# A generalized buchdahl model for compact stars in $f(R,T)$ gravity

J. Kumar<EMAIL_ADDRESS>H. D. Singh<EMAIL_ADDRESS>A. K.
Prasad<EMAIL_ADDRESS>Department of Applied Mathematics, Central
University of Jharkhand, Ranchi-835205, India

###### Abstract

In this paper, we study the stellar structure in terms of alternative theory
of gravity specially by $f(R,T)$ gravity theory. Here, we consider the
function $f(R,T)=R+2\varsigma T$ where $R$ is the Ricci scalar, $T$ is the
stress-energy momentum and $\varsigma$ is the coupling constant. Using it we
developed a stellar model that briefly explains the isotropic matter
distribution within the compact object filled with perfect fluid. The
stability of the model is shown by several physical and stability conditions.
With the accecptibility of our theory, we were able to collect data for
compact stars like PSR-B0943+10, CEN X-3, SMC X-4, Her X-1 and 4U1538-52 with
great accuracy.

###### keywords:

Perfect fluids; Modified theory of relativity; Compact stars; General
relativity.

##  1 Introduction

Recently, it has been discovered that our universe was expanding at an
accelerated rate. Over past few years, more-n-more studies were done to
understand these late-time acceleration of the universe and evidences were
found that the cause of this is due to the presence of some dark-energy matter
[1, 2, 3, 4, 5, 6, 7] which has huge negative impact and pressure. Surveys on
cosmic radiation and dark matters [8, 9, 10, 11] shown that $4\%$, $20\%$ and
$76\%$ of ordinary baryonic matter, dark matter and dark energy are
distributed respectively over the universe. As the gravitational connection to
this is least understood and Einstein’s general theory on relativity was
unable to provide decent reasons behind such causes.

So, different approaches were made for investigating such cases. And one
approach is the consideration of modified theory of relativity which is a
classical generalization of general relativity that were able to explain late-
time acceleration and early-time inflation along with dark-energy. Suggestion
were made that by replacing/ substituting curvature invariant function in
Einstein-Hilbert action we can achieve cosmic acceleration. Many studies were
done in past years on modified gravity like $f(R)$, $f(G)$ or $f(R,G)$ in
different contexts [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Similarly,
another modified theory is $f(R,T)$ theory of gravity [22]. In this theory,
gravitational Lagrangian couples with trace $T$ of the energy-momentum tensor
of the matter is added to general Einstein-Hilbert action. Now, with its
introduction, many researchers tried their best in providing information and
genuine explaination of various cosmological and astrophysical events [23, 24,
25, 26, 27, 28, 29, 30, 31, 32]. In paper [33], $f(R,T)$ has been used in the
studies of factors that affect the stability of isotropic spherical symmetry.
And by applying perturbation scheme [34, 35, 36] instability of spherically
symmetric under different condition was studied. In a recent aricle, stellar
model in the framework of $f(R,T)$ gravity has been studied [37] that shows
this model has more impressive behaviour when compared to the general
relativity theory. In another paper [38], the effect of modified $f(R,T)$ has
been studied and compared with the results of standard general relativity.
Futher, investigation was done by applying Krori and Barua metric [39]. In
article [40], they have distinguish between general relativity and its
extended version for relativistic stars which has firm gravitational field
system.

However, in this paper we did an investigation of stellar model in the
reference of $f(R,T)$ behaviour using Buchdahl metric for different compact
stars. First of all, for a given star we determined the energy density and
isotropic pressure in the context of $f(R,T)$, then the velocity of sound and
Energy condition (weak, null and strong). Later, we further investigated the
Redshift and also the Adiabatic Index of the star. This paper is arranged in
the following sequences : In sect 1, we have given brief introduction
regarding General Relativity and Modified Theory of Relativity. In sect 2, we
have presented the basic formulation of $f(R,T)$ theory of gravity. In sect 3,
we have shown stellar structure in the framework of $f(R,T)$ Theory. In sect
4, we did a detail investigation of compact stars using Buchdahl metric for
cases $K<0$ and $K>1$. Further in sect 5 and 6, we did observation regarding
the physical quantities and stabilities of compact stars. Finally, in sect 7,
we discussed and concluded our results.

