Chapter 13
A Model for Anisotropic Strange Stars Debabrata Deb, Sourav Roy Chowdhury, Saibal Ray, Farook Rahaman and B. K. Guha
13.1 Introduction The interior of the compact stars may have differential fluid pressures which are directionally dependent. Ruderman [1] argued that the nuclear matter may have anisotropic features with very high density ranges (>1015 gm/cm3 ), where the nuclear interaction must be treated relativistically. On the contrary, Bowers and Liang [2] investigated that anisotropy might have non-negligible effects on different physical parameters. However, one can note the recent literature regarding anisotropic matter distribution where several authors have considered anisotropy in connection to compact stars [3–7]. Our motivation in the present investigation is to construct a model for strange star by assuming that (i) the compact star is made up of anisotropic fluid, and (ii) the
D. Deb (B) · S. Roy Chowdhury · B. K. Guha Department of Physics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India e-mail:
[email protected] S. Roy Chowdhury e-mail:
[email protected] B. K. Guha e-mail:
[email protected] S. Ray Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700010, West Bengal, India e-mail:
[email protected] F. Rahaman Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India e-mail:
[email protected] © Springer International Publishing AG, part of Springer Nature 2018 Md. Naimuddin (ed.), XXII DAE High Energy Physics Symposium, Springer Proceedings in Physics 203, https://doi.org/10.1007/978-3-319-73171-1_13
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MIT Bag model is applicable in this environment. We have studied several physical parameters, viz., the radius, redshift and anisotropy among which it can be observed that anisotropy is increasing with the radial coordinate inside the stellar system from zero at the center to it’s maximum at the surface for all the strange stars.
13.2 The Einstein Field Equations and Their Solutions We consider the metric of the stellar system as follows ds 2 = −eν(r ) dt 2 + eλ(r ) dr 2 + r 2 (dθ2 + sin 2 θdφ2 ),
(13.1)
where the metric potentials ν and λ are functions of the radial coordinate r only. The energy-momentum tensor can be provided as Tνμ = (ρ + pr )u μ u ν − pt gνμ + ( pr − pt )η μ ην ,
(13.2)
where ρ, pr and pt of (13.2) represents the energy density, radial and tangential pressures respectively of the fluid sphere. According to the phenomenological MIT bag model, the EOS of strange matter can be written in a linear form pr =
1 (ρ − 4 Bg ), 3
(13.3)
where ρ, pr and Bg are the energy density, the radial pressure and the bag constant respectively. Again, assuming that the matter within the spherical system follows the density profile proposed by Mak and Harko [8]: ρ0 r 2 , ρ(r ) = ρc 1 − 1 − ρc R 2
(13.4)
where ρc and ρ0 are the central and surface densities respectively. Now, from the well known Einstein field equations one can find out expression for the anisotropy as [ 12 c1 2 r 4 − 15 c1 c5r 2 + c6 π − 29 c1 R 2 ]r 2 , Δ(r ) = pt − pr = 36R 2 [r 2 35 c1r 2 − ρc R 2 π + 38 R 2 ] where c1 = (ρc − ρ0 ), ρ0 )R 4 .
c5 =
1 5
(156 ρc − 84 ρ0 ) R 2 ,
(13.5)
c6 = (6 ρc − 3 ρ0 ) (4 ρc −
13 A Model for Anisotropic Strange Stars
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Fig. 13.1 Variation of anisotropy as a function of the radial coordinate r for the strange star L MC X − 4
Table 13.1 Physical parameters as derived from the proposed model for L MC X − 4 [9] Observed Predicted Redshift (Z ) Δ (0) (in Δ (R) (in Δ (R) (in −2 −2 km−2 ) mass (in M ) radius (km) km ) km ) 1.29 ± 0.05
9.48
0.29
0
0.000027
−0.0000031
13.3 Discussion and Conclusion Based on the present model for compact stars let us consider a particular candidate L MC X − 4 which is a representative of the strange stars. For this star, we have found out numerical values of the several physical parameters with the Bag constant 83 MeV/(fm)3 . The model yields the values of the maximum mass and maximum radius as Mmax = 3.54M and Rmax = 11.811 km respectively. We also find that anisotropy is minimum at the center and maximum at the surface, which is clear from Fig. 13.1 as well as Table 13.1.
References 1. 2. 3. 4. 5.
R. Ruderman, Annu. Rev. Astron. Astrophys. 10, 427 (1972) R. Bowers, E. Liang, Astrophys. J. 188, 657 (1974) V. Varela, F. Rahaman, S. Ray, K. Chakraborty, M. Kalam, Phys. Rev. D 82, 044052 (2010) F. Rahaman, S. Ray, A.K. Jafry, K. Chakraborty, Phys. Rev. D 82, 104055 (2010) F. Rahaman, P.K.F. Kuhfittig, M. Kalam, A.A. Usmani, S. Ray, Class. Quantum Gravity 28, 155021 (2011) 6. F. Rahaman, R. Maulick, A.K. Yadav, S. Ray, R. Sharma, Gen. Relativ. Gravit. 44, 107 (2012)
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7. M. Kalam, F. Rahaman, S. Ray, S.M. Hossein, I. Karar, J. Naskar, Eur. Phys. J. C 72, 2248 (2012) 8. M.K. Mak, T. Harko, Chin. J. Astron. Astrophys. 2, 248 (2002) 9. M.L. Rawls, J.A. Orosz, J.E. McClintock, M.A.P. Torres, C.D. Bailyn, M.M. Buxton, Astrophys. J. 730, 25 (2011)
