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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.

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 this dense astrophysical object, it is needed to find out the exact solution of Einstein’s relativistic field equation. Therefore exploration of anisotropic solution of Einstein’s relativistic field equation attracts the researchers to do work in this field. In 1916 Schwarzschild [

Buchdahl [

The radial pressure of fluid distribution may be different from the tangential pressure which may lead anisotropy in pressure. In the study of the properties of internal structure of a super dense astrophysical object, the local anisotropy is commonly used by the researchers in their solutions. The phase transition of fluid, existence of a solid core and other physical phenomena create local anisotropy in the pressure of the fluid. The surface red-shift of an astrophysical object, its stable mass and other physical properties may be affected by by anisotropy present in fluid distribution within that object. Combination of effect of anisotropy with equation of hydrostatic equilibrium investigates role of equations of state for fluid distribution.

Some researchers, Sah and Chandra [

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 [

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

R i j − 1 2 R g i j = − 8 π G c 4 T i j (1)

where T i j , the energy momentum tensor for spherically anisotropic fluid distribution is defined as

T i j = ( ρ c 2 + p r ) u i u j − p t g i j + ( p r − p t ) x i x j (2)

where ρ is the proper density, p r and p t 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 ν μ metric tensor such that

g i j u i u j = 1

For spherically symmetric stable fluid distribution, the metric element can be expressed as

d s 2 = − B 2 d r 2 − r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) + A 2 d t 2 (3)

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

8 π G c 4 p r = 1 B 2 ( 2 A ′ r A + 1 r 2 ) − 1 r 2 (4)

8 π G c 4 p t = 1 B 2 ( A ″ A + A ′ r A − A ′ B ′ A B − B ′ r B ) (5)

8 π G c 2 ρ = − 1 B 2 ( 2 B ′ r B − 1 r 2 ) + 1 r 2 (6)

8 π G c 4 ( p t − p r ) = Δ ( r ) = δ ( r ) B 2 (7)

where

δ ( r ) = ( A ″ A − B ′ r B − A ′ B ′ A B − A ′ r A − 1 r 2 ) + B 2 r 2 (8)

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 + Z = g 00 − 1 2 (9)

which turns out the gravitational redshift ( Z ) for compact stellar object and it’s surface redshift ( Z Σ ) as

Z = c A − 1 (10)

and

Z Σ = 1 ( 1 − 2 S p ) 1 2 − 1 (11)

where Schwarzchild parameter or compactness, S p = G M c 2 r Σ , r Σ is the radius of

spherically symmetric stellar fluid configuration of mass M .

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 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. A , B ≥ 0 , ρ 0 > 0 and p 0 > 0 .

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 ( d p r d r ) 0 = 0 = ( d p t d r ) 0 = ( d ρ d r ) 0 and ( d 2 p r d r 2 ) 0 , ( d 2 p t d r 2 ) 0 and ( d 2 ρ d r 2 ) 0 < 0 for maximum value of ρ , p r and p t . such that the radial pressure gradient, d p r d r is negative for 0 ≤ r ≤ r Σ .

b) For monotonically decreasing density and pressures ( d p r d r ) 0 , ( d p t d r ) 0 , ( d ρ d r ) 0 ≤ 0 for 0 ≤ r r Σ ≤ 1 .

3) At boundary radial pressure, p r shold be equal to zero while tangential pressure, p t may not be equal to zero i.e. pressure anisotropy is zero at the center i.e. Δ 0 = 0 . The pressure anisotropy Δ should be increased with radial distance from the center of fluid distribution and at the surface it must be Δ Σ = ( p t ) Σ for zero surface radial pressure.

4) The null energy condition (NEC) ρ ≥ 0 , weak energy condition (WEC) 0 ≤ p r ρ c 2 ≤ 1 , 0 ≤ p t ρ c 2 ≤ 1 and strong energy condition (SEC) 0 ≤ p r ρ c 2 + 2 p t ρ c 2 ≤ 1 with p r ρ c 2 > 0 and p t ρ c 2 > 0 should be satisfied throughout within the fluid object.

