Anisotropic Compact Star Model: a Brief Study Via Embedding

Eur. Phys. J. C (2019) 79:138 https://doi.org/10.1140/epjc/s10052-019-6642-6

Regular Article - Theoretical Physics

Anisotropic compact star model: a brief study via embedding

Piyali Bhara Department of Mathematics, Government General Degree College, Singur, Hooghly, West Bengal 712 409, India

Received: 16 May 2018 / Accepted: 3 February 2019 / Published online: 14 February 2019 © The Author(s) 2019

Abstract In present article a new model of compact star is and other one is transverse pressure pt orthogonal to the for- = − 2 obtained in the framework of general relativity which does mer. pt pr is known as anisotropic factor and r not suffer from any kinds of singularity. We assume that the is anisotropic force which becomes attractive or repulsive underlying fluid distribution is anisotropic in nature along according to pr < pt or pr > pt . Bowers and Liang [3], with a new form for the metric potential eλ which is phys- in 1974, wrote a paper on the study of anisotropic distribu- ically reasonable. Though the model parameters depend on tion of matter which got wide attention. Anisotropy can be four constants a, b, A and B but we have shown that the revealed due to the existence of solid stellar core or by phase solutions depend on two free constants since these four con- transitions, pion condensation in a star [4], the presence of stants are correlated to one another. Our proposed model type-IIIA superfluid [5,6], rotation, electromagnetic field [7Ð of anisotropic compact star obeys all the necessary physi- 9] etc. The effect of pressure anisotropic has been studied in cal requirements which have been analyzed with the help details in the Refs. [10,11]. By using uniform matter density, of the graphical representation where n lies in the range of anisotropic model has been investigated in [12]. Chan et al. − 200 ≤ n ≤ 200. We have shown that the model satisfies [13] studied the role of the local pressure anisotropy in detail all the energy conditions as well as the causality condition. and showed that small anisotropies might in principle dras- The model is potentially stable and also satisfy HarrisonÐ tically change the stability of the system. Some anisotropic ZeldovichÐNovikov’s stability condition. compact star models are obtained which admit conformal motion. The solutions depend on the conformal factor and matter density ρ [14,15] or a relation between radial and 1 Introduction transverse pressure etc. [14]. In Ref. [15], a new class of interior solutions for anisotropic stars are obtained by choos- A well known fact is that the compact objects such as white ing a particular density distribution function of Lorentzian dwarfs, neutron stars or black hole are formed at the end type as provided by Nazari and Mehdipour [16,17] which point of the gravitational collapse of a star when most of their admits conformal motion in higher dimensional noncommu- nuclear fuel has been consumed. They are supported by either tative spacetime. Some researchers obtained the polytropic nuclear or electromagnetic forces which act as an internal quark star model [21] in order to establish a relation between pressures, but gravity plays an important role in both the theory and observations. Heintzmann and Hillebrandt [19] process of the gravitation collapse and in the final equilibrium described a model of a relativistic, anisotropic neutron star configuration. The factor which is reasonable behind the type model at high densities by means of several simple assump- of the ends up object is the star’s mass. White dwarfs are tions and have shown that for an arbitrary large anisotropy formed from light stars with masses M < 4M and radius is there is no limiting mass for neutron stars, however the maxi- − ∼ 10 2 R Moreover it has shown that the maximum allowed mum mass of a neutron star still lies beyond 3−4M. Sharma mass for white dwarfs is around 1.4M.[1](M, R are et al. [20] considered the theoretical possibility of anisotropy the solar mass and solar radius respectively). in strange stars, with densities greater than that of neutron Ruderman [2] was the pioneer who proposed that the pres- stars but less than that of black holes. Lai and Xu [21]pro- sure inside a highly compact objects shows anisotropic in posed that the polytropic equations of state are stiffer than the nature, i.e., it has two components: one is radial pressure pr conventional realistic models, i.e., the MIT bag model. In the framework of a polytropic model , they show that a very low a e-mail: [email protected] massive quark star can also exist and be still gravitationally 123 138 Page 2 of 13 Eur. Phys. J. C (2019) 79 :138 stable even if the polytropic index n > 3. The properties of 1 Rij − Rgij = Gij, (3) compact stars depends on the assumed description of matter 2 in their interiors. Xu et al. [22] and Azam et al. [23] studied where Rij, R and gij are the Ricci tensors and Ricci scalar the behavior and the physical properties of several compact and metric tensor respectively. Tij is the energy-momentum objects. A general scheme for compact astrophysical objects tensor of the underlying fluid distribution. which are not composed of neutron matter, but where, given Let us assume that the matter involved in the distribution the conditions of very high density in their interiors are pre- is anisotropic in nature, by using the general expression, we, sented by Alcock et al. [24] and Haensel et al. [25]. therefore, get the expression for energy-momentum tensor as The RandallÐSundrum (RS) second brane-world model follows: [26] is stand on the concept that our 4-dimensional space- μ μ μ μ μ μ T = ρv vξ + p χξ χ + p (v vξ − χξ χ − g ), (4) time is a hypersurface embedded into the 5-dimensional ξ r t ξ μ μ bulk. After the work done by Randall and Sundrum [26]on with v vξ = 1 =−χξ χ , χξ is the unit space-like vector brane theory, the study on embedding spacetime attracted the and vμ is the fluid-4 velocity of the rest frame and there- μ researchers more. If a n-dimensional space Vn can be embed- fore v χξ = 0. The above formula gives the components of ded in (n+k)-dimensional space, where k is a minimum num- the energy-momentum tensor of an anisotropic fluid at any ber of extra dimensions, then Vn is said to be of embedding point in terms of the density ρ, the anisotropic and transverse class k of n-dimensional space. Two very well known impor- pressures pr , pt respectively. With the simple form of line μ tant solutions, e.g. Friedmann universe and Schwarzschild’s element, Tξ takes the form: interior solutions are of class I, (so in this case k = 1), on 0 = ρ, 1 =− , 2 = 3 =− , the other hand, Schwarzschild exterior solution is of class II T0 T1 pr T2 T3 pt (5) ( = ) ( = ) k 2 and the Kerr metric [27]isofclassV k 5 .The and Karmarkar condition relates to class one spacetimes. Pandey k = = , and Sharma [39] showed that the Karmarkar condition is Tj 0 if j k only a necessary condition for a spacetime to be of class 1. using (5), for the line element (1), (2) takes the form: A further requirement has to be imposed for sufficiency of −λ −λ the Karmarkar condition. The derivation of the Karmarkar e − 1 e ν πp = + , 8 r 2 (6) Both charged and uncharged star model of embedding class- r r 2 I spacetime extensively study in the Refs. [28Ð38]. −λ ν ν ν λ ν − λ 8πpt = e + − + , (7) We have organized the paper as follows: in Sect. 2 the 2 4 4 2r basic field equations have been discussed, in Sect. 3 we have 1 − e−λ e−λλ given a short discussion about embedding class-I spacetime 8πρ = + , (8) r 2 r and also obtained a new model. In next section we match our interior space-time with the exterior Schwarzschild line ele- where differentiations with respect to r are denoted by ‘ = = ment and junction condition is also discussed. The physical and we have chosen G c 1. Here G is the gravitational analysis of the model are discussed in Sect. 5. In Sect. 6 we constant and c is the speed of light. The gravitational mass in a sphere of radius ‘r’ is given by, discussed the stability conditions of the present model and some discussions are made in the final section. r m(r) = 4π ρ(ω)ω2dω. (9) 0 In next section we shall solve the Eqs. (6)Ð(8) to obtain the 2 Basic field equations model of compact star.

