# Isotropic uncharged model with compactness and stable configurations

Amit Kumar Prasad<EMAIL_ADDRESS>Department of Mathematics, Central
University of Jharkhand,Ranchi-835205 India. Jitendra Kumar
<EMAIL_ADDRESS>Department of Mathematics, Central University of
Jharkhand,Ranchi-835205 India. Ashok Kumar<EMAIL_ADDRESS>Department of
Mathematics, Hemwati Nandan Bhuguna University, Srinagar (Uttarakhand), India.

## Abstract

In present work, we have studied a new stellar distribution model with
spherically symmetric matter and an uncharged isotropic distribution in
general relativity. In this model we have considered a particular metric
potential. The model is capable to represents some known compact stars like
Her X-1,4U 1538-52 and SAX J1808.4-3658.The model satisfy the energy condition
and hydrostatic equilibrium equation, i.e., the modified Tolman-Oppenheimer-
Volkoff (TOV) equation for uncharged matter. In addition to this,we also
present the velocity of sound, surface redshift and pressure density ratio.
The physical quantities such as pressure,density, redshift etc.,are compared
with graphical representations that are important from theoretical and
astrophysical scale.
Keywords:Isotropic Fluids; Compact star; General Relativity.

##  1 Introduction

An analysis of the solution of Einstein field equation shows that the exact
solution plays an important role in development of many areas of gravitational
field such as black hole solution,solar system test,gravitational collapse and
so on. Generally in astrophysics compact stars formed due to gradual
gravitational collapse are considered to fall the three categories, white
dwarfs, neutron stars and black holes. This classification is based on the
internal structure and composition of stars, where the formers contain matter
in one of the densest forms found in the universe. According the strange
matter hypothesis strange quark matter could be more stable than nuclear
matter and thus neutrons star should largely be composed of pure quark matter.
Possible observational signatures associated with theoretically proposed
states of matter inside the compact stars therefore have remained an active
research area in astrophysics , different types of mathematical modeling of
such compact objects being considered.The singularity free interior solutions
of compact object have an important consequences in relativistic
astrophysics.The study of high density object like neutron stars,quark stars
and white dwarfs,form their microscopic composition and properties of dense
matter is one of the most fundamental problem in modern astrophysics.

In general, it is important to measure the mass and radiusBurikham (2015,
2007) of compact stars which depends on the equation of state Ray1 (2004,
2009, 2010, 2012).The motivation to undertake such a task, because the
interior structure of such compact stars can vary with mass.On other hand,
BuchdahlBuchdahl (1959) proposed a method on the mass radius ratio of
relativistic fluid spheres which is an important contribution in order to
study the stability of the fluid spheres.Thus the motivation of this study is
to prediction of mass and radius of compact stars.The mass and radius of
compact stars such as Her X-1,4U 1538-52 and SAX J1808.4-3658 has been
analyzed by Gangopadhyay et algangopadhya (2013a).

The exact solution of Einstein-Maxwell field equations for static isotropic
astrophysical object is of continuous interest to mathematician as well as
physicistsamit (2018, 2018a). A large number of solutions have studied
inPratibha (2011, 2017, 2016, 2016a, 2017a, 2017b, 2015a, 2015b, 2010, 2010a,
2017c, 2012a, 2010b, 2013, 2014) for exact solution of Einstein-Maxwell field
equation.Some pioneer work in relativity is given by IvanovIvanov (2002),Ray
et al.Ray (2003a),Stettnerstettner (1973),Krori and Baruakrori (1975),Ray and
Dasdas (2004a),Pant and Negipant (2012b), FloridesFlorides (1983),
DionysiouDionysiou (1982),Pant et al.pant1 (2011a) etc.gave well behaved
solution for charged fluid sphere.

The equation of state(EOS) is an important features to describe a self
gravitating fluid when it comes to solving the field equations. IvanovIvanov
(2002) has showed that the analytical solutions in the static, spherically
symmetric uncharged case of perfect fluid with linear EOS is an extremely
difficult problem. Sharma and Maharajmaharaj (2007) have demonstrated this
complexity in the case of a static, spherically symmetric uncharged
anisotropic fluid. In the resent model we choose Buchdahl metric and solve the
system of field equations and obtained a linear EOS.

The algorithm for an anisotropic uncharged fluid has been done by Lake and
HerreraLake (2003, 2004, 2008).In this work LakeLake (2003, 2004) has
considered an algorithm and choose a single monotonic function which is
generates a static spherically symmetric perfect fluids solutions of
Einstein’s equation.Against the above studies we have considered an uncharged
isotropic fluid distribution in the context of the formation of the compact
stars and find a new solution in section 2, section 3 consists of physical
conditions for well behaved solutions.In section 4 the matching condition of
interior metric to an exterior Reissner-Nordstrom line element and determine
the constant coefficient. Stability analysis of compact objects and for better
illustration of our result, the relevant physical quantities are presented by
table and figure in section 5.Finally in section 6 we have drawn contains
about present model.

##  2 Field equations for Uncharged Fluid Sphere in Schwarzschild Coordinates

Let us consider the spherically symmetric metric in Schwarzschild Coordinates

$\displaystyle ds^{2}=-e^{\lambda(r)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})+e^{\nu(r)}dt^{2}$ (1)

where $\lambda(r)$ and $\nu(r)$ are the functions of $r$ only.The Einstein
equation for a perfect fluid distribution is given by

$\displaystyle
R^{i}_{j}-\dfrac{1}{2}R\delta^{i}_{j}=-\kappa\big{[}(c^{2}\rho+p)\nu^{i}\nu_{j}-p\delta^{i}_{j}\big{]}$
(2)

where $\kappa=\dfrac{8\pi G}{c^{4}}$,with $G$ and $c$ are the gravitational
constant and speed of light in vacuum respectively. Here $\rho$ and $p$ denote
matter density and fluid pressure respectively. The $\nu^{i}$ is the time-like
4-velocity vector such that

