Anisotropic models for compact stars

In the present paper we obtain an anisotropic analogue of Durgapal-Fuloria (1985) perfect fluid solution. The methodology consists of contraction of anisotropic factor $\Delta$ by the help of both metric potentials $e^{\nu}$ and $e^{\lambda}$. Here we consider $e^{\lambda}$ same as Durgapal-Fuloria (1985) whereas $e^{\nu}$ is that given by Lake (2003). The field equations are solved by the change of dependent variable method. The solutions set mathematically thus obtained are compared with the physical properties of some of the compact stars, strange star as well as white dwarf. It is observed that all the expected physical features are available related to stellar fluid distribution which clearly indicate validity of the model.


Introduction
Few decades ago a new analytic relativistic model was obtained by Durgapal and Fuloria [1] for superdense stars in the framework of Einstein's General Theory of Relativity. They showed that the model in connection to neutron star stands all the tests of physical reality with the maximum mass 4.17 M ⊙ and the surface redshift 0.63. Very recently Gupta and Maurya [3] presented a class of charged analogues of superdense star model due to Durgapal and Fuloria [1] under the Einstein-Maxwell spacetimes. The members of this class have been shown to satisfy various physical conditions and exhibit features (i) with the maximum mass 3.2860 M ⊙ and the radius 18.3990 km for a particular interval of the parameter 1 < K ≤ 1.7300, and (ii) with the maximum mass 1.9672 M ⊙ and the radius 15.9755 km for another interval of the parameter 1 < K ≤ 1.1021. Later on a family of well behaved charged analogues of Durgapal and Fuloria [1] perfect fluid exact solution was also a e-mail: sunil@unizwa.edu.om b e-mail: kumar001947@gmail.com c e-mail: saibal@iucaa.ernet.in d e-mail: baijudayanand@yahoo.co.in obtained by Murad and Fatema [4] where they have studied the Crab pulsar with radius 13.21 km.
In a similar way we have considered a generalization of Durgapal and Fuloria [1] with anisotropic fluid sphere such that pr = p t , where pr and p t respectively are radial and tangential pressures of fluid distribution. The present work is a sequel of the paper [5] where we have developed a general algorithm in the form of metric potential ν for all spherically symmetric charged anisotropic solutions in connection to compact stars. However, in the present study without considering any anisotropic function we can develop algorithm by the help of metric potentials only and here lies the beauty of the investigation. Another point we would like to add here that till now, as far as our knowledge is concerned, no alternative anisotropic analogue of Duragapal-Fuloria [1] solution is available in the literature.
In connection to anisotropy we note that it was Ruderman [6] who argued that the nuclear matter may have anisotropic features at least in certain very high density ranges (> 10 15 gm/cm 3 ) and thus the nuclear interaction can be treated under relativistic background. Later on Bowers and Liang [7] specifically investigated the non-negligible effects of anisotropy on maximum equilibrium mass and surface redshift. In this regard several recently performed anisotropic compact star models may be consulted for further reference [8,9,10,11,12,13,14,15,16]. We also note some special works with anisotropic aspect in the physical system like Globular Clusters, Galactic Bulges and Dark Halos in the Refs. [17,18].
As a special feature of anisotropy we note that for small radial increase the anisotropic parameter increases. However, after reaching a maximum in the interior of the star it becomes a decreasing function of the radial distance as shown by Mak and Harko [19,20]. Obviously at the centre of the fluid sphere the anisotropy is expected to vanish.
We would like to mention that algorithm for perfect fluid and anisotropic uncharged fluid is already available in the literature [2,21,22]. As for example, we note that in his work Lake [2,21] has considered an algorithm based on the choice of a single monotone function which generates all regular static spherically symmetric perfect as well as anisotropic fluid solutions under the Einstein spacetimes. It is also observed that Herrera et al. [22] have extended the algorithm to the case of locally anisotropic fluids. Thus we opt for an algorithm to a more general case with anisotropic fluid distribution. However, in this context it is to note that in the Ref. [5] we developed an algorithm in the Einstein-Maxwell spacetimes.
The outline of the present paper can be put as follows: in Sec. 2 the Einstein field equations for anisotropic stellar source are given whereas the general solutions are shown in Sec. 3 along with the necessary matching condition. In Sec. 4 we represent interesting features of the physical parameters which include density, pressure, stability, charge, anisotropy and redshift. As a special study we provide several data sheets in connection to compact stars. Sec. 5 is used as a platform for some discussions and conclusions.