##  2  General mathematical formulation of $f(R,T)$ gravity theory

In this section, we introduced the formation of $f(R,T)$ [41] gravity theory.
While obtaining the Einstein’s field equations, the ricci scalar in Einstein-
Hilbert action is integrated over $d^{4}x$ as

$\displaystyle S_{EH}=\frac{1}{16}\int{R\sqrt{-g}d^{4}x}$ (1)

Now one can get $f(R,T)$ field equation from the above equation if instead of
Ricci scalar $R$, we select $f(R,T)$. Thus, the full action for the modified
theory of gravity is given by

$\displaystyle
S=\frac{1}{16}\int{f(R,T)\sqrt{-g}d^{4}x}+\int{L_{M}\sqrt{-g}d^{4}x}$ (2)

where in the first term $f(R,T)$ is arbitrary function of Ricci scalar $R$
with $g$ as the determinant of the metric tensor $g_{\mu\nu}$ and $T$ being
the trace of the stress-energy tensor,$T_{\mu\nu}$. And in the second term
$L_{M}$ is the matter Lagrangian density which is related to stress-momentum
tensor given as

$\displaystyle
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{M})}{\delta
g^{\mu\nu}}$ (3)

with trace $T=g^{\mu\nu}T_{\mu\nu}$. Assuming that the Lagrangian density
$L_{M}$ [42] only depends on the metric tensor components $g_{\mu\nu}$, we
obtain eqn.(3) as

$\displaystyle T_{\mu\nu}=g_{\mu\nu}L_{M}-\frac{2\delta(L_{M})}{\delta
g^{\mu\nu}}$ (4)

By varying eqn.(2) with respect to $g_{\mu\nu}$, we obtained the field
equation

$\displaystyle(R_{\mu\nu}-\bigtriangledown_{\mu}\bigtriangledown_{\nu})f_{R}(R,T)+g_{\mu\nu}\diamondsuit
f_{R}(R,T)-\frac{1}{2}f(R,T)g_{\mu\nu}$ $\displaystyle=$ $\displaystyle 8\pi
T_{\mu\nu}-f_{T}(R,T)(T_{\mu\nu}+\varPsi_{\mu\nu})$ (5)

where $f_{R}(R,T)=\frac{\delta f(R,T)}{\delta R}$ and $f_{T}(R,T)=\frac{\delta
f(R,T)}{\delta T}$. The covariant derivative is denoted by
$\bigtriangledown_{\mu}$ and the diamond operature $\diamondsuit$ is defined
as

$\displaystyle\diamondsuit=\frac{1}{\sqrt{-g}}\frac{\delta}{\delta
x^{\mu}}(\sqrt{-g}g^{\mu\nu}\frac{\delta}{\delta x^{\nu}})$

$\displaystyle\varPsi_{\mu\nu}=g^{\alpha\beta}\frac{\delta
T_{\alpha\beta}}{\delta g^{\mu\nu}}$

Now we perform the covariant derivative of eqn.(5) by which we obtain the
covariant derivative of the energy-momentum tensor as

$\displaystyle\bigtriangledown^{\mu}T_{\mu\nu}=\frac{f_{T}(R,T)}{8\pi-
f_{T}(R,T)}[(T_{\mu\nu}+\varPsi_{\mu\nu})\bigtriangledown^{\mu}lnf_{T}(R,T)+\varPsi_{\mu\nu}\bigtriangledown^{\mu}-\frac{1}{2}g^{\mu\nu}\bigtriangledown^{\mu}T]$
(6)

Clearly from eqn.(6) we can see that the stress-energy momentum tensor
$T_{\mu\nu}$ in $f(R,T)$ theory does not follow the conservation law as in
Einstein general relativity (GR) due to the presence of some nonminimal matter
geometry coupling in its formulation. By using eqn.(4), the tensor
$\varPsi_{\mu\nu}$ is given by

