In 1915 Einstein proposed General Theory of Relativity and the famous field equations. In the year 1916 Schwarzschild solved Einstein’s field equation for the interior of compact star in Hydrostatic equilibrium. After that have been investigated the relativistic astrophysical objects relating to static isotropic and anisotropic stellar type configuration in both theoretical and observational point of view. Exploration of astrophysical may provide an explanation to the queries about the fundamental laws of nature dealing with cosmic events and with the properties of matter under extreme conditions.

The study of compact object is always an interesting topic to the researchers. Theoretically there exist a number of comprehensive collections^[1] of static, spherically symmetric solutions. These studies reveal that out of 127 published solutions only 16 of these satisfy all the physical conditions. According to Ruderman^[2] it is found that the density of the core beyond the density ∼ 10¹⁵ gm/cc. provides the anisotropic nature of the stellar object i.e., the pressure can be decomposed into two parts: radial pressure p_r and transverse pressure p_t, where p_t is in the orthogonal to p_r. Anisotropy can be introduced by the existence of a solid stellar core or by the presence of a type 3A superfluid as indicated by Ref. [3], different kinds of phase transitions^[4] or pion condensation.^[5] According to Dev and Gleiser^[6] it is revealed that pressure anisotropy affects the physical properties, stability and structure of stellar matter. The stability of stellar bodies is improved for positive measure of anisotropy when compared to configurations of isotropic stellar objects.^[7] A natural question concerned to spherically symmetric relativistic static objects is to determine an upper bound on the compactness ratio m/r, where m is the ADM mass and r is the radius of the boundary of the static object. Reference [8] shows that a spherically symmetric isotropic object for which the energy density is non-increasing outwards satisfies the bound m/r ˂ 4/9. Since this bound was obtained for a class of isotropic fluid sphere violating the dominant energy condition, several researchers have worked in this area to find a sharp bound of maximum allowable ratio of mass to the radius for anisotropic spheres satisfying all the energy conditions. Sharp bounds on 2m/r for static spherical objects under a variety of assumptions on the eigenvalues of the Einstein tensor was obtained by Ref. [9]. Bounds on m/r for static objects with a positive cosmological constant was obtained by Ref. [10].

The consequence of mathematical modelling for the formation of compact object in cosmological context has great interest in recent time. It is very difficult to solve the Einstein’s field equation (EFE) due to its highly non-linear nature. Though it is unknown to us how the pressure related to the matter density inside a compact object, several researchers used linear and non-linear relation between pressure and matter density to find the solutions of the EFE. Reference [11] provided a new class of interior solution of a (2+1)-dimensional anisotropic star in Finch and Skea space-time corresponding to the BTZ black hole. The model is developed by considering the MIT bag model EOS and a particular ansatz for the metric function g_rr proposed by Ref. [12]. In the background of KB metric^[13] proposed a new Singularity-free anisotropic strange quintessence star model by assuming a linear relationship between the matter density ρ and radial pressure p_r. Reference [14] proposed a new model of dark energy star consisting of five zones, namely, the solid core of constant energy density, the thin shell between core and interior, an inhomogeneous interior region with anisotropic pressures, a thin shell, and the exterior vacuum region. Possibility of higher-dimensional anisotropic compact star was studied by Ref. [15]. Analytical models for quark stars assuming liner equation of state was studied by Ref. [16]. Reference [17] obtained an exact solution for a static anisotropic fluid sphere in static pseudo-spheroidal space-time. A model of compact relativistic body with anisotropic pressures in the presence of the electromagnetic field was studied by Ref. [18] by assuming a linear relationship between the pressure and density. Some recent works on compact star have been performed by Maurya et al.^[19] and Banerjee et al.^[20] by assuming specific mass function and conformal killing vector respectively. Reference [21] studied the behavior of static spherically symmetric relativistic objects with locally anisotropic matter distribution considering the Tolman VII form for the gravitational potential g_rr in curvature coordinates together with the linear relation between the energy density and the radial pressure. Reference [22] proposed a model of super dense star admitting conformal motion in presence of quintessence field by taking Ref. [23] ansatz motivated by accelerating phase of our present universe.

In spite of the execution of different modelling systems, we can apply mathematical modelling to get a relativistic solution of complicated astrophysical problem theoretically. Inspired by all of these earlier works, we obtain a new model of singularity free compact star by assuming a liner equation of state between the matter density and radial pressure and using a new expression for the metric potential g_rr.

The plan of investigation for this paper is as follows: In Sec. 2 Einstein’s field equations have been discussed. We have generated a new model in Sec. 3. We have matched our interior solution to the exterior Schwarzschild vacuum solution in Sec. 4. Physical acceptability conditions and physical analysis are described in the next two sections. Stability of the present model is studied in Sec. 7 and finally some concluding remarks are discussed in Sec. 8.

The line element for a static spherically symmetry 4D space time has the form

Assuming G = 1 = c the Einstein field equations are given by

Introducing a new independent variable x and defining new functions y and Z so that

The field equations (7)–(9), are reduced to a system of three algebraic equations with five unknowns (ρ, p_r, p_t, Z, y) for the anisotropic pressures, which we are going to solve in next section.

We seek solutions to the Einstein’s field equations (7)–(9), by making an explicit choices for the gravitational potential Z and a relation between energy density and pressure on physical grounds. For simplification of the field equations for an anisotropic compact star satisfying a linear EOS was discussed by Sharma & Maharaj^[24] and the same EOS was applied for charged perfect fluid by Ivanov.^[25] Therefore, to obtain a complete solution of the field equations we assume a linear equation of state (EOS) between the radial pressure p_r and the matter density ρ, which is given by

Using Eqs. (7), (8), and (11), we get

Fig. 1

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	Fig. 1 (Color online) Matter density is plotted against r inside the stellar interior by taking a = 915, b = 0.0042, α = 0.25, β = 0.000 258.

