Effects of electric field and anisotropy on the mass-radius relationship of a particular class of compact stars

For a static, spherically symmetric, anisotropic and charged distribution of matter, we present a new class of exact solutions to the Einstein-Maxwell system. By assuming specific forms of the electric field and anisotropy, we transform the master equation of the Einstein-Maxwell system to Bessel and modified Bessel differential equations. The subsequent solutions turn out to be generalizations of the isotropic and uncharged stellar model of Finch and Skea (1989), the isotropic and charged stellar model of Hansraj and Maharaj (2006) and the anisotropic and charged stellar model of Maharaj et al. (2017). We analyze the physical viability of the solutions and utilize one particular class of solutions to examine the effects of charge and anisotropy on the mass-radius relationship of compact stars.


Introduction
A pulsar is a compact object where GR finds its applications. With technological advancements, accuracy in observational data from pulsars has improved immensely over the years since its first discovery. Hence, pulsars continue attracting huge research interests to theoretical astrophysicists. Studies of highly dense compact stars like pulsars, in the presence of an electromagnetic field, is possible by solving the relevant Einstein-Maxwell field equations. In a series of papers, two of the current authors have published new class of exact solutions to the Einstein-Maxwell system by employing different techniques and exploring their physical applicability in the context of superdense stars (Komathi-raj and Sharma 2018; Komathiraj et al. 2019;Sharma 2020, 2021). Motivation for the consideration of electromagnetic field with or without anisotropic stress has been dealt with in details in these papers. In an Einstein-Maxwell system, the gravitational attraction is counterbalanced by the Coulomb repulsion which prevents the system from collapsing to a point singularity. A wide range of solutions to the Einstein-Maxwell system was compiled by Ivanov (2002). On the other hand, Bowers and Liang (1974) for the first time examined the effects of anisotropy on compact stars which eventually prompted many investigators to develop and analyze anisotropic stellar models. Stability analysis and determination of bounds on the mass to radius ratio M/R had been some of the key objectives in such investigations Gleiser 2002, 2003;Harko 2002, 2003;Maharaj 2005, 2006;Maharaj and Chaisi 2006).
In this paper, in continuation with our previous works, we propose a new set of exact solutions to the Einstein-Maxwell system and utilize them to analyze the effects of charge and anisotropy on the mass-radius relationship of relativistic compact stars.
The paper has been organized as follows: In Sect. 2, we write down the Einstein-Maxwell field equations for a static, spherically symmetric, charged and anisotropic matter distribution. We obtain an equivalent system of the equations by adopting the Durgapal and Bannerji (1983) transformations. In Sect. 3, we choose a physically reasonable form of the gravitational potential g rr , the electric field intensity and the measure of anisotropy and obtain a master equation which becomes integrable for particular choices of one of the model parameters α. In Sect. 4, we consider the case α = 1 and solve the Einstein-Maxwell system in terms of elementary functions. In Sect. 5, we treat the case 0 ≤ α < 1 and generate the solutions in terms of Bessel functions. We show that a plethora of physically reasonable stellar solutions can be regained by suitable parametrization of the new class of solutions. In Sect. 6, we present solutions for α > 1 case. In Sect. 7, we analyze physical viability of the solutions and discuss the effects of charge and anisotropy on the mass-radius relationship of such class of stars. We conclude by summarizing our results in Sect. 8.

Einstein-Maxwell system
The exterior space-time of a static spherically symmetric relativistic charged fluid distribution is uniquely described by the Reissner-Nordström metric where M and Q are the total mass and charge of the configuration. The interior space-time of the charged sphere, however, is not unique. Let that the interior space-time be described by the line element (in Schwarzchild coordinates (x a ) = (t, r, θ, φ)) ds 2 = −e 2ν(r) dt 2 + e 2λ(r) dr 2 + r 2 d 2 , where ν(r) and λ(r) are yet to be specified. The matter composition of the charged sphere is assumed to be an anisotropic imperfect fluid having energy momentum tensor where ρ is the energy density, p r is the radial pressure, p t is the tangential pressure and E electric field intensity that is measured relative to the comoving fluid 4-velocity u i = e −ν δ i 0 . For the above distribution, the Einstein-Maxwell field equations are obtained as where σ is the charge density and a prime denotes derivative with respect to the radial coordinate r. Here we use the unit system where the constant 8πG = 1 and the speed of light c = 1. The system of equations determines the behaviour of the gravitational field for an anisotropic charged perfect fluid source.
The mass of the gravitating object within a stellar radius r is defined as To solve the system, we invoke the following transformation where A and C are arbitrary constants. The above transformation was first used by Durgapal and Bannerji (1983) for the development of a compact relativistic star. Under the transformation, the system (4)-(7) takes the form where a dot (.) denotes differentiation with respect to the variable x. The quantity = p t − p r is defined as the measure of anisotropy. The mass function (8) takes the form As the system (10)-(13) comprises four equations in seven unknowns Z, y, ρ, p r , , E and σ , one needs to fix three of these variables at this stage to solve the system.

