arXiv:2012.00361v1 [gr-qc] 1 Dec 2020

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020

A new class of viable and exact solutions of EFE’s

with Karmarkar conditions: An application to cold

star modeling

Neeraj Pant1, Megandhren Govender2, & Satyanarayana

Gedela3

1,3Department of Mathematics, National Defence Academy, Khadakwasla,Pune-411023, India2 Department of Mathematics, Faculty of Applied Sciences, Durban Universityof Technology, Durban, South Africa3 Department of Mathematics, SSJ Campus, Kumaun University,Almora-263601, India.

E-mail: [email protected], [email protected],

[email protected]

August 2020

Abstract. In this work we present a theoretical framework within Einstein’sclassical general relativity which models stellar compact objects such as PSRJ1614-2230 and SAX J1808.4-3658. The Einstein field equations are solvedby assuming that the interior of the compact object is described by a class Ispacetime. The so-called Karmarkar condition arising from this requirement isintegrated to reduce the gravitational behaviour to a single generating function.By appealing to physics we adopt a form for the gravitational potential which issufficiently robust to accurately describe compact objects. Our model satisfies allthe requirements for physically realistic stellar structures.

Keywords: Compact star, anisotropy, embedding class, Einstein field equations, adia-batic index.

Submitted to: Res. Astron. Astrophys.

http://arxiv.org/abs/2012.00361v1

Pant et al; A new class of viable and exact solutions of EFE’s with..... 2

1. Introduction

Since the publication of Einstein’s general relativityin 1914, researchers were captivated by the searchfor exact solutions of the field equations. Overthe past century a myriad of exact solutions wereobtained which attempted to explain observations incosmology and astrophysics. The gravitational fieldexterior to a static, spherically symmetric star wasfirst obtained by Schwarzschild in 1916. This vacuumsolution was followed by the interior Schwarzschildsolution which describes the gravitational field ofa uniform density sphere [Schwarzschild (1916a,b)].Causality is one of the cornerstones of relativitywhich requires 0 < dp

dρ < 1 [Dev & Gleiser (2002),

Dev & Gleiser (2003)]. It is clear that causality isviolated at each interior point of the Schwarzschildconstant density sphere. This prompted researchersto consider more realistic matter configurations whichincluded inhomogeneous density profiles, anisotropicpressures, electric charge, bulk viscosity and scalarfields. Generalization of the perfect fluid interiormatter distribution to include anisotropic stresses hasyielded interesting physical characteristics of suchmodels. It was shown that physical properties suchas surface tension, compactness and surface red-shiftof these stars are sensitive to the anisotropy parameter[Sharma & Maharaj (2007), Bowers & Liang (1974),Maurya & Govender (2017), Pant et al (2016)]. Theimpact of electric charge in compact objects has beenwidely studied within the context of stability andphysical viability. It was shown that the presenceof electric charged alters the Buchdahl limit requiredfor stability of a self-gravitating, bounded matterdistribution [Singh et al (2016), Andreasson et al(2012)]. Departure from spherical symmetry has alsobeen pursued in the context of slowly rotating stars andin the description of gravitational waves [Herrera et al(2005a,b)]. Various techniques ranging from ad-hocassumptions, imposition of pressure isotropy, use of anequation of state, use of the condition of conformalflatness, Lie symmetry analysis, to name just a few,were used to solve the field equations [Manjonjo et al(2018), Ivanov (2018)]. While these methods yieldsolutions, there is no guarantee that the ensuing modelsare physically viable. An extensive review of exactsolutions of the Einstein field equations describingstatic objects show that a very small subset of thesesatisfy all the requirements for realistic stellar models[Stephani et al (2003)].

A natural question which arises in astrophysics iswhat happens when a star loses hydrostatic equilib-rium and undergoes continued gravitational collapse?Oppenheimer and Snyder tackled this problem by con-sidering a spherically symmetric dust cloud under-

going gravitational collapse [Oppenheimer & Snyder(1939)]. Their model served as a catalyst inunderstanding end-states of gravitational collapse.The Cosmic Censorship Conjecture which ruled outthe formation of naked singularities for collapsingmatter configurations with reasonable initial stateswas shown to be violated under various assump-tions [Guo & Joshi (2015), Ghosh & Maharaj (2015),Sherif et al (2019)]. The study of black holes hasmoved into the observable realm making it a pop-ular research topic [Akiyama et al (2019)]. Blackhole physics has evolved immensely from the simpleOppenheimer-Snyder dust model to include anisotropicpressures, electromagnetic field, cosmological constantas well as higher dimensions.