Fig. 2

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	Fig. 2 (Color online) Variation of radial and transverse pressure are plotted against r by taking the same values of the constant mentioned in Fig. 1. Green color dashed line represents the transverse pressure and red color solid line represents the radial pressure.

Plugging Eq. (13) into Eq. (12) we get,

The matter density, radial and transverse pressure are obtained as,

The interior spacetime for our proposed model continuously match to the Schwarzschild space time

Using the continuity of e^ν and e^λ across the boundary r = r_Σ,

From Eqs. (15) and (21) we get,

The central density and central pressure is obtained by,

Differentiating Eqs. (1)–(3) with respect to x we get the following equations.

For our model e^λ|_r=0 = 1 and e^ν|_r=0 = B, a positive constant. From Figs. 1 and 2 it is observed that matter density, radial and transverse pressure all are positive and monotonic decreasing function of r. The radial pressure p_r vanishes at the boundary of the star whereas the matter density and transverse pressure are positive at the boundary of the star. The profile of the anisotropic factor Δ is shown in Fig. 3. Figure shows that Δ ˃ 0 for our model, i.e., anisotropic force is repulsive in nature. Moreover at the center of the star Δ vanishes.

Fig. 3

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	Fig. 3 (Color online) Variation of anisotropic factor is plotted against r by taking the same values of the constant mentioned in Fig. 1.

5.2 Mass Radius Relationship

The mass function for our model is obtained as,

by using the relationship,

The compactness factor (u(r)) and surface redshift (z_s(r_Σ)) is obtained as,

The profile of mass function is plotted against r in Fig. 4, which is monotonic increasing and regular inside the stellar interior. The profile of the compactness factor is shown in Fig. 6.

Fig. 4

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	Fig. 4 (Color online) Variation of mass function is plotted against r by taking the same values of the constant mentioned in Fig. 1.

Fig. 5

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	Fig. 5 (Color online) The NEC, WEC, SEC and DEC are plotted against r for the compact star Her X-1 by taking the same values of the constant mentioned in Fig. 1.

Fig. 6

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	Fig. 6 (Color online) Compactness factor is plotted against r by taking the same values of the constant mentioned in Fig. 1.

5.3 Junction Conditions

In this article, the interior solution of a static, anisotropic compact star, governed by a linear equation of state p_r = α_ρ − β with 0 ˂ α ˂ 1 and positive β, is matched with an exterior Schwarzschild vacuum solution with p = ρ = 0 at a junction surface Σ, with junction radius r = R. The metric coefficients are continuous at the junction surface S where r = R, but their derivatives may not be continuous at the junction. We shall now utilize Darmois-Israel^[26–27] formalism to determine the surface stresses at the junction interface. The intrinsic surface stress energy tensor S_ij at the junction surface Σ is obtained from the Lanczos equation^[28–31] in the following form

The discontinuity of the second fundamental form or extrinsic curvatures , across the junction interface is given by,

where the second fundamental form is defined as,

where are the unit normal vector to S and can be written as

with n^νn_ν = 1, where ξⁱ is the intrinsic coordinate in S. The parametric equation for S is given by f(x^α(ξⁱ)) = 0. Here + and − corresponds to respectively exterior (Schwarzschild spacetime) and interior space time of the star. Using Lanczos equation^[28–31] the surface energy density Σ and surface pressure P are obtained as

In the present study of mathematical model of relativistic anisotropic compact stellar model provides some interesting and physically acceptable solutions of the different metric potentials and the other physical parameters like density, pressure, mass etc. of the system.

(i) From Eqs. (16), (17) and (18) we can observe that all the solutions are non singular at the centre (i.e., r = 0).

(ii) We have obtained the value of the constant D using matching condition.

(iii) From the graphical representation, it is found that the matter density, radial pressure (p_r) and transverse pressure (p_t) are all positive and monotonic decreasing function of r.

(iv) Taking a = 915, b = 0.0042, α = 0.25, β = 0.000 258 and using the conditions p_r(r = r_Σ) = 0, we obtain the radius of the star 6.7 km and mass of the star is obtained as 1.148 M_⊙, which is very close to the observational data of compact star Her X-1 (Mass = 1.07 ± 0.36 M_⊙).^[35]

(v) Plugging G and c in the expression of ρ and p_r we obtained the central density and central pressure as 2.6 × 10¹⁵ gm/cc. 2.72 × 10³⁵ dyne/cm², 1.39 × 10¹⁵ gm/cc., which indicates the physical stability of our model.

(vi) From graphical representation, it is found that the anisotropic factor Δ ˃ 0, i.e., anisotropic force is repulsive in nature, which implies that more compact object to be formed as discussed by Ref. [7]. Moreover it vanishes at the center of the star.

(vii) With the help of graphical representation, all energy conditions are satisfied by our proposed model.

(viii) By using the value of parameters we have obtained compactness factor = 0.253 and surface redshift z_s = 0.423.

(ix) Figure 7 shows three forces acting on our model are counterbalancing to keep the system in static equilibrium and condition^[36] is well satisfied.

(x) The adiabatic index Γ ˃ 4/3 and the causality conditions are satisfied by our model, which ensures that our model is potentially stable.

Finally, we can say that all these results that we have obtained for our present investigation of compact stars are theoretically as well as physically admissible. So we can conclude that this model can be used for the description of ultra compact stars in future.