a core-envelope model of compact stars

Gravitation & Cosmology, Vol. 11 (2005), No. 3 (43), pp. 244–248 c 2005 Russian Gravitational Society

A CORE-ENVELOPE MODEL OF COMPACT STARS B.C. Paul1 † and R. Tikekar2 ‡ † Physics Department, North Bengal University, Dist. Darjeeling, Pin 734 430, India ‡ Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar, Pin 388120, Gujarat, India. Received 5 November 2004 Received in final form 18 March 2005 We present a core-envelope model of compact stars. The core, described by an anistropic fluid, is surrounded by an isotropic envelope. Assuming an ansatz that prescribes a spheroidal geometry of the space inside the star, we find solutions for the core and the envelope. The parameter which determines the order of spheroidicity of space ( λ ) is found to play an important role. For a physically interesting core-envelope model, the parameter has a lower bound ( λ > 2 ) which is higher than in the case of a relativistic sphere of perfect fluid without a core ( λ > 3/17 ). In our model, the isotropic fluid distribution of the envelope requires a large λ for a physically interesting dense core of a compact star.

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1. Introduction The work of Ruderman [1] and Canuto [2] on compact stars having matter distributions with densities much greater than the nuclear density indicate that superdense stars are likely to develop an anisotropic pressure. According to these views, in such massive stellar objects the radial pressure differs from the tangential pressure inside the core. The origin of anisotropy in the fluid pressure could be due to a number of physical processes that may take place inside the star. For example, it may be due to the existence of a solid core [3] or the presence of a type P superfluid, or a boson star [4], different kinds of exotic phase transitions due to gravitational collapse [5], pion condensation [6] or other physical phenomena [7]. These expectations provide a motivation for studying relativistic superdense star models with anisotropic fluids as its interior. The anisotropy is also expected to have a non-negligible effect on the maximum equilibrium mass and surface redshift of the configurations [8]. Maharaj and Maartens [9] obtained a solution for 1 e-mail: 2 e-mail:

[email protected] [email protected]

an anisotropic fluid sphere with uniform energy density. Later, Gokhroo and Mehra [10] extended their work to a variable density distribution. Precise information on the behaviour of matter in superdense stars is not known. However, a compact object with such high-energy matter in it may be studied using different approaches. The study of a relativistic core-envelope model is an attempt in this direction. In this approach, the interior of a superdense star is considered to be comprised of two regions: (i) the core region and (ii) the envelope region. We seek a compact object with an anisotropic pressure region as its core, whose surrounding contains a different fluid distribution, in fact, one may assume it to be a perfect fluid distribution. Iyer et al. [11] studied a core-envelope model and showed that this approach leads to information about bounds on various parameters of ultra-compact objects in general relativity (e.g., neutron star) such as mass, size and their ratio. In this paper we present a core-envelope model of a compact star in which the core region is described as matter with anistropic pressure and the surroundeding envelope as a distribution of a fluid with isotropic pressure. We here follow a different approch for finding

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a solution in the envelope region, previously adopted by Mukherjee et al. [12]. The conventional approach for stellar models is to prescribe an equation of state for the fluid forming the interior of a star for solving the Einstein equations. In view of their non-linearity as well as hydrodynamical complexity, one always has in realistic situations to resort to numerical manipulations. In the case of superdense compact objects like neutron stars, the equation of state is uncertain and not well understood. In such a situation, apart from the conventional physical approach, it may be worthwhile to explore an alternative approach in which one assumes a simple geometry for the 3-space to make the Einstein equations tractable. This will lead to an equation of state which may be useful and physically acceptable. We consider a spacetime geometry that was given by Vaidya and Tikekar [13] for a superdense star by proposing an ansatz for the geometry of a 3-surface embedded in four-dimensional Euclidean space. The ansatz prescribes for it a spheroidal geometry, described by two parameters λ and R ; λ = 0 gives a spherical geometry while λ = −1 corresponds to flat space.

