# Analytical model on mass limit of strange stars

Sajahan Molla11affiliationmark:  Masum Murshid22affiliationmark:  Mehedi
Kalam22affiliationmark:

###### Abstract

In this paper, we present a new kind of stellar model using the Nariai IV
metric.This model can be used to study the strange/quark stars(which is our
present interest, though it can also be applicable to neutron stars). We
present a mass-radius region where all the regularity conditions, energy
conditions, the TOV equation, and stability conditions are satisfied.
According to our model, strange stars having mass up to
$1.9165M_{\odot}(=2.81km)$ is stable. A strange star having a mass greater
than $1.9165M_{\odot}$ violates the stability conditions.This model can be
very useful to predict the radius of strange stars of mass greater than
$1M_{\odot}$.

00footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, New Alipore
College, L Block, New Alipore, Kolkata 700053, India00footnotetext: Department
of Physics, Aliah University, IIA/27, New Town, Kolkata 700160, India

Keywords Relativistic stars Structure Stability Radius Compactness Redshift

##  1 Introduction

The study of the compact stars drew attention to many scientists for numerous
years. The compact stars are the end product of the thermonuclear powered
stars. The mass of the thermonuclear powered stars decides which state of the
compact star would it be. The degeneracy pressure of the constituting fermions
(electrons for the white dwarf and neutrons for the neutron star) of the
compact stars balances the self-gravitational force. A neutron star is the
extreme state of the compact star before it turns into a black hole while the
gravitational force overwhelms the degeneracy pressure. Neutron star also has
several substates like hyperon star, hybrid star or strange star. These three
sub-states of the neutron star are just hypothetical states, no direct
observation yet found. The strange star (made of strange quarks) and a normal
neutron star (fundamentally constructed of neutrons) can be distinguished in
three different ways Xu (2001), which are based on the mass-radius relation,
differences of viscosity and surface condition. The neutron and strange stars
have almost same radius while the mass of the star is about $1M_{\odot}$.
Whereas the low mass neutron and strange stars radius significantly differ
from each other Alcock (1986); Bombaci (1997); Li (1999); Xu (2005). A neutron
star having mass $\sim 0.2M_{\odot}$ has a radius of $>15km$, whereas a
strange star with mass $\sim 0.2M_{\odot}$ is only $<5km$. Another difference
is that a neutron star has a lower mass limit of about $\sim 0.2M_{\odot}$,
whereas a strange star does not have any lower mass limit Xu (2005).

In four-dimensional spacetime general relativity describes the gravitational
interaction and its consequences very nicely. Einstein’s theory of general
relativity placed the ground of our understanding of compact stars. In 1916
Schwarzschild Schwarzschild (1916) first solved the exact solution of
Einstein’s field equations. He had calculated the gravitational field equation
of a homogeneous incompressible fluid sphere which had finite radius.
Oppenheimer, Volkoff and Tolman Oppenheimer (1939); Tolman (1939) in 1939
successfully derived the balancing equations of relativistic stellar
structures from Einstein’s field equations. They also calculated the limit of
the mass of a stable relativistic incompressible fluid sphere. After the
discovery of neutron stars, a connecting bridge is build up between
theoretical and observational astrophysics. Though few studies related to
compact star masses Heap (1992); Lattimer (2005); Stickland (1997); Orosz
(1999); Kerkwijk (1995), which are in binaries, have been proposed (with
negligible error), there is no information about the radius. Therefore, the
theoretical study of the stellar structure is required to follow the correct
direction for the newly observed stellar masses to get some new pieces of
information. By the compact star modeling some interesting results found in
terms of the important parameters like stability factor, radius, compactness,
redshift, etc., which cannot be inferred from direct observation. The study of
the interior region of the compact stars is not straightforward. Numerous
authors Rahaman (2012a, b, 2014); Hossein (2012); Kalam (2012, 2013, 2014,
2016, 2017, 2018); Maurya (2016); Dayanandan (2016); Maharaja (2014);
Ngubelanga (2015); Paul (2015); Pant (2014); Bhar (2017); Lobo (2006);
Bronnikov (2006); Egeland (2007); Dymnikova (2002); Chakraborty (2017); Singh
(2021); Sharma (2021); Paul (2018); Goswami (2020); Biswas (2020); Aziz
(2019); Chowdhury (2019); Deb (2017); Jasim (2021) established different
models to enlight several features of the interior of the compact objects
using various types of metric pleasing Einstein equations as well as all the
conditions which are required for a stellar model to be causal.

