An isotropic compact stellar model in curvature coordinate system consistent with observational data
NNoname manuscript No. (will be inserted by the editor)
An isotropic compact stellar model in curvature coordinatesystem consistent with observational data
Jitendra Kumar a,1 , Puja Bharti b,1 Central University of Jharkhhand, Cheri-Manatu, Ranchi, India.Received: date / Accepted: date
Abstract
This paper investigates a spherically symmetric compact relativistic body with isotropicpressure profiles within the framework of general relativity. In order to solve the Einstein’sfield equations, we have considered the Vaidya–Tikekar type metric potential, which de- pends upon parameter K . We have presented a perfect fluid model, considering K / ∈ [ , ] ,which represent compact stars like SMC X-1, Her X-1, 4U 1538-52, SAX J1808.4-3658,LMC X-4, EXO 1785-248 and 4U1820-30, to an excellent degree of accuracy. We haveinvestigated the physical features such as the energy conditions, velocity of sound, surfaceredshift, adiabatic index of the model in detail and shown that our model obeys all thephysical requirements for a realistic stellar model. Using the Tolman-Oppenheimer-Volkoffequations, we have explored the hydrostatic equilibrium and the stability of the compact ob-jects. This model also fulfils the Harrison-Zeldovich-Novikov stability criterion. The resultsobtained in this paper can be used in analyzing other isotropic compact objects. Key words:
Compact stars; General Relativity; Pressure isotropy; Field equations; Perfectfluid, Exact solutions. a e-mail: [email protected] b e-mail: [email protected] a r X i v : . [ a s t r o - ph . GA ] F e b General relativity is a great medium for understanding and exploring the gravitational sys-tem. Ultra-compact objects like pulsars, neutron stars and black holes have helped scien-tists to look for exact solutions of the Einstein field equations by modelling physical objectsbased on observational data, rather than by using mere mathematical excursions. Several the-oretical investigations, laboratory experiments and observational tests have been performedduring the previous couple of decades. But, it has been difficult to obtain a reliable descrip-tion of dense compact object. The observational data forming compact stars might be ableto provide information about the largest uncertainties in nuclear physics that rely heavily onthe equations of state (EoS) at nuclear and supranuclear densities. We might achieve this byestimating their mass and radius which depends on EoS.The first solution of the Einstein field equations describing a self-gravitating, boundedobject was obtained by Schwarzschild [1] about a century ago. The Schwarzschild interiorsolution describes a uniform density sphere. It was the first approximation in describingthe gravitational field of a static, spherically symmetric object. Although this model is notrealistic as the propagation speed within the object exceeds the speed of light, the efforts putby Schwarzschild motivated the researchers to search for exact solutions of the Einstein fieldequations describing self-gravitating objects. As a result, we have a large number of exactsolutions of the field equations describing an outsized number of stellar objects. The analysis of available exact solutions indicates that many of them are unable to describe physicallyrealizable stellar structures [2]. While few of these solutions are only valid in some regionof the object, some other solutions display unphysical behaviour in the density and pressureprofiles.A large number of currently existing exact solutions were obtained through various as-sumptions on the space-time geometry and/or matter content inside the compact object [3].Spherical symmetry is the most common assumption, while modelling static stars. But, thereis more freedom in choosing the matter content of the stellar fluid. History tells us that re-searchers have already worked with perfect fluids, charged interiors, pressure anisotropy,bulk viscosity and scalar fields. Developments in cosmology, inspired the researchers tomodel stellar structures which includes dark energy, dark matter and phantom energy [4, 5].There is no doubt that an astrophysical structure is not composed of a perfect fluid.However, we may consider relativistic static perfect fluid spheres as first approximation tocompact star models. The perfect fluid model necessarily requires that the pressure inside astar should be isotropic, i.e., it should have equal radial ( p r ) and tangential ( p t ) pressures.Recent developments in cosmological survey have made us understand the origin anddistribution of matter and evolution of compact objects in the Universe. We can measuresome of their properties like mass, rotation frequency and emission of radiation. Whereas,measurement of parameters which determines the nature of compact stars is still a greatchallenge. Properties such as internal composition, mass and radius, which are not directlylinked to observations, requires theoretical models. These theoretical mass and radius aredetermined by solving the hydrostatic equilibrium equation which convey the equilibriumbetween gravitational force and pressure. Limited knowledge of nuclear EoS leads to unpre-dictability of Mass-radius relation. This limits the mass of compact stars. As per Buchdahl[9], for a regular fluid sphere with a non increasing mass density, the ratio of its gravita-tional mass M to that of coordinate radius R satisfies MR < ∼
