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e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Department of Mathematics and Natural Sciences, BRAC
University, Dhaka, 1212, Bangladesh 22institutetext: Department of Natural
Sciences, Daffodil International University, Dhaka, 1207, Bangladesh

# Some new Wyman-Adler type static relativistic charged anisotropic fluid
spheres compatible to _self-bound_ stellar modeling

Mohammad Hassan Murade1,addr1  Saba Fatemae2,addr2

(Received: date / Accepted: date)

###### Abstract

In this work some families of relativistic anisotropic charged fluid spheres
have been obtained by solving Einstein-Maxwell field equations with the
preferred form of one of the metric potentials, a suitable forms of electric
charge distribution and pressure anisotropy functions. The resulting equation
of state (EOS) of the matter distribution has been obtained. Physical analysis
shows that the relativistic stellar structure for matter distribution obtained
in this work may reasonably model an electrically charged compact star whose
energy density associated with the electric fields is on the same order of
magnitude as the energy density of fluid matter itself (e.g. electrically
charged _bare_ strange stars). These models permit a simple method of
systematically fixing bounds on the maximum possible mass of cold compact
electrically charged self-bound stars. It has been demonstrated numerically
that the maximum compactness and mass increase in the presence of electric
field and anisotropic pressures.

Based on the analytic model developed in this present work, the values of the
relevant physical quantities have been calculated by assuming the estimated
masses and radii of some well known potential strange star candidates like PSR
J1614-2230, PSR J1903+327, Vela X-1, and 4U 1820-30.

###### Keywords:

General relativityEinstein-MaxwellReissner-NordströmExact
solutionSchwarzschild coordinatesCharged fluid spherePerfect fluid
sphereAnisotropic fluid sphereCompact starSelf-bound starRelativistic
starEquation of state.

††journal: Eur. Phys. J. C

##  1 Introduction

The subject of modeling relativistic compact stellar objects through the
analytical solution of Einstein’s gravitational field equations has a long
history and still the interest remains as one of the key issue to the present
researchers. Since the work of Schwarzschild Schwarzschild03 , Tolman Tolman39
and Oppenheimer and Volkoff Oppenheimer-Volkoff39  the determination of
maximum mass of very compact astrophysical objects has been a key issue in
relativistic astrophysics. Such findings are important in because analytical
solutions enable the distribution of matter in the interior of stellar object
to be modeled in terms of simple algebraic relations.

The central energy density of compact stellar object could be of the order of
$10^{15}$g cm-3, several times higher the normal nuclear matter density and
due to the absence of reliable information about behavior of matter at such
ultrahigh density, insight into the structure can be obtained by reference to
applicable analytic solutions to the equation of relativistic stellar
structure Lattimer-Prakash01 . The known analytic solutions of Einstein’s
gravitational field equations fall into two classes. The first class that
describes “normal” matter neutron stars for which density vanishes at the
surface where the pressure vanishes. The Tolman VII solution with vanishing
surface energy density falls into this class and hence is useful approximation
to realistic neutron star models. And the class that describes stars for which
density is finite, about 2-3 times the normal nuclear matter saturation
density Postnikovetal.10 , at the surface where the pressure vanishes includes
Tolman IV solution and the solutions discussed in Wyman49 ; Heintzmann69 ;
Adler74 ; Adams-Cohen75 ; Kuchowicz75 ; Vaidya-Tikekar82 ; Korkina81 ;
Durgapal82 ; Orlyansky97 ; Durgapal-Bannerji83 ; Durgapaletal.84 ; Durgapal-
Fuloria85 ; Pant-Pant93 ; Pant94 ; Pant96 ; Maurya-Gupta12a . This type of
solutions is useful approximation to realistic models of “self-bound” strange
quark star Lattimer04 . The best-known example of self-bound stars results
from the Bodmer-Witten hypothesis also known as the strange quark matter
hypothesis asserts that strange quark matter (SQM for short) is the ultimate
ground state of matter. Still the fundamental significance of this hypothesis
remains as a serious possibility in physics and astrophysics Weber99 ; Weber05
; Glendenning00 ; Haensel03 ; Haenseletal.07 .

An important distinction between quark stars and conventional neutron stars is
that the quark stars are self-bound by the strong interaction, gravity just
make them massive, whereas neutron stars are bound by gravity. This allows a
quark star to rotate faster than would be possible for a neutron star. A quark
star can also be _bare_. The surfaces of a bare strange star and that of
normal matter neutron star have striking differences. The very properties of
the quark surface, e.g., strong bounding of particles, abrupt density change
from $4\times 10^{14}$g cm-3 to $\sim 0$ in $\sim 1$ fm.

In a very recent past a polytropic quark star model has been suggested Lai-
Xu09 ; Thirukkanesh-Ragel12  in order to establish a general framework in
which theoretical quark star models could be tested by observations. The key
difference between polytropic quark stars and the polytropic model studied
previously for normal (i.e. non-quarkian) stars is that the quark star models
with non-vanishing density at the stellar surface may not be avoidable due to
the strong interaction between quarks which is relevant to the effect of color
confinement. As discussed in Lai-Xu09  the polytropic equations of state are
stiffer than the conventional realistic models (e.g. the MIT bag model) for
quark matter, and pulsar-like stars calculated with a polytropic equation of
state could then have high maximum masses $>2M_{\odot}$. In this framework of
polytropic model a very low massive quark star can also be, and be still
gravitationally stable even if the polytropic index, $n>3$.

Apart from the constituents of these types of compact stars, the most
fascinating distinction between a strange star and a normal neutron star is
the surface electric fields associated with it. Bare strange stars possess
ultra-strong electric fields on their surfaces, which, for ordinary strange
matter, is around $10^{18}$ V/cm and $10^{20}$ V/cm for color superconducting
strange matter Usov04 ; Usovetal.05 ; Negreirosetal.10 . The influence of
energy densities of ultra-high electric fields on the bulk properties of
compact stars was explored in Rayetal.03 ; Malheiroetal.04 ; Weberetal.07 ;
Weberetal.09 ; Weberetal.10 ; Negreirosetal.09 . It also has been shown that
electric fields of this magnitude, generated by charge distributions located
near the surfaces of strange quark stars, increase the stellar mass by up to
30% depending on the strength of the electric field. In contrast to the
strange star the surface electric field in the case of neutron star is absent
Weberetal.12 . These features may allow one to observationally distinguish
quark stars from neutron stars.

The principal motivation of this work is to develop some new analytical
relativistic stellar models by obtaining closed-form solutions of Einstein-
Maxwell field equations. In order to obtain a realistic charged stellar model
one can start with an explicit EOS and suitable form of electric charge
distribution and then integrating the equation of hydrostatic equilibrium,
also known charged generalization of Tolman-Oppenheimer-Volkoff (TOV) equation
Bekenstein71  which is obtained by requiring the conservation of mass-energy,
as that determines the global structure of electrically charged stars. The
integration starts at the center of the star with a prescribed central
pressure and ends where the pressure decreases to zero, indicates the surface
of the star. Some recent studies include Feliceetal.95 ; Anninos-Rothman01 ;
Siffertetal.07 ; Negreiros-Malheiro07 . Such input equations of state do not
normally allow for closed-form solutions.

In the second approach one can have insight into such structures by solving
the Einstein-Maxwell equations which represent an under-determined system of
nonlinear ordinary differential equations of the second order. Due to the high
nonlinearity it is difficult to obtain exact solutions to this system. For the
special case of a static isotropic perfect fluid, the system of field
equations can be reduced to a set of four coupled ordinary differential
equations in five unknowns and arrive to exact solutions by making an _ad hoc_
assumption for one of the metric functions or for the energy density. The EOS
can then be extracted from the resulting metric. The first exact solutions of
field equations, in this approach, known to have astrophysical significance,
may have been discovered by Tolman Tolman39 . Out of the different types of
exact solutions obtained by Tolman, model V and VI are not considered
physically viable, as they correspond to singular solutions (infinite values
of central density and pressure). Except these models, all other solutions are
known as regular solutions (finite and positive pressure and density at the
origin). Models IV and VII are found physically viable in the study of compact
astrophysical stellar objects. Out of numerous works done following Tolman’s
approach some include Nduka76 ; Nduka78 ; Mak-Fung95 ; Maketal.96 ;
Pateletal.97 ; Harko-Mak00 ; Sharmaetal.01 ; Sharmaetal.06 ; Gupta-Kumar05a ;
Gupta-Kumar05b ; Gupta-Kumar05c ; Maharaj-Komathiraj07 ; Thirukkanesh-
Maharaj09 ; Gupta-Kumar11a ; Gupta-Kumar11b ; Hansraj-Maharaj06 ; Pant11a ;
Pant11b ; Pant-Tewari11 ; Pantetal.11a ; Pantetal.11b ; Pant12 ; Pant-Maurya12
; Pant-Faruqi12 ; Pant-Negi12 ; Maurya-Gupta12b ; Mauryaetal.12 ; Bijalwan12 ;
Thirukkanesh-Ragel12 ; Takisa-Maharaj12 ; Takisa-Maharaj13a ; Takisa-
Maharaj13b ; Hansrajetal.14 . As might be expected with Tolman’s method,
unphysical pressure-density configurations are found more frequently than
physical ones.

