Charged Analogues of Isotropic Compact Stars Model with Buchdahl Metric in General Relativity
CCharged Analogues of Isotropic Compact Stars Modelwith Buchdahl Metric in General Relativity
Amit Kumar Prasad a , Jitendra Kumar a, ∗ a Department of Mathematics,Central University of Jharkhand,Ranchi-835205 India.
Abstract
In this work, we examine a spherically symmetric compact body with isotropicpressure profile, in this context we obtain a new class of exact solutions ofEinstein’s-Maxwell field equation for compact stars with uniform charged dis-tributions on the basis of Pseudo-spheroidal space time with a particular formof electric field intensity and the metric potential g rr . Indicating these twoparameters takes into account further examination to be done in deciding un-known constants and depicts the compact strange star candidates likes PSRJ1614-2230, 4U 1608-52, SAX J1808.4-3658, 4U 1538-52, SMC X-1, Her X-1and Cen X-3.By the isotropic Tolman-Oppenhimer-Volkoff(TOV)equation, Weexplore the equilibrium among hydrostatic, gravitational and electric forces.Then, we analyze the stability of model through adiabatic index( γ ) and ve-locity of sound (0 < dpc dρ < Keywords:
Isotropic Fluids, Electric Intensity, Reissner-Nordstrom Metric,Compact star, General Relativity.
1. Introduction
The first exact solution of the Einstein field equations obtained by KarlSchwarzschild[1].In a theoretical sense, stars are confined in gas and dust cloudswith non-uniform matter circulation and scattered all through generally cosmicsystems.In astronomy, compact objects are typically alluded on the whole to ∗ Corresponding author
Email address: [email protected] (Jitendra Kumar)
Preprint submitted to Elsevier October 24, 2019 a r X i v : . [ phy s i c s . g e n - ph ] S e p hite dwarfs or neutron stars. From the time of Sir Isaac Newton, our compre-hension of the idea of gravity has progressed however mysteries in physics stillremain. Einsteins theory of General Relativity (GR) is one of the best essentialspeculations of gravity in physics. In spite of the fact that its success, numerousexpansions of the first Einstein conditions has been researched to satisfactorypresent observational information on both cosmological and astrophysical scales.Observational information has been educated us that the universe is experienc-ing a period of quickened extension.The microscopic structure and properties ofa dense matter on phenomenal conditions are necessary to examine for compactobject. Also, the high exactness information from Type Ia supernovae [2], thecosmic microwave background anisotropies [3], baryonic acoustic motions [4] andfrom gravitational lensing are giving help to stellar. The information appearsto show that the universe is directly overwhelmed by two obscure componentsone pressure less dark matter (DM) and second dark energy (DE). This is inlight of the fact that at such uncommon densities nuclear matter may includenucleons and leptons just as a couple of fascinating segments in their differentstructures also, stages, for example, mesons, hyperons, baryon resonances simi-larly as strange quark matter (SQM). Be that as it may, it is as yet impracticalto find a far-reaching portrayal of the very dense matter in a firmly cooperatingsystem. So it is valuable to examine a definite composition and the idea ofmolecule connections in the inside of such object.In context of general theory ofrelativity, a broadly the pursued course is to indicate an equation of state andafterward solve the Einstein field equations for the study of the compositionof a compact star.This is also useful for Tolman-Oppenheimer-Volkoff (TOV)equation (see [5, 6]) or the condition of hydro dynamical equilibrium. It is con-ceivable that anisotropic matter is a significant fixing in numerous astrophysicalobjects for example, stars, gravastars, and so on. Generally, extensive exertionhas been committed to picking up a far reaching comprehension of propertiesof the anisotropic matte with the expectation of delivering physically suitablemodels of compact stars. Specifically, compact stars may before long give dataabout the gravitational connection in an extraordinary gravitational condition.Most of the compact stars divided between a strange star and a normalneutron star. Many authors[7, 8, 9] studied that the strange