In the paper Vaidya (1951) presented an exactsolution describing the exterior gravitational fieldof a radiating star. This solution is a uniquesolution of the Einstein field equations describing aspherically symmetric atmosphere composed of nullradiation. The Vaidya solution made it possible tomodel dissipative collapse in which the collapsing coreradiates energy to the exterior spacetime in the formof a radial heat flux or null radiation. There wereseveral early attempts at modeling a radiating starwith a Vaidya exterior. The problem was the matchingof the interior and exterior spacetimes across theboundary of the star. The junction conditions requiredfor the smooth matching of a spherically symmetric,shear-free line element to Vaidya’s outgoing solutionwas provided by Santos (1985). It was shown thatfor a radiating spherical body dissipating energy inthe form of a radial heat flux, the pressure on theboundary is proportional to the magnitude of theheat flux. This condition ensures conservation ofmomentum across the boundary of the collapsing body.Since the publication of the Santos junction conditions,there has been an explosion of models describingdissipative collapse starting with simple solutions andthus rapidly developing into more sophisticated stellarmodels. The authors Herrera et al (1989), Chan et al(1994), Di Prisco et al (2007), Herrera & Martinez(1998), Di Prisco et al (1997) have been instrumentalin investigating the nature of collapse with dissipationwithin a general framework thus giving researchersrich insights into these problems especially with theinclusion of shear, inhomogeneity and anisotropy.The thermodynamics of radiating stars was developedby Govender and co-workers since the early 1990’s.Relaxational effects due to heat dissipation and shearviscosity predict temperature and luminosity profileswhich are significantly different from the Eckart theoryof thermodynamics [Govender et al (2010), Govender(2013), Govender & Govinder (2001)]. Recently,there has been a resurgence in seeking exact


solutions to the Einstein field equations describingstatic, compact objects by employing the conceptof embedding. The Karmarkar condition whichneeds to be satisfied if the spacetime has to be ofclass I embedding has been widely used to generatevarious stellar models describing anisotropic spheres[Karmarkar (1948)]. These models have been shownto satisfy all the stringent stability and physical testsimposed by the behaviour of the thermodynamic andgravitational variables [Bhar (2019), TellOrtiz et al(2019), Maurya et al (2019a), Jasim et al (2020),Sing et al (2020), Gedela et al (2020), Ivano (2020),Sarkar et al (2020)]. Many of these solutions have beenreconciled with observational data of compact objectsincluding strange stars, pulsars and neutron stars[Gedela et al (2018, 2019a,b), Upreti et al (2020),Fuloria (2017), Pant et al (2020)]. By utilisinga quadratic equation of state together with theKarmarkar condition a model for the strange starcandidate SAX J1808.4-3658 was obtained. It wasshown that this model agrees with observationalcharacteristics of this star. Furthermore, a comparisonof the quadratic EoS model with modified Bose-Einstein condensation EoS and linear EoS was carriedout Gedela et al (2019c). The Karmarkar conditionhas also been utilised to model dissipative collapseensuing from an initially static configuration losinghydrostatic equilibrium and starts to radiate energyto the exterior spacetime. The Karmarkar conditiontogether with the junction condition which representsconservation of momentum across the collapsingboundary determine the temporal and gravitationalevolution of the model [Naidu et al (2018)]. Manyof these models indicate their robustness under thescrutiny of physical viability. To this end weemploy the Karmarkar condition to seek a modelwhich accurately describes two stellar compact objects,namely, PSR J1614-2230 and SAX J1808.4-3658.

This paper is structured as follows: In section I,we present the Einstein field equations describing theinterior spacetime of the stellar model. The Karmarkarand embedding class I conditions are introduced insection III. By adopting a parametric form for one ofthe metric potentials we generate a stellar model insection IV. The matching of the interior and exteriorspacetimes is accomplished in section V. The physicalfeatures of the model is discussed in section VI. Weinvestigate the stability of our model in section VII.The paper concludes with a discussion and finding ofour main results in section VIII.