$\displaystyle\varPsi_{\mu\nu}=-2T_{\mu\nu}+g_{\mu\nu}L_{M}-2g^{\alpha\beta}\frac{\delta^{2}L_{M}}{\delta
g^{\mu\nu}\delta g^{\alpha\beta}}$ (7)

To find the field equations, we consider the interior of star to be filled
with a perfect fluid source along with the energy-momentum tensor of the form

$\displaystyle T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}$ (8)

provided the four velocity $u_{\nu}$ satifies $u_{\mu}u^{\mu}=-1$ and
$u_{\nu}\bigtriangledown^{\mu}u_{\nu}=0$, $\rho$ and $p$ is the matter density
and isotropic pressure respectively . Taking the matter Lagrangian as
$L_{M}=-p$, we obtain eqn.(7) as

$\displaystyle\varPsi_{\mu\nu}=-2T_{\mu\nu}-pg_{\mu\nu}$ (9)

In this paper, we assumed the function $f(R,T)$as $f(R,T)=R+2f(T)$ where
$f(T)$ is the function of trace $(T)$ of the stress-energy tensor of matter.
Here, we choose $f(T)=\varsigma T$ in order to determine the modified theory
of gravity, where $\varsigma$ is a coupling constant.
Using the expression in eqn.(5), we obtain

$\displaystyle G_{\mu\nu}=8\pi T_{\mu\nu}+\varsigma
Tg_{\mu\nu}+2\varsigma(T_{\mu\nu}+pg_{\mu\nu})$ (10)

When $f(R,T)\equiv R$, then eqn.(5) reduces to Einstein field equations. By
putting $f(R,T)=R+2\varsigma T$ and eqn.(9) in eqn.(6) we get

$\displaystyle\bigtriangledown^{\mu}T_{\mu\nu}=-\frac{\varsigma}{2(4\pi+\varsigma)}[g_{\mu\nu}\bigtriangledown^{\mu}T+2\bigtriangledown^{\mu}(pg_{\mu\nu})]$
(11)

Thus, one can easily obtain the conservation equation in Einstein’s gravity by
putting $\varsigma=0$. For studying cosmological and astrophysical problems
$f(R,T)$ has been widely considered due to its advantages in explaining such
space problems.

##  3  Einstein’s field equation in $f(R,T)$ gravity theory

Defining the interior space-time of static stellar configuration for
spherically symmetric metric, by the line element as

$\displaystyle ds^{2}=-e^{\lambda(r)}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})+e^{\nu(r)}dt^{2}$ (12)

where $(t,r,\theta,\phi)$ are the time and space cordinates respectively and
$\lambda(r)$ and $\nu(r)$ are the functions of radial cordinate, $r$ only.
Now, we have to find the field equations of eqn.(12) as [43]

$\displaystyle\rho_{E}=\frac{e^{-\lambda}}{8\pi}\bigg{(}-\frac{1}{r^{2}}+\frac{\lambda^{\prime}}{r}+\frac{e^{\lambda}}{r^{2}}\bigg{)}$
(13) $\displaystyle
p_{E}=\frac{e^{-\lambda}}{8\pi}\bigg{(}\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}-\frac{e^{\lambda}}{r^{2}}\bigg{)}$
(14) $\displaystyle
p_{E}=\frac{e^{-\lambda}}{32\pi}\bigg{(}2\nu^{\prime\prime}+\nu^{\prime
2}-\lambda^{\prime}\nu^{\prime}+2\frac{\nu^{\prime}-\lambda^{\prime}}{r}\bigg{)}$
(15)

where $\rho_{E}$ and $p_{E}$ are the Einstein’s density and pressure
respectively and prime refers to differentiation with respect to $r$. Along
with

$\displaystyle\rho_{E}=\rho_{F}+\frac{\varsigma}{8\pi}(3\rho_{F}-p_{F})$
$\displaystyle p_{E}=p_{F}+\frac{\varsigma}{8\pi}(\rho_{F}-3p_{F})$

where $p_{F}$ and $\rho_{F}$ denotes the $f(R,T)$ pressure and density
respectively. Here, for our convenience, we have assumed $G=c=1$. On solving
eqns.(13)-(15), we obtain