stars possess ultra-strong electric fields on their surfaces. The impact of energy densities on ultra-high electric fields of compact stars was investigated in[10, 11, 12, 13, 14, 15]. Itadditionally has been demonstrated that the electric fields increment the stellarmass by up to 30% relying upon the quality of it. As opposed to the strange starthe surface electric field on account of neutron star is absent[16].Based on theseproperties, one to observationally recognize quark stars from neutron stars.The important characteristic of many astrophysical objects, like compactstars, gravastars etc.,is isotropic matter.In an extreme gravitational environ-ments, compact stars provide information about the gravitational interaction.The extreme internal density and strong gravity of compact star indicate thatthe pressure within such objects have two different types of pressures, namelythe radial and tangential pressure, and these are to be equal. These informa-tion of isotropic matter may producing physically valid models of compact stars.Bowers and Liang [17] have noticed about the structure and evolution of rela-tivistic compact objects in general relativity.They investigated the changes inthe gravitational mass and surface redshift by generalisation of the equation ofhydrostatic equilibrium and obtained a static spherically symmetric configura-2ion. Ruderman [18] analyzed that at high densities of order 10 g/cm nuclearmatter transformed in anisotropic in nature. Also, they point out that the ra-dial pressure may not be equal to the tangential one in massive stellar objects.Based on above physical condition, many contentions have been presented forthe presence of anisotropy in star models for example, by the presence of type3A superfluid [19], various types of phase transitions [20],mixture of two flu-ids,the presence of solid core or by other different physical marvels.Also exactsolution of Einstein’s field is important for study of astrophysical object, be-cause many exact solutions of Einstein’s field equations have been found butsome of them satisfied all the physical plausibility conditions. This shows thecomplexity in getting exact solutions of Einstein’s field equations describingphysically realizable astrophysical objects. Several workers have charged starson spheroidal space-time have been studied by Patel and Kopper[21], Sharma etal.[22], Gupta and Kumar[23], Komatiraj and Maharaj[24]. Many projects aresuggested by Ivanov[25] for constructing charged fluid spheres.Recently Naveenand Bijalwan et al.[26][27] for all K except for 0 < K < < K < . ± . M (cid:12) (Elebert et al.[33]).Abubekerov et al.[34]reported the mass of Her X-1 to be 0 . ± . M (cid:12) .Rawls et al.[35] reported themass of 4U 1538-52 to be 0 . ± . M (cid:12) ,Cen X-3 to be 1 . ± . M (cid:12) and SMCX-1 to be 1 . ± . M (cid:12) .Demorest et al.[36] reported the mass of PSR J1614-2230 to be 1 . ± . M (cid:12) .Guver et al.[37] reported the mass of 4U 1608-52 tobe 1 . ± . M (cid:12) .In the present problem we have constructed a charged fluid sphere startingwith a specific metric potential g and generalized charge intensity. Delgaty-Lake[38] and Pant et al.[39] have proposed that the physically valid solution incurvature coordinates, the following conditions should be satisfied1. At the boundary r = a , pressure p should be zero.2. c ρ should always be grater than p within the range 0 ≤ r ≤ a.
3. The pressure gradient dp/dr should be negative for 0 < r ≤ a , i.e,( dp/dr ) r =0 =0 and ( d p/dr ) r =0 < .
4. The density gradient dρ/dr should also be negative for 0 < r ≤ a , i.e,( dρ/dr ) r =0 = 0 and ( d ρ/dr ) r =0 < dp/c dρ ) / < γ = (cid:18)(cid:18) c ρ + pp (cid:19) (cid:18) dpc dρ (cid:19)(cid:19) > / , is conditionfor stability of a fluid sphere.7. The surface redshift Za should be positive and finite.These features, positive density and positive pressure are the most importantfeatures characterizing a star. The task is now to check the well-behaved geom-etry and capability of describing realistic stars, we plot Figs. 1-6. Our stellarmodel is depending on the different values of K, η, b, and Ca . Such analyticalrepresentations have been performed by using recent measurements of mass and3adius of neutron stars,PSR J1614-2230, 4U 1608-52, SAX J1808.4-3658, 4U1538-52, SMC X-1, Her X-1 and Cen X-3.