2. The Einstein Field Equations

The line element within the spherically symmetricanisotropic fluid matter distribution in Schwarzschild

coordinates (xi) = (t, r, θ, φ) is delineated in thefollowing form:

ds2 = eν(r)dt2 − eλ(r)dr2 − r2(dθ2 + sin2 θdφ2). (1)

where the gravitational potentials ν(r) and λ(r) areyet unknown. The energy-momentum tensor foranisotropic matter takes the form

Tjk = [(pt + ρ)vjvk − ptgjk + (pr − pt)χjχk], (2)

where ρ, pr and pt are the energy density, radialand transverse pressures respectively and pt is in theperpendicular direction to pr. The normalized 4-

velocity vector vj =√

1gtt

δjt and the unit spacelike

vector χj =√

− 1grr

δjr along r provide gjkvjvk = 1

and gjkχjχk = −1 respectively.

The line element (1) and momentum tensor Tjk

(2) give rise to the following system of equations[Maurya et al (2019c)]

8πρ =

(

1− e−λ(r))

r2+

λ′(r)e−λ(r)

r, (3)

8πpr =ν′(r)e−λ(r)

r−(

1− e−λ(r))

r2, (4)

8πpt =e−λ

4

(

2ν′′ + ν′2 − ν′λ′ +

2ν′

r− 2λ′

r

)

, (5)

where (′) denotes the derivative with respect to theradial coordinate r.

Using the field equations Eqs.(4) and (5), theanisotropic factor (∆) takes the form

∆ = pt − pr

= e−λ

[

ν′′

2− λ′ν′

4+

ν′2

4− ν′ + λ′

2r+

eλ − 1

r2

]

. (6)

Here we choose the gravitational constant G and speedof sound c to be unity.

3. The Karmarkar condition

The Karmarkar condition required for the spacetimeto be of class I embedding is

R1414 =R1212R3434 +R1224R1334

R2323, (7)

subject to R2323 6= 0 [Pandey & Sharma (1981)].The non-zero Riemann tensor components for the

line element (1) are

R1414 = − eν(r)(ν

′′

(r)

2+

ν′2(r)

4− λ′(r)ν′(r)

4), (8)

R2323 = − e−λ(r)r2sin2θ(eλ(r) − 1), (9)

R1212 =1

2rλ′(r), (10)

R3434 = − 1

2r sin2 θν′(r)eν(r)−λ(r). (11)


The differential equation derived using theKarmarkar condition (7) assumes the form

2ν′′

ν′+ ν′ =

λ′eλ(r)

eλ(r) − 1. (12)

Solving eqn.(12), we find the following relationbetween eλ(r) and eν(r)

eλ(r) =(

P +Q

∫ r

0

√

eλ(r) − 1dr)2

, (13)

where P and Q are integration constants.In view of (6), the anisotropy of the fluid ∆

[Maurya et al (2016)] is obtained as

∆ =ν′(r)

4eλ(r)

[2

r− λ′(r)

eλ(r) − 1

][ν′(r)eν(r)

2rB2− 1]

. (14)

At this juncture we should point out that when ∆ = 0,the only bounded solution simultaneously satisfyingthe Karmarkar condition and pressure isotropy is theinterior Schwarzschild solution. This solution suffersvarious shortcomings including superluminal speedswithin the interior of the fluid. To this end we considera solution describing an anisotropic fluid distributionwhich will be taken up in the next section.

4. A new parametric class solutions

In this paper, we assumed the following metricpotential

eλ(r) = 1 + ar2αn(r), (15)

where

αn(r) = cscn(

br2 + c)

,

and a, b and c are positive constants and n ≥ 0. Wehave selected eλ(r) such that at center eλ(r) = 1, whichemphasizes that at the center the tangent 3 space isflat and the Einstein field equations (EFEs) can beintegrated. Substituting the eλ(r) from (15) in (13),we obtain the remaining metric potential eν(r) as

eν(r) =

(

P − Qh1(r)h2(r)√

aαn(r)

4b

)2

, (16)

where P and Q are integration constants.Using the metric potentials given by Eqs. (15) and

(16), the expressions of ρ, pr, ∆ and pt can be cast as

ρ =aαn(r)

(

r2(

aαn(r) − 2bn cot(

br2 + c))

+ 3)

(ar2αn(r) + 1) 2,

(17)

pr =h2(r)

√

aαn(r)

h3(r) (ar2αn(r) + 1), (18)

