Abstract
We study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state, considering structures with the polytropic indices n = 1(0.5)3 and illustrate the results for the relativistic parameters sigma = 0 - 0.75. We determine 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 sigma less than the critical values 0.42, 0.20, 0.10, and 0.04, respectively, while unstable relativistic polytropes are obtained when a is greater than the same values. When n = 3.0 and sigma > 0.5, energetically unstable solutions occur. The results of critical values 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 10(-)(3).