A Spherical Relativistic Anisotropic Compact Star Model

We provide solutions to Einsteins field equations for a model of a spherically symmetric anisotropic fluid distribution, relevant to the description of compact stars. The central matter-energy density, radial and tangential pressures, red shift and speed of sound are positive definite and are decreasing monotonically with increasing radial distance from the center of matter distribution of astrophysical object. The causality condition is satisfied for complete fluid distribution. The central value of anisotropy is zero and is increasing monotonically with increasing radial distance from the center of the distribution. The adiabatic index is increasing with increasing radius of spherical fluid distribution. The stability conditions in relativistic compact star are also discussed in our investigation. The solution is representing the realistic objects such as SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432 with suitable conditions.


Introduction
The General Theory of Relativity provides the mathematical platform for describing the physical world generated by highly compact astrophysical objects.According to Einstein's relativistic theory, the existence of matter and its gravitational field causes to exist space-time gets curvature.When an incoherent matter like dust is contracted and condensed, a condition is reached where gas degeneracy pressure, thermal pressure (negligible with respect to gas degeneracy pressure) and gravitational pressure are in equilibrium.This equilibrium state forms a dense astrophysical object like neutron star, white dwarf, quark star, strange star etc.In order to understand the structural and physical properties of Some researchers, Sah and Chandra [20], Komathiraj and Maharaj [21], Herrera et al. [22], Thirukkanesh and Regel [23], Chaisi and Maharaj [24], Sunzu et al. [25], Maurya and Gupta [26] and Bhar et al. [27] etc. gave physically accepted solutions of Einstein's field equations for anisotropic fluid distribution.The researchers Sah and Chandra [28], Whitman and Burch [29], Tikekar [30], Ivanov [31], Gupta and Kumar [32], Herrera et al. [33] [34] [35], Tewari and Charan [36] [37] [38], Tewari [39], Ivanov [40] [41] [42] [43], Maurya and Gupta [44] etc gave remarkable work in relativistic astrophysics.The stellar model given in this paper is the contribution of the work related to charged fluid distribution, done by Sah and Chandra.In this work, we discuss physically accepted solutions for anisotropic fluid distribution in addition with energy conditions, variation of mass with central density, equation of state and mass-radius relationship in contest of some astrophysical objects, SAXJ1808.4-3658,HerX-1, 4U1538-52 and LMC X-4.
The whole work of this paper is divided into nine sections.The second section is comprised of Einstein's non-linear differential equations of field theory.Besides this the physical parameters like matter-energy density, radial and tangential pressures, anisotropy, surface and central red shifts are expressed in this section.In third section the parameters of physically acceptable non-singular solutions are given.A non-singular physically acceptable solution of Einstein's field equations given by Sah and Chandra [28] for anisotropic fluid distribution is incooperated in fourth section.The physical properties of the solution are given in fifth section.The matching of interior metric of the distribution with the exterior metric given by Schwarzschild is included in sixth section.In seventh section the tabular and graphical representation of physical parameters of our model of super dense star corresponding non-singular and physically accepted solutions are presented.The stability criteria are given in eighth section.In ninth section, some conclusions of whole work presented in this paper are given.

Einstein's Field Equation of General Relativity
The relationship between gravity due to existing fluid material and geometry of space-time, contained in the Einstein's non-linear differential equations for relativistic field theory is given by where ij T , the energy momentum tensor for spherically anisotropic fluid distribution is defined as where ρ is the proper density, r p and t p are radial and tangential pressures of an anisotropic fluid along and perpendicular to u µ (time-like four-velocity vector) respectively, x µ is the unit space like vector along radial vector and g µ ν metric tensor such that For spherically symmetric stable fluid distribution, the metric element can be expressed as ( ) where A and B are metric coefficients depending on radial distance r only.
Incorporating Equation (2) and Equation (3) with Equation (1), we get the following equations where ( ) In the above equations, (') denotes the derivative against radial distance r.
The redshift due to gravity of dense spherically symmetric fluid distribution is given by 1 2 00 which turns out the gravitational redshift ( Z ) for compact stellar object and it's surface redshift ( Σ Z ) as ( ) where Schwarzchild parameter or compactness, , r Σ is the radius of spherically symmetric stellar fluid configuration of mass M .