∆ =h5(r)r

2 (2bh6(r) − h7(r))

h8(r) (1 + ar2αn(r))2 ,

(19)

pt = pr +∆, (20)

where

h1(r) = 2F1

(

1

2,n+ 2

4;3

2; cos2

(

br2 + c)

)

,

h2(r) = sin(

2(

br2 + c))

sin2(

br2 + c)

n−2

4 ,

h3(r) = 2Pb√aαn − aQh1(r)

√αn cos

(

br2 + c)

− 4bQ,

h4(r) =√aQh1(r) cos

(

br2 + c)

− 2Pb

h5(r) = aαn(r) + bn cot(

br2 + c)

h6(r) = aPαn(r) −Q√

aαn(r)

h7(r) = aBh4(r) cos(

br2 + c)

cscn

2

(

br2 + c)

h8(r) = 2Pb−√aQh1(r) cos

(

br2 + c)

The mass function m(r), gravitational red-shiftz(r) and compactification factor u(r) at the surfaceand within the interior of the stellar system are givenby

m(r) =ar3αn(r)

2 (ar2αn(r) + 1), (21)

z(r) =1

P − Qh1(r)h2(r)√

aαn(r)

4b

− 1, (22)

u(r) =m(r)

r=

ar2αn(r)

2 (ar2αn(r) + 1). (23)

5. Matching of interior and exterior spacetime

over the boundary

To determine the constants a, b, c, P, Q appearingin our class of solutions, the interior metric mustbe matched smoothly across the boundary with theexterior Schwarz-schild solution

ds2 =(

1− 2M

r

)

dt2 −(

1− 2M

r

)−1

dr2

− r2(dθ2 + sin2 θdφ2). (24)

By comparing the interior solution (1) with exteriorsolution (24) at the boundary r = R (Darmois-Isrealiconditions), we obtain

eνb = 1− 2M

R

=

(

P +Q(

n√1− γ + 2bR2 + 2c

)√

aαn(R)

b (n2 + 4)

)2

,

(25)

e−λ(r)b = 1− 2M

R=

1

1 + aR2αn(R), (26)

pr(R) = 0. (27)


With the help of the boundary conditions (25-27), weobtain

a = − 2M csc−n(

bR2 + c)

R2(2M −R), (28)

P =

√

1− 2MR

(

ah1(R) cos(

bR2 + c)√

αn(R) + 4b)

4b,

(29)

Q =1

2

√

1− 2M

R

√

a cscn (bR2 + c), (30)

where γ =(

bR2 + c)2.

The constants b and c are free parameters and areselected such a way that all the physical propertiesof the assumed stars for a suitable range of n arewell-behaved and satisfy the Darmois-Israel conditions.The values of P and Q are expressed in Eqs. (29) and(30) respectively.

6. Discussion of physical features for

well-behaved solutions

6.1. Geometrical regularity

The metric potentials (geometrical parameters) forthe stars PSR J1614-2230 and SAX J1808.4-3658 forthe range of n mentioned in Table 1 at the center(r = 0), give the values eν |r=0 = positive constantand e−λ(r)|r=0 = 1. This shows that the metric poten-tials are regular and free from geometric singularitiesinside the stars. Also, both metric potentials eν(r) ande−λ(r) are monotonically increasing and decreasing re-spectively, with r (Fig.1).

6.2. Viable trends of physical parameters

6.2.1. Density and pressure trends The matterdensity ρ, radial pressure pr and transverse pressurept for stars PSR J1614-2230 and SAX J1808.4-3658are non-negative inside the stars and monotonicallydecrease from center to surface of these stars forthe range of n mentioned in Table 1(Fig.2,Fig.3)[Zeldovich & Novikov (1971), Ivano (2002)].

6.2.2. Relation between pressure-density ratios (Equa-tion of state) We plot the graphs of the equation ofstate parameters (pr/ρ, pt/ρ) to establish some con-nection between density and the pressures. Using thetrend of plots, we establish a linear, quadratic or CFLEoS for our model. An example of starting off with themetric functions and then establishing an EoS is theclassic paper by Mukherjee et al (1997). In this workthey show that the Vaidya-Tikekar geometry leads toa linear EoS. From the plots of figures, we observe thedecreasing trend of pressure to density ratios with r

