS becomes stiffer. Later, Illyas [72] also discussed the charged anisotropic fluid for compact stars by considering some viable fG gravity models. In particular, either to demonstrate the core of the astrophysical elements or to explore different aspects of the astrophysical elements, charged anisotropic fluid is always considered as the most favourable condition.The concept of the black hole is considered as a crucial topic in literature. However, one can believe that there are two different classifications of black holes, one consisting of a horizon having a singularity inside it and the other without any singularity. In general, regular black holes are those black holes which are singularity free. The Bardeen black hole [73] was the very first example of the regular black hole, and this idea came to light by Bardeen in 1968. Later, the stability features of the Bardeen model were discussed by Moreno and Sarbach [74]. Later, Zhou et al. [75] analyzed the geodesic configuration of test objects in the context of the Bardeen spacetime. Further, the Bardeen model in the frame of GR has been studied by Shamir et al. [76] and they provided some interesting observations of a viable celestial system in the light of charged perfect fluid. The theory about the Bardeen model states that this model can be defined as a gravitational collapsing magnetic monopole produced by the certain structure of nonlinear electrodynamics and was introduced by Ayon-Beato and Garcia [77]. Some literature is available in the context of the Bardeen model [78–81]. In addition, the charged stellar structure along with the Bardeen sphere in the background of conformal motion was explored [82]. In a recent paper [83], compact star solutions have been discussed for the Bardeen sphere by using the famous Karmarkar approach. Being motivated from the above literature, here, we explore the studies of Mustafa et al. [83] in the context of modified fR theory by assuming the charged compact structures along with the Bardeen model.The aim of this study is to explore some realistic stellar charged anisotropic model in the context offR gravity, by utilizing the Krori-Barua [66] spacetime. The illustration of metric potential confirms that the metric tensors exhibit continuous, nonsingular, and well-behaved nature. In particular, we study the physical attributes of the compact objects against various values of M for different ranges of R, i.e., R=10.10km with M=1.2,1.3,1.4,1.5Me, R=9.10km with M=1.2,1.4Me, and R=8.10km with M=1.2Me. For this purpose, we take specific electric fields E2=kQr [83] and establish physically appropriate models of compact stars.The arrangement of our current study is as follows: in the following portion, the field equations for thefR gravity have been presented for anisotropic configuration. A realistic fR theory of gravity model is also presented in the same section. Section 3 is dedicated to the Bardeen charged model and junction conditions. In segment IV, we explore the physical features and graphical responses of the compact stars. In the last segment, we summarized our work.