2. Einstein field equations
Let us consider the static spherically symmetric metric in curvature coordi-nates ds = − e λ ( r ) dr − r ( dθ + sin θdφ ) + e ν ( r ) dt (1)where λ ( r ) and ν ( r ) are satisfy the Einstein-Maxwell equation for charged fluiddistribution R ij − Rδ ij = − κ (cid:2) ( c ρ + p ) ν i ν j − pδ ij + 14 π ( − F im F jm + 14 δ ij F mn F mn ) (cid:3) (2)with κ = 8 πGc while ρ, p, ν i denote matter density,fluid pressure and the unittime-like flow vector respectively and F ij denote the skew symmetric electro-magnetic field tensor.In view of (1) the equation (2) reduce to (Landau and Lifshitz[40]) ν (cid:48) r e − λ − (1 − e − λ ) r = κp − q r (3) (cid:18) ν (cid:48)(cid:48) − λ (cid:48) ν (cid:48) ν (cid:48) ν (cid:48) − λ (cid:48) r (cid:19) e − λ = κp + q r (4) λ (cid:48) r e − λ + (1 − e − λ ) r = κc ρ + q r (5)where (’) prime denotes the differentiation with respect to r and q ( r ) = 4 π (cid:90) r σr e λ/ dr = r (cid:112) − F F = r F e ( λ + ν ) / (6) q ( r ) represents the total charge contained within the sphere of radius r in viewof (1). Additionally,the component F (cid:54) = 0. On the further side of the pressurefree interface ‘ r = a (cid:48) the charged fluid sphere is expected to join with theReissner-Nordstrom metric: ds = − (cid:18) − Mr + e r (cid:19) − dr − r ( dθ + sin θdφ ) + (cid:18) − Mr + e r (cid:19) dt (7)where M is the gravitational mass of the fluid sphere such that M = ζ ( a ) + ξ ( a ) (8)where ζ ( a ) = κ (cid:90) a ρr dr, ξ ( a ) = (cid:90) a rσqe λ/ dr, e = q ( a ) (9)where ξ ( a ) is the mass equivalence to electromagnetic energy of distribution, ζ ( a ) is the mass and ‘ e (cid:48) is the total charge interior of the sphere (Florides[41]).4n this model we propose a charged fluid distributions by considering thegeneralized electric field intensity against[31] q r = C r Cr ) ( f + f ) (10)where f = Cr (4 K − − ( K + 2)4 K (1 + Cr ) , f = − η (1 − K )4 K ( Y η + Y b )(1 + Cr )and the metric potential e λ = K (1 + Cr ) K + Cr , < K < C, K, η being constants.The consistency of the field Eqs. (3)-(5) using Eqs. (10) and (11) yield theequation (1 + Y ) d ZdY − Y dZdY − [1 − K + K ( f + f )] = 0 (12)where Y = (cid:114) K + x − K , Cr = x and e ν = Z .The expression for energy density and pressure can be had from (3), (5), (10)and (11) as( K + x ) √ xCK (1 + x ) 2 Z (cid:48) Z + (1 − K ) K (1 + x ) + x x ) ( f + f ) = κpC (13)( K − x ) K (1 + x ) − x x ) ( f + f ) = κc ρC (14)Let Z = (1 + Y ) / Φ( Y ) (15)Put the values of Z from equation (15) into equation (12) we get d Φ dY + τ Φ = 0 (16)where τ = − η ( Y η + Y b )Hence the solution of the differential equation (16) isΦ( Y ) = ( Y η + Y b ) Aη b (cid:20) sin (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − csc (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − log sin (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19)(cid:21) + B ( Y η + Y b )In the equation (15) Z becomes Z = A (1 + Y ) / (cid:34) ( Y η + Y b ) η b G ( Y ) + B ( Y η + Y b ) (cid:35) (17)5here G ( Y ) = (cid:20) sin (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − csc (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − log sin (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19)(cid:21) Now put the equation (17) into equations (13)-(14),we get the expressions ofdensity and pressure κc ρC = ( K − x ) K (1 + x ) − x K (1 + x ) (cid:20) K − − Y Y ) + 2 η (1 + Y )( Y η + Y b ) (cid:21) (18) κpC = (cid:18) K + x ) K (1 + x ) (cid:112) (1 − K )( K + x ) (cid:19)(cid:20) E × E E × E E × E (cid:21) − ( K − K (1 + x ) + E E (cid:18) K + x − K (cid:19) / η + (cid:18) K + x − K (cid:19) b (cid:18) x − K (cid:19) / + (cid:18) x − K (cid:19) / (cid:18) (cid:114) K + x − K η + b (cid:19) E η b (cid:20) sin (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − csc (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − log sin (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19)(cid:21) + BA , E η (cid:18) Yb (cid:19) / (cid:18) Y (cid:19) / E (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) +2 csc (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) × cot (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) ,E (cid:18) Y (cid:19) / (cid:18) Y η + Y b (cid:19) , E x K (1 + x ) (cid:20) K −