Fig.(4) for both the stars PSR J1614-2230 and SAXJ1808.4-3658 for the range of n mentioned in Table 1.Based on the trends of the plots, we calculate equationof state (EOS) for neutron star PSR J1614-2230 as

pr = 0.861538ρ2 + 0.206369ρ− 0.00223306, (31)

pr = 69.1848ρ2 − 1.27803ρ+ 0.00560289, (32)

for n = 13.5, n = 28.98 respectively and for the strangestar SAX J1808.4-3658 as

pr = 0.276979ρ2 + 0.155325ρ− 0.00151322, (33)

pr = 48.6746ρ2 − 0.639035ρ+ 0.00149093, (34)

for n = 9.56, n = 20.3 respectively, using the methodof least of square technique (elaborated in appendix).The profiles of equation of state for PSR J1614-2230(n = 13.5 ), SAX J1808.4-3658 (n = 9.56) are exhibitedin the Fig.(5). The trends of EOS for other values of nin their corresponding ranges of the stars remain sameas in the Fig.(5).

6.2.3. Mass-radius relation, red-shift and compactifi-cation factor The mass function m(r) and gravita-tional red-shift z(r) function of stars PSR J1614-2230and SAX J1808.4-3658 for the range of n mentionedin Table 1 are increasing and decreasing respectivelywith r. The variation of m(r) and z(r) is shown inFigs.(6,7). Also, compactification parameter u(r) forboth the stars are increasing functions with r, shown inFig.(8) and lies within the Buchdahl limit [Buchdahl(1959)].

6.2.4. Anisotropic parameter In Fig.(9), the radialpressures (pr) coincides with tangential pressure (pt)at the center of stars PSR J1614-2230 and SAXJ1808.4-3658 for the range n mentioned in Table 1, i.e,pressure anisotropies vanish at the center, ∆(0) = 0and increase outwards [Bowers & Liang (1974), Ivano(2002)].

7. Physical Stability analysis

7.1. Zeldovich’s condition

The values of pr, pt and ρ at the center are given by

8πprc = 8πptc

= a cscn(c)

(

−2Pb√

a cscn(c) + 4bQ+ β1β2Q)

(

2Pb√

a cscn(c)− β1β2Q) > 0,

(35)

and

8πρc = 3a cscn(c) > 0 if a > 0. (36)

Using Zeldovich’s condition [Zeldovich & Novikov(1971)], i.e., prc/ρc ≤ 1, we get


−2Pb√

a cscn(c) + 4bQ+ β1β2Q

3(

2Pb√

a cscn(c)− β1β2Q) ≤ 1, (37)

In view of (36) and (37), we get the followinginequality

2Ab√

a cscn(c)

4b+ β1β2≤ Q

P≤ 2Ab

√

a cscn(c)

b+ β1β2, (38)

where

β1 = 2F1

(

1

2,n+ 2

4;3

2; cos2(c)

)

,

β2 = a cos(c) sinn

2 (c) cscn(c).

7.1.1. Hererra cracking stability of an anisotropicfluid sphere The Hererra cracking method [Herrera(1992)] is used to analyze the stability of anisotropicstars under radial perturbations. We also employthe concept of cracking due to Abreu et al (2007) toanalyze potentially stable regions within the stellarconfiguration by subjecting our model to the condition−1 < v2t − v2r ≤ 0

dptdρ

=dprdρ

+d∆

dρ=

dprdρ

+d∆

dρ

dr

dρ, (39)

v2r − v2t = −d∆

dρ

dr

dρ. (40)

For a physically feasible model of anisotropic fluidsphere the radial and transverse velocities of soundshould be less than 1, which are referred to as causalityconditions in the literature. The profiles of v2r and v2t ofstars PSR J1614-2230 and SAX J1808.4-3658 for therange n mentioned in Table 1 are given in Fig.(10),which shows that 0 < v2r ≤ 1 and 0 < v2t ≤ 1everywhere within the stellar configuration. Therefore,both the speeds satisfy the causality conditions andmonotonically decreasing nature. Here, we usethe Herrera cracking method [Herrera (1992)] foranalyzing the stability of anisotropic stars under theradial perturbations. Using the concept of cracking,Abreu et al (2007) gave the idea that the region of theanisotropic fluid sphere where −1 < v2t − v2r ≤ 0 ispotentially stable. Fig.(11) clearly depicts that ourmodel is potentially stable inside the both stars PSRJ1614-2230 and SAX J1808.4-3658 for the range nmentioned in Table 1.

