# A spherically symmetric model of anisotropic fluid for strange quark spheres

## Abstract

In the present work, we try to find a solution without singularity of Einstein’s field equations for the spherically symmetric perfect fluid objects, accurately strange quark spheres, taking into consideration Schwarzschild metric as the outside space-time. An ensemble of inside solutions found on the basis of the simplest linear state equation in the specific form \(p_r =\alpha \rho -\beta \). The energy density \(\rho (r)\), the radial pressure \(p_r (r)\) and the tangential pressure \(p_t (r)\) are devoid of any singularity and exhibit a well-behaved nature within the generalized anisotropic solution for compact spherical object. The generalized TOV equation is very much preserved inside the system and all energy conditions are excellent. The stability of the matter distribution of our system is checked by the concept of Herrera’s cracking and the condition of causality is all around fulfilled for our models. The adiabatic index of our specific configuration is greater than 4 / 3 in all interior points of the system and the mass-to-radius ratio in our situation is determined also lies within the Buchdahl limit i.e. \(\hbox {M}/\hbox {R}\le 4/3\,\left( {\approx 0.444} \right) \). We explore the physical characteristics based on the analytical model developed for relativistic compact stellar spheres inside the framework of the general theory of relativity. The evaluated mass and radius are in close concurrence with the observational information. We show that various physical characteristics of the known strange spherical object, viz. PSR J1614-2230, Vela X-1, 4U 1608-52, PSR J1903+327, 4U 1820-30, Cen X-3, Her X-1, and SAX J1808.4-3658, can be described by the current model.

## 1 Introduction

In the midst of the sphere’s main sequence, hydrogen fuel combustion happens in the central zone in order to maintain the equilibrium of the star corresponding to the internal gravitational force. Burned hydrogen is the most productive wellspring of energy for any sphere assembled in principle and continues easily with no ferocious occasions, providing the star with the primary gauge of external pressure as well as the development of the helium center. However, beyond the combustion of Hydrogen, all other atomic combustion phases such as Helium combustion, Carbon combustion may happen on some cataclysmic occasions, for instance, planetary cloud, an explosion of supernovae joined by huge mass expulsion. In these occasions, the sphere center is too much packed prompting the configuration of relativistic compacts stellar spheres such as neutron star, white dwarfs, and black holes. Besides, the destiny of the packed center will be dictated by the underlying mass of its sphere of ancestor. The matter is to a great degree thick for the previously mentioned relativistic compacts stellar spheres in this way showing some extraordinary conduct and brings about unusual variation in space-time structure throughout it. The extraordinary states of energy density and pressure interior the sphere of the neutron can cause the neutron stage change into bosons, hyperons, and other strange quark matter. The more rigid states equation is necessary to clarify the conduct of the strange matter of the quark and other exotic states of matter [1, 2, 3, 4]. The modeling of ultra-dense configurations in astrophysics of relativistic has aroused huge concern for astrophysicists in recent decades. The ongoing advances in observational procedures have uncovered numerous characteristics of the relativistic compacts stellar spheres yet at the same time; numerous lacunae are there to be settled. The hypothetical investigation of the nature and exact constitution of compact relativistic stellar spheres within the framework of the general relativity hypothesis is of fundamental importance for astrophysics. It encourages researchers to understand the interactions of particles with high density and to study their different stages of development as well as their internal characteristics. The compactification parameter of dead objects exits them from the condition of isotropic and matter demonstrates the anisotropic behavior which explains the difference in many of the characteristics of the compact objects.

In order to comprehend the inside configuration and transformative phases of relativistic compact stellar spheres, the anisotropic fluid appropriation plays a principal goal. As the relativistic compact stellar spheres have heavy centers and their densities surpass the density of nuclear, along these lines, pressure should be anisotropic interior the relativistic compact stellar spheres [5, 6]. In the anisotropic distribution of matter, it is seen that pressure is parceled into radial and tangential constituents. In this case, numerous specialists have studied the features of dense relativistic compact stellar spheres with anisotropic fluid geometry. Hossein et al. [7] examined the highlights of anisotropic relativistic spheres with constant of cosmology and discovered stable astrophysical spheres configuration. Kalam et al. [8] analyzed the relativistic modeling of anisotropic neutron spheres and verified the availability of stability, anisotropy estimation and energy conditions with quintessence dark energy. Paul and Deb [9] detailed new exact solutions of relativistic compact stellar spheres for anisotropic matter distribution, which keep up the equilibrium of hydrostatic.