Technique used to generate new solutions
We plan to solve the system by specifying physically reasonable forms of one of the gravitational potentials Z, radial variation of the electric field E 2 and measure of anisotropy in such a manner that the system becomes integrable and provides a viable model of an anisotropic charged superdense star. In our approach, we choose where a, b, α, β and γ are real constants. Note that for γ = 0, (15)-(17) are similar to the choices studied previously by Maharaj et al. (2017) which is a generalization of the stellar models developed earlier by Hansraj and Maharaj (2006) and Finch and Skea (1989). It is important to note that the solutions of Maharaj et al. (2017) are restricted for specific values of a such that a 2 + 1 is a halfinteger (i.e., a = −1, 1, 3, . . .). We demonstrate in following sections that one can accommodate a much wider range of values than previously used to generate solutions. It should also be stressed that the choice of Z is a generalization of some earlier approaches. For example, setting a = 1, one regains the potential form considered earlier by Sharma et al. (2017). The form of E 2 in (16) is physically reasonable as it remains regular and continuous throughout the sphere. Similarly, the form (17) chosen for ensures that anisotropy also vanishes at the centre of the star (i.e., p r = p t at the centre).
Substituting (15)-(17) into (12), we obtain which, under the transformation helps us to write the equation in a more convenient form Equation (20) is the master equation of the system which should be integrable so as to find an exact solution for an anisotropic charged sphere. We note that it is possible to generate three categories of solutions to (20) in terms of different values of the parameter α. The three possible cases are In the following section, we take up these cases separately.

Case I: α = 1
In this case, equation (20) is easily integrable and the solution is obtained as where c 1 and c 2 are constants. The complete set of solutions to the Einstein-Maxwell system along with physical quantities are given below: It is interesting to note that for β = 1, the electric field vanishes and we obtain an uncharged anisotropic model. The form of the exact solution (22)-(29) has a similar form as that of Maharaj et al. (2017) which follows for γ = 0. In other words, this solution may be regarded as a generalization of the Maharaj et al. (2017) charged stellar model.

Case II: 0 ≤ α < 1
For 0 ≤ α < 1, equation (20) becomes relatively difficult to solve. However, the master equation can be transformed to a standard Bessel differential equation if the following transformation is introduced: where d is a constant. A similar kind of transformation may be found in references (Hansraj and Maharaj 2006;Komathiraj and Maharaj 2007;Maharaj and Thirukkanesh 2009;Komathiraj and Sharma 2018) where solutions for charged stars were obtained without considering any anisotropic stress. With the help of (30), the differential equation (20) can be written as By introducing yet another transform W = X k , where k is a constant, we obtain A substantial simplification of the above is possible if we choose Equation (32) then takes the form By introducing the transformation which is the Bessel equation of order 6+γ 4 . In general, the differential equation (34) has linearly independent solutions J 6+γ 4 (V ) and J − 6+γ 4 (V ) which are Bessel functions. The general solution to (34) can, therefore, be written as where b 1 and b 2 are constants. It is well known that the Bessel functions of half-integer order can be written in terms of the elementary functions. Consequently, for γ = −4, 0, 4, . . ., the solution (34) can be written as , . . . Let us consider the cases γ = −4, 0, 4 as under:

Model I: γ = −4
For γ = −4, the solution (35) can be written as where The general solution to (20) is obtained as where, we set Subsequently, the solution to the Einstein-Maxwell system takes the form This is a new solution. We note that for α = β, the model (38)-(45) reduces to a solution for an uncharged anisotropic star. Extensive analyses of anisotropic stellar models ( = 0) have been carried out in the past by investigators which includes the works of Gleiser (2002, 2003), Harko (2002, 2003), Maharaj (2005, 2006) and Maharaj and Chaisi (2006), amongst others.