7.1.2. Bondi stability condition for adiabatic indexFor a relativistic anisotropic sphere the stabilitydepends on the adiabatic index Γr, the ratio of twospecific heats, defined by Heintzmann & Hillebrandt(1975),

Γr =ρ+pr

pr

∂pr

∂ρ .

Bondi (1964) suggested that for a stable Newto-nian sphere, Γ value should be greater than 4

3 . For ananisotropic relativistic sphere the stability condition isgiven by Chan et al (1993),

Γ > 43 +

[ 4(pt0−pr0)

3|p′

r0|r

+ ρ0pr0

2|p′

r0|r]

,

where pr0, pt0 and ρ0 represent the initial radialpressure, tangential pressure and energy densityrespectively in static equilibrium. The first andlast term inside the square brackets represent theanisotropic and relativistic corrections respectively.Moreover, both the quantities are positive and increasethe unstable range of Γ.

Chandrasekhar (1964a) established a conditionon Γ to study the stability of interior of Schwarzschildmetric and it is defined as

Γ > Γcr =4

3+

19

42(2δ), (41)

where δ is compactification factor and Γcr is the criticaladiabatic index which is determined from neutralconfiguration.

Moustakidis (2017) suggested that in the interiorof fluid sphere Γcr should linearly depend on thepressure and density rations at center and Γ >Γcr . For stable Newtonian sphere, Bondi andChandrasekhar suggested that Γ > 4

3 [Bondi (1964),Chandrasekhar (1964a,b)].

The present class of models satisfy Bondi,Chandrasekhar, Moustakidis conditions for both thecompact stars PSR J1614-2230 and SAX J1808.4-3658for the range of n mentioned in Table 1 and Γcr linearly

depend on the ratio pr(0)ρ(0) .

7.1.3. Energy conditions For a physically stablestatic model the interior of the star should satisfy (i)null energy condition ρ+pr ≥ 0 (NEC) (ii) weak energyconditions ρ + pr ≥ 0, ρ ≥ 0 (WECr) and ρ + pt ≥0, ρ ≥ 0 (WECt) and (iii) strong energy conditionρ + pr + 2pt ≥ 0 (SEC) [Maurya et al (2019b)]. Theprofiles of energy conditions i.e. NEC, WEC, SEC aredisplayed in Fig.(13) and our models satisfy all theenergy conditions for both the stars PSR J1614-2230and SAX J1808.4-3658 for the range n mentioned inTable 1.

7.2. Tolman-Oppenheimer-Volkoff condition forequilibrium under three forces

The Tolman-Oppenheimer-Volkoff (TOV) equation[Ponce de Leon (1987)] for anisotropic fluid matterdistribution is given as

−Mg(r)(ρ + pr)

r2e(λ(r)−ν(r))/2 − dpr

dr+

2∆(r)

r= 0,(42)

where Fg, Fh, Fa are gravitational, hydrostaticand anisotropic forces respectively and Mg(r) is the


gravitational mass can be obtained from the Tolman-Whittaker formula

Mg(r) =1

2r2ν′(r)e(ν(r)−λ(r))/2. (43)

The TOV equation (42) can be expressed in thefollowing balanced force equation

Fg + Fh + Fa = 0, . (44)

In an equilibrium state the three forces Fg, Fh andFa satisfy TOV equation. The profiles of the threeforces of the stars PSR J1614-2230, SAX J1808.4-3658are exhibited in Fig.(14) and in which Fg overshadowsthe other two forces Fh and Fa such that the systemto be in a static equilibrium.

7.3. Harrison-Zeldovich-Novikov Static stabilitycriterion

The Harrison-Zeldovich-Novikov static stability crite-ria for non-rotating spherically symmetric equilibriumstellar models provides that the mass of compact starsmust be an increasing function of its central densityunder small radial pulsation i.e.

∂M

∂ρc> 0. (45)

This criteria ensures that the model is static andstable. It was proposed by Harrison et al (1965) andZeldovich & Novikov (1971) independently for stablestellar models. With the help of (36) and total mass

M = m(R) =aR3 cscn

(

bR2 + c)

2 (aR2 cscn (bR2 + c) + 1), (46)

The expression of the mass in terms of the centraldensity is given by

M(ρc) =ρR3 csc−n(c) cscn

(

bR2 + c)

2 (ρR2 csc−n(c) cscn (bR2 + c) + 3).