In recent years, however, several authors followed an alternative approach to
present analytical stellar models of electrically neutral/charged compact
strange stars within the framework of linear equation of state (EOS) based on
MIT bag model together with a particular choice of metric potentials/mass
function Mak-Harko04 ; Sharma-Maharaj07 ; Esculpi-Aloma10 ; Komathiraj-
Maharaj11 ; Takisa-Maharaj12 ; Maharaj-Takisa12 ; Rahamanetal.12 ;
Kalametal.13 ; Thirukkanesh-Ragel13b . Some works also studied the viability
of nonlinear EOS based on suitable geometry for the description in the
interior 3-spaces of such compact star Vaidya-Tikekar82 ; Tikekar90 . This
approach leads to physically viable and easily tractable models of superdense
stars in equilibrium. Tikekar and Jotania Tikekar-Jotania05 ; Tikekar-
Jotania07 , Jotania and Tikekar Jotania-Tikekar06  showed that the ansatz
suggested by Tikekar and Thomas Tikekar-Thomas98  has these features and the
general three-parameter solution based on it also leads to physically
plausible relativistic models of strange stars. Several aspects of physical
relevance and the maximum mass of class of compact star models, based on
Vaidya-Tikekar ansatz, for the both isotropic and anisotropic pressures have
been investigated in Tikekar-Thomas99 ; Sharmaetal.06 ; Debetal.12 ; Sharma-
Ratanpal13 .

Out of the 127 known analytical solutions to Einstein’s equations, compiled in
Delgaty-Lake98 , only a few satisfy elementary tests of physical relevance
and, hence, are viable in the description of relativistic compact stellar
objects. For strange quark stars, the energy density does not vanish at the
surface. Known applicable analytic solutions include Lattimer-Prakash01 ;
Postnikovetal.10 ; Lattimer04

  * •

Schwarzschild interior solution or the incompressible fluid solution (constant
density solution).

  * •

Generalized Tolman IV solution.

  * •

Matese and Whitman I.

In contrast, so far the literature known to present authors, the charged
analogues of Tolman’s models (V-VI) obtained in Pant-Sah79 ; Patino-Rago89 ;
Singhetal.95 ; Ray-Das02 ; Ray-Das04 ; Ray-Das07 ; Rayetal.07  are not
physically viable in the description of compact astrophysical objects as the
infinite values of central density and pressure. Though the Schwarzschild
constant density solution is physically unrealistic, the charged analogues,
obtained in Guptaetal.12 ; Gupta-Kumar05c ; Bijalwan-Gupta08 ; Bijalwan11 ,
and the charged analogue of Matese and Whitman solution obtained in Maurya-
Gupta12c  may be relevant in the description of self-bound electrically
charged strange quark stars. Charged analogues of Tolman IV and VII models
Gupta-Maurya11 ; Maurya-Gupta11c ; Kiess12 ), as the neutral ones, exhibit the
physical features required for the construction of physically realizable
relativistic compact stellar structure. The charged analogues of Vaidya-
Tikekar models have been derived in Patel-Koppar87 ; Kopparetal.91 .
Astrophysical consequences of the charged analogues of Vaidya-Tikekar
solutions in modeling electrically charged compact star have been discussed in
Gupta-Kumar11a ; Patel-Pandya86 ; Komathiraj-Maharaj07 ; Bijalwan-Gupta11 ;
Chattopadhyayetal.12 .

It was shown by Bonnor Bonnor60 ; Bonnor65  that a spherical body can remain
in equilibrium under its own gravitation and electric repulsion if the matter
present in the sphere carries certain modest electric charge density. The
problem of the stability of a homogeneous distribution of matter containing a
net surface charge was considered by Stettner Stettner73 . Stettner showed
that a fluid sphere of uniform density with a modest surface charge is more
stable than the same system without charge. The electric charge weakens
gravity to the extent of turning it into a repulsive field, as happens in the
vicinity of a Reissner-Nordström singularity. Thus the gravitational collapse
of spherical matter distribution to a point singularity may be avoided if the
matter acquires large amounts of electric charge during an accretion process
onto a compact object. The gravitational attraction may then be balanced by
electrostatic repulsion due to the same electric charge and by the pressure
gradient Bekenstein71 ; Ghezzi05 . And hence the study of the gravitational
behavior of compact charged stellar object has raised the possibility of
modeling such compact astrophysical objects in terms of simple algebraic
relations between the matter pressure and its energy density.

Of course no astrophysical object is entirely composed of perfect fluid. The
theoretical investigations of Ruderman Ruderman72  about more realistic
stellar models show that the nuclear matter may be locally anisotropic at
least in certain very high density ranges $(\rho>10^{15}$ g cm${}^{-3})$,
where the nuclear interactions in the stellar matter must be treated
relativistically. According to these views, in such massive stellar objects
the radial pressure may not be equal to the tangential pressure. Since the
pioneering work of Bowers and Liang Bowers-Liang74 , there has been an
extensive literature devoted to the study of anisotropic spherically symmetric
static general relativistic con figurations (See Cosenzaetal.81 ; Bayin82 ;
Herrera-Leon85 ; Leon87a ; Leon87b ; Bondi92 ; Herrera-Santos97 ;
Herreraetal.01 ; Dev-Gleiser02 ; Dev-Gleiser03 ; Gleiser-Dev04 ; Mak-Harko02a
; Mak-Harko02b ; Maketal.02 ; Mak-Harko03 ; Chaisi-Maharaj06a ; Chaisi-
Maharaj06b ; Sharma-Maharaj07 ; Herreraetal.08 ; Viaggiu09 ; Maharaj-Takisa12
; Maurya-Gupta12d ; Maurya-Gupta13 ; Takisa-Maharaj13a ; Takisa-Maharaj13b ;
Maurya-Gupta14 ; Maharaj-Sunzu-Ray14 ; Sunzu-Maharaj-Ray14  and the references
therein).

Following the approach of Durgapal Durgapal82 , Maurya and Gupta Maurya-
Gupta11b ; Maurya-Gupta11c  some new analytical relativistic stellar models
have been developed by obtaining closed-form solutions of Einstein-Maxwell
field equations. Our analysis depends on several mathematical key assumptions.
First, we choose a particular functional form for one of the metric
potentials. The form chosen ensures that the metric function is nonsingular,
continuous, and well behaved in the interior of the star. On physical basis
this is one of the desirable features for any well-behaved model. Further, we
assume particular forms of electric charge distribution and pressure
anisotropy. The maximum allowable mass and corresponding values of physical
quantities have been determined. The solutions obtained in this work are
expected to provide simplified but easy to mathematically analyzed charged
stellar models with nonzero super-high surface density which could reasonably
model the stellar core of an electrically charged strange quark star by
satisfying applicable physical boundary conditions.

The presentation of this work is as follows. The next section, Sect. 2, is
devoted for the solution of Einstein-Maxwell field equations of perfect fluid
and derives the pressure and density relation. In Sect. 3 we present the
elementary criteria to be satisfied the obtained solution as to present a
realistic stellar model. Sect. 4 develops the important ratios by matching the
obtained metric components with the space-time exterior to the charged object
which is described by the unique Reissner-Nordström metric. Physical analysis
has been made on the obtained models in Sect. 5. It has been demonstrated
numerically that the maximum compactness, redshift and mass increase in the
presence of electric field and anisotropic pressures; which are in agreement
with some other workKarmakaretal.07 . In Sect. 6 some explicit numerical
models of relativistic anisotropic stars, with a possible astrophysical
relevance, are also presented and we also apply our “toy” model to some well
known potential strange star candidates to calculate various physical
quantities by assuming the estimated masses and predicted radii. And finally
Section 7 discusses and concludes the work.

##  2 Interior Solutions of Einstein-Maxwell Field Equations

###  2.1 Field equations

In this work we intend to study a static, spherically symmetric matter
distribution whose interior metric is given in Schwarzschild
coordinatesTolman39 ; Oppenheimer-Volkoff39
$x^{\mu}=\left(t,r,\theta,\varphi\right)$111Throughout the work we will use
$c=G=1$, except in tables and figures.,

$ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)$ (2.1.1)

The functions $\nu$ and $\lambda$ satisfy the Einstein-Maxwell field
equations,

$G_{\nu}^{\mu}=R_{\nu}^{\mu}-\frac{1}{2}\delta_{\nu}^{\mu}=\kappa\left(T_{\nu}^{\mu}+E_{\nu}^{\mu}\right)$
(2.1.2)

where $\kappa=8\pi$ is Einstein’s constant. The matter within the star is
assumed to be locally anisotropic fluid in nature and consequently
$T_{\nu}^{\mu}$ and $E_{\nu}^{\mu}$ are the energy-momentum tensor of fluid
distribution and electromagnetic field defined by, Dionysiou82 ,

$\displaystyle T_{\nu}^{\mu}$ $\displaystyle=$ $\displaystyle\left(P_{t}+\mu
c^{2}\right)v^{\mu}v_{\nu}-P_{t}\delta^{\mu}_{\nu}+(P_{r}-P_{t})\chi^{\mu}\chi_{\nu}$
$\displaystyle E_{\nu}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left(-F^{\mu m}F_{\nu
m}+\frac{1}{4}\delta_{\nu}^{\mu}F^{mn}F_{mn}\right)$

where $\rho,\,P_{r},\,P_{t},\,v^{\mu}$, denote energy density, radial
pressure, tangential pressure of the fluid distribution respectively.
$v^{\mu}$ and $F_{\mu\nu}$ denote the velocity vector and anti-symmetric
electromagnetic field strength tensor defined by,

$F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial A_{\mu}}-\frac{\partial
A_{\mu}}{\partial A_{\nu}}$ (2.1.3)

which satisfies Maxwell equations,

$\displaystyle F^{\mu\nu}_{;\nu}=\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\nu}}\left(\sqrt{-g}F^{\mu\nu}\right)$ $\displaystyle=$ $\displaystyle-4\pi
j^{\mu}$ (2.1.4a) $\displaystyle
F_{\mu\nu;\lambda}+F_{\nu\lambda;\mu}+F_{\lambda\mu;\nu}$ $\displaystyle=$
$\displaystyle 0$ (2.1.4b)

where $g$ is the determinant of quantities $g_{\mu\nu}$ in eq. (2.1.1),
defined by,

$g=\begin{pmatrix}e^{\nu}&0&0&0\\\ 0&-e^{\lambda}&0&0\\\ 0&0&-r^{2}&0\\\
0&0&0&-r^{2}\sin^{2}\theta\\\
\end{pmatrix}=-e^{\nu+\lambda}r^{4}\sin^{2}\theta$

where, $A_{\nu}=(\varphi(r),0,0,0)$ is four potential and $j^{\mu}$ is the
four current vector defined by

$j^{\mu}=\frac{\rho_{\mathrm{ch}}}{\sqrt{g_{00}}}\frac{dx^{\mu}}{dx^{0}}$

where $\rho_{\mathrm{ch}}$ denotes the proper charge density.