We study the field equations and the mathematical solutions of strange star
within Nariai spacetime. Several studies have been performed using the Nariai
spacetime by other authors to understand various astrophysical aspects. Nariai
IV metric has also been used to describe neutron stars Lattimer (2001, 2005,
2005a) by Moustakidis (2017). Shaikh et al. Shaikh (2020) studied the
curvature properties of a charged Nariai type spacetime and found that such a
metric is not locally symmetric but semi symmetric, and its Ricci tensor is
neither Codazzi nor cyclic parallel or recurrent but generalized recurrent.
Carlos Batista Batista (2016) presented some solutions which are made of the
direct product of several 2-spaces of constant curvature. These solutions turn
out to have many magnetic charges, contrary to the usual higher-dimensional
generalization of the Nariai spacetime, which has no magnetic charge at all.
Fennen and Giulini Fennen (2015) investigated exact solutions to Einstein’s
equations corresponding to the two spherically symmetric stars, which are made
of an incompressible perfect fluid, possibly oppositely charged. The negative
pressure of the positive cosmological constant keeps these two stars
separated. But there is no cosmological horizon separating these two stars.
They showed that the solution gives rise to the Nariai metric or an oversight
generalization thereof in the charged case.

In this paper, we want to study the physical behavior of the strange star. For
this, we have considered the isotropic stellar model to study the fluid
sphere. We found a solution of the fluid sphere where metric constants are
absolutely depend on star’s mass and radius. In this paper we have also
discussed the dynamical stability analysis of the system. Our main objective
in this study is to give an estimate of the possible radius, central pressure,
compactness and redshift of the wide mass range of the strange star.

The material of the paper is organised as follows: In Sect. 2 we give the
basic field equations in connection to the compact star of Narai IV metric. In
Sect. 3 we have discussed matching condition. In Sect. 4 we have talk over the
physical conditions required of the star model. In Sect. 5 we review: (5.1)
Density and isotropic pressure behavior at the interior of the star, (5.2)
Energy conditions, (5.3) TOV equations, (5.4) Stability condition, (5.5)
Compactness and Surface Redshift. In Sect. 6, we give our concluding remark.

##  2 Interior solution

We consider the static and spherically symmetric metric in isotropic co-
ordinates as

$\displaystyle
ds^{2}=c^{2}e^{\alpha}dt^{2}-e^{\beta}\left(dr^{2}+r^{2}d\Omega^{2}\right)$
(1)

where $\alpha$ and $\beta$ are function of r.

Einstein’s field equations of gravitation for a non empty space-time are

$\displaystyle R_{ij}-\frac{1}{2}Rg_{ij}=-\frac{8\pi G}{c^{4}}T_{ij}$ (2)

where $R_{ij}$ is a Ricci tensor, $T_{ij}$ is energy momentum tensor and R is
the scalar curvature.

The energy-momentum tensor $T_{ij}$ is defined as

$\displaystyle T_{ij}=\left(p+\rho c^{2}\right)v_{i}v_{j}-pg_{ij}$ (3)

where $p$ denotes the isotropic pressure and $\rho$ is the density
distribution.

