# Compact stellar model in Tolman space-time in presence of pressure
anisotropy

Piyali Bhar<EMAIL_ADDRESS>Department of Mathematics, Government
General Degree College, Singur, Hooghly, West Bengal 712 409, India  Shyam Das
<EMAIL_ADDRESS>Department of Physics, P. D. Women’s College,
Jalpaiguri, West Bengal 735101, India  Bikram Keshari Parida
<EMAIL_ADDRESS>Department of Physics, Pondicherry University,
Kalapet, Puducherry 605014, India

###### Abstract

In this paper, we develop a new relativistic compact stellar model for a
spherically symmetric anisotropic matter distribution. The model has been
obtained through generating a new class of solutions by invoking the Tolman
ansatz for one of the metric potentials $g_{rr}$ and a physically reasonable
selective profile of radial pressure. We have matched our obtained interior
solution to the Schwarzschild exterior spacetime over the bounding surface of
the compact star. These matching conditions together with the condition of
vanishing the radial pressure across the boundary of the star have been
utilized to determine the model parameters. We have shown that the central
pressure of the star depends on the parameter $p_{0}$. We have estimated the
range of $p_{0}$ by using the recent data of compact stars 4U 1608-52 and Vela
X-1. The effect of $p_{0}$ on different physical parameters e.g., pressure
anisotropy, the subliminal velocity of sound, relativistic adiabatic index
etc. have also been discussed. The developed model of the compact star is
elaborately discussed both analytically and graphically to justify that it
satisfies all the criteria demanded a realistic star. From our analysis, we
have shown that the effect of anisotropy becomes small for higher values of
$p_{0}$. The mass-radius (M-R) relationship which indicates the maximum mass
admissible for observed pulsars for a given surface density has also been
investigated in our model. Moreover, the variation of radius and mass with
central density has been shown which allow us to estimate central density for
a given radius (or mass) of a compact star.

##  I Introduction

Compact stars are the ultra-high dense objects exist in nature where a huge
mass gets compacted in a small region producing the mass density beyond that
of the nuclear density Weber1 ; Glendenning ; Shapiro . A compact star is born
through a non equilibrium process of the gravitational collapse refers
collectively to a neutron star, made out of neutron-rich nuclear matter or, a
hybrid star made of nuclear matter with a quark matter core or, strange star
exclusively composed of quark matter. White dwarfs which are less dense than
neutron stars and black holes may be included in this compact star category.
Compact stars are unique objects that manifest themselves across a wide range
of multi-messenger signals like electromagnetic radiation from radio to gamma-
rays, cosmic rays, neutrinos, and gravitational waves. It has been observed
that compact stars may have huge magnetic fields, especially at the surface.
Their extreme density, gravity and magnetic fields make them exceptional
astrophysical laboratories for exploring the fundamental theories and
interactions of elementary particles and testing general relativity at extreme
conditions. Compact stars are not only extreme concerning their density but
some of them also possess rotation in the millisecond regime. In fact, the
first compact star has been discovered as pulsars, by observing pulsating
radio signals, for the first time in 1967 Hewish .

The general theory of relativity (GR) attracts a lot of interest in the fields
of astrophysics, cosmology and gravitational-wave astronomy and it potentially
leads to major breakthroughs since its introduction in 1915. Adoption of GR by
the scientific community has increased dramatically in understanding the
physics of compact stellar objects over the years. Following the discovery of
quasars in the 1960s, and other very high energy phenomena in the universe
such as gamma-ray bursts gravitation theory and relativistic astrophysics have
gone through extensive developments in recent decades and the study of a
compact star has got tremendous momentum. In relativistic astrophysics,
studies of compact stars remain a field of active research since it can be
used as a testbed for general relativity as well as particle behavior in the
extreme conditions. Understanding the behavior and properties of stellar
structure in the strong-field regime remains one of the fundamental questions.

Einstein field Equation is the cornerstone of General Relativity (GR). Exact
solutions of the field equations are essential for describing the physical
behavior of an astrophysical compact object. Unfortunately, the high
nonlinearity associated with the field equations makes the underlying
calculations complicated with mathematical difficulties. There are very few
exact interior solutions of the field equations satisfying the required
general physical conditions inside the star. Delgaty and Lake Delgaty  have
found 127 solutions out of which only 16 qualify the test to meets all the
conditions required for describing a physically realistic system. The study of
general relativistic compact stellar objects via finding exact solutions of
the field equations compatible with observational data has remained one of the
major research areas in relativistic astrophysics. Observational data
available with the advent of gravitational wave astronomy (LIGO, Virgo, KAGRA,
LISA), as well as to high-angular-resolution observations of black hole
vicinity (EHT, VLTI/GRAVITY) stimulate many researchers to contribute
immensely to the theoretical development of the physics of compact objects.

For a compact star, its mass and radius are considered as the basic
properties. One can connect these with the microscopic properties of nuclear
and quark matter. This link is made by the equation of state (EOS) of the
matter phase(s) inside the star which relates the pressure with the energy
density of dense matter. For the study of compact stars, one requires proper
understanding of the equation of state (EOS) corresponding to the material
compositions of the star. The equation of state used to study the compact
star, in particular, for an estimation of the accurate size, maximum mass and
other physical features. With the knowledge of the EOS with proper boundary
conditions, physical properties (such as mass-radius relationships) of the
star can be analyzed by integrating the equation of hydrostatic equilibrium
known as Tolman-Oppenheimer-Volkoff OV  equations. In particular, a stiff
equation of state produces a large maximum mass.

In literature, assuming the different form of an equation of state (EOS)
researchers generated many exact solutions to the field equations for
describing a realistic compact object. For example, a linear EOS have been
found to develop stellar structure by many researchers MAK ; Sharma1 ; Thiru1
; Maharaj ; Thiru2 ; Govender1 . Exact solutions to Einstein field equations
for an anisotropic sphere admitting a quadratic type of EOS Feroze ; Maharaj2
; Sunzu , polytropic type Thiru3 ; Mafa ; Ngubelanga ; Isayev ; Nasim  or Van
der Waals type EOS Thiru4  have also been found in the literature.

One of the main difficulty is the uncertainty estimation in the EOS, which is
core to the modeling of compact stellar objects and to understand the physical
behavior of a compact object. Such limitation on EOS, encourage many
researchers to adopt various ad-hoc approaches for a wide variety scenario of
astrophysical systems. Amongst many alternative techniques, one either assumes
the geometry or the fall-off behavior of density or pressure of the matter
source. Towards this direction the geometrical approach suggested by Vaidya
and Tikekar VT and Tikekar Tikekar  having compact three-spheroidal geometry
or Finch-Skea ansatz Finch  where the associated space-time is paraboloidal
are very useful for describing a super-dense compact object. Such alternative
methods are useful for the development of stellar models as shown by many
investigators such as MaharajL ; Mukherjee ; GuptaJ ; TT ; Jotania ; Sharma02S
; JohnSD ; ThomasVO05 ; ThomasVO07 ; TikVO ; SharmaBS ; PandyaVO ; SharmaSDS ;
PatelSS ; MuradS13 ; Murad2S13 ; FatemaHM13 ; Fatema2HM13 ; MuradS15 ;
ThirukkaneshSD06 ; MaharajS06 ; KomathirajSD ; IvanovBV02 ; Tikekar84 ;
Sharma06SS ; Tik90 ; Maharaj96 ; Tik98 ; Mukherjee97 ; Tikekar99VO ;
Chattopadhyay10 ; Chattopadhyay12 . Several simple but effective approaches to
solve the field equations have been also used where the two metric potentials
$g_{tt}$ and $g_{rr}$ are in general linked through an equation. The
dependency of the metric potentials describes as Karmakar embedding class one
method, conformally flat geometry and conformal motion respectively.

The first interior solution Karl  corresponding to a spherically symmetric
stellar object was given by Karl Schwarzschild by imposing the isotropic
condition on the Einstein equations. The matter under consideration was
treated to be perfect fluid, which has an equal radial ($p_{r}$) and
tangential ($p_{t}$) pressures. Later, Lemaitre in $1933$ Lemaitre  developed
a constant density anisotropic stellar model supported with tangential
pressure only. This model was generalized for variable density by Florides
Florides . In $1972$, Ruderman Ruderman  and Canuto Canuto  pointed out that
at very high-density nuclear matter may become anisotropic in general. From
the theoretical point of view, it is important to include the pressure
anisotropy in the energy-momentum tensor describing the matter distribution of
the system for describing the relativistic stellar structure. Bowers and Liang
Bowers  generalized the equations of hydrostatic equilibrium to include the
effects of local anisotropy on relativistic fluid spheres. Their work suggests
that the anisotropy effects on the maximum equilibrium mass and surface
redshift of a compact star. Interestingly, it was shown by Ivanov BVIvanov
that by considering a compact system to be anisotropic the effects of shear,
electromagnetic field, etc. can be automatically taken into account. The
different factors that have been identified for the justification of the
existence of anisotropy within a stellar interior such as the presence of a
solid core Kippenhahn , phase transitions Sokolov , a type III superfluid, a
pion condensation Sawyer  or the presence of the electrical field Usov , slow
rotation HerreraSantos . Strong magnetic fields can also generate an
anisotropic pressure inside a self-gravitating body as pointed out by Weber
Weber . It has been shown that the mixture of two gases can be represented by
effective anisotropic fluid models Letelier ; Bayin . On the galactic scale,
the existence of the anisotropy has been pointed out Binney . Several
anisotropic models have been investigated by Maurya and Gupta Maurya14 ,
Maurya et al. Maurya15 , Pandya et al. Pandya15 , Bhar et al. Bhar14 , Murad
Murad13 , Maharaj and Mafa Takisa Maharaj12 , Mafa Takisa et al. Mafa14a ;
Mafa14b , Sunzu et al. Sunzu14as ; Sunzu14bs . General algorithms for
generating static anisotropic solutions was also found by LakeKLake09 . Some
recent work may be found in Refs. Thiru8 ; Thiru6 ; Kumar ; Thiru7 ; Maurya1 ;
THarko ; Harko1 ; Harko2 ; Harko3 ; Mak02 ; MakD02 . A complete review on
anisotropic fluid spheres can be found by the work of Herrera and Santos
Herrera97San .