For static matter distribution the only non-zero component of the four-current
is $j^{0}$. Because of spherical symmetry, the four-current component is only
a function of radial distance, $r$. The only nonvanishing components of
electromagnetic field tensor are $F^{01}$ and $F^{10}$, related by
$F^{01}=-F^{10}$, which describe the radial component of the electric field.
From eq. (2.1.4a) one obtains the following expression for the electric field:

$F^{01}=-e^{-\frac{\nu+\lambda}{2}}\frac{q(r)}{r^{2}}$

where $q(r)$ represents the total charge contained within the sphere of radius
$r$ defined by,

$q(r)=4\pi\int_{0}^{r}{e^{\frac{\lambda}{2}}\rho_{\mathrm{ch}}u^{2}}du$
(2.1.5)

Equation (2.1.5) can be treated as the relativistic version of Gauss’s law.

For the metric (2.1.1), the Einstein-Maxwell field equations may be expressed
as the following system of ordinary differential equations Dionysiou82 ,

$\displaystyle\frac{\nu^{^{\prime}}}{r}e^{-\lambda}-\frac{\left(1-e^{-\lambda}\right)}{r^{2}}$
$\displaystyle=$ $\displaystyle\kappa P_{r}-\frac{q^{2}}{r^{4}}$ (2.1.6)
$\displaystyle\left(\frac{\nu^{{}^{\prime\prime}}}{2}-\frac{\nu^{^{\prime}}\lambda^{^{\prime}}}{4}+\frac{\nu^{{}^{\prime}2}}{4}+\frac{\nu^{^{\prime}}-\lambda^{^{\prime}}}{2r}\right)e^{-\lambda}$
$\displaystyle=$ $\displaystyle\kappa P_{t}+\frac{q^{2}}{r^{4}}$ (2.1.7)
$\displaystyle\frac{\lambda^{\prime}}{r}e^{-\lambda}+\frac{\left(1-e^{-\lambda}\right)}{r^{2}}$
$\displaystyle=$ $\displaystyle\kappa\rho+\frac{q^{2}}{r^{4}}$ (2.1.8)

where prime $\left({}^{\prime}\right)$ denotes the $r$-derivative.

In analogy to the electrically uncharged case, it is usually introduced a
quantity $m(r)$ by the following expression,

$e^{-\lambda}=1-\frac{2m(r)}{r}+\frac{q^{2}}{r^{2}}$ (2.1.9)

If $R$ represents the radius of the fluid distribution then it can be showed
that $m$ is constant $m(r=R)=M$ outside the fluid distribution where $M$ is
the gravitational mass. Thus the function $m(r)$ represents the gravitational
mass of the matter contained in a sphere of radius $r$. Using eqs. (2.1.8),
(2.1.6), and (2.1.7) respectively, one can arrive at,

$m(r)=\frac{\kappa}{2}\int\rho
r^{2}dr+\frac{q^{2}}{2r}+\frac{1}{2}\int\frac{q^{2}}{r^{2}}dr$ (2.1.10)
$\nu^{\prime}=\frac{(\kappa
rP_{r}+\frac{2m}{r^{2}}-\frac{2q^{2}}{r^{3}})}{(1-\frac{2m}{r}+\frac{q^{2}}{r^{2}})}$
(2.1.11)
$\displaystyle\frac{dP_{r}}{dr}=-\frac{(P_{r}+\rho)}{2}\nu^{\prime}+\frac{q}{4\pi
r^{4}}\frac{dq}{dr}+\frac{2\Delta}{r}$ (2.1.12)

Finally combining (2.1.11) and (2.1.12), we get,

$\displaystyle\frac{dP_{r}}{dr}$ $\displaystyle=$
$\displaystyle-\frac{(P_{r}+\rho)}{2}\frac{(\kappa
rP_{r}+\frac{2m}{r^{2}}-\frac{2q^{2}}{r^{3}})}{(1-\frac{2m}{r}+\frac{q^{2}}{r^{2}})}$
(2.1.13) $\displaystyle+\frac{q}{4\pi r^{4}}\frac{dq}{dr}+\frac{2\Delta}{r}$

which is the charged generalization of Tolman-Oppenheimer-Volkoff (TOV)
equation of hydrostatic equilibrium for anisotropic stellar configuration
Leon87b . In eq. (2.1.13) the additional term, $2\Delta/r$, representing
“force” which is due to the anisotropic nature of the fluid. This force is
directed outward when $P_{t}>P_{r}\,(\Delta>0)$ and inward when
$P_{t}<P_{r}\,(\Delta<0)$. The existence of repulsive force (in the case
$\Delta>0$) allows the construction of more compact distribution when using
anisotropic fluid than when using isotropic fluid.

Instead of solving eq. (2.1.13), for any prescribed equation of state, we
rather interested in solving eqs. (2.1.6)-(2.1.8) with the help of following
ansatz Korkina81 ; Durgapal82 ,

$e^{\nu}=B_{N}\left(1+Cr^{2}\right)^{N}$ (2.1.14)

where $N$ is a positive integer and $B_{N},\,C>0$ are two constants to be
determined by the appropriate physical boundary conditions. Subtracting
(2.1.6) from (2.1.7) one obtain the equation of “pressure anisotropy”,

$\displaystyle\left(\frac{\nu^{{}^{\prime\prime}}}{2}-\frac{\nu^{^{\prime}}\lambda^{^{\prime}}}{4}+\frac{\nu^{{}^{\prime}2}}{4}-\frac{\nu^{^{\prime}}+\lambda^{^{\prime}}}{2r}\right)e^{-\lambda}+\frac{\left(1-e^{-\lambda}\right)}{r^{2}}$
$\displaystyle=\kappa(P_{t}-P_{r})+\frac{2q^{2}}{r^{4}}$ (2.1.15)

Equation (2.1) is a second order nonlinear differential equation in $\nu$ but
first order linear in $\lambda$. At this moment it is convenient to introduce
the following transformations

$e^{-\lambda}=Z,\,\,x=Cr^{2}$ (2.1.16)

which transform eqs. (2.1.6)-(2.1.8) to the following,

$\displaystyle\frac{\kappa}{C}P_{r}$ $\displaystyle=$
$\displaystyle\frac{[1+(2N+1)x]}{x(1+x)}Z-\frac{1}{x}+\frac{Cq^{2}}{x^{2}}$
(2.1.17) $\displaystyle\frac{\kappa}{C}P_{t}$ $\displaystyle=$
$\displaystyle\frac{(2N+N^{2}x)}{(1+x)^{2}}Z+\frac{[1+(1+N)x]}{(1+x)}\frac{dZ}{dx}-\frac{Cq^{2}}{x^{2}}$
(2.1.18) $\displaystyle\frac{\kappa}{C}\rho$ $\displaystyle=$
$\displaystyle-2\frac{dZ}{dx}-\frac{Z}{x}+\frac{1}{x}-\frac{Cq^{2}}{x^{2}}$
(2.1.19)

And the eq. (2.1) can be written in terms of auxiliary variable $x$ as,

$\displaystyle\frac{dZ}{dx}$ $\displaystyle+$
$\displaystyle\left[\frac{(N^{2}-2N-1)x^{2}-2x-1}{x(1+x)(1+(1+N)x)}\right]Z$
(2.1.20) $\displaystyle=$
$\displaystyle\frac{(1+x)}{x(1+(1+N)x)}\left(\frac{2Cq^{2}}{x}+\Delta
x-1\right)$

where, $\displaystyle\Delta=\frac{\kappa}{C}(P_{t}-P_{r})$ is the measure of
pressure anisotropy. Eq. (2.1.20) yields the following solution Fatema-Murad13
,

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\frac{x}{(1+x)^{N-2}\left[1+(1+N)x\right]^{\frac{2}{1+N}}}\times$
(2.1.21)
$\displaystyle\int{\frac{(1+x)^{N-1}\left[1+(1+N)x\right]^{\frac{1-N}{1+N}}}{x^{2}}\left(\frac{2Cq^{2}}{x}+\Delta
x\right)}\,dx$
$\displaystyle+\frac{1}{(1+x)^{N-2}}-\frac{1}{2}\binom{N-1}{2}\frac{x}{(1+x)^{N-2}}$
$\displaystyle-\sum_{i=0}^{N-4}\sum_{j=0}^{i+1}\frac{(-1)^{j}}{(N+1)^{i+2}}\binom{N-1}{i+3}\binom{i+1}{j}\times$
$\displaystyle\frac{x[1+(N+1)x]^{i-j+1}}{(i-j+1)(1+x)^{n-2}}$
$\displaystyle+A_{N}\frac{x}{(1+x)^{N-2}\left[1+(1+N)x\right]^{\frac{2}{1+N}}}\text{$N\geq
4$}$

where $A_{N}$ is the constant of integration may be determined by imposing
appropriate physical boundary conditions.