The velocity vector $v_{i}$ satisfying the relation

$\displaystyle g_{ij}v^{i}v^{j}=1$ (4)

Since the field is static, therefore

$\displaystyle
v^{1}=v^{2}=v^{3}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}v^{4}=\frac{1}{\sqrt{g_{44}}}$
(5)

Thus for matter distribution with isotropic pressure the field equation eqn.2
reduces as Pant (2012):

$\displaystyle\frac{8\pi G\rho}{c^{2}}$ $\displaystyle=$
$\displaystyle-e^{-\beta}[\beta^{\prime\prime}+\frac{(\beta^{\prime})^{2}}{4}+\frac{2\beta^{\prime}}{r}]$
(6) $\displaystyle\frac{8\pi Gp}{c^{4}}$ $\displaystyle=$ $\displaystyle
e^{-\beta}[\frac{(\beta^{\prime})^{2}}{4}+\frac{\beta^{\prime}}{r}+\frac{\alpha^{\prime}\beta^{\prime}}{2}+\frac{\alpha^{\prime}}{r}]$
(7) $\displaystyle\frac{8\pi Gp}{c^{4}}$ $\displaystyle=$ $\displaystyle
e^{-\beta}[\frac{\beta^{\prime\prime}}{2}+\frac{\alpha^{\prime\prime}}{2}+\frac{(\alpha^{\prime})^{2}}{4}+\frac{\beta^{\prime}}{2r}+\frac{\alpha^{\prime}}{2r}]$
(8)

where, prime($\prime$) denotes differentiation with respect to r. Now from
eqn.7 and eqn.8 we get the following differential equation in term of $\alpha$
and $\beta$ as

$\displaystyle\beta^{\prime\prime}+\alpha^{\prime\prime}+\frac{(\alpha^{\prime})^{2}}{2}-\frac{(\beta^{\prime})^{2}}{2}-\alpha^{\prime}\beta^{\prime}-(\frac{\beta^{\prime}}{r}+\frac{\alpha^{\prime}}{r})=0$
(9)

Our task is to obtain the physical properties ($\rho$ and $p$) of the fluid
sphere from the above equations.

We assume that the interior space-time of a star is described by the metric
propose by H. Nariai Nariai (1950, 1951)

$\displaystyle
ds^{2}=A\cos^{2}\left(a-\frac{\sqrt{2}Mr^{2}}{4}\right)\cos^{-2}\left(b+\frac{Mr^{2}}{4}\right)dt^{2}$
$\displaystyle-A\cos^{-2}\left(b+\frac{Mr^{2}}{4}\right)\left(dr^{2}+r^{2}d\Omega^{2}\right)$
(10)

where $A$, $M$, $a$, $b$ are constants.
Now using the metric eqn.10 we get,

$8\pi\rho=\frac{M\left(-Mr^{2}\left(\sin^{2}\left(b+\frac{Mr^{2}}{4}\right)+2\right)-6\sin\left(2b+\frac{Mr^{2}}{2}\right)\right)}{4A}$
(11) $8\pi
p=\frac{M\cos^{2}\left(b+\frac{Mr^{2}}{4}\right)\left(2\sqrt{2}\tan\left(a-\frac{Mr^{2}}{2\sqrt{2}}\right)\left(Mr^{2}\tan\left(b+\frac{Mr^{2}}{4}\right)+2\right)+\tan\left(b+\frac{Mr^{2}}{4}\right)\left(3Mr^{2}\tan\left(b+\frac{Mr^{2}}{4}\right)+8\right)\right)}{4A}$
(12)

##  3 Matching Conditions

It is well known to us that the interior metric of a compact star should be
matched to the Schwarzschild exterior solution at the boundary.

$ds^{2}=\left(1-\frac{2m}{r_{s}}\right)dt^{2}-\left(1-\frac{2m}{r_{s}}\right)^{-1}dr_{s}^{2}-r_{s}^{2}d\Omega^{2}$
(13)

It is always possible to transform this standard form of the Schwarzschild
metric into the isotropic form Nariai (1950) (see Eq-1) by putting

$r_{s}=m+\frac{m^{2}}{4r}+r$ (14)

where $m=$ mass , $r=$ radial co-ordinate. After using this transformation,
the standard Schwarzchild metric transforms into isotropic form as Hajj (1986)