As far as the dynamical evolution of a self-gravitating system is concerned
the anisotropic effects on the evolution was investigated by Herrera and co-
workers HerreraP98 ; Herrera04T ; Herrera11I , Chan et al. Chan03J , Govinder
et al. Govinder98ks , Herrera and Santos Herrera02O , Chan et al. Chan94R ,
Naidu et al. Naidu06NF  and Rajah and Maharaj RajahSD . Conformally flat
solutions corresponding to anisotropic compact self-gravitating objects were
developed by Herrera et al. HerreraAdi ; HerreraCon . Raposo et al. also
raposo  studied on the dynamical properties of anisotropic self-gravitating
fluids in a covariant framework.

Inspired by the previous work done by several researchers, in the present
paper we develop a model of a compact star by assuming a physically reasonable
choice for the radial pressure. Our paper has been organized as follows. In
sect. II Einstein field equations have been described. Sect. III deals with
the solutions of the field equation. In the next section (sect. IV) we have
matched our interior solution to the exterior Schwarzschild spacetime at the
boundary. The physical attributes and the stability condition of the model are
respectively given in sect. V and VI. Some discussions and concluding remarks
have been given in sect. VIII.

##  II Interior Spacetime and Einstein field Equations

It is well known that the energy-momentum tensor for an anisotropic model of
the compact object can be described by,

$T_{\chi}^{\xi}=(\rho+p_{t})u^{\xi}u_{\chi}+p_{t}g_{\chi}^{\xi}+(p_{r}-p_{t})\eta^{\xi}\eta_{\chi},$
(1)

where $\eta^{i}$ is the space-like vector and the vector $u_{i}$ represent
fluid $4$-velocity and which is orthogonal to $\eta^{i}$ with
$-u^{i}u_{j}=\eta^{i}\eta_{j}=1$ and $u^{i}\eta_{j}=0$. $\rho,\,p_{r}$ and
$p_{t}$ are the matter density, radial pressure and tangential pressure
respectively.
In curvature co-ordinate $(t,\,r,\,\theta,\,\phi)$ for a static spherically
symmetry configuration in (3+1)-dimension, the interior spacetime is described
by the line element,

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

where $\lambda$ and $\nu$ being metric potentials and are the functions of the
radial coordinate ‘r’.
Using the Einstein field equations $G_{\mu\nu}=\kappa T_{\mu\nu}$, we get the
following set of equations

$\displaystyle\kappa c^{2}\rho$ $\displaystyle=$
$\displaystyle\frac{\lambda^{\prime}}{r}e^{-\lambda}+\frac{1}{r^{2}}(1-e^{-\lambda}),$
(3) $\displaystyle\kappa p_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}(e^{-\lambda}-1)+\frac{\nu^{\prime}}{r}e^{-\lambda},$
(4) $\displaystyle\kappa p_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-\lambda}\left[2\nu^{\prime\prime}+\nu^{\prime
2}-\lambda^{\prime}\nu^{\prime}+\frac{2}{r}(\nu^{\prime}-\lambda^{\prime})\right],$
(5)

where $\kappa=\frac{8\pi G}{c^{4}}$, $G_{\mu\nu}$ being Einstein tensor, $G$
is the universal gravitational constant and $c$ is the speed of the light. In
the above equations ‘prime’ denotes differentiation with respect to radial co-
ordinate $r$. In this respect we want to mention that the mass function $m(r)$
inside the radius ‘r’ can be obtained from the relation
$e^{-\lambda}=1-\frac{2m(r)}{r}$ which can directly follow from eq. (3).

Figure 1: Energy density $\rho$ is plotted against $r$ inside the stellar
interior for a possible modelling of the compact stars Vela X-1 and 4U
1608-52.

Figure 2: The metric potentials $e^{\nu}$ and $e^{\lambda}$ are plotted
against $r$ inside the stellar interior for a possible modelling of the
compact stars Vela X-1 and 4U 1608-52.

##  III Choice of metric potential and new anisotropic solution

It is well known that the solutions of the field equation which could describe
a compact stellar structure that satisfies all the physically reasonable
conditions are a very difficult job. To reduce the difficulties we apply a
geometrical approach by prescribing a suitable form of the metric function. To
solve the above set of Eqs. (3)-(5), let us take the metric potential $g_{rr}$
in the following form:

$e^{\lambda}=1+ar^{2}+br^{4},$ (6)

where ‘$a$’ and ‘$b$’ are constant parameters having the units km-2 and km-4
respectively. This metric potential was proposed by Tolmantolman1  for
describing a compact stellar object. This metric potential is free from
central singularity and monotonic increasing function of $r$. This type of
ansatz were used earlier by several authors to model compact star in both GTR
and modified gravity x1 ; x2 .
Substituting the expression for the metric potential $e^{\lambda}$ from (6)
into (3) and assuming $G=1=c$, the matter density can be obtained as,

$8\pi\rho=\frac{3a+(a^{2}+5b)r^{2}+2abr^{4}+b^{2}r^{6}}{(1+ar^{2}+br^{4})^{2}},$
(7)

Figure 3: Radial pressure $p_{r}$, transverse pressure $p_{t}$ and anisotropic
factor $\Delta$ are plotted against $r$ inside the stellar interior for a
possible modelling of the compact stars Vela X-1 and 4U 1608-52 for
$p_{0}=0.0023$, $p_{0}=0.0026$, $p_{0}=0.0029$ and $p_{0}=0.0032$ by
considering different values of ‘a’ and ‘b’ mentioned in Table II.

To find the other metric potential $e^{\nu}$ let us assume the radial pressure
in the form

$\displaystyle 8\pi p_{r}=\frac{p_{0}(1-ar^{2})}{(1+ar^{2}+br^{4})^{2}},$ (8)

where $p_{0}$ is a non-negative constant. The expression of $p_{r}$ is
reasonable due to the fact that it is monotonic decreasing function of ’$r$’
and vanishes at $r=\frac{1}{\sqrt{a}}$. Therefore, the radius of the star is
obtained as $R=\frac{1}{\sqrt{a}}$, which is a finite quantity. Moreover, it
possess a finite value of central pressure equal to $p_{0}/8\pi$ for all
$p_{0}>0$. The same type of choice of pressure was earlier used by Sharma and
Ratanpal SharmaRatan  for describing a static spherically symmetric
anisotropic stellar configuration.

Substituting the expression for the radial pressure $p_{r}$ mentioned above
and the metric potential $e^{\lambda}$ into ( 4) we get,

$\displaystyle\frac{d\nu}{dr}$ $\displaystyle=$
$\displaystyle\frac{p_{0}\,r(1-ar^{2})}{1+ar^{2}+br^{4}}+r(a+br^{2}),$ (9)

On integrating equation (9) we get,

$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{[}2ar^{2}+br^{4}-\frac{2p_{0}(a^{2}+2b)}{b\sqrt{a^{2}-4b}}\tan^{-1}\Big{(}\frac{a+2br^{2}}{\sqrt{a^{2}-4b}}\Big{)}$
(10) $\displaystyle-\frac{ap_{0}}{b}\log(1+ar^{2}+br^{4})\Big{]}+B,$

where $B$ is the constant of integration, which will be determined from the
boundary conditions. Employing the expression of the metric potentials in Eq.
(5) we get the expression of the transverse pressure $p_{t}$ as,

$\displaystyle 8\pi p_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{4\Psi^{3}}\big{[}p_{0}^{2}r^{2}(1-ar^{2})^{2}+r^{2}(a+br^{2})\Psi\xi_{1}$
(11) $\displaystyle-2p_{0}(-2+4ar^{2}++\xi_{2})\big{]},$

where $\xi_{1},\,\xi_{2}$ defined as,

$\displaystyle\xi_{1}$ $\displaystyle=3a+(a^{2}+5b)r^{2}+2abr^{4}+b^{2}r^{6},$
$\displaystyle\xi_{2}$
$\displaystyle=(a^{2}+3b)r^{4}+a(a^{2}-3b)r^{6}+(2a^{2}-b)br^{8}+ab^{2}r^{10},$
$\displaystyle\Psi$ $\displaystyle=1+ar^{2}+br^{4}.$

The anisotropic factor $\Delta=p_{t}-p_{r}$ is given by,

$\displaystyle 8\pi\Delta$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{4\Psi^{3}}\big{[}A_{1}+A_{2}r^{2}+A_{3}r^{4}+A_{4}r^{6}+A_{5}r^{8}$
(12) $\displaystyle+4ab^{3}r^{10}+b^{4}r^{12}\big{]},$

where $A_{i}$’s are constants given by,

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
3a^{2}-8ap_{0}+p_{0}^{2},$ $\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle 2(2a^{3}+4ab+a^{2}p_{0}-5bp_{0}-ap_{0}^{2}),$ $\displaystyle
A_{3}$ $\displaystyle=$ $\displaystyle
a^{4}+5b^{2}-2a^{3}p_{0}+10abp_{0}+a^{2}(14b+p_{0}^{2}),$ $\displaystyle
A_{4}$ $\displaystyle=$ $\displaystyle 2b(2a^{3}+8ab-2a^{2}p_{0}+bp_{0}),$
$\displaystyle A_{5}$ $\displaystyle=$ $\displaystyle
2b^{2}(3(a^{2}+b)-ap_{0}).$

The anisotropy factor $\Delta$ measures the anisotropy of the system and
$\frac{2\Delta}{r}$ is termed as the anisotropic force which will be repulsive
in nature if $p_{t}>p_{r}$ and attractive if the inequalities is in reverse
direction. In the coming sections we shall check the physical features of our
proposed model.

