Analytical model of strange star in the low-mass X-ray binary 4U 1820-30

In this article, we have proposed a model for a realistic strange star under Tolman VII metric\citep{Tolman1939}. Here the field equations are reduced to a system of three algebraic equations for anisotropic pressure. Mass, central density and surface density of strange star in the low-mass X-ray binary 4U 1820-30 has been matched with the observational data according to our model. Strange materials clearly satisfies the stability condition (i.e. sound velocities<1) and TOV-equation. Here also surface red shift of the star has been found to be within reasonable limit.


I. INTRODUCTION
Compact objects are of great attention for a long time. Several researchers [2,3,4,5,6,7,8,9,10,11] investigated compact stars analytically or numerically. Stars, in general, are evolved by burning lighter elements into heavier nuclei from the time of birth. In the end of nuclear burnning white-dwarf, neutron stars, quark stars, dark stars and eventually black holes may formed due to strong gravity. To include the effects of local anisotropy, Bowers and Liang (1974) [12] stressed on the importance of local anisotropic equations of state for relativistic fluid sphere. They showed that anisotropy may have effects on such parameter like maximum equilibrium mass and surface redshift. In stellar system, Ruderman (1972) [13] argued that, in very high density range (∼ 10 15 gm/cc) nuclear matter may have anisotropic features and nuclear interaction should be treated relativistically. Anisotropy in matter indicates that radial pressure (p r ) is not same as the tangential pressure (p t ). A star becomes anisotropic, if its matter density exceeds the nuclear density [12,14,17]. This phenomenon may occur for existence of solid core, phase transition, presence of electromagnetic field etc. 4U 1820-30 resides in the glob- * Electronic address: kalam@iucaa.ernet.in † Electronic address: rahaman@iucaa.ernet.in ‡ Electronic address: sajahan.phy@gmail.com § Electronic address: akjafry@yahoo.com ¶ Electronic address: hossein@iucaa.ernet.in ular cluster NGC 6624. It is an ultra-compact binary and has an orbital period of 11.4 minutes [20]. During Rossi X-ray Timing Explorer(RXTE) observations, it has been obseved that 4U 1820-30 exhibits a super-burst. Possibly this is due to burning of a large mass of carbon [21]. In 1939, Tolman [1] proposed static solutions for a sphere of fluid. In that article, he pointed out that due to complexicity of the VII-th solution (among the eight different solutions), it is not a feasible one for physical consideration(obviously there is a misptint in the article in Tolman VII solution). We have some doubt about the conclusion. We, here, want to check the feasibility of our model by taking the corrected metric. Motivated by the above fact, we are specifically interested for modeling strange star in the low-mass X-ray binary 4U 1820-30. We compare our measurements of mass, radius , central density,surface density and surface red-shifts with the strange star in the low-mass X-ray binary 4U 1820-30 and it is found to be consistent with standard data [22]. We organize our paper as follows: In Sec II, we have provided the basic equations in connection to the Tolman VII metric. In Sec. III, we have studied the physical behaviors of the star namely, Anisotropic behavior, Matching conditions, TOV equations, Energy conditions, Stability and Mass-radius relation & Surface redshift in different sub-sections. The article concluded with a short discussion.

II. INTERIOR SOLUTION
We assume that the interior space-time of a star is described by the metric where R, C, A, B are constants. Such type of metric (1) was proposed by Tolman [1](known as Tolman VII metric)to develop a viable model for a star. We assume that the energy-momentum tensor for the interior of the star has the standard form where ρ is the energy-density, p r and p t are the radial and transverse pressure respectively. Einstein's field equations accordingly are obtained as (c = 1, G = 1) Where and

III. ANALYSIS OF PHYSICAL BEHAVIOUR
In this section we will discuss the following features of the anisotropic strange star : A. Density and Pressure Behavior of the star Now from eqn.(3) and eqn. (6)we get where we have assumed that b is the radius of the star and ρ 0 and ρ b is the matter density at center and surface of the star. Now, we will check, whether at the centre of the star, matter density dominates or not. Here,we see that Clearly, at the centre of the star, density is maximum and it decreases radially outward. Similarly, from Eq.(4), we get dp r dr = 8r Now, at the centre(r=0), dp r dr (r = 0) = 0 Therefore, at the centre, we also see that the radial pressure is maximum and it decreases from the centre towards the boundary. Thus, the energy density and the radial pressure are well behaved in the interior of the stellar structure. Variations of the energy-density and two pressures have been shown in Fig. 1 and Fig. 2, respectively. The anisotropic parameter ∆(r) = (p t − p r ) representing the anisotropic stress is given by Fig.3. The 'anisotropy' will be directed outward when p t > p r i.e. ∆ > 0, and inward when p t < p r i.e. ∆ < 0. It is apparent from the Fig.(3) of our model that a repulsive 'anisotropic' force (∆ > 0) allows the construction of more massive distributions.