$\displaystyle\frac{\nu^{\prime}}{2}(p_{F}+\rho_{F})+\frac{dp}{dr}=\frac{\varsigma}{8\pi+2\varsigma}(p^{\prime}-\rho^{\prime})$
(16)

which provides the hydrostatic equillibrium condition for general relativity
when we take $\varsigma=0$.
Now rewriting modified pressure and density i.e $p_{F}$ and $\rho_{F}$ by
using eqn.(13) and eqn.(14), we get

$\displaystyle
p_{F}=\frac{\pi}{(8\pi^{2}+6\pi\varsigma+\varsigma^{2})}[(8\pi+3\varsigma)p_{E}+\varsigma\rho_{E}]$
(17)
$\displaystyle\rho_{F}=\frac{\pi}{(8\pi^{2}+6\pi\varsigma+\varsigma^{2})}[(8\pi+3\varsigma)\rho_{E}+\varsigma
p_{E}]$ (18)

Now, it has been assumed that all the cosmological phenomena are engulfed in
the vacuum space-time, so we need to check our solution for interior space-
time with the exterior Schwarzschild line element as

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

where $M$ is the total mass with radius $R$. At boundary when $r=R$

$\displaystyle e^{-\lambda(R)}=e^{\nu(R)}=1-\frac{2M}{R}$ (20)

Now, we need to solve the field equation i.e eqn.(17) and eqn.(18) for which
we approached with Buchdahl ansatz and study its impact on the physical
systems.

##  4  Buchdahl ansatz in $f(R,T)$ gravity theory

In order to find the solution of the modified field equation, we use Buchdahl
anastz [44] which easily investigate all the physical parameters of a stellar
model. The most broadly studied Buchdahl metric anastz is given by

$\displaystyle e^{\lambda}=\dfrac{K(1+Cr^{2})}{K+Cr^{2}}$ (21)

where $K$ and $C$ are constant parameters that depends on the geometry of the
given star. Now we consider few transformations for solving field equations ,

$\displaystyle\chi=\sqrt{\frac{K+Cr^{2}}{K-1}}~{}~{}~{}~{}\textrm{and}~{}~{}~{}~{}~{}e^{\nu}=W^{2}$
(22)

Substituting eqn.(21) and eqn.(22) in eqn.(13) and eqn.(14) and doing some
simple algebraic calculations we obtained the following equations which will
be later used in solving exact solutions,

$\displaystyle-\kappa p_{E}$
$\displaystyle=\frac{(K+Cr^{2})}{K(1+Cr^{2})}\bigg{[}-\frac{2W^{\prime}}{rW}+\frac{C(K-1)}{K+Cr^{2}}\bigg{]}$
(23) $\displaystyle-\kappa p_{E}$
$\displaystyle=\frac{(K+Cr^{2})}{K(1+Cr^{2})}\bigg{[}-\frac{W^{\prime\prime}}{W}+\frac{W^{\prime}}{rW}-\frac{C(K-1)}{(K+Cr^{2})(1+Cr^{2})}\bigg{(}1+\frac{rW^{\prime}}{W}\bigg{)}\bigg{]}$
(24) $\displaystyle\kappa c^{2}\rho_{E}$
$\displaystyle=\frac{C(K-1)(3+Cr^{2})}{K(1+Cr^{2})^{2}}$ (25)

Now subtracting eqn.(24) from eqn.(23), we get

$\displaystyle\dfrac{K+Cr^{2}}{K(1+Cr^{2})}\Bigg{[}\dfrac{W^{\prime\prime}}{W}-\dfrac{W^{\prime}}{rW}+\dfrac{C(K-1)r(Cr-W^{\prime}/W)}{(K+Cr^{2})(1+Cr^{2})}\Bigg{]}=0$
(26)

Using eqn.(22) in the above equation, we obtained the differential equation of
the form,

$\displaystyle(1-\chi^{2})\frac{d^{2}W}{d\chi^{2}}+\chi\frac{dW}{d\chi}+(1-K)W=0$
(27)

