Strange star admitting Chaplygin equation of state in Finch-skea spacetime Piyali Bhar
1,*
Abstract In present paper we propose a new model of anisotropic strange star which admits Chaplygin equation of state. The exterior spacetime is described by Schwarzschild line element. The model is developed by assuming Finch-skea ansatz [M.R. Finch and J.E.F. Skea, Class. Quantum. Grav. 6 (1989) 467]. We obtain the model parameters in closed form. Our model is free from central singularity. Choosing some particular values to the parameter we show that our model corroborate with the observational data of the strange star SAX J 1808.4-3658(SS2). Keywords General Relativity, Finch-skea spacetime, Chaplygin equation of state, Strange star 1 Introduction In studies of relativistic models of compact stars are an intersecting topic to the researchers for last few decades. The possible compact objects are strange stars and Neutron stars . Strange stars are composed of quark or strange matter consisting of u, d, and s quarks where as on the other hand neutron stars are composed of neutrons. Stars composed of strange matter is called the strange star. The formation of strange matter, can be classified into two ways: the quark hadron phase transition in the early universe and conversion of neutron stars into strange stars at ultrahigh densities (Witten 1984). Bodmer (1971) proposed that a phase transition between hadronic and strange quark matter may occur in the universe at a density higher than the nuclear density when a massive star explodes as a supernova. Piyali Bhar 1 Department
of Mathematics, Gopinathpur High School (H.S), Haripal, Hooghly- 712 403, West Bengal, India * Email:
[email protected]
So it is expected to find the strange quark at the inner core of neutron star or quark star. According to Ruderman (1972) the pressure inside the highly compact astrophysical objects like X-ray pulsar,Her-X-1, X-ray buster 4U 1820-30,millisecond pulsar SAXJ1804.4-3658 etc that have core density beyond the nuclear density (∼ 1015 gm/cc) show anisotropy ,i.e, the pressure inside these compact objects can be decomposed into two parts radial pressure pr and transverse pressure pt where pt is in the perpendicular direction to pr . ∆ = pt − pr is called the anisotropic factor. The reason behind these anisotropic nature are the existence of solid core, in presence of type P superfluid, phase transition, rotation, magnetic field, mixture of two fluid, existence of external field etc. Local anisotropy in selfgravitating systems were studied by Herrera & Santos (1997) (please see the references there in for a review of anisotropic fluid.) A large number of works have done earlier by assuming anisotropic pressure of the underlying fluid. By assuming a special type of matter density Dev & Gleiser (2004) proposed an model of anisotropic star. From their analysis authors have shown that the absolute 8 stability bound 2M R < 9 can be violated and the star’s surface redshift may be arbitrarily large. Strange Star in Krori-Barua Spacetime is described by Rahaman et al. (2011). Strange quintessence star and singularity free quintessence star in Krori-Barua spacetime are obtained by Bhar (2014b,c). A new class of interior solution of a (2+1)-dimensional anisotropic star in Finch and Skea spacetime corresponding to the exterior BTZ black hole was developed by Bhar et al. (2014). The model is obtained by considering the MIT bag model Equation of state and a particular ansatz for the metric function grr proposed by Finch & Skea (1989). Relativistic stellar model admitting a quadratic equation of state was proposed by Sharma et al. (2013) in
2
finch-skea spacetime. The earlier work is generalized in modified Finch-Skea spacetime by Pandya et al. (2014) by incorporating a dimensionless parameter n. In a very recent work Bhar (2015) obtained a new model of an anisotropic superdense star which admits conformal motions in the presence of a quintessence field which is characterized by a parameter ωq with −1 < ωq < −1/3. The model has been developed by choosing Vaidya & Tikekar (1982) ansatz. Recently researchers are using Chaplygin equation of state with great interest since it describes the accelerating phase of the present Universe. Chaplygin equation of state is a special form of polytropic equation of state which is used to describe dark star. Lobo (2006) has given a model of a stable dark energy star by assuming two spatial types of mass function: one is of constant energy density and the other mass function is a Matese & Whitman (1980) mass. All the features of the dark energy star have been discussed and the system is stable under a small linear perturbation. Very recently Bhar & Rahaman (2015) proposed a new model of dark energy star consisting of five zones, namely, the solid core of constant energy density, the thin shell between core and interior, an inhomogeneous interior region with anisotropic pressures, a thin shell, and the exterior vacuum region. They discussed various physical properties. The model satisfies all the physical requirements. The stability condition under a small linear perturbation is also discussed. Bertolami & Paramos (2005) used the generalized Chaplygin gas (GCG) EOS with special reference to anisotropic pressure, where as Rahaman et al. (2010) has described anisotropic charged fluids with Chaplygin equation of state. In this paper the authors deal with a nonlinear EOS and were able to find solutions using an algebraic method without solving any differential equations. Inspired by all of the previous works in the present paper we want to model an anisotropic star in Finch-skea spacetime incorporating Chaplygin equation of state. Our paper is planned as follows: In Sect. 2 we discuss the Einstein field equations. The solutions of the system are treated in Sect. 3. The other features are given in Sects. 4 -10 and finally some concluding remarks are given in Sect. 11.