The interior of a static spherically symmetric spacetime in standard co-ordinate xa = (t, r,θ,φ)is described by the 3 A particular model following line element: So form the above field equations (6)Ð(8), it is noted that ν( ) λ( ) ds2 = e r dt2 − e r dr2 − r 2(dθ 2 + sin2 θdφ2), (1) we have three field equations for five unknown functions: ν λ e , e ,ρ,pr and pt . So to generate a model of compact where λ and ν are the functions of radial coordinate r only. star, we are free to chose any two of them, though to be a The Einstein field equation is, physically realistic model a number of physical conditions to be satisfied by our present model. Gij = 8πTij. (2) To generate a particular model of compact star, let us 2 Here Gij is the Einstein’s tensor having the following expres- assume that the co-efficient of dr , i.e., grr has the following sions, form: 123 Eur. Phys. J. C (2019) 79 :138 Page 3 of 13 138

λ − e = 1 + a2r 2(1 + b5r 5) n, (10) with eλ = 1. Solving Eq. (16) we get, where a,b are some positive constants having the unit 2 ν λ length−1 and n ∈ R, R is the set of real numbers. We are e = A + B e − 1 dr , (17) the first to use this metric potential. Now using the relation between the mass function and the where A and B being the constants of integration. metric potential, By using (16) and (17), from Eqs. (6) and (7), we obtain the pressure anisotropic = pt − pr as, −λ 2m(r) e = 1 − , (11) r ν 2 λ νeν 8π = − − 1 . (18) the mass function of the star is obtained as, 4eλ r eλ − 1 2rB2 a2r 3 m(r) = . (12) From the Eq. (18), it follows that the anisotropic factor 2 a2r 2 + (1 + b5r 5)n will vanish if the following two case arises: The metric potential that we use in this paper gives a mass function which is monotonic increasing in nature and regular • Case 1: at the center of the star. At the same time it provides a matter λ density which gives a profile of monotonic decreasing in = 2, λ (19) nature and gives a finite value at the center of the star. So our e − 1 r chosen metric potential is physically reasonable. Now for a symmetric tensor bμν of a 4-dimensional Rie- which on integrating gives, mannian space satisfying the Gauss and Codazzi equations −λ 2 given as e = 1 − A1r , (20)

Rμναβ = (bμαbνβ − bμβ bνα) (13) A1 is the constant of integration. bμν;α − bμα;ν = 0, (14) Now using (20), from Eq. (17), we obtain the other metric can be embedded in 5-dimensional Pseudo-Euclidean space. co-efﬁcient as follows: where ‘(;)’ represents covariant derivatives and takes the − + 2 value 1or 1 according to the normal to the manifold ν B 2 e = A − √ 1 − A1r . (21) being time-like or space-like respectively. For the line ele- A1 ment given in (1), the non-zero components of the Riemann curvature tensor are given as, Now using the expression of eλ in (20), from Eq. (8), we 2 2 −λ R2323 = r sin θ 1 − e obtain the constant matter density and this solution is nothing but the Schwarzschild’s interior solution [40]. 1 R1212 = λ r • Case 2: 2 R = 0 1224 νeν ν 1 1 2 1 − 1 = 0, R1414 = e ν + ν − λ ν 2rB2 2 4 4 r R = sin2θνeν−λ. which on integrating, we get, 3434 2 It is clear that the non-zero components of the symmetry ν 2 2 e = B r + A2. (22) tensor bμν are b11, b22, b33, b44 and also b14(= b41) due 2 to its symmetry nature and b33 = b22 sin θ. By substituting And like previous case we obtain, the components of bμν from (13) we get,

R1212 R3434 + R1224 R1334 2 2 R = , (15) λ A2 + 2B r 1414 e = . R2323 2 2 (23) A2 + B r with R2323 = 0[39]. The space-time that satisﬁes the condition (15) represents the space time of embedding class I. Where A2 is the constant of integration. For the condition (15), the line element (1) gives the fol- The solution obtained in (22) and (23) gives, KohlerÐ lowing differential equation Chao solution [41] and it gives non-vanishing Weyl ten- λ ν sor which is suitable for cosmological mode. So we are =−2(ν + ν 2) + ν 2 + λ ν , (16) 1 − eλ not interested to consider isotropic pressure. 123 138 Page 4 of 13 Eur. Phys. J. C (2019) 79 :138

