AA Relativistic Stellar Model for Pulsar PSR
B0943+10 with Vaidya-Tikekar Ansatz
Jitendra Kumar ∗ and Puja Bharti † Department of Mathematics, Central University of Jharkhand, Cheri-Manatu, Ranchi-835222India.
This paper presents an exact solution of the Einstein-Maxwell ﬁeld equations in a staticand spherically symmetric Schwarzschild canonical coordinate system in the presence ofcharged perfect ﬂuid. We have employed the Vaidya–Tikekar ansatz for the metric po-tential. Using graphical analysis and tabular information we have shown that our modelobeys all the physical requirements and stability conditions required for a realistic stellarmodel. This theoretical model approximates observations of pulsar PSR B0943+10 to anexcellent degree of accuracy.
General relativity, Exact solution, Perfect ﬂuid sphere, Relativistic stars,Equation of state.
In the general theory of relativity, the Einstein’s ﬁeld equations establish a relation be-tween the geometry of space-time and the distribution of matter within it. It has beena compelling ﬁeld for both mathematicians and physicists to discover new analytical so-lutions for these equations. The very ﬁrst exact solution of Einstein’s ﬁeld equationsrepresenting a bounded matter distribution was provided by Schwarzschild (1916). This ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] F e b ncouraged the researchers to search for physically applicable solutions for the same. Itis still an interesting topic to researchers these days. The exact solution plays a crucialrole within the development of varied areas of a gravitational ﬁeld like black hole solution,gravitational collapse, solar system test, stellar modelling of pulsars like PSR B0943+10and so on.Pulsar PSR B0943+10 (discovered at Pushchino in December 1968), is 2000 light-years from Earth. The pulsar is about 5 million years old, which is relatively older fora pulsar. PSR B0943+10 is one of the brightest pulsars at low frequency (2016b). Ithas an interesting characteristic that it emits both radio waves and X-rays. In the radioband, PSR B0943+10 is one of the most studied pulsars showing the mode-switchingphenomenon. In fact, at irregular intervals, every few hours or less, PSR B0943+10switches between a radio bright mode with highly organized pulsations and a quietermode with a rather chaotic temporal structure (1984b). Its 0.1–10 GHz spectrum is verysteep (2000). PSR B0943+10 exhibits a very interesting behavior in the X-ray band aswell. It was the ﬁrst rotation-powered pulsar exhibiting variations in its X-ray emission(2017d), contradicting the common view that rotation powered neutron stars are sourceof constant X-ray emission.Due to the robust nonlinearity of Einstein’s ﬁeld equations and thus the shortage of acomprehensive rule to get all solutions, it becomes diﬃcult to locate any new exact solu-tion. Thousands of exact solution of the ﬁeld equations describing an outsized number ofstellar objects varying between perfect ﬂuids, charged bodies, anisotropic matter distribu-tions, higher-dimensional stars, exotic matter conﬁgurations, etc. are present so far. Butmost of them are physically irrelevant within the relativistic structure of compact stellarobjects. For obtaining exact solutions describing static compact objects, some impositionslike space-time dimensionality, symmetry requirements, an equation of state relating thepressure and energy density of the stellar ﬂuid, the behavior of the pressure anisotropyor isotropy, vanishing of the Weyl stresses are made to make the problem mathematicallymore identiﬁable (2017a).Though there is a good range of stellar solutions exhibiting deviation from sphericity,spherical symmetry is the closest natural assumption to elucidate stellar objects. Thereexists a collection (1998a, 2003a) of static, spherically symmetric solutions which providesuseful guide to the literature. In this regard, the primary model had been proposed byTolman (1939a), which was followed by some generalizations made by Wyman (1949),Leibovitz (1969), and Whitman (1977). Bayin (1978) then used the strategy of quadra-tures and gave new astrophysical solutions for the static ﬂuid spheres. The study ofSharma et