## 2. Some Basic Modified Field Equations

Firstly, we evolve the field equations of thefR gravity. In this regard, we assume the action of fR theory of gravity [10] shown as
(1)S=∫fR2κ+Lm−gd4x.Here,f is a function of Ricci scalar and Lm is the matter of the Lagrangian field. By varying the action (1) respecting the metric potential gηζ, we develop the following fR field equation:
(2)FRηζ−12fRgηζ−∇η∇ζF+gηζ□F=−κTηζ,where ∇η and □ indicate covariant derivative and D’Alembertian symbol, i.e., □≡∇η∇η and F=dfR/dR. Furthermore, for the investigation of the stellar objects, we assume the static spherically symmetric spacetime as
(3)ds2=−eνrdt2+eλrdr2+r2dθ2+sin2θdϕ2.Here,νr and λr are functions of r only. We divide the energy-momentum tensor into two parts, i.e.,
(4)Tηζ=Tηζaf+Tηζef.Anisotropic fluid is considered as the basic generalization of isotropic perfect fluid, in which the transverse pressure is not equal to radial pressure. In comparison with isotropic fluid, anisotropic spheres are assumed to describe more compact stellar objects like neutrons and gravastars [84, 85]. In case of necessity, anisotropic fluid narrates the inner structure of spherical symmetric spheres and the solid crust of the compact stars [86]. In the field of stellar objects, local pressure anisotropy is considered to be the most realistic assumption for demonstrating the nature of matter distribution [87–99]. The stress-energy tensor corresponding to an anisotropic matter distribution, in an orthonormal basis, is presented by the density and pressure (radial and transverse) of the compact star which is correlated with the metric potential functions νr and λr given in Equation (3).
(5)Tηζ=ρ+pruηuζ+ptgηζ+pr−ptXηXζ,where ρ, pr, and pt designate the energy density, radial, and transverse pressure component, respectively. Further, uη and Xη are its timelike four velocity and a spacelike unit vectors orthogonal to uη, respectively, which fulfill the following correlations:
(6)uη=e−νr/2δ0η,Xη=e−λr/2δ1η,uηuη=−XηXη=1.The energy-momentum tensor for electromagnetic field is given as(7)Tηζef=14πgηηFημFζμ−14gηηδζηFμνFμν.Moreover,Fημ is the Maxwell stress tensor and its equation can be represented as
(8)Fημ=βμ,η−βη,μ,Fημ;μ=−4πJη.Here,β is the four potential and Jη is the four current which is presented as Jη=σνη, where σ indicates as charge density. Now, for spherically symmetric static line element, the nonzero components of four potential are J0 and for Maxwell tensor is F01, described as
(9)F01=−F10=qr2eν+λ/2,where
(10)q=16π∫0rσρ2eλ/2dρ.The termq denotes charge within the core of compact objects corresponding to r. Furthermore, the expression for electric field can be considered as E2=kQr [83] and E2=F10F01, which is expressed as E2=q2/r4. The fR field equations for (3) are given as
(11)8πρ+E2=e−λFν″2+ν′r+ν′ν′−λ′4+12fR−e−λF″−e−λF′λ′2+2r,(12)8πpr−E2=e−λF−ν″2+λ′r−ν′ν′−λ′4−12fR+e−λF′ν′+2λ′2+2r,(13)8πpt+E2=F1−e−λr2+e−λλ′−ν′2r−12fR+e−λF″+e−λF′ν′+λ′2+1r,(14)σ=e−λ/24πr2r2E′,here, “prime” symbolizes the r derivative. Further, the anisotropic factor Δ is presented as
(15)Δ=8πpt−8πpr.Next, by utilizing Equations (12) and (13), the following equation is generated:
(16)Δ=F1−e−λr2+e−λν″2−ν′+λ′2r+ν′ν′−λ′4+e−λF′′−e−λF′λ′2+1r.It is worthwhile to mention here that to simplify these Equations (11)–(14), we have considered Krori-Barua [66] spacetime, i.e., νr=Br2+C and λr=Ar2, where A, B, and C are any arbitrary constants. The formulation of singularities inside the compact star is believed as the key feature in the investigation of stellar objects. The geometry of any spacetime is actually presented by its metric potentials. For the realistic model of the fR theory of gravity, the metric potentials grr and gtt should exhibit positive and regular behavior. Thus, in our case, Figure 1 shows that these metric potentials narrate the monotonically increasing, singularity-free graphical illustration and attain maximum value at the surface boundary which confirm that the model under consideration reveals physically realistic behavior. Further, to investigate the nature of the compact stars, here, we consider a realistic model of fR theory of gravity [100].
(17)fR=R−1−γξ2Rξ2γ,here, ξ and γ are any constant. This model (17) actually relates fR gravity with scalar-tensor gravity and is employed to explore the gravitational impacts of cosmic acceleration [100]. Faulkner et al. [100] argued that conformal coupling is the significant aspect of this model (17). Moreover, this model also showed consistent outcomes for the Chameleon mechanism and solar system constraints [101]. To probe the stellar object existence for (17) model, the constant parameters selected in such a way that ρ, pr, and pt reveal the positive and finite nature.Figure 1
Variation ofgrr (a) and gtt (b).
(a)(b)