###  2.2 Models of Electric Charge Distribution and Pressure Anisotropy

As the “realistic” charge distribution inside the fluid sphere is not known
Giuliani-Rothman08 , but it seems intuitively reasonable that due to
electrical repulsion the charge distribution should be weighted toward the
surface Anninos-Rothman01 . One can imagine several plausible mathematical
forms of $2Cq^{2}/x^{2}$, to integrate the eq. (2.1.21). Various authors
presented variety of solutions previously for different suitable choices of
charge distributions with isotropic pressure. Some of the solutions are
compiled in table 1 (Also see Murad-Fatema13c ). In this work we consider the
following forms of electric charge distribution and pressure anisotropy:

$\frac{2Cq^{2}}{x^{2}}=Kx^{n+1}(1+x)^{1-N}(1+mx)^{p}(1+(1+N)x)^{\frac{N-1}{N+1}}$
(2.2.1) $\Delta=\delta x(1-2ax)(1+x)^{1-N}(1+(1+N)x)^{\frac{N-1}{N+1}}$
(2.2.2)

where $K,\,\delta\geq 0$, $n$ is a nonnegative integer, and $m,\,p$ are
nonzero and $a$ is any real numbers. It must be emphasized that these
hypothetical models of electric charge distribution and pressure anisotropy
are chosen, in term of $x$, in such a way that these allow us to integrate the
eq. (2.1.21) rather than for any particular physical reasons. Moreover, the
electric field intensity and anisotropy vanish at the center and remains
continuous and bounded in the interior of the star for a wide range of values
of the parameters. Thus these choices may be physically reasonable and useful
in the study of the gravitational behavior of anisotropic charged stellar
objects.

Table 1:  Exact static spherically symmetric perfect fluid solutions of Einstein-Maxwell equations obtained by different charge distributions for isotropic pressure $(\Delta=0)$. The second column shows the models which may be rediscovered from the present model (2.2.1). $N$ | Generated by | $n$ | $m$ | $p$ | Charge Distribution | Reference
---|---|---|---|---|---|---
present model | $2Cq^{2}/x^{2}$
1 | Y | 0 | $-$ | 0 | $Kx$ | Pant-Rajasekhara11
1 | Y | 0 | 1 | 1 | $Kx(1+x)$ | Pant-Negi12
1 | Y | 0 | 1 | $n$ | $Kx(1+x)^{n}$ | Pant-Negi12
2 | Y | 0 | 1 | 1 | $Kx(1+3x)^{\frac{1}{3}}$ | Pant-Rajasekhara11 ; Pant-Tewari11
2 | Y | 0 | 1 | 2 | $Kx(1+x)(1+3x)^{\frac{1}{3}}$ | Pantetal.11a
2 | Y | 0 | 1 | 3 | $Kx(1+x)^{2}(1+3x)^{\frac{1}{3}}$ | Pant-Faruqi12 ; Murad13
2 | Y | 0 | 1 | $n$ | $Kx(1+x)^{n}(1+3x)^{\frac{1}{3}}$ | Fatema-Murad13
2 | Y | $n$ | $3$ | $-1/3$ | $K\displaystyle\frac{x^{n+1}}{(1+x)}$ | Murad-Fatema13c
2 | N | $-$ | $-$ | $-$ | $Kx^{n+1}(1+mx)^{p}(1+3x)^{\frac{1}{3}}$
2 | Y | $n$ | $m$ | $p$ | $K\displaystyle\frac{x^{n+1}(1+mx)^{p}(1+3x)^{\frac{1}{3}}}{(1+x)}$
2 | N | $-$ | $-$ | $-$ | $Kx^{n+1}(1+x)^{l-1}(1+3x)^{m+\frac{1}{3}}$ | Murad-Fatema14
2 | N | $-$ | $-$ | $-$ | $Kx(1+mx)^{\frac{1}{3}}(1+3x)^{\frac{1}{3}}$ | Rahman-Murad14
2 | Y | 0 | $m$ | $1/3$ | $\displaystyle K\frac{x(1+mx)^{\frac{1}{3}}(1+3x)^{\frac{1}{3}}}{(1+x)}$
3 | Y | 0 | 1 | 2 | $Kx\sqrt{(1+4x)}$ | Pantetal.11a
3 | Y | 0 | 1 | $n+2$ | $Kx(1+x)^{n}\sqrt{(1+4x)}$ | Pant-Maurya12
4 | Y | 0 | 1 | 3 | $Kx(1+5x)^{\frac{3}{5}}$ | Pant11a
4 | Y | 0 | 1 | 4 | $Kx(1+x)(1+5x)^{\frac{3}{5}}$ | Mehtaetal.13
4 | Y | 0 | 1 | $n+3$ | $Kx(1+x)^{n}(1+5x)^{\frac{3}{5}}$ | Murad-Fatema13a
4 | N | $-$ | 0 | 0 | $\displaystyle\frac{Kx^{r}}{(1+x)^{2}}$ | Mauryaetal.11
4 | Y | 2 | 5 | $-3/5$ | $\displaystyle\frac{Kx^{3}}{(1+x)^{3}}$ | Faruqi-Pant12
5 | Y | 0 | 1 | 4 | $Kx(1+6x)^{\frac{2}{3}}$ | Gupta-Maurya11
5 | Y | 0 | 1 | 5 | $Kx(1+x)(1+6x)^{\frac{2}{3}}$ | Fuloriaetal.11
5 | Y | 0 | 1 | 6 | $Kx(1+x)^{2}(1+6x)^{\frac{2}{3}}$ | Fuloria-Tewari12
5 | Y | 0 | 1 | $n+3$ | $Kx(1+x)^{n}(1+6x)^{\frac{2}{3}}$ | Murad-Fatema13b
6 | Y | 0 | 1 | 5 | $Kx(1+7x)^{\frac{5}{7}}$ | Maurya-Gupta11a
$N$ | Y | 0 | 1 | $N-1$ | $n^{2}Kx[1+(N+1)x]^{\frac{(N-1)}{(N+1)}}$ | Maurya-Gupta11c

###  2.3 Anisotropic Charged Stellar Models

Ishak et al. Ishaketal.01 , Lake Lake03 , and recently Maurya and Gupta
Maurya-Gupta11b ; Maurya-Gupta11c  showed that the ansatz for the metric
function (2.1.14) produces an infinite family of analytic solutions of the
self-bound type. Four of these were previously known ($N=$1, 3, 4, and 5). The
most relevant case is for $N=2$, for which the speed of sound $\approx
1/\sqrt{3}$ throughout most of the star, similar to the behavior of strange
quark matter Lattimer-Prakash05 . For $N=2$ the solution of the Einstein-
Maxwell system (2.1.6)-(2.1.8) for the model charge distribution and pressure
anisotropy considered in this work are then given by,

Case I: $p\neq-1$. $n=$ nonnegative integer

$e^{\nu}=B_{2}(1+x)^{2}$ (2.3.1) $\displaystyle e^{-\lambda}$ $\displaystyle=$
$\displaystyle\frac{K}{m^{n+1}}\sum_{i=0}^{n}\frac{(-1)^{i}}{(n-i+p+1)}\binom{n}{i}\times$
(2.3.2)
$\displaystyle\frac{x(1+mx)^{(n-i+p+1)}}{(1+3x)^{\frac{2}{3}}}+\delta\frac{x^{2}(1-ax)}{(1+3x)^{\frac{2}{3}}}$
$\displaystyle+1+A_{2}\frac{x}{(1+3x)^{\frac{2}{3}}}$
$\frac{2Cq^{2}}{x^{2}}=K\frac{x^{n+1}(1+mx)^{p}(1+3x)^{\frac{1}{3}}}{(1+x)}$
(2.3.3) $\Delta=\frac{\delta x(1-2ax)(1+3x)^{\frac{1}{3}}}{(1+x)}$ (2.3.4)
$\displaystyle\frac{\kappa}{C}P_{r}$ $\displaystyle=$
$\displaystyle\frac{K}{m^{n+1}}\sum_{i=0}^{n}\frac{(-1)^{i}}{(n-i+p+1)}\binom{n}{i}\times$
(2.3.5)
$\displaystyle\frac{(1+mx)^{(n-i+p+1)}(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}$
$\displaystyle+\frac{K}{2}\frac{x^{n+1}(1+mx)^{p}(1+3x)^{\frac{1}{3}}}{(1+x)}$
$\displaystyle+\delta\frac{x(1-ax)(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}$
$\displaystyle+\frac{4}{(1+x)}+A_{2}\frac{(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}$
$\frac{\kappa}{C}P_{t}=\frac{\kappa}{C}P_{r}+\Delta$ (2.3.6)
$\displaystyle\frac{\kappa}{C}\rho$ $\displaystyle=$
$\displaystyle-\frac{K}{m^{n+1}}\sum_{i=0}^{n}\frac{(-1)^{i}}{(n-i+p+1)}\binom{n}{i}(1+mx)^{(n-i+p)}\times$
(2.3.7)
$\displaystyle\frac{\Pi_{m,n,i,p}(x)}{(1+3x)^{\frac{5}{3}}}-\frac{K}{2}\frac{x^{n+1}(1+mx)^{p}(1+3x)^{\frac{1}{3}}}{(1+x)}$
$\displaystyle-\delta x\frac{(5+(11-7a)x-17ax^{2})}{(1+3x)^{\frac{5}{3}}}$
$\displaystyle-A_{2}\frac{(3+5x)}{(1+3x)^{\frac{5}{3}}}$

