Physics & Astronomy International Journal Research Article Open Access A new class of anisotropic charged compact star Abstract Volume 1 Issue 5 - 2017 A new model of charged compact star is reported by solving the Einstein-Maxwell field 1 2 equations by choosing a suitable form of radial pressure. The model parameters ρ,,pp Ratanpal BS, Bhar P 2 r ⊥ and E are in closed form and all are well behaved inside the stellar interior. A comparative 1Department of Applied Mathematics, The MS University of study of charged and uncharged model is done with the help of graphical analysis. Baroda, India 2Department of Mathematics, Government General Degree Keywords: general relativity, exact solutions, anisotropy, relativistic compact stars, College, India charged distribution Correspondence: BS Ratanpal, Department of Applied Mathematics, Faculty of Technology & Engineering, The MS University of Baroda, Vadodara-390 001, India, Tel +919825185736, Email [email protected] Received: July 29, 2017 | Published: November 24, 2017 Introduction solid core, in presence of type 3A super fluid,12 phase transition,13 pion condensation,14 rotation, magnetic field, mixture of two fluid, To find the exact solution of Einstein’s field equations is difficult existence of external field etc. Local anisotropy in self gravitating due to its non-linear nature. A large number of exact solutions of systems was studied by.15,16 Demonstrated that pressure anisotropy Einstein’s field equations in literature but not all of them are physically affects the physical properties, stability and structure of stellar matter. relevant. A comprehensive collection of static, spherically symmetric Relativistic stellar model admitting a quadratic equation of state was 1,2 solutions are found in. A large collection of models of stellar objects proposed by17 in finch-skea space-time.18 Has generalized earlier work 3 incorporating charge can be found in literature. Proposed that a fluid in modified Finch-Skea spacetime by incorporating a dimensionless sphere of uniform density with a net surface charge is more stable than parameter n. In a very recent work19 obtained a new model of an 4 without charge. An interesting observation of is that in the presence anisotropic super dense star which admits conformal motions in the of charge, the gravitational collapse of a spherically symmetric presence of a quintessence field which is characterized by a parameter distribution of matter to a point singularity may be avoided. Charged ω with −−1<ω < 1/3. The model has been developed by 5,6 q q anisotropic matter with linear equation of state is discussed by. choosing ansatz.20,21 Have studied the behavior of static spherically Found that the solutions of Einstein-Maxwell system of equations symmetric relativistic objects with locally anisotropic matter are important to study the cosmic censorship hypothesis and the distribution considering the Tolman VII form for the gravitational formation of naked singularities. The presence of charge affects potential g in curvature coordinates together with the linear relation rr the values for redshifts, luminosities, and maximum mass for stars. between the energy density and the radial pressure. Charged perfect fluid sphere satisfying a linear equation of state was discussed by.7 Regular models with quadratic equation of state were Charged anisotropic star on paraboloidal space-time was studied discussed by.8 They obtained exact and physically reasonable solution by.22,23 Studied anisotropic star on pseudo-spheroidal space time. of Einstein-Maxwell system of equations. Their model is well behaved Charged anisotropic star on pseudo-spheroidal space time was studied and regular. In particular there is no singularity in the proper charge by.24 The study of compact stars having Matese and Whitman mass density.9 Considered a self gravitating, charged and anisotropic fluid function was carried out by.25 Motivated by these earlier works in the sphere. To solve Einstein-Maxwell field equation they have assumed present paper we develop a model of compact star by incorporating both linear and nonlinear equation of state and discussed the result charge. Our paper is organized as follows: In section 2, interior space analytically.10 Extend the work of5 by considering quadratic equation time and the Einstein-Maxwell system is discussed. Section 3 deals of state for the matter distribution to study the general situation of with solution of