## 3. Matching Conditions

In this section, we assume the Bardeen black hole model as an exterior spacetime of the compact objects given by [73].
(18)ds2=−hrdt2+hr−1dr2+r2dθ2+sin2θdϕ2,where hr=1−2Mr2/q2+r23/2. The Bardeen model can be derived as exact solutions of suitable nonlinear electrodynamics coupled to gravity. The nonzero Einstein tensor in the Bardeen black hole can be related to the energy-momentum tensor of a nonlinear electromagnetic Lagrangian [74]. Moreover, the existence of Bardeen model solutions does not deny the singularity theorems [102]. It is notable that the sphere asymptotically responses as
(19)hr=1−2Mr+3Mq2r3+O1r5.One can observe from (19) that the expression 1/r relates with the mass M of the compact star configuration, although the term relating 1/r3 increased the importance in this discussion and became different from the usual Reissner-Nordstrom solution [103]. Here, in our present work, we take hr=1−2M/r+3Mq2/r3. The continuity condition for (3) and (18) at r=R takes the form
(20)grr+=grr−,gtt+=gtt−,∂gtt+∂r=∂gtt−∂r,here, “+” indicates the exterior metric and “-” identifies the interior. Next, we utilize Equations (3), (18), and (20) and obtain the values of A, B, and C, defined as
(21)A=1R2lnRR−2M+3MR2E2,B=2M+3MR2E22R2R−2M+3MR2E2,C=lnR−2M+3MR2E2ReBR2.Further, we determine the mathematical values of the parameters by assigning specific values to the free parametersR, M, and ξ and fixing some parameters, such that γ=2,k=0.000001, and Q=0.00001, as represented in Table 1.Table 1
Constant values ofξ, A, B, and C for considered compact stars.
(a)MM⨀R=8.10km [82]ξABC1.20.050.0088220.0059742-0.970785(b)MM⨀R=9.10km[82]ξABC1.20.0430.0059920.003879-0.8174041.40.0440.0073590.005068-1.02913(c)MM⨀R=10.10km[82]ξABC1.20.0350.00425960.002667-0.7066251.30.0360.0047150.003027-0.7897171.40.0370.0051920.003422-0.8787261.50.0380.0056930.003859-0.974462Further, here, we mention the mandatory requirements, which have to be fulfilled for the well-behaved nature of the stellar system.(i)
Thegrr and gtt should be singularity free. Also, the anisotropic matter configuration must fulfill eλr=0=1 and eνr=0=constant(ii)
ρ,pr, and pt must be positive and maximum at the center of the star(iii)
The gradient ofρ,pr, and pt should be negative inside the limit 0≤r≤R(iv)
The obtained solution must fulfill the energy bonds NEC, WEC, SEC, and DEC(v)
The condition of EoS must satisfy, i.e.,0<wr,wt<1(vi)
The velocity sounds must lie within0,1, i.e., 0<vr2,vt2<1(vii)
Redshift function must show monotonic decreasing behavior while mass and compactness should narrate monotonic increasing illustration(viii)
Adiabatic index for anisotropic matter configuration must be greater than4/3


## 4. Physical Attributes of the Stellar Structure

Here, in this section, we will examine the graphical responses of the considered compact objects in the framework of thefR gravity model. For this, we will perform different physical tests such as energy density, pressure components, energy conditions, equilibrium constraints, causality condition, mass function, and redshift. All these attributes play a valuable role in illustrating the nature of compact stars.


### 4.1. Energy Density and Pressure Progression

Here, we explore the graphical responses of the density, redial pressure, and transverse pressure in the light of thefR gravity model. It is clearly shown from Figure 2 that all these plots illustrate positive and singularity-free nature. Also, these plots attain the highest values at the center and touch the surface at the boundary which shows that these plots exhibit satisfactory behavior. We also discuss the graphical illustration of gradients of ρ,pr, and pt, and their plots are shown in Figure 3.Figure 2
Evolution ofρ, pr, and pt.Figure 3
Variation ofdρ/dr,dpr/dr, and dpt/dr.The gradient ofρ,pr, and pt presented negative behavior, which can be seen from Figure 3. These aspects indicate the apex compactness behavior of the compact stars.


### 4.2. Charge Density and Electric Field

Further, we examined the graphical response impact ofq, σ, and E2. In Figure 4, the plot of electric charge shows increasing responses, whereas the nature of charge density is decreasing. One can easily observe that σ attains its highest value at center. Moreover, the electric field illustrates that its graph vanishes when r=0 and then increases in a positive direction.Figure 4
Variation ofq, σ, and E2.