49 . This constraint arises fromthe condition that, to prevent gravitational collapse, isotropic pressure does not become in-finity at the center of the sphere. In general relativity, the equilibrium of a spherical object is described by the Tolman–Oppenheimer–Volkoff (TOV) equations, and the equation of stateis required for its completeness.To determine the structure of a compact star, the most common route is to specify anequation of state and then solve the Einstein field equations. Traditionally, this approach hasbeen proved beneficial while using the law of energy conservation in the form of the TOVequation or the equation of hydrodynamical equilibrium.The equation of state of a compact star is not very clear yet. If one starts with EoS, shegenerally lands into numerical methods leading to graphical results which lacks in the anal-ysis of local properties of the matter close to the centre of such relativistic stars. Therefore,most researchers prefer to obtain exact solutions of the concerned Einstein’s field equationsusing ad-hoc methods such as assuming one of the metric potential. The remaining met-ric potential is obtained using isotropic conditions for perfect fluid. After this, the physicalquantities including pressure, energy density, velocity of sound and adiabatic indices areexamined for the reality as well as stability conditions inside the fluid sphere. Examples ofsome remarkable perfect fluid solutions by assuming metric potential g can be found in[1, 6–17].To find the exact solution of the Einstein– Maxwell field equations, Komathiraj andMaharaj [18] have used Vaidya and Tikekar ansatz [14] for metric potential with a specifiedform of electric field intensity. Bijalwan and Gupta [19, 20] have obtained a charged perfectfluid model with generalized electric intensity for all K / ∈ ( , ) . By extending this work, Kumar and Gupta [25, 26] obtained another solution for K ∈ ( , ) . Using this approach,a large number of solutions have been obtained in [21–24]. Recently, Kumar et. al. [29]has obtained perfect fluid charged analogues models with generalized electric intensity for K ∈ ( , ) .In this paper, we are going to use a physically viable Vaidya and Tikekar metric potential[14] to obtain a closed-form solution of the Einstein field equations for a spherically sym-metric isotropic matter distribution. We will use this solution to develop feasible models forcompact stars with some standard observed mass and radius as proposed in [30, 31]. To findout the model parameters, we will utilize the boundary conditions, which says that interiorspacetime metric matches the exterior Schwarzschild metric at surface and radial pressureis zero across the boundary. Due to the complexity of the solution, we will use graphicalapproach to verify if the matter variables of the model satisfy criteria for realistic star.This paper has been organized as mentioned below:In Sect. 2, the Einstein field equations for the isotropic system of the compact object has beenpresented. In Sect. 3, by assuming the Vaidya-Tikekar metric potential, the relevant fieldequations has been solved to develop a new model. In Sect. 4, an analytical and graphicalrepresentations has been performed to check the physical acceptability and stability of themodel. For this we have used recent measurements of mass and radius of stars SMC X-1, Her X-1, 4U 1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 and 4U1820-30.Finally, Sect. 5 is devoted to conclusion. Let us consider the line element to describe the static and spherically symmetric stellar sys-tem in curvature coordinates ( x i ) = ( t , r , θ , φ ) ds = e ν ( r ) dt − e λ ( r ) dr − r ( d θ + sin θ d φ ) , (1) where the metric potentials ν ( r ) and λ ( r ) are arbitrary functions of radial coordinate r .These potentials plays a key role in determining the surface redshift and gravitational massfunction respectively.The Einstein-Maxwell field equations for obtaining the hydrostatic stellar structure of thecharged sphere can be written as − κ ( T ij + E ij ) = R ij − R δ ij = G ij , (2)where κ = π Gc , G here stands for gravitational constant and c is the speed of light, R ij and R represent Ricci Tensor and Ricci Scalar respectively. Throughout the discussion wewill take G = c =
1, as geometrized units. Since we are assuming that matter within the staris a perfect fluid, the corresponding energy-momentum tensor T ij and electromagnetic fieldtensor E ij will be T ij = ( ρ + p ) v i v j − p δ ij (3)and E ij = π ( − F im F jm + F mn F mn ) , (4)where, ρ ( r ) is the energy density, p ( r ) is the isotropic pressure, F i j is anti-symmetric elec-tromagnetic field strength tensor defined as F i j = ∂ A j ∂ x i − ∂ A i ∂ x j which satisfies Maxwells equa- tions, F ik , j + F k j , i + F ji , k = [ √− gF ik ] , k = π J i √− g Here A j = ( φ ( r ) , , , ) is the potential and J i is the electromagnetic current vector de-fined as J i = σ √ g dx i dx = σ ν i , where σ = e ( ν / ) J represents the charge density, g is thedeterminant of the metric g i j which is defined by g = − e ν + λ r sin θ and J is the only non-vanishing component of the electromagnetic current J i for the static spherically symmetricstellar system. Since the field is static, we have ν = ( , , , √ g ) .Also, the total charge within a sphere of radius r is given by q ( r ) = r E ( r ) = 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-Maxwell field equation(2) provides the following relationship [27]: λ (cid:48) r e − λ + − e − λ r = c κρ + q r , (6) ν (cid:48) r e − λ − − e − λ r = κ p − 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 = κ p + q r (8)Here prime denotes differentiation with respect to r . Using Eqs. (7) and (8), we can obtain (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 + r = q r (9) We can get the definition of charged density σ by substituting this value in eq. (5).The mass contained within a radius r of the sphere is defined as, m ( r ) = κ (cid:90) R ρ r dr (10)Consider r = R as the outer boundary of the fluid sphere. The unique exterior metric for aspherically symmetric charged distribution of matter is 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) d θ + sin θ d φ (cid:1) (11)where, M = m ( R ) , is total gravitational mass. We are going to uncover the solutions to Einstein’s field equations for isotropic fluid matter.To achieve this, we have to solve 3 equations (6,7& 9) for 5 unknown functions. Let’s spec-ify two variables a priori to solve these equations analytically.