5) The causality conditions 0 ≤ d p r c 2 d ρ < 1 and 0 ≤ d p t c 2 d ρ < 1 must be

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. d d r ( d p r d ρ ) < 0 or ( d 2 p r d ρ 2 ) > 0 and d d r ( d p t d ρ ) < 0 or ( d 2 p t d ρ 2 ) > 0 .

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 r = ρ c 2 + p r p r d p r c 2 d ρ and Γ t = ρ c 2 + p t p t d p t c 2 d ρ should be positive and radial adiabatic index, Γ r 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 v t 2 c 2 − v r 2 c 2 should lie between −1 and 0 for the matter within the object.

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 , p r and p t . The two metric components are so related to allow simple forms for the physical variables. There is no integration in fact.

The non-singular solution of Einstein’s field Equations (4) to (7) presented by Sah and Chandra [

A = β + γ ( 1 − α r 2 ) − 1 (12)

B = ( 1 − α r 2 ) − 2 (13)

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

8 π G c 4 p r = α { 4 γ ( 1 − α r 2 ) 3 β ( 1 − α r 2 ) + γ + α 3 r 6 − 4 α 2 r 4 + 6 α r 2 − 4 } (14)

Equation (12) and Equation (13) with Equation (5) give tangential pressure as

8 π G c 4 p t = 4 α { γ ( 1 − α r 2 ) 3 β ( 1 − α r 2 ) + γ − ( 1 − α r 2 ) 3 } (15)

Equation (6) gives the matter energy density for our solution as

8 π G c 2 ρ = α ( 12 − 30 α r 2 + 28 α 2 r 4 − 9 α 3 r 6 ) (16)

The red shift and anisotropy of astrophysical fluid distribution are given by

Z = ( c − β ) ( 1 − α r 2 ) − γ β ( 1 − α r 2 ) + γ (17)

Δ = α ( 6 α r 2 − 8 α 2 r 4 + 3 α 3 r 6 ) (18)

The radial derivatives of pressures p ′ r and p ′ t given by Equations (14) and (15) are

8 π G c 4 p ′ r = − α 2 r [ 8 γ ( 1 − α r 2 ) 2 { 2 β ( 1 − α r 2 ) + 3 γ } { β ( 1 − α r 2 ) + γ } 2 − 12 + 16 α r 2 − 6 α 2 r 4 ] (19)

8 π G c 4 p ′ t = − α 2 r [ γ ( 1 − α r 2 ) 2 { 2 β ( 1 − α r 2 ) + 3 γ } { β ( 1 − α r 2 ) + γ } 2 − 3 ( 1 − α r 2 ) 2 ] (20)

The radial derivative of matter energy density given by Equation (16) is

8 π G c 2 ρ ′ = − α 2 r ( 60 − 112 α r 2 + 54 α 2 r 4 ) (21)

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

Z ′ = − 2 c α γ r { β ( 1 − α r 2 ) + γ } 2 (22)

Δ ′ = 2 α 2 r ( 6 − 16 α r 2 + 9 α 2 r 4 ) (23)

For positive value of metric coefficient A i.e. A 0 > 0 is possible if ( β + γ ) > 0 . 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

8 π G c 4 ( p r ) 0 = − 4 α β β + γ = 8 π G c 4 ( p t ) 0 (24)

8 π G c 2 ρ 0 = 12 α (25)

Z 0 = c − β − γ β + γ (26)

For positive values of pressure p 0 , density ρ 0 and redshift Z 0 , α > 0 , β + γ < c and β < 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

8 π G c 4 ( p ″ r ) 0 = − 4 α 2 ( 3 γ 2 − 4 β γ − 3 β 2 ) ( β + γ ) 2 (27)

8 π G c 4 ( p ″ t ) 0 = 8 α 2 β ( 3 β + 4 γ ) ( β + γ ) 2 (28)

For maximum values of radial and tangential pressures at the center, the conditions of model parameters are ( p ″ r ) 0 < 0 i.e. − 2 − 13 3 < β γ < − 2 + 13 3 and ( p ″ t ) 0 < 0 i.e. − 4 3 < β γ .