To find the solution of the above equation we use transformation for two cases
:
Case 1 for $K<0$ given as

$\displaystyle W=(1-\chi^{2})^{1/4}U$ (28)

Thus eqn.(27) reduces to

$\displaystyle\dfrac{d^{2}U}{dX^{2}}+\psi U=0$ (29)

where

$\displaystyle\psi=\frac{1-K}{1-\chi^{2}}-\frac{2+3\chi^{2}}{4(1-\chi^{2})^{2}}$

In order to solve the eqn.(29) easily, we choose $\psi$ as

$\displaystyle\psi=\frac{A_{1}}{\chi^{2}}$ (30)

where $A_{1}$ be any arbitrary constant.
Thus eqn.(29) becomes

$\displaystyle\dfrac{d^{2}U}{dX^{2}}+\frac{A_{1}}{\chi^{2}}U=0$ (31)

Solving the differential equation we obtain the solution as

$\displaystyle U=c_{1}\frac{\chi^{1-A_{1}}}{1-A_{1}}+c_{2}$

Therefore, we get

$\displaystyle
W=(1-\chi^{2})^{1/4}\bigg{[}c_{1}\frac{\chi^{1-A_{1}}}{1-A_{1}}+c_{2}\bigg{]}$
(32)

where $c_{1}$ and $c_{2}$ are arbitrary constants.
Using eqn.(32) in eqns.(23) and (25) we obtained equations,

$\displaystyle p_{E}$ $\displaystyle=\frac{c}{8\pi
K(1+Cr^{2})}\bigg{[}-\frac{\chi^{2}}{1-\chi^{2}}-(K-1)+\frac{2\chi^{1-A_{1}}}{\bigg{[}\frac{\chi^{1-A_{1}}}{1-A_{1}}+\frac{c_{2}}{c_{1}}\bigg{]}}\bigg{]}$
(33) $\displaystyle\rho_{E}$ $\displaystyle=\frac{C(K-1)(3+Cr^{2})}{8\pi
K(1+Cr^{2})^{2}}$ (34)

Substituting the above equations in eqns.(17)-(18), we obtained the required
modified pressure and density, respectively

$\displaystyle p_{F}$
$\displaystyle=\frac{\pi}{(8\pi^{2}+6\pi\varsigma+\varsigma^{2})}\bigg{[}\frac{(8\pi+3\varsigma)}{8\pi
K(1+Cr^{2})}\bigg{[}-\frac{\chi^{2}}{1-\chi^{2}}-(K-1)+$ $\displaystyle\ \ \ \
\frac{2\chi^{1-A_{1}}}{\bigg{[}\frac{\chi^{1-A_{1}}}{1-A_{1}}+\frac{c_{2}}{c_{1}}\bigg{]}}\bigg{]}+\varsigma\frac{(K-1)(3+Cr^{2})}{8\pi
K(1+Cr^{2})^{2}}\bigg{]}$ (35) $\displaystyle\rho_{F}$
$\displaystyle=\frac{\pi}{(8\pi^{2}+6\pi\varsigma+\varsigma^{2})}\bigg{[}(8\pi+3\varsigma)\frac{(K-1)(3+Cr^{2})}{8\pi
K(1+Cr^{2})^{2}}+\frac{\varsigma}{8\pi K(1+Cr^{2})}$ $\displaystyle\ \ \ \
\bigg{[}-\frac{\chi^{2}}{1-\chi^{2}}-(K-1)+\frac{2\chi^{1-A_{1}}}{\bigg{[}\frac{\chi^{1-A_{1}}}{1-A_{1}}+\frac{c_{2}}{c_{1}}\bigg{]}}\bigg{]}$
(36)

Case 2 for $K>1$ , we consider the transformation,

$\displaystyle
W=(\chi^{2}-1)^{1/4}\bigg{[}c_{1}\frac{\chi^{1-A_{1}}}{1-A_{1}}+c_{2}\bigg{]}$
(37)

where $c_{1}$ and $c_{2}$ are arbitrary constants. Here, we find the pressure
and density is same as for $K<0$. When $0<K<1$ , it is observed that due to
the presence of the term $K-1$ the density becomes negative. Where as for
pressure the transformation $\chi$ has no real solution in $0<K<1$. Hence, for
given value of $K$, physical validation for general relativity and modified
theory is not possible. Now, the interior of a stellar model has to follow
some standard restrictions as follows which can be seen in fig 1.

