S r in unit of R

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1 International Journal of Modern Physics D fc World Scientific Publishing Comany A CORE-ENVELOPE MODEL OF COMPACT STARS B. C. Paul Λ Physics Deartment, North Bengal University Dist : Darjeeling, Pin : , West Bengal, India. R. Tikekar y Deartment of Mathematics, Sardar Patel University Vallabh Vidyanagar, Pin : 38810, Gujarat, India. Received (received date) Revised (revised date) We resent a core enveloe model of comact stars. The core of a comact star is described by anistroic fluid which is surrounded by an isotroic enveloe. Assuming an ansatz that describe the sheroidal geometry of the sace inside the the star we found solutions of the core and that of the enveloe. The arameter which determines the order of sheroidicity of the sace ( ) is found to lay here an imortant role and to obtain core-enveloe model which is found to have alower bound ( >). However, in the case of relativistic fluid shere with erfect fluid distribution and without core enveloe model the bound is lowered to the limit > PACS numbers : 040, 040J, 0440D, 9530L 1. INTRODUCTION The work of Ruderman [1] and Canuto [] on comact star having matter distributions with densities much greater than the nuclear regime indicate that the suerdense stars are likely to develo anisotroic ressure. According to these views in such massive stellar objects the radial ressure differs from the tangential ressure inside the core. The origin of anisotroy in fluid ressure could be due to a number of hysical rocesses that may take lace inside the star. For examle, it may be due to the existence of a solid core [3] or the resence of tye P suerfluid or boson star [4], different kinds of exotic hase transitions due to gravitational collase [5], ion condensation [6] or other hysical henomena [7]. Maharaj and Maartens [8] obtained a solution for anisotroic fluid shere with uniform energy density. Later, Gokhroo and Mehra [9] extended their work for variable density distribution. A definite information about the behavior of matter in suerdense stars is not known. However, a comact object with such high energy matter in it may be studied from different aroaches. The study of relativistic core-enveloe model is an attemt in this direction. In this aroach interior of the suerdense star is considered to be Λ bcaul@iucaa.ernet.in y rameshtik@yahoo.co.uk

2 B. C. Paul and R. Tikekar comrising of two regions : (i) core region and (ii) enveloe region. It is imortant to look for a comact object with the anisotroic ressure region to be the core and surrounding the core contains the fluid distribution different from that of it, in fact one may assume erfect fluid distribution to study the enveloe region. Iyer et al. [ 10 ] studied a core-enveloe model in which itisshown that the core-enveloe aroach leads to information about bounds on the various arameters of ultra comact objects in general relativity (e.g., neutron star) such as mass, size and their ratio. In this aer we resent a core-enveloe model of a comact star in which the core region is described by matter with anistroic ressure surrounded by a distribution of a fluid with isotroic ressure in the enveloe regions. We follow here a different aroch for the solution of the enveloe region reviously adoted by Mukherjee et al. [11]. The conventional aroach for stellar models is to rescribe an equation of state for the fluid forming the interior of a star for solving the Einstein equation. In view of the non-linearity of equations as well as hydrodynamical comlexity, one has in realistic situations always resort to numerical maniulations. In the case of suerdense comact objects like neutron stars, the equation of state is uncertain and not well understood. In such a situation, aart from the conventional hysical aroach, it may beworthwhile to exlore an alternative aroach in which one assumes a simle geometry for the 3-sace so as to make the Einstein equation tractable. This will lead to an equation of state which may be useful and hysically accetable. We consider sacetime geometry that was given by Vaidya and Tikekar [1] for a suerdense star by roosing an ansatz for the geometry of the 3-surface embedded in a 4-Euclidean sace. The ansatz rescribes a sheroidal geometry for the 3-surface, described by two arameters and R; = 0 gives sherical while = 1 corresonds to flat sace.. THE FIELD EQUATIONS We begin with a static sherically symmetric sacetime described by the metric, with an ansatz ds = e ν(r) dt + e μ(r) dr + r (d + sin d' ) (1) e μ(r) = 1+r =R 1 r =R : () Note that the t = const: hyersurface has the geometry of a 3-sheroidal sace immersed in a 4-Euclidean sace and is characterised by the two curvature arameters and R. The suitability of the above metric has already been investigated by one of us [13]. The Einstein field equation is R μν 1 g μνr = 8ßGT μν (3) where g μν, R μν, R are the metric temsor, Ricci tensor and scalar curvature resectively and T μν is the energy momentum tensor. For an anisotroic fluid distribution following Maharaj and Maartens [8] we consider the energy momentum tensor given by T μν =(ρ + )u μ u ν g μν ß μν (4)