Physical Acceptability Conditions for Stable Stellar Configuration
The non-singular solution of Einstein's field equation must satisfy the following physically accepted conditions 1) There is no singularity in the solution interior and on the surface of the International Journal of Astronomy and Astrophysics astrophysical object.For these metric coefficients A and B , central pressure and central density should be positive definite throughout the interior of the object i.e. , 0 ≥ A B , 0 0 ρ > and 0 0 p > .
2) The density, radial and tangential pressures should be maximum at the center and decreased monotonically on moving from center to the surface of the fluid object i.e. a) at center 3) At boundary radial pressure, r p shold be equal to zero while tangential pressure, t p may not be equal to zero i.e. pressure anisotropy is zero at the center i.e. 0 0 ∆ = .The pressure anisotropy ∆ should be increased with radial distance from the center of fluid distribution and at the surface it must be ( ) for zero surface radial pressure.4) The null energy condition (NEC) 0 ρ ≥ , weak energy condition (WEC) Thus the equation of state for highly dense astrophysical object indicates that the speed of sound should be decreased with increasing radial distance.
6) The realistic adiabatic index Γ should be greater than 4 3 .

7)
The gravitational red shift must have positive finite value and be decreased monotonically with increasing the radial distance.
8) The difference of the square of the radial speed of sound and tangential speed of sound The method of solving the Einstein equations is to choose ansatze for the two metric functions.Then the three Einstein Equations ( 4)-( 6) give expressions for the matter components ρ , r p and t p .The two metric components are so related to allow simple forms for the physical variables.There is no integration in fact.

A Physically Acceptable Non-Singular Solution
The non-singular solution of Einstein's field Equations ( 4) to (7) presented by Sah and Chandra [28] is given by ( ) Here α has dimension (length) −2 and β , γ are dimensionless constants.
Using Equation ( 12) and Equation ( 13), Equation ( 4) gives radial pressure of the fluid inside the object as ( ) ( ) Equation ( 12) and Equation ( 13) with Equation ( 5) give tangential pressure as Equation ( 6) gives the matter energy density for our solution as ( ) The red shift and anisotropy of astrophysical fluid distribution are given by ( ) The radial derivatives of pressures r p′ and t p′ given by Equations ( 14) and ( 15) are The radial derivative of matter energy density given by Equation ( 16) is ( )

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The radial derivatives of redshift and anisotropy given by Equation (17) and Equation ( 18) respectively are

Conditions for Model Parameters for Physical Acceptable Solution
For positive value of metric coefficient A i.e.
The monotonically increasing nature of metric coefficients A and B with increasing radial distance r for suitable choice of constants , α β , and γ are shown in Figure 1(a) and Figure 1(b).The values of pressure p, density ρ and red shift Z at the center are given by ( ) ( ) For positive values of pressure 0 p , density 0 ρ and redshift 0 From Equations ( 19) to (23), the radial derivatives of pressure, density, red shift and anisotropy at the center of fluid distribution are zero.
From Equation (19) and Equation (20), at the center of fluid distribution   From Equation (21) ( ) which shows that for all values of parameters, the matter energy density is maximum at the center of the fluid distribution i.e. ( ) 0 0 ρ′′ < .
At the center, the equations of state for matter distribution are given by ( ) ( ) The central equation of state should obey the conditions From Equation (19) and Equation ( 21) and from Equation (20) and Equation ( 22) The causality conditions at the center which gives the condition for model parameters From Equation (22) and Equation ( 23) ( ) For maximum value of red shift at the center, ( ) 0 0 ( ) 0 0 ′′ ∆ > , the anisotropy is minimum at the center for all values of model parameters.
It is observed that for model parameters given by International Journal of Astronomy and Astrophysics the metric coefficients A and B increase monotonically with increasing radial distance r as illustrated by (Figure 1 value increases with increasing radial distance r (Figure 5, Figure 6).

Model Parameters by Matching Conditions
For the stable astrophysical fluid distribution the interior metric must be matched with the Schwarzschild exterior metric given by ( ) For this the metric coefficients must be continuous (First fundamental form) and differentiable (Second fundamental form) at the surface at r r Σ = and Thus where 1 2 p Ψ = − S ;
With the help of model parameters for astrophysical objects SAXJ1808.4-3658,HerX-1, 4U1538-52, LMC-X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432 given in Table 3 with compactness p T , a super dense compact star model can be made which may be useful for further study of the different properties of various astrophysical objects.

Stability Criterion
In this section our aim is to determine the physical requirement of realistic solution featuring the stability of compact stellar or astrophysical objects.
Weak Energy Condition (WEC) Strong Energy Condition The principles of energy condition are plotted graphically against radial distance for realistic stellar objects in Figure 7(a), Figure 7(b) and Figure 7(c).