al. (2001c) and Ivanov (2002d) shows that the presence of an electrical ﬁeld af-fects the values of surface redshifts, luminosities, and maximum mass of compact objects.Ray and Das (2003c) performed the charged generalization of Bayin’s work (1978). Mak2nd Harko (2004a) and Komathiraj and Maharaj (2007a,b) highlighted the particularincontrovertible fact that the electromagnetic ﬁeld features an important role in describ-ing the gravitational behavior of stars composed of quark matter. Models constructedin this manner are proven to be useful in describing the physical properties of compactrelativistic objects with diﬀerent matter distributions. There are several investigationson the Einstein-Maxwell system of equations for static charged spherically symmetricgravitational ﬁelds for eg. (2012a, 2013a,b).Exact solutions of the ﬁeld equations for various ‘neutral as well as charged static spher-ically symmetric conﬁgurations’ for anisotropic pressure compatible to compact stellarmodeling have been obtained in numerous works (1985, 1997a, 2001a, 2002b,c, 2003b,2006a, 2008, 2012b,c, 2013c, 2014,a,b). In recent times various models of relativistic starshave been found with anisotropic pressures (2017b,c, 2018a, 2019a,b). However, in addi-tion to this, it is necessary to keep in mind the compact stars with isotropic pressure, asat times this may be typically thought to be the equilibrium state of gravitating matter.Physical analysis indicates that isotropic models may even be accustomed to describecompact charged spheres. Some samples of isotropic stars with an electromagnetic ﬁeldare often seen in (2006b, 2007c,d, 2009). Various comprehensive investigations of chargedisotropic spheres are contained among the works (2002d, 2012a, 2013a,b, 2018c,d).There is not enough information regarding the equation of state of matter containedwithin the interior of compact stars. This makes it diﬃcult to apply analytic solutions tothe equation of relativistic stellar structure to acquire its understanding (2001b, 2004b).Oppenheimer-Volkoﬀ technique (1939b) and Tolman method (1939a) are two customarymethodologies which are generally followed to acquire a realistic stellar model. In theﬁrst approach, we start with an explicit equation of state. The integration starts atthe center of the star with a prescribed central pressure and iterated till the surface ofthe star has been reached, i.e., where pressure diminishes to zero. Normally such inputequations of state does not yield closed-form solutions. In the second approach, Einstein’sgravitational ﬁeld equations need to be solved. For a static isotropic perfect ﬂuid case,the ﬁeld equations can be reduced to a set of three coupled ordinary diﬀerential equationsin four unknowns. After getting exact solutions, one can solve the ﬁeld equations byconsidering one of the metric functions or the energy density as priori. Consequently, theequation of state can be computed from the resulting metric. Since non-physical pressure-density conﬁgurations are found more frequently than physical ones, a new solution thatought to be regular, well behaved, and can reasonably model a compact astrophysicalstellar object is always appreciated (2018b). We are going to follow Tolman’s methodologyin this paper and specify one of the gravitational potentials as the Vaidya-Tikekar (1982)potential which has been shown to model superdense stars in several papers.The presence of ﬁve unknown functions and only three basic ﬁeld equations permits one3o specify the metrics and solve for the ﬂuid attributes (1975a). Vaidya and Tikekar (1982)proposed a static spherically symmetric model of a superdense star based on an exactsolution of Einstein’s equations by prescribing an ansatz (Vaidya–Tikekar ansatz) for themetric functions. It was for the geometry of ‘t = constant’ hypersurface and the physical3-space of the star was spheroidal. Using the Vaidya–Tikekar ansatz several studies havebeen performed. Gupta and Kumar (2005) observed a particular form of electric ﬁeldintensity, having positive gradient. He used Vaidya–Tikekar ansatz to generate exactsolutions of the ﬁeld equation in charge analog. Later this form of electric ﬁeld intensitywas used