$\displaystyle\nu^{i}\nu_{i}=1$

In view of the metric (1) the Einstein field equations are given by

$\displaystyle\dfrac{\nu^{\prime}}{r}e^{-\lambda}-\dfrac{(1-e^{-\lambda})}{r^{2}}=\kappa
p$ (3)
$\displaystyle\bigg{(}\dfrac{\nu^{\prime\prime}}{2}-\dfrac{\lambda^{\prime}\nu^{\prime}}{4}+\dfrac{\nu^{{}^{\prime}2}}{4}+\dfrac{\nu^{\prime}-\lambda^{\prime}}{2r}\bigg{)}e^{-\lambda}=\kappa
p$ (4)
$\displaystyle\dfrac{\lambda^{\prime}}{r}e^{-\lambda}+\dfrac{(1-e^{-\lambda})}{r^{2}}=\kappa
c^{2}\rho$ (5)

where prime $(^{\prime})$ denotes the differentiation with respect to $r$. Now
we consider a well known form of metric potential,which was proposed by
BuchdahlBuchdahl (1959) of the form as

$\displaystyle e^{\lambda}=\dfrac{K(1+Cr^{2})}{K+Cr^{2}},~{}~{}~{}~{}K>1$ (6)

where $K$ and $C$ are arbitrary constant.The metric function(6) is regular and
non-singular at the center of the star which satisfies the primary physical
requirements for a realistic star. Now using (6),the equations (3)-(5) reduce
to the following form

$\displaystyle\dfrac{(K+Cr^{2})}{K(1+Cr^{2})}\bigg{[}\dfrac{-2y^{\prime}}{ry}+\dfrac{C(K-1)}{K+Cr^{2}}\bigg{]}=-\kappa
p$ (7) $\displaystyle\dfrac{C(K-1)(3+Cr^{2})}{K(1+Cr^{2})^{2}}=\kappa
c^{2}\rho$ (8)
$\displaystyle\dfrac{(K+Cr^{2})}{K(1+Cr^{2})}\bigg{[}\dfrac{y^{\prime\prime}}{y}-\dfrac{y^{\prime}}{ry}+\dfrac{C(K-1)r(Cr-y^{\prime}/y)}{(K+Cr^{2})(1+Cr^{2})}\bigg{]}=0$
(9)

where $e^{\nu}=y^{2}$. Now to solve the equation (9) we introduce the new
variables define by

$\displaystyle X=\sqrt{\dfrac{K+Cr^{2}}{K-1}},~{}~{}~{}~{}K>1$ (10)

and

$\displaystyle y(X)=(X^{2}-1)^{1/4}\Psi(X)$ (11)

Using the equations (10) and (11), the equation(9) reduce to the following
form of second order differential equation

$\displaystyle\dfrac{d^{2}\Psi}{dX^{2}}+I\Psi=0$ (12)

where

$\displaystyle I=\dfrac{2(1-2K)(1-X^{2})-5X^{2}}{4(1-X^{2})^{2}}$ (13)

In order to solve equation (12) more easily if we set $K=\frac{7}{4}$, then
the equation (12) takes the form

$\displaystyle\Psi^{\prime\prime}-\dfrac{5}{4(1-X^{2})^{2}}\Psi=0$ (14)

So the solution of (14) is given as

$\displaystyle\Psi(X)=(X+1)\left[A_{1}\bigg{\rvert}\dfrac{X+1}{X-1}\bigg{\rvert}^{(0.25)}+A_{2}\bigg{\rvert}\dfrac{X+1}{X-1}\bigg{\rvert}^{(-1.25)}\right]$
(15)

where $A_{1}$ and $A_{2}$ are arbitrary constants of integration. Now put the
value of $\Psi(X)$ from equation (15) and $X=\sqrt{\dfrac{K+Cr^{2}}{K-1}}$
into the equation (11),we get

$\displaystyle
y(r)=\left[\dfrac{4(1+Cr^{2})}{3}\right]^{1/4}\Big{(}g(r)+1\Big{)}\Bigg{[}A_{1}\Big{\rvert}F(r)\Big{\rvert}^{(0.25)}+A_{2}\Big{\rvert}F(r)\Big{\rvert}^{(-1.25)}\Bigg{]}$
(16)

where
$g(r)=\sqrt{\frac{7+4Cr^{2}}{3}},~{}~{}~{}F(r)=\dfrac{5+2Cr^{2}+3g(r)}{2(1+Cr^{2})}$
. Therefore the expressions of density and pressure are given by

$\displaystyle\kappa c^{2}\rho=\dfrac{3C(3+Cr^{2})}{7(1+Cr^{2})^{2}}$ (17)
$\displaystyle\kappa
p=\dfrac{2(7+4Cr^{2})}{7ry(r)(1+Cr^{2})}\Bigg{[}N_{1}(r)\Bigg{(}A_{1}\Big{\rvert}F(r)\Big{\rvert}^{(0.25)}+A_{2}\Big{\rvert}F(r)\Big{\rvert}^{(-1.25)}\Bigg{)}+N_{2}(r)N_{3}(r)N_{4}(r)\Bigg{]}-\dfrac{3C}{7(1+Cr^{2})}$
(18)

where
$N_{1}(r)=\dfrac{2Cr}{3}\Bigg{[}\big{(}g(r)+1\big{)}\Bigg{(}\frac{4(1+Cr^{2})}{3}\Bigg{)}^{-3/4}+2\Bigg{(}\frac{4(1+Cr^{2})}{3}\Bigg{)}^{1/4}g(r)\Bigg{]},~{}~{}~{}~{}~{}~{}~{}N_{4}(r)=\dfrac{-8Cr}{3g(r)\big{(}g(r)-1\big{)}^{2}}\\\
N_{2}(r)=\Bigg{(}\frac{4(1+Cr^{2})}{3}\Bigg{)}^{1/4}\big{(}g(r)+1\big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}N_{3}(r)=\Bigg{[}A_{1}(0.25)\Big{\rvert}F(r)\Big{\rvert}^{(-0.75)}+A_{2}(-1.25)\Big{\rvert}F(r)\Big{\rvert}^{(-2.25)}\Bigg{]}\\\
$