### 4.3. Anisotropy Evolution

In this segment, we investigate the graphical responses of anisotropy parameterΔ, given in Equation (15) [104], since the repulsive nature of Δ validates the existence of compact objects. It can be notable from Figure 5 that for a considered fR gravity model, anisotropy parameter exhibits repulsive nature as Δ>0. This implies that our system is well-behaved and stable.Figure 5
Behavior ofΔ.


### 4.4. Energy Conditions

To verify the existence of the compact objects, energy constraints’ role is very crucial. The famous energy constraints [105] are categories as follows: null energy (NEC), weak energy (WEC), strong energy (SEC), and dominant energy conditions (DEC), and defined as
(i)
NEC:ρ+E2≥0(ii)
WEC:ρ+pr≥0,ρ+pt+E2≥0(iii)
SEC:ρ+pr+2pt+E2≥0(iv)
DEC:ρ−pr+E2≥0,ρ−pt≥0It has been observed from Figure6 that all energy bonds showed well-fitted nature for the considered fR gravity model.Figure 6
Evolution of energy conditions for our proposed model.


### 4.5. Equilibrium Constraint

Further, we examine the equilibrium state of thefR gravity model for the succeeding forces: gravitational force, hydrostatics force, anisotropic force, and electric force. For this purpose, we explore the Tolman-Oppenheimer-Volkoff equation [45, 46] along electric charge, given as
(22)MGrρ+prreλ−ν/2+dprdr−2rpt−pr−σrEreλr/2=0.The effective gravitational massMGr represented by
(23)MGr=12ν′reν−λ/2.By replacingMGr in Equation (22), we get
(24)ν′2ρ+pr+dprdr−2rpt−pr−σrEreλr/2=0,where Fg=−ν′/2ρ+pr,Fh=−dpr/dr,Fa=,andFe=σrEreλr/2. Here, Fg symbolizes as gravitational force, Fh indicates hydrostatic force, Fa defines as anisotropic force, and Fe identifies as electric force. For a realistic model of fR gravity, the joint sum of the above-mentioned four forces should be exactly equal to zero. This implies that for an equilibrium system, all these forces cancel out the effect of each other and maintain the balancing state, i.e.,
(25)Fg+Fh+Fa+Fe=0.From Figure7, it can be clearly noticed that all forces are satisfied by the necessary constraints of equilibrium.Figure 7
Behavior ofFh, Fg, Fa, and Fe for our proposed model.


### 4.6. Equation of State

Next, we determine the EoS for ther component and t component. The two ratios of EoS are mentioned as
(26)wr=prρ,wt=ptρ.In the graphical illustration shown in Figure8, it can be noticed that 0<wrandwt<1. This implies that our system exhibits well-behaved nature.Figure 8
Evolution ofwr and wt for our proposed model.


### 4.7. Stability Condition

The velocity sound forr component and t component is denoted by vr2 and vt2 and is defined as
(27)vr2=dprdρ,vt2=dptdρ.To prove the physical stability of the consideredfR model, we employ the Herrera technique [87] presented as 0≤vr2andvt2≤1. In this regard, we discussed one more worthy approach presented by Abreu et al. [106] to investigate the potentially unstable (stable) configuration of the stellar objects. It is important to mention that potentially unstable (stable) realms within the compact structure are managed by the modification of sound propagation. The anisotropic realm is considered as potentially stable where −1≤vt2−vr2≤0. This means that the horizon in which the transverse speed component is less than the radial speed sound component is recognized as potentially stable otherwise not. It is worthy to highlight here that the considered fR model is entirely stable as shown in Figure 9.Figure 9
Behavior ofvr2, vt2, and vt2−vr2 for fR gravity model.