Let’s consider the metric ansatz, given by Vaidya and Tikekar [14] e λ = K ( + Cr ) K + Cr , (12)and a new variable as e ν = Z ( r ) (13)where C and K are some constant parameters. This choice of metric potential provides asingularity free solution at r = e λ ( ) =
1. Vaidya and Tikekar [14] had consideredthis metric potential to study spheroidal spacetimes governing the behavior of superdensestars. Several works utilizing this form of metric potential can be found in literature [28–30].In order to get the exact solutions more efficiently, we will use the above substitutions,so that we can transform the field equations to an equivalent form as, c κρ = C ( K − )( + Cr ) K ( + Cr ) − q r (14) κ p = K + Cr K ( + Cr ) Z (cid:48) rZ + C ( − K ) K ( + Cr ) + q r (15) and d Zdr − (cid:104) K + KCr + C r r ( + Cr )( K + Cr ) (cid:105) dZdr + (cid:104) C r ( K − )( K + Cr )( + Cr ) − Kq ( + Cr ) r ( K + Cr ) (cid:105) Z = K / ∈ [ , ] , i.e., for K < K > To get a convinient form of the above equations let’s introduce the transformation X = (cid:115) K + Cr K − < C < | K | R is a parameter, which characterizes the geometry of star.Substituting the value of X into eq. (16), we get, d ZdX − XX − dZdX + ( K − ) (cid:104) X − − K ( + Cr ) q C r (cid:105) Z (18)It is obvious from eq. (17) that when K is negetive X is less than 1 and when K >
1, we get X > Z = ( − X ) / Y when K < Z = ( X − ) / Y when K > d Y dX + φ Y = , (20) where, φ = − X (cid:104) − K + Kq ( + Cr ) C r + X + ( X − ) (cid:105) (21)It is difficult to solve the second order differential equation (20) using standard techniques.In order to solve this differential equation, let’s take φ = − a X ( a + a X ) (22)where, a , a ∈ R such that a is non-zero.We have made this choice for φ , as it will later become evident that it simplifies the analysis.For the stars which we have considered here, such a choice gives physically viable electricfield intensity.Putting this value of φ from eq. (22) to eq. (20), the resulting differential equation becomes X ( a + a X ) d YdX − a Y = Y = a + a XX (cid:104) A a a S ( X ) + A (cid:105) (24)where, A and A are arbitrary constants and S ( X ) =
12 cos (cid:16) tan − (cid:114) a Xa (cid:17) − cos (cid:16) tan − (cid:114) a Xa (cid:17) + | cos (cid:16) tan − (cid:114) a Xa (cid:17) | (25) Using eqs. (19) and (24) we get the value of Z as, Z = A ( − X ) / a + a XX (cid:104) a a S ( X ) + A A (cid:105) , when K < Z = A ( X − ) / a + a XX (cid:104) a a S ( X ) + A A (cid:105) , when K > Z = e ν / is finite, freefrom singularity at centre with ν (cid:48) ( ) = E = q r = C r K ( + Cr ) (cid:104)
54 1 ( − X ) − a X ( a + a X ) ( − X ) + K − (cid:105) (27)Putting eqs. (27) and (26) into eqs. (14) and (15) respectively, we obtain the expressions forenergy density and pressure as: c κρ = C ( K − )( + Cr ) K ( + Cr ) − C r K ( + Cr ) (cid:104) ( − X ) − a X ( a + a X ) ( − X ) + K − (cid:105) (28) κ p = CX K ( X − ) (cid:104) P · P + P · P P · P (cid:105) − CK ( X − ) + C r K ( + Cr ) (cid:104) ( − X ) − a X ( a + a X ) ( − X ) + K − (cid:105) (29)From eqs (28) and (29), graidiant of density and pressure can be obtained as, c κ d ρ dr = C r (cid:104) D − D − D (cid:105) (30) κ d pdr = C r (cid:104) X K ( X − ) DP · P + K ( − K )( X − ) (cid:16) P · P + P · P P · P − (cid:17) + D + D (cid:105) (31)respectively. See Appendix A for values of P i ( ≤ i ≤ ) , D and D j ( ≤ j ≤ ) . In this section, we are going to perform some analytical calculations to ensure that this modelis obeying essential physics for a stellar structure throughout the interior and outer surface.We will do the stability analysis of the model by studying general physical properties andplotting several figures for some of the compact star candidates. The solutions found in thispaper might be useful in study of relativistic compact stellar objects.