Figure 1: Variation of metric potential $e^{\lambda}$(left) and
$e^{\nu}$(right) with respect to fractional radius $r/R$ for compact star SAX
J1808.4-3658,4U 1538-52 and Her X-1. For plotting this figure the numerical
values of physical parameters and constants are as follows:
(i)$K=1.75,CR^{2}=0.4137,M$ =$0.9M_{\odot}$ and $R=14.35\,km$ for SAX
J1808.4-3658,(ii)$K=1.75,CR^{2}=0.4138,M=$ $0.87M_{\odot}$ and $R=13.87\,km$
for 4U 1538-52 ,(iii)$K=1.75,CR^{2}=0.414,M=$ $0.85M_{\odot}$ and
$R=13.548\,km$ for Her X-1

##  3 Physical Features for Well Behaved Solution

  1. 1.

From equation (6),we observe $(e^{\lambda})_{(r=0)}=1$ and
$(e^{\nu})_{(r=0)}>0$. This show that metric potentials are singularity free
and positive at center.It is monotonically increasing with increasing the
radius of the compact star(see Fig.1).

  2. 2.

Pressure $p$ should be zero at the boundary $r=R.$

  3. 3.

$(dp/dr)_{r=0}=0$ and $(d^{2}p/dr^{2})_{r=0}<0,$ so that pressure gradient
$dp/dr$ is negative for $0<r\leq R$.

  4. 4.

$(d\rho/dr)_{r=0}=0$ and $(d^{2}\rho/dr^{2})_{r=0}<0,$ so that density
gradient $d\rho/dr$ is negative for $0<r\leq R$.
The above two conditions imply that pressure and density should be maximum at
the center and monotonically decreasing towards the surface(see Fig.2).

  5. 5.

The velocity of sound $(dp/c^{2}d\rho)^{1/2}$ should be less than that of
light throughout the charged fluid sphere $(0\leq r\leq R)$. This is called
casual condition.

  6. 6.

The ratio of pressure to the density $(p/c^{2}\rho)$ should be monotonically
decreasing with the increasing of $r.$(see Fig.2)

  7. 7.

$c^{2}\rho\geq p>0$ or $c^{2}\rho\geq 3p>0,0\leq r\leq R$,where former
inequality denotes weak energy condition (WEC) and later inequality denotes
strong energy condition (SEC).

Figure 2: Behaviour of pressure(p in $km^{-2}$ left), density($\rho$ in
$km^{-2}$ middle) and $p/\rho$(right) vs. fractional radius r/R for SAX
J1808.4-3658, 4U 1538-52 and Her X-1. For plotting this figure the numerical
values of physical parameters and constants with $G=c=1$ are as follows:
(i)$K=1.75,CR^{2}=0.4137,M$ =$0.9M_{\odot}$ and $R=14.35\,km$ for SAX
J1808.4-3658,(ii)$K=1.75,CR^{2}=0.4138,M=$ $0.87M_{\odot}$ and $R=13.87\,km$
for for 4U 1538-52 ,(iii)$K=1.75,CR^{2}=0.414,M=$ $0.85M_{\odot}$ and
$R=13.548\,km$ Her X-1

##  4 Matching Conditions of Boundary

The solution is smoothly connected to the pressure free boundary with the
Schwarzschild exterior metric

$\displaystyle
ds^{2}=-\bigg{(}1-\dfrac{2M}{r}\bigg{)}^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})+\bigg{(}1-\dfrac{2M}{r}\bigg{)}dt^{2}$ (19)

Beside the above the smooth joining with the Schwarzschild metric which
requires the continuity of $e^{\lambda}$ and $e^{\nu}$ across the boundary
$r=R$ and we get

$\displaystyle e^{-\lambda}=1-\dfrac{2M}{R}$ (20) $\displaystyle
y^{2}=1-\dfrac{2M}{R}$ (21) $\displaystyle p(R)=0.$ (22)

Using the equations (21) and (22),we get the expressions of arbitrary constant
$A_{1}$ and $A_{2}$ as follow

$\displaystyle\dfrac{A_{1}}{A_{2}}=\dfrac{M_{1}(R)N_{1}(R)G(R)+M_{1}(R)N_{4}(R)N_{2}(R)G_{1}(R)-M_{2}(R)N_{2}(R)G(R)}{M_{2}(R)N_{2}(R)G_{2}(R)+M_{1}(R)N_{2}(R)N_{4}(R)G_{3}(R)-M_{1}(R)N_{1}(R)G_{2}(R)}$
(23) $\displaystyle
A_{2}=\dfrac{\sqrt{\frac{7+4CR^{2}}{7(1+CR^{2})}}}{\left[\dfrac{4(1+Cr^{2})}{3}\right]^{1/4}\Big{(}g(R)+1\Big{)}\Bigg{[}\frac{A_{1}}{A_{2}}\Big{\rvert}F(R)\Big{\rvert}^{(0.25)}+\Big{\rvert}F(R)\Big{\rvert}^{(-1.25)}\Bigg{]}}$
(24)

The expression of mass $M$ is given as

$\displaystyle M=\frac{R}{2}\Bigg{[}1-\dfrac{7+4CR^{2}}{7(1+CR^{2})}\Bigg{]}$
(25)