$ds^{2}=\left(\frac{1-\frac{m}{2r}}{1+\frac{m}{2r}}\right)^{2}dt^{2}-\left(1+\frac{m}{2r}\right)^{4}\left(dr^{2}+r^{2}d\Omega^{2}\right)$
(15)

The matching of Nariai IV metric at the boundary ($r=R$, where $R$ is radius
of the star) with the exterior Schwarzschild metric gives rise to

$A\cos^{2}\left(a-\frac{\sqrt{2}MR^{2}}{4}\right)\cos^{-2}\left(b+\frac{MR^{2}}{4}\right)=\left(\frac{1-\frac{m}{2R}}{1+\frac{m}{2R}}\right)^{2}$
(16)
$A\cos^{-2}\left(b+\frac{MR^{2}}{4}\right)=\left(1+\frac{m}{2R}\right)^{4}$
(17)

and the continuity of the metric components $g_{tt}$ and $g_{rr}$ at the
boundary gives

$\displaystyle
MR\left(\sqrt{2}\tan\left(a-\frac{MR^{2}}{2\sqrt{2}}\right)+\tan\left(b+\frac{MR^{2}}{4}\right)\right)$
$\displaystyle=-\frac{8m}{4m^{2}-R^{2}}$ (18)

$MR\tan\left(b+\frac{MR^{2}}{4}\right)=-\frac{8m}{2mR+R^{2}}$ (19)

We solve the above four eqns.(16,17,3 & 19) to get the metric coefficient in
terms of the mass (m) and radius (R) of the star as follows

$A=\frac{(m-4R)^{2}(m+2R)^{4}}{2(m+4R)\left(m^{5}+8m^{4}R+28m^{3}R^{2}+48m^{2}R^{3}+40mR^{4}+32R^{5}\right)}$
(20)

$M=\pm\frac{8\sqrt{2}m(m-4R)}{(m+2R)\sqrt{m\left(m^{2}+6mR+16R^{2}\right)\left(m^{3}+6m^{2}R+8mR^{2}+16R^{3}\right)}}$
(21)

$\displaystyle
b=\mp\frac{2\sqrt{2}mR^{2}(m-4R)}{(m+2R)\sqrt{m\left(m^{2}+6mR+16R^{2}\right)\left(m^{3}+6m^{2}R+8mR^{2}+16R^{3}\right)}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(22)
$\displaystyle+\cos^{-1}\left(\mp\frac{2\sqrt{2}R^{2}(m-4R)}{\sqrt{(m+4R)\left(m^{5}+8m^{4}R+28m^{3}R^{2}+48m^{2}R^{3}+40mR^{4}+32R^{5}\right)}}\right)$

$a=\pm\frac{4mR^{2}(m-4R)}{(m+2R)\sqrt{m\left(m^{2}+6mR+16R^{2}\right)\left(m^{3}+6m^{2}R+8mR^{2}+16R^{3}\right)}}+\cos^{-1}\left(\pm\frac{4R^{2}(m-2R)}{(m+2R)^{3}}\right)$
(23)

These two set of solutions also satisfy $p_{|r=R}=0$ equation i.e. pressure
get vanished at the boundary.

##  4 Conditions for well behaved solution

For well-behaved nature of the solution in spherical coordinates, the
following conditions should be satisfied Delgaty (1998),Pant (2010, 2012);
Pant& Maurya (2012):

  1. (i)

The solution should be free from physical and geometrical singularities i.e.
finite and positive values of central density and central pressure i.e.
$\rho_{c}>0$ and $p_{c}>0$.

  2. (ii)

The solution should give rise to positive and monotonically decreasing
expressions for density($\rho$) and pressure(p) with the increase of r.

  3. (iii)

The pressure at the boundary should vanish.