Boundary Conditions
To explore the boundary conditions, we are going to use the fact that all astrophysical objectsare immersed in vacuum or almost vacuum space-time. Also, the interior metric (1) joinssmoothly at the surface of spheres ( r = R ) to the exterior metric (11). In order to matchsmoothly on the boundary surface r = R , we will impose the boundary conditions which areequivalent to the following two conditions: e ν ( R ) = Z ( R ) = − MR + Q R , & e − λ ( R ) = − MR + Q R , (32)and P ( R ) = Q = q ( R ) . Using these boundary conditions (32,33), we can easily obtain the con-stants A and A (see Appendix B).For a given radius R , we can determine the total mass M of the star and vice-versa. Keep-ing in mind the constraints on the mass-radius ratio (cid:16) MR ≤ (cid:17) [9, 32], we have demon-strated that for some particular values of the parameters, we can generate specific mass and radius of some well known pulsars. In this process we have used true values of c and G atappropriate places. Some of such possibilities are tabulated in Table 1. Table 1
The approximate values of the masses M , radii R , and the constants A , A , C and K for the compactstars Compact Star a a C ( / km ) K A A M / M (cid:12) R ( km ) M / R SMC X-1 [30] 1.2033 29 0.0003274 -0.069 -82.834082 0.028222 1.04 8.301 0.1847Her X-1 [30] 1.1770 29.2 0.000304832 -0.072 -86.475757 0.029931 0.85 8.1 0.154754U 1538-52 [30] 1.2094 29.5 0.000323238 -0.06851 -87.211173 0.028868 7.866 0.87 0.16314SAX J1808.4-3658 [30] 1.20827 29.58 0.000316364 -0.06704 -87.656350 0.028542 0.9 7.951 0.1669LMC X-4 [30] 1.21118 29 0.000307745 -0.064 -80.025483 0.026362 1.29 8.831 0.2154EXO 1785-248 [30] 1.22416 29.01 0.000241 -0.064 -79.487376 0.026218 1.3 10 0.21664U1820-30 [31] 0.07 4.1 3.299803448 2.246 -0.000067 0.040332 2.25 10.023 0.3311
Table 2
Numerical values of surface charge ( q s ), central density ( ρ ), surface density ( ρ s ), central pressure( p ) and surface redshift ( z s ) of compact star candidates.Compact star q s ( C ) ρ ( g / cc ) ρ s ( g / cc ) p ( Pa ) z s SMC X-1 1 . × . × . × . × . . × . × . × . × . . × . × . × . × . . × . × . × . × . . × . × . × . × . . × . × . × . × . . × . × . × . × . e l r / R S M C X - 1 H e r X - 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 01 . 01 . 21 . 41 . 61 . 82 . 02 . 22 . 4 e l r / R 4 U 1 8 2 0 - 3 0 Fig. 1
Behavior of e λ within the stellar configuration of star candidates SMC X-1, Her X-1, 4U 1538-52,SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > e n r / R S M C X - 1 H e r X - 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8 e n r / R 4 U 1 8 2 0 - 3 0 Fig. 2
Behavior of e ν within the stellar configuration of star candidates SMC X-1, Her X-1, 4U 1538-52,SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > Regularity and Reality Conditions
It is clear from fig (1) and (2) that the obtained metric potentials e λ and e ν are free fromphysical and geometrical singularities. Additionally, they are finite and monotonically in-creasing throughout the stellar interior. Thus, the behavior of metric functions is consistentwith the requirements.For physical feasibility of the model it is also required that• the energy density is positive definite and its gradient is negative everywhere within theradius.• for an isotropic fluid distribution pressure is positive definite and the pressure gradient isnegative within the stellar interior.Graphs in Fig. (3) and (4) indicate that the energy density is positive with a maximum valueat the centre and the pressure is finite and vanishes at the boundaries for each considered starcandidates. Also, both pressure as well as density are monotonically decreasing in nature to-wards the surface of star. We have taken the same values of the constants as mentioned inTable 1. Density(/km2) r / R
S M C X - 1 H e r X - 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 L M C X - 4 E X 0 1 7 5 8 - 2 4 8
Density (\km2) r / R 4 U 1 8 2 0 - 3 0
Fig. 3
Variation of density with respect to fractional radius (r/R) for star candidates SMC X-1, Her X-1, 4U1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > pressure (\km2) r / R S M C X - 1 H e r X - 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4
E X O 1 7 8 5 - 2 4 8
Pressure (\km2) r / R 4 U 1 8 2 0 - 3 0
Fig. 4