where
$g(R)=\sqrt{\frac{7+4CR^{2}}{3}},~{}~{}~{}F(R)=\dfrac{5+2Cr^{2}+3g(R)}{2(1+CR^{2})},~{}~{}~{}~{}~{}~{}N_{4}(R)=\dfrac{-8}{3g(R)\big{(}g(R)-1\big{)}^{2}}\\\
~{}~{}~{}N_{1}(R)=(2/3)\Bigg{[}\big{(}g(R)+1\big{)}\Bigg{(}\frac{4(1+CR^{2})}{3}\Bigg{)}^{-3/4}+2\Bigg{(}\frac{4(1+CR^{2})}{3}\Bigg{)}^{1/4}g(R)\Bigg{]},~{}~{}~{}~{}~{}~{}~{}M_{1}(R)=\dfrac{2C(7+4CR^{2})}{7(1+CR^{2})}\\\
N_{2}(R)=\Bigg{(}\frac{4(1+CR^{2})}{3}\Bigg{)}^{1/4}\big{(}g(R)+1\big{)},~{}~{}~{}~{}~{}~{}~{}G(R)=\big{(}F(R)\big{)}^{1/4}~{}~{}~{}~{}G_{1}(R)=\frac{1}{4}\big{(}F(R)\big{)}^{-3/4},\\\
~{}~{}~{}~{}~{}~{}G_{2}(R)=\big{(}F(R)\big{)}^{-5/4},~{}~{}~{}~{}~{}G_{3}(R)=\frac{5}{4}\big{(}F(R)\big{)}^{-9/4},~{}~{}~{}~{}M_{2}(R)=\dfrac{3C}{7(1+CR^{2})}$

##  5 Stability Analysis of Compact Objects

In this section we have studied physical properties of interior of the fluid
sphere and equilibrium conditions under different forces.

###  5.1 Causality Condition

The speed of sound $\dfrac{dp}{c^{2}d\rho}$ is less than the velocity of
light.Here we fix $c=1$, and obtained speed of sound for uncharged fluid
matter.Herreralherrera (1992) states that for the stability the value of sound
belongs to region of the interval $0<v^{2}=\frac{dp}{d\rho}<1$ and should be
monotonically decreasing away from the center. Now from the equations (17) and
(18),we get expression of speed of sound

Figure 3: Behaviour of velocity of sound vs. fractional radius r/R for SAX
J1808.4-3658,4U 1538-52 and Her X-1.For plotting this figure we have employed
data set values of physical parameters and constants which are the same as
used in Fig.2

$\displaystyle\dfrac{dp}{d\rho}=\frac{L_{1}(r)\bigg{(}L_{2}(r)+L_{3}(r)\bigg{)}-L_{4}(r)L_{5}(r)-L_{6}(r)\bigg{(}L_{1}(r)\bigg{)}^{2}}{\bigg{(}L_{1}(r)\bigg{)}^{2}L_{7}(r)}$
(26)

where $\\\
L_{1}(r)=\left[\dfrac{4(1+Cr^{2})}{3}\right]^{1/4}\Big{(}g(r)+1\Big{)}\Bigg{[}\Big{\rvert}F(r)\Big{\rvert}^{(0.25)}+\frac{A_{2}}{A_{1}}\Big{\rvert}F(r)\Big{\rvert}^{(-1.25)}\Bigg{]}\\\
L_{2}(r)=\dfrac{-12C^{2}r}{7(1+Cr^{2})^{2}}\Bigg{[}N_{1}(r)S(r)+N_{2}(r)S_{1}(r)N_{4}(r)\Bigg{]},~{}~{}~{}L_{4}(r)=\dfrac{2C(7+4Cr^{2})}{7(1+Cr^{2})}\Bigg{[}N_{1}(r)S(r)+N_{2}(r)S_{1}(r)N_{4}(r)\Bigg{]}\\\
L_{3}(r)=\dfrac{2C(7+4Cr^{2})}{7(1+Cr^{2})}\Bigg{[}S_{2}(r)S(r)+2N_{1}(r)S_{1}(r)N_{4}(r)+N_{2}(r)S_{3}(r)S_{1}(r)+N_{2}(r)N_{4}(r)S_{4}(r)\Bigg{]}\\\
L_{5}(r)=N_{1}(r)S(r)+N_{2}(r)S_{1}(r)N_{4}(r),~{}~{}~{}~{}L_{6}(r)=\dfrac{-6C^{2}r}{7(1+Cr^{2})^{2}},~{}~{}~{}~{}~{}~{}L_{7}(r)=\dfrac{-6C^{2}r(5+Cr^{2})}{7(1+Cr^{2})^{3}}\\\
S(r)=\Bigg{[}\Big{\rvert}F(r)\Big{\rvert}^{1/4}+\frac{A_{2}}{A_{1}}\Big{\rvert}F(r)\Big{\rvert}^{-5/4}\Bigg{]},~{}~{}~{}~{}S_{1}(r)=\Bigg{[}(0.25)\Big{\rvert}F(r)\Big{\rvert}^{-3/4}+\frac{A_{2}}{A_{1}}(-1.25)\Big{\rvert}F(r)\Big{\rvert}^{-9/4}\Bigg{]}\\\
\\\ \\\
S_{2}(r)=\frac{8Cr}{9}\left[-\dfrac{6\big{(}g(r)+1\big{)}}{\Big{(}4\frac{4(1+Cr^{2})}{3}\Big{)}^{7/4}}+\dfrac{2}{g(r)\Big{(}\frac{4(1+Cr^{2})}{3}\Big{)}^{3/4}}-\dfrac{2{\Big{(}\frac{4(1+Cr^{2})}{3}\Big{)}^{1/4}}}{\big{(}g(r)\big{)}^{3}}\right],~{}~{}~{}~{}~{}~{}S_{3}(r)=\dfrac{16Cr}{3g(r)\bigg{(}g(r)-1\bigg{)}^{3}}\\\
S_{4}(r)=\Bigg{[}(-3/16)\Big{\rvert}F(r)\Big{\rvert}^{-7/4}-\frac{A_{2}}{A_{1}}(-45/16)\Big{\rvert}F(r)\Big{\rvert}^{-13/4}\Bigg{]}N_{4}(r)$
For better understanding we use the graphical representation to represent on
Fig.(3).Thus,it is clear that in the Fig.(3), the speed of sound lies within
the proposed interval and therefore this result maintains stability.