by Sharma et. al. (2001c). Komathiraj and Maharaj (2007b) additionallyaccepted a similar articulation to show another kind of Vaidya–Tikekar type star. Bijalwanand Gupta (2011, 2012d) obtained a charged perfect ﬂuid model of Vaidya–Tikekar typestars with more generalized electric intensity. Additionally, some of the other researches onVaidya–Tikekar stars can be found in literature (1996, 1997b, 1998b). Recently Kumaret. al. (2018d) used the Vaidya-Tikekar metric potential to explore a class of chargedcompact objects ﬁlled with self-gravitating, charged, isotropic ﬂuids.Above mentioned literature survey motivate us to perform this research work on thesubsequent line of action. In this research paper, our objective is to get an exact solution ofthe ﬁeld equations for a static spherically symmetric ﬂuid sphere. The matter distributionis charged with isotropic pressures. This paper is organized as follows: Following a quickintroduction in section 1, we have introduced the Einstein-Maxwell ﬁeld equations forthe static charged ﬂuid spheres in general relativity in section 2. In section 3, we haveproposed a new model to solve the system of equations analytically. For this, we haveused Vaidya–Tikekar (1982) ansatz for the metric potential and acquired the expressionfor density and pressure. In section 4, we have discussed the requirements for a well-behaved solution. Boundary conditions are discussed in section 6. In section 7, we aregoing to investigate the applicability of the model and obtained results will be comparedwith observational data. We have done the stability analysis of the obtained model insection 8. Finally, section 9 is dedicated to concluding remarks. Let us consider the metric in Schwarzschild coordinates ( x i ) = ( t, r, θ, φ ) to describe theinterior of static and spherically symmetric stellar system ds = e ν ( r ) dt − e λ ( r ) dr − r ( dθ + sin θdφ ) , (1)4here the metric potentials ν ( r ) and λ ( r ) are arbitrary functions of radial coordinate r .These potentials uniquely determine the surface redshift and gravitational mass functionrespectively. The signature of the space time taken here is (+,-,-,-). The Einstein-Maxwellﬁeld equations for obtaining the hydrostatic stellar structure of the charged sphere canbe written as − κ ( T ij + E ij ) = R ij − Rδ ij = G ij , (2)where κ = 8 πGc , G here stands for gravitational constant and c is the speed of light, R ij and R represent Ricci Tensor and Ricci Scalar respectively. Since we are assuming thatmatter within the star is a perfect ﬂuid, the corresponding energy-momentum tensor T ij and electromagnetic ﬁeld tensor E ij will be T ij = ( ρ + p ) v i v j − pδ ij (3)and E ij = 14 π ( − F im F jm + 14 F mn F mn ) , (4)where, ρ ( r ) is the energy density, p ( r ) is the isotropic pressure, F ij is anti-symmetricelectromagnetic ﬁeld strength tensor deﬁned as F ij = ∂A j ∂x i − ∂A i ∂x j which satisﬁes Maxwellsequations, F ik,j + F kj,i + F ji,k = 0 and [ √− gF ik ] ,k = 4 πJ i √− g Here A j = ( φ ( r ) , , ,
0) is the potential and J i is the electromagnetic current vectordeﬁned as J i = σ √ g dx i dx = σν i , where σ = e ( ν/ J represents the charge density, g isthe determinant of the metric g ij which is deﬁned by g = − e ν + λ r sin θ and J is theonly non-vanishing component of the electromagnetic current J i for the static sphericallysymmetric stellar system. Since the ﬁeld is static, we have ν = (0 , , , √ g ).Also, the total charge within a sphere of radius r is given by q ( r ) = r E ( r ) = 4 π (cid:90) r J r e ( ν + λ ) / dr, (5)where, E ( r ) is the intensity of the electric ﬁeld.Thus, for the spherically symmetric metric of Eq. (1) the Einstein ﬁeld equation (2)provides the following relationship: λ (cid:48) r e − λ + 1 − e − λ r = c κρ + q r , (6)5 (cid:48) r e − λ − − e − λ r = κp − q r , (7) (cid:16) ν (cid:48)(cid:48) − λ (cid:48) ν (cid:48) ν (cid:48) ν (cid:48) − λ (cid:48) r (cid:17) e − λ = κp + q r (8)Here prime denotes diﬀerentiation with respect to r . By taking radial and transversestresses equal at each interior point of the stellar conﬁguration and then using Eqs. (7)and (8), we can obtain the condition of pressure isotropy as (cid:16) ν (cid:48)(cid:48) − λ (cid:48) ν (cid:48) ν (cid:48) − ν (cid:48) + λ (cid:48) r − r (cid:17) e − λ + 1 r = 2 q r (9)We can get the deﬁnition of charged density σ by substituting this value in eq. (5).We are using geometrized units, and thus we have taken G = c = 1 throughout thediscussion. In seeking solutions to Einstein’s ﬁeld equations for isotropic ﬂuid matter, we have ﬁveunknown functions of r , namely, q ( r ), ρ ( r ), p ( r ), ν ( r ) and λ ( r ). Since we have 3 equations(6,7& 9) and 5 unknown functions, one has to specify two variables a priori to solve theseequations analytically. Let’s consider the widely used Vaidya–Tikekar metric potential(1982) e λ = K (1 + Cr ) K + Cr , (10)where C and K are two parameters. Let’s introduce a new variable as e ν = Z ( r ) (11)The substitution, followed by some computations, leads us to an equivalent form ofﬁeld equations, which might be helpful to ﬁnd the exact solutions more eﬃciently. Using(6), (7) and (9), we have: c κρ = C ( K − Cr ) K (1 + Cr ) − q r , (12) κp = K + Cr K (1 + Cr ) 2 Z (cid:48) rZ + C (1 − K ) K (1 + Cr ) + q r , (13)6nd 2 q r = K + Cr K (1 + Cr ) (cid:104) Z (cid:48)(cid:48) Z − Z (cid:48) rZ + Cr ( K − K + Cr )(1 + Cr ) (cid:16) cr − Z (cid:48) Z (cid:17)(cid:105) (14)Our aim is to consider the perfect ﬂuid distribution represented by metric (1) when0 < K < X = (cid:115) K + Cr − K , (15)so as to get a convinient form of the above equations. Substituting the value of X intoeq. (14), we get, d ZdX − X X dZdX − (1 − K ) (cid:104)
11 + X + 2 K (1 + Cr ) q C r (cid:105) Z, (16)Let’s use the transformation Z = (1 + X ) / Y (17)to convert eq. (16) into the normal form d YdX + ψY = 0 , (18)where, ψ = −
11 + X (cid:104) − K + 2 Kq (1 + Cr ) C r + 3 X − X ) (cid:105) (19)In order to solve the diﬀerential equation (18), we have considered ψ = − aX ( a + bX ) , (20)where, a ( (cid:54) = 0) and b are real constants. It will later become evident that, for the starwhich we have considered here, such a choice of ψ gives physically viable electric ﬁeldintensity. Comparision of eqs. (19) and (20) provides the following deﬁnition of electricﬁeld intensity: E = q r = C r K (1 + Cr ) (cid:104)
54 (1 − K )(1 + Cr ) + 2 aX ( a + bX ) 1 + Cr (1 − K ) + K − (cid:105) (21)If we put the value of ψ from eq. (20) to eq. (18), the resulting diﬀerential equation willbe X ( a + bX ) d YdX − aY = 0 , (22)7earranging eq. (22) in a way such that its comparision with diﬀerntial equation P d YdX + P dYdX + P Y = R , leads us to the relationship: P = a + bXX , P = 0 , P = − aX and R = 0. One can easily verify that this diﬀerential equation is exact. Hence, we can writeit in the form P − dP dX + d P dX = 0. Premitive of this equation is, P dψdX + (cid:16) P − dP dX (cid:17) ψ = (cid:82) RdX + A . This implies, dψdX + aX ( a + bX ) ψ = A, (23)Finally, solution of diﬀerential equation (23) is given by the expression Y = a + bXX (cid:104) A ab H ( X ) + B (cid:105) , (24)where, A and B are arbitrary constants and H ( X ) = 12 cos (cid:16) tan − (cid:114) bXa (cid:17) − cos (cid:16) tan − (cid:114) bXa (cid:17) | cos (cid:16) tan − (cid:114) bXa (cid:17) | (25)Together eqs. (17) and (24) yields, Z = (1 + X ) / a + bXX (cid:104) A ab H ( X ) + B (cid:105) (26)To achieve the expression for energy density and pressure, let’s put eqs. (21) and (26)into eqs. (12) and (13), respectively. Hereby, we obtain the following expressions: c κρ = C ( K − Cr ) K (1 + Cr ) − C r K (1 + Cr ) (cid:104)
54 (1 − K )(1 + Cr ) + 2 aX ( a + bX ) 1 + Cr (1 − K ) + K − (cid:105) (27) κp = C ( K + Cr ) K (1 + Cr ) (cid:104) A · A + A · A A · A (cid:105) + C r K (1 + Cr ) (cid:104)
54 (1 − K )(1 + Cr ) + 2 aX ( a + bX ) · Cr (1 − K ) + K − (cid:105) + C (1 − K ) K (1 + Cr )(28)On diﬀerentiating eqs (27) and (28) with respect to r , we get graidiant of density andpressure respectively as, c κ dρdr = C r (cid:104) N − N + N (cid:105) (29) κ dpdr = C r (cid:104) K + Cr K (1 + Cr ) N ( r ) A · A + 2(1 − K ) K (1 + Cr ) (cid:16) A · A + A · A A · A − (cid:17) + N − N (cid:105) (30) See Appendix A for A i ( i = 1 , , ..., N and N j ( j = 1 , , ..., Physical Acceptability Conditions for Well be-haved Solution
