A Parametric Model to Study the Mass Radius Relationship of Stars
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A parametric model to study the mass radius relationship of stars Safiqul Islam∗, Satadal Dattay and Tapas K Das z Harish-Chandra research Institute, Chhatnag Road, Jhunsi, Allahabad-211019, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai - 400094, India ∗safi[email protected] [email protected] [email protected] Abstract In static and spherically symmetric spacetime, we solve the Einstein Maxwell equations. The effective gravitational potential and the electric field for charged anisotropic fluid are defined in terms of two free parameters. For such configuration, the mass of the star as a function of stellar radius is found in terms of two aforementioned parameters, subjected to certain stability criteria. For various values of these two parameters one finds that such mass radius relationship can model stellar objects located at various regions of Hertzsprung-Russel diagram. 1 Introduction: fluid considered in our model, embedded within a sphere of radius R. Hence our model here provides For relativistic charged fluid with the signature the mass M(R) of star of radius R. M(R) in our of pressure anisotropy, where the anisotropy is calculations, however, is characterized by (a,b), defined by the finite non zero difference between and there remains a specific relationship between the radial and the tangential fluid pressure, the a and b, which are obtained by using some prede- Einstein Maxwell field equations are solved for fined stability criterion. Various values of a and static spherically symmetric spacetime. Certain b provides various [M(R)-R] measurements. For functional form of the electric field as well as different values of a and b, one can find M(R) for the effective gravitational potential have been different values of R, and hence using our model, introduced in our model, where such field and po- we can study the mass radius relation for different tential are characterized by two free parameters a categories of stellar objects located at various and b, with certain relationships defined between regions of the Hertzsprung-Russel diagram. arXiv:1702.05171v1 [astro-ph.SR] 16 Feb 2017 these two parameters, where such relationships are obtained using a particular form of stability criteria. The charge and the mass energy density 2 Einstein-Maxwell equa- have been expressed (as a consequence of the tions: interior solution) as a function of the radial distance. From there, we obtain the mass-radius We consider the interior spacetime of a (3 + 1)-D relationship for the interior solution. Once such star in Schwarzschild coordinates (t; r; θ; φ) as mass-radius relationship is integrated for a par- ticular limit defined by the radius of the star, one can obtain what will be the mass of the charged ds2 = −e2ν(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2θdφ2) (1) Here ν and λ are the metric potentials which 1 have functional dependence on the radial coordi- Therefore, the energy-momentum tensors in the nate r and ν(r) is to be determined. interior of the star can be expressed in the follow- The Hilbert action coupled to electromagnetism ing form: is given by 0 1 2 Z p R 1 T0 = 8πρ + E (9) I = dx3 −g − F cF + L ; (2) 2 16π 4 a bc m 1 1 2 where Lm is the Lagrangian for matter. The T1 = 8πpr − E (10) variation with respect to the metric gives the fol- 2 lowing self consistent Einstein-Maxwell equations 1 for a charged anisotropic fluid distribution, T 2 = T 3 = 8πp − E2 (11) 2 3 t 2 1 The Einstein-Maxwell field equations with mat- G = R − Rg = −8πT ab ab 2 ab ab ter distribution as equation 3. are analogous with (m) EM the transformations, = −8π(Tab + Tab ); (3) 1 1 The explicit forms of the energy momentum ten- 8πρ + E2 = [r(1 − e−2λ)]0 (12) sor (EMT) components for the matter source (we 2 r2 assumed that the matter distribution at the inte- rior of the star is anisotropic) and electromagnetic 1 1 2ν0 fields are given by, 8πp − E2 = − (1 − e−2λ) + e−2λ (13) r 2 r2 r (m) Tab = (ρ + pt)uaub − ptgab + (pr − pt)vavb; (4) 1 ν0 λ0 8πp − E2 = e−2λ(ν00 +ν02 + −ν0λ0 − ) (14) t 2 