Eur. Phys. J. A (2017) 53: 89 DOI 10.1140/epja/i2017-12280-y

THE EUROPEAN PHYSICAL JOURNAL A

Regular Article – Theoretical Physics

Relativistic modeling of compact stars for anisotropic matter distribution S.K. Mauryaa Department of Mathematical and Physical Sciences, College of Arts and Science, University of Nizwa, Nizwa, Oman Received: 18 January 2017 / Revised: 25 February 2017 c Societ` Published online: 11 May 2017 –  a Italiana di Fisica / Springer-Verlag 2017 Communicated by D. Blaschke Abstract. In this paper we have solved Einstein’s field equations of spherically symmetric spacetime for anisotropic matter distribution by assuming physically valid expressions of the metric function eλ and radial pressure (pr ). Next we have discussed the physical properties of the model in details by taking the radial pressure pr equal to zero at the boundary of the star. The physical analysis of the star indicates that its model parameters such as density, redshift, radial pressure, transverse pressure and anisotropy are well behaved. Also we have obtained the mass and radius of our compact star which are 2.29M and 11.02 km, respectively. It is observed that the model obtained here for compact stars is compatible with the mass and radius of the strange star PSR 1937 +21.

1 Introduction The gravitational collapse of stars produces astronomical compact objects with extremely high interior densities like neutron stars, and black holes are the most common end product in the evolution of stars. One of the most fundamental tasks in general relativity is the physical modeling of a relativistic compact star collapsing under its own gravity. After the discovery of pulsars and explanation of their properties by assuming them to be rotating neutron stars, the theoretical investigation of super dense stars has been done using both numerical and analytical methods and the parameters of compact astronomical objects have been worked out by general relativistic treatment. The future of the compact astrophysical objects (the outcome of many astrophysical processes, including supernova explosions and the merger of binaries) is decided by the internal mass distribution of the star. The properties of compact stars are strongly affected by the assumed description of matter in their interiors. Xu et al. [1] and Azam et al. [2] studied the behavior and the physical properties of several compact objects. Alcock et al. [3] and Haensel et al. [4] presented a general scheme for compact astrophysical objects which are not composed of neutron matter, but where, given the conditions of very high density in their interiors, there could rather be a phase transition from nuclear to quark matter. The extreme conditions prevailing in the interior of compact stars seem to be however beyond the reach of the presently planned terrestrial laboratories. So that the a

e-mail: [email protected]

theoretical description of compact star matter is currently one of the most challenging issues of nuclear and particle physics. Despite substantial work, the nature of matter at the extreme densities in the core of compact star remains uncertain. Many scenarios, from nuclear and hadronic matter to exotic states involving Bose-Einstein condensation of pions or kaons, to bulk quark matter and quark matter in droplets, have been proposed. Recent experiments in the last decades on relativistic nuclear collisions at the Brookhaven Relativistic Heavy Ion Collider (RHIC), and at LHC in CERN, probe hot dense matter shedding light on the phenomena of hot plasma formed by dense quarks and gluons [5,6]. On top of the possibilities of getting high-density nuclear matter in terrestrial laboratories, FRIB in USA, FAIR in Germany, J-Parc in Japan and RAON in Korea in the near future, the possible detections of gravitational waves from binary neutron stars (or binary neutron star black holes) are believed to be promising probes of the ultra-high density interior of compact stars. This will open up a new window of study of compact objects, giving us a deeper understanding of the uncertainties in the properties at ultrahigh-densities nuclear matter. The uncertainties in the properties at ultrahigh densities are reflected in uncertainties in the maximum possible mass of a compact star. At present, some of the best-known characteristics of stellar objects are their masses and radii. The microscopic matter distribution in the star is a key factor in determining the relationship between the mass and the radius. In the general-relativistic case the pressure inside compact stars is most likely nearly isotropic, however, various physical phenomena can lead to local anisotropies. At the

Page 2 of 11

final stage of stellar evolution, the normal baryonic matter is intensely compressed by gravity in the core of massive stars during supernova events. In such type of compact stars, the internal pressure could be anisotropic in ultrahigh-density matter because the radial pressure could not be balanced by the tangential pressure. The origin of this anisotropic pressure could arise due to the existence of a solid stellar core or by the presence of type-IIIA superfluid [7,8], phase transitions, pion condensation [9], electromagnetic field [10–12], rotation, etc. As a consequence of these effects, the equation of state (EOS) of the stellar matter becomes essentially anisotropic in an ultrahighdensity situation. So the study of anisotropic fluid spheres in spherically symmetric space-time is a key factor in relativistic astrophysics. Bowers and Liang [13] suggested that a compact star has anisotropic relativistic matter distribution. Anisotropy cannot be neglected in stellar clusters and galaxies, as well as in individual stars. Heintzmann and Hillebrandt [14] presented a simple discussion on the properties of a relativistic, anisotropic neutron star model at high densities by means of several simple assumptions and have shown that for an arbitrary large anisotropy there is no limiting mass for neutron stars, however the maximum mass of a neutron star still lies beyond 3–4 M . In recent years there have been several investigators who extensively studied the dynamical stability of an anisotropic sphere [15–19]. Hillebrandt and Steinmetz [20] numerically studied the stability of an anisotropic sphere in general relativity. Sharma et al. [21] considered the theoretical possibility of anisotropy in strange stars, with densities greater than that of neutron stars but less than that of black holes. From the above motivation we adopt a simple model for pressure anisotropy where the matter stress-energy tensor is diagonal for a spherically symmetric space-time but the tangential pressure differs from the radial one. Chan et al. [22] studied in detail the role played by the local pressure anisotropy and radiation in the onset of dynamical instabilities. They showed that small anisotropies might in principle drastically change the stability of the system. Herrera and Santos [23] have extended the Jeans instability criterion in Newtonian gravity to systems with anisotropic pressure. Esculpi et al. [24] obtained a family of static solutions of the Einsteins field equations with spherical symmetry for an anisotropic fluid with homogeneous energy density. Recent reviews on isotropic and anisotropic fluid spheres can be found in [23,25]. Mak and Harko [26] numerically studied a class of exact solutions of Einstein’s gravitational field equations describing spherically symmetric and static anisotropic stellar-type configurations by assuming a particular form of the anisotropy factor. Moreover Malaver [27,28] has also studied a new class of analytical solutions specifying the form of the gravitational potential Z(x) and of the electrical field for isotropic and anisotropic matter, respectively. Also an important observational parameter, the anisotropy, is studied in detail by Mak and Harko [26] and they have shown that for a dust-filled universe the cosmological evolution always ends into an anisotropic phase, while for highdensity-matter–filled universes, anisotropy can exist not