For a well-behaved nature of the solution, the pre-requisites are:(i) The solution should be free from physical and geometrical singularities, i.e., valuesof central pressure ( p ) and central density ( ρ ) must be ﬁnite and positive, and e λ and e ν must have a non-zero positive value.(ii) The solution should have positive and monotonically decreasing expressions for en-ergy density and pressure with rise of radius r . Mathematically, ρ ≥ p ≥ dρdr ≤ dpdr ≤
0. At the stellar boundary ( r = R ) the radial pressure p shouldvanish, i.e., p ( R ) = 0.(iii) The casualty condition should be obeyed, i.e. velocity of sound should be lessthan that of light throughout the model. Also, it should be decreasing towards thesurface. Besides this, at the center, (cid:16) dρdr (cid:17) and (cid:16) dpdr (cid:17) must be zero and (cid:16) d ρdr (cid:17) and (cid:16) d pdr (cid:17) must have negative value at the center so that the gradient of density andpressure shall be negative within the radius. The condition pρ < dpdρ should be validthroughout within the sphere.(iv) The Redshift z should be positive, ﬁnite, and monotonically decreasing in naturewith the increase of r .(v) The adiabatic constant γ should increase from its lowest value 43 at center to inﬁnityas we move outwards, for a stable model.(vi) The solution must satisfy the Tolman-Oppenheimer-Volkoﬀ (TOV) equation.(vii) The solution is required to fulﬁll all the energy conditions simultaneously.(viii) The interior metric functions should match smoothly to the exterior Schwarzschildspace-time metric at the boundary. The unique exterior metric for a spherically symmetric charged distribution of matter isthe Reissner-N¨ordstro solution. To explore the boundary conditions, we are going to use9he principle that, the metric coeﬃcients and their ﬁrst derivatives in interior solutionand exterior solution are continuous up to and on the boundary.Consider r = R as the outer boundary of the ﬂuid sphere. The ﬁeld in the exterior region( r ≥ R ) is described by the Reissner-N¨ordstro metric ds = − (cid:16) − Mr + q r (cid:17) dt + (cid:16) − Mr + q r (cid:17) − dr + r (cid:0) dθ + sin θdφ (cid:1) , (31)where, M = m ( R ), total gravitational mass, is actually a constant, and deﬁned by M = ξ ( R ) + ζ ( R ), with ζ ( R ) = κ (cid:90) R ρr dr and ξ ( R ) = κ (cid:90) R rσqe λ/ dr Here, ξ ( R ) is the mass and ζ ( R ) is the mass equivalence to electromagnetic energy ofdistribution (1983).Applicable boundary conditions are :1. The interior metric (1) should join smoothly at the surface of spheres (r = R) tothe exterior metric (31).2. Pressure p ( r ) should vanish at r = R .Arbitrary constants A and B can be obtained using the boundary conditions. The conti-nuity of e ν , e λ and Q at r = R implies that, e ν ( R ) = Z ( R ) = 1 − MR + Q R , e − λ ( R ) = 1 − MR + Q R , Q = q ( R ) , p ( R ) = 0 (32)Using P ( R ) = 0, we can easily obtain BA . Then, along with condition e ν ( R ) = e − λ ( R ) itwill give us A and B ( see Appendix B ). The solutions discussed in this paper can be used to model a relativistic star. In thisSection, we have critically veriﬁed our model by performing mathematical analysis andplotting several graphs. This indicates that the result overcomes all the barriers of phys-ical tests. 10he expression for the energy density in eq. (27) implies that at r = 0, ρ = 3 C ( K − πK (33)As, 0 < K <
1, the central density ρ will be positive iﬀ we take C < C so that the transformation (15) remains physicallyacceptable throughout the conﬁguration. For this, we require | C | ≤ KR . However, thefunction we have chosen for ψ in eq. (20) requires that C (cid:54) = − KR . Thus, in our model,we have to consider | C | < KR . Also, we can say that K and R characterize the geometryof the star.The physical acceptability of the model has been examined by plugging the mass andradius of observed pulsar as input parameters. To validate our model, we have con-sidered the pulsar PSR B0943+10, a low-mass bare quark star of radius r ∼ . km and mass M ∼ . M (cid:12) (2006c). Using these values of mass and radius as an inputparameter, the boundary conditions have been utilized to determine the constants as C = − . ∗ − km − , K = 0 . a = 0 .