2 Einstein Field Equations To describe the interior of a static spherically symmetry distribution of matter the line element is given by, ds2 = −e2ν(r) dt2 + e2λ(r) dr2 + r2 (dθ2 + sin2 θdφ2 ) (1)
in Schwarzschild co-ordinate xa = (t, r, θ, φ). Where λ and ν are functions of radial co-ordinate ‘r’ only. Let us assume that the matter distribution inside the compact star is anisotropic in nature whose energy momentum tensor given by the following equation Tνµ = (ρ + pr )uµ uν − pt gνµ + (pr − pt )η µ ην
(2)
with ui uj = −η i ηj = 1 and ui ηj = 0. Here the vector ui is the fluid 4-velocity and η i is the spacelike vector which is orthogonal to ui , ρ is the matter density, pr and pt are respectively the radial and the transversal pressure of the fluid and pt lies in the orthogonal direction to pr . Taking G = 1 = c Einstein Field equations can be written as, 8πρ =
′ 1 r(1 − e−2λ ) 2 r
(3)
1 2ν ′ −2λ (1 − e−2λ ) + e 2 r r λ′ ν′ ′ ′ ′′ ′2 −2λ −ν λ − ν +ν + 8πpt = e r r
8πpr = −
(4) (5)
Where ‘prime’ denotes differentiation with respect to the radial co-ordinate ‘r’. Now the mass contained within the sphere of radius ‘r’ is given by, Z r 4πρ(ω)ω 2 dω (6) m(r) = 0
We assume that the radial pressure (pr ) and the matter density ρ is related as, pr = α 1 ρ +
α2 ρ
(7)
Where α1 and α2 are some constants. The above equation is known as the generalized Chaplygin gas Equation of state for the fluid sphere. Now let us introduce the transformation x = r2 ,
Z(x) = e−2λ(r)
and
y 2 (x) = e2ν(r) (8)
Using this transformation Einstein field equations becomes, 8πρ =
1−Z − 2Z˙ x
8πpr = 4Z
Z −1 y˙ + y x
(9)
(10)
3
pt = pr + ∆ 8π∆ = 4xZ
(11) y¨ 1−Z y˙ + + Z˙ 1 + 2x y y x
(12)
Using (7),(9) and (10) we obtain y˙ α1 = y 4Z
1−Z − 2Z˙ x
α2 + (8π)2 4Z
1−Z − 2Z˙ x
−1
Z −1 (13) 4Zx Where ‘dot’ denotes derivative with respect to ‘x’. The mass function becomes, Z x √ ωρ(ω)dω (14) m(x) = 2π −
0
pt = pr + ∆
3 Solution To solve the Einstein field equations let us take the ansatz proposed by Finch & Skea (1989) as e2λ = 1 +
r2 R2
(15)
Using (8) and (15) we get, Z(x) = (1 + ax)−1
(16)
Where a = R12 Using the expression of Z(x) from equation (9)-(13) we obtain (α1 a) 3 + ax a α2 (1 + ax)3 y˙ = + (8π)2 + y 4 1 + ax 4a 3 + ax 4
(17)
Integrating the above equations we get,
where ‘C’ is the constant of integration, which will be determined from the boundary condition. The matter density, radial and transverse pressure is obtained as,
α2 pr = α 1 ρ + ρ
and the anisotropic factor ∆ is obtained as, (8π)2 α2 (1 + ax)2 α1 a 3 + ax × + 8π∆ = 4x 4 (1 + ax)2 4a 3 + ax α1 a 3 + ax (8π)2 α2 (1 + ax)2 a + + 4 (1 + ax)2 4a 3 + ax 4 α1 a2 (8π)2 α2 (1 + ax)(4 + ax) − +4x 2 (3 + ax)2 2(1 + ax)3 (8π)2 α2 1 + ax a α1 a 3 + ax + + −2ax 4 (1 + ax)3 4a 3 + ax 4(1 + ax)2 −
a a + (1 + ax)2 1 + ax
In this section we match our interior solution to the exterior Schwarzschild metric at the boundary r = rb where rb > 2M . The exterior spacetime is given by the line element
(8π)2 α2 [12ax − 8 ln(3 + ax)] + C 4a2
3a + a2 x (1 + ax)2
(20)
4 Exterior Spacetime And Matching Condition
α1 ax ln(y) = [ax + 2 ln(1 + ax)] + 4 4 3 2 (8π) α2 (3 + ax) − 3(3 + ax)2 + 4a2 3
ρ=
Fig. 1 The matter density ρ is shown against ‘r’ inside the stellar interior.