In relativistic compact stellar spheres, the more dense neutron spheres can additionally be fell to frame a black hole while for less dense neutron spheres, it is possible to transform into a quark sphere. It is predicted that the inside matter appropriation of quark spheres is expected to be processed using the state equation of the MIT quark bag model [10, 11]. In recent years, the investigation of quark spheres has roused numerous physicists to examine its characteristics and in addition, their inside configuration. In this way, Bhar [12] analyzed features of PSR J1614-2230, 4U1820-30 and SAX J 1808.4-3658 spheres using the state equation of MIT quark bag model and gotten stability structure of these spheres. Rahaman et al. [13] suggested another mass insertion expression for the candidates of strange sphere with the state equation of MIT-Bag model and researched the physical conduct of spheres from 6 km to the limit surface. Different all around carried on solutions of Einstein’s field equations have been investigated by forcing the barotropic state equation on the inside of the spherical object. Sharma and Ratanpal [14] have been proposed a relativistic astrophysical model conceding a quadratic state equation. Malaver [15, 16] examined a quadratic state equation for the distribution of matter and proposed special shapes for line element potentials in order to build compact spherical object models. a few latest models of anisotropy for compact spherical objects with a distribution of strange matter incorporate the consequences of Sharma and Maharaj [17] and Lobo [18] with a barotropic state equation. Komathiraj and Maharaj [19] studied the anisotropic compact spherical object models of quark spheres obeying the state equation of the MIT bag model. However, can be noted that recent works concerning the distribution of the anisotropic matter, where a few specialists have considered the anisotropy is related to compact stellar objects [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].

Our objective in this paper is to present a new interior anisotropic solution to the Einstein field equations relativistic compact stellar sphere by using a linear state equation of the form \(p_r =\alpha \rho -\beta \). We investigate the physical properties from the analytical model developed for compact spheres inside the framework of the general theory of relativity. The estimated mass and radius are in close agreement with the observational data. We show that various physical properties of the known strange spherical object, such as PSR J1614-2230, Vela X-1, 4U 1608-52, PSR J1903+327, 4U 1820-30, Cen X-3, Her X-1, and SAX J1808.4-3658, can be portrayed by our model.

The outline of this article is as follows: In Sect. 2, we present the interior metric and the corresponding Einstein field equations of anisotropic fluid distributions. The Generalized anisotropic solutions for compact spherical object are discussed in Sect. 3. The other characteristics are given in the Sects. 4 and 5, and a comparative study are given in Sect. 6. Concluding remarks close this paper.

## 2 Interior solution and Einstein field equations of anisotropic fluid distributions

*r*yet to be established. Also, having a momentum-energy tensor depicted for the astrophysical fluid filling the inside of the object has the standard form

## 3 Generalized anisotropic solution for compact spherical object

We notice that the physical properties of the model rely upon the constants *R*, \(\alpha \) and \(\beta \). We have to put proper limits on these parameters aiming that the model can depict a reasonable compact spherical object. To this end, in view of different physical prerequisites, let us currently break down the conduct of the physical parameters.

*a*is the radius of the compact spherical object. At that point, from Eq. (8), the surface and focal densities are respectively acquired as

*r*. Clearly, the density is greatest at the core of the compact spherical object and it diminishes radially outward.

## 4 Matching conditions

## 5 Physical properties of the spherical objects

In this section, we will discuss various physical characteristics of strange spherical objects using the previously proposed model.

### 5.1 Stability of the system

#### 5.1.1 The Tolman–Oppenheimer–Volkoff equation

*r*and is given by the Tolman–Whittaker procedure as follows

#### 5.1.2 The status of the speed of sound within the system

To check the potential stability or instability of our system we have used the concept of Herrera’s cracking [52]. The causality condition determines the physical acceptability admitted for the fluid distribution which requires \(0<v_{sr}^2 \le 1\) and \(0<v_{st}^2 \le 1\). According to the condition of Herrera [52] and Andréasson [53] for the stability of the distribution of matter, the region for which \(\left| {v_{st}^2 -v_{sr}^2 } \right| \le 1\) i.e. no cracking is a potentially stable region. In Fig. 6, we observe that \(\left| {v_{st}^2 -v_{sr}^2 } \right| \le 1\) in the region for which \(r\le 9.819\). This implies that our system meets all these conditions and that it is potentially stable in this region. The tangential and radial speeds of sound are defined respectively as \(v_{st} =\sqrt{dp_t /d\rho }\) and \(v_{sr} =\sqrt{dp_r /d\rho }\).

#### 5.1.3 Adiabatic index

*r*of our model for spherical object Cen X-3 is motivated by the fact that it gives a monotonically increasing function in the stellar inside. Similar forms for the adiabatic index have earlier been considered by Maurya et al. [32] for anisotropic fluid stars. Figure 7 clearly shows that the adiabatic index of our specific configuration is greater than 4 / 3 in all interior points of the specific configuration. This characteristic obviously indicates that the system in this work is stable.