1+ 2 − Y Y ) +2 η (1 + Y )( Y η + Y b ) (cid:21) The expression of velocity of sound is as follows, dpc dρ = L ( E × E − √ CxK (1 + Y ) + E × E E × E (cid:18) E − E × E − E × E (cid:19) (20)where, L = ( E × E × M × ( E × E E × E M (cid:20) ( E × E (cid:18) ( B B × E E × B E × B E × B (cid:19) − ( E × E E × E B × E B × E (cid:21) M √ Cx (1 − K − x ) K (1 + x ) (cid:112) (1 − K )( K + x ) , M K + x ) K (1 + x ) (cid:112) (1 − K )( K + x ) ,E (cid:20) K − − Y Y ) + 2 η (1 + Y )( Y η + Y b ) (cid:21) , E
10 = x K (1 + x ) , .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .00 .0 0 0 0 0 00 .0 0 0 0 0 50 .0 0 0 0 1 00 .0 0 0 0 1 50 .0 0 0 0 2 00 .0 0 0 0 2 50 .0 0 0 0 3 0 Pressure(p) r / a
P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3
Density( r ) r / a P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3
Figure 1: Behavior of pressure(p in km − ) and Density ( ρ in km − ) vs. fractional radius r/afor the compact objects PSR J1614-2230,4U 1608-52, SAX J1808.4-3658, 4U 1538-52,SMCX-1,Her X-1 and Cen X-3. For this figure, we have used the numerical values of physicalparameters and constants are as follows: (i) K=0.0000135,b = 0.09, C = − . × − km − , η =6, M=01 . M (cid:12) and a = 9.69 km for PSR J1614-2230,(ii)K = 0.0000209, b = 0.0903, C = − . × − km − , η =4.9, M =01 . M (cid:12) , and a =9.3 km for 4U 1608-52,(iii)K=0.0000319,b=0.09, C = − . × − km − , η =3.9, M =0 . M (cid:12) , and a =7.951 km for SAXJ1808.4-3658,(iv)K = 0.0000296, b = 0.0903, C = − . × − km − , η =4, M =0 . M (cid:12) ,and a =7.866 km for 4U 1538-52,(v) K = 0.000023, b = 0.1, C = − . × − km − , η =5,M =1 . M (cid:12) , and a =8.831 km for SMC X-1,(vi) K = 0.0000277, b = 0.06, C = − . × − km − , η =2.9, M = 0 . M (cid:12) and a = 8.1 km for Her X-1,(vii) K = 0.000018, b =0.09, C = − . × − km − , η =5, M =1 . M (cid:12) , and a =9.178 km for Cen X-3 E K − √ Cx K (1 + x ) − − η ( Y η + b ) 2 √ CxK (1 − K ) − (1 + Y ) K η √ Cx (2 η + b/Y )(1 − K )( Y η + b ) ,E √ Cx (1 − x )(1 + x ) , B √ Cx (1 − K ) (1 + Y ) (cid:32) Y η + b (cid:33) − Y (cid:32) Y η + b (cid:33) (1 + Y ) / ,B √ Cx (1 − K )(1 + Y ) / (cid:34)(cid:32) Y η + b (cid:33) + (1 + Y ) η Y (cid:35) , E
11 = 2 √ Cx (1 − K )(5 − x ) K (1 + x ) B η √ Cx b / (1 − K ) (cid:32) b + Y η (cid:33) × E , B η √ Cx b / (1 − K ) (cid:2) (1 + Y ) − / Y / + (1 + Y ) / Y / (cid:3) B η √ CxbY / (1 − K )( b + Y η ) (cid:34) cos 2 (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) × cot (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) − (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19) +csc (cid:18) arctan (cid:18)(cid:114) Y η b (cid:19)(cid:19)(cid:35) B √ Cx − K )(1 + Y ) / ( Y η + bY ) + 2(1 + Y ) / √ Cx (1 − K ) (cid:32) η + bY (cid:33) Observing the Fig.5(top right),it shows that velocity of sound lies within theproposed range for different compact stars as labeled in figure.7 . Boundary conditions
The charged fluid sphere is expected to join smoothly with the Reissner-Nordstrom metric (7).The continuity of e λ , e ν and q across the boundary r = a gives following equations e λ = 1 − m ( a ) a + e a , (21) y = 1 − m ( a ) a + e a , (22) q ( a ) = e, (23) p ( a ) = 0 . (24)The conditions (22) and (24) can be used to compute the values of arbitraryconstants A/B.