From Equation (21)

8 π G c 4 ( ρ ″ ) 0 = − 60 α 2 (29)

which shows that for all values of parameters, the matter energy density is maximum at the center of the fluid distribution i.e. ( ρ ″ ) 0 < 0 .

At the center, the equations of state for matter distribution are given by

( p r c 2 ρ ) 0 = − β 3 ( β + γ ) (30)

( p t c 2 ρ ) 0 = − β 3 ( β + γ ) (31)

The central equation of state should obey the conditions 0 < ( p r c 2 ρ ) 0 ≤ 1 and 0 < ( p t c 2 ρ ) 0 ≤ 1 which gives − 4 3 ≤ β γ .

From Equation (19) and Equation (21)

( d p r c 2 d ρ ) 0 = ( 3 γ 2 − 2 β γ − 3 β 2 ) 15 ( β + γ ) 2 (32)

and from Equation (20) and Equation (22)

( d p t c 2 d ρ ) 0 = − ( 4 γ 2 − 25 β γ − 20 β 2 ) 60 ( β + γ ) 2 (33)

The causality conditions 0 < ( d p r c 2 d ρ ) 0 ≤ 1 and 0 < ( d p t c 2 d ρ ) 0 ≤ 1 are satisfied at the center which gives the condition for model parameters − 6 7 < β γ < − 10 9 .

From Equation (22) and Equation (23)

( Z ″ ) 0 = − 2 α c γ ( β + γ ) 2 (34)

( Δ ″ ) 0 = 12 α 2 (35)

For maximum value of red shift at the center, ( Z ″ ) 0 < 0 i.e. γ > 0 . Since ( Δ ″ ) 0 > 0 , the anisotropy is minimum at the center for all values of model parameters.

It is observed that for model parameters given by

α > 0 , β < 0 , γ > 0 , c > β + γ > 0 , − 4 3 < β γ < − 10 3 and − 6 7 < β γ < 0 , (36)

the metric coefficients A and B increase monotonically with increasing radial distance r as illustrated by (

For the stable astrophysical fluid distribution the interior metric must be matched with the Schwarzschild exterior metric given by

d s 2 = ( 1 − 2 G M c 2 r ) c 2 d t 2 − ( 1 − 2 G M c 2 r ) − 1 d r 2 − r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) (37)

For this the metric coefficients must be continuous (First fundamental form) and differentiable (Second fundamental form) at the surface at r = r Σ and ( p r ) r Σ = 0 .

Thus

α = 1 r Σ 2 ( 1 − Ψ 1 2 ) ; 0 < X < 1 (38)

β = − c ( 1 + Ψ 1 2 + Ψ − 3 Ψ 3 2 ) 4 Ψ (39)

β γ = − 1 + Ψ 1 2 + Ψ − 3 Ψ 3 2 Ψ 1 2 ( 1 + Ψ 1 2 + Ψ + Ψ 3 2 ) (41)

where Ψ = 1 − 2 S p ; S p = G M c 2 r Σ , Schwarzchild parameter. Equation (38) to

Equation (41) represent the model parameters obtaining from these boundary conditions.

From Equation (38) to Equation (41) the model parameters Ψ , β , γ and

β γ with α r Σ 2 = 0.12 or S p = 0.1133 for an astrophysical object

SAXJ1808.4-3658 are 0.7734, −0.1979c, 0.9475c and −0.2089 respectively. The physical quantities estimated in our model for astrophysical objects SAXJ1808.4-3658, HerX-1, 4U1538-52 and LMC X-4 are illustrated by means of

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

In this section our aim is to determine the physical requirement of realistic solution featuring the stability of compact stellar or astrophysical objects.