###  5.2 Tolman-Oppenheimer-Volkoff (TOV) equations

The general-relativistic hydrostatic equations were developed and used to
models of compact stars by Tolman, Oppenhiemer and volkoff in 35 (1939). These
equations are obtained from Einstein-Maxwell field equations when metric is
static and isotropic.The latter hypothesis is predicted to be a well
approximation for densest interior of static compact star, because the strong
gravitational force is balanced by a huge pressure and rigid body forces have
a negligible effect on the structure. In the connection of the microscopic
theory for the relation between pressure and energy density, and the mass,
this equation gives a equilibrium solution.The Tolman-Oppenheimer-Volkoff(TOV)
equation 35 (1939, 1939a) in the presence of charge is given by

$\displaystyle-\frac{M_{G}(\rho+p)}{r^{2}}e^{\frac{\lambda-\nu}{2}}-\frac{dp}{dr}+\sigma\frac{q}{r^{2}}e^{\frac{\lambda}{2}}=0,$
(27)

where $\sigma$ is charge density,$q$ is charge and $M_{G}$ is the effective
gravitational mass which is given by:

Figure 4: Behaviour of different forces(in $km^{-3}$ with $G=c=1$ ) vs.
fractional radius $r/R$. For plotting this figure the numerical values of
physical parameters and constants are as follows:(i) $K=1.75,CR^{2}=0.4137,M=$
$0.9M_{\odot}$ and $R=14.35\,km$ for SAX J1808.4-3658(left), (ii)Her X-1(left)
$K=1.74,CR^{2}=0.4138,M$ =$0.87M_{\odot}$ and $R=13.87\,km$ for 4u
1538-52(middle),(iii) $K=1.74,CR^{2}=0.414,M$ =$0.85M_{\odot}$ and
$R=13.458\,km$ for Her X-1(left)

$\displaystyle M_{G}(r)=\frac{1}{2}r^{2}\nu^{\prime}e^{(\nu-\lambda)/2}.$ (28)

Plugging the value of $M_{G}(r)$ in equation (27), we get

$\displaystyle-\frac{\nu^{\prime}}{2}(\rho+p)-\frac{dp}{dr}+\sigma\frac{q}{r^{2}}e^{\frac{\lambda}{2}}=0,$
(29)

But in our model we have considered an uncharged isotropic fluid distribution
i.e., the charge($q$ ) is vanish so equation (29) becomes

$\displaystyle-\frac{\nu^{\prime}}{2}(\rho+p)-\frac{dp}{dr}=0,$ (30)

The above equation can be expressed into three different components
gravitational force $(F_{g})$, hydrostatic force $(F_{h})$ and electric force
$(F_{e})$, which are defined as:

$\displaystyle F_{g}=-\frac{\nu^{\prime}}{2}(\rho+p)=\dfrac{Z^{\prime}}{8\pi
Z}(\rho+p)$ (31) $\displaystyle F_{h}=-\frac{dp}{8\pi
dr}=-\frac{1}{8\pi}\left[\frac{L_{1}(r)\bigg{(}L_{2}(r)+L_{3}(r)\bigg{)}-L_{4}(r)L_{5}(r)}{\bigg{(}L_{1}(r)\bigg{)}^{2}}-L_{6}(r)\right]$
(32) $\displaystyle F_{e}=0$ (33)

where we use the same notation as above. Fig.(4) represents the behavior of
the generalized TOV equations. We observe from these figures that the system
is counterbalanced by the components the gravitational force $(F_{g})$ and
hydrostatic force$(F_{h})$ and the system attains a static equilibrium.

###  5.3 Energy Condition

Figure 5: Behaviour of different energy conditions(in $km^{-2}$ with $G=c=1$)
vs. fractional radius Curves are plotted for the SEC,WEC and NEC for the
compact objects SAX J1808.4-3658, 4U 1538-52 and Her X-1. For plotting this
figure we have employed data set values of physical parameters and constants
which are the same as used in Fig.2

The energy conditions depend on the matter density and pressure.That follow
certain restrictions.Basic information about the energy condition inPratibha
(2011).Here we focus on the (i) Null energy condition, (ii) Weak energy
conditions and(ii)Strong energy condition , which have the following
inequalities

$\displaystyle\rho\geq 0$ (34) $\displaystyle\rho-p\geq 0$ (35)
$\displaystyle\rho-3p\geq 0$ (36)

Using these inequalities we justify the nature of energy conditions for the
specific stellar configuration as shown in Fig.(5), that satisfy our result.

###  5.4 Adiabatic index

In order to have an equilibrium configuration the matter must be stable
against the collapse of local regions. This requires Le Chateliers principle,
also known as local or microscopic stability condition, that the pressure must
be monotonically decreasing function of $r$ such that $\dfrac{dp}{d\rho}\geq
0.$ Heintzmann and HillebrandtHeint (1975) also proposed that compact star
with the equation of state are stable for adiabatic index
$\Gamma=\big{(}\frac{p+\rho}{p}\big{)}\frac{dp}{d\rho}>4/3.$ Fig.6 show that
$\Gamma>4/3$, so model developed in this paper is stable.

Figure 6: Behaviour of adiabatic constant$(\Gamma)$ vs. fractional radius r/R for SAX J1808.4-3658,4U 1538-52 and Her X-1. For plotting this figure we have employed data set values of physical parameters and constants which are the same as used in Fig.2 Table 1: Numerical value of parameters $C,M_{\odot},R$ and Buchdahl limit for different compact stars. Compact star candidates |  M($M_{\odot}$) |  Predicted radius $R(km)$ | $C(km^{-2})$ | $2M/R\leq 8/9$
---|---|---|---|---
SAX J1808.4-3658 | 0.9 | 14.35 | $2.009\times 10^{-13}$ | 0.18501
4U 1538-52 | 0.87 | 13.87 | $2.151\times 10^{-13}$ | 0.18503
Her X-1 | 0.85 | 13.548 | $2.255\times 10^{-13}$ | 0.18508

###  5.5 Surface Redshift

The gravitational redshift $Z_{s}$ within a static line element can be
obtained as