λ Substituting the expression for e in (17) and integrating Where the expressions for φ2(r), φ3(r), ξ(r) are given by, it, the metric coefficient eν(r) becomes, 5 5 φ2(r) =−4 + b (−4 + 5n)r , 2 ν 1 2 φ ( ) =− + 5(− + ) 5, e = A + aBr φ1(r) , 3 r 2 b 2 5n r 2 2 ξ(r) = 2A + aBr φ1(r). where 2 n 7 φ (r) = hypergeometric F , , , −b5r 5 , 1 2 1 5 2 5 4 Exterior spacetime and matching conditions and hypergeometric F is defined as 2 1 In this section we match our interior spacetime to the ∞ k (a)k(b)k x Schwarzschild exterior solution hypergeometric F1(a, b; c; x) = , (24) 2 ( ) ! −1 = c k k 2M 2M k 0 ds2 = 1 − dt2 − 1 − dr2 where, |x| < 1; a, b, c are real numbers and c = r r − 2( θ 2 + 2 θ φ2), 0, −1, −2,... Here (a)n (n is a positive integer) being r d sin d (29) Pochhammer symbol defined by, at the boundary r = r, r being the radius of the star and (a)n = a(a + 1) ···(a + n − 1), therefore it is obvious that r > 2M, M is the mass of the black hole. ( ) = with a 0 1. Now for an isotropic fluid sphere at a stellar surface, ( ) With the help of simple algebra, a n takes the form, O’Brien–Synge [42] and Robson [43] proposed a proper (a + n) junction condition given by, (a) = . n (a) p(r + 0) = p(r − 0), (30) Now the series expansion of r.h.s of (24)is, gab(r + 0) = gab(r − 0), (31) a · b a(a + 1)b(b + 1) ∂ ( + ) = ∂ ( − ), 1 + x + x2 +··· . r gab r 0 r gab r 0 (32) 1 · c 1 · 2 · c(c + 1) where r is the stellar radius, p is the isotropic pressure, gab Once we have the two metric potentials, by substituting is the metric components (a, b = r, t,θ,φ). Now conditions the expressions of metric co-efficient in the field equations (52) and (32) provide the following relationship, (6)Ð(8), the matter density, radial and transverse pressure −1 becomes, 2m 2 2 5 5 −n 1 − = 1 + a r(1 + b r) , (33) r a2 (1 + b5r 5)n 3 + b5(3 − 5n)r 5 + a2(r 2 + b5r 7) ρ = 2 2 2m aBrφ(r) 8π (1 + b5r 5) a2r 2 + (1 + b5r 5)n 2 1 − = A + , (34) r 2 (25) 2m 5 5 − n 2 n = aBr(1+b r ) 2 2A+aBr φ (r) , a −2aA+ 4B(1 + b5r 5) 2 − a2 Br2φ (r) 2 1 p = 1 (26) r r π 2 2 + ( + 5 5)n + 2φ ( ) 8 a r 1 b r 2A aBr 1 r (35) −1+ n ( + 5 5) 2 2m − −n a 1 b r = a2r(1 + b5r5 ) 1 2 + b5(2 − 5n)r5 . pt = 2 16π a2r 2 + (1 + b5r 5)n 2 2A + aBr2φ (r) r2 1 − 2m 1 r 2 2 5 5 5 5 n (36) × 4a Br (1 + b r ) − 2B(1 + b r ) φ2(r) 5 5 n 2 Since we are considering the anisotropic fluid sphere the + a(1 + b r ) 2 φ3(r) 2A + aBr φ1(r) . (27) condition given in Eq. (30) changes in our present paper And the anisotropic factor is given by, in presence of both radial and tangential pressure. Now pr (r = r) = 0gives, = (pt − pr ) 5 5 n 2 2 1 − 2aA+ 4B(1 + b r ) 2 − a Br φ (r) = 0, (37) = 1 16π(1 + b5r 5) a2r 2 + (1 + b5r 5)n 2 ξ(r) but there is a discontinuous tangential pressure. Now to avoid × ar2 2a2(1 + b5r 5) + 5b5nr3(1 + b5r 5)n {2aA the discontinuity we calculate the surface stresses at the junction boundary by using the DarmoisÐIsrael [44,45]forma- 5 5 n 2 2 − 2B(1 + b r ) 2 + a Br φ1(r) . (28) tion. 123 Eur. Phys. J. C (2019) 79 :138 Page 5 of 13 138

The expression for surface energy density σ and the sur- and radial and transverse pressure gradients are obtained face pressure P at the junction surface r = r are obtained as, as, + π dpr = ar 1 −λ 8 σ =− e dr (1 + b5r 5) a2r 2 + (1 + b5r 5)n 2 χ(r) 4πr − n × + 2φ ( ) 2 ( + 5 5) 2 φ ( ) 5 5 n 2A aBr 1 r 2a B 1 b r 2 r 1 2m (1 + b r) =− 1 − − , (38) 3n π 2 2 5 5 n − 5 3( + 5 5) 2 + 3( + 5 5) 4 r r a r + (1 + b r) 10b Bnr 1 b r 2a 1 b r + 2 1 aν × 2A − 2Br + aBr φ1(r) P = 1 + e−λ 8πr 2 × 2(A + Br) + aBr2φ (r) ⎡ − ⎤ 1 n 2 5 5 − 5 5 n 2aBr (1+b r ) 2 + a(1 + b r ) φ (r) (43) ⎢ − m 1 + ⎥ 6 1 1 2A+aBr2 φ (r ) = ⎢ r − 1 ⎥ . dp ar π ⎣ ⎦ 8π t = 8 r − 2m 2 2 5 5 −n 3 1 1 + a r(1 + b r) dr 2(1 + b5r 5)2 a2r 2 + (1 + b5r 5)n χ(r) r 5 4 5 5 2 5 5 2n (39) × −8a r (B + b Br ) + 4a(1 + b r ) φ8(r)

5 3 5 5 5n + 10Ab Bnr (1 + b r ) 2 φ7(r) 4 2 5 5 1+ n + 4a ABr (1 + b r ) 2 φ2(r) 5 Physical analysis 2 5 5 3n − 2a AB(1 + b r ) 2 φ9(r) λ( ) ν( ) + 3( + 5 5)nφ ( ) • For our present model, e 0 = 1 and e 0 = A2,a 4a 1 b r 10 r + 3 2 4( + 5 5)nφ ( )ψ( ) positive constant, and a B r 1 b r 11 r r + 2( + 5 5)φ ( ) ( ) . aBr 1 b r 12 r g1 r (44) ν − − (e ) = a2r(1 + b5r 5) 1 n 2 + b5(2 − 5n)r 5 , χ( ), φ ( ) ( ), ν −n Where the expressions for r i r (i=1,2,…,12), g1 r ( ) = ( + 5 5) 2 + 2φ ( ) . e aBr 1 b r 2A aBr 1 r ψ(r) are given by,