2. Field equations and the core-envelope model We begin with a static, spherically symmetric spacetime described by the metric ds2 = −e2ν(r) dt2 + e2µ(r) dr2 + r2 (dθ2 + sin2 θdϕ2 ) (1) with the ansatz e2µ(r) =

1 + λr2 /R2 . 1 − r2 /R2

(2)

Note that the t = const hypersurface has the geometry of a 3-spheroidal space immersed in a 4-Euclidean space and is characterised by two curvature parameters, λ and R . The suitability of the above metric has already been investigated by one of us [14]. The Einstein field equations are 1 Rµν − gµν R = −8πGTµν , 2

(3)

where gµν , Rµν and R are the metric tensor, Ricci tensor and scalar curvature, respectively, and Tµν is the energy-momentum tensor. For an anisotropic fluid distribution, following Maharaj and Maartens [9], we consider the energy-momentum tensor given by Tµν = (ρ + p)uµ uν − pgµν − πµν ,

(4)

where ρ and p are the energy density and isotropic pressure and uµ denotes the unit four-velocity field of matter; πµν is the anisotropic stress tensor given by √ 1 π µν = 3S C µ C ν − (uµ uν − g µν ) (5) 3

For a radially symmeteric anisotropic fluid distribution, S = S(r) denotes the magnitude of the anisotropic stress tensor and C µ = (0, e−λ , 0, 0) is a radial vector. The corresponding energy-momentum tensor has the following non-vanishing components: √ T11 = − p + 2S/ 3 , T00 = ρ, √ T22 = T33 = − p − S/ 3 . The pressure along the radial and tangential directions are different and are denoted as √ (6) Pr = −T11 = p + 2S/ 3 , √ P⊥ = −T22 = p − S/ 3 . (7) The difference between the radial and transverse pressures is √ (8) S = (Pr − P⊥ )/ 3 which is a measure of anisotropy of the fluid distribution. The field equations (3) corresponding to the metric (1) with the ansatz (2) lead to a set of three equations (we choose 8πG = c = 1) (1 + λ)(3 + λr2 /R2 ) , (9) R2 (1 + λr2 /R2 )2 2(ν /r)(1 − r2 /R2 ) − (λ + 1)/R2 Pr = − , (10) (1 + λr2 /R2 ) √ 3S 1+λ ν 2 − 2 =− ν +ν − r R (1 − r2 /R2 )(1 + λr2 /R2 ) 1+λ . (11) − 2 R (1 + λr2 /R2 ) ρ=

j The consistency condition T1;j = 0 [15] implies that Eq. (11) is equivalent to

√ S P.r = −(ρ − Pr )ν − 2 3 . r. r

(12)

Regularity of this equation at the centre requires that the anisotropy parameter S(r) should vanish there. in the limit r → 0 will be This ensures that S(r) r regular at r = 0. Accordingly, one expects the anisotropy of fluid pressure to grow gradually, reach an optimum and then diminish radially outward, thus accommodating a core with anisotropic fluid pressure in the central region of the configuration. We thus consider a star with an anisotropic core having a radial pressure (Pr ) different from the transverse pressure (P⊥ ) with Pr − P⊥ vanishing at the centre r = 0. The radial pressure Pr is expected to decrease outward while the transverse pressure P⊥ will initially increase (decrease) radially outward, attain an optimum value and then decrease

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(increase). Subsequently, the transverse pressure attains the same value as Pr on some spherical surface r = a which will be adopted as the core boundary. The fluid distribution in the envelope region is considered to be isotropic. The pressure decreases in the envelope region and becomes zero at the surface (say, at r = b , where b is the stellar radius). So we choose a compact star of size r = b and divide it into two parts: I: 0 ≤ r ≤ a, the anisotropic fluid core; II: a ≤ r ≤ b , the outer isotropic fluid envelope. In the next section we obtain a specific model making a suitable choice of S(r) complying with the requirements discussed above.

S 1 0.8 0.6 0.4 0.2 0.020.040.060.08 0.1 0.120.14

in of r R unit

Figure 1: Variations of anisotropy with core size for λ = 100, 40, 10 are represented by dark, broken and thin lines respectively

2.1. Core A solution to the field equations (9)–(11) is obtained here by introducing the new variables eν (13) x = 1 − r2 /R2 , φ = (1 + λ − λx2 )1/4 whence one obtains √ 3S(1 + λ − λx2 )R2 d2 φ + η− φ = 0, dx2 1 − x2

(14)

where η=

2λ(λ + 1)(2λ + 1) − (4λ + 7)λ2 x2 . 4(1 + λ − λx2 )2

We prescribe the anisotropy parameter as S=

(1 − x2 )(2λ(λ + 1)(2λ + 1) − (4λ + 7)λ2 x2 ) √ .(15) 3R2 (1 + λ − λx2 )3

Since S = 0 at the centre x = 1, the prescription meets the requirements discussed above and ensures regularity of the TOV (Tolman-Oppenheimer-Volkov) equation. With this choice, the second term in Eq. (14) vanishes, and the resulting equation admits the simple general solution φ = Cx + D