3 A Core-Enveloe model of comact stars where ρ, are the energy density and isotroic ressure and u μ denotes unit four velocity field of matter, ß μν is the anisotroic stress tensor. The anisotroic stress tensor is given by» ß μν = 3S C μ C ν 1 3 (uμ u ν g μν ) For radially symmeteric anisotroic fluid distribution of matter, S = S(r) denotes the magnitude of the anisotroic stress tensor and C μ =(0;e ; 0; 0), which isa radial vector. The energy momentum tensor corresonding to the exression (4) has the following non-vanishing comonents T o o = ρ; T 1 1 = + 3 S ; T = T 3 3 = 3 S : The ressure along the radial and tengential direction are not same whichwe denote here by P r = T 1 1 = + 3 S P? = T = 3 S The difference between the radial ressure and the transverse ressure is (5) (6) (7) S = P r P? (8) 3 which in fact reresents the measure of the anisotroy of the fluid distribution. The field equation (3) corresonding to the metric (1) using ansatz () is given by a set of three equations (we choose 8ßG = c =1) ρ = (1 + )(3 + r =R ) ; (9) R (1 + r =R ) P r = h ν 0 (1 r r =R ) +1 R (1 + r =R ) 3S =»ν 00 + ν 0 ν0 r 1+ R (1 r =R )(1 + r =R ) i (10) 1+ R (1 + r =R ) (11) We consider a star with an anisotroic core having radial ressure (P r ) different from the transverse ressure (P? ). However at the boundary of the core, let us say at radius r = a the two ressures coincide and the fluid distribution in the enveloe region is described by isotroic fluid distribution. The ressure decreases in the enveloe region and it becomes zero at the surface (say at r = b, where b is the radius of the star under consideration). In the next section we describe a comact star with core described by an anistroic fluid distribution and the enveloe of the star is described by a erfect fluid distribution. The equation of state of a comact object whose core is described by an anistroic fluid distribution and outside the core it is described by an istroic fluid distribution are also evaluated. We describe

4 B. C. Paul and R. Tikekar the core u to the radius where S(r = a) =0. We choose a star of size r = b and divide it into two arts : I:0» r» a as the CORE of the star described by an anisotroic fluid distribution. II : a» r» b as the outer ENVELOPE of the CORE which can be described by an isotroic fluid distribution..1. CORE OF THE STAR The solution of the field equation (9)-(11) is obtained here by introducing a new variable x = e 1 ν r =R ; ffi = (1) (1 + x ) 1=4 where one obtains d ffi dx + " ( + 1)( +1) (4 +7) x 4(1 + x ) 3S(1 + x )R 1 x # ffi =0: (13) On rescribing the anisotroy arameter as S =» (1 x )(( + 1)( +1)+(4 +7) x ) 3R (1 + x ) 3 (14) the second term in the second derivative differential equation (13) vanishes and the resulting equation ermits a simle general solution given by ffi = Cx + D (15) with C and D as arbitrary constants of integration leading to a simle solution e ν(r) =(1+r =R ) 1=4 (C 1 r =R + D) (16) where the anistroy is described by eq. (14). Thus the sace-time metric of the core of a comact star is described by ds = (1 + r =R ) 1= (C 1 r =R + D) dt + 1+r =R 1 r =R dr +r d + r sin dffi The radial ressure and the anisotroy arameter S(r) are now given by (17) P r = C 1 r =R [3 + ( +4)r =R ]+D[1 + ( +)r =R ] R (1 + r =R ) (C ; (18) 1 r =R + D) r S = 4 3R 4 (1 + r =R ) 3» + + r R (43 +7 ) The variation of anisotroy arameter for various are shown in fig. 1. (19)

5 A Core-Enveloe model of comact stars S r in unit of R Figure 1: Variation of anisotroy with core size for = 100; 40; 10 are reresented by dark, broken and thin lines resectively It is evident that the anisotroy vanishes at different core sizes for different values of, as one increases the core size decreases. Here we find that the core size is determined by. One obtains that the anisotroy vanishes at s r = (4 +7) R (0) which reresents the size of the core and we denote it by r = a. For a ositive we note that our solution is valid for >. Thus a core is found to exist with anisotroic fluid distributionq decided by satisfying the lower limit. Thus the size of the core is given by a = R at which P (4+7) r(r = a) =P? (r = a) i.e., both the radial ressure and the transverse ressure converge to the same value. The radial ressure at r = a is given by (4 + 7)(4 +6 +)=( ) + D( )(4 +7) P r=a = C 5R ( +1) (C : 1 a =R + D) (1) The anisotroy vanishes at the core boundary and the enveloe is thus reresented by erfect fluid distribution, which we discuss in the next section... ENVELOPE We now determine the equation of state of the enveloe which is described by the radial limit a» r» b. In this case the isotroic fluid distribution in the enveloe leads to a very simle relation from the condition of ressure isotroy : If we write ν 00 + ν 0 μ 0 ν 0 ν0 r μ0 r 1 r 1 e μ =0: () ψ = e ν ;

6 B. C. Paul and R. Tikekar x =1 r R ; r z = +1 x (3) the ressure isotroy condition now gives rise to a second order differential equation (1 z ) ψ 00 (z)+zψ 0 (z)+( +1)ψ(z) =0: (4) This equation admits a general solution [11]» cos[(n +1) + fl] ψ = A n +1 cos[(n 1) + fl] ; (5) n 1 where = cos 1 z and A and fl are constants to be determined by matching the solution with the exterior Schwarzschild solution ds = 1 M(r) dt + 1 M(r) 1 dr + r (d + sin d' ): (6) r r At the boundary r = b one gets e ν(b) =1 M b e μ(b) =1 M b In this model the energy density and the ressure are given by» 1+ ρ = 1 R (1 z ) = 1 R (1 z ) ( + 1)(1 z )»1+ zψ0 ( +1)ψ (7) ; (8) : (9) We note that ρ is obviously ositive for >1. Thus inside the core of a comact star energy density is always ositive as we require >. The mass contained inside a radius r is given by M(r) = 1 which on integration for r = b yields M(b) b Z r 0 r 0 ρ(r 0 )dr 0 (30) = (1 + )b =R (1 + b =R ) : (31) Now one can determine the radius of a comact star from the condition that the ressure should vanish at the boundary r = b. Thus for a given mass, the reduced radius b is determined from the condition P (b) = 0 which leads to R ψ 0 (z b ) ψ(z b ) = +1 z b (3)

7 A Core-Enveloe model of comact stars q with z b = +1 1, b R for a given. Thus determines the equation of state for the CORE as well as for the enveloe too. There are four unknowns, R, A and fl in the CORE region and we have another four unknowns, R, C and D in the enveloe region. If the values of mass and radius are given, we have two free arameters, one of which is utilized to match the exterior Schwarzschild metric. However, for a given mass of a star one determines the size of the star and vice versa for a given. Thus it now leads to determination of only two unknowns C and D as, R, A and fl are determined from the boundary matching condition of the enveloe. Our model is valid for >. Thus we find that to describe a core model with anisotroic fluid found here, the class of general solutions with erfect fluid obtained for» are not accetable. The surface condition that the ressure is zero determines fl and the constant A is determined from the equation(7). Knowing A and fl one can determines the two other unknowns C and D from the matching conditions at the core-enveloe boundary ( i.e., r = a ). Thus one obtains 1=4 h 1+ a C 1 a =R + Di R q where = cos 1 z a, z a = +1» cos((n +1) + fl) = A n +1 cos((n 1) + fl) n 1 (33) 1 a R Λ and n = +. The other condition is q C 1 a 3+( +4) R a + D 1+( +) R a R R q 1+ a R (C 1 a + D) R 1 = R (1 za)» 1+ z a +1 ψza ψ r=a (34) 3. DISCUSSIONS To conclude we resent a core-enveloe model of comact stars which follows from the exact general solutions of the Einstein equations for a suerdense star in hydrostatic equilibrium satisfying all hysical constraints for the CORE and the EN- VELOPE. The matter content for the core is described by an anisotroic fliud distribution and that of the enveloe is described by erfect fluid assumtions. The equation of state for the core is determined and is found that it requires > for consistency. This lower bound on is different from that if one considers a erfect fluid which is> 3 [11]. Thus the the geometrical arameter decides the equation 17 of state inside the comact star with sheroidal geometry. Inside the core of star we introduce density variation arameter ρ(a) = Q (where ρ(a) and ρ(0) reresent ρ(0) the density of the core and centre resesectively) to know the density rofile in the model and we get a R = 1 6Q ± 4Q +1» 1: 6Q

8 B. C. Paul and R. Tikekar Consequently, one obtains a restriction on Q given by Q» 3+ 3(1 + ) which is determined by once again. It is evident that large values of (>> 3) leads to Q» 1 whereas for lower values, say =3(>) one gets Q» 1. Thus 3 the anisotroic core shows a high degree of density variation as one moves from the centre to the core boundary. In the enveloe region we denote the density variation arameter by μq = ρ(b) ρ(a) = (3 + b )(1 + R a ) R (3 + a )(1 + R b ) R (where ρ(b) reresents the density of the star) which may not admit high density variation as in our model we have both a and b are ossesing values less than R R unity. Table 1. b in Km. R in Km. M(b) in Km. a in Km. Q μ Q ρ(a) : : = : : : : = : : : : = : : Table 1. The variation of arameter R, mass of a star M(b), corresonding core size (a), density rofile inside the core (Q), density rofile inside the enveloe ( Q) μ and core density with different size of the star (b) are shown for a given and star density ρ(b) = gm/cc. It is evident from the table that a dense core of a star is obtained once one takes larger values of, also in that case the size of the core diminishes. Acknowledgments The authors would like to thank IUCAA, Pune for hosatility for carrying out this work under the Visiting Associatesho Program.

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