Eur. Phys. J. A (2017) 53: 89

only in ordinary compact stars but also in hypothetical objects like boson stars and dark-energy star [29,30]. Besides in the isolated objects in astrophysics, anisotropy is present in globular clusters, galactic bulges and dark halos [31–33]. In addition to that, several authors have also obtained solutions of the Einstein field equations in different approaches [34–57]. In this connection, some other useful solutions of Einstein-Maxwell equations can also be seen in refs. [58–64]. The content of the present article is as follows: In sect. 2, we have mentioned the spherical symmetric metric, Einstein field equations and generalized the TOV equation for anisotropic matter distribution. The new solution for the anisotropic stars by taking the metric function and radial pressure has been presented in sect. 3 including the physical properties of metric functions (λ(r) and ν(r)), radial pressure (pr ), tangential pressure (pt ), anisotropy factor (Δ), radial velocity (Vr ), tangential velocity (Vt ) in subsects. 3.1–3.4. In sect. 4, we have determined arbitrary constants by using matching conditions. Some other physical features of the anisotropic model have been mentioned in sect. 5 which is divided into six subsections. The details are as follows: in subsect. 5.1, we have shown dominant energy conditions for anisotropic matter. The equilibrium condition for anisotropic stars by using the Tolman-Oppenheimer-Volkoff equation is given in subsect. 5.2. Next in subsect. 5.3, we have checked the stability of the model by using the adiabatic index, and the stability via cracking concept is discussed in subsect. 5.4. In subsect. 5.5, we have defined the effective mass-radius relation while the surface reshift of compact stars by using compactness u is presented in subsect. 5.6. At last, we have written the conclusion part of the paper in sect. 6.

2 Spherical symmetric line element, Einstein’s field equations and generalized Tolman-Oppenheimer-Volkoff equation for anisotropic fluid distribution 2.1 Spherical symmetric line element Let us consider the static spherically symmetric line element in curvature coordinates (by assuming G = c = 1), ds2 = eν(r) dt2 − eλ(r) dr2 − r2 (dθ2 + sin2 θdφ2 ).

(1)

The energy momentum tensor (T i j ) for the anisotropic fluid distribution can be written as [65]:   (2) T i j = (ρ + pt )v i vj − pt δ i j + (pr − pt )θi θj , where v i is the four-velocity eν(r)/2 v i = δ i 4 , while θi is the unit spacelike vector in the direction of the radial vector, θi = e−λ(r)/2 δ i 1 . However ρ, pr and pt correspond to mass density, radial and tangential pressure, respectively. However the components of T i j are given as follows: T 1 1 = −pr , T 2 2 = T 3 3 = −pt and T 4 4 = ρ.

Eur. Phys. J. A (2017) 53: 89

Page 3 of 11

2.8

2.2 Einstein’s field equations

eλ

2.6

Einstein’s field equation for the anisotropic matter distribution is given by

2.4

2.2

1 −8πT i j = Ri j − Rg i j 2

(3)

where the prime denotes differential with respect to r. 2.3 Generalized Tolman-Oppenheimer-Volkoff equation

eλ

where we suppose G = 1 = c in relativistic geometrized units. Using eq. (2) and eq. (3), we get the following differential equation for the metric (1) [65]:    1 ν 1 + 2 e−λ − 2 , (4) 8πpr = r r r   2 λ ν  ν ν  − λ −λ ν  − + + e (5) 8πpt = 2 4 4 2r    λ 1 −λ 1 8πρ = − 2 e + 2 (6) r r r

2.0 1.8 1.6 1.4 1.2 1.0 0.8

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

r/R λ

Fig. 1. Metric coefficient e versus fractional radius r/R for the compact star PSR 1937 +21. We have employed the data values in this figure as: a = −0.002, b = 0.00823 and R = 11.02 km.

3.1 Physical motivation of eλ

From eqs. (4) and (6), we obtain λ + ν  −λ e (7) 8 π (ρ + pr ) = r    ν ν  λ ν λ 2(1 − e−λ ) dpr = − − 2 − 2 e−λ + . 8π dr r r r r r3 (8) Then by using eqs. (4)-(8), we can write, 2(pt − pr ) ν dpr (ρ + pr ) + = . 2 dr r

(9)

The above eq. (9) represent generalized TolmanOppenheimer-Volkoff (TOV) equation for anisotropic fluid distribution which gives the equilibrium condition for the anisotropic stellar system.

