The search of new solutions for the Einstein-Maxwell field equations is an important area of research because it allows describe compact objects with strong gravitational fields as neutron stars, white dwarfs and quark stars [

In the development of the first stellar models it is important to mention the pioneering research of Schwarzschild [

A great number of exact models from the Einstein-Maxwell field equations have been generated by Gupta and Maurya [

The analysis of compact objects with anisotropic matter distribution is very important, because that the anisotropy plays a significant role in the studies of relativistic spheres of fluid [

In this paper we generated new classes of exact solutions for anisotropic charged distribution with a consistent with quark matter. New models have been obtained by specifying a particular form for one of the metric potentials and for the electric field intensity. The paper has been organized as follows: In section 2, we present the Einstein-Maxwell field equations. In section 3, we have chosen a particular form for the metric potential and for the electric field intensity. In Section.4, physical requirements for the new models are described. In section 5 we present the physical analysis of the new models. In section 6, we conclude.

We consider a spherically symmetric, static and homogeneous spacetime. In Schwarzschild coordinates the metric is given by:

where $\nu (r)$and a$\lambda (r)$are two arbitrary functions.

Using the transformations,$x=C{r}^{2}$, $Z(x)={e}^{-\text{2\lambda}(r)}$ and ${A}^{\text{2}}{y}^{\text{2}}(x)={e}^{\text{2\nu}(r)}$ with arbitrary constants A and c>0, suggested by Durgapal and Bannerji [

$\rho $ is the energy density, ${p}_{r}$ is the radial pressure, $E$ is electric field intensity,${p}_{t}$ is the tangential pressure, σ is the charge density, $\Delta ={p}_{t}-{p}_{r}$ is the measure of anisotropy and dots denote differentiations with respect to x.

With the transformations of [

In this paper, we asume the following linear equation of state in the bag model

where B is the bag constant. We can write the Einstein-Maxwell field equations with the eq. (8) in the following form

The equations (9), (10), (11), (12), (13) governs the gravitational behavior of an anisotropic charged quark star.

In this research, we have chosen the Thirukanesh-Ragel-Malaver ansatz [

This electric field is finite at the centre of the star and remains continuous in the interior. For the case

Using eq. (15) in eq. (8), the radial pressure can be written in the form

and for the mass function we obtain

Substituting (14) and

Integrating eq. (18)

where ${c}_{1}$_{ }is the constant within integration procedures.

For convenience we have let

For the metric functions ${e}^{2\lambda}$, ${e}^{2\nu}$

With eq. (14) and

and the anisotropy F044 can be written as

$\Delta =4xC\left(1-ax\right)\left[\frac{\left({D}^{2}-D\right){a}^{2}}{{\left(ax-1\right)}^{2}}+\frac{Da}{3\left({a}^{2}{x}^{2}-1\right)}+\frac{1-6a}{36{\left(ax+1\right)}^{2}}\right]-\frac{2xD{a}^{2}}{1-ax}-\frac{7ax}{1+ax}$With

replacing eq. (25) in eq. (8), we have for the radial pressure

and the mass function is

Substituting eq. (14) and

Integrating eq. (28)

where ${c}_{2}$is the constant of integration

Again for convenience

The charge density can be written as

and for the metric functions${e}^{2\lambda}$, ${e}^{2\nu}$ and anisotropy F044 we have

For a model to be physically acceptable, the following conditions should be satisfied [

The metric potentials ${e}^{2\lambda}$and ${e}^{2\nu}$assume finite values throughout the stellar interior and are singularity-free at the center

The energy density

The radial pressure also should be positive and a decreasing function of radial parameter.

The radial pressure and density gradients $\raisebox{1ex}{$d{p}_{r}$}\!\left/ \!\raisebox{-1ex}{$dr$}\right.\le 0$ and $\raisebox{1ex}{$d\rho $}\!\left/ \!\raisebox{-1ex}{$dr$}\right.\le 0$for $0\le r\le R$ .

The anisotropy is zero at the center

The interior solution should match with the exterior of the Reissner-Nordstrom spacetime, for which the metric is given by

through the boundary

The conditions (ii) and (iv) imply that the energy density must reach a maximum at the centre and decreasing towards the surface of the sphere.

With

The energy density and radial pressure decrease from the centre to the surface of the star. From eq.(17), the mass function can be written as

and the total mass of the star is

On the boundary

and therefore, the continuity of ${e}^{2\lambda}$and ${e}^{2\nu}$across the boundary r=R is

Then for the matching conditions, we obtain:

For the case

The energy density and radial pressure decrease from the centre to the surface of the star. For the mass function we have

The total mass of the star is

On the boundary

The figures 1, 2, 3, 4, 5, 6 and 7 present the dependence of $\rho $ , $\frac{d\rho}{dr}$ , ${p}_{r}$, $\frac{d{p}_{r}}{dr}$,

* *Energy density against radial coordinate. It has been considered that *n*=1(solid line) *; **n**=2* (long-dash line).

** **Density gradient against radial coordinate. It has been considered that *n*=1(solid line) *; **n**=2* (long-dash line).

Radial pressure against radial coordinate. It has been considered that *n*=1(solid line) *; **n**=2* (long-dash line).

** **Radial pressure against radial coordinate. It has been considered that *n*=1(solid line) *; **n**=2* (long-dash line).

** **Mass function against radial coordinate. It has been considered that *n*=1(solid line) *; **n**=2* (long-dash line).

** **Anisotropy against radial coordinate. It has been considered that *n*=1(solid line) *; **n**=2* (long-dash line).

Charge density against radial coordinate. It has been considered that *n*=1(solid line) *; **n**=2* (long-dash line).

In theFigure

In this paper we have generated new models of anisotropic stars considering the Thirukkanesh-Ragel-Malaver ansatz for the gravitational potential and the MIT bag model equation of state. These models may be used in the description of compact objects in absence of charge and in the study of internal structure of strange quark stars. We show that the developed configuration obeys the physical conditions required for the physical viability of the stellar model. A graphical analysis shows that the radial pressure, energy density, mass function and anisotropy are regular at the origin and well behaved in the interior. The new solutions match smoothly with the Reissner–Nordström exterior metric at the boundary