Case II: $p=-1$. $n=0$

$e^{\nu}=B_{2}\left(1+x\right)^{2}$ (2.3.8) $\displaystyle e^{-\lambda}$
$\displaystyle=$
$\displaystyle\frac{K}{m}\frac{x\ln(1+x)}{(1+3x)^{\frac{2}{3}}}+\delta\frac{x^{2}(1-ax)}{(1+3x)^{\frac{2}{3}}}+1$
(2.3.9) $\displaystyle+A_{2}\frac{x}{(1+3x)^{\frac{2}{3}}}$
$\Delta=\frac{\delta x(1-2ax)(1+3x)^{\frac{1}{3}}}{(1+x)}$ (2.3.10)
$\frac{2Cq^{2}}{x^{2}}=K\frac{x(1+3x)^{\frac{1}{3}}}{(1+x)(1+mx)}$ (2.3.11)
$\displaystyle\frac{\kappa}{C}P_{r}$ $\displaystyle=$
$\displaystyle\frac{K}{m}\frac{\ln(1+mx)(1+5x)}{(1+3x)^{\frac{1}{3}}(1+x)}+\frac{K}{2}\frac{x(1+3x)^{\frac{1}{3}}}{(1+x)(1+mx)}$
(2.3.12)
$\displaystyle+\delta\frac{x(1-ax)(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}+\frac{4}{(1+x)}$
$\displaystyle+A_{2}\frac{(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}$
$\frac{\kappa}{C}P_{t}=\frac{\kappa}{C}P_{r}+\Delta$ (2.3.13)
$\displaystyle\frac{\kappa}{C}\rho$ $\displaystyle=$
$\displaystyle-\frac{K}{m}\frac{(3+5x)\ln(1+mx)}{(1+3x)^{\frac{5}{3}}}-\frac{2Kx}{(1+3x)^{\frac{2}{3}}(1+mx)}$
(2.3.14) $\displaystyle-\frac{K}{2}\frac{x(1+3x)^{\frac{1}{3}}}{(1+x)(1+mx)}$
$\displaystyle-\delta x\frac{(5+(11-7a)x-17ax^{2})}{(1+3x)^{\frac{5}{3}}}$
$\displaystyle-A_{2}\frac{(3+5x)}{(1+3x)^{\frac{5}{3}}}$

Case III: $p=-1$. $n=$ positive integer

$e^{\nu}=B_{2}\left(1+x\right)^{2}$ (2.3.15) $\displaystyle e^{-\lambda}$
$\displaystyle=$
$\displaystyle\frac{K}{m^{n+1}}\sum_{i=0}^{n-1}\left[\frac{(-1)^{i}}{(n-i)}\binom{n}{i}\frac{x(1+mx)^{(n-i)}}{(1+3x)^{\frac{2}{3}}}\right]$
(2.3.16)
$\displaystyle+\frac{K}{m}(-1)^{n}\frac{x\ln(1+mx)}{(1+3x)^{\frac{1}{3}}}+\delta\frac{x^{2}(1-ax)}{(1+3x)^{\frac{2}{3}}}$
$\displaystyle+1+A_{2}\frac{x}{(1+3x)^{\frac{2}{3}}}$
$\frac{2Cq^{2}}{x^{2}}=K\frac{x^{n+1}(1+3x)^{\frac{1}{3}}}{(1+x)(1+mx)}$
(2.3.17) $\Delta=\frac{\delta x(1-2ax)(1+3x)^{\frac{1}{3}}}{(1+x)}$ (2.3.18)
$\displaystyle\frac{\kappa}{C}P_{r}$ $\displaystyle=$
$\displaystyle\frac{K}{m^{n+1}}\sum_{i=0}^{n-1}\left[\frac{(-1)^{i}}{(n-i)}\binom{n}{i}\frac{(1+mx)^{(n-i)}(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}\right]$
(2.3.19)
$\displaystyle+\frac{(-1)^{n}K}{m^{n+1}}\frac{\ln(1+mx)(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}$
$\displaystyle+\frac{K}{2}\frac{x^{n+1}(1+3x)^{\frac{1}{3}}}{(1+x)(1+mx)}$
$\displaystyle+\delta\frac{x(1-ax)(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}$
$\displaystyle+\frac{4}{(1+x)}+A_{2}\frac{(1+5x)}{(1+3x)^{\frac{2}{3}}(1+x)}$
$\frac{\kappa}{C}P_{t}=\frac{\kappa}{C}P_{r}+\Delta$ (2.3.20)
$\displaystyle\frac{\kappa}{C}\rho$ $\displaystyle=$
$\displaystyle-\frac{K}{m^{n+1}}\sum_{i=0}^{n-1}\frac{(-1)^{i}}{(n-i)}\binom{n}{i}(1+mx)^{(n-i-1)}\times$
(2.3.21)
$\displaystyle\frac{\Pi_{m,n,-1,i}(x)}{(1+3x)^{\frac{5}{3}}}-\frac{(-1)^{n}K}{m^{n+1}}\frac{(3+5x)\ln(1+mx)}{(1+3x)^{\frac{5}{3}}}$
$\displaystyle-\frac{(-1)^{n}K}{m^{n}}\frac{2x}{(1+mx)(1+3x)^{\frac{5}{3}}}$
$\displaystyle-\frac{K}{2}\frac{x^{n+1}(1+3x)^{\frac{1}{3}}}{(1+mx)(1+x)}$
$\displaystyle-\delta x\frac{(5+(11-7a)x-17ax^{2})}{(1+3x)^{\frac{5}{3}}}$
$\displaystyle-A_{2}\frac{(3+5x)}{(1+3x)^{\frac{5}{3}}}$

where,

$\displaystyle\Pi_{m,n,p,i}(x)$ $\displaystyle=$ $\displaystyle
3+(2mn-2mi+2mp+5m+5)x$ $\displaystyle+(6mn-6mi+6mp+11m)x^{2}.$

In the absence of electric field intensity $(K=0)$ and pressure anisotropy
$(\delta=0)$ eqs. (2.3.1)-(2.3.21) reduce to the well-known Wyman
solutionWyman49 , also known as Wyman IIa metric222Wyman IIa metric is the
generalization of Tolman VI solution. Leibovitz rediscovered Wyman IIa metric
in Leibovitz69 . Adler Adler74 , Kuchowicz Kuchowicz75  and Adams-Cohen Adams-
Cohen75  also rediscovered a particular case of Wyman IIa metric respectively,
but none of those works cited Wyman’s work! according to the classification
made in Delgaty-Lake98 . Hence, the models presented by the eqs.
(2.3.1)-(2.3.21) represent the anisotropic charged analogues of Wyman-Adler-
Kuchowicz solution. Equations (2.3.5), (2.3.7), (2.3.12), (2.3.14), and
(2.3.19), (2.3.21) constitute the equations of state of each case.

##  3 Elementary Criteria for Physical Acceptability

Due to the high nonlinearity of Einstein field equations (2.1.2) not many
realistic physical solutions are known for the description of static
spherically symmetric perfect fluid spheres. Out of 127 solutions only 16 were
found to pass elementary tests of physical relevance Delgaty-Lake98 . A
physically acceptable interior solution of the gravitational field equations
must comply with the certain (not necessarily mutually independent) physical
conditions Kuchowicz72 ; Buchdahl79 :

  1. (a)

Regularity conditions

     * (i)

The solution should be free from physical and geometric singularities i.e.
$e^{\nu}>0$ and $e^{\lambda}>0$ in the range $0\leq r\leq R$

     * (ii)

The radial and tangential pressures and density are positive,
$P_{r},\,P_{t},\,\rho>0$.

     * (iii)

Radial pressure $P_{r}$ should be zero at boundary $r=R$ i.e. $P_{r}(r=R)=0$,
the energy density and tangential pressure may follow $\rho(r=R)\geq 0$ and
$P_{r}(r=R)\geq 0$.

  2. (b)

Stability conditions

     * (iv)

In order to have an equilibrium configuration the matter must be stable
against the collapse of local regions. This requires, _Le Chatelier’s
principle_ also known as local or microscopic stability condition, that the
radial pressure $P_{r}$ must be a monotonically non-decreasing function of
$\rho$ Bayin82 ,

$dP_{r}/d\rho\geq 0.$

     * (v)

The relativistic adiabatic index is given by
$\Gamma=\frac{(P_{r}+\rho)}{P_{r}}\frac{dP_{r}}{d\rho}$. The necessary
condition for this exact solution to serve as a model of a relativistic star
is that $\Gamma>\frac{4}{3}$.

  3. (c)

Causality condition

     * (vi)

The condition $0\leq\sqrt{dP_{r}/d\rho}\leq 1$, $0\leq\sqrt{dP_{t}/d\rho}\leq
1$ be the condition that the speed of sound not exceeds that of light.

  4. (d)

Energy conditions

     * (vii)

A physically reasonable energy-momentum tensor has to obey the conditions
$\rho\geq P_{r}+2P_{t}$ and $\rho+P_{r}+2P_{t}\geq 0$.

  5. (e)

Monotone decrease of physical parameters

     * (ix)

Pressure and density, should maximum at the center and monotonically
decreasing towards the pressure free interface (i.e. boundary of the fluid
sphere). Mathematically,

$\left(\frac{dP_{r}}{dr}\right)_{r=0}=0,\,\left(\frac{dP_{t}}{dr}\right)_{r=0}=0,\,\left(\frac{d\rho}{dr}\right)_{r=0}=0,$
$\left(\frac{d^{2}P_{r}}{dr^{2}}\right)_{r=0}<0,\left(\frac{d^{2}P_{t}}{dr^{2}}\right)_{r=0}<0,\,\left(\frac{d^{2}\rho}{dr^{2}}\right)_{r=0}<0$

So that,

$\frac{dP_{r}}{dr}<0,\,\frac{dP_{t}}{dr}<0,\,\frac{d\rho}{dr}<0,\,0<r\leq R.$

     * (x)

Additionally, the velocity of sound should be monotonically decreasing towards
the surface, i.e.,

$\frac{d}{dr}\left(\frac{dP_{r}}{d\rho}\right)<0,\,\frac{d}{dr}\left(\frac{dP_{t}}{d\rho}\right)<0$

for $0\leq r\leq R$.