##  IV Exterior spacetime and matching condition

To fix the model parameters $a,\,b$ and $D$ we match our interior spacetime to
the exterior Schwarzschild spacetime at the boundary $r=R$, where $R$ is the
radius of the star. For our present case the interior and exterior line
element is given by,

$\displaystyle ds^{2}_{-}$ $\displaystyle=$
$\displaystyle-\exp\Big{\\{}\frac{1}{4}\Big{\\{}2ar^{2}+br^{4}+\frac{2p_{0}(a^{2}+2b)}{b\sqrt{a^{2}-4b}}\times$
(13)
$\displaystyle\tan^{-1}\Big{(}\frac{a+2br^{2}}{\sqrt{a^{2}-4b}}\Big{)}-\frac{ap_{0}}{b}\log(1+ar^{2}+br^{4})\Big{\\}}$
$\displaystyle+B\Big{\\}}dt^{2}+(1+ar^{2}+br^{4})dr^{2}$
$\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$ $\displaystyle
ds^{2}_{+}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}$
(14) $\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$

Now using the matching condition at the boundary of the star, one can get two
fundamental form. The first fundamental from consists in the continuity of the
metric potential across the boundary $r=R$. Explicitly

$e^{\lambda^{-}}|_{r=R}=e^{\lambda^{+}}|_{r=R}\quad\mbox{and}\quad
e^{\nu^{-}}|_{r=R}=e^{\nu^{+}}|_{r=R}.$ (15)

The second fundamental form gives,

$p_{r}(R)=0,$ (16)

which determines the size of the compact object. The second fundamental form
tells us that the size of a compact star can not be arbitrarily large, i.e.,
it is finite.
Therefore, from first fundamental form we obtain

$\displaystyle\ln\left(1-\frac{2M}{R}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{\\{}2aR^{2}+bR^{4}+\frac{2p_{0}(a^{2}+2b)}{b\sqrt{a^{2}-4b}}\times$
$\displaystyle\tan^{-1}\Big{(}\frac{a+2bR^{2}}{\sqrt{a^{2}-4b}}\Big{)}$
$\displaystyle-\frac{ap_{0}}{b}\log(1+aR^{2}+bR^{4})\Big{\\}}+B,$
$\displaystyle\left(1-\frac{2M}{R}\right)^{-1}$ $\displaystyle=$
$\displaystyle 1+aR^{2}+bR^{4}.$ (18)

Using the expression of radial pressure (8) from the Eq. (16) we have,

$\displaystyle R^{2}=\frac{1}{a}.$ (19)

Now solving the Eqs. (LABEL:eqm3)-(19) we get,

$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{1}{R^{2}},$ (20)
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{R^{4}}\left[\left(1-\frac{2M}{R}\right)^{-1}-2\right],$
(21) $\displaystyle B$ $\displaystyle=$
$\displaystyle\ln\left(1-\frac{2M}{R}\right)-\frac{1}{4}\Big{\\{}2aR^{2}+bR^{4}+\frac{2p_{0}(a^{2}+2b)}{b\sqrt{a^{2}-4b}}\times$
$\displaystyle\tan^{-1}\Big{(}\frac{a+2bR^{2}}{\sqrt{a^{2}-4b}}\Big{)}-\frac{ap_{0}}{b}\log(1+aR^{2}+bR^{4})\Big{\\}}.$

The above set of equations determines the constants $a$, $b$ and $B$ in terms
of mass, radius of the compact object. So we can calculate the model
parameters $a$, $b$ and $B$ numerically for a particular compact star from its
estimated mass and radius data. For our present work we have estimated the
mass and radius for the two compact stars Vela X-1vela  and 4U 1608-52 4u
which has been presented in Table I and the values of the constants $a,\,b$
are obtained in Table II where as the numerical values of $B$ is obtained in
Table III. One interesting thing we can note that here $a,\,b$ do not depends
on $p_{0}$ but $B$ does.

Figure 4: The equation of state parameters are plotted against radial
parameter $r$ for a possible modelling of the compact star Vela X-1 and 4U
1608-52 for different values of $p_{0}$ mentioned in the figures.

##  V Analysis of the solution

###  V.1 Nature of pressure and density

The density and pressure at the core of the star is obtained as,

$\rho_{c}=\frac{3a}{\kappa};~{}~{}~{}p_{c}=\frac{p_{0}}{\kappa}$

. Now it is well known that for a physically acceptable model, both the
central density and central pressure are non-negative, which gives the
following two inequalities:

$a>0;~{}~{}~{}~{}~{}p_{0}>0.$ (23)

It is an important task to find a physically reasonable bound for the central
density $p_{0}$ which we shall discuss in the coming sections.

The density and pressure gradients are obtained as,

$\displaystyle\kappa\frac{d\rho}{dr}$ $\displaystyle=$
$\displaystyle-\frac{2r}{\Psi^{3}}(5a^{2}-5b+B_{1}r^{2}+B_{2}r^{4}+3ab^{2}r^{6}+b^{3}r^{8}),$
$\displaystyle\kappa\frac{dp_{r}}{dr}$ $\displaystyle=$
$\displaystyle-\frac{2p_{0}}{\Psi^{2}}\Big{(}2br^{3}+ar(2-br^{4})\Big{)},$
$\displaystyle\kappa\frac{dp_{t}}{dr}$ $\displaystyle=$
$\displaystyle\frac{r}{2\Psi^{4}}\times\Big{[}B_{3}+2B_{4}r^{2}-B_{5}r^{4}-4B_{6}r^{6}+B_{7}r^{8}$
$\displaystyle+4bB_{8}r^{10}+B_{9}r^{12}+B_{10}r^{14}+B_{11}r^{16}\Big{]},$

where $B_{i}^{\prime}s$ are constants given by,

$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle a(a^{2}+13b),$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle 3b(a^{2}+4b),$
$\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle
3a^{2}-16ap_{0}+p_{0}^{2},$ $\displaystyle B_{4}$ $\displaystyle=$
$\displaystyle a^{3}+8ab+3a^{2}p_{0}-14bp_{0}-2ap_{0}^{2},$ $\displaystyle
B_{5}$ $\displaystyle=$ $\displaystyle
a^{4}-19a^{2}b-15b^{2}+a(17a^{2}-58b)p_{0}$
$\displaystyle+2(a^{2}+b)p_{0}^{2},$ $\displaystyle B_{6}$ $\displaystyle=$
$\displaystyle-8ab^{2}+(a^{4}+10a^{2}b-9b^{2})p_{0}$
$\displaystyle+a(-a^{2}+b)p_{0}^{2},$ $\displaystyle B_{7}$ $\displaystyle=$
$\displaystyle b(a^{4}+4a^{2}b+15b^{2})-a(a^{4}+8a^{2}b+49b^{2})p_{0}$
$\displaystyle+(a^{4}+6a^{2}b-3b^{2})p_{0}^{2},$ $\displaystyle B_{8}$
$\displaystyle=$ $\displaystyle
ab(a^{2}+b)-(a^{4}+a^{2}b+2b^{2})p_{0}+abp_{0}^{2},$ $\displaystyle B_{9}$
$\displaystyle=$ $\displaystyle b^{2}(b-ap_{0})\big{(}b+a(6a+p_{0})\big{)},$
$\displaystyle B_{10}$ $\displaystyle=$ $\displaystyle 4ab^{3}(b-ap_{0}),$
$\displaystyle B_{11}$ $\displaystyle=$ $\displaystyle b^{4}(b-ap_{0}).$

Now at the center of the star,

$\displaystyle\kappa\rho^{\prime\prime}$ $\displaystyle=$
$\displaystyle-10(a^{2}-b)<0,$ (24) $\displaystyle\kappa p_{r}^{\prime\prime}$
$\displaystyle=$ $\displaystyle-4ap_{0}<0,$ (25) $\displaystyle\kappa
p_{t}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{3a^{2}-16ap_{0}+p_{0}^{2}}{2}<0.$ (26)

From Eq. (24), we have,

$\displaystyle b<a^{2},$ (27)

and Eq. (25) is automatically satisfied. Eq. (26) implies,

$p_{0}\in(0.189a,15.81a).$ (28)

The equation of state parameters $\omega_{r}$ and $\omega_{t}$ are given by,

$\displaystyle\omega_{r}=\frac{p_{r}}{\rho}$ $\displaystyle=$
$\displaystyle\frac{p_{0}(1-ar^{2})\Psi}{\xi_{1}},$
$\displaystyle\omega_{t}=\frac{p_{t}}{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{4\Psi\xi_{1}}\Big{[}p_{0}^{2}r^{2}(1-ar^{2})^{2}\Psi++\varphi\Big{]},$

Where,

$\displaystyle\varphi$ $\displaystyle=$ $\displaystyle
p_{0}(4-a^{4}r^{8}-3a^{3}\xi_{3}-a^{2}\xi_{4}-ar^{2}\xi_{5}+br^{4}\xi_{6})$
$\displaystyle+r^{2}(a+br^{2})\Psi\xi_{1},$ $\displaystyle\xi_{3}$
$\displaystyle=$ $\displaystyle r^{6}(2+br^{4}),$ $\displaystyle\xi_{4}$
$\displaystyle=$ $\displaystyle r^{4}+11br^{8}+3b^{2}r^{12},$
$\displaystyle\xi_{5}$ $\displaystyle=$ $\displaystyle
4+br^{4}(-2+br^{4})(6+br^{4}),$ $\displaystyle\xi_{6}$ $\displaystyle=$
$\displaystyle-2+br^{4}(7+br^{4}).$

Table 1: The observed and estimated mass and radius for a possible modelling of the compact star 4U 1608-52 4u  and Vela X-1 vela . Compact Star | $M/M_{\odot}$ | R(km) | $M/M_{\odot}$ | R(km)
---|---|---|---|---
| (observed) | (observed) | (estimated) | (estimated)
4U 1608-52 | $1.74\pm 0.14$ | $10.811\pm 0.197$ | 1.74 | 10.6
Vela X-1 | $1.77\pm 0.08$ | $10.852\pm 0.108$ | 1.77 | 10.8
Table 2: The numerical values of $a,\,b$ central density $(\rho_{c})$ and surface density $(\rho_{s})$ are obtained for a possible modelling of the compact star 4U 1608-52 4u  and Vela X-1 vela . Compact Star | $a$ | $b$ | $\rho_{c}$ | $\rho_{s}$
---|---|---|---|---
units | km-2 | km-4 | gm.cm-3 | gm.cm-3
4U 1608-52 | $0.00889996$ | $-4.8392\times 10^{-6}$ | $1.43346\times 10^{15}$ | $4.54525\times 10^{14}$
Vela X-1 | $0.00857339$ | $-4.70387\times 10^{-6}$ | $1.38086\times 10^{15}$ | $4.36711\times 10^{14}$