3 New solution for anisotropic compact star In this section, we have to determine five physical unknowns namely λ, ν, pr , pt and ρ (which are given in eqs. (4)–(6)). To solve the system of equations we need to suppose some unknowns due to consistency of the equations. For this purpose we take the following ansatz λ as eλ =

(1 + b r2 )2 , (1 − a r2 )2

(10)

where a, b are constants with a = b = 0. Moreover Takisa et al. [66] have proposed realistic model PSR J1614-2230 with quadratic equation of state by using the metric function eλ = (1 + ar2 )/(1 + br2 ) and Newton et al. [67] have used the same proposed ansatz (10) in embedding class one spacetimes.

For the physical validity of the solution, the metric coefficient eλ must be nonsingular at the centre and positive finite everywhere inside the star. From equation (10), the metric coefficient eλ is 1 at the centre and it is monotonically increasing away from the centre (fig. 1). This implies that our supposition of λ is physically valid. Since the mass density (ρ) is purely dependent on the metric coefficient λ, then from eqs. (6) and (10), we obtain 8πρ =

a(6−2br2 )−a2 r2 (5+br2 )+b(6+3br2 +b2 r4 ) . (1+br2 )3

(11) For realistic physical solution, the matter density must be positive throughout the star and also maximum at the centre. We can see from fig. 2 the density is maximum at centre and decreasing throughout the star. Now our next aim to determine other two unknowns ν and pt . For this purpose, we take the following form of radial pressure pr : 8π pr =

α (1 − b r2 ) , (1 + b r2 )2

(12)

where, α ≥ 0. 3.2 Physical motivation of pr For any physical compact star model, the radial pressure pr should be positive, finite throughout the star and it must also vanish on the boundary of the star (which gives the radius  of the star). From eq. (12), pr will become zero at r =

1 b,

which implies that the radius of the star is

Page 4 of 11

Eur. Phys. J. A (2017) 53: 89

0.0016

Integrating eq. (14) w.r.t. r, we obtain the metric coefficient ν as

ρ

0.0014

A1

eν = A e (1−a r2 ) (1 − a r2 )A2 ,

0.0012

(15)

where A is arbitrary constant of integration and A1 = 2 2 a2 +2 a b+b2 +a α−b α −b α , A2 = b −a . 2 a2 2 a2 The tangential pressure (pt ) and anisotropic factor (Δ = pt − pr ) are given by the following expressions (taking x = r2 ):

0.0010

ρ 0.0008 0.0006 0.0004

8π pt =

p1 + p2 + p3 , 4(1 − a x)2 (1 + b x)3

(16)

8π Δ =

x [ Δ1 + (a + b)2 Δ2 + Δ3 ] , 4(1 + b x)3 (1 − a x)2

(17)

0.0002 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

where

r/R

p1 = α2 x (b x − 1)2 (1 + b x),

Fig. 2. Mass density (ρ) with respect to fractional radius r/R for the compact star PSR 1937 +21. The numerical values of the arbitrary constants can see from the tables 1 and 2.

p2 = (a + b)2 x [12 + 12 b x + 5 b2 x2 + b3 x3 +a2 x2 (5 + b x) − 2 a x (8 + 5 b x + b2 x2 )], p3 = −2 α [a2 x2 − 2 + 2 b3 x3 + b4 x4

0.00040

+2 b x (1 − 4 a x + 2 a2 x2 ) + b2 (x2 − a2 x4 )],

pr

Δ1 = [4 α (2 a + b2 x − a2 x − 2 a b2 x2

0.00035

+a2 b2 x3 ) + α2 (b x − 1)2 (1 + b x)],

0.00030

Δ2 = [12 + 12 b x + 5 b2 x2 + b3 x3 + 5 a2 x2

0.00025

pr

+b a2 x3 − 16 a x − 10 a b x2 − 2 a b2 x3 ],

0.00020

Δ3 = [−2 α (a2 x + 2 b3 x2 + b4 x3 + 2 b −8 a b x + 4 a2 b x2 + b2 x − a2 b2 x3 )].

0.00015 0.00010

3.3 Physical behavior of ν, pt and Δ

0.00005 0.00000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/R Fig. 3. Behavior of radial pressure (pr ) with respect to fractional radius r/R for the compact star PSR 1937 +21. The numerical values of the arbitrary constants can be seen from tables 1 and 2.

 R = 1b . We plot fig. 3 for the radial pressure which shows that pr is maximum as well as positive at the centre and monotonically decreases towards the surface. From eq. (4), the metric coefficient ν  is given by ν =

8π r pr (1 − a r2 )2 2(b + a) r + (b2 − a2 ) r3 − . (1 + b r2 )2 (1 + b r2 )2

r [(b2 − a2 − α b) r2 + (2 b + 2 a + α)] . (1 − a r2 )2

dρ 2r[4 a b (b x − 5)−b2 (15 + 4 b x+b2 x2 )+ρ1 (x)] = , dr (1 + b x)4 (18) dpr 2br α(b x − 3) = , (19) 8π dr (1 + b x)3 dpt 2r [pt1 + pt2 + pt3 + pt4 + pt5 + pt6 + pt7 ] = , (20) 8π dr 4 (1 − a x)3 (1 + b x)4 8π

(13) The value of ν can be obtained by inserting the value of pr form eq. (12) into eq. (13), which is given as ν =

i) Lake [68] has proposed that the metric function eν must increase monotonically with r and be positive finite at the centre. As we can observe from fig. 4, it is monotonically increasing throughout the star and positive finite at the centre. ii) For any physically acceptable anisotropic model, the tangential pressure pt should always be greater than the radial pressure pr throughout the star and must have the same value at the centre. This implies that Δ must be zero at the centre (as pr and pt are equal at the centre) and positive everywhere inside the star. The behaviors of pt and Δ are shown in fig. 5 and fig. 6. Next we have determined the gradient for the pressures and density to discuss other physical properties of the anisotropic star which are given by

(14)

Eur. Phys. J. A (2017) 53: 89

0.40

Page 5 of 11

0.000044

eν

0.000040

0.35

Δ

eν

0.000036

0.30

0.000032

0.25

0.000028

Δ

0.20

0.000024 0.000020

0.15

0.000016

0.10

0.000012

0.000008

0.05

0.000004

0.00

0.000000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/R

r/R ν

Fig. 4. Metric coefficients e versus the fractional radius r/R for the compact star PSR 1937 +21. We have employed the data values of the constants in this figure as: a = −0.002, b = 0.00823 and R = 11.02 km.