### 4.8. Mass, Compactness, and Redshift Evolution

For the existence of a realistic and viable model, its mass function denoted byMr, compactness factor symbolizes by Ur, and the redshift function identifies as Zr are considered as the necessary constraints in the analysis of a stellar system. Generally, one can believe that objects having anisotropic fluid should not be arbitrarily massive for a viable modified theory model. In the frame of isotropic fluid, the concept of maximum permissible mass-radius ratio, i.e., 2M/R<8/9, was studied by Buchdahl [107]. Later, Mak and Harko [108] generalized the concept of Buchdahl and argued that this mass-radius ratio holds for isotropic as well as anisotropic fluid. Hence, the mass function for anisotropic fluid is given by
(28)Mr=4π∫0Rr2ρdr=R21−e−λ.This response ofMr implies that Mr is regular and attains maximum values at the boundary. Further, we explore the compactness factor which is presented as a mass-radius ratio [108] and is defined as
(29)Ur=2Mrr=2e−2Ar2−1+eAr2−1+eAr2−3Br2−B2r4+Ar22+Br2r3.Next, the redshiftZr is given as [104]
(30)Zr=e−ν/2−1=−1+e−C/2−Br2/2.From Figure10, it is notable that the graphs of Mr and Ur illustrate the monotonically increasing nature, whereas the behavior of Zr is decreasing.Figure 10
Behavior ofMr, Ur, and Zr.


### 4.9. Adiabatic Index

Adiabatic index plays one of the most important roles in the stability of the stars. In general, it expresses the stiffness of EoS for energy density and categorizes the relativistic and nonrelativistic celestial objects according to their stability condition. The concept of the dynamical stability against infinitesimal radial adiabatic perturbation of the celestial structures was proposed by Chandrasekhar [109]. Later, his idea became much popular among cosmologists and can be tested for both isotropic and anisotropic fluid [110–115]. For a physically acceptable stellar model, the value of adiabatic index should be greater than 4/3. Moreover, the formula of the adiabatic index for r component and t component is shown as
(31)γr=ρ+prprdprdρ=ρ+prprvr2,γt=ρ+ptptdptdρ=ρ+ptptvt2.Hence, from Figure11, it is clear that this condition is fulfilled at each point of our stellar system.Figure 11
Variation ofγr and γt.


## 4.1. Energy Density and Pressure Progression

Here, we explore the graphical responses of the density, redial pressure, and transverse pressure in the light of thefR gravity model. It is clearly shown from Figure 2 that all these plots illustrate positive and singularity-free nature. Also, these plots attain the highest values at the center and touch the surface at the boundary which shows that these plots exhibit satisfactory behavior. We also discuss the graphical illustration of gradients of ρ,pr, and pt, and their plots are shown in Figure 3.Figure 2
Evolution ofρ, pr, and pt.Figure 3
Variation ofdρ/dr,dpr/dr, and dpt/dr.The gradient ofρ,pr, and pt presented negative behavior, which can be seen from Figure 3. These aspects indicate the apex compactness behavior of the compact stars.


## 4.2. Charge Density and Electric Field

Further, we examined the graphical response impact ofq, σ, and E2. In Figure 4, the plot of electric charge shows increasing responses, whereas the nature of charge density is decreasing. One can easily observe that σ attains its highest value at center. Moreover, the electric field illustrates that its graph vanishes when r=0 and then increases in a positive direction.Figure 4
Variation ofq, σ, and E2.


## 4.3. Anisotropy Evolution

In this segment, we investigate the graphical responses of anisotropy parameterΔ, given in Equation (15) [104], since the repulsive nature of Δ validates the existence of compact objects. It can be notable from Figure 5 that for a considered fR gravity model, anisotropy parameter exhibits repulsive nature as Δ>0. This implies that our system is well-behaved and stable.Figure 5
Behavior ofΔ.


## 4.4. Energy Conditions

To verify the existence of the compact objects, energy constraints’ role is very crucial. The famous energy constraints [105] are categories as follows: null energy (NEC), weak energy (WEC), strong energy (SEC), and dominant energy conditions (DEC), and defined as
(i)
NEC:ρ+E2≥0(ii)
WEC:ρ+pr≥0,ρ+pt+E2≥0(iii)
SEC:ρ+pr+2pt+E2≥0(iv)
DEC:ρ−pr+E2≥0,ρ−pt≥0It has been observed from Figure6 that all energy bonds showed well-fitted nature for the considered fR gravity model.Figure 6
Evolution of energy conditions for our proposed model.