The Einstein field equations
In this work we intend to study a static and spherically symmetric matter distribution whose interior metric is given in Schwarzschild coordinates, x i = (r, θ, φ, t) [23,24] ds 2 = −e λ(r) dr 2 − r 2 (dθ 2 + sin 2 θdφ 2 ) + e ν(r) dt 2 . (1) The functions ν and λ satisfy the Einstein field equations, where κ = 8π is the Einstein constant with G = 1 = c in relativistic geometrized unit, G and c respectively being the Newtonian gravitational constant and velocity of photon in vacua. The matter within the star is assumed to be locally anisotropic fluid in nature and consequently T i j is the energy-momentum tensor of fluid distribution defined by where v i is the four-velocity as e λ(r)/2 v i = δ i 4 , θ i is the unit space like vector in the direction of radial vector, θ i = e λ(r)/2 δ i 1 is the energy density, pr is the pressure in direction of θ i (normal pressure) and p t is the pressure orthogonal to θ i (transverse or tangential pressure).
For the spherically symmetric metric (1), the Einstein field equations may be expressed as the following system of ordinary differential equations [25] − κT 1 where the prime denotes differential with respect to radial coordinate r. The pressure anisotropy condition for the system can be provided as Now let us consider the metric potentials [1] in the following forms: where C is a positive constant and ψ is a function which depends on radial coordinate r. The nature of plots for these quantities are shown in Fig. 1. The above Eq. (7) together with Eqs. (8) and (9) becomes Here our initial aim is to find out the pressure anisotropic function ∆, which is zero at the centre and monotonic increasing for suitable choices of ψ. However, Lake [2] imposes condition that ψ should be regular and monotonic increasing function of radial coordinate r. Let us therefore take the form of ψ as follows: where α > 0. Substituting the value of ψ from Eq. (11) in Eq. (10), we get , the pressure anisotropy is finite as well as positive everywhere as can be seen in Fig. 2.
By inserting the above value of ∆ in the Eq. (12), we get Now our next task is to obtain the most general solution of the differential Eq. (13). Here we shall use the change of dependent variable method. We consider the differential equation of the form Let y = y 1 be the particular solution of the differential Eq. (14). Then y = y 1 U will be complete solution of the differential Eq. (14), where where a 1 and b 1 are arbitrary constants. Again let us consider here that ψ = (1−α+Cr 2 ) 2 = ψαr is a particular solution of Eq. (13). So, the most general solution of the differential Eq. (13) can be given by whereÃ andB are arbitrary constants.
After integrating it, we get and A and B are arbitrary constants with Using Eqs. (8), (12) and (16) the expressions for energy-density and pressure read as where

Matching condition
The above system of equations is to be solved subject to the boundary condition that radial pressure pr = 0 at r = R (where r = R is the outer boundary of the fluid sphere). It is clear that m(r = R) = M is a constant and, in fact, the interior metric (2.1) can be joined smoothly at the surface of spheres (r = R), to an exterior Schwarzschild metric whose mass is same as above i.e. m(r = R) = M [26].
The exterior spacetime of the star will be described by the Schwarzschild metric given by Continuity of the metric coefficients g tt , grr across the boundary surface r = R between the interior and the exterior regions of the star yields the following conditions: where ψ(r = R) = ψ R . Equations (35) and (36) respectively give The radial pressure pr is zero at the boundary (r = R) provides 4 Some physical features of the model

Regularity at centre
The density ρ and radial pressure pr and tangential pressure p t should be positive inside the star. The central density at centre for the present model is The metric Eq. (22) implies that C = 7ρ0 72 is positive finite. Again, from Eq. (20), we obtain where pr(r = 0) > 0. This immediately implies that

Causality conditions
Inside the fluid sphere the speed of sound should be less than the speed of light i.e. 0 ≤ Vsr = dpr dρ < 1 and 0 ≤ V st = dpt dρ < 1. Therefore st is plotted with marker continuous line for Her X-1 (iv) V 2 st is plotted with continuous line for white dwarf where The physical quantities related to the above equations are plotted in Figs. 6 and 7.