0.00040

r/R

gradients of pressures and density

pt

0.00035 0.00030 0.00025

pt

Fig. 6. Anisotropic factor (Δ) versus the fractional radius r/R for the compact star PSR 1937 +21. For this figure, we have taken the same values of the constants a, b and α as in figs. 4 and 5.

0.00020 0.00015 0.00010 0.00005 0.00000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/R Fig. 5. Behavior of tangential pressure (pt ) versus the fractional radius r/R for the compact star PSR 1937 +21. For this figure, we have taken the same values of the constants a, b and α as used in fig. 4.

0.00000 -0.00002

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.00004 -0.00006 -0.00008 -0.00010 -0.00012 -0.00014 -0.00016 dpr/dr dρ dpr , dr dr

dpt/dr

dρ/dr

dpt dr

Fig. 7. Variation of and with fractional radius r/R for the compact star PSR 1937 +21. The numerical values of the constants can be seen in tables 1 and 2.

pt4 = −2 a2 [−6 + 2 α x + b4 x4 − b3 x3 (8 + α x) where

+ b x (20 + 11 α x) − b2 x2 (43 + 28 α x)], 2

x=r , ρ1 (x) = a2 (8 b x + b2 x2 − 5), pt1 = [α2 + b6 x4 − 2 b5 x3 (α x − 2) − 4 b α (4 + k x) +2 α b3 x2 (−5 + 2 α x)], pt2 = [a4 x2 (15 − 44 b x + b2 x2 ) + a5 x3 (−5 + 8 b x +b2 x2 ) + b2 (12 + 4 α x − 2 α2 x2 )], pt3 = b4 x2 (3 − 8 α x + α2 x2 ) − 2 a3 x [10 + b4 x4 +b3 x3 (8 − α x) − b x (32 + α x)],

pt5 = a [b6 x5 − 2 b5 x4 (α x − 4) + k (8 + α x) + b4 x3 (31 − 4 α x + α2 x2 )], pt6 = a [−2 b2 x (10 + 16 α x + α2 x2 ) − 2 b3 x2 (−20 + α x + 2α2 x2 )], pt7 = −10 a3 b2 x3 (5 + 2 α x) + 4 a b (6 + 12 α x + α2 x2 ). As we can see from eqs. (15)–(17), ( dρ dr )r=0 = 0, dpr dpt ( dr )r=0 = 0 and ( dr )r=0 = 0. dpr dpt However from fig. 7, it is clear that dρ dr , dr and dr are negative throughout the star. This implies that pr , ρ

Page 6 of 11

Eur. Phys. J. A (2017) 53: 89

0.60

0.57

Vr

0.59

Vt

0.56

0.58 0.55

vr 0.57

vt 0.54

0.56 0.53

0.55 0.54

0.52 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/R

r/R

Fig. 8. Behavior of radial velocity (left panel) and tangential velocity (right panel) versus the fractional radius r/R for the compact star PSR 1937 +21. The numerical values of the constants can see in tables 1 and 2.

and pt are maximum at the centre and decrease outward (figs. 2, 3 and 5). However the expressions for the velocity of sound can be determined by dpr dpt , vt = . (21) vr = dρ dρ

However the constant A has been determined by using condition e−λ(R) = eν(R) as: A=

(1 − aR2 )2−A2 A1

.

(24)

(1 + bR2 )2 e (1−aR2 ) Now the total mass M can be provided via. condition e−λ(R) = 1 − 2M R as

3.4 Causality condition It is well known that the velocity of sound for any anisotropic model should be less than the velocity of light, i.e. 0 ≤ vr < 1 and 0 ≤ vt < 1 at all points of inside the star. It is observed in fig. 8 that vr and vt are less than 1 everywhere inside the star. This implies that causality condition is also satisfied for this anisotropic solution.

  R (b2 − a2 )R4 + 2(b + a)R2 M= . 2 (1 + bR2 )2

(25)

5 Some other physical features of the anisotropic model 5.1 Dominant energy conditions

4 Matching conditions for anisotropic solution

In this section, we will discuss about the energy conditions for the anisotropic compact star model. For a physically acceptable model, the following conditions must be satisfied throughout the star:

In this section we have determined the arbitrary constants by using the following matching conditions: i) We must join the interior solution of metric (1) with the exterior Schwarzschild solution at the boundary r = i) Null energy condition (NEC): ρ ≥ 0, R, where the exterior Schwarzschild solution is given as ii) Weak-energy condition (WECr ): ρ − pr ≥ 0, 

2M iii) Weak-energy condition (WECt ): ρ − pt ≥ 0, 2 2 2 2 2 2 dt − r (dθ + sin θ dφ ) ds = 1 − r iv) Strong-energy condition (SEC): ρ − pr − 2pt ≥ 0. −1

2M dr2 . (22) It is observed from fig. 9 that all the dominant energy − 1− r conditions are satisfied everywhere within the star. ii) The radial pressure pr must vanish at boundary of the star r = R (second fundamental form), which has same Schwarzschild mass M at r = R [69]. We obtain the radius of star via pr = 0 at r = R as 1 R= √ . b

(23)

5.2 Generalized Tolman-Oppenheimer-Volkoff (TOV) equation The anisotropic stellar system is in stable equilibrium if it satisfies the Tolman-Oppenheimer-Volkoff equation. For

Eur. Phys. J. A (2017) 53: 89

0.0016

0.0014

Page 7 of 11

NEC WECt

0.00006

WEcr SEC

0.00004

Energy conditions

0.0012

0.00002

0.0010 0.0008

Fi

0.00000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.0006

r/R

-0.00002 0.0004

-0.00004

0.0002

0.0000

-0.00006 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/R

-0.00008

Fg

Fh

Fa

Fig. 9. Energy conditions versus the fractional radius r/R for the compact star PSR 1937 +21. For this figure, we have employed the values of the constants as: a = −0.002, b = 0.00823, α = 0.0099 and R = 11.02 km.

