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Stability Analysis of Relativistic Polytropes

A main question in astrophysics and cosmology has been the severe stability of the astrophysical objects, whether a particular equilibrium configuration is stable. In this article, we study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state , considering structures with the polytropic indices and illustrates the results for the relativistic parameters . We determined the critical relativistic parameter at which the mass of the polytrope has a maximum value and represents the first mode of radial instability. For n=1 (0.5)-2.5, stable relativistic polytropes occur for less than the critical values 0.42, 0.20, 0.10, and 0.04 respectively, while unstable relativistic polytropes are obtained when the relativistic parameter is greater than the same values. When n=3.0 and, energetically unstable solutions have occurred. The results of critical values obtained in this paper for different polytropic indices are in full agreement with those evaluated by several authors. Comparisons between analytical and numerical solutions of the given relativistic functions provide a maximum relative error of order.