4. Tolman-Oppenheimer-Volkoff (TOV) equations
In the presence of charge, the Tolman-Oppenheimer-Volkoff(TOV) equation[5, 6] is given by − M G ( ρ + p ) r e λ − ν − dpdr + σ qr e λ = 0 , (25)where M G represents the gravitational mass and defined as: M G ( r ) = 12 r ν (cid:48) e ( ν − λ ) / . (26)Substituting the value of M G ( r ) in equation (25), we get − ν (cid:48) ρ + p ) − dpdr + σ qr e λ = 0 , (27)The equation (27) can be expressed into three unique segments, gravitational( F g ), hydrostatic ( F h ) and electric ( F e ), which are defined as: F g = − ν (cid:48) ρ + p ) = Z (cid:48) πZ ( ρ + p ) (28) F h = − dpdr = − π (cid:34) L ( E × E − √ CxK (1 + Y ) + E × E E × E (cid:35) (29) F e = σ qr e λ = 18 π r dq dr = 12 π [ E × E E × E
10 ] (30)We can see the behavior of the generalized TOV equations by figure (2) and thesystem is counterbalanced by three different forces, e.g, gravitational force ( F g ),hydrostatic force( F h ) and electric force ( F e ). This conclude that the systemattains a static equilibrium. 8 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0-0 .0 0 1 5-0 .0 0 1 0-0 .0 0 0 50 .0 0 0 00 .0 0 0 50 .0 0 1 00 .0 0 1 5 Fi r /a F g F h F e Fi r /a F g F h F e Fi r /a F g F h F e Fi r /a F g F h F e Fi r /a F g F h F e Fi r /a F g F h F e Fi r /a F g F h F e Figure 2: Behavior of different forces(in km − ) vs. fractional radius r/a for the com-pact objects PSR J1614-2230,4U 1608-52, SAX J1808.4-3658, 4U 1538-52,SMC X-1,Her X-1 and Cen X-3. For this figure we have used the numerical values of physical parame-ters and constants are as follows: (i) K=0.0000135,b = 0.09, C = − . × − km − , η =6, M=01 . M (cid:12) and a = 9.69 km for PSR J1614-2230(first row left),(ii) K = 0.0000209,b = 0.0903, C = − . × − km − , η =4.9, M =01 . M (cid:12) , and a =9.3 km for 4U1608-52(first row middle),(iii)K =0.0000319, b = 0.09, C = − . × − km − , η =3.9,M =0 . M (cid:12) , and a =7.951 km for SAX J1808.4-3658(first row right),(iv)K = 0.0000296,b = 0.0903, C = − . × − km − , η =4, M =0 . M (cid:12) , and a =7.866 km for 4U1538-52(second row left),(v) K = 0.000023, b = 0.1, C = − . × − km − , η =5, M=1 . M (cid:12) , and a =8.831 km for SMC X-1(second row middle),(vi) K = 0.0000277, b =0.06, C = − . × − km − , η =2.9, M = 0 . M (cid:12) and a = 8.1 km for Her X-1(secondrow right),(vii) K = 0.000018, b = 0.09, C = − . × − km − , η =5, M =1 . M (cid:12) , and a =9.178 km for Cen X-3(bottom) able 1: Values of different physical parameters of PSR J1614-2230 K = 0 . , C = − . × − km − , η = 6 , b = 0 . , Zo = 0 . , Za = 0 . km − ) ρ ( km − ) q( km ) dp/c dρ p/ρ γ × − × − × − × − × − × − × − × − × − × −
5. Energy Conditions
Here we analyze the energy conditions according to relativistic classical fieldtheories of gravitation. In the context of GR the energy conditions are localinequalities that process a relation between matter density and pressure obey-ing certain restrictions.The charged fluid sphere should satisfy the three energyconditions (i) strong energy condition (SEC),(ii) weak energy condition (WEC)and (iii) null energy condition (NEC). For satisfying the above energy condi-tions, the following inequalities must hold simultaneously inside the chargedfluid sphere:Null energy condition (NEC): ρ + q r π ≥ ρ + p + q r π ≥ ρ + 3 p + q r π ≥ . The nature of energy conditions for the specific stellar configuration as shownin Fig.3, that are satisfied for our proposed model.