S.N. | r r Σ | A c | B | 8 π G c 2 ρ r Σ 2 | 8 π G c 4 p r r Σ 2 | 8 π G c 4 p t r Σ 2 | Z | Δ |
---|---|---|---|---|---|---|---|---|

1 | 0.0 | 0.6061 | 1.0000 | 1.4400 | 0.1267 | 0.1267 | 0.3340 | 0.0000 |

2 | 0.1 | 0.6072 | 1.0024 | 1.4357 | 0.1252 | 0.1261 | 0.3320 | 1.2000 × 10 − 11 |

3 | 0.2 | 0.6105 | 1.0096 | 1.4228 | 0.1207 | 0.1241 | 0.3259 | 4.7502 × 10 − 11 |

4 | 0.3 | 0.6162 | 1.0219 | 1.4015 | 0.1133 | 0.1210 | 0.3158 | 1.0602 × 10 − 10 |

5 | 0.4 | 0.6242 | 1.0395 | 1.3721 | 0.1032 | 0.1167 | 0.3018 | 1.8637 × 10 − 10 |

6 | 0.5 | 0.6347 | 1.0628 | 1.3350 | 0.0905 | 0.1113 | 0.2838 | 2.8698 × 10 − 10 |

7 | 0.6 | 0.6479 | 1.0923 | 1.2906 | 0.0756 | 0.1050 | 0.2620 | 4.0589 × 10 − 10 |

8 | 0.7 | 0.6640 | 1.1288 | 1.2397 | 0.0588 | 0.0979 | 0.2364 | 5.4074 × 10 − 10 |

9 | 0.8 | 0.6832 | 1.1733 | 1.1828 | 0.0404 | 0.0902 | 0.2071 | 6.8885 × 10 − 10 |

10 | 0.9 | 0.7058 | 1.2269 | 1.1208 | 0.0208 | 0.0821 | 0.1742 | 8.4721 × 10 − 10 |

11 | 1.0 | 0.7324 | 1.2913 | 1.0545 | 0.0000 | 0.0736 | 0.1379 | 1.0125 × 10 − 9 |

S.N. | r r Σ | p r c 2 ρ | p t c 2 ρ | d p r c 2 d ρ | d p t c 2 d ρ | ( ρ c 2 + p r ) p r d p r c 2 d ρ | ( ρ c 2 + p t ) p t d p t c 2 d ρ | v t 2 c 2 − v r 2 c 2 |
---|---|---|---|---|---|---|---|---|

1 | 0.0 | 0.0880 | 0.0880 | 0.3501 | 0.0187 | 4.3279 | 0.2319 | −0.3313 |

2 | 0.1 | 0.0872 | 0.0878 | 0.93496 | 0.0187 | 4.3581 | 0.2320 | −0.3309 |

3 | 0.2 | 0.0848 | 0.0872 | 0.3483 | 0.0186 | 4.4525 | 0.2321 | −0.3296 |

4 | 0.3 | 0.0808 | 0.0863 | 0.3460 | 0.0184 | 4.6239 | 0.2323 | −0.3275 |

5 | 0.4 | 0.0752 | 0.0850 | 0.3427 | 0.0182 | 4.8988 | 0.2326 | −0.3245 |

6 | 0.5 | 0.0678 | 0.0833 | 0.3385 | 0.0179 | 5.3285 | 0.2331 | −0.3206 |

7 | 0.6 | 0.0586 | 0.0813 | 0.3334 | 0.0175 | 6.0193 | 0.2337 | −0.3158 |

8 | 0.7 | 0.0474 | 0.0790 | 0.3272 | 0.0171 | 7.2211 | 0.2344 | −0.3101 |

9 | 0.8 | 0.0342 | 0.0763 | 0.3201 | 0.01767 | 9.6814 | 0.2354 | −0.3034 |

10 | 0.9 | 0.186 | 0.0732 | 0.3118 | 0.0161 | 17.0758 | 0.2367 | −0.2596 |

11 | 1.0 | 0.0000 | 0.0689 | 0.3024 | 0.0155 | ∞ | 0.2384 | −0.2868 |

Our proposed model of anisotropic fluid satisfies the following energy conditions within the framework of general relativity.

