A New Class of Anisotropic Charged Compact Star;

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 jjj The reason behind these anisotropic nature are the existence of RR− δππ=8 ( Tiii++ E) , ( 2) ii2

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 1 πδ=3S ccj−− uu jj,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 4 mn i 2(1+ar ) 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 ∂A 30 F =− i ,6 ( ) state and to model a charged anisotropic matter with linear equation ij ∂∂xx ij 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 2λ /2 of E into (21) we get, F =− ∫r 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 ) 2λ /2 q( r )=4πσ∫r r e dr. (11) 0 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 +22 +− r r (1ar ) 1 ( a b ) 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 2 2S (b ar) 1 22b− 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. 153

  +α 22 2 2 22 2 3 22 2bp0 +α −+ α − αα + 2  C= ab p 2 abp ab 44 ab a a . C1+ ar 2b 2 30 0 2( ) 2 1+ar 22 22 ds = dt −dr −+ r( dθsin2 θφ d ). (27) 22 2 b−2 ab p +− bα a 1+−(a br) ( ) 0 Exterior space time and matching condition 2  (b−− ar) 1 22b2 − ab  We match our interior space time (27) to the exterior Reissner- Nordström space time at the boundary rr= (where r is the radius From (14), (15) and (18), we have b b of the star.). The exterior space time is given by the line element

2 24 −1 r A++ Ar Ar 22 12 3 2 22Mq2 Mq 22 22  ds =11−+dt −−+dr − r dθ +sin2 θφ d . ( 30) π ( ) 22( ) 8 3=S , 28 rrrr −+2468 + + + 4 Br1234 Br Br Br  By using the continuity of the metric potential g and g at the 22 2 2 rr tt Where, A= b p+ 14 b p − 12 abp +− 3 b 12α a , boundary rr= we get, 10 0 0 b 2 22 2 2 2 2 ν ()r 2Mq A=− 2 ab p + 8 ab p − 8 a bp − 2α abp ++− 2 ab 8 αα ab 16 a e b =1,−+ ( 31) 2 0 00 0 2 rb r

22222 3 2 22 2 322 −1 A30=, ab p− 4 ab p 00 + 4 abp + 2α abp 0 −+ ab 44 α ab − αα a + a 2 λ ()r 2Mq , b −+ e =1. ( 32) 2 23 r 2 B= 12 ab− 24 a 2 − b r 2 , B= 12 ab− 24 a , B= 12 ab 16 a and 2 3 The radial pressure should vanish at the boundary of the star, hence 34. B=4 ab− 4 a 4 from equation (24) we obtain From (18) we obtain, 1 a = . (33) 2 rb 4bpCrCrCr+++246 01 2 3 Using (33) & (19) we obtain 8=π p , (29) ⊥ 4−−−−Br2468 Br Br Br 4m 1234 b = . ( 34) 3 rb 22 2 Where, C= b p− 8 abp +− 3 b 12α a , We compute the values of ‘a’ and ‘b’ for different compact stars 10 0 which is given in Table 1. 22 2 2 2 2 C=− 2 ab p + 8 ab p − 12 a bp − 2α abp ++− 2 ab 8 αα ab 16 a 2 00 00 ,

Table 1 The values of ‘a’ and ‘b’ obtained from the equation (33) and (34)

Compact star Mass(km) Radius(km) −2 −2 u MM() a( km ) b( km ) zs

U 1820-30 1.58 2.33050 9.1 0.012076 0.012370 0.256099 0.431786

PSR J1903+327 1.667 2.45882 9.438 0.011226 0.011699 0.260524 0.444954

U 1608-52 1.74 2.56650 9.31 0.011537 0.012722 0.275671 0.492941

Vela X-1 1.77 2.61075 9.56 0.010942 0.011952 0.273091 0.484428

PSR J1614-2230 1.97 2.90575 9.69 0.01065 0.012775 0.299871 0.580629

Cen X-3 1.49 2.19775 9.178 0.011871 0.011371 0.239458 0.385309

Physical analysis To be a physically acceptable model matter density ()ρ , radial pressure ( p ), transverse pressure ( p ) all should be non-negative r r inside the stellar interior. It is clear from equations (22) and (24) it is clear that p is positive throughout the distribution. The profile of p r ⊥ and p are shown in Figures 1 & 2 respectively. From the figure it is ⊥ clear that all are positive inside the stellar interior.

dρ dp dp Figure 3 , ⊥ and ⊥ are plotted against r for the star PSR J1614- dr dr dr 2230.