Physical parameters for different anisotropic astrophysical models for \(\upalpha =0.333\)

Observed objects | Radius | | Radius | | \(\beta \) | \(\rho (0)\) | \(p_r (0)\) |
---|---|---|---|---|---|---|---|

| \(\left( {M_\odot } \right) \) |
| \(\left( {\mathrm{MeV \, fm}^{-3}} \right) \) | \(\left( {{\begin{array}{l} {\times 10^{15} } \\ {\mathrm{gm\, cm}^{-3}} \\ \end{array} }} \right) \) | \(\left( {{\begin{array}{l} {\times 10^{35} } \\ {\mathrm{dyne\, cm}^{-3}} \\ \end{array} }} \right) \) | ||

PSR J1614-2230 | 10.300 | 1.970 | 9.69 | 0.280 | 106.66 | 1.421 | 2.482 |

Vela X-1 | 10.143 | 1.770 | 9.56 | 0.260 | 108.00 | 1.299 | 2.116 |

4U 1608-52 | 10.105 | 1.740 | 9.52 | 0.255 | 110.60 | 1.272 | 2.035 |

PSR J1903+327 | 10.017 | 1.667 | 9.43 | 0.245 | 112.07 | 1.198 | 1.813 |

4U 1820-30 | 10.000 | 1.580 | 9.31 | 0.235 | 114.90 | 1.113 | 1.558 |

Cen X-3 | 9.819 | 1.490 | 9.17 | 0.225 | 118.26 | 1.042 | 1.347 |

Her X-1 | 6.700 | 0.880 | 8.10 | 0.168 | 132.96 | 1.381 | 4.145 |

SAX J1808.4-3658 | 7.070 | 1.435 | 7.95 | 0.299 | 217.08 | 4.808 | 14.425 |

### 5.2 Energy conditions

### 5.3 Mass function

*a*is obtained as

*a*.

### 5.4 Mass-radius ratio

*a*. We remark that the requirement on the greatest permitted mass-to-radius ratio in our situation is determined as \(M/R=0.323\), which is like the isotropic fluid spherical object, i.e., \(M/R\le 4/9 \left( {\approx 0.444} \right) \).

### 5.5 Surface redshift function

*a*, as shown in Fig. 10. The greatest value permitted surface redshift \(z_s \) in our case is 0.317. The figure in like manner shows that the surface redshift function is limited within the stellar configuration.

## 6 A comparative study

To examine the physical characteristics based on the analytical model developed so far, we choose the spherical object Cen X-3 as a delegate of the strange spherical objects possessing the following parameters: mass \(m(a)=1.49\, M_\odot \) and radius \(a=9.819\, \mathrm{km}\) for \(\beta =118.26\, \mathrm{MeV\, fm}^{-3}\) and \(\alpha =0.333\).

Using the chosen mass and radius values shown in Table 1, we have shown various physical characteristics of the proposed configuration of strange compact objects. The observed masses shown in Table 1 are available in the following references [56, 57, 58, 59, 60]. We have highlighted the values of the constant \(\beta \), especially for astrophysical configurations of strange spherical object candidates such as PSR J1614-2230, Vela X-1, 4U 1608-52, PSR J1903+327, 4U 1820-30, Her X-1, and SAX J1808.4-3658. We find that when the values of \(\beta \) increase with \(\alpha \) fixed at 0.333, the astrophysical system becomes more compact and the density in the spherical object increases progressively with the increasing values of \(\beta \). The observed value of the mass of Cen X-3 [56] is obtained for the astrophysical system to contract, that is to say the progressively decreasing radius values. However, from the point of view of a mathematically steady model, it gives the idea that an extensive variety of estimation of the constant \(\beta \) is conceivable, which is coherent with the data of Relativistic Heavy Ion Collider (RHIC) and Super Proton Synchrotron (SPS). The obtained solutions are free of any singularity and exhibit a well-behaved nature within the generalized anisotropic solution for compact spherical object. This agrees with the results reported by Maurya et al. [32, 39, 61]. We have demonstrated that the anisotropy effects lead a wide range of results in the geometry and composition of an object. For instance, this may either yield a gentler equation of state or an object with various mass and radius. Our examination depends basically on the choice of a specific metric function such one given by the Eq. (7) yielding a linear equation of state.

## 7 Concluding remarks

We have proposed a spherically symmetric model of anisotropic fluid for strange quark spheres which satisfies all conditions of physical reality and is devoid of focal singularity with the outer space-time being the Schwarzschild metric. We introduced this formalism to determine exact estimations of the radius of the different candidates of the strange sphere by using their observational mass and constant \(\beta \) gave in Table 1. We also discussed the stability analysis which verified that our model was completely satisfied with the conditions of causality and the concept of Herrera cracking. In a similar idea, in order to analyze the meaning of our model in the search for compact objects, we have considered various compact objects, including PSR J1614-2230, Vela X-1, 4U 1608-52, PSR J1903+327, 4U 1820-30, Her X-1, and SAX J1808.4-3658 and showed that, for the estimated radius, the masses of the object dictated by our model are close to the observed observation (see Refs. [56, 57, 58, 59, 60]). This generally suggests that the solution proposed in this work can be used as a feasible model for describing ultra-dense compact strange spheres.

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