  4. (iv)

The ratio of pressure to the density (p/$\rho$) should be monotonically
decreasing with the increase of r. $(\frac{d\rho}{dr})_{r=0}=0$ and
$(\frac{d^{2}\rho}{dr^{2}})_{r=0}<0$ so that density gradient
$\frac{d\rho}{dr}$ is negative for $0<r\leq R$. $(\frac{dp}{dr})_{r=0}=0$ and
$(\frac{d^{2}p}{dr^{2}})_{r=0}<0$ so that pressure gradient $(\frac{dp}{dr})$
is negative for $0<r\leq R$. These above conditions imply that pressure and
density should be maximum at the centre and monotonically decreasing towards
the pressure-free interface.

  5. (v)

The solution should satisfy all the energy conditions i.e. weak energy
condition, strong energy condition, null energy condition and dominant energy
condition.

  6. (vi)

The solution should obey the casualty condition
$v_{s}^{2}=(\frac{dp}{d\rho})\leq 1$ i.e. speed of sound should be less than
that of light throughout the model.

  7. (vii)

Stellar equilibrium exist everywhere within the star.

  8. (viii)

When the observer and the source of light are in relative motion, the
observer, instead of receiving the light at its original wavelength, receives
the light of a slightly shifted wavelength. The shift causes an increase in
the wavelength and a decrease in the frequency or energy of electromagnetic
radiation called redshift. The redshift is written as follow

$Z(r)=\frac{1}{\sqrt{1-\frac{2M(r)}{r})}}-1$ (24)

.

The redshift ($Z$) should be positive and finite for the stellar model.The
maximum allowed value of surface redshift is $Z\leq 0.85$ Haensel (2000).

Under these above conditions, the solution should be well behaved.

##  5 Analysis of Physical properties

###  5.1 Density and Pressure Behavior of the star

The internal metric is free from physical and geometrical singularities. Thus
central density and central pressure must be positive and finite i.e.
$\rho_{c}>0$ and $p_{c}>0$. The central density and pressure are given by

$\rho_{c}=-\frac{3M\sin(b)\cos(b)}{8\pi A}>0$ (25)

$p_{c}=\frac{M\cos^{2}(b)\left(\sqrt{2}\tan(a)+2\tan(b)\right)}{8\pi A}>0$
(26)

and since the energy density and pressure is monotonically decreasing
function, we have

$\displaystyle\frac{d\rho}{dr}\bigg{|}_{r=0}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{dp}{dr}\bigg{|}_{r=0}=0$
(27)

and

$\displaystyle\frac{d^{2}\rho}{dr^{2}}\bigg{|}_{r=0}$ $\displaystyle=$
$\displaystyle-\frac{5M^{2}\cos^{2}(b)}{16\pi A}<0$ (28)
$\displaystyle\frac{d^{2}p}{dr^{2}}\bigg{|}_{r=0}$ $\displaystyle=$
$\displaystyle\frac{M^{2}\left(2-2\sec^{2}(a)\cos^{2}(b)-\sin^{2}(b)\right)}{16\pi
A}<0$

Fig. 1 : Graphical presentation of the allowed the mass-radius region Fig. 2 :
Variation of the density $\rho$ against radial parameter $r$ of the star Fig.
3 : Variation of the pressure $p$ against radial parameter $r$. Fig. 4 :
Equation of state of four strange stars, where $\alpha,\beta,\gamma$ are
constants. Fig. 5 : Energy conditions at the stellar interior Fig. 6 :
Behaviors of gravitational and hydrostatic force at the stellar interior of
the star Fig. 7 : Variation of the sound speed at the stellar interior. Fig. 8
:  Variation of the adiabatic index at the stellar interior. Fig. 9 :
Variation of the compactness $u$ against radial parameter $r$. Fig. 10 :
Variation of the redshift $Z_{s}$ against radial parameter $r$ of the star

###  5.2 Energy conditions

From Fig. 5, we see that all the energy conditions such as null energy
condition (NEC), weak energy condition (WEC), strong energy condition (SEC)
and dominant energy condition (DEC) satisfies at every point in the interior
of the compact star simultaneously. These energy conditions are as follows:

$\displaystyle\mbox{i)
NEC:}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho+p\geq
0$ $\displaystyle\mbox{ii)
WEC:}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho+p\geq 0,\rho\geq
0$ $\displaystyle\mbox{iii) SEC:}~{}~{}~{}~{}~{}~{}~{}~{}\rho+p\geq
0,\rho+3p\geq 0$ $\displaystyle\mbox{iv)
DEC:}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho>|p|$
(30)

###  5.3 TOV equation

The stellar equation (some authors considered it as TOV equation) can be
written as

$\frac{dp}{dr}+\frac{1}{2}\nu^{\prime}\left(\rho+p\right)=0$ (31)

The stellar equation (TOV equation) describes the equilibrium condition for
the strange star subject to the effective gravitational ($F_{g}$) and
effective hydrostatic ($F_{h}$) force nature of the stellar object as

$F_{h}+F_{g}=0,$ (32)

where,

$\displaystyle F_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nu^{\prime}\left(\rho+p\right)$ (33) $\displaystyle
F_{h}$ $\displaystyle=$ $\displaystyle\frac{dp}{dr}$ (34)

Therefore, from Fig. 6 we see that the static equilibrium do exist in the
presence of gravitational and hydrostatic forces.

###  5.4 Stability

For a physically acceptable model, speed of sound should be within the range
$0\leq v^{2}=(\frac{dp}{d\rho})\leq 1$ Herrera (1992); Abreu (2007); Chengjun
Xia (2021); Ma (2019); Bedaque (2015); Cherman (2009); Hohler (2009);
Benincasa (2006); Mateos (2007); Benincasa (2006a); Borsanyi (2010); Alford
(2013). Fig-7 indicates that the sound speed is monotonically decreasing
function of the radial distance. The sound speed at the center is given by

$0\leq
v_{c}^{2}=\frac{1}{10}\left(4\sec^{2}(a)+2\tan^{2}(b)-4\sec^{2}(b)\right)\leq
1$ (35)

The adiabatic Index should be Chandrasekhar (1964),

$\gamma=\frac{p+\rho}{p}\frac{dp}{d\rho}>\frac{4}{3}$ (36)

Fig-8 shows that the adiabatic index within the star is always greater than
$\frac{4}{3}$. The adiabatic index at the centre is give by

$\gamma_{c}=\frac{\left(\tan(b)-\sqrt{2}\tan(a)\right)\left(-2\sec^{2}(a)+\sec^{2}(b)+1\right)}{5\left(\sqrt{2}\tan(a)+2\tan(b)\right)}>\frac{4}{3}$
(37)

###  5.5 Compactness and Surface Redshift

From eqn.17, we get the compactness and redshift written as follow

$u(r)=\frac{m(r)}{r}=2\left(\sqrt[4]{A\sec^{2}\left(b+\frac{Mr^{2}}{4}\right)}-1\right)$
(38)

and

$Z(r)=\frac{1}{\sqrt{1-4\left(\sqrt[4]{A\sec^{2}\left(b+\frac{Mr^{2}}{4}\right)}-1\right)}}-1$
(39)

According to Buchdahl Buchdahl (1959), for a static spherically symmetric
perfect fluid allowable mass-radius ratio should be $\frac{2M}{R}<\frac{8}{9}$
and Mak Mak (2001) also gave a more generalized expression for the same
manner. The maximum allowed value of surface redshift is $Z\leq 0.85$ Haensel
(2000). The central compactness and the central redshift are given by

$u_{c}=2\left(\sqrt[4]{A\sec^{2}(b)}-1\right)<\frac{4}{9}$ (40)
$Z_{c}=\frac{1}{\sqrt{1-4\left(\sqrt[4]{A\sec^{2}(b)}-1\right)}}-1\leq 0.85$
(41)

Figs. 9 & 10 show that the compactness and redshift within the star are always
less than $\frac{4}{9}$ and less than to the value of $0.85$.