## 4.5. Equilibrium Constraint

Further, we examine the equilibrium state of thefR gravity model for the succeeding forces: gravitational force, hydrostatics force, anisotropic force, and electric force. For this purpose, we explore the Tolman-Oppenheimer-Volkoff equation [45, 46] along electric charge, given as
(22)MGrρ+prreλ−ν/2+dprdr−2rpt−pr−σrEreλr/2=0.The effective gravitational massMGr represented by
(23)MGr=12ν′reν−λ/2.By replacingMGr in Equation (22), we get
(24)ν′2ρ+pr+dprdr−2rpt−pr−σrEreλr/2=0,where Fg=−ν′/2ρ+pr,Fh=−dpr/dr,Fa=,andFe=σrEreλr/2. Here, Fg symbolizes as gravitational force, Fh indicates hydrostatic force, Fa defines as anisotropic force, and Fe identifies as electric force. For a realistic model of fR gravity, the joint sum of the above-mentioned four forces should be exactly equal to zero. This implies that for an equilibrium system, all these forces cancel out the effect of each other and maintain the balancing state, i.e.,
(25)Fg+Fh+Fa+Fe=0.From Figure7, it can be clearly noticed that all forces are satisfied by the necessary constraints of equilibrium.Figure 7
Behavior ofFh, Fg, Fa, and Fe for our proposed model.


## 4.6. Equation of State

Next, we determine the EoS for ther component and t component. The two ratios of EoS are mentioned as
(26)wr=prρ,wt=ptρ.In the graphical illustration shown in Figure8, it can be noticed that 0<wrandwt<1. This implies that our system exhibits well-behaved nature.Figure 8
Evolution ofwr and wt for our proposed model.


## 4.7. Stability Condition

The velocity sound forr component and t component is denoted by vr2 and vt2 and is defined as
(27)vr2=dprdρ,vt2=dptdρ.To prove the physical stability of the consideredfR model, we employ the Herrera technique [87] presented as 0≤vr2andvt2≤1. In this regard, we discussed one more worthy approach presented by Abreu et al. [106] to investigate the potentially unstable (stable) configuration of the stellar objects. It is important to mention that potentially unstable (stable) realms within the compact structure are managed by the modification of sound propagation. The anisotropic realm is considered as potentially stable where −1≤vt2−vr2≤0. This means that the horizon in which the transverse speed component is less than the radial speed sound component is recognized as potentially stable otherwise not. It is worthy to highlight here that the considered fR model is entirely stable as shown in Figure 9.Figure 9
Behavior ofvr2, vt2, and vt2−vr2 for fR gravity model.


## 4.8. Mass, Compactness, and Redshift Evolution

For the existence of a realistic and viable model, its mass function denoted byMr, compactness factor symbolizes by Ur, and the redshift function identifies as Zr are considered as the necessary constraints in the analysis of a stellar system. Generally, one can believe that objects having anisotropic fluid should not be arbitrarily massive for a viable modified theory model. In the frame of isotropic fluid, the concept of maximum permissible mass-radius ratio, i.e., 2M/R<8/9, was studied by Buchdahl [107]. Later, Mak and Harko [108] generalized the concept of Buchdahl and argued that this mass-radius ratio holds for isotropic as well as anisotropic fluid. Hence, the mass function for anisotropic fluid is given by
(28)Mr=4π∫0Rr2ρdr=R21−e−λ.This response ofMr implies that Mr is regular and attains maximum values at the boundary. Further, we explore the compactness factor which is presented as a mass-radius ratio [108] and is defined as
(29)Ur=2Mrr=2e−2Ar2−1+eAr2−1+eAr2−3Br2−B2r4+Ar22+Br2r3.Next, the redshiftZr is given as [104]
(30)Zr=e−ν/2−1=−1+e−C/2−Br2/2.From Figure10, it is notable that the graphs of Mr and Ur illustrate the monotonically increasing nature, whereas the behavior of Zr is decreasing.Figure 10
Behavior ofMr, Ur, and Zr.