ν λ 2 Clearly (e ) = = 0 and (e ) = = 0, indicates that χ(r) = 2A + aBr2φ (r) ; r 0 r 0 1 metric co-efﬁcients are regular at the center. 5 5 φ4(r) = −8 + b (−3 + 5n)r ; • The central density of the star is given by, 5 5 5 5 φ5(r) = 2 + b r 4 + b (2 + 5(−1 + n)n)r πρ = πρ( = ) = 2 > , 8 c 8 r 0 3a 0 (40) φ (r) = 20A2b5nr3 − 8B2(1 + b5r 5) 6 5 5 2 + 5ab Bnr φ1(r) 4A + aBr φ1(r) and density gradient is obtained as, φ ( ) =− + 5(− + ) 5 7 r 14 b 4 5n r 2 5 3 5 5 dρ 1 φ8(r) = 5A b nr 7 + b (2 − 5n)r π = 8 3 dr (1 + b5r 5)2 a2r 2 + (1 + b5r 5)n + 2( + 5 5) − + 5(− + ) 5 B 1 b r 4 b 4 5n r × − 6 3( + 5 5)2 − 4 ( + 5 5)nφ ( ) 2a r 1 b r 5a r 1 b r 5 r φ (r) = 24 + b5r 5(48 − 20n + b5(24 − 70n + 75n2)r 5) 9 n φ ( ) = 2 2( + 5 5) − + 5(− + ) 5 + 5a2b5nr4(1 + b5r 5)2 φ (r) . (41) 10 r B r 1 b r 6 b 6 5n r 4 + A2 8 + b5r 5 {16 − 5n At the point r = 0, + b5(−4 + 5n)(−2 + 5n)r 5 2 4 dρ d ρ −10a φ (r) =− b5nr3( + b5r 5)n − + b5(− + n)r 5 = 0; = < 0. 11 5 1 7 2 5 dr dr2 κ + a2 8 + b5r 5 {16 − 5n • The central pressure of the star is given by, + b5(−4 + 5n)(−2 + 5n)r 5 φ ( ) = 5 3( + 5 5)2n ( ) a 2B 12 r 5b Bnr 1 b r f7 r pc = pr (r = 0) = pt (r = 0) = −a + > 0, + 4 2( + 5 5)φ ( ) − 5 3 8π A 2a Br 1 b r 2 r 20aAb nr 5 5 3n 5 5 (42) × (1 + b r ) 2 −7 + b (−2 + 5n)r 123 138 Page 6 of 13 Eur. Phys. J. C (2019) 79 :138

3 5 5 n 5 5 + 4a A(1 + b r ) 2 8 + b r {16 − 5n and the central redshift is given by, + b5(−4 + 5n)(−2 + 5n)r 5 −ν(0) 1 Zc = e − 1 = − 1. (49) −a2 B(1 + b5r 5)n 24 + b5r 5 {48 − 20n A + b5(24 − 70n + 75n2)r 5 Inside the stellar interior this should be non-negative, i.e., 1 − > < < 7 n 7 A 1 0 implies, 0 A 1 , this range of constant A g (r) = hypergeometric F 1, − , , −b5r 5 1 2 1 5 2 5 is well satisﬁed with the numerical values of A obtained 2 in Table 1. 2 n 7 5 5 ψ(r) = hypergeometric F1 , , , −b r . Now, 2 5 2 5 − n Now at the point r = 0, dZ 4aBr(1 + b5r 5) 2 =− , (50) • 2 2 From Eq. (42), we get, dr 2A + aBr φ1(r) B a > . (45) at r = 0, A 2 2 Again, in the interior of a compact star model, the inequal- dZ = , d Z =−aB < . 0 2 2 0 ities ρ − pr ≥ 0,ρ− pt ≥ 0 should satisfy what dr dr A is called the dominant energy condition, proposed by It shows that the proﬁle of gravitational redshift is a Zeldovich [46]. Therefore, the above two inequalities monotonic decreasing function of ‘r’. should hold at the center of the star, which implies, • The surface redshift is given by, pc ≤ 1, i.e., ρc λ(r ) 2 2 5 5 −n B Zs = e − 1 = 1 + a r(1 + b r) − 1. < 2a. (46) A (51) Combining the inequalities in (45) and (46), we get a B In literature, we found that in the absence of a cosmo- bound for A as, logical constant the surface redshift (zs) lies in the range a B z ≤ 2[47Ð49]. On the other hand, Bohmer and Harko < < 2a. (47) s 2 A [49] showed that for an anisotropic star in the presence of a cosmological constant the surface redshift obeys the • The equation of state parameters ωr and ωt are obtained inequality zs ≤ 5. For our present model, the surface as: redshift is obtained as 0.5333 and it is interesting to note pr pt ωr = ; ωt = . that the surface redshift of the model do not depend on ρ ρ n. • Energy condition • The gravitational redshift is given by, The model of an anisotropic compact star will satisfy the √ 1 all the energy conditions namely, (i) the Null energy con- Z = e−ν − 1 = − 1, (48) + 1 2φ ( ) dition (NEC), (ii) Weak energy condition (WEC), (iii) A 2 aBr 1 r

Table 1 The numerical values of a, A, B, central density (ρc),sur- km for different values of n. The units of a, A, B, b,ρc,ρs , pc −1 −2 −2 −1 −3 −3 −2 face density (ρs ), central pressure (pc) are obtained by ﬁxing b = are km , km , km , km , gm · cm , gm · cm , dyne· cm 0.0095 for a compact star of mass M = 1.85M and radius = 9.5 respectively na A B ρc ρs pc

50.122308 0.432156 0.0398916 2.40939 × 1015 6.32563 × 1014 3.68233 × 1035 50 0.122324 0.432127 0.0398916 2.41004 × 1015 6.32367 × 1014 3.6826 × 1035 3000 0.12341 0.430173 0.0398916 2.45299 × 1015 6.19524 × 1014 3.70055 × 1035 − 250 0.122215 0.432324 0.0398916 2.40572 × 1015 6.33673 × 1014 3.68079 × 1035 − 500 0.122123 0.432489 0.0398916 2.40212 × 1015 6.34761 × 1014 3.67929 × 1035 − 2500 0.121394 0.433801 0.0398916 2.37353 × 1015 6.43468 × 1014 3.66735 × 1035