Now if we solve all these conditions
(Eqs-25,26,27,28,5.1,35,37,40,41) alongside the demand that the central
density of the star Bombaci (2018) has at least of the order of the nuclear
density i.e. $2.7\times 10^{14}g/cc$, then we get the important mass-radius
relation (Fig-1). Fig-1 presents the allowed region of the mass-radius graph.
The minimum radius for a given mass ($m$) is $R_{min}=4.4857m$ km ($m$ in
$km$). Table-1 enlists the maximum radius $R_{max}$, minimum central pressure
$p_{c(max)}$, minimum compactness $u_{min}$ and minimum redshift $Z_{min}$ for
a given mass. We see from the Figs-2-3 that the energy density and the
pressure are monotonically decrease with the radial distance.

Table 1 : List of the maximum radius for a given mass of a strange star, and the corresponding minimum central pressure, minimum compactness and minimum redshift Mass ($M_{\odot}$) |  Maximum Radius ($R_{max}$ in km) |  Minimum Central pressure ($p_{c(min)}$ in $10^{-6}km^{-2}$) |  Minimum Compactness ($u_{min}$) |  Minimum Redshift ($Z_{min}$)
---|---|---|---|---
0.01 |  2.78414 |  0.50505 |  0.005304 |  0.005346
0.05 |  4.70399 |  1.51260 |  0.015695 |  0.016075
0.1 |  5.86247 |  2.45425 |  0.025187 |  0.026181
0.2 |  7.25713 |  4.03779 |  0.040694 |  0.043359
0.3 |  8.18254 |  5.45781 |  0.054137 |  0.058972
0.4 |  8.88272 |  6.80319 |  0.066493 |  0.073957
0.5 |  9.44583 |  8.11042 |  0.078161 |  0.088709
0.7 |  10.3150 |  10.6820 |  0.100205 |  0.118320
0.8 |  10.6616 |  11.9675 |  0.110797 |  0.133436
0.9 |  10.9653 |  13.2622 |  0.121195 |  0.148886
1. |  11.2366 |  14.5567 |  0.131409 |  0.164696
1.2 |  11.6854 |  17.2502 |  0.151634 |  0.198029
1.3 |  11.8765 |  18.6274 |  0.161627 |  0.215590
1.4 |  12.0482 |  20.0345 |  0.171580 |  0.233871
1.5 |  12.2027 |  21.4746 |  0.181508 |  0.252957
1.6 |  12.3417 |  22.9511 |  0.191428 |  0.272937
1.7 |  12.4668 |  24.4672 |  0.201352 |  0.293914
1.8 |  12.5791 |  26.0261 |  0.211292 |  0.315999
1.9 |  12.6799 |  27.6314 |  0.221258 |  0.339318
1.91 |  12.6894 |  27.7946 |  0.222256 |  0.341723
1.915 |  12.6941 |  27.8764 |  0.222755 |  0.342930
1.916 |  12.6950 |  27.8927 |  0.222855 |  0.343172
1.91675 |  12.6957 |  27.9050 |  0.222930 |  0.343354
Table 2 : List of the maximum value of some parameters corresponding to minimum radii $R_{min}=4.4857m$ km ($m$ in $km$) Maximum Central Energy Density ($\rho_{c(max)}$) in $km^{-2}$ |  Maximum Surface Energy Density ($\rho_{R(max)}$) in $km^{-2}$ |  Maximum Central Pressure ($p_{c(max)}$) in $km^{-2}$ |  Maximum Compactness ($u_{max}$) |  Maximum Redshift ($Z_{max}$)
---|---|---|---|---
$\frac{0.00160562}{m^{2}}$ |  $\frac{0.0012333}{m^{2}}$ |  $\frac{0.000223533}{m^{2}}$ |  $0.222841$ |  $0.343188$