Model II: γ = 0
For γ = 0, (35) yields where, The general solution in this case takes the form where f , c 1 and c 2 are as in Sect. 5.1. The complete set of exact solutions to the Einstein-Maxwell system (10)- (13) can be written as It should be stressed that the new solution (48) (47) can be written as The solution (56) corresponds to the stellar model of Maharaj et al. (2017) which is the generalization of neutron star model of Finch and Skea (1989).

Hansraj and Maharaj (2006) stellar model
If we set a = 1 and β = 0 in (47), we obtain which is the Hansraj and Maharaj (2006) stellar model for a superdense charged star. The form (57) has a similar structure to (56); however, it is important to note that the solution (48)-(55) is different from Maharaj et al. (2017) stellar model due to = 0 (or β = 0).

Finch and Skea (1989) stellar model
If we set a = 1, β = 0 and α = 0, (47) reduces to The exact solution (58) is the neutron star model of Finch and Skea (1989). The Finch and Skea neutron star model has been shown to satisfy all the requirements for the description of an isolated spherically symmetric uncharged star and has been extensively used by many researchers to study the physical properties of a neutron star.

Model III: γ = 4
For γ = 4, we have where The general solution to the differential equation (20) in this case is obtained as where, f , c 1 and c 2 are as given in previous sections. Subsequently, the complete set of solutions (10)-(13) can be written as This is a new solution to the Einstein-Maxwell system in terms of elementary functions. The charged and anisotropic solution (61)-(68) does not have an isotropic analogue as the measure of anisotropy cannot be vanished as in Sect. 5.1.

Case III: α > 1
We now consider the case α > 1 and write the differential equation (20) as The integration process of (69) is similar to that in Sect. 5. Equation (69) can be written as where ψ 2 = α − 1. If we now introduce a new independent variable ψW =V , equation (70) takes the form which is the modified Bessel equation of order 6+γ 4 . In general, the differential equation (71) has linearly independent solutions I 6+γ 4 (V ) and I − 6+γ 4 (V ) which are modified Bessel functions. The general solution to (71) can, therefore, be written as where b 1 and b 2 are constants. It is well known that the modified Bessel functions of half-integer order can be written in terms of the hyperbolic functions. We consider the cases γ = −4, 0, 4 as in Sect. 5.

Model V: γ = 0
For γ = 0, the solution (72) becomes where, The general solution of equation (20) is obtained as and the complete set of solutions to the Einstein-Maxwell system (10)-(13) can be written as where y is given in (84). Equations (85)-(92) provide new exact solutions to the Einstein-Maxwell system expressed in terms of hyperbolic functions. This, in fact, is a generalization of the Maharaj et al. (2017) model which can be regained by setting a = 1. For a = 1 and β = 0, the system becomes isotropic and we regain the charged isotropic stellar model developed by Hansraj and Maharaj (2006).

Model VI: γ = 4
For γ = 4, (72) takes the form where, The general solution to (20) is obtained as Subsequently, the complete solution to the Einstein-Maxwell system (10)-(13) takes the form where y is given by (94). This is also a new category of solutions.
In the following section, we explore the physical viability of the new solutions.