Variation of pressure with respect to fractional radius (r/R) for star candidates SMC X-1, Her X-1, 4U1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > Electric charge
The fig (5), clearly states that the electric field given by eq. (27) is positive and increasingtowards the surface for each star candidate. Along with this, the charge at centre is zero andattains its maximum value at the boundary.Ray et al. [40] have demonstrated that the global balance of the forces allows a hugecharge(10 C) to be available inside a compact star. Referring to the Table 2, we can saythat, in this model the net charge is effective to balance the mechanism of the force.4.4
Surface Gravitational Redshift
Let’s consider the surface gravitational redshift z s of compact objects with help of the defi-nition, z s = λ − λ e λ e , where λ is the observed wavelength and λ e is the emitted wavelengthat the surface of a non-rotating star. Thus, the gravitational redshift from the surface of the Charge (km) r / R
S M C X - 1 H e r X - 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 .4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8 charge (km) r / R 4 U 1 8 2 0 - 3 0
Fig. 5
Variation of charge with respect to fractional radius (r/R) for star candidates SMC X-1, Her X-1, 4U1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > star, as measured by a distant observer, is given by z s = | e ν ( R ) | − / − = (cid:16) − MR + Q R (cid:17) − / − Redshift r / R
S M C X 1 H e r X - 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 00123456
Redshift r / R 4 U 1 8 2 0 - 3 0
Fig. 6
Variation of redshift with respect to fractional radius (r/R) for star candidates SMC X-1, Her X-1, 4U1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > For an isotropic star a constraint on the gravitational redshift for perfect fluid spheres isgiven by z s < Energy Conditions
It is justifiable to expect this model to satisfy the energy conditions within the frameworkof general relativity. There exists a linear relationship between energy density and pressure, obeying certain restrictions. This is termed as energy conditions. To enhance our investiga-tion on the structure of relativistic space-time, let’s examine the following conditions [34]:1. Dominant energy condition (DEC): ρ − p ≥
02. Null energy condition (NEC): ρ + q π r ≥
03. Weak energy condition (WEC): ρ − p + q π r ≥
04. Strong energy condition (SEC): ρ − p + q π r ≥ Generalized Tolman-Oppenheimer-Volkov Equation
A star remains in hydrostatic equilibrium under different forces, namely, gravitational force( F g ), hydrostatic force ( F h ) and electric force ( F e ). Let’s consider the generalized Tolman-Oppenheimer-Volkoff equation in the presence of charge [35] − M G ( ρ + p ) r e ( λ − ν ) / − d pdr + σ qr e λ / = , (35)where M G ( r ) is the effective gravitational mass of the star within radius r and is defined by M G ( r ) = r ν (cid:48) e ( ν − λ ) / (36)Substituting the value of M G ( r ) in eq. (35), we obtain, F g + F h + F e = F g = − ν (cid:48) ( ρ + p ) = − Z (cid:48) Z ( ρ + p ) = − C r π (cid:104) P · P + P · P P · P (cid:105)(cid:104) K ( K − )( X − ) + (38) X K ( X − ) P · P + P · P P · P (cid:105) , F h = − d pdr = − C r π (cid:104) X K ( X − ) DP · P + K ( − K )( X − ) (cid:16) P · P + P · P P · P − (cid:17) + D + D (cid:105) , (39) F e = σ qr e λ / = π r dq dr = C r π (cid:104) + Cr K ( + Cr ) (cid:110) ( − X ) − a ( − X ) X ( a + a X ) + K − (cid:111) + D (cid:105) (40)We have drawn figures for each compact star candidates to show the behaviour of theseforces. It is evident from fig (8) that F g nullifies the combined effect of F h and F e . In otherwords, the static equilibrium is attainable under these three different forces for this model. SMC X-1 r / R
D E CN E C W E C S E C0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 00 . 0 3 00 . 0 3 50 . 0 4 00 . 0 4 50 . 0 5 00 . 0 5 50 . 0 6 00 . 0 6 50 . 0 7 0
Her X-1 r / R
D E C N E C W E C S E C
4U 1538-52 r / R
S E C N E C W E C S E C0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 00 . 0 30 . 0 40 . 0 50 . 0 60 . 0 70 . 0 8
SAX J1808.4-3658 r / R
D E C N E C W E C S E C 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 00 . 0 00 . 0 30 . 0 60 . 0 90 . 1 20 . 1 5
LMC X-4 r / R
D E C N E C W E C S E C0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 00 . 0 00 . 0 30 . 0 60 . 0 90 . 1 20 . 1 5
EXO1785-248 r / R