Figure 1 The matter density is plotted against r for the star PSR J1614-2230.

Figure 4 ρ−−pprt2 is plotted against r for the star PSR J1614-2230. For a physically acceptable model of anisotropic fluid sphere the radial and transverse velocity of sound should be less than 1 which is Figure 2 The transverse pressure p is plotted against r for the star PSR known as causality conditions. t 2 2 J1614-2230. Where the radial velocity ()v and transverse velocity ()v of dp st st The profile of c and ⊥ are shown in Figure 3, it is clearly sound can be obtained as dr indicates that p , p and p are decreasing in radially outward 2 r r ⊥ dpr bp0 (3− ar ) 31 = . ( 35) direction. According to for an anisotropic fluid spheres the trace of 2 dρ 5)b++αα ab( − r the energy tensor should be positive. To check this condition for our 23 2 4 6 8 model we plot ρ −−pp2 against r in Figure 4. From the figure (1+ar ) D12345 ++++ Dr Dr Dr Dr r ⊥ dp⊥  = . (36) it is clear that our proposed model of compact star satisfies Bondi’s dρ −10ab −− 2 aαα 2 a2 ( b − ) r 2  2 ++ Er2 Er 4 + Er 6 + Er 8 + Er 10 + Er 12  conditions.   12345 6

Where,

22 2 2 2 2 D=− 6 ab p + 32 ab p − 24 a bp − 4α abp −+ 2 ab 16 αα ab − 8 a , 2 0 0 00

3 2 3 2 22 3 2 3 22 2 2 3 22 D= 5 ab p−+ 8 ab p 2α ab p − 12 a b p + 24 a bp + 6α a bp +−−−++ 7 ab 12 a b 8α ab 8 α a b 24 αα a 3 a , 3 00 0 0 0 0

3b 2 23 2 23 32 22 4 3 23 22 32 3 22 4 23 D= 6 abp− 6 abp + 16 abp − 40 ab p − 8α ab p + 24 abp + 8α abp ++ 6 ab 8 α ab −− 6 ab 32 α ab − 2 α ab + 24 αα a + 2 a , 4 00 0 0 000

33 2 42 2 32 4 33 42 32 23 4 5 2 4 D50= ab p− ab p 0 + 2α ab p 0 − 2 α abp 0 −++ ab ab4α ab + α ab − 84 α ab + αα a − a ,

2222 3 4 22 3 4 E=12 ab− 4 , E= 2 b−+ 20 ab 30 a , E= 12 ab−+ 40 ab 30 a , E= 12 ab−+ 40 ab 30 a , 1 2 4 4

A relativistic star will be stable if the relativistic adiabatic index 4 32 4 5 42 5 6 Γ > . Where Γ is given by E= 8 ab−+ 20 ab 12 a and E=242 ab−+ ab a . 5 6 3 ρ+ prr dp 1 Γ = (3 7) Due to the complexity of the expression of

Figure 7 The adiabatic index is plotted against r for the star PSR J1614-2230. For an anisotropic fluid sphere all the energy conditions namely Weak Energy Condition (WEC), Null Energy Condition (NEC), 2 dp Figure 5 v = r is plotted against r for the star PSR J1614-2230. Strong Energy Condition (SEC) and Dominant Energy Condition sr dρ (DEC) are satisfied if and only if the following inequalities hold simultaneously in every point inside the fluid sphere. ()i NEC:0ρ +≥ p ( 3 8) r (ii ) WEC : p +≥ρρ 0, > 0 ( 39) r

(iv ) DEC :ρρ > pr , > p⊥ ( 41)

(iv ) DEC :ρρ > pr , > p⊥ ( 41) Due to the complexity of the expression of p we will prove the ⊥ inequality (38)-(41) with the help of graphical representation. The profiles of the L.H.S of the above inequalities are depicted in Figure 8 for the compact star PSR J1614-2230. The figure shows that all the energy conditions are satisfied by our model of compact star (Figures 9 & 10).