Well behaved condition
The velocity of sound is monotonically decreasing away from the centre and it is increasing with the increase of density i.e. d dr ( dpr dρ ) < 0 or ( d 2 pr dρ 2 ) > 0 and d dr ( dpt dρ ) < 0 or ( d 2 pt dρ 2 ) > 0 for 0 ≤ r ≤ R. In this context it is worth mentioning that the equation of state at ultra-high distribution has the property that the sound speed is decreasing outwards [27] as can be observed from Fig. 6.

Energy conditions
The anisotropic fluid sphere composed of strange matter will satisfy the null energy condition (NEC), weak energy condition (WEC) and strong energy condition (SEC), if the following inequalities hold simultaneously at all points in the star: NEC: ρ ≥ 0, WEC: ρ + pr ≥ 0, WEC: ρ + p t ≥ 0, SEC: ρ + pr + 2p t ≥ 0. We have shown the energy conditions in Fig. 8 for Her X-1 under (i) and for white dwarf under (ii).

Case-1:
In order to have an equilibrium configuration the matter must be stable against the collapse of local regions. This requires Le Chateliers principle, also known as local or microscopic stability condition, that the radial pressure pr must be a monotonically non-decreasing function of r such that dpr dρ ≥ 0 [28]. Heintzmann and Hillebrandt [29] also proposed that neutron star with anisotropic equation of state are stable for γ > 4/3 as is observed from Fig. 9 and also shown in Tables 1 and 2 of our model related to compact stars.

Case-2:
For physically acceptable model, one expects that the velocity of sound should be within the range 0 = V 2 si = (dp i /dρ) ≤ 1 [30,31]. We plot the radial and transverse velocity of sound in Fig. 7 and conclude that all parameters satisfy the inequalities 0 = V 2 sr = (dp i /dρ) ≤ 1 and 0 = V 2 st = (dp i /dρ) ≤ 1 everywhere inside the star models. Also 0 = V 2 st ≤ 1 and 0 = V 2 sr ≤ 1, therefore |V 2 sr | ≤ 1. Now, to examine the stability of local anisotropic fluid distribution, we follow the cracking (also known as overturning) concept of Herrera [30] which states that the region for which radial speed of sound is greater than the transverse speed of sound is a potentially stable region.
For this we calculate the difference of velocities as follows: It can be seen that |V 2 sr | at the centre lies between 0 and 1 (see Fig.  10). This implies that we must have 0 ≤ α(10α+4) 50(1−α) 2 ≤ 1. Then α should satisfy the following condition: 0 ≤ α ≤ 52− where M G = M G (r) is the effective gravitational mass which can be given by Substituting the value of M G (r) in Eq. (46), we get Equation (48) basically describes the equilibrium condition for an anisotropic fluid subject to gravitational (Fg), hydrostatic (F h ) and anisotropic stress (Fa) which can, in a compact form, be expressed as where The above forces can be expressed in the following explicit forms: Variation of different forces and attainment of equilibrium has been drawn in Fig. 11.

Effective mass-radius relation and surface redshift
Let us now turn our attention towards the effective mass to radius relationship. For static spherically symmetric perfect fluid star, Buchdahl [32] has proposed an absolute constraint on the maximally allowable mass-to-radius ratio (M/R) for isotropic fluid spheres as 2M/R ≤ 8/9 (in the unit c = G = 1). This basically states that for a given radius a static isotropic fluid sphere cannot be arbitrarily massive. However, for more generalized expression for mass-to-radius ratio one may look at the paper by Mak and Harko [9].
For the present compact star model, the effective mass is written as The compactness of the star is therefore can be given by Therefore, the surface redshift (Z) corresponding to the above compactness factor (u) is obtained as We have shown the variation of physical quantities related to Buchdahl's massto-radius ratio (2M/R) for isotropic fluid spheres and also surface redshift are plotted in Figs. 12 and 13.