NEC r /a
P S R J 1 6 1 4 -2 2 3 0 4 U 1 6 0 8 -5 2 S A X J 1 8 0 8 .4 -3 6 5 8 4 U 1 5 3 8 -5 2 S M C X -1 H e r X -1 C e n X -3
WEC r /a
P S R J 1 6 1 4 -2 2 3 0 4 U 1 6 0 8 -5 2 S A X J 1 8 0 8 .4 -3 6 5 8 4 U 1 5 3 8 -5 2 S M C X -1 H e r X -1 C e n X -3
SEC r /a
P S R J 1 6 1 4 -2 2 3 0 4 U 1 6 0 8 -5 2 S A X J 1 8 0 8 .4 -3 6 5 8 4 U 1 5 3 8 -5 2 S M C X -1 H e r X -1 C e n X -3
Figure 3: Behavior of Energy conditions( in km − ) vs. fractional radius r/a for the compactstars PSR J1614-2230,4U 1608-52, SAX J1808.4-3658, 4U 1538-52,SMC X-1,Her X-1 and CenX-3. In this figure we have used same data set values of physical parameters and constantswhich are the same in Fig.1 able 2: Value of different physical parameter of 4U 1608-52 K = 0 . , C = − . × − km − , η = 4 . , b = 0 . , Zo = 0 . , Za = 0 . . km − ) ρ ( km − ) q( km ) dp/c dρ p/ρ γ . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − Table 3: Value of different physical parameter of SAX J1808.4-3658 K = 0 . , C = − . × − km − , η = 3 . , b = 0 . , Zo = 0 . , Za = 0 . km − ) ρ ( km − ) q( km ) dp/c dρ p/ρ γ . × − . × − . × − . × − . × − . × − . × − . × − . × − . × −
6. Electric Charge
Varela et al.[42] have shown that fluid spheres with net charge contain fluidelements with unbounded proper charge density located at the fluid-vacuuminterface and net charge can be huge(10 C ). Ray et al.[10] have analyzed theimpact of charge in compact stars considering the limit of the most extrememeasure of the charge. They have demonstrated the global balance of the forcesallow a huge charge(10 C ) to be available in compact star.In this model we have found that the maximum charge on the boundary is6 . × C and at the center is zero. We have plot the Fig.4 for the charge q in the relativistic units(km).For coulombs unit, one has multiply these valueby 1 . × C .Thus in this model the net amount of charge is effective tobalance the mechanism of the force. 11 able 4: Value of different physical parameter of 4U 1538-52 K = 0 . , C = − . × − km − , η = 4 , b = 0 . , Zo = 0 . , Za = 0 . km − ) ρ ( km − ) q( km ) dp/c dρ p/ρ γ . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − Table 5: Value of different physical parameter of SMC X-1 K = 0 . , C = − . × − km − , η = 5 , b = 0 . , Zo = 0 . , Za = 0 . km − ) ρ ( km − ) q( km ) dp/c dρ p/ρ γ . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − Table 6: Value of different physical parameter of Her X-1 K = 0 . , C = − . × − km − , η = 2 . , b = 0 . , Zo = 0 . , Za = 0 . km − ) ρ ( km − ) q( km ) dp/c dρ p/ρ γ . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − able 7: Value of different physical parameter of Cen X-3 K = 0 . , C = − . × − km − , η = 5 , b = 0 . , Zo = 0 . , Za = 0 . km − ) ρ ( km − ) q( km ) dp/c dρ p/ρ γ . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − Table 8: Numerical values of radius( a ) M ( M (cid:12) ) , central density, surface density,central pres-sure and mass-radius ratio of compact star candidates. Compact star a( km ) M( M (cid:12) ) Central density Surface density Central pressure M/a( g/cm ) ( g/cm ) ( dyne/cm )PSR J1614-2230 9.69 1.97 8.886 × × × × × × × × × × × × × × × × × × × × × .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .00123456 charge(q) r / a P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3
Figure 4: Behavior of Charge(q in km ) vs. fractional radius r/a for the compact stars PSRJ1614-2230,4U 1608-52, SAX J1808.4-3658, 4U 1538-52,SMC X-1,Her X-1 and Cen X-3.Inthis figure we have used same data set values of physical parameters and constants which arethe same in Fig.1
7. Surface Redshift