123 Eur. Phys. J. C (2019) 79 :138 Page 7 of 13 138

Strong energy condition and (iv) Dominant energy con- which is a monotonic decreasing function of ‘r’. For different ditions if and only if the following inequalities hold: values of ‘n’, the numerical values of the matter density in −3 MeV fm unit are obtained in table 3. The radial pressure pr ρ − pr ≥ 0,ρ− pt ≥ 0,ρ− pr − 2pt ≥ 0. (52) and transverse pressure pt are plotted in Fig. 2(left) and (middle) respectively. Both are monotonic decreasing function of We shall check the above inequalities with the help of ‘r’ and pr (r = r) = 0 where as pr (r = r)>0. The mea- graphical representation. sure of anisotropy = pt − pr is plotted in Fig. 2(right). From the graph of one can note that >0for0< r ≤ r Before going to plot the model parameters of the present and the positive nature of the anisotropic factor helps to con- λ ν struct more compact object which was prosed in [50] because model one can note that the expressions of e , e ,ρ,pr , pt etc. contains four constants a, b, A and B. So first we have for positive anisotropy offers an repulsive force which helps to fix some reasonable values to these constants satisfying to hold the model against the collapse. Moreover at the cen- = the Eqs. (33)Ð(36). For drawing the plots we have consider a ter of the star the anisotropic factor vanishes, i.e., 0at = compact star of mass 1.85M and radius = 9.5 km and we fix r 0,which is also a necessary condition to construct a com- ω b = 0.0095. For these values, by using boundary conditions, pact star model [51,52]. The equation of state parameters r ω we obtain the other constants a, A, B for different values of and t are plotted in Fig. 3(top) and (bottom) respectively. n which have shown in Table 1. Both the profiles are monotonic decreasing functions of ‘r’ ≤ ω ,ω ≤ Using the values of the constant mentioned in the table, as well as 0 r t 1 and it verifies that the underly- the metric potential eλ and eν are plotted in Fig. 1(left) and ing fluid distribution is in-exotic in nature [53]. The profile (middle). We can see that both the metric potentials are mono- of the mass function and gravitational redshift are shown tonic increasing function of ‘r’. For different values of ‘n’, the in Fig. 4(top) and (bottom) respectively. The mass function ( ) = numerical values of the metric potentials are obtained in table is monotonic increasing of ‘r’ and m r r=0 0 and the 2. The profile of the matter density ρ is shown in Fig. 1(right) gravitational redshift is monotonic decreasing function of

λ ν Table 2 The numerical values of the metric potential e and e are obtained by ﬁxing b = 0.0095 for a compact star of mass M = 1.85M and radius = 9.5km for different values of n r = . r = . r = . r = . r = . r = . Metric potential n r 0 0 r 0 2 r 0 4 r 0 6 r 0 8 r 1 0 eλ 51 1.054 1.21601 1.48602 1.86404 2.35003 50 1 1.05402 1.21607 1.48615 1.86419 2.35003 3000 1 1.05498 1.21988 1.49413 1.87452 2.35003 − 250 1 1.05392 1.21569 1.48534 1.86315 2.35003 − 500 1 1.05384 1.21537 1.48467 1.86228 2.35003 − 2500 1 1.0532 1.21283 1.47935 1.85537 2.35003 eν 50.186759 0.18676 0.186763 0.186767 0.186774 0.186782 50 0.186733 0.186734 0.186737 0.186742 0.186748 0.186757 3000 0.185049 0.18505 0.185053 0.185057 0.185064 0.185073 − 250 0.186904 0.186905 0.186908 0.186913 0.186919 0.186928 − 500 0.187047 0.187047 0.18705 0.187055 0.187061 0.18707 − 2500 0.188183 0.188184 0.188187 0.188192 0.188198 0.188207

− Table 3 Matter density ρ in MeV fm 3 unit are obtained by ﬁxing b = 0.0095 for a compact star of mass M = 1.85M and radius = 9.5km.for different values of n r = . r = . r = . r = . r = . r = . n r 0 0 r 0 2 r 0 4 r 0 6 r 0 8 r 1 0

5 1353.59 1353.58 1353.53 1353.46 1353.35 1353.22 50 1353.96 1353.94 1353.9 1353.82 1353.72 1353.58 3000 1378.08 1378.07 1378.02 1377.94 1377.84 1377.7 − 250 1351.53 1351.51 1351.47 1351.39 1351.29 1351.15 − 500 1349.5 1349.49 1349.45 1349.37 1349.27 1349.13 − 2500 1333.44 1333.43 1333.38 1333.31 1333.21 1333.08

123 138 Page 8 of 13 Eur. Phys. J. C (2019) 79 :138

Fig. 1 (Left) In this figure, along x-axis, y-axis and z-axis we have dimensionless quantity. (Right) Along x-axis, y-axis and z-axis we have respectively plotted the radius of the star in km unit, the dimensionless respectively plotted the radius of the star in km unit, the dimensionless quantity n which lies in the range − 200 ≤ n ≤ 200 and the met- quantity n which lies in the range − 200 ≤ n ≤ 200 and the matter ric co-efficient eλ which is a dimensionless quantity. (middle) along density ρ in MeV fm−3 unit. All the profiles are drawn for a compact x-axis, y-axis and z-axis we have respectively plotted the radius of star of mass M = 1.85M and radius of the star = 9.5 km. by fixing the star in km unit, the dimensionless quantity n which lies in the b = 0.0095 km−1 range − 200 ≤ n ≤ 200 and the metric co-efficient eν which is a

−3 Fig. 2 (Left) In this figure, along x-axis, y-axis and z-axis we have transverse pressure pt in MeV fm unit. (Right) along x-axis, y-axis respectively plotted the radius of the star in km unit, the dimensionless and z-axis we have respectively plotted the radius of the star in km unit, quantity n which lies in the range −200 ≤ n ≤ 200 and the radial the dimensionless quantity n which lies in the range −200 ≤ n ≤ 200 −3 −3 pressure pr in MeV fm unit. (Middle) along x-axis, y-axis and z-axis and the anisotropic factor in MeV fm unit. All the profiles are we have respectively plotted the radius of the star in km unit, the dimen- drawn for a compact star of mass M = 1.85M and radius of the star sionless quantity n which lies in the range −200 ≤ n ≤ 200 and the = 9.5 km. by fixing b = 0.0095 km−1

‘r’. However Buchdahl [54] gave a clear predictions about may consult [56].) ρc is the central density of the star whose the maximum possible mass of relativistic stars in the form expression has given in (40) and M = m(r = r). For our < 4R M of the limit M 9 . For our present model the ratio of R present model the expression is obtained as, is obtained as 0.287 lies in the range proposed by Buchdahl ∂ M 4πr 3 (1 + b5r 5 )−n [54]. The profiles of ρ− pr ,ρ− pt ,ρ− pr −2pt are shown in = . 2 (53) Fig. 5 and it verifies that all the inequalities stated in Eq. (52) ∂ρc + 8 ρ π 2 ( + 5 5 )−n 3 1 3 c r 1 b r are holds and consequently our model satisfies all the energy conditions. With the help of the graphical representation, in Fig. 6, we have shown that the above inequality is satisfied by our present model. 6 Stability 6.2 Causality condition and method of cracking 6.1 The static stability criterion due to HarrisonÐ ZeldovichÐNovikov For any relativistic anisotropic stellar model, some basic physical conditions should be satisfied, some of them have Based on the work done by Harrison et al. [55] and been discussed in the previous section. One of the most ZeldovichÐNovikov [46], for a star to be stable, it has to important properties among the physical conditions is the ∂ satisfy the inequality M > 0 (for a details discussion one causality condition, i.e., the radial and transverse veloc- ∂ρc 123 Eur. Phys. J. C (2019) 79 :138 Page 9 of 13 138