  1. 1.

Pressure and density should be positive definite at the centre.

  2. 2.

The pressure should be maximum at the centre and decreasing monotonically
within $0<r<R$.

  3. 3.

The density should be maximum at the centre and decreasing monotonically
within $0<r<R$.

  4. 4.

The ratio of pressure and density should be less than unity within $0<r<R$.

Figure 1: Above two graphs represent the behaviour of pressure and Denity
against fractional radius $r/R$ of compact star PSR B0943+10 for $K=-1.53$,
$C=0.002071Km^{-2}$, $A_{1}=0$, $M=0.02M_{\odot}$, $R=2.6Km$ where as the
below two graphs represent compact star Her X-1 for $K=2.1$,
$C=0.022018Km^{-2}$, $A_{1}=0$, $M=0.85M_{\odot}$, $R=8.1Km$

##  5 Determination of constant

Using eqn.(20) with boundary condition i.e when $r=R$ and $p_{F}=0$, we
obtained the value of the arbitary constant as

$\displaystyle F=\frac{M1(8\pi+3\varsigma)(2\chi^{1-A_{1}})(8\pi
K(1+CR^{2})^{2})}{[M1(8\pi+3\varsigma)(8\pi
K(1+CR^{2})^{2})](K-1+\chi^{2})-M2}-M3$ (38)

where $F=\frac{c_{2}}{c_{1}}$, $M1=\frac{1}{8\pi K(1+CR^{2})}$,
$M2=\varsigma(K-1)(3+CR^{2})$ and $M3=\frac{\chi^{1-A_{1}}}{1-A_{1}}$.

##  6 Physical properties to be satisfied by the stellar structure of $f(R,T)$
gravity theory

To show the effectiveness of the our stellar model, we studied and analyzed
some physical properties and depicted it graphically. Here, we study the
Velocity of Sound, Energy Conditions, Redshift, Adiabetic Index etc. in the
following subsections.

###  6.1 Velocity of Sound

In order to establish the stability of the stellar system, it must satisfy the
causality condition which is the velocity of sound should be less than the
velocity of light i.e, $0<V^{2}<1$.
The velocity of sound is obtained as

$\displaystyle V^{2}_{F}=\frac{dp_{F}}{d\rho_{F}}$ (39)

In fig 2, shows that for the stellar structure the velocity of sound is less
than the speed of light inside compact stars and with increase in $r$, it
decreases.

Figure 2: Left and Right graphs represents the behaviour of Velocity of sound
against fractional radius $r/R$ of compact stars PSR B0943+10 and Her X-1
respectively.

###  6.2 Redshift

The expression for gravitational Redshift $Z_{S}$ for the stellar model is
given by

$\displaystyle Z_{S}=e^{-\nu(r)/2}-1$ (40)

Figure 3: Behaviour of Redshift against fractional radius $r/R$ of compact
stars PSR B0943+10 and Her X-1 respectively.

which should be positive inside the astrophysical objects. In fig 3, we have
shown the gravitational redshift inside the compact stars which indicate the
physical validity of the matter distribution of our stellar model.

###  6.3 Energy Condition

In order to show the consistency of the stellar system, it must satisfy the
energy condition as given in the following inequalities,

$\displaystyle NullEnergyCondition:$ $\displaystyle NEC=$
$\displaystyle\rho_{F}\geq 0$ $\displaystyle WeakEnergyCondition:$
$\displaystyle WEC=$ $\displaystyle\rho_{F}-p_{F}\geq 0$ $\displaystyle
StrongEnergyCondition:$ $\displaystyle SEC=$ $\displaystyle\rho_{F}-3p_{F}\geq
0$

In fig 4, we can see the behaviour of these energy conditions and observed
that throughout the stellar model all the energy conditions are satisfied and
evenly distributed over the interior region of the compact stars.

