# Vaidya–Tikekar type superdense star admitting conformal motion in presence of quintessence field

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## Abstract

To explain the accelerated expansion of our universe, dark energy is a suitable candidate. Motivated by this concept in the present paper we have obtained a new model of an anisotropic superdense star which admits conformal motions in the presence of a quintessence field which is characterized by a parameter \(\omega _q \) with \(-1<\omega _q<-\frac{1}{3}\). The model has been developed by choosing the Vaidya–Tikekar ansatz (J Astrophys Astron 3:325, 1982). Our model satisfies all the physical requirements. We have analyzed our result analytically as well as with the help of a graphical representation.

### Keywords

Dark Energy Interior Solution Radial Pressure Conformal Killing Null Energy Condition## 1 Introduction

The study of dark matter and dark energy has become a topic of considerable interest in present decades. This study is not only important from a theoretical point of view but also from a physical point of view. The reason is that much observational evidence suggests that the expansion of our universe is accelerating. Dark energy is the most acceptable hypothesis to explain this. Work done based on the cosmic microwave background (CMB) estimated that our universe is made up of 68.3 % dark energy, 26.8 % dark matter and 4.9 % ordinary matter. Dark matter cannot be seen by telescopes but one can infer its evidence from gravitational effects on visible matter and gravitational lensing of background radiation. On the other hand, the evidence of dark energy may be inferred from measures of large scale wave patterns of the mass density. One notable feature of dark energy is that it has a strong negative pressure, i.e., the ratio of pressure to density, which is termed the equation of state parameter \((\omega )\), is negative. The dark energy equation of state is given by \(p=\omega \rho \) with \(\omega <-\frac{1}{3}\). The dark energy star model has been studied by several authors [1, 2, 3, 4, 5, 6, 7]. If we choose \(\omega =-1\) we will get the model of a gravastar [8, 9, 10, 11, 12, 13], which is also a dark energy star. \(\omega <-1\) is for the phantom energy and it violates the null energy condition. Several authors have used a phantom equation of state to describe the wormhole model [14, 15, 16, 17]. Motivated by this previous work we have chosen quintessence dark energy to develop our present model. Here the quintessence field is characterized by a parameter \(\omega _q\) with \(-1<\omega _q<-\frac{1}{3}\). We have assumed that the underlying fluid is a mixture of ordinary matter and a still unknown form of matter i.e. of dark energy type, which is repulsive in nature. These two fluids are non-interacting and we have considered the combined effect of these two fluids in our model. Let us assume that the pressure distribution inside the fluid sphere is not isotropic in nature, but that it can be decomposed into two parts: the radial pressure \(p_\mathrm{r}\) and the transverse pressure \(p_\mathrm{t}\). Here \(p_\mathrm{t}\) is in the perpendicular direction to \(p_\mathrm{r}\) and \(\Delta =p_\mathrm{t}-p_\mathrm{r}\) is defined as an anisotropic factor. The choice of an anisotropic pressure is inspired by the fact that at the core of the superdense star, where the density \(\sim \)10\(^{15}\) gm/cc, the matter distribution shows anisotropy.

In 1982 Vaidya and Tikekar [18] proposed a static spherically symmetric model of a superdense star based on an exact solution of Einstein’s equations. The physical 3-space \(\{t = \mathrm{constant}\}\) of the star is spheroidal, the density of the star is \({\sim }2 \times 10^{14}\) gm/cc, and the mass is about four times the solar mass. Several studies have been performed by using the Vaidya–Tikekar ansatz. Gupta and Kumar have studied a charged Vaidya–Tikekar star in [19]. In this paper the authors have considered a particular form of electric field intensity, which has a positive gradient. This particular form of the electric field intensity was used by Sharma et al. [20]. Komathiraj and Maharaj [21] have also assumed the same expression to model a new type of Vaidya–Tikekar type star. Some new closed form solutions of Vaidya–Tikekar type star were obtained by Gupta et al. [22]. Bijalwan and Gupta [23] have taken a more general form of the electric intensity to obtain a new solution of Vaidya–Tikekar type stars with a charge analog. In this paper the authors have matched their interior solution to the exterior R-N metric and they have analyzed their result numerically by assuming suitable values of the chosen parameter. Some other works on Vaidya–Tikekar stars can be found in [24, 25, 26]. Very recently Bhar [27, 28] proposed a new model of a strange star in the presence of a quintessence field.