Fig. 4 (Top) Along x-axis, y-axis and z-axis we have respectively plotted the radius of the star in km unit, the dimensionless quantity n which Fig. 3 (Top) In this figure, along x-axis, y-axis and z-axis we have lies in the range − 200 ≤ n ≤ 200 and mass of the star in km unit. respectively plotted the radius of the star in km unit, the dimensionless (Bottom) Along x-axis, y-axis and z-axis we have respectively plotted quantity n which lies in the range − 200 ≤ n ≤ 200 and the ratio the radius of the star in km unit, the dimensionless quantity n which lies of radial pressure p to the matter density ρ which is a dimensionless r in the range − 200 ≤ n ≤ 200 and the gravitational redshift which is quantity. (bottom) along x-axis, y-axis and z-axis we have respectively also a dimensionless quantity. Both the profiles are drawn for a compact plotted the radius of the star in km unit, the dimensionless quantity n star of mass M = 1.85M and radius of the star = 9.5km by fixing which lies in the range − 200 ≤ n ≤ 200 and the ratio of transverse − b = 0.0095 km 1 pressure pt to the matter density ρ which is a dimensionless quantity. Both the profiles are drawn for a compact star of mass M = 1.85M and radius of the star= 9.5km by fixing b = 0.0095 km−1

of pressure waves vr then the model is potentially stable and this fact is known as “cracking method” and mathematically ity of sound should not exceed the speed of the light, i.e., it implies v2 − v2 < 0 is needed throughout the interior of v2,v2 ≤ 1[57,58]. At the same time, Le Chatelier’s prin- t r r t the stellar conﬁguration for star’s stability. Since we are deal- ciple requires that speed of sound must be positive i.e., ing here very complicated expression for v and v , it is not v2,v2 > 0. Using the above two inequalities one gets, r t r t possible for us to check this inequality by analytically rather 0

Fig. 5 (Left) In this figure, along x-axis, y-axis and z-axis we have in MeV fm−3 unit. (Right) Along x-axis, y-axis and z-axis we have respectively plotted the radius of the star in km unit, the dimension- respectively plotted the radius of the star in km unit, the dimensionless less quantity n which lies in the range − 200 ≤ n ≤ 200 and ρ − pr quantity n which lies in the range − 200 ≤ n ≤ 200 and ρ − pr − 2pt in MeV fm−3 unit. (Middle) Along x-axis, y-axis and z-axis we have in MeV fm−3 unit. All the profiles are drawn for a compact star of mass − respectively plotted the radius of the star in km unit, the dimension- M = 1.85M and radius of the star = 9.5k. by fixing b = 0.0095 km 1 less quantity n which lies in the range − 200 ≤ n ≤ 200 and ρ − pt

values of r ,t are obtained in Table 4. We note that both , > 4 r t 3 everywhere inside the anisotropic stellar model for all values of n.

7 Discussion

In our present paper, a new model of compact star has been obtained by assuming the underlying fluid distribution is anisotropic in nature. The model is obtained by solving the Einstein’s field equations which contains three non linear ordinary differential equations with five unknown functions (ρ, p , p , eλ, eν). So to solve the equations one have to Fig. 6 In this figure, along x-axis, y-axis and z-axis we have respec- r t tively plotted the matter density ρ in MeV fm−3 unit, the dimensionless choose any two of them and by our knowledge of algebra we ∂ M = quantity n which lies in the range − 200 ≤ n ≤ 200 and ∂ρ .The can do it in 5C2 10 different ways. If one choose KroriÐ c λ ν profile is drawn for a compact star of mass M = 1.85M and radius of Barua ansatz [60], both e and e are known functions of = . = . −1 the star 9 5km.byfixingb 0 0095 km ‘r’ and by choosing this ansatz one need not solve any differential equations to model a compact star rather one has to just replace the expressions for the metric co-efficients 6.3 Adiabatic index and has to perform some algebraic calculations. In literature several number of papers have done by choosing KB The adiabatic index for an isotropic fluid sphere was pro- ansatz [53,61Ð68]. Some compact star models are developed posed by Chan et al. [13] given by, by choosing metric co-efficients eλ along with a reasonable ρ ρ + p dp equation of state (relation between the matter density and = , (56) radial pressure p )[69Ð74], by choosing eλ along with p p dρ r r [75Ð77]. and the above definition for anisotropic fluid sphere changes In present paper we obtain a new class of compact star as, model in embedding class one spacetime. The study has been done by choosing a new expression for eλ and the reason of ρ + pr dpr ρ + pt dpt = and = (57) adopting such an metric potential has been discussed earlier. r p dρ t p dρ. r t Since our spacetime satisfy the Karmakar’s condition the Bondi [59] proposed that, >4/3, is the condition for other metric co-efficient has been obtained very easily from the stability of a Newtonian sphere and = 4/3 being the a well known relation, as in case of embedding class-I space- condition for a neutral equilibrium. The profile of the adia- time the two metric co-efficients are related to each other batic index r and t are shown in Fig. 8 and the numerical by a differential equation. So this model has been obtained 123 Eur. Phys. J. C (2019) 79 :138 Page 11 of 13 138

Fig. 7 (Left) Along x-axis, y-axis and z-axis we have respectively dimensionless quantity. (Right) Along x-axis, y-axis and z-axis we have plotted the radius of the star in km unit, the dimensionless quantity respectively plotted the radius of the star in km unit, the dimensionless n which lies in the range − 200 ≤ n ≤ 200 and the radial velocity quantity n which lies in the range − 200 ≤ n ≤ 200 and the difference dpr dpt − dpr of sound dρ which is a dimensionless quantity. (Middle) Along x- between transverse and radial velocity of sound dρ dρ which axis, y-axis and z-axis we have respectively plotted the radius of the is a dimensionless quantity. All the proﬁles are drawn for a compact star in km unit, the dimensionless quantity n which lies in the range star of mass M = 1.85M and radius of the star = 9.5km by ﬁxing −1 − ≤ ≤ dpt b = 0.0095 km 200 n 200 and the transverse velocity of sound dρ which is a

without assuming any equation of state (a relation between the matter density and the pressure). But we have shown the relation between the pressure and density through graphical representation in Fig. 9. Figure 9 shows that pr vs. ρ and pt vs. ρ follow almost linear equation of state. The value of the constant a and both central density ρc and central pressure pc increase as n increases on the contrary the values of the integrating constant A decreases as n increases and B remains unchanged for all values of n. In 2008, Herrera et al. [79] proposed an algorithm to obtain static spherically symmetric anisotropic solutions of Einstein’s ﬁeld equations and they showed that this solution can be generated from EFEs by two generating functions ς(r), (r) given as, ν( ) 2 e r = exp 2ς(r) − dr , (58) r (r) = pr − pt . (59)