Equilibrium of Different Forces
There are three kind of forces viz gravitational force g F , hydrostatic force h F and anisotropical force a F acting on a compact star which are in equilibrium. Thus The generalized Tolman-Oppenheimer-Volkoff equaion for equilibrium condition under these forces for anisotropic fluid distribution is given by M is effective gravitational mass and is defined by Tolman-Whittaker as ( ) Now TOV equation reduces to  In view of Equation ( 45) and Equation ( 48) the three forces are given by ( ) ( )

Causality Conditions
The causality condition states that the radial and tangential speeds of sound should not be more than one i.e.The condition for potentially stable anisotropic fluid distribution is

Relativistic Adiabatic Index
The relativistic adiabatic index for anisotropic fluid distribution of an astrophysical object is given by

Harrison Zeldovich Novikov Static Stability Criterion
Harrison [45] Zeldovich Novikov [46] static stability criterion are given by The rate of change of mass of stellar configuration with respect to central density is always positive.Consequently our solution gives stable stellar configuration.The variation of masses of astrophysical objects SAXJ1808.4-3658,HerX-1, 4U1538-52 and LMC-X-4 with their central density is graphically represented in Figure 12.

Conclusions
We have given a well behaved analytic charge free solution for spherical and symmetric anisotropic fluid distribution given by Sah and Chandra [28]      always less than 4 9 . The red shift is also satisfied the upper bound limit for the realistic star models i.e. 1 s ≤ Z . For the EoS, we plot a graph for p r and p t versus against r as shown in Figure 13 It is found that the present model is very close to the observed data of a number of compact stars like SAXJ1808.4-3658,HerX-1, 4U1538-52, LMC X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432 and many more given by Elebert et al. [47], Abubekerov et.al. [48], Rawls et al. [49], Demorest et al. [50] and Gangopadhyay et al. [51].Thus the solution obtained by us is very helpful in constructing the models of super dense astrophysical object like Neutron stars, Strange star, quark stars and many more.
satisfied throughout the stellar object.The speed of sound must be decreased monotonically on increasing the radial distance and increased with increasing density i.e.
and radial adiabatic index, r should lie between −1 and 0 for the matter within the object.International Journal of Astronomy and Astrophysics

Figure 1 .
Figure 1.(a): The variation of Metric Potential " A "; (b): The variation of Metric Potential "B " against r r Σ .
maximum values of radial and tangential pressures at the center, the conditions of model parameters are ( ) (a) and Figure1(b)).The matter energy density ρ reduces monotonically with increasing radial distance r as shown in the (Figure2) while radial and pressures decreases monotonically with increasing radial distance r such that the radial pressure becomes zero at the surface of the fluid distribution (Figure3(a) and Figure3(b)).The pressure density ratios are always positive and less than one throughout the distribution (Figure4(a) and Figure4(b)).The red shift and anisotropy with zero central
Figure 3. (a): Variation of Radial Pressure r p against r r Σ ; (b): Variation of tangential Pressure t p with r r Σ .
represent the model parameters obtaining from these boundary conditions.
p = S for an astrophysical object International Journal of Astronomy and Astrophysics

Figure 5 .
Figure 5. Variation of Red Shift with r r Σ .

Figure 6 .
Figure 6.Variation of Anisotropy factor with r r Σ .

( 48 )
International Journal of Astronomy and Astrophysics

Figure 7 .
Figure 7. (a): Variation of NEC with r r Σ ; (b): Variation of WEC with r r Σ ; (c): Variation of SEC with r r Σ .
within the stellar object.The graphical representations of radial and tangential speeds of sound with respect to r r Σ in the Figure9(a) and Figure9(b) which shows that the causality conditions are obeyed throughout the stellar configuration.
is the condition for potentially unstable anisotropic fluid distribution.In our relativistic stellar model the stability factor satisfies the condition for potentially stable anisotropic fluid distribution everywhere inside fluid sphere depicted by Figure10.

Figure 8 .
Figure 8.(a) Variation of three forces g F , h F and a F with r r Σ ; (b): Variation of three forces g F , h F and a F with r r Σ .

Figure 10 .
Figure 10.Variation of Stability factor

Figure 12 .
Figure 12.Variation of mass with central density.

Figure 13 .
Figure 13.(a): Variation of radial pressure r p with density ρ against r; (b): Variation of tangential pressure t p with density (a) and Figure 13(b).
Our proposed model of anisotropic fluid satisfies the following energy conditions within the framework of general relativity.P. C. Fulara, A. Sah DOI: 10.4236/ijaa.2018.8100458 International Journal of Astronomy and Astrophysics

Table 1 and
Table 2 are made for astrophysical object SAXJ1808.4-3658.The clear that calculated values of mass and radius of these stars is well fitted with the observational data.We have also obtained the central density, surface density, central pressure, and surface redshift given in Table5.The compactness factor is International Journal of Astronomy and Astrophysics
35 2.149 10 × 0.312 International Journal of Astronomy and Astrophysics