field equations. Section 4 contains exterior space time a compact relativistic body in presence of electromagnetic field and and matching conditions. Physical analysis of the model is discussed anisotropy. in section 5. Section 6 contains conclusion. Ruderman R11 investigated that for highly compact astrophysical Interior spacetime objects like X-ray pulsar, Her-X-1, X-ray buster 4U 1820-30, millisecond pulsar SAX J 1804.4-3658, PSR J1614-2230, LMC X-4 We consider the static spherically symmetric spacetime metric as, 15 3 2νλ ()rr 2 () 2 2 22 etc. having core density beyond the nuclear density ( 10gm / cm ) ds= e dt− e dr −+ r( dθsin2 θφ d ). (1) there can be pressure anisotropy, i.e, the pressure inside these compact objects can be decomposed into two parts radial pressure p and Where ν and λ are functions of the radial coordinate ‘r’ only. ⊥ transverse pressure p perpendicular direction to p . ∆−= pp ⊥ r r ⊥ Einstein-Maxwell Field Equations is given by is called the anisotropic factor which measures the anisotropy. jj1 The reason behind these anisotropic nature are the existence of RR− δππ=8 Tjjj++ E, (2) ii2 ( iii) Submit Manuscript | http://medcraveonline.com Phys Astron Int J. 2017;1(5):151‒157. 151 ©2017 Ratanpal et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: A new class of anisotropic charged compact star ©2017 Ratanpal et al. 152 Where, j jj 83π Sp= − p. (18) T=(ρδ+− p) uu p ,3 ( ) r ⊥ i ii Solution of field equations j j1 jj πδ=3S cc−− uu ,4 ( ) To solve the above set of equations (13)-(15) we take the mass i i( ii) 2 function of the form And br3 j 1 jk 1 mn j mr( ) = , ( 19) E=−+ FF F F δ .5 ( ) 2 i 4π ik mn i 2(1+ar ) 4 Where ‘a’ and ‘b’ are two positive constants. The mass function Here ρ is proper density, p is fluid pressure, u is unit four 26 i i given in (19) is known as Matese & Whitman mass function that velocities, S denotes magnitude of anisotropic tensor and C is 27 −λ gives a monotonic decreasing matter density which was used by to − /2 radial vector given by (0,e ,0,0) . F Denotes the anti-symmetric 28 ij model an anisotropic fluid star, to develop a model of dark energy electromagnetic field strength tensor defined by star,29 to model a class of relativistic stars with a linear equation of ∂ Aj ∂Ai 30 F =− ,6 ( ) state and to model a charged anisotropic matter with linear equation ij ∂∂xxij of state. That satisfies the Maxwell equations −λ 2m Using the relationship e =1− and equation (19) we get, FFF++= 0, (7) r ij, k jk ,, i ki j 2 λ 1+ar And e = . ( 20) 2 ∂ ik i +− (F−− g) =4,π gJ (8) 1(a br ) ∂ k x From equation (13) and (20) we obtain 2 3b+ abr 2 Where g denotes the determinant of g , Ar=(φ ( ),0,0,0) is 8πρ = − E . ( 21) ij i 22 four-potential and (1+ar ) ii 2 Ju=,9σ ( ) We choose E of the form 2 F Is the four-current vector where ij denotes the charge density. 2 αar E = , (22) 22 The only non-vanishing components of F is FF= − . Here (1+ar ) ij 01 10 νλ+ Which is regular at the center of the star. Substituting the expression 2 e 2 r 2λ /2 of E into (21) we get, F =− ∫ 4πσr e dr , ( 10) 01 2 0 2 r 3(b+− abα ) r And the total charge inside a radius r is given by 8πρ = . ( 23) (1+ar22 ) r 2λ /2 q( r )=4πσ∫0 r e dr. (11) 2 01 To integrate the equation (14) we take radial pressure of the form, 2 01 E= − FF The electric field intensity 01 can be obtained from E= − FF 01 bp(1− ar2 ) , which subsequently reduces to 8π p = 0 , ( 24) qr() r + 22 E = . (12) (1ar ) r2 Where p is a positive constant, the choice of p is reasonable The field equations given by (2) are now equivalent to the following 0 r due to the fact that it is monotonic decreasing function of ‘r’ and the set of the non-linear ODE’s 1 radial pressure vanishes at r = which gives the radius of the star. −−λλ a 1−eeλ′ 2 ++= 8πρ E , ( 13) From (24) and (14) we get, r2 r −−λλ (bp+ b )( r − a bp +−α b ) r3 ee−1 ν ′ 2 00 +−=8π pE, ( 14) ν ′ = . (25) 2 r r (1+ar22 ) 1 +− ( a b ) r r 2 −λ ν′′′′′′′ ν νλ ν− λ 2 Integrating we get, e+− + = 8π pE+ , ( 15) ⊥ 24 4 2r 2bp +α 0 2 2b C(1+ ar ) ν = log , (26) 22 b−2 ab p +− bα a Where we have taken ( ) 0 2 2S (b−− ar) 1 22b2 − ab pp=+ , ( 16) r 3 Where C is constant of integration, and the space time metric in S the interior is given by pp=− . ( 17) ⊥ 3 Citation: Ratanpal BS, Bhar P. A new class of anisotropic charged compact star. Phys Astron Int J. 2017;1(5):151‒157. DOI: 10.15406/paij.2017.01.00027 Copyright: A new class of anisotropic charged compact star ©2017 Ratanpal et al.