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Obtaining exact solutions of the spherically symmetric general relativisticgravitational field equations describing the interior structure of an isotropicfluid sphere is a long standing problem in theoretical and mathematicalphysics. The usual approach to this problem consists mainly in the numericalinvestigation of the Tolman-Oppenheimer-Volkoff and of the mass continuityequations, which describes the hydrostatic stability of the dense stars. In thepresent paper we introduce an alternative approach for the study of therelativistic fluid sphere, based on the relativistic mass equation, obtained byeliminating the energy density in the Tolman-Oppenheimer-Volkoff equation.Despite its apparent complexity, the relativistic mass equation can be solvedexactly by using a power series representation for the mass, and the Cauchyconvolution for infinite power series. We obtain exact series solutions forgeneral relativistic dense astrophysical objects described by the linearbarotropic and the polytropic equations of state, respectively. For thepolytropic case we obtain the exact power series solution corresponding toarbitrary values of the polytropic index $n$. The explicit form of the solutionis presented for the polytropic index $n=1$, and for the indexes $n=1/2$ and$n=1/5$, respectively. The case of $n=3$ is also considered. In each case theexact power series solution is compared with the exact numerical solutions,which are reproduced by the power series solutions truncated to seven termsonly. The power series representations of the geometric and physical propertiesof the linear barotropic and polytropic stars are also obtained.

astrophysics and space science

Exact power series solutions of the structure equations of the general  relativistic isotropic fluid stars with linear barotropic and polytropic  equations of state