Fig. 10. The behavior of different forces Fi versus the fractional radius r/R for the compact star PSR 1937 +21. For plotting this figure, we have taken the values of constants as: a = −0.002, b = 0.00823, α = 0.0099 and R = 11.02 km.

anisotropic fluid distribution the generalized TOV equation can be written as [70,71]

In the above we have used

e

λ−ν 2

dpr MG (ρ + pr ) 2 = 0, + (pr − pt ) + r2 r dr

(26)

Fg1 = [6b + α + b3 r4 + a(6 − 2br2 ) − a2 r2 (5 + br2 ) Fa1

where the effective gravitational mass MG is defined as ν−λ 1 MG (r) = r2 e 2 ν  . 2

Fa2 (27) Fa3

By inserting eq. (27) into eq. (26), we get 2 dpr 1 = 0. − ν  (ρ + pr ) + (pt − pr ) − 2 r dr

(28)

Now the above equation can be represented in terms of different forces which are the gravitational force (Fg ), the hydrostatic force (Fh ) and the anisotropic factor (Fa ). Then the system is under stable equilibrium condition if

+b2 r2 (3 − αr2 )], = [4 α (2 a + b2 x − a2 x − 2 a b2 x2 + a2 b2 x3 ) +α2 (b x − 1)2 (1 + b x)], = [12 + 12 b x + 5 b2 x2 + b3 x3 + 5 a2 x2 + b a2 x3 −16 a x − 10 a b x2 − 2a b2 x3 ], = [−2 α (a2 x + 2 b3 x2 + b4 x3 + 2 b − 8 a b x +4 a2 b x2 + b2 x − a2 b2 x3 )].

The variation of different forces is shown by fig. 10. This variation of different forces indicates that the system is in equilibrium condition due to the fact that the gravitational force is counterbalanced by the joint action of hydrostatic force and anisotropic stress. 5.3 Adiabatic index

Fg + Fh + Fa = 0,

(29)

where the explicit forms of these forces are given by 1 Fg = − ν  (ρ + pr ) 2 r [b2 − a2 − bαr2 + (2b + 2a + α)] Fg1 , (30) =− 16π (1 − ar2 )2 (1 + br2 )3 )   r b α(br2 − 3) dpr = (31) Fh = − dr 4π (1 + br2 )3 and Fa =

2 r [Fa1 + (a + b)2 Fa2 + Fa3 ] (pt − pr ) = . r 4π(1 + br2 )3 (1 − ar2 )2

(32)

It is well known that an anisotropic stellar system is stable under the Newtonian theory for Γi > 4/3. Later on the stability of the anisotropic star has been discussed together with the proposed limit on the adiabatic index Γi by Herrera et al. [72] and Chan et al. [22], which is given by   4 (pt0 − pr0 ) ρ0 pr0 4 + 4π  r , (33) Γr , Γt > + 3 3 |pr0 |r |pr0 | where, pr0 , pt0 , and ρ0 are the initial radial, tangential, and energy density in static equilibrium satisfying eq. (28). The first and last term inside the square brackets are positive quantities and represent the anisotropic and relativistic corrections, respectively, which increase the unstable

Page 8 of 11

Eur. Phys. J. A (2017) 53: 89

7

3.0

6

2.8 2.6

5

2.4 4

Гr

Гt 2.2

3

2.0 2

1.8

1

1.6

0

1.4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

r/R

1

r/R

Fig. 11. Behavior of the adiabatic index Γr (left panel) and Γt (right panel) versus the fractional radius r/R for the compact star PSR 1937 +21. For plotting this figure, we have taken the values of constants as: a = −0.002, b = 0.00823, α = 0.0099 and R = 11.02 km.

range of the adiabatic index. The behavior of the adiabatic index, Γr and Γt , can be seen in fig. 11. Let us note that both Γr , Γt > 43 everywhere inside the star.

0.36

5.4 Stability of anisotropic stars via the cracking method

0.32

0.34

v 2i 0.30

For the anisotropic compact star model, we plot the square of radial (vr2 = dpr /dρ) and transverse (vt2 = dpt /dρ) velocity to examine the stability of the model. Figure 12 shows that vr2 and vt2 satisfy the inequalities 0 ≤ vr2 ≤ 1 and 0 ≤ vt2 ≤ 1 everywhere within the anisotropic stellar object [73,74]. Now to discuss about whether the anisotropic matter distribution is stable or not, we will use the cracking concept of Herrera [73] which states that the region is potentially stable where the radial velocity of sound is greater than the tangential velocity of sound. From fig. 13 we can clearly see that the radial velocity of sound is always greater than the tangential velocity of sound, which implies that there is no change in sign of vr2 −vt2 and vt2 −vr2 throughout the star.

