A Relativistic Stellar Model for Pulsar PSR B0943+10 with Vaidya-Tikekar Ansatz
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.
Abstract
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.
Keywords:
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.
Acknowledgments
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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