$\displaystyle Z_{s}=\sqrt{g_{tt}(R)}-1=\sqrt{1-\frac{2M}{R}}-1$ (37)

where $g_{tt}(R)=e^{\nu(R)}=1-\frac{2M}{R}$

Figure 7: Behaviour of Redshift vs. fractional radius r/R for SAX J1808.4-3658, 4U 1538-52 and Her X-1. For plotting this figure we have employed data set values of physical parameters and constants which are the same as used in Fig.2 Table 2: Energy densities, central pressure and Buchdahl condition for different compact star candidates for the above parameter values of Tables 1. Compact star candidates | Central Density | Surface density | Central pressure
---|---|---|---
| $(gm/cm^{3})$ | $(gm/cm^{3})$ | $(dyne/cm^{2})$
SAX J1808.4-3658 | $1.383\times 10^{14}$ | $7.873\times 10^{13}$ | $4.422\times 10^{34}$
4U 1538-52 | $1.48\times 10^{14}$ | $8.428\times 10^{13}$ | $4.304\times 10^{34}$
Her X-1 | $1.552\times 10^{14}$ | $8.836\times 10^{13}$ | $0.452\times 10^{34}$

The maximum possible value of redshift should be at the center of the star and
decrease with the increase of radius.BuchdahlBuchdahl (1959) and
Straumannstraumann (1984) have shown that for an isotropic star the surface
redshift $Z_{s}\leq 2$.For an anisotropic star Bohmer and Harko bohmer (2006)
showed that the surface redshift could be increased up to $Z_{s}\leq 5$.
IvanovIvanov (2002) modified the maximum value of redshift and showed that it
could be as high as $Z_{s}=5.211$. In this model we have $Z_{s}\leq 1$ for
compact stars SAX J1808.4-3658,4U 1538-52 and Her X-1. Also it is decreasing
towards the boundary(see Fig.7).

Table 3: The numerical values of physical parameters of the star SAX J1808.4-3658 for $C=2.009\times 10^{-13}km^{-2},K=7/4$ $r/R$ | $p(km^{-2})$ | $\rho(km^{-2})$ | $p/\rho$ | $dp/d\rho$ | Redshift
---|---|---|---|---|---
0 | $2.66225\times 10^{-5}$ | 0.000823 | 0.032363 | 0.098283 | 0.114842
0.1 | $2.62111\times 10^{-5}$ | 0.000817 | 0.032083 | 0.097498 | 0.114301
0.2 | $2.50048\times 10^{-5}$ | 0.000801 | 0.031238 | 0.095115 | 0.11269
0.3 | $2.30747\times 10^{-5}$ | 0.000774 | 0.029807 | 0.09106 | 0.110046
0.4 | $2.05321\times 10^{-5}$ | 0.000739 | 0.02776 | 0.085206 | 0.106427
0.5 | $1.75132\times 10^{-5}$ | 0.000699 | 0.025056 | 0.077382 | 0.10191
0.6 | $1.41603\times 10^{-5}$ | 0.000654 | 0.021649 | 0.067378 | 0.096586
0.7 | $1.06171\times 10^{-5}$ | 0.000607 | 0.017489 | 0.054947 | 0.090557
0.8 | $7.00901\times 10^{-6}$ | 0.000559 | 0.012525 | 0.039815 | 0.083927
0.9 | $3.44264\times 10^{-6}$ | 0.000513 | 0.00671 | 0.021685 | 0.076806
1 | 0 | 0.000468 | 0 | 0.000246 | 0.069299

###  5.6 Equation of state (EOS)

The equation of state(EoS),i.e. a relation between pressure and density is an
important features of neutron star. So in this section we have discuss about
EoS.It is wroth-wile to mentioned that different equation of state (EoS) of
the neutron star lead to different mass-radius (M-R) relation. Many authors
Dey1998 (1998, 2002, 2000) have suggested that the EoS $P=P(\rho)$ can be well
approximated by a linear function of the energy density $\rho$ of compact star
at high densities. Some researchers have also expressed more approximated form
of the equation of state (EOS) $P=P(\rho)$ as a linear function of energy
density $\rho$ (in details see Haensel1989 (1989, 1989a, 1990)). Here we find
the EoS in a linear function of form $P=P(\rho)$ as,

$\displaystyle\kappa\,p_{r}(r)$ $\displaystyle=$
$\displaystyle\frac{2\,(7+4\,{\tilde{\rho_{1}}})}{7\,G(\rho_{1})\,(1+{\tilde{\rho_{1}}})}\times\left[f_{2}(\rho_{1})\,\Bigg{(}~{}G_{1}(\rho_{1s})\,\bigg{|}\phi(\rho_{1})\bigg{|}^{0.25}+\,\bigg{|}\phi(\rho_{1})\bigg{|}^{-1.25}~{}\Bigg{)}-f_{1}(\rho_{1})\,f_{3}(\rho_{1})\,f_{4}(\rho_{1})\,\right]\,$
(38) $\displaystyle-\frac{3C}{7\,(1+{\tilde{\rho_{1}}})},$