B. Matching Conditions
Interior metric of the star should be matched to the Schwarzschild exterior metric at the boundary (r = b). Assuming the continuity of the metric functions g tt , g rr and ∂gtt ∂r at the boundary, we get and (12) Now from equation (11) , we get the compactification factor as

C. TOV equation
For an anisotropic fluid distribution, the generalized TOV equation has the form dp r dr Following [16], we write the above equation as where M G (r) is the gravitational mass inside a sphere of radius r and is given by and e λ(r) = (1 − r 2 R 2 + 4 r 4 A 4 ) −1 which can easily be derived from the Tolman-Whittaker formula and the Einstein's field equations. The modified TOV equation describes the equilibrium condition for the strange star subject to effective gravitational(F g ) and effective hydrostatic(F h ) plus another force due to the effective anisotropic(F a ) nature of the stellar object as where the force components are given by We plot ( Fig. 4 ) the behaviors of pressure anisotropy, gravitational and hydrostatic forces in the interior region which shows sharply that the static equilibrium configurations do exist due to the combined effect of pressure anisotropy, gravitational and hydrostatic forces.

E. Stability
For a physically acceptable model, one expects that the velocity of sound should be within the range 0 ≤ v 2 s = ( dp dρ ) ≤ 1 [17,18]. According to Herrera's [17] cracking (or overturning) condition : The region for which radial speed of sound is greater than the transverse speed of sound is a potentially stable region. In our case(anisotropic strange stars), we plot the radial and transverse sound speeds in Fig.5 and observe that these parameters satisfy the inequalities 0 ≤ v 2 sr ≤ 1 and 0 ≤ v 2 st ≤ 1 everywhere within the stellar object. We also note that v 2 st − v 2 sr ≤ 1. Since, 0 ≤ v 2 sr ≤ 1 and 0 ≤ v 2 st ≤ 1, therefore, | v 2 st − v 2 sr |≤ 1. In Fig.6, we have plotted | v 2 st − v 2 sr |. We notice that v 2 st < v 2 sr throughout the interior region. In other words, v 2 st < v 2 sr keeps the same sign everywhere within the matter distribution i.e. no cracking will occur. These results show that our anisotropic compact stars model is stable.

F. Mass-Radius relation and Surface redshift
In this section, we study the maximum allowable massradius ratio in our model. According to Buchdahl [15], for a static spherically symmetric perfect fluid allowable mass-radius ratio is given by 2Mass Radius < 8 9 . Mak [19] also gave more generalized expression. In our model the gravitational mass in terms of the energy density ρ can be expressed as The compactness of the star is given by The nature of the Mass and Compactness of the star from the centre are shown in Fig. 7 and Fig.8. The surface redshift (Z s ) corresponding to the above compactness (u) is obtained as where Thus, the maximum surface redshift for the anisotropic strange stars of different radius could be found very easily from the Fig. 9. We calculate the maximum surface redshift for our configuration using the numerical values of the parameters as b = 9.5, R = 16.9, A = 24.18 and we get Z s = 0.375. The nature of surface redshift of the star is shown in Fig. 9.

IV. CONCLUSION
In this work we have investigated the nature of anisotropic strange stars in the low-mass X-ray binary 4U 1820-30 by taking following considerations : (a) The stars are anisotropic in nature i.e. p r = p t . (b) The space-time of the strange stars can be described by Tolman VII metric.
The results are quite interesting, which are as follows: (i) Though the radial pressure(p r ) vanishes at the boundary (r = b), tangential pressure(p t ) does not. However, at the centre of the star, it's anisotropic behavior vanishes. (ii) Our model is well stable according to Herrera stability condition [17]. (iii) From mass-radius relation, any interior features of the star can be evaluated. Therefore, our overall observations of anisotropic strange stars under Tolman VII metric satisfies all physical requirements of a stable star. It is to be noted that while solving Einstein's equations as well as for plotting, we have set c=G=1. Now, plugging G and c into relevant equations, the values of the central density and surface density of our strange star turn out to be ρ 0 = .55 × 10 15 gm cm −3 and ρ b = .27 × 10 15 gm cm −3 for the numerical values of the parameters as b = 9.5, R = 16.9, A = 24.18. Also, the mass of our strange star is calculated as 1.01M ⊙ . Interestingly, we observe that the measurement of the mass, radius and central density of our strange star are almost consistent with the strange star in the low-mass X-ray binary 4U 1820-30 [22].
Recently, Cackett et al. [23] reported that the gravitational redshift of strange star in the low-mass X-ray binary 4U 1820-30, based on the modeling of the relativistically broadened iron line in the X-ray spectrum of the source observed with Suzaku is Z s = 0.43. The surface redshift of our strange star with radius 9.5 km turns out to be 0.375. This indicates that the measurement of redshift of our strange star is nearly reliable with the strange star in the low-mass X-ray binary 4U 1820-30.
Finally, we conclude by pointing that spacetime comprising Tolman VII metric with anisotropy may be used to construct a suitable model of a strange star in the low-mass X-ray binary 4U 1820-30.