Also,

∂M

∂ρc=

R3 csc−n(c) cscn(

bR2 + c)

6(

13ρR

2 csc−n(c) cscn (bR2 + c) + 1)2 > 0,

satisfies (Fig.15) the static stability criterion (45).The Harrison-Zeldovich-Novikov condition is sat-

isfied for both the stars PSR J1614-2230 and SAXJ1808.4-3658 for the range n mentioned in Table 1.

8. Discussion and Conclusion

Our aim in this paper is to use the Karmarkar condition(which is purely geometric) to establish a physicallyviable stellar model (albeit a toy model). Toy modelsare important as they give a sense of the behaviour ofthe various physical and thermodynamical propertiesof the star and assist in setting up numerical codesand simulations.

In this paper, we have explored a new parametricclass of solutions for anisotropic matter distribution tomodel the compact star PSR J1614-2230 and strangestar SAX J1808.4-3658 by invoking the Karmarkarcondition and adopting a form for one of the metricpotentials, eλ(r). We find a range for one of theparameters, n for the both stars such that the solutionsare well behaved for particular choices of the freeconstants b, c. We have analyzed all the geometricaland physical properties of these two stars and verifiedthe physically viability of the solutions for the samerange of n.

The graphs of the two stars for different models i.e.(i) n = 13.5, 18.66, 23.82, 28.98 for PSR J1614-2230;(ii) n = 9.56, 13.14, 16.72, 20.3 for SAX J1808.4-3658for parameters values of b = 0.0001/km2, c = 2.5/km2

are plotted to find the range of n such that the solutionsare well behaved. Furthermore, we concluded thatthe range of well behaved solutions for PSR J1614-2230 is n = 13.5 to 28.98 and for SAX J1808.4-3658is n = 9.56 to 20.3 corresponding to same parametervalues b, c.

For any value in the range of n the geometricalparameters (e−λ(r) and eν(r)) are decreasing andincreasing respectively throughout interior of the starsand both curves meet at their boundary (Fig.1).The physical parameters such as density, radial andtangential pressures, pressures to density ratios, red-shift, both the velocities in that range of n are non-negative at the center and monotonically decreasingfrom center to surface of the stars Figs. (2,3,4,7,10).Physical parameters mass, compactification factor,anisotropy and adiabatic index are increasing outwardwhich is required for a physically viable stellarconfiguration Figs.(6,8,9,12).

Our models satisfy all the stability conditions forthe two stars for any value of n in that range, i.e,Herrera cracking condition (−1 < v2t − v2r < 0, 0 <v2r , v2t < 1), Bondi condition (Γ > 4/3), Zeldovich’scondition (0 < pr

ρ , pr

ρ < 1) and Harrison-Zeldovich-

Novikov criterion (∂M∂ρc

> 0) Figs.(11,12,15). For thesame range of n of the both stars the present modelshold all the energy conditions (ρ > 0, ρ + pr > 0,ρ + pt > 0, ρ + pr + 2pt > 0) which are required for aphysically viable configuration (Fig.13). Furthermore,our models represent a static anisotropic stellar fluid inequilibrium configuration as the gravitational force, thehydrostatic force and the anisotropic force are actingin the interior stars through the TOV equation arecounter-balancing each other (Fig.14).

The physical quantities i.e., central adiabaticindex (Γc), central density (ρc), central pressure (prc),central red-shift (zc(r)), surface red-shift (zs(c)) andcompactness factor (u(r) = GM

cR2 ) are given in Table1. From Table 1 we conclude that the larger the value


Table 1. The variation in physical parameters, i.e., central adiabatic index, central density, central red-shift, surface red-shift and compactness factor for different models of (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km forparameters n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for parametersn = 9.56, 13.14, 16.72, 20.3 for the values of b = 0.0001/km2, c = 2.5, G = 6.67 × 10−11m3kg−1s−2, M⊙ = 2 × 1030kg andC = 3× 108ms−1.

n = 13.5 n =18.66

n =23.82

n =28.98

n =9.56

n =13.14

n =16.72

n =20.3

Central adiabaticindex(Γc)

2.5881 3.2254 4.5 8.0523 4.2634 5.1352 6.7296 9.76

Central density(ρcg/cm3 × 1014)

4.8075 4.3597 3.9575 3.5959 3.6093 3.4397 3.2792 3.127

Central radialPressure (Prc)(dyne/cm2 ×1034)