where,

$G(\rho_{1})=\bigg{[}\frac{4(1+\tilde{\rho_{1}})}{3}\bigg{]}^{\frac{1}{4}}[f(\rho_{1})+1)]\,\Bigg{[}~{}G_{1}(\rho_{2s})\,\bigg{|}\phi(\rho_{1})\bigg{|}^{0.25}+\bigg{|}\phi(\rho_{1})\bigg{|}^{-1.25}~{}\Bigg{]},\\\
\\\
f_{2}(\rho_{1})=\dfrac{2C}{3}\Bigg{[}\big{(}f(\rho_{1})+1\big{)}\Bigg{(}\frac{4(1+\tilde{\rho_{1}})}{3}\Bigg{)}^{-3/4}+2\Bigg{(}\frac{4(1+\tilde{\rho_{1}})}{3}\Bigg{)}^{1/4}f(\rho_{1})\Bigg{]},\\\
\\\
f_{3}(\rho_{1})=\Bigg{[}G_{1}(\rho_{1s})(0.25)\bigg{|}\phi(\rho_{1})\bigg{|}^{(-0.75)}+(-1.25)\bigg{|}\phi(\rho_{1})\bigg{|}^{(-2.25)}\Bigg{]},~{}~{}~{}f_{4}(\rho_{1})=\dfrac{-8C}{3f(\rho_{1})\big{(}f(\rho_{1})-1\big{)}^{2}}\\\
f_{1}(\rho_{1})=\Bigg{(}\frac{4(1+\tilde{\rho_{1}})}{3}\Bigg{)}^{1/4}\big{(}f(\rho_{1})+1\big{)},~{}~{}~{}~{}~{}~{}~{}\phi(\rho_{1})=\dfrac{5+2\tilde{\rho_{1}}+3f(\rho_{1})}{2(1+\tilde{\rho_{1}})},\\\
\\\
\tilde{\rho_{1}}=\frac{(1-2\,\rho_{1})\pm\,\sqrt{1+8\,\rho_{1}}}{2\,\rho_{1}},~{}~{}~{}\rho_{1}=\frac{7\,\kappa\,\rho}{3\,C},~{}~{}~{}f(\rho_{1})=\sqrt{\frac{7+4\,\tilde{\rho_{1}}}{3}},\\\
\\\
G_{1}(\rho_{1s})=\frac{f_{2}(\rho_{1s})\,L_{2}(\rho_{1s})\,\bigg{|}\phi(\rho_{1s})\bigg{|}^{-5/4}-\frac{L(\rho_{1s})}{f^{2}(\rho_{1s})}\,\bigg{|}\phi(\rho_{1s})\bigg{|}^{-5/4}+S(\rho_{1s})}{\frac{L(\rho_{1s})}{f^{2}(\rho_{1s})}\,\bigg{|}\phi(\rho_{1s})\bigg{|}^{1/4}-f_{2}(\rho_{1s})\,L_{2}(\rho_{1s})\,\bigg{|}\phi(\rho_{1s})\bigg{|}^{1/4}+P(\rho_{1s})},\\\
\\\
L(\rho_{1s})=\bigg{[}\frac{4(1+\tilde{\rho_{1s}})}{3}\bigg{]}^{\frac{1}{4}}[f(\rho_{1s})+1],~{}~{}~{}L_{2}(\rho_{1s})=\frac{8\,f(\rho_{1s})}{9\,(1+\tilde{\rho_{1s}})}\\\
\\\
f_{2}(\rho_{1s})=\dfrac{2C}{3}\Bigg{[}\big{(}f(\rho_{1s})+1\big{)}\Bigg{(}\frac{4(1+\tilde{\rho_{1s}})}{3}\Bigg{)}^{-3/4}+\,2\Bigg{(}\frac{4(1+\tilde{\rho_{1s}})}{3}\Bigg{)}^{1/4}f(\rho_{1s})\Bigg{]},~{}~{}~{}\\\
\\\
P(\rho_{2s})=f_{1}(\rho_{2s})\,L_{2}(\rho_{1s})\,(-1/4)\,\bigg{|}\phi(\rho_{1s})\bigg{|}^{-3/4},~{}~{}~{}~{}\phi(\rho_{1s})=\dfrac{5+2\tilde{\rho_{1s}}+3f(\rho_{1s})}{2(1+\tilde{\rho_{1s}})},\\\
\\\
f_{1}(\rho_{1s})=2\,\bigg{[}\frac{4(1+\tilde{\rho_{1s}})}{3}\bigg{]}^{\frac{1}{4}}\,\frac{[f(\rho_{1s})+1]}{[f(\rho_{1s})-1]^{2}},~{}~{}~{}S(\rho_{1s})=f_{1}(\rho_{1s})\,L_{2}(\rho_{1s})\,(7/4)\,\bigg{|}\phi(\rho_{1s})\bigg{|}^{-9/2},\\\
\\\
\tilde{\rho_{1s}}=\frac{(1-2\,\rho_{1s})\pm\,\sqrt{1+8\,\rho_{1s}}}{2\,\rho_{1s}},~{}~{}~{}\rho_{1s}=\frac{7\,\kappa\,\rho_{s}}{3\,C},~{}~{}~{}f(\rho_{2s})=\sqrt{\frac{7+4\,\rho_{1s}}{3}}.$
$\rho$ and $\rho_{s}$

Figure 8: Behaviour of pressure$p$ (in $km^{-2}$) vs. energy density$\rho$
($km^{-2}$) for SAX J1808.4-3658, 4U 1538-52 and Her X-1

From equation(38), we can observe that the pressures is a function of density,
which describe the an EoS for SAX J1808.4-3658,4U 1538-52 and Her X-1. In an
argument, Dey et al Dey1998 (1998) have proposed a new type of EoSs mainly
describe strange matter. This was later generalised by Gondek-Rosinska et
alGondek2000 (2000) in a linear function of density ($\rho$), as

$\displaystyle p=\alpha\,(\rho-\rho_{s}),$ (39)

where $\rho_{s}$ denotes surface density and $\alpha$ is non-negative
constant. Harko and Cheng harko2002 (2002) have demonstrated that the
equation(39), gives the maximum mass of a strange star which is $M_{max}$ =
$1.83M_{\odot}$ when $\rho_{s}=4B$ ( $B=56MeVfm^{3}$). In the present paper,
we have developed same relation as considered by Gondek2000 (2000). In that
work, we have showed that the equation(39) corresponds to self-bound matter at
the surface density $\rho_{s}$. Fig.8 represents the behavior of pressure
verses density for compact stars with realistic EoS . In the Fig. 8, we
observe that the pressure $p$ vanishes at surface density $\rho_{s}$ i.e. at
the boundary of our model. This implies that $p$ can be expressed by
interpolation in power of $\rho-\rho_{s}$. Such parametrization is very
convenient for stellar modelling, which also significant to the interior of
stable stellar configurations Gondek2000 (2000).