001 and b = 0 . A = − . B = 29 . r/R q ( km ) ρ ( km − ) p ( km − ) p/ρ dp/c dρ z γ . . . ∗ − . ∗ − . . . . . . . ∗ − . ∗ − . . . . . . . ∗ − . ∗ − . . . . . . . ∗ − . ∗ − . . . . . . . ∗ − . ∗ − . . . . . . . ∗ − . . . . Inf.
Table 2: Obtained numerical values for charge at surface, central density, surface density,central pressure and mass-radius ratio of compact star PSR “B0943+10”. q s (C) ρ ( g/cm ) ρ s ( g/cm ) p (Pa) M/R . ∗ . ∗ . ∗ . ∗ . G = 6 . ∗ − N/ms , c = 3 ∗ m/s, M (cid:12) = 1 . km to obtainnumerical values of physical quantities and have multiplied charge by 1 . ∗ toconvert it from relativistic unit ( km ) to coulomb. e l r / R e n r / R Figure 1: Variation of e λ (left) and e ν (right ) with respect to fractional radius (r/R) r ( /km2 ) r / R Figure 2: Variation of density with respect to fractional radius (r/R).(i) The proﬁles of the metric potentials e λ and e ν for PSR B0943+10 in Fig. (1) showthat they are free from physical and geometrical singularities, regular at the centre12i.e., the metric potentials are positive and ﬁnite at the centre) and monotonicallyincreasing with the radius inside the star. The behavior of metric functions isconsistent with the requirements for a physically acceptable model.(ii) The graph in Fig. (2) indicates that the energy density is positive with a maximumvalue at the centre and decreasing in nature throughout the star. Also, We can seein Fig. (3) that the pressure is monotonically decreasing towards the surface. Atthe centre, it is ﬁnite and vanishes at the boundary of the star. p (/km2) r / R Figure 3: Variation of pressure with respect to fractional radius (r/R). p/ r r / R Figure 4: Variation of ratio of pressure to that of density with increase in radius (r/R).13 .2 Causality condition
For a physically acceptable isotropic model, the square in sound speed v s must be lessthan 1 in the star’s interior, i.e., 0 ≤ v s = dpdρ ≤ r . vs2 r / R Figure 5: Variation of square of velocity of sound with respect to fractional radius (r/R). R3 k p' r / R R3 kr ' r / R Figure 6: Variation of pressure gradient (left) and density gradient (right) with respectto fractional radius (r/R).It is very clear from ﬁg (4) and ﬁg (5) that the ratio pρ is less than dpdρ throughout the14tellar model. One can also verify this through Table 1. As we can see in ﬁg (6), gradientof pressure and density are zero at centre and have negative values at every other pointin the region. In the study of stellar conﬁgurations describing charged isotropic matter distributions, itis necessary to check whether the energy-momentum tensor is well behaved, i.e. positivedeﬁned everywhere within the star. For this, the fulﬁllment of below energy conditionsare required (2016a):1. Dominant energy condition (DEC): ρ − p ≥
02. Null energy condition (NEC): ρ + q πr ≥
03. Weak energy condition (WEC): ρ − p + q πr ≥
04. Strong energy condition (SEC): ρ − p + q πr ≥ Energy Conditions r / R D E C N E C W E C S E C
Figure 7: Energy Conditions within stellar conﬁguration of PSR B0943+10.15 .4 Redshift
The gravitational redshift z within the static line element is given by z = | e ν ( r ) | − / − | Z ( r ) | − z s = | e ν ( R ) | − / − (cid:16) − MR + Q R (cid:17) − / − Redshift r / R
Figure 8: Variation of redshift with respect to fractional radius (r/R).We have central redshift z = 0 . z s = 0 . z s <
2, which is required for spherically symmetric isotropic ﬂuid spheres as suggested in(1959, 1984a, 2007e).