2 dp Figure 6 v = ⊥ is plotted against r for the star PSR J1614-2230. st dρ

Figure 8 The left and middle figures show the dominant energy conditions where as the right figure shows the weak null and strong energy conditions Figure 10 The variation of electric field is shown against r for the star PSR are satisfied by our model for the star PSR J1614-2230. J1614-2230. The ratio of mass to the radius of a compact star cannot be arbitrarily large.32 showed that for a (3+1)-dimensional fluid sphere 28M < . To see the maximum ratio of mass to the radius for our r 9 modelb we calculate the compactness of the star given by

m( r) br2 ur( )= = , ( 42) r 2(1)+ar2 and the corresponding surface redshift z is obtained by, s −1/2 1+−z ( r )= 1 2 ur ( ) sb b Therefore z can be obtained as, s 1 − 1(+−a br )2 2 zr( ) = b − 1. (43) sb 2 1+arb Figure 9 Variation of anisotropy is shown against r for the star PSR J1614- The surface redshift of different compact stars is given in Table 2. 2230.

Table 2 The values of central density, surface density, central pressure and radial velocity of the sound at the origin for different compact stars are obtained

Central Surface Surface Central Compact star dpr ρ Density ()0 Density density Pressure ()p0 ρ d |r=0

(uncharged) (charged) (charged) gm. cm−3 dyne. cm−2

U 1820-30 15 14 14 35 0.295227 1.994 ×10 6.648 ×10 6.514 ×10 2.989 ×10

PSR J1903+327 15 14 35 35 0.294958 1.886 ×10 6.287 ×10 2.827 ×10 2.827 ×10

U 1608-52 15 35 35 35 0.295357 2.051×10 3.074 ×10 3.074 ×10 3.074 ×10

15 14 35 35 Vela X-1 1.927 ×10 6.423 ×10 2.888 ×10 2.888 ×10 0.295063

PSR J1614-2230 15 14 35 35 0.295376 2.059 ×10 6.865 ×10 3.087 ×10 3.087 ×10

15 14 35 35 Cen X-3 1.833 ×10 6.111×10 2.748 ×10 2.748 ×10 0.294815

Conclusion 13. Sokolov AI. Phase transitions in a superfluid neutron liquid. Journal of Experimental and Theoretical Physics. 1980;79(4):1137–1138. We have obtained a new class of solution for charged compact 14. Sawyer RF. Condensed π−Phase in Neutron-Star Matter. Physical stars having26 mass function. The electric field intensity is increasing 4 Review Letters. 1972;29(6). in radially outward direction and the adiabatic index Γ > . The 3 physical requirements are checked for the star PSR J1614-2230 and 15. Herrera L, Santos NO. Local anisotropy in self-gravitating systems. model satisfies all the physical conditions. Some salient features of Physics Reports. 1972;286(2):53–130. the model are 16. Dev K, Gleiser M. Anisotropic Stars: Exact Solutions. General Relativity and Gravitation. 2002;34(11):1793–1818. In present model ifα =0, the model corresponds to23 model. 1 17. Sharma R, Ratanpal BS. Relativistic stellar model admitting a α ab== In present model if =0, 2 , where R is geometric quadratic equation of state. International Journal of Modern Physics D. R 17 parameter then the model corresponds to model, which is stable for 2013;22(13). 1

Pulsars: Structure and Dynamics. Annual Review of Astronomy and Astrophysics. 1972;10:427–476. 32. Buchdahl HA. General Relativistic Fluid Spheres. Physical Review journal Achieve. 1959;116(4). 12. Kippenhahn R, Weigert A. Steller Structure and Evolution. Springer, Berlin, Germany. 1990.

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