r r EM 1 c 1 cd T = − F Fbc − gabFcdF ; (5) and ab 4π a 4 (r2E)0 = 4πr2σeλ (15) where ρ, pr, pt, ua, va and Fab are, respectively, matter-energy density, radial fluid pressure, trans- where a `0' denotes differentiation with respect to verse fluid pressure, four velocity, radial four vec- the radial parameter r. When E=0, the Einstein- tor of the fluid element and electromagnetic field Maxwell system given above reduces to the un- tensor. The case pt = pr, corresponds to the charged Einstein system. The equation (15) yields isotropic fluid when the anisotropic force vanishes. the expressed for E in the form We also consider G = c = 1 in our observations. In our consideration, the four velocity and radial Z r a −ν a a a 4π 2 λ(x) q(r) four vector satisfy, u = e δ0 , u ua = 1, v = E(r) = x σ(x)e dx = (16) −λ a a r2 r2 e δ1 , v va = −1. 0 Also, the electromagnetic field is related to cur- rent four vector as, where q(r) is total charge of the sphere under consideration and σ(r) is the proper charge den- J c = σ(r)uc; (6) sity. The mass of a star in an uncharged system is as generally defined by, ab a F;b = −4πJ ; (7) Z r where, σ(r) is the proper charge density of the dis- M(r) = 4π ρ(x)x2dx (17) tribution. Hence the electromagnetic field tensor 0 can be given as, Here R is taken as the radius of our star model. t r r t The equation of state is considered as, Fab = E(r)(δaδb − δaδb); (8) where E(r) is the electric field. pr = !ρ (18) 2 3 A particular class of solu- tions: x x We consider the the electric field intensity as 4a2r2 E2(r) = (19) (1 + 2ar2)2 We observe that the function is regular if a > 0 A similar form of E can be used as shown by Tikekar et al., [5], Komathiraj et al., [3] and Is- lam S. et al., [2]. We consider the gravitational potential Z(r) [6] as, (1 + ar2)(1 − br2) Z(r) = (20) (1 + 2ar2) where a and b are real constants. Hence from above we observe, (1 + ar2)(1 − br2) −2λ(r) Figure 1: The density parameter ρ is shown e = 2 (21) (1 + 2ar ) against r, a and b having unit km−2 which on solving we get integration as zero without any loss of generality 2ar br ar as, λ0(r) = + − (22) (1 + 2ar2) (1 − br2) (1 + ar2) (3!a + !b + a) and ν(r) = log(1 + 2ar2) 2(2a + b) 1 λ(r) = [log(1+2ar2)−log(1+ar2)−log(1−br2)] (3!a + 2!b + a) 2 − log(1 + ar2) (23) 4(a + b) Using equations, (12), (19) and (21) we obtain (10!ab + 3!b2 + 6!a2 + 2a2 + 4ab + b2) − 4(a + b)(2a + b) 3(a + b) + abr2(6ar2 + 7) ρ = (24) 2 8π(1 + 2ar2)2 ×log(1 − br ) (27) Also for a positive density we must have, Hence the following relation is evident, a − < b; (25) (3!a+!b+a) 7 2 4 2ν(r) 2 1 + 3 ar + 2ar e = (1 + 2ar ) 2(2a+b) −(3!a+2!b+a) It is clearly evident from figure 1. that the den- ×(1 + ar2) 4(a+b) sity decreases gradually from the centre where it −(10!ab+3!b2+6!a2+2a2+4ab+b2) 2 is maximum and at the surface of the star of ra- ×(1 − br ) 4(a+b)(2a+b) (28) dius R, it becomes minimum. The variation is very small for the considered range of parameter. Hence the star is of almost uniform density having a value of 1:04926 × 105 kg=m3. [We have assumed that the radius of the star is 0:1 × R = 69570 km]. Equations, (18) and (24) above yield 3(a + b) + abr2(6ar2 + 7) p = ! (26) r 8π(1 + 2ar2)2 Using equations, (14), (19) and (21), we get the value of the other parameter taking the constant of 3 We also plot the metric potentials as below, we obtain, a(3!a + !b + a)(1 + ar2) p = t 4π(2a + b)(1 + 2ar2)3 ×(1 − br2)(1 − 2ar2) b(10!ab + 3!b2 + 6!a2 + 2a2 + 4ab + b2) + 16π(a + b)(2a + b)(1 + 2ar2)(1 − br2) ×(1 + ar2)(1 + br2) a(3!a + 2!b + a)(1 − ar2)(1 − br2) − 16π(a + b)(1 + ar2)(1 + 2ar2) a2(3!a + !b + a)2r2(1 + ar2)(1 − br2) x x + 2π(2a + b)2(1 + 2ar2)3 a2(3!a + 2!b + a)2r2(1 − br2) + 32π(a + b)2(1 + ar2)(1 + 2ar2) b2(10!ab + 3!b2 + 6!a2 + 2a2 + 4ab + b2)2 + 32π(a + b)2(2a + b)2(1 − br2)(1 + 2ar2) ×r2(1 + ar2) a2(3!a + !b + a)(3!a + 2!b + a) − 4π(2a + b)(a + b)(1 + 2ar2)2 ×r2(1 − br2) ab(3!a + 2!b + a)r2 − 16π(a + b)2(2a + b)(1 + 2ar2) ×(10!ab + 3!b2 + 6!a2 + 2a2 + 4ab + b2) ab(3!a + !b + a)r2(1 + ar2) + 4π(a + b)(2a + b)2(1 + 2ar2)2 Figure 2: The metric potentials e2ν and e2λ are ×(10!ab + 3!b2 + 6!a2 + 2a2 + 4ab + b2) shown against r.