Now, we will examine the stability of our isotropic, charged ﬂuid conﬁguration in theEinstein-Maxwell space-time in the following subsections.16 .1 Electric charge in the sphere
In Fig. (9), we observe that the electric ﬁeld given by eq. (21) is positive and an increasingfunction with increasing radius. The charge starts from zero at the centre and acquiresthe maximum value at the boundary. q (km) r / R
Figure 9: Behavior of charge within the stellar conﬁguration.Any ﬂuid sphere with net charge, contains ﬂuid elements with unbounded propercharged density located at the ﬂuid-vaccum interface. This net charge can be huge (10 C)(2010). Ray et al. (2003c) have analyzed the impact of charge in compact stars by con-sidering the limit of the very extreme measure of the charges. They have demonstratedthat the global balance of the forces allows a huge charge(10 C) to be available in acompact star.In this model, the charge on the boundary is 3 . ∗ C. Thus, we can say that,in this model the net charge is eﬀective to balance the mechanism of the force.
A star remains in static equilibrium under the forces, namely, gravitational force ( F g ),hydrostatic force ( F h ) and electric force ( F e ). This condition is formulated mathematicallyas the TOV equation by Tolman–Oppenheimer–Volkoﬀ (1939a,b). In the presence of acharge, the same takes the following form (1939c) − M G ( ρ + p ) r e ( λ − ν ) / − dpdr + σ qr e λ/ = 0 , (36)17here M G ( r ) is the gravitational mass of the star within radius r and is deﬁned by M G ( r ) = 12 r ν (cid:48) e ( ν − λ ) / (37)Substituting the value of M G ( r ) in eq. (36), we obtain, − ν (cid:48) ρ + p ) − dpdr + σ qr e λ/ = 0 (38)which is equivalent to F g + F h + F e = 0 , (39)where , F g = − ν (cid:48) ρ + p ) = − Z (cid:48) Z ( ρ + p ) = − C r π (cid:104) A · A + A · A A · A (cid:105)(cid:104) K − K (1 + Cr ) + (40) K + Cr K (1 + Cr ) A · A + A · A A · A (cid:105) ,F h = − dpdr = − C r π (cid:104) K + Cr K (1 + Cr ) N ( r ) A · A + 2(1 − K ) K (1 + Cr ) (cid:16) A · A + A · A A · A − (cid:17) + N − N (cid:105) , (41)
0 R F g R F h R F e r / R Figure 10: Variations of gravitational force ( F g ), hydrostatic force ( F h ) and electric force( F e ) acting on the system with respect to fractional radius ( r/R ). F e = σ qr e λ/ = 18 πr dq dr = C r π (cid:104) Cr K (1 + Cr ) (cid:110)
54 (1 − K )(1 + Cr ) + 2 aX ( a + bX ) 1 + Cr (1 − K ) + K − (cid:111) − N (cid:105) . (42)18he graph in ﬁg. (10) shows that F h and F e are positive and are nulliﬁed by F g , whichis negative to keep the system in static equilibrium. The adiabatic index, deﬁned as γ = (cid:16) c ρ + pp (cid:17)(cid:16) dpc dρ (cid:17) , (43)is related to the stability of an isotropic stellar conﬁguration.We have demonstrated the behavior of adiabatic index γ in Fig. (11), which showsthe desirable features. The value of γ at the center is 4047 . γ increases drastically. Adiabatic Index r / R
Figure 11: Nature of adiabatic index with increase in radius.If we consider a Newtonian sphere to be in stable equilibrium, γ must have valuesstrictly greater than 43 throughout the region, and γ = 43 is the condition for a neutralequilibrium (1975b). This condition changes for a relativistic isotropic sphere due toregenerative eﬀect of pressure, which makes the sphere more unstable.19 .4 Harrison-Zeldovich-Novikov Stability Criterion According to Harrison-Zeldovich-Novikov criterion (1965, 1971), for a compact star to bestable, it’s mass should increase with rise in central density. Mathematically, dMdρ > M = 4 πR ρ M · (cid:104) K − πKR ρ · M M (cid:105) (44)where, M = 3( K −
1) + 8 πKρ R , M = (cid:104) −