−1 2M 2M ds2 = − 1 − dt2 + 1 − dr2 r r + r2 (dθ2 + sin2 θdφ2 )
(21)
Now at the boundary r = rb the coefficient of grr , gtt all are continuous. Which implies,
(18)
(19)
1−
2M = e2ν(rb ) rb
(22)
4
•
•
Fig. 2 The radial and transverse pressure are shown against ‘r’ inside the stellar interior.
2M 1− rb
−1
= 1 + arb2
(23)
From equation (22) we obtain, 2M α1 2 1 − arb + 2 ln(1 + arb2 ) C = ln 1 − 2 rb 4 (8π)2 α2 (3 + arb2 )3 ar2 2 2 − − 3(3 + ar ) − b − b 4 4a2 3 (8π)2 α2 12arb2 − 8 ln(3 + arb2 ) 2 4a
•
•
stellar interior. The profile of ρ,pr and pt are shown in fig. 1 and fig. 2 respectively. From the figure it is clear that our proposed model of strange star satisfies this condition. At the boundary of the star the radial pressure vanishes. dpr Both dρ dr and dr < 0 (see fig. 4)i.e., ρ and pr are monotonic decreasing function of ‘r’, they have maximum value at the center of the star and it decreases radially outwards. For an anisotropic fluid sphere the trace of the energy tensor should be positive suggested by Bondi (1999). To check this condition for our model we plot ρ − pr − 2pt vs ‘r’ in Fig.5. From the figure it is clear that ρ − pr − 2pt ≥ 0. The profile of the anisotropic factor ∆ = pt − pr is shown in fig. 3. The anisotropic factor ∆ > 0 for our model. So the anisotropic force is repulsive in nature, which helps to construct more compact objects (Gokhroo & Mehra (1994)). At the center of the star the anisotropic factor vanishes which is expected. Moreover for an anisotropic fluid sphere the radial and transverse velocity of sound should be less than 1, which is known as causality conditions. This condition is discussed separately in sec. 7.
(24)
Fig. 4 interior
Fig. 3 The anisotropic factor ∆ = pt − pr is shown against ‘r’ inside the stellar interior.
dρ dr
and
dpr dr
are shown against ‘r’ inside the stellar
6 adiabatic index For an anisotropic fluid sphere the adiabatic index (γ) is given by
5 Physical Analysis γ= • Matter density (ρ), radial pressure (pr ) and transverse pressure (pt ) should be non-negative inside the
ρ + pr dpr pr dρ
(25)
Heintzmann & Hillebrandt (1975) proposed that for
5
a neutron star with anisotropic equation of state will be stable if γ > 34 . To see the characteristics of the adiabatic index for our model the profile of γ is shown in fig. 6. From the figure we see that γ > 34 everywhere within the interior of the fluid sphere. So according to Heintzmann & Hillebrandt (1975) our model is stable.
7 Stability Our proposed model of anisotropic strange star will be physically acceptable if the radial and transverse velocity of sound should be less than 1 which is known as causality conditions. 2 Where the radial velocity (vsr ) and transverse velocity 2 (vst ) of sound can be obtained as
Fig. 5 ρ − pr − 2pt are shown against ‘r’ inside the stellar interior.