D E C N E C W E C S E C 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0051 01 52 02 5 r / R
D E C N E C W E C S E C
Fig. 7
Energy conditions on the system with respect to fractional radius ( r / R ) for star candidates SMC X-1,Her X-1, 4U 1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > SMC X-1 r / R R F g R F h R F e Her X-1 r / R R F g R F h R F e
4U 1538-52 r / R R F g R F h R F e SAX J1808.4-3658 r / R R F g R F h R F e LMC X-4 r / R R F g R F h R F e EXO1785-248 r / R R F g R F g R F g r / R R F g R F h R F e Fig. 8
Variations of gravitational force ( F g ), hydrostatic force ( F h ) and electric force ( F e ) acting on the systemwith respect to fractional radius ( r / R ) for star candidates SMC X-1, Her X-1, 4U 1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > vs2 r / R S M C X 1 H e r X 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 00 . 00 . 10 . 20 . 30 . 40 . 5 vs2 r / R 4 U 1 8 2 0 - 3 0
Fig. 9
Variation of velocity of sound of compact star candidates SMC X-1, Her X-1, 4U 1538-52, SAXJ1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K >
1) with respect to fractional radius(r/R) . p/ r r / R S M C X 1 H e r X 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8 p/ r r / R 4 U 1 8 2 0 - 3 0 Fig. 10
Ratio of pressure to that of density with respect to fractional radius (r/R) for star candidates SMCX-1, Her X-1, 4U 1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > Casuality Condition
Now, we are going to analyse the speed of sound propagation v s , which is given by v s = d pd ρ = X K ( X − ) DP · P + K ( − K )( X − ) (cid:16) P · P + P · P P · P − (cid:17) + D + D D − D − D (41)Naturally the velocity of sound does not exceed the velocity of light. Thus, the sound speedmust have value less than 1, as we have taken c =
1. For a physically acceptable isotropicfluid distribution, the causality condition, i.e., 0 ≤ v s ≤
1, must be satisfied to achieve astable equilibrium. It was stated by Canuto [36] that for an ultra-high distribution of matter,the speed of sound should decrease monotonically towards the surface of the star.We have shown in Fig. (9) that, for our charged isotropic model, the speed of sound remainsless than the speed of light and for each star candidate, it decreases with increase in r . Adiabatic Index r / R
S M C X 1 H e r X 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8
Adiabatic Index r / R 4 U 1 8 2 0 - 3 0
Fig. 11
Variation of adiabatic index with respect to fractional radius (r/R) for star candidates SMC X-1, HerX-1, 4U 1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > We can compare fig (9) and fig (10) to see that the ratio p ρ is less than d pd ρ throughoutthe stellar structure of stars.4.8 Relativistic Adiabatic Index
The adiabatic index γ = (cid:16) c ρ + pp (cid:17)(cid:16) d pc d ρ (cid:17) (42)is related to the stability of a stellar configuration. For an isotropic star to be in stable equi-librium, γ must have values strictly greater than 43 throughout the region. Graphs in fig. (11)represent the behavior of adiabatic index γ . We can see that the desirable features have beenobtained for each star candidate that we have considered.4.9 Harrison-Zeldovich-Novikov Stability Criterion
Harrison-Zeldovich-Novikov criterion [37, 38] states the condition for stability of a com-pact object. According to this criterion, to have a stable configuration, mass of a compactstar should increase with increase in central density throughout the stellar region. Mathe-matically, dMd ρ > ρ = C ( K − ) π K (43)Let’s write M in terms of ρ using eqs. (32) and (43) as M = π R ρ M · (cid:104) K − + π KR ρ · M M (cid:105) (44)where, M = ( K − ) + π K ρ R , M = (cid:104) − ( − K ) M − a M ( − K ) X ( a + a X ) + K − (cid:105) , M (km) r ( \ k m ) S M C X 1 H e r X 1 4 U 1 5 3 8 - 5 2 S A X J 1 8 0 8 . 4 - 3 6 5 8 L M C X - 4 E X O 1 7 8 5 - 2 4 8 0 5 1 0 1 5 2 0 2 50 . 00 . 51 . 01 . 52 . 02 . 53 . 03 . 54 . 0
M (km) r ( k m ) Fig. 12
Variation of mass function with respect to density for star candidates SMC X-1, Her X-1, 4U 1538-52, SAX J1808.4-3658, LMC X-4, EXO 1785-248 ( K <
0) and 4U1820-30 ( K > We can observe in fig (12) that mass of every star is positive definite and it increaseswith increase in central density. Thus, we can conclude that the presented model satisfiesHarrison-Zeldovich-Novikov criterion of stability.
