We study the static equilibrium configurations of uncharged and charged spheres composed by a relativistic polytropic fluid, and we compare with those of spheres composed by a nonrelativistic polytropic fluid, the later case being already studied in a previous work [J. D. Arbañil, P. S. Lemos, and V. T. Zanchin, Phys. Rev. D 88, 084023 (2013)PRVDAQ1550-799810.1103/PhysRevD.88.084023]. An equation of state connecting the pressure p and the energy density ρ is assumed. In the nonrelativistic fluid case, the connection is through a nonrelativistic polytropic equation of state, p=ωργ, with ω and γ being respectively the polytropic constant and the polytropic exponent. In the relativistic fluid case, the connection is through a relativistic polytropic equation of state, p=ωδγ, with δ=ρ-p/(γ-1), and δ being the rest-mass density of the fluid. For the electric charge distribution, we assume that the charge density ρe is proportional to the energy density ρ, ρe=αρ, with α being a constant such that 0≤|α|≤1. The study is developed by integrating numerically the hydrostatic equilibrium equation. Some properties of the charged spheres such as the gravitational mass, the total electric charge, the radius, the surface redshift, and the speed of sound are analyzed by varying the central rest-mass density, the charge fraction, and the polytropic exponent. In addition, some limits that arise in general relativity, such as the Chandrasekhar limit, the Oppenheimer-Volkoff limit, the Buchdahl bound, and the Buchdahl-Andréasson bound are studied. It is confirmed that charged relativistic polytropic spheres with γ→∞ and α→1 saturate the Buchdahl-Andréasson bound, thus indicating that it reaches the quasiblack hole configuration. We show by means of numerical analysis that, as expected, the major differences between the two cases appear in the high energy density region.
Bibliographical noteFunding Information:
V. T. Z. thanks Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP, Grant No. 2011/18729-1; Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, Grant No. 308346/2015-7; and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—CAPES, Brazil, Grant No. 88881.064999/2014-01.
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