###  V.2 Mass function and redshift

Introducing the relation between the mass function $m(r)$ and the metric
potential $e^{\lambda}$, $e^{-\lambda}=1-\frac{2m}{r}$, the expression for
mass function can be obtained as,

$m(r)=\frac{r^{3}(a+br^{2})}{2(1+ar^{2}+br^{4})}.$ (29)

Using the formulae of the compactness factor $u(r)$ and surface redshift
($z_{s}$)

$\displaystyle u(r)$ $\displaystyle=$ $\displaystyle\frac{m}{r},$
$\displaystyle 1+z_{s}(R)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-2u(R)}},$

the expression of these two quantities in our present model are obtained as,

$\displaystyle u(r)$ $\displaystyle=$
$\displaystyle\frac{r^{2}(a+br^{2})}{2(1+ar^{2}+br^{4})},$ $\displaystyle
z_{s}$ $\displaystyle=$ $\displaystyle\sqrt{1+aR^{2}+bR^{4}}-1,$

where ‘$R$’ being the radius of the star. The gravitational redshift is
obtained as,

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{e^{\nu}}}-1,$
$\displaystyle=$
$\displaystyle\bigg{[}\exp\Big{\\{}2ar^{2}+\frac{1}{4}\Big{[}\frac{2p_{0}(a^{2}+2b)}{b\sqrt{a^{2}-4b}}\tan^{-1}\Big{(}\frac{a+2br^{2}}{\sqrt{a^{2}-4b}}\Big{)}$
$\displaystyle+br^{4}-\frac{ap_{0}}{b}\log(1+ar^{2}+br^{4})\Big{]}+B\Big{\\}}\bigg{]}^{-\frac{1}{2}}-1.$

Now the central value of the gravitational redshift is obtained as,

$\displaystyle
Z_{0}=\frac{1}{\sqrt{\bigg{[}\exp\Big{\\{}\frac{1}{4}\Big{[}\frac{2p_{0}(a^{2}+2b)}{b\sqrt{a^{2}-4b}}\tan^{-1}\Big{(}\frac{a}{\sqrt{a^{2}-4b}}\Big{)}\Big{]}+B\Big{\\}}\bigg{]}}}-1.$

The gradient of the gravitational redshift is obtained as,

$\displaystyle\frac{dZ}{dr}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{e^{\nu}}}\Big{[}r\frac{p_{0}(1-ar^{2})}{1+ar^{2}+br^{4}}+r(a+br^{2})\Big{]}.$

Now at the center of the star $\frac{dZ}{dr}=0$, and

$\left(\frac{d^{2}Z}{dr^{2}}\right)_{r=0}=-\frac{(a+p_{0})}{2\sqrt{e^{B+\frac{(a^{2}+2b)p_{0}\tan^{-1}\left(\frac{a}{\sqrt{a^{2}-4b}}\right)}{2b\sqrt{a^{2}-4b}}}}}<0.$

It verifies that the gravitational redshift has maximum value at the center of
the star.

###  V.3 Mass-radius relationship

We have generated the mass-radius ($M-R$) relationship for our developed model
as shown in Fig. 5. The mass-radius relationship obtained for an assumed
surface density $8.5\times 10^{14}$ gm/cc. The chosen surface density is
roughly close to that considered by Sharma and Maharaj Sharma1 . The maximum
mass allowed in this model is found to be $2.7M_{\odot}$ with the
corresponding radius of value $8.9$ km. This limit on maximum mass for a
neutron star is approximately $3.2M_{\odot}$ Ruffini .

Figure 5: Mass-radius relationship.

###  V.4 Radius-Central density and mass-central density relation

For compact objects we have plotted the variation of the radius and mass with
the central density in Fig. 6. This plot allows us to determine the central
density of compact star once the radius or the mass of the corresponding star
is known. Similar type of study can be found in the work of Deb et al. Deb18 .

Figure 6: Variation of radius and mass with central density.

###  V.5 Energy Conditions

It is well known that for a compact star model, the energy conditions should
be satisfied and in this subsection we are in a position to study about it.
For an anisotropic compact star, all the energy conditions like, Weak Energy
Condition (WEC), Null Energy Condition (NEC), Strong Energy Condition (SEC)
and dominant energy conditions (DEC) are satisfied if and only if the
following inequalities hold simultaneously for every points inside the stellar
configuration.

Figure 7: NEC, WEC, SEC and DEC are plotted against the radius $r$ for a
possible modelling of the compact star Vela X-1 and 4U 1608-52 for different
values of $p_{0}$ mentioned in the figures.

$\displaystyle WEC$ $\displaystyle:$ $\displaystyle
T_{\mu\nu}t^{\mu}t^{\nu}\geq 0~{}\mbox{or}~{}\rho\geq 0,~{}\rho+p_{i}\geq 0$
(30) $\displaystyle NEC$ $\displaystyle:$ $\displaystyle
T_{\mu\nu}l^{\mu}l^{\nu}\geq 0~{}\mbox{or}~{}\rho+p_{i}\geq 0$ (31)
$\displaystyle DEC$ $\displaystyle:$ $\displaystyle
T_{\mu\nu}t^{\mu}t^{\nu}\geq 0~{}\mbox{or}~{}\rho\geq|p_{i}|$ (32)
$\displaystyle SEC$ $\displaystyle:$ $\displaystyle
T_{\mu\nu}t^{\mu}t^{\nu}-\frac{1}{2}T^{\lambda}_{\lambda}t^{\sigma}t_{\sigma}\geq
0~{}\mbox{or}~{}\rho+\sum_{i}p_{i}\geq 0,$

where $i$ takes the value $r$ and $t$ for radial and transverse pressure.
$~{}t^{\mu}$ and $l^{\mu}$ are time-like vector and null vector respectively
and $T^{\mu\nu}t_{\mu}$ is nonspace-like vector. To check all the inequality
stated above we have drawn the profiles of l.h.s of (30)-(V.5) in fig 7 in the
interior of the compact star PSR J 1614-2230 and 4U1608-52. The figure shows
that all the energy conditions are well satisfied by our present model of
compact star.

##  VI Stability analysis

In this section we want to check the stability of the present model with the
help of (i) causality condition, (ii) relativistic adiabatic index (iii)
Harrison-Zeldovich-Novikov’s stability condition and (iv) tov equation.

###  VI.1 Causality Condition

For a compact star model, the radial and transverse velocity of sound is
obtained as,

$V_{r}^{2}=\frac{dp_{r}}{d\rho},\hskip
28.45274ptV_{t}^{2}=\frac{dp_{t}}{d\rho}.$

Figure 8: $V_{r}^{2},\,~{}V_{t}^{2}$ and $V_{t}^{2}-V_{r}^{2}$ are plotted
against $r$ for $p_{0}=0.0023$, $p_{0}=0.0026$, $p_{0}=0.0029$ and
$p_{0}=0.0032$ respectively for different values of ‘a’ and ‘b’ mentioned in
Table II for a possible modelling of the compact stars Vela X-1 and 4U
1608-52.

Using the above formulae, for our present model, the radial and transverse
velocity of sound are calculated as,

$\displaystyle V_{r}^{2}$ $\displaystyle=$
$\displaystyle\frac{p_{0}\Psi}{\zeta}\Big{(}2br^{2}+a(2-br^{4})\Big{)},$ (34)
$\displaystyle V_{t}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4\Psi\zeta}\Big{[}C_{3}-2C_{4}r^{2}+C_{5}r^{4}+4C_{6}r^{6}+C_{7}r^{8}$
(35) $\displaystyle-C_{8}r^{10}+C_{9}r^{12}+C_{10}r^{14}+C_{11}r^{16}\Big{]},$

where

$\displaystyle\zeta=5a^{2}-5b+C_{1}r^{2}+C_{2}r^{4}+3ab^{2}r^{6}+b^{3}r^{8},$

and $C_{i}^{\prime}s$ are constants given by,

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle a(a^{2}+13b),$
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle 3b(a^{2}+4b),$
$\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle-3a^{2}+16ap_{0}-p_{0}^{2},$ $\displaystyle C_{4}$
$\displaystyle=$ $\displaystyle a^{3}+8ab+3a^{2}p_{0}-14bp_{0}-2ap_{0}^{2},$
$\displaystyle C_{5}$ $\displaystyle=$ $\displaystyle
a^{4}-19a^{2}b-15b^{2}+a(17a^{2}-58b)p_{0}$
$\displaystyle+2(a^{2}+b)p_{0}^{2},$ $\displaystyle C_{6}$ $\displaystyle=$
$\displaystyle-8ab^{2}+(a^{4}+10a^{2}b-9b^{2})p_{0}+a(-a^{2}+b)p_{0}^{2},$
$\displaystyle C_{7}$ $\displaystyle=$ $\displaystyle
a^{5}p_{0}+8a^{3}bp_{0}+49ab^{2}p_{0}+3b^{2}(-5b+p_{0}^{2})$
$\displaystyle-a^{4}(b+p_{0}^{2})-2a^{2}b(2b+3p_{0}^{2}),$ $\displaystyle
C_{8}$ $\displaystyle=$ $\displaystyle
4b\big{(}ab(a^{2}+b)-(a^{4}+a^{2}b+2b^{2})p_{0}+abp_{0}^{2}\big{)},$
$\displaystyle C_{9}$ $\displaystyle=$ $\displaystyle
b^{2}(-b+ap_{0})\big{(}b+a(6a+p_{0})\big{)},$ $\displaystyle C_{10}$
$\displaystyle=$ $\displaystyle 4ab^{3}(-b+ap_{0}),$ $\displaystyle C_{11}$
$\displaystyle=$ $\displaystyle b^{4}(-b+ap_{0}).$

Now at the point $r=0$, the square of radial and transverse velocity of sound
are obtained as,

$\displaystyle
V_{r}^{2}=\frac{3ap_{0}}{5(a^{2}-b)},~{}~{}V_{t}^{2}=-\frac{3a^{2}-20ap_{0}+p_{0}^{2}}{20(a^{2}-b)}.$