Fig. 12. Variation of the square of radial velocity vr2 and tangential velocity vt2 with the fractional coordinate r/R for PSR 1937 +21. For plotting this figure, we have taken the values of constants as: a = −0.002, b = 0.00823, α = 0.0099 and R = 11.02 km.

5.5 The effective mass-radius relation

5.6 Surface red-shift

It is well known that in any physically acceptable model, anisotropic stars cannot be arbitrary massive. Buchdahl [75] has given the maximum allowable mass-radius ratio for the isotropic case, which is 2M/R < 89 . Furthermore Mak and Harko [26] have also proposed a more generalized expression for this mass-to-radius ratio. The effective mass of a star is given by   2 eλ(R) − 1 Mef f [(b − a2 )R4 + 2(b + a)R2 ] = . = R 2 (1 + bR2 )2 2eλ(R) (34)

For the above mass-radius ratio the compactness of the star can be written as   2 (b − a2 )R4 + 2(b + a)R2 Mef f 1 −λ(R) = [1 − e . ]= u= R 2 2 (1 + bR2 )2 (35) Then the surface redshift corresponding to the above compactness u can be defined as √ 1 − 1 − 2u (b + a)r2 . (36) = eλ(R)/2 − 1 = Zs = √ (1 − ar2 ) 1 − 2u

0.28

0.26 V2r

V2t

0.24 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

r/R

Eur. Phys. J. A (2017) 53: 89

Page 9 of 11 0.000

0

0.045

-0.005

0.040

-0.010

0.035

-0.015

0.030

v 2t − v 2r

v 2r − v 2t

0.050

0.025

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

r/R

-0.020 -0.025

0.020 -0.030

0.015 0.010

-0.035

0.005

-0.040

0.000

-0.045

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 -0.050

r/R

Fig. 13. Variation of vr2 − vt2 (left panel) and vr2 − vt2 (right panel) with the fractional coordinate r/R for PSR 1937 +21. For this graph, the parameter values of constants are as follows: a = −0.002, b = 0.00823, α = 0.0099 and R = 11.02 km.

As we observed from eq. (36) the surface redshift of the compact star depends on the compactness u. The surface redshift increases with the increase of u, however the compactness u satisfies the Buchdahl condition. This implies that the surface redshift of any compact star cannot be arbitrarily large. The behavior of the redshift throughout the star can be determined by the formula Z = e−ν/2 − 1. From fig. 14, it is clear that the redshift attains the maximum at the centre and the minimum at the boundary with corresponding values Z0 = 1.7118 and Zs = 0.6090.

1.8000

Z

1.6000 1.4000 1.2000

Z

1.0000 0.8000 0.6000 0.4000

6 Conclusion In this paper we have obtained the anisotropic solution for the relativistic compact star model by solving Einstein field equations. For this purpose we have taken eλ = (1 + b r2 )2 /(1 − a r2 )2 and radial pressure pr = α(1 − b r2 )/(1 + b r2 )2 ) to solve the Einstein field equations. After that we have discussed the physical properties of the anisotropic solution via values of parameters a and b. In order to examine the suitability of our model to fit into the observational data, we have also obtained the mass and radius of the our compact star model. The mass and radius of anisotropic compact star are M = 2.29M and R = 11.02 km, respectively. We found that this mass and radius quite match with those of the really observed strange star PSR 1937 +21. Further details of the physical properties of anisotropic solution are as follows. The metric potentials eλ and eν are regular at the centre and monotonically increasing (fig. 1 and fig. 4). The variation of radial pressure and density from the centre to the boundary of the star is shown graphically in fig. 2 and fig. 3, respectively. It can be seen that the density and pressures are monotonically decreasing functions of the radial variable r. However both density and pressure attain the maximum value at the center. Moreover It is well known that the anisotropic factor Δ = pt − pr should

0.2000 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r/R Fig. 14. Red-shift (Z) versus the fractional radius r/R for the compact star PSR 1937 +21. The numerical values of compact star is given in tables 1 and 2.

vanish at the origin and at positive throughout the star. In fig. 6, we have shown the variation of anisotropy throughout the star. It has been found that the anisotropy is directed outward because Δ ≥ 0 i.e. pt ≥ pr , which implies that the anisotropic force allows the construction of more massive stars. However from fig. 8, we have found that both sound velocities (radial and tangential) are less than the velocity of light throughout the star while the tangential velocity is decreasing outward. The generalized Tolman-Oppenheimer-Volkoff (TOV) equation defines the equilibrium condition for the anisotropic fluid subject to gravitational force (Fg ), hydrostatic force (Fh ), and anisotropic stress (Fa ). The nature of these forces indicates that the matter distribution inside the compact star is in equilibrium stage. The profiles of

Page 10 of 11

Eur. Phys. J. A (2017) 53: 89

Table 1. Values of the different physical parameters of PSR 1937 +21 for a = −0.002, b = 0.00823, α = 0.0099, M = 2.29M , Solar mass (M ) = 1.475 km, R = 11.02 km [1, 2]. r/R