The gravitational redshift Z s within a static line element can be obtained as Z s = (cid:112) g tt ( a ) − (cid:114) − Ma + q a − g tt ( a ) = e ν ( a ) = 1 − Ma + q a The maximum possible value of redshift should be at the center of the starand decrease with the increase of radius.Buchdahl[43] and Straumann[44] haveshown that for an isotropic star the surface redshift Z s ≤ Z s ≤
5. Ivanov[25] modified the maximum value of redshift and showedthat it could be as high as Z s = 5 . Z s ≤
8. Causality and Well behaved condition
Inside the fluid sphere the velocity of sound is less than the light, i.e. 0 ≤ v ≤ dpdρ <
1. According to Canuto[46], for well behaved nature of the chargesolution, the velocity of sound should be monotonically decreasing towards theboundary with an ultra-high distribution of matter.From Fig.5(top right) it isverified that velocity of sound should monotonically decreasing.This imply ourmodel for charge compact star is well behaved.
9. Adiabatic Index
The stability of relativistic isotropic fluid sphere depends on the adiabaticindex γ . Heintzmann and Hillebrandt[47] proposed that isotropic compact star14 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .00 .10 .20 .30 .40 .50 .60 .70 .80 .9 Velocity of sound(v2) r / a
P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3 p/ r r / a P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3 g r / a P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3
Redshift r / a
P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3
Figure 5: Behaviour of Velocity of sound,density-pressure ratio,adiabatic constant and redshiftvs. fractional radius r/a for the compact stars PSR J1614-2230,4U 1608-52, SAX J1808.4-3658, 4U 1538-52,SMC X-1,Her X-1 and Cen X-3. For this figure we have used same data setvalues of physical parameters and constants which are the same in Fig.1 models are stable if γ > / γ = (cid:0) p + ρp (cid:1) dpdρ (32)From Fig.5(bottom left) we have seen that the value of γ is greater than 4 /
10. Harrison-Zeldovich-Novikov Stability Criterion
The Harrison-Zeldovich-Novikov[48, 49] criterion states that the compactstar to be stable if the its mass is increasing with increasing central density,i.e. dMdρ >
0, however it is unstable if dMdρ <
0. For this purpose we calculate thedensity of star at the center ( ρ ), mass ( M ( a )) and its gradient in terms ofcentral density as ρ = 3 C ( K − πK (33)15 ( a ) = 4 πρ a M ( a ) (cid:20) K − πρ K a M ( a ) M ( a ) (cid:21) (34) dM ( a ) dρ = 12 π a ( K − M ( a )) (cid:20) K − π ρ K a M ( a ) M ( a ) + 4 π ( ρ K a ) K − (cid:18) − π a (1 − K ) M ( a ) − M ( a ) (cid:19)(cid:21) (35)where M ( a ) = 8 π ρ K a + 3( K − M ( a ) = (cid:18) − − K ) M ( a ) + 8 η M ( a )( η M ( a ) − b (1 − K ) (cid:112) − M ( a ) /
3) + 4 K − (cid:19) M ( a ) = − π η a M ( a ) M ( a )(1 − K ) (cid:18) η M ( a ) + (cid:112) M ( a )( η M ( a ) + (cid:112) M ( a )) (cid:19) − π a ( η M ( a ) − b (1 − K ) (cid:112) − M ( a ) / M ( a ) = +3 K ( K − − π ρ K a (1 − K ) , M ( a ) = 3 K (1 − K ) − π ρ K a (1 − K ) From Fig.6 we see that the mass M ( M Θ ) of isotropic compact star increases Mass(M q ) r P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3 dM/d r r P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3
Figure 6: Behavior of Mass ( M (cid:12) )(left panel) and dMdρ vs. central density ρ for the compactstars PSR J1614-2230,4U 1608-52, SAX J1808.4-3658, 4U 1538-52,SMC X-1,Her X-1 and CenX-3.In this figure we have used same data set values of physical parameters and constantswhich are the same in Fig.1 with central density ( ρ ). On other hand we observe that dMdρ is positive withrespect to central density ( ρ ). Hence present isotropic compact star model isstable.