Moreover, at the center of the star,

$\displaystyle V_{t}^{2}-V_{r}^{2}$ $\displaystyle=$
$\displaystyle-\frac{3a^{2}-8ap_{0}+p_{0}^{2}}{20(a^{2}-b)}.$ (36)

Now by using Le Chatelier’s principle we have the speed of sound must be
positive inside the stellar interior, i.e.,
$0<\frac{p_{r}^{\prime}}{\rho^{\prime}},\frac{p_{t}^{\prime}}{\rho^{\prime}}$.
We also know that for a physically acceptable model, the velocity of the sound
(both radial and transverse) should be less than the speed of the light i.e.,
both
$\frac{p_{r}^{\prime}}{\rho^{\prime}},\frac{p_{t}^{\prime}}{\rho^{\prime}}<1$
which is known as the causality condition. Here $\rho^{\prime}$ and
$p_{r}^{\prime}$ represent differentiation with respect to r. Combining the
above two inequalities we have, $0\leq V_{r}^{2},~{}V_{t}^{2}\leq 1.$

To find a potentially stable region, in 1992, Herrera proposed a method which
is known as “cracking method”. This method tells us that a stellar model will
be potentially stable if the square of radial velocity of sound exceeds the
square of transverse velocity of sound everywhere within the stellar model,
otherwise the stellar model will be potentially unstable i.e.,

$\displaystyle
V_{t}^{2}-V_{r}^{2}\left\\{\begin{array}[]{ll}<0~{}\text{for}~{}0\leq r\leq
R~{}\Rightarrow{}&\hbox{potentially stable}\\\ >0~{}\text{for}~{}0\leq r\leq
R~{}\Rightarrow{}&\hbox{unstable}.\end{array}\right.$

Now at the center of the star, $V_{t}^{2}-V_{r}^{2}~{}<0$ gives,

$\displaystyle-\frac{3a^{2}-8ap_{0}+p_{0}^{2}}{20(a^{2}-b)}<0$ (38)

Using, Eq. (27) from Eq. (38), we further obtain the following inequality:

$\displaystyle 3a^{2}-8ap_{0}+p_{0}^{2}>0.$

The above inequality gives,

$\displaystyle p_{0}\not\in(0.3944a,~{}7.6055a).$ (39)

Combining (28) and (39), we further get,

$\displaystyle p_{0}\in(0.18975a,~{}0.3944a)\cup(7.6055a,~{}15.8102a).$ (40)

Hence we have obtained a range for $p_{0}$ that could describe a physically
reasonable stellar structure.

###  VI.2 Relativistic Adiabatic index

The adiabatic index determines the stability of a compact object and for an
anisotropic stellar configuration it is defined as,

$\displaystyle\Gamma_{r}=\frac{\rho(r)+p_{r}(r)}{p_{r}(r)}\,\frac{dp_{r}(r)}{d\rho(r)}.$
(41)

Figure 9: Adiabatic Index is plotted against $r$ inside the stellar interior
for a possible modelling of the compact star Vela X-1 and 4U 1608-52 for
different values of $p_{0}$ mentioned in the figures.

For our model we have

$\displaystyle\Gamma_{r}$ $\displaystyle=$
$\displaystyle\frac{3a+p_{0}+(a^{2}+5b)r^{2}+D_{1}r^{4}+D_{2}r^{6}}{p_{0}(1-ar^{2})\Psi}V_{r}^{2},$

where $D_{1}$ and $D_{2}$ are constants given by,

$D_{1}=2ab-a^{2}p_{0}+bp_{0}~{};~{}~{}D_{2}=b(b-ap_{0}).$

Any stellar configuration will maintain its stability if adiabatic index
$\Gamma_{r}>4/3$ Heintzmann . For our solution, the adiabatic index
$\Gamma_{r}$ takes the value more than $4/3$ throughout the interior of the
compact star, as evident from Fig. 9.

Table 3: The numerical values of central pressure ($p_{c}$) in dyne$\cdot$cm-2 unit, surface anisotropy in dyne$\cdot$cm-2 unit, central radial velocity ($V_{r}(0)$), surface radial velocity ($V_{r}(R)$), central transverse velocity ($V_{t}(0)$), surface transverse velocity ($V_{t}(R)$) for a possible modelling of the compact star 4U 1608-52 4u . $p_{0}$ | $p_{c}$ | $\Delta_{s}=p_{t}(R)$ | $V_{r}(0)$ | $V_{r}(R)$ | $V_{t}(0)$ | $V_{t}(R)$ | $\Gamma_{r}(0)$
---|---|---|---|---|---|---|---
$0.0023$ | $1.11134\times 10^{35}$ | $6.64583\times 10^{34}$ | $0.382269$ | $0.305075$ | $0.314703$ | $0.185608$ | $1.84249$
$0.0026$ | $1.2563\times 10^{35}$ | $6.26024\times 10^{34}$ | $0.406435$ | $0.324362$ | $0.360459$ | $0.212585$ | $1.86155$
$0.0029$ | $1.40126\times 10^{35}$ | $5.87465\times 10^{34}$ | $0.429243$ | $0.342564$ | $0.400895$ | $0.236505$ | $1.88061$
$0.0032$ | $1.54621\times 10^{35}$ | $5.48905\times 10^{34}$ | $0.450899$ | $0.359847$ | $0.437487$ | $0.258218$ | $1.89967$

###  VI.3 Tov equation

Now we are ready to check the stability of our present model under three
different forces $viz$ gravitational force $F_{g}$, hydrostatics force $F_{h}$
and anisotropic force $F_{a}$. The effect of the above three forces can be
described by the conservation equation given by

$\nabla^{\mu}T_{\mu\nu}=0,$ (42)

known as TOV equation. Now using the expression given in (1) into (42) one can
obtain the following equation:

$-\frac{\nu^{\prime}}{2}(\rho+p_{r})+\frac{2}{r}(p_{t}-p_{r})=p_{r}^{\prime}.$
(43)

The eqn. (43) can be written as,

$F_{g}+F_{h}+F_{a}=0,$ (44)

Figure 10: Different forces are plotted against radial parameter $r$ for a
possible modelling of the compact star Vela X-1 and 4U 1608-52 for different
values of $p_{0}$ mentioned in the figures.

where the expression for $F_{g},\,F_{h}$ and $F_{a}$ are obtained as:

$\displaystyle F_{g}$ $\displaystyle=$
$\displaystyle-\frac{\nu^{\prime}}{2}(\rho+p_{r})$ (45) $\displaystyle=$
$\displaystyle\frac{r}{2\kappa\Psi^{3}}\Big{\\{}\Big{(}a+p_{0}+(a^{2}+b-ap_{0})r^{2}+2abr^{4}$
$\displaystyle+b^{2}r^{6}\Big{)}\times\Big{(}3a+p_{0}+(a^{2}+5b-ap_{0})r^{2}$
$\displaystyle+2abr^{4}+b^{2}r^{6}\Big{)}\Big{\\}},$ $\displaystyle F_{h}$
$\displaystyle=$
$\displaystyle-\frac{dp_{r}}{dr}=\frac{2p_{0}}{\kappa\Psi^{2}}\Big{(}2br^{3}+ar(2-br^{4})\Big{)},$
(46) $\displaystyle F_{a}$ $\displaystyle=$
$\displaystyle\frac{2\Delta}{r}=\frac{r}{8\kappa\Psi^{3}}\big{[}A_{1}+A_{2}r^{2}+A_{3}r^{4}+A_{4}r^{6}+A_{5}r^{8}$
(47) $\displaystyle+4ab^{3}r^{10}+b^{4}r^{12}\big{]}.$

The three different forces acting on the system are shown in fig. 10 for the
compact stars Vela X-1 and 4U 1608-52 for different values of $p_{0}$.

###  VI.4 Harrison-Zeldovich-Novikov’s stability condition

Depending on the mass and central density of the star, Harrison et al. harri
and Zeldovich-Novikov novi  proposed the stability condition for the model of
compact star. From their investigation they suggested that for stable
configuration $\frac{\partial M}{\partial\rho_{c}}>0$, where $M,\,\rho_{c}$
denotes the mass and central density of the compact star.

Figure 11: Variation of mass (M/$M_{\odot}$) and $\frac{\partial
M}{\partial\rho_{c}}$ with respect to the central density $\rho_{c}$ for a
possible modelling of the compact star Vela X-1 and 4U 1608-52.

For our present model,

$\displaystyle\frac{\partial M}{\partial\rho_{c}}=\frac{12\pi
R^{3}\left(1+b(R^{4}-R^{6})\right)}{(3+3bR^{4}+8\pi R^{2}\rho_{c})^{2}}.$ (48)

Above expression of $\frac{\partial M}{\partial\rho_{c}}$ is positive and
hence the stability condition is well satisfied. The variation of the mass
function as well as $\frac{\partial M}{\partial\rho_{c}}$ with respect to the
central density is depicted in fig. 11.