In this paper, we have investigated the nature of isotropic compact stars. By employingthe Vaidya and Tikekar ansatz for metric potential, we have simplified the Einstein fieldequations and obtained exact solutions for isotropic compact stars. Based on physical re-quirements, we equated the interior solution to the exterior one (Reissner-N¨ordstro solution)at surface to fix the value of constants A and A . Using these values of constants and freeparameters C and K it is possible to determine mass and radius for compact stars. To re-fine the model further, we have shown through graphs, that metric potentials are regular,energy density and pressure are finite at the center and monotonically decreasing towardsthe boundary. The pressure vanishes at the boundary. The electric field intensity is zero atcentre and it increases towards the surface. We have shown that the model is compatiblewith the compact objects such as, SMC X-1, Her X-1, 4U 1538-52, SAX J1808.4-3658,LMC X-4, EXO 1785-248 and 4U1820-30. As shown in Table 2, the gravitational redshiftis bounded above function and satisfies z s <
2. Adiabatic index is strictly greater than 43throughout the model. The model satisfies the TOV equation, energy conditions, the casual-ity condition and it also fullfills Harrison-Zeldovich-Novikov criterion. This shows that theobtained model is stable. To obtain numerical values of physical quantities, we have taken G = . × − N / ms , c = × m / s , M (cid:12) = . km and have multiplied charge by1 . × to convert it from relativistic unit ( km ) to coulomb.As a future scope, we can look for other forms of metric potentials which could possessmore general behaviour and thus it might be able to describe other types of compact objects. Acknowledgments
The Authors are sincerely grateful towards Science and Engineering Research Board (SERB),DST, New Delhi for providing the needed financial support. They are also very humbledtowards the Department of Mathematics, Central University of Jharkhand, Ranchi, India,where the paper has been written and finalized, for the much needed support.
Appendix A: Notations used in eqs 29, 30 and 31 P = ( K − ) X (cid:16) a + a X ( X − ) − a X (cid:17) P = a a S ( X ) + A A P = a a ( K − ) X P = sec (cid:16) tan − (cid:114) a Xa (cid:17) + cos (cid:16) tan − (cid:114) a Xa (cid:17) − P = (cid:16) a + a XX (cid:17) D ( r ) = D · P + D · P + D · P + D · P − (cid:16) P · P + P · P P · P (cid:17) · (cid:16) D · P + D · P (cid:17) D ( r ) = − P ( K − ) X + ( K − ) X (cid:104) a ( X − ) − X ( a + a X )( X − ) + a X (cid:105) D ( r ) = a a ( K − ) P X ( a + a X ) D ( r ) = − a ( K − ) a X D ( r ) = a ( K − ) X ( a + a X ) (cid:110) sec (cid:16) tan − (cid:114) a Xa (cid:17) − cos (cid:16) tan − (cid:114) a Xa (cid:17)(cid:111) D ( r ) = − a ( K − ) X , D ( r ) = ( − K )( + Cr ) K ( + Cr ) D ( r ) = − Cr K ( + Cr ) (cid:104) ( − X ) − a ( − X ) X ( a + a X ) + K − (cid:105) D ( r ) = Cr K ( − K )( + Cr ) (cid:104) − ( − X ) + a X ( a + a X ) + a ( − X )( a + a X ) X ( a + a X ) (cid:105) Appendix B : Finding constants A & A Here we are going to calculate the values of arbitrary constants A and A , used in eq. (26),using boundary conditions (32,33).First we are going to determine the value of A A . Using boundary conditon p ( R ) = − P · P · ( K + CR ) P · ( K + CR ) + ( − K ) · ( J + ) · P = P = a a S ( X ) + A A (45) Thus, we have A A = (cid:104) P · P · X ( J + ) · P − P · X − a a S ( X ) (cid:105) (46)where, X = (cid:114) K + CR − K , J = CR ( − X ) ( + CR ) (cid:104) ( − X ) − a ( − X ) X ( a + a X ) + K − (cid:105) , P = ( K − ) X (cid:16) a + a X ( X − ) − a X (cid:17) , P = a a ( K − ) X , P = sec (cid:16) tan − (cid:114) a X a (cid:17) + cos (cid:16) tan − (cid:114) a X a (cid:17) − P = (cid:16) a + a X X (cid:17) To find the value of A , we will use the condition Z ( R ) = K + CR K ( + CR ) . After a little bitof computation, we can obtain the values of A and A as, A = √ K ( − X ) / · (cid:104) ( J + ) · P − P · X ( a + bX ) · P · P (cid:105) , when K < A = √ K ( X − ) / · (cid:104) ( J + ) · P − P · X ( a + bX ) · P · P (cid:105) , when K > A = √ K ( − X ) / · (cid:104) X a + a X − a a ( J + ) · P − P · X ( a + a X ) · P · P S ( X ) (cid:105) , when K < A = √ K ( X − ) / · (cid:104) X a + a X − a a ( J + ) · P − P · X ( a + a X ) · P · P S ( X ) (cid:105) , when K > . (48) Appendix C: Structural properties of compact stars in relativistic units