(16)

with C and D arbitrary constants. Subsequently, (17) eν(r) = (1 + λr2 /R2 )1/4 (C 1 − r2 /R2 + D) where the anistropy is described by Eq. (15). The radial pressure and the anisotropy parameter S(r) are now given by

2 2 2 C 1 − Rr 2 [3 + λ(λ+4)r ] + D[1 + λ(λ+2)r ] R2 R2

Pr = , (18) 2 2 R2 (1 + λ Rr 2 )2 (C 1 − Rr 2 + D) r2 r2 2 3 2 S= √ λ − 2λ − 2 (4λ + 7λ ) . 2 R 48R4 (1 + λ Rr 2 )3 (19) Variation of the anisotropy parameter for different λ is shown in Fig. 1.

It is evident that the anisotropy vanishes at different core sizes for different values of λ . We note that the core size decreases with increasing λ . We also note that the anisotropy parameter vanishes at the radial distance λ−2 R. (20) r= λ(4λ + 7) This distance gives the core to be denoted r = a. For positive λ our solution is valid for λ > 2. Thus a core is found to exist with an anisotropic fluid distribution governed by λ satisfying the lower limit. The size of the core a is given by Eq. (20), at which Pr (r = a) = P⊥ (r = a) i.e., the radial and transverse pressures coincide. The radial pressure at r = a is given by Pr=a = where C = C

25R2 (λ

+

C + D . 1 − a2 /R2 + D)

1)2 (C

(21)

(4λ + 7)(4λ2 + 6λ + 2) λ(λ2 + 14λ + 13)

and D = D(λ2 + 4λ + 3)(4λ + 7). 2.2. Envelope We now determine the equation of state of the envelope described by a ≤ r ≤ b . In this case the isotropic fluid distribution leads to a very simple relation from the condition of pressure isotropy:

µ 1 ν − − 2 1 − e−2µ = 0. (22) ν + ν 2 − µ ν − r r r There are two unknowns in the equation; however, we also take the ansatz (2) to describe the geometry. Then the above equation leads to a second-order differential equation in ν only. Now we write r2 x2 = 1 − 2 , ψ = eν , R λ x (23) z= λ+1

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and the pressure isotropy condition can be rewritten as (1 − z 2 ) ψ (z) + z ψ (z) + (λ + 1) ψ(z) = 0. This equation admits the general solution [12] cos[(n + 1)ζ + γ] cos[(n − 1)ζ + γ] − ψ=A , n+1 n−1

(24)

(25)

where ζ = cos−1 z and A and γ are constants to be determined by matching the solution with the exterior Schwarzschild solution −1 2M 2M dr2 + r2 dΩ2 , (26) ds2 = − 1− dt2 + 1− r r where dΩ2 = (dθ2 + sin2 θdϕ2 ). At the boundary r = b one gets 2M 2M , e−2µ(b) = 1 − . e2ν(b) = 1 − b b In this model the energy density and the pressure 1 2 ρ = 2 1 + , R (1 − z 2 ) (λ + 1)(1 − z 2 ) 2zψ 1 1 + . p = − 2 R (1 − z 2 ) (λ + 1)ψ

where

(27) ζ = cos are (28) (29)

We note that ρ is positive for λ > −1. Thus inside the core of a compact star the energy density is always positive which is evident from the constraint λ > 2. The mass contained inside a radius r is 1 r 2 r ρ(r )dr , (30) M (r) = 2 0 which on integration for r = b yields (1 + λ)b2 /R2 M (b) = . b 2(1 + λb2 /R2 )

Our model is valid for λ > 2. Thus we find that to describe a core model with the anisotropic fluid found here, the class of general solutions with perfect fluid obtained for λ ≤ 2 is not acceptable. The surface condition that the pressure is zero determines γ , and the constant A is determined from Eq. (27). Knowing A and γ , one can determine the two other unknowns C and D from matching conditions at the core-envelope boundary (i.e., r = a). Thus one obtains 1 a2 a2 4 C 1− 2 +D 1+λ 2 R R cos((n + 1)ζ + γ) cos((n − 1)ζ + γ) − A , (33) n+1 n−1

(31)

Now one can determine the radius of a compact star from the condition that the pressure should vanish at the boundary r = b . Thus, for given mass, the reduced radius b/R is determined from the condition P (b) = 0 which leads to λ+1 λ ψ (zb ) b2 =− , zb = 1− 2 , ψ(zb ) 2zb λ+1 R (32) for a given value of λ . Thus λ determines the equations of state for both the core and the envelope. There are four unknowns λ , R , A and γ in the core region, and we have other four unknowns λ , R , C and D in the envelope region. If the values of mass and radius are given, we have two free parameters, one of which is utilized to match the exterior Schwarzschild metric. However, for given stellar mass one determines the size of the star and vice versa for given λ . Thus it now leads to determination of only two unknowns C and D since λ , R , A and γ are determined from the boundary matching condition of the envelope.