Table 4: The numerical values of central pressure ($p_{c}$) in dyne$\cdot$cm-2 unit, surface anisotropy in dyne$\cdot$cm-2 unit, central radial velocity ($V_{r}(0)$), surface radial velocity ($V_{r}(R)$), central transverse velocity ($V_{t}(0)$), surface transverse velocity ($V_{t}(R)$) for a possible modelling of the compact star Vela X-1 vela . $p_{0}$ | $p_{c}$ | $\Delta_{s}=p_{t}(R)$ | $V_{r}(0)$ | $V_{r}(R)$ | $V_{t}(0)$ | $V_{t}(R)$ | $\Gamma_{r}(0)$
---|---|---|---|---|---|---|---
$0.0023$ | $1.11134\times 10^{35}$ | $6.2321\times 10^{34}$ | $0.38895$ | $0.311366$ | $0.328293$ | $0.200689$ | $1.84302$
$0.0026$ | $1.2563\times 10^{35}$ | $5.84535\times 10^{34}$ | $0.413539$ | $0.33105$ | $0.373797$ | $0.227046$ | $1.86275$
$0.0029$ | $1.40126\times 10^{35}$ | $5.45861\times 10^{34}$ | $0.436746$ | $0.349628$ | $0.414194$ | $0.250647$ | $1.88248$
$0.0032$ | $1.54621\times 10^{35}$ | $5.07186\times 10^{34}$ | $0.45878$ | $0.367267$ | $0.450859$ | $0.272209$ | $1.90222$

###  VI.5 Linearized stability analysis

We have already matched the interior space-time continuously to an exterior
schwarzschild vacuum solution with $p_{r}=0$ at the junction interface
$\Sigma$, with junction radius R.
Now by using the standard Darmois-Israel formalism israel1 ; israel2  and with
the help of Lanczos equations the surface stresses can be obtained, as follows
:

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle-\frac{1}{4\pi
R}\Big{[}\sqrt{1-\frac{2M}{R}+\dot{R}^{2}}$ (49)
$\displaystyle-\sqrt{\frac{1}{(1+aR^{2}+bR^{4})}+\dot{R}^{2}}~{}\Big{]},$

and

$\displaystyle\mathcal{P}$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
R}\Big{[}\frac{1-\frac{M}{R}+\dot{R}+R\ddot{R}}{\sqrt{1-\frac{2M}{R}+\dot{R}^{2}}}$
(50)
$\displaystyle-\frac{(1+aR^{2}+bR^{4})^{-1}+\dot{R}^{2}\frac{1-bR^{4}}{1+aR^{2}+bR^{4}}+R\ddot{R}}{\sqrt{(1+aR^{2}+bR^{4})^{-1}+\dot{R}^{2}}}\Big{]}$

Where $\sigma$ and $\mathcal{P}$ represent the surface density and tangential
surface pressure of the internal forces on the junction surface.

We now proceed with the surface mass of the thin shell is given by

$m_{s}=4\pi R^{2}\sigma.$ (51)

When one substitutes Eq. (49) into Eq. (51), one can get

$m_{s}=-R\left[\sqrt{1-\frac{2M}{R}+\dot{R}^{2}}-\sqrt{\dot{R}^{2}+(1+aR^{2}+bR^{4})^{-1}}\right],$
(52)

Now, differentiating twice the expression in (51), and taking into account the
radial derivative of $\sigma^{\prime}$, we can obtain,

$\left(\frac{m_{s}}{2R}\right)^{\prime\prime}=\Upsilon-4\pi~{}\sigma^{\prime}\eta,$
(53)

where the parameters $\eta$ and $\Upsilon$ are given by,

$\eta=\frac{\mathcal{P}^{\prime}}{\sigma^{\prime}},~{}~{}~{}~{}\Upsilon=\frac{4\pi}{R}(\sigma+\mathcal{P})+2\pi~{}R\Xi^{\prime}.$
(54)

The expression found above will play an important role in stability analysis
of static solutions. Here the parameter $\eta$ is used to determine the
stability of the system. $\eta$ can be interpreted as the square of velocity
of sound on the shell and it lies in the range $0<\eta\leq 1$. For our present
model,

$\Xi=\frac{aR+2bR^{3}}{4\pi+4a\pi R^{2}+4b\pi
R^{4}}\sqrt{\dot{R}^{2}+(1+aR^{2}+bR^{4})^{-1}}.$

Now we are going to check for the stability analysis of the solution. In order
to study the dynamical stability of the stellar model we consider a linear
perturbation around those static solutions.
Rearranging (49), we get,

$\dot{R}^{2}+V(R)=0,$ (55)

with $V(R)$ given by

$V(R)=1-\frac{m(R)+M}{R}-\left(\frac{m_{s}(R)}{2R}\right)^{2}-\left(\frac{M-m(R)}{m_{s}(R)}\right)^{2}.$
(56)

Note that the potential function $V(R)$ helps us to determine the stability
region for the thin shell under the linear perturbation. By considering the
Taylor series expansion around the static solution $R_{0}$, up to second order
we get,

$\displaystyle V(R)$ $\displaystyle=$ $\displaystyle
V(R_{0})+(R-R_{0})V^{\prime}(R_{0})+\frac{(R-R_{0})^{2}}{2}V^{\prime\prime}(R_{0})$
(57) $\displaystyle+\mathcal{O}[(R-R_{0})^{3}],$

where prime corresponding to a derivative with respect to R. According to the
standard method we are linearizing around the static radius $R=R_{0}$, we must
have $V(R_{0})=0$ and $V^{\prime}(R_{0})=0$. Now, $V^{\prime}(R_{0})=0$ gives
the following relation

$\displaystyle\left(\frac{m_{s}(R_{0})}{2R_{0}}\right)^{\prime}=\Phi$
$\displaystyle=$
$\displaystyle\frac{R_{0}}{m_{s}(R_{0})}\left[-\left(\frac{m(R_{0})+M}{R_{0}}\right)^{\prime}\right.$
$\displaystyle\left.-2\left(\frac{M-m(R_{0})}{m_{s}(R_{0})}\right)\left(\frac{M-m(R_{0})}{m_{s}(R_{0})}\right)^{\prime}~{}\right]$
. (58)

With this definition the second derivative $V^{\prime\prime}(R_{0})$ can be
written as

$\displaystyle V^{\prime\prime}(R_{0})$ $\displaystyle=$
$\displaystyle-\left(\frac{m+M}{R_{0}}\right)^{\prime\prime}-2\left[\left(\frac{m_{s}}{2R_{0}}\right)^{\prime}\right]^{2}$
(59)
$\displaystyle-2\left(\frac{m_{s}}{2R_{0}}\right)\left(\frac{m_{s}}{2R_{0}}\right)^{\prime\prime}-2\left[\left(\frac{M-m}{m_{s}}\right)^{\prime}~{}\right]^{2}$
$\displaystyle-2\left(\frac{M-m}{m_{s}}\right)\left(\frac{M-m}{m_{s}}\right)^{\prime\prime}.$

Thus, for a static configuration $V(R_{0})=0,\,V^{\prime}(R_{0})=0$. Now from
(55) and (57), we get,

$\displaystyle\dot{R}^{2}+\frac{1}{2}V^{\prime\prime}(R_{0})\left(R-R_{0}\right)^{2}+\mathcal{O}[(R-R_{0})^{3}]=0,$
(60)

To ensure the stability of static configuration at R = $R_{0}$, we must have,
$V^{\prime\prime}(R_{0})>0$. For the sake of simplicity we rearrange the Eq.
(60), which turns out

$\Rightarrow~{}V^{\prime\prime}=\Pi-2\Phi^{2}-\frac{m_{s}}{R_{0}}\left(\Upsilon-4\pi~{}\sigma^{\prime}\eta\right)\Big{|}_{R_{0}}.$
(61)

where,

$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle-\left(\frac{m+M}{R_{0}}\right)^{\prime\prime}-2\left[\left(\frac{M-m}{m_{s}}\right)^{\prime}\right]^{2}$
(62)
$\displaystyle-2\left(\frac{M-m}{m_{s}}\right)\left(\frac{M-m}{m_{s}}\right)^{\prime\prime}.$

(For details calculations please refer ref pb15 .)

Now by assuming $\eta(R_{0})=\eta_{0}$ and using Eq. (61) for
$V^{\prime\prime}(R_{0})>0$ we have

$\eta_{0}\frac{d\sigma^{2}}{dR}\Big{|}_{R_{0}}>\frac{\sigma}{2\pi}\left[\Upsilon+\frac{R_{0}}{m_{s}}(2\Phi^{2}-\Pi)\right]=\Omega~{}(\mathrm{say}).$
(63)

From the above inequality we get the following two cases,

$\eta_{0}>\Omega\left(\frac{d\sigma^{2}}{dR}\Big{|}_{R_{0}}\right)^{-1},~{}~{}~{}\mathrm{if}~{}~{}~{}\frac{d\sigma^{2}}{dR}\Big{|}_{R_{0}}>0,$
(64)
$\eta_{0}<\Omega\left(\frac{d\sigma^{2}}{dR}\Big{|}_{R_{0}}\right)^{-1},~{}~{}~{}\mathrm{if}~{}~{}~{}\frac{d\sigma^{2}}{dR}\Big{|}_{R_{0}}<0.$
(65)

Now to make a comment regarding the stability region of the stellar
configuration, the profile of $\frac{d\sigma_{0}^{2}}{dR_{0}}$ is plotted in
fig 12. So, we can conclude that the stability region is given by eq. (65)
since $\frac{d\sigma_{0}^{2}}{dR_{0}}<0$ for our present paper.

Figure 12: $\frac{d\sigma_{0}^{2}}{dR_{0}}$ has been shown against $R_{0}$

##  VII Calculation of tidal love number of compact neutron star with
anisotropic pressure

Gravitational perturbation are very much important, though very much difficult
to handle, since they convey information about the structure of the
gravitational field equations. So let’s start by trying to answer these
question: Given a metric for instance Schwarzschild geometry or metric of a
compact neutron star, how does a small perturburtation in the metric evolve?
Here small perturbation means anything- a wave, a celestial body, a falling
particle - that disturb the background metric slightly. Mathematically
consider a background metric ${}^{(0)}g_{\mu\nu}(x^{\nu})$\- metric of a
neutron star. With small perturbation $h_{\mu\nu}(x^{\nu})$, the modified new
metric can be written as

$\displaystyle
g_{\mu\nu}\left(x^{\nu}\right)=^{(0)}g_{\mu\nu}\left(x^{\nu}\right)+h_{\mu\nu}\left(x^{\nu}\right),$
(66)

where the background geometry of spacetime of a spherical static star can be
written as

${}^{\left(0\right)}ds^{2}$
$\displaystyle=^{\left(0\right)}g_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (67)

For the linearized metric perturbation $h_{\alpha\beta}$, using the method as
in Regge1957 , Biswas2019 , we restrict ourselves to static $l=2,\,m=0$ even
parity perturbation. With these assumptions the perturbed metric becomes

$\displaystyle
h_{\alpha\beta}=\operatorname{diag}\left[H_{0}(r)e^{\nu},H_{2}(r)e^{\lambda},r^{2}K(r),r^{2}\sin^{2}\theta
K(r)\right]Y_{2m}(\theta,\phi),$ (68)

Because of these external perturbation the equalibrium configuration of a star
gets deformed tidally. As a cosequence spherically symmetric star develpes a
quadrapole moment $Q_{ij}$. And $Q_{ij}$ can be related with the linear order
external tidal field $\varepsilon_{ij}$ as Hinderer2008

$\displaystyle Q_{ij}=-\Lambda\,\varepsilon_{ij},$ (69)

where $\Lambda$ is the tidal deformability of the neutron star and it is
related to the tidal love number $k_{2}$ as $\Lambda=\dfrac{2}{3}k_{2}\,R^{5}$
Hinderer2008 .