     * (xi)

The ratio of pressure to density, $P_{r,t}/\rho$, should be monotonically
decreasing with the increase of $r$, i.e.,

$\frac{d}{dr}\left(\frac{P_{r}}{\rho}\right)_{r=0}=0,\,\,\frac{d^{2}}{dr^{2}}\left(\frac{P_{r}}{\rho}\right)_{r=0}<0,$
$\frac{d}{dr}\left(\frac{P_{t}}{\rho}\right)_{r=0}=0,\,\,\frac{d^{2}}{dr^{2}}\left(\frac{P_{t}}{\rho}\right)_{r=0}<0.$

  6. (f)

Matching condition

     * (xii)

The interior solution should match continuously with an exterior Reissner-
Nordström solution,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}\times$
$\displaystyle dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\text{$r\geq R.$}$

This requires the continuity of $e^{\nu},\,e^{\lambda}$ and $q$ across the
boundary $r=R$,

$e^{\nu(R)}=e^{{-\lambda(R)}}=\left(1-\frac{2M}{R}+\frac{Q^{2}}{R^{2}}\right)$

and $q(R)=Q$, where $M$ and $Q$ represent the total mass and charge inside the
fluid sphere respectively.

  7. (g)

Charge distribution

     * (xiii)

Electric field intensity $E$, such that $E(0)=0$, is taken to be monotonically
increasing, i.e., $dE/dr>0$ for $0<r\leq R$.

  8. (g)

Pressure anisotropy

     * (xiv)

Pressure anisotropy, $\Delta$, vanishes at the center, i.e., $\Delta(0)=0$
Bowers-Liang74 ; Ivanov02a .

  9. (h)

Allowable mass to radius ratio

     * (xv)

Buchdahl Buchdahl59  obtained an absolute constraint of the maximally
allowable mass-to-radius ratio $(M/R)$ for isotropic fluid spheres of the form
$2M/R\leq 8/9$ (in the unit, $c=G=1$) which states that, for a given radius a
static isotropic fluid sphere cannot be arbitrarily massive. Böhmer and Harko
Bohmer-Harko07  proved that for a compact object with charge, $Q(<M)$, there
is a lower bound for the mass-radius ratio,

$\frac{3Q^{2}}{2R^{2}}\frac{\left(1+\frac{Q^{2}}{18R^{2}}\right)}{\left(1+\frac{Q^{2}}{12R^{2}}\right)}\leq\frac{2M}{R}$

Upper bound of the mass of charged sphere was generalized by Andréasson
Andreasson09  and proved that

$\sqrt{M}\leq\frac{\sqrt{R}}{3}+\sqrt{\frac{R}{9}+\frac{Q^{2}}{3R}}$

##  4 Physical Boundary Conditions

###  4.1 Determination of the Arbitrary Constant $A_{2}$

The boundary condition $P_{r}(R)=0$, can be utilized to specify $A_{2}$. For
the _Case I_ :

$A_{2}=-\frac{K}{m^{n+1}}\sum_{i=0}^{n}\left[\frac{(-1)^{i}}{(n-i+p+1)}\binom{n}{i}(1+mX)^{(n-i+p+1)}\right]-\frac{K}{2}\frac{X^{n+1}(1+mX)^{p}(1+3X)}{(1+5X)}-\delta
X(1-aX)-4\frac{(1+3X)^{\frac{2}{3}}}{(1+5X)}$

where $X=CR^{2}$.

###  4.2 Total Charge to Radius Ratio $Q/R$

Using $X=CR^{2}$ in eq. (2.2.1) we obtain the square of ratio $Q/R$,

$\frac{Q^{2}}{R^{2}}=\frac{K}{2}\frac{X^{n+2}(1+mX)^{p}(1+3X)^{\frac{1}{3}}}{(1+X)}$
(4.2.1)

###  4.3 Total Mass to Radius Ratio $M/R$

By matching the metric coefficients obtained in (2.3.1)-(2.3.2) with the
exterior Reissner-Nordström metric at the boundary and with reference to the
eq. (4.2.1) one can establish the equation of compactness,

$\frac{2M}{R}=\left(1-e^{-\lambda(X)}+\frac{Q^{2}}{R^{2}}\right)$ (4.3.1)

###  4.4 Total Charge to Mass Ratio $Q/M$

By the use of the equations (4.2.1) and (4.3.1) we obtain the charge to mass
ratio $Q/M$,

###  4.5 Determination of the Constant $B_{2}$

The constant $B_{2}$ can be specified by the boundary condition
$e^{\nu(R)}=e^{-\lambda(R)}$, which gives,

$B_{2}=(1+X)^{-2}e^{-\lambda(X)}$ (4.5.1)

###  4.6 Central and Surface Redshifts

The central and surface redshifts of the charged fluid sphere are given by

$z_{c}=\sqrt{e^{-\nu(0)}}-1,\,\,z_{s}=\sqrt{e^{-\nu(R)}}-1=\frac{(1+X)^{-1}}{\sqrt{B_{2}}}-1$

##  5 Construction of Physically Realistic Fluid Spheres

###  5.1 Pressure and Density Gradients

Differentiating the pressure and density equations (2.3.5)-(2.3.7),
(2.3.12)-(2.3.14), (2.3.19)-(2.3.21) with respect to the auxiliary variable
$x$ one obtains the pressure and density gradients respectively for each of
the model EOS.

Case I: $p\neq-1,\,n=0$

$\displaystyle\frac{\kappa}{C}\frac{dP_{r}}{dx}$ $\displaystyle=$
$\displaystyle\frac{K}{m(p+1)}(1+mx)^{p}\frac{\Sigma(x)}{(1+3x)^{\frac{5}{3}}(1+x)^{2}}$
(5.1.1)
$\displaystyle+\frac{K}{2}(1+mx)^{p-1}\frac{\Phi(x)}{(1+3x)^{\frac{5}{3}}(1+x)^{2}}$
$\displaystyle+\delta\frac{\Psi_{1}(x)}{(1+x)^{2}(1+3x)^{\frac{5}{3}}}-\frac{4}{(1+x)^{2}}$
$\displaystyle+2A_{2}\frac{(1-5x^{2})}{(1+3x)^{\frac{5}{3}}(1+x)^{2}}$
$\displaystyle\frac{\kappa}{C}\frac{dP_{t}}{dx}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{C}\frac{dP_{r}}{dx}+\frac{d\Delta}{dx}$ (5.1.2)
$\displaystyle\frac{\kappa}{C}\frac{d\rho}{dx}$ $\displaystyle=$
$\displaystyle-\frac{K}{m(p+1)}(1+mx)^{p-1}\frac{\Theta(x)}{(1+3x)^{\frac{8}{3}}}$
(5.1.3)
$\displaystyle+\frac{K}{2}(1+mx)^{p-1}\frac{\Phi(x)}{(1+3x)^{\frac{5}{3}}(1+x)^{2}}$
$\displaystyle-\delta\frac{\Psi_{2}(x)}{(1+3x)^{\frac{8}{3}}}+10A_{2}\frac{(1+x)}{(1+3x)^{\frac{8}{3}}}$

where,

$\displaystyle\Sigma(x)$ $\displaystyle=$ $\displaystyle(mp+m-1)+(9mp+8m-20)x$
$\displaystyle+(23mp+3m-35)x^{2}+(15mp-20m)x^{3},$ $\displaystyle\Phi(x)$
$\displaystyle=$ $\displaystyle 1+(mp+m+4)x+(4mp+4m+1)x^{2}$
$\displaystyle+(3mp+m)x^{3},$ $\displaystyle\Theta(x)$ $\displaystyle=$
$\displaystyle(5mp+5m-10)+(2m^{2}p^{2}+7m^{2}p+5m^{2}$
$\displaystyle+22mp+2m-10)x+(12m^{2}p^{2}+34m^{2}p$
$\displaystyle+12m^{2}+21mp+m)x^{2}+(18m^{2}p^{2}+39m^{2}p$
$\displaystyle+11m^{2})x^{3},$ $\displaystyle\Psi_{1}(x)$ $\displaystyle=$
$\displaystyle 1+(11-2a)x+(23-20a)x^{2}$ $\displaystyle+(5-46a)x^{3}-25ax^{4}$
$\displaystyle\Psi_{2}(x)$ $\displaystyle=$ $\displaystyle
5+(12-14a)x+(11-58a)x^{2}-68ax^{3}$ $\displaystyle\frac{d\Delta}{dx}$
$\displaystyle=$
$\displaystyle\delta\frac{(1+(4-4a)x+(1-16a)x^{2}-8ax^{3})}{(1+x)^{2}(1+3x)^{\frac{2}{3}}}$

###  5.2 Specifying the Maximum Mass and Radius

A fluid sphere satisfying conditions (a) and (e) of Sect. 3 will be termed as
_well-behaved_. For a particular set $(m,n,p,a,\delta)$ the values of $K,\,X$
have been plugged-in to the eqs. (2.3.5)-(2.3.7) and (5.1.2)-(5.1.3) for which
the fluid distribution satisfies the elementary criteria for physical
acceptability. Once the compactness $M/R$ and the ratio $Q/R$ of the compact
fluid sphere are obtained the maximum mass can then be calculated by using one
of the following quantities: (i) radius, (ii) central density, (iii) surface
density, (iv) central pressure or, (v) total charge as parameter. In this
subsection we describe how to calculate the values various physical variables.
For the _Case I_ this can be accomplished in the following way:

####  5.2.1 For a given radius

  * (a)

Total Mass

$M=\frac{R}{2}\left(1-e^{-\lambda(X)}+\frac{Q^{2}}{R^{2}}\right)$ (5.2.1.1)

where the mass $M$ is in the unit km 333The following physical constants, in
their conventional values, have been used for the numerical calculation:
$c=1=2.997\times 10^{8}$m s-1, $G=1=6.674\times 10^{-11}$N m2 kg-2,
$M_{\odot}=1.486$ km$\,=2\times 10^{30}$kg.

  * (b)

Total Charge

$Q=R\sqrt{\frac{K}{2}\frac{X^{2}(1+mX)^{p}(1+3X)^{\frac{1}{3}}}{(1+X)}}$
(5.2.1.2)

where the charge $Q$ is in the unit of radius444$Q$ km$=\left(Q\times
1000\times c^{2}/\sqrt{\frac{G}{4\pi\epsilon_{0}}}\right)\,C$.