Physical viability
To examine the physical viability of the new class of solutions, let us consider a particular class of solutions (48)-(55) obtained in the Sect. 5.2 as it already contains some of the previously developed realistic stellar models (Maharaj et al. 2017;Hansraj and Maharaj 2006;Finch and Skea 1989).
First, we need to choose the value of a in such a manner that the energy density ρ, the radial pressure p r and the tangential pressure p t remain positive. The choice of a must also ensure that the gravitational potential e 2λ remains positive as the other potential e 2ν is necessarily positive.
Using Equation (50), we obtain the central density ρ 0 = (ρ) r=0 = 3aC which implies that aC > 0. Using (51) and (52), we obtain the radial and tangential pressures at r = 0 as The density and pressures should be positive at the centre of the star which puts a bound on the model parameters as At the boundary of the star (r = R), we impose the condition that the radial pressure vanishes, i.e., p R = p(x = CR 2 ) = 0, which yields where Matching the interior solution to the exterior Reissner-Nordström metric across the boundary r = R, we get The matching conditions determine the constants A as Utilizing the above results, we analyze physical viability of the solution (48)-(55) graphically for a given set of choices a = 0.9, c 1 = C = 1, and α = 0.5, β = 0.2 over the interval 0 ≤ r ≤ 1. Using these values in (104) and (105), we determine the remaining constants as c 2 = 0.799491 and A = 0.31395 which are consistent with the bound (103). Figures 1-2 show that the gravitational potentials are continuous, regular and well-behaved in the interior of the star. The energy density is a decreasing function of r i.e., dρ dr < 0 within the star as shown in Fig. 3. Figure 4 shows that the radial pressure and the tangential pressure are continuous and monotonically decreasing towards the surface of the star and the radial pressure vanishes at the boundary r = 1. Radial variation of the electric field intensity E 2 is shown in Fig. 5 which is similar to our observation in Komathiraj et al. (2019), Sharma (2020, 2021). The proper charge density σ 2 is given in Fig. 6 which is positive and monotonically decreasing. In Fig. 7, we show the fall-off behaviour of the anisotropy factor . The anisotropic factor is positive and monotonically increases from the centre until it attains a maximum value at the boundary of the stellar object as expected (Lemaitre 1933). Fulfillment of energy conditions ρ + p r + 2p t > 0 and ρ − p r − 2p t > 0 are also shown to be satisfied in Fig. 8. In Fig. 9 we show that respective values of the radial and tangential speed of sound v 2 r = dp r dρ and v 2 t = dp t dρ remain less than the speed of light c = 1 throughout the interior of the star which implies that the causality condition is satisfied in this model (Delgaty and Lake 1998). The quantity v 2 r − v 2 t in Fig. 9 is always positive and bounded by unity which shows that the model is stable (Herrera 1992; Abreu et al. 2007). In Fig. 10, we note that both radial adiabatic index r = ρ+p r p r dp r dρ and transverse adiabatic index t = ρ+p t p t dp t dρ are greater than 4/3 which is the requirement for a stable configuration. The mass function is zero at the centre and increases outward as shown in Fig. 11. In summary, we demonstrate that there exists particular set of values for which the solution (48)-(55) satisfies all the necessary requirements of a realistic star.

Mass-radius relationship
A particular objective of the current investigation is to analyze the effects of charge and anisotropy on the mass-radius (M − R) relationship of compact stars. With observational data providing more and more precisional measurements of  In an attempt to examine the impacts of electric field and anisotropy on compactness, we use a particular solution obtained in this paper and adopt numerical techniques to generate the mass-radius relationship.  We assume a reasonable value of the surface density (we choose ρ(r = R) = 7.5 × 10 1 4 gm/cm 3 ) to obtain the massradius (M − R) relationship for two sets of values (Set-I and Set-II) as shown in Figs. 12 and 13, respectively. We note that the presence of charge leads to an accumulation of more mass within a given radius both in isotropic and anisotropic cases, the compactness being greater in isotropic cases. For an isotropic object, the compactness appears to  increase with the increase in the electric field strengths. On the other hand, the presence of anisotropy reduces the compactness of charged compact objects. The effects become more pronounced for comparatively larger values of electric field and anisotropic stress, as shown in Fig. 13. Our study reconfirms Ratanpal et al. (2017) earlier claims that electric field and/or anisotropy can serve as tuning parameters to fine-tune the mass-radius relationship of compact stars.

Discussion
In this paper, by introducing a more general form of the electric field and the anisotropic factor than considered earlier, we have managed to generate a much broader class of solutions to the Einstein-Maxwell system. The advantage of the new class of solutions is that the general form of the closed-form solutions can be used to study all possible compositions (isotropic and uncharged, isotropic and charged, anisotropic and uncharged and anisotropic and charged). This facilitates an analysis of the impacts of electric field and anisotropy on the mass-radius relationship of compact stars.