Table 3
Structural properties of “SMC X-1”within radius. r / R q ( km ) ρ ( km − ) p ( km − ) p / ρ v s z γ . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . . In f . Table 4
Structural properties of “Her X-1”within radius. r / R q ( km ) ρ ( km − ) p ( km − ) p / ρ v s z γ . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . . In f . Table 5
Structural properties of “4U 1538-52”within radius. r / R q ( km ) ρ ( km − ) p ( km − ) p / ρ v s z γ . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . . In f . Table 6
Structural properties of “SAX J1808.4-3658”within radius. r / R q ( km ) ρ ( km − ) p ( km − ) p / ρ v s z γ . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . . In f . Table 7
Structural properties of “LMC X-4”within radius. r / R q ( km ) ρ ( km − ) p ( km − ) p / ρ v s z γ . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . . In f . Table 8
Structural properties of “EXO1785-248”within radius. r / R q ( km ) ρ ( km − ) p ( km − ) p / ρ v s z γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In f . Table 9
Structural properties of “4U1820-30”within radius. r / R q ( km ) ρ ( km − ) p ( km − ) p / ρ v s z γ . . . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . × − . . . . . . . . . In f . References
1. K. Schwarzschild: Sitzer. Preuss. Akad. Wiss. Berlin , 424 (1916). Republished inGen. Relativ. Gravit. , 951 (2003)2. M. S. R. Delgaty, K. Lake: Comput. Phys. Commun. , 395 (1998)3. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt: Exact Solutionsof the Einstein Field Equations (Cambridge University Press, Cambridge, 2003)4. A. DeBenedictis, R. Garattini, F.S.N. Lobo: Phys. Rev. D , 104003 (2008)5. F. Rahaman, A.K. Yadav, S. Ray, R. Maulick, R. Sharma: Gen. Relativ. Grav. , 107(2011)
6. R. C. Tolman: Phys. Rev. , 364 (1939)7. M. Wyman,: Phys. Rev. 75, 1930 (1949)8. G. K. Patwardhan, P. C. Vaidya: J. Univ. Bombay , 23 (1943). Part III9. H. A. Buchdahl: Phys. Rev. , 1027 (1959)10. B. Kuchowicz: Phys. Lett. A , 419 (1967b)11. B. Kuchowicz: Acta Phys. Pol. , 541 (1968a)12. B. Kuchowicz: Acta Phys. Pol. , 131 (1968b)13. S. Bayin : Phys. Rev. D , 2745 (1978)14. P. C. Vaidya, R. Tikekar: J. Astrophys. Astron. , 325 (1982)15. M. R. Finch, J. E. F. Skea: Class. Quantum Gravity , 467 (1989)16. S. Mukherjee, B. C. Paul, N. K. Dadhich: Class. Quantum Gravity , 3475 (1997)17. Y. K. Gupta, M. K. Jasim: Astrophys. Space Sci. , 403 (2000)18. K. Komathiraj, S.D. Maharaj: J. Math. Phys. , 042501 (2007)19. N. Bijalwan, Y.K. Gupta: Astrophys. Sp. Sci. , 223–229 (2011)20. N. Bijalwan, Y.K. Gupta: Astrophys. Sp. Sci. , 455462 (2012)21. L. K. Patel, Kopper: Aust. J. Phys. , 441 (1987)22. R. Sharma, S. Mukherjee, S. D. Maharaj: Gen. Relat. Gravity , 999 (2001)23. Y.K. Gupta, N. Kumar: Gen. Relat. Gravity , 575 (2005)24. K. Komathiraj, S. D. Maharaj: Int. J. Mod. Phys. D. , 1803 (2007)25. J. Kumar, Y. K. Gupta: Astrophys.Space Sci. , 331–337 (2013).
26. J. Kumar, Y. K. Gupta: Astrophys Space Sci. ,243–250 (2014)27. L. D. Landau, E. M. Lifshitz: Pergamon Press, Oxford, England. 225 (1975).28. M. P. Korkina and O. Y. Orlyanskii: Ukr. Fiz. Zh. , 127 (1991) [Ukr. J. Phys. 36, 885(1991)].29. J. Kumar, A. K. Prasad, S. K. Maurya, A. Banerjee: Eur. Phys. J. C. , 540 (2018)30. A. K. Prasad, J. Kumar, S. K. Maurya, B. Dayanandan: Astrophysics and Space Science. , 66 (2019)31. G. Abbas, M. R. Shahzad: The European Physical Journal A. , 211 (2018)32. H. Bondi: Proc. R. Soc. A , 303 (1964).33. N. Straumann: General Relativity and Relativistic Astrophysics (Springer, Berlin,1984), p. 43.34. S. K. Maurya, Y. K. Gupta, S. Ray, V. Chatterjee: Astrophys Space Sci , 351 (2016)35. J. P. de Leon: Gen. Rel. Grav. , 1123 (1939)36. V. Canuto: in Solvay Conference on Astrophysics and Gravitation, Brussels, 1973 ISBN10: 2800405961.37. B. K. Harrison et al.: Gravitational Theory and Gravitational Collapse. University ofChicago Press, Chicago (1965)38. Ya. B. Zeldovich, I. D. Novikov: Relativistic Astrophysics Vol. : Stars and Relativity.University of Chicago Press, Chicago (1971)39. K. Lake: Phys. Rev. D. , 104015 (2003)40. S. Ray, A. L. Espindola, M. Malheiro, J. P. S. Lemos, V. T. Zanchin: Phys. Rev. D.68