  * (c)

Central Density

$\rho_{c}=\frac{3X}{\kappa R^{2}}\left[\frac{-K}{m(p+1)}-A_{2}\right]$
(5.2.1.3)

  * (d)

Surface Density

$\rho_{s}=\frac{X}{\kappa R^{2}}\Omega(X)$ (5.2.1.4)

where,

$\displaystyle\Omega(X)$ $\displaystyle=$
$\displaystyle-\frac{K}{m(p+1)}(1+mX)^{p}\frac{\Pi_{m,0,p,0}(X)}{(1+3X)^{\frac{5}{3}}}$
$\displaystyle-\frac{K}{2}\frac{X(1+mX)^{p}(1+3X)^{\frac{1}{3}}}{(1+X)}$
$\displaystyle-\delta
X\frac{(5+(11-7a)X-17aX^{2})}{(1+3X)^{\frac{5}{3}}}-A_{2}\frac{(3+5X)}{(1+3X)^{\frac{5}{3}}}$

####  5.2.2 For a given surface density

$R^{2}=\frac{X}{\kappa\rho_{s}}\Omega(X)$ (5.2.2.1)

And the total mass, total charge, and the central density can be calculated by
eqs. (5.2.1.1)-(5.2.1.3).

####  5.2.3 For a given central density

The radius of the charged fluid sphere for a prescribed central density can be
calculated by the following equation,

$R^{2}=\frac{3X}{\kappa\rho_{c}}\left[\frac{-K}{m(p+1)}-A_{2}\right]$
(5.2.3.1)

where the central energy density $\rho_{c}$ is given in the unit kg m-3 and
the radius in m. The total mass, total charge, and the surface density then
can be calculated by eqs. (5.2.1.1), (5.2.1.2), and (5.2.1.4) respectively.

####  5.2.4 For a given central pressure

The radius of the charged fluid sphere for a prescribed central pressure can
be calculated by the following equation.

$R^{2}=\frac{X}{\kappa P_{c}}\left[\frac{K}{m(p+1)}+4+A_{2}\right]$ (5.2.4.1)

where the central pressure $P_{c}$ is given in the unit N m-2 5551N
m${}^{-2}=10$ dyne cm-2 and $1$ MeV fm${}^{-3}=1.6022\times 10^{33}$ dyne
cm-2. The total mass, total charge, the central and surface densities then can
be calculated by eqs. (5.2.1.1)-(5.2.1.4).

####  5.2.5 For a given electric charge

The radius of the charged fluid sphere for a prescribed total charge can be
calculated by the following equation

$R=Q/\sqrt{\frac{K}{2}\frac{X^{2}(1+mX)^{p}(1+3X)^{\frac{1}{3}}}{(1+X)}}$
(5.2.5.1)

where the charge $Q$ is given in the unit km. Then the total mass, total
charge, the central and surface densities then can be calculated by eqs.
(5.2.1.1)-(5.2.1.4).

###  5.3 Physical Analysis of the Models

For each choice of constant parameters $(K,\,m,\,n,\,p,\,\delta,\,a)$, the
maximum mass of charged star depends on the corresponding set of maximum
values of $X=X_{\mathrm{max}}$ upto which the pressure and density and their
gradients satisfy $P_{r}\geq 0$, $P_{t}\geq 0$, $\rho\geq 0$, $dP_{r}/dx<0$,
$dP_{t}/dx<0$, $d\rho/dx<0$ and the speed of sound satisfy
$0\leq\sqrt{dP_{r}/d\rho}\leq 1$, $0\leq\sqrt{dP_{t}/d\rho}\leq 1$ and
monotonically decreasing with increasing radius.

Case Ia:

_Isotropic pressure_
We set $(m,\,n,\,p,\,\delta,\,a)=(10^{4},\,0,\,0.24,\,0,\,0)$. For this choice
the range of values, $K\geq 0.093,\,0<X\leq 0.672$ are obtained over which the
fluid distribution satisfies the above mentioned inequalities. With the
decrease of $K$, $X$ increases. The maximum value of compactness parameter is
obtained $(2M/R)_{\mathrm{max}}=0.8246$, using (4.3.1), at
$K_{\mathrm{min}}=0.093,\,X_{\mathrm{max}}=0.672$. Corresponding to the values
of $(K_{\mathrm{min}},\,X_{\mathrm{max}})$ the total charge to radius ratio,
and total charge to total mass ratio are found to be $Q/R=0.3878$ and
$Q/M=0.9406$ using eq. (4.2.1). For a particular choice of stellar surface
density $\rho_{s}=4.68\times 10^{14}$ g cm-3 666The surface density of bare
strange stars is equal to that of strange quark matter (SQM) at zero pressure.
By using the formula given in Zdunik00  the SQM density with $m_{s}c^{2}=150$
MeV, $\alpha_{c}=0.17$, $B=60$ MeV fm-3 is calculated to be
$\rho_{s}=4.6\times 10^{14}$ g cm-3. It is therefore some fourteen orders of
magnitude larger than the surface density of normal neutron stars. as
parameter the total mass and other physical quantities are calculated by the
use of eqs. (5.2.1.1)-(5.2.1.3) and found $M_{\mathrm{max}}=2.8740M_{\odot}$,
$R=10.35$ km, $P_{c}=279.73$ MeV fm-3, $\rho_{c}=2.52\times 10^{15}$ g cm-3,
$Q=4.66\times 10^{20}\,C$.

Case Ib:

_Anisotropic pressure_
Due to the presence of pressure anisotropy the range of values of $K$ and $X$
are increased. For the input parameters $m=10^{4}$, $n=0$, $p=0.23$,
$\delta=0.2$, $a=0.71$ the maximum value of compactness parameter is obtained
$(2M/R)_{\mathrm{max}}=0.8307$, at
$K_{\mathrm{min}}=0.094,\,X_{\mathrm{max}}=0.7$. The charge-radius ratio, and
charge-mass ratio are found to be $Q/R=0.3890$ and $Q/M=0.9366$. For
$\rho_{s}=4.68\times 10^{14}$ g cm-3 the maximum mass found
$M_{\mathrm{max}}=2.8999M_{\odot}$, with radius $R=10.37$ km, central pressure
$P_{c}=284.30$ MeV fm-3, central energy density $\rho_{c}=2.65\times 10^{15}$
g cm-3, and total charge $Q=4.68\times 10^{20}\,C$. Details are reported in
Tables 2-3.

Table 2: Some values of parameters $(K,X_{\mathrm{max}})$ for which well-behaved charge fluid sphere can be generated $p$ | $m$ | $\delta$ | $a$ | $\left(K_{\mathrm{min}},X_{\mathrm{max}}\right)$ | $A_{2}$ | $B_{2}$ | $\left(P_{r}/c^{2}\rho\right)_{c}$ | $\sqrt{\left(dP_{r}/c^{2}d\rho\right)_{c}}$ | $\sqrt{\left(dP_{t}/c^{2}d\rho\right)_{c}}$ | $2M/R$ | $Q/R$ | $Q/M$
---|---|---|---|---|---|---|---|---|---|---|---|---
_Case I_ $n=0$
0.20 | $10^{4}$ | 0 | 0 | (0.093, 0.672) | -2.5122 | 0.1165 | 0.1974 | 0.5887 | 0.5887 | 0.8246 | 0.3878 | 0.9406
0.22 | $10^{4}$ | 0.2 | 0.71 | (0.103, 0.699) | -2.5488 | 0.1111 | 0.1898 | 0.5684 | 0.5619 | 0.8306 | 0.3891 | 0.9368
0.23 | $10^{4}$ | 0.2 | 0.71 | (0.094, 0.700) | -2.5439 | 0.1109 | 0.1908 | 0.5696 | 0.5631 | 0.8307 | 0.3891 | 0.9366
0.24 | $10^{4}$ | 0.2 | 0.71 | (0.086, 0.700) | -2.5403 | 0.1109 | 0.1915 | 0.5706 | 0.5641 | 0.8306 | 0.3889 | 0.9365
_Case II_ $n=0$
$-1$ | $2$ | 0 | 0 | (2.063, 0.612) | -2.9967 | 0.1295 | 0.1116 | 0.4086 | 0.4086 | 0.8042 | 0.3751 | 0.9327
$2$ | 0.2 | 0.64 | (1.866, 0.708) | -2.9714 | 0.0988 | 0.1154 | 0.4116 | 0.4056 | 0.8638 | 0.3899 | 0.9028
Table 3: Maximum mass and the various physical variables of charged fluid
spheres for a given surface density

$p$ | $m$ | $\delta$ | $a$ | $\left(K,X_{\mathrm{max}}\right)$ |  $\rho_{s,14}$a $=4.68$
---|---|---|---|---|---
$M(M_{\odot})$ |  $R$(km) |  $P_{c,35}$b |  $\rho_{c,15}$c |  $Q_{20}$d
_Case I_ $n=0$
0.24 | $10^{4}$ | 0 | 0 | $(0.093,\,0.672)$ | 2.8741 | 10.35 | 4.48 | 2.52 | 4.66
0.22 | $10^{4}$ | 0.2 | 0.71 | $(0.103,\,0.699)$ | 2.8988 | 10.37 | 4.53 | 2.66 | 4.68
0.23 | $10^{4}$ | 0.2 | 0.71 | $(0.094,\,0.700)$ | 2.8999 | 10.37 | 4.55 | 2.66 | 4.68
0.24 | $10^{4}$ | 0.2 | 0.71 | $(0.086,\,0.700)$ | 2.8998 | 10.37 | 4.56 | 2.65 | 4.68
_Case II_ $n=0$
$-1$ | 2 | 0 | 0 | (2.063, 0.612) | 2.7739 | 10.25 | 2.81 | 2.80 | 4.46
2 | 0.2 | 0.64 | (1.866, 0.708) | 2.8613 | 10.14 | 3.40 | 3.28 | 4.59