pr

pt

0.0

3.94 × 10

3.94 × 10

0.1

3.82 × 10

3.83 × 10

0.2

3.50 × 10

3.54 × 10

0.3

3.02 × 10

3.11 × 10

0.4 0.5

4 4 4

4 4 4

ρ

Δ

Vr

Vt

eν

eλ

Z

0.0015

0.0000

0.5881

0.5637

0.1360

1.0000

1.7118

0.0015

9.67 × 10

0.5888

0.5631

0.1378

1.0152

1.6935

0.0014

3.92 × 10

0.5906

0.5617

0.1435

1.0609

1.6399

7 6

0.0012

8.83 × 10

0.5931

0.5598

0.1532

1.1378

1.5550

2.46 × 10

4

2.61 × 10

0.0011

5

1.53 × 10

0.5954

0.5579

0.1674

1.2468

1.4444

1.89 × 104

2.12 × 104

9.06 × 104

2.25 × 105

0.5966

0.5564

0.1866

1.3886

1.3150

0.6

1.36 × 10

1.66 × 10

7.57 × 10

2.94 × 10

0.5956

0.5555

0.2116

1.5640

1.1740

0.7

9.05 × 10

1.26 × 10

6.27 × 10

3.52 × 10

0.5913

0.5550

0.2432

1.7728

1.0278

0.8

5.28 × 10

9.20 × 10

5.18 × 10

3.93 × 10

0.5830

0.5542

0.2823

2.0143

0.8822

0.9

5

2.29 × 10

6.45 × 10

4.28 × 10

4.17 × 10

0.5696

0.5521

0.3297

2.2871

0.7416

0.0000

4.26 × 10

3.55 × 10

4.26 × 10

0.5505

0.5474

0.3863

2.5889

0.6090

4 4

4 5 5

1.0

4

4

4

4

4

5

4

5

4

5

4

6

5 5 5 5 5

Table 2. Central density, surface density and central pressure for PSR 1937 +21 for the above parameter values of table 1 [1, 2]. Compact star candidates PSR 1937 +21

Central density

Surface density

Central pressure

Mass

3

3

2

(M )

35

2.29

gm/cm 2.0073 × 10

15

gm/cm

dyne/cm

4.7916 × 10

4.7857 × 10

14

these forces show that the gravitational force is balanced by the joint action of hydrostatic and anisotropic forces to attain the required stability of the model. However, the effect of an anisotropic force is less than the that of the hydrostatic force (fig. 10). Also for a stable relativistic star, the adiabatic index Γi should be greater than 4/3 throughout the star. We have plotted the graph of Γi against r/R in fig. 11. This graph clearly indicates that Γi > 4/3 throughout the star. On the other hand we have also employed the cracking concept of Herrera to examine the stability of the model. From figs. 12 and 13, it is clear that the radial velocity is dominating the tangential velocity everywhere inside the star and there is no change in sign of vr2 − vt2 and vt2 − vr2 throughout the star, which implies that there is no cracking inside the star. According to the physical interpretation of Buchdahl [75], the maximum allowable compactness for a fluid sphere is given by 2M/R < 8/9. Our compactness value for the chosen model (see table 2) is within this acceptable range and hence provides a stable stellar configuration. Also in connection to the isotropic case and in the absence of the cosmological constant it has been shown that the surface redshift is Z ≤ 2 [76,77]. Moreover, it is observed that in an anisotropic strange star in the presence of a cosmological constant the surface redshift must obey the general restriction Z ≤ 5 [77], whereas the maximum surface redshift for the strange star is 5.211 [78]. Therefore, the redshift for the present anisotropic star without cosmological constant turns out to be Z = 1.7118 and it seems to be within an acceptable range (fig. 14). Other numerical values are well listed in table 1 and table 2. The solution so obtained can be useful to model the interior

Radius (km)

2M/R

11.02

0.6137

structure of anisotropic relativistic objects because it satisfies all necessary physical conditions and requirements and shows as well a good agreement with the observed values. The author acknowledges continuous support and encouragement from the administration of the University of Nizwa. The author is also thankful to the anonymous referee for raising several pertinent issues which have helped to improve the paper substantially.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13.

R.X. Xu et al., Chin. Phys. Lett. 18, 837 (2001). M. Azam et al., Chin. Phys. Lett. 33, 070401 (2016). C. Alcock et al., Astrophys. J. 310, 261 (1986). P. Haensel et al., Astron. Astrophys. 160, 121 (1986). M. Harrison, S.G. Peggs, T. Roser, Annu. Rev. Nucl. Part. Sci. 52, 425 (2002). V. Begun, W. Florkowski, Phys. Rev. C 91, 054909 (2015). R. Kippenhahn, A. Weigert, Steller Structure and Evolution (Springer, Berlin, 1990). A.I. Sokolov, J. Exp. Theor. Phys. 79, 1137 (1980). R.F. Sawyer, Phys. Rev. Lett. 29, 382(E) (1972). D. Reimers et al., Astron. Astrophys. 311, 572 (1996). A.P. Martinez, R.G. Felipe, D.M. Paret, Int. J. Mod. Phys. D 19, 1511 (2010). G.P. Horedt, Polytropes - Applications in Astrophysics and Related Fields, in Astrophysics and Space Science Library, Vol. 306 (Springer, 2004). R.L. Bowers, E.P.T. Liang, Astrophys. J. 188, 657 (1971).