Model parameters and comparison with some of the compact stars
In this Section we prepare several data sheets for the model parameters in the following Tables 1-3 and compare those with some of the compact stars, e.g. Strange Practically what we have done in the tables are as follows: In Tables 1-3 values of different physical parameters of Strange star Her X-1 and White dwarf have been provided. Under this data set then we calculate some physical parameters of compact star, say central density, surface density, central pressure etc in Table  4. It can be observed that these data are quite satisfactory for the compact stars whether it is strange star with central density 1.0913 × 10 15 gm/cm −3 or white In the present work we have investigated about an anisotropic analogue of Durgapal-Fuloria [1] and possibilities of interesting physical properties of the proposed model. As a necessary step we have contracted the anisotropic factor ∆ by the help of both metric potentials e ν and e λ . However, e λ is considered here same as Durgapal-Fuloria [1] whereas e ν is that given by Lake [2]. The field equations are solved by the change of dependent variable method and under suitable boundary condition the interior metric (2.1) has been joined smoothly at the surface of spheres (r = R), to an exterior Schwarzschild metric whose mass is same as m(r = R) = M [26]. The solutions set thus obtained are correlated with the physical properties of some of the compact stars which include strange star as well as white dwarf. It is observed that the model is viable in connection to several physical features which are quite interesting and acceptable as proposed by other researchers within the framework of General Theory of Relativity.
As a detailed discussion we would like to put forward here that several verification scheme of the model have been performed and extract expected results some of which are as follows: (1) Regularity at centre: The density ρ and radial pressure pr and tangential pressure p t should be positive inside the star. It is shown that the central density at centre is ρ 0 = ρ(r = 0) = 72C 7 and pr(r = 0) > 0. This means that the density ρ as well as radial pressure pr and tangential pressure p t all are positive inside the star.
(2) Causality conditions: It is shown that inside the fluid sphere the speed of sound is less than the speed of light i.e. 0 ≤ Vsr = dpr dρ < 1, 0 ≤ V st = dpt dρ < 1.
(3) Well behaved condition: The velocity of sound is monotonically decreasing away from the centre and it is increasing with the increase of density as can be observed from Fig. 6.
(4) Energy conditions: From Fig. 9 we observe that the anisotropic fluid sphere composed of strange matter satisfy the null energy condition (NEC), weak energy condition (WEC) and strong energy condition (SEC) simultaneously at all points in the star.
(5) Stability conditions: Following Heintzmann and Hillebrandt [29] we note that neutron star with anisotropic equation of state are stable for γ > 4/3 as is observed in Tables 1 and 2 of our model. Also, it is expected that the velocity of sound should be within the range 0 = V 2 si = (dp i /dρ) ≤ 1 [30,31]. The plots for the radial and transverse velocity of sound in Fig. 7 everywhere inside the star models.
(6) Generalized TOV equation: The generalized Tolman-Oppenheimer-Volkoff equation describes the equilibrium condition for the anisotropic fluid subject to gravitational (Fg), hydrostatic (F h ) and anisotropic stress (Fa). Fig. 8 shows that the gravitational force is balanced by the joint action of hydrostatic and anisotropic forces to attain the required stability of the model. However, effect of anisotropic force is very less than the hydrostatic force.
(7) Effective mass-radius relation and surface redshift: For static spherically symmetric perfect fluid star, the Buchdahl [32] absolute constraint on the maxi-mally allowable mass-to-radius ratio (M/R) for isotropic fluid spheres as 2M/R ≤ 8/9 = 0.8888 is seen to be maintained in the present model as can be observed from the Table 4.
In Sec. 5 we have made a comparative study by using model parameters and data of two of the compact stars which are, in general, very satisfactory as compared to the observational results. However, at this point we would like to comment that the sample data used for verifying the present model are to be increased to obtain more satisfactory and exhaustive features in the realm of physical reality.