S.N. | Stellar Objects | α | β | γ | T p |
---|---|---|---|---|---|

1 | SAXJ1808.4-3658 | 1.66 × 10 − 9 | − 0.1979 c | 0.9475 c | 0.1133 |

2 | HerX-1 | 1.7825 × 10 − 9 | − 0.215 c | 0.9441 c | 0.1215 |

3 | 4U1538-52 | 2.435 × 10 − 9 | − 0.269 c | 0.9344 c | 0.1460 |

4 | LMC-X-4 | 2.552 × 10 − 9 | − 0.3206 c | 0.9267 c | 0.1674 |

5 | CenX-3 | 3.112 × 10 − 9 | − 0.4976 c | 0.9092 c | 0.2287 |

6 | VelaX-1 | 3.719 × 10 − 9 | − 0.7959 c | 0.9034 c | 0.2850 |

7 | PSRJ1614-2230 | 3.6589 × 10 − 9 | − 0.7548 c | 0.9028 c | 0.2920 |

8 | PSRJ0348+0432 | 1.213 × 10 − 9 | − 0.4369 c | 0.9137 c | 0.2096 |

Null Energy Condition (NEC)

ρ ≥ 0 (42)

Weak Energy Condition (WEC)

0 ≤ p r ρ c 2 ≤ 1 , 0 ≤ p t ρ c 2 ≤ 1 (43)

Strong Energy Condition (SEC)

0 ≤ p r ρ c 2 + 2 p t ρ c 2 ≤ 1 (44)

The principles of energy condition are plotted graphically against radial distance for realistic stellar objects in

There are three kind of forces viz gravitational force F g , hydrostatic force F h and anisotropical force F a acting on a compact star which are in equilibrium.

Thus

F g + F h + F a = 0 (45)

The generalized Tolman-Oppenheimer-Volkoff equaion for equilibrium condition under these forces for anisotropic fluid distribution is given by

− M G ( r ) ( ρ c 2 + p r ) B r 2 A − d p r d r + 2 ( p t − p r ) r = 0 (46)

where M G ( r ) is effective gravitational mass and is defined by Tolman-Whittaker as

M G ( r ) = r 2 A ′ B (47)

Now TOV equation reduces to

− A ′ ( ρ c 2 + p r ) A − d p r d r + 2 ( p t − p r ) r = 0 (48)

In view of Equation (45) and Equation (48) the three forces are given by

F g = − A ′ ( ρ c 2 + p r ) A

Or

F g = − 8 α 2 γ r ( 1 − α r 2 ) { 2 β ( 1 − α r 2 ) + 3 γ } { β ( 1 − α r 2 ) + γ } (49)

F h = − d p r d r

Or

F h = − α 2 r [ 8 γ ( 1 − α r 2 ) 2 { 2 β ( 1 − α r 2 ) + 3 γ } { β ( 1 − α r 2 ) + γ } 2 − 12 + 16 α r 2 − 6 α 2 r 4 ] (50)

F a = 2 ( p t − p r ) r

Or

F a = 2 α ( 6 α r 2 − 8 α 2 r 4 + 3 α 3 r 6 ) r (51)

The graphical representations of above mentioned forces and their resultant for astrophysical objects SAXJ1808.4-3658, HerX-1, 4U1538-52 and LMC-X-4 are shown in the

The causality condition states that the radial and tangential speeds of sound

should not be more than one i.e. 0 < v r 2 c 2 < 1 and 0 < v t 2 c 2 < 1 anywhere within

the stellar object. The graphical representations of radial and tangential speeds of sound with respect to r / r Σ in the

− 1 < v t 2 c 2 − v r 2 c 2 < 0 and 0 < v t 2 c 2 − v r 2 c 2 < 1 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

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

Γ r = ρ c 2 + p r p r d p r c 2 d ρ and Γ t = ρ c 2 + p t p t d p t c 2 d ρ

In our model the variation of radial and tangential adiabatic indices with r r Σ are represented graphically in