11. Novelty of Present Model
The novelty of our model is that the density( ρ ) is always positive and inmonotonically decreasing order for 0 < K <
1. If we remove the electricintensity( q ( r ) = 0) or adopt the anisotropic approach then the density( ρ ) isincreasing towards the boundary, i.e. dρdr > < K <
1. An anisotropicapproach we obtained ρ and dρdr as follows- ρ = C ( K − Cr )8 π K (1 + Cr ) ,dρdr = − C r ( K − Cr )8 π K (1 + Cr ) > − K C r ( K − > < K < .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .00 .0 0 0 5 20 .0 0 0 5 40 .0 0 0 5 60 .0 0 0 5 80 .0 0 0 6 00 .0 0 0 6 20 .0 0 0 6 40 .0 0 0 6 6 r r / R P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .00 .0 0 0 0 00 .0 0 0 0 10 .0 0 0 0 20 .0 0 0 0 30 .0 0 0 0 40 .0 0 0 0 5 d r /dr r / R P S R J 1 6 1 4 - 2 2 3 0 4 U 1 6 0 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 4 U 1 5 3 8 - 5 2 S M C X - 1 H e r X - 1 C e n X - 3
Figure 7: Behaviour of ρ and dρ /dr vs. fractional radius for PSR J1614-2230, 4U 1608-52SAX J1808.4-3658,4U 1538-52, SMC X-1,Her X-1 and Cen X-3.In this figure we have usedsame data set values of physical parameters and constants which are the same in Fig.1 − . × − < C <
0. Also both term (5+ Cr ) and (1+ Cr ) are pos-itive. Hence the density gradient is also positive. From Fig.7, we have concludethat our approach is valid with charged compare to anisotropic approach.
12. Final Remarks
In this article, we obtained a new class of charged super-dense star modelsby solving Einstein-Maxwell field equation for a static symmetric distributionof perfect fluid based on a suitable metric potential and considering a particu-lar form of electric intensity.The boundary conditions required for the smoothmatching of the interior space-time to the exterior Reissner-Nordstrom space-time which is fixes the constants( see Table 1 to 8 ) in our solution and de-termines the mass contained within the charged sphere.In particular, we havedemonstrated that the radii and masses measurements for the seven observedcompact stars namely,PSR J1614-2230, 4U 1608-52, SAX J1808.4-3658, 4U1538-52, SMC X-4, Her X-1 and Cen X-3.It could additionally restrict the arbi-trary chosen constant parameters and the nature of the stars has been discussedusing values of these constants.Graphical analysis of the solution shows that thepressure,density and ratio of p/ρ are monotonically decreasing towards the sur-face, when K lies between 0 . . K = 0 . , . , . , . , . , . , . . Causality condition is obeyed at each interior point of the configuration. Stabil-ity analysis via the Zeldovich stability criterion indicate that our model satisfycausality condition i.e,( dp/c dρ ) < K as above.The stabilityof the charged fluid models depends on the adiabatic index γ . Heintzmannand Hillebrandt [47] proposed that a neutron star model with EOS is stableif γ > K (as above) is greater than 4 / Zo = redshift at the center, Za = redshift at the surface,solar mass M (cid:12) =1 . km, G = 6 . × − cm /gs , c = 2 . × cm/s. ReferencesReferences [1] K. Schwarzschild, Sitzer. Preuss. Akad. Wiss. Berlin, 424 (1916) 189 .[2] M.Betoule, R.Kessler, J.Guy, J.Mosher, D.Hardin, R.Biswas et al., As-tron.Astrophys. 568 (2014) A22.[3] M.White, D.Scott,J.Silk, Ann.Rev.Astron.Astrophys. 32(194) 319-370.[4] S.Alam, M.Ata, S.Baily, F.Beutler, D.Bizyaev, J.A.Blazek et al.,arXiv:1607.03155(2016).[5] R.C. Tolman, Phys. Rev.55