## 4.9. Adiabatic Index

Adiabatic index plays one of the most important roles in the stability of the stars. In general, it expresses the stiffness of EoS for energy density and categorizes the relativistic and nonrelativistic celestial objects according to their stability condition. The concept of the dynamical stability against infinitesimal radial adiabatic perturbation of the celestial structures was proposed by Chandrasekhar [109]. Later, his idea became much popular among cosmologists and can be tested for both isotropic and anisotropic fluid [110–115]. For a physically acceptable stellar model, the value of adiabatic index should be greater than 4/3. Moreover, the formula of the adiabatic index for r component and t component is shown as
(31)γr=ρ+prprdprdρ=ρ+prprvr2,γt=ρ+ptptdptdρ=ρ+ptptvt2.Hence, from Figure11, it is clear that this condition is fulfilled at each point of our stellar system.Figure 11
Variation ofγr and γt.


## 5. Conclusion

Here, we investigate the existence of stellar structure by considering the charged static spherical symmetric spacetime along with the Bardeen sphere in the background of a viable model for thefR theory of gravity. For this purpose, we assume the metric potentials by adopting the Krori-Barua [66] model, i.e., νr=Br2+C and λr=Ar2, where A, B, and C are any constant parameters. Moreover, in order to determine the values of these constant parameters, we relate the interior geometry with the exterior Bardeen sphere [73]. To validate the existence of our system, we evaluate different mandatory features related to the celestial structure. To do so, we discuss the graphical behavior of various figures of M for different values of R, i.e., R=10.10km with M=1.2,1.3,1.4,1.5M⊙, R=9.10km with M=1.2,1.4M⊙, and R=8.10km with M=1.2M⊙.The main goal of our investigation is to generate a new realistic family offR theory of gravity solutions in the presence of charge anisotropic fluid. The significant outcomes are enlisted beneath.
(i)
Metric potentials are generally used to narrate the nature of spacetime. It can be easily seen from Figure1 that the graphs of the metric potentials grr=eλ and gtt=eν are positive, singularity free, and fulfill the necessity, i.e., eλr=0=1 and eνr=0=eC. It is noticed that grr and gtt attain the maximum values and show monotonically increasing behavior, which indicates that the chosen system exhibits a justifiable behavior(ii)
The variation of density and pressure components corresponding to radial coordinater for the considered model is positive and regular inside the sphere. From Figure 2, it is observed that the graphs of ρ,pr, and pt attain maximum value at the center and decreasing downward, which confirms that the considered fR gravity model is well-behaved in nature. Also, the graphical representation of gradients of ρ,pr, and pt, shown in Figure 3, yields satisfactory outcomes(iii)
Next, discussion is based on the graphs ofq, σ, and E2. One can clearly see in Figure 4 that the graphical response of E2 is maximum at r=R, while on the other hand, σ demonstrates the decreasing behavior. Moreover, the plot of charge q shows positive behavior and remained consistent for the entire system. Further, Figure 5 clarifies that Δ reveals consistently positive nature for the considered fR gravity model. Thus, the repulsive nature of anisotropy confirms the physically acceptable behavior of the chosen stars(iv)
Figure6 shows that all the energy constraints are fulfilled(v)
Figure7 represents the balancing behavior of (Fg), (Fh), (Fa), and (Fe) forces for our proposed viable fR gravity model(vi)
For the well-behaved nature of the compact objects, the graphical response of EoS must fulfill the condition0<wrandwt<1. The corresponding plots given in Figure 8 show the consistent behavior of our proposed model of the fR gravity(vii)
For stellar objects, the values of velocity of soundsvr2 for r and vt2 for t should be furnished within limits of 0,1. It is clear from Figure 11 that the causality constraints exhibit satisfactory nature for considered stellar structure. Furthermore, the Abreu approach outcomes are also consistent for the considered fR gravity model(viii)
It has been noted from Figure9 that the plots of Mr and Ur exhibit monotonically increasing nature while the plot of Zr is monotonically decreasing; this implies that we establish a stable system(ix)
The radial and transverse adiabatic index values denoted byγr and γt for the chosen model are greater than 4/3, as shown in Figure 10, which confirms the stability of our systemThus, our presented work satisfied all those results proposed by Mustafa et al. [83] under the frame of GR. This confirms that our considered fR gravity model is entirely stable and physically acceptable.



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