In the recent past many researchers have worked on conformal motion. Anisotropic stars admitting conformal motion have been studied by Rahaman et al. [29]. A charged gravastar admitting conformal motion has been studied by Usmani et al. [8]. Relativistic stars admitting conformal motion have been analyzed in [30]. Isotropic and anisotropic charged spheres admitting a one parameter group of conformal motions were analyzed in [31, 32, 33]. A charged fluid sphere with a linear equation of state admitting conformal motion has been studied in [34]. In this paper the authors have also discussed the dynamical stability analysis of the system. Ray et al. [35, 36] have given an electromagnetic mass model admitting a conformal Killing vector. By assuming the existence of a one parameter group of conformal motion Mak and Harko [37] have described a charged strange quark star model. The authors have also discussed conformally symmetric vacuum solutions of the gravitational field equations in brane-world models [38]. In a very recent work Rahaman et al. [39] and Bhar [40] have described conformal motion in higher dimensional spacetimes.

The plan of our paper is as follows: In Sect. 2 we discuss the interior solution and the Einstein field equation. The conformal Killing vector and solution of the system are given in Sects. 3 and 4, respectively. Some physical properties of the model is given in Sects. 5, 6, 7, 8, 9, 10 and 11 and we make some concluding remarks in Sect. 12.

## 2 Interior solutions and Einstein field equation

## 3 Conformal Killing equation

## 4 Solution

## 5 Physical analysis

## 6 Exterior spacetime and matching condition

## 7 TOV equation

## 8 Energy condition

## 9 Stability

For a physically acceptable model one must have the velocity of sound in the range \(0<v^{2}=\frac{\mathrm{d}p}{\mathrm{d}\rho }\le 1\).

## 10 Some features

### 10.1 Mass radius relation

### 10.2 Compactness

### 10.3 Surface redshift

## 11 Junction condition

## 12 Discussion and concluding remarks

In the present paper we have proposed a new model of a superdense star by choosing the Vaidya–Tikekar spacetime which admits CKV in the presence of a quintessence field which is characterized by a parameter \(\omega _q\) with \(-1<\omega _q<-\frac{1}{3}\). For our model the effective density is regular at the center but the radial and transverse pressure suffers from a central singularity like other CKV models. The profile of both the density function and the radial pressure are monotonically decreasing, which indicates that the density and radial pressure of the star is maximum at the center and it decreases from the center to the surface of the star. By taking \(R=10\) and \(K=-0.049\) from the relation \(p_{\mathrm{r~eff}}(a)=0\) (where \(a\) is the radius of the star) we obtain \(a=8. 1\) km. Plugging in \(G\) and \(c\), the central density of the star is calculated as \(\rho _\mathrm{c}=1. 69\times 10^{15}\) gm/cc and \(M_{\mathrm{eff}}=1. 83 M_{\bigodot }\). The mass function is regular at the center and the maximum allowable ratio of mass to radius is \(0. 333<\frac{4}{9}\), which lies in the Buchdahl [48] limit; and the maximum value of the surface redshift is calculated as \(0. 732\). For our model the radial and transverse speed of sound is less than \(1\), which gives the stability condition. According to Herrera’s [44] concept if for a model the radial speed of sound is greater than the transverse speed of sound, the model is potentially stable. With the help of a graphical representation we have shown that \(v_{\mathrm{sr}}^{2}-v_{\mathrm{st}}^{2}>0\). So our model is potentially stable. All the energy conditions are satisfied inside the fluid sphere. Our model is also in static equilibrium under anisotropic, gravitational, and hydrostatic forces. We have matched our interior solution to the exterior Schwarzschild metric in the presence of a thin shell and also obtained the mass of the quintessence star in terms of the thin shell mass. A relation among the radial pressure, surface pressure, and surface density has been obtained.

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