Now for the spherically static metic (67), the stress-energy tensor is given
as

${}^{(0)}T_{\chi}^{\xi}=\left(\rho+p_{t}\right)u^{\xi}u_{\chi}+p_{t}g_{\chi}^{\xi}+\left(p_{r}-p_{t}\right)\eta^{\xi}\eta_{\chi}.$
(70)

Furthermore the energy momentum tensor is perturbed by a perturbation tensor
$\delta T_{\chi}^{\xi}$. The perturbed tensor is defined by

$\displaystyle T_{\chi}^{\xi}=^{(0)}T_{\chi}^{\xi}+\delta T_{\chi}^{\xi},$
(71)

where the non-zero components of $T_{\chi}^{\xi}$ are

$\displaystyle T_{t}^{t}$ $\displaystyle=-\frac{d\rho}{dp_{r}}\delta
p_{r}(r)Y(\theta,\phi)-\rho(r),$ (72) $\displaystyle T_{r}^{r}$
$\displaystyle=\delta p_{r}(r)Y(\theta,\phi)+p_{r}(r),$ (73) $\displaystyle
T_{\theta}^{\theta}$ $\displaystyle=\frac{dp_{t}}{dp_{r}}\delta
p_{r}(r)Y(\theta,\phi)+p_{t}(r),$ (74) $\displaystyle T_{\phi}^{\phi}$
$\displaystyle=\frac{dp_{t}}{dp_{r}}\delta p_{r}(r)Y(\theta,\phi)+p_{t}(r).$
(75)

Note that in Eq. (72)-(75), and hereafter, we use $p\equiv p_{r}$ to denote
the radial pressure. With these perturbed quantities we can write down the
perturbed Einstein Field Equations

$\displaystyle G_{\chi}^{\xi}=8\pi T_{\chi}^{\xi},$ (76)

where the Einstein tensor $G_{\chi}^{\xi}$ is calculated using the metric
$g_{\chi\xi}$.

From the various components of background Einstein field equation
${}^{(0)}G_{\chi}^{\xi}=8\pi^{(0)}T_{\chi}^{\xi}$, we can have the following
things (which will be used later on):

${}^{(0)}G_{t}^{t}$ $\displaystyle=8\pi^{(0)}T_{t}^{t}$
$\displaystyle\Rightarrow\lambda^{\prime}(r)$ $\displaystyle=\frac{8\pi
r^{2}e^{\lambda(r)}\rho(r)-e^{\lambda(r)}+1}{r},$ (77) ${}^{(0)}G_{r}^{r}$
$\displaystyle=8\pi^{(0)}T_{r}^{r}$ $\displaystyle\Rightarrow\nu^{\prime}(r)$
$\displaystyle=\frac{8\pi r^{2}p_{r}(r)e^{\lambda(r)}+e^{\lambda(r)}-1}{r}.$
(78)

Also we know that $\nabla_{\xi}^{(0)}T_{\chi}^{\xi}=0$. Choosing $\xi=r$, by
expanding and solving the equation, we can find the expression as

$\displaystyle
p_{r}^{\prime}(r)=\frac{1}{2r}\left[-4p_{r}(r)+4p_{t}(r)-rp_{r}(r)\nu^{\prime}(r)-r\rho(r)\nu^{\prime}(r)\right].$
(79)

Now from the various components of perturbed Einstein equation (76), we get
the following relations

$\displaystyle G_{\theta}^{\theta}-G_{\phi}^{\phi}=0\Rightarrow
H_{0}(r)=H_{2}(r)=H(r),$ (80) $\displaystyle G_{r}^{\theta}=0\Rightarrow
K^{\prime}=H^{\prime}+H\nu^{\prime},$ (81) $\displaystyle
G_{\theta}^{\theta}+G_{\phi}^{\phi}=8\pi(T_{\theta}^{\theta}+T_{\phi}^{\phi})\Rightarrow\delta
p=\frac{H(r)e^{-\lambda(r)}\left(\lambda^{\prime}(r)+\nu^{\prime}(r)\right)}{16\pi
r\frac{dp_{t}}{dp_{r}}}.$ (82)

Using the identity

$\dfrac{\partial^{2}Y(\theta,\phi)}{\partial\theta^{2}}+cot(\theta)\dfrac{\partial
Y(\theta,\phi)}{\partial\theta}+\csc^{2}(\theta)\dfrac{\partial^{2}Y(\theta,\phi)}{\partial\phi^{2}}=-6Y(\theta,\phi)$

, eqn (77), (78) ,(79) ,(81), (82) we have the master equation for $H(r)$ as

$\displaystyle-\frac{1}{e^{-\lambda(r)}Y(\theta,\phi)}\left[G_{t}^{t}-G_{r}^{r}\right]=-\frac{8\pi}{e^{-\lambda(r)}Y(\theta,\phi)}\left[T_{t}^{t}-T_{r}^{r}\right]$
$\displaystyle\Rightarrow
H^{\prime\prime}(r)+\mathcal{R}H^{\prime}(r)+\mathcal{S}H(r)=0$ (83)

Where,

$\displaystyle\mathcal{R}=-\left[\frac{-e^{\lambda(r)}-1}{r}-4\pi
re^{\lambda(r)}(p_{r}-\rho(r))\right]$ (84)

$\displaystyle\mathcal{S}=-\left[16\pi
e^{\lambda(r)}\left(p_{r}\left(e^{\lambda(r)}-2\right)-p_{t}(r)-\rho(r)\right)+64\pi^{2}r^{2}p_{r}^{2}e^{2\lambda(r)}+\frac{4e^{\lambda(r)}+e^{2\lambda(r)}+1}{r^{2}}\right.$
$\displaystyle+\left.\frac{-4\pi\frac{d\rho}{dp_{r}}e^{\lambda(r)}(p_{r}+\rho(r))-4\pi
e^{\lambda(r)}(p_{r}+\rho(r))}{\frac{dp_{t}}{dp_{r}}}\right]$ (85)

$e^{\lambda(r)}=1+ar^{2}+br^{4},\,a=1/R^{2},\,b=\frac{1}{R^{4}}\left[\left(1-\frac{2M}{R}\right)^{-1}-2\right]$.
$M$ & $R$ are mass and radius of the star respectively. And the expression for
$p_{r},\,p_{t}$ & $\rho$ are given in the equation (7),(8),(11) respectively.

Outside the star, considering Schwarzschild metric and by setting
$\rho=0,p_{r}=0,p_{t}=0,\,e^{\lambda}=1/(1-2M/r)$, equation (83) becomes:

$\displaystyle-H^{\prime\prime}(r)+\left(\frac{1}{2M-r}-\frac{1}{r}\right)H^{\prime}(r)+\frac{2H(r)\left(2M^{2}-6Mr+3r^{2}\right)}{r^{2}(r-2M)^{2}}=0$
(86)

Solution to the equation (86) is

$\displaystyle
H(r)=\frac{3c_{1}r(2M-r)}{M^{2}}+\frac{c_{2}\left(-2M\left(2M^{3}+4M^{2}r-9Mr^{2}+3r^{3}\right)-3r^{2}(r-2M)^{2}\log\left(\frac{r}{M}-2\right)+3r^{2}(r-2M)^{2}\log\left(\frac{r}{M}\right)\right)}{2M^{2}r(2M-r)}$
(87)

Where $c_{1}$ & $c_{2}$ are integration constant which yet to be determined.
In order to get these constants, let’s expand the equation (87).

$\displaystyle
H(r)=-\frac{3c_{1}r^{2}}{M^{2}}+\frac{6c_{1}r}{M}-\frac{c_{2}\left(8M^{3}\right)}{5r^{3}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right)$
(88)

Now in the star’s local asymptotic rest frame (asymptotically mass-centered
Cartesian coordinates) at large $r$ the metric coefficient $g_{tt}$ is given
by Thorne1998 , Hinderer2008

$\displaystyle\frac{\left(1-g_{tt}\right)}{2}$ $\displaystyle=$
$\displaystyle-\frac{M}{r}-\frac{3Q_{ij}}{2r^{3}}\left(n^{i}n^{j}-\frac{1}{3}\delta^{ij}\right)+\mathcal{O}(\frac{1}{r^{3}})$
(89) $\displaystyle+\frac{1}{2}\mathcal{E}_{ij}x^{i}x^{j}+\mathcal{O}(r^{3}),$

where $n^{i}=x^{i}/r$. Matching the asymptotic solution from equation (88) to
the expansion from equation (89) and using the equation (69) we have

$\displaystyle c_{1}=-\frac{M^{2}\mathcal{E}}{3},\quad
c_{2}=\frac{15\mathcal{Q}}{8M^{3}}$ (90)

We now solve for $k_{2}$ in terms of $H$ and its derivative at the star’s
surface $r=R$ using equations (90) and (87), and use equation
$k_{2}=\frac{3}{2}\Lambda R^{-5}$ to obtain the expression for tidal love
number

$\displaystyle
k_{2}=[8(1-2\mathcal{C})^{2}\mathcal{C}^{5}(2\mathcal{C}(\mathit{y}-1)-\mathit{y}+2)]/X$
(91)

Where,

$\displaystyle X$
$\displaystyle=5(2\mathcal{C}(\mathcal{C}(2\mathcal{C}(\mathcal{C}(2\mathcal{C}(\mathit{y}+1)+3\mathit{y}-2)-11\mathit{y}+13)+3(5\mathit{y}-8))-3\mathit{y}+6)$
$\displaystyle\left.+3(1-2\mathcal{C})^{2}(2\mathcal{C}(\mathit{y}-1)-\mathit{y}+2)\log\left(\frac{1}{\mathcal{C}}-2\right)-3(1-2\mathcal{C})^{2}(2\mathcal{C}(\mathit{y}-1)-\mathit{y}+2)\log\left(\frac{1}{\mathcal{C}}\right)\right)$
(92)

Here $\mathcal{C}=\frac{M}{R}$ and $\mathit{y}$ depends on $r,\,H$ and it’s
derivatives

$\displaystyle\mathit{y}=\dfrac{rH^{\prime}(r)}{H(r)}_{R}.$ (93)

To get the numerical value of $k_{2}$ for a particular star, first of all we
have to find the numerical value of $\mathit{y}$. To do this let’s modify
master differential equation (83) using the equation (93) as Rahmansyah2020

$\displaystyle
r\mathit{y}^{\prime}+\mathit{y}^{2}+(r\mathcal{R}-1)\mathit{y}+r^{2}\mathcal{S}=0$
(94)

At $\mathcal{C}\rightarrow 0$ we expect $k_{2}\rightarrow 0$ and hence
$\mathit{y}(0)=2$. Also one can check numerically that at $\mathcal{C}=1/2$
the tidal love number vanishes for all value of $\mathit{y}$. It is because
the tidal deformability of any order $l$, both electric and magnetic, vanish
in the Black hole limit. In other words, the multipolar structure of a black
hole is not affected by the tidal field. This can be view as a corollary of
the no-hair theorem, and recently it has been proved beyond the perturbative
level Gurlebeck2015 .