9.008 9.5265 9.9761 10.362 2.5136 2.7694 3.0125 3.2432

Central red-shift(zc)

0.5531 0.5482 0.5435 0.5389 0.22474 0.22402 0.22332 0.22262

Surface red-shift(zb)

0.29815 0.29815 0.29815 0.29815 0.13694 0.13694 0.13694 0.13694

Compactness factorGMC2R

0.30134 0.30134 0.30134 0.30134 0.16777 0.16777 0.16777 0.16777

of n, the central adiabatic index and central pressureare increasing, whereas the central density and centralred-shift are decreasing with increasing the value ofn. Other physical parameters i.e. compactificationfactor and red-shift at the surface remain constant forany value of the range n. This work has provideda family of parametric solutions of the Einstein fieldequations obeying the Karmarkar condition. We showthat these solutions are sufficiently useful to modelcompact objects and predict their observed stellarcharacteristics within very good approximation.

Appendix: Generating function

All the spherically symmetric solutions can begenerated from the two generating functions given byHerrera et al (2008). The two primitive generatingfunctions η(r) and Π(r) are given as

eν(r) = e

[∫

(2η(r)− 2

r)dr]

, Π(r) = 8π(pr − pt). (47)

The two generating functions pertaining to the presentclass of solutions are obtained as

η(r) =√aQh1(r) cos

(

br2 + c)

− 2b(√

aQr2 cscn

2

(

br2 + c)

+ P)

r (√aQh1(r) cos (br2 + c)− 2Pb)

and

Π(r) = 8π(pr − pt) = −8π∆.

Appendix: Equation of state

The equation of state is defined as the relation betweenradial pressure (pr) and density (ρ) within the star.

Since the presence of cumbersome transformation ofpr in terms of ρ, here we use curve fitting technique ofapproximation to get equation of state. Further, from

Fig.(10), we observe that the plot of vr =√

dpr

dρ is not

a straight line (i.e. dpr

dρ is not a constant), therefore,it is necessary that the relation between pr and ρ isparabolic in nature. Consequently, in order to get theequation of state we consider the curve fitting methodfor quadratic form

pr(r) = U + Tρ(r) + Sρ2(r), (48)

Σpr(r) = 11U + TΣρ(r) + SΣρ2(r), (49)

Σρ(r) pr(r) = UΣρ(r) + TΣρ2(r) + SΣρ3(r) (50)

Σρ2(r) pr(r) = UΣρ2(r) + TΣρ3(r) + SΣρ4(r), (51)

where, r varies from central to boundary of the star.To find the curve via least square method, we considerthe points with the differences 0.969, 0.7951 for PSRJ1614-2230, SAX J1808.4-3658 respectively. Solvingthe Eqns.(49,50,51) for S, T , U and substituting thevalues in Eq.(48) , we get required equation of state.


Figure 1. Variation of e−λ(r), eν(r) with r for (i) PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 2. Variation of ρ with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for the modelsn = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 3. Variation of pr, pt with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.


Figure 4. Variation of pr/ρ and pt/ρ with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 5. Variation of equation of state parameters with ρ for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69kmfor the model n = 13.5; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the model n = 9.56 and thevalues of b = 0.0001/km2, c = 2.5.

Figure 6. Variation of mass (m(r)) with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.


Figure 7. Variation of red-shift with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 8. Variation of the compactification factor u(r) with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69kmfor the models n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 9. Variation of anistropy ∆(r) with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.


Figure 10. Variation of v2r , v2t with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 11. Variation of vt2 − vr2 with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for themodels n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 12. Variation of Γ(r) with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km for the modelsn = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.


Figure 13. Variation of energy conditions with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km forthe models n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 14. Variation of balancing forces Fa, Fg, Fa, Fa+Fg+Fh with r for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radiusR = 9.69km for the models n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951kmfor the models n = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.

Figure 15. Variation of mass with central density ρc for (i)PSR J1614-2230 with mass M = 1.97M⊙ and radius R = 9.69km forthe models n = 13.5, 18.66, 23.82, 28.98; (ii) SAX J1808.4-3658 with mass M = 0.9M⊙ and radius R = 7.951km for the modelsn = 9.56, 13.14, 16.72, 20.3 and the values of b = 0.0001/km2, c = 2.5.


Acknowledgments The authors are thankful to thelearned referee for the valuable comments and suggestions toimprove the paper.

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