Eur. Phys. J. A (2017) 53: 89 14. H. Heintzmann, W. Hillebrandt, Astron. Astrophys. 38, 51 (1975). 15. J. Binney, S. Tremaine, Galactic dynamics (Princeton University Press, Princeton, 1987). 16. P. Cuddeford, Mon. Not. R. Astron. Soc. 253, 414 (1991). 17. R.W. Michie, Mon. Not. R. Astron. Soc. 125, 127 (1963). 18. K. Dev, M. Gleiser, Gen. Relativ. Gravit. 34, 1793 (2002). 19. M.K. Mak, T. Harko, Proc. R. Soc. Lond. A 459, 393 (2002). 20. W. Hillebrandt, K.O. Steinmetz, Astron. Astrophys. 53, 283 (1976). 21. R. Sharma et al., Gen. Relativ. Gravit. 33, 999 (2001). 22. R. Chan, L. Herrera, N.O. Santos, Mon. Not. R. Astron. Soc. 265, 533 (1993). 23. L. Herrera, N.O. Santos, Phys. Rep. 286, 53 (1997). 24. M. Esculpi et al., Gen. Relativ. Gravit. 39, 633 (2007). 25. M.S.R. Delgaty, K. Lake, Comput. Phys. Commun. 115, 395 (1998). 26. M.K. Mak, T. Harko, Proc. R. Soc. A 459, 393 (2003). 27. M. Malaver, AASCIT Commun. 1, 48 (2014). 28. M. Malaver, Res. J. Model. Simulat. 1, 65 (2014). 29. F.E. Schunck, E.W. Mielke, Class. Quantum. Grav. 20, R301 (2003). 30. A.K. Yadav, F. Rahaman, S. Ray, Int. J. Theor. Phys. 50, 871 (2011). 31. M. Visser, D.L. Wiltshire, Class. Quantum. Grav. 21, 1135 (2004). 32. P.H. Nguyen, J.F. Pedraza, Phys. Rev. D 88, 1135 (2013). 33. P.H. Nguyen, M. Lingam, Mon. Not. R. Astron. Soc. 436, 2014 (2013). 34. L. Herrera, A. Di Prisco, J. Martin et al., Phys. Rev. D 69, 084026 (2004). 35. V. Varela et al., Phys. Rev. D 82, 044052 (2010). 36. F. Rahaman et al., Eur. Phys. J. C 75, 564 (2015). 37. K. Komathiraj, S.D. Maharaj, Int. J. Mod. Phys. D 16, 1803 (2011). 38. M.H. Murad, Astrophys. Space Sci. 361, 20 (2016). 39. M. Malaver, Front. Math. Appl. 1, 9 (2014). 40. M. Malaver, Int. J. Mod. Phys. Appl. 2, 1 (2015). 41. M. Malaver, Open Sci. J. Mod. Phys. 2, 65 (2015). 42. S.K. Maurya, Y.K. Gupta, Phys. Scr. 86, 025009 (2012). 43. S.K. Maurya, Y.K. Gupta, Astrophys. Space Sci. 344, 243 (2013). 44. S.K. Maurya, Y.K. Gupta, Astrophys. Space Sci. 353, 657 (2014). 45. S.K. Maurya, Y.K. Gupta, M.K. Jasim, Rep. Math. Phys. 76, 21 (2015). 46. S.K. Maurya, Y.K. Gupta, S. Ray, B. Dayanandan, Eur. Phys. J. C 75, 225 (2015).

Page 11 of 11 47. S.K. Maurya, Y.K. Gupta, S. Ray, arXiv:1502.01915 (2015). 48. S.K. Maurya, Y.K. Gupta, B. Dayanandan, S. Ray, Eur. Phys. J. C 76, 266 (2016). 49. S.K. Maurya, Y.K. Gupta, B. Dayanandan, M.K. Jasim, A. Al-Jamal, Int. J. Mod. Phys. 26, 1750002 (2017). 50. S.K. Maurya et al., Eur. Phys. J. A 52, 191 (2016). 51. K.N. Singh et al., Int. J. Mod. Phys. D 25, 1650099 (2016). 52. K.N. Singh, N. Pant, Astrophys. Space Sci. 361, 177 (2016). 53. K.N. Singh, N. Pant, arXiv:1607.05971v2 (2016). 54. K.N. Singh et al., Can. J. Phys. 94, 1017 (2016). 55. P. Bhar et al., Astrophys. Space Sci. 360, 32 (2015). 56. P. Bhar et al., Astrophys. Space Sci. 359, 13 (2015). 57. P. Bhar, S.K. Maurya, Y.K. Gupta, T. Manna, Eur. Phys. J. A 52, 312 (2016). 58. S.K. Maurya, Y.K. Gupta, Pratibha, Int. J. Mod. Phys. D 20, 1289 (2011). 59. S.K. Maurya, Y.K. Gupta, Astrophys. Space Sci. 332, 415 (2011). 60. S.K. Maurya, Y.K. Gupta, Int. J. Theor. Phys. 51, 943 (2012). 61. S.K. Maurya, Y.K. Gupta, Nonlinear Anal. Real World Appl. 13, 677 (2012). 62. S.K. Maurya, Y.K. Gupta, S. Ray, S.R. Choudhary, Eur. Phys. J. C 75, 1 (2015). 63. N. Pant, S.K. Maurya, Appl. Math. Comput. 218, 8260 (2012). 64. B.V. Ivanov, Int. J. Theor. Phys. 49, 1236 (2010). 65. D.D. Dionysiou, Astrophys. Space Sci. 85, 331 (1982). 66. P. Mafa Takisa, S.D. Maharaj, Subharthi Ray, Astrophys. Space Sci. 354, 463 (2014). 67. Ksh. Newton Singh, Neeraj Pant, M. Govender, Chin. Phys. C 41, 15103 (2017). 68. K. Lake, Phys. Rev. D 67, 104015 (2003). 69. C.W. Misner, D.H. Sharp, Phys. Rev. B 136, 571 (1964). 70. R.C. Tolman, Phys. Rev. 55, 364 (1939). 71. J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55, 374 (1939). 72. L. Herrera et al., Astrophys. J. 234, 1094 (1979). 73. L. Herrera, Phys. Lett. A 165, 206 (1992). 74. H. Abreu, H. Hernandez, L.A. Nunez, Class. Quantum Grav. 24, 4631 (2007). 75. H.A. Buchdahl, Phys. Rev. 116, 1027 (1959). 76. N. Straumann, General Relativity and Relativistic Astrophysics (Springer Verlag, Berlin, 1984). 77. C.G. Bohmer, T. Harko, Class. Quantum Grav. 23, 6479 (2006). 78. B.V. Ivanov, Phys. Rev. D 65, 104011 (2002).