##  6 Conclusions

In this article, we present a strange star model(having mass up to
$1.9165M_{\odot}(=2.81km)$) by using the Nariai IV metric. Using this model,
we study the strange stars physical properties. We present a mass-radius
region where all the regularity conditions, energy conditions, the TOV
equation, and stability conditions are satisfied. We show, in Fig. 4, how the
pressure depends on the energy in four cases; and found that the mathematical
relation between them has a form like $p=\alpha+\beta\rho+\gamma\rho^{2}$.
From the EOS graph (see Fig. 4), one can see that the slope of these plots is
high, which indicates that our EOS is stiff. This kind of stiffness in EOS is
found in SQM(1-3) EOS Prakash et al. (1995); Lattimer (2001). The maximum mass
($1.9165M_{\odot}$) found in our model is also very similar to the maximum
mass (around $2M_{\odot}$) found for SQM3 EOS (see Fig.2 of Lattimer (2001)).
That’s why we are claiming that strange stars are made of, mainly, quarks with
a little bit mixture of baryons. According to the Ref. Chengjun Xia (2021), we
see that our model fulfil the causality limit ( $\frac{c_{s}}{c}<1$ ). We see
in the Ref. Chakraborty (2017) that for hybrid star ( which is made by quark
matter and nuclear(baryonic) matter, considered as two different fluids)
square of sound speed is less than 1/3. We also see in this Ref. Alford (2013)
that the square of sound speed may be greater than 1/3 when the quark matter
is strongly coupled. Therefore, we can argue that our strange star model is
applicable for both cases: (i) strange star made by quark matter which is
strongly coupled in which square of sound speed greater than 1/3 (ii) Also
hybrid star made by quark matter and nuclear(baryonic) matter in which square
of sound speed less than 1/3. According to our model, strange stars having a
mass greater than $1.9165M_{\odot}$ violates stability conditions. This upper
limit of the mass of a strange star comes out as a consequence of imposing
regularity conditions, stability conditions and taking the minimum central
density of the value of nuclear density ($2.7\times 10^{14}g/cc$) Bombaci
(2018). For a given mass ($m$), we get a range of radius by imposing the
regularity conditions, energy conditions, the TOV equation, and stability
conditions. The minimum radius for a given mass ($m$) of a strange star is
$R_{min}=4.4857m$ km ($m$ in $km$) , and the upper limit of the mass is
enlisted in Table-1. This maximum radius is the immediate result of imposing
the fact that the central density is at least of the order of nuclear density.
If we change the minimum central density from the order of the nuclear
density, then the maximum radius for a given mass also deviates from the
Table-1. We see that as we increase the minimum central density (here, we take
the minimum central density as $2.7\times 10^{14}g/cc$ ), the upper bound of
radius for a given mass get squeezed. However, the minimum radius of that
strange star does not depend on the minimum central density at all. It is
determined by the fact that causality condition must be obeyed. We substitute
the value of $R_{min}$ in eqn.-25 & 26 to get the maximum value of the central
energy density and the central pressure, and we get the value of maximum
surface energy density, surface compactness and surface redshift by
substituting the value of $R_{min}$ in eqn.-11,38 & 39 respectively which is
depicted in Table-2. Here we see interesting results. The maximum central
energy density and pressure and the maximum surface energy density are
inversely proportional to the square of the mass of a strange star
corresponding to the minimum radius. But the maximum compactness and the
maximum redshift is independent of the mass of the strange star. For a given
mass, we get a range of the central energy density as well as a range of the
radius of the star from our model. Therefore, given the mass, one can get the
radius by fixing the central density or vice versa. As we approach the maximum
mass, the range of radius for a given mass gets shrink down. For greater than
one solar mass, this range of radius is very narrow, which is useful to
predict the radius of a strange star from an only input parameter that is the
mass of the strange star. Therefore, we think that this model can be very
beneficial to predict the radius for a strange star of mass greater than one
solar mass.

## Acknowledgments

MM is thankful to CSIR (Grand No.-09/1157(0007)/ 2019-EMR-I) for providing
financial support. SM and MK is grateful to the Inter-University Centre for
Astronomy and Astrophysics (IUCAA), Pune, India for providing Associateship
programme under which a part of this work was carried out.

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