After solving the the differential equation (94) using initial value
$\mathit{y}(0)=2$, equation (84) & eqn (85), we can calculate $k_{2}$ from the
equation (91).

Figure 13: $y$ is plotted over $p_{0}$ for four different compact star
$SAXJ1748.9-2021$, $4U1608-52$, $VelaX-I$, $KS1731-260$. The value of $M$ &
$R$ of these stars are take from the paper Roupas2020 . Only allowed range of
$p_{0}$ are considered.

In the fig. 13 the variation of $\mathit{y}$ w.r.t. $p_{0}$ are shown for four
different compact stars. In the figure 14 the variation of tidal love number
$k_{2}$ against the central pressure $p_{0}$ are shown for four different
compact objects. We found that tidal love number $k_{2}$ decreases
monotonically with increasing $p_{0}$. Here the maximum range of $p_{0}$ is
$0.3944\,a$ as in the equation (40). Where as the minimum range of $p_{0}$
varies for different compact stars. The minimum possible value of $p_{0}$ for
some compact star is given in the table 5.

Table 5: The numerical value of Mass, Radius, minimum and maxium range of $p_{0}$ for different compact star. $a=1/R^{2}$. Numerical values of mass and radius are taken from Roupas2020  Name | min. $p_{0}$ | max. $p_{0}$ | M($M_{\odot}$) |  $R$ km
---|---|---|---|---
SAX J1748.9-2021 | $0.27305a$ | $0.3944a$ | $1.81\pm 0.25$ | $11.7\pm 1.7$
4U 1608-52 | $0.30750a$ | $0.3944a$ | $1.74\pm 0.14$ | $10.811\pm 0.19$
Vela X-I | $0.318500a$ | $0.3944a$ | $1.77\pm 0.08$ | $10.852\pm 0.1$
KS 1731-260 | $0.30755a$ | $0.3944a$ | $1.61\pm 0.35$ | $10\pm 2.2$

Figure 14: Tidal love number $k_{2}$ is plotted against $p_{0}$ for compact
star $SAXJ1748.9-2021$, $4U1608-52$, $VelaX-I$, $KS1731-260$. The value of $M$
& $R$ of these stars are take from the paper Roupas2020 . Only allowed range
of $p_{0}$ are considered.

##  VIII Discussion

In our present paper we proposed a new model of compact star in the background
of general theory of relativity. In this work we have focused on the compact
object 4U 1608-52 and Vela X-1 whose estimated masses and radii are obtained
in Table I. Our present model satisfies the following conditions:

  * •

Regularity of metric potential: Both the metric potentials
$e^{\nu},\,e^{\lambda}$ are regular i.e., free from all kinds of singularity
inside the stellar interior.
$e^{\lambda}|_{r=0}=1,\,e^{\nu}|_{r=0}=\exp\bigg{[}\frac{2p_{0}(a^{2}+2b)}{b\sqrt{4b-a^{2}}}\tan^{-1}\Big{(}\frac{a}{\sqrt{4b-a^{2}}}\Big{)}+B\bigg{]}$.

  * •

The matter density, both radial and transverse pressures take maximum value at
the center of the star and gradually decreases towards the boundary, i.e., all
are monotonic decreasing functions of r and at the same time they do not
suffer from any kinds of singularity. Both the pressures depend on
dimensionless quantity $p_{0}$ and the central pressure increases for the
increasing values of $p_{0}$.

  * •

The anisotropic factor vanishes at the center of the star and $\Delta>0$ for
$0\leq r\leq R$. For positive anisotropic factor makes the system more stable
and it helps to construct more compact object gm . The equation of state
parameters are plotted against ‘r’ inside the stellar interior in fig. 4 and
one can note that $0~{}<\omega_{r},\,\omega_{t}~{}<1$.

  * •

We have verified that all the energy conditions are satisfied by our model
with the help of graphical representation and the model is stable under the
effect of the three different forces. From the fig. 10 we see that the
gravitational force is attractive but both the anisotropic and hydrostatic
forces are repulsive and the combine effect of these two forces is
counterbalanced by the gravitational force which makes the system in static
equilibrium. One interesting thing we can also notice that $F_{h}$ and $F_{a}$
intersects at some point inside the fluid sphere. The causality conditions and
the the Herrera’s cracking conditions are also satisfied as well. Our model is
potentially stable as well since $V_{t}^{2}-V_{r}^{2}<0$ everywhere inside the
fluid sphere.

  * •

We have obtained a reasonable bound for the dimensionless quantity $p_{0}$
from the physical analysis and we have shown that it depends on ’a’.

  * •

We have successfully demonstrated the possibility that the anisotropic stars
could be ultra compact objects due to the additional support from anisotropy
and could be sources of gravitational wave echoes. In this model, we have
calculated tidal deformability of the anisotropic stars and show how the tidal
love number varies with the compactness factor of a star. As an interesting
consequence, we can see that in the black hole limit the love number vanishes.
Also, the existence of the non zero value of tidal love number at zero
compactness factor agrees with some of the recent findings.

  * •

Generating function: Lake lake1  proposed an algorithm based on the choice of
a single monotone function subject to boundary conditions which generates all
regular static spherically symmetric perfect-fluid solutions of Einstein
equations. Herrera et al. h1  extended this work to the case of locally
anisotropic fluids and proved that two functions instead of one is required to
generate all possible solutions for anisotropic fluid. To find these two
generating function we first consider pressure anisotropy for our present
model given as,

$\displaystyle p_{t}-p_{r}$ $\displaystyle=$
$\displaystyle\frac{e^{-\lambda}}{\kappa}\bigg{[}\frac{-\lambda^{\prime}\nu^{\prime}}{4}+\frac{\nu^{\prime
2}}{4}+\frac{\nu^{\prime\prime}}{2}-\frac{\lambda^{\prime}-\nu^{\prime}}{2r}$
(95)
$\displaystyle-\frac{\nu^{\prime}}{r}+\frac{e^{\lambda}-1}{r^{2}}\bigg{]}.$

Now by introducing DB transformation

$x=r^{2},~{}~{}~{}V(x)=e^{-\lambda},~{}~{}\text{ and}~{}~{}~{}y(x)=e^{\nu},$

the Eqn. (95), transform to,

$\displaystyle\frac{dV}{dx}\Big{(}1+x\frac{\dot{y}}{y}\Big{)}+V\Big{[}\Big{(}2\frac{\ddot{y}}{y}-\frac{\dot{y}^{2}}{y^{2}}\Big{)}x-\frac{1}{x}\Big{]}=\kappa\Delta-\frac{1}{x},$
(96)

The above equation can be denoted as,

$\displaystyle\frac{dV}{dx}+F(x)V=S(x),$ (97)

which is linear equation of $x$. The integrating factor of the above equation
is,

$e^{\int F(x)dx}$

and the solution of the above equation is,

$V(x)=e^{-\int F(x)dx}\int S(x)e^{\int F(x)dx}dx+k,$

where $k$ is the constant of integration and

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\frac{\Big{(}2\frac{\ddot{y}}{y}-\frac{\dot{y}^{2}}{y^{2}}\Big{)}x-\frac{1}{x}}{1+x\frac{\dot{y}}{y}},$
(98) $\displaystyle S(x)$ $\displaystyle=$
$\displaystyle\frac{\kappa\Delta-\frac{1}{x}}{1+x\frac{\dot{y}}{y}}.$ (99)

From the above discussion it is clear that the model of the compact star
stands on two functions $F(x)$ and $S(x)$, where as $F(x)$ and $S(x)$ depends
on $y(x)$ and $\Delta$. Therefore for our model the two generating functions
are therefore,

$\displaystyle y(x)$ $\displaystyle=$
$\displaystyle\exp\Big{[}\frac{1}{4}\Big{\\{}2ax+bx^{2}+\frac{2p_{0}(a^{2}+2b)}{b\sqrt{a^{2}-4b}}\times$
(100)
$\displaystyle\tan^{-1}\Big{(}\frac{a+2bx}{\sqrt{a^{2}-4b}}\Big{)}-\frac{ap_{0}}{b}\log(1+ax+bx^{2})\Big{\\}}$
$\displaystyle+B\Big{]},$ $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{x}{4\Psi^{3}}\big{[}A_{1}+A_{2}x+A_{3}x^{2}+A_{4}x^{3}+A_{5}x^{4}$
(101) $\displaystyle+4ab^{3}x^{5}+b^{4}x^{6}\big{]}.$

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## Acknowledgement

PB is thankful to IUCAA , Pune, Govt of India for providing visiting
associateship. SD gratefully acknowledges support from the Inter-university
Centre for Astronomy and Astrophysics (IUCAA), Pune, India, where a part of
this work was carried out under its Visiting Research Associateship Programme.