  * a

$\rho_{s}=\rho_{s,14}\times 10^{14}$g cm-3

  * b

$P_{c}=(P_{r})_{c}=(P_{t})_{c}=P_{c,35}\times 10^{35}$dyne cm-2

  * c

$\rho_{c}=\rho_{c,15}\times 10^{15}$g cm-3

  * d

$Q=Q_{20}\times 10^{20}\,C$

The behaviors of various physical variables in the interior of the star have
been investigated and found regular and well behaved throughout the fluid
sphere. It has been observed that the radial and tangential speeds of sound
$\sqrt{dP_{r}/d\rho}$, $\sqrt{dP_{t}/d\rho}$ always remain less than the speed
of light and the condition of causality is satisfied. The mass-radius relation
for a sequence of charged fluid spheres generated by the _Case I_ with the
input $m=10^{4}$, $n=0$, $p=0.2$, $\delta=0.2$, $a=0.71$,
$K_{\mathrm{min}}=0.094$, $0<X\leq 0.7$ and _Case II_ with input $m=2$, $n=0$,
$p=-1$, $\delta=0.2$, $a=0.64$, $K=1.866$ $0<X\leq 0.708$ with surface density
$\rho_{s}=4.69\times 10^{14}$ g cm-3 have been demonstrated in Fig. 1. The
behavior of Fig. 1 reproduces that of other quark star models Negreirosetal.09
. The behaviors of pressure anisotropy the these cases have been demonstrated
in fig 2. In Fig. 3 the mass versus central density for the same sequences of
configurations are plotted and this shows the necessary condition of stability
is satisfied $(dP_{r}/d\rho_{c}>0)$.

The pressure-density profiles together with various other physical variables
given by our analytical model, are plotted in Fig. 4-9. We have demonstrated
that the models obtained in subsection 2.3 satisfy the physical requirements
for wide range of values of $m,\,n,\,p,\,\delta,\,a$ and $K$, giving us a
possibility for different charge variations and anisotropy within the fluid
spheres. The resulting spheres can be utilized to construct physically
reasonable compact self-bound charged stellar model such as charged strange
quark star.

##  6 An Application of the Model for Some Well Known Strange Star Candidates

The analysis of very compact astrophysical objects has been a key issue in
relativistic astrophysics for the last few decades. Recent observations show
that the estimated mass and radius of several compact objects such as X-ray
pulsar Her X-1, X-ray burster 4U 1820-30, millisecond pulsar SAX J
1808.4-3658, X-ray sources 4U 1728-34, PSR 0943+10 and RX J185635-3754 are not
compatible with the standard neutron star models Deyetal.98 ; Lietal.99 . For
a recent review on this may readers are referred to Weber05 .

Based on the analytic model developed so far, to get an estimate of the range
of various physical parameters of some potential strange star candidates we
have calculated the values of the relevant physical quantities, such as
central/surface pressure and density, by using the refined mass and predicted
radius of 12 pulsars recently reported in Gangopadhyayetal.13 . The values are
reported in Table 4.

Table 4: Physical values of energy density and pressure for different strange stars calculated by _Case I_ with $m=10^{4},\,n=0$. Strange |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
Star | $(p,\delta,a,K,X)$ | $M(M_{\odot})$ |  $R$ (km) | $P_{c,35}$ | $\rho_{c,15}$ | $\rho_{s,14}$ | $Q_{20}$
Candidate |  |  |  |  |  |  |
PSR J1614-2230 | (0.24, 0, 0, 0.093, 0.396) | 1.97 | 9.69 | 2.88 | 1.74 | 6.20 | 2.50
(0.23, 0.2, 0.71, 0.094, 0.403) | 2.87 | 1.79 | 6.15 | 2.45
PSR J1903+327 | (0.24, 0, 0, 0.093,0.304) | 1.667 | 9.438 | 2.17 | 1.46 | 6.44 | 1.83
(0.23, 0.2, 0.71, 0.094, 0.307) | 2.14 | 1.49 | 6.39 | 1.78
Vela X-1 | (0.24, 0, 0, 0.093, 0.333) | 1.77 | 9.56 | 2.38 | 1.54 | 6.33 | 2.05
(0.23, 0.2, 0.71, 0.094, 0.337) | 2.36 | 1.58 | 6.29 | 2.00
4U 1820-30 | (0.24, 0, 0, 0.093, 0.295) | 1.58 | 9.1 | 2.25 | 1.54 | 6.89 | 1.71
(0.23, 0.2, 0.71, 0.094, 0.298) | 2.22 | 1.57 | 6.85 | 1.67
4U 1608-52 | (0.24, 0, 0, 0.093, 0.324) | 1.74 | 9.528 | 2.32 | 1.52 | 6.36 | 1.98
(0.23, 0.2, 0.71, 0.094, 0.328) | 2.29 | 1.55 | 6.32 | 1.94
Figure 1:  Mass-radius relation for a sequence of charged fluid spheres. The
solid (blue) line corresponds to the _Case I_ with the input
$(m,n,p,\delta,a,K_{\mathrm{min}})=$ $(10^{4},0,0.23,0.2,0.71,0.094)$,
$0<X\leq 0.7$. And the dashed-doted (red) line corresponds to the _Case II_
$(2,\,0,\,-1,\,0.2,\,0.64,\,1.866)$, $0<X\leq 0.708$ with surface density
$\rho_{s}=4.69\times 10^{14}$ g cm-3. Figure 2:  Behaviour of pressure
anisotropy $\Delta$ for the same sequences of charged fluid spheres in fig. 1.
Figure 3:  Mass vs. central density for same sequences of charged fluid
spheres in fig. 1. Figure 4:  Pressure-density profiles for the same sequences
of charged fluid spheres in fig. 1. Figure 5:  Behavior of radial pressure
$P_{r}$ and tangential pressure $P_{t}$ (MeV fm-3) for the _Case I_ with the
input $(m,n,p,\delta,a,K_{\mathrm{min}},X_{\mathrm{max}})=$
$(10^{4},0,0.23,0.2,0.71,0.094,0.7)$. The solid (blue) line corresponds to
radial pressure and the dashed-doted (red) line corresponds to the tangential
pressure. Figure 6:  Behavior of energy density $c^{2}\rho$ (MeV fm-3) for the
same charged fluid sphere as in fig. 5. Figure 7:  Behavior of radial and
tangential pressure gradients $\kappa dP_{r}/Cdx,\,\kappa dP_{t}/Cdx$ for the
same charged fluid sphere as in fig. 5. Figure 8:  Behavior of energy density
gradient $\kappa d\rho/Cc^{2}dx$ for the same charged fluid sphere as in fig.
5. Figure 9:  Speed of adiabatic sound
$\sqrt{dP_{r}/c^{2}d\rho},\,\sqrt{dP_{t}/c^{2}d\rho}$ for the strange star
candidates in fig. 5.

##  7 Concluding Remarks

In this work we have studied particular simple families of relativistic
charged stellar models obtained by solving Einstein-Maxwell field equations
for a static spherically symmetric distribution of perfect fluid distribution
based on three _ad hoc_ assumptions, one for metric potential and others for
the forms of electric charge distribution and pressure anisotropy. The
analytical equation of state has been computed from the resulting metric.
These families of analytical relativistic stellar models may be considered as
_anisotropic charged analogues_ of Wyman-Adler solution.

A wide range of values of constant parameters are allowed to specify the
maximum mass of charged fluid spheres. Various authors usually have chosen
$2\times 10^{14}$g cm-3 as stellar surface density to calculate the mass and
radius of the charged fluid spheres which may have given rise to the stellar
configuration as massive as $4-6M_{\odot}$ with much lower central density.
Such massive configuration may not serve as a realistic model for a strange
quark star. This choice is, therefore, not a physical one. Modeling a compact
(quark) star requires the use of a higher surface density. Certainly, the
value of the surface density affects the calculated value of the stellar mass
- to see this, we observe that the method employed in the present work one can
obtain arbitrarily large maximum mass just by inserting vanishing small
surface density (e.g., $0.1-1$ g cm-3 to model a thin crust). In our model
calculation, the density at the stellar radius is chosen within the range
$4-10\times 10^{14}$g cm-3 Bombaci01  and drops abruptly to zero, as with all
stellar models matching an interior metric to the external Reissner-Nordström
form. This sharp drop in density is a reasonable model approximation since the
thickness of the “quark surface” is of order $1$ fm, a negligibly small
dimension compared to the stellar radius.

In this work we assumed $P_{t}>P_{r}\,(\Delta>0)$ and have shown that the
upper bound on the maximum mass increases in the presence of anisotropy.
Moreover, for the some models (Case I), the speed of sound is obtained
$\approx 1/\sqrt{3}$ at the center and remains almost the same throughout most
of the fluid sphere. This behavior is like MIT bag model. An analytical
stellar model with such physical features is most likely to present realistic
model of strange quark stars. And hence the EOS given by our models, besides
the usual linear EOS based on phenomenological MIT bag model, could play a
significant role in the description of internal structure of electrically
charged _bare_ strange quark stars.

###### Acknowledgements.

One of the authors (M.H. Murad) expresses his sincere gratitude to Prof. Mofiz
Uddin Ahmed, and Prof. A A Ziauddin Ahmad, Chair, Department of Mathematics
and Natural Sciences, BRAC University, Dhaka, Bangladesh for their motivations
and encouragements.

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  *[MMW]: matrix multiplicative weights
  *[3MW]: minimal-information matrix multiplicative weights
  *[QRD]: quantum replicator dynamics
  *[FTQL]: “follow the quantum leader”
  *[POVM]: positive operator-valued measure
  *[NE]: Nash equilibrium
  *[IWE]: importance weighted estimator
  *[2PE]: two-point estimator
  *[]:
  *[1PE]: one-point estimator
  *[FTRL]: “follow the regularized leader”
  *[LHS]: left-hand side
  *[RHS]: right-hand side