Figure 4: Top graphs represent the Null energy condition, whereas the middle
graphs represent the Weak energy condition and the end graphs represent the
Strong energy condition of compact stars PSR B0943+10 and Her X-1
respectively.

###  6.4 Adiabatic Index

In order to investigate the stability of the model, we have to study the
adiabatic index $\Gamma_{I}$ of the system, which is defined by

$\displaystyle\Gamma_{I}=\bigg{(}\frac{c^{2}\rho_{F}+p_{F}}{p_{F}}\bigg{)}\bigg{(}\frac{dp_{F}}{c^{2}d\rho_{F}}\bigg{)}$
(41)

Figure 5: Behaviour of Adiabatic Index against fractional radius $r/R$ of
compact stars PSR B0943+10 and Her X-1 respectively.

In fig 5, we presented the Adiabatic Index $\Gamma$ of our stellar model where
the value of $\Gamma>4/3$ which shows that our model has steady range with
respect to the curvature term present in $f(R,T)$ gravity. But this condition
may change which makes the astrophysical object more unstable due to
redemptive effect of pressure.

Table 1: Central Pressure for different compact stars of case 1 and 2 in unit $dyne/cm^{2}$ |  |  | Case 1 |  |  |  |
---|---|---|---|---|---|---|---
Compact Star | $\varsigma=0$ | $\varsigma=0.2$ | $\varsigma=0.4$ | $\varsigma=0.6$ | $\varsigma=0.8$ | $\varsigma=1$ |
$PSR-B0943+10$ | $1.24193\times 10^{32}$ | $1.2039\times 10^{32}$ | $1.16888\times 10^{32}$ | $1.13585\times 10^{32}$ | $1.10583\times 10^{32}$ | $1.07681\times 10^{32}$ | $A_{1}=0$
CEN X-3 | $1.59908\times 10^{34}$ | $1.61123\times 10^{34}$ | $1.62134\times 10^{34}$ | $1.62978\times 10^{34}$ | $1.63656\times 10^{34}$ | $1.6418\times 10^{34}$ | $A_{1}=2$
SMC X-4 | $1.37287\times 10^{44}$ | $1.30413\times 10^{44}$ | $1.24138\times 10^{44}$ | $1.18395\times 10^{44}$ | $1.13122\times 10^{44}$ | $1.08267\times 10^{44}$ | $A_{1}=-1$
|  |  | Case 2 |  |  |  |
Compact Star | $\varsigma=0$ | $\varsigma=0.2$ | $\varsigma=0.4$ | $\varsigma=0.6$ | $\varsigma=0.8$ | $\varsigma=1$ |
Her X-1 | $1.60709\times 10^{44}$ | $1.60283\times 10^{44}$ | $1.59762\times 10^{44}$ | $1.59134\times 10^{44}$ | $1.58443\times 10^{44}$ | $1.57666\times 10^{44}$ | $A_{1}=0$
$4U1538-52$ | $1.17804\times 10^{44}$ | $1.15809\times 10^{44}$ | $1.13919\times 10^{44}$ | $1.12123\times 10^{44}$ | $1.1042\times 10^{44}$ | $1.08789\times 10^{44}$ | $A_{1}=2$
Table 2: Central Density for different compact stars of case 1 and 2 in unit $gm/cm^{3}$ |  |  | Case 1 |  |  |  |
---|---|---|---|---|---|---|---
Compact Star | $\varsigma=0$ | $\varsigma=0.2$ | $\varsigma=0.4$ | $\varsigma=0.6$ | $\varsigma=0.8$ | $\varsigma=1$ |
$PSR-B0943+10$ | $2.19479\times 10^{13}$ | $2.14372\times 10^{13}$ | $2.09497\times 10^{13}$ | $2.04839\times 10^{13}$ | $2.00384\times 10^{13}$ | $1.96117\times 10^{13}$ | $A_{1}=0$
CEN X-3 | $2.57064\times 10^{14}$ | $2.51209\times 10^{14}$ | $2.45623\times 10^{14}$ | $2.40287\times 10^{14}$ | $2.35185\times 10^{14}$ | $2.30299\times 10^{14}$ | $A_{1}=2$
SMC X-4 | $4.96258\times 10^{13}$ | $4.85814\times 10^{13}$ | $4.75739\times 10^{13}$ | $4.66021\times 10^{13}$ | $4.56652\times 10^{13}$ | $4.47616\times 10^{13}$ | $A_{1}=-1$
|  |  | Case 2 |  |  |  |
Compact Star | $\varsigma=0$ | $\varsigma=0.2$ | $\varsigma=0.4$ | $\varsigma=0.6$ | $\varsigma=0.8$ | $\varsigma=1$ |
Her X-1 | $7.39146\times 10^{13}$ | $7.23293\times 10^{13}$ | $7.08161\times 10^{13}$ | $6.93692\times 10^{13}$ | $6.79837\times 10^{13}$ | $6.66549\times 10^{13}$ | $A_{1}=0$
$4U1538-52$ | $8.4603\times 10^{13}$ | $8.27299\times 10^{13}$ | $8.09403\times 10^{13}$ | $7.92263\times 10^{13}$ | $7.75854\times 10^{13}$ | $7.60111\times 10^{13}$ | $A_{1}=2$
Table 3: Value of different parameters like $M$, $R$ and $C$ |  | Case 1 |  |  |
---|---|---|---|---|---
Compact Star | $M(M_{\odot})$ | $R(Km)$ | $C(Km^{-2})$ | M/R |
$PSR-B0943+10$ | 0.02 | 2.6 | 0.002071 | 0.011415 | $A_{1}=0$
CEN X-3 | 1.49 | 4.178 | 0.026467 | 0.462008 | $A_{1}=2$
SMC X-4 | 1.29 | 8.831 | 0.005145 | 0.215479 | $A_{1}=-1$
|  | Case 2 |  |  |
Compact Star | $M(M_{\odot})$ | $R(Km)$ | $C(Km^{-2})$ | M/R |
Her X-1 | 0.85 | 8.1 | 0.022018 | 0.154768 | $A_{1}=0$
$4U1538-52$ | 0.87 | 7.866 | 0.024294 | 0.163151 | $A_{1}=2$