Harrison [

d M d ρ 0 > 0 for stable fluid distribution,

d M d ρ 0 < 0 for unstable fluid distribution,

In our model the mass of configuration in terms of central density is given by

( 1 − 2 G M c 2 r Σ ) 3 2 [ − 4 + 5 ( 1 − 2 G M c 2 r Σ ) 1 2 ] = 8 π G c 2 ρ 0 r Σ 2 − 1 (52)

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

We have given a well behaved analytic charge free solution for spherical and symmetric anisotropic fluid distribution given by Sah and Chandra [

By using suitable model parameters given in

S.N. | Stellar Objects | M / M ⊙ | r Σ (in km) | M / M ⊙ | r Σ (in km) | Referance |
---|---|---|---|---|---|---|

(Observed) | (Observed) | (Calculated) | (Calculated) | |||

1 | SAXJ1808.4-3658 | 0.9 ± 0.3 | 7.951 ± 1.0 | 0.65 | 8.50 | Elebert et al. [ |

2 | HerX-1 | 0.85 ± 0.15 | 8.1 ± 0.45 | 0.70 | 8.54 | Abubekerov et al. [ |

3 | 4U1538-52 | 0.87 ± 0.07 | 7.866 ± 0.21 | 0.80 | 8.07 | Rawls et al. [ |

4 | LMC-X-4 | 1.04 ± 0.09 | 8.301 ± 0.2 | 0.96 | 8.50 | Rawls et al. [ |

5 | CenX-3 | 1.49 ± 0.08 | 9.178 ± 0.13 | 1.42 | 9.20 | Rawls et al. [ |

6 | VelaX-1 | 1.77 ± 0.08 | 9.56 ± 0.08 | 1.85 | 9.62 | Rawls et al. [ |

7 | PSRJ1614-2230 | 1.97 ± 0.04 | 9.69 ± 0.2 | 1.94 | 9.85 | Demorest et al. [ |

8 | PSRJ0348+0432 | 2.01 ± 0.04 | 13.0 ± 2.0 | 1.98 | 14.00 | Demorest et al. [ |

S.N. | StellarObjects | Central Density (in gm/cm^{3}) | Surface Density (in gm/cm^{3}) | Central Pressure (in dyne/cm^{2}) | Red Shift |
---|---|---|---|---|---|

1 | SAXJ1808.4-3658 | 1.07 × 10 15 | 0.784 × 10 15 | 0.847 × 10 35 | 0.138 |

2 | HerX-1 | 1.15 × 10 15 | 0.819 × 10 15 | 1.027 × 10 35 | 0.149 |

3 | 4U1538-52 | 1.57 × 10 15 | 1.034 × 10 15 | 1.093 × 10 35 | 0.188 |

4 | LMC-X-4 | 1.645 × 10 15 | 1.009 × 10 15 | 1.582 × 10 35 | 0.226 |

5 | CenX-3 | 2.006 × 10 15 | 0.982 × 10 15 | 6.938 × 10 35 | 0.357 |

6 | VelaXv1 | 2.431 × 10 15 | 0.923 × 10 15 | 3.956 × 10 35 | 0.525 |

7 | PSRJ1614-2230 | 2.358 × 10 15 | 0.879 × 10 15 | 3.608 × 10 35 | 0.551 |

8 | PSRJ0348+0432 | 0.781 × 10 15 | 0.412 × 10 15 | 2.149 × 10 35 | 0.312 |

always less than 4 9 . The red shift is also satisfied the upper bound limit for the

realistic anisotropic star models i.e. Z s ≤ 1 . For the EoS, we plot a graph for p_{r} and p_{t} versus against r as shown in

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. [

Fulara, P.C. and Sah, A. (2018) A Spherical Relativistic Anisotropic Compact Star Model. International Journal of Astronomy and Astrophysics, 8, 46-67. https://doi.org/10.4236/ijaa.2018.81004