154 (1 − K ) M − aM − K ) X ( a + bX ) + K − (cid:105) , M(km) r ( / k m ) Figure 12: Behavior of Mass M vs. central density ρ for PSR “B0943+10”.We observed in ﬁg 12 that mass of the star is positive and increasing with increasein central density. Thus, we can conclude that the presented model satisﬁes Harrison-Zeldovich-Novikov criterion. Hence, the model is stable. In this paper, a static spherically symmetric isotropic ﬂuid model has been developedby using well-known Vaidya-Tikekar metric potential. The presented solution satisﬁes20ll the physical criteria of a relativistic compact object. All the physical quantities areregular and well-behaved throughout the stellar interior. Energy density and pressure aredecreasing functions as we move towards the surface of the star from centre. This modelsatisﬁes causality conditions, stability conditions and energy conditions. A thoroughphysical analysis has been accomplished for the star PSR B0943+10.If we remove the electric intensity (i.e., when q ( r ) = 0) then we will have the expressionfor density ρ A and its gradient as, c κρ A = C ( K − Cr ) K (1 + Cr ) , and c κ dρ A dr = C r (cid:104) − K )(5 + Cr ) K (1 + Cr ) (cid:105) , respectively. It can be veriﬁed that for 0 < K <
1, density( ρ A ) is positive throughoutthe structure and dρ A dr >
0, i.e., ρ A is increasing towards the boundary. This shows thatfor the parameter K , 0 < K < K / ∈ [0 ,
1] will be considered. Themodel could even be useful to elucidate more compact objects apart from the star PSRB0943+10.
The Authors would like to express their sincere gratitude towards Science and EngineeringResearch Board (SERB), DST, New Delhi for providing the ﬁnancial support neededto complete this work. They are also very grateful to the Department of Mathematics,Central University of Jharkhand, Ranchi, India for the necessary support where the paperhas been written and ﬁnalized. 21 ppendix A: Notations used in paper
To express equations in a simpliﬁed manner, notations used are as follows A = 2 (cid:112) (1 − K )( K + Cr ) (cid:16) a + bX X ) − aX (cid:17) A = ab H ( X ) + BAA = 2 (cid:112) (1 − K )( K + Cr ) (cid:16) a + bXX (cid:17) A = a b ( a + bX ) (cid:34) sec (cid:16) tan − (cid:114) bXa (cid:17) + cos (cid:16) tan − (cid:114) bXa (cid:17) − (cid:35) A = (cid:16) a + bXX (cid:17) N ( r ) = − ( A · A + A · A )( N · A + N · A ) A · A + N · A + N · A + N · A + N · A N ( r ) = − A ( K + Cr + 2(1 − K )( K + Cr ) (cid:104) b X ) − X ( a + bX )(1 + X ) + 2 aX (cid:105) N ( r ) = A (cid:112) (1 − K )( K + Cr ) N ( r ) = − K + Cr (cid:104) A + 2 aK + Cr (cid:105) N ( r ) = 1 (cid:112) (1 − K )( K + Cr ) (cid:104) − b ( a + bX ) A + a b ( a + bX ) (cid:110) sec (cid:16) tan − (cid:114) bXa (cid:17) − cos (cid:16) tan − (cid:114) bXa (cid:17)(cid:111)(cid:105) N ( r ) = − (cid:112) (1 − K )( K + Cr ) aX N ( r ) = 2(1 − K )(5 + Cr ) K (1 + Cr ) N ( r ) = 1 − Cr K (1 + Cr ) (cid:104)
54 1 − K Cr + 1 + Cr − K aX ( a + bX ) + K − (cid:105) N ( r ) = Cr K (1 − K )(1 + Cr ) (cid:104) (cid:16) − K Cr (cid:17) − aX ( a + bX ) + (cid:16) Cr − K (cid:17) a (2 a + 3 bX ) X ( a + bX ) (cid:105) ppendix B : Finding constants A & B This appendix is devoted in ﬁnding values of arbitrary constants A and B , used in eq.(26), using boundary conditions (32).First we are going to determine the value of BA . Using the fact that P ( R ) = 0, thefollowing relation can be obtained from eq. (28) − A · A · ( K + CR ) A · ( K + CR ) + (1 − K ) · ( J + 1) · A = A = ab H ( X ) + BA (45)Thus, we have BA = − (cid:104) A · A · X A · X + ( J + 1) · A + ab H ( X ) (cid:105) (46)where, J = CR − K )(1 + CR ) (cid:104)
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