##  7 Conclusion

In this paper, we presented isotropic matter inside compact stars in the
framework of modified theory of relativity i.e $f(R,T)$ gravity theory by
considering Buchdahl assumptions. Extending our model to study the analytic
solution for $K<0$ and $K>1$ of compact stars like PSR-B0943+10, SMC X-4, CEN
X-3, Her X-1 and 4U1538-52. Futhermore, pressure inside compact stars is
mostly isotropic that vanishes at the surface and density decreases with
increase in radius $r$ as shown in fig 1. We have shown the central density
and pressure of compact stars for different parameters in Tab 1. Overall here
we find two solution for both the cases which are found to give equal results.

At boundary $r=R$, we matched the interior with exterior Schwarzchild
spacetime to find constant $\frac{c_{2}}{c_{1}}$. Applying it we find all
physical quantities, we did simultaneous comparison between compact stars
PSR-B0943+10 and Her X-1 at $A_{1}=0$ and illustrate this physical properties
against fractional radius $r/R$ in Km like velocity of sound (fig 2) which is
less than the speed of light, energy conditions (fig4) which are non-negative,
gravitational Redshift (fig3) which show satisfactory results for both stars
and finally Adiabatic Index (fig 5) which is greater than $4/3$. Aside this we
present total mass M (normalized in solar mass $M_{\odot}$) with total radius
$R$ along with different parameters like $K$ and $C$ for various compact stars
in Tab 3 which are regular and well defined throughout the stellar structure,
which shows the validity of our model. So, it can be concluded that our
stellar model in reference to $f(R,T)$ Gravity Theory has admissible results
that represents astrophysical objects more precisely and accurately.

## Acknowledgements

The Authors would like to express their sincere gratitude towards Department
of Mathematics, Central University of Jharkhand, Ranchi, India for the
necessary support where the paper has been written and finalized.

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