Table 4: The numerical values of physical parameters of the star 4U 1538-52 for $C=2.151\times 10^{-13}km^{-2},K=7/4$ $r/R$ | $p(km^{-2})$ | $\rho(km^{-2})$ | $p/\rho$ | $dp/d\rho$ | Redshift
---|---|---|---|---|---
0 | $2.85086\times 10^{-5}$ | 0.000881 | 0.032368 | 0.098284 | 0.114866
0.1 | $2.80682\times 10^{-5}$ | 0.000875 | 0.032089 | 0.097499 | 0.114325
0.2 | $2.67753\times 10^{-5}$ | 0.000857 | 0.031244 | 0.095116 | 0.112714
0.3 | $2.47093\times 10^{-5}$ | 0.000829 | 0.029813 | 0.09106 | 0.110069
0.4 | $2.19861\times 10^{-5}$ | 0.000792 | 0.027765 | 0.085204 | 0.106449
0.5 | $1.8753\times 10^{-5}$ | 0.000748 | 0.025061 | 0.077378 | 0.101931
0.6 | $1.5164\times 10^{-5}$ | 0.000701 | 0.021653 | 0.067371 | 0.096606
0.7 | $1.13697\times 10^{-5}$ | 0.000649 | 0.017492 | 0.054937 | 0.090574
0.8 | $7.50583\times 10^{-6}$ | 0.000599 | 0.012527 | 0.039801 | 0.083944
0.9 | $3.6867\times 10^{-6}$ | 0.000549 | 0.006711 | 0.021666 | 0.076821
1 | 0 | 0.000502 | 0 | 0.000221 | 0.069312

###  5.7 Static stability criterion

The most important feature of stability for stellar configuration is static
stability criterion harrison (1965, 1971). In this criterion, it is postulate
that the any stellar configuration has an increasing mass with increasing
central density, i.e. $dM/d\rho_{0}>0$ represents stable configuration and
vice versa. If the mass remains constant with increasing central density, i.e.
$dM/d\rho_{0}=0$ we get the turning point between stable and unstable region.
For this model, we obtained $M(R)$ and $dM/d\rho_{0}$ as follows-

Figure 9: Behaviour of Mass vs. central density(left) and $dM/d\rho_{0}$ vs.
central density(right) for SAX J1808.4-3658, 4U 1538-52 and Her X-1

$M(R)=\frac{12\,\pi\,\rho_{0}\,R^{3}}{(9+56\,\pi\,\rho_{0}\,R^{2})}~{}~{}~{}~{}~{}\textrm{and}~{}~{}~{}~{}\frac{dM}{d\rho_{0}}=\frac{108\,\pi\,R^{3}}{(9+56\,\pi\,\rho_{0}\,R^{2})^{2}}$
(40)

Hence from Fig.9,we can conclude that presenting model represents static
stable configuration.

Table 5: The numerical values of physical parameters of the star 4U 1538-52 for $C=2.151\times 10^{-13}km^{-2},K=7/4$ $r/R$ | $p(km^{-2})$ | $\rho(km^{-2})$ | $p/\rho$ | $dp/d\rho$ | Redshift
---|---|---|---|---|---
0 | $2.99041\times 10^{-5}$ | 0.000924 | 0.03238 | 0.098287 | 0.114915
0.1 | $2.94426\times 10^{-5}$ | 0.000917 | 0.0321 | 0.097501 | 0.114373
0.2 | $2.80875\times 10^{-5}$ | 0.000899 | 0.031255 | 0.095117 | 0.112761
0.3 | $2.59187\times 10^{-5}$ | 0.000869 | 0.029824 | 0.091059 | 0.110115
0.4 | $2.30627\times 10^{-5}$ | 0.000831 | 0.027776 | 0.085201 | 0.106493
0.5 | $1.96706\times 10^{-5}$ | 0.000785 | 0.02507 | 0.077371 | 0.101972
0.6 | $1.59055\times 10^{-5}$ | 0.000734 | 0.021661 | 0.067359 | 0.096644
0.7 | $1.19252\times 10^{-5}$ | 0.000682 | 0.017498 | 0.054918 | 0.09061
0.8 | $7.87207\times 10^{-6}$ | 0.000628 | 0.012532 | 0.039773 | 0.083976
0.9 | $3.86576\times 10^{-6}$ | 0.000576 | 0.006714 | 0.021628 | 0.076851
1 | 0 | 0.000526 | 0 | 0.000171 | 0.069338

##  6 Conclusion

In this article, we have discuss a new solution of Vaidya-Tikekar model for
spherically symmetric uncharged fluid ball and found,it is physically valid
solution.The Fluid ball contain an uncharged perfect fluid matter and
Schwarzschild exterior metric. Mainly, we perform a detailed investigation of
the physical result of high density system like uncharged fluid Lake (2003,
2004, 2008) and observe that the physical viability and acceptable of the our
model in connection with compact star like Her X-1,4U 1538-52 and SAX
J1808.4-3658.

It has been observe that the energy density and pressure are positive at the
center i.e $\rho_{0}>0,p_{0}>0$ and monotonically decreasing throughout the
fluid ball,see Fig.2. The energy conditions are very important to understand
many theorem of general relativity such as singularity theorem of stellar
collapse. Fig.5 shows that the energy conditions are positive throughout the
star and model satisfy (i) strong energy condition(SEC) and (ii) weak energy
condition (WEC)YK (2012a).We have also studied about the surface redshift. It
should be maximum at the center and monotonically decreasing from the center
to surfaceIvanov (2002) see Fig.7. The modified TOV equation describes the
equilibrium condition see Fig.3 and observe that the gravitational force is
balanced by the hydrostatic force. For stability analysis the adiabatic
constant($\Gamma$) is an important physical parameter and compact star will be
stable if $\Gamma>4/3$Heint (1975). Fig.6 show that $\Gamma>4/3$, so model
developed in this paper is stable.The mass-radius relation must be less than
8/9 Buchdahl (1959). Our model also satisfy this condition. The numerical
values of physical quantities are shown in the Table 1-5.We have obtained the
EoS for the present compact star model, which is the significant physical
property to describe structure of any realistic matter. We can see from
equation(38) the pressure is purely function of density. Hence we conclude
that this approach may help to describe the structure of compact star.

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