2 vsr =
dpr dρ
(26)
2 = vst
dpt dρ
(27)
2 2 Fig. 7 vsr , vst are shown against ‘r’ inside the stellar interior
Fig. 6 The adiabatic index γ is shown against ‘r’ inside the stellar interior
For the complexity of the expression of pt we prove the above inequalities with the help of graphical repre2 2 are shown in fig. 7. and vst sentation. The profile of vsr 2 2 ≤ 1 ev, vst From the figure it is clear that 0 ≤ vsr erywhere within the anisotropic fluid sphere . Now for 2 2 a potentially stable region one must have vst − vsr 0
(35)
(iii) SEC : ρ + pr ≥ 0
ρ + pr + 2pt ≥ 0
(36)
(iv) DEC : ρ > |pr |,
10 Some Features
ρ > |pt |
(37) Fig. 12 The mass function m(r) is plotted against ‘r’ for the anisotropic strange star.
The mass function is regular at the center of the star since asr → 0,m(r) → 0. Moreover mass function is monotonic increasing function of ‘r’ and it is positive inside the stellar configuration. 10.2 Compactness
Fig. 11 The energy conditions are plotted against ‘r’ inside the stellar interior.
For the complexity of the expression of pt we will prove the above inequalities with the help of graphical representation. The L.H.S of the above inequations are plotted in fig. 11. The figure shows that all the energy conditions are satisfied by our model of anisotropic strange star.
The ratio of mass to the radius of a strange star, known as compactification factor, can not be arbitrarily large. Buchdahl (1959) showed that for a (3+1)-dimensional 8 fluid sphere 2M rb < 9 . To see the maximum allowable ratio of mass to the radius for our model we calculate the compactness of the star given by u(r) =
ar2 m(r) = r 2(1 + ar2 )
(39)
The profile of u(r) is shown in fig 13. The figure shows that u(r) is a monotonic increasing function of ‘r’. To see twice the maximum allowable ratio of the mass to the radius we have calculated the values of 2m rb = 0.616 for the strange star SAX J 1808.4-3658(SS2) from our model, which lies in the expected range of Buchdahl (1959).
8
11 Discussion and concluding remarks
Fig. 13 The compactification factor u(r) is plotted against ‘r’ for the anisotropic strange star.
10.3 Surface Redshift The surface redshift function zs of our model of strange star is defined by, 1
1 + zs = (1 − 2u)− 2 Therefore the surface redshift function zs can be obtained as
zs =
1 1 + ar2
− 21
−1
(40)
Fig. 14 The surface redshift function zs (r) is plotted against ‘r’ for the anisotropic strange star.
The profile of the surface redshift is shown in fig. 14. The maximum value of the surface redshift for the strange star SAX J 1808.4-3658(SS2) is obtained as 0.614.
In the present paper we have solved the Einstein field equations in closed form by choosing Chaplygin equation of state. The model is developed by choosing Finch-skea ansatz. Our model is free from central singularities.The model parameters ρ, pr , pt all are nonnegative inside the stellar interior. Both ρ, pr are monotonic decreasing function of ‘r’, i.e., they have maximum value at the center of the star and it decreases radially outwards. By choosing some reasonable values to the parameters we obtain the central density of the star is 6.41 × 1015 gm/cc. . The pressure anisotropy is chosen motivated by the fact that the central density is beyond the nuclear density. The anisotropic factor ∆ > 0 for our model and according to Gokhroo & Mehra (1994) it helps to construct more compact object. Our proposed model is in static equilibrium under gravitational,hydrostatics and anisotropic forces. For 2 2 our model 0 ≤ vsr , vst ≤ 1 ,i.e., the causality conditions 2 2 hold. Moreover 0 ≤ |vst − vsr | ≤ 1, which ensures that our model is potentially stable (Andreasson 1992). The adiabatic index for our model of strange star > 43 . Plugging ‘G’ and ‘c’ in the expression of pr the central radial pressure is obtained as 5.47 × 1035 dyne/cm2 . Taking ‘a = 0.0398′ , ‘α1 = 0.1′ , ‘α2 = − 0.0001156′ and by using the conditions pr (rb ) = 0 we obtain the radius of the star is 6.35 km and mass=1.326MJ , which is very much consistence with the observational data of strange star SAX J 1808.4-3658(SS2)(Gondek-Rosinska et. al 2000; Glendenning 1997). The maximum allowable ratio of mass to the radius is obtained as 0.308 and the maximum surface redshift calculated from our model is 0.614 ≤ 1, lies in the expected range of Barraco & Hamity (2002). From the analysis we can conclude that our proposed model satisfies all the requirements to be physically acceptable.
9
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