−1

za ,

za =

λ λ+1

a2 1− 2 R

√ and n = λ2 + 2. The other condition is

a2 a2 a2 3 + λ(λ + 4) R + D 1 + λ(λ + 2) R C 1− R 2 2 2

a2 a2 (C 1 − R R2 1 + λ R 2 2 + D) 1 ψza 2za =− 2 . (34) 1+ R (1 − za2 ) λ+1 ψ r=a

3. Discussion To conclude, we have presented a core-envelope model of compact stars which follows from exact general solutions to the Einstein equations for a superdense star in hydrostatic equilibrium satisfying all physical constraints for the core and the envelope. The core is described by an anisotropic fliud distribution, while the envelope is described as a perfect fluid. The equation of state for the core is determined and found to require λ > 2 for consistency. This lower bound on λ is different from the one for a perfect fluid which is λ > 3/17 [12]. Thus a geometric parameter governs the equation of state inside a compact star. Inside the core, we introduce the density variation parameter ρ(a)/ρ(0) = Q (where ρ(a) and ρ(0) represent the core and centre densities, respectively) to know the density profile, and we get √ 1 − 6Q ± 24Q + 1 a2 ≤ 1. = R2 6Qλ Consequently, one obtains a restriction on Q given by Q≤

3+λ 3(1 + λ)

which is determined by λ once again. It is evident that large values of λ (>> 3) lead to Q ≤ 1/3 whereas for lower values, say, λ = 3 (λ > 2) one gets Q ≤ 1/2. Thus an anisotropic core shows a high degree of

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Table 1: Variations of the parameter R , stellar mass M (b), the corresponding core size (a), the density profile ¯ and the core density with different size of the star inside the core (Q), the density profile inside the envelope ( Q) 14 3 (b ) are shown for given λ and ρ(b) = 2 × 10 g/cm

λ=5

λ = 10

λ = 1000

b in km

R in km

M (b) in km

a in km

Q

¯ Q

ρ(a)

18.635 17.931 15.814 6.669

34.022 40.095 50.008 66.689

6.708 5.379 3.163 0.191

0.0027 0.0030 0.0033 0.0039

0.84 0.84 0.84 0.84

0.21 0.29 0.46 0.81

9.4 × 1014 6.8 × 1014 4.4 × 1014 2.5 × 1014

18.209 18.096 17.168 8.690

33.245 40.465 54.289 86.896

7.511 6.635 4.721 0.435

0.0024 0.0026 0.0030 0.0039

0.8 0.8 0.8 0.8

0.12 0.18 0.32 0.82

1.7 × 1015 1.1 × 1015 0.6 × 1014 0.2 × 1015

16.404 16.417 16.456 16.975

29.950 36.701 52.039 169.751

8.183 8.176 8.155 7.724

0.00027 0.00030 0.00036 0.00065

0.7 0.7 0.7 0.7

0.001 0.002 0.004 0.038

1.6 × 1017 1.1 × 1017 5.6 × 1016 5.2 × 1015

density variation as one moves from the centre to the core boundary. In the envelope region we denote the density variation parameter by 2 2 2 2 2 ¯ = ρ(b) = (3 + λb /R )(1 + λa /R ) Q 2 2 2 ρ(a) (3 + λa /R )(1 + λb /R2 )2

(where ρ(r) represents the stellar density) which may not admit high density variation since in our model we have both a/R and b/R possesing values smaller than unity. It is evident from Table 1 that a dense core is obtained if one considers stellar models with larger values of λ . The core size is then found to be very small compared to that with lower values of λ . Acknowledgement The authors thank IUCAA, Pune for hospitality in carrying out this work under the Visiting Associateshop Program. The authors would also like to thank the referee for valuable suggestions and remarks. BCP thanks UGC, New Delhi for financial support (project No. 3016/2004(SR)

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