For our present model, these two generating functions are obtained as,

− n Fig. 8 (Top) In this figure, along x-axis, y-axis and z-axis we have 1 2aBr(1 + b5r 5) 2 ς(r) = + (60) respectively plotted the radius of the star in km unit, the dimensionless r 2A + aBr2φ (r) quantity n which lies in the range − 200 ≤ n ≤ 200 and the radial 1 (r) =−. adiabatic index r which is a dimensionless quantity. (Bottom) Along (61) x-axis, y-axis and z-axis we have respectively plotted the radius of the star in km unit, the dimensionless quantity n which lies in the range The most important characteristic of the present study is that − 200 ≤ n ≤ 200 and the transverse adiabatic index t which is a dimensionless quantity. Both the profiles are drawn for a compact all the results have been discussed through 3- dimensional − ≤ ≤ star of mass M = 1.85M and radius of the star= 9.5km by fixing analysis when 200 n 200. Similar plot can be drawn b = 0.0095 km−1 for n > 200 as well as n < −200. 123 138 Page 12 of 13 Eur. Phys. J. C (2019) 79 :138

Table 4 The numerical values of r and t are obtained by ﬁxing b = 0.0095 for a compact star of mass M = 1.85M and radius = 9.5km.for different values of n r = . r = . r = . r = . r = . r = . Adiabatic index n r 0 0 r 0 2 r 0 4 r 0 6 r 0 8 r 1 0

r 5 1.48667 1.48667 1.4867 1.48674 1.48679 1.48686 50 1.48667 1.48668 1.4867 1.48674 1.4868 1.48687 3000 1.48722 1.48723 1.48726 1.4873 1.48735 1.48742 − 250 1.48662 1.48663 1.48665 1.48669 1.48675 1.48682 − 500 1.48658 1.48659 1.48661 1.48665 1.4867 1.48677 − 2500 1.48627 1.48628 1.4863 1.48634 1.48639 1.48646

t 51.14873 1.14873 1.14873 1.14874 1.14875 1.14876 50 1.14861 1.14861 1.14862 1.14862 1.14863 1.14864 3000 1.14093 1.14093 1.14093 1.14094 1.14095 1.14096 − 250 1.14939 1.1494 1.1494 1.14941 1.14941 1.14943 − 500 1.15005 1.15005 1.15005 1.15006 1.15007 1.15008 − 2500 1.15529 1.15529 1.1553 1.1553 1.15531 1.15532

n 100 pr References 0.007 0.006 0.005 1. S.L. Shapiro, S.A. Teukolsky, Black Holes, White Dwarfs and Neu- 0.004 0.003 tron Stars: The Physics of Compact Objects (Cornell University, 0.002 Ithaca, 1983) 0.001 ρ 2. M. Ruderman, Annu. Rev. Astron. Astrophys. 10, 427Ð476 (1972) 0.020 0.025 0.030 0.035 0.040 0.045 3. R.L. Bowers, E.P.T. Liang, Astrophys. J. 188, 657 (1974) n 100 4. R.F. Sawyer, Phys. Rev. Lett. 29(6), 382Ð385 (1972) pt 5. R. Kippenhahn, A. Weigert, Stellar Structure and Evolution 0.007 (Springer, New York, 1990) 0.006 6. A.I. Sokolov, Sov. Phys. JETP 52(4), 575Ð576 (1980) 0.005 0.004 7. A. Putney, ApJL 451, L67 (1995) 0.003 ρ 8. D. Reimers, S. Jordan, D. Koester, N. Bade, T. Kohler, L. Wisotzki, 0.020 0.025 0.030 0.035 0.040 0.045 Astron. Astrophys. 311, 572Ð578 (1996) 9. A.P. Martinez, R.G. Felipe, D.M. Paret, Int. J. Mod. Phys. D 19, Fig. 9 In this figure we have shown the variation of the radial and 1511 (2010) transverse pressure with respect to the matter density. (Top) Along x- 10. L. Herrera, N.O. Santos, Phys. Rep. 286, 53 (1997) axis and y-axis we have respectively plotted the matter density ρ in 11. L. Herrera, A. Di Prisco, J. Martin, J. Ospino, N.O. Santos, O. −3 −3 MeV fm unit and radial pressure pr in MeV fm unit. (Bottom) Troconis, Phys. Rev. D 69, 084026 (2004) Along x-axis and y-axis we have respectively plotted the matter density 12. S.D. Maharaj, R. Maartens, Gen. Relativ. Gravit. 21(9), 899Ð905 −3 −3 ρ in MeV fm unit and transverse pressure pt in MeV fm unit. Both (1989) the profiles are drawn for a compact star of mass M = 1.85M and 13. R. Chan, L. Herrera, N.O. Santos, Mon. Not. R. Astron. Soc. 265, − radius of the star= 9.5km by fixing b = 0.0095 km 1. Both the figures 533 (1993) are drawn for n = 100 14. F. Rahaman, M. Jamil, R. Sharma, K. Chakraborty, Astrophys. Space Sci. 330, 249 (2010) 15. P. Bhar, F. Rahaman, S. Ray, V. Chatterjee, Eur. Phys. J. C 75, 190 (2015) Data Availability Statement This manuscript has no associated data 16. K. Nozari, S.H. Mehdipour, JHEP 0903, 061 (2009) or the data will not be deposited. [Authors’ comment: There is no data.] 17. S.H. Mehdipour, Eur. Phys. J. Plus 127, 80 (2012) 18. S. Thirukkanesh, F.C. Ragel, Pramana J. Phys. 78, 687 (2012) Open Access This article is distributed under the terms of the Creative 19. H. Heintzmann, W. Hillebrandt, Astron. Astrophys. 38, 51 (1975) Commons Attribution 4.0 International License (http://creativecomm 20. R. Sharma et al., Gen. Relativ. Gravit. 33, 999 (2001) ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 21. X.Y. Lai, R.X. Xu, Astropart. Phys. 31, 128 (2009) and reproduction in any medium, provided you give appropriate credit 22. R.X. Xu et al., Chin. Phys. Lett. 18, 837 (2001) to the original author(s) and the source, provide a link to the Creative 23. M. Azam et al., Chin. Phys. Lett. 33, 070401 (2016) Commons license, and indicate if changes were made. 24. C. Alcock et al., Astrophys. J. 310, 261 (1986) Funded by SCOAP3. 25. P. Haensel et al., Astron. Astrophys. 160, 121 (1986)

123 Eur. Phys. J. C (2019) 79 :138 Page 13 of 13 138

26. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) 52. R.L. Bowers, E.P.T. Liang, Astrophys. J. 188, 657 (1974) 27. R.P. Kerr, Phys. Rev. Lett. 11, 237 (1963) 53. F. Rahaman et al., Phys. Rev. D 82, 104055 (2010) 28. P. Bhar, K.N. Singh, N. Sarkar, F. Rahaman, Eur. Phys. J. C 77, 54. H.A. Buchdahl, Phys. Rev. D 116, 1027 (1959) 596 (2017) 55. B.K. Harrison, K.S. Thorne, M. Wakano, J.A. Wheeler, Gravitation 29. P. Bhar, Eur. Phys. J. Plus 132, 274 (2017) Theory and Gravitational Collapse (1965) 30. P. Bhar, K.N. Singh, F. Rahaman, N. Pant, S. Banerjee, Int. J. Mod. 56. P.Haensel, A.Y.Potekin, D.G. Yakovlev, Neutron Stars 1: Equation Phys. D 26, 1750078 (2017) of State and Structure, vol. 1 (Springer, Berlin, 2007) 31. P. Bhar, S.K. Maurya, Y.K. Gupta, T. Manna, Eur. Phys. J. A 52, 57. L. Herrera, Phys. Lett. A 165, 206 (1992) 312 (2016) 58. H. Abreu, H. Hernandez, L.A. Nunes, Class. Quantum Gravity 24, 32. K.N. Singh, P. Bhar, N. Pant, Int. J. Mod. Phys. D 25(11), 1650099 4631 (2007) (2016) 59. H. Bondi, Proc. R. Soc. Lond. A 281, 39 (1964) 33. P. Bhar, K.N. Singh, T. Manna, Int. J. Mod. Phys. D 26, 1750090 60. K.D. Krori, J. Barua, J. Phys. A Math. Gen. 8, 508 (1975) (2017) 61. P. Bhar, F. Rahaman, A. Jawad, S. Islam, Astrophys. Space Sci. 34. P. Bhar, M. Govender, Int. J. Mod. Phys. D 26, 1750053 (2017) 360, 32 (2015) 35. K.N. Singh, P. Bhar, F. Rahaman, N. Pant, M. Rahaman, Mod. 62. P. Bhar, Astrophys. Space Sci. 357, 46 (2015) Phys. Lett. A 32, 1750093 (2017) 63. P. Bhar, Astrophys. Space Sci. 356, 365Ð373 (2015) 36. K.N. Singh, P.Bhar, N. Pant, Astrophys. Space Sci. 361, 339 (2016) 64. P. Bhar, Astrophys. Space Sci. 356, 309Ð318 (2015) 37. S. Thakadiyil, M.K. Jasim, Int. J. Theor. Phys. 52, 3960 (2013) 65. V. Varela, F. Rahaman, S. Ray, K. Chakraborty, M. Kalam, Phys. 38. K.N. Singh, N. Pant, N. Pradhan, Astrophys. Space Sci. 361, 173 Rev. D 82, 044052 (2010) (2016) 66. P. Bhar, M. Govender, R. Sharma, Eur. Phys. J. C 77, 109 (2017) 39. S.N. Pandey, S.P. Sharma, Gen. Relativ. Gravit. 14, 113 (1981) 67. M. Kalam, F. Rahaman, S. Monowar Hossein, S. Ray, Eur. Phys. 40. K. Schwarzschild, Sitz. Deut. Akad. Wiss. Math. Phys. Berlin 24, J. C 73, 2409 (2013) 424 (1916) [Scbw Int] 68. M. Hossein, F. Rahaman, J. Naskar, M. Kalam, S. Ray, Int. J. Mod. 41. M. Kohler, K.L. Chao, Z. Naturforchg 20, 1537 (1965) Phys. D 21, 1250088 (2012) 42. S. O’Brien, J.L. Synge, Jump Conditions at Discontinuities in 69. P. Bhar, F. Rahaman, R. Biswas, H.I. Fatima, Commun. Theor. General Relativity, in Communications of the Dublin Institute for Phys. 62, 221Ð226 (2014) Advanced Studies, Dublin (1952) 70. P. Bhar, Eur. Phys. J. C 75, 123 (2015) 43. E.H. Robson, Ann. Inst. Henri Poincare 16, 41 (1972) 71. P.Bhar, M.H. Murad, N. Pant, Astrophys. Space Sci. 359, 13 (2015) 44. W. Israel, Nuovo Cimento B 44, 48 (1966) 72. P. Bhar, Astrophys. Space Sci. 359, 41 (2015) 45. W. Israel, Nuovo Cimento B 48, 463 (1967) (Erratum) 73. P. Bhar, M.H. Murad, Astrophys. Space Sci. 361, 334 (2016) 46. Y.B. Zeldivich, I.D. Navikov, Relativistic Astrophysics, vol 1, Stars 74. P. Bhar, K.N. Singh, N. Pant, Indian J. Phys. 91(6), 701Ð709 (2017) and Relativity (University of Chicago Press, Chicago, 1971) 75. P. Bhar, K.N. Singh, T. Manna, Astrophys. Space Sci. 361, 284 47. H.A. Buchdahl, Phys. Rev. 116, 1027 (1959) (2016) 48. N. Straumann, General Relativity and Relativistic Astrophysics 76. P. Bhar, B.S. Ratanpal, Astrophys. Space Sci. 361, 217 (2016) (Springer, Berlin, 1984) 77. R. Sharma, B.S. Ratanpal, Int. J. Mod. Phys. D 22, 1350074 (2013) 49. C.G. Böhmer, T. Harko, Class. Quantum Gravity 23, 6479 (2006) 78. R. Sharma, S.D. Maharaj, Mon. Not. R. Astron. Soc. 375, 1265Ð 50. M.K. Gokhroo, A.L. Mehra, Gen. Relativ. Gravit. 26, 75 (1994) 1268 (2007) 51. B.V. Ivanov, Phys. Rev. D 65, 104011 (2002) 79. L. Herrera, J. Ospino